$( This is the Metamath database set.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/mpeuni/conventions.html
   to understand the label naming conventions used in set.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 set.mm" followed by "MM> WRITE SOURCE set.mm / SPLIT".  To
   recombine, omit "/ SPLIT". $)

$( The database set.mm was created by Norman Megill on 30-Sep-1992 and has
   been continuously enriched since then (list of contributors below). $)


$( !
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
  Metamath source file for 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
FC  Filip Cernatescu
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
II  Igor Ieskov
AI  Asger C. Ipsen
JJ  Jerry James
SJ  Szymon Jaroszewicz
BJ  Benoit Jubin
JK  Jim Kingdon
ML  M L
WL  Wolf Lammen
GL  Gerard Lang
BL  Brendan Leahy
LL  Larry Lesyna
RL  Raph Levien
FL  Frederic Line
RFL Roy F. Longton
JPM Jeffrey P. Machado
JM  Jeff Madsen
GM  Giovanni Mascellani
PM  Peter Mazsa
RM  Rodolfo Medina
NM  Norman Megill
MKU metakunt
DM  David Moews
MM  Mykola Mostovenko
SN  Steven Nguyen
MO  Mel L. O'Cat
OAI OpenAI
SO  Stefan O'Rear
JO  Jason Orendorff
KP  K P
NP  Noam Pasman
JPP Jon Pennant
RP  Richard Penner
SP  Stanislas Polu
JP  Josh Purinton
RMI Remi
RR  Rohan Ridenour
SR  Steve Rodriguez
ATS Andrew Salmon
AS  Alan Sare
ES  Eric Schmidt
GS  Glauco Siliprandi
SS  Saveliy Skresanov
BT  BTernaryTau
ET  Ender Ting
JU  Jarvin Udandy
ADH Stijn "Adhemar" Vandamme
AV  Alexander van der Vekens
JV  Jannik Vierling
ZW  Zhi Wang
EW  Emmett Weisz
DAW David A. Wheeler
RW  Roger Witte
KW  Kyle Wyonch
JY  Jonathan Yan
FZ  Fan Zheng
KZ  Kunhao Zheng

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 $[ set-header.mm $] $)
$( !
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Contents of this header
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

* Quick "How To"
* Bibliography
* Metamath syntax summary
* Other notes


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

How to use this file under Windows 95/98/NT/2K/XP/Vista/7/10:

1. Download the Metamath program metamath.exe following 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 set.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 identifier 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/mpeuni/mmset.html#bib
  Hilbert space - see https://us.metamath.org/mpeuni/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. 111--114 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.  The comment opening and closing delimiters must be
      surrounded by whitespace (space, tab, CR, LF, or FF).
  $[ <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, also known as backticks
(` `). These tildes, tokens, math symbols and backticks should be surrounded by
spaces.  This way the LaTeX and HTML rendition of comments will be accurate,
and tools to globally change labels and math symbols will also change them in
comments.  Note that inside of backticks a pair of backticks is interpreted as
a single backtick.  A special comment containing $ t (with no space after the
dollar sign) 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 Appendix B 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.  Theorems can be written in different forms, including "closed form",
"deduction form", and "inference form" (for details, see ~ conventions ).  For
basic theorems, all three forms are generally given, but for more advanced
theorems, we prefer to use the deduction form, since it permits to write proofs
in the "deduction style", and we do not add theorems in inference form unless
there are reasonable grounds for it (for instance, shortening sufficiently many
proofs to counterbalance their addition).

3.  On providing new definitions and theorems, the conventions provided in the
comment of ~ conventions should be obeyed.

4.  For a chronological list of changes to label names and label deletions, see
the changes-set.txt file.  This should help if you have a proof not checked
into the main repository and want to update it for recent changes.

$)

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


$( Begin $[ set-main.mm $] $)
$( Begin $[ set-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. $)


$(
###############################################################################
  CLASSICAL 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 classical first-order logic (often abbreviated as FOL) with
  equality and no terms (the most common logic system used by mathematicians).

  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.

$)


$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
  Pre-logic
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#

  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 the color of wff variables. $)
  $( $j
    varcolorcode "wff" as "0000FF";
    altvarcolorcode "wff" as "337DFF";
  $)

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

       This can also be used to apply subproofs from other theorems.  In step
       2, simply assign the theorem to a1ii.2, and run
       <HTML><code>MM-PA&gt; EXPAND &lt;theorem&gt;</code></HTML>
       to "import" a subproof
       from another theorem.

       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 $.
  $}


$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
  Propositional calculus
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#

  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 three basic axioms and
  another axiom that defines the modus-ponens inference rule.  It is attributed
  to Jan Lukasiewicz (pronounced woo-kah-SHAY-vitch) and was popularized by
  Alonzo Church, who called it system P2.  (Thanks to Ted Ulrich for this
  information.)  These axioms are ~ ax-1 , ~ ax-2 , ~ ax-3 , and (for modus
  ponens) ~ ax-mp . Some closely followed texts include [Margaris] for the
  axioms and [WhiteheadRussell] for the theorems.

  The propositional calculus used here is the classical system widely used by
  mathematicians.  In particular, this logic system accepts the "law of the
  excluded middle" as proven in ~ exmid , which says that a logical statement
  is either true or not true.  This is an essential distinction of classical
  logic and is not a theorem of intuitionistic logic.

  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 ~ mmbiblio.html
  for a complete cross-reference from sources used to its formalization in the
  Metamath Proof 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).  In classical logic
     (which is our logic), a wff is interpreted as either true or false.  So if
     ` ph ` is true, then ` -. ph ` is false; if ` ph ` is false, then
     ` -. ph ` is true.  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 resulting wff
     is (interpreted as) false when ` ph ` is true and ` ps ` is false; it is
     true otherwise.  Think of the truth table for an OR gate with input ` ph `
     connected through an inverter.  After we state the axioms of propositional
     calculus ( ~ ax-1 , ~ ax-2 , ~ ax-3 , and ~ ax-mp ) and define the
     biconditional ( ~ df-bi ), the constant true ` T. ` ( ~ df-tru ), and the
     constant false ` F. ` ( ~ df-fal ), we will be able to prove these truth
     table values: ` ( ( T. -> T. ) <-> T. ) ` ( ~ truimtru ),
     ` ( ( T. -> F. ) <-> F. ) ` ( ~ truimfal ), ` ( ( F. -> T. ) <-> T. ) `
     ( ~ falimtru ), and ` ( ( F. -> F. ) <-> T. ) ` ( ~ falimfal ).  These
     have straightforward meanings, for example, ` ( ( T. -> T. ) <-> T. ) `
     just means "the value of ` ( T. -> T. ) ` is ` T. ` ".

     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.  Contrast with ` <-> ` ( ~ df-bi ),
     ` /\ ` ( ~ df-an ), and ` \/ ` ( ~ df-or ).

     This is called "material implication" and the arrow is usually read as
     "implies".  However, material implication is not identical to the meaning
     of "implies" in natural language.  For example, the word "implies" may
     suggest a causal relationship in natural language.  Material implication
     does not require any causal relationship.  Also, note that in material
     implication, if the consequent is true then the wff is always true (even
     if the antecedent is false).  Thus, if "implies" means material
     implication, it is true that "if the moon is made of green cheese that
     implies that 5=5" (because 5=5).  Similarly, if the antecedent is false,
     the wff is always true.  Thus, it is true that, "if the moon is made of
     green cheese that implies that 5=7" (because the moon is not actually made
     of green cheese).  A contradiction implies anything ( ~ pm2.21i ).  In
     short, material implication has a very specific technical definition, and
     misunderstandings of it are sometimes called "paradoxes of logical
     implication". $)
  wi $a wff ( ph -> ps ) $.

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


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  The axioms of propositional calculus
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  Propositional calculus (Axioms ~ ax-1 through ~ ax-3 and rule ~ ax-mp ) can
  be thought of as asserting formulas that are universally "true" when their
  variables are replaced by any combination of "true" and "false".
  Propositional calculus was first formalized by Frege in 1879, using as his
  axioms (in addition to rule ~ ax-mp ) the wffs ~ ax-1 , ~ ax-2 , ~ pm2.04 ,
  ~ con3 , ~ notnot , and ~ notnotr .  Around 1930, Lukasiewicz simplified the
  system by eliminating the third (which follows from the first two, as you can
  see by looking at the proof of ~ pm2.04 ) and replacing the last three with
  our ~ ax-3 .  (Thanks to Ted Ulrich for this information.)

  The theorems of propositional calculus are also called _tautologies_.
  Tautologies can be proved very simply using truth tables, based on the
  true/false interpretation of propositional calculus.  To do this, we assign
  all possible combinations of true and false to the wff variables and verify
  that the result (using the rules described in ~ wi and ~ wn ) always
  evaluates to true.  This is called the _semantic_ approach.  Our approach is
  called the _syntactic_ approach, in which everything is derived from axioms.
  A metatheorem called the Completeness Theorem for Propositional Calculus
  shows that the two approaches are equivalent and even provides an algorithm
  for automatically generating syntactic proofs from a truth table.  Those
  proofs, however, tend to be long, since truth tables grow exponentially with
  the number of variables, and the much shorter proofs that we show here were
  found manually.

$)

  ${
    $( 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 3 axioms of
     propositional calculus.  The 3 axioms are also given as Definition 2.1 of
     [Hamilton] p. 28.  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".  It is
     Proposition 1 of [Frege1879] p. 26, its first axiom.  (Contributed by NM,
     30-Sep-1992.) $)
  ax-1 $a |- ( ph -> ( ps -> ph ) ) $.

  $( Axiom _Frege_.  Axiom A2 of [Margaris] p. 49.  One of the 3 axioms of
     propositional calculus.  It "distributes" an antecedent over two
     consequents.  This axiom was part of Frege's original system and is known
     as _Frege_ in the literature; see Proposition 2 of [Frege1879] p. 26.  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, 30-Sep-1992.) $)
  ax-2 $a |- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) $.

  $( Axiom _Transp_.  Axiom A3 of [Margaris] p. 49.  One of the 3 axioms of
     propositional calculus.  It 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 axiom
     is called _Transp_ or "the principle of transposition" in _Principia
     Mathematica_ (Theorem *2.17 of [WhiteheadRussell] p. 103).  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 NM, 30-Sep-1992.)  Use its
     alias ~ con4 instead.  (New usage is discouraged.) $)
  ax-3 $a |- ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) $.


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

  The results in this section are based on implication only, and avoid ~ ax-3 ,
  so are intuitionistic.  The system { ~ ax-mp , ~ ax-1 , ~ ax-2 } axiomatizes
  what is sometimes called "intuitionistic implicational calculus" or "minimal
  implicational calculus".

  In an implication, the wff before the arrow is called the "antecedent" and
  the wff after the arrow is called the "consequent".

$)

  ${
    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.) $)
    mp2 $p |- ch $=
      ( wi ax-mp ) BCEABCGDFHH $.
  $}

  ${
    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 $.
  $}

  ${
    a1i.1 $e |- ph $.
    $( Inference introducing an antecedent.  Inference associated with ~ ax-1 .
       Its associated inference is ~ a1ii .  See ~ conventions for a definition
       of "associated inference".  (Contributed by NM, 29-Dec-1992.) $)
    a1i $p |- ( ps -> ph ) $=
      ( wi ax-1 ax-mp ) ABADCABEF $.
  $}

  ${
    2a1i.1 $e |- ph $.
    $( Inference introducing two antecedents.  Two applications of ~ a1i .
       Inference associated with ~ 2a1 .  (Contributed by Jeff Hankins,
       4-Aug-2009.) $)
    2a1i $p |- ( ps -> ( ch -> ph ) ) $=
      ( wi a1i ) CAEBACDFF $.
  $}

  ${
    mp1i.1 $e |- ph $.
    mp1i.2 $e |- ( ph -> ps ) $.
    $( Inference detaching an antecedent and introducing a new one.
       (Contributed by Stefan O'Rear, 29-Jan-2015.) $)
    mp1i $p |- ( ch -> ps ) $=
      ( ax-mp a1i ) BCABDEFG $.
  $}

  ${
    a2i.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Inference distributing an antecedent.  Inference associated with
       ~ ax-2 .  Its associated inference is ~ mpd .  (Contributed by NM,
       29-Dec-1992.) $)
    a2i $p |- ( ( ph -> ps ) -> ( ph -> ch ) ) $=
      ( wi ax-2 ax-mp ) ABCEEABEACEEDABCFG $.
  $}

  ${
    mpd.1 $e |- ( ph -> ps ) $.
    mpd.2 $e |- ( ph -> ( ps -> ch ) ) $.
    $( A modus ponens deduction.  A translation of natural deduction rule
       ` -> ` E ( ` -> ` elimination), see ~ natded .  Deduction form of
       ~ ax-mp .  Inference associated with ~ a2i .  Commuted form of ~ mpcom .
       (Contributed by NM, 29-Dec-1992.) $)
    mpd $p |- ( ph -> ch ) $=
      ( wi a2i ax-mp ) ABFACFDABCEGH $.
  $}

  ${
    imim2i.1 $e |- ( ph -> ps ) $.
    $( Inference adding common antecedents in an implication.  Inference
       associated with ~ imim2 .  Its associated inference is ~ syl .
       (Contributed by NM, 28-Dec-1992.) $)
    imim2i $p |- ( ( ch -> ph ) -> ( ch -> ps ) ) $=
      ( wi a1i a2i ) CABABECDFG $.
  $}

  ${
    $( 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 (and ~ imim1i and ~ imim2i ), which Russell and Whitehead call
       "the principle of the syllogism ... because ... the syllogism in Barbara
       is derived from [[ ~ syl ]" (quote after Theorem *2.06 of
       [WhiteheadRussell] p. 101).  Some authors call this law a "hypothetical
       syllogism".  Its associated inference is ~ mp2b .

       (A bit of trivia: this is the most commonly referenced assertion in our
       database (13449 times as of 22-Jul-2021).  In second place is ~ eqid
       (9597 times), followed by ~ adantr (8861 times), ~ syl2anc (7421 times),
       ~ adantl (6403 times), and ~ simpr (5829 times).  The Metamath program
       command 'show usage' shows the number of references.)

       (Contributed by NM, 30-Sep-1992.)  (Proof shortened by Mel L. O'Cat,
       20-Oct-2011.)  (Proof shortened by Wolf Lammen, 26-Jul-2012.) $)
    syl $p |- ( ph -> ch ) $=
      ( wi a1i mpd ) ABCDBCFAEGH $.
  $}

  ${
    3syl.1 $e |- ( ph -> ps ) $.
    3syl.2 $e |- ( ps -> ch ) $.
    3syl.3 $e |- ( ch -> th ) $.
    $( Inference chaining two syllogisms ~ syl .  Inference associated with
       ~ imim12i .  (Contributed by NM, 28-Dec-1992.) $)
    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 ~ syl .  (Contributed by BJ,
       14-Jul-2018.)  The use of this theorem is marked "discouraged" because
       it can cause the Metamath program "MM-PA> MINIMIZE__WITH *" command to
       have very long run times.  However, feel free to use "MM-PA>
       MINIMIZE__WITH 4syl / OVERRIDE" if you wish.  Remember to update the
       "discouraged" file if it gets used.  (New usage is discouraged.) $)
    4syl $p |- ( ph -> ta ) $=
      ( 3syl syl ) ADEABCDFGHJIK $.
  $}

  ${
    mpi.1 $e |- ps $.
    mpi.2 $e |- ( ph -> ( ps -> ch ) ) $.
    $( A nested modus ponens inference.  Inference associated with ~ com12 .
       (Contributed by NM, 29-Dec-1992.)  (Proof shortened by Stefan Allan,
       20-Mar-2006.) $)
    mpi $p |- ( ph -> ch ) $=
      ( a1i mpd ) ABCBADFEG $.
  $}

  ${
    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 $.
  $}

  $( Principle of identity.  Theorem *2.08 of [WhiteheadRussell] p. 101.  For
     another version of the proof directly from axioms, see ~ idALT .  Its
     associated inference, ~ idi , requires no axioms for its proof, contrary
     to ~ id .  Note that the second occurrences of ` ph ` in Steps 1 and 2 may
     be simultaneously replaced by any wff ` ps ` , which may ease the
     understanding of the proof.  (Contributed by NM, 29-Dec-1992.)  (Proof
     shortened by Stefan Allan, 20-Mar-2006.) $)
  id $p |- ( ph -> ph ) $=
    ( wi ax-1 mpd ) AAABZAAACAECD $.

  $( Alternate proof of ~ id .  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.appstate.edu/~~hirstjl/primer/hirst.pdf .
     Note that the second occurrences of ` ph ` in Steps 1 to 4 and the sixth
     in Step 3 may be simultaneously replaced by any wff ` ps ` , which may
     ease the understanding of the proof.  For a shorter version of the proof
     that takes advantage of previously proved theorems, see ~ id .
     (Contributed by NM, 30-Sep-1992.)  (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 ~ id 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.  Deduction form of ~ ax-1
       and ~ a1i .  (Contributed by NM, 5-Jan-1993.)  (Proof shortened by
       Stefan Allan, 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.  Deduction form of
       ~ ax-2 .  (Contributed by NM, 23-Jun-1994.) $)
    a2d $p |- ( ph -> ( ( ps -> ch ) -> ( ps -> th ) ) ) $=
      ( wi ax-2 syl ) ABCDFFBCFBDFFEBCDGH $.
  $}

  ${
    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 Mel L. 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 $.
  $}

  ${
    com12.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Inference that swaps (commutes) antecedents in an implication.
       Inference associated with ~ pm2.04 .  Its associated inference is
       ~ mpi .  (Contributed by NM, 29-Dec-1992.)  (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 first premise is used to replace the
       second antecedent of the second premise.  (Contributed by NM,
       27-Dec-1992.)  (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-Jan-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.  Commuted form
       of ~ mpd .  (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, sometimes called "Assertion" or "Pon" (for "ponens"), can be
     thought of as a closed form of modus ponens ~ ax-mp .  Theorem *2.27 of
     [WhiteheadRussell] p. 104.  (Contributed by NM, 15-Jul-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.  Double deduction associated with
       ~ ax-mp .  Deduction associated with ~ mpd .  (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.  Deduction associated with ~ mpi .
       (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 Mel L. 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.  Deduction associated with ~ syl .  See
       ~ conventions for the meaning of "associated deduction" or "deduction
       form".  (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Mel L.
       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.  Deduction associated with ~ mp2 .
       (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 ) ) $.
    $( Double deduction introducing an antecedent.  Deduction associated with
       ~ a1d .  Double deduction associated with ~ ax-1 and ~ a1i .
       (Contributed by NM, 17-Dec-2004.)  (Proof shortened by Mel L. O'Cat,
       15-Jan-2008.) $)
    a1dd $p |- ( ph -> ( ps -> ( th -> ch ) ) ) $=
      ( wi ax-1 syl6 ) ABCDCFECDGH $.
  $}

  ${
    2a1dd.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Double deduction introducing two antecedents.  Two applications of
       ~ 2a1dd .  Deduction associated with ~ 2a1d .  Double deduction
       associated with ~ 2a1 and ~ 2a1i .  (Contributed by Jeff Hankins,
       5-Aug-2009.) $)
    2a1dd $p |- ( ph -> ( ps -> ( th -> ( ta -> ch ) ) ) ) $=
      ( wi a1dd ) ABECGDABCEFHH $.
  $}

  ${
    pm2.43i.1 $e |- ( ph -> ( ph -> ps ) ) $.
    $( Inference absorbing redundant antecedent.  Inference associated with
       ~ pm2.43 .  (Contributed by NM, 10-Jan-1993.)  (Proof shortened by Mel
       L. 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.  Deduction associated with
       ~ pm2.43 and ~ pm2.43i .  (Contributed by NM, 18-Aug-1993.)  (Proof
       shortened by Mel L. 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 Mel L. 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, 10-Jan-1993.)  (Proof shortened by Mel L. 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.  Deduction associated with ~ imim2
       and ~ imim2i .  (Contributed by NM, 10-Jan-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.  Its associated inference is ~ imim2i .  Its
     associated deduction is ~ imim2d .  An alternate proof from more basic
     results is given by ~ ax-1 followed by ~ a2d .  (Contributed by NM,
     29-Dec-1992.)  (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.  Deduction associated with ~ 3syld .
       (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.  Inference associated with
       ~ imim12 .  Its associated inference is ~ 3syl .  (Contributed by NM,
       12-Mar-1993.)  (Proof shortened by Mel L. 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.  Inference
       associated with ~ imim1 .  Its associated inference is ~ syl .
       (Contributed by NM, 28-Dec-1992.)  (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 ) ) $.
    $( A syllogism inference.  (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 $.
  $}

  ${
    mpsylsyld.1 $e |- ph $.
    mpsylsyld.2 $e |- ( ps -> ( ch -> th ) ) $.
    mpsylsyld.3 $e |- ( ph -> ( th -> ta ) ) $.
    $( Modus ponens combined with a double syllogism inference.  (Contributed
       by Alan Sare, 22-Jul-2012.) $)
    mpsylsyld $p |- ( ps -> ( ch -> ta ) ) $=
      ( a1i sylsyld ) BACDEABFIGHJ $.
  $}

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  ${
    syldd.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
    syldd.2 $e |- ( ph -> ( ps -> ( th -> ta ) ) ) $.
    $( Nested syllogism deduction.  Deduction associated with ~ syld .  Double
       deduction associated with ~ syl .  (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.  Deduction associated with ~ syl5 .
       (Contributed by NM, 14-May-1993.)  (Proof shortened by Josh Purinton,
       29-Dec-2000.)  (Proof shortened by Mel L. 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 first premise is used to replace the
       third antecedent of the second premise.  (Contributed by NM,
       12-Jan-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.  Deduction associated with ~ syl6 .
       (Contributed by NM, 11-May-1993.)  (Proof shortened by Josh Purinton,
       29-Dec-2000.)  (Proof shortened by Mel L. O'Cat, 2-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, 13-May-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, 14-May-1993.) $)
    syl9r $p |- ( th -> ( ph -> ( ps -> ta ) ) ) $=
      ( wi syl9 com12 ) ADBEHABCDEFGIJ $.
  $}

  ${
    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 $.
  $}

  ${
    imim12d.1 $e |- ( ph -> ( ps -> ch ) ) $.
    imim12d.2 $e |- ( ph -> ( th -> ta ) ) $.
    $( Deduction combining antecedents and consequents.  Deduction associated
       with ~ imim12 and ~ imim12i .  (Contributed by NM, 7-Aug-1994.)  (Proof
       shortened by Mel L. 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.  Deduction associated with ~ imim1
       and ~ imim1i .  (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.  Its associated inference is ~ imim1i .
     (Contributed by NM, 29-Dec-1992.)  (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.  Closed form of ~ syld .
     (Contributed by NM, 3-Jan-2005.) $)
  pm2.83 $p |- ( ( ph -> ( ps -> ch ) )
      -> ( ( ph -> ( ch -> th ) ) -> ( ph -> ( ps -> th ) ) ) ) $=
    ( wi imim1 imim3i ) BCECDEBDEABCDFG $.

  $( Over minimal implicational calculus, Peirce's axiom ~ peirce implies an
     axiom sometimes called "Roll",
     ` ( ( ( ph -> ps ) -> ch ) -> ( ( ch -> ph ) -> ph ) ) ` , of which
     ~ looinv is a special instance.  The converse also holds: substitute
     ` ( ph -> ps ) ` for ` ch ` in Roll and use ~ id and ~ ax-mp .
     (Contributed by BJ, 15-Jun-2021.) $)
  peirceroll $p |- ( ( ( ( ph -> ps ) -> ph ) -> ph )
                   -> ( ( ( ph -> ps ) -> ch ) -> ( ( ch -> ph ) -> ph ) ) ) $=
    ( wi imim1 id syl9r ) ABDZCDCADHADZIADZAHCAEJFG $.

  ${
    com3.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
    $( Commutation of antecedents.  Swap 2nd and 3rd.  Deduction associated
       with ~ com12 .  (Contributed by NM, 27-Dec-1992.)  (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.  This was
     the third axiom in Frege's logic system, specifically Proposition 8 of
     [Frege1879] p. 35.  Its associated inference is ~ com12 .  (Contributed by
     NM, 27-Dec-1992.)  (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.  Deduction associated
       with ~ com23 .  Double deduction associated with ~ com12 .  (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 Mel L. 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.  Deduction associated
       with ~ com13 .  (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.  Deduction associated
       with ~ com34 .  Double deduction associated with ~ com23 .  Triple
       deduction associated with ~ com12 .  (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.  Deduction associated
       with ~ com24 .  Double deduction associated with ~ com13 .  (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.  Deduction associated
       with ~ com14 .  (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 $.
  $}

  $( Closed form of ~ imim12i and of ~ 3syl .  (Contributed by BJ,
     16-Jul-2019.) $)
  imim12 $p |- ( ( ph -> ps ) ->
                      ( ( ch -> th ) -> ( ( ps -> ch ) -> ( ph -> th ) ) ) ) $=
    ( wi imim2 imim1 syl9r ) CDEBCEBDEABEADECDBFABDGH $.

  $( Elimination of a nested antecedent.  Sometimes called "Syll-Simp" since it
     is a syllogism applied to ~ ax-1 ("Simplification").  (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 .  Partial converse of ~ ja (the other
       partial converse being ~ jarli ).  (Contributed by Wolf Lammen,
       20-Sep-2013.) $)
    jarri $p |- ( ps -> ch ) $=
      ( wi ax-1 syl ) BABECBAFDG $.
  $}

  ${
    pm2.86d.1 $e |- ( ph -> ( ( ps -> ch ) -> ( ps -> th ) ) ) $.
    $( Deduction associated with ~ 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 $.

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

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


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Logical negation
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  This section makes our first use of the third axiom of propositional
  calculus, ~ ax-3 .  It introduces logical negation.

$)

  $( Alias for ~ ax-3 to be used instead of it for labeling consistency.  Its
     associated inference is ~ con4i and its associated deduction is ~ con4d .
     (Contributed by BJ, 24-Dec-2020.) $)
  con4 $p |- ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) $=
    ( ax-3 ) ABC $.

  ${
    con4i.1 $e |- ( -. ph -> -. ps ) $.
    $( Inference associated with ~ con4 .  Its associated inference is ~ mt4 .

       Remark: this can also be proved using ~ notnot followed by ~ nsyl2 ,
       giving a shorter proof but depending on more axioms (namely, ~ ax-1 and
       ~ ax-2 ).  (Contributed by NM, 29-Dec-1992.) $)
    con4i $p |- ( ps -> ph ) $=
      ( wn wi con4 ax-mp ) ADBDEBAECABFG $.
    $( $j usage 'con4i' avoids 'ax-1' 'ax-2'; $)
  $}

  ${
    con4d.1 $e |- ( ph -> ( -. ps -> -. ch ) ) $.
    $( Deduction associated with ~ con4 .  (Contributed by NM, 26-Mar-1995.) $)
    con4d $p |- ( ph -> ( ch -> ps ) ) $=
      ( wn wi con4 syl ) ABECEFCBFDBCGH $.
  $}

  ${
    mt4.1 $e |- ph $.
    mt4.2 $e |- ( -. ps -> -. ph ) $.
    $( The rule of modus tollens.  Inference associated with ~ con4i .
       (Contributed by Wolf Lammen, 12-May-2013.) $)
    mt4 $p |- ps $=
      ( con4i ax-mp ) ABCBADEF $.
  $}

  ${
    mt4d.1 $e |- ( ph -> ps ) $.
    mt4d.2 $e |- ( ph -> ( -. ch -> -. ps ) ) $.
    $( Modus tollens deduction.  Deduction form of ~ mt4 .  (Contributed by NM,
       9-Jun-2006.) $)
    mt4d $p |- ( ph -> ch ) $=
      ( con4d mpd ) ABCDACBEFG $.
  $}

  ${
    mt4i.1 $e |- ch $.
    mt4i.2 $e |- ( ph -> ( -. ps -> -. ch ) ) $.
    $( Modus tollens inference.  (Contributed by Wolf Lammen, 12-May-2013.) $)
    mt4i $p |- ( ph -> ps ) $=
      ( a1i mt4d ) ACBCADFEG $.
  $}

  ${
    pm2.21i.1 $e |- -. ph $.
    $( A contradiction implies anything.  Inference associated with ~ pm2.21 .
       Its associated inference is ~ pm2.24ii .  (Contributed by NM,
       16-Sep-1993.) $)
    pm2.21i $p |- ( ph -> ps ) $=
      ( wn a1i con4i ) BAADBDCEF $.
  $}

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

  ${
    pm2.21d.1 $e |- ( ph -> -. ps ) $.
    $( A contradiction implies anything.  Deduction associated with ~ pm2.21 .
       (Contributed by NM, 10-Feb-1996.) $)
    pm2.21d $p |- ( ph -> ( ps -> ch ) ) $=
      ( wn a1d con4d ) ACBABECEDFG $.
  $}

  ${
    pm2.21ddALT.1 $e |- ( ph -> ps ) $.
    pm2.21ddALT.2 $e |- ( ph -> -. ps ) $.
    $( Alternate proof of ~ pm2.21dd .  (Contributed by Mario Carneiro,
       9-Feb-2017.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    pm2.21ddALT $p |- ( ph -> ch ) $=
      ( pm2.21d mpd ) ABCDABCEFG $.
  $}

  $( From a wff and its negation, anything follows.  Theorem *2.21 of
     [WhiteheadRussell] p. 104.  Also called the Duns Scotus law.  Its commuted
     form is ~ pm2.24 and its associated inference is ~ pm2.21i .  (Contributed
     by NM, 29-Dec-1992.)  (Proof shortened by Wolf Lammen, 14-Sep-2012.) $)
  pm2.21 $p |- ( -. ph -> ( ph -> ps ) ) $=
    ( wn id pm2.21d ) ACZABFDE $.

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

  $( 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 $.

  ${
    jarli.1 $e |- ( ( ph -> ps ) -> ch ) $.
    $( Inference associated with ~ jarl .  Partial converse of ~ ja (the other
       partial converse being ~ jarri ).  (Contributed by Wolf Lammen,
       4-Oct-2013.) $)
    jarli $p |- ( -. ph -> ch ) $=
      ( wn wi pm2.21 syl ) AEABFCABGDH $.
  $}

  ${
    pm2.18d.1 $e |- ( ph -> ( -. ps -> ps ) ) $.
    $( Deduction form of the Clavius law ~ pm2.18 .  (Contributed by FL,
       12-Jul-2009.)  (Proof shortened by Andrew Salmon, 7-May-2011.)  Shorten
       ~ pm2.18 .  (Revised by Wolf Lammen, 17-Nov-2023.) $)
    pm2.18d $p |- ( ph -> ps ) $=
      ( id wn pm2.21 sylcom mt4d ) AABADABEBAEZCBIFGH $.
  $}

  $( Clavius law, or "consequentia mirabilis" ("admirable consequence").  If a
     formula is implied by its negation, then it is true.  Can be used in
     proofs by contradiction.  Theorem *2.18 of [WhiteheadRussell] p. 103.  See
     also the weak Clavius law ~ pm2.01 .  (Contributed by NM, 29-Dec-1992.)
     (Proof shortened by Wolf Lammen, 17-Nov-2023.) $)
  pm2.18 $p |- ( ( -. ph -> ph ) -> ph ) $=
    ( wn wi id pm2.18d ) ABACZAFDE $.

  ${
    pm2.18i.1 $e |- ( -. ph -> ph ) $.
    $( Inference associated with the Clavius law ~ pm2.18 .  (Contributed by
       BJ, 30-Mar-2020.) $)
    pm2.18i $p |- ph $=
      ( wn wi pm2.18 ax-mp ) ACADABAEF $.
  $}

  $( Double negation elimination.  Converse of ~ notnot and one implication of
     ~ notnotb .  Theorem *2.14 of [WhiteheadRussell] p. 102.  This was the
     fifth axiom of Frege, specifically Proposition 31 of [Frege1879] p. 44.
     In classical logic (our logic) this is always true.  In intuitionistic
     logic this is not always true, and formulas for which it is true are
     called "stable".  (Contributed by NM, 29-Dec-1992.)  (Proof shortened by
     David Harvey, 5-Sep-1999.)  (Proof shortened by Josh Purinton,
     29-Dec-2000.) $)
  notnotr $p |- ( -. -. ph -> ph ) $=
    ( wn pm2.18 jarli ) ABAAACD $.

  ${
    notnotri.1 $e |- -. -. ph $.
    $( Inference associated with ~ notnotr .  For a shorter proof using
       ~ ax-2 , see ~ notnotriALT .  (Contributed by NM, 27-Feb-2008.)  (Proof
       shortened by Wolf Lammen, 15-Jul-2021.)  Remove dependency on ~ ax-2 .
       (Revised by Steven Nguyen, 27-Dec-2022.) $)
    notnotri $p |- ph $=
      ( wn pm2.21i mt4 ) ACZCZABFGCBDE $.
    $( $j usage 'notnotri' avoids 'ax-2'; $)

    $( Alternate proof of ~ notnotri .  The proof via ~ notnotr and ~ ax-mp
       also has three essential steps, but has a total number of steps equal to
       8, instead of the present 7, because it has to construct the formula
       ` ph ` twice and the formula ` -. -. ph ` once, whereas the present
       proof has to construct the formula ` ph ` twice and the formula
       ` -. ph ` once, and therefore makes only one use of ~ wn instead of two.
       This can be checked by running the Metamath command "MM> SHOW PROOF
       notnotri / NORMAL".  (Contributed by NM, 27-Feb-2008.)  (Proof shortened
       by Wolf Lammen, 15-Jul-2021.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    notnotriALT $p |- ph $=
      ( wn pm2.21i pm2.18i ) AACABDE $.
  $}

  ${
    notnotrd.1 $e |- ( ph -> -. -. ps ) $.
    $( Deduction associated with ~ notnotr and ~ notnotri .  Double negation
       elimination rule.  A translation of the natural deduction rule ` -. -. `
       C , ` _G |- -. -. ps => _G |- ps ` ; see ~ natded .  This is Definition
       NNC in [Pfenning] p. 17.  This rule is valid in classical logic (our
       logic), but not in intuitionistic logic.  (Contributed by DAW,
       8-Feb-2017.) $)
    notnotrd $p |- ( ph -> ps ) $=
      ( wn notnotr syl ) ABDDBCBEF $.
  $}

  ${
    con2d.1 $e |- ( ph -> ( ps -> -. ch ) ) $.
    $( A contraposition deduction.  (Contributed by NM, 19-Aug-1993.) $)
    con2d $p |- ( ph -> ( ch -> -. ps ) ) $=
      ( wn notnotr syl5 con4d ) ABEZCIEBACEBFDGH $.
  $}

  $( Contraposition.  Theorem *2.03 of [WhiteheadRussell] p. 100.  (Contributed
     by NM, 29-Dec-1992.)  (Proof shortened by Wolf Lammen, 12-Feb-2013.) $)
  con2 $p |- ( ( ph -> -. ps ) -> ( ps -> -. ph ) ) $=
    ( wn wi id con2d ) ABCDZABGEF $.

  ${
    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 $.
  $}

  ${
    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 $.
  $}

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

  ${
    con2i.a $e |- ( ph -> -. ps ) $.
    $( A contraposition inference.  Its associated inference is ~ mt2 .
       (Contributed by NM, 10-Jan-1993.)  (Proof shortened by Mel L. 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 $.
  $}

  ${
    nsyl2.1 $e |- ( ph -> -. ps ) $.
    nsyl2.2 $e |- ( -. ch -> ps ) $.
    $( A negated syllogism inference.  (Contributed by NM, 26-Jun-1994.)
       (Proof shortened by Wolf Lammen, 14-Nov-2023.) $)
    nsyl2 $p |- ( ph -> ch ) $=
      ( wn nsyl3 con4i ) CAABCFDEGH $.
  $}

  $( Double negation introduction.  Converse of ~ notnotr and one implication
     of ~ notnotb .  Theorem *2.12 of [WhiteheadRussell] p. 101.  This was the
     sixth axiom of Frege, specifically Proposition 41 of [Frege1879] p. 47.
     (Contributed by NM, 28-Dec-1992.)  (Proof shortened by Wolf Lammen,
     2-Mar-2013.) $)
  notnot $p |- ( ph -> -. -. ph ) $=
    ( wn id con2i ) ABZAECD $.

  ${
    notnoti.1 $e |- ph $.
    $( Inference associated with ~ notnot .  (Contributed by NM,
       27-Feb-2008.) $)
    notnoti $p |- -. -. ph $=
      ( wn notnot ax-mp ) AACCBADE $.
  $}

  ${
    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 $.
  $}

  ${
    con1d.1 $e |- ( ph -> ( -. ps -> ch ) ) $.
    $( A contraposition deduction.  (Contributed by NM, 27-Dec-1992.) $)
    con1d $p |- ( ph -> ( -. ch -> ps ) ) $=
      ( wn notnot syl6 con4d ) ABCEZABECIEDCFGH $.
  $}

  $( Contraposition.  Theorem *2.15 of [WhiteheadRussell] p. 102.  Its
     associated inference is ~ con1i .  (Contributed by NM, 29-Dec-1992.)
     (Proof shortened by Wolf Lammen, 12-Feb-2013.) $)
  con1 $p |- ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) $=
    ( wn wi id con1d ) ACBDZABGEF $.

  ${
    con1i.1 $e |- ( -. ph -> ps ) $.
    $( A contraposition inference.  Inference associated with ~ con1 .  Its
       associated inference is ~ mt3 .  (Contributed by NM, 3-Jan-1993.)
       (Proof shortened by Mel L. O'Cat, 28-Nov-2008.)  (Proof shortened by
       Wolf Lammen, 19-Jun-2013.) $)
    con1i $p |- ( -. ps -> ph ) $=
      ( wn id nsyl2 ) BDZBAGECF $.
  $}

  ${
    mt3d.1 $e |- ( ph -> -. ch ) $.
    mt3d.2 $e |- ( ph -> ( -. ps -> ch ) ) $.
    $( Modus tollens deduction.  (Contributed by NM, 26-Mar-1995.) $)
    mt3d $p |- ( ph -> ps ) $=
      ( wn con1d mpd ) ACFBDABCEGH $.
  $}

  ${
    mt3i.1 $e |- -. ch $.
    mt3i.2 $e |- ( ph -> ( -. ps -> ch ) ) $.
    $( Modus tollens inference.  (Contributed by NM, 26-Mar-1995.)  (Proof
       shortened by Wolf Lammen, 15-Sep-2012.) $)
    mt3i $p |- ( ph -> ps ) $=
      ( wn a1i mt3d ) ABCCFADGEH $.
  $}

  ${
    pm2.24i.1 $e |- ph $.
    $( Inference associated with ~ pm2.24 .  Its associated inference is
       ~ pm2.24ii .  (Contributed by NM, 20-Aug-2001.) $)
    pm2.24i $p |- ( -. ph -> ps ) $=
      ( wn a1i con1i ) BAABDCEF $.
  $}

  ${
    pm2.24d.1 $e |- ( ph -> ps ) $.
    $( Deduction form of ~ pm2.24 .  (Contributed by NM, 30-Jan-2006.) $)
    pm2.24d $p |- ( ph -> ( -. ps -> ch ) ) $=
      ( wn a1d con1d ) ACBABCEDFG $.
  $}

  ${
    con3d.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( A contraposition deduction.  Deduction form of ~ con3 .  (Contributed by
       NM, 10-Jan-1993.) $)
    con3d $p |- ( ph -> ( -. ch -> -. ps ) ) $=
      ( wn notnotr syl5 con1d ) ABEZCIEBACBFDGH $.
  $}

  $( Contraposition.  Theorem *2.16 of [WhiteheadRussell] p. 103.  This was the
     fourth axiom of Frege, specifically Proposition 28 of [Frege1879] p. 43.
     Its associated inference is ~ con3i .  (Contributed by NM, 29-Dec-1992.)
     (Proof shortened by Wolf Lammen, 13-Feb-2013.) $)
  con3 $p |- ( ( ph -> ps ) -> ( -. ps -> -. ph ) ) $=
    ( wi id con3d ) ABCZABFDE $.

  ${
    con3i.a $e |- ( ph -> ps ) $.
    $( A contraposition inference.  Inference associated with ~ con3 .  Its
       associated inference is ~ mto .  (Contributed by NM, 3-Jan-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 $.
  $}

  ${
    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 NM, 15-Feb-1996.) $)
    nsyl4 $p |- ( -. ch -> ps ) $=
      ( wn con1i syl ) CFABACEGDH $.

    $( A negated syllogism inference.  (Contributed by Wolf Lammen,
       20-May-2024.) $)
    nsyl5 $p |- ( -. ps -> ch ) $=
      ( nsyl4 con1i ) CBABCDEFG $.
  $}

  $( Theorem *3.2 of [WhiteheadRussell] p. 111, expressed with primitive
     connectives (see ~ pm3.2 ).  (Contributed by NM, 29-Dec-1992.)  (Proof
     shortened by Josh Purinton, 29-Dec-2000.) $)
  pm3.2im $p |- ( ph -> ( ps -> -. ( ph -> -. ps ) ) ) $=
    ( wn wi pm2.27 con2d ) AABCZDBAGEF $.

  ${
    jc.1 $e |- ( ph -> ps ) $.
    jc.2 $e |- ( ph -> ch ) $.
    $( Deduction joining the consequents of two premises.  A deduction
       associated with ~ pm3.2im .  (Contributed by NM, 28-Dec-1992.) $)
    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 $.
  $}

  ${
    impi.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( An importation inference.  (Contributed by NM, 29-Dec-1992.)  (Proof
       shortened by Wolf Lammen, 20-Jul-2013.) $)
    impi $p |- ( -. ( ph -> -. ps ) -> ch ) $=
      ( wn wi con3rr3 con1i ) CABEFABCDGH $.
  $}

  ${
    expi.1 $e |- ( -. ( ph -> -. ps ) -> ch ) $.
    $( An exportation inference.  (Contributed by NM, 29-Dec-1992.)  (Proof
       shortened by Mel L. O'Cat, 28-Nov-2008.) $)
    expi $p |- ( ph -> ( ps -> ch ) ) $=
      ( wn wi pm3.2im syl6 ) ABABEFECABGDH $.
  $}

  $( Simplification.  Similar to Theorem *3.27 (Simp) of [WhiteheadRussell]
     p. 112.  (Contributed by NM, 3-Jan-1993.)  (Proof shortened by Wolf
     Lammen, 13-Nov-2012.) $)
  simprim $p |- ( -. ( ph -> -. ps ) -> ps ) $=
    ( idd impi ) ABBABCD $.

  $( Simplification.  Similar to Theorem *3.26 (Simp) of [WhiteheadRussell]
     p. 112.  (Contributed by NM, 3-Jan-1993.)  (Proof shortened by Wolf
     Lammen, 21-Jul-2012.) $)
  simplim $p |- ( -. ( ph -> ps ) -> ph ) $=
    ( wi pm2.21 con1i ) AABCABDE $.

  $( General instance of Theorem *2.5 of [WhiteheadRussell] p. 107.
     (Contributed by NM, 3-Jan-2005.)  (Proof shortened by Wolf Lammen,
     9-Oct-2012.) $)
  pm2.5g $p |- ( -. ( ph -> ps ) -> ( -. ph -> ch ) ) $=
    ( wi wn simplim pm2.24d ) ABDEACABFG $.

  $( Theorem *2.5 of [WhiteheadRussell] p. 107.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.5 $p |- ( -. ( ph -> ps ) -> ( -. ph -> ps ) ) $=
    ( pm2.5g ) ABBC $.

  $( 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.)  (Proof shortened by Wolf Lammen, 8-Oct-2012.) $)
  pm2.52 $p |- ( -. ( ph -> ps ) -> ( -. ph -> -. ps ) ) $=
    ( wn conax1k ) ABACD $.

  $( A general instance of Theorem *2.521 of [WhiteheadRussell] p. 107.
     (Contributed by BJ, 28-Oct-2023.) $)
  pm2.521g $p |- ( -. ( ph -> ps ) -> ( ps -> ch ) ) $=
    ( wi wn conax1 pm2.21d ) ABDEBCABFG $.

  $( A general instance of Theorem *2.521 of [WhiteheadRussell] p. 107.
     (Contributed by NM, 3-Jan-2005.)  (Proof shortened by Wolf Lammen,
     8-Oct-2012.) $)
  pm2.521g2 $p |- ( -. ( ph -> ps ) -> ( ch -> ph ) ) $=
    ( wi wn simplim a1d ) ABDEACABFG $.

  $( Theorem *2.521 of [WhiteheadRussell] p. 107.  Instance of ~ pm2.521g and
     of ~ pm2.521g2 .  (Contributed by NM, 3-Jan-2005.) $)
  pm2.521 $p |- ( -. ( ph -> ps ) -> ( ps -> ph ) ) $=
    ( pm2.521g ) ABAC $.

  $( Exportation theorem ~ pm3.3 (closed form of ~ ex ) expressed with
     primitive connectives.  (Contributed by NM, 28-Dec-1992.) $)
  expt $p |- ( ( -. ( ph -> -. ps ) -> ch ) -> ( ph -> ( ps -> ch ) ) ) $=
    ( wn wi pm3.2im imim1d com12 ) AABDEDZCEBCEABICABFGH $.

  $( Importation theorem ~ pm3.1 (closed form of ~ imp ) expressed with
     primitive connectives.  (Contributed by NM, 25-Apr-1994.)  (Proof
     shortened by Wolf Lammen, 20-Jul-2013.) $)
  impt $p |- ( ( ph -> ( ps -> ch ) ) -> ( -. ( ph -> -. ps ) -> ch ) ) $=
    ( wi wn simprim simplim imim1i mpdi ) ABCDZDABEZDEZBCABFLAJAKGHI $.

  ${
    pm2.61d.1 $e |- ( ph -> ( ps -> ch ) ) $.
    pm2.61d.2 $e |- ( ph -> ( -. ps -> ch ) ) $.
    $( Deduction eliminating an antecedent.  (Contributed by NM, 27-Apr-1994.)
       (Proof shortened by Wolf Lammen, 12-Sep-2013.) $)
    pm2.61d $p |- ( ph -> ch ) $=
      ( wn con1d syld pm2.18d ) ACACFBCABCEGDHI $.
  $}

  ${
    pm2.61d1.1 $e |- ( ph -> ( ps -> ch ) ) $.
    pm2.61d1.2 $e |- ( -. ps -> ch ) $.
    $( Inference eliminating an antecedent.  (Contributed by NM,
       15-Jul-2005.) $)
    pm2.61d1 $p |- ( ph -> ch ) $=
      ( wn wi a1i pm2.61d ) ABCDBFCGAEHI $.
  $}

  ${
    pm2.61d2.1 $e |- ( ph -> ( -. ps -> ch ) ) $.
    pm2.61d2.2 $e |- ( ps -> ch ) $.
    $( Inference eliminating an antecedent.  (Contributed by NM,
       18-Aug-1993.) $)
    pm2.61d2 $p |- ( ph -> ch ) $=
      ( wi a1i pm2.61d ) ABCBCFAEGDH $.
  $}

  ${
    pm2.61i.1 $e |- ( ph -> ps ) $.
    pm2.61i.2 $e |- ( -. ph -> ps ) $.
    $( Inference eliminating an antecedent.  (Contributed by NM, 5-Apr-1994.)
       (Proof shortened by Wolf Lammen, 19-Nov-2023.) $)
    pm2.61i $p |- ps $=
      ( nsyl4 pm2.18i ) BABBCDEF $.
  $}

  ${
    pm2.61ii.1 $e |- ( -. ph -> ( -. ps -> ch ) ) $.
    pm2.61ii.2 $e |- ( ph -> ch ) $.
    pm2.61ii.3 $e |- ( ps -> ch ) $.
    $( Inference eliminating two antecedents.  (Contributed by NM, 4-Jan-1993.)
       (Proof shortened by Josh Purinton, 29-Dec-2000.) $)
    pm2.61ii $p |- ch $=
      ( wn pm2.61d2 pm2.61i ) ACEAGBCDFHI $.
  $}

  ${
    pm2.61nii.1 $e |- ( ph -> ( ps -> ch ) ) $.
    pm2.61nii.2 $e |- ( -. ph -> ch ) $.
    pm2.61nii.3 $e |- ( -. ps -> ch ) $.
    $( Inference eliminating two antecedents.  (Contributed by NM,
       13-Jul-2005.)  (Proof shortened by Andrew Salmon, 25-May-2011.)  (Proof
       shortened by Wolf Lammen, 13-Nov-2012.) $)
    pm2.61nii $p |- ch $=
      ( pm2.61d1 pm2.61i ) ACABCDFGEH $.
  $}

  ${
    pm2.61iii.1 $e |- ( -. ph -> ( -. ps -> ( -. ch -> th ) ) ) $.
    pm2.61iii.2 $e |- ( ph -> th ) $.
    pm2.61iii.3 $e |- ( ps -> th ) $.
    pm2.61iii.4 $e |- ( ch -> th ) $.
    $( Inference eliminating three antecedents.  (Contributed by NM,
       2-Jan-2002.)  (Proof shortened by Wolf Lammen, 22-Sep-2013.) $)
    pm2.61iii $p |- th $=
      ( wn wi a1d pm2.61ii pm2.61i ) CDHABCIZDJEADNFKBDNGKLM $.
  $}

  ${
    ja.1 $e |- ( -. ph -> ch ) $.
    ja.2 $e |- ( ps -> ch ) $.
    $( Inference joining the antecedents of two premises.  For partial
       converses, see ~ jarri and ~ jarli .  (Contributed by NM, 24-Jan-1993.)
       (Proof shortened by Mel L. O'Cat, 19-Feb-2008.) $)
    ja $p |- ( ( ph -> ps ) -> ch ) $=
      ( wi imim2i pm2.61d1 ) ABFACBCAEGDH $.
  $}

  ${
    jad.1 $e |- ( ph -> ( -. ps -> th ) ) $.
    jad.2 $e |- ( ph -> ( ch -> th ) ) $.
    $( Deduction form of ~ ja .  (Contributed by Scott Fenton, 13-Dec-2010.)
       (Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
    jad $p |- ( ph -> ( ( ps -> ch ) -> th ) ) $=
      ( wi wn com12 ja ) BCGADBCADGABHDEIACDFIJI $.
  $}

  $( Weak Clavius law.  If a formula implies its negation, then it is false.  A
     form of "reductio ad absurdum", which can be used in proofs by
     contradiction.  Theorem *2.01 of [WhiteheadRussell] p. 100.  Provable in
     minimal calculus, contrary to the Clavius law ~ pm2.18 .  (Contributed by
     NM, 18-Aug-1993.)  (Proof shortened by Mel L. O'Cat, 21-Nov-2008.)  (Proof
     shortened by Wolf Lammen, 31-Oct-2012.) $)
  pm2.01 $p |- ( ( ph -> -. ph ) -> -. ph ) $=
    ( wn id ja ) AABZEECZFD $.

  ${
    pm2.01i.1 $e |- ( ph -> -. ph ) $.
    $( Inference associated with the weak Clavius law ~ pm2.01 .  (Contributed
       by BJ, 30-Mar-2020.) $)
    pm2.01i $p |- -. ph $=
      ( wn wi pm2.01 ax-mp ) AACZDGBAEF $.
  $}

  ${
    pm2.01d.1 $e |- ( ph -> ( ps -> -. ps ) ) $.
    $( Deduction based on reductio ad absurdum.  (Contributed by NM,
       18-Aug-1993.)  (Proof shortened by Wolf Lammen, 5-Mar-2013.) $)
    pm2.01d $p |- ( ph -> -. ps ) $=
      ( wn id pm2.61d1 ) ABBDZCGEF $.
  $}

  $( Theorem *2.6 of [WhiteheadRussell] p. 107.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.6 $p |- ( ( -. ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) $=
    ( wn wi id idd jad ) ACBDZABBHEHBFG $.

  $( Theorem *2.61 of [WhiteheadRussell] p. 107.  Useful for eliminating an
     antecedent.  (Contributed by NM, 4-Jan-1993.)  (Proof shortened by Wolf
     Lammen, 22-Sep-2013.) $)
  pm2.61 $p |- ( ( ph -> ps ) -> ( ( -. ph -> ps ) -> ps ) ) $=
    ( wn wi pm2.6 com12 ) ACBDABDBABEF $.

  $( Theorem *2.65 of [WhiteheadRussell] p. 107.  Proof by contradiction.
     (Contributed by NM, 21-Jun-1993.)  (Proof shortened by Wolf Lammen,
     8-Mar-2013.) $)
  pm2.65 $p |- ( ( ph -> ps ) -> ( ( ph -> -. ps ) -> -. ph ) ) $=
    ( wi wn idd con3 jad ) ABCZABDADZHIEABFG $.

  ${
    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 con2i con3i pm2.61i ) BAEABDFABCGH $.
  $}

  ${
    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.)  (Proof shortened by Wolf
       Lammen, 22-Jul-2019.) $)
    pm2.21dd $p |- ( ph -> ch ) $=
      ( pm2.65i pm2.21i ) ACABDEFG $.
  $}

  ${
    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 $.
  $}

  ${
    mto.1 $e |- -. ps $.
    mto.2 $e |- ( ph -> ps ) $.
    $( The rule of modus tollens.  The rule says, "if ` ps ` is not true, and
       ` ph ` implies ` ps ` , then ` ph ` must also be not true".  Modus
       tollens is short for "modus tollendo tollens", a Latin phrase that means
       "the mode that by denying denies" - remark in [Sanford] p. 39.  It is
       also called denying the consequent.  Modus tollens is closely related to
       modus ponens ~ ax-mp .  Note that this rule is also valid in
       intuitionistic logic.  Inference associated with ~ con3i .  (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 $.
  $}

  ${
    mt2.1 $e |- ps $.
    mt2.2 $e |- ( ph -> -. ps ) $.
    $( A rule similar to modus tollens.  Inference associated with ~ con2i .
       (Contributed by NM, 19-Aug-1993.)  (Proof shortened by Wolf Lammen,
       10-Sep-2013.) $)
    mt2 $p |- -. ph $=
      ( a1i pm2.65i ) ABBACEDF $.
  $}

  ${
    mt3.1 $e |- -. ps $.
    mt3.2 $e |- ( -. ph -> ps ) $.
    $( A rule similar to modus tollens.  Inference associated with ~ con1i .
       (Contributed by NM, 18-May-1994.)  (Proof shortened by Wolf Lammen,
       11-Sep-2013.) $)
    mt3 $p |- ph $=
      ( wn mto notnotri ) AAEBCDFG $.
  $}

  $( Peirce's axiom.  A non-intuitionistic implication-only statement.  Added
     to intuitionistic (implicational) propositional calculus, it gives
     classical (implicational) propositional calculus.  For another
     non-intuitionistic positive statement, see ~ curryax .  When ` F. ` is
     substituted for ` ps ` , then this becomes the Clavius law ~ pm2.18 .
     (Contributed by NM, 29-Dec-1992.)  (Proof shortened by Wolf Lammen,
     9-Oct-2012.) $)
  peirce $p |- ( ( ( ph -> ps ) -> ph ) -> ph ) $=
    ( wi simplim id ja ) ABCAAABDAEF $.

  $( The Inversion Axiom of the infinite-valued sentential logic (L-infinity)
     of Lukasiewicz.  Using ~ dfor2 , we can see that this essentially
     expresses "disjunction commutes".  Theorem *2.69 of [WhiteheadRussell]
     p. 108.  It is a special instance of the axiom "Roll", see ~ peirceroll .
     (Contributed by NM, 12-Aug-2004.) $)
  looinv $p |- ( ( ( ph -> ps ) -> ps ) -> ( ( ps -> ph ) -> ph ) ) $=
    ( wi imim1 peirce syl6 ) ABCZBCBACGACAGBADABEF $.

  $( A self-implication (see ~ id ) does not imply its own negation.  The
     justification theorem ~ bijust is one of its instances.  (Contributed by
     NM, 11-May-1999.)  (Proof shortened by Josh Purinton, 29-Dec-2000.)
     Extract ~ bijust0 from proof of ~ bijust .  (Revised by BJ,
     19-Mar-2020.) $)
  bijust0 $p |- -. ( ( ph -> ph ) -> -. ( ph -> ph ) ) $=
    ( wi wn id pm2.01 mt2 ) AABZGCBGADGEF $.

  $( Theorem used to justify the definition of the biconditional ~ df-bi .
     Instance of ~ bijust0 .  (Contributed by NM, 11-May-1999.) $)
  bijust $p |- -. ( ( -. ( ( ph -> ps ) -> -. ( ps -> ph ) )
                   -> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) )
              -> -. ( -. ( ( ph -> ps ) -> -. ( ps -> ph ) )
                   -> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) ) ) $=
    ( wi wn bijust0 ) ABCBACDCDE $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Logical equivalence
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  Definition ~ df-bi in this section is our first definition, which
  introduces and defines the biconditional connective ` <-> ` used to denote
  logical equivalence.  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 ( ~ df-an is its first use), as it
  allows to use logic to manipulate definitions directly.  This greatly
  simplifies many proofs since it eliminates the need for a separate mechanism
  for introducing and eliminating definitions.

  A note on definitions: definitions are required to be eliminable (that is, a
  theorem stated in terms of the defined symbol can also be stated without it)
  and conservative (that is, a theorem whose statement does not contain the
  defined symbol can be proved without using that definition).  This means that
  a definition does not increase the expressive power nor the deductive power,
  respectively, of a theory.  On the other hand, definitions are often useful
  to write shorter proofs, so in (i)set.mm we will generally not try to avoid
  them.  This is why, for instance, some theorems which do not contain
  disjunction in their statement are placed after the section on disjunction
  because a shorter proof using disjunction is possible.

$)

  $( Declare the biconditional connective. $)
  $c <-> $.  $( Bidirectional arrow (read:  "if and only if" or
                "is logically equivalent to") $)

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

  $( Define the biconditional (logical "iff" or "if and only if"), also called
     biimplication.

     Definition ~ df-bi in this section is our first definition, which
     introduces and defines the biconditional connective ` <-> ` .  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 ( ~ df-or is its
     first use), as it allows to use logic to manipulate definitions directly.
     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 ~ bijust .  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 ~ bijust 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 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 a naming convention for human readability.)

     After we define the constant true ` T. ` ( ~ df-tru ) and the constant
     false ` F. ` ( ~ df-fal ), we will be able to prove these truth table
     values: ` ( ( T. <-> T. ) <-> T. ) ` ( ~ trubitru ),
     ` ( ( T. <-> F. ) <-> F. ) ` ( ~ trubifal ), ` ( ( F. <-> T. ) <-> F. ) `
     ( ~ falbitru ), and ` ( ( F. <-> F. ) <-> T. ) ` ( ~ falbifal ).

     See ~ dfbi1 , ~ dfbi2 , and ~ dfbi3 for theorems suggesting typical
     textbook definitions of ` <-> ` , showing that our definition has the
     properties we expect.  Theorem ~ dfbi1 is particularly useful if we want
     to eliminate ` <-> ` from an expression to convert it to primitives.
     Theorem ~ dfbi shows this definition rewritten in an abbreviated form
     after conjunction is introduced, for easier understanding.

     Contrast with ` \/ ` ( ~ df-or ), ` -> ` ( ~ wi ), ` -/\ ` ( ~ df-nan ),
     and ` \/_ ` ( ~ df-xor ).  In some sense ` <-> ` returns true if two truth
     values are equal; ` = ` ( ~ df-cleq ) returns true if two classes are
     equal.  (Contributed by NM, 27-Dec-1992.) $)
  df-bi $a |- -. ( ( ( ph <-> ps ) -> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) )
        -> -. ( -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) -> ( ph <-> ps ) ) ) $.

  $( $j justification 'bijust' for 'df-bi'; $)

  $( Property of the biconditional connective.  (Contributed by NM,
     11-May-1999.) $)
  impbi $p |- ( ( ph -> ps ) -> ( ( ps -> ph ) -> ( ph <-> ps ) ) ) $=
    ( wi wb wn df-bi simprim ax-mp expi ) ABCZBACZABDZLJKECEZCZMLCZECEOABFNOGHI
    $.

  ${
    impbii.1 $e |- ( ph -> ps ) $.
    impbii.2 $e |- ( ps -> ph ) $.
    $( Infer an equivalence from an implication and its converse.  Inference
       associated with ~ impbi .  (Contributed by NM, 29-Dec-1992.) $)
    impbii $p |- ( ph <-> ps ) $=
      ( wi wb impbi mp2 ) ABEBAEABFCDABGH $.
  $}

  ${
    impbidd.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
    impbidd.2 $e |- ( ph -> ( ps -> ( th -> ch ) ) ) $.
    $( Deduce an equivalence from two implications.  Double deduction
       associated with ~ impbi and ~ impbii .  Deduction associated with
       ~ impbid .  (Contributed by Rodolfo Medina, 12-Oct-2010.) $)
    impbidd $p |- ( ph -> ( ps -> ( ch <-> th ) ) ) $=
      ( wi wb impbi 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 wb impbi syl2imc ) BCDGADCGCDHEFCDIJ $.
  $}

  ${
    impbid.1 $e |- ( ph -> ( ps -> ch ) ) $.
    impbid.2 $e |- ( ph -> ( ch -> ps ) ) $.
    $( Deduce an equivalence from two implications.  Deduction associated with
       ~ impbi and ~ impbii .  (Contributed by NM, 24-Jan-1993.)  Prove it from
       ~ impbid21d .  (Revised by Wolf Lammen, 3-Nov-2012.) $)
    impbid $p |- ( ph -> ( ps <-> ch ) ) $=
      ( wb impbid21d pm2.43i ) ABCFAABCDEGH $.
  $}

  $( Relate the biconditional connective to primitive connectives.  See
     ~ dfbi1ALT for an unusual version proved directly from axioms.
     (Contributed by NM, 29-Dec-1992.) $)
  dfbi1 $p |- ( ( ph <-> ps ) <-> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) ) $=
    ( wb wi wn df-bi impbi con3rr3 mt3 ) ABCZABDBADEDEZCZJKDZKJDZEDABFMNLJKGHI
    $.

  $( Alternate proof of ~ dfbi1 .  This proof, discovered by Gregory Bush on
     8-Mar-2004, has several curious properties.  First, it has only 17 steps
     directly from the axioms and ~ df-bi , compared to over 800 steps were the
     proof of ~ dfbi1 expanded into axioms.  Second, step 2 demands only the
     property of "true"; any axiom (or theorem) could be used.  It might be
     thought, therefore, that it is in some sense redundant, but in fact no
     proof is shorter than this (measured by number of steps).  Third, it
     illustrates how intermediate steps can "blow up" in size even in short
     proofs.  Fourth, the compressed proof is only 182 bytes (or 17 bytes in
     D-proof notation), but the generated web page is over 200kB with
     intermediate steps that are essentially incomprehensible to humans (other
     than Gregory Bush).  If there were an obfuscated code contest for proofs,
     this would be a contender.  This "blowing up" and incomprehensibility of
     the intermediate steps vividly demonstrate the advantages of using many
     layered intermediate theorems, since each theorem is easier to understand.
     (Contributed by Gregory Bush, 10-Mar-2004.)  (New usage is discouraged.)
     (Proof modification is discouraged.) $)
  dfbi1ALT $p |-
                ( ( ph <-> ps ) <-> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) ) $=
    ( wch wth wb wi wn df-bi ax-1 ax-mp ax-3 ax-2 ) ABEZABFBAFGFGZFNMFGFGZMNEZA
    BHCDCFFZOPFZCDIRGZQGZFZQRFSPOFZSFZFZUASUBISUCTFZFZUDUAFUEUFTGZUCGZFZUEUHUIM
    NHUHUGIJTUCKJUESIJSUCTLJJRQKJJJ $.

  $( Property of the biconditional connective.  (Contributed by NM,
     11-May-1999.) $)
  biimp $p |- ( ( ph <-> ps ) -> ( ph -> ps ) ) $=
    ( wb wi wn df-bi simplim ax-mp syl ) ABCZABDZBADEZDEZKJMDZMJDEZDENABFNOGHKL
    GI $.

  ${
    biimpi.1 $e |- ( ph <-> ps ) $.
    $( Infer an implication from a logical equivalence.  Inference associated
       with ~ biimp .  (Contributed by NM, 29-Dec-1992.) $)
    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 $.
  $}

  $( 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 wn dfbi1 simprim sylbi ) ABCABDZBADZEDEJABFIJGH $.

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

  $( Commutative law for the biconditional.  Theorem *4.21 of
     [WhiteheadRussell] p. 117.  (Contributed by NM, 11-May-1993.) $)
  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, 14-May-1993.) $)
    bicomd $p |- ( ph -> ( ch <-> ps ) ) $=
      ( wb bicom sylib ) ABCECBEDBCFG $.
  $}

  ${
    bicomi.1 $e |- ( ph <-> ps ) $.
    $( Inference from commutative law for logical equivalence.  (Contributed by
       NM, 3-Jan-1993.) $)
    bicomi $p |- ( ps <-> ph ) $=
      ( wb bicom1 ax-mp ) ABDBADCABEF $.
  $}

  ${
    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 $.
  $}

  ${
    impcon4bid.1 $e |- ( ph -> ( ps -> ch ) ) $.
    impcon4bid.2 $e |- ( ph -> ( -. ps -> -. ch ) ) $.
    $( A variation on ~ impbid with contraposition.  (Contributed by Jeff
       Hankins, 3-Jul-2009.) $)
    impcon4bid $p |- ( ph -> ( ps <-> ch ) ) $=
      ( con4d impbid ) ABCDABCEFG $.
  $}

  ${
    biimpri.1 $e |- ( ph <-> ps ) $.
    $( Infer a converse implication from a logical equivalence.  Inference
       associated with ~ biimpr .  (Contributed by NM, 29-Dec-1992.)  (Proof
       shortened by Wolf Lammen, 16-Sep-2013.) $)
    biimpri $p |- ( ps -> ph ) $=
      ( bicomi biimpi ) BAABCDE $.
  $}

  ${
    biimpd.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Deduce an implication from a logical equivalence.  Deduction associated
       with ~ biimp and ~ biimpi .  (Contributed by NM, 11-Jan-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, 11-May-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, 28-Dec-1992.) $)
    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, 21-Jun-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, 16-May-1993.)  (Proof shortened by Wolf Lammen,
       25-Oct-2012.) $)
    mpbii $p |- ( ph -> ch ) $=
      ( a1i mpbid ) ABCBADFEG $.
  $}

  ${
    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, 3-Jan-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, 3-Jan-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 $.
  $}

  ${
    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, 12-Jan-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, 21-Jun-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.) $)
    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.  Deduction
       associated with ~ biimpr and ~ biimpri .  (Contributed by NM,
       11-Jan-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, 21-Jun-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, 14-May-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, 21-Jun-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.  This is part of Frege's eighth axiom per
     Proposition 54 of [Frege1879] p. 50; see also ~ eqid .  (Contributed by
     NM, 2-Jun-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.) $)
    2thd $p |- ( ph -> ( ps <-> ch ) ) $=
      ( wb pm5.1im sylc ) ABCBCFDEBCGH $.
  $}

  $( Two self-implications (see ~ id ) are equivalent.  This theorem, rather
     trivial in our axiomatization, is (the biconditional form of) a standard
     axiom for monothetic BCI logic.  This is the most general theorem of which
     ~ trujust is an instance.  Relatedly, this would be the justification
     theorem if the definition of ` T. ` were ~ dftru2 .  (Contributed by BJ,
     7-Sep-2022.) $)
  monothetic $p |- ( ( ph -> ph ) <-> ( ps -> ps ) ) $=
    ( wi id 2th ) AACBBCADBDE $.

  ${
    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 ) $=
      ( id mpbid ) AABADCE $.
  $}

  ${
    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.  Deduction associated with ~ ibi .  (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, 3-Jan-1993.)  (Proof shortened by Wolf Lammen, 13-Oct-2012.) $)
    bitri $p |- ( ph <-> ch ) $=
      ( sylbb sylbbr impbii ) ACABCDEFABCDEGH $.
  $}

  ${
    bitr2i.1 $e |- ( ph <-> ps ) $.
    bitr2i.2 $e |- ( ps <-> ch ) $.
    $( An inference from transitive law for logical equivalence.  (Contributed
       by NM, 12-Mar-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, 2-Jun-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, 3-Jan-1993.) $)
    bitr4i $p |- ( ph <-> ch ) $=
      ( bicomi bitri ) ABCDCBEFG $.
  $}

  $( Register '<->' as an equality for its type (wff). $)
  $( $j
    equality 'wb' from 'biid' 'bicomi' 'bitri';
    definition 'dfbi1' for 'wb';
  $)

  ${
    bitrd.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    bitrd.2 $e |- ( ph -> ( ch <-> th ) ) $.
    $( Deduction form of ~ bitri .  (Contributed by NM, 12-Mar-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, 14-May-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, 30-Jun-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,
       1-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,
       12-Mar-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,
       12-Mar-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, 3-Jan-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, 3-Jan-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, 21-Jun-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, 3-Jan-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, 11-May-1993.) $)
    3bitr4g $p |- ( ph -> ( th <-> ta ) ) $=
      ( bitrid bitr4di ) ADCEDBACGFIHJ $.
  $}

  $( Double negation.  Theorem *4.13 of [WhiteheadRussell] p. 117.
     (Contributed by NM, 3-Jan-1993.) $)
  notnotb $p |- ( ph <-> -. -. ph ) $=
    ( wn notnot notnotr impbii ) AABBACADE $.

  $( A biconditional form of contraposition.  Theorem *4.1 of
     [WhiteheadRussell] p. 116.  (Contributed by NM, 11-May-1993.) $)
  con34b $p |- ( ( ph -> ps ) <-> ( -. ps -> -. ph ) ) $=
    ( wi wn con3 con4 impbii ) ABCBDADCABEBAFG $.

  ${
    con4bid.1 $e |- ( ph -> ( -. ps <-> -. ch ) ) $.
    $( A contraposition deduction.  (Contributed by NM, 21-May-1994.) $)
    con4bid $p |- ( ph -> ( ps <-> ch ) ) $=
      ( wn biimprd con4d biimpd impcon4bid ) ABCACBABEZCEZDFGAJKDHI $.
  $}

  ${
    notbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Deduction negating both sides of a logical equivalence.  (Contributed by
       NM, 21-May-1994.) $)
    notbid $p |- ( ph -> ( -. ps <-> -. ch ) ) $=
      ( wn notnotb 3bitr3g con4bid ) ABEZCEZABCIEJEDBFCFGH $.
  $}

  $( Contraposition.  Theorem *4.11 of [WhiteheadRussell] p. 117.  (Contributed
     by NM, 21-May-1994.)  (Proof shortened by Wolf Lammen, 12-Jun-2013.) $)
  notbi $p |- ( ( ph <-> ps ) <-> ( -. ph <-> -. ps ) ) $=
    ( wb wn id notbid con4bid impbii ) ABCZADBDCZIABIEFJABJEGH $.

  ${
    notbii.1 $e |- ( ph <-> ps ) $.
    $( Negate both sides of a logical equivalence.  (Contributed by NM,
       3-Jan-1993.)  (Proof shortened by Wolf Lammen, 19-May-2013.) $)
    notbii $p |- ( -. ph <-> -. ps ) $=
      ( wb wn notbi mpbi ) ABDAEBEDCABFG $.

    $( Theorem notbii is the congruence law for negation. $)
    $( $j congruence 'notbii'; $)
  $}

  ${
    con4bii.1 $e |- ( -. ph <-> -. ps ) $.
    $( A contraposition inference.  (Contributed by NM, 21-May-1994.) $)
    con4bii $p |- ( ph <-> ps ) $=
      ( wb wn notbi mpbir ) ABDAEBEDCABFG $.
  $}

  ${
    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 ) $=
      ( biimpri nsyl ) ABCDBCEFG $.
  $}

  ${
    sylnibr.1 $e |- ( ph -> -. ps ) $.
    sylnibr.2 $e |- ( ch <-> ps ) $.
    $( A mixed syllogism inference from an implication and a biconditional.
       Useful for substituting a 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 $.
  $}

  ${
    imbi2i.1 $e |- ( ph <-> ps ) $.
    $( Introduce an antecedent to both sides of a logical equivalence.  This
       and the next three rules are useful for building up wff's around a
       definition, in order to make use of the definition.  (Contributed by NM,
       3-Jan-1993.)  (Proof shortened by Wolf Lammen, 6-Feb-2013.) $)
    imbi2i $p |- ( ( ch -> ph ) <-> ( ch -> ps ) ) $=
      ( wb a1i pm5.74i ) CABABECDFG $.
  $}

  ${
    bibi2i.1 $e |- ( ph <-> ps ) $.
    $( Inference adding a biconditional to the left in an equivalence.
       (Contributed by NM, 26-May-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, 26-May-1993.) $)
    bibi1i $p |- ( ( ph <-> ch ) <-> ( ps <-> ch ) ) $=
      ( wb bicom bibi2i 3bitri ) ACECAECBEBCEACFABCDGCBFH $.

    ${
      bibi12i.2 $e |- ( ch <-> th ) $.
      $( The equivalence of two equivalences.  (Contributed by NM,
         26-May-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, 11-May-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, 11-May-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, 11-May-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, 11-May-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, 16-May-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, 26-May-1993.) $)
    bibi12d $p |- ( ph -> ( ( ps <-> th ) <-> ( ch <-> ta ) ) ) $=
      ( wb bibi1d bibi2d bitrd ) ABDHCDHCEHABCDFIADECGJK $.
  $}

  $( Closed form of ~ imbi12i .  Was automatically derived from its "Virtual
     Deduction" version and the Metamath program "MM-PA> MINIMIZE__WITH *"
     command.  (Contributed by Alan Sare, 18-Mar-2012.) $)
  imbi12 $p |- ( ( ph <-> ps ) ->
                    ( ( ch <-> th ) -> ( ( ph -> ch ) <-> ( ps -> th ) ) ) ) $=
    ( wb wi wn simplim simprim imbi12d expi ) ABEZCDEZACFBDFELMGZFGABCDLNHLMIJK
    $.

  $( 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, 3-Jan-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, 1-Aug-1993.) $)
    imbi12i $p |- ( ( ph -> ch ) <-> ( ps -> th ) ) $=
      ( wb wi imbi12 mp2 ) ABGCDGACHBDHGEFABCDIJ $.

    $( Theorem imbi12i is the congruence law for implication. $)
    $( $j congruence 'imbi12i'; $)
  $}

  $( 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 $.

  $( Closed nested implication form of ~ bitr3i .  Derived automatically from
     ~ bitr3VD .  (Contributed by Alan Sare, 31-Dec-2011.) $)
  bitr3 $p |- ( ( ph <-> ps ) -> ( ( ph <-> ch ) -> ( ps <-> ch ) ) ) $=
    ( wb bibi1 biimpd ) ABDACDBCDABCEF $.

  $( Contraposition.  Theorem *4.12 of [WhiteheadRussell] p. 117.  (Contributed
     by NM, 15-Apr-1995.)  (Proof shortened by Wolf Lammen, 3-Jan-2013.) $)
  con2bi $p |- ( ( ph <-> -. ps ) <-> ( ps <-> -. ph ) ) $=
    ( wn wb notbi notnotb bibi2i bicom 3bitr2i ) ABCZDACZJCZDKBDBKDAJEBLKBFGKBH
    I $.

  ${
    con2bid.1 $e |- ( ph -> ( ps <-> -. ch ) ) $.
    $( A contraposition deduction.  (Contributed by NM, 15-Apr-1995.) $)
    con2bid $p |- ( ph -> ( ch <-> -. ps ) ) $=
      ( wn wb con2bi sylibr ) ABCEFCBEFDCBGH $.
  $}

  ${
    con1bid.1 $e |- ( ph -> ( -. ps <-> ch ) ) $.
    $( A contraposition deduction.  (Contributed by NM, 9-Oct-1999.) $)
    con1bid $p |- ( ph -> ( -. ch <-> ps ) ) $=
      ( wn bicomd con2bid ) ABCEACBABECDFGF $.
  $}

  ${
    con1bii.1 $e |- ( -. ph <-> ps ) $.
    $( A contraposition inference.  (Contributed by NM, 12-Mar-1993.)  (Proof
       shortened by Wolf Lammen, 13-Oct-2012.) $)
    con1bii $p |- ( -. ps <-> ph ) $=
      ( wn notnotb xchbinx bicomi ) ABDAADBAECFG $.
  $}

  ${
    con2bii.1 $e |- ( ph <-> -. ps ) $.
    $( A contraposition inference.  (Contributed by NM, 12-Mar-1993.) $)
    con2bii $p |- ( ps <-> -. ph ) $=
      ( wn notnotb xchbinxr ) BBDABECF $.
  $}

  $( Contraposition.  Bidirectional version of ~ con1 .  (Contributed by NM,
     3-Jan-1993.) $)
  con1b $p |- ( ( -. ph -> ps ) <-> ( -. ps -> ph ) ) $=
    ( wn wi con1 impbii ) ACBDBCADABEBAEF $.

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

  $( 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 $.
  $}

  ${
    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.)  (Proof
     shortened by Wolf Lammen, 12-Nov-2012.) $)
  mtt $p |- ( -. ph -> ( -. ps <-> ( ps -> ph ) ) ) $=
    ( wn wi biimt con34b bitr4di ) ACZBCZHIDBADHIEBAFG $.

  $( If a proposition is false, then implying it is equivalent to being false.
     One of four theorems that can be used to simplify an implication
     ` ( ph -> ps ) ` , the other ones being ~ ax-1 (true consequent), ~ pm2.21
     (false antecedent), ~ pm5.5 (true antecedent).  (Contributed by Mario
     Carneiro, 26-Apr-2019.)  (Proof shortened by Wolf Lammen, 26-May-2019.) $)
  imnot $p |- ( -. ps -> ( ( ph -> ps ) <-> -. ph ) ) $=
    ( wn wi mtt bicomd ) BCACABDBAEF $.

  $( Theorem *5.501 of [WhiteheadRussell] p. 125.  (Contributed by NM,
     3-Jan-2005.) $)
  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 a 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 $.
  $}

  $( The negation of a wff is equivalent to the wff's equivalence to falsehood.
     (Contributed by Juha Arpiainen, 19-Jan-2006.)  (Proof shortened by Wolf
     Lammen, 28-Jan-2013.) $)
  nbn2 $p |- ( -. ph -> ( -. ps <-> ( ph <-> ps ) ) ) $=
    ( wn wb pm5.501 notbi bitr4di ) ACZBCZHIDABDHIEABFG $.

  $( 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, 21-Jun-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 $.
  $}

  $( 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.) $)
  pm5.21im $p |- ( -. ph -> ( -. ps -> ( ph <-> ps ) ) ) $=
    ( wn wb nbn2 biimpd ) ACBCABDABEF $.

  ${
    2false.1 $e |- -. ph $.
    2false.2 $e |- -. ps $.
    $( Two falsehoods are equivalent.  (Contributed by NM, 4-Apr-2005.)  (Proof
       shortened by Wolf Lammen, 19-May-2013.) $)
    2false $p |- ( ph <-> ps ) $=
      ( wn 2th con4bii ) ABAEBECDFG $.
  $}

  ${
    2falsed.1 $e |- ( ph -> -. ps ) $.
    2falsed.2 $e |- ( ph -> -. ch ) $.
    $( Two falsehoods are equivalent (deduction form).  (Contributed by NM,
       11-Oct-2013.)  (Proof shortened by Wolf Lammen, 11-Apr-2024.) $)
    2falsed $p |- ( ph -> ( ps <-> ch ) ) $=
      ( wn 2thd con4bid ) ABCABFCFDEGH $.
  $}

  ${
    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.) $)
      pm5.21nii $p |- ( ph <-> ch ) $=
        ( wb pm5.21ni pm2.61i ) BACGFABCDEHI $.
    $}
  $}

  ${
    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.)  (Proof
       shortened by Wolf Lammen, 6-Oct-2013.) $)
    pm5.21ndd $p |- ( ph -> ( ch <-> th ) ) $=
      ( wb wn con3d pm5.21im syl6c pm2.61d ) ABCDHZGABICIDINACBEJADBFJCDKLM $.
  $}

  ${
    bija.1 $e |- ( ph -> ( ps -> ch ) ) $.
    bija.2 $e |- ( -. ph -> ( -. ps -> ch ) ) $.
    $( Combine antecedents into a single biconditional.  This inference,
       reminiscent of ~ ja , is reversible:  The hypotheses can be deduced from
       the conclusion alone (see ~ pm5.1im and ~ pm5.21im ).  (Contributed by
       Wolf Lammen, 13-May-2013.) $)
    bija $p |- ( ( ph <-> ps ) -> ch ) $=
      ( wb biimpr syli wn biimp con3d pm2.61d ) ABFZBCBMACABGDHBIMAICMABABJKEHL
      $.
  $}

  $( Theorem *5.18 of [WhiteheadRussell] p. 124.  This theorem says that
     logical equivalence is the same as negated "exclusive or".  (Contributed
     by NM, 28-Jun-2002.)  (Proof shortened by Andrew Salmon, 20-Jun-2011.)
     (Proof shortened by Wolf Lammen, 15-Oct-2013.) $)
  pm5.18 $p |- ( ( ph <-> ps ) <-> -. ( ph <-> -. ps ) ) $=
    ( wb wn pm5.501 con1bid bitr2d nbn2 pm2.61i ) AABCZABDZCZDZCAMBJABLAKEFABEG
    ADZMKJNKLAKHFABHGI $.

  $( Two ways to express "exclusive or".  (Contributed by NM, 1-Jan-2006.) $)
  xor3 $p |- ( -. ( ph <-> ps ) <-> ( ph <-> -. ps ) ) $=
    ( wn wb pm5.18 con2bii bicomi ) ABCDZABDZCIHABEFG $.

  $( Move negation outside of biconditional.  Compare Theorem *5.18 of
     [WhiteheadRussell] p. 124.  (Contributed by NM, 27-Jun-2002.)  (Proof
     shortened by Wolf Lammen, 20-Sep-2013.) $)
  nbbn $p |- ( ( -. ph <-> ps ) <-> -. ( ph <-> ps ) ) $=
    ( wb wn xor3 con2bi bicom 3bitrri ) ABCDABDCBADZCIBCABEABFBIGH $.

  $( Associative law for the biconditional.  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 .
     Interestingly, this law was not included in _Principia Mathematica_ but
     was apparently first noted by Jan Lukasiewicz circa 1923.  (Contributed by
     NM, 8-Jan-2005.)  (Proof shortened by Juha Arpiainen, 19-Jan-2006.)
     (Proof shortened by Wolf Lammen, 21-Sep-2013.) $)
  biass $p |- ( ( ( ph <-> ps ) <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) ) $=
    ( wb pm5.501 bibi1d bitr3d wn nbbn nbn2 bitr3id pm2.61i ) AABDZCDZABCDZDZDA
    ONPABMCABEFAOEGAHZOHZNPRBHZCDQNBCIQSMCABJFKAOJGL $.

  $( Lukasiewicz's shortest axiom for equivalential calculus.  Storrs McCall,
     ed., _Polish Logic 1920-1939_ (Oxford, 1967), p. 96.  (Contributed by NM,
     10-Jan-2005.) $)
  biluk $p |- ( ( ph <-> ps ) <-> ( ( ch <-> ps ) <-> ( ph <-> ch ) ) ) $=
    ( wb bicom bibi1i biass bitri mpbi bitr4i ) ABDZCBACDZDZDZCBDLDKCDZMDKNDOBA
    DZCDMKPCABEFBACGHKCMGICBLGJ $.

  $( Theorem *5.19 of [WhiteheadRussell] p. 124.  (Contributed by NM,
     3-Jan-2005.) $)
  pm5.19 $p |- -. ( ph <-> -. ph ) $=
    ( wb wn biid pm5.18 mpbi ) AABAACBCADAAEF $.

  $( Logical equivalence of commuted antecedents.  Part of Theorem *4.87 of
     [WhiteheadRussell] p. 122.  (Contributed by NM, 11-May-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 pm5.5 pm5.74i ) AABCBABDE $.

  $( 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 $.

  $( Theorem *4.8 of [WhiteheadRussell] p. 122.  (Contributed by NM,
     3-Jan-2005.) $)
  pm4.8 $p |- ( ( ph -> -. ph ) <-> -. ph ) $=
    ( wn wi pm2.01 ax-1 impbii ) AABZCGADGAEF $.

  $( A formula is equivalent to its negation implying it.  Theorem *4.81 of
     [WhiteheadRussell] p. 122.  Note that the second step, using ~ pm2.24 ,
     could also use ~ ax-1 .  (Contributed by NM, 3-Jan-2005.) $)
  pm4.81 $p |- ( ( -. ph -> ph ) <-> ph ) $=
    ( wn wi pm2.18 pm2.24 impbii ) ABACAADAAEF $.

  $( 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 $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Logical conjunction
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  This section defines conjunction of two formulas, denoted by infix " ` /\ ` "
  and read "and".  It is defined in terms of implication and negation, which is
  possible in classical logic (but not in intuitionistic logic: see iset.mm).

  After the definition, we briefly introduce conversion of simple expressions
  to and from conjunction.  Two simple operations called importation ( ~ imp )
  and exportation ( ~ ex ) follow.  In the propositions-as-types
  interpretation, they correspond to uncurrying and currying respectively. They
  are foundational for this section.  Most of the theorems proved here trace
  back to them, mostly indirectly, in a layered fashion, where more complex
  expressions are built from simpler ones.  Here are some of these successive
  layers:
  importation and exportation,
  commutativity and associativity laws,
  adding antecedents and simplifying,
  conjunction of consequents,
  syllogisms,
  etc.

  As indicated in the "note on definitions" in the section comment for logical
  equivalence, some theorems containing only implication, negation and
  conjunction are placed in the section after disjunction since theirs proofs
  use disjunction (although this is not required since definitions are
  conservative, see said section comment).

$)

  $( Declare connective for conjunction ("and"). $)
  $c /\ $.  $( Wedge (read:  "and") $)

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

  $( Define conjunction (logical "and").  Definition of [Margaris] p. 49.  When
     both the left and right operand are true, the result is true; when either
     is false, the result is false.  For example, it is true that
     ` ( 2 = 2 /\ 3 = 3 ) ` .  After we define the constant true ` T. `
     ( ~ df-tru ) and the constant false ` F. ` ( ~ df-fal ), we will be able
     to prove these truth table values: ` ( ( T. /\ T. ) <-> T. ) `
     ( ~ truantru ), ` ( ( T. /\ F. ) <-> F. ) ` ( ~ truanfal ),
     ` ( ( F. /\ T. ) <-> F. ) ` ( ~ falantru ), and
     ` ( ( F. /\ F. ) <-> F. ) ` ( ~ falanfal ).

     This is our first use of the biconditional connective in a definition; we
     use the biconditional connective in place of the traditional "<=def=>",
     which means the same thing, except that we can manipulate the
     biconditional connective directly in proofs rather than having to rely on
     an informal definition substitution rule.  Note that if we mechanically
     substitute ` -. ( ph -> -. ps ) ` for ` ( ph /\ ps ) ` , we end up with an
     instance of previously proved theorem ~ biid .  This is the justification
     for the definition, along with the fact that it introduces a new symbol
     ` /\ ` .  Contrast with ` \/ ` ( ~ df-or ), ` -> ` ( ~ wi ), ` -/\ `
     ( ~ df-nan ), and ` \/_ ` ( ~ df-xor ).  (Contributed by NM,
     5-Jan-1993.) $)
  df-an $a |- ( ( ph /\ ps ) <-> -. ( ph -> -. ps ) ) $.

  $( Theorem *4.63 of [WhiteheadRussell] p. 120.  (Contributed by NM,
     3-Jan-2005.) $)
  pm4.63 $p |- ( -. ( ph -> -. ps ) <-> ( ph /\ ps ) ) $=
    ( wa wn wi df-an bicomi ) ABCABDEDABFG $.

  $( Theorem *4.67 of [WhiteheadRussell] p. 120.  (Contributed by NM,
     3-Jan-2005.) $)
  pm4.67 $p |- ( -. ( -. ph -> -. ps ) <-> ( -. ph /\ ps ) ) $=
    ( wn pm4.63 ) ACBD $.

  $( Express an implication in terms of a negated conjunction.  (Contributed by
     NM, 9-Apr-1994.) $)
  imnan $p |- ( ( ph -> -. ps ) <-> -. ( ph /\ ps ) ) $=
    ( wa wn wi df-an con2bii ) ABCABDEABFG $.

  ${
    imnani.1 $e |- -. ( ph /\ ps ) $.
    $( Infer an implication from a negated conjunction.  (Contributed by Mario
       Carneiro, 28-Sep-2015.) $)
    imnani $p |- ( ph -> -. ps ) $=
      ( wn wi wa imnan mpbir ) ABDEABFDCABGH $.
  $}

  $( Implication in terms of conjunction and negation.  Theorem 3.4(27) of
     [Stoll] p. 176.  (Contributed by NM, 12-Mar-1993.)  (Proof shortened by
     Wolf Lammen, 30-Oct-2012.) $)
  iman $p |- ( ( ph -> ps ) <-> -. ( ph /\ -. ps ) ) $=
    ( wi wn wa notnotb imbi2i imnan bitri ) ABCABDZDZCAJEDBKABFGAJHI $.

  $( Law of noncontradiction.  Theorem *3.24 of [WhiteheadRussell] p. 111 (who
     call it the "law of contradiction").  (Contributed by NM, 16-Sep-1993.)
     (Proof shortened by Wolf Lammen, 24-Nov-2012.) $)
  pm3.24 $p |- -. ( ph /\ -. ph ) $=
    ( wi wn wa id iman mpbi ) AABAACDCAEAAFG $.

  $( Express a conjunction in terms of a negated implication.  (Contributed by
     NM, 2-Aug-1994.) $)
  annim $p |- ( ( ph /\ -. ps ) <-> -. ( ph -> ps ) ) $=
    ( wi wn wa iman con2bii ) ABCABDEABFG $.

  $( Theorem *4.61 of [WhiteheadRussell] p. 120.  (Contributed by NM,
     3-Jan-2005.) $)
  pm4.61 $p |- ( -. ( ph -> ps ) <-> ( ph /\ -. ps ) ) $=
    ( wn wa wi annim bicomi ) ABCDABECABFG $.

  $( Theorem *4.65 of [WhiteheadRussell] p. 120.  (Contributed by NM,
     3-Jan-2005.) $)
  pm4.65 $p |- ( -. ( -. ph -> ps ) <-> ( -. ph /\ -. ps ) ) $=
    ( wn pm4.61 ) ACBD $.

  ${
    imp.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Importation inference.  (Contributed by NM, 3-Jan-1993.)  (Proof
       shortened by Eric Schmidt, 22-Dec-2006.) $)
    imp $p |- ( ( ph /\ ps ) -> ch ) $=
      ( wa wn wi df-an impi sylbi ) ABEABFGFCABHABCDIJ $.

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

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  ${
    impd.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 $.
  $}

  ${
    ex.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.)  A translation of natural deduction
       rule ` -> ` I ( ` -> ` introduction), see ~ natded .  (Contributed by
       NM, 3-Jan-1993.)  (Proof shortened by Eric Schmidt, 22-Dec-2006.) $)
    ex $p |- ( ph -> ( ps -> ch ) ) $=
      ( wn wi wa df-an sylbir expi ) ABCABEFEABGCABHDIJ $.

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

  ${
    expd.1 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
    $( Commuted form of ~ expd .  (Contributed by Alan Sare, 18-Mar-2012.)
       Shorten ~ expd .  (Revised by Wolf Lammen, 28-Jul-2022.) $)
    expdcom $p |- ( ps -> ( ch -> ( ph -> th ) ) ) $=
      ( wi wa com12 ex ) BCADFABCGDEHI $.

    $( Exportation deduction.  (Contributed by NM, 20-Aug-1993.)  (Proof
       shortened by Wolf Lammen, 28-Jul-2022.) $)
    expd $p |- ( ph -> ( ps -> ( ch -> th ) ) ) $=
      ( expdcom com3r ) BCADABCDEFG $.
  $}

  ${
    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 $.
  $}

  ${
    imp31.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
    $( 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 $.
  $}

  ${
    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 $.
  $}

  ${
    imp4.1 $e |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $.
    $( An importation inference.  (Contributed by NM, 26-Apr-1994.)  Shorten
       ~ imp4a .  (Revised by Wolf Lammen, 19-Jul-2021.) $)
    imp4b $p |- ( ( ph /\ ps ) -> ( ( ch /\ th ) -> ta ) ) $=
      ( wa wi imp impd ) ABGCDEABCDEHHFIJ $.

    $( An importation inference.  (Contributed by NM, 26-Apr-1994.)  (Proof
       shortened by Wolf Lammen, 19-Jul-2021.) $)
    imp4a $p |- ( ph -> ( ps -> ( ( ch /\ th ) -> ta ) ) ) $=
      ( wa wi imp4b ex ) 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 $.
  $}

  ${
    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.)  Shorten ~ exp4a .  (Revised by
       Wolf Lammen, 20-Jul-2021.) $)
    exp4b $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
      ( wi wa expd ex ) ABCDEGGABHCDEFIJ $.
  $}

  ${
    exp4a.1 $e |- ( ph -> ( ps -> ( ( ch /\ th ) -> ta ) ) ) $.
    $( An exportation inference.  (Contributed by NM, 26-Apr-1994.)  (Proof
       shortened by Wolf Lammen, 20-Jul-2021.) $)
    exp4a $p |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) $=
      ( wa wi imp exp4b ) 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 $.
  $}

  ${
    imp5.1 $e |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) ) $.
    $( 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.)
       (Proof shortened by Wolf Lammen, 2-Aug-2022.) $)
    imp5a $p |- ( ph -> ( ps -> ( ch -> ( ( th /\ ta ) -> et ) ) ) ) $=
      ( wa wi imp5d exp31 ) ABCDEHFIABCDEFGJK $.

    $( An importation inference.  (Contributed by Jeff Hankins, 7-Jul-2009.) $)
    imp5g $p |- ( ( ph /\ ps ) -> ( ( ( ch /\ th ) /\ ta ) -> et ) ) $=
      ( wa wi imp4b impd ) ABHCDHEFABCDEFIGJK $.

    $( 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 $.
  $}

  ${
    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 $.
  $}

  ${
    exp5j.1 $e |- ( ph -> ( ( ( ( ps /\ ch ) /\ th ) /\ ta ) -> et ) ) $.
    $( An exportation inference.  (Contributed by Jeff Hankins, 7-Jul-2009.) $)
    exp5j $p |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) ) $=
      ( wi wa expd exp4c ) ABCDEFHABCIDIEFGJK $.
  $}

  ${
    exp5l.1 $e |- ( ph -> ( ( ( ps /\ ch ) /\ ( th /\ ta ) ) -> et ) ) $.
    $( An exportation inference.  (Contributed by Jeff Hankins, 7-Jul-2009.) $)
    exp5l $p |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) ) $=
      ( wa expd exp5c ) ABCDEFABCHDEHFGIJ $.
  $}

  ${
    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 $.
  $}

  $( 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, 10-Jan-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 $.

  ${
    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 $.
  $}

  ${
    expdimp.1 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
    $( A deduction version of exportation, followed by importation.
       (Contributed by NM, 6-Sep-2008.) $)
    expdimp $p |- ( ( ph /\ ps ) -> ( ch -> th ) ) $=
      ( wi expd imp ) ABCDFABCDEGH $.
  $}

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  $( 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 id ancoms ) BABACZFDE $.

  $( 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 $.
  $}

  ${
    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 $.
  $}

  $( Closed form of ~ ancoms .  (Contributed by Alan Sare, 31-Dec-2011.) $)
  ancomst $p |- ( ( ( ph /\ ps ) -> ch ) <-> ( ( ps /\ ph ) -> ch ) ) $=
    ( wa ancom imbi1i ) ABDBADCABEF $.

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

  ${
    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, 21-Jun-1993.)  (Proof shortened by Wolf
     Lammen, 24-Nov-2012.) $)
  anass $p |- ( ( ( ph /\ ps ) /\ ch ) <-> ( ph /\ ( ps /\ ch ) ) ) $=
    ( wa id anassrs anasss impbii ) ABDCDZABCDDZABCJJEFABCIIEGH $.

  $( Join antecedents with conjunction ("conjunction introduction").  Theorem
     *3.2 of [WhiteheadRussell] p. 111.  Its associated inference is ~ pm3.2i
     and its associated deduction is ~ jca (and the double deduction is
     ~ jcad ).  See ~ pm3.2im for a version using only implication and
     negation.  (Contributed by NM, 5-Jan-1993.)  (Proof shortened by Wolf
     Lammen, 12-Nov-2012.) $)
  pm3.2 $p |- ( ph -> ( ps -> ( ph /\ ps ) ) ) $=
    ( wa id ex ) ABABCZFDE $.

  ${
    pm3.2i.1 $e |- ph $.
    pm3.2i.2 $e |- ps $.
    $( Infer conjunction of premises.  Inference associated with ~ pm3.2 .  Its
       associated deduction is ~ jca (and the double deduction is ~ jcad ).
       (Contributed by NM, 21-Jun-1993.) $)
    pm3.2i $p |- ( ph /\ ps ) $=
      ( wa pm3.2 mp2 ) ABABECDABFG $.
  $}

  $( 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 id expcom ) BABACZFDE $.

  $( Nested conjunction of antecedents.  (Contributed by NM, 4-Jan-1993.) $)
  pm3.43i $p |- ( ( ph -> ps )
      -> ( ( ph -> ch ) -> ( ph -> ( ps /\ ch ) ) ) ) $=
    ( wa pm3.2 imim3i ) BCBCDABCEF $.

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

  $( A theorem similar to the standard definition of the biconditional.
     Definition of [Margaris] p. 49.  (Contributed by NM, 24-Jan-1993.) $)
  dfbi2 $p |- ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) ) $=
    ( wb wi wn wa dfbi1 df-an bitr4i ) ABCABDZBADZEDEJKFABGJKHI $.

  $( Definition ~ df-bi rewritten in an abbreviated form to help intuitive
     understanding of that definition.  Note that it is a conjunction of two
     implications; one which asserts properties that follow from the
     biconditional and one which asserts properties that imply the
     biconditional.  (Contributed by NM, 15-Aug-2008.) $)
  dfbi $p |- ( ( ( ph <-> ps ) -> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
        /\ ( ( ( ph -> ps ) /\ ( ps -> ph ) ) -> ( ph <-> ps ) ) ) $=
    ( wb wi wa dfbi2 mpbi ) ABCZABDBADEZCHIDIHDEABFHIFG $.

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

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

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

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

  ${
    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 $.
  $}

  $( Elimination of a conjunct.  Theorem *3.26 (Simp) of [WhiteheadRussell]
     p. 112.  (Contributed by NM, 3-Jan-1993.)  (Proof shortened by Wolf
     Lammen, 14-Jun-2022.) $)
  simpl $p |- ( ( ph /\ ps ) -> ph ) $=
    ( id adantr ) AABACD $.

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

  $( Elimination of a conjunct.  Theorem *3.27 (Simp) of [WhiteheadRussell]
     p. 112.  (Contributed by NM, 3-Jan-1993.)  (Proof shortened by Wolf
     Lammen, 14-Jun-2022.) $)
  simpr $p |- ( ( ph /\ ps ) -> ps ) $=
    ( id adantl ) BBABCD $.

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

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  $( 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 $.

  ${
    simpld.1 $e |- ( ph -> ( ps /\ ch ) ) $.
    $( Deduction eliminating a conjunct.  A translation of natural deduction
       rule ` /\ ` EL ( ` /\ ` elimination left), see ~ natded .  (Contributed
       by NM, 26-May-1993.) $)
    simpld $p |- ( ph -> ps ) $=
      ( wa simpl syl ) ABCEBDBCFG $.
  $}

  ${
    simprd.1 $e |- ( ph -> ( ps /\ ch ) ) $.
    $( Deduction eliminating a conjunct.  (Contributed by NM, 14-May-1993.)  A
       translation of natural deduction rule ` /\ ` ER ( ` /\ ` elimination
       right), see ~ natded .  (Proof shortened by Wolf Lammen, 3-Oct-2013.) $)
    simprd $p |- ( ph -> ch ) $=
      ( ancomd simpld ) ACBABCDEF $.
  $}

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

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

  ${
    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 $.
  $}

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

  $( Closed form of ~ simplbi2com .  (Contributed by Alan Sare,
     22-Jul-2012.) $)
  simplbi2comt $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 $.
  $}

  ${
    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.)  (Proof shortened by Wolf Lammen,
       26-Mar-2022.) $)
    simpl2im $p |- ( ph -> th ) $=
      ( simprd syl ) ACDABCEGFH $.
  $}

  ${
    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.)  (Proof shortened by
       Wolf Lammen, 26-Mar-2022.) $)
    simplbiim $p |- ( ph -> th ) $=
      ( simprbi syl ) ACDABCEGFH $.
  $}

  ${
    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 $.
  $}

  ${
    sylan9.1 $e |- ( ph -> ( ps -> ch ) ) $.
    sylan9.2 $e |- ( th -> ( ch -> ta ) ) $.
    $( Nested syllogism inference conjoining dissimilar antecedents.
       (Contributed by NM, 14-May-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, 14-May-1993.) $)
    sylan9r $p |- ( ( th /\ ph ) -> ( ps -> ta ) ) $=
      ( wi syl9r imp ) DABEHABCDEFGIJ $.
  $}

  ${
    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 $.
  $}

  ${
    jca.1 $e |- ( ph -> ps ) $.
    jca.2 $e |- ( ph -> ch ) $.
    $( Deduce conjunction of the consequents of two implications ("join
       consequents with 'and'").  Deduction form of ~ pm3.2 and ~ pm3.2i .  Its
       associated deduction is ~ jcad .  Equivalent to the natural deduction
       rule ` /\ ` I ( ` /\ ` introduction), see ~ natded .  (Contributed by
       NM, 3-Jan-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.  Deduction
       form of ~ jca and double deduction form of ~ pm3.2 and ~ pm3.2i .
       (Contributed by NM, 15-Jul-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,
       15-Jul-1993.) $)
    jcai $p |- ( ph -> ( ps /\ ch ) ) $=
      ( mpd jca ) ABCDABCDEFG $.
  $}

  $( 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 $.

  ${
    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 $.
  $}

  ${
    jccir.1 $e |- ( ph -> ps ) $.
    jccir.2 $e |- ( ps -> ch ) $.
    $( Inference conjoining a consequent of a consequent to the right of the
       consequent in an implication.  See also ~ ex-natded5.3i .  (Contributed
       by Mario Carneiro, 9-Feb-2017.)  (Revised by AV, 20-Aug-2019.) $)
    jccir $p |- ( ph -> ( ps /\ ch ) ) $=
      ( syl jca ) ABCDABCDEFG $.

    $( Inference conjoining a consequent of a consequent to the left of the
       consequent in an implication.  Remark:  One can also prove this theorem
       using ~ syl and ~ jca (as done in ~ jccir ), which would be 4 bytes
       shorter, but one step longer than the current proof.
       (Proof modification is discouraged.)  (Contributed by AV,
       20-Aug-2019.) $)
    jccil $p |- ( ph -> ( ch /\ ps ) ) $=
      ( jccir ancomd ) ABCABCDEFG $.
  $}

  ${
    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 $.
  $}

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

  $( Introduction of antecedent as conjunct.  (Contributed by NM,
     5-Dec-1995.) $)
  ibar $p |- ( ph -> ( ps <-> ( ph /\ ps ) ) ) $=
    ( iba biancomd ) ABABABCD $.

  ${
    biantru.1 $e |- ph $.
    $( A wff is equivalent to its conjunction with truth.  (Contributed by NM,
       26-May-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 ) ) $=
      ( biantru biancomi ) BABABCDE $.
  $}

  ${
    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 $.
  $}

  ${
    bianfi.1 $e |- -. ph $.
    $( A wff conjoined with falsehood is false.  (Contributed by NM,
       21-Jun-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 $.
  $}

  ${
    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.)  (Proof shortened by Wolf Lammen,
       19-Jan-2020.) $)
    rbaibr $p |- ( ch -> ( ps <-> ph ) ) $=
      ( biancomi baibr ) ACBACBDEF $.

    $( Move conjunction outside of biconditional.  (Contributed by Mario
       Carneiro, 11-Sep-2015.)  (Proof shortened by Wolf Lammen,
       19-Jan-2020.) $)
    rbaib $p |- ( ch -> ( ph <-> ps ) ) $=
      ( rbaibr bicomd ) CBAABCDEF $.
  $}

  ${
    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 ) ) $=
      ( biancomd baibd ) ABDCABDCEFG $.
  $}

  ${
    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 $.
  $}

  $( 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 $.

  $( 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 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 $.

  $( 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 $.
  $}

  ${
    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 $.
  $}

  $( 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 $.
  $}

  $( Implication in terms of biconditional and conjunction.  Theorem *4.71 of
     [WhiteheadRussell] p. 120.  (Contributed by NM, 21-Jun-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 ) ) $=
      ( pm4.71i biancomi ) ABAABCDE $.
  $}

  ${
    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 ) ) ) $=
      ( pm4.71d biancomd ) ABCBABCDEF $.
  $}

  $( Theorem *4.24 of [WhiteheadRussell] p. 117.  (Contributed by NM,
     11-May-1993.) $)
  pm4.24 $p |- ( ph <-> ( ph /\ ph ) ) $=
    ( id pm4.71i ) AAABC $.

  $( Idempotent law for conjunction.  (Contributed by NM, 8-Jan-2004.)  (Proof
     shortened by Wolf Lammen, 14-Mar-2014.) $)
  anidm $p |- ( ( ph /\ ph ) <-> ph ) $=
    ( wa pm4.24 bicomi ) AAABACD $.

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

  ${
    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 $.
  $}

  $( 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 $.
  $}

  $( 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 simpl biantrurd pm5.74i ) ABDZCACDHACABEFG $.

  $( Distribution of implication over biconditional.  Theorem *5.32 of
     [WhiteheadRussell] p. 125.  (Contributed by NM, 1-Aug-1994.) $)
  pm5.32 $p |- ( ( ph -> ( ps <-> ch ) ) <->
               ( ( ph /\ ps ) <-> ( ph /\ ch ) ) ) $=
    ( wb wi wn wa notbi imbi2i pm5.74 3bitri df-an bibi12i bitr4i ) ABCDZEZABFZ
    EZFZACFZEZFZDZABGZACGZDPAQTDZERUADUCOUFABCHIAQTJRUAHKUDSUEUBABLACLMN $.

  ${
    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 $.
  $}

  ${
    bianim.1 $e |- ( ph <-> ( ps /\ ch ) ) $.
    bianim.2 $e |- ( ch -> ( ps <-> th ) ) $.
    $( Exchanging conjunction in a biconditional.  (Contributed by Peter Mazsa,
       31-Jul-2023.) $)
    bianim $p |- ( ph <-> ( th /\ ch ) ) $=
      ( wa pm5.32ri bitri ) ABCGDCGECBDFHI $.
  $}

  ${
    pm5.32d.1 $e |- ( ph -> ( ps -> ( ch <-> th ) ) ) $.
    $( Distribution of implication over biconditional (deduction form).
       (Contributed by NM, 29-Oct-1996.) $)
    pm5.32d $p |- ( ph -> ( ( ps /\ ch ) <-> ( ps /\ th ) ) ) $=
      ( wb wi wa pm5.32 sylib ) ABCDFGBCHBDHFEBCDIJ $.

    $( 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 $.
  $}

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  ${
    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 $.
  $}

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

  ${
    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 ) $=
      ( a1i sylanbrc ) ABCDECAFHGI $.
  $}

  ${
    syldan.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    syldan.2 $e |- ( ( ph /\ ch ) -> th ) $.
    $( A syllogism deduction with conjoined antecedents.  (Contributed by NM,
       24-Feb-2005.)  (Proof shortened by Wolf Lammen, 6-Apr-2013.) $)
    syldan $p |- ( ( ph /\ ps ) -> th ) $=
      ( wa simpl syl2anc ) ABGACDABHEFI $.
  $}

  ${
    sylbida.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    sylbida.2 $e |- ( ( ph /\ ch ) -> th ) $.
    $( A syllogism deduction.  (Contributed by SN, 16-Jul-2024.) $)
    sylbida $p |- ( ( ph /\ ps ) -> th ) $=
      ( biimpa syldan ) ABCDABCEGFH $.
  $}

  ${
    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.  For an implication-only version, see
       ~ syl2im .  (Contributed by NM, 31-Jan-1997.) $)
    syl2an $p |- ( ( ph /\ ta ) -> th ) $=
      ( sylan sylan2 ) EACDGABCDFHIJ $.

    $( A double syllogism inference.  For an implication-only version, see
       ~ syl2imc .  (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 $.
  $}

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  ${
    anim12d1.1 $e |- ( ph -> ( ps -> ch ) ) $.
    anim12d1.2 $e |- ( th -> ta ) $.
    $( Variant of ~ anim12d where the second implication does not depend on the
       antecedent.  (Contributed by Rodolfo Medina, 12-Oct-2010.) $)
    anim12d1 $p |- ( ph -> ( ( ps /\ th ) -> ( ch /\ ta ) ) ) $=
      ( wi a1i anim12d ) ABCDEFDEHAGIJ $.
  $}

  ${
    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, 14-May-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, 3-Jan-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,
       3-Jan-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 ) ) ) $=
      ( wi wa pm3.43 syl2an ) ABCHBEHBCEIHDFGBCEJK $.
  $}

  ${
    anim12dan.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    anim12dan.2 $e |- ( ( ph /\ th ) -> ta ) $.
    $( Conjoin antecedents and consequents in a deduction.  (Contributed by
       Jeff Madsen, 16-Jun-2011.) $)
    anim12dan $p |- ( ( ph /\ ( ps /\ th ) ) -> ( ch /\ ta ) ) $=
      ( wa ex anim12d imp ) ABDHCEHABCDEABCFIADEGIJK $.
  $}

  ${
    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 adantrd adantld anim12ii ) ABEICDFABCEGJDEFBHKL $.

    $( 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 $.
  $}

  $( 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 $.

  ${
    anbi.1 $e |- ( ph <-> ps ) $.
    $( Introduce a left conjunct to both sides of a logical equivalence.
       (Contributed by NM, 3-Jan-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, 12-Mar-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 biancomi ) ACECBABCDFG $.

    $( Variant of ~ anbi1i with commutation.  (Contributed by Peter Mazsa,
       7-Mar-2020.) $)
    anbi1ci $p |- ( ( ch /\ ph ) <-> ( ps /\ ch ) ) $=
      ( wa anbi2i biancomi ) CAEBCABCDFG $.
  $}

  ${
    bianbi.1 $e |- ( ph <-> ( ps /\ ch ) ) $.
    bianbi.2 $e |- ( ps <-> th ) $.
    $( Exchanging conjunction in a biconditional.  (Contributed by Peter Mazsa,
       31-Jul-2023.) $)
    bianbi $p |- ( ph <-> ( th /\ ch ) ) $=
      ( wa anbi1i bitri ) ABCGDCGEBDCFHI $.
  $}

  ${
    anbi12.1 $e |- ( ph <-> ps ) $.
    anbi12.2 $e |- ( ch <-> th ) $.
    $( Conjoin both sides of two equivalences.  (Contributed by NM,
       12-Mar-1993.) $)
    anbi12i $p |- ( ( ph /\ ch ) <-> ( ps /\ th ) ) $=
      ( wa anbi2i bianbi ) ACGADBCDAFHEI $.

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

  ${
    anbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Deduction adding a left conjunct to both sides of a logical equivalence.
       (Contributed by NM, 11-May-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, 11-May-1993.)  (Proof shortened by
       Wolf Lammen, 16-Nov-2013.) $)
    anbi1d $p |- ( ph -> ( ( ps /\ th ) <-> ( ch /\ th ) ) ) $=
      ( wb a1d pm5.32rd ) ADBCABCFDEGH $.
  $}

  ${
    anbi12d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    anbi12d.2 $e |- ( ph -> ( th <-> ta ) ) $.
    $( Deduction joining two equivalences to form equivalence of conjunctions.
       (Contributed by NM, 26-May-1993.) $)
    anbi12d $p |- ( ph -> ( ( ps /\ th ) <-> ( ch /\ ta ) ) ) $=
      ( wa anbi1d anbi2d bitrd ) ABDHCDHCEHABCDFIADECGJK $.
  $}

  $( 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 $.
  $}

  $( Double commutation in conjunction.  (Contributed by Peter Mazsa,
     27-Jun-2019.) $)
  an2anr $p |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) <->
                 ( ( ps /\ ph ) /\ ( th /\ ch ) ) ) $=
    ( wa ancom anbi12i ) ABEBAECDEDCEABFCDFG $.

  $( 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 ) ) ) $=
      ( wb wa pm4.38 syl2an ) ABCIEFIBEJCFJIDGHBECFKL $.

    $( 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 $.
  $}

  ${
    anbiim.1 $e |- ( ph -> ( ch -> th ) ) $.
    anbiim.2 $e |- ( ps -> ( th -> ch ) ) $.
    $( Adding biconditional when antecedents are conjuncted.  (Contributed by
       metakunt, 16-Apr-2024.)  (Proof shortened by Wolf Lammen,
       7-May-2025.) $)
    anbiim $p |- ( ( ph /\ ps ) -> ( ch <-> th ) ) $=
      ( wa wi adantr adantl impbid ) ABGCDACDHBEIBDCHAFJK $.
  $}

  ${
    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 bianbi ) DAFDBFCBDFABCDEGDBHI $.
  $}

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

  $( 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.)  (Proof shortened by Peter
     Mazsa, 18-Sep-2022.) $)
  an12 $p |- ( ( ph /\ ( ps /\ ch ) ) <-> ( ps /\ ( ph /\ ch ) ) ) $=
    ( wa ancom bianass biancomi ) ABCDZDBACDHCBABCEFG $.

  $( 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 an21 biancomi ) ABDCDACDBABCEF $.

  $( 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 an21 ancom bitr3i ) ABCDDBADZCDCHDBACEHCFG $.

  $( 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 $.
  $}

  $( Rearrangement of 4 conjuncts.  (Contributed by NM, 10-Jul-1994.) $)
  an4 $p |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) <->
              ( ( ph /\ ch ) /\ ( ps /\ th ) ) ) $=
    ( wa anass an12 bianass bitri ) ABECDEZEABJEZEACEBDEZEABJFKCLABCDGHI $.

  $( 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 $.
  $}

  $( 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.) $)
    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 simpl mpcom ) AABECABFDG $.
  $}

  ${
    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.) $)
    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 $.
  $}

  $( 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 $.
  $}

  ${
    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 ) $=
      ( adantl syldanl ) BACDEACBFHGI $.
  $}

  ${
    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 $.
  $}

  ${
    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 ) ) $=
      ( ex sylsyld ) ABCDEFGBDEHIJ $.
  $}

  ${
    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.)  (Proof shortened by Wolf Lammen,
       28-Mar-2022.) $)
    syl2an2r $p |- ( ( ph /\ ch ) -> ta ) $=
      ( sylan syldan ) ACDEGABDEFHIJ $.
  $}

  ${
    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 adantl syl2anc ) CAIBDEABCFJGHK $.
  $}

  ${
    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-2010.) $)
    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 ) ) $=
      ( biancomd mpbirand ) ABDCEABDCFGH $.
  $}

  ${
    mpbiran.1 $e |- ps $.
    mpbiran.2 $e |- ( ph <-> ( ps /\ ch ) ) $.
    $( Detach truth from conjunction in biconditional.  (Contributed by NM,
       27-Feb-1996.) $)
    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.) $)
    mpbiran2 $p |- ( ph <-> ps ) $=
      ( biancomi mpbiran ) ACBDACBEFG $.
  $}

  ${
    mpbir2an.1 $e |- ps $.
    mpbir2an.2 $e |- ch $.
    mpbir2an.maj $e |- ( ph <-> ( ps /\ ch ) ) $.
    $( Detach a conjunction of truths in a biconditional.  (Contributed by NM,
       10-May-2005.) $)
    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 $.
  $}

  ${
    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 ) $=
      ( adantl adantrr ) CABDABCEFG $.

    $( 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.)  (Proof shortened by Wolf Lammen, 5-Apr-2022.) $)
    ad3antlr $p |- ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) -> ps ) $=
      ( wa adantl ad2antrr ) CAGBDEABCFHI $.

    $( Deduction adding 4 conjuncts to antecedent.  (Contributed by Mario
       Carneiro, 4-Jan-2017.)  (Proof shortened by Wolf Lammen, 5-Apr-2022.) $)
    ad4antr $p |- ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et ) -> ps ) $=
      ( wa adantr ad3antrrr ) ACHBDEFABCGIJ $.

    $( Deduction adding 4 conjuncts to antecedent.  (Contributed by Mario
       Carneiro, 5-Jan-2017.)  (Proof shortened by Wolf Lammen, 5-Apr-2022.) $)
    ad4antlr $p |- ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et ) -> ps ) $=
      ( wa adantl ad3antrrr ) CAHBDEFABCGIJ $.

    $( Deduction adding 5 conjuncts to antecedent.  (Contributed by Mario
       Carneiro, 4-Jan-2017.)  (Proof shortened by Wolf Lammen, 5-Apr-2022.) $)
    ad5antr $p |- ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et )
      /\ ze ) -> ps ) $=
      ( wa adantr ad4antr ) ACIBDEFGABCHJK $.

    $( Deduction adding 5 conjuncts to antecedent.  (Contributed by Mario
       Carneiro, 5-Jan-2017.)  (Proof shortened by Wolf Lammen, 5-Apr-2022.) $)
    ad5antlr $p |- ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et )
      /\ ze ) -> ps ) $=
      ( wa adantl ad4antr ) CAIBDEFGABCHJK $.

    $( Deduction adding 6 conjuncts to antecedent.  (Contributed by Mario
       Carneiro, 4-Jan-2017.)  (Proof shortened by Wolf Lammen, 5-Apr-2022.) $)
    ad6antr $p |- ( ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et )
      /\ ze ) /\ si ) -> ps ) $=
      ( wa adantr ad5antr ) ACJBDEFGHABCIKL $.

    $( Deduction adding 6 conjuncts to antecedent.  (Contributed by Mario
       Carneiro, 5-Jan-2017.)  (Proof shortened by Wolf Lammen, 5-Apr-2022.) $)
    ad6antlr $p |- ( ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et )
      /\ ze ) /\ si ) -> ps ) $=
      ( wa adantl ad5antr ) CAJBDEFGHABCIKL $.

    $( Deduction adding 7 conjuncts to antecedent.  (Contributed by Mario
       Carneiro, 4-Jan-2017.)  (Proof shortened by Wolf Lammen, 5-Apr-2022.) $)
    ad7antr $p |- ( ( ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et )
      /\ ze ) /\ si ) /\ rh ) -> ps ) $=
      ( wa adantr ad6antr ) ACKBDEFGHIABCJLM $.

    $( Deduction adding 7 conjuncts to antecedent.  (Contributed by Mario
       Carneiro, 5-Jan-2017.)  (Proof shortened by Wolf Lammen, 5-Apr-2022.) $)
    ad7antlr $p |- ( ( ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et )
      /\ ze ) /\ si ) /\ rh ) -> ps ) $=
      ( wa adantl ad6antr ) CAKBDEFGHIABCJLM $.

    $( Deduction adding 8 conjuncts to antecedent.  (Contributed by Mario
       Carneiro, 4-Jan-2017.)  (Proof shortened by Wolf Lammen, 5-Apr-2022.) $)
    ad8antr $p |- ( ( ( ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et )
      /\ ze ) /\ si ) /\ rh ) /\ mu ) -> ps ) $=
      ( wa adantr ad7antr ) ACLBDEFGHIJABCKMN $.

    $( Deduction adding 8 conjuncts to antecedent.  (Contributed by Mario
       Carneiro, 5-Jan-2017.)  (Proof shortened by Wolf Lammen, 5-Apr-2022.) $)
    ad8antlr $p |- ( ( ( ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et )
      /\ ze ) /\ si ) /\ rh ) /\ mu ) -> ps ) $=
      ( wa adantl ad7antr ) CALBDEFGHIJABCKMN $.

    $( Deduction adding 9 conjuncts to antecedent.  (Contributed by Mario
       Carneiro, 4-Jan-2017.)  (Proof shortened by Wolf Lammen, 5-Apr-2022.) $)
    ad9antr $p |- ( ( ( ( ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et )
      /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) -> ps ) $=
      ( wa adantr ad8antr ) ACMBDEFGHIJKABCLNO $.

    $( Deduction adding 9 conjuncts to antecedent.  (Contributed by Mario
       Carneiro, 5-Jan-2017.)  (Proof shortened by Wolf Lammen, 5-Apr-2022.) $)
    ad9antlr $p |- ( ( ( ( ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et )
      /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) -> ps ) $=
      ( wa adantl ad8antr ) CAMBDEFGHIJKABCLNO $.

    $( Deduction adding 10 conjuncts to antecedent.  (Contributed by Mario
       Carneiro, 4-Jan-2017.)  (Proof shortened by Wolf Lammen, 5-Apr-2022.) $)
    ad10antr $p |- ( ( ( ( ( ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et )
      /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) /\ ka ) -> ps ) $=
      ( wa adantr ad9antr ) ACNBDEFGHIJKLABCMOP $.

    $( Deduction adding 10 conjuncts to antecedent.  (Contributed by Mario
       Carneiro, 5-Jan-2017.)  (Proof shortened by Wolf Lammen, 5-Apr-2022.) $)
    ad10antlr $p |- ( ( ( ( ( ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et )
      /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) /\ ka ) -> ps ) $=
      ( wa adantl ad9antr ) CANBDEFGHIJKLABCMOP $.
  $}

  ${
    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.)  (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 $.
  $}

  $( 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 $.

  $( 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.)  (Proof shortened by Wolf Lammen, 6-Apr-2022.) $)
  simplll $p |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ph ) $=
    ( id ad3antrrr ) AABCDAEF $.

  $( Simplification of a conjunction.  (Contributed by Jeff Hankins,
     28-Jul-2009.)  (Proof shortened by Wolf Lammen, 6-Apr-2022.) $)
  simpllr $p |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ps ) $=
    ( id ad3antlr ) BBACDBEF $.

  $( 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.)  (Proof shortened by Wolf Lammen, 24-May-2022.) $)
  simp-4l $p |- ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta ) -> ph ) $=
    ( id ad4antr ) AABCDEAFG $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.)  (Proof shortened by Wolf Lammen, 24-May-2022.) $)
  simp-4r $p |- ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta ) -> ps ) $=
    ( id ad4antlr ) BBACDEBFG $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.)  (Proof shortened by Wolf Lammen, 24-May-2022.) $)
  simp-5l $p |- ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
    /\ et ) -> ph ) $=
    ( id ad5antr ) AABCDEFAGH $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.)  (Proof shortened by Wolf Lammen, 24-May-2022.) $)
  simp-5r $p |- ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
    /\ et ) -> ps ) $=
    ( id ad5antlr ) BBACDEFBGH $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.)  (Proof shortened by Wolf Lammen, 24-May-2022.) $)
  simp-6l $p |- ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
    /\ et ) /\ ze ) -> ph ) $=
    ( id ad6antr ) AABCDEFGAHI $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.)  (Proof shortened by Wolf Lammen, 24-May-2022.) $)
  simp-6r $p |- ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
    /\ et ) /\ ze ) -> ps ) $=
    ( id ad6antlr ) BBACDEFGBHI $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.)  (Proof shortened by Wolf Lammen, 24-May-2022.) $)
  simp-7l $p |- ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
    /\ et ) /\ ze ) /\ si ) -> ph ) $=
    ( id ad7antr ) AABCDEFGHAIJ $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.)  (Proof shortened by Wolf Lammen, 24-May-2022.) $)
  simp-7r $p |- ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
    /\ et ) /\ ze ) /\ si ) -> ps ) $=
    ( id ad7antlr ) BBACDEFGHBIJ $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.)  (Proof shortened by Wolf Lammen, 24-May-2022.) $)
  simp-8l $p |- ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
    /\ et ) /\ ze ) /\ si ) /\ rh ) -> ph ) $=
    ( id ad8antr ) AABCDEFGHIAJK $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.)  (Proof shortened by Wolf Lammen, 24-May-2022.) $)
  simp-8r $p |- ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
    /\ et ) /\ ze ) /\ si ) /\ rh ) -> ps ) $=
    ( id ad8antlr ) BBACDEFGHIBJK $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.)  (Proof shortened by Wolf Lammen, 24-May-2022.) $)
  simp-9l $p |- ( ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
    /\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) -> ph ) $=
    ( id ad9antr ) AABCDEFGHIJAKL $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.)  (Proof shortened by Wolf Lammen, 24-May-2022.) $)
  simp-9r $p |- ( ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
    /\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) -> ps ) $=
    ( id ad9antlr ) BBACDEFGHIJBKL $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.)  (Proof shortened by Wolf Lammen, 24-May-2022.) $)
  simp-10l $p |- ( ( ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
    /\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) -> ph ) $=
    ( id ad10antr ) AABCDEFGHIJKALM $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.)  (Proof shortened by Wolf Lammen, 24-May-2022.) $)
  simp-10r $p |- ( ( ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
    /\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) -> ps ) $=
    ( id ad10antlr ) BBACDEFGHIJKBLM $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.)  (Proof shortened by Wolf Lammen, 24-May-2022.) $)
  simp-11l $p |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
    /\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) /\ ka ) -> ph ) $=
    ( wa simpl ad10antr ) ABMACDEFGHIJKLABNO $.

  $( Simplification of a conjunction.  (Contributed by Mario Carneiro,
     4-Jan-2017.)  (Proof shortened by Wolf Lammen, 24-May-2022.) $)
  simp-11r $p |- ( ( ( ( ( ( ( ( ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) /\ ta )
    /\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) /\ ka ) -> ps ) $=
    ( wa simpr ad10antr ) ABMBCDEFGHIJKLABNO $.

$( Restating theorems using conjunction. $)

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

  ${
    pm2.18da.1 $e |- ( ( ph /\ -. ps ) -> ps ) $.
    $( Deduction based on reductio ad absurdum.  See ~ pm2.18 .  (Contributed
       by Mario Carneiro, 9-Feb-2017.) $)
    pm2.18da $p |- ( ph -> ps ) $=
      ( wn ex pm2.18d ) ABABDBCEF $.
  $}

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

  ${
    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.
       Variant of ~ pm5.21ndd .  (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 $.
  $}

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

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

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

  $( 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 $.

  $( Theorem *4.14 of [WhiteheadRussell] p. 117.  Related to ~ con34b .
     (Contributed by NM, 3-Jan-2005.)  (Proof shortened by Wolf Lammen,
     23-Oct-2012.) $)
  pm4.14 $p |- ( ( ( ph /\ ps ) -> ch ) <-> ( ( ph /\ -. ch ) -> -. ps ) ) $=
    ( wi wn wa con34b imbi2i impexp 3bitr4i ) ABCDZDACEZBEZDZDABFCDALFMDKNABCGH
    ABCIALMIJ $.

  $( Theorem *3.37 (Transp) of [WhiteheadRussell] p. 112.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Wolf Lammen, 23-Oct-2012.) $)
  pm3.37 $p |- ( ( ( ph /\ ps ) -> ch ) -> ( ( ph /\ -. ch ) -> -. ps ) ) $=
    ( wa wi wn pm4.14 biimpi ) ABDCEACFDBFEABCGH $.

  $( Conjoin antecedents and consequents of two premises.  This is the closed
     theorem form of ~ anim12d .  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 id im2anan9 ) ABEZABCDEZCDHFIFG $.

$( Replacing conjunction. $)

  $( 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 $.

  ${
    exbiri.1 $e |- ( ( ph /\ ps ) -> ( ch <-> th ) ) $.
    $( Inference form of ~ exbir .  This proof is ~ exbiriVD automatically
       translated and minimized.  (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 $.
  $}

$( Contradiction using conjunction. $)

  ${
    pm2.61ian.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    pm2.61ian.2 $e |- ( ( -. ph /\ ps ) -> ch ) $.
    $( Elimination of an antecedent.  (Contributed by NM, 1-Jan-2005.) $)
    pm2.61ian $p |- ( ps -> ch ) $=
      ( wi ex wn pm2.61i ) ABCFABCDGAHBCEGI $.
  $}

  ${
    pm2.61dan.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    pm2.61dan.2 $e |- ( ( ph /\ -. ps ) -> ch ) $.
    $( Elimination of an antecedent.  (Contributed by NM, 1-Jan-2005.) $)
    pm2.61dan $p |- ( ph -> ch ) $=
      ( ex wn pm2.61d ) ABCABCDFABGCEFH $.
  $}

  ${
    pm2.61ddan.1 $e |- ( ( ph /\ ps ) -> th ) $.
    pm2.61ddan.2 $e |- ( ( ph /\ ch ) -> th ) $.
    pm2.61ddan.3 $e |- ( ( ph /\ ( -. ps /\ -. ch ) ) -> th ) $.
    $( Elimination of two antecedents.  (Contributed by NM, 9-Jul-2013.) $)
    pm2.61ddan $p |- ( ph -> th ) $=
      ( wn wa adantlr anassrs pm2.61dan ) ABDEABHZICDACDMFJAMCHDGKLL $.
  $}

  ${
    pm2.61dda.1 $e |- ( ( ph /\ -. ps ) -> th ) $.
    pm2.61dda.2 $e |- ( ( ph /\ -. ch ) -> th ) $.
    pm2.61dda.3 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
    $( Elimination of two antecedents.  (Contributed by NM, 9-Jul-2013.) $)
    pm2.61dda $p |- ( ph -> th ) $=
      ( wa anassrs wn adantlr pm2.61dan ) ABDABHCDABCDGIACJDBFKLEL $.
  $}

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  ${
    condan.1 $e |- ( ( ph /\ -. ps ) -> ch ) $.
    condan.2 $e |- ( ( ph /\ -. ps ) -> -. ch ) $.
    $( Proof by contradiction.  (Contributed by NM, 9-Feb-2006.)  (Proof
       shortened by Wolf Lammen, 19-Jun-2014.) $)
    condan $p |- ( ph -> ps ) $=
      ( wn pm2.65da notnotrd ) ABABFCDEGH $.
  $}

$( Relation of conjunction to biconditional. $)

  $( An implication is equivalent to the equivalence of some implied
     equivalence and some other equivalence involving a conjunction.  A utility
     lemma as illustrated in ~ biadanii and ~ elelb .  (Contributed by BJ,
     4-Mar-2023.)  (Proof shortened by Wolf Lammen, 8-Mar-2023.) $)
  biadan $p |- ( ( ph -> ps ) <->
                   ( ( ps -> ( ph <-> ch ) ) <-> ( ph <-> ( ps /\ ch ) ) ) ) $=
    ( wi wa wb pm4.71r bicom pm5.32 bibi12i biluk 3bitr4ri 3bitri ) ABDABAEZFNA
    FZBACFDZABCEZFZFZABGANHRPFQAFZNQFZFSORTPUAAQHBACIJPRHNAQKLM $.

  ${
    biadani.1 $e |- ( ph -> ps ) $.
    $( Inference associated with ~ biadan .  (Contributed by BJ,
       4-Mar-2023.) $)
    biadani $p |- ( ( ps -> ( ph <-> ch ) ) <-> ( ph <-> ( ps /\ ch ) ) ) $=
      ( wi wb wa biadan mpbi ) ABEBACFEABCGFFDABCHI $.

    $( Alternate proof of ~ biadani not using ~ biadan .  (Contributed by BJ,
       4-Mar-2023.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    biadaniALT $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 $.
  $}

  $( 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 $.

  $( Two propositions are equivalent if they are both false.  Theorem *5.21 of
     [WhiteheadRussell] p. 124.  (Contributed by NM, 21-May-1994.) $)
  pm5.21 $p |- ( ( -. ph /\ -. ps ) -> ( ph <-> ps ) ) $=
    ( wn wb pm5.21im imp ) ACBCABDABEF $.

  $( Theorem *5.35 of [WhiteheadRussell] p. 125.  Closed form of ~ 2thd .
     (Contributed by NM, 3-Jan-2005.) $)
  pm5.35 $p |- ( ( ( ph -> ps ) /\ ( ph -> ch ) ) ->
                ( ph -> ( ps <-> ch ) ) ) $=
    ( wi wa pm5.1 pm5.74rd ) ABDZACDZEABCHIFG $.

  $( 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 $.

  $( 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 ax-1 pm4.71i ) ABACABDE $.

  $( An implication and its reverse are equivalent exactly when both operands
     are equivalent.  The right hand side resembles that of ~ dfbi2 , but
     ` <-> ` is a weaker operator than ` /\ ` .  Note that an implication and
     its reverse can never be simultaneously false, because of ~ pm2.521 .
     (Contributed by Wolf Lammen, 18-Dec-2023.) $)
  impimprbi $p |- ( ( ph <-> ps ) <-> ( ( ph -> ps ) <-> ( ps -> ph ) ) ) $=
    ( wb wi wa dfbi2 pm5.1 sylbi impbi wn pm2.521 pm2.24d bija impbii ) ABCZABD
    ZBADZCZOPQERABFPQGHPQOABIPJQOABKLMN $.

$( Moving subexpressions in/out of a conjunction. $)

  $( 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 $.

  $( Theorem *5.31 of [WhiteheadRussell] p. 125.  (Contributed by NM,
     3-Jan-2005.) $)
  pm5.31 $p |- ( ( ch /\ ( ph -> ps ) ) -> ( ph -> ( ps /\ ch ) ) ) $=
    ( wi wa simpr simpl jctird ) CABDZEABCCIFCIGH $.

  $( Variant of ~ pm5.31 .  (Contributed by Rodolfo Medina, 15-Oct-2010.) $)
  pm5.31r $p |- ( ( ch /\ ( ph -> ps ) ) -> ( ph -> ( ch /\ ps ) ) ) $=
    ( wi ax-1 id anim12ii ) CACABDZBCAEHFG $.

  $( 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 $.

  $( 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 $.

$( Absorption in conjunction. $)

  $( 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 $.

  $( 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 $.

$( Miscellaneous theorems using conjunction. $)

  ${
    syl12anc.1 $e |- ( ph -> ps ) $.
    syl12anc.2 $e |- ( ph -> ch ) $.
    syl12anc.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 jca syl2anc ) ABCDJEFACDGHKIL $.
    $}

    ${
      syl21anc.4 $e |- ( ( ( ps /\ ch ) /\ th ) -> ta ) $.
      $( Syllogism combined with contraction.  (Contributed by Jeff Hankins,
         1-Aug-2009.) $)
      syl21anc $p |- ( ph -> ta ) $=
        ( wa jca syl2anc ) ABCJDEABCFGKHIL $.
    $}

    ${
      syl22anc.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 $.
    $}
  $}

  ${
    bibiad.1 $e |- ( ( ph /\ ps ) -> th ) $.
    bibiad.2 $e |- ( ( ph /\ ch ) -> th ) $.
    bibiad.3 $e |- ( ( ph /\ th ) -> ( ps <-> ch ) ) $.
    $( Eliminate an hypothesis ` th ` in a biconditional.  (Contributed by
       Thierry Arnoux, 4-May-2025.) $)
    bibiad $p |- ( ph -> ( ps <-> ch ) ) $=
      ( wa simpl simpr biimpa syl21anc biimpar impbida ) ABCABHADBCABIEABJADHZB
      CGKLACHADCBACIFACJOBCGMLN $.
  $}

  ${
    syl1111anc.1 $e |- ( ph -> ps ) $.
    syl1111anc.2 $e |- ( ph -> ch ) $.
    syl1111anc.3 $e |- ( ph -> th ) $.
    syl1111anc.4 $e |- ( ph -> ta ) $.
    syl1111anc.5 $e |- ( ( ( ( ps /\ ch ) /\ th ) /\ ta ) -> et ) $.
    $( Four-hypothesis elimination deduction for an assertion with a singleton
       virtual hypothesis collection.  Similar to ~ syl112anc except the
       unification theorem uses left-nested conjunction.  (Contributed by Alan
       Sare, 17-Oct-2017.) $)
    syl1111anc $p |- ( ph -> et ) $=
      ( wa jca syl21anc ) ABCLDEFABCGHMIJKN $.
  $}

  ${
    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 $.
  $}

  ${
    mpsyl4anc.1 $e |- ph $.
    mpsyl4anc.2 $e |- ps $.
    mpsyl4anc.3 $e |- ch $.
    mpsyl4anc.4 $e |- ( th -> ta ) $.
    mpsyl4anc.5 $e |- ( ( ( ( ph /\ ps ) /\ ch ) /\ ta ) -> et ) $.
    $( An elimination deduction.  (Contributed by Alan Sare, 17-Oct-2017.) $)
    mpsyl4anc $p |- ( th -> et ) $=
      ( a1i syl1111anc ) DABCEFADGLBDHLCDILJKM $.
  $}

  $( 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 $.

  $( Removal of conjunct from one side of an equivalence.  (Contributed by NM,
     21-Jun-1993.) $)
  bimsc1 $p |- ( ( ( ph -> ps ) /\ ( ch <-> ( ps /\ ph ) ) )
               -> ( ch <-> ph ) ) $=
    ( wa wb wi id simpr ancr impbid2 sylan9bbr ) CBADZEZCLABFZAMGNLABAHABIJK $.

  ${
    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 imim1 com3l syl6c com23 ) ABFIZBCIZEJZDAQEDJZRSDJG
      ABFCABFCHKLSTRDREDMNOP $.
  $}

  ${
    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 $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Logical disjunction
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  This section defines disjunction of two formulas, denoted by infix " ` \/ ` "
  and read "or".  It is defined in terms of implication and negation, which is
  possible in classical logic (but not in intuitionistic logic: see iset.mm).
  This section contains only theorems proved without ~ df-an (theorems that are
  proved using ~ df-an are deferred to the next section).  Basic theorems that
  help simplifying and applying disjunction are ~ olc , ~ orc , and ~ orcom .

  As mentioned in the "note on definitions" in the section comment for logical
  equivalence, all theorems in this and the previous section can be stated in
  terms of implication and negation only.  Additionally, in classical logic
  (but not in intuitionistic logic: see iset.mm), it is also possible to
  translate conjunction into disjunction and conversely via the De Morgan law
  ~ anor : conjunction and disjunction are dual connectives.  Either is
  sufficient to develop all propositional calculus of the logic (together with
  implication and negation).  In practice, conjunction is more efficient, its
  big advantage being the possibility to use it to group antecedents in a
  convenient way, using ~ imp and ~ ex as noted in the previous section.

  An illustration of the conservativity of ~ df-an is given by ~ orim12dALT ,
  which is an alternate proof of ~ orim12d not using ~ df-an .

$)

  $( Declare connectives for disjunction ("or"). $)
  $c \/ $.  $( Vee (read:  "or") $)

  $( Extend wff definition to include disjunction ("or"). $)
  wo $a wff ( ph \/ ps ) $.

  $( Define disjunction (logical "or").  Definition of [Margaris] p. 49.  When
     the left operand, right operand, or both are true, the result is true;
     when both sides are false, the result is false.  For example, it is true
     that ` ( 2 = 3 \/ 4 = 4 ) ` ( ~ ex-or ).  After we define the constant
     true ` T. ` ( ~ df-tru ) and the constant false ` F. ` ( ~ df-fal ), we
     will be able to prove these truth table values:
     ` ( ( T. \/ T. ) <-> T. ) ` ( ~ truortru ), ` ( ( T. \/ F. ) <-> T. ) `
     ( ~ truorfal ), ` ( ( F. \/ T. ) <-> T. ) ` ( ~ falortru ), and
     ` ( ( F. \/ F. ) <-> F. ) ` ( ~ falorfal ).

     Contrast with ` /\ ` ( ~ df-an ), ` -> ` ( ~ wi ), ` -/\ ` ( ~ df-nan ),
     and ` \/_ ` ( ~ df-xor ).  (Contributed by NM, 27-Dec-1992.) $)
  df-or $a |- ( ( ph \/ ps ) <-> ( -. ph -> ps ) ) $.

  $( Theorem *4.64 of [WhiteheadRussell] p. 120.  (Contributed by NM,
     3-Jan-2005.) $)
  pm4.64 $p |- ( ( -. ph -> ps ) <-> ( ph \/ ps ) ) $=
    ( wo wn wi df-or bicomi ) ABCADBEABFG $.

  $( Theorem *4.66 of [WhiteheadRussell] p. 120.  (Contributed by NM,
     3-Jan-2005.) $)
  pm4.66 $p |- ( ( -. ph -> -. ps ) <-> ( ph \/ -. ps ) ) $=
    ( wn pm4.64 ) ABCD $.

  $( Theorem *2.53 of [WhiteheadRussell] p. 107.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.53 $p |- ( ( ph \/ ps ) -> ( -. ph -> ps ) ) $=
    ( wo wn wi df-or biimpi ) ABCADBEABFG $.

  $( Theorem *2.54 of [WhiteheadRussell] p. 107.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.54 $p |- ( ( -. ph -> ps ) -> ( ph \/ ps ) ) $=
    ( wo wn wi df-or biimpri ) ABCADBEABFG $.

  $( Implication in terms of disjunction.  Theorem *4.6 of [WhiteheadRussell]
     p. 120.  (Contributed by NM, 3-Jan-1993.) $)
  imor $p |- ( ( ph -> ps ) <-> ( -. ph \/ ps ) ) $=
    ( wi wn wo notnotb imbi1i df-or bitr4i ) ABCADZDZBCJBEAKBAFGJBHI $.

  ${
    imori.1 $e |- ( ph -> ps ) $.
    $( Infer disjunction from implication.  (Contributed by NM,
       12-Mar-2012.) $)
    imori $p |- ( -. ph \/ ps ) $=
      ( wi wn wo imor mpbi ) ABDAEBFCABGH $.
  $}

  ${
    imorri.1 $e |- ( -. ph \/ ps ) $.
    $( Infer implication from disjunction.  (Contributed by Jonathan Ben-Naim,
       3-Jun-2011.) $)
    imorri $p |- ( ph -> ps ) $=
      ( wi wn wo imor mpbir ) ABDAEBFCABGH $.
  $}

  $( Theorem *4.62 of [WhiteheadRussell] p. 120.  (Contributed by NM,
     3-Jan-2005.) $)
  pm4.62 $p |- ( ( ph -> -. ps ) <-> ( -. ph \/ -. ps ) ) $=
    ( wn imor ) ABCD $.

  ${
    jaoi.1 $e |- ( ph -> ps ) $.
    jaoi.2 $e |- ( ch -> ps ) $.
    $( Inference disjoining the antecedents of two implications.  (Contributed
       by NM, 5-Apr-1994.) $)
    jaoi $p |- ( ( ph \/ ch ) -> ps ) $=
      ( wo wn pm2.53 syl6 pm2.61d2 ) ACFZABKAGCBACHEIDJ $.
  $}

  ${
    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 $.
  $}

  ${
    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.) $)
    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 $.
  $}

  ${
    ori.1 $e |- ( ph \/ ps ) $.
    $( Infer implication from disjunction.  (Contributed by NM,
       11-Jun-1994.) $)
    ori $p |- ( -. ph -> ps ) $=
      ( wo wn wi df-or mpbi ) ABDAEBFCABGH $.
  $}

  ${
    orri.1 $e |- ( -. ph -> ps ) $.
    $( Infer disjunction from implication.  (Contributed by NM,
       11-Jun-1994.) $)
    orri $p |- ( ph \/ ps ) $=
      ( wo wn wi df-or mpbir ) ABDAEBFCABGH $.
  $}

  ${
    orrd.1 $e |- ( ph -> ( -. ps -> ch ) ) $.
    $( Deduce disjunction from implication.  (Contributed by NM,
       27-Nov-1995.) $)
    orrd $p |- ( ph -> ( ps \/ ch ) ) $=
      ( wn wi wo pm2.54 syl ) ABECFBCGDBCHI $.
  $}

  ${
    ord.1 $e |- ( ph -> ( ps \/ ch ) ) $.
    $( Deduce implication from disjunction.  (Contributed by NM,
       18-May-1994.) $)
    ord $p |- ( ph -> ( -. ps -> ch ) ) $=
      ( wo wn wi df-or sylib ) ABCEBFCGDBCHI $.
  $}

  ${
    orci.1 $e |- ph $.
    $( Deduction introducing a disjunct.  (Contributed by NM, 19-Jan-2008.)
       (Proof shortened by Wolf Lammen, 14-Nov-2012.) $)
    orci $p |- ( ph \/ ps ) $=
      ( pm2.24i orri ) ABABCDE $.

    $( Deduction introducing a disjunct.  (Contributed by NM, 19-Jan-2008.)
       (Proof shortened by Wolf Lammen, 14-Nov-2012.) $)
    olci $p |- ( ps \/ ph ) $=
      ( wn a1i orri ) BAABDCEF $.
  $}

  $( Introduction of a disjunct.  Theorem *2.2 of [WhiteheadRussell] p. 104.
     (Contributed by NM, 30-Aug-1993.) $)
  orc $p |- ( ph -> ( ph \/ ps ) ) $=
    ( pm2.24 orrd ) AABABCD $.

  $( Introduction of a disjunct.  Axiom *1.3 of [WhiteheadRussell] p. 96.
     (Contributed by NM, 30-Aug-1993.) $)
  olc $p |- ( ph -> ( ps \/ ph ) ) $=
    ( wn ax-1 orrd ) ABAABCDE $.

  $( Axiom *1.4 of [WhiteheadRussell] p. 96.  (Contributed by NM,
     3-Jan-2005.) $)
  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, 3-Jan-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 $.
  $}

  ${
    orcd.1 $e |- ( ph -> ps ) $.
    $( Deduction introducing a disjunct.  A translation of natural deduction
       rule ` \/ ` IR ( ` \/ ` insertion right), see ~ natded .  (Contributed
       by NM, 20-Sep-2007.) $)
    orcd $p |- ( ph -> ( ps \/ ch ) ) $=
      ( wo orc syl ) ABBCEDBCFG $.

    $( Deduction introducing a disjunct.  A translation of natural deduction
       rule ` \/ ` IL ( ` \/ ` insertion left), see ~ natded .  (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 $.
  $}

  ${
    olcnd.1 $e |- ( ph -> ( ps \/ ch ) ) $.
    olcnd.2 $e |- ( ph -> -. ch ) $.
    $( A lemma for Conjunctive Normal Form unit propagation, in deduction form.
       (Contributed by Giovanni Mascellani, 15-Sep-2017.)  (Proof shortened by
       Wolf Lammen, 13-Apr-2024.) $)
    olcnd $p |- ( ph -> ps ) $=
      ( ord mt3d ) ABCEABCDFG $.
  $}

  ${
    orcnd.1 $e |- ( ph -> ( ps \/ ch ) ) $.
    orcnd.2 $e |- ( ph -> -. ps ) $.
    $( A lemma for Conjunctive Normal Form unit propagation, in deduction form.
       (Contributed by Giovanni Mascellani, 15-Sep-2017.) $)
    orcnd $p |- ( ph -> ch ) $=
      ( orcomd olcnd ) ACBABCDFEG $.
  $}

  ${
    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.) $)
    mtord $p |- ( ph -> -. ps ) $=
      ( wo wn pm2.53 syl6ci mtod ) ABDFABCDHCIDGECDJKL $.
  $}

  ${
    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 $.
  $}

  $( 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 $.

  $( If neither of two propositions is true, then these propositions are
     equivalent.  (Contributed by BJ, 26-Apr-2019.) $)
  norbi $p |- ( -. ( ph \/ ps ) -> ( ph <-> ps ) ) $=
    ( wo orc olc pm5.21ni ) AABCBABDBAEF $.

  $( If two propositions are not equivalent, then at least one is true.
     (Contributed by BJ, 19-Apr-2019.)  (Proof shortened by Wolf Lammen,
     19-Jan-2020.) $)
  nbior $p |- ( -. ( ph <-> ps ) -> ( ph \/ ps ) ) $=
    ( wo wb norbi con1i ) ABCABDABEF $.

  $( 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 $.

  $( Theorem *2.25 of [WhiteheadRussell] p. 104.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.25 $p |- ( ph \/ ( ( ph \/ ps ) -> ps ) ) $=
    ( wo wi orel1 orri ) AABCBDABEF $.

  $( 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 $.

  $( Slight generalization of Theorem *2.67 of [WhiteheadRussell] p. 107.
     (Contributed by NM, 3-Jan-2005.) $)
  pm2.67-2 $p |- ( ( ( ph \/ ch ) -> ps ) -> ( ph -> ps ) ) $=
    ( wo orc imim1i ) AACDBACEF $.

  $( Theorem *2.67 of [WhiteheadRussell] p. 107.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.67 $p |- ( ( ( ph \/ ps ) -> ps ) -> ( ph -> ps ) ) $=
    ( pm2.67-2 ) ABBC $.

  $( A non-intuitionistic positive statement, sometimes called a paradox of
     material implication.  Sometimes called Curry's axiom.  Similar to ~ exmid
     (obtained by substituting ` F. ` for ` ps ` ) but positive.  For another
     non-intuitionistic positive statement, see ~ peirce .  (Contributed by BJ,
     4-Apr-2021.) $)
  curryax $p |- ( ph \/ ( ph -> ps ) ) $=
    ( wi pm2.21 orri ) AABCABDE $.

  $( Law of excluded middle, 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.  This is
     an essential distinction of our classical logic and is not a theorem of
     intuitionistic logic.  In intuitionistic logic, if this statement is true
     for some ` ph ` , then ` ph ` is decidable.  (Contributed by NM,
     29-Dec-1992.) $)
  exmid $p |- ( ph \/ -. ph ) $=
    ( wn id orri ) AABZECD $.

  $( Law of excluded middle in a context.  (Contributed by Mario Carneiro,
     9-Feb-2017.) $)
  exmidd $p |- ( ph -> ( ps \/ -. ps ) ) $=
    ( wn wo exmid a1i ) BBCDABEF $.

  $( Theorem *2.1 of [WhiteheadRussell] p. 101.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Wolf Lammen, 23-Nov-2012.) $)
  pm2.1 $p |- ( -. ph \/ ph ) $=
    ( id imori ) AAABC $.

  $( Theorem *2.13 of [WhiteheadRussell] p. 101.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.13 $p |- ( ph \/ -. -. -. ph ) $=
    ( wn notnot orri ) AABZBBECD $.

  $( Theorem *2.621 of [WhiteheadRussell] p. 107.  (Contributed by NM,
     3-Jan-2005.) $)
  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 $.

  $( Theorem *2.68 of [WhiteheadRussell] p. 108.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.68 $p |- ( ( ( ph -> ps ) -> ps ) -> ( ph \/ ps ) ) $=
    ( wi jarl orrd ) ABCBCABABBDE $.

  $( Logical 'or' expressed in terms of implication only.  Theorem *5.25 of
     [WhiteheadRussell] p. 124.  (Contributed by NM, 12-Aug-2004.)  (Proof
     shortened by Wolf Lammen, 20-Oct-2012.) $)
  dfor2 $p |- ( ( ph \/ ps ) <-> ( ( ph -> ps ) -> ps ) ) $=
    ( wo wi pm2.62 pm2.68 impbii ) ABCABDBDABEABFG $.

  $( Theorem *2.07 of [WhiteheadRussell] p. 101.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.07 $p |- ( ph -> ( ph \/ ph ) ) $=
    ( olc ) AAB $.

  $( Axiom *1.2 of [WhiteheadRussell] p. 96, which they call "Taut".
     (Contributed by NM, 3-Jan-2005.) $)
  pm1.2 $p |- ( ( ph \/ ph ) -> ph ) $=
    ( id jaoi ) AAAABZDC $.

  $( Idempotent law for disjunction.  Theorem *4.25 of [WhiteheadRussell]
     p. 117.  (Contributed by NM, 11-May-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 $.

  $( 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 *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 $.

  ${
    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 $.
  $}

  ${
    orim12dALT.1 $e |- ( ph -> ( ps -> ch ) ) $.
    orim12dALT.2 $e |- ( ph -> ( th -> ta ) ) $.
    $( Alternate proof of ~ orim12d which does not depend on ~ df-an .  This is
       an illustration of the conservativity of definitions (definitions do not
       permit to prove additional theorems whose statements do not contain the
       defined symbol).  (Contributed by Wolf Lammen, 8-Aug-2022.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    orim12dALT $p |- ( ph -> ( ( ps \/ th ) -> ( ch \/ ta ) ) ) $=
      ( wo wn wi pm2.53 con3d imim12d pm2.54 syl56 ) BDHBIZDJACIZEJCEHBDKAQPDEA
      BCFLGMCENO $.
  $}

  ${
    orbi2i.1 $e |- ( ph <-> ps ) $.
    $( Inference adding a left disjunct to both sides of a logical equivalence.
       (Contributed by NM, 3-Jan-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, 3-Jan-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,
       3-Jan-1993.) $)
    orbi12i $p |- ( ( ph \/ ch ) <-> ( ps \/ th ) ) $=
      ( wo orbi2i orbi1i bitri ) ACGADGBDGCDAFHABDEIJ $.
  $}

  ${
    bid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Deduction adding a left disjunct to both sides of a logical equivalence.
       (Contributed by NM, 21-Jun-1993.) $)
    orbi2d $p |- ( ph -> ( ( th \/ ps ) <-> ( th \/ ch ) ) ) $=
      ( wn wi wo imbi2d df-or 3bitr4g ) ADFZBGLCGDBHDCHABCLEIDBJDCJK $.

    $( Deduction adding a right disjunct to both sides of a logical
       equivalence.  (Contributed by NM, 21-Jun-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 $.

  ${
    bi12d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    bi12d.2 $e |- ( ph -> ( th <-> ta ) ) $.
    $( Deduction joining two equivalences to form equivalence of disjunctions.
       (Contributed by NM, 21-Jun-1993.) $)
    orbi12d $p |- ( ph -> ( ( ps \/ th ) <-> ( ch \/ ta ) ) ) $=
      ( wo orbi1d orbi2d bitrd ) ABDHCDHCEHABCDFIADECGJK $.
  $}

  $( 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 $.

  $( 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 $.

  $( 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 $.

  $( Disjunction distributes over implication.  (Contributed by Wolf Lammen,
     5-Jan-2013.) $)
  orimdi $p |- ( ( ph \/ ( ps -> ch ) )
        <-> ( ( ph \/ ps ) -> ( ph \/ ch ) ) ) $=
    ( wn wi wo imdi df-or imbi12i 3bitr4i ) ADZBCEZEKBEZKCEZEALFABFZACFZEKBCGAL
    HOMPNABHACHIJ $.

  $( Theorem *2.76 of [WhiteheadRussell] p. 108.  (Contributed by NM,
     3-Jan-2005.) $)
  pm2.76 $p |- ( ( ph \/ ( ps -> ch ) )
      -> ( ( ph \/ ps ) -> ( ph \/ ch ) ) ) $=
    ( wi wo orimdi biimpi ) ABCDEABEACEDABCFG $.

  $( Theorem *2.85 of [WhiteheadRussell] p. 108.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Wolf Lammen, 5-Jan-2013.) $)
  pm2.85 $p |- ( ( ( ph \/ ps ) -> ( ph \/ ch ) )
      -> ( ph \/ ( ps -> ch ) ) ) $=
    ( wi wo orimdi biimpri ) ABCDEABEACEDABCFG $.

  $( 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 $.

  $( Implication distributes over disjunction.  Theorem *4.78 of
     [WhiteheadRussell] p. 121.  (Contributed by NM, 3-Jan-2005.)  (Proof
     shortened by Wolf Lammen, 19-Nov-2012.) $)
  pm4.78 $p |- ( ( ( ph -> ps ) \/ ( ph -> ch ) ) <->
                ( ph -> ( ps \/ ch ) ) ) $=
    ( wn wo wi orordi imor orbi12i 3bitr4ri ) ADZBCEZEKBEZKCEZEALFABFZACFZEKBCG
    ALHOMPNABHACHIJ $.

  $( 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 $.

  $( 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 $.
    $( The dual of ~ biorf is not ~ biantr but ~ iba (and ~ ibar ).  So there
       should also be a "biorfr".  (Note that these four statements can
       actually be strengthened to biconditionals.)  (Contributed by BJ,
       26-Oct-2019.) $)
    biorfi $p |- ( ps <-> ( ph \/ ps ) ) $=
      ( wn wo wb biorf ax-mp ) ADBABEFCABGH $.

    $( A wff is equivalent to its disjunction with falsehood.  (Contributed by
       NM, 23-Mar-1995.)  (Proof shortened by Wolf Lammen, 16-Jul-2021.)
       (Proof shortened by AV, 10-Aug-2025.) $)
    biorfri $p |- ( ps <-> ( ps \/ ph ) ) $=
      ( wo biorfi orcom bitri ) BABDBADABCEABFG $.

    $( Obsolete version of ~ biorfri as of 10-Aug-2025.  A wff is equivalent to
       its disjunction with falsehood.  (Contributed by NM, 23-Mar-1995.)
       (Proof shortened by Wolf Lammen, 16-Jul-2021.)
       (New usage is discouraged.)  (Proof modification is discouraged.) $)
    biorfriOLD $p |- ( ps <-> ( ps \/ ph ) ) $=
      ( wo orc pm2.53 mt3i impbii ) BBADZBAEIBACBAFGH $.
  $}

  $( Rewriting implication based theorems using disjunction. $)

  $( Theorem *2.26 of [WhiteheadRussell] p. 104.  See ~ pm2.27 .  (Contributed
     by NM, 3-Jan-2005.)  (Proof shortened by Wolf Lammen, 23-Nov-2012.) $)
  pm2.26 $p |- ( -. ph \/ ( ( ph -> ps ) -> ps ) ) $=
    ( wi pm2.27 imori ) AABCBCABDE $.

  $( Contradiction and disjunction. $)

  $( 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 orel2 jao1i com12 ) ABCZDABDZAAHIBAEFG $.

  $( 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 $.

$( Some expressions connecting implication and disjunction. $)

  $( A general instance of Theorem *5.11 of [WhiteheadRussell] p. 123.
     (Contributed by NM, 3-Jan-2005.) $)
  pm5.11g $p |- ( ( ph -> ps ) \/ ( -. ph -> ch ) ) $=
    ( wi wn pm2.5g orri ) ABDAECDABCFG $.

  $( Theorem *5.11 of [WhiteheadRussell] p. 123.  (Contributed by NM,
     3-Jan-2005.) $)
  pm5.11 $p |- ( ( ph -> ps ) \/ ( -. ph -> ps ) ) $=
    ( pm5.11g ) ABBC $.

  $( Theorem *5.12 of [WhiteheadRussell] p. 123.  (Contributed by NM,
     3-Jan-2005.) $)
  pm5.12 $p |- ( ( ph -> ps ) \/ ( ph -> -. ps ) ) $=
    ( wi wn pm2.51 orri ) ABCABDCABEF $.

  $( Theorem *5.14 of [WhiteheadRussell] p. 123.  (Contributed by NM,
     3-Jan-2005.) $)
  pm5.14 $p |- ( ( ph -> ps ) \/ ( ps -> ch ) ) $=
    ( wi pm2.521g orri ) ABDBCDABCEF $.

  $( Theorem *5.13 of [WhiteheadRussell] p. 123.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Wolf Lammen, 14-Nov-2012.) $)
  pm5.13 $p |- ( ( ph -> ps ) \/ ( ps -> ph ) ) $=
    ( pm5.14 ) ABAC $.

  $( Theorem *5.55 of [WhiteheadRussell] p. 125.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Wolf Lammen, 20-Jan-2013.) $)
  pm5.55 $p |- ( ( ( ph \/ ps ) <-> ph ) \/ ( ( ph \/ ps ) <-> ps ) ) $=
    ( wo wb biort bicomd wn biorf nsyl5 orri ) ABCZADZKBDZALMAAKABEFAGBKABHFIJ
    $.

  $( 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 $.

  $( Simplify an implication between implications.  (Contributed by Paul
     Chapman, 17-Nov-2012.)  (Proof shortened by Wolf Lammen, 3-Apr-2013.) $)
  imimorb $p |- ( ( ( ps -> ch ) -> ( ph -> ch ) ) <->
                  ( ph -> ( ps \/ ch ) ) ) $=
    ( wi wo bi2.04 dfor2 imbi2i bitr4i ) BCDZACDDAJCDZDABCEZDJACFLKABCGHI $.

  $( 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 norbi id ja ax-1 impbii ) ABCZABDZELKLLABFLGHLKIJ $.

  $( Disjunction distributes over the biconditional.  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 NM, 8-Jan-2005.)  (Proof shortened by Wolf Lammen,
     4-Feb-2013.) $)
  orbidi $p |- ( ( ph \/ ( ps <-> ch ) ) <->
                ( ( ph \/ ps ) <-> ( ph \/ ch ) ) ) $=
    ( wn wb wi wo pm5.74 df-or bibi12i 3bitr4i ) ADZBCEZFLBFZLCFZEAMGABGZACGZEL
    BCHAMIPNQOABIACIJK $.

  $( Disjunction distributes over the biconditional.  Theorem *5.7 of
     [WhiteheadRussell] p. 125.  This theorem is similar to ~ orbidi .
     (Contributed by Roy F. Longton, 21-Jun-2005.) $)
  pm5.7 $p |- ( ( ( ph \/ ch ) <-> ( ps \/ ch ) ) <->
               ( ch \/ ( ph <-> ps ) ) ) $=
    ( wb wo orbidi orcom bibi12i bitr2i ) CABDECAEZCBEZDACEZBCEZDCABFJLKMCAGCBG
    HI $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Mixed connectives
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  This section gathers theorems of propositional calculus which use (either in
  their statement or proof) mixed connectives (at least conjunction and
  disjunction).

  As noted in the "note on definitions" in the section comment for logical
  equivalence, some theorem statements may contain for instance only
  conjunction or only disjunction, but both definitions are used in their
  proofs to make them shorter (this is exemplified in ~ orim12d versus
  ~ orim12dALT ).  These theorems are mostly grouped at the beginning of this
  section.

  The family of theorems starting with ~ animorl focus on the relation between
  conjunction and disjunction and can be seen as the starting point of mixed
  connectives in statements.  This sectioning is not rigorously true, since for
  instance the section begins with ~ jaao and related theorems.

$)

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  ${
    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), see ~ natded .
       (Contributed by Mario Carneiro, 29-May-2016.) $)
    mpjaodan $p |- ( ph -> ch ) $=
      ( wo jaodan mpdan ) ABDHCGABCDEFIJ $.
  $}

  $( 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 ) ) $=
    ( wi id jaao ) BADZBACADZCGEHEF $.

  $( 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 $.

  $( Disjunction of antecedents.  Compare Theorem *4.77 of [WhiteheadRussell]
     p. 121.  (Contributed by NM, 30-May-1994.)  (Proof shortened by Wolf
     Lammen, 9-Dec-2012.) $)
  jaob $p |- ( ( ( ph \/ ch ) -> ps ) <-> ( ( ph -> ps ) /\ ( ch -> ps ) ) ) $=
    ( wo wi wa pm2.67-2 olc imim1i jca pm3.44 impbii ) ACDZBEZABEZCBEZFNOPABCGC
    MBCAHIJBACKL $.

  $( 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 *3.48 of [WhiteheadRussell] p. 114.  (Contributed by NM,
     28-Jan-1997.) $)
  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.  See ~ orim12dALT
       for a proof which does not depend on ~ df-an .  (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 $.

  $( 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 *2.81 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 *2.8 of [WhiteheadRussell] p. 108.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Wolf Lammen, 5-Jan-2013.) $)
  pm2.8 $p |- ( ( ph \/ ps ) -> ( ( -. ps \/ ch ) -> ( ph \/ ch ) ) ) $=
    ( wo wn pm2.53 con1d orim1d ) ABDZBEACIABABFGH $.

  $( 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 Andrew Salmon, 7-May-2011.) $)
  pm2.74 $p |- ( ( ps -> ph )
      -> ( ( ( ph \/ ps ) \/ ch ) -> ( ph \/ ch ) ) ) $=
    ( wi wo orel2 ax-1 ja orim1d ) BADABEZACBAJADBAFAJGHI $.

  $( 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 pm2.24 orim2d jao1i orim1d ) ABEZCEACFZEZKDKCMCLBACBGHIJ $.

  $( 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 $.

  $( Negated conjunction in terms of disjunction (De Morgan's law).  Theorem
     *4.51 of [WhiteheadRussell] p. 120.  (Contributed by NM, 14-May-1993.)
     (Proof shortened by Andrew Salmon, 13-May-2011.) $)
  ianor $p |- ( -. ( ph /\ ps ) <-> ( -. ph \/ -. ps ) ) $=
    ( wa wn wi wo imnan pm4.62 bitr3i ) ABCDABDZEADJFABGABHI $.

  $( Conjunction in terms of disjunction (De Morgan's law).  Theorem *4.5 of
     [WhiteheadRussell] p. 120.  (Contributed by NM, 3-Jan-1993.)  (Proof
     shortened by Wolf Lammen, 3-Nov-2012.) $)
  anor $p |- ( ( ph /\ ps ) <-> -. ( -. ph \/ -. ps ) ) $=
    ( wa wn wo notnotb ianor xchbinx ) ABCZIDADBDEIFABGH $.

  $( Negated disjunction in terms of conjunction (De Morgan's law).  Compare
     Theorem *4.56 of [WhiteheadRussell] p. 120.  (Contributed by NM,
     3-Jan-1993.)  (Proof shortened by Andrew Salmon, 7-May-2011.) $)
  ioran $p |- ( -. ( ph \/ ps ) <-> ( -. ph /\ -. ps ) ) $=
    ( wn wi wa wo pm4.65 pm4.64 xchnxbi ) ACZBDJBCEABFABGABHI $.

  $( Theorem *4.52 of [WhiteheadRussell] p. 120.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Wolf Lammen, 5-Nov-2012.) $)
  pm4.52 $p |- ( ( ph /\ -. ps ) <-> -. ( -. ph \/ ps ) ) $=
    ( wn wa wi wo annim imor xchbinx ) ABCDABEACBFABGABHI $.

  $( Theorem *4.53 of [WhiteheadRussell] p. 120.  (Contributed by NM,
     3-Jan-2005.) $)
  pm4.53 $p |- ( -. ( ph /\ -. ps ) <-> ( -. ph \/ ps ) ) $=
    ( wn wo wa pm4.52 con2bii bicomi ) ACBDZABCEZCJIABFGH $.

  $( Theorem *4.54 of [WhiteheadRussell] p. 120.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Wolf Lammen, 5-Nov-2012.) $)
  pm4.54 $p |- ( ( -. ph /\ ps ) <-> -. ( ph \/ -. ps ) ) $=
    ( wn wa wi wo df-an pm4.66 xchbinx ) ACZBDJBCZEAKFJBGABHI $.

  $( Theorem *4.55 of [WhiteheadRussell] p. 120.  (Contributed by NM,
     3-Jan-2005.) $)
  pm4.55 $p |- ( -. ( -. ph /\ ps ) <-> ( ph \/ -. ps ) ) $=
    ( wn wo wa pm4.54 con2bii bicomi ) ABCDZACBEZCJIABFGH $.

  $( 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 (De Morgan's law).  Compare Theorem
     *4.57 of [WhiteheadRussell] p. 120.  (Contributed by NM, 3-Jan-1993.)
     (Proof shortened by Andrew Salmon, 7-May-2011.) $)
  oran $p |- ( ( ph \/ ps ) <-> -. ( -. ph /\ -. ps ) ) $=
    ( wn wa wo pm4.56 con2bii ) ACBCDABEABFG $.

  $( Theorem *4.57 of [WhiteheadRussell] p. 120.  (Contributed by NM,
     3-Jan-2005.) $)
  pm4.57 $p |- ( -. ( -. ph /\ -. ps ) <-> ( ph \/ ps ) ) $=
    ( wo wn wa oran bicomi ) ABCADBDEDABFG $.

  $( Theorem *3.1 of [WhiteheadRussell] p. 111.  (Contributed by NM,
     3-Jan-2005.) $)
  pm3.1 $p |- ( ( ph /\ ps ) -> -. ( -. ph \/ -. ps ) ) $=
    ( wa wn wo anor biimpi ) ABCADBDEDABFG $.

  $( Theorem *3.11 of [WhiteheadRussell] p. 111.  (Contributed by NM,
     3-Jan-2005.) $)
  pm3.11 $p |- ( -. ( -. ph \/ -. ps ) -> ( ph /\ ps ) ) $=
    ( wa wn wo anor biimpri ) ABCADBDEDABFG $.

  $( Theorem *3.12 of [WhiteheadRussell] p. 111.  (Contributed by NM,
     3-Jan-2005.) $)
  pm3.12 $p |- ( ( -. ph \/ -. ps ) \/ ( ph /\ ps ) ) $=
    ( wn wo wa pm3.11 orri ) ACBCDABEABFG $.

  $( Theorem *3.13 of [WhiteheadRussell] p. 111.  (Contributed by NM,
     3-Jan-2005.) $)
  pm3.13 $p |- ( -. ( ph /\ ps ) -> ( -. ph \/ -. ps ) ) $=
    ( wn wo wa pm3.11 con1i ) ACBCDABEABFG $.

  $( Theorem *3.14 of [WhiteheadRussell] p. 111.  (Contributed by NM,
     3-Jan-2005.) $)
  pm3.14 $p |- ( ( -. ph \/ -. ps ) -> -. ( ph /\ ps ) ) $=
    ( wa wn wo pm3.1 con2i ) ABCADBDEABFG $.

  $( 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 *4.45 of [WhiteheadRussell] p. 119.  (Contributed by NM,
     3-Jan-2005.) $)
  pm4.45 $p |- ( ph <-> ( ph /\ ( ph \/ ps ) ) ) $=
    ( wo orc pm4.71i ) AABCABDE $.

  $( Absorption of redundant internal disjunct.  Compare Theorem *4.45 of
     [WhiteheadRussell] p. 119.  (Contributed by NM, 21-Jun-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 $.

  $( 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 orel2 orc impbid1 pm5.32ri ) BCZABDZAIJABAEABFGH $.

  $( Conjunction in antecedent versus disjunction in consequent.  Theorem *5.6
     of [WhiteheadRussell] p. 125.  (Contributed by NM, 8-Jun-1994.) $)
  pm5.6 $p |- ( ( ( ph /\ -. ps ) -> ch ) <-> ( ph -> ( ps \/ ch ) ) ) $=
    ( wn wa wi wo impexp df-or imbi2i bitr4i ) ABDZECFALCFZFABCGZFALCHNMABCIJK
    $.

  ${
    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 $.
  $}

  $( Theorem *4.79 of [WhiteheadRussell] p. 121.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Wolf Lammen, 27-Jun-2013.) $)
  pm4.79 $p |- ( ( ( ps -> ph ) \/ ( ch -> ph ) ) <->
                ( ( ps /\ ch ) -> ph ) ) $=
    ( wi wo wa id jaoa wn simplim pm3.3 syl5 orrd impbii ) BADZCADZEBCFADZOBAPC
    OGPGHQOPOIBQPBAJBCAKLMN $.

  $( 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 bianbi ) ABEZCEDFJDFCDFADFBDFGJDCHADBHI $.

  $( Distributive law for disjunction.  Theorem *4.41 of [WhiteheadRussell]
     p. 119.  (Contributed by NM, 5-Jan-1993.)  (Proof shortened by Andrew
     Salmon, 7-May-2011.)  (Proof shortened by Wolf Lammen, 28-Nov-2013.) $)
  ordi $p |- ( ( ph \/ ( ps /\ ch ) ) <-> ( ( ph \/ ps ) /\ ( ph \/ ch ) ) ) $=
    ( wn wa wi wo jcab df-or anbi12i 3bitr4i ) ADZBCEZFLBFZLCFZEAMGABGZACGZELBC
    HAMIPNQOABIACIJK $.

  $( 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, 21-Jun-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 $.

$( Theorems relating XOR to conjunction or disjunction. $)

  $( Theorem *5.17 of [WhiteheadRussell] p. 124.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Wolf Lammen, 3-Jan-2013.) $)
  pm5.17 $p |- ( ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) <-> ( ph <-> -. ps ) ) $=
    ( wn wb wi wa wo bicom dfbi2 orcom df-or bitr2i imnan anbi12i 3bitrri ) ABC
    ZDPADPAEZAPEZFABGZABFCZFAPHPAIQSRTSBAGQABJBAKLABMNO $.

  $( Theorem *5.15 of [WhiteheadRussell] p. 124.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Wolf Lammen, 15-Oct-2013.) $)
  pm5.15 $p |- ( ( ph <-> ps ) \/ ( ph <-> -. ps ) ) $=
    ( wb wn xor3 biimpi orri ) ABCZABDCZHDIABEFG $.

  $( Theorem *5.16 of [WhiteheadRussell] p. 124.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Wolf Lammen, 17-Oct-2013.) $)
  pm5.16 $p |- -. ( ( ph <-> ps ) /\ ( ph <-> -. ps ) ) $=
    ( wb wn wi wa pm5.18 biimpi imnan mpbi ) ABCZABDCZDZEKLFDKMABGHKLIJ $.

  $( Two ways to express exclusive disjunction ( ~ df-xor ).  Theorem *5.22 of
     [WhiteheadRussell] p. 124.  (Contributed by NM, 3-Jan-2005.)  (Proof
     shortened by Wolf Lammen, 22-Jan-2013.) $)
  xor $p |- ( -. ( ph <-> ps ) <->
                ( ( ph /\ -. ps ) \/ ( ps /\ -. ph ) ) ) $=
    ( wn wa wo wb wi iman anbi12i dfbi2 ioran 3bitr4ri con1bii ) ABCDZBACDZEZAB
    FZABGZBAGZDNCZOCZDQPCRTSUAABHBAHIABJNOKLM $.

  $( Two ways to express "exclusive or".  (Contributed by NM, 3-Jan-2005.)
     (Proof shortened by Wolf Lammen, 24-Jan-2013.) $)
  nbi2 $p |- ( -. ( ph <-> ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) ) $=
    ( wb wn wo wa xor3 pm5.17 bitr4i ) ABCDABDCABEABFDFABGABHI $.

  $( Conjunction distributes over exclusive-or, using ` -. ( ph <-> ps ) ` to
     express exclusive-or.  This is one way to interpret the distributive law
     of multiplication over addition in modulo 2 arithmetic.  This is not
     necessarily true in intuitionistic logic, though ~ anxordi does hold in
     it.  (Contributed by NM, 3-Oct-2008.) $)
  xordi $p |- ( ( ph /\ -. ( ps <-> ch ) ) <->
                -. ( ( ph /\ ps ) <-> ( ph /\ ch ) ) ) $=
    ( wb wn wa wi annim pm5.32 xchbinx ) ABCDZEFAKGABFACFDAKHABCIJ $.

  $( Theorem *5.54 of [WhiteheadRussell] p. 125.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Wolf Lammen, 7-Nov-2013.) $)
  pm5.54 $p |- ( ( ( ph /\ ps ) <-> ph ) \/ ( ( ph /\ ps ) <-> ps ) ) $=
    ( wa wb iba bicomd adantl pm5.21ni orri ) ABCZADZJBDJKBBKABAJBAEFZGLHI $.

  $( Theorem *5.62 of [WhiteheadRussell] p. 125.  (Contributed by Roy F.
     Longton, 21-Jun-2005.) $)
  pm5.62 $p |- ( ( ( ph /\ ps ) \/ -. ps ) <-> ( ph \/ -. ps ) ) $=
    ( wa wn wo exmid ordir mpbiran2 ) ABCBDZEAIEBIEBFABIGH $.

  $( Theorem *5.63 of [WhiteheadRussell] p. 125.  (Contributed by NM,
     3-Jan-2005.)  (Proof shortened by Wolf Lammen, 25-Dec-2012.) $)
  pm5.63 $p |- ( ( ph \/ ps ) <-> ( ph \/ ( -. ph /\ ps ) ) ) $=
    ( wn wa wo exmid ordi mpbiran bicomi ) AACZBDEZABEZKAJELAFAJBGHI $.

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  $( 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 biorfri 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.  (Contributed by NM,
     3-Jan-2005.) $)
  pm4.83 $p |- ( ( ( ph -> ps ) /\ ( -. ph -> ps ) ) <-> ps ) $=
    ( wn wo wi wa exmid a1bi jaob bitr2i ) BAACZDZBEABEKBEFLBAGHABKIJ $.

  $( Negation inferred from embedded conjunct.  (Contributed by NM,
     20-Aug-1993.)  (Proof shortened by Wolf Lammen, 25-Nov-2012.) $)
  pclem6 $p |- ( ( ph <-> ( ps /\ -. ph ) ) -> -. ps ) $=
    ( wn wa wb ibar nbbn sylib con2i ) BABACZDZEZBJKELCBJFAKGHI $.

  $( 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 $.

  $( Theorem *5.71 of [WhiteheadRussell] p. 125.  (Contributed by Roy F.
     Longton, 23-Jun-2005.) $)
  pm5.71 $p |- ( ( ps -> -. ch ) -> ( ( ( ph \/ ps ) /\ ch ) <->
                ( ph /\ ch ) ) ) $=
    ( wn wo wa wb orel2 orc impbid1 anbi1d pm2.21 pm5.32rd ja ) BCDZABEZCFACFGB
    DZPACQPABAHABIJKOCPACPAGLMN $.

  $( 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.)  (Proof shortened by Kyle Wyonch,
     12-Feb-2021.) $)
  pm5.75 $p |- ( ( ( ch -> -. ps ) /\ ( ph <-> ( ps \/ ch ) ) )
      -> ( ( ph /\ -. ps ) <-> ch ) ) $=
    ( wo wb wn wa wi anbi1 biorf bicomd pm5.32ri bitrdi abai rbaib sylan9bbr )
    ABCDZEZABFZGZCSGZCSHZCRTQSGUAAQSISQCSCQBCJKLMUACUBCSNOP $.

  ${
    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 ) $=
      ( wn mpnanrd wo ord mtord notnotrd ) AEAEJCDABCFGKABDFHKAECDLIMNO $.
  $}

  ${
    ecase3.1 $e |- ( ph -> ch ) $.
    ecase3.2 $e |- ( ps -> ch ) $.
    ecase3.3 $e |- ( -. ( ph \/ ps ) -> ch ) $.
    $( Inference for elimination by cases.  (Contributed by NM, 23-Mar-1995.)
       (Proof shortened by Wolf Lammen, 26-Nov-2012.) $)
    ecase3 $p |- ch $=
      ( wo jaoi pm2.61i ) ABGCACBDEHFI $.
  $}

  ${
    ecase.1 $e |- ( -. ph -> ch ) $.
    ecase.2 $e |- ( -. ps -> ch ) $.
    ecase.3 $e |- ( ( ph /\ ps ) -> ch ) $.
    $( Inference for elimination by cases.  (Contributed by NM,
       13-Jul-2005.) $)
    ecase $p |- ch $=
      ( ex pm2.61nii ) ABCABCFGDEH $.
  $}

  ${
    ecase3d.1 $e |- ( ph -> ( ps -> th ) ) $.
    ecase3d.2 $e |- ( ph -> ( ch -> th ) ) $.
    ecase3d.3 $e |- ( ph -> ( -. ( ps \/ ch ) -> th ) ) $.
    $( Deduction for elimination by cases.  (Contributed by NM, 2-May-1996.)
       (Proof shortened by Andrew Salmon, 7-May-2011.) $)
    ecase3d $p |- ( ph -> th ) $=
      ( wo jaod pm2.61d ) ABCHDABDCEFIGJ $.
  $}

  ${
    ecased.1 $e |- ( ph -> ( -. ps -> th ) ) $.
    ecased.2 $e |- ( ph -> ( -. ch -> th ) ) $.
    ecased.3 $e |- ( ph -> ( ( ps /\ ch ) -> th ) ) $.
    $( Deduction for elimination by cases.  (Contributed by NM, 8-Oct-2012.) $)
    ecased $p |- ( ph -> th ) $=
      ( wn wo wa pm3.11 syl5 ecase3d ) ABHZCHZDEFNOIHBCJADBCKGLM $.
  $}

  ${
    ecase3ad.1 $e |- ( ph -> ( ps -> th ) ) $.
    ecase3ad.2 $e |- ( ph -> ( ch -> th ) ) $.
    ecase3ad.3 $e |- ( ph -> ( ( -. ps /\ -. ch ) -> th ) ) $.
    $( Deduction for elimination by cases.  (Contributed by NM, 24-May-2013.)
       (Proof shortened by Wolf Lammen, 20-Sep-2024.) $)
    ecase3ad $p |- ( ph -> th ) $=
      ( imp wn wa pm2.61ddan ) ABCDABDEHACDFHABICIJDGHK $.
  $}

  ${
    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 $.
  $}

  ${
    4cases.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    4cases.2 $e |- ( ( ph /\ -. ps ) -> ch ) $.
    4cases.3 $e |- ( ( -. ph /\ ps ) -> ch ) $.
    4cases.4 $e |- ( ( -. ph /\ -. ps ) -> ch ) $.
    $( Inference eliminating two antecedents from the four possible cases that
       result from their true/false combinations.  (Contributed by NM,
       25-Oct-2003.) $)
    4cases $p |- ch $=
      ( pm2.61ian wn pm2.61i ) BCABCDFHABICEGHJ $.
  $}

  ${
    4casesdan.1 $e |- ( ( ph /\ ( ps /\ ch ) ) -> th ) $.
    4casesdan.2 $e |- ( ( ph /\ ( ps /\ -. ch ) ) -> th ) $.
    4casesdan.3 $e |- ( ( ph /\ ( -. ps /\ ch ) ) -> th ) $.
    4casesdan.4 $e |- ( ( ph /\ ( -. ps /\ -. ch ) ) -> th ) $.
    $( Deduction eliminating two antecedents from the four possible cases that
       result from their true/false combinations.  (Contributed by NM,
       19-Mar-2013.) $)
    4casesdan $p |- ( ph -> th ) $=
      ( wi wa expcom wn 4cases ) BCADIABCJDEKABCLZJDFKABLZCJDGKAONJDHKM $.
  $}

  ${
    cases.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    cases.2 $e |- ( -. ph -> ( ps <-> th ) ) $.
    $( Case disjunction according to the value of ` ph ` .  (Contributed by NM,
       25-Apr-2019.) $)
    cases $p |- ( ps <-> ( ( ph /\ ch ) \/ ( -. ph /\ th ) ) ) $=
      ( wn wo wa exmid biantrur andir pm5.32i orbi12i 3bitri ) BAAGZHZBIABIZPBI
      ZHACIZPDIZHQBAJKAPBLRTSUAABCEMPBDFMNO $.
  $}

  $( Lemma for an alternate version of weak deduction theorem.  (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 biimt jarri bitrd ) ABBADZCAEZKEZABFCAKMGLKHIJ $.

  $( Lemma for an alternate version of weak deduction theorem.  (Contributed by
     NM, 2-Apr-1994.) $)
  dedlem0b $p |- ( -. ph -> ( ps <-> ( ( ps -> ph ) -> ( ch /\ ph ) ) ) ) $=
    ( wn wi wa pm2.21 imim2d com23 simpr imim12i con1d com12 impbid ) ADZBBAEZC
    AFZEZOPBQOAQBAQGHIROBRBABDPQABAGCAJKLMN $.

  $( Lemma for weak deduction theorem.  See also ~ ifptru .  (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 weak deduction theorem.  See also ~ ifpfal .  (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 $.

  $( Case disjunction according to the value of ` ph ` .  (Contributed by BJ,
     6-Apr-2019.)  (Proof shortened by Wolf Lammen, 28-Feb-2022.) $)
  cases2 $p |- ( ( ( ph /\ ps ) \/ ( -. ph /\ ch ) )
                                   <-> ( ( ph -> ps ) /\ ( -. ph -> ch ) ) ) $=
    ( wa wn wo wi pm4.83 dedlema pm5.74i dedlemb anbi12i ancom orbi12i 3bitr4ri
    ) ABADZCAEZDZFZGZQSGZDSABGZQCGZDABDZQCDZFASHUBTUCUAABSABCIJQCSABCKJLUDPUERA
    BMQCMNO $.

  $( Alternate proof of ~ cases2 , not using ~ dedlema or ~ dedlemb .
     (Contributed by BJ, 6-Apr-2019.)  (Proof shortened by Wolf Lammen,
     2-Jan-2020.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  cases2ALT $p |- ( ( ( ph /\ ps ) \/ ( -. ph /\ ch ) )
                                   <-> ( ( ph -> ps ) /\ ( -. ph -> ch ) ) ) $=
    ( wa wn wo wi pm3.4 pm2.24 adantr pm2.21 jaoi pm2.27 imdistani orcd adantrr
    jca olcd adantrl pm2.61ian impbii ) ABDZAEZCDZFZABGZUCCGZDZUBUHUDUBUFUGABHA
    UGBACIJQUDUFUGUCUFCABKJUCCHQLAUHUEAUFUEUGAUFDUBUDAUFBABMNOPUCUGUEUFUCUGDUDU
    BUCUGCUCCMNRSTUA $.

  $( An alternate definition of the biconditional.  Theorem *5.23 of
     [WhiteheadRussell] p. 124.  (Contributed by NM, 27-Jun-2002.)  (Proof
     shortened by Wolf Lammen, 3-Nov-2013.)  (Proof shortened by NM,
     29-Oct-2021.) $)
  dfbi3 $p |- ( ( ph <-> ps ) <-> ( ( ph /\ ps ) \/ ( -. ph /\ -. ps ) ) ) $=
    ( wi wa wn wb wo con34b anbi2i dfbi2 cases2 3bitr4i ) ABCZBACZDMAEZBEZCZDAB
    FABDOPDGNQMBAHIABJABPKL $.

  $( Theorem *5.24 of [WhiteheadRussell] p. 124.  (Contributed by NM,
     3-Jan-2005.) $)
  pm5.24 $p |- ( -. ( ( ph /\ ps ) \/ ( -. ph /\ -. ps ) ) <->
                ( ( ph /\ -. ps ) \/ ( ps /\ -. ph ) ) ) $=
    ( wb wn wa wo xor dfbi3 xchnxbi ) ABCABDZEBADZEFABEKJEFABGABHI $.

  $( The disjunction of the four possible combinations of two wffs and their
     negations is always true.  A four-way excluded middle (see ~ exmid ).
     (Contributed by David Abernethy, 28-Jan-2014.)  (Proof shortened by NM,
     29-Oct-2021.) $)
  4exmid $p |- ( ( ( ph /\ ps ) \/ ( -. ph /\ -. ps ) )
              \/ ( ( ph /\ -. ps ) \/ ( ps /\ -. ph ) ) ) $=
    ( wa wn wo pm5.24 biimpi orri ) ABCADZBDZCEZAJCBICEZKDLABFGH $.

  $( The consensus theorem.  This theorem and its dual (with ` \/ ` and ` /\ `
     interchanged) are commonly used in computer logic design to eliminate
     redundant terms from Boolean expressions.  Specifically, we prove that the
     term ` ( ps /\ ch ) ` on the left-hand side is redundant.  (Contributed by
     NM, 16-May-2003.)  (Proof shortened by Andrew Salmon, 13-May-2011.)
     (Proof shortened by Wolf Lammen, 20-Jan-2013.) $)
  consensus $p |- ( ( ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) \/ ( ps /\ ch ) ) <->
                      ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) ) $=
    ( wa wn wo id orc adantrr olc adantrl pm2.61ian jaoi impbii ) ABDZAEZCDZFZB
    CDZFRRRSRGASRABRCOQHIPCRBQOJKLMRSHN $.

  $( Theorem *4.42 of [WhiteheadRussell] p. 119.  See also ~ ifpid .
     (Contributed by Roy F. Longton, 21-Jun-2005.) $)
  pm4.42 $p |- ( ph <-> ( ( ph /\ ps ) \/ ( ph /\ -. ps ) ) ) $=
    ( wa wn wo wb dedlema dedlemb pm2.61i ) BAABCABDCEFBAAGBAAHI $.

  ${
    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, 21-Jun-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.) $)
    oplem1 $p |- ( ph -> ps ) $=
      ( wn wa notbii ord biimtrrid jcad biimpar syl6 pm2.18d sylibr ) ADBADADJZ
      CEKDATCETBJACBDHLABCFMNADEGMOCDEIPQRHS $.
  $}

  $( A single axiom for Boolean algebra known as DN_1.  See McCune, Veroff,
     Fitelson, Harris, Feist, Wos, _Short single axioms for Boolean algebra_,
     Journal of Automated Reasoning, 29(1):1--16, 2002.
     ( ~ https://www.cs.unm.edu/~~mccune/papers/basax/v12.pdf ).  (Contributed
     by Jeff Hankins, 3-Jul-2009.)  (Proof shortened by Andrew Salmon,
     13-May-2011.)  (Proof shortened by Wolf Lammen, 6-Jan-2013.) $)
  dn1 $p |- ( -. ( -. ( -. ( ph \/ ps ) \/ ch ) \/
            -. ( ph \/ -. ( -. ch \/ -. ( ch \/ th ) ) ) ) <-> ch ) $=
    ( wo wn wa pm2.45 imnan mpbi biorfri orcom ordir 3bitri pm4.45 bitri orbi2i
    wi anor anbi2i 3bitrri ) CABEFZCEZACEZGZUCACFCDEZFEFZEZGUCFUHFEFCCUBAGZEUIC
    EUEUICUBAFRUIFABHUBAIJKCUILUBACMNUDUHUCCUGACCUFGUGCDOCUFSPQTUCUHSUA $.

  $( A closed form of ~ mpbir , analogous to ~ pm2.27 (assertion).
     (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)  (Proof shortened by Roger
     Witte, 17-Aug-2020.) $)
  bianir $p |- ( ( ph /\ ( ps <-> ph ) ) -> ps ) $=
    ( wb biimpr impcom ) BACABBADE $.

  ${
    jaoi2.1 $e |- ( ( ph \/ ( -. ph /\ ch ) ) -> ps ) $.
    $( Inference removing a negated conjunct in a disjunction of an antecedent
       if this conjunct is part of the disjunction.  (Contributed by Alexander
       van der Vekens, 3-Nov-2017.)  (Proof shortened by Wolf Lammen,
       21-Sep-2018.) $)
    jaoi2 $p |- ( ( ph \/ ch ) -> ps ) $=
      ( wo wn wa pm5.63 sylbi ) ACEAAFCGEBACHDI $.
  $}

  ${
    jaoi3.1 $e |- ( ph -> ps ) $.
    jaoi3.2 $e |- ( ( -. ph /\ ch ) -> ps ) $.
    $( Inference separating a disjunct of an antecedent.  (Contributed by
       Alexander van der Vekens, 25-May-2018.) $)
    jaoi3 $p |- ( ( ph \/ ch ) -> ps ) $=
      ( wn wa jaoi jaoi2 ) ABCABAFCGDEHI $.
  $}

  $( Selecting one statement from a disjunction if one of the disjuncted
     statements is false.  (Contributed by AV, 6-Sep-2018.)  (Proof shortened
     by AV, 13-Oct-2018.)  (Proof shortened by Wolf Lammen, 19-Jan-2020.) $)
  ornld $p |- ( ph -> ( ( ( ph -> ( th \/ ta ) ) /\ -. th ) -> ta ) ) $=
    ( wo wi wn wa pm3.35 ord expimpd ) AABCDZEZBFCALGBCAKHIJ $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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 ~ dfifp2 , ~ dfifp3 , ~ dfifp4 , ~ dfifp5 , ~ dfifp6 and ~ dfifp7 for
     alternate definitions.

     This definition (in the form of ~ dfifp2 ) 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 ~ dfifp2 for
     compatibility with intuitionistic logic development: with this form, it is
     clear that ` if- ( ph , ps , ch ) ` implies decidability of ` ph ` , which
     is most often what is wanted.

     Church uses the conditional operator as an intermediate step to prove
     completeness of some systems of connectives.  The first result is that the
     system ` { if- , T. , F. } ` is complete: for the induction step, consider
     a formula of n+1 variables; single out one variable, say ` ph ` ; when one
     sets ` ph ` to True (resp.  False), then what remains is a formula of n
     variables, so by the induction hypothesis it is equivalent to a formula
     using only the connectives ` if- , T. , F. ` , say ` ps ` (resp. ` ch ` );
     therefore, the formula ` if- ( ph , ps , ch ) ` is equivalent to the
     initial formula of n+1 variables.  Now, since ` { -> , -. } ` and similar
     systems suffice to express the connectives ` if- , T. , F. ` , they are
     also complete.

     (Contributed by BJ, 22-Jun-2019.) $)
  df-ifp $a |-
            ( if- ( ph , ps , ch ) <-> ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) ) $.

  $( Alternate definition of the conditional operator for 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.) $)
  dfifp2 $p |-
            ( if- ( ph , ps , ch ) <-> ( ( ph -> ps ) /\ ( -. ph -> ch ) ) ) $=
    ( wif wa wn wo wi df-ifp cases2 bitri ) ABCDABEAFZCEGABHLCHEABCIABCJK $.

  $( Alternate definition of the conditional operator for propositions.
     (Contributed by BJ, 30-Sep-2019.) $)
  dfifp3 $p |- ( if- ( ph , ps , ch ) <-> ( ( ph -> ps ) /\ ( ph \/ ch ) ) ) $=
    ( wif wi wn wa wo dfifp2 pm4.64 anbi2i bitri ) ABCDABEZAFCEZGMACHZGABCINOMA
    CJKL $.

  $( Alternate definition of the conditional operator for propositions.
     (Contributed by BJ, 30-Sep-2019.) $)
  dfifp4 $p |-
            ( if- ( ph , ps , ch ) <-> ( ( -. ph \/ ps ) /\ ( ph \/ ch ) ) ) $=
    ( wif wi wo wn dfifp3 imor bianbi ) ABCDABEACFAGBFABCHABIJ $.

  $( Alternate definition of the conditional operator for propositions.
     (Contributed by BJ, 2-Oct-2019.) $)
  dfifp5 $p |-
         ( if- ( ph , ps , ch ) <-> ( ( -. ph \/ ps ) /\ ( -. ph -> ch ) ) ) $=
    ( wif wi wn wo dfifp2 imor bianbi ) ABCDABEAFZCEKBGABCHABIJ $.

  $( Alternate definition of the conditional operator for propositions.
     (Contributed by BJ, 2-Oct-2019.) $)
  dfifp6 $p |-
            ( if- ( ph , ps , ch ) <-> ( ( ph /\ ps ) \/ -. ( ch -> ph ) ) ) $=
    ( wif wa wn wo wi df-ifp ancom annim bitri orbi2i ) ABCDABEZAFZCEZGNCAHFZGA
    BCIPQNPCOEQOCJCAKLML $.

  $( Alternate definition of the conditional operator for propositions.
     (Contributed by BJ, 2-Oct-2019.) $)
  dfifp7 $p |- ( if- ( ph , ps , ch ) <-> ( ( ch -> ph ) -> ( ph /\ ps ) ) ) $=
    ( wa wi wn wo wif orcom dfifp6 imor 3bitr4i ) ABDZCAEZFZGOMGABCHNMEMOIABCJN
    MKL $.

  $( Define the biconditional as conditional logic operator.  (Contributed by
     RP, 20-Apr-2020.)  (Proof shortened by Wolf Lammen, 30-Apr-2024.) $)
  ifpdfbi $p |- ( ( ph <-> ps ) <-> if- ( ph , ps , -. ps ) ) $=
    ( wi wa wn wb wif con34b anbi2i dfbi2 dfifp2 3bitr4i ) ABCZBACZDMAEBEZCZDAB
    FABOGNPMBAHIABJABOKL $.

  $( The conditional operator is implied by the conjunction of its possible
     outputs.  Dual statement of ~ ifpor .  (Contributed by BJ,
     30-Sep-2019.) $)
  anifp $p |- ( ( ps /\ ch ) -> if- ( ph , ps , ch ) ) $=
    ( wa wn wo wif olc anim12i dfifp4 sylibr ) BCDAEZBFZACFZDABCGBMCNBLHCAHIABC
    JK $.

  $( The conditional operator implies the disjunction of its possible outputs.
     Dual statement of ~ anifp .  (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.) $)
  ifpn $p |- ( if- ( ph , ps , ch ) <-> if- ( -. ph , ch , ps ) ) $=
    ( wn wo wi wa wif ancom dfifp5 dfifp3 3bitr4i ) ADZBEZMCFZGONGABCHMCBHNOIAB
    CJMCBKL $.

  $( 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 ) ) $=
    ( wi wif biimt wo wa orc biantrud dfifp3 bitr4di bitr2d ) ABABDZABCEZABFANN
    ACGZHOAPNACIJABCKLM $.

  $( 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 ) ) $=
    ( wif wn ifpn ifptru bitrid ) ABCDAEZCBDICABCFICBGH $.

  $( Value of the conditional operator for propositions when the same
     proposition is returned in either case.  Analogue for propositions of
     ~ ifid .  This is essentially ~ pm4.42 .  (Contributed by BJ,
     20-Sep-2019.) $)
  ifpid $p |- ( if- ( ph , ps , ps ) <-> ps ) $=
    ( wif wb ifptru ifpfal pm2.61i ) AABBCBDABBEABBFG $.

  ${
    casesifp.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    casesifp.2 $e |- ( -. ph -> ( ps <-> th ) ) $.
    $( Version of ~ cases expressed using ` if- ` .  Case disjunction according
       to the value of ` ph ` .  One can see this as a proof that the two
       hypotheses characterize the conditional operator for propositions.  For
       the converses, see ~ ifptru and ~ ifpfal .  (Contributed by BJ,
       20-Sep-2019.) $)
    casesifp $p |- ( ps <-> if- ( ph , ch , th ) ) $=
      ( wa wn wo wif cases df-ifp bitr4i ) BACGAHDGIACDJABCDEFKACDLM $.
  $}

  ${
    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 ) ) ) $=
      ( wi wo wa wif imbi12d orbi12d anbi12d dfifp3 3bitr4g ) ABCKZBDLZMEFKZEGL
      ZMBCDNEFGNATUBUAUCABECFHIOABEDGHJPQBCDREFGRS $.
  $}

  ${
    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 $.
  $}

  ${
    ifpimpda.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    ifpimpda.2 $e |- ( ( ph /\ -. ps ) -> th ) $.
    $( Separation of the values of the conditional operator for propositions.
       (Contributed by AV, 30-Dec-2020.)  (Proof shortened by Wolf Lammen,
       27-Feb-2021.) $)
    ifpimpda $p |- ( ph -> if- ( ps , ch , th ) ) $=
      ( wi wn wif ex dfifp2 sylanbrc ) ABCGBHZDGBCDIABCEJAMDFJBCDKL $.
  $}

  ${
    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 $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  The weak deduction theorem for propositional calculus
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  This subsection contains a few results related to the weak deduction theorem
  in propositional calculus.  For the weak deduction theorem in set theory, see
  the section beginning with ~ dedth .  For more information on the weak
  deduction theorem, see the Weak Deduction Theorem page ~ mmdeduction.html .

$)

  ${
    elimh.1 $e |- ( ( if- ( ch , ph , ps ) <-> ph ) -> ( ta <-> ch ) ) $.
    elimh.2 $e |- ( ( if- ( ch , ph , ps ) <-> ps ) -> ( ta <-> th ) ) $.
    elimh.3 $e |- th $.
    $( Hypothesis builder for the weak deduction theorem.  For more
       information, see the Weak Deduction Theorem page ~ mmdeduction.html .
       (Contributed by NM, 26-Jun-2002.)  Revised to use the conditional
       operator.  (Revised by BJ, 30-Sep-2019.)  Commute consequent.  (Revised
       by Steven Nguyen, 27-Apr-2023.) $)
    elimh $p |- ta $=
      ( wif wb ifptru syl ibir wn ifpfal mpbiri pm2.61i ) CECECCABIZAJECJCABKFL
      MCNZEDHSRBJEDJCABOGLPQ $.
  $}

  ${
    dedt.1 $e |- ( ( if- ( ch , ph , ps ) <-> ph ) -> ( ta <-> th ) ) $.
    dedt.2 $e |- ta $.
    $( The weak deduction theorem.  For more information, see the Weak
       Deduction Theorem page ~ mmdeduction.html .  (Contributed by NM,
       26-Jun-2002.)  Revised to use the conditional operator.  (Revised by BJ,
       30-Sep-2019.)  Commute consequent.  (Revised by Steven Nguyen,
       27-Apr-2023.) $)
    dedt $p |- ( ch -> th ) $=
      ( wif wb ifptru mpbii syl ) CCABHAIZDCABJMEDGFKL $.
  $}

  $( Proof of ~ con3 from its associated inference ~ con3i that illustrates the
     use of the weak deduction theorem ~ dedt .  (Contributed by NM,
     27-Jun-2002.)  Revised to use the conditional operator.  (Revised by BJ,
     30-Sep-2019.)  Revised ~ dedt and ~ elimh .  (Revised by Steven Nguyen,
     27-Apr-2023.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  con3ALT $p |- ( ( ph -> ps ) -> ( -. ps -> -. ph ) ) $=
    ( wi wn wif wb id notbid imbi1d imbi2 elimh con3i dedt ) BAABCZBDZADZCNBAEZ
    DZPCQBFZROPSQBSGHIAQBANAACAQCQBAJQAAJAGKLM $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Abbreviated conjunction and disjunction of three wff's
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Extend wff definition to include three-way disjunction ('or'). $)
  w3o $a wff ( ph \/ ps \/ ch ) $.
  $( Extend wff definition to include three-way conjunction ('and'). $)
  w3a $a wff ( ph /\ ps /\ ch ) $.

  $( These abbreviations help eliminate parentheses to aid readability. $)

  $( Define disjunction ('or') of three 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 three 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 $.

  $( Partial elimination of a triple disjunction by denial of a disjunct.
     (Contributed by Scott Fenton, 26-Mar-2011.) $)
  3orel1 $p |- ( -. ph -> ( ( ph \/ ps \/ ch ) -> ( ps \/ ch ) ) ) $=
    ( w3o wo wn 3orass orel1 biimtrid ) ABCDABCEZEAFJABCGAJHI $.

  $( 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 disjunction.  (Contributed by Mario Carneiro,
     4-Sep-2016.) $)
  3orcoma $p |- ( ( ph \/ ps \/ ch ) <-> ( ps \/ ph \/ ch ) ) $=
    ( wo w3o or12 3orass 3bitr4i ) ABCDDBACDDABCEBACEABCFABCGBACGH $.

  $( Commutation law for triple disjunction.  (Contributed by Scott Fenton,
     20-Apr-2011.)  (Proof shortened by Wolf Lammen, 8-Apr-2022.) $)
  3orcomb $p |- ( ( ph \/ ps \/ ch ) <-> ( ph \/ ch \/ ps ) ) $=
    ( w3o 3orcoma 3orrot bitri ) ABCDBACDACBDABCEBACFG $.

  $( 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 $.

  $( Convert triple conjunction to conjunction, then commute.  (Contributed by
     Jonathan Ben-Naim, 3-Jun-2011.)  (Proof shortened by Andrew Salmon,
     14-Jun-2011.)  (Revised to shorten ~ 3ancoma by Wolf Lammen,
     5-Jun-2022.) $)
  3anan12 $p |- ( ( ph /\ ps /\ ch ) <-> ( ps /\ ( ph /\ ch ) ) ) $=
    ( w3a wa 3anass an12 bitri ) ABCDABCEEBACEEABCFABCGH $.

  $( 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 $.

  $( Commutation law for triple conjunction.  (Contributed by NM, 21-Apr-1994.)
     (Proof shortened by Wolf Lammen, 5-Jun-2022.) $)
  3ancoma $p |- ( ( ph /\ ps /\ ch ) <-> ( ps /\ ph /\ ch ) ) $=
    ( w3a wa 3anan12 3anass bitr4i ) ABCDBACEEBACDABCFBACGH $.

  $( Commutation law for triple conjunction.  (Contributed by NM, 21-Apr-1994.)
     (Revised to shorten ~ 3anrot by Wolf Lammen, 9-Jun-2022.) $)
  3ancomb $p |- ( ( ph /\ ps /\ ch ) <-> ( ph /\ ch /\ ps ) ) $=
    ( w3a wa df-3an 3anan32 bitr4i ) ABCDABECEACBDABCFACBGH $.

  $( Rotation law for triple conjunction.  (Contributed by NM, 8-Apr-1994.)
     (Proof shortened by Wolf Lammen, 9-Jun-2022.) $)
  3anrot $p |- ( ( ph /\ ps /\ ch ) <-> ( ps /\ ch /\ ph ) ) $=
    ( w3a 3ancoma 3ancomb bitri ) ABCDBACDBCADABCEBACFG $.

  $( Reversal law for triple conjunction.  (Contributed by NM, 21-Apr-1994.) $)
  3anrev $p |- ( ( ph /\ ps /\ ch ) <-> ( ch /\ ps /\ ph ) ) $=
    ( w3a 3ancoma 3anrot bitr4i ) ABCDBACDCBADABCECBAFG $.

  $( 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 $.

  $( Idempotent law for conjunction.  (Contributed by Peter Mazsa,
     17-Oct-2023.) $)
  3anidm $p |- ( ( ph /\ ph /\ ph ) <-> ph ) $=
    ( w3a wa df-3an anabs1 anidm 3bitri ) AAABAACZACHAAAADAAEAFG $.

  $( Associative law for four conjunctions with a triple conjunction.
     (Contributed by Alexander van der Vekens, 24-Jun-2018.) $)
  3an4anass $p |- ( ( ( ph /\ ps /\ ch ) /\ th )
                    <-> ( ( ph /\ ps ) /\ ( ch /\ th ) ) ) $=
    ( w3a wa df-3an anbi1i anass bitri ) ABCEZDFABFZCFZDFLCDFFKMDABCGHLCDIJ $.

  $( 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 $.

  $( Negated triple conjunction expressed in terms of triple disjunction.
     (Contributed by Jeff Hankins, 15-Aug-2009.)  (Proof shortened by Andrew
     Salmon, 13-May-2011.)  Shorten with ~ xchnxbir .  (Revised by Wolf Lammen,
     8-Apr-2022.) $)
  3ianor $p |- ( -. ( ph /\ ps /\ ch ) <-> ( -. ph \/ -. ps \/ -. ch ) ) $=
    ( wa wn wo w3a w3o ianor orbi1i df-3an xchnxbir df-3or 3bitr4i ) ABDZEZCEZF
    ZAEZBEZFZQFABCGZESTQHPUAQABIJOCDRUBOCIABCKLSTQMN $.

  $( Triple conjunction expressed in terms of triple disjunction.  (Contributed
     by Jeff Hankins, 15-Aug-2009.)  (Proof shortened by Wolf Lammen,
     8-Apr-2022.) $)
  3anor $p |- ( ( ph /\ ps /\ ch ) <-> -. ( -. ph \/ -. ps \/ -. ch ) ) $=
    ( wn w3o w3a 3ianor con1bii bicomi ) ADBDCDEZDABCFZKJABCGHI $.

  $( Triple disjunction in terms of triple conjunction.  (Contributed by NM,
     8-Oct-2012.) $)
  3oran $p |- ( ( ph \/ ps \/ ch ) <-> -. ( -. ph /\ -. ps /\ -. ch ) ) $=
    ( wn w3a w3o 3ioran con1bii bicomi ) ADBDCDEZDABCFZKJABCGHI $.

  ${
    3impa.1 $e |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $.
    $( Importation from double to triple conjunction.  (Contributed by NM,
       20-Aug-1995.)  (Revised to shorten ~ 3imp by Wolf Lammen,
       20-Jun-2022.) $)
    3impa $p |- ( ( ph /\ ps /\ ch ) -> th ) $=
      ( w3a wa df-3an sylbi ) ABCFABGCGDABCHEI $.
  $}

  ${
    3imp.1 $e |- ( ph -> ( ps -> ( ch -> th ) ) ) $.
    $( Importation inference.  (Contributed by NM, 8-Apr-1994.)  (Proof
       shortened by Wolf Lammen, 20-Jun-2022.) $)
    3imp $p |- ( ( ph /\ ps /\ ch ) -> th ) $=
      ( imp31 3impa ) ABCDABCDEFG $.

    $( 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 $.
  $}

  ${
    bi23imp13.1 $e |- ( ph -> ( ps <-> ( ch -> th ) ) ) $.
    $( ~ 3imp with middle implication of the hypothesis a biconditional.
       (Contributed by Alan Sare, 6-Nov-2017.) $)
    bi23imp13 $p |- ( ( ph /\ ps /\ ch ) -> th ) $=
      ( wi biimpd 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 $.
  $}

  ${
    3impia.1 $e |- ( ( ph /\ ps ) -> ( ch -> th ) ) $.
    $( Importation to triple conjunction.  (Contributed by NM, 13-Jun-2006.)
       (Proof shortened by Wolf Lammen, 21-Jun-2022.) $)
    3impia $p |- ( ( ph /\ ps /\ ch ) -> th ) $=
      ( expimpd 3impib ) ABCDABCDEFG $.
  $}

  ${
    3exp.1 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
    $( Exportation from triple to double conjunction.  (Contributed by NM,
       20-Aug-1995.)  (Revised to shorten ~ 3exp and ~ pm3.2an3 by Wolf Lammen,
       22-Jun-2022.) $)
    3expa $p |- ( ( ( ph /\ ps ) /\ ch ) -> th ) $=
      ( wa w3a df-3an sylbir ) ABFCFABCGDABCHEI $.

    $( Exportation inference.  (Contributed by NM, 30-May-1994.)  (Proof
       shortened by Wolf Lammen, 22-Jun-2022.) $)
    3exp $p |- ( ph -> ( ps -> ( ch -> th ) ) ) $=
      ( 3expa exp31 ) 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.)
       (Proof shortened by Wolf Lammen, 22-Jun-2022.) $)
    3expia $p |- ( ( ph /\ ps ) -> ( ch -> th ) ) $=
      ( 3expb expr ) ABCDABCDEFG $.

    $( 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 2nd.  (Contributed by NM,
       28-Jan-1996.)  (Proof shortened by Andrew Salmon, 13-May-2011.)  (Proof
       shortened by Wolf Lammen, 22-Jun-2022.) $)
    3com12 $p |- ( ( ps /\ ph /\ ch ) -> th ) $=
      ( 3exp 3imp21 ) ABCDABCDEFG $.

    $( Commutation in antecedent.  Swap 1st and 3rd.  (Contributed by NM,
       28-Jan-1996.)  (Proof shortened by Wolf Lammen, 22-Jun-2022.) $)
    3com13 $p |- ( ( ch /\ ps /\ ph ) -> th ) $=
      ( 3exp 3imp31 ) ABCDABCDEFG $.

    $( Commutation in antecedent.  Rotate right.  (Contributed by NM,
       28-Jan-1996.)  Theorems shortened and reordered.  (Revised by Wolf
       Lammen, 9-Apr-2022.) $)
    3comr $p |- ( ( ch /\ ph /\ ps ) -> th ) $=
      ( 3com12 3com13 ) BACDABCDEFG $.

    $( Commutation in antecedent.  Swap 2nd and 3rd.  (Contributed by NM,
       28-Jan-1996.)  (Proof shortened by Wolf Lammen, 9-Apr-2022.) $)
    3com23 $p |- ( ( ph /\ ch /\ ps ) -> th ) $=
      ( 3comr 3com12 ) CABDABCDEFG $.

    $( Commutation in antecedent.  Rotate left.  (Contributed by NM,
       28-Jan-1996.) $)
    3coml $p |- ( ( ps /\ ch /\ ph ) -> th ) $=
      ( 3com23 3com13 ) ACBDABCDEFG $.
  $}

  ${
    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 $.
  $}

  ${
    3adant.1 $e |- ( ( ph /\ ps ) -> ch ) $.
    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       16-Jul-1995.)  (Proof shortened by Wolf Lammen, 21-Jun-2022.) $)
    3adant1 $p |- ( ( th /\ ph /\ ps ) -> ch ) $=
      ( adantll 3impa ) DABCABCDEFG $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       16-Jul-1995.) $)
    3adant2 $p |- ( ( ph /\ th /\ ps ) -> ch ) $=
      ( adantlr 3impa ) ADBCABCDEFG $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       16-Jul-1995.)  (Proof shortened by Wolf Lammen, 21-Jun-2022.) $)
    3adant3 $p |- ( ( ph /\ ps /\ th ) -> ch ) $=
      ( adantrr 3impb ) ABDCABCDEFG $.
  $}

  ${
    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, 21-Apr-1994.)
     (Proof shortened by Wolf Lammen, 22-Jun-2022.) $)
  simp1 $p |- ( ( ph /\ ps /\ ch ) -> ph ) $=
    ( id 3ad2ant1 ) ABACADE $.

  $( Simplification of triple conjunction.  (Contributed by NM, 21-Apr-1994.)
     (Proof shortened by Wolf Lammen, 22-Jun-2022.) $)
  simp2 $p |- ( ( ph /\ ps /\ ch ) -> ps ) $=
    ( id 3ad2ant2 ) BABCBDE $.

  $( Simplification of triple conjunction.  (Contributed by NM, 21-Apr-1994.)
     (Proof shortened by Wolf Lammen, 22-Jun-2022.) $)
  simp3 $p |- ( ( ph /\ ps /\ ch ) -> ch ) $=
    ( id 3ad2ant3 ) CACBCDE $.

  ${
    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 $.
  $}

  $( Simplification of triple conjunction.  (Contributed by NM, 21-Apr-1994.)
     (Proof shortened by Wolf Lammen, 21-Jun-2022.) $)
  3simpa $p |- ( ( ph /\ ps /\ ch ) -> ( ph /\ ps ) ) $=
    ( wa id 3adant3 ) ABABDZCGEF $.

  $( Simplification of triple conjunction.  (Contributed by NM, 21-Apr-1994.)
     (Proof shortened by Wolf Lammen, 21-Jun-2022.) $)
  3simpb $p |- ( ( ph /\ ps /\ ch ) -> ( ph /\ ch ) ) $=
    ( wa id 3adant2 ) ACACDZBGEF $.

  $( Simplification of triple conjunction.  (Contributed by NM, 21-Apr-1994.)
     (Proof shortened by Andrew Salmon, 13-May-2011.)  (Proof shortened by Wolf
     Lammen, 21-Jun-2022.) $)
  3simpc $p |- ( ( ph /\ ps /\ ch ) -> ( ps /\ ch ) ) $=
    ( wa id 3adant1 ) BCBCDZAGEF $.

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  ${
    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.)  (Proof
       shortened by Wolf Lammen, 26-Jun-2022.) $)
    syl3an2 $p |- ( ( ps /\ ph /\ th ) -> ta ) $=
      ( w3a 3anim2i syl ) BADHBCDHEACBDFIGJ $.
  $}

  ${
    syl3an3.1 $e |- ( ph -> th ) $.
    syl3an3.2 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
    $( A syllogism inference.  (Contributed by NM, 22-Aug-1995.)  (Proof
       shortened by Wolf Lammen, 26-Jun-2022.) $)
    syl3an3 $p |- ( ( ps /\ ch /\ ph ) -> ta ) $=
      ( w3a 3anim3i syl ) BCAHBCDHEADBCFIGJ $.
  $}

  ${
    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 $.
  $}

  ${
    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 $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       8-Jan-2006.)  (Proof shortened by Wolf Lammen, 23-Jun-2022.) $)
    3adant1l $p |- ( ( ( ta /\ ph ) /\ ps /\ ch ) -> th ) $=
      ( wa simpr syl3an1 ) EAGABCDEAHFI $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       8-Jan-2006.)  (Proof shortened by Wolf Lammen, 23-Jun-2022.) $)
    3adant1r $p |- ( ( ( ph /\ ta ) /\ ps /\ ch ) -> th ) $=
      ( wa simpl syl3an1 ) AEGABCDAEHFI $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       8-Jan-2006.)  (Proof shortened by Wolf Lammen, 25-Jun-2022.) $)
    3adant2l $p |- ( ( ph /\ ( ta /\ ps ) /\ ch ) -> th ) $=
      ( wa simpr syl3an2 ) EBGABCDEBHFI $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       8-Jan-2006.)  (Proof shortened by Wolf Lammen, 25-Jun-2022.) $)
    3adant2r $p |- ( ( ph /\ ( ps /\ ta ) /\ ch ) -> th ) $=
      ( wa simpl syl3an2 ) BEGABCDBEHFI $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       8-Jan-2006.)  (Proof shortened by Wolf Lammen, 25-Jun-2022.) $)
    3adant3l $p |- ( ( ph /\ ps /\ ( ta /\ ch ) ) -> th ) $=
      ( wa simpr syl3an3 ) ECGABCDECHFI $.

    $( Deduction adding a conjunct to antecedent.  (Contributed by NM,
       8-Jan-2006.)  (Proof shortened by Wolf Lammen, 25-Jun-2022.) $)
    3adant3r $p |- ( ( ph /\ ps /\ ( ch /\ ta ) ) -> th ) $=
      ( wa simpl syl3an3 ) CEGABCDCEHFI $.

    $( 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 $.
  $}

  ${
    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 conjuncts to antecedent.  (Contributed by NM,
       25-Dec-2007.) $)
    3ad2antr1 $p |- ( ( ph /\ ( ch /\ ps /\ ta ) ) -> th ) $=
      ( adantrr 3adantr3 ) ACBDEACDBFGH $.

    $( Deduction adding conjuncts to antecedent.  (Contributed by NM,
       27-Dec-2007.) $)
    3ad2antr2 $p |- ( ( ph /\ ( ps /\ ch /\ ta ) ) -> th ) $=
      ( adantrl 3adantr3 ) ABCDEACDBFGH $.

    $( Deduction adding conjuncts to antecedent.  (Contributed by NM,
       30-Dec-2007.) $)
    3ad2antr3 $p |- ( ( ph /\ ( ps /\ ta /\ ch ) ) -> th ) $=
      ( adantrl 3adantr1 ) AECDBACDEFGH $.
  $}

  $( Simplification of conjunction.  (Contributed by Jeff Hankins,
     17-Nov-2009.)  (Proof shortened by Wolf Lammen, 23-Jun-2022.) $)
  simpl1 $p |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ph ) $=
    ( simpl 3ad2antl1 ) ABDACADEF $.

  $( Simplification of conjunction.  (Contributed by Jeff Hankins,
     17-Nov-2009.)  (Proof shortened by Wolf Lammen, 23-Jun-2022.) $)
  simpl2 $p |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ps ) $=
    ( simpl 3ad2antl2 ) BADBCBDEF $.

  $( Simplification of conjunction.  (Contributed by Jeff Hankins,
     17-Nov-2009.)  (Proof shortened by Wolf Lammen, 23-Jun-2022.) $)
  simpl3 $p |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ch ) $=
    ( simpl 3ad2antl3 ) CADCBCDEF $.

  $( Simplification of conjunction.  (Contributed by Jeff Hankins,
     17-Nov-2009.)  (Proof shortened by Wolf Lammen, 23-Jun-2022.) $)
  simpr1 $p |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ps ) $=
    ( simpr 3ad2antr1 ) ACBBDABEF $.

  $( Simplification of conjunction.  (Contributed by Jeff Hankins,
     17-Nov-2009.)  (Proof shortened by Wolf Lammen, 23-Jun-2022.) $)
  simpr2 $p |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> ch ) $=
    ( simpr 3ad2antr2 ) ABCCDACEF $.

  $( Simplification of conjunction.  (Contributed by Jeff Hankins,
     17-Nov-2009.)  (Proof shortened by Wolf Lammen, 23-Jun-2022.) $)
  simpr3 $p |- ( ( ph /\ ( ps /\ ch /\ th ) ) -> th ) $=
    ( simpr 3ad2antr3 ) ABDDCADEF $.

  $( 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.)  (Proof
     shortened by Wolf Lammen, 23-Jun-2022.) $)
  simpll1 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th ) /\ ta ) -> ph ) $=
    ( w3a simp1 ad2antrr ) ABCFADEABCGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 23-Jun-2022.) $)
  simpll2 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th ) /\ ta ) -> ps ) $=
    ( w3a simp2 ad2antrr ) ABCFBDEABCGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 23-Jun-2022.) $)
  simpll3 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th ) /\ ta ) -> ch ) $=
    ( w3a simp3 ad2antrr ) ABCFCDEABCGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 23-Jun-2022.) $)
  simplr1 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) ) /\ ta ) -> ph ) $=
    ( w3a simp1 ad2antlr ) ABCFADEABCGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 23-Jun-2022.) $)
  simplr2 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) ) /\ ta ) -> ps ) $=
    ( w3a simp2 ad2antlr ) ABCFBDEABCGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 23-Jun-2022.) $)
  simplr3 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) ) /\ ta ) -> ch ) $=
    ( w3a simp3 ad2antlr ) ABCFCDEABCGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 23-Jun-2022.) $)
  simprl1 $p |- ( ( ta /\ ( ( ph /\ ps /\ ch ) /\ th ) ) -> ph ) $=
    ( w3a simp1 ad2antrl ) ABCFAEDABCGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 23-Jun-2022.) $)
  simprl2 $p |- ( ( ta /\ ( ( ph /\ ps /\ ch ) /\ th ) ) -> ps ) $=
    ( w3a simp2 ad2antrl ) ABCFBEDABCGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 23-Jun-2022.) $)
  simprl3 $p |- ( ( ta /\ ( ( ph /\ ps /\ ch ) /\ th ) ) -> ch ) $=
    ( w3a simp3 ad2antrl ) ABCFCEDABCGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 23-Jun-2022.) $)
  simprr1 $p |- ( ( ta /\ ( th /\ ( ph /\ ps /\ ch ) ) ) -> ph ) $=
    ( w3a simp1 ad2antll ) ABCFAEDABCGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 23-Jun-2022.) $)
  simprr2 $p |- ( ( ta /\ ( th /\ ( ph /\ ps /\ ch ) ) ) -> ps ) $=
    ( w3a simp2 ad2antll ) ABCFBEDABCGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 23-Jun-2022.) $)
  simprr3 $p |- ( ( ta /\ ( th /\ ( ph /\ ps /\ ch ) ) ) -> ch ) $=
    ( w3a simp3 ad2antll ) ABCFCEDABCGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 23-Jun-2022.) $)
  simpl1l $p |- ( ( ( ( ph /\ ps ) /\ ch /\ th ) /\ ta ) -> ph ) $=
    ( wa simpll 3ad2antl1 ) ABFCEADABEGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 23-Jun-2022.) $)
  simpl1r $p |- ( ( ( ( ph /\ ps ) /\ ch /\ th ) /\ ta ) -> ps ) $=
    ( wa simplr 3ad2antl1 ) ABFCEBDABEGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 23-Jun-2022.) $)
  simpl2l $p |- ( ( ( ch /\ ( ph /\ ps ) /\ th ) /\ ta ) -> ph ) $=
    ( wa simpll 3ad2antl2 ) ABFCEADABEGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 23-Jun-2022.) $)
  simpl2r $p |- ( ( ( ch /\ ( ph /\ ps ) /\ th ) /\ ta ) -> ps ) $=
    ( wa simplr 3ad2antl2 ) ABFCEBDABEGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 23-Jun-2022.) $)
  simpl3l $p |- ( ( ( ch /\ th /\ ( ph /\ ps ) ) /\ ta ) -> ph ) $=
    ( wa simpll 3ad2antl3 ) ABFCEADABEGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 23-Jun-2022.) $)
  simpl3r $p |- ( ( ( ch /\ th /\ ( ph /\ ps ) ) /\ ta ) -> ps ) $=
    ( wa simplr 3ad2antl3 ) ABFCEBDABEGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 24-Jun-2022.) $)
  simpr1l $p |- ( ( ta /\ ( ( ph /\ ps ) /\ ch /\ th ) ) -> ph ) $=
    ( wa simprl 3ad2antr1 ) ECABFADEABGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 24-Jun-2022.) $)
  simpr1r $p |- ( ( ta /\ ( ( ph /\ ps ) /\ ch /\ th ) ) -> ps ) $=
    ( wa simprr 3ad2antr1 ) ECABFBDEABGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 24-Jun-2022.) $)
  simpr2l $p |- ( ( ta /\ ( ch /\ ( ph /\ ps ) /\ th ) ) -> ph ) $=
    ( wa simprl 3ad2antr2 ) ECABFADEABGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 24-Jun-2022.) $)
  simpr2r $p |- ( ( ta /\ ( ch /\ ( ph /\ ps ) /\ th ) ) -> ps ) $=
    ( wa simprr 3ad2antr2 ) ECABFBDEABGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 24-Jun-2022.) $)
  simpr3l $p |- ( ( ta /\ ( ch /\ th /\ ( ph /\ ps ) ) ) -> ph ) $=
    ( wa simprl 3ad2antr3 ) ECABFADEABGH $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 24-Jun-2022.) $)
  simpr3r $p |- ( ( ta /\ ( ch /\ th /\ ( ph /\ ps ) ) ) -> ps ) $=
    ( wa simprr 3ad2antr3 ) ECABFBDEABGH $.

  $( 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.)  (Proof
     shortened by Wolf Lammen, 24-Jun-2022.) $)
  simpl11 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ et ) -> ph ) $=
    ( w3a simpl1 3ad2antl1 ) ABCGDFAEABCFHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 24-Jun-2022.) $)
  simpl12 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ et ) -> ps ) $=
    ( w3a simpl2 3ad2antl1 ) ABCGDFBEABCFHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 24-Jun-2022.) $)
  simpl13 $p |- ( ( ( ( ph /\ ps /\ ch ) /\ th /\ ta ) /\ et ) -> ch ) $=
    ( w3a simpl3 3ad2antl1 ) ABCGDFCEABCFHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 24-Jun-2022.) $)
  simpl21 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ et ) -> ph ) $=
    ( w3a simpl1 3ad2antl2 ) ABCGDFAEABCFHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 24-Jun-2022.) $)
  simpl22 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ et ) -> ps ) $=
    ( w3a simpl2 3ad2antl2 ) ABCGDFBEABCFHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 24-Jun-2022.) $)
  simpl23 $p |- ( ( ( th /\ ( ph /\ ps /\ ch ) /\ ta ) /\ et ) -> ch ) $=
    ( w3a simpl3 3ad2antl2 ) ABCGDFCEABCFHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 24-Jun-2022.) $)
  simpl31 $p |- ( ( ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ et ) -> ph ) $=
    ( w3a simpl1 3ad2antl3 ) ABCGDFAEABCFHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 24-Jun-2022.) $)
  simpl32 $p |- ( ( ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ et ) -> ps ) $=
    ( w3a simpl2 3ad2antl3 ) ABCGDFBEABCFHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 24-Jun-2022.) $)
  simpl33 $p |- ( ( ( th /\ ta /\ ( ph /\ ps /\ ch ) ) /\ et ) -> ch ) $=
    ( w3a simpl3 3ad2antl3 ) ABCGDFCEABCFHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 24-Jun-2022.) $)
  simpr11 $p |- ( ( et /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) ) -> ph ) $=
    ( w3a simpr1 3ad2antr1 ) FDABCGAEFABCHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 24-Jun-2022.) $)
  simpr12 $p |- ( ( et /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) ) -> ps ) $=
    ( w3a simpr2 3ad2antr1 ) FDABCGBEFABCHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 24-Jun-2022.) $)
  simpr13 $p |- ( ( et /\ ( ( ph /\ ps /\ ch ) /\ th /\ ta ) ) -> ch ) $=
    ( w3a simpr3 3ad2antr1 ) FDABCGCEFABCHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 24-Jun-2022.) $)
  simpr21 $p |- ( ( et /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) ) -> ph ) $=
    ( w3a simpr1 3ad2antr2 ) FDABCGAEFABCHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 24-Jun-2022.) $)
  simpr22 $p |- ( ( et /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) ) -> ps ) $=
    ( w3a simpr2 3ad2antr2 ) FDABCGBEFABCHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 24-Jun-2022.) $)
  simpr23 $p |- ( ( et /\ ( th /\ ( ph /\ ps /\ ch ) /\ ta ) ) -> ch ) $=
    ( w3a simpr3 3ad2antr2 ) FDABCGCEFABCHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 24-Jun-2022.) $)
  simpr31 $p |- ( ( et /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) ) -> ph ) $=
    ( w3a simpr1 3ad2antr3 ) FDABCGAEFABCHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 24-Jun-2022.) $)
  simpr32 $p |- ( ( et /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) ) -> ps ) $=
    ( w3a simpr2 3ad2antr3 ) FDABCGBEFABCHI $.

  $( Simplification of conjunction.  (Contributed by NM, 9-Mar-2012.)  (Proof
     shortened by Wolf Lammen, 24-Jun-2022.) $)
  simpr33 $p |- ( ( et /\ ( th /\ ta /\ ( ph /\ ps /\ ch ) ) ) -> ch ) $=
    ( w3a simpr3 3ad2antr3 ) FDABCGCEFABCHI $.

  $( 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 $.

  ${
    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 simp3 mpbirand ) ABCGDCEABCHFI $.
  $}

  $( 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 $.
  $}

  $( Version of ~ pm3.2 for a triple conjunction.  (Contributed by Alan Sare,
     24-Oct-2011.)  (Proof shortened by Kyle Wyonch, 24-Apr-2021.)  (Proof
     shortened by Wolf Lammen, 21-Jun-2022.) $)
  pm3.2an3 $p |- ( ph -> ( ps -> ( ch -> ( ph /\ ps /\ ch ) ) ) ) $=
    ( w3a id 3exp ) ABCABCDZGEF $.

  ${
    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.) $)
    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.)  (Revised by Mario Carneiro,
       9-Jan-2015.) $)
    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 $.
  $}

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  $( Version of ~ impexp for a triple conjunction.  (Contributed by Alan Sare,
     31-Dec-2011.) $)
  3impexp $p |- ( ( ( ph /\ ps /\ ch ) -> th ) <->
                ( ph -> ( ps -> ( ch -> th ) ) ) ) $=
    ( w3a wi id 3expd 3impd impbii ) ABCEDFZABCDFFFZKABCDKGHLABCDLGIJ $.

  ${
    3an1rs.1 $e |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta ) $.
    $( Swap conjuncts.  (Contributed by NM, 16-Dec-2007.)  (Proof shortened by
       Wolf Lammen, 14-Apr-2022.) $)
    3an1rs $p |- ( ( ( ph /\ ps /\ th ) /\ ch ) -> ta ) $=
      ( 3exp1 com34 3imp1 ) ABDCEABCDEABCDEFGHI $.
  $}

  ${
    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 $.
  $}

  $( An equivalence of two four-terms conjunctions with the terms regrouped
     (here, the second sub-conjunct of the first term is pulled separately).
     (Contributed by Zhi Wang, 4-Sep-2024.) $)
  4anpull2 $p |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) )
          <-> ( ( ph /\ ch /\ th ) /\ ps ) ) $=
    ( wa w3a anass 3anass anbi1i ancom anbi2i 3bitr4ri bitri ) ABECDEZEABNEZEZA
    CDFZBEZABNGANEZBEANBEZERPANBGQSBACDHIOTABNJKLM $.

  ${
    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 $.
  $}

  ${
    ad5ant2345.1 $e |- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ta ) $.
    $( Deduction adding conjuncts to antecedent.  (Contributed by Alan Sare,
       17-Oct-2017.) $)
    ad5ant2345 $p |- ( ( ( ( ( et /\ ph ) /\ ps ) /\ ch ) /\ th ) -> ta ) $=
      ( wa wi exp41 adantl imp41 ) FAHBCDEABCDEIIIFABCDEGJKL $.
  $}

  ${
    syl3anc.1 $e |- ( ph -> ps ) $.
    syl3anc.2 $e |- ( ph -> ch ) $.
    syl3anc.3 $e |- ( ph -> th ) $.
    ${
      syl3anc.4 $e |- ( ( ps /\ ch /\ th ) -> ta ) $.
      $( Syllogism combined with contraction.  (Contributed by NM,
         11-Mar-2012.) $)
      syl3anc $p |- ( ph -> ta ) $=
        ( w3a 3jca syl ) ABCDJEABCDFGHKIL $.
    $}

    syl3Xanc.4 $e |- ( ph -> ta ) $.
    ${
      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 $.
    $}

    syl23anc.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 $.
    $}

    syl33anc.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 $.
    $}

    syl133anc.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 $.
    $}

    syl233anc.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 $.
    $}

    syl333anc.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 $.
    $}
  $}

  ${
    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 $.
  $}

  ${
    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.)  (Proof
       shortened by Wolf Lammen, 26-Jun-2022.) $)
    syld3an1 $p |- ( ( ch /\ ps /\ th ) -> ta ) $=
      ( w3a simp2 simp3 syl3anc ) CBDHABDEFCBDICBDJGK $.
  $}

  ${
    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 ) $=
      ( w3a simp1 simp3 syl3anc ) ACDHABDEACDIFACDJGK $.
  $}

  ${
    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.)  (Proof
       shortened by Wolf Lammen, 27-Jun-2022.) $)
    syl3anl2 $p |- ( ( ( ps /\ ph /\ th ) /\ ta ) -> et ) $=
      ( w3a 3anim2i sylan ) BADIBCDIEFACBDGJHK $.
  $}

  ${
    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.)  (Proof
       shortened by Wolf Lammen, 27-Jun-2022.) $)
    syl3anr2 $p |- ( ( ch /\ ( ps /\ ph /\ ta ) ) -> et ) $=
      ( w3a 3anim2i sylan2 ) BAEICBDEIFADBEGJHK $.
  $}

  ${
    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 $.
  $}

  ${
    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.)  (Proof shortened by
       Wolf Lammen, 28-Jun-2022.) $)
    syl2an23an $p |- ( ( th /\ ph ) -> et ) $=
      ( wa syl2an3an anabss7 ) DAFABCDAKEFGHIJLM $.
  $}

  ${
    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 three antecedents.  (Contributed by NM, 8-Apr-1994.) $)
  3jao $p |- ( ( ( ph -> ps ) /\ ( ch -> ps ) /\ ( th -> ps ) ) ->
              ( ( ph \/ ch \/ th ) -> ps ) ) $=
    ( wi w3o wo jao df-3or syl7bi syl6 3imp ) ABEZCBEZDBEZACDFZBEZMNACGZBEZOQEA
    BCHPRDGSOBACDIRBDHJKL $.

  $( Disjunction of three antecedents.  (Contributed by NM, 13-Sep-2011.)
     (Proof shortened by Hongxiu Chen, 29-Jun-2025.) $)
  3jaob $p |- ( ( ( ph \/ ch \/ th ) -> ps ) <->
              ( ( ph -> ps ) /\ ( ch -> ps ) /\ ( th -> ps ) ) ) $=
    ( wo wi wa w3o w3a pm5.53 df-3or imbi1i df-3an 3bitr4i ) ACEDEZBFABFZCBFZGD
    BFZGACDHZBFPQRIACDBJSOBACDKLPQRMN $.

  $( Obsolete version of ~ 3jaob as of 29-Jun-2025.  (Contributed by NM,
     13-Sep-2011.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  3jaobOLD $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 three 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 three 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 three 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 three 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 three-way disjunction in a deduction.  (Contributed by
       Thierry Arnoux, 13-Apr-2018.)  (Proof shortened by Wolf Lammen,
       20-Apr-2024.) $)
    mpjao3dan $p |- ( ph -> ch ) $=
      ( w3o 3jaodan mpdan ) ABDEJCIABCDEFGHKL $.
  $}

  ${
    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 $.
  $}

  ${
    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 ) ) ) $=
    ( wa w3a an4 bianbi df-3an anbi12i 3bitr4i ) ABGZCGZDEGZFGZGZADGZBEGZGZCFGZ
    GABCHZDEFHZGSTUBHRNPGUBUANCPFIABDEIJUCOUDQABCKDEFKLSTUBKM $.

  $( Analogue 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 $.

  $( Analogue 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.) $)
    3anandirs $p |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> ta ) $=
      ( w3a wa simpl1 simpr simpl2 simpl3 syl222anc ) ABCGZDHADBDCDEABCDINDJZAB
      CDKOABCDLOFM $.
  $}

  ${
    ecase23d.1 $e |- ( ph -> -. ch ) $.
    ecase23d.2 $e |- ( ph -> -. th ) $.
    ecase23d.3 $e |- ( ph -> ( ps \/ ch \/ th ) ) $.
    $( Deduction for elimination by cases.  (Contributed by NM,
       22-Apr-1994.) $)
    ecase23d $p |- ( ph -> ps ) $=
      ( wo w3o 3orass sylib wn ioran sylanbrc olcnd ) ABCDHZABCDIBPHGBCDJKACLDL
      PLEFCDMNO $.
  $}

  ${
    3ecase.1 $e |- ( -. ph -> th ) $.
    3ecase.2 $e |- ( -. ps -> th ) $.
    3ecase.3 $e |- ( -. ch -> th ) $.
    3ecase.4 $e |- ( ( ph /\ ps /\ ch ) -> th ) $.
    $( Inference for elimination by cases.  (Contributed by NM,
       13-Jul-2005.) $)
    3ecase $p |- th $=
      ( wi 3exp wn 2a1d pm2.61i pm2.61nii ) BCDABCDIIABCDHJAKDBCELMFGN $.
  $}

  ${
    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 $.
  $}

  $( Distribution of conjunction over threefold conjunction.  (Contributed by
     Thierry Arnoux, 8-Apr-2019.) $)
  an3andi $p |- ( ( ph /\ ( ps /\ ch /\ th ) ) <->
               ( ( ph /\ ps ) /\ ( ph /\ ch ) /\ ( ph /\ th ) ) ) $=
    ( wa w3a anandi bianbi df-3an anbi2i 3bitr4i ) ABCEZDEZEZABEZACEZEZADEZEABC
    DFZEOPRFNALERQALDGABCGHSMABCDIJOPRIK $.

  $( Rearrange a 9-fold conjunction.  (Contributed by Thierry Arnoux,
     14-Apr-2019.)  (Proof shortened by Wolf Lammen, 21-Apr-2024.) $)
  an33rean $p |- (
        ( ( ph /\ ps /\ ch ) /\ ( th /\ ta /\ et ) /\ ( ze /\ si /\ rh ) ) <->
    ( ( ph /\ ta /\ rh ) /\ ( ( ps /\ th ) /\ ( et /\ si ) /\ ( ch /\ ze ) ) )
    ) $=
    ( w3a wa 3anass 3anan12 3anrev bitri 3anbi123i anass anbi2i 3bitr4i df-3an
    3an6 an42 anbi1i 3bitr3i 3bitri ) ABCJZDEFJZGHIJZJABCKZKZEDFKZKZIHGKZKZJAEI
    JZUIUKUMJZKUOBDKZFHKZCGKZJZKUFUJUGULUHUNABCLDEFMUHIHGJUNGHINIHGLOPAUIEUKIUM
    UAUPUTUOUIUKKZUMKZUQURKZUSKZUPUTVAHKZGKVCCKZGKVBVDVEVFGUIUKHKZKZUQURCKKZVEV
    FVHUIDURKZKVIVGVJUIDFHQRBCDURUBOUIUKHQUQURCQSUCVAHGQVCCGQUDUIUKUMTUQURUSTSR
    UE $.

  $( Partial elimination of a triple disjunction by denial of a disjunct.
     (Contributed by Scott Fenton, 26-Mar-2011.)  (Proof shortened by Andrew
     Salmon, 25-May-2011.)  (Proof shortened by Eric Schmidt, 8-Oct-2025.) $)
  3orel2 $p |- ( -. ps -> ( ( ph \/ ps \/ ch ) -> ( ph \/ ch ) ) ) $=
    ( w3o wn wo 3orcoma 3orel1 biimtrid ) ABCDBACDBEACFABCGBACHI $.

  $( Obsolete version of ~ 3orel2 as of 8-Oct-2025.  (Contributed by Scott
     Fenton, 26-Mar-2011.)  (Proof shortened by Andrew Salmon, 25-May-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  3orel2OLD $p |- ( -. ps -> ( ( ph \/ ps \/ ch ) -> ( ph \/ ch ) ) ) $=
    ( w3o wn wo 3orrot 3orel1 orcom imbitrdi biimtrid ) ABCDBCADZBEZACFZABCGMLC
    AFNBCAHCAIJK $.

  $( Partial elimination of a triple disjunction by denial of a disjunct.
     (Contributed by Scott Fenton, 26-Mar-2011.) $)
  3orel3 $p |- ( -. ch -> ( ( ph \/ ps \/ ch ) -> ( ph \/ ps ) ) ) $=
    ( w3o wo wn df-3or orel2 biimtrid ) ABCDABEZCECFJABCGCJHI $.

  $( Elimination of two disjuncts in a triple disjunction.  (Contributed by
     Scott Fenton, 9-Jun-2011.) $)
  3orel13 $p |- ( ( -. ph /\ -. ch ) -> ( ( ph \/ ps \/ ch ) -> ps ) ) $=
    ( wn w3o wo 3orel3 orel1 sylan9r ) CDABCEABFADBABCGABHI $.

  ${
    3pm3.2ni.1 $e |- -. ph $.
    3pm3.2ni.2 $e |- -. ps $.
    3pm3.2ni.3 $e |- -. ch $.
    $( Triple negated disjunction introduction.  (Contributed by Scott Fenton,
       20-Apr-2011.) $)
    3pm3.2ni $p |- -. ( ph \/ ps \/ ch ) $=
      ( w3o wo pm3.2ni df-3or mtbir ) ABCGABHZCHLCABDEIFIABCJK $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Logical "nand" (Sheffer stroke)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare connective for alternative denial ("nand"). $)
  $c -/\ $.  $( Overlined "wedge" (read:  "nand") $)

  $( Extend wff definition to include alternative denial ("nand"). $)
  wnan $a wff ( ph -/\ ps ) $.

  $( Define incompatibility, or alternative denial ("not-and" or "nand").  See
     ~ dfnan2 for an alternative.  This is also called the Sheffer stroke,
     represented by a vertical bar, but we use a different symbol to avoid
     ambiguity with other uses of the vertical bar.  In the second edition of
     Principia Mathematica (1927), Russell and Whitehead used the Sheffer
     stroke and suggested it as a replacement for the "or" and "not" operations
     of the first edition.  However, in practice, "or" and "not" are more
     widely used.  After we define the constant true ` T. ` ( ~ df-tru ) and
     the constant false ` F. ` ( ~ df-fal ), we will be able to prove these
     truth table values: ` ( ( T. -/\ T. ) <-> F. ) ` ( ~ trunantru ),
     ` ( ( T. -/\ F. ) <-> T. ) ` ( ~ trunanfal ), ` ( ( F. -/\ T. ) <-> T. ) `
     ( ~ falnantru ), and ` ( ( F. -/\ F. ) <-> T. ) ` ( ~ falnanfal ).
     Contrast with ` /\ ` ( ~ df-an ), ` \/ ` ( ~ df-or ), ` -> ` ( ~ wi ), and
     ` \/_ ` ( ~ df-xor ).  (Contributed by Jeff Hoffman, 19-Nov-2007.) $)
  df-nan $a |- ( ( ph -/\ ps ) <-> -. ( ph /\ ps ) ) $.

  $( Conjunction in terms of alternative denial.  (Contributed by Mario
     Carneiro, 9-May-2015.) $)
  nanan $p |- ( ( ph /\ ps ) <-> -. ( ph -/\ ps ) ) $=
    ( wnan wa df-nan con2bii ) ABCABDABEF $.

  $( Alternative denial in terms of our primitive connectives (implication and
     negation).  (Contributed by WL, 26-Jun-2020.) $)
  dfnan2 $p |- ( ( ph -/\ ps ) <-> ( ph -> -. ps ) ) $=
    ( wnan wa wn wi df-nan imnan bitr4i ) ABCABDEABEFABGABHI $.

  $( Alternative denial in terms of disjunction and negation.  This explains
     the name "alternative denial".  (Contributed by BJ, 19-Oct-2022.) $)
  nanor $p |- ( ( ph -/\ ps ) <-> ( -. ph \/ -. ps ) ) $=
    ( wnan wa wn wo df-nan ianor bitri ) ABCABDEAEBEFABGABHI $.

  $( Alternative denial is commutative.  Remark: alternative denial is not
     associative, see ~ nanass .  (Contributed by Mario Carneiro, 9-May-2015.)
     (Proof shortened by Wolf Lammen, 26-Jun-2020.) $)
  nancom $p |- ( ( ph -/\ ps ) <-> ( ps -/\ ph ) ) $=
    ( wn wi wnan con2b dfnan2 3bitr4i ) ABCDBACDABEBAEABFABGBAGH $.

  $( Nested alternative denials.  (Contributed by Jeff Hoffman, 19-Nov-2007.)
     (Proof shortened by Wolf Lammen, 26-Jun-2020.) $)
  nannan $p |- ( ( ph -/\ ( ps -/\ ch ) ) <-> ( ph -> ( ps /\ ch ) ) ) $=
    ( wnan wn wi wa dfnan2 nanan imbi2i bitr4i ) ABCDZDALEZFABCGZFALHNMABCIJK
    $.

  $( Implication in terms of alternative denial.  (Contributed by Jeff Hoffman,
     19-Nov-2007.) $)
  nanim $p |- ( ( ph -> ps ) <-> ( ph -/\ ( ps -/\ ps ) ) ) $=
    ( wnan wa wi nannan anidmdbi bitr2i ) ABBCCABBDEABEABBFABGH $.

  $( Negation in terms of alternative denial.  (Contributed by Jeff Hoffman,
     19-Nov-2007.)  Use ~ dfnan2 .  (Revised by Wolf Lammen, 26-Jun-2020.) $)
  nannot $p |- ( -. ph <-> ( ph -/\ ph ) ) $=
    ( wnan wn wi dfnan2 pm4.8 bitr2i ) AABAACZDHAAEAFG $.

  $( Biconditional in terms of alternative denial.  (Contributed by Jeff
     Hoffman, 19-Nov-2007.)  (Proof shortened by Wolf Lammen, 27-Jun-2020.) $)
  nanbi $p |- ( ( ph <-> ps ) <->
          ( ( ph -/\ ps ) -/\ ( ( ph -/\ ph ) -/\ ( ps -/\ ps ) ) ) ) $=
    ( wb wnan wa wi wn dfbi3 df-nan bicomi nannot anbi12i imbi12i 3bitri nannan
    wo df-or bitr4i ) ABCZABDZAADZBBDZEZFZTUAUBDDSABEZAGZBGZEZPUEGZUHFUDABHUEUH
    QUITUHUCTUIABIJUFUAUGUBAKBKLMNTUAUBOR $.

  $( Introduce a right anti-conjunct to both sides of a logical equivalence.
     (Contributed by Anthony Hart, 1-Sep-2011.)  (Proof shortened by Wolf
     Lammen, 27-Jun-2020.) $)
  nanbi1 $p |- ( ( ph <-> ps ) -> ( ( ph -/\ ch ) <-> ( ps -/\ ch ) ) ) $=
    ( wb wn wi wnan imbi1 dfnan2 3bitr4g ) ABDACEZFBKFACGBCGABKHACIBCIJ $.

  $( Introduce a left anti-conjunct to both sides of a logical equivalence.
     (Contributed by Anthony Hart, 1-Sep-2011.)  (Proof shortened by SF,
     2-Jan-2018.) $)
  nanbi2 $p |- ( ( ph <-> ps ) -> ( ( ch -/\ ph ) <-> ( ch -/\ ps ) ) ) $=
    ( wb wnan nanbi1 nancom 3bitr4g ) ABDACEBCECAECBEABCFCAGCBGH $.

  $( Join two logical equivalences with anti-conjunction.  (Contributed by SF,
     2-Jan-2018.) $)
  nanbi12 $p |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) ->
                                       ( ( ph -/\ ch ) <-> ( ps -/\ th ) ) ) $=
    ( wb wnan nanbi1 nanbi2 sylan9bb ) ABEACFBCFCDEBDFABCGCDBHI $.

  ${
    nanbii.1 $e |- ( ph <-> ps ) $.
    $( Introduce a right anti-conjunct to both sides of a logical equivalence.
       (Contributed by SF, 2-Jan-2018.) $)
    nanbi1i $p |- ( ( ph -/\ ch ) <-> ( ps -/\ ch ) ) $=
      ( wb wnan nanbi1 ax-mp ) ABEACFBCFEDABCGH $.

    $( Introduce a left anti-conjunct to both sides of a logical equivalence.
       (Contributed by SF, 2-Jan-2018.) $)
    nanbi2i $p |- ( ( ch -/\ ph ) <-> ( ch -/\ ps ) ) $=
      ( wb wnan nanbi2 ax-mp ) ABECAFCBFEDABCGH $.

    nanbi12i.2 $e |- ( ch <-> th ) $.
    $( Join two logical equivalences with anti-conjunction.  (Contributed by
       SF, 2-Jan-2018.) $)
    nanbi12i $p |- ( ( ph -/\ ch ) <-> ( ps -/\ th ) ) $=
      ( wb wnan nanbi12 mp2an ) ABGCDGACHBDHGEFABCDIJ $.
  $}

  ${
    nanbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Introduce a right anti-conjunct to both sides of a logical equivalence.
       (Contributed by SF, 2-Jan-2018.) $)
    nanbi1d $p |- ( ph -> ( ( ps -/\ th ) <-> ( ch -/\ th ) ) ) $=
      ( wb wnan nanbi1 syl ) ABCFBDGCDGFEBCDHI $.

    $( Introduce a left anti-conjunct to both sides of a logical equivalence.
       (Contributed by SF, 2-Jan-2018.) $)
    nanbi2d $p |- ( ph -> ( ( th -/\ ps ) <-> ( th -/\ ch ) ) ) $=
      ( wb wnan nanbi2 syl ) ABCFDBGDCGFEBCDHI $.

    nanbi12d.2 $e |- ( ph -> ( th <-> ta ) ) $.
    $( Join two logical equivalences with anti-conjunction.  (Contributed by
       Scott Fenton, 2-Jan-2018.) $)
    nanbi12d $p |- ( ph -> ( ( ps -/\ th ) <-> ( ch -/\ ta ) ) ) $=
      ( wb wnan nanbi12 syl2anc ) ABCHDEHBDICEIHFGBCDEJK $.
  $}

  $( A characterization of when an expression involving alternative denials
     associates.  Remark: alternative denial is commutative, see ~ nancom .
     (Contributed by Richard Penner, 29-Feb-2020.)  (Proof shortened by Wolf
     Lammen, 23-Oct-2022.) $)
  nanass $p |- ( ( ph <-> ch ) <->
                 ( ( ( ph -/\ ps ) -/\ ch ) <-> ( ph -/\ ( ps -/\ ch ) ) ) ) $=
    ( wb bicom1 nanbi2 nanbi12d wa wi nannan simpr imim2i sylbi wn nanan sylbir
    wnan simpl nancom bitri impbid21d pm5.1im syl2imc impbii nanbi2i bibi1i
    bija ) ACDZCBAQZQZABCQZQZDZABQZCQZULDUHUMUHCAUIUKACEACBFGUJULUHUJULACULABCH
    ZIACIABCJUPCABCKLMUJCBAHZICAICBAJUQACBAKLMUAULNZAUJNZCUHURAUKHAAUKOAUKRPUSC
    UIHCCUIOCUIRPACUBUCUGUDUJUOULUJCUNQUOUIUNCBASUECUNSTUFT $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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.  After we define the constant true
     ` T. ` ( ~ df-tru ) and the constant false ` F. ` ( ~ df-fal ), we will be
     able to prove these truth table values: ` ( ( T. \/_ T. ) <-> F. ) `
     ( ~ truxortru ), ` ( ( T. \/_ F. ) <-> T. ) ` ( ~ truxorfal ),
     ` ( ( F. \/_ T. ) <-> T. ) ` ( ~ falxortru ), and
     ` ( ( F. \/_ F. ) <-> F. ) ` ( ~ falxorfal ).  Contrast with ` /\ `
     ( ~ df-an ), ` \/ ` ( ~ df-or ), ` -> ` ( ~ wi ), and ` -/\ `
     ( ~ df-nan ).  (Contributed by FL, 22-Nov-2010.) $)
  df-xor $a |- ( ( ph \/_ ps ) <-> -. ( ph <-> ps ) ) $.

  $( Two ways to write XNOR (exclusive not-or).  (Contributed by Mario
     Carneiro, 4-Sep-2016.) $)
  xnor $p |- ( ( ph <-> ps ) <-> -. ( ph \/_ ps ) ) $=
    ( wxo wb df-xor con2bii ) ABCABDABEF $.

  $( The connector ` \/_ ` is commutative.  (Contributed by Mario Carneiro,
     4-Sep-2016.)  (Proof shortened by Wolf Lammen, 21-Apr-2024.) $)
  xorcom $p |- ( ( ph \/_ ps ) <-> ( ps \/_ ph ) ) $=
    ( wxo wb wn df-xor bicom xchbinx bitr4i ) ABCZBADZEBACJABDKABFABGHBAFI $.

  $( The connector ` \/_ ` is associative.  (Contributed by FL, 22-Nov-2010.)
     (Proof shortened by Andrew Salmon, 8-Jun-2011.)  (Proof shortened by Wolf
     Lammen, 20-Jun-2020.) $)
  xorass $p |- ( ( ( ph \/_ ps ) \/_ ch ) <-> ( ph \/_ ( ps \/_ ch ) ) ) $=
    ( wxo wb xor3 biass xnor bibi1i bibi2i 3bitr3i nbbn 3bitr2ri df-xor 3bitr4i
    wn ) ABDZCEPZABCDZEPZQCDASDTASPZEZQPZCEZRASFABEZCEABCEZEUDUBABCGUEUCCABHIUF
    UAABCHJKQCLMQCNASNO $.

  $( This tautology shows that xor is really exclusive.  (Contributed by FL,
     22-Nov-2010.) $)
  excxor $p |- ( ( ph \/_ ps ) <-> ( ( ph /\ -. ps ) \/ ( -. ph /\ ps ) ) ) $=
    ( wxo wb wn wa wo df-xor xor ancom orbi2i 3bitri ) ABCABDEABEFZBAEZFZGMNBFZ
    GABHABIOPMBNJKL $.

  $( Two ways to express "exclusive or".  (Contributed by Mario Carneiro,
     4-Sep-2016.) $)
  xor2 $p |- ( ( ph \/_ ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) ) $=
    ( wxo wb wn wo wa df-xor nbi2 bitri ) ABCABDEABFABGEGABHABIJ $.

  $( Exclusive disjunction implies disjunction ("XOR implies OR").
     (Contributed by BJ, 19-Apr-2019.) $)
  xoror $p |- ( ( ph \/_ ps ) -> ( ph \/ ps ) ) $=
    ( wxo wo wa wn xor2 simplbi ) ABCABDABEFABGH $.

  $( Exclusive disjunction implies alternative denial ("XOR implies NAND").
     (Contributed by BJ, 19-Apr-2019.) $)
  xornan $p |- ( ( ph \/_ ps ) -> -. ( ph /\ ps ) ) $=
    ( wxo wo wa wn xor2 simprbi ) ABCABDABEFABGH $.

  $( XOR implies NAND (written with the ` -/\ ` connector).  (Contributed by
     BJ, 19-Apr-2019.) $)
  xornan2 $p |- ( ( ph \/_ ps ) -> ( ph -/\ ps ) ) $=
    ( wxo wa wn wnan xornan df-nan sylibr ) ABCABDEABFABGABHI $.

  $( The connector ` \/_ ` is negated under negation of one argument.
     (Contributed by Mario Carneiro, 4-Sep-2016.)  (Proof shortened by Wolf
     Lammen, 27-Jun-2020.) $)
  xorneg2 $p |- ( ( ph \/_ -. ps ) <-> -. ( ph \/_ ps ) ) $=
    ( wn wxo wb df-xor pm5.18 xnor 3bitr2i ) ABCZDAJECABEABDCAJFABGABHI $.

  $( The connector ` \/_ ` is negated under negation of one argument.
     (Contributed by Mario Carneiro, 4-Sep-2016.)  (Proof shortened by Wolf
     Lammen, 27-Jun-2020.) $)
  xorneg1 $p |- ( ( -. ph \/_ ps ) <-> -. ( ph \/_ ps ) ) $=
    ( wn wxo xorcom xorneg2 xchbinx bitri ) ACZBDBIDZABDZCIBEJBADKBAFBAEGH $.

  $( The connector ` \/_ ` is unchanged under negation of both arguments.
     (Contributed by Mario Carneiro, 4-Sep-2016.) $)
  xorneg $p |- ( ( -. ph \/_ -. ps ) <-> ( ph \/_ ps ) ) $=
    ( wn wxo xorneg1 xorneg2 con2bii bitr4i ) ACBCZDAIDZCABDZAIEJKABFGH $.

  ${
    xorbi12.1 $e |- ( ph <-> ps ) $.
    xorbi12.2 $e |- ( ch <-> th ) $.
    $( Equality property for exclusive disjunction.  (Contributed by Mario
       Carneiro, 4-Sep-2016.)  (Proof shortened by Wolf Lammen,
       21-Apr-2024.) $)
    xorbi12i $p |- ( ( ph \/_ ch ) <-> ( ps \/_ th ) ) $=
      ( wxo wb wn df-xor bibi12i xchbinx bitr4i ) ACGZBDHZIBDGNACHOACJABCDEFKLB
      DJM $.
  $}

  ${
    xor12d.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    xor12d.2 $e |- ( ph -> ( th <-> ta ) ) $.
    $( Equality property for exclusive disjunction.  (Contributed by Mario
       Carneiro, 4-Sep-2016.) $)
    xorbi12d $p |- ( ph -> ( ( ps \/_ th ) <-> ( ch \/_ ta ) ) ) $=
      ( wb wn wxo bibi12d notbid df-xor 3bitr4g ) ABDHZICEHZIBDJCEJAOPABCDEFGKL
      BDMCEMN $.
  $}

  $( Conjunction distributes over exclusive-or.  In intuitionistic logic this
     assertion is also true, even though ~ xordi does not necessarily hold, in
     part because the usual definition of xor is subtly different in
     intuitionistic logic.  (Contributed by David A. Wheeler, 7-Oct-2018.) $)
  anxordi $p |- ( ( ph /\ ( ps \/_ ch ) ) <->
      ( ( ph /\ ps ) \/_ ( ph /\ ch ) ) ) $=
    ( wb wn wa wxo xordi df-xor anbi2i 3bitr4i ) ABCDEZFABFZACFZDEABCGZFMNGABCH
    OLABCIJMNIK $.

  $( Exclusive-or variant of the law of the excluded middle ( ~ exmid ).  This
     statement is ancient, going back to at least Stoic logic.  This statement
     does not necessarily hold in intuitionistic logic.  (Contributed by David
     A. Wheeler, 23-Feb-2019.) $)
  xorexmid $p |- ( ph \/_ -. ph ) $=
    ( wn wxo wb pm5.19 df-xor mpbir ) AABZCAHDBAEAHFG $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Logical "nor"
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare connective for joint denial ("nor"). $)
  $c -\/ $.  $( Overlined "vee" (read:  "nor") $)

  $( Extend wff definition to include joint denial ("nor"). $)
  wnor $a wff ( ph -\/ ps ) $.

  $( Define joint denial ("not-or" or "nor").  After we define the constant
     true ` T. ` ( ~ df-tru ) and the constant false ` F. ` ( ~ df-fal ), we
     will be able to prove these truth table values:
     ` ( ( T. -\/ T. ) <-> F. ) ` ( ~ trunortru ), ` ( ( T. -\/ F. ) <-> F. ) `
     ( ~ trunorfal ), ` ( ( F. -\/ T. ) <-> F. ) ` ( ~ falnortru ), and
     ` ( ( F. -\/ F. ) <-> T. ) ` ( ~ falnorfal ).  Contrast with ` /\ `
     ( ~ df-an ), ` \/ ` ( ~ df-or ), ` -> ` ( ~ wi ), ` -/\ ` ( ~ df-nan ),
     and ` \/_ ` ( ~ df-xor ).  (Contributed by Remi, 25-Oct-2023.) $)
  df-nor $a |- ( ( ph -\/ ps ) <-> -. ( ph \/ ps ) ) $.

  $( The connector ` -\/ ` is commutative.  (Contributed by Remi, 25-Oct-2023.)
     (Proof shortened by Wolf Lammen, 23-Apr-2024.) $)
  norcom $p |- ( ( ph -\/ ps ) <-> ( ps -\/ ph ) ) $=
    ( wnor wo wn df-nor orcom xchbinx bitr4i ) ABCZBADZEBACJABDKABFABGHBAFI $.

  $( ` -. ` is expressible via ` -\/ ` .  (Contributed by Remi, 25-Oct-2023.)
     (Proof shortened by Wolf Lammen, 8-Dec-2023.) $)
  nornot $p |- ( -. ph <-> ( ph -\/ ph ) ) $=
    ( wnor wn wo df-nor oridm xchbinx bicomi ) AABZACIAADAAAEAFGH $.

  $( ` /\ ` is expressible via ` -\/ ` .  (Contributed by Remi, 26-Oct-2023.)
     (Proof shortened by Wolf Lammen, 8-Dec-2023.) $)
  noran $p |- ( ( ph /\ ps ) <-> ( ( ph -\/ ph ) -\/ ( ps -\/ ps ) ) ) $=
    ( wa wnor wo wn anor nornot orbi12i xchbinx df-nor bitr4i ) ABCZAADZBBDZEZF
    NODMAFZBFZEPABGQNROAHBHIJNOKL $.

  $( ` \/ ` is expressible via ` -\/ ` .  (Contributed by Remi, 26-Oct-2023.)
     (Proof shortened by Wolf Lammen, 8-Dec-2023.) $)
  noror $p |- ( ( ph \/ ps ) <-> ( ( ph -\/ ps ) -\/ ( ph -\/ ps ) ) ) $=
    ( wo wnor wn df-nor con2bii nornot bitri ) ABCZABDZEKKDKJABFGKHI $.

  $( This lemma shows the equivalence of two expressions, used in ~ norass .
     (Contributed by Wolf Lammen, 18-Dec-2023.) $)
  norasslem1 $p |- ( ( ( ph \/ ps ) -> ch ) <-> ( ( ph -\/ ps ) \/ ch ) ) $=
    ( wo wi wn wnor imor df-nor orbi1i bitr4i ) ABDZCELFZCDABGZCDLCHNMCABIJK $.

  $( This lemma specializes ~ biimt suitably for the proof of ~ norass .
     (Contributed by Wolf Lammen, 18-Dec-2023.) $)
  norasslem2 $p |- ( ph -> ( ps <-> ( ( ph \/ ch ) -> ps ) ) ) $=
    ( wo wi wb biimt orcs ) ACBACDZBEFIBGH $.

  $( This lemma specializes ~ biorf suitably for the proof of ~ norass .
     (Contributed by Wolf Lammen, 18-Dec-2023.) $)
  norasslem3 $p |- ( -. ph -> ( ( ps -> ch ) <-> ( ( ph \/ ps ) -> ch ) ) ) $=
    ( wn wo biorf imbi1d ) ADBABECABFG $.

  $( A characterization of when an expression involving joint denials
     associates.  This is identical to the case when alternative denial is
     associative, see ~ nanass .  Remark:  Like alternative denial, joint
     denial is also commutative, see ~ norcom .  (Contributed by RP,
     29-Oct-2023.)  (Proof shortened by Wolf Lammen, 17-Dec-2023.) $)
  norass $p |- ( ( ph <-> ch ) <->
                 ( ( ( ph -\/ ps ) -\/ ch ) <-> ( ph -\/ ( ps -\/ ch ) ) ) ) $=
    ( wnor wo wb wn notbi wi norasslem1 bibi12i bicom norasslem2 bibi12d bitrid
    impimprbi norasslem3 pm2.61i 3bitr4i df-nor norcom orbi1i orcom ) ABDZCEZAB
    CDZEZFZUEGZUGGZFACFZUDCDZAUFDZFUEUGHBAECIZBCEAIZFZBADZCEZUFAEZFUKUHUNURUOUS
    BACJBCAJKBUKUPFUKCAFBUPACLBCUNAUOBCAMBACMNOUKACIZCAIZFBGZUPACPVBUTUNVAUOBAC
    QBCAQNORUEURUGUSUDUQCABUAUBAUFUCKSULUIUMUJUDCTAUFTKS $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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 Definition ~ df-ex 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'; $)

  $( Declare the color of setvar variables. $)
  $( $j
    varcolorcode "setvar" as "FF0000";
    altvarcolorcode "setvar" as "FF0000";
  $)

  ${
    $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 $.

    $( Register the universal quantifier 'A.' as a primitive expression
       (lacking a definition). $)
    $( $j primitive 'wal'; $)
  $}


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  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 Theorem ~ weq 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'; $)

  $( Declare the color of class variables. $)
  $( $j
    varcolorcode "class" as "CC33CC";
    altvarcolorcode "class" as "CC33CC";
  $)

  ${
    $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 ~ 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 to express, i.e., "prove", the ~ weq of
       predicate calculus from the ~ wceq of set theory, so that we do not
       overload the ` = ` connective with two syntax definitions.  This is done
       to prevent ambiguity that would complicate some Metamath parsers.) $)
    cv $a class x $.

    $( Register class-to-set promotion as a primitive expression (lacking a
       definition). $)
    $( $j primitive 'cv'; $)
  $}

  $( 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
       ~ mmset.html#class .

       (The purpose of introducing ` wff A = B ` here, and not in set theory
       where it belongs, is to allow to express, i.e., "prove", the ~ weq of
       predicate calculus in terms of the ~ wceq of set theory, so that we do
       not "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 $.

    $( Register class equality as a primitive expression (lacking a
       definition). $)
    $( $j primitive 'wceq'; $)
  $}


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  The true constant
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  $c T. $.  $( Symbol for the true constant. $)

  $( The constant ` 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 .  Instance of
       ~ monothetic .  (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 monothetic ) ACZGDAEBCZHDBEF $.

    $( Definition of the truth value "true", or "verum", denoted by ` T. ` .
       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 instance of ~ id 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 use ~ tru 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.)  Use ~ tru instead.
       (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 $.
  $}

  $( An alternate definition of "true" (see comment of ~ df-tru ).  The
     associated justification theorem is ~ monothetic .  (Contributed by
     Anthony Hart, 13-Oct-2010.)  (Revised by BJ, 12-Jul-2019.)  Use ~ tru
     instead.  (New usage is discouraged.) $)
  dftru2 $p |- ( T. <-> ( ph -> ph ) ) $=
    ( wtru wi tru id 2th ) BAACDAEF $.

  $( A proposition is equivalent to it being implied by ` T. ` .  Closed form
     of ~ mptru .  Dual of ~ dfnot .  It is to ~ tbtru what ~ a1bi is to
     ~ tbt .  (Contributed by BJ, 26-Oct-2019.) $)
  trut $p |- ( ph <-> ( T. -> ph ) ) $=
    ( wtru tru a1bi ) BACD $.

  ${
    mptru.1 $e |- ( T. -> ph ) $.
    $( Eliminate ` T. ` as an antecedent.  A proposition implied by ` T. ` is
       true.  This is modus ponens ~ ax-mp when the minor hypothesis is ` T. `
       (which holds by ~ tru ).  (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 $.

  ${
    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 $.
  $}

  $( Anything implies ` T. ` .  Dual statement of ~ falim .  Deduction form of
     ~ tru .  Note on naming: in 2022, the theorem now known as ~ mptru was
     renamed from trud so if you are reading documentation written before that
     time, references to trud refer to what is now ~ mptru .  (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 $.


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  The false constant
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  $c F. $.  $( Symbol for the false constant. $)

  $( The constant ` 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 $.

  $( 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 $.

  ${
    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 [[sequitur]] quodlibet" (Latin for "from
     falsehood, anything [[follows]]").  Dual statement of ~ trud .
     (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 $.

  $( Given falsum ` F. ` , we can define the negation of a wff ` ph ` as the
     statement that ` F. ` 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 $.
  $}

  ${
    efald.1 $e |- ( ( ph /\ -. ps ) -> F. ) $.
    $( Deduction based on reductio ad absurdum.  (Contributed by Mario
       Carneiro, 9-Feb-2017.) $)
    efald $p |- ( ph -> ps ) $=
      ( wn inegd notnotrd ) ABABDCEF $.
  $}

  ${
    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 $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Truth tables
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  Some sources define logical connectives by their truth tables.  These are
  tables that give the truth value of the composed expression for all possible
  combinations of the truth values of their arguments.  In this section, we
  show that our definitions and axioms produce equivalent results for all the
  logical connectives we have introduced (either axiomatically or by a
  definition): implication ~ wi , negation ~ wn , biconditional ~ df-bi ,
  conjunction ~ df-an , disjunction ~ df-or , alternative denial ~ df-nan ,
  exclusive disjunction ~ df-xor .

$)


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Implication
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  $( A ` -> ` identity.  (Contributed by Anthony Hart, 22-Oct-2010.)  An
     alternate proof is possible using ~ trud instead of ~ id but the principle
     of identity ~ id is more basic, and the present proof indicates that the
     result still holds in relevance logic.
     (Proof modification is discouraged.) $)
  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 trut bicomi ) ABACADE $.

  $( A ` -> ` identity.  (Contributed by Anthony Hart, 22-Oct-2010.)  An
     alternate proof is possible using ~ falim instead of ~ trud but the
     present proof using ~ trud emphasizes that the result does not require the
     principle of explosion.  (Proof modification is discouraged.) $)
  falimtru $p |- ( ( F. -> T. ) <-> T. ) $=
    ( wfal wtru wi trud bitru ) ABCADE $.

  $( A ` -> ` identity.  (Contributed by Anthony Hart, 22-Oct-2010.)  An
     alternate proof is possible using ~ falim instead of ~ id but the present
     proof using ~ id emphasizes that the result does not require the principle
     of explosion.  (Proof modification is discouraged.) $)
  falimfal $p |- ( ( F. -> F. ) <-> T. ) $=
    ( wfal wi id bitru ) AABACD $.


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Negation
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  $( 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 $.


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Equivalence
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  $( 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 Anthony Hart, 22-Oct-2010.)  (Proof
     shortened by Andrew Salmon, 13-May-2011.)  (Proof shortened by Wolf
     Lammen, 10-Jul-2020.) $)
  falbitru $p |- ( ( F. <-> T. ) <-> F. ) $=
    ( wfal wtru wb tbtru bicomi ) AABCADE $.

  $( A ` <-> ` identity.  (Contributed by Anthony Hart, 22-Oct-2010.)  (Proof
     shortened by Andrew Salmon, 13-May-2011.)  (Proof shortened by Wolf
     Lammen, 10-Jul-2020.) $)
  trubifal $p |- ( ( T. <-> F. ) <-> F. ) $=
    ( wtru wfal wb bicom falbitru bitri ) ABCBACBABDEF $.

  $( 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 $.


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Conjunction
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  $( 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 Anthony Hart, 22-Oct-2010.) $)
  falantru $p |- ( ( F. /\ T. ) <-> F. ) $=
    ( wfal wtru wa fal intnanr bifal ) ABCABDEF $.

  $( A ` /\ ` identity.  (Contributed by Anthony Hart, 22-Oct-2010.) $)
  falanfal $p |- ( ( F. /\ F. ) <-> F. ) $=
    ( wfal anidm ) AB $.


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Disjunction
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  $( 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 $.


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Alternative denial
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  $( A ` -/\ ` identity.  (Contributed by Anthony Hart, 22-Oct-2010.)  (Proof
     shortened by Andrew Salmon, 13-May-2011.) $)
  trunantru $p |- ( ( T. -/\ T. ) <-> F. ) $=
    ( wtru wnan wn wfal nannot nottru bitr3i ) AABACDAEFG $.

  $( A ` -/\ ` identity.  (Contributed by Anthony Hart, 23-Oct-2010.)  (Proof
     shortened by Andrew Salmon, 13-May-2011.)  (Proof shortened by Wolf
     Lammen, 10-Jul-2020.) $)
  trunanfal $p |- ( ( T. -/\ F. ) <-> T. ) $=
    ( wtru wfal wnan wn wa df-nan truanfal xchbinx notfal bitri ) ABCZBDAKABEBA
    BFGHIJ $.

  $( A ` -/\ ` identity.  (Contributed by Anthony Hart, 23-Oct-2010.)  (Proof
     shortened by Andrew Salmon, 13-May-2011.) $)
  falnantru $p |- ( ( F. -/\ T. ) <-> T. ) $=
    ( wfal wtru wnan nancom trunanfal bitri ) ABCBACBABDEF $.

  $( A ` -/\ ` identity.  (Contributed by Anthony Hart, 22-Oct-2010.)  (Proof
     shortened by Andrew Salmon, 13-May-2011.) $)
  falnanfal $p |- ( ( F. -/\ F. ) <-> T. ) $=
    ( wfal wnan wn wtru nannot notfal bitr3i ) AABACDAEFG $.


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Exclusive disjunction
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  $( A ` \/_ ` identity.  (Contributed by David A. Wheeler, 8-May-2015.) $)
  truxortru $p |- ( ( T. \/_ T. ) <-> F. ) $=
    ( wtru wxo wn wfal wb df-xor trubitru xchbinx nottru bitri ) AABZACDKAAEAAA
    FGHIJ $.

  $( A ` \/_ ` identity.  (Contributed by David A. Wheeler, 8-May-2015.) $)
  truxorfal $p |- ( ( T. \/_ F. ) <-> T. ) $=
    ( wtru wfal wxo wn wb df-xor trubifal xchbinx notfal bitri ) ABCZBDAKABEBAB
    FGHIJ $.

  $( A ` \/_ ` identity.  (Contributed by David A. Wheeler, 9-May-2015.)
     (Proof shortened by Wolf Lammen, 10-Jul-2020.) $)
  falxortru $p |- ( ( F. \/_ T. ) <-> T. ) $=
    ( wfal wtru wxo xorcom truxorfal bitri ) ABCBACBABDEF $.

  $( A ` \/_ ` identity.  (Contributed by David A. Wheeler, 9-May-2015.) $)
  falxorfal $p |- ( ( F. \/_ F. ) <-> F. ) $=
    ( wfal wxo wtru wn wb df-xor falbifal xchbinx nottru bitri ) AABZCDAKAAECAA
    FGHIJ $.


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Joint denial
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  $( A ` -\/ ` identity.  (Contributed by Remi, 25-Oct-2023.)  (Proof shortened
     by Wolf Lammen, 7-Dec-2023.) $)
  trunortru $p |- ( ( T. -\/ T. ) <-> F. ) $=
    ( wtru wnor wn wfal wo df-nor truortru xchbinx df-fal bitr4i ) AABZACDKAAEA
    AAFGHIJ $.

  $( A ` -\/ ` identity.  (Contributed by Remi, 25-Oct-2023.)  (Proof shortened
     by Wolf Lammen, 17-Dec-2023.) $)
  trunorfal $p |- ( ( T. -\/ F. ) <-> F. ) $=
    ( wtru wfal wnor wn wo df-nor truorfal xchbinx df-fal bitr4i ) ABCZADBKABEA
    ABFGHIJ $.

  $( A ` -\/ ` identity.  (Contributed by Remi, 25-Oct-2023.) $)
  falnortru $p |- ( ( F. -\/ T. ) <-> F. ) $=
    ( wfal wtru wnor norcom trunorfal bitri ) ABCBACAABDEF $.

  $( A ` -\/ ` identity.  (Contributed by Remi, 25-Oct-2023.)  (Proof shortened
     by Wolf Lammen, 17-Dec-2023.) $)
  falnorfal $p |- ( ( F. -\/ F. ) <-> T. ) $=
    ( wfal wnor wn wtru wo df-nor falorfal xchbinx notfal bitri ) AABZACDKAAEAA
    AFGHIJ $.

  $( Add support for natural deduction, indicating the constants used to
     indicate implication, context conjunction, and empty context. $)
  $( $j natded_init 'wi' 'wa' 'wtru'; $)

  $( These are variations on the assumption rule of natural deduction.
     They have the form ` Gamma -> ph ` where ` Gamma ` is composed entirely
     of (binary) conjunctions and one of the conjuncts is ` ph ` . $)
  $( $j natded_assume 'id' 'simpl' 'simpr'
          'simpll' 'simplr' 'simprl' 'simprr'
          'simplll' 'simpllr' 'simplrl' 'simplrr'
          'simprll' 'simprlr' 'simprrl' 'simprrr'
          'simp-4l' 'simp-4r' 'simp-5l' 'simp-5r'
          'simp-6l' 'simp-6r' 'simp-7l' 'simp-7r'
          'simp-8l' 'simp-8r' 'simp-9l' 'simp-9r'
          'simp-10l' 'simp-10r' 'simp-11l' 'simp-11r'; $)

  $( These are variations on the weakening lemma of natural deduction.
     They have the form ` _D -> ph |- _G -> ph ` where ` _G `
     and ` _D ` are composed entirely of (binary) conjunctions and
     ` _D C_ _G ` . $)
  $( $j natded_weak 'a1i' 'adantl' 'adantr'
          'adantll' 'adantlr' 'adantrl' 'adantrr'
          'adantlll' 'adantllr' 'adantlrl' 'adantlrr'
          'adantrll' 'adantrlr' 'adantrrl' 'adantrrr'
          'ad2antll' 'ad2antlr' 'ad2antrl' 'ad2antrr'
          'ad2ant2l' 'ad2ant2lr' 'ad2ant2rl' 'ad2ant2r'
          'ad3antlr' 'ad3antrrr' 'ad4antlr' 'ad4antr'
          'ad4antlr' 'ad4antr' 'ad5antlr' 'ad5antr'
          'ad6antlr' 'ad6antr' 'ad7antlr' 'ad7antr'
          'ad8antlr' 'ad8antr' 'ad9antlr' 'ad9antr'
          'ad10antlr' 'ad10antr'
          'anidms' 'ancoms' 'anasss' 'anassrs' 'an12s'  'ancom2s' 'an13s'
          'an32s' 'ancom1s' 'an31s' 'anass1rs' 'anabsan'  'anabss1' 'anabss4'
          'anabss5' 'anabss7' 'anabsan2' 'anabss3' 'an4s' 'an42s'
          'anandis' 'anandirs'; $)

  $( These are variations on the cut axiom (modus ponens) of natural deduction.
     All assumptions and the consequent have the form ` ph ` or ` _G -> ph `
     where ` _G ` is composed entirely of (binary) conjunctions, and possibly
     some intermediate statements are cancelled. (This does not include the
     "natded_assume" and "natded_weak" lists, which also count in this
     category.) $)
  $( $j natded_cut 'mp2b' 'mp1i' 'syl' '3syl' '4syl' 'syldan' 'sylan' 'sylan2'
          'syl2an' 'syl2anr' 'sylanl1' 'sylanl2' 'sylanr1' 'sylanr2' 'syl2anc'
          'sylancl' 'sylancr' 'sylancom' 'mpdan' 'mpancom' 'mpan' 'mpan2'
          'mp2an' 'mp4an' 'mpanl1' 'mpanl2' 'mpanl12' 'mpanr1' 'mpanr2'
          'mpanr12' 'mpanlr1' 'syl12anc' 'syl21anc' 'syl22anc'; $)

  $( Natural deduction rules for the logical connectives, defining the
     connective itself and natural deduction rules pertaining to it.
     These don't have a fixed form, except that all the statements should be
     parseable as natural deduction statements, i.e. ` ph ` or ` _G -> ph `
     where _G is a nested conjunction. $)
  $( $j natded_true 'wtru' with 'mptru' 'tru' 'trud';
        natded_imp 'wi' with 'ex' 'imp' 'mpd' 'ax-mp' 'syl' 'mpi';
        natded_and 'wa' with 'jca' 'pm3.2i' 'jca31' 'jca32' 'jctil' 'jctir'
          'jctl' 'jctr' 'ancli' 'ancri';
        natded_or 'wo' with 'olc' 'orc' 'olci' 'orci' 'olcd' 'orcd' 'olcs'
          'orcs' 'jaoi' 'jaod' 'jaodan' 'mpjaodan';
        natded_not 'wn' 'wfal' with 'falim' 'falimd' 'inegd' 'mtand' 'pm2.65da'
          'pm2.01da' 'pm2.18da' 'pm2.21dd' 'pm2.21fal' 'efald' 'pm2.18da'
          'notnotr' 'notnotrd'; $)


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Half adder and full adder in propositional calculus
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  Propositional calculus deals with truth values, which can be interpreted as
  bits.  Using this, we can define the half adder and the full adder in pure
  propositional calculus, and show their basic properties.

  The half adder adds two 1-bit numbers.  Its two outputs are the "sum" S and
  the "carry" C.  The real sum is then given by 2C+S.  The sum and carry
  correspond respectively to the logical exclusive disjunction ( ~ df-xor ) and
  the logical conjunction ( ~ df-an ).

  The full adder takes into account an "input carry", so it has three inputs
  and again two outputs, corresponding to the "sum" ( ~ df-had ) and "updated
  carry" ( ~ df-cad ).

  Here is a short description.  We code the bit 0 by ` F. ` and 1 by ` T. ` .
  Even though ` hadd ` and ` cadd ` are invariant under permutation of their
  arguments, assume for the sake of concreteness that ` ph ` (resp. ` ps ` ) is
  the i^th bit of the first (resp. second) number to add (with the convention
  that the i^th bit is the multiple of 2^i in the base-2 representation), and
  that ` ch ` is the i^th carry (with the convention that the 0^th carry is 0).
  Then, ` hadd ( ph , ps , ch ) ` gives the i^th bit of the sum, and
  ` cadd ( ph , ps , ch ) ` gives the (i+1)^th carry.  Then, addition is
  performed by iteration from i = 0 to i = 1 + (max of the number of digits of
  the two summands) by "updating" the carry.

$)


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Full adder: sum
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  $c hadd $.

  $( Syntax for the "sum" output of the full adder.  (Contributed by Mario
     Carneiro, 4-Sep-2016.) $)
  whad $a wff hadd ( ph , ps , ch ) $.

  $( Definition of the "sum" output of the full adder (triple exclusive
     disjunction, or XOR3, or testing whether an odd number of parameters are
     true).  (Contributed by Mario Carneiro, 4-Sep-2016.) $)
  df-had $a |- ( hadd ( ph , ps , ch ) <-> ( ( ph \/_ ps ) \/_ ch ) ) $.

  ${
    hadbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    hadbid.2 $e |- ( ph -> ( th <-> ta ) ) $.
    hadbid.3 $e |- ( ph -> ( et <-> ze ) ) $.
    $( Equality theorem for the adder sum.  (Contributed by Mario Carneiro,
       4-Sep-2016.) $)
    hadbi123d $p |- ( ph ->
      ( hadd ( ps , th , et ) <-> hadd ( ch , ta , ze ) ) ) $=
      ( wxo whad xorbi12d df-had 3bitr4g ) ABDKZFKCEKZGKBDFLCEGLAPQFGABCDEHIMJM
      BDFNCEGNO $.
  $}

  ${
    hadbii.1 $e |- ( ph <-> ps ) $.
    hadbii.2 $e |- ( ch <-> th ) $.
    hadbii.3 $e |- ( ta <-> et ) $.
    $( Equality theorem for the adder sum.  (Contributed by Mario Carneiro,
       4-Sep-2016.) $)
    hadbi123i $p |- ( hadd ( ph , ch , ta ) <-> hadd ( ps , th , et ) ) $=
      ( whad wb wtru a1i hadbi123d mptru ) ACEJBDFJKLABCDEFABKLGMCDKLHMEFKLIMNO
      $.
  $}

  $( Associative law for the adder sum.  (Contributed by Mario Carneiro,
     4-Sep-2016.) $)
  hadass $p |- ( hadd ( ph , ps , ch ) <-> ( ph \/_ ( ps \/_ ch ) ) ) $=
    ( whad wxo df-had xorass bitri ) ABCDABECEABCEEABCFABCGH $.

  $( The adder sum is the same as the triple biconditional.  (Contributed by
     Mario Carneiro, 4-Sep-2016.) $)
  hadbi $p |- ( hadd ( ph , ps , ch ) <-> ( ( ph <-> ps ) <-> ch ) ) $=
    ( wxo wb wn whad df-xor df-had xnor bibi1i nbbn bitri 3bitr4i ) ABDZCDOCEFZ
    ABCGABEZCEZOCHABCIROFZCEPQSCABJKOCLMN $.

  $( Commutative law for the adder sum.  (Contributed by Mario Carneiro,
     4-Sep-2016.)  (Proof shortened by Wolf Lammen, 17-Dec-2023.) $)
  hadcoma $p |- ( hadd ( ph , ps , ch ) <-> hadd ( ps , ph , ch ) ) $=
    ( wb whad bicom bibi1i hadbi 3bitr4i ) ABDZCDBADZCDABCEBACEJKCABFGABCHBACHI
    $.

  $( Commutative law for the adders sum.  (Contributed by Mario Carneiro,
     4-Sep-2016.) $)
  hadcomb $p |- ( hadd ( ph , ps , ch ) <-> hadd ( ph , ch , ps ) ) $=
    ( wxo whad biid xorcom xorbi12i hadass 3bitr4i ) ABCDZDACBDZDABCEACBEAAKLAF
    BCGHABCIACBIJ $.

  $( Rotation law for the adder sum.  (Contributed by Mario Carneiro,
     4-Sep-2016.) $)
  hadrot $p |- ( hadd ( ph , ps , ch ) <-> hadd ( ps , ch , ph ) ) $=
    ( whad hadcoma hadcomb bitri ) ABCDBACDBCADABCEBACFG $.

  $( The adder sum distributes over negation.  (Contributed by Mario Carneiro,
     4-Sep-2016.)  (Proof shortened by Wolf Lammen, 11-Jul-2020.) $)
  hadnot $p |- ( -. hadd ( ph , ps , ch ) <->
    hadd ( -. ph , -. ps , -. ch ) ) $=
    ( wb wn whad notbi bibi1i xor3 hadbi xchnxbir 3bitr4i ) ABDZCEZDZAEZBEZDZND
    ABCFZEPQNFMRNABGHMCDOSMCIABCJKPQNJL $.

  $( If the first input is true, then the adder sum is equivalent to the
     biconditionality of the other two inputs.  (Contributed by Mario Carneiro,
     4-Sep-2016.)  (Proof shortened by Wolf Lammen, 11-Jul-2020.) $)
  had1 $p |- ( ph -> ( hadd ( ph , ps , ch ) <-> ( ps <-> ch ) ) ) $=
    ( whad wb hadrot hadbi bitri biass mpbir biimpri ) ABCDZBCEZEZANAELMAEZELBC
    ADOABCFBCAGHLMAIJK $.

  $( If the first input is false, then the adder sum is equivalent to the
     exclusive disjunction of the other two inputs.  (Contributed by Mario
     Carneiro, 4-Sep-2016.)  (Proof shortened by Wolf Lammen, 12-Jul-2020.) $)
  had0 $p |- ( -. ph -> ( hadd ( ph , ps , ch ) <-> ( ps \/_ ch ) ) ) $=
    ( wn whad wxo wb had1 hadnot xnor notbi bitr3i 3bitr4g con4bid ) ADZABCEZBC
    FZOOBDZCDZERSGZPDQDZORSHABCIUABCGTBCJBCKLMN $.

  $( The value of the adder sum is, if the first input is true, the
     biconditionality, and if the first input is false, the exclusive
     disjunction, of the other two inputs.  (Contributed by BJ,
     11-Aug-2020.) $)
  hadifp $p |-
    ( hadd ( ph , ps , ch ) <-> if- ( ph , ( ps <-> ch ) , ( ps \/_ ch ) ) ) $=
    ( whad wb wxo had1 had0 casesifp ) AABCDBCEBCFABCGABCHI $.


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Full adder: carry
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  $c cadd $.

  $( Syntax for the "carry" output of the full adder.  (Contributed by Mario
     Carneiro, 4-Sep-2016.) $)
  wcad $a wff cadd ( ph , ps , ch ) $.

  $( Definition of the "carry" output of the full adder.  It is true when at
     least two arguments are true, so it is equal to the "majority" function on
     three variables.  See ~ cador and ~ cadan for alternate definitions.
     (Contributed by Mario Carneiro, 4-Sep-2016.) $)
  df-cad $a |- ( cadd ( ph , ps , ch ) <->
    ( ( ph /\ ps ) \/ ( ch /\ ( ph \/_ ps ) ) ) ) $.

  $( The adder carry in disjunctive normal form.  (Contributed by Mario
     Carneiro, 4-Sep-2016.)  (Proof shortened by Wolf Lammen, 11-Jul-2020.) $)
  cador $p |- ( cadd ( ph , ps , ch ) <->
        ( ( ph /\ ps ) \/ ( ph /\ ch ) \/ ( ps /\ ch ) ) ) $=
    ( wa wxo wo wcad w3o wn xor2 rbaib anbi1d ancom andir 3bitr3g pm5.74i df-or
    wi 3bitr4i df-cad 3orass ) ABDZCABEZDZFZUBACDZBCDZFZFZABCGUBUFUGHUBIZUDRUJU
    HRUEUIUJUDUHUJUCCDABFZCDUDUHUJUCUKCUCUKUJABJKLUCCMABCNOPUBUDQUBUHQSABCTUBUF
    UGUAS $.

  $( The adder carry in conjunctive normal form.  (Contributed by Mario
     Carneiro, 4-Sep-2016.)  (Proof shortened by Wolf Lammen, 25-Sep-2018.) $)
  cadan $p |- ( cadd ( ph , ps , ch ) <->
        ( ( ph \/ ps ) /\ ( ph \/ ch ) /\ ( ps \/ ch ) ) ) $=
    ( wcad wo w3a w3o df-3or cador andi orbi1i 3bitr4i ordir ordi orcom animorl
    wa wi wb bitr4i pm4.72 mpbi anbi12i 3bitri df-3an ) ABCDZABEZACEZQZBCEZQZUG
    UHUJFUFAUJQZBCQZEZAUMEZUJUMEZQUKABQZACQZUMGUQUREZUMEUFUNUQURUMHABCIULUSUMAB
    CJKLAUJUMMUOUIUPUJABCNUPUMUJEZUJUJUMOUMUJRUJUTSBCCPUMUJUAUBTUCUDUGUHUJUET
    $.

  ${
    cadbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    cadbid.2 $e |- ( ph -> ( th <-> ta ) ) $.
    cadbid.3 $e |- ( ph -> ( et <-> ze ) ) $.
    $( Equality theorem for the adder carry.  (Contributed by Mario Carneiro,
       4-Sep-2016.) $)
    cadbi123d $p |- ( ph ->
      ( cadd ( ps , th , et ) <-> cadd ( ch , ta , ze ) ) ) $=
      ( wa wxo wo wcad anbi12d xorbi12d orbi12d df-cad 3bitr4g ) ABDKZFBDLZKZMC
      EKZGCELZKZMBDFNCEGNATUCUBUEABCDEHIOAFGUAUDJABCDEHIPOQBDFRCEGRS $.
  $}

  ${
    cadbii.1 $e |- ( ph <-> ps ) $.
    cadbii.2 $e |- ( ch <-> th ) $.
    cadbii.3 $e |- ( ta <-> et ) $.
    $( Equality theorem for the adder carry.  (Contributed by Mario Carneiro,
       4-Sep-2016.) $)
    cadbi123i $p |- ( cadd ( ph , ch , ta ) <-> cadd ( ps , th , et ) ) $=
      ( wcad wb wtru a1i cadbi123d mptru ) ACEJBDFJKLABCDEFABKLGMCDKLHMEFKLIMNO
      $.
  $}

  $( Commutative law for the adder carry.  (Contributed by Mario Carneiro,
     4-Sep-2016.) $)
  cadcoma $p |- ( cadd ( ph , ps , ch ) <-> cadd ( ps , ph , ch ) ) $=
    ( wa wxo wo wcad ancom xorcom anbi2i orbi12i df-cad 3bitr4i ) ABDZCABEZDZFB
    ADZCBAEZDZFABCGBACGNQPSABHORCABIJKABCLBACLM $.

  $( Commutative law for the adder carry.  (Contributed by Mario Carneiro,
     4-Sep-2016.)  (Proof shortened by Wolf Lammen, 11-Jul-2020.) $)
  cadcomb $p |- ( cadd ( ph , ps , ch ) <-> cadd ( ph , ch , ps ) ) $=
    ( wcad wo w3a cadan 3ancoma orcom 3anbi3i 3bitri bitr4i ) ABCDZACEZABEZCBEZ
    FZACBDMONBCEZFNORFQABCGONRHRPNOBCIJKACBGL $.

  $( Rotation law for the adder carry.  (Contributed by Mario Carneiro,
     4-Sep-2016.) $)
  cadrot $p |- ( cadd ( ph , ps , ch ) <-> cadd ( ps , ch , ph ) ) $=
    ( wcad cadcoma cadcomb bitri ) ABCDBACDBCADABCEBACFG $.

  $( The adder carry distributes over negation.  (Contributed by Mario
     Carneiro, 4-Sep-2016.)  (Proof shortened by Wolf Lammen, 11-Jul-2020.) $)
  cadnot $p |- ( -. cadd ( ph , ps , ch ) <->
    cadd ( -. ph , -. ps , -. ch ) ) $=
    ( wa wn w3a wo wcad ianor 3anbi123i w3o 3ioran cador xchnxbir cadan 3bitr4i
    ) ABDZEZACDZEZBCDZEZFZAEZBEZGZUDCEZGZUEUGGZFABCHZEUDUEUGHRUFTUHUBUIABIACIBC
    IJQSUAKUCUJQSUALABCMNUDUEUGOP $.

  $( If (at least) two inputs are true, then the adder carry is true.
     (Contributed by Mario Carneiro, 4-Sep-2016.) $)
  cad11 $p |- ( ( ph /\ ps ) -> cadd ( ph , ps , ch ) ) $=
    ( wa wxo wo wcad orc df-cad sylibr ) ABDZKCABEDZFABCGKLHABCIJ $.

  $( If one input is true, then the adder carry is true exactly when at least
     one of the other two inputs is true.  (Contributed by Mario Carneiro,
     8-Sep-2016.)  (Proof shortened by Wolf Lammen, 19-Jun-2020.) $)
  cad1 $p |- ( ch -> ( cadd ( ph , ps , ch ) <-> ( ph \/ ps ) ) ) $=
    ( wcad wo wa w3a cadan 3anass bitri olc jca biantrud bitr4id ) CABCDZABEZAC
    EZBCEZFZFZPOPQRGTABCHPQRIJCSPCQRCAKCBKLMN $.

  $( If one input is false, then the adder carry is true exactly when both of
     the other two inputs are true.  (Contributed by Mario Carneiro,
     8-Sep-2016.)  (Proof shortened by Wolf Lammen, 21-Sep-2024.) $)
  cad0 $p |- ( -. ch -> ( cadd ( ph , ps , ch ) <-> ( ph /\ ps ) ) ) $=
    ( wn wcad wa wxo wo df-cad idd pm2.21 adantrd jaod biimtrid cad11 impbid1 )
    CDZABCEZABFZRSCABGZFZHQSABCIQSSUAQSJQCSTCSKLMNABCOP $.

  $( The value of the carry is, if the input carry is true, the disjunction,
     and if the input carry is false, the conjunction, of the other two inputs.
     (Contributed by BJ, 8-Oct-2019.) $)
  cadifp $p |-
      ( cadd ( ph , ps , ch ) <-> if- ( ch , ( ph \/ ps ) , ( ph /\ ps ) ) ) $=
    ( wcad wo wa cad1 cad0 casesifp ) CABCDABEABFABCGABCHI $.

  $( The adder carry is true as soon as its first two inputs are the truth
     constant.  (Contributed by Mario Carneiro, 4-Sep-2016.) $)
  cadtru $p |- cadd ( T. , T. , ph ) $=
    ( wtru wcad tru cad11 mp2an ) BBBBACDDBBAEF $.


$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
  Other axiomatizations related to classical propositional calculus
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Minimal implicational calculus
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  Minimal implicational calculus, or intuitionistic implicational calculus, or
  positive implicational calculus, is the implicational fragment of minimal
  calculus (which is also the implicational fragment of intuitionistic calculus
  and of positive calculus).  It is sometimes called "C-pure intuitionism"
  since the letter C is sometimes used to denote implication, especially in
  prefix notation.  It can be axiomatized by the inference rule of modus ponens
  ~ ax-mp together with the axioms { ~ ax-1 , ~ ax-2 } (sometimes written KS),
  or with { ~ imim1 , ~ ax-1 , ~ pm2.43 } (written B'KW), or with { ~ imim2 ,
  ~ pm2.04 , ~ ax-1 , ~ pm2.43 } (written BCKW), or with the single axiom
  ~ minimp .  This section proves ~ minimp from { ~ ax-1 , ~ ax-2 }, and then
  the converse, due to Ivo Thomas.

  Sources for this section are the webpage
  ~ https://web.ics.purdue.edu/~~dulrich/C-pure-intuitionism-page.htm
  on Ted Ulrich's website, and the articles
  C. A. Meredith,
  _A single axiom of positive logic_,
  Journal of computing systems, vol. 1 (1953), 169--170,
  and
  C. A. Meredith, A. N. Prior,
  _Notes on the axiomatics of the propositional calculus_,
  Notre Dame Journal of Formal Logic, vol. 4 (1963), 171--187.

  We may use a compact notation for derivations known as the D-notation where
  "D" stands for "condensed Detachment".  For instance, "D21" means detaching
  ~ ax-1 from ~ ax-2 , that is, using modus ponens ~ ax-mp with ~ ax-1 as minor
  premise and ~ ax-2 as major premise.  D-strings are accepted by the grammar
  Dstr := digit | "D" Dstr Dstr.

  (Contributed by BJ, 11-Apr-2021.)

$)

  $( A single axiom for minimal implicational calculus, due to Meredith.  Other
     single axioms of the same length are known, but it is thought to be the
     minimal length.  Among single axioms of this length, it is the one with
     simplest antecedents (i.e., in the corresponding ordering of binary trees
     which first compares left subtrees, it is the first one).  (Contributed by
     BJ, 4-Apr-2021.) $)
  minimp $p |- ( ph ->
  ( ( ps -> ch ) -> ( ( ( th -> ps ) -> ( ch -> ta ) ) -> ( ps -> ta ) ) ) ) $=
    ( wi jarr a2d com12 a1i ) BCFZDBFCEFZFZBEFZFFAMKNMBCEDBLGHIJ $.
  $( $j usage 'minimp' avoids 'ax-3'; $)

  $( Derivation of Syll-Simp ( ~ jarr ) from ~ ax-mp and ~ minimp .
     (Contributed by BJ, 4-Apr-2021.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  minimp-syllsimp $p |- ( ( ( ph -> ps ) -> ch ) -> ( ps -> ch ) ) $=
    ( wi minimp ax-mp mp2 mp2b ) ABDZIDZJJDJDDZICDZDZLDZILCDZDZLBCDZDZJLIDZJDJD
    DZKDZITDZMDTLDDZNTIIIIETTTDZUDUDDUDDDDZUAUCDTTTTTEUETKILEFMMMDZUFUFDUFDDDZM
    TDUCNDMMMMMEMIILIEUGMTUBLEGHIKDZINDSNODDZPDDZNPDZIIIIIEUHINLOENNNDZULULDULD
    DDZNUIDUJUKDNNNNNENLIKCEUMNUIIPEGHLLLDZUNUNDUNDDDZLPDBLDPQDDZRDDZPRDZLLLLLE
    UOLPBQEPPPDZUSUSDUSDDDZPUPDUQURDPPPPPEPBLACEUTPUPLREGHH $.

  $( Derivation of ~ ax-1 from ~ ax-mp and ~ minimp .  (Contributed by BJ,
     4-Apr-2021.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  minimp-ax1 $p |- ( ph -> ( ps -> ph ) ) $=
    ( wi minimp-syllsimp ax-mp ) ABCZACBACZCAGCABADFAGDE $.

  $( Derivation of a commuted form of ~ ax-2 from ~ ax-mp and ~ minimp .
     (Contributed by BJ, 4-Apr-2021.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  minimp-ax2c $p |- ( ( ph -> ps ) ->
                                ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) ) $=
    ( wi minimp ax-mp mp2 minimp-syllsimp ) ABDZAADZJJDJDDZADZBCDZDZACDZDZDZPAM
    DZODZDZISDZAKDZQAAAAAEUBABKCEFRRDZPDSDZTRUCUCUCDUCDDDZRNDZUDRRRRRELLDZRDNDZ
    UFLUGUGUGDUGDDDZLADZUHLLLLLEZLKDZUBLDLDZUJLAAAAEKKKDZUNUNDUNDDDZUNUMKKKKKEK
    AAAAEUOKKAAEGUIULUMUJDDUKUILKUBAEFGUILALMEGUGRNHFUERNROEGUCPSHFSQDZIIDZTDUA
    DZTUADZDZDQUSDZDZUTVAQQQDZVCVCDVCDDDZQURDZVBQQQQQEIUQUQUQDUQDDDZVEIIIIIEVFI
    PISEFVDQURSUSEGUQTUAHUPUTVAHGG $.

  $( Derivation of ~ ax-2 from ~ ax-mp and ~ minimp .  (Contributed by BJ,
     4-Apr-2021.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  minimp-ax2 $p |- ( ( ph -> ( ps -> ch ) ) ->
                                          ( ( ph -> ps ) -> ( ph -> ch ) ) ) $=
    ( wi minimp-ax2c minimp-syllsimp ax-mp mp2 ) ABDZABCDDZACDZDDZJLIKDZDZDZJMD
    ZABCEIJDNDOIJKEIJNFGJLDOPDZDLQDJLMEJLQFGH $.

  $( Derivation of ~ pm2.43 (also called "hilbert" or W) from ~ ax-mp and
     ~ minimp .  It uses the classical derivation from ~ ax-1 and ~ ax-2
     written DD22D21 in D-notation (see head comment for an explanation) and
     shortens the proof using ~ mp2 (which only requires ~ ax-mp ).
     (Contributed by BJ, 31-May-2021.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  minimp-pm2.43 $p |- ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) ) $=
    ( wi minimp-ax2 minimp-ax1 ax-mp mp2 ) AABCZCZAACZHCCIJCZIHCAABDAHACCKAHEAH
    ADFIJHDG $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Implicational Calculus
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  Implicational calculus is the fragment of propositional logic that uses only
  material implication, and not negation.  It can be axiomatized by inference
  rule modus ponens ~ ax-mp together with the axioms { ~ ax-1 , ~ ax-2 ,
  ~ peirce } or the Tarski-Bernays axioms { ~ ax-1 , ~ imim1 , ~ peirce } or
  with the single axiom { ~ impsingle }.  From either one of these three axiom
  sets, all tautologies containing only material implication may be proved.  In
  contrast, minimal implicational calculus, which is presented just before this
  section, cannot prove certain tautologies ( ~ peirce , for example ).  The
  class of theorems proved by minimal implicational calculus is thus a subset
  of the class of theorems proved by implicational calculus.

  The primary source for this section is the paper by Jan Lukasiewicz, _The
  Shortest Axiom of the Implicational Calculus of Propositions_, Proceedings
  of the Royal Irish Academy, Section A, vol. 52 (1948-1950), 25--33.  It will
  be cited as simply "Lukasiewicz" in the remainder of this section.

  This section proves that the above three distinct axiom sets for
  implicational calculus all produce the same class of theorems.

  (Contributed by Larry Lesyna and Jeffrey P. Machado, 1-Aug-2023.)

$)

  $( The shortest single axiom for implicational calculus, due to Lukasiewicz.
     It is now known to be the _unique_ shortest axiom.  The axiom is proved
     here starting from { ~ ax-1 , ~ ax-2 and ~ peirce }.  (Contributed by
     Larry Lesyna and Jeffrey P. Machado, 18-Jul-2023.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  impsingle $p |- ( ( ( ph -> ps ) -> ch )
     -> ( ( ch -> ph ) -> ( th -> ph ) ) ) $=
    ( wi imim1 peirce a1d syl6 ) ABEZCECAEJAEZDAEJCAFKADABGHI $.

  $( Derivation of impsingle-step4 from ~ ax-mp and ~ impsingle .  It is used
     as a lemma in proofs of ~ imim1 and ~ peirce from ~ impsingle .  It is
     Step 4 in Lukasiewicz, where it appears as 'CCCpqpCsp' using
     parenthesis-free prefix notation.  (Contributed by Larry Lesyna and
     Jeffrey P. Machado, 2-Aug-2023.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  impsingle-step4 $p |- ( ( ( ph -> ps ) -> ph ) -> ( ch -> ph ) ) $=
    ( wta wet wze wsi wth wi impsingle ax-mp ) DEIFIFDIGDIIIZABIZAICAIIZDEFGJAH
    IZMIZNIZLNIZAHMCJNMIPIZQRIMMINISABMCJMMNOJKNMPLJKKK $.

  $( Derivation of impsingle-step8 from ~ ax-mp and ~ impsingle .  It is used
     as a lemma in proofs of ~ ax-1 ~ imim1 and ~ peirce from ~ impsingle .  It
     is Step 8 in Lukasiewicz, where it appears as 'CCCsqpCqp' using
     parenthesis-free prefix notation.  (Contributed by Larry Lesyna and
     Jeffrey P. Machado, 2-Aug-2023.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  impsingle-step8 $p |- ( ( ( ph -> ps ) -> ch ) -> ( ps -> ch ) ) $=
    ( wta wet wze wsi wth wi impsingle ax-mp ) DEIFIFDIGDIIIZABIZCIZBCIZIZDEFGJ
    ZCHIZMIZPIZLPIZCHMBJPMISIZTUAIMOIPIZUBOBIMIZUCLUDQBHIZOIZUDIZLUDIZBHOAJUDOI
    UFIZUGUHIOOIUDIUIBCOAJOOUDUEJKUDOUFLJKKKOBMNJKMOPRJKPMSLJKKK $.

  $( Derivation of impsingle-ax1 ( ~ ax-1 ) from ~ ax-mp and ~ impsingle .
     Lukasiewicz was used to construct this proof.  Every formula corresponding
     to a detachment step was converted to its corresponding Metamath formula.
     mmj2 was used to unify each formula using ~ ax-mp , which in turn produced
     this proof.  With extremely high confidence, this result shows that the
     Lukasiewicz proof of ~ ax-1 (step 27) is correct and that Metamath
     correctly verifies the proof.  The same comments apply to the proofs that
     follow this one.  (Contributed by Larry Lesyna and Jeffrey P. Machado,
     2-Aug-2023.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  impsingle-ax1 $p |- ( ph -> ( ps -> ph ) ) $=
    ( wch wi impsingle-step8 ax-mp ) CBDZADBADZDAHDCBAEGAHEF $.

  $( Derivation of impsingle-step15 from ~ ax-mp and ~ impsingle .  It is used
     as a lemma in proofs of ~ imim1 and ~ peirce from ~ impsingle .  It is
     Step 15 in Lukasiewicz, where it appears as 'CCCrqCspCCrpCsp' using
     parenthesis-free prefix notation.  (Contributed by Larry Lesyna and
     Jeffrey P. Machado, 2-Aug-2023.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  impsingle-step15 $p |- ( ( ( ph -> ps ) -> ( ch -> th ) )
                              -> ( ( ph -> th ) -> ( ch -> th ) ) ) $=
    ( wla wta wsi wrh wmu wet wze wi impsingle impsingle-step8 ax-mp ) DELZALZA
    DLZCDLZLZLZABLZSLZTLZDEACMFGLHLHFLIFLLLZUAUDLZFGHIMTJLZQLZUFLZUEUFLZTJQUCMU
    FQLUHLZUIUJLQUDLUFLZUKUDALQLZULUBUDLZUMSKLZUBLUDLUNSKUBRMUOUBUDNOABUDPMOUDA
    QUAMOQUDUFUGMOUFQUHUEMOOOO $.

  $( Derivation of impsingle-step18 from ~ ax-mp and ~ impsingle .  It is used
     as a lemma in proofs of ~ imim1 and ~ peirce from ~ impsingle .  It is
     Step 18 in Lukasiewicz, where it appears as 'CCCCrpCspCCCpqrtCuCCCpqrt'
     using parenthesis-free prefix notation.  (Contributed by Larry Lesyna and
     Jeffrey P. Machado, 2-Aug-2023.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  impsingle-step18 $p |- ( ( ( ( ph -> ps ) -> ( ch -> ps ) )
                               -> ( ( ( ps -> th ) -> ph ) -> ta ) )
                 -> ( et -> ( ( ( ps -> th ) -> ph ) -> ta ) ) ) $=
    ( wrh wi impsingle impsingle-step8 ax-mp impsingle-step15 ) BDHAHZEHZCBHZHZ
    ABHZOHZHZRNHFNHHMRHZSBDACINSHZTSHOGHZNHSHUAOGNQIUBNSJKMEPRLKKNORFIK $.

  $( Derivation of impsingle-step19 from ~ ax-mp and ~ impsingle .  It is used
     as a lemma in proofs of ~ imim1 and ~ peirce from ~ impsingle .  It is
     Step 19 in Lukasiewicz, where it appears as 'CCCCspqCrpCCCpqrCsp' using
     parenthesis-free prefix notation.  (Contributed by Larry Lesyna and
     Jeffrey P. Machado, 2-Aug-2023.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  impsingle-step19 $p |- ( ( ( ( ph -> ps ) -> ch ) -> ( th -> ps ) )
                        -> ( ( ( ps -> ch ) -> th ) -> ( ph -> ps ) ) ) $=
    ( wta wet wze wsi wrh wmu wi impsingle-step18 ax-mp ) EFKGFKKFHKEKIKZKJNKKZ
    ABKZCKDBKZKZBCKDKZPKZKZEFGHIJLQPKTKUAKOUAKDBACPRLQPSCTOLMM $.

  $( Derivation of impsingle-step20 from ~ ax-mp and ~ impsingle .  It is used
     as a lemma in proofs of ~ imim1 and ~ peirce from ~ impsingle .  It is
     Step 20 in Lukasiewicz, where it appears as 'CCCCrppCspCCCpqrCsp' using
     parenthesis-free prefix notation.  (Contributed by Larry Lesyna and
     Jeffrey P. Machado, 2-Aug-2023.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  impsingle-step20 $p |- ( ( ( ( ph -> ps ) -> ps ) -> ( ch -> ps ) )
                          -> ( ( ( ps -> th ) -> ph ) -> ( ch -> ps ) ) ) $=
    ( wta wze wsi wrh wet wi impsingle-step19 impsingle impsingle-step8 ax-mp )
    CBJZDJZABJZJZBDJAJZOJZJZQBJZOJZTJZCBDAKEFJGJGEJHEJJJZUAUDJZEFGHLTIJZRJZUFJZ
    UEUFJZTIRUCLUFRJUHJZUIUJJRUDJUFJZUKUDQJRJZULUBUDJZUMOEJZUBJUDJUNOEUBSLUOUBU
    DMNQBUDPLNUDQRUALNRUDUFUGLNUFRUHUELNNNN $.

  $( Derivation of impsingle-step21 from ~ ax-mp and ~ impsingle .  It is used
     as a lemma in the proof of ~ imim1 from ~ impsingle .  It is Step 21 in
     Lukasiewicz, where it appears as 'CCCCprqqCCqrCpr' using parenthesis-free
     prefix notation.  (Contributed by Larry Lesyna and Jeffrey P. Machado,
     2-Aug-2023.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  impsingle-step21 $p |- ( ( ( ( ph -> ps ) -> ch ) -> ch )
                           -> ( ( ch -> ps ) -> ( ph -> ps ) ) ) $=
    ( wi impsingle-step15 impsingle-step20 ax-mp ) CABDZDHDCBDZHDZDHCDCDJDCHABE
    CHICFG $.

  $( Derivation of impsingle-step22 from ~ ax-mp and ~ impsingle .  It is used
     as a lemma in proofs of ~ imim1 and ~ peirce from ~ impsingle .  It is
     Step 22 in Lukasiewicz, where it appears as 'Cpp' using parenthesis-free
     prefix notation.  (Contributed by Larry Lesyna and Jeffrey P. Machado,
     2-Aug-2023.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  impsingle-step22 $p |- ( ph -> ph ) $=
    ( wth wmu wla wps wi impsingle-step4 impsingle ax-mp ) BCFBFDBFFZAAFZBCDGAE
    FZAFZKFZJKFZAEAGKAFMFNOFAALGKAMJHIII $.

  $( Derivation of impsingle-step25 from ~ ax-mp and ~ impsingle .  It is used
     as a lemma in the proof of ~ imim1 from ~ impsingle .  It is Step 25 in
     Lukasiewicz, where it appears as 'CCpqCCCprqq' using parenthesis-free
     prefix notation.  (Contributed by Larry Lesyna and Jeffrey P. Machado,
     2-Aug-2023.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  impsingle-step25 $p |- ( ( ph -> ps )
                         -> ( ( ( ph -> ch ) -> ps ) -> ps ) ) $=
    ( wth wi impsingle-step22 impsingle-step20 impsingle-step8 impsingle-step15
    ax-mp ) ACEZKBEZBEZEZABEMEBDEZKEMEZNMMEPMFKBLDGJOKMHJACLBIJ $.

  $( Derivation of impsingle-imim1 ( ~ imim1 ) from ~ ax-mp and ~ impsingle .
     It is step 29 in Lukasiewicz.  (Contributed by Larry Lesyna and Jeffrey P.
     Machado, 2-Aug-2023.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  impsingle-imim1 $p |- ( ( ph -> ps )
                          -> ( ( ps -> ch ) -> ( ph -> ch ) ) ) $=
    ( wi impsingle-step21 impsingle-step25 ax-mp ) ACDZBDBDZBCDHDZDZABDZJDZACBE
    MIDIDZKMDLIDNABCFLIJFGLJIEGG $.

  $( Derivation of impsingle-peirce ( ~ peirce ) from ~ ax-mp and ~ impsingle .
     It is step 28 in Lukasiewicz.  (Contributed by Larry Lesyna and Jeffrey P.
     Machado, 2-Aug-2023.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  impsingle-peirce $p |- ( ( ( ph -> ps ) -> ph ) -> ph ) $=
    ( wi impsingle-step22 impsingle-step25 ax-mp ) AACABCACACADAABEF $.

  $( Derivation of ~ ax-2 from the Tarski-Bernays axiom set { ~ ax-1 ,
     ~ imim1 , ~ peirce }.  Note that no inference rule other than ~ ax-mp is
     used in this proof.  This proof establishes a circle of equivalence:  From
     { ~ impsingle }, { ~ ax-1 , ~ imim1 , ~ peirce } was proved.  From {
     ~ ax-1 , ~ imim1 , ~ peirce }, { ~ ax-1 , ~ ax-2 and ~ peirce } was
     proved.  From { ~ ax-1 , ~ ax-2 and ~ peirce }, { ~ impsingle } was
     proved.  Therefore, the theorems that can be proved by selecting any one
     of these three distinct axiom sets must be equivalent.  Prover9 and mmj2
     were used to help constuct this proof.  (Contributed by Larry Lesyna and
     Jeffrey P. Machado, 1-Aug-2023.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  tarski-bernays-ax2 $p |- ( ( ph -> ( ps -> ch ) )
                             -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) $=
    ( wi peirce ax-1 ax-mp imim1 ) ABDZABCDZDZACDZDZDZKILDZDZJLDZMDZNKQLDZDZRJC
    DZLDZSDZTQUBLDZDZUCLCDZUADZUDDZUEUBUFLDZDZUDDZUHUILDZUDDZUDDZUKUMUNDZUNUMUL
    DZUOULUPLCEULUMFGUMULUDHGUNUDDUNDZUNDZUOUNDZUNUDEUOUQDURUSDUMUNUDHUOUQUNHGG
    GUJUMDUNUKDUBUILHUJUMUDHGGUGUJDUKUHDUFUALHUGUJUDHGGQUGDUHUEDJLCHQUGUDHGGUBU
    DLDZDZUEUCDZUDUBDZUTDZVAUDUTDZUTDZVDUTLDUTDZUTDZVFUTLEVEVGDVHVFDUDUTLHVEVGU
    THGGVCVEDVFVDDUDUBLHVCVEUTHGGUBVCDVDVADUBUDFUBVCUTHGGUTSDZUCDZVBDZVAVBDZUEV
    IDVKQUDLHUEVIUCHGVAVJDVKVLDUBUTSHVAVJVBHGGGGKUBDUCTDAJCHKUBSHGGQSLDZDZTRDZS
    QDZVMDZVNSVMDZVMDZVQVMLDVMDZVMDZVSVMLEVRVTDWAVSDSVMLHVRVTVMHGGVPVRDVSVQDSQL
    HVPVRVMHGGQVPDVQVNDQSFQVPVMHGGVMMDZRDZVODZVNVODZTWBDWDKSLHTWBRHGVNWCDWDWEDQ
    VMMHVNWCVOHGGGGIQDRNDABCHIQMHGGKMLDZDZNPDZMKDZWFDZWGMWFDZWFDZWJWFLDWFDZWFDZ
    WLWFLEWKWMDWNWLDMWFLHWKWMWFHGGWIWKDWLWJDMKLHWIWKWFHGGKWIDWJWGDKMFKWIWFHGGWF
    ODZPDZWHDZWGWHDZNWODWQIMLHNWOPHGWGWPDWQWRDKWFOHWGWPWHHGGGG $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Derive the Lukasiewicz axioms from Meredith's sole axiom
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Carew Meredith's sole axiom for propositional calculus.  This amazing
     formula is thought to be the shortest possible single axiom for
     propositional calculus with inference rule ~ ax-mp , where negation and
     implication are primitive.  Here we prove Meredith's axiom from ~ ax-1 ,
     ~ ax-2 , and ~ ax-3 .  Then from it we derive the Lukasiewicz axioms
     ~ luk-1 , ~ luk-2 , and ~ luk-3 .  Using these we finally rederive our
     axioms as ~ ax1 , ~ ax2 , and ~ ax3 , thus proving the equivalence of all
     three systems.  C. A. Meredith, "Single Axioms for the Systems (C,N),
     (C,O) and (A,N) of the Two-Valued Propositional Calculus", _The Journal of
     Computing Systems_ vol. 1 (1953), pp. 155-164.  Meredith claimed to be
     close to a proof that this axiom is the shortest possible, but the proof
     was apparently never completed.

     An obscure Irish lecturer, Meredith (1904-1976) became enamored with logic
     somewhat late in life after attending talks by Lukasiewicz and produced
     many remarkable results such as this axiom.  From his obituary:  "He did
     logic whenever time and opportunity presented themselves, and he did it on
     whatever materials came to hand: in a pub, his favored pint of porter
     within reach, he would use the inside of cigarette packs to write proofs
     for logical colleagues."  (Contributed by NM, 14-Dec-2002.)  (Proof
     shortened by Andrew Salmon, 25-Jul-2011.)  (Proof shortened by Wolf
     Lammen, 28-May-2013.) $)
  meredith $p |- ( ( ( ( ( ph -> ps ) -> ( -. ch -> -. th ) ) -> ch ) ->
       ta ) -> ( ( ta -> ph ) -> ( th -> ph ) ) ) $=
    ( wi wn pm2.21 con4 imim12i com13 con1d com12 a1d ax-1 imim1d ja ) ABFZCGDG
    FZFZCFZEEAFZDAFZFUAGZUCUBDUDADAUATAGZDCUERSDCFABHCDIJKLMNEDEAEDOPQ $.

  $( Step 3 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
     (The step numbers refer to Meredith's original paper.)  (Contributed by
     NM, 14-Dec-2002.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  merlem1 $p |- ( ( ( ch -> ( -. ph -> ps ) ) -> ta ) -> ( ph -> ta ) ) $=
    ( wn wi meredith ax-mp ) DAEZFIBFZEZIFFZJFCJFZFZMDFADFFJDECEFZEKEFZFOFDFLFN
    IBOKDGJPDCLGHDIJAMGH $.

  $( Step 4 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
     (Contributed by NM, 14-Dec-2002.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  merlem2 $p |- ( ( ( ph -> ph ) -> ch ) -> ( th -> ch ) ) $=
    ( wi wn merlem1 meredith ax-mp ) BBDZAECEZDDADAADZDKBDCBDDAJIAFBBACKGH $.

  $( Step 7 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
     (Contributed by NM, 14-Dec-2002.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  merlem3 $p |- ( ( ( ps -> ch ) -> ph ) -> ( ch -> ph ) ) $=
    ( wi wn merlem2 ax-mp meredith ) AADZCEZJDZDZCDBCDZDZMADCADZDOBEZPDDBDZLDZN
    KKDLDRJKIFKLQFGCABBLHGAACCMHG $.

  $( Step 8 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
     (Contributed by NM, 14-Dec-2002.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  merlem4 $p |- ( ta -> ( ( ta -> ph ) -> ( th -> ph ) ) ) $=
    ( wi wn meredith merlem3 ax-mp ) AADBEZIDDBDZCDCADBADDZDCKDAABBCFKJCGH $.

  $( Step 11 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
     (Contributed by NM, 14-Dec-2002.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  merlem5 $p |- ( ( ph -> ps ) -> ( -. -. ph -> ps ) ) $=
    ( wi wn meredith merlem1 merlem4 ax-mp ) BBCZBDZJCCBCBCIICCZABCZADZDZBCCZBB
    BBBEIJNDCCBCZACZOCZKOCZBBBNAEOKDZCMTCCZACQCZRSCUAUBMBLTFAPUAGHOTAKQEHHH $.

  $( Step 12 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
     (Contributed by NM, 14-Dec-2002.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  merlem6 $p |- ( ch -> ( ( ( ps -> ch ) -> ph ) -> ( th -> ph ) ) ) $=
    ( wi merlem4 merlem3 ax-mp ) BCEZIAEDAEEZECJEADIFJBCGH $.

  $( Between steps 14 and 15 of Meredith's proof of Lukasiewicz axioms from his
     sole axiom.  (Contributed by NM, 22-Dec-2002.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  merlem7 $p |- ( ph -> ( ( ( ps -> ch ) -> th ) -> ( ( ( ch -> ta ) ->
                  ( -. th -> -. ps ) ) -> th ) ) ) $=
    ( wi wn merlem4 merlem6 meredith ax-mp ) BCFZLDFZCEFDGBGFFZDFZFZFZAPFZDNLHP
    AGZFCGZSFFZCFLFZQRFOUAFUBSMOTICEDBUAJKPSCALJKK $.

  $( Step 15 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
     (Contributed by NM, 22-Dec-2002.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  merlem8 $p |- ( ( ( ps -> ch ) -> th ) -> ( ( ( ch -> ta ) ->
                  ( -. th -> -. ps ) ) -> th ) ) $=
    ( wph wi wn meredith merlem7 ax-mp ) EEFZEGZLFFEFEFKKFFZABFCFBDFCGAGFFCFFEE
    EEEHMABCDIJ $.

  $( Step 18 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
     (Contributed by NM, 22-Dec-2002.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  merlem9 $p |- ( ( ( ph -> ps ) -> ( ch -> ( th -> ( ps -> ta ) ) ) ) ->
                    ( et -> ( ch -> ( th -> ( ps -> ta ) ) ) ) ) $=
    ( wi wn merlem6 merlem8 ax-mp meredith ) CDBEGZGZGZFHZGBHZPGGZBGABGZGZSOGFO
    GGMRHDHGZHAHGZGUAGRGZTNRGUCPCNQIDMRUBJKBEUAARLKOPBFSLK $.

  $( Step 19 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
     (Contributed by NM, 14-Dec-2002.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  merlem10 $p |- ( ( ph -> ( ph -> ps ) ) -> ( th -> ( ph -> ps ) ) ) $=
    ( wi wn meredith merlem9 ax-mp ) AADZAEZJDDADADIIDDZAABDZDZCLDDZAAAAAFLADJC
    EDDADZADNDKNDLAACAFOAMCBKGHH $.

  $( Step 20 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
     (Contributed by NM, 14-Dec-2002.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  merlem11 $p |- ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) ) $=
    ( wi wn meredith merlem10 ax-mp ) AACZADZICCACACHHCCZAABCZCZKCZAAAAAELMCJMC
    ABLFLKJFGG $.

  $( Step 28 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
     (Contributed by NM, 14-Dec-2002.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  merlem12 $p |- ( ( ( th -> ( -. -. ch -> ch ) ) -> ph ) -> ph ) $=
    ( wn wi merlem5 merlem2 ax-mp merlem4 merlem11 ) CBDDBEZEZAEZMAEZEZNLOBBEKE
    LBBFBKCGHAMLIHMAJH $.

  $( Step 35 of Meredith's proof of Lukasiewicz axioms from his sole axiom.
     (Contributed by NM, 14-Dec-2002.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  merlem13 $p |- ( ( ph -> ps ) ->
              ( ( ( th -> ( -. -. ch -> ch ) ) -> -. -. ph ) -> ps ) ) $=
    ( wi wn merlem12 merlem5 ax-mp merlem6 meredith merlem11 ) BBEZAFZDCFFCEEZN
    FZEZFZEZEAEZAEZABEQBEETUAEZUASUBOREZREZSRCDGRBEZRFPEZEREUCEZUDSEUFUGQPEUFPC
    DGQPHIRUEUFOJIRBRNUCKIIAMSTJITALIBBAQAKI $.

  $( 1 of 3 axioms for propositional calculus due to Lukasiewicz, derived from
     Meredith's sole axiom.  (Contributed by NM, 14-Dec-2002.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  luk-1 $p |- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) ) $=
    ( wi wn meredith merlem13 ax-mp ) CCDZAEZEZEJDDKDBDZBCDACDDZDZABDZMDZCCKABF
    MADZOEZEZERDDSDLDZNPDOLDTABJIGOLRQGHMASOLFHH $.

  $( 2 of 3 axioms for propositional calculus due to Lukasiewicz, derived from
     Meredith's sole axiom.  (Contributed by NM, 14-Dec-2002.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  luk-2 $p |- ( ( -. ph -> ph ) -> ph ) $=
    ( wn wi merlem5 merlem4 ax-mp merlem11 meredith ) ABZACZJACZCZKAJBZCIBMCCZI
    CZICZLOPCZPNQAMDIONEFOIGFAMIJIHFJAGF $.

  $( 3 of 3 axioms for propositional calculus due to Lukasiewicz, derived from
     Meredith's sole axiom.  (Contributed by NM, 14-Dec-2002.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  luk-3 $p |- ( ph -> ( -. ph -> ps ) ) $=
    ( wn wi merlem11 merlem1 ax-mp ) ACZHBDZDIDAIDHBEABHIFG $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Derive the standard axioms from the Lukasiewicz axioms
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    luklem1.1 $e |- ( ph -> ps ) $.
    luklem1.2 $e |- ( ps -> ch ) $.
    $( Used to rederive standard propositional axioms from Lukasiewicz'.
       (Contributed by NM, 23-Dec-2002.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    luklem1 $p |- ( ph -> ch ) $=
      ( wi luk-1 ax-mp ) BCFZACFZEABFIJFDABCGHH $.
  $}

  $( Used to rederive standard propositional axioms from Lukasiewicz'.
     (Contributed by NM, 22-Dec-2002.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  luklem2 $p |- ( ( ph -> -. ps ) ->
                ( ( ( ph -> ch ) -> th ) -> ( ps -> th ) ) ) $=
    ( wn wi luk-1 luk-3 ax-mp luklem1 ) ABEZFZBACFZFZMDFBDFFLKCFZMFZNAKCGBOFPNF
    BCHBOMGIJBMDGJ $.

  $( Used to rederive standard propositional axioms from Lukasiewicz'.
     (Contributed by NM, 22-Dec-2002.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  luklem3 $p |- ( ph -> ( ( ( -. ph -> ps ) -> ch ) -> ( th -> ch ) ) ) $=
    ( wn wi luk-3 luklem2 luklem1 ) AAEZDEZFJBFCFDCFFAKGJDBCHI $.

  $( Used to rederive standard propositional axioms from Lukasiewicz'.
     (Contributed by NM, 22-Dec-2002.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  luklem4 $p |- ( ( ( ( -. ph -> ph ) -> ph ) -> ps ) -> ps ) $=
    ( wn wi luk-2 luklem3 ax-mp luk-1 luklem1 ) ACADADZBDZBCZBDZBLJDZKMDJCJDJDZ
    NJEJONDAEJJJLFGGLJBHGBEI $.

  $( Used to rederive standard propositional axioms from Lukasiewicz'.
     (Contributed by NM, 22-Dec-2002.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  luklem5 $p |- ( ph -> ( ps -> ph ) ) $=
    ( wn wi luklem3 luklem4 luklem1 ) AACADADBADZDHAAABEAHFG $.

  $( Used to rederive standard propositional axioms from Lukasiewicz'.
     (Contributed by NM, 22-Dec-2002.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  luklem6 $p |- ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) ) $=
    ( wi luk-1 wn luklem5 luklem2 luklem4 luklem1 ax-mp ) AABCZCKBCZKCZKAKBDKEZ
    KCZKCMKCZCZPMOCZQNLCRNBEZNCZLNSFTSBCBCLCLSKBBGBLHIINLKDJMOKDJKPHJI $.

  $( Used to rederive standard propositional axioms from Lukasiewicz'.
     (Contributed by NM, 22-Dec-2002.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  luklem7 $p |- ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) ) $=
    ( wi luk-1 luklem5 luklem1 luklem6 ax-mp ) ABCDZDJCDZACDZDZBLDZAJCEBKDMNDBJ
    KDZKBJBDOBJFJBCEGJCHGBKLEIG $.

  $( Used to rederive standard propositional axioms from Lukasiewicz'.
     (Contributed by NM, 22-Dec-2002.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  luklem8 $p |- ( ( ph -> ps ) -> ( ( ch -> ph ) -> ( ch -> ps ) ) ) $=
    ( wi luk-1 luklem7 ax-mp ) CADZABDZCBDZDDIHJDDCABEHIJFG $.

  $( Standard propositional axiom derived from Lukasiewicz axioms.
     (Contributed by NM, 22-Dec-2002.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  ax1 $p |- ( ph -> ( ps -> ph ) ) $=
    ( luklem5 ) ABC $.

  $( Standard propositional axiom derived from Lukasiewicz axioms.
     (Contributed by NM, 22-Dec-2002.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  ax2 $p |- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) $=
    ( wi luklem7 luklem8 luklem6 ax-mp luklem1 ) ABCDDBACDZDZABDZJDZABCEKLAJDZD
    ZMBJAFNJDOMDACGNJLFHII $.

  $( Standard propositional axiom derived from Lukasiewicz axioms.
     (Contributed by NM, 22-Dec-2002.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  ax3 $p |- ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) $=
    ( wn wi luklem2 luklem4 luklem1 ) ACZBCDHADADBADZDIHBAAEAIFG $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Derive Nicod's axiom from the standard axioms
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  Prove Nicod's axiom and implication and negation definitions.

$)

  $( This theorem "defines" implication in terms of 'nand'.  Analogous to
     ~ nanim .  In a pure (standalone) treatment of Nicod's axiom, this theorem
     would be changed to a definition ($a statement).  (Contributed by NM,
     11-Dec-2008.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  nic-dfim $p |- ( ( ( ph -/\ ( ps -/\ ps ) ) -/\ ( ph -> ps ) ) -/\
                   ( ( ( ph -/\ ( ps -/\ ps ) ) -/\ ( ph -/\ ( ps -/\ ps ) ) )
                                   -/\ ( ( ph -> ps ) -/\ ( ph -> ps ) ) ) ) $=
    ( wnan wi wb nanim bicomi nanbi mpbi ) ABBCCZABDZEJKCJJCKKCCCKJABFGJKHI $.

  $( This theorem "defines" negation in terms of 'nand'.  Analogous to
     ~ nannot .  In a pure (standalone) treatment of Nicod's axiom, this
     theorem would be changed to a definition ($a statement).  (Contributed by
     NM, 11-Dec-2008.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  nic-dfneg $p |- ( ( ( ph -/\ ph ) -/\ -. ph ) -/\
           ( ( ( ph -/\ ph ) -/\ ( ph -/\ ph ) ) -/\ ( -. ph -/\ -. ph ) ) ) $=
    ( wnan wn wb nannot bicomi nanbi mpbi ) AABZACZDIJBIIBJJBBBJIAEFIJGH $.

  ${
    $( Minor premise. $)
    nic-jmin $e |- ph $.
    $( Major premise. $)
    nic-jmaj $e |- ( ph -/\ ( ch -/\ ps ) ) $.
    $( Derive Nicod's rule of modus ponens using 'nand', from the standard one.
       Although the major and minor premise together also imply ` ch ` , this
       form is necessary for useful derivations from ~ nic-ax .  In a pure
       (standalone) treatment of Nicod's axiom, this theorem would be changed
       to an axiom ($a statement).  (Contributed by Jeff Hoffman, 19-Nov-2007.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    nic-mp $p |- ps $=
      ( wnan wa wi nannan mpbi simprd ax-mp ) ABDACBACBFFACBGHEACBIJKL $.

    $( A direct proof of ~ nic-mp .  (Contributed by NM, 30-Dec-2008.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    nic-mpALT $p |- ps $=
      ( wa wi wn wnan df-nan anbi2i xchbinx mpbi iman mpbir simprd ax-mp ) ABDA
      CBACBFZGARHZFZHZACBIZIZUAEUCAUBFTAUBJUBSACBJKLMARNOPQ $.
  $}

  $( Nicod's axiom derived from the standard ones.  See _Introduction to
     Mathematical Philosophy_ by B. Russell, p. 152.  Like ~ meredith , the
     usual axioms can be derived from this and vice versa.  Unlike ~ meredith ,
     Nicod uses a different connective ('nand'), so another form of modus
     ponens must be used in proofs, e.g., ` { ` ~ nic-ax , ~ nic-mp ` } ` is
     equivalent to ` { ` ~ luk-1 , ~ luk-2 , ~ luk-3 , ~ ax-mp ` } ` .  In a
     pure (standalone) treatment of Nicod's axiom, this theorem would be
     changed to an axiom ($a statement).  (Contributed by Jeff Hoffman,
     19-Nov-2007.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  nic-ax $p |- ( ( ph -/\ ( ch -/\ ps ) ) -/\ ( ( ta -/\ ( ta -/\ ta ) ) -/\
               ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) ) ) $=
    ( wnan wa wi nannan biimpi simpl imim2i wn imnan df-nan bitr4i imim2d con2b
    con3 mpbir 3bitr4ri imbitrrdi biimtrrid nanim sylib 3syl pm4.24 jctil ) ACB
    FFZEEEFFZDCFZADFZULFFZFFUIUJUMGHUIUMUJUIACBGZHZACHZUMUIUOACBIJUNCACBKLUPUKU
    LHUMUKDCMZHZUPULURDCGMUKDCNDCOPUPURDAMZHZULUPUQUSDACSQADMHADGMUTULADNDARADO
    UAUBUCUKULUDUEUFUJEEEGZHEVAEUGJEEEITUHUIUJUMIT $.

  $( A direct proof of ~ nic-ax .  (Contributed by NM, 11-Dec-2008.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  nic-axALT $p |- ( ( ph -/\ ( ch -/\ ps ) ) -/\ ( ( ta -/\ ( ta -/\ ta ) )
           -/\ ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) ) ) $=
    ( wnan wa wn anidm df-nan anbi2i notbii iman 3bitr4i bitr4i xchbinx anbi12i
    wi imnan mpbir simpl imim2i con3 imim2d biimpri jctil con2b bitr3i 3bitri
    syl ) ACBFZFZEEEFZFZDCFZADFZUPFZFZFZFULUSGZHZVAACBGZRZEEEGZRZDCHZRZDAHZRZRZ
    GZRZVCVJVEVCACRZVJVBCACBUAUBVMVFVHDACUCUDUJVDEEIUEUFVAVCVKHZGZHVLUTVOULVCUS
    VNAUKGZHAVBHZGZHULVCVPVRUKVQACBJKLAUKJAVBMNUSUNURGVKUNURJUNVEURVJEUMGZHEVDH
    ZGZHUNVEVSWAUMVTEEEJKLEUMJEVDMNUOUQGZHVGVIHZGZHURVJWBWDUOVGUQWCUODCGHVGDCJD
    CSOUQUPUPGZVIUPUPJWEUPADGHZVIUPIADJWFADHRVIADSADUGUHUIPQLUOUQJVGVIMNQPQLVCV
    KMOTULUSJT $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Derive the Lukasiewicz axioms from Nicod's axiom
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $( Minor premise. $)
    nic-imp.1 $e |- ( ph -/\ ( ch -/\ ps ) ) $.
    $( Inference for ~ nic-mp using ~ nic-ax as major premise.  (Contributed by
       Jeff Hoffman, 17-Nov-2007.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    nic-imp $p |- ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) $=
      ( wta wnan nic-ax nic-mp ) ACBGGDCGADGZJGGFFFGGEABCDFHI $.
  $}

  $( Lemma for ~ nic-id .  (Contributed by Jeff Hoffman, 17-Nov-2007.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  nic-idlem1 $p |- ( ( th -/\ ( ta -/\ ( ta -/\ ta ) ) ) -/\
                 ( ( ( ph -/\ ( ch -/\ ps ) ) -/\ th ) -/\
                   ( ( ph -/\ ( ch -/\ ps ) ) -/\ th ) ) ) $=
    ( wnan nic-ax nic-imp ) ACBFFACFAAFZIFFEEEFFDABCAEGH $.

  ${
    nic-idlem2.1 $e |- ( et -/\ ( ( ph -/\ ( ch -/\ ps ) ) -/\ th ) ) $.
    $( Lemma for ~ nic-id .  Inference used by ~ nic-id .  (Contributed by Jeff
       Hoffman, 17-Nov-2007.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    nic-idlem2 $p |- ( ( th -/\ ( ta -/\ ( ta -/\ ta ) ) ) -/\ et ) $=
      ( wnan nic-ax nic-imp nic-mp ) FACBHHZDHZHDEEEHHZHZFHZPGOMMFLACHAAHZQHHND
      ABCAEIJJK $.
  $}

  $( Theorem ~ id expressed with ` -/\ ` .  (Contributed by Jeff Hoffman,
     17-Nov-2007.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  nic-id $p |- ( ta -/\ ( ta -/\ ta ) ) $=
    ( wph wps wch wth wnan nic-ax nic-idlem2 nic-idlem1 nic-mp ) BCFZCBFZLFFZDD
    DFZFZFZCCCFFZFAAAFFZOEEEMDQCCCBEGHMNDPCORFKLLOAIHJ $.

  $( The connector ` -/\ ` is symmetric.  (Contributed by Jeff Hoffman,
     17-Nov-2007.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  nic-swap $p |- ( ( th -/\ ph ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) $=
    ( wta wnan nic-id nic-ax nic-mp ) AAADDBADABDZHDDCCCDDAEAAABCFG $.

  ${
    nic-isw1.1 $e |- ( th -/\ ph ) $.
    $( Inference version of ~ nic-swap .  (Contributed by Jeff Hoffman,
       17-Nov-2007.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    nic-isw1 $p |- ( ph -/\ th ) $=
      ( wnan nic-swap nic-mp ) BADABDZGCABEF $.
  $}

  ${
    nic-isw2.1 $e |- ( ps -/\ ( th -/\ ph ) ) $.
    $( Inference for swapping nested terms.  (Contributed by Jeff Hoffman,
       17-Nov-2007.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    nic-isw2 $p |- ( ps -/\ ( ph -/\ th ) ) $=
      ( wnan nic-swap nic-imp nic-mp nic-isw1 ) BACEZBCAEZEJBEZLDJKKBCAFGHI $.
  $}

  ${
    nic-iimp1.1 $e |- ( ph -/\ ( ch -/\ ps ) ) $.
    nic-iimp1.2 $e |- ( th -/\ ch ) $.
    $( Inference version of ~ nic-imp using right-handed term.  (Contributed by
       Jeff Hoffman, 17-Nov-2007.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    nic-iimp1 $p |- ( th -/\ ph ) $=
      ( wnan nic-imp nic-mp nic-isw1 ) DADCGADGZKFABCDEHIJ $.
  $}

  ${
    nic-iimp2.1 $e |- ( ( ph -/\ ps ) -/\ ( ch -/\ ch ) ) $.
    nic-iimp2.2 $e |- ( th -/\ ph ) $.
    $( Inference version of ~ nic-imp using left-handed term.  (Contributed by
       Jeff Hoffman, 17-Nov-2007.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    nic-iimp2 $p |- ( th -/\ ( ch -/\ ch ) ) $=
      ( wnan nic-isw1 nic-iimp1 ) CCGZBADJABGEHFI $.
  $}

  ${
    nic-idel.1 $e |- ( ph -/\ ( ch -/\ ps ) ) $.
    $( Inference to remove the trailing term.  (Contributed by Jeff Hoffman,
       17-Nov-2007.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    nic-idel $p |- ( ph -/\ ( ch -/\ ch ) ) $=
      ( wnan nic-id nic-isw1 nic-imp nic-mp ) CCEZCEAJEZKJCCFGABCJDHI $.
  $}

  ${
    nic-ich.1 $e |- ( ph -/\ ( ps -/\ ps ) ) $.
    nic-ich.2 $e |- ( ps -/\ ( ch -/\ ch ) ) $.
    $( Chained inference.  (Contributed by Jeff Hoffman, 17-Nov-2007.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    nic-ich $p |- ( ph -/\ ( ch -/\ ch ) ) $=
      ( wnan nic-isw1 nic-imp nic-mp ) CCFZBFAJFZKJBEGABBJDHI $.
  $}

  ${
    nic-idbl.1 $e |- ( ph -/\ ( ps -/\ ps ) ) $.
    $( Double the terms.  Since doubling is the same as negation, this can be
       viewed as a contraposition inference.  (Contributed by Jeff Hoffman,
       17-Nov-2007.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    nic-idbl $p |- ( ( ps -/\ ps ) -/\ ( ( ph -/\ ph ) -/\ ( ph -/\ ph ) ) ) $=
      ( wnan nic-imp nic-ich ) BBDABDAADABBBCEABBACEF $.
  $}

  $( Biconditional justification from Nicod's axiom.  For nic-* definitions,
     the biconditional connective is not used.  Instead, definitions are made
     based on this form. ~ nic-bi1 and ~ nic-bi2 are used to convert the
     definitions into usable theorems about one side of the implication.
     (Contributed by Jeff Hoffman, 18-Nov-2007.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  nic-bijust $p |- ( ( ta -/\ ta ) -/\ ( ( ta -/\ ta ) -/\ ( ta -/\ ta ) ) ) $=
    ( nic-swap ) AAB $.

  ${
    $( 'Biconditional' premise. $)
    nic-bi1.1 $e |- ( ( ph -/\ ps ) -/\ ( ( ph -/\ ph )
         -/\ ( ps -/\ ps ) ) ) $.
    $( Inference to extract one side of an implication from a definition.
       (Contributed by Jeff Hoffman, 18-Nov-2007.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    nic-bi1 $p |- ( ph -/\ ( ps -/\ ps ) ) $=
      ( wnan nic-id nic-iimp1 nic-isw2 nic-idel ) AABBAAABDBBDAADACAEFGH $.
  $}

  ${
    $( 'Biconditional' premise. $)
    nic-bi2.1 $e |- ( ( ph -/\ ps ) -/\ ( ( ph -/\ ph )
         -/\ ( ps -/\ ps ) ) ) $.
    $( Inference to extract the other side of an implication from a
       'biconditional' definition.  (Contributed by Jeff Hoffman, 18-Nov-2007.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    nic-bi2 $p |- ( ps -/\ ( ph -/\ ph ) ) $=
      ( wnan nic-isw2 nic-id nic-iimp1 nic-idel ) BBAABDZAADZBBDZBKIJCEBFGH $.
  $}

  ${
    $( Minor premise. $)
    nic-smin $e |- ph $.
    $( Major premise. $)
    nic-smaj $e |- ( ph -> ps ) $.
    $( Derive the standard modus ponens from ~ nic-mp .  (Contributed by Jeff
       Hoffman, 18-Nov-2007.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    nic-stdmp $p |- ps $=
      ( wi wnan nic-dfim nic-bi2 nic-mp ) ABBCABEZABBFFZKDKJABGHII $.
  $}

  $( Proof of ~ luk-1 from ~ nic-ax and ~ nic-mp (and Definitions ~ nic-dfim
     and ~ nic-dfneg ).  Note that the standard axioms ~ ax-1 , ~ ax-2 , and
     ~ ax-3 are proved from the Lukasiewicz axioms by Theorems ~ ax1 , ~ ax2 ,
     and ~ ax3 .  (Contributed by Jeff Hoffman, 18-Nov-2007.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  nic-luk1 $p |- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) ) $=
    ( wta wi nic-dfim nic-bi2 nic-ax nic-isw2 nic-idel nic-bi1 nic-idbl nic-imp
    wnan nic-swap nic-ich nic-mp ) ABEZBCEZACEZEZUANNZRUAEZUCRABBNNZUAUDRABFGUD
    STTNZNZUAUDCCNZBNZAUGNZUINZNZUFUDDDDNNZUKUKUDULABBUGDHIJUKUEUHNUFUEUJUJUHUI
    TUITACFKLMSUHUHUESBUGNZUHUMSBCFGUGBOPMPPUFUASTFKPPUBUCRUAFKQ $.

  $( Proof of ~ luk-2 from ~ nic-ax and ~ nic-mp .  (Contributed by Jeff
     Hoffman, 18-Nov-2007.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  nic-luk2 $p |- ( ( -. ph -> ph ) -> ph ) $=
    ( wn wi wnan nic-dfim nic-bi2 nic-dfneg nic-iimp1 nic-isw2 nic-isw1 nic-bi1
    nic-id nic-mp ) ABZACZAADZDZOACZROPONPDZSPSONAEFNPPPNDNNDPPDPAGPLHIHJQROAEK
    M $.

  $( Proof of ~ luk-3 from ~ nic-ax and ~ nic-mp .  (Contributed by Jeff
     Hoffman, 18-Nov-2007.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  nic-luk3 $p |- ( ph -> ( -. ph -> ps ) ) $=
    ( wnan nic-dfim nic-bi1 nic-dfneg nic-bi2 nic-id nic-iimp1 nic-iimp2 nic-mp
    wn wi ) AALZBMZOCCZAOMZQNBBCZOANRCONBDENAACZSASNAFGAHIJPQAODEK $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( This alternative axiom for propositional calculus using the Sheffer Stroke
     was discovered by Lukasiewicz in his Selected Works.  It improves on
     Nicod's axiom by reducing its number of variables by one.

     This axiom also uses ~ nic-mp for its constructions.

     Here, the axiom is proved as a substitution instance of ~ nic-ax .
     (Contributed by Anthony Hart, 31-Jul-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  lukshef-ax1 $p |- ( ( ph -/\ ( ch -/\ ps ) ) -/\ ( ( th -/\ ( th -/\ th ) )
          -/\ ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) ) ) $=
    ( nic-ax ) ABCDDE $.

  $( Lemma for ~ renicax .  (Contributed by NM, 31-Jul-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  lukshefth1 $p |- ( ( ( ( ta -/\ ps ) -/\ ( ( ph -/\ ta ) -/\ ( ph
          -/\ ta ) ) ) -/\ ( th -/\ ( th -/\ th ) ) ) -/\ ( ph -/\ ( ps
          -/\ ch ) ) ) $=
    ( wnan lukshef-ax1 nic-mp ) ABCFFZEEEFFZEBFAEFZKFFZFZFZLDDDFFZFZIFZQACBEGPM
    MFFZNQQFFIIIFFJODEFEDFZSFFZFFRLLLFFEEEDGJTOLGHPMMIGHH $.

  $( Lemma for ~ renicax .  (Contributed by NM, 31-Jul-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  lukshefth2 $p |- ( ( ta -/\ th ) -/\ ( ( th -/\ ta ) -/\ ( th
          -/\ ta ) ) ) $=
    ( wps wch wph wnan lukshef-ax1 nic-mp lukshefth1 ) AAAFFZBAFABFZKFFBBBFFAJF
    ZCDEFFZAFZNFFZJBEFEBFZPFFZMJADFCAFZRFFZFFOJCEDAGMSJAGHQJFZEEEFFZFZOTFZUCEEE
    ABIOUAENFLEFZUDFFZFFUBUCUCFFTTTFFLNNEGOUEUATGHHHAAABGH $.

  $( A rederivation of ~ nic-ax from ~ lukshef-ax1 , proving that ~ lukshef-ax1
     with ~ nic-mp can be used as a complete axiomatization of propositional
     calculus.  (Contributed by Anthony Hart, 31-Jul-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  renicax $p |- ( ( ph -/\ ( ch -/\ ps ) ) -/\ ( ( ta -/\ ( ta -/\ ta ) )
          -/\ ( ( th -/\ ch ) -/\ ( ( ph -/\ th ) -/\ ( ph -/\ th ) ) ) ) ) $=
    ( wnan lukshefth1 lukshefth2 nic-mp lukshef-ax1 ) EEEFFZDCFADFZLFFZFZACBFFZ
    FZONFZQOMKFZFZPPROFSSACBEDGORHINRRFFSPPFFOOOFFMKHNRROJIIONHI $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Justification for ~ tbw-negdf .  (Contributed by Anthony Hart,
     15-Aug-2011.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  tbw-bijust $p |- ( ( ph <-> ps ) <->
                      ( ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) -> F. ) ) $=
    ( wb wi wn wfal dfbi1 pm2.21 imim2i falim impbii notbii ax-1 pm2.43i 3bitri
    id ja ) ABCABDZBADZEZDZERSFDZDZEZUCFDZABGUAUCUAUCTUBRSFHIUBTRSFTTPTJQIKLUDU
    EUCFHUEUDUCFUEUDDZUDUEMUFJQNKO $.

  $( The definition of negation, in terms of ` -> ` and ` F. ` .  (Contributed
     by Anthony Hart, 15-Aug-2011.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  tbw-negdf $p |- ( ( ( -. ph -> ( ph -> F. ) )
    -> ( ( ( ph -> F. ) -> -. ph ) -> F. ) ) -> F. ) $=
    ( wn wfal wi wb pm2.21 ax-1 falim ja pm2.43i impbii tbw-bijust mpbi ) ABZAC
    DZENODONDZCDDCDNOACFONACPNOGPHIJKNOLM $.

  $( The first of four axioms in the Tarski-Bernays-Wajsberg system.
     (Contributed by Anthony Hart, 13-Aug-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  tbw-ax1 $p |- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) ) $=
    ( imim1 ) ABCD $.

  $( The second of four axioms in the Tarski-Bernays-Wajsberg system.
     (Contributed by Anthony Hart, 13-Aug-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  tbw-ax2 $p |- ( ph -> ( ps -> ph ) ) $=
    ( ax-1 ) ABC $.

  $( The third of four axioms in the Tarski-Bernays-Wajsberg system.
     (Contributed by Anthony Hart, 13-Aug-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  tbw-ax3 $p |- ( ( ( ph -> ps ) -> ph ) -> ph ) $=
    ( peirce ) ABC $.

  $( The fourth of four axioms in the Tarski-Bernays-Wajsberg system.

     This axiom was added to the Tarski-Bernays axiom system (see ~ tb-ax1 ,
     ~ tb-ax2 , and ~ tb-ax3 ) by Wajsberg for completeness.  (Contributed by
     Anthony Hart, 13-Aug-2011.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  tbw-ax4 $p |- ( F. -> ph ) $=
    ( falim ) AB $.

  ${
    tbwsyl.1 $e |- ( ph -> ps ) $.
    tbwsyl.2 $e |- ( ps -> ch ) $.
    $( Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'.
       (Contributed by Anthony Hart, 16-Aug-2011.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    tbwsyl $p |- ( ph -> ch ) $=
      ( wi tbw-ax1 ax-mp ) BCFZACFZEABFIJFDABCGHH $.
  $}

  $( Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'.
     (Contributed by Anthony Hart, 16-Aug-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  tbwlem1 $p |- ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) ) $=
    ( wi tbw-ax1 tbw-ax2 tbwsyl tbw-ax3 ax-mp ) ABCDZDJCDZACDZDZBLDZAJCEBKDMNDB
    JKDZKBJBDOBJFJBCEGOKCDKDKJKCEKCHGGBKLEIG $.

  $( Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'.
     (Contributed by Anthony Hart, 16-Aug-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  tbwlem2 $p |- ( ( ph -> ( ps -> F. ) ) -> ( ( ( ph -> ch ) -> th )
    -> ( ps -> th ) ) ) $=
    ( wfal wi tbw-ax1 tbw-ax4 tbwlem1 ax-mp tbwsyl ) ABEFZFZBACFZFZNDFBDFFMLCFZ
    NFZOALCGBPFZQOFLBCFZFZRECFZTCHLUASFFUATFBECGLUASIJJLBCIJBPNGJKBNDGK $.

  $( Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'.
     (Contributed by Anthony Hart, 16-Aug-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  tbwlem3 $p |- ( ( ( ( ( ph -> F. ) -> ph ) -> ph ) -> ps ) -> ps ) $=
    ( wfal wi tbw-ax3 tbw-ax2 tbw-ax1 tbwsyl ax-mp ) ACDADADZBDZKBDZDZLJMACEJKJ
    DMJKFKJBGHIMLBDLDLKLBGLBEHI $.

  $( Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'.
     (Contributed by Anthony Hart, 16-Aug-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  tbwlem4 $p |- ( ( ( ph -> F. ) -> ps ) -> ( ( ps -> F. ) -> ph ) ) $=
    ( wfal wi tbw-ax4 tbw-ax1 tbwlem1 ax-mp tbwlem2 tbwlem3 tbwsyl ) ACDZBDZLBC
    DZCDZDZNADZBODZMPDZNNDZRCCDZTCENUANDDUATDBCCFNUANGHHNBCGHMRPDDRSDLBOFMRPGHH
    PLADADQDQLNAAIAQJKK $.

  $( Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'.
     (Contributed by Anthony Hart, 16-Aug-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  tbwlem5 $p |- ( ( ( ph -> ( ps -> F. ) ) -> F. ) -> ph ) $=
    ( wfal wi tbw-ax2 tbw-ax1 tbwsyl tbwlem1 ax-mp tbwlem4 ) ACDZABCDZDZDZMCDAD
    AKLDZDNABADOABEBACFGAKLHIAMJI $.

  $( ~ luk-1 derived from the Tarski-Bernays-Wajsberg axioms.  (Contributed by
     Anthony Hart, 16-Aug-2011.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  re1luk1 $p |- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) ) $=
    ( tbw-ax1 ) ABCD $.

  $( ~ luk-2 derived from the Tarski-Bernays-Wajsberg axioms.  (Contributed by
     Anthony Hart, 16-Aug-2011.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  re1luk2 $p |- ( ( -. ph -> ph ) -> ph ) $=
    ( wn wi wfal tbw-negdf tbw-ax2 tbwlem4 ax-mp tbw-ax1 tbw-ax3 tbwsyl ) ABZAC
    ZADCZACZANLCZMOCLNCZPDCZCZDCZPAERSCTPCRQFPSGHHNLAIHADJK $.

  $( ~ luk-3 derived from the Tarski-Bernays-Wajsberg axioms.

     This theorem, along with ~ re1luk1 and ~ re1luk2 proves that ~ tbw-ax1 ,
     ~ tbw-ax2 , ~ tbw-ax3 , and ~ tbw-ax4 , with ~ ax-mp can be used as a
     complete axiom system for all of propositional calculus.  (Contributed by
     Anthony Hart, 16-Aug-2011.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  re1luk3 $p |- ( ph -> ( -. ph -> ps ) ) $=
    ( wfal wi wn tbw-ax4 tbw-ax1 tbwlem1 ax-mp tbw-negdf tbwlem5 tbwsyl ) AACDZ
    BDZAEZBDZMABDZDZANDCBDZRBFMSQDDSRDACBGMSQHIIMABHIOMDZNPDTMODZCDDCDTAJTUAKIO
    MBGIL $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( A single axiom for propositional calculus discovered by C. A. Meredith.

     This axiom is worthy of note, due to it having only 19 symbols, not
     counting parentheses.  The more well-known ~ meredith has 21 symbols, sans
     parentheses.

     See ~ merco2 for another axiom of equal length.  (Contributed by Anthony
     Hart, 13-Aug-2011.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  merco1 $p |- ( ( ( ( ( ph -> ps ) -> ( ch -> F. ) ) -> th ) -> ta )
         -> ( ( ta -> ph ) -> ( ch -> ph ) ) ) $=
    ( wi wfal wn ax-1 falim ja imim2i imim1i meredith syl ) ABFZCGFZFZDFZEFPDHZ
    CHZFZFZDFZEFEAFCAFFUDSERUCDQUBPCGUBUATIUBJKLMMABDCENO $.

  $( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
     (Contributed by Anthony Hart, 17-Sep-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  merco1lem1 $p |- ( ph -> ( F. -> ch ) ) $=
    ( wfal wi merco1 ax-mp ) AACADZDZDZACBDZDZHGDZHDZIGCDACDZDZGDHDZMGNDZNDZGDO
    DPCAANGEGNAGOEFGCAGHEFHCDZNDZGDLDZMIDQSDHDTDUACAASHEGNHHTEFHCAGLEFFHJDZKDZI
    KDZJCDNDZGDHDZUCRJDUEDUFCAANJEGNAJUEEFJCAGHEFKCDICDZDZJDUBDZUCUDDJUGDSDKDUH
    DUICBISKEJUGHKUHEFKCIJUBEFFF $.

  $( ~ tbw-ax4 rederived from ~ merco1 .  (Contributed by Anthony Hart,
     17-Sep-2011.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  retbwax4 $p |- ( F. -> ph ) $=
    ( wfal wi merco1lem1 ax-mp ) ABACZCZFAADGADE $.

  $( ~ tbw-ax2 rederived from ~ merco1 .  (Contributed by Anthony Hart,
     17-Sep-2011.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  retbwax2 $p |- ( ph -> ( ps -> ph ) ) $=
    ( wi wfal merco1lem1 merco1 ax-mp ) AAAACZCZCZABACZCZDACZHCZICZJHACADCZCACZ
    MCOQAEHAAAMFGIPCPCDCNCOJCAHAPDFIPADNFGGMKCZLCZJLCZKACPCACZMCSUAAEKAAAMFGLBD
    CZCJDCZCDCRCSTCAKBUCDFLUBJDRFGGG $.

  $( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
     (Contributed by Anthony Hart, 17-Sep-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  merco1lem2 $p |- ( ( ( ph -> ps ) -> ch ) -> ( ( ( ps -> ta ) -> ( ph ->
    F. ) ) -> ch ) ) $=
    ( wi wfal retbwax2 merco1 ax-mp ) CAEZBDEAFEEZFEZEZBEABEZEZNCEKCEELMEOLJGBD
    AFMHICAKBNHI $.

  $( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
     (Contributed by Anthony Hart, 17-Sep-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  merco1lem3 $p |- ( ( ( ph -> ps ) -> ( ch -> F. ) ) -> ( ch -> ph ) ) $=
    ( wi wfal merco1lem2 retbwax2 ax-mp ) AAADZAEDDZIDZDZABDCEDDZCADZDZIEDJEDDZ
    LAAEAFKLDPLDKAGJILEFHHNEDMEDDZLODZCAEBFORDQRDOLGMNREFHHH $.

  $( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
     (Contributed by Anthony Hart, 17-Sep-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  merco1lem4 $p |- ( ( ( ph -> ps ) -> ch ) -> ( ps -> ch ) ) $=
    ( wi wfal merco1lem3 merco1 ax-mp ) CADZBEDZDZBDABDZDZLCDBCDDJAEDZDIEDZDKDM
    JNIFBEAOKGHCABBLGH $.

  $( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
     (Contributed by Anthony Hart, 17-Sep-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  merco1lem5 $p |- ( ( ( ( ph -> F. ) -> ch ) -> ta ) -> ( ph -> ta ) ) $=
    ( wi wfal merco1lem4 merco1 ax-mp ) CADZAEDZDBDJBDZDKCDACDDIJBFCAABKGH $.

  $( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
     (Contributed by Anthony Hart, 17-Sep-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  merco1lem6 $p |- ( ( ph -> ( ph -> ps ) ) -> ( ch -> ( ph -> ps ) ) ) $=
    ( wi wfal merco1lem5 merco1lem3 ax-mp merco1 ) ABDZEDCEDZDZEDZADZAJDCJDDJME
    DZDZNLODZPOEDMDQLEEFOELGHJKOFHABMGHJECEAIH $.

  $( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
     (Contributed by Anthony Hart, 17-Sep-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  merco1lem7 $p |- ( ph -> ( ( ( ps -> ch ) -> ps ) -> ps ) ) $=
    ( wi wfal merco1lem5 merco1 ax-mp merco1lem6 ) BCDZBDZKBDZDZALDBEDKEDZDCDJD
    MBNCFBEKCJGHKBAIH $.

  $( ~ tbw-ax3 rederived from ~ merco1 .  (Contributed by Anthony Hart,
     17-Sep-2011.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  retbwax3 $p |- ( ( ( ph -> ps ) -> ph ) -> ph ) $=
    ( wi retbwax2 merco1lem7 ax-mp ) AAACCZABCACACAADGABEF $.

  $( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
     (Contributed by Anthony Hart, 17-Sep-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  merco1lem8 $p |- ( ph -> ( ( ps -> ( ps -> ch ) ) -> ( ps -> ch ) ) ) $=
    ( wi merco1lem6 ax-mp ) BBCDZDZHGDZDAIDBCHEHGAEF $.

  $( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
     (Contributed by Anthony Hart, 18-Sep-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  merco1lem9 $p |- ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) ) $=
    ( wfal wi merco1lem8 ax-mp ) CADZAABDZDHDZDZIGABEJABEF $.

  $( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
     (Contributed by Anthony Hart, 18-Sep-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  merco1lem10 $p |- ( ( ( ( ( ph -> ps ) -> ch ) -> ( ta -> ch ) ) -> ph ) ->
    ( th -> ph ) ) $=
    ( wi wfal merco1 merco1lem2 ax-mp ) ABFZDGFZFCAFEGFFAFZGFZFKCFECFFZFZOAFDAF
    FMKFOFPCAEAKHMKOLIJABDNOHJ $.

  $( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
     (Contributed by Anthony Hart, 18-Sep-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  merco1lem11 $p |- ( ( ph -> ps ) -> ( ( ( ch -> ( ph -> ta ) ) -> F. ) -> ps
    ) ) $=
    ( wi wfal merco1lem5 merco1lem3 ax-mp merco1lem4 merco1 merco1lem2 ) ADEZBA
    EZCMEZFEZFEZEZFEZFEZEZABEPBEEZOTEZUAQTEZUCRTEZUDTFESEUERFFGTFRHINQTJIOFTGIC
    MTJISAEUBEUAUBEBAPFAKSAUBDLII $.

  $( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
     (Contributed by Anthony Hart, 18-Sep-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  merco1lem12 $p |- ( ( ph -> ps ) -> ( ( ( ch -> ( ph -> ta ) ) -> ph ) -> ps
    ) ) $=
    ( wi wfal merco1lem3 merco1 ax-mp merco1lem9 merco1lem11 ) BAEZCADEZEZAEZFE
    ZEFEAEZABEOBEEOAEZQOREZRMPECFEZENESMPCGADOTNHIOAJIOALFKIBAOFAHI $.

  $( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
     (Contributed by Anthony Hart, 18-Sep-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  merco1lem13 $p |- ( ( ( ( ph -> ps ) -> ( ch -> ps ) ) -> ta ) -> ( ph ->
    ta ) ) $=
    ( wi wfal merco1 merco1lem4 ax-mp merco1lem12 ) DAEZAFEEAEABECBEEZEZLDEADEE
    ALEZMBAECFEEAEZAELENBACAAGOALHIALKFJIDAAALGI $.

  $( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
     (Contributed by Anthony Hart, 18-Sep-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  merco1lem14 $p |- ( ( ( ( ph -> ps ) -> ps ) -> ch ) -> ( ph -> ch ) ) $=
    ( wi wfal merco1lem13 merco1lem8 merco1 ax-mp merco1lem9 merco1lem12 ) CADZ
    AEDDADABDZBDZDZNCDACDDANDZOMNDNDZPDZPABMNFRRPDZDZSPADREDDADZQDTUAMBGPARAQHI
    RPJIIANLEKICAAANHI $.

  $( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
     (Contributed by Anthony Hart, 18-Sep-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  merco1lem15 $p |- ( ( ph -> ps ) -> ( ph -> ( ch -> ps ) ) ) $=
    ( wi merco1lem14 merco1lem13 ax-mp ) ABDZBDCBDZDAIDZDHJDABIEHBCJFG $.

  $( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
     (Contributed by Anthony Hart, 18-Sep-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  merco1lem16 $p |- ( ( ( ph -> ( ps -> ch ) ) -> ta ) -> ( ( ph -> ch ) -> ta
    ) ) $=
    ( wi wfal merco1lem15 merco1lem11 ax-mp merco1 ) DAEZACEZFEEFEABCEEZEZMDELD
    EELMENACBGLMKFHIDALFMJI $.

  $( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
     (Contributed by Anthony Hart, 18-Sep-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  merco1lem17 $p |- ( ( ( ( ( ph -> ps ) -> ph ) -> ch ) -> ta ) -> ( ( ph ->
    ch ) -> ta ) ) $=
    ( wi merco1lem11 merco1lem7 merco1 ax-mp merco1lem9 merco1lem16 merco1lem4
    wfal ) DAEZACEZMEZECEZABEAEZCEZEZSDEODEEPCEZSEZTPOEZSEZUBOSEZUDCAEZRMEEMEAE
    ZUERAEZUGEZUGRAUFMFUIUIUGEZEZUJUGAEUIMEEAEZUHEUKULABGUGAUIAUHHIUIUGJIICARMA
    HISAEZUCMEEMEOEZUEUDEUCOEZUNEZUNUCOUMMFUPUPUNEZEZUQUNAEUPMEEAEZUOEURUSOMGUN
    AUPAUOHIUPUNJIISAUCMOHIIPACSKIUMQMEEMEUAEZUBTEQUAEUTNPCLQUAUMMFISAQMUAHIIDA
    OCSHI $.

  $( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco1 .
     (Contributed by Anthony Hart, 18-Sep-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  merco1lem18 $p |- ( ( ph -> ( ps -> ch ) ) -> ( ( ps -> ph ) -> ( ps ->
    ch ) ) ) $=
    ( wfal merco1 merco1lem17 ax-mp merco1lem5 merco1lem3 merco1lem4 merco1lem2
    wi merco1lem9 ) BALZABCLZLZNOLZLZLZROBLZALRLZSTNDLZLTLALRLUAOBNTAETUBARFGBC
    ARFGSSRLZLZUCQRDLSDLZLZDLZDLZLZUDRUHLZUIUFUHLZUJUHDLUGLUKUFDDHUHDUFIGRUEUHH
    GPQUHJGUGNLUDLUIUDLRDSDNEUGNUDOKGGSRMGG $.

  $( ~ tbw-ax1 rederived from ~ merco1 .

     This theorem, along with ~ retbwax2 , ~ retbwax3 , and ~ retbwax4 , shows
     that ~ merco1 with ~ ax-mp can be used as a complete axiomatization of
     propositional calculus.  (Contributed by Anthony Hart, 18-Sep-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  retbwax1 $p |- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) ) $=
    ( wi merco1lem18 merco1lem16 ax-mp merco1lem15 merco1lem14 wfal merco1lem10
    merco1 merco1lem9 merco1lem13 ) BCDZABDZACDZDZDZPOQDZDZBQDRDSBACEBACRFGOSUA
    DZDZUBSRDUBDZUCRUBDZUDRUADUEPQOHRUASHGRSUAEGORUBIGUCUBDZJDZUADZUFUGTDZUHUFQ
    DZTDZUIOUBQITADZUGJDZDZQDUJDZUKUIDQADZUGDZUMDUNDZUOUMJDULJDDUGDUQDURUGJJUPU
    LKUMJULUGUQLGQAUFUMUNLGTAUGQUJLGGUGTPHGUHUBDUFDZUFDZUHUFDUSUTDUTUFJUAUSSKUS
    UFMGUHUBUCUFNGGGG $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( A single axiom for propositional calculus discovered by C. A. Meredith.

     This axiom has 19 symbols, sans auxiliaries.  See notes in ~ merco1 .
     (Contributed by Anthony Hart, 7-Aug-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  merco2 $p |- ( ( ( ph -> ps ) -> ( ( F. -> ch ) -> th ) ) -> ( ( th
         -> ph ) -> ( ta -> ( et -> ph ) ) ) ) $=
    ( wi wfal falim pm2.04 mpi jarl idd jad looinv 3syl a1dd a1i com4l ) FABGZH
    CGZDGGZDAGZEAUBUCEAGGGFUBUCAEUBTDGZADGDGUCAGUBUAUDCITUADJKUDADDABDLUDDMNADO
    PQRS $.

  $( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco2 .
     (Contributed by Anthony Hart, 16-Aug-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  mercolem1 $p |- ( ( ( ph -> ps ) -> ch ) -> ( ps -> ( th -> ch ) ) ) $=
    ( wi wfal merco2 ax-mp ) AAEZFAEZAEEIAIEEEZABEZCEZBDCEZEZEZAAAAAAGZKKPEZQCA
    EZJLEZEZPEZKREZCAALBDGPTEJUAEEZUBUCETOEJPEEZUDOJFEZEFBETEEUEBNAFJAGOUFBTJMG
    HTOAPJSGHPTAUAKKGHHHH $.

  $( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco2 .
     (Contributed by Anthony Hart, 16-Aug-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  mercolem2 $p |- ( ( ( ph -> ps ) -> ph ) -> ( ch -> ( th -> ph ) ) ) $=
    ( wi wfal merco2 ax-mp ) AAEZFAEZAEEIAIEEEZABEZAEZCDAEEZEZAAAAAAGZKKOEZPIJL
    EZEZOEZKQEZAAALCDGOREJSEEZTUAERNEJOEEZUBNLEJREEZUCLJFEZEJNEEUDABAFCDGLUEANJ
    JGHNLARJMGHRNAOJIGHORASKKGHHHH $.

  $( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco2 .
     (Contributed by Anthony Hart, 16-Aug-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  mercolem3 $p |- ( ( ps -> ch ) -> ( ps -> ( ph -> ch ) ) ) $=
    ( wi wfal merco2 mercolem2 ax-mp ) AADZEADZADDIAIDDDZBCDZBACDZDZDZAAAAAAFZK
    KODZPCADZJBDZDZODZKQDZCAABBAFOSDJTDDZUAUBDSNDJODDZUCNBDJSDDUDBMJJGNBASJLFHS
    NAOJRFHOSATKKFHHHH $.

  $( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco2 .
     (Contributed by Anthony Hart, 16-Aug-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  mercolem4 $p |- ( ( th -> ( et -> ph ) ) -> ( ( ( th -> ch )
  -> ph ) -> ( ta -> ( et -> ph ) ) ) ) $=
    ( wi wfal merco2 mercolem1 ax-mp mercolem3 ) AAFZGAFZAFFLALFFFZCEAFZFZCBFZA
    FZDOFFZFZAAAAAAHZNNTFZUAOAFZMCFFZTFZNUBFZOAACRDHTCFZMUDFFZUEUFFUGUDFZUHQMTF
    FZUIMQFZTFZUJLUKFSFULAAAQDEHLUKSPIJMQTMIJCBATUCMHJMUGUDKJTCAUDNNHJJJJ $.

  $( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco2 .
     (Contributed by Anthony Hart, 16-Aug-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  mercolem5 $p |- ( th -> ( ( th -> ph ) -> ( ta -> ( ch -> ph ) ) ) ) $=
    ( wi wfal merco2 mercolem1 ax-mp mercolem2 ) AAEZFAEZAEEKAKEEEZCCAEDBAEEEZE
    ZAAAAAAGZMMOEZPLCEZOEZMQEZKRENESAAACDBGKRNCHIOCELREESTECNLLJOCARMMGIIII $.

  $( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco2 .
     (Contributed by Anthony Hart, 16-Aug-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  mercolem6 $p |- ( ( ph -> ( ps -> ( ph -> ch ) ) )
  -> ( ps -> ( ph -> ch ) ) ) $=
    ( wi wfal merco2 mercolem1 ax-mp mercolem5 mercolem4 ) AADZEADADDKAKDDDZABA
    CDZDZDZNDZAAAAAAFZLLPDZQLLRDZQORDZLSDZLTQMRDZLTDZAODZMDPDUBAOMBGUDMPLGHATDU
    BUCDNOALIRCALOJHHHLTUADZQPUADZLUEDZALDZPDSDUFALPLGUHPSLGHOUEDUFUGDRLOLIUANO
    LTJHHHHHHH $.

  $( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco2 .
     (Contributed by Anthony Hart, 16-Aug-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  mercolem7 $p |- ( ( ph -> ps ) -> ( ( ( ph -> ch )
  -> ( th -> ps ) ) -> ( th -> ps ) ) ) $=
    ( wi wfal merco2 mercolem3 mercolem6 ax-mp mercolem5 mercolem4 ) AAEZFAEAEE
    MAMEEEZABEZACEZDBEZEZQEZEZAAAAAAGPSEZNTEZRUAEUARPQHRPQIJATEUAUBEBDARKSCANOL
    JJJ $.

  $( Used to rederive the Tarski-Bernays-Wajsberg axioms from ~ merco2 .
     (Contributed by Anthony Hart, 16-Aug-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  mercolem8 $p |- ( ( ph -> ps ) -> ( ( ps -> ( ph -> ch ) )
  -> ( ta -> ( th -> ( ph -> ch ) ) ) ) ) $=
    ( wi wfal merco2 mercolem3 ax-mp mercolem7 ) AAFZGAFZAFFLALFFFZABFZBACFZFED
    PFFFZFZAAAAAAHZNNRFZSPMBFZFUAFZRFZNTFZUBQFUCPUAABEDHOUBQIJRMUBFZFUEFZUCUDFO
    UBFUFABCMKOUBQMKJRUEAUBNNHJJJJ $.

  $( ~ tbw-ax1 rederived from ~ merco2 .  (Contributed by Anthony Hart,
     16-Aug-2011.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  re1tbw1 $p |- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) ) $=
    ( wi mercolem3 mercolem8 mercolem6 ax-mp ) BCDZABDZIACDZDZDZDZMJNDZNIBKDZDZ
    OABCEPNDZQODZPMDZRJTDTABCIJFJPLGHIPMEHQRSDZDUAIPMJQFQROGHHHJILGHIJKGH $.

  $( ~ tbw-ax2 rederived from ~ merco2 .  (Contributed by Anthony Hart,
     16-Aug-2011.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  re1tbw2 $p |- ( ph -> ( ps -> ph ) ) $=
    ( wi mercolem1 ax-mp mercolem6 ) BABACZCZCZHAICZIAACZACHCJAAABDKAHBDEABGFEB
    AAFE $.

  $( ~ tbw-ax3 rederived from ~ merco2 .  (Contributed by Anthony Hart,
     16-Aug-2011.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  re1tbw3 $p |- ( ( ( ph -> ps ) -> ph ) -> ph ) $=
    ( wi mercolem2 mercolem6 ax-mp ) AACZACAGCCZABCACZACZAAAADIHJCZCKABHIDIHAEF
    F $.

  $( ~ tbw-ax4 rederived from ~ merco2 .

     This theorem, along with ~ re1tbw1 , ~ re1tbw2 , and ~ re1tbw3 , shows
     that ~ merco2 , along with ~ ax-mp , can be used as a complete
     axiomatization of propositional calculus.  (Contributed by Anthony Hart,
     16-Aug-2011.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  re1tbw4 $p |- ( F. -> ph ) $=
    ( wi wfal re1tbw3 re1tbw2 re1tbw1 ax-mp mercolem3 merco2 ) AABZCABZJABZABZJ
    AADALBMJBAJEALAFGGZJJKBZNKKBZJOBZKABZKBZKBZPKADKSBTPBKREKSKFGGRPBPQBCKAHKAA
    KJJIGGGG $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Justification for ~ rb-imdf .  (Contributed by Anthony Hart, 17-Aug-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  rb-bijust $p |- ( ( ph <-> ps ) <->
                           -. ( -. ( -. ph \/ ps ) \/ -. ( -. ps \/ ph ) ) ) $=
    ( wb wi wn wo dfbi1 imor notbii imbi12i pm4.62 3bitri ) ABCABDZBADZEZDZEAEB
    FZBEAFZEZDZEQESFZEABGPTMQOSABHNRBAHIJITUAQRKIL $.

  $( The definition of implication, in terms of ` \/ ` and ` -. ` .
     (Contributed by Anthony Hart, 17-Aug-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  rb-imdf $p |- -. ( -. ( -. ( ph -> ps ) \/ ( -. ph \/ ps ) )
    \/ -. ( -. ( -. ph \/ ps ) \/ ( ph -> ps ) ) ) $=
    ( wi wn wo wb imor rb-bijust mpbi ) ABCZADBEZFJDKEDKDJEDEDABGJKHI $.

  ${
    anmp.min $e |- ph $.
    anmp.maj $e |- ( -. ph \/ ps ) $.
    $( Modus ponens for ` { \/ , -. } ` axiom systems.  (Contributed by Anthony
       Hart, 12-Aug-2011.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    anmp $p |- ps $=
      ( imorri ax-mp ) ABCABDEF $.
  $}

  $( The first of four axioms in the Russell-Bernays axiom system.
     (Contributed by Anthony Hart, 13-Aug-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  rb-ax1 $p |- ( -. ( -. ps \/ ch ) \/ ( -. ( ph \/ ps ) \/ ( ph \/ ch ) ) ) $=
    ( wn wo wi orim2 imor 3imtr3i imori ) BDCEZABEZDACEZEZBCFLMFKNABCGBCHLMHIJ
    $.

  $( The second of four axioms in the Russell-Bernays axiom system.
     (Contributed by Anthony Hart, 13-Aug-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  rb-ax2 $p |- ( -. ( ph \/ ps ) \/ ( ps \/ ph ) ) $=
    ( wo wn pm1.4 con3i con1i orri ) ABCZDZBACZKJIKABEFGH $.

  $( The third of four axioms in the Russell-Bernays axiom system.
     (Contributed by Anthony Hart, 13-Aug-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  rb-ax3 $p |- ( -. ph \/ ( ps \/ ph ) ) $=
    ( wn wo pm2.46 con1i orri ) ACZBADZIHBAEFG $.

  $( The fourth of four axioms in the Russell-Bernays axiom system.
     (Contributed by Anthony Hart, 13-Aug-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  rb-ax4 $p |- ( -. ( ph \/ ph ) \/ ph ) $=
    ( wo wn pm1.2 con3i con1i orri ) AABZCZAAIHAADEFG $.

  ${
    rbsyl.1 $e |- ( -. ps \/ ch ) $.
    rbsyl.2 $e |- ( ph \/ ps ) $.
    $( Used to rederive the Lukasiewicz axioms from Russell-Bernays'.
       (Contributed by Anthony Hart, 18-Aug-2011.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    rbsyl $p |- ( ph \/ ch ) $=
      ( wo wn rb-ax1 anmp ) ABFZACFZEBGCFJGKFDABCHII $.
  $}

  ${
    rblem1.1 $e |- ( -. ph \/ ps ) $.
    rblem1.2 $e |- ( -. ch \/ th ) $.
    $( Used to rederive the Lukasiewicz axioms from Russell-Bernays'.
       (Contributed by Anthony Hart, 18-Aug-2011.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    rblem1 $p |- ( -. ( ph \/ ch ) \/ ( ps \/ th ) ) $=
      ( wo wn rb-ax1 anmp rb-ax2 rbsyl ) ACGHZBCGZBDGZCHDGNHOGFBCDIJMCBGZNCBKMC
      AGZPAHBGQHPGECABIJACKLLL $.
  $}

  $( Used to rederive the Lukasiewicz axioms from Russell-Bernays'.
     (Contributed by Anthony Hart, 18-Aug-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  rblem2 $p |- ( -. ( ch \/ ph ) \/ ( ch \/ ( ph \/ ps ) ) ) $=
    ( wn wo rb-ax2 rb-ax3 rbsyl rb-ax1 anmp ) ADZABEZECAEDCLEEKBAELBAFABGHCALIJ
    $.

  $( Used to rederive the Lukasiewicz axioms from Russell-Bernays'.
     (Contributed by Anthony Hart, 18-Aug-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  rblem3 $p |- ( -. ( ch \/ ph ) \/ ( ( ch \/ ps ) \/ ph ) ) $=
    ( wo wn rb-ax2 rblem2 rbsyl ) CADEZACBDZDZJADAJFIACDKCBAGCAFHH $.

  ${
    rblem4.1 $e |- ( -. ph \/ th ) $.
    rblem4.2 $e |- ( -. ps \/ ta ) $.
    rblem4.3 $e |- ( -. ch \/ et ) $.
    $( Used to rederive the Lukasiewicz axioms from Russell-Bernays'.
       (Contributed by Anthony Hart, 18-Aug-2011.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    rblem4 $p |- ( -. ( ( ph \/ ps ) \/ ch ) \/ ( ( et \/ ta ) \/ th ) ) $=
      ( wo wn rblem1 rb-ax2 rb-ax1 anmp rbsyl rb-ax4 rblem2 rb-ax3 ) ABJZCJKZCB
      JZAJZFEJZDJUBUDADCFBEIHLGLUABCJZAJZUCUFKZAUBJZUCAUBMUGAUEJZUHUEKUBJUIKUHJ
      BCMAUEUBNOUEAMPPUAUFUFJUFUFQTUFCUFTKUIUFAUEMBCARPCKZUEJUJUFJCBSUEAUJROLPP
      P $.
  $}

  $( Used to rederive the Lukasiewicz axioms from Russell-Bernays'.
     (Contributed by Anthony Hart, 19-Aug-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  rblem5 $p |- ( -. ( -. -. ph \/ ps ) \/ ( -. -. ps \/ ph ) ) $=
    ( wn wo rb-ax2 rb-ax4 rb-ax3 rbsyl anmp rblem1 ) ACZCZBDCABCZCZDNADANELABNK
    ADLCZADKAADAAFAAGHZKOAAOLDLODOLLDLLFLLGHOLEIPJINMDMNDNMMDMMFMMGHNMEIJH $.

  ${
    rblem6.1 $e |- -. ( -. ( -. ph \/ ps ) \/ -. ( -. ps \/ ph ) ) $.
    $( Used to rederive the Lukasiewicz axioms from Russell-Bernays'.
       (Contributed by Anthony Hart, 19-Aug-2011.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    rblem6 $p |- ( -. ph \/ ps ) $=
      ( wn wo rb-ax4 rb-ax3 rbsyl rb-ax2 anmp rblem3 rblem5 ) ADBEZDZBDAEDZEZDZ
      MCNDZPEZQDMEPREZSNREZTRNEUARNNENNFNNGHRNIJRONKJPRIJMPLJJ $.
  $}

  ${
    rblem7.1 $e |- -. ( -. ( -. ph \/ ps ) \/ -. ( -. ps \/ ph ) ) $.
    $( Used to rederive the Lukasiewicz axioms from Russell-Bernays'.
       (Contributed by Anthony Hart, 19-Aug-2011.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    rblem7 $p |- ( -. ps \/ ph ) $=
      ( wn wo rb-ax3 rblem5 anmp ) ADBEDZBDAEZDZEZDZJCKDLEMDJEKIFJLGHH $.
  $}

  ${
    re1axmp.min $e |- ph $.
    re1axmp.maj $e |- ( ph -> ps ) $.
    $( ~ ax-mp derived from Russell-Bernays'.  (Contributed by Anthony Hart,
       19-Aug-2011.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    re1axmp $p |- ps $=
      ( wi wn wo rb-imdf rblem6 anmp ) ABCABEZAFBGZDKLABHIJJ $.
  $}

  $( ~ luk-1 derived from Russell-Bernays'.  (Contributed by Anthony Hart,
     19-Aug-2011.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  re2luk1 $p |- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) ) $=
    ( wi wn rb-imdf rblem7 rblem6 rb-ax2 rb-ax4 rb-ax3 rbsyl anmp rblem1 rb-ax1
    wo rblem4 ) ABDZEZBCDZACDZDZPZRUBDZSTEZUAPZUBUBUFTUAFGSAEZBPZUFUHEZBECPZEZU
    GCPZPZUFUKUEULUAUEUJPZUKEZUEPZTUJBCFHUNEUEUOPUPUEUOIUEUEUJUOUEEUEUEPUEUEJUE
    UEKLUOUKPUKUOPUOUKUKPUKUKJUKUKKLZUOUKIMNLMUAULACFGNUKUIULPZPZUIUMPZUGBCOUSE
    ZUMUIPZUTUMUIIVAURUKPVBUIULUKUIULUKUIEUIUIPUIUIJUIUIKLULEULULPULULJULULKLUQ
    QUKURILLMLRUHABFHLLUDUCRUBFGM $.

  $( ~ luk-2 derived from Russell-Bernays'.  (Contributed by Anthony Hart,
     19-Aug-2011.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  re2luk2 $p |- ( ( -. ph -> ph ) -> ph ) $=
    ( wn wi wo rb-ax4 rb-ax3 rbsyl rb-ax2 anmp rblem1 rb-imdf rblem6 rblem7 ) A
    BZACZBZADZOACZPNBZADZATBAADZAAEZSAAANADSBZADNUAAUBAAFGZNUCAAUCSDSUCDUCSSDSS
    ESSFGUCSHIUDJIUDJGOTNAKLGRQOAKMI $.

  $( ~ luk-3 derived from Russell-Bernays'.

     This theorem, along with ~ re1axmp , ~ re2luk1 , and ~ re2luk2 shows that
     ~ rb-ax1 , ~ rb-ax2 , ~ rb-ax3 , and ~ rb-ax4 , along with ~ anmp , can be
     used as a complete axiomatization of propositional calculus.  (Contributed
     by Anthony Hart, 19-Aug-2011.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  re2luk3 $p |- ( ph -> ( -. ph -> ps ) ) $=
    ( wn wi wo rb-imdf rblem7 rb-ax4 rb-ax3 rbsyl rb-ax2 anmp rblem2 ) ACZNBDZE
    ZAODZNNCZBEZOOSNBFGNREZNSERNETRNNENNHNNIJRNKLRBNMLJQPAOFGL $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Stoic logic non-modal portion (Chrysippus of Soli)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  The Greek Stoics developed a system of logic called Stoic 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 ).

  In this section we show that the propositional logic system we use (which is
  non-modal) is at least as strong as the non-modal portion of Stoic logic.  We
  show this by showing that our system assumes or proves all of key features of
  Stoic logic's non-modal portion (specifically the Stoic logic
  indemonstrables, themata, and principles).

  "In terms of contemporary logic, Stoic syllogistic is best understood as a
  substructural backwards-working Gentzen-style natural-deduction system that
  consists of five kinds of axiomatic arguments (the indemonstrables) and four
  inference rules, called themata.  An argument is a syllogism precisely if it
  either is an indemonstrable or can be reduced to one by means of the themata
  (Diogenes Laertius (D. L. 7.78))." (_Ancient Logic_, Stanford Encyclopedia of
  Philosophy ~ https://plato.stanford.edu/entries/logic-ancient/ ).  There are
  also a few "principles" that support logical reasoning, discussed below.  For
  more information, see "Stoic Logic" by Susanne Bobzien, especially
  [Bobzien] p. 110-120, especially for a discussion about the themata
  (including how they were reconstructed and how they were used).  There are
  differences in the systems we can only partly represent, for example, in
  Stoic logic "truth and falsehood are temporal properties of assertibles...
  They can belong to an assertible at one time but not at another"
  ([Bobzien] p. 87).  Stoic logic also included various kinds of modalities,
  which we do not include here since our basic propositional logic does not
  include modalities.

  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.  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 .

  After we prove the indemonstratables, we then prove all the Stoic logic
  themata (the inference rules of Stoic logic; "thema" is singular).  This is
  straightforward for thema 1 ( ~ stoic1a and ~ stoic1b ) and thema 3
  ( ~ stoic3 ).  However, while Stoic logic was once a leading logic system,
  most direct information about Stoic logic has since been lost, including the
  exact texts of thema 2 and thema 4.  There are, however, enough references
  and specific examples to support reconstruction.  Themata 2 and 4 have been
  reconstructed; see statements T2 and T4 in [Bobzien] p. 110-120 and our
  proofs of them in ~ stoic2a , ~ stoic2b , ~ stoic4a , and ~ stoic4b .

  Stoic logic also had a set of principles involving assertibles.  Statements
  in [Bobzien] p. 99 express the known principles.  The following paragraphs
  discuss these principles and our proofs of them.

  "A principle of double negation, expressed by saying that a double-negation
  (Not: not: p) is equivalent to the assertible that is doubly negated (p) (DL
  VII 69)."  In other words, ` ( ph <-> -. -. ph ) ` as proven in ~ notnotb .

  "The principle that all conditionals that are formed by using the same
  assertible twice (like 'If p, p') are true (Cic. Acad. II 98)."  In other
  words, ` ( ph -> ph ) ` as proven in ~ id .

  "The principle that all disjunctions formed by a contradiction (like 'Either
  p or not: p') are true (S. E. M VIII 282)."  Remember that in Stoic logic,
  'or' means 'exclusive or'.  In other words, ` ( ph \/_ -. ph ) ` as proven in
  ~ xorexmid .

  [Bobzien] p. 99 also suggests that Stoic logic may have dealt with
  commutativity (see ~ xorcom and ~ ancom ) and the principle of contraposition
  ( ~ con4 ) (pointing to DL VII 194).

  In short, the non-modal propositional logic system we use is at least as
  strong as the non-modal portion of Stoic logic.

  For more about Aristotle's system, see ~ barbara and related theorems.

$)

  ${
    $( 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, 3-Jul-2016.)
       (Proof shortened by Wolf Lammen, 12-Nov-2017.)  (Proof shortened by BJ,
       19-Apr-2019.) $)
    mptxor $p |- -. ps $=
      ( wxo wa wn xornan ax-mp mptnan ) ABCABEABFGDABHIJ $.
  $}

  ${
    $( 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 assumption, another follows, then that 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
      $.
  $}


$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
  Predicate calculus with equality:  Tarski's system S2 (1 rule, 6 schemes)
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#

  Here we extend the language of wffs with predicate calculus, which allows
  to talk about individual objects in a domain of discourse (which for us will
  be the universe of all sets, so we call them "setvar variables") and make
  true/false statements about predicates, which are relationships between
  objects, such as whether or not two objects are equal.  In addition, we
  introduce universal quantification ("for all", e.g., ~ ax-4 ) in order to
  make statements about whether a wff holds for every object in the domain of
  discourse.  Later we introduce existential quantification ("there exists",
  ~ df-ex ) which is defined in terms of universal quantification.

  Our axioms are really axiom _schemes_, and our wff and setvar variables are
  metavariables ranging over expressions in an underlying "object language".
  This is explained here:  ~ mmset.html#axiomnote .

  Our axiom system starts with the predicate calculus axiom schemes system S2
  of Tarski defined in his 1965 paper, "A Simplified Formalization of Predicate
  Logic with Identity" [Tarski].  System S2 is defined in the last paragraph on
  p. 77, and repeated on p. 81 of [KalishMontague].  We do not include scheme
  B5 (our ~ sp ) of system S2 since [KalishMontague] shows it to be logically
  redundant (Lemma 9, p. 87, which we prove as Theorem ~ spw below).

  Theorem ~ spw can be used to prove any _instance_ of ~ sp having mutually
  distinct setvar variables and no wff metavariables.  However, it seems that
  ~ sp in its general form cannot be derived from only Tarski's schemes.  We do
  not include B5 i.e. ~ sp as part of what we call "Tarski's system" because we
  want it to be the smallest set of axioms that is logically complete with
  no redundancies.  We later prove ~ sp as Theorem ~ axc5 using the auxiliary
  axiom schemes that make our system metalogically complete.

  Our version of Tarski's system S2 consists of propositional calculus
  ( ~ ax-mp , ~ ax-1 , ~ ax-2 , ~ ax-3 ) plus ~ ax-gen , ~ ax-4 , ~ ax-5 ,
  ~ ax-6 , ~ ax-7 , ~ ax-8 , and ~ ax-9 .  The last three are equality axioms
  that represent three sub-schemes of Tarski's scheme B8.  Due to its
  side-condition ("where ` ph ` is an atomic formula and ` ps ` is obtained by
  replacing an occurrence of the variable ` x ` by the variable ` y ` "), we
  cannot represent his B8 directly without greatly complicating our scheme
  language, but the simpler schemes ~ ax-7 , ~ ax-8 , and ~ ax-9 are sufficient
  for set theory and much easier to work with.

  Tarski's system is exactly equivalent to the traditional axiom system in most
  logic textbooks but has the advantage of being easy to manipulate with a
  computer program, and its simpler metalogic (with no built-in notions of
  "free variable" and "proper substitution") is arguably easier for a
  non-logician human to follow step by step in a proof (where "follow" means
  being able to identify the substitutions that were made, without necessarily
  a higher-level understanding).  In particular, it is logically complete in
  that it can derive all possible object-language theorems of predicate
  calculus with equality, i.e., the same theorems as the traditional system can
  derive.

  However, for efficiency (and indeed a key feature that makes Metamath
  successful), our system is designed to derive reusable theorem schemes
  (rather than object-language theorems) from other schemes.  From this
  "metalogical" point of view, Tarski's S2 is not complete.  For example, we
  cannot derive scheme ~ sp , even though (using ~ spw ) we can derive all
  instances of it that do not involve wff metavariables or bundled setvar
  variables.  (Two setvar variables are "bundled" if they can be substituted
  with the same setvar variable, i.e., do not have a "$d" disjoint variable
  condition.)  Later we will introduce auxiliary axiom schemes ~ ax-10 ,
  ~ ax-11 , ~ ax-12 , and ~ ax-13 that are metatheorems of Tarski's system
  (i.e. are logically redundant) but which give our system the property of
  "scheme completeness", allowing us to prove directly (instead of, say,
  by induction on formula length) all possible schemes that can be expressed in
  our language.

$)


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Universal quantifier (continued); define "exists" and "not free"
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  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 $.


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Existential quantifier
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  $( 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 $.

  $( Define existential quantification. ` E. x ph ` means "there exists at
     least one set ` x ` such that ` ph ` is true".  Dual of ~ alex .  See also
     the dual pair ~ alnex / ~ exnal .  Definition of [Margaris] p. 49.
     (Contributed by NM, 10-Jan-1993.) $)
  df-ex $a |- ( E. x ph <-> -. A. x -. ph ) $.

  $( Universal quantification of negation is equivalent to negation of
     existential quantification.  Dual of ~ exnal (but does not depend on
     ~ ax-4 contrary to it).  See also the dual pair ~ df-ex / ~ alex .
     Theorem 19.7 of [Margaris] p. 89.  (Contributed by NM, 12-Mar-1993.) $)
  alnex $p |- ( A. x -. ph <-> -. E. x ph ) $=
    ( wex wn wal df-ex con2bii ) ABCADBEABFG $.

  $( An equivalence between an implication with an existentially quantified
     antecedent and an implication with a universally quantified consequent.
     An interesting case is when the same formula is substituted for both
     ` ph ` and ` ps ` , since then both implications express a type of
     nonfreeness.  See also ~ alimex .  (Contributed by BJ, 12-May-2019.) $)
  eximal $p |- ( ( E. x ph -> ps ) <-> ( -. ps -> A. x -. ph ) ) $=
    ( wex wi wn wal df-ex imbi1i con1b bitri ) ACDZBEAFCGZFZBEBFMELNBACHIMBJK
    $.


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Nonfreeness predicate
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  $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 ~ nf5 for an alternate definition which involves nested
     quantifiers on the same variable.

     Not-free is a commonly used constraint, 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 not-free notion
     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 formula ` 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 .

     This definition of "not free" tightly ties to the quantifier ` A. x ` .
     At this state (no axioms restricting quantifiers yet) "nonfree" appears
     quite arbitrary.  Its intended semantics expresses single-valuedness
     (constness) across a parameter, but is only evolved as much as later
     axioms assign properties to quantifiers.  It seems the definition here is
     best suited in situations, where axioms are only partially in effect.  In
     particular, this definition more easily carries over to other logic models
     with weaker axiomization.

     The reverse implication of the definiens (the right hand side of the
     biconditional) always holds, see ~ 19.2 .

     This predicate only applies to wffs.  See ~ df-nfc for a not-free
     predicate for class variables.  (Contributed by Mario Carneiro,
     24-Sep-2016.)  Convert to definition.  (Revised by BJ, 6-May-2019.) $)
  df-nf $a |- ( F/ x ph <-> ( E. x ph -> A. x ph ) ) $.

  $( Alternate definition of nonfreeness.  (Contributed by BJ, 16-Sep-2021.) $)
  nf2 $p |- ( F/ x ph <-> ( A. x ph \/ -. E. x ph ) ) $=
    ( wnf wex wal wi wn wo df-nf imor orcom 3bitri ) ABCABDZABEZFMGZNHNOHABIMNJ
    ONKL $.

  $( Alternate definition of nonfreeness.  (Contributed by BJ, 16-Sep-2021.) $)
  nf3 $p |- ( F/ x ph <-> ( A. x ph \/ A. x -. ph ) ) $=
    ( wnf wal wex wn wo nf2 alnex orbi2i bitr4i ) ABCABDZABEFZGLAFBDZGABHNMLABI
    JK $.

  $( Alternate definition of nonfreeness.  This definition uses only primitive
     symbols ( ` -> , -. , A. ` ).  (Contributed by BJ, 16-Sep-2021.) $)
  nf4 $p |- ( F/ x ph <-> ( -. A. x ph -> A. x -. ph ) ) $=
    ( wnf wal wn wo wi nf3 df-or bitri ) ABCABDZAEBDZFKELGABHKLIJ $.

  ${
    nfi.1 $e |- ( E. x ph -> A. x ph ) $.
    $( Deduce that ` x ` is not free in ` ph ` from the definition.
       (Contributed by Wolf Lammen, 15-Sep-2021.) $)
    nfi $p |- F/ x ph $=
      ( wnf wex wal wi df-nf mpbir ) ABDABEABFGCABHI $.
  $}

  ${
    nfri.1 $e |- F/ x ph $.
    $( Consequence of the definition of not-free.  (Contributed by Wolf Lammen,
       16-Sep-2021.) $)
    nfri $p |- ( E. x ph -> A. x ph ) $=
      ( wnf wex wal wi df-nf mpbi ) ABDABEABFGCABHI $.
  $}

  ${
    nfd.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.) $)
    nfd $p |- ( ph -> F/ x ps ) $=
      ( wex wal wi wnf df-nf sylibr ) ABCEBCFGBCHDBCIJ $.
  $}

  ${
    nfrd.1 $e |- ( ph -> F/ x ps ) $.
    $( Consequence of the definition of not-free in a context.  (Contributed by
       Wolf Lammen, 15-Oct-2021.) $)
    nfrd $p |- ( ph -> ( E. x ps -> A. x ps ) ) $=
      ( wnf wex wal wi df-nf sylib ) ABCEBCFBCGHDBCIJ $.
  $}

  $( Closed form of ~ nfth .  (Contributed by Wolf Lammen, 19-Aug-2018.)
     (Proof shortened by BJ, 16-Sep-2021.)  (Proof shortened by Wolf Lammen,
     3-Sep-2022.) $)
  nftht $p |- ( A. x ph -> F/ x ph ) $=
    ( wal wex ax-1 nfd ) ABCZABGABDEF $.

  $( Closed form of ~ nfnth .  (Contributed by BJ, 16-Sep-2021.)  (Proof
     shortened by Wolf Lammen, 4-Sep-2022.) $)
  nfntht $p |- ( -. E. x ph -> F/ x ph ) $=
    ( wex wn wal pm2.21 nfd ) ABCZDABHABEFG $.

  $( Closed form of ~ nfnth .  (Contributed by BJ, 16-Sep-2021.)  (Proof
     shortened by Wolf Lammen, 4-Sep-2022.) $)
  nfntht2 $p |- ( A. x -. ph -> F/ x ph ) $=
    ( wn wal wex wnf alnex nfntht sylbi ) ACBDABECABFABGABHI $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Rule scheme ax-gen (Generalization)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    ax-gen.1 $e |- ph $.
    $( Rule of (universal) generalization.  In our axiomatization, this is the
       only postulated (that is, axiomatic) rule of inference of predicate
       calculus (together with the rule of modus ponens ~ ax-mp of
       propositional 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 ` ,
       then we can conclude ` A. x x = x ` or even ` A. y x = x ` .  Theorem
       ~ altru shows the special case ` A. x T. ` .  The converse rule of
       inference ~ spi (universal instantiation, or universal specialization)
       shows that we can also go the other way: in other words, we can add or
       remove universal quantifiers from the beginning of any theorem as
       required.  Note that the closed form ` ( ph -> A. x ph ) ` need not hold
       (but may hold in special cases, see ~ ax-5 ).  (Contributed by NM,
       3-Jan-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 $.
  $}

  ${
    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 $.
  $}

  ${
    nfth.1 $e |- ph $.
    $( No variable is (effectively) free in a theorem.  (Contributed by Mario
       Carneiro, 11-Aug-2016.) ~ df-nf changed.  (Revised by Wolf Lammen,
       12-Sep-2021.) $)
    nfth $p |- F/ x ph $=
      ( wnf nftht mpg ) AABDBABECF $.
  $}

  ${
    nfnth.1 $e |- -. ph $.
    $( No variable is (effectively) free in a non-theorem.  (Contributed by
       Mario Carneiro, 6-Dec-2016.) ~ df-nf changed.  (Revised by Wolf Lammen,
       12-Sep-2021.) $)
    nfnth $p |- F/ x ph $=
      ( wn wnf nfntht2 mpg ) ADABEBABFCG $.
  $}

  ${
    hbth.1 $e |- ph $.
    $( No variable is (effectively) free in a theorem.

       This and later "hypothesis-building" lemmas, with labels starting
       "hb...", allow 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, 11-May-1993.)  This hb* idiom
       is generally being replaced by the nf* idiom (see ~ nfth ), but keeps
       its interest in some cases.  (Revised by BJ, 23-Sep-2022.) $)
    hbth $p |- ( ph -> A. x ph ) $=
      ( wal ax-gen a1i ) ABDAABCEF $.
  $}

  $( The true constant has no free variables.  (This can also be proven in one
     step with ~ nfv , but this proof does not use ~ ax-5 .)  (Contributed by
     Mario Carneiro, 6-Oct-2016.) $)
  nftru $p |- F/ x T. $=
    ( wtru tru nfth ) BACD $.

  $( The false constant has no free variables (see ~ nftru ).  (Contributed by
     BJ, 6-May-2019.) $)
  nffal $p |- F/ x F. $=
    ( wfal fal nfnth ) BACD $.

  ${
    sptruw.1 $e |- ph $.
    $( Version of ~ sp when ` ph ` is true.  Instance of ~ a1i .  Uses only
       Tarski's FOL axiom schemes.  (Contributed by NM, 23-Apr-2017.) $)
    sptruw $p |- ( A. x ph -> ph ) $=
      ( wal a1i ) AABDCE $.
  $}

  $( For all sets, ` T. ` is true.  (Contributed by Anthony Hart,
     13-Sep-2011.) $)
  altru $p |- A. x T. $=
    ( wtru tru ax-gen ) BACD $.

  $( For all sets, ` -. F. ` is true.  (Contributed by Anthony Hart,
     13-Sep-2011.) $)
  alfal $p |- A. x -. F. $=
    ( wfal wn fal ax-gen ) BCADE $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Axiom scheme ax-4 (Quantified Implication)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Axiom of Quantified Implication.  Axiom C4 of [Monk2] p. 105 and Theorem
     19.20 of [Margaris] p. 90.  It is restated as ~ alim for labeling
     consistency.  It should be used only by ~ alim .  (Contributed by NM,
     21-May-2008.)  Use ~ alim instead.  (New usage is discouraged.) $)
  ax-4 $a |- ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) ) $.

  $( Restatement of Axiom ~ ax-4 , for labeling consistency.  It should be the
     only theorem using ~ ax-4 .  (Contributed by NM, 10-Jan-1993.) $)
  alim $p |- ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) ) $=
    ( ax-4 ) ABCD $.

  ${
    alimi.1 $e |- ( ph -> ps ) $.
    $( Inference quantifying both antecedent and consequent.  (Contributed by
       NM, 5-Jan-1993.) $)
    alimi $p |- ( A. x ph -> A. x ps ) $=
      ( wi wal alim 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 $.
  $}

  $( Add an antecedent in a universally quantified formula.  (Contributed by
     BJ, 6-Oct-2018.) $)
  ala1 $p |- ( A. x ph -> A. x ( ps -> ph ) ) $=
    ( wi ax-1 alimi ) ABADCABEF $.

  $( Closed form of ~ al2imi .  Version of ~ alim for a nested implication.
     (Contributed by Alan Sare, 31-Dec-2011.) $)
  al2im $p |- ( A. x ( ph -> ( ps -> ch ) ) ->
                                     ( A. x ph -> ( A. x ps -> A. x ch ) ) ) $=
    ( wi wal alim syl6 ) ABCEZEDFADFIDFBDFCDFEAIDGBCDGH $.

  ${
    al2imi.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Inference quantifying antecedent, nested antecedent, and consequent.
       (Contributed by NM, 10-Jan-1993.) $)
    al2imi $p |- ( A. x ph -> ( A. x ps -> A. x ch ) ) $=
      ( wi wal al2im mpg ) ABCFFADGBDGCDGFFDABCDHEI $.
  $}

  ${
    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 $.
  $}

  ${
    alimdh.1 $e |- ( ph -> A. x ph ) $.
    alimdh.2 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction form of Theorem 19.20 of [Margaris] p. 90, see ~ alim .
       (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, 24-Jan-1993.) $)
  albi $p |- ( A. x ( ph <-> ps ) -> ( A. x ph <-> A. x ps ) ) $=
    ( wb wal biimp al2imi biimpr impbid ) ABDZCEACEBCEJABCABFGJBACABHGI $.

  ${
    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 $.

    $( Theorem albii is the congruence law for universal quantification. $)
    $( $j congruence 'albii'; $)

    $( Inference adding two 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 $.

    $( Inference adding three universal quantifiers to both sides of an
       equivalence.  (Contributed by Peter Mazsa, 10-Aug-2018.) $)
    3albii $p |- ( A. x A. y A. z ph <-> A. x A. y A. z ps ) $=
      ( wal 2albii albii ) AEGDGBEGDGCABDEFHI $.
  $}

  $( Closed form of ~ sylg .  (Contributed by BJ, 2-May-2019.) $)
  sylgt $p |- ( A. x ( ps -> ch ) ->
                                ( ( ph -> A. x ps ) -> ( ph -> A. x ch ) ) ) $=
    ( wi wal alim imim2d ) BCEDFBDFCDFABCDGH $.

  ${
    sylg.1 $e |- ( ph -> A. x ps ) $.
    sylg.2 $e |- ( ps -> ch ) $.
    $( A syllogism combined with generalization.  Inference associated with
       ~ sylgt .  General form of ~ alrimih .  (Contributed by NM, 9-Jan-1993.)
       Extract from proof of ~ alrimih .  (Revised by BJ, 4-Oct-2019.) $)
    sylg $p |- ( ph -> A. x ch ) $=
      ( wal alimi syl ) ABDGCDGEBCDFHI $.
  $}

  ${
    alrimih.1 $e |- ( ph -> A. x ph ) $.
    alrimih.2 $e |- ( ph -> ps ) $.
    $( Inference form of Theorem 19.21 of [Margaris] p. 90.  See ~ 19.21 and
       ~ 19.21h .  Instance of ~ sylg .  (Contributed by NM, 9-Jan-1993.) $)
    alrimih $p |- ( ph -> A. x ps ) $=
      ( sylg ) AABCDEF $.
  $}

  ${
    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.  See ~ hbxfreq for equality version.  (Contributed by
       Jonathan Ben-Naim, 3-Jun-2011.) $)
    hbxfrbi $p |- ( ph -> A. x ph ) $=
      ( wal albii 3imtr4i ) BBCFAACFEDABCDGH $.
  $}

  $( Universal quantifier in terms of existential quantifier and negation.
     Dual of ~ df-ex .  See also the dual pair ~ alnex / ~ exnal .  Theorem
     19.6 of [Margaris] p. 89.  (Contributed by NM, 12-Mar-1993.) $)
  alex $p |- ( A. x ph <-> -. E. x -. ph ) $=
    ( wal wn wex notnotb albii alnex bitri ) ABCADZDZBCJBEDAKBAFGJBHI $.

  $( Existential quantification of negation is equivalent to negation of
     universal quantification.  Dual of ~ alnex .  See also the dual pair
     ~ df-ex / ~ alex .  Theorem 19.14 of [Margaris] p. 90.  (Contributed by
     NM, 12-Mar-1993.) $)
  exnal $p |- ( E. x -. ph <-> -. A. x ph ) $=
    ( wal wn wex alex con2bii ) ABCADBEABFG $.

  $( Part of theorem *11.5 in [WhiteheadRussell] p. 164.  (Contributed by
     Andrew Salmon, 24-May-2011.) $)
  2nalexn $p |- ( -. A. x A. y ph <-> E. x E. y -. ph ) $=
    ( wn wex wal df-ex alex albii xchbinxr bicomi ) ADCEZBEZACFZBFZDMLDZBFOLBGN
    PBACHIJK $.

  $( Theorem *11.22 in [WhiteheadRussell] p. 160.  (Contributed by Andrew
     Salmon, 24-May-2011.) $)
  2exnaln $p |- ( E. x E. y ph <-> -. A. x A. y -. ph ) $=
    ( wex wn wal df-ex alnex albii xchbinxr ) ACDZBDKEZBFAECFZBFKBGMLBACHIJ $.

  $( Theorem *11.25 in [WhiteheadRussell] p. 160.  (Contributed by Andrew
     Salmon, 24-May-2011.) $)
  2nexaln $p |- ( -. E. x E. y ph <-> A. x A. y -. ph ) $=
    ( wn wal wex 2exnaln bicomi con1bii ) ADCEBEZACFBFZKJDABCGHI $.

  $( An equivalence between an implication with a universally quantified
     consequent and an implication with an existentially quantified antecedent.
     An interesting case is when the same formula is substituted for both
     ` ph ` and ` ps ` , since then both implications express a type of
     nonfreeness.  See also ~ eximal .  (Contributed by BJ, 12-May-2019.) $)
  alimex $p |- ( ( ph -> A. x ps ) <-> ( E. x -. ps -> -. ph ) ) $=
    ( wal wi wn wex alex imbi2i con2b bitri ) ABCDZEABFCGZFZEMAFELNABCHIAMJK $.

  ${
    aleximi.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( A variant of ~ al2imi : instead of applying ` A. x ` quantifiers to the
       final implication, replace them with ` E. x ` .  A shorter proof is
       possible using ~ nfa1 , ~ sps and ~ eximd , but it depends on more
       axioms.  (Contributed by Wolf Lammen, 18-Aug-2019.) $)
    aleximi $p |- ( A. x ph -> ( E. x ps -> E. x ch ) ) $=
      ( wal wex wn con3d al2imi alnex 3imtr3g con4d ) ADFZCDGZBDGZNCHZDFBHZDFOH
      PHAQRDABCEIJCDKBDKLM $.
  $}

  ${
    alexbii.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Biconditional form of ~ aleximi .  (Contributed by BJ, 16-Nov-2020.) $)
    alexbii $p |- ( A. x ph -> ( E. x ps <-> E. x ch ) ) $=
      ( wal wex biimpd aleximi biimprd impbid ) ADFBDGCDGABCDABCEHIACBDABCEJIK
      $.
  $}

  $( Theorem 19.22 of [Margaris] p. 90.  (Contributed by NM, 10-Jan-1993.)
     (Proof shortened by Wolf Lammen, 4-Jul-2014.) $)
  exim $p |- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ps ) ) $=
    ( wi id aleximi ) ABDZABCGEF $.

  ${
    eximi.1 $e |- ( ph -> ps ) $.
    $( Inference adding existential quantifier to antecedent and consequent.
       (Contributed by NM, 10-Jan-1993.) $)
    eximi $p |- ( E. x ph -> E. x ps ) $=
      ( wi wex exim mpg ) ABEACFBCFECABCGDH $.

    $( Inference adding two 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 $.
  $}

  $( Add an antecedent in an existentially quantified formula.  (Contributed by
     BJ, 6-Oct-2018.) $)
  exa1 $p |- ( E. x ph -> E. x ( ps -> ph ) ) $=
    ( wi ax-1 eximi ) ABADCABEF $.

  $( Theorem 19.38 of [Margaris] p. 90.  The converse holds under nonfreeness
     conditions, see ~ 19.38a and ~ 19.38b .  (Contributed by NM, 12-Mar-1993.)
     Allow a shortening of ~ 19.21t .  (Revised by Wolf Lammen, 2-Jan-2018.) $)
  19.38 $p |- ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) ) $=
    ( wex wal wi wn alnex pm2.21 alimi sylbir ala1 ja ) ACDZBCEABFZCEZNGAGZCEPA
    CHQOCABIJKBACLM $.

  $( Under a nonfreeness hypothesis, the implication ~ 19.38 can be
     strengthened to an equivalence.  See also ~ 19.38b .  (Contributed by BJ,
     3-Nov-2021.)  (Proof shortened by Wolf Lammen, 9-Jul-2022.) $)
  19.38a $p |-
             ( F/ x ph -> ( ( E. x ph -> A. x ps ) <-> A. x ( ph -> ps ) ) ) $=
    ( wnf wex wal wi 19.38 id nfrd alim syl9 impbid2 ) ACDZACEZBCFZGABGCFZABCHN
    OACFQPNACNIJABCKLM $.

  $( Under a nonfreeness hypothesis, the implication ~ 19.38 can be
     strengthened to an equivalence.  See also ~ 19.38a .  (Contributed by BJ,
     3-Nov-2021.)  (Proof shortened by Wolf Lammen, 9-Jul-2022.) $)
  19.38b $p |-
             ( F/ x ps -> ( ( E. x ph -> A. x ps ) <-> A. x ( ph -> ps ) ) ) $=
    ( wnf wex wal wi 19.38 exim id nfrd syl9r impbid2 ) BCDZACEZBCFZGABGCFZABCH
    QOBCENPABCINBCNJKLM $.

  $( Quantified implication in terms of quantified negation of conjunction.
     (Contributed by BJ, 16-Jul-2021.) $)
  imnang $p |- ( A. x ( ph -> -. ps ) <-> A. x -. ( ph /\ ps ) ) $=
    ( wn wi wa imnan albii ) ABDEABFDCABGH $.

  $( 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 imnang alnex bitri ) ABDECFABGZDCFLCHDABCILCJK $.

  $( Existential quantification of a conjunction expressed with only primitive
     symbols ( ` -> ` , ` -. ` , ` A. ` ).  (Contributed by NM, 10-May-1993.)
     State the most general instance.  (Revised by BJ, 29-Sep-2019.) $)
  exnalimn $p |- ( E. x ( ph /\ ps ) <-> -. A. x ( ph -> -. ps ) ) $=
    ( wn wi wal wa wex alinexa con2bii ) ABDECFABGCHABCIJ $.

  $( A relationship between two quantifiers and negation.  (Contributed by NM,
     18-Aug-1993.) $)
  alexn $p |- ( A. x E. y -. ph <-> -. E. x A. y ph ) $=
    ( wn wex wal exnal albii alnex bitri ) ADCEZBFACFZDZBFLBEDKMBACGHLBIJ $.

  $( Theorem *11.51 in [WhiteheadRussell] p. 164.  (Contributed by Andrew
     Salmon, 24-May-2011.)  (Proof shortened by Wolf Lammen, 25-Sep-2014.) $)
  2exnexn $p |- ( E. x A. y ph <-> -. A. x E. y -. ph ) $=
    ( wn wex wal alexn con2bii ) ADCEBFACFBEABCGH $.

  $( Theorem 19.18 of [Margaris] p. 90.  (Contributed by NM, 12-Mar-1993.) $)
  exbi $p |- ( A. x ( ph <-> ps ) -> ( E. x ph <-> E. x ps ) ) $=
    ( wb id alexbii ) ABDZABCGEF $.

  ${
    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 $.
  $}

  ${
    2exbii.1 $e |- ( ph <-> ps ) $.
    $( Inference adding two 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 $.
  $}

  ${
    3exbii.1 $e |- ( ph <-> ps ) $.
    $( Inference adding three 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 $.
  $}

  $( Equivalence theorem for the nonfreeness predicate.  Closed form of
     ~ nfbii .  (Contributed by Giovanni Mascellani, 10-Apr-2018.)  Reduce
     axiom usage.  (Revised by BJ, 6-May-2019.) $)
  nfbiit $p |- ( A. x ( ph <-> ps ) -> ( F/ x ph <-> F/ x ps ) ) $=
    ( wb wal wex wi wnf exbi albi imbi12d df-nf 3bitr4g ) ABDCEZACFZACEZGBCFZBC
    EZGACHBCHNOQPRABCIABCJKACLBCLM $.

  ${
    nfbii.1 $e |- ( ph <-> ps ) $.
    $( Equality theorem for the nonfreeness predicate.  (Contributed by Mario
       Carneiro, 11-Aug-2016.) ~ df-nf changed.  (Revised by Wolf Lammen,
       12-Sep-2021.) $)
    nfbii $p |- ( F/ x ph <-> F/ x ps ) $=
      ( wb wnf nfbiit mpg ) ABEACFBCFECABCGDH $.

    ${
      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 $.
    $}
  $}

  $( A variable is nonfree in a proposition if and only if it is so in its
     negation.  (Contributed by BJ, 6-May-2019.)  (Proof shortened by Wolf
     Lammen, 6-Oct-2024.) $)
  nfnbi $p |- ( F/ x ph <-> F/ x -. ph ) $=
    ( wn wex wal wi wnf exnal imbi1i df-nf nf4 3bitr4ri ) ACZBDZMBEZFABECZOFMBG
    ABGNPOABHIMBJABKL $.

  $( If a variable is nonfree in a proposition, then it is nonfree in its
     negation.  (Contributed by Mario Carneiro, 24-Sep-2016.)  (Proof shortened
     by Wolf Lammen, 28-Dec-2017.)  (Revised by BJ, 24-Jul-2019.) ~ df-nf
     changed.  (Revised by Wolf Lammen, 4-Oct-2021.) $)
  nfnt $p |- ( F/ x ph -> F/ x -. ph ) $=
    ( wnf wn nfnbi biimpi ) ABCADBCABEF $.

  ${
    nfn.1 $e |- F/ x ph $.
    $( Inference associated with ~ nfnt .  (Contributed by Mario Carneiro,
       11-Aug-2016.) ~ df-nf changed.  (Revised by Wolf Lammen,
       18-Sep-2021.) $)
    nfn $p |- F/ x -. ph $=
      ( wnf wn nfnt ax-mp ) ABDAEBDCABFG $.
  $}

  ${
    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 $.
  $}

  $( A transformation of quantifiers and logical connectives.  (Contributed by
     NM, 25-Mar-1996.)  (Proof shortened by Wolf Lammen, 4-Sep-2014.) $)
  exanali $p |- ( E. x ( ph /\ -. ps ) <-> -. A. x ( ph -> ps ) ) $=
    ( wn wa wex wi wal annim exbii exnal bitri ) ABDEZCFABGZDZCFNCHDMOCABIJNCKL
    $.

  $( Theorem *11.521 in [WhiteheadRussell] p. 164.  (Contributed by Andrew
     Salmon, 24-May-2011.) $)
  2exanali $p |- ( -. E. x E. y ( ph /\ -. ps ) <->
        A. x A. y ( ph -> ps ) ) $=
    ( wi wn wex wal wa 2nalexn con1bii annim 2exbii xchnxbir ) ABEZFZDGCGZODHCH
    ZABFIZDGCGRQOCDJKSPCDABLMN $.

  $( 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 $.

  ${
    exan.1 $e |- E. x ph $.
    exan.2 $e |- 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.)  (Proof shortened by Wolf Lammen, 13-Jan-2018.)  Reduce
       axiom dependencies.  (Revised by BJ, 7-Jul-2021.)  (Proof shortened by
       Wolf Lammen, 6-Nov-2022.)  Expand hypothesis.  (Revised by Steven
       Nguyen, 19-Jun-2023.) $)
    exan $p |- E. x ( ph /\ ps ) $=
      ( wa jctr eximii ) AABFCDABEGH $.
  $}

  ${
    alrimdh.1 $e |- ( ph -> A. x ph ) $.
    alrimdh.2 $e |- ( ps -> A. x ps ) $.
    alrimdh.3 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction form of Theorem 19.21 of [Margaris] p. 90, see ~ 19.21 and
       ~ 19.21h .  (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 $.
  $}

  ${
    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 ) ) $=
      ( wal wex wi aleximi syl ) AADGBDHCDHIEABCDFJK $.
  $}

  ${
    nexdh.1 $e |- ( ph -> A. x ph ) $.
    nexdh.2 $e |- ( ph -> -. ps ) $.
    $( Deduction for generalization rule for negated wff.  (Contributed by NM,
       2-Jan-2002.) $)
    nexdh $p |- ( ph -> -. E. x ps ) $=
      ( wn wal wex alrimih alnex sylib ) ABFZCGBCHFALCDEIBCJK $.
  $}

  ${
    albidh.1 $e |- ( ph -> A. x ph ) $.
    albidh.2 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for universal quantifier (deduction form).
       (Contributed by NM, 26-May-1993.) $)
    albidh $p |- ( ph -> ( A. x ps <-> A. x ch ) ) $=
      ( wb wal alrimih albi syl ) ABCGZDHBDHCDHGALDEFIBCDJK $.
  $}

  ${
    exbidh.1 $e |- ( ph -> A. x ph ) $.
    exbidh.2 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for existential quantifier (deduction form).
       (Contributed by NM, 26-May-1993.) $)
    exbidh $p |- ( ph -> ( E. x ps <-> E. x ch ) ) $=
      ( wal wex wb alexbii syl ) AADGBDHCDHIEABCDFJK $.
  $}

  $( 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 19.26 of [Margaris] p. 90.  Also Theorem *10.22 of
     [WhiteheadRussell] p. 147.  (Contributed by NM, 12-Mar-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 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 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 df-3an albii bitri 3bitr4i ) ABEZDFZCDFZEZADFZBDF
    ZEZPEABCGZDFZRSPGOTPABDHIUBNCEZDFQUAUCDABCJKNCDHLRSPJM $.

  $( Theorem 19.29 of [Margaris] p. 90.  See also ~ 19.29r .  (Contributed by
     NM, 21-Jun-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 pm3.2 aleximi imp ) ACDBCEABFZCEABJCABGHI $.

  $( Variation of ~ 19.29 .  (Contributed by NM, 18-Aug-1993.)  (Proof
     shortened by Wolf Lammen, 12-Nov-2020.) $)
  19.29r $p |- ( ( E. x ph /\ A. x ps ) -> E. x ( ph /\ ps ) ) $=
    ( wal wex wa pm3.21 aleximi impcom ) BCDACEABFZCEBAJCBAGHI $.

  $( Variation of ~ 19.29r 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 ~ 19.29 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 $.

  $( Theorem 19.35 of [Margaris] p. 90.  This theorem is useful for moving an
     implication (in the form of the right-hand side) into the scope of a
     single existential quantifier.  (Contributed by NM, 12-Mar-1993.)  (Proof
     shortened by Wolf Lammen, 27-Jun-2014.) $)
  19.35 $p |- ( E. x ( ph -> ps ) <-> ( A. x ph -> E. x ps ) ) $=
    ( wi wex wal pm2.27 aleximi com12 wn exnal pm2.21 eximi sylbir exa1 impbii
    ja ) ABDZCEZACFZBCEZDTSUAARBCABGHITUASTJAJZCESACKUBRCABLMNBACOQP $.

  ${
    19.35i.1 $e |- E. x ( ph -> ps ) $.
    $( Inference associated with ~ 19.35 .  (Contributed by NM,
       21-Jun-1993.) $)
    19.35i $p |- ( A. x ph -> E. x ps ) $=
      ( wi wex wal 19.35 mpbi ) ABECFACGBCFEDABCHI $.
  $}

  ${
    19.35ri.1 $e |- ( A. x ph -> E. x ps ) $.
    $( Inference associated with ~ 19.35 .  (Contributed by NM,
       12-Mar-1993.) $)
    19.35ri $p |- E. x ( ph -> ps ) $=
      ( wi wex wal 19.35 mpbir ) ABECFACGBCFEDABCHI $.
  $}

  $( Theorem 19.25 of [Margaris] p. 90.  (Contributed by NM, 12-Mar-1993.) $)
  19.25 $p |- ( A. y E. x ( ph -> ps ) ->
              ( E. y A. x ph -> E. y E. x ps ) ) $=
    ( wi wex wal 19.35 biimpi aleximi ) ABECFZACGZBCFZDKLMEABCHIJ $.

  $( Theorem 19.30 of [Margaris] p. 90.  (Contributed by NM, 12-Mar-1993.)
     (Proof shortened by Andrew Salmon, 25-May-2011.) $)
  19.30 $p |- ( A. x ( ph \/ ps ) -> ( A. x ph \/ E. x ps ) ) $=
    ( wo wal wex wn exnal pm2.53 aleximi biimtrrid orrd ) ABDZCEZACEZBCFZOGAGZC
    FNPACHMQBCABIJKL $.

  $( Theorem 19.43 of [Margaris] p. 90.  (Contributed by NM, 12-Mar-1993.)
     (Proof shortened by Wolf Lammen, 27-Jun-2014.) $)
  19.43 $p |- ( E. x ( ph \/ ps ) <-> ( E. x ph \/ E. x ps ) ) $=
    ( wo wex wn wi wal df-or exbii 19.35 alnex imbi1i 3bitri bitr4i ) ABDZCEZAC
    EZFZBCEZGZRTDQAFZBGZCEUBCHZTGUAPUCCABIJUBBCKUDSTACLMNRTIO $.

  $( Obsolete proof of ~ 19.43 .  Do not delete as it is referenced on the
     ~ mmrecent.html page and in ~ conventions-labels .  (Contributed by NM,
     5-Aug-1993.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  19.43OLD $p |- ( E. x ( ph \/ ps ) <-> ( E. x ph \/ E. x ps ) ) $=
    ( wo wn wal wex wa ioran albii 19.26 alnex anbi12i 3bitri notbii df-ex oran
    3bitr4i ) ABDZEZCFZEACGZEZBCGZEZHZESCGUBUDDUAUFUAAEZBEZHZCFUGCFZUHCFZHUFTUI
    CABIJUGUHCKUJUCUKUEACLBCLMNOSCPUBUDQR $.

  $( Theorem 19.33 of [Margaris] p. 90.  (Contributed by NM, 12-Mar-1993.) $)
  19.33 $p |- ( ( A. x ph \/ A. x ps ) -> A. x ( ph \/ ps ) ) $=
    ( wal wo orc alimi olc jaoi ) ACDABEZCDBCDAJCABFGBJCBAHGI $.

  $( The antecedent provides a condition implying the converse of ~ 19.33 .
     (Contributed by NM, 27-Mar-2004.)  (Proof shortened by Andrew Salmon,
     25-May-2011.)  (Proof shortened by Wolf Lammen, 5-Jul-2014.) $)
  19.33b $p |- ( -. ( E. x ph /\ E. x ps ) ->
               ( A. x ( ph \/ ps ) <-> ( A. x ph \/ A. x ps ) ) ) $=
    ( wex wa wn wo wal wi ianor alnex pm2.53 al2imi biimtrrid olc syl6com 19.30
    orcomd ord orc jaoi sylbi 19.33 impbid1 ) ACDZBCDZEFZABGZCHZACHZBCHZGZUGUEF
    ZUFFZGUIULIZUEUFJUMUOUNUIUMUKULUMAFZCHUIUKACKUHUPBCABLMNUKUJOPUIUNUJULUIUFU
    JUIUJUFABCQRSUJUKTPUAUBABCUCUD $.

  $( Theorem 19.40 of [Margaris] p. 90.  (Contributed by NM, 26-May-1993.) $)
  19.40 $p |- ( E. x ( ph /\ ps ) -> ( E. x ph /\ E. x ps ) ) $=
    ( wa wex exsimpl exsimpr jca ) ABDCEACEBCEABCFABCGH $.

  $( Theorem *11.42 in [WhiteheadRussell] p. 163.  Theorem 19.40 of [Margaris]
     p. 90 with two 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 $.

  $( The antecedent provides a condition implying the converse of ~ 19.40 .
     This is to ~ 19.40 what ~ 19.33b is to ~ 19.33 .  (Contributed by BJ,
     6-May-2019.)  (Proof shortened by Wolf Lammen, 13-Nov-2020.) $)
  19.40b $p |- ( ( A. x ph \/ A. x ps ) ->
                          ( ( E. x ph /\ E. x ps ) <-> E. x ( ph /\ ps ) ) ) $=
    ( wal wo wex wa wi pm3.21 aleximi pm3.2 jaoa orcoms 19.40 impbid1 ) ACDZBCD
    ZEACFZBCFZGZABGZCFZQPTUBHQRUBPSBAUACBAIJABUACABKJLMABCNO $.

  $( 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 two 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 $.

  $( 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 wa abai rbaibr alexbii ) ABDZAABEZCJAIABFGH $.

  $( Introduce a conjunct in the scope of an existential quantifier.
     (Contributed by NM, 11-Aug-1993.)  (Proof shortened by BJ,
     16-Sep-2022.) $)
  exintr $p |- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ( ph /\ ps ) ) ) $=
    ( wi wa ancl aleximi ) ABDAABECABFG $.

  $( Universally quantified and uncurried (imported) form of syllogism.
     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 $.

  ${
    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 ) ` .  Deduction form of ~ nfim .  (Contributed by
       Mario Carneiro, 24-Sep-2016.)  (Proof shortened by Wolf Lammen,
       30-Dec-2017.) ~ df-nf changed.  (Revised by Wolf Lammen, 18-Sep-2021.)
       Eliminate curried form of ~ nfimt .  (Revised by Wolf Lammen,
       10-Jul-2022.) $)
    nfimd $p |- ( ph -> F/ x ( ps -> ch ) ) $=
      ( wi wex wal 19.35 biimpi nfrd imim12d 19.38 syl56 nfd ) ABCGZDQDHZBDIZCD
      HZGZABDHZCDIZGQDIRUABCDJKAUBSTUCABDELACDFLMBCDNOP $.
  $}

  $( Closed form of ~ nfim and ~ nfimd .  (Contributed by BJ, 20-Oct-2021.)
     Eliminate curried form, former name nfimt2.  (Revised by Wolf Lammen,
     6-Jul-2022.) $)
  nfimt $p |- ( ( F/ x ph /\ F/ x ps ) -> F/ x ( ph -> ps ) ) $=
    ( wnf wa simpl simpr nfimd ) ACDZBCDZEABCIJFIJGH $.

  ${
    nfim.1 $e |- F/ x ph $.
    nfim.2 $e |- F/ x ps $.
    $( If ` x ` is not free in ` ph ` and ` ps ` , then it is not free in
       ` ( ph -> ps ) ` .  Inference associated with ~ nfimt .  (Contributed by
       Mario Carneiro, 11-Aug-2016.)  (Proof shortened by Wolf Lammen,
       2-Jan-2018.) ~ df-nf changed.  (Revised by Wolf Lammen, 17-Sep-2021.) $)
    nfim $p |- F/ x ( ph -> ps ) $=
      ( wnf wi nfimt mp2an ) ACFBCFABGCFDEABCHI $.
  $}

  ${
    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 ` , then it is not
       free in ` ( ps /\ ch ) ` .  (Contributed by Mario Carneiro,
       7-Oct-2016.) $)
    nfand $p |- ( ph -> F/ x ( ps /\ ch ) ) $=
      ( wa wn wi df-an nfnd nfimd nfxfrd ) BCGBCHZIZHADBCJAODABNDEACDFKLKM $.

    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 $.
  $}

  ${
    nfan.1 $e |- F/ x ph $.
    nfan.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, 13-Jan-2018.)  (Proof shortened by Wolf
       Lammen, 9-Oct-2021.) $)
    nfan $p |- F/ x ( ph /\ ps ) $=
      ( wa wnf wtru a1i nfand mptru ) ABFCGHABCACGHDIBCGHEIJK $.

    $( If ` x ` is not free in ` ph ` and ` ps ` , then it is not free in
       ` ( ph -/\ ps ) ` .  (Contributed by Scott Fenton, 2-Jan-2018.) $)
    nfnan $p |- F/ x ( ph -/\ ps ) $=
      ( wnan wa wn df-nan nfan nfn nfxfr ) ABFABGZHCABIMCABCDEJKL $.

    nfan.3 $e |- F/ x ch $.
    $( If ` x ` is not free in ` ph ` , ` ps ` , and ` ch ` , then 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 $.
  $}

  ${
    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 $.
  $}

  ${
    nf.1 $e |- F/ x ph $.
    nf.2 $e |- F/ x ps $.
    $( If ` x ` is not free in ` ph ` and ` ps ` , then it is not free in
       ` ( ph <-> ps ) ` .  (Contributed by NM, 26-May-1993.)  (Revised 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 $.

    $( If ` x ` is not free in ` ph ` and ` ps ` , then it is not free in
       ` ( ph \/ ps ) ` .  (Contributed by NM, 5-Aug-1993.)  (Revised by Mario
       Carneiro, 11-Aug-2016.) $)
    nfor $p |- F/ x ( ph \/ ps ) $=
      ( wo wn wi df-or nfn nfim nfxfr ) ABFAGZBHCABIMBCACDJEKL $.

    nf.3 $e |- F/ x ch $.
    $( If ` x ` is not free in ` ph ` , ` ps ` , and ` ch ` , then it is not
       free in ` ( ph \/ ps \/ ch ) ` .  (Contributed by Mario Carneiro,
       11-Aug-2016.) $)
    nf3or $p |- F/ x ( ph \/ ps \/ ch ) $=
      ( w3o wo df-3or nfor nfxfr ) ABCHABIZCIDABCJMCDABDEFKGKL $.
  $}


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  The empty domain of discourse
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-

  This database develops mathematics from first-order logic, which has only
  nonempty models.  Before stating axioms excluding the empty model
  (typically, ~ ax-6 in logic and ~ ax-nul in set theory), we state in this
  short subsection a few results relative to the empty domain, which we
  characterize by the assumption ` -. E. x T. ` .  As expected, on the empty
  domain, every universally quantified formula is true ( ~ emptyal ) and every
  existential formula is false ( ~ emptyex ), and every variable is effectively
  nonfree in any formula ( ~ emptynf ).

$)

  $( Two characterizations of the empty domain.  (Contributed by G&eacute;rard
     Lang, 5-Feb-2024.) $)
  empty $p |- ( -. E. x T. <-> A. x F. ) $=
    ( wfal wal wtru wn wex df-fal albii alnex bitr2i ) BACDEZACDAFEBKAGHDAIJ $.

  $( On the empty domain, any existentially quantified formula is false.
     (Contributed by Wolf Lammen, 21-Jan-2024.) $)
  emptyex $p |- ( -. E. x T. -> -. E. x ph ) $=
    ( wex wtru trud eximi con3i ) ABCDBCADBAEFG $.

  $( On the empty domain, any universally quantified formula is true.
     (Contributed by Wolf Lammen, 12-Mar-2023.) $)
  emptyal $p |- ( -. E. x T. -> A. x ph ) $=
    ( wtru wex wn wal emptyex alex sylibr ) CBDEAEZBDEABFJBGABHI $.

  $( On the empty domain, any variable is effectively nonfree in any formula.
     (Contributed by Wolf Lammen, 12-Mar-2023.) $)
  emptynf $p |- ( -. E. x T. -> F/ x ph ) $=
    ( wtru wex wn wal wnf emptyal nftht syl ) CBDEABFABGABHABIJ $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Axiom scheme ax-5 (Distinctness) - first use of $d
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x ph $.
    $( Axiom of Distinctness.  This axiom quantifies 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.

       (See comments in ~ ax5ALT about the logical redundancy of ~ ax-5 in the
       presence of our obsolete axioms.)

       This axiom essentially says that if ` x ` does not occur in ` ph ` ,
       i.e. ` ph ` does not depend on ` x ` in any way, then we can add the
       quantifier ` A. x ` to ` ph ` with no further assumptions.  By ~ sp , we
       can also remove the quantifier (unconditionally).

       For an explanation of disjoint variable conditions, see
       ~ https://us.metamath.org/mpeuni/mmset.html#distinct .  (Contributed by
       NM, 10-Jan-1993.) $)
    ax-5 $a |- ( ph -> A. x ph ) $.
  $}

  ${
    $d x ps $.
    $( Version of ~ ax-5 with antecedent.  Useful in proofs of deduction
       versions of bound-variable hypothesis builders.  (Contributed by NM,
       1-Mar-2013.) $)
    ax5d $p |- ( ph -> ( ps -> A. x ps ) ) $=
      ( wal wi ax-5 a1i ) BBCDEABCFG $.
  $}

  ${
    $d x ph $.
    $( A rephrasing of ~ ax-5 using the existential quantifier.  (Contributed
       by Wolf Lammen, 4-Dec-2017.) $)
    ax5e $p |- ( E. x ph -> ph ) $=
      ( wex wi wn wal ax-5 eximal mpbir ) ABCADAEZJBFDJBGAABHI $.
  $}

  ${
    $d x ph $.
    $( If a formula holds for some value of a variable not occurring in it,
       then it holds for all values of that variable.  (Contributed by BJ,
       28-Dec-2020.) $)
    ax5ea $p |- ( E. x ph -> A. x ph ) $=
      ( wex wal ax5e ax-5 syl ) ABCAABDABEABFG $.
  $}

  ${
    $d x ph $.
    $( If ` x ` is not present in ` ph ` , then ` x ` is not free in ` ph ` .
       (Contributed by Mario Carneiro, 11-Aug-2016.)  Definition change.
       (Revised by Wolf Lammen, 12-Sep-2021.) $)
    nfv $p |- F/ x ph $=
      ( ax5ea 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 $.
  $}

  ${
    $d x ph $.
    alimdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction form of Theorem 19.20 of [Margaris] p. 90, see ~ alim .  See
       ~ alimdh and ~ alimd for versions without a distinct variable condition.
       (Contributed by NM, 3-Apr-1994.) $)
    alimdv $p |- ( ph -> ( A. x ps -> A. x ch ) ) $=
      ( ax-5 alimdh ) ABCDADFEG $.

    $( Deduction form of Theorem 19.22 of [Margaris] p. 90, see ~ exim .  See
       ~ eximdh and ~ eximd for versions without a distinct variable condition.
       (Contributed by NM, 27-Apr-1994.) $)
    eximdv $p |- ( ph -> ( E. x ps -> E. x ch ) ) $=
      ( ax-5 eximdh ) ABCDADFEG $.
  $}

  ${
    $d x ph $.  $d y ph $.
    2alimdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction form of Theorem 19.20 of [Margaris] p. 90 with two
       quantifiers, see ~ alim .  (Contributed by NM, 27-Apr-2004.) $)
    2alimdv $p |- ( ph -> ( A. x A. y ps -> A. x A. y ch ) ) $=
      ( wal alimdv ) ABEGCEGDABCEFHH $.

    $( Deduction form of Theorem 19.22 of [Margaris] p. 90 with two
       quantifiers, see ~ exim .  (Contributed by NM, 3-Aug-1995.) $)
    2eximdv $p |- ( ph -> ( E. x E. y ps -> E. x E. y ch ) ) $=
      ( wex eximdv ) ABEGCEGDABCEFHH $.
  $}

  ${
    $d x ph $.
    albidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for universal quantifier (deduction form).  See
       also ~ albidh and ~ albid .  (Contributed by NM, 26-May-1993.) $)
    albidv $p |- ( ph -> ( A. x ps <-> A. x ch ) ) $=
      ( ax-5 albidh ) ABCDADFEG $.

    $( Formula-building rule for existential quantifier (deduction form).  See
       also ~ exbidh and ~ exbid .  (Contributed by NM, 26-May-1993.) $)
    exbidv $p |- ( ph -> ( E. x ps <-> E. x ch ) ) $=
      ( ax-5 exbidh ) ABCDADFEG $.

    $( An equality theorem for nonfreeness.  See ~ nfbidf for a version without
       disjoint variable condition but requiring more axioms.  (Contributed by
       Mario Carneiro, 4-Oct-2016.)  Remove dependency on ~ ax-6 , ~ ax-7 ,
       ~ ax-12 by adapting proof of ~ nfbidf .  (Revised by BJ,
       25-Sep-2022.) $)
    nfbidv $p |- ( ph -> ( F/ x ps <-> F/ x ch ) ) $=
      ( wex wal wi wnf exbidv albidv imbi12d df-nf 3bitr4g ) ABDFZBDGZHCDFZCDGZ
      HBDICDIAOQPRABCDEJABCDEKLBDMCDMN $.
  $}

  ${
    $d x ph $.  $d y ph $.
    2albidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for two universal 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 two 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 three 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 four 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 $.
    alrimiv.1 $e |- ( ph -> ps ) $.
    $( Inference form of Theorem 19.21 of [Margaris] p. 90.  See ~ 19.21 and
       ~ 19.21v .  (Contributed by NM, 21-Jun-1993.) $)
    alrimiv $p |- ( ph -> A. x ps ) $=
      ( ax-5 alrimih ) ABCACEDF $.
  $}

  ${
    $d x ph $.  $d y ph $.
    alrimivv.1 $e |- ( ph -> ps ) $.
    $( Inference form of Theorem 19.21 of [Margaris] p. 90.  See ~ 19.21 and
       ~ 19.21v .  (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 form of Theorem 19.21 of [Margaris] p. 90.  See ~ 19.21 and
       ~ 19.21v .  (Contributed by NM, 10-Feb-1997.) $)
    alrimdv $p |- ( ph -> ( ps -> A. x ch ) ) $=
      ( ax-5 alrimdh ) ABCDADFBDFEG $.
  $}

  ${
    $d x ps $.
    exlimiv.1 $e |- ( ph -> ps ) $.
    $( Inference form of Theorem 19.23 of [Margaris] p. 90, see ~ 19.23 .

       See ~ exlimi for a more general version requiring more axioms.

       This inference, along with its many variants such as ~ rexlimdv , 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.appstate.edu/~~hirstjl/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 (fictitious) 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 ` , see ~ exlimiiv .  (Contributed by NM,
       21-Jun-1993.)  Remove dependencies on ~ ax-6 and ~ ax-8 .  (Revised by
       Wolf Lammen, 4-Dec-2017.) $)
    exlimiv $p |- ( E. x ph -> ps ) $=
      ( wex eximi ax5e syl ) ACEBCEBABCDFBCGH $.

    exlimiiv.2 $e |- E. x ph $.
    $( Inference (Rule C) associated with ~ exlimiv .  (Contributed by BJ,
       19-Dec-2020.) $)
    exlimiiv $p |- ps $=
      ( wex exlimiv ax-mp ) ACFBEABCDGH $.
  $}

  ${
    $d x ps $.  $d y ps $.
    exlimivv.1 $e |- ( ph -> ps ) $.
    $( Inference form of Theorem 19.23 of [Margaris] p. 90, see ~ 19.23 .
       (Contributed by NM, 1-Aug-1995.) $)
    exlimivv $p |- ( E. x E. y ph -> ps ) $=
      ( wex exlimiv ) ADFBCABDEGG $.
  $}

  ${
    $d x ch $.  $d x ph $.
    exlimdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction form of Theorem 19.23 of [Margaris] p. 90, see ~ 19.23 .
       (Contributed by NM, 27-Apr-1994.)  Remove dependencies on ~ ax-6 ,
       ~ ax-7 .  (Revised by Wolf Lammen, 4-Dec-2017.) $)
    exlimdv $p |- ( ph -> ( E. x ps -> ch ) ) $=
      ( wex eximdv ax5e syl6 ) ABDFCDFCABCDEGCDHI $.
  $}

  ${
    $d x ch $.  $d x ph $.  $d y ch $.  $d y ph $.
    exlimdvv.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction form of Theorem 19.23 of [Margaris] p. 90, see ~ 19.23 .
       (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 (Rule C, explained in
       ~ exlimiv ).  (Contributed by Mario Carneiro, 15-Jun-2016.) $)
    exlimddv $p |- ( ph -> ch ) $=
      ( wex ex exlimdv mpd ) ABDGCEABCDABCFHIJ $.
  $}

  ${
    $d x ph $.
    nexdv.1 $e |- ( ph -> -. ps ) $.
    $( Deduction for generalization rule for negated wff.  (Contributed by NM,
       5-Aug-1993.)  Reduce dependencies on axioms.  (Revised by Wolf Lammen,
       13-Jul-2020.)  (Proof shortened by Wolf Lammen, 10-Oct-2021.) $)
    nexdv $p |- ( ph -> -. E. x ps ) $=
      ( ax-5 nexdh ) ABCACEDF $.
  $}

  ${
    $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.) $)
    2ax5 $p |- ( ph -> A. x A. y ph ) $=
      ( id alrimivv ) AABCADE $.
  $}

  ${
    $d x ph $.
    $( Version of ~ stdpc5 with a disjoint variable condition, requiring fewer
       axioms.  (Contributed by BJ, 7-Mar-2020.)  Revised to shorten ~ 19.21v .
       (Revised by Wolf Lammen, 12-Jul-2020.) $)
    stdpc5v $p |- ( A. x ( ph -> ps ) -> ( ph -> A. x ps ) ) $=
      ( wal wi ax-5 alim syl5 ) AACDABECDBCDACFABCGH $.

    $( Version of ~ 19.21 with a disjoint variable condition, requiring fewer
       axioms.

       _Notational convention_:  We sometimes suffix with "v" the label of a
       theorem using a distinct variable ("dv") condition instead of a
       nonfreeness hypothesis such as ` F/ x ph ` .  Conversely, we sometimes
       suffix with "f" the label of a theorem introducing such a nonfreeness
       hypothesis ("f" stands for "not free in", see ~ df-nf ) instead of a
       disjoint variable condition.  For instance, ~ 19.21v versus ~ 19.21 and
       ~ vtoclf versus ~ vtocl .  Note that "not free in" is less restrictive
       than "does not occur in".  Note that the version with a disjoint
       variable condition is easily proved from the version with the
       corresponding nonfreeness hypothesis, by using ~ nfv .  However, the dv
       version can often be proved from fewer axioms.  (Contributed by NM,
       21-Jun-1993.)  Reduce dependencies on axioms.  (Revised by Wolf Lammen,
       2-Jan-2020.)  (Proof shortened by Wolf Lammen, 12-Jul-2020.) $)
    19.21v $p |- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) $=
      ( wi wal stdpc5v wex ax5e imim1i 19.38 syl impbii ) ABDCEZABCEZDZABCFOACG
      ZNDMPANACHIABCJKL $.
    $( $j usage '19.21v' avoids 'ax-6' 'ax-7' 'ax-12'; $)
  $}

  ${
    $d x ph $.
    $( Version of ~ 19.32 with a disjoint variable condition, requiring fewer
       axioms.  (Contributed by BJ, 7-Mar-2020.) $)
    19.32v $p |- ( A. x ( ph \/ ps ) <-> ( ph \/ A. x ps ) ) $=
      ( wn wi wal wo 19.21v df-or albii 3bitr4i ) ADZBEZCFLBCFZEABGZCFANGLBCHOM
      CABIJANIK $.
    $( $j usage '19.32v' avoids 'ax-6' 'ax-7' 'ax-12'; $)
  $}

  ${
    $d x ps $.
    $( Version of ~ 19.31 with a disjoint variable condition, requiring fewer
       axioms.  (Contributed by BJ, 7-Mar-2020.) $)
    19.31v $p |- ( A. x ( ph \/ ps ) <-> ( A. x ph \/ ps ) ) $=
      ( wo wal 19.32v orcom albii 3bitr4i ) BADZCEBACEZDABDZCEKBDBACFLJCABGHKBG
      I $.
    $( $j usage '19.31v' avoids 'ax-6' 'ax-7' 'ax-12'; $)
  $}

  ${
    $d x ps $.
    $( Version of ~ 19.23 with a disjoint variable condition instead of a
       nonfreeness hypothesis.  (Contributed by NM, 28-Jun-1998.)  Reduce
       dependencies on axioms.  (Revised by Wolf Lammen, 11-Jan-2020.)  Remove
       dependency on ~ ax-6 .  (Revised by Rohan Ridenour, 15-Apr-2022.) $)
    19.23v $p |- ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) $=
      ( wi wal wex exim ax5e syl6 ax-5 imim2i 19.38 syl impbii ) ABDCEZACFZBDZO
      PBCFBABCGBCHIQPBCEZDOBRPBCJKABCLMN $.
    $( $j usage '19.23v' avoids 'ax-6' 'ax-7' 'ax-12'; $)
  $}

  ${
    $d x ps $.  $d y ps $.
    $( Theorem ~ 19.23v 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 ph y $.  $d ps x $.  $d x y $.
    $( Version of ~ pm11.53 with a disjoint variable condition, requiring fewer
       axioms.  (Contributed by BJ, 7-Mar-2020.) $)
    pm11.53v $p |- ( A. x A. y ( ph -> ps ) <-> ( E. x ph -> A. y ps ) ) $=
      ( wi wal wex 19.21v albii 19.23v bitri ) ABEDFZCFABDFZEZCFACGMELNCABDHIAM
      CJK $.
  $}

  ${
    $d x ps $.
    $( One direction of ~ 19.36v that can be proven without ~ ax-6 .
       (Contributed by Rohan Ridenour, 16-Apr-2022.)  (Proof shortened by Wolf
       Lammen, 22-Sep-2024.) $)
    19.36imv $p |- ( E. x ( ph -> ps ) -> ( A. x ph -> ps ) ) $=
      ( wal wi wex pm2.27 aleximi ax5e syl6com ) ACDABEZCFBCFBAKBCABGHBCIJ $.
  $}

  ${
    $d x ps $.
    19.36iv.1 $e |- E. x ( ph -> ps ) $.
    $( Inference associated with ~ 19.36v .  Version of ~ 19.36i with a
       disjoint variable condition.  (Contributed by NM, 5-Aug-1993.)  Reduce
       dependencies on axioms.  (Revised by Wolf Lammen, 17-Jan-2020.)  Remove
       dependency on ~ ax-6 .  (Revised by Rohan Ridenour, 15-Apr-2022.) $)
    19.36iv $p |- ( A. x ph -> ps ) $=
      ( wi wex wal 19.36imv ax-mp ) ABECFACGBEDABCHI $.
  $}

  ${
    $d x ph $.
    $( One direction of ~ 19.37v that can be proven without ~ ax-6 .
       (Contributed by Rohan Ridenour, 16-Apr-2022.) $)
    19.37imv $p |- ( E. x ( ph -> ps ) -> ( ph -> E. x ps ) ) $=
      ( wal wi wex ax-5 19.35 biimpi syl5 ) AACDZABECFZBCFZACGLKMEABCHIJ $.
  $}

  ${
    $d x ph $.
    19.37iv.1 $e |- E. x ( ph -> ps ) $.
    $( Inference associated with ~ 19.37v .  (Contributed by NM, 5-Aug-1993.)
       Remove dependency on ~ ax-6 .  (Revised by Rohan Ridenour,
       15-Apr-2022.) $)
    19.37iv $p |- ( ph -> E. x ps ) $=
      ( wi wex 19.37imv ax-mp ) ABECFABCFEDABCGH $.
  $}

  ${
    $d x ps $.
    $( Version of ~ 19.41 with a disjoint variable condition, requiring fewer
       axioms.  (Contributed by NM, 21-Jun-1993.)  Remove dependency on
       ~ ax-6 .  (Revised by Rohan Ridenour, 15-Apr-2022.) $)
    19.41v $p |- ( E. x ( ph /\ ps ) <-> ( E. x ph /\ ps ) ) $=
      ( wa wex 19.40 ax5e anim2i syl pm3.21 eximdv impcom impbii ) ABDZCEZACEZB
      DZOPBCEZDQABCFRBPBCGHIBPOBANCBAJKLM $.
  $}

  ${
    $d x ps $.  $d y ps $.
    $( Version of ~ 19.41 with two quantifiers and a disjoint variable
       condition requiring fewer axioms.  (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 $.
    $( Version of ~ 19.41 with three quantifiers and a disjoint variable
       condition requiring fewer axioms.  (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 $.
    $( Version of ~ 19.41 with four quantifiers and a disjoint variable
       condition requiring fewer axioms.  (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 $.
    $( Version of ~ 19.42 with a disjoint variable condition requiring fewer
       axioms.  (Contributed by NM, 21-Jun-1993.) $)
    19.42v $p |- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) ) $=
      ( wa wex 19.41v exancom ancom 3bitr4i ) BADCEBCEZADABDCEAJDBACFABCGAJHI
      $.
  $}

  ${
    $d y ph $.
    $( Distribution of existential quantifiers.  See also ~ exdistrv .
       (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 w ph $.  $d z ph $.  $d y ps $.  $d x ps $.  $d w y $.  $d y z $.
    $d w x $.  $d x z $.
    $( Distribute two pairs of existential quantifiers (over disjoint
       variables) over a conjunction.  For a version with fewer disjoint
       variable conditions but requiring more axioms, see ~ ee4anv .
       (Contributed by BJ, 5-Jan-2023.) $)
    4exdistrv $p |- ( E. x E. z E. y E. w ( ph /\ ps ) <->
                                          ( E. x E. y ph /\ E. z E. w ps ) ) $=
      ( wa wex exdistrv 2exbii bitri ) ABGFHDHZEHCHADHZBFHZGZEHCHMCHNEHGLOCEABD
      FIJMNCEIK $.
  $}

  ${
    $d x ph $.  $d y ph $.
    $( Version of ~ 19.42 with two quantifiers and a disjoint variable
       condition requiring fewer axioms.  (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 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 x ph $.  $d y ph $.  $d z ph $.
    $( Version of ~ 19.42 with three quantifiers and a disjoint variable
       condition requiring fewer axioms.  (Contributed by NM, 21-Sep-2011.)
       (Proof shortened by Wolf Lammen, 27-Aug-2023.) $)
    19.42vvv $p |- ( E. x E. y E. z ( ph /\ ps )
                       <-> ( ph /\ E. x E. y E. z ps ) ) $=
      ( wa wex exdistr2 19.42v bitri ) ABFEGDGCGABEGDGZFCGAKCGFABCDEHAKCIJ $.
  $}

  ${
    $d y ph $.  $d z ph $.  $d z ps $.
    $( Distribution of existential quantifiers in a triple conjunction.
       (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 in a quadruple conjunction.
       (Contributed by NM, 9-Mar-1995.)  (Proof shortened by Wolf Lammen,
       20-Jan-2018.) $)
    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 w3a 19.42v anbi2i df-3an 3bitr4i 3exbii 3exdistr bitri ) ABIZCDI
      ZIHJZGJFJEJABCDHJIZKZGJFJEJABUBGJIFJIEJUAUCEFGSTHJZISUBIUAUCUDUBSCDHLMSTH
      LABUBNOPABUBEFGQR $.
  $}


$(
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  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 its
  first "real" use.

$)

  $( 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 "MM>
     SHOW PROOF weq / ALL" in the Metamath program.)  (Contributed by NM,
     24-Jan-2006.) $)
  weq $p wff x = y $=
    ( cv wceq ) ACBCD $.

  ${
    speimfw.2 $e |- ( x = y -> ( ph -> ps ) ) $.
    $( Specialization, with additional weakening (compared to ~ 19.2 ) to allow
       bundling of ` x ` and ` y ` .  Uses only Tarski's FOL axiom schemes.
       (Contributed by NM, 23-Apr-2017.)  (Proof shortened by Wolf Lammen,
       5-Dec-2017.) $)
    speimfw $p |- ( -. A. x -. x = y -> ( A. x ph -> E. x ps ) ) $=
      ( weq wn wal wex df-ex biimpri com12 aleximi syl5com ) CDFZGCHGZOCIZACHBC
      IQPOCJKAOBCOABELMN $.

    $( Alternate proof of ~ speimfw (longer compressed proof, but fewer
       essential steps).  (Contributed by NM, 23-Apr-2017.)  (Proof shortened
       by Wolf Lammen, 5-Aug-2017.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    speimfwALT $p |- ( -. A. x -. x = y -> ( A. x ph -> E. x ps ) ) $=
      ( weq wex wi wn wal eximi df-ex 19.35 3imtr3i ) CDFZCGABHZCGOICJIACJBCGHO
      PCEKOCLABCMN $.
  $}

  ${
    spimfw.1 $e |- ( -. ps -> A. x -. ps ) $.
    spimfw.2 $e |- ( x = y -> ( ph -> ps ) ) $.
    $( Specialization, with additional weakening (compared to ~ sp ) to allow
       bundling of ` x ` and ` y ` .  Uses only Tarski's FOL axiom schemes.
       (Contributed by NM, 23-Apr-2017.)  (Proof shortened by Wolf Lammen,
       7-Aug-2017.) $)
    spimfw $p |- ( -. A. x -. x = y -> ( A. x ph -> ps ) ) $=
      ( weq wn wal wex speimfw df-ex con1i sylbi syl6 ) CDGHCIHACIBCJZBABCDFKPB
      HCIZHBBCLBQEMNO $.
  $}

  ${
    ax12i.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    ax12i.2 $e |- ( ps -> A. x ps ) $.
    $( Inference that has ~ ax-12 (without ` A. y ` ) as its conclusion.  Uses
       only Tarski's FOL axiom schemes.  The hypotheses may be eliminable
       without using ~ ax-12 in special cases.  Proof similar to Lemma 16 of
       [Tarski] p. 70.  (Contributed by NM, 20-May-2008.) $)
    ax12i $p |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) $=
      ( weq wi wal biimprcd alrimih biimtrdi ) CDGZABMAHZCIEBNCFMABEJKL $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Axiom scheme ax-6 (Existence)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Axiom of Existence.  One of the equality and substitution axioms of
     predicate calculus with equality.  This axiom tells us that at least one
     thing exists.  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.)  It is equivalent to axiom scheme
     C10' in [Megill] p. 448 (p. 16 of the preprint); the equivalence is
     established by ~ axc10 and ~ ax6fromc10 .  A more convenient form of this
     axiom is ~ ax6e , which has additional remarks.

     Raph Levien proved the independence of this axiom from the other logical
     axioms on 12-Apr-2005.  See item 16 at
     ~ https://us.metamath.org/award2003.html .

     ~ ax-6 can be proved from the weaker version ~ ax6v requiring that the
     variables be distinct; see Theorem ~ ax6 .

     ~ ax-6 can also be proved from the Axiom of Separation (in the form that
     we use that axiom, where free variables are not universally quantified).
     See Theorem ~ ax6vsep .

     Except by ~ ax6v , this axiom should not be referenced directly.  Instead,
     use Theorem ~ ax6 .  (Contributed by NM, 10-Jan-1993.)
     (New usage is discouraged.) $)
  ax-6 $a |- -. A. x -. x = y $.

  ${
    $d x y $.
    $( Axiom B7 of [Tarski] p. 75, which requires that ` x ` and ` y ` be
       distinct.  This trivial proof is intended merely to weaken Axiom ~ ax-6
       by adding a distinct variable restriction ($d).  From here on, ~ ax-6
       should not be referenced directly by any other proof, so that Theorem
       ~ ax6 will show that we can recover ~ ax-6 from this weaker version if
       it were an axiom (as it is in the case of Tarski).

       Note:  Introducing ` x , y ` as a distinct variable group "out of the
       blue" with no apparent justification has puzzled some people, but it is
       perfectly sound.  All we are doing is adding an additional prerequisite,
       similar to adding an unnecessary logical hypothesis, that results in a
       weakening of the theorem.  This means that any _future_ theorem that
       references ~ ax6v must have a $d specified for the two variables that
       get substituted for ` x ` and ` y ` .  The $d does not propagate
       "backwards", i.e., it does not impose a requirement on ~ ax-6 .

       When possible, use of this theorem rather than ~ ax6 is preferred since
       its derivation is much shorter and requires fewer axioms.  (Contributed
       by NM, 7-Aug-2015.) $)
    ax6v $p |- -. A. x -. x = y $=
      ( ax-6 ) ABC $.
  $}

  ${
    $d x y $.
    $( At least one individual exists.  Weaker version of ~ ax6e .  When
       possible, use of this theorem rather than ~ ax6e is preferred since its
       derivation is much shorter and requires fewer axioms.  (Contributed by
       NM, 3-Aug-2017.) $)
    ax6ev $p |- E. x x = y $=
      ( weq wex wn wal ax6v df-ex mpbir ) ABCZADJEAFEABGJAHI $.
  $}

  ${
    $d x y $.
    spimw.1 $e |- ( -. ps -> A. x -. ps ) $.
    spimw.2 $e |- ( x = y -> ( ph -> ps ) ) $.
    $( Specialization.  Lemma 8 of [KalishMontague] p. 87.  Uses only Tarski's
       FOL axiom schemes.  (Contributed by NM, 19-Apr-2017.)  (Proof shortened
       by Wolf Lammen, 7-Aug-2017.) $)
    spimw $p |- ( A. x ph -> ps ) $=
      ( weq wn wal wi ax6v spimfw ax-mp ) CDGHCIHACIBJCDKABCDEFLM $.
  $}

  ${
    $d x y $.
    spimew.1 $e |- ( ph -> A. x ph ) $.
    spimew.2 $e |- ( x = y -> ( ph -> ps ) ) $.
    $( Existential introduction, using implicit substitution.  Compare Lemma 14
       of [Tarski] p. 70.  (Contributed by NM, 7-Aug-1994.)  (Proof shortened
       by Wolf Lammen, 22-Oct-2023.) $)
    spimew $p |- ( ph -> E. x ps ) $=
      ( weq wn wal wex ax6v speimfw mpsyl ) CDGHCIHAACIBCJCDKEABCDFLM $.
  $}

  ${
    $d x y $.
    speiv.1 $e |- ( x = y -> ( ps -> ph ) ) $.
    speiv.2 $e |- ps $.
    $( Inference from existential specialization.  (Contributed by NM,
       19-Aug-1993.)  Use ~ spimew .  (Revised by Wolf Lammen, 22-Oct-2023.) $)
    speiv $p |- E. x ph $=
      ( wex hbth spimew ax-mp ) BACGFBACDBCFHEIJ $.
  $}

  ${
    $d x y $.
    speivw.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    speivw.2 $e |- ps $.
    $( Version of ~ spei with a disjoint variable condition, which does not
       require ~ ax-13 (neither ~ ax-7 nor ~ ax-12 ).  (Contributed by BJ,
       31-May-2019.) $)
    speivw $p |- E. x ph $=
      ( weq biimprd speiv ) ABCDCDGABEHFI $.
  $}

  ${
    $d x y $.
    exgen.1 $e |- ph $.
    $( Rule of existential generalization, similar to universal generalization
       ~ ax-gen , but valid only if an individual exists.  Its proof requires
       ~ ax-6 in our axiomatization but the equality predicate does not occur
       in its statement.  Some fundamental theorems of predicate calculus can
       be proven from ~ ax-gen , ~ ax-4 and this theorem alone, not requiring
       ~ ax-7 or excessive distinct variable conditions.  (Contributed by Wolf
       Lammen, 12-Nov-2017.)  (Proof shortened by Wolf Lammen, 20-Oct-2023.) $)
    exgen $p |- E. x ph $=
      ( vy weq idd speiv ) AABDBDEAFCG $.
  $}

  $( There exists a variable such that ` T. ` holds; that is, there exists a
     variable.  This corresponds under the standard translation to one of the
     formulations of the modal axiom (D), the other being ~ 19.2 .
     (Contributed by Anthony Hart, 13-Sep-2011.)  (Proof shortened by BJ,
     12-May-2019.) $)
  extru $p |- E. x T. $=
    ( wtru tru exgen ) BACD $.

  $( Theorem 19.2 of [Margaris] p. 89.  This corresponds to the axiom (D) of
     modal logic (the other standard formulation being ~ extru ).  Note:  This
     proof is very different from Margaris' because we only have Tarski's FOL
     axiom schemes available at this point.  See the later ~ 19.2g for a more
     conventional proof of a more general result, which uses additional axioms.
     The reverse implication is the defining property of effective nonfreeness
     (see ~ df-nf ).  (Contributed by NM, 2-Aug-2017.)  Remove dependency on
     ~ ax-7 .  (Revised by Wolf Lammen, 4-Dec-2017.) $)
  19.2 $p |- ( A. x ph -> E. x ph ) $=
    ( wi id exgen 19.35i ) AABAACBADEF $.

  ${
    19.2d.1 $e |- ( ph -> A. x ps ) $.
    $( Deduction associated with ~ 19.2 .  (Contributed by BJ, 12-May-2019.) $)
    19.2d $p |- ( ph -> E. x ps ) $=
      ( wal wex 19.2 syl ) ABCEBCFDBCGH $.
  $}

  ${
    19.8w.1 $e |- ( ph -> A. x ph ) $.
    $( Weak version of ~ 19.8a and instance of ~ 19.2d .  (Contributed by NM,
       1-Aug-2017.)  (Proof shortened by Wolf Lammen, 4-Dec-2017.) $)
    19.8w $p |- ( ph -> E. x ph ) $=
      ( 19.2d ) AABCD $.
  $}

  ${
    $d x y $.  $d y ph $.
    spnfw.1 $e |- ( -. ph -> A. x -. ph ) $.
    $( Weak version of ~ sp .  Uses only Tarski's FOL axiom schemes.
       (Contributed by NM, 1-Aug-2017.)  (Proof shortened by Wolf Lammen,
       13-Aug-2017.) $)
    spnfw $p |- ( A. x ph -> ph ) $=
      ( vy weq idd spimw ) AABDCBDEAFG $.
  $}

  ${
    $d x ph $.
    $( Version of ~ sp when ` x ` does not occur in ` ph ` .  Converse of
       ~ ax-5 .  Uses only Tarski's FOL axiom schemes.  (Contributed by NM,
       10-Apr-2017.)  (Proof shortened by Wolf Lammen, 4-Dec-2017.)  Shorten
       ~ 19.3v .  (Revised by Wolf Lammen, 20-Oct-2023.) $)
    spvw $p |- ( A. x ph -> ph ) $=
      ( wn ax-5 spnfw ) ABACBDE $.
    $( $j usage 'spvw' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)

    $( Version of ~ 19.3 with a disjoint variable condition, requiring fewer
       axioms.  Any formula can be universally quantified using a variable
       which it does not contain.  See also ~ 19.9v .  (Contributed by Anthony
       Hart, 13-Sep-2011.)  Remove dependency on ~ ax-7 .  (Revised by Wolf
       Lammen, 4-Dec-2017.)  (Proof shortened by Wolf Lammen, 20-Oct-2023.) $)
    19.3v $p |- ( A. x ph <-> ph ) $=
      ( wal spvw ax-5 impbii ) ABCAABDABEF $.
    $( $j usage '19.3v' avoids 'ax-12'; $)

    $( Version of ~ 19.8a with a disjoint variable condition, requiring fewer
       axioms.  Converse of ~ ax5e .  (Contributed by BJ, 12-Mar-2020.) $)
    19.8v $p |- ( ph -> E. x ph ) $=
      ( ax-5 19.8w ) ABABCD $.
    $( $j usage '19.8v' avoids 'ax-12'; $)

    $( Version of ~ 19.9 with a disjoint variable condition, requiring fewer
       axioms.  Any formula can be existentially quantified using a variable
       which it does not contain.  See also ~ 19.3v .  (Contributed by NM,
       28-May-1995.)  Remove dependency on ~ ax-7 .  (Revised by Wolf Lammen,
       4-Dec-2017.) $)
    19.9v $p |- ( E. x ph <-> ph ) $=
      ( wex ax5e 19.8v impbii ) ABCAABDABEF $.
    $( $j usage '19.9v' avoids 'ax-12'; $)
  $}

  $( Theorem 19.39 of [Margaris] p. 90.  (Contributed by NM, 12-Mar-1993.) $)
  19.39 $p |- ( ( E. x ph -> E. x ps ) -> E. x ( ph -> ps ) ) $=
    ( wex wi wal 19.2 imim1i 19.35 sylibr ) ACDZBCDZEACFZLEABECDMKLACGHABCIJ $.

  $( Theorem 19.24 of [Margaris] p. 90.  (Contributed by NM, 12-Mar-1993.) $)
  19.24 $p |- ( ( A. x ph -> A. x ps ) -> E. x ( ph -> ps ) ) $=
    ( wal wi wex 19.2 imim2i 19.35 sylibr ) ACDZBCDZEKBCFZEABECFLMKBCGHABCIJ $.

  $( Theorem 19.34 of [Margaris] p. 90.  (Contributed by NM, 12-Mar-1993.) $)
  19.34 $p |- ( ( A. x ph \/ E. x ps ) -> E. x ( ph \/ ps ) ) $=
    ( wal wex wo 19.2 orim1i 19.43 sylibr ) ACDZBCEZFACEZLFABFCEKMLACGHABCIJ $.

  ${
    $d x ps $.
    $( Version of ~ 19.36 with a disjoint variable condition instead of a
       nonfreeness hypothesis.  (Contributed by NM, 18-Aug-1993.)  Reduce
       dependencies on axioms.  (Revised by Wolf Lammen, 17-Jan-2020.) $)
    19.36v $p |- ( E. x ( ph -> ps ) <-> ( A. x ph -> ps ) ) $=
      ( wi wex wal 19.35 19.9v imbi2i bitri ) ABDCEACFZBCEZDKBDABCGLBKBCHIJ $.
  $}

  ${
    $d x ps $.  $d y ph $.  $d x y $.
    $( Version of ~ 19.12vv with a disjoint variable condition, requiring fewer
       axioms.  See also ~ 19.12 .  (Contributed by BJ, 18-Mar-2020.) $)
    19.12vvv $p |- ( E. x A. y ( ph -> ps ) <-> A. y E. x ( ph -> ps ) ) $=
      ( wi wal wex 19.21v exbii 19.36v albii bitr2i 3bitri ) ABEZDFZCGABDFZEZCG
      ACFZPEZNCGZDFZOQCABDHIAPCJUARBEZDFSTUBDABCJKRBDHLM $.
  $}

  ${
    $d x ps $.
    $( Version of ~ 19.27 with a disjoint variable condition, requiring fewer
       axioms.  (Contributed by NM, 3-Jun-2004.) $)
    19.27v $p |- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ ps ) ) $=
      ( wa wal 19.26 19.3v anbi2i bitri ) ABDCEACEZBCEZDJBDABCFKBJBCGHI $.
  $}

  ${
    $d x ph $.
    $( Version of ~ 19.28 with a disjoint variable condition, requiring fewer
       axioms.  (Contributed by NM, 25-Mar-2004.) $)
    19.28v $p |- ( A. x ( ph /\ ps ) <-> ( ph /\ A. x ps ) ) $=
      ( wa wal 19.26 19.3v bianbi ) ABDCEACEBCEAABCFACGH $.
  $}

  ${
    $d x ph $.
    $( Version of ~ 19.37 with a disjoint variable condition, requiring fewer
       axioms.  (Contributed by NM, 21-Jun-1993.) $)
    19.37v $p |- ( E. x ( ph -> ps ) <-> ( ph -> E. x ps ) ) $=
      ( wi wex wal 19.35 19.3v imbi1i bitri ) ABDCEACFZBCEZDALDABCGKALACHIJ $.
  $}

  ${
    $d x ps $.
    $( Version of ~ 19.44 with a disjoint variable condition, requiring fewer
       axioms.  (Contributed by NM, 12-Mar-1993.) $)
    19.44v $p |- ( E. x ( ph \/ ps ) <-> ( E. x ph \/ ps ) ) $=
      ( wo wex 19.43 19.9v orbi2i bitri ) ABDCEACEZBCEZDJBDABCFKBJBCGHI $.
  $}

  ${
    $d x ph $.
    $( Version of ~ 19.45 with a disjoint variable condition, requiring fewer
       axioms.  (Contributed by NM, 12-Mar-1993.) $)
    19.45v $p |- ( E. x ( ph \/ ps ) <-> ( ph \/ E. x ps ) ) $=
      ( wo wex 19.43 19.9v orbi1i bitri ) ABDCEACEZBCEZDAKDABCFJAKACGHI $.
  $}

  ${
    $d x y $.  $d x ph $.
    spimevw.1 $e |- ( x = y -> ( ph -> ps ) ) $.
    $( Existential introduction, using implicit substitution.  This is to
       ~ spimew what ~ spimvw is to ~ spimw .  Version of ~ spimev and
       ~ spimefv with an additional disjoint variable condition, using only
       Tarski's FOL axiom schemes.  (Contributed by NM, 10-Jan-1993.)  (Revised
       by BJ, 17-Mar-2020.) $)
    spimevw $p |- ( ph -> E. x ps ) $=
      ( ax-5 spimew ) ABCDACFEG $.
  $}

  ${
    $d x y $.  $d x ps $.
    spimvw.1 $e |- ( x = y -> ( ph -> ps ) ) $.
    $( A weak form of specialization.  Lemma 8 of [KalishMontague] p. 87.  Uses
       only Tarski's FOL axiom schemes.  For stronger forms using more axioms,
       see ~ spimv and ~ spimfv .  (Contributed by NM, 9-Apr-2017.) $)
    spimvw $p |- ( A. x ph -> ps ) $=
      ( wn ax-5 spimw ) ABCDBFCGEH $.
  $}

  ${
    $d x y $.  $d x ps $.
    spvv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Specialization, using implicit substitution.  Version of ~ spv with a
       disjoint variable condition, which does not require ~ ax-7 , ~ ax-12 ,
       ~ ax-13 .  (Contributed by NM, 30-Aug-1993.)  (Revised by BJ,
       31-May-2019.) $)
    spvv $p |- ( A. x ph -> ps ) $=
      ( weq biimpd spimvw ) ABCDCDFABEGH $.
  $}

  ${
    spfalw.1 $e |- -. ph $.
    $( Version of ~ sp when ` ph ` is false.  Uses only Tarski's FOL axiom
       schemes.  (Contributed by NM, 23-Apr-2017.)  (Proof shortened by Wolf
       Lammen, 25-Dec-2017.) $)
    spfalw $p |- ( A. x ph -> ph ) $=
      ( wn hbth spnfw ) ABADBCEF $.
  $}

  ${
    $d x y $.  $d x ps $.
    chvarvv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    chvarvv.2 $e |- ph $.
    $( Implicit substitution of ` y ` for ` x ` into a theorem.  Version of
       ~ chvarv with a disjoint variable condition, which does not require
       ~ ax-13 .  (Contributed by NM, 20-Apr-1994.)  (Revised by BJ,
       31-May-2019.) $)
    chvarvv $p |- ps $=
      ( spvv mpg ) ABCABCDEGFH $.
  $}

  ${
    $d x y $.
    $( Version of ~ equs4 with a disjoint variable condition, which requires
       fewer axioms.  (Contributed by NM, 10-May-1993.)  (Revised by BJ,
       31-May-2019.) $)
    equs4v $p |- ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) ) $=
      ( weq wi wal wex wa ax6ev exintr mpi ) BCDZAEBFLBGLAHBGBCILABJK $.
    $( $j usage 'equs4v' avoids 'ax-5' 'ax-7' 'ax-12' 'ax-13'; $)
  $}

  ${
    $d x y $.
    $( Version of ~ equs4v with its consequence simplified by ~ exsimpr .
       (Contributed by BJ, 9-Nov-2021.) $)
    alequexv $p |- ( A. x ( x = y -> ph ) -> E. x ph ) $=
      ( weq wi wal wex ax6ev exim mpi ) BCDZAEBFKBGABGBCHKABIJ $.
  $}

  ${
    $d x y $.  $d y ph $.
    $( One direction of the equivalence in ~ exsb is based on fewer axioms.
       (Contributed by Wolf Lammen, 2-Mar-2023.) $)
    exsbim $p |- ( E. y A. x ( x = y -> ph ) -> E. x ph ) $=
      ( weq wi wal wex alequexv exlimiv ) BCDAEBFABGCABCHI $.
  $}

  ${
    $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 ax6ev a1bi bitr4i ) BCDZAEBFLBGZAEALABHMABCIJK $.
  $}

  ${
    $d x y $.  $d x ps $.
    equsalvw.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Version of ~ equsalv with a disjoint variable condition, and of ~ equsal
       with two disjoint variable conditions, which requires fewer axioms.  See
       also the dual form ~ equsexvw .  (Contributed by BJ, 31-May-2019.) $)
    equsalvw $p |- ( A. x ( x = y -> ph ) <-> ps ) $=
      ( weq wi wal pm5.74i albii equsv bitri ) CDFZAGZCHMBGZCHBNOCMABEIJBCDKL
      $.
    $( $j usage 'equsalvw' avoids 'ax-7' 'ax-12' 'ax-13'; $)

    $( Version of ~ equsexv with a disjoint variable condition, and of ~ equsex
       with two disjoint variable conditions, which requires fewer axioms.  See
       also the dual form ~ equsalvw .  (Contributed by BJ, 31-May-2019.)
       (Proof shortened by Wolf Lammen, 23-Oct-2023.) $)
    equsexvw $p |- ( E. x ( x = y /\ ph ) <-> ps ) $=
      ( weq wa wex wn wi wal alinexa notbid equsalvw bitr3i con4bii ) CDFZAGCHZ
      BRIQAIZJCKBIZQACLSTCDQABEMNOP $.
    $( $j usage 'equsexvw' avoids 'ax-7' 'ax-12' 'ax-13'; $)
  $}

  ${
    $d x y $.
    cbvaliw.1 $e |- ( A. x ph -> A. y A. x ph ) $.
    cbvaliw.2 $e |- ( -. ps -> A. x -. ps ) $.
    cbvaliw.3 $e |- ( x = y -> ( ph -> ps ) ) $.
    $( Change bound variable.  Uses only Tarski's FOL axiom schemes.  Part of
       Lemma 7 of [KalishMontague] p. 86.  (Contributed by NM, 19-Apr-2017.) $)
    cbvaliw $p |- ( A. x ph -> A. y ps ) $=
      ( wal spimw alrimih ) ACHBDEABCDFGIJ $.
    $( $j usage 'cbvaliw' avoids 'ax-5' 'ax-7' 'ax-12' 'ax-13'; $)
  $}

  ${
    $d x y $.  $d x ps $.  $d y ph $.
    cbvalivw.1 $e |- ( x = y -> ( ph -> ps ) ) $.
    $( Change bound variable.  Uses only Tarski's FOL axiom schemes.  Part of
       Lemma 7 of [KalishMontague] p. 86.  (Contributed by NM, 9-Apr-2017.) $)
    cbvalivw $p |- ( A. x ph -> A. y ps ) $=
      ( wal spimvw alrimiv ) ACFBDABCDEGH $.
    $( $j usage 'cbvalivw' avoids 'ax-7' 'ax-12' 'ax-13'; $)
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Axiom scheme ax-7 (Equality)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Axiom of Equality.  One of the equality and substitution axioms of
     predicate calculus with equality.  It states that equality is a
     right-Euclidean binary relation (this is similar, but not identical, to
     being transitive, which is proved as ~ equtr ).  This axiom scheme is a
     sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose
     general form cannot be represented with our notation.  Also appears as
     Axiom C7 of [Monk2] p. 105 and Axiom Scheme C8' in [Megill] p. 448 (p. 16
     of the preprint).

     The equality symbol was invented in 1557 by Robert Recorde.  He chose a
     pair of parallel lines of the same length because "noe .2. thynges, can be
     moare equalle".

     We prove in ~ ax7 that this axiom can be recovered from its weakened
     version ~ ax7v where ` x ` and ` y ` are assumed to be disjoint variables.
     In particular, the only theorem referencing ~ ax-7 should be ~ ax7v .  See
     the comment of ~ ax7v for more details on these matters.  (Contributed by
     NM, 10-Jan-1993.)  (Revised by BJ, 7-Dec-2020.)  Use ~ ax7 instead.
     (New usage is discouraged.) $)
  ax-7 $a |- ( x = y -> ( x = z -> y = z ) ) $.

  ${
    $d x y $.
    $( Weakened version of ~ ax-7 , with a disjoint variable condition on
       ` x , y ` .  This should be the only proof referencing ~ ax-7 , and it
       should be referenced only by its two weakened versions ~ ax7v1 and
       ~ ax7v2 , from which ~ ax-7 is then rederived as ~ ax7 , which shows
       that either ~ ax7v or the conjunction of ~ ax7v1 and ~ ax7v2 is
       sufficient.

       In ~ ax7v , it is still allowed to substitute the same variable for
       ` x ` and ` z ` , or the same variable for ` y ` and ` z ` .  Therefore,
       ~ ax7v "bundles" (a term coined by Raph Levien) its "principal instance"
       ` ( x = y -> ( x = z -> y = z ) ) ` with ` x , y , z ` distinct, and its
       "degenerate instances" ` ( x = y -> ( x = x -> y = x ) ) ` and
       ` ( x = y -> ( x = y -> y = y ) ) ` with ` x , y ` distinct.  These
       degenerate instances are for instance used in the proofs of ~ equcomiv
       and ~ equid respectively.  (Contributed by BJ, 7-Dec-2020.)  Use ~ ax7
       instead.  (New usage is discouraged.) $)
    ax7v $p |- ( x = y -> ( x = z -> y = z ) ) $=
      ( ax-7 ) ABCD $.
  $}

  ${
    $d x y $.  $d x z $.
    $( First of two weakened versions of ~ ax7v , with an extra disjoint
       variable condition on ` x , z ` , see comments there.  (Contributed by
       BJ, 7-Dec-2020.) $)
    ax7v1 $p |- ( x = y -> ( x = z -> y = z ) ) $=
      ( ax7v ) ABCD $.
  $}

  ${
    $d x y $.  $d y z $.
    $( Second of two weakened versions of ~ ax7v , with an extra disjoint
       variable condition on ` y , z ` , see comments there.  (Contributed by
       BJ, 7-Dec-2020.) $)
    ax7v2 $p |- ( x = y -> ( x = z -> y = z ) ) $=
      ( ax7v ) ABCD $.
  $}

  ${
    $d x y $.
    $( Identity law for equality.  Lemma 2 of [KalishMontague] p. 85.  See also
       Lemma 6 of [Tarski] p. 68.  (Contributed by NM, 1-Apr-2005.)  (Revised
       by NM, 9-Apr-2017.)  (Proof shortened by Wolf Lammen, 22-Aug-2020.) $)
    equid $p |- x = x $=
      ( vy weq ax7v1 pm2.43i ax6ev exlimiiv ) BACZAACZBHIBAADEBAFG $.
  $}

  $( 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 $.

  ${
    $d x y $.
    $( Weaker form of ~ equcomi with a disjoint variable condition on
       ` x , y ` .  This is an intermediate step and ~ equcomi is fully
       recovered later.  (Contributed by BJ, 7-Dec-2020.) $)
    equcomiv $p |- ( x = y -> y = x ) $=
      ( weq equid ax7v2 mpi ) ABCAACBACADABAEF $.
  $}

  ${
    $d x y $.
    $( A commuted form of ~ ax6ev .  (Contributed by BJ, 7-Dec-2020.) $)
    ax6evr $p |- E. x y = x $=
      ( weq ax6ev equcomiv eximii ) ABCBACAABDABEF $.
  $}

  ${
    $d t x $.  $d t y $.  $d t z $.
    $( Proof of ~ ax-7 from ~ ax7v1 and ~ ax7v2 (and earlier axioms), proving
       sufficiency of the conjunction of the latter two weakened versions of
       ~ ax7v , which is itself a weakened version of ~ ax-7 .

       Note that the weakened version of ~ ax-7 obtained by adding a disjoint
       variable condition on ` x , z ` (resp. on ` y , z ` ) does not permit,
       together with the other axioms, to prove reflexivity (resp. symmetry).
       (Contributed by BJ, 7-Dec-2020.) $)
    ax7 $p |- ( x = y -> ( x = z -> y = z ) ) $=
      ( vt weq wa wi ax7v2 ax7v1 imp a1i syl2and ax6evr exlimiiv ex ) ABEZACEZB
      CEZADEZPQFRGDSPDBEZQDCEZRADBHADCHTUAFRGSTUARDBCIJKLDAMNO $.
  $}

  $( Commutative law for equality.  Equality is a symmetric relation.  Lemma 3
     of [KalishMontague] p. 85.  See also Lemma 7 of [Tarski] p. 69.
     (Contributed by NM, 10-Jan-1993.)  (Revised by NM, 9-Apr-2017.) $)
  equcomi $p |- ( x = y -> y = x ) $=
    ( weq equid ax7 mpi ) ABCAACBACADABAEF $.

  $( Commutative law for equality.  Equality is a symmetric relation.
     (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, 10-Jan-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 ax7 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 $.

  $( Commuted version of ~ equeucl (equality is left-Euclidean).  (Contributed
     by BJ, 12-Apr-2021.) $)
  equeuclr $p |- ( x = z -> ( y = z -> y = x ) ) $=
    ( weq wi equtrr equcoms ) BCDBADECACABFG $.

  $( Equality is a left-Euclidean binary relation.  (Right-Euclideanness is
     stated in ~ ax-7 .)  Curried (exported) form of ~ equtr2 .  (Contributed
     by BJ, 11-Apr-2021.) $)
  equeucl $p |- ( x = z -> ( y = z -> x = y ) ) $=
    ( weq equeuclr com12 ) BCDACDABDBACEF $.

  $( An equivalence law for equality.  (Contributed by NM, 1-Aug-1993.)  (Proof
     shortened by Wolf Lammen, 10-Dec-2017.) $)
  equequ1 $p |- ( x = y -> ( x = z <-> y = z ) ) $=
    ( weq ax7 equtr impbid ) ABDACDBCDABCEABCFG $.

  $( An equivalence law for equality.  (Contributed by NM, 21-Jun-1993.)
     (Proof shortened by Wolf Lammen, 4-Aug-2017.)  (Proof shortened by BJ,
     12-Apr-2021.) $)
  equequ2 $p |- ( x = y -> ( z = x <-> z = y ) ) $=
    ( weq equtrr equeuclr impbid ) ABDCADCBDABCEACBFG $.

  $( Equality is a left-Euclidean binary relation.  Uncurried (imported) form
     of ~ equeucl .  (Contributed by NM, 12-Aug-1993.)  (Proof shortened by
     Andrew Salmon, 25-May-2011.)  (Proof shortened by BJ, 11-Apr-2021.) $)
  equtr2 $p |- ( ( x = z /\ y = z ) -> x = y ) $=
    ( weq equeucl imp ) ACDBCDABDABCEF $.

  $( 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 $.

  ${
    $d x z $.  $d y z $.
    $( A variable introduction law for equality.  Lemma 15 of [Monk2] p. 109.
       (Contributed by NM, 9-Jan-1993.)  Remove dependencies on ~ ax-10 ,
       ~ ax-13 .  (Revised by Wolf Lammen, 10-Jun-2019.)  Move the quantified
       variable ( ` z ` ) to the left of the equality signs.  (Revised by Wolf
       Lammen, 11-Apr-2021.)  (Proof shortened by Wolf Lammen, 12-Jul-2022.) $)
    equvinv $p |- ( x = y <-> E. z ( z = x /\ z = y ) ) $=
      ( weq wa wex equequ1 equsexvw bicomi ) CADCBDZECFABDZJKCACABGHI $.
    $( $j usage 'equvinv' avoids 'ax-10' 'ax-12' 'ax-13'; $)

    $( A modified version of the forward implication of ~ equvinv adapted to
       common usage.  (Contributed by Wolf Lammen, 8-Sep-2018.) $)
    equvinva $p |- ( x = y -> E. z ( x = z /\ y = z ) ) $=
      ( weq wex wa ax6evr equtr ancrd eximdv mpi ) ABDZBCDZCEACDZMFZCECBGLMOCLM
      NABCHIJK $.
    $( $j usage 'equvinva' avoids 'ax-10' 'ax-12' 'ax-13'; $)

    $( A biconditional form of ~ equvel with disjoint variable conditions and
       proved from Tarski's FOL axiom schemes.  (Contributed by Andrew Salmon,
       2-Jun-2011.)  Reduce axiom usage.  (Revised by Wolf Lammen,
       10-Apr-2021.)  (Proof shortened by Wolf Lammen, 12-Jul-2022.) $)
    equvelv $p |- ( A. z ( z = x -> z = y ) <-> x = y ) $=
      ( weq equequ1 equsalvw ) CBDABDCACABEF $.
    $( $j usage 'equvelv' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12'
       'ax-13'; $)
  $}

  $( An equivalence between two ways of expressing ~ ax-13 .  See the comment
     for ~ ax-13 .  (Contributed by NM, 2-May-2017.)  (Proof shortened by Wolf
     Lammen, 26-Feb-2018.)  (Revised by BJ, 15-Sep-2020.) $)
  ax13b $p |- ( ( -. x = y -> ( y = z -> ph ) )
                       <-> ( -. x = y -> ( -. x = z -> ( y = z -> ph ) ) ) ) $=
    ( weq wn wi ax-1 equeuclr con3rr3 imim1d pm2.43 syl6 impbid2 pm5.74i ) BCEZ
    FZCDEZAGZBDEZFZSGZQSUBSUAHQUBRSGSQRUASRTPCBDIJKRALMNO $.
  $( $j usage 'ax13b' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)

  ${
    $d x y $.
    spfw.1 $e |- ( -. ps -> A. x -. ps ) $.
    spfw.2 $e |- ( A. x ph -> A. y A. x ph ) $.
    spfw.3 $e |- ( -. ph -> A. y -. ph ) $.
    spfw.4 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Weak version of ~ sp .  Uses only Tarski's FOL axiom schemes.  Lemma 9
       of [KalishMontague] p. 87.  This may be the best we can do with minimal
       distinct variable conditions.  (Contributed by NM, 19-Apr-2017.)  (Proof
       shortened by Wolf Lammen, 10-Oct-2021.) $)
    spfw $p |- ( A. x ph -> ph ) $=
      ( wal weq biimpd cbvaliw wi biimprd equcoms spimw syl ) ACIBDIAABCDFECDJZ
      ABHKLBADCGBAMCDRABHNOPQ $.
    $( $j usage 'spfw' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    $d x y $.  $d x ps $.  $d y ph $.
    spw.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Weak version of the specialization scheme ~ sp .  Lemma 9 of
       [KalishMontague] p. 87.  While it appears that ~ sp in its general form
       does not follow from Tarski's FOL axiom schemes, from this theorem we
       can prove any _instance_ of ~ sp having mutually distinct setvar
       variables and no wff metavariables (see ~ ax12wdemo for an example of
       the procedure to eliminate the hypothesis).  Other approximations of
       ~ sp are ~ spfw (minimal distinct variable requirements), ~ spnfw (when
       ` x ` is not free in ` -. ph ` ), ~ spvw (when ` x ` does not appear in
       ` ph ` ), ~ sptruw (when ` ph ` is true), ~ spfalw (when ` ph ` is
       false), and ~ spvv (where ` ph ` is changed into ` ps ` ).  (Contributed
       by NM, 9-Apr-2017.)  (Proof shortened by Wolf Lammen, 27-Feb-2018.) $)
    spw $p |- ( A. x ph -> ph ) $=
      ( wn ax-5 wal spfw ) ABCDBFCGACHDGAFDGEI $.
    $( $j usage 'spw' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    $d x y $.
    cbvalw.1 $e |- ( A. x ph -> A. y A. x ph ) $.
    cbvalw.2 $e |- ( -. ps -> A. x -. ps ) $.
    cbvalw.3 $e |- ( A. y ps -> A. x A. y ps ) $.
    cbvalw.4 $e |- ( -. ph -> A. y -. ph ) $.
    cbvalw.5 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Change bound variable.  Uses only Tarski's FOL axiom schemes.
       (Contributed by NM, 9-Apr-2017.) $)
    cbvalw $p |- ( A. x ph <-> A. y ps ) $=
      ( wal weq biimpd cbvaliw wi biimprd equcoms impbii ) ACJBDJABCDEFCDKZABIL
      MBADCGHBANCDRABIOPMQ $.
    $( $j usage 'cbvalw' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12'
       'ax-13'; $)
  $}

  ${
    $d x y $.  $d x ps $.  $d y ph $.
    cbvalvw.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Change bound variable.  Uses only Tarski's FOL axiom schemes.  See
       ~ cbvalv for a version with fewer disjoint variable conditions but
       requiring more axioms.  (Contributed by NM, 9-Apr-2017.)  (Proof
       shortened by Wolf Lammen, 28-Feb-2018.) $)
    cbvalvw $p |- ( A. x ph <-> A. y ps ) $=
      ( wal ax-5 wn cbvalw ) ABCDACFDGBHCGBDFCGAHDGEI $.
    $( $j usage 'cbvalvw' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12'
       'ax-13'; $)

    $( Change bound variable.  Uses only Tarski's FOL axiom schemes.  See
       ~ cbvexv for a version with fewer disjoint variable conditions but
       requiring more axioms.  (Contributed by NM, 19-Apr-2017.) $)
    cbvexvw $p |- ( E. x ph <-> E. y ps ) $=
      ( wn wal wex weq notbid cbvalvw notbii df-ex 3bitr4i ) AFZCGZFBFZDGZFACHB
      DHPROQCDCDIABEJKLACMBDMN $.
    $( $j usage 'cbvexvw' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12'
       'ax-13'; $)
  $}

  ${
    $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, requiring fewer axioms.  (Contributed by
       David Moews, 1-May-2017.)  Avoid ~ ax-13 .  (Revised by GG,
       10-Jan-2024.)  Reduce axiom usage, along an idea of GG. (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, requiring fewer axioms.  (Contributed by
       David Moews, 1-May-2017.)  Avoid ~ ax-13 .  (Revised by GG,
       10-Jan-2024.)  Reduce axiom usage.  (Revised by Wolf Lammen,
       10-Feb-2024.) $)
    cbvexdvaw $p |- ( ph -> ( E. x ps <-> E. y ch ) ) $=
      ( wex wn wal weq wa notbid cbvaldvaw alnex 3bitr3g con4bid ) ABDGZCEGZABH
      ZDICHZEIQHRHASTDEADEJKBCFLMBDNCENOP $.
  $}

  ${
    $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.
       Version of ~ cbval2vv with more disjoint variable conditions, which
       requires fewer axioms .  (Contributed by NM, 4-Feb-2005.)  Avoid
       ~ ax-13 .  (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.
       Version of ~ cbvex2vv with more disjoint variable conditions, which
       requires fewer axioms .  (Contributed by NM, 26-Jul-1995.)  Avoid
       ~ ax-13 .  (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. $)
    cbvex4vw.vf $f setvar f $.
    cbvex4vw.vg $f setvar g $.
    $d w z ch $.  $d u v ph $.  $d x y ps $.  $d f g ps $.  $d f g w z $.
    $d u v w x y z $.
    cbvex4vw.1 $e |- ( ( x = v /\ y = u ) -> ( ph <-> ps ) ) $.
    cbvex4vw.2 $e |- ( ( z = f /\ w = g ) -> ( ps <-> ch ) ) $.
    $( Rule used to change bound variables, using implicit substitution.
       Version of ~ cbvex4v with more disjoint variable conditions, which
       requires fewer axioms.  (Contributed by NM, 26-Jul-1995.)  Avoid
       ~ ax-13 .  (Revised by GG, 10-Jan-2024.) $)
    cbvex4vw $p |- ( E. x E. y E. z E. w ph <-> E. v E. u E. f E. g ch ) $=
      ( wex weq wa 2exbidv cbvex2vw 2exbii bitri ) AGNFNZENDNBGNFNZINHNCKNJNZIN
      HNUAUBDEHIDHOEIOPABFGLQRUBUCHIBCFGJKMRST $.
  $}

  ${
    $d y z $.  $d x y $.  $d z ph $.  $d y ps $.
    alcomimw.1 $e |- ( y = z -> ( ph <-> ps ) ) $.
    $( Weak version of ~ ax-11 .  See ~ alcomw for the biconditional form.
       Uses only Tarski's FOL axiom schemes.  (Contributed by NM, 10-Apr-2017.)
       (Proof shortened by Wolf Lammen, 28-Dec-2023.) $)
    alcomimw $p |- ( A. x A. y ph -> A. y A. x ph ) $=
      ( wal cbvalvw biimpi alimi ax-5 wi weq biimprd equcoms spimvw 2alimi 3syl
      ) ADGZCGBEGZCGZUADGACGDGSTCSTABDEFHIJUADKTADCBAEDBALDEDEMABFNOPQR $.
    $( $j usage 'alcomimw' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12'
       'ax-13'; $)
  $}

  ${
    $d ph z $.  $d ps x $.  $d x y $.  $d x z $.
    excomimw.1 $e |- ( x = z -> ( ph <-> ps ) ) $.
    $( Weak version of ~ excomim .  Uses only Tarski's FOL axiom schemes.
       (Contributed by BTernaryTau, 23-Jun-2025.) $)
    excomimw $p |- ( E. x E. y ph -> E. y E. x ph )
        $=
      ( wn wal wex weq notbid alcomimw con3i 2exnaln 3imtr4i ) AGZDHCHZGPCHDHZG
      ADICIACIDIRQPBGDCECEJABFKLMACDNADCNO $.
    $( $j usage 'excomimw' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12'
       'ax-13'; $)
  $}

  ${
    $d ph z $.  $d ph w $.  $d ps x $.  $d ch y $.  $d x y $.  $d y z $.
    $d w x $.
    alcomw.1 $e |- ( x = w -> ( ph <-> ps ) ) $.
    alcomw.2 $e |- ( y = z -> ( ph <-> ch ) ) $.
    $( Weak version of ~ alcom and biconditional form of ~ alcomimw .  Uses
       only Tarski's FOL axiom schemes.  (Contributed by BTernaryTau,
       28-Dec-2024.) $)
    alcomw $p |- ( A. x A. y ph <-> A. y A. x ph ) $=
      ( wal alcomimw impbii ) AEJDJADJEJACDEFIKABEDGHKL $.
    $( $j usage 'alcomw' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12'
       'ax-13'; $)
  $}

  ${
    $d x y $.
    hbn1fw.1 $e |- ( A. x ph -> A. y A. x ph ) $.
    hbn1fw.2 $e |- ( -. ps -> A. x -. ps ) $.
    hbn1fw.3 $e |- ( A. y ps -> A. x A. y ps ) $.
    hbn1fw.4 $e |- ( -. ph -> A. y -. ph ) $.
    hbn1fw.5 $e |- ( -. A. y ps -> A. x -. A. y ps ) $.
    hbn1fw.6 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Weak version of ~ ax-10 from which we can prove any ~ ax-10 instance not
       involving wff variables or bundling.  Uses only Tarski's FOL axiom
       schemes.  (Contributed by NM, 19-Apr-2017.)  (Proof shortened by Wolf
       Lammen, 28-Feb-2018.) $)
    hbn1fw $p |- ( -. A. x ph -> A. x -. A. x ph ) $=
      ( wal wn cbvalw notbii hbxfrbi ) ACKZLBDKZLCPQABCDEFGHJMNIO $.
    $( $j usage 'hbn1fw' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12'
       'ax-13'; $)
  $}

  ${
    $d y ph $.  $d x ps $.  $d x y $.
    hbn1w.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Weak version of ~ hbn1 .  Uses only Tarski's FOL axiom schemes.
       (Contributed by NM, 9-Apr-2017.) $)
    hbn1w $p |- ( -. A. x ph -> A. x -. A. x ph ) $=
      ( wal ax-5 wn hbn1fw ) ABCDACFDGBHCGBDFZCGAHDGJHCGEI $.
    $( $j usage 'hbn1w' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12'
       'ax-13'; $)

    $( Weak version of ~ hba1 .  See comments for ~ ax10w .  Uses only Tarski's
       FOL axiom schemes.  (Contributed by NM, 9-Apr-2017.)  (Proof shortened
       by Wolf Lammen, 10-Oct-2021.) $)
    hba1w $p |- ( A. x ph -> A. x A. x ph ) $=
      ( wal wn wb weq cbvalvw notbii a1i spw con2i hbn1w con1i alimi 3syl ) ACF
      ZSGZCFZGZUBCFSCFUASTBDFZGZCDTUDHCDISUCABCDEJKLZMNTUDCDUEOUBSCSUAABCDEOPQR
      $.
    $( $j usage 'hba1w' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12'
       'ax-13'; $)

    $( Weak version of ~ hbe1 .  See comments for ~ ax10w .  Uses only Tarski's
       FOL axiom schemes.  (Contributed by NM, 19-Apr-2017.) $)
    hbe1w $p |- ( E. x ph -> A. x E. x ph ) $=
      ( wex wn wal df-ex weq notbid hbn1w hbxfrbi ) ACFAGZCHGCACINBGCDCDJABEKLM
      $.
    $( $j usage 'hbe1w' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12'
       'ax-13'; $)
  $}

  ${
    $d x z $.  $d x y $.  $d z ph $.  $d x ps $.
    hbalw.1 $e |- ( x = z -> ( ph <-> ps ) ) $.
    hbalw.2 $e |- ( ph -> A. x ph ) $.
    $( Weak version of ~ hbal .  Uses only Tarski's FOL axiom schemes.  Unlike
       ~ hbal , this theorem requires that ` x ` and ` y ` be distinct, i.e.,
       not be bundled.  (Contributed by NM, 19-Apr-2017.) $)
    hbalw $p |- ( A. y ph -> A. x A. y ph ) $=
      ( wal alimi alcomimw syl ) ADHZACHZDHLCHAMDGIABDCEFJK $.
    $( $j usage 'hbalw' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12'
       'ax-13'; $)
  $}

  ${
    $d x y $.  $d ps x $.  $d ph y $.
    19.8aw.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( If a formula is true, then it is true for at least one instance.  This
       is to ~ 19.8a what ~ spw is to ~ sp .  (Contributed by SN,
       26-Sep-2024.) $)
    19.8aw $p |- ( ph -> E. x ph ) $=
      ( wex wn wal alnex weq notbid spw sylbir con4i ) ACFZAOGAGZCHPACIPBGCDCDJ
      ABEKLMN $.
  $}

  ${
    $d x y $.  $d ps x $.  $d ph y $.
    exexw.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Existential quantification over a given variable is idempotent.  Weak
       version of ~ bj-exexbiex , requiring fewer axioms.  (Contributed by GG,
       4-Nov-2024.) $)
    exexw $p |- ( E. x ph <-> E. x E. x ph ) $=
      ( wal wex weq notbid hba1w spw alimi impbii notbii df-ex 2exnaln 3bitr4i
      wn ) ARZCFZRTCFZRACGZUBCGTUATUASBRZCDCDHABEIZJTSCSUCCDUDKLMNACOACCPQ $.
    $( $j usage 'exexw' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    $d x y z $.
    $( A special instance of ~ sp applied to an equality with a disjoint
       variable condition.  Unlike the more general ~ sp , we can prove this
       without ~ ax-12 .  Instance of ~ aeveq .

       The antecedent ` A. x x = y ` with distinct ` x ` and ` y ` is a
       characteristic of a degenerate universe, in which just one object
       exists.  Actually more than one object may still exist, but if so, we
       give up on equality as a discriminating term.

       Separating this degenerate case from a richer universe, where inequality
       is possible, is a common proof idea.  The name of this theorem follows a
       convention, where the condition ` A. x x = y ` is denoted by 'aev', a
       shorthand for 'all equal, with a distinct variable condition'.
       (Contributed by Wolf Lammen, 14-Mar-2021.) $)
    spaev $p |- ( A. x x = y -> x = y ) $=
      ( vz weq equequ1 spw ) ABDCBDACACBEF $.
    $( $j usage 'spaev' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12'
       'ax-13'; $)
  $}

  ${
    $d x y t $.  $d y z t $.
    $( Change bound variable in an equality with a disjoint variable condition.
       Instance of ~ aev .  (Contributed by NM, 22-Jul-2015.)  (Revised by BJ,
       18-Jun-2019.) $)
    cbvaev $p |- ( A. x x = y -> A. z z = y ) $=
      ( vt weq wal ax7 cbvalivw syl ) ABEZAFDBEZDFCBEZCFJKADADBGHKLDCDCBGHI $.
    $( $j usage 'cbvaev' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12'
       'ax-13'; $)
  $}

  ${
    $d x y z $.
    $( Lemma for ~ aevlem .  Instance of ~ aev .  (Contributed by NM,
       8-Jul-2016.)  (Proof shortened by Wolf Lammen, 17-Feb-2018.)  Remove
       dependency on ~ ax-12 .  (Revised by Wolf Lammen, 14-Mar-2021.)  Extract
       from proof of a former lemma for ~ axc11n and add DV condition to reduce
       axiom usage.  (Revised by BJ, 29-Mar-2021.)  (Proof shortened by Wolf
       Lammen, 30-Mar-2021.) $)
    aevlem0 $p |- ( A. x x = y -> A. z z = x ) $=
      ( weq wal spaev alrimiv cbvaev equeuclr al2imi sylc ) ABDZAEZLCECBDZCECAD
      ZCEMLCABFGABCHLNOCACBIJK $.
  $}

  ${
    $d x y u $.  $d z t u $.
    $( Lemma for ~ aev and ~ axc16g .  Change free and bound variables.
       Instance of ~ aev .  (Contributed by NM, 22-Jul-2015.)  (Proof shortened
       by Wolf Lammen, 17-Feb-2018.)  Remove dependency on ~ ax-13 , along an
       idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.)  Reduce axiom usage.
       (Revised by BJ, 29-Mar-2021.) $)
    aevlem $p |- ( A. x x = y -> A. z z = t ) $=
      ( vu weq wal cbvaev aevlem0 4syl ) ABFAGEBFEGAEFAGDEFDGCDFCGABEHEBAIAEDHD
      ECIJ $.
  $}

  ${
    $d x y $.  $d u z $.  $d u t $.
    $( The antecedent ` A. x x = y ` with a disjoint variable condition
       (typical of a one-object universe) forces equality of everything.
       (Contributed by Wolf Lammen, 19-Mar-2021.) $)
    aeveq $p |- ( A. x x = y -> z = t ) $=
      ( vu weq wal wex aevlem ax6ev ax7 aleximi mpi ax5e 3syl ) ABFAGECFZEGZCDF
      ZEHZRABECIQEDFZEHSEDJPTREECDKLMRENO $.
    $( $j usage 'aeveq' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12'
       'ax-13'; $)
  $}

  ${
    $d x y $.  $d v w z $.
    $( A "distinctor elimination" lemma with no disjoint variable conditions on
       variables in the consequent.  (Contributed by NM, 8-Nov-2006.)  Remove
       dependency on ~ ax-11 .  (Revised by Wolf Lammen, 7-Sep-2018.)  Remove
       dependency on ~ ax-13 , inspired by an idea of BJ. (Revised by Wolf
       Lammen, 30-Nov-2019.)  Remove dependency on ~ ax-12 .  (Revised by Wolf
       Lammen, 19-Mar-2021.) $)
    aev $p |- ( A. x x = y -> A. z t = u ) $=
      ( vv vw weq wal aevlem aeveq alrimiv syl ) ABHAIFGHFIZEDHZCIABFGJNOCFGEDK
      LM $.
    $( $j usage 'aev' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    $v s $.  $d w s z $.
    $( Define temporary individual variable s. $)
    v.vs $f setvar s $.
    $d x y $.  $d w s $.
    $( A version of ~ aev with two universal quantifiers in the consequent.
       One can prove similar statements with arbitrary numbers of universal
       quantifiers in the consequent (the series begins with ~ aeveq , ~ aev ,
       ~ aev2 ).

       Using ~ aev and ~ alrimiv , one can actually prove (with no more axioms)
       any scheme of the form ` ( A. x x = y -> ` PHI) , DV ` ( x , y ) ` where
       PHI involves only setvar variables and the connectors ` -> ` , ` <-> ` ,
       ` /\ ` , ` \/ ` , ` T. ` , ` = ` , ` A. ` , ` E. ` , ` E* ` , ` E! ` ,
       ` F/ ` .  An example is given by ~ aevdemo .  This list cannot be
       extended to ` -. ` or ` F. ` since the scheme ` A. x x = y ` is
       consistent with ~ ax-mp , ~ ax-gen , ~ ax-1 -- ~ ax-13 (as the
       one-element universe shows), so for instance ` ( A. x x = y -> F. ) , `
       DV ` ( x , y ) ` is not provable from these axioms alone (indeed, ~ dtru
       uses non-logical axioms as well).  (Contributed by BJ, 23-Mar-2021.) $)
    aev2 $p |- ( A. x x = y -> A. z A. t u = v ) $=
      ( vw v.vs weq wal aev alrimiv syl ) ABIAJGHIGJZEDIFJZCJABGHGKNOCGHFDEKLM
      $.
    $( $j usage 'aev2' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    $d x y $.
    $( All variables are effectively bound in an identical variable specifier.
       Version of ~ hbae with a disjoint variable condition, requiring fewer
       axioms.  Instance of ~ aev2 .  (Contributed by NM, 13-May-1993.)  Reduce
       axiom usage.  (Revised by Wolf Lammen, 22-Mar-2021.) $)
    hbaev $p |- ( A. x x = y -> A. z A. x x = y ) $=
      ( aev2 ) ABCBAAD $.
    $( $j usage 'hbaev' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12'
       'ax-13'; $)
  $}

  ${
    $d u v $.
    $( If some set variables can assume different values, then any two distinct
       set variables cannot always be the same.  (Contributed by Wolf Lammen,
       10-Aug-2019.) $)
    naev $p |- ( -. A. x x = y -> -. A. u u = v ) $=
      ( weq wal aev con3i ) DCEDFABEAFDCABAGH $.
    $( $j usage 'naev' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    $d t u $.  $d v w z $.
    $( Generalization of ~ hbnaev .  (Contributed by Wolf Lammen,
       9-Apr-2021.) $)
    naev2 $p |- ( -. A. x x = y -> A. z -. A. t t = u ) $=
      ( vv vw weq wal wn naev ax-5 alimi 3syl ) ABHAIJFGHFIJZOCIEDHEIJZCIABGFKO
      CLOPCFGDEKMN $.
    $( $j usage 'naev2' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12'
       'ax-13'; $)
  $}

  ${
    $d x y $.
    $( Any variable is free in ` -. A. x x = y ` , if ` x ` and ` y ` are
       distinct.  This condition is dropped in ~ hbnae , at the expense of more
       axiom dependencies.  Instance of ~ naev2 .  (Contributed by NM,
       13-May-1993.)  (Revised by Wolf Lammen, 9-Apr-2021.) $)
    hbnaev $p |- ( -. A. x x = y -> A. z -. A. x x = y ) $=
      ( naev2 ) ABCBAD $.
    $( $j usage 'hbnaev' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12'
       'ax-13'; $)
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Define proper substitution
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x y z $.  $d t y z $.  $d ph y z $.
    $( Justification theorem for ~ df-sb proved from Tarski's FOL axiom
       schemes.  (Contributed by BJ, 22-Jan-2023.) $)
    sbjust $p |- ( A. y ( y = t -> A. x ( x = y -> ph ) ) <->
                   A. z ( z = t -> A. x ( x = z -> ph ) ) ) $=
      ( weq wi wal equequ1 equequ2 imbi1d albidv imbi12d cbvalvw ) CEFZBCFZAGZB
      HZGDEFZBDFZAGZBHZGCDCDFZOSRUBCDEIUCQUABUCPTACDBJKLMN $.
  $}

  $c [ $.  $( Left bracket $)
  $c / $.  $( Slash. $)
  $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 $.

  $( Indicate that the variable "y" is free in wsb even though it could
     potentially bind occurrences in "ph". $)
  $( $j free_var 'wsb' with 'y'; $)

  ${
    $d y x $.  $d y t $.  $d y ph $.
    $( Define proper substitution.  For our notation, we use ` [ t / x ] ph `
       to mean "the wff that results from the proper substitution of ` t ` for
       ` x ` in the wff ` ph ` ".  That is, ` t ` properly replaces ` x ` .
       For example, ` [ t / x ] z e. x ` is the same as ` z e. t ` (when ` x `
       and ` z ` are distinct), as shown in ~ elsb2 .

       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 ( t ) ` is the wff that results
       when ` t ` is properly substituted for ` x ` in ` ph ( x ) ` ".  For
       example, if the original ` ph ( x ) ` is ` x = t ` , then ` ph ( t ) `
       is ` t = t ` , from which we obtain that ` ph ( x ) ` is ` x = x ` .  So
       what exactly does ` ph ( x ) ` mean?  Curry's notation solves this
       problem.

       A very similar notation, namely ` ( y | x ) ph ` , was introduced in
       Bourbaki's Set Theory (Chapter 1, Description of Formal Mathematic,
       1953).

       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 ` t ` are replaced
       with the same variable, as ~ sbid shows.  We achieve this by applying
       twice Tarski's definition ~ sb6 which is valid for disjoint variables,
       and introducing a dummy variable ` y ` which isolates ` x ` from ` t ` ,
       as in ~ dfsb7 with respect to ~ sb5 .  We can also achieve this by
       having ` x ` free in the first conjunct and bound in the second, as the
       alternate definition ~ dfsb1 shows.  Another version that mixes free and
       bound variables is ~ dfsb3 .  When ` x ` and ` t ` are distinct, we can
       express proper substitution with the simpler expressions of ~ sb5 and
       ~ sb6 .

       Note that the occurrences of a given variable in the definiens are
       either all bound ( ` x , y ` ) or all free ( ` t ` ).  Also note that
       the definiens uses only primitive symbols.

       This double level definition will make several proofs using it appear as
       doubled.  Alternately, one could often first prove as a lemma the same
       theorem with a disjoint variable condition on the substitute and the
       substituted variables, and then prove the original theorem by applying
       this lemma twice in a row.  (Contributed by NM, 10-May-1993.)  Revised
       from the original definition ~ dfsb1 .  (Revised by BJ, 22-Dec-2020.) $)
    df-sb $a |- ( [ t / x ] ph <-> A. y ( y = t -> A. x ( x = y -> ph ) ) ) $.
  $}

  ${
    $d x y $.  $d t y $.  $d ph y $.
    sbt.1 $e |- ph $.
    $( A substitution into a theorem yields a theorem.  See ~ sbtALT for a
       shorter proof requiring more axioms.  See ~ chvar and ~ chvarv for
       versions using implicit substitution.  (Contributed by NM, 21-Jan-2004.)
       (Proof shortened by Andrew Salmon, 25-May-2011.)  (Proof shortened by
       Wolf Lammen, 20-Jul-2018.)  Revise ~ df-sb .  (Revised by Steven Nguyen,
       6-Jul-2023.) $)
    sbt $p |- [ t / x ] ph $=
      ( vy wsb weq wi wal a1i ax-gen df-sb mpbir ) ABCFECGZBEGZAHZBIZHZEIREQNPB
      AODJKJKABECLM $.
  $}

  $( The result of substituting in the truth constant "true" is true.
     (Contributed by BJ, 2-Sep-2023.) $)
  sbtru $p |- [ y / x ] T. $=
    ( wtru tru sbt ) CABDE $.

  ${
    $d y x $.  $d y t $.  $d y ph $.
    $( 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 ` t ` (properly) substituted for ` x ` .  Translated to
       traditional notation, it can be read:  " ` A. x ph ( x ) -> ph ( t ) ` ,
       provided that ` t ` is free for ` x ` in ` ph ( x ) ` ".  Axiom 4 of
       [Mendelson] p. 69.  See also ~ spsbc and ~ rspsbc .  (Contributed by NM,
       14-May-1993.)  Revise ~ df-sb .  (Revised by BJ, 22-Dec-2020.) $)
    stdpc4 $p |- ( A. x ph -> [ t / x ] ph ) $=
      ( vy wal weq wi wsb ala1 a1d alrimiv df-sb sylibr ) ABEZDCFZBDFZAGBEZGZDE
      ABCHNRDNQOAPBIJKABDCLM $.
  $}

  ${
    sbtALT.1 $e |- ph $.
    $( Alternate proof of ~ sbt , shorter but using ~ ax-4 and ~ ax-5 .
       (Contributed by NM, 21-Jan-2004.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    sbtALT $p |- [ y / x ] ph $=
      ( wsb stdpc4 mpg ) AABCEBABCFDG $.
  $}

  $( A double specialization using explicit substitution.  This is Theorem
     PM*11.1 in [WhiteheadRussell] p. 159.  See ~ stdpc4 for the analogous
     single specialization.  See ~ 2sp for another double specialization.
     (Contributed by Andrew Salmon, 24-May-2011.)  (Revised by BJ,
     21-Oct-2018.) $)
  2stdpc4 $p |- ( A. x A. y ph -> [ z / x ] [ w / y ] ph ) $=
    ( wal wsb stdpc4 alimi syl ) ACFZBFACEGZBFLBDGKLBACEHILBDHJ $.

  ${
    $d y z $.  $d x z $.  $d ph z $.  $d ps z $.
    $( Distribute substitution over implication.  (Contributed by NM,
       14-May-1993.)  Remove dependencies on axioms.  (Revised by Steven
       Nguyen, 24-Jul-2023.) $)
    sbi1 $p |- ( [ y / x ] ( ph -> ps ) ->
                                          ( [ y / x ] ph -> [ y / x ] ps ) ) $=
      ( vz wi wsb weq wal df-sb ax-2 al2imi imim3i 3imtr4g sylbi ) ABFZCDGEDHZC
      EHZPFZCIZFZEIZACDGZBCDGZFPCEDJUBQRAFZCIZFZEIQRBFZCIZFZEIUCUDUAUGUJETUFUIQ
      SUEUHCRABKLMLACEDJBCEDJNO $.
  $}

  $( Distribute substitution over implication.  Closed form of ~ sbimi .
     Specialization of implication.  (Contributed by NM, 5-Aug-1993.)  (Proof
     shortened by Andrew Salmon, 25-May-2011.)  Revise ~ df-sb .  (Revised by
     BJ, 22-Dec-2020.)  (Proof shortened by Steven Nguyen, 24-Jul-2023.) $)
  spsbim $p |- ( A. x ( ph -> ps ) -> ( [ t / x ] ph -> [ t / x ] ps ) ) $=
    ( wi wal wsb stdpc4 sbi1 syl ) ABEZCFKCDGACDGBCDGEKCDHABCDIJ $.

  $( Biconditional property for substitution.  Closed form of ~ sbbii .
     Specialization of biconditional.  (Contributed by NM, 2-Jun-1993.)  Revise
     ~ df-sb .  (Revised by BJ, 22-Dec-2020.) $)
  spsbbi $p |- ( A. x ( ph <-> ps ) -> ( [ t / x ] ph <-> [ t / x ] ps ) ) $=
    ( wb wal wsb wi biimp alimi spsbim syl biimpr impbid ) ABEZCFZACDGZBCDGZPAB
    HZCFQRHOSCABIJABCDKLPBAHZCFRQHOTCABMJBACDKLN $.

  ${
    sbimi.1 $e |- ( ph -> ps ) $.
    $( Distribute substitution over implication.  (Contributed by NM,
       25-Jun-1998.)  Revise ~ df-sb .  (Revised by BJ, 22-Dec-2020.)  (Proof
       shortened by Steven Nguyen, 24-Jul-2023.) $)
    sbimi $p |- ( [ t / x ] ph -> [ t / x ] ps ) $=
      ( wi wsb sbt sbi1 ax-mp ) ABFZCDGACDGBCDGFKCDEHABCDIJ $.
  $}

  ${
    sb2imi.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Distribute substitution over implication.  Compare ~ al2imi .
       (Contributed by Steven Nguyen, 13-Aug-2023.) $)
    sb2imi $p |- ( [ t / x ] ph -> ( [ t / x ] ps -> [ t / x ] ch ) ) $=
      ( wsb wi sbimi sbi1 syl ) ADEGBCHZDEGBDEGCDEGHALDEFIBCDEJK $.
  $}

  ${
    sbbii.1 $e |- ( ph <-> ps ) $.
    $( Infer substitution into both sides of a logical equivalence.
       (Contributed by NM, 14-May-1993.) $)
    sbbii $p |- ( [ t / x ] ph <-> [ t / x ] ps ) $=
      ( wsb biimpi sbimi biimpri impbii ) ACDFBCDFABCDABEGHBACDABEIHJ $.

    $( Infer double substitution into both sides of a logical equivalence.
       (Contributed by AV, 30-Jul-2023.) $)
    2sbbii $p |- ( [ t / x ] [ u / y ] ph <-> [ t / x ] [ u / y ] ps ) $=
      ( wsb sbbii ) ADEHBDEHCFABDEGII $.
  $}

  ${
    $d x ph $.
    sbimdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction substituting both sides of an implication, with ` ph ` and
       ` x ` disjoint.  See also ~ sbimd .  (Contributed by Wolf Lammen,
       6-May-2023.)  Revise ~ df-sb .  (Revised by Steven Nguyen,
       6-Jul-2023.) $)
    sbimdv $p |- ( ph -> ( [ t / x ] ps -> [ t / x ] ch ) ) $=
      ( wi wal wsb alrimiv spsbim syl ) ABCGZDHBDEICDEIGAMDFJBCDEKL $.
  $}

  ${
    $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 $.
  $}

  $( Conjunction inside and outside of a substitution are equivalent.  Compare
     ~ 19.26 .  (Contributed by NM, 14-May-1993.)  (Proof shortened by Steven
     Nguyen, 13-Aug-2023.) $)
  sban $p |- ( [ y / x ] ( ph /\ ps ) <-> ( [ y / x ] ph /\ [ y / x ] ps ) ) $=
    ( wa wsb simpl sbimi simpr jca pm3.2 sb2imi imp impbii ) ABEZCDFZACDFZBCDFZ
    EPQROACDABGHOBCDABIHJQRPABOCDABKLMN $.

  $( Threefold 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 ) ) $=
    ( wa wsb w3a sban anbi1i df-3an sbbii bitri 3bitr4i ) ABFZDEGZCDEGZFZADEGZB
    DEGZFZQFABCHZDEGZSTQHPUAQABDEIJUCOCFZDEGRUBUDDEABCKLOCDEIMSTQKN $.

  ${
    $d y x $.  $d y t $.  $d y ph $.
    $( Existential generalization: if a proposition is true for a specific
       instance, then there exists an instance where it is true.  (Contributed
       by NM, 29-Jun-1993.)  (Proof shortened by Wolf Lammen, 3-May-2018.)
       Revise ~ df-sb .  (Revised by BJ, 22-Dec-2020.)  (Proof shortened by
       Steven Nguyen, 11-Jul-2023.) $)
    spsbe $p |- ( [ t / x ] ph -> E. x ph ) $=
      ( vy wsb weq wi wal wex df-sb alequexv sylbi exsbim syl ) ABCEZBDFAGBHZDI
      ZABIODCFPGDHQABDCJPDCKLABDMN $.
  $}

  ${
    $d u x $.  $d u y $.  $d u z $.  $d u ph $.
    $( Equality property for substitution, from Tarski's system.  Used in proof
       of Theorem 9.7 in [Megill] p. 449 (p. 16 of the preprint).  (Contributed
       by NM, 14-May-1993.)  Revise ~ df-sb .  (Revised by BJ, 30-Dec-2020.) $)
    sbequ $p |- ( x = y -> ( [ x / z ] ph <-> [ y / z ] ph ) ) $=
      ( vu weq wi wal wsb equequ2 imbi1d albidv df-sb 3bitr4g ) BCFZEBFZDEFAGDH
      ZGZEHECFZQGZEHADBIADCIORTEOPSQBCEJKLADEBMADECMN $.
  $}

  $( An equality theorem for substitution.  (Contributed by NM, 14-May-1993.)
     (Proof shortened by Wolf Lammen, 15-Sep-2018.)  (Proof shortened by Steven
     Nguyen, 7-Jul-2023.) $)
  sbequi $p |- ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) $=
    ( weq wsb sbequ biimpd ) BCEADBFADCFABCDGH $.

  ${
    $d y x t $.  $d y ph $.
    $( Alternate definition of substitution when variables are disjoint.
       Compare Theorem 6.2 of [Quine] p. 40.  Also proved as Lemmas 16 and 17
       of [Tarski] p. 70.  The implication "to the left" also holds without a
       disjoint variable condition ( ~ sb2 ).  Theorem ~ sb6f replaces the
       disjoint variable condition with a nonfreeness hypothesis.  Theorem
       ~ sb4b replaces it with a distinctor antecedent.  (Contributed by NM,
       18-Aug-1993.)  (Proof shortened by Wolf Lammen, 21-Sep-2018.)  Revise
       ~ df-sb .  (Revised by BJ, 22-Dec-2020.)  Remove use of ~ ax-11 .
       (Revised by Steven Nguyen, 7-Jul-2023.)  (Proof shortened by Wolf
       Lammen, 16-Jul-2023.) $)
    sb6 $p |- ( [ t / x ] ph <-> A. x ( x = t -> ph ) ) $=
      ( vy wsb weq wi wal df-sb equequ2 imbi1d albidv equsalvw bitri ) ABCEDCFZ
      BDFZAGZBHZGDHBCFZAGZBHZABDCIRUADCOQTBOPSADCBJKLMN $.
  $}

  ${
    $d x y z $.  $d w y $.
    $( 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 x y $.
    $( One direction of ~ sb5 , provable from fewer axioms.  Version of ~ sb1
       with a disjoint variable condition using fewer axioms.  (Contributed by
       NM, 13-May-1993.)  (Revised by Wolf Lammen, 20-Jan-2024.) $)
    sb1v $p |- ( [ y / x ] ph -> E. x ( x = y /\ ph ) ) $=
      ( wsb weq wi wal wa wex sb6 equs4v sylbi ) ABCDBCEZAFBGMAHBIABCJABCKL $.
  $}

  ${
    $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 ) $=
      ( wsb wex spsbe ax5e syl wal ax-5 stdpc4 impbii ) ABCDZAMABEAABCFABGHAABI
      MABJABCKHL $.
  $}

  ${
    $d ph x $.  $d ph z $.
    $( Commutativity law for substitution.  This theorem was incorrectly used
       as our previous version of ~ pm11.07 but may still be useful.
       (Contributed by Andrew Salmon, 17-Jun-2011.)  (Proof shortened by Jim
       Kingdon, 22-Jan-2018.) $)
    sbcom4 $p |- ( [ w / x ] [ y / z ] ph <-> [ y / x ] [ w / z ] ph ) $=
      ( wsb sbv sbbii bitri 3bitr4i ) ABEFAADCFZBEFADEFZBCFZABEGKABEADCGHMABCFA
      LABCADEGHABCGIJ $.
  $}

  ${
    pm11.07.1 $e |- ph $.
    $( Axiom *11.07 in [WhiteheadRussell] p. 159.  The original reads: *11.07
       "Whatever possible argument ` x ` may be, ` ph ( x , y ) ` is true
       whatever possible argument ` y ` may be" implies the corresponding
       statement with ` x ` and ` y ` interchanged except
       in " ` ph ( x , y ) ` ".  Under our formalism this appears to correspond
       to ~ idi and not to ~ sbcom4 as earlier thought.  See
       ~ https://groups.google.com/g/metamath/c/iS0fOvSemC8/m/M1zTH8wxCAAJ .
       (Contributed by BJ, 16-Sep-2018.)  (New usage is discouraged.) $)
    pm11.07 $p |- ph $=
      (  ) B $.
  $}

  ${
    $d x y $.  $d x ph $.
    $( Substitution in an implication with a variable not free in the
       antecedent affects only the consequent.  Version of ~ sbrim based on
       fewer axioms, but with more disjoint variable conditions.  Based on an
       idea of GG. (Contributed by Wolf Lammen, 29-Jan-2024.) $)
    sbrimvw $p |- ( [ y / x ] ( ph -> ps ) <-> ( ph -> [ y / x ] ps ) ) $=
      ( wi wsb weq wal sb6 bi2.04 albii 19.21v 3bitr2i imbi2i bitr4i ) ABEZCDFZ
      ACDGZBEZCHZEZABCDFZEQRPEZCHASEZCHUAPCDIUDUCCARBJKASCLMUBTABCDINO $.
  $}

  ${
    $d t x $.
    sbbiiev.1 $e |- ( x = t -> ( ph <-> ps ) ) $.
    $( An equivalence of substitutions (as in ~ sbbii ) allowing the additional
       information that ` x = t ` .  Version of ~ sbiev and ~ sbievw without a
       disjoint variable condition on ` ps ` , useful for substituting only
       part of ` ph ` .  (Contributed by SN, 24-Aug-2025.) $)
    sbbiiev $p |- ( [ t / x ] ph <-> [ t / x ] ps ) $=
      ( weq wi wal wsb pm5.74i albii sb6 3bitr4i ) CDFZAGZCHNBGZCHACDIBCDIOPCNA
      BEJKACDLBCDLM $.
  $}

  ${
    $d x y $.  $d x ps $.
    sbievw.is $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Conversion of implicit substitution to explicit substitution.  Version
       of ~ sbie and ~ sbiev with more disjoint variable conditions, requiring
       fewer axioms.  (Contributed by NM, 30-Jun-1994.)  (Revised by BJ,
       18-Jul-2023.)  (Proof shortened by SN, 24-Aug-2025.) $)
    sbievw $p |- ( [ y / x ] ph <-> ps ) $=
      ( wsb sbbiiev sbv bitri ) ACDFBCDFBABCDEGBCDHI $.

    $( Obsolete version of ~ sbievw as of 24-Aug-2025.  (Contributed by NM,
       30-Jun-1994.)  (Revised by BJ, 18-Jul-2023.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    sbievwOLD $p |- ( [ y / x ] ph <-> ps ) $=
      ( wsb weq wi wal sb6 equsalvw bitri ) ACDFCDGAHCIBACDJABCDEKL $.
  $}

  ${
    $d x ph $.  $d x ch $.  $d x y $.
    sbiedvw.1 $e |- ( ( ph /\ x = y ) -> ( ps <-> ch ) ) $.
    $( Conversion of implicit substitution to explicit substitution (deduction
       version of ~ sbievw ).  Version of ~ sbied and ~ sbiedv with more
       disjoint variable conditions, requiring fewer axioms.  (Contributed by
       NM, 30-Jun-1994.)  (Revised by GG, 29-Jan-2024.) $)
    sbiedvw $p |- ( ph -> ( [ y / x ] ps <-> ch ) ) $=
      ( wsb wi sbrimvw weq wb expcom pm5.74d sbievw bitr3i pm5.74ri ) ABDEGZCAQ
      HABHZDEGACHZABDEIRSDEDEJZABCATBCKFLMNOP $.
  $}

  ${
    $d x y ps $.  $d x y t $.  $d u y $.
    2sbievw.1 $e |- ( ( x = t /\ y = u ) -> ( ph <-> ps ) ) $.
    $( Conversion of double implicit substitution to explicit substitution.
       Version of ~ 2sbiev with more disjoint variable conditions, requiring
       fewer axioms.  (Contributed by AV, 29-Jul-2023.)  Avoid ~ ax-13 .
       (Revised by GG, 10-Jan-2024.) $)
    2sbievw $p |- ( [ t / x ] [ u / y ] ph <-> ps ) $=
      ( wsb weq sbiedvw sbievw ) ADEHBCFCFIABDEGJK $.
  $}

  ${
    $d y z $.
    $( Substituting ` y ` for ` x ` and then ` z ` for ` y ` is equivalent to
       substituting ` z ` for both ` x ` and ` y ` .  Version of ~ sbcom3 with
       a disjoint variable condition using fewer axioms.  (Contributed by NM,
       27-May-1997.)  (Revised by Giovanni Mascellani, 8-Apr-2018.)  (Revised
       by BJ, 30-Dec-2020.)  (Proof shortened by Wolf Lammen, 19-Jan-2023.) $)
    sbcom3vv $p |- ( [ z / y ] [ y / x ] ph <-> [ z / y ] [ z / x ] ph ) $=
      ( wsb sbequ sbbiiev ) ABCEABDECDACDBFG $.
  $}

  ${
    $d x w $.  $d w y $.  $d w ph $.  $d w ps $.  $d x ch $.
    sbievw2.1 $e |- ( x = w -> ( ph <-> ch ) ) $.
    sbievw2.2 $e |- ( w = y -> ( ch <-> ps ) ) $.
    $( ~ sbievw applied twice, avoiding a DV condition on ` x ` , ` y ` .
       Based on proofs by Wolf Lammen.  (Contributed by Steven Nguyen,
       29-Jul-2023.) $)
    sbievw2 $p |- ( [ y / x ] ph <-> ps ) $=
      ( wsb sbcom3vv sbievw sbbii sbv 3bitr3i bitr3i ) ADEIZCFEIZBADFIZFEIPFEIQ
      PADFEJRCFEACDFGKLPFEMNCBFEHKO $.
  $}

  ${
    $d x z w $.  $d z ph w $.  $d y w $.
    $( A composition law for substitution.  Version of ~ sbco2 with disjoint
       variable conditions and fewer axioms.  (Contributed by NM, 30-Jun-1994.)
       (Revised by BJ, 22-Dec-2020.)  (Proof shortened by Wolf Lammen,
       29-Apr-2023.) $)
    sbco2vv $p |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph ) $=
      ( vw wsb sbequ sbievw2 ) ABDFABCFABEFDCEADEBGAECBGH $.
  $}

  ${
    $d x y $.  $d ph y $.  $d ps x $.
    cbvsbv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Change the bound variable (i.e. the substituted one) in wff's linked by
       implicit substitution.  The proof was extracted from a former ~ cbvabv
       version.  (Contributed by Wolf Lammen, 16-Mar-2025.) $)
    cbvsbv $p |- ( [ z / x ] ph <-> [ z / y ] ps ) $=
      ( wsb sbco2vv sbievw sbbii bitr3i ) ACEGACDGZDEGBDEGACEDHLBDEABCDFIJK $.
    $( $j usage 'cbvsbv' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    $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.)  (Proof shortened by Wolf Lammen, 12-Oct-2024.)  Avoid
       ~ ax-11 .  (Revised by SN, 3-Sep-2025.) $)
    sbco4lem $p |- ( [ x / v ] [ y / x ] [ v / y ] ph <->
                     [ x / w ] [ y / x ] [ w / y ] ph ) $=
      ( wsb weq sbequ sbbidv cbvsbv ) ACEFZBCFACDFZBCFEDBEDGKLBCAEDCHIJ $.
    $( $j usage 'sbco4lem' avoids 'ax-11'; $)
  $}

  ${
    $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.)  Avoid ~ ax-11 .  (Revised by SN, 3-Sep-2025.) $)
    sbco4 $p |- ( [ y / u ] [ x / v ] [ u / x ] [ v / y ] ph <->
        [ x / w ] [ y / x ] [ w / y ] ph ) $=
      ( vt wsb weq sbequ sbbidv sbievw sbco4lem bitri ) ACEHZBFHZEBHZFCHOBCHZEB
      HZACDHBCHDBHZQSFCFCIPREBOFCBJKLSACGHBCHGBHTABCGEMABCDGMNN $.
    $( $j usage 'sbco4' avoids 'ax-11'; $)
  $}

  ${
    $d x w z $.  $d y w $.
    $( Substitution in an equality.  (Contributed by Raph Levien and FL,
       4-Dec-2005.)  Reduce axiom usage.  (Revised by Wolf Lammen,
       23-Jul-2023.) $)
    equsb3 $p |- ( [ y / x ] x = z <-> y = z ) $=
      ( vw weq equequ1 sbievw2 ) ACEBCEDCEABDADCFDBCFG $.

    $( Substitution applied to the atomic wff with equality.  Variant of
       ~ equsb3 .  (Contributed by AV, 29-Jul-2023.)  (Proof shortened by Wolf
       Lammen, 2-Sep-2023.) $)
    equsb3r $p |- ( [ y / x ] z = x <-> z = y ) $=
      ( vw weq equequ2 sbievw2 ) CAECBECDEABDADCFDBCFG $.
  $}

  ${
    $d x y $.
    $( Substitution applied to an atomic wff.  Version of ~ equsb1 with a
       disjoint variable condition, which neither requires ~ ax-12 nor
       ~ ax-13 .  (Contributed by NM, 10-May-1993.)  (Revised by BJ,
       11-Sep-2019.)  Remove dependencies on axioms.  (Revised by Wolf Lammen,
       30-May-2023.)  (Proof shortened by Steven Nguyen, 19-Jun-2023.)  Revise
       ~ df-sb .  (Revised by Steven Nguyen, 11-Jul-2023.)  (Proof shortened by
       Steven Nguyen, 22-Jul-2023.) $)
    equsb1v $p |- [ y / x ] x = y $=
      ( weq wsb equid equsb3 mpbir ) ABCABDBBCBEABBFG $.
  $}

  $( Any substitution in an always false formula is false.  (Contributed by
     Steven Nguyen, 3-May-2023.) $)
  nsb $p |- ( A. x -. ph -> -. [ t / x ] ph ) $=
    ( wn wal wex wsb alnex biimpi spsbe nsyl ) ADBEZABFZABCGLMDABHIABCJK $.

  $( One direction of ~ sbn , using fewer axioms.  Compare ~ 19.2 .
     (Contributed by Steven Nguyen, 18-Aug-2023.) $)
  sbn1 $p |- ( [ t / x ] -. ph -> -. [ t / x ] ph ) $=
    ( wn wsb wfal nsb fal mpg pm2.21 sb2imi mtoi ) ADZBCEABCEFBCEZFDNDBFBCGHIMA
    FBCAFJKL $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Membership predicate
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( 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 to prove the
       ~ wel of predicate calculus in terms of the ~ wcel 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 $.

    $( Register class membership as a primitive expression (lacking a
       definition). $)
    $( $j primitive 'wcel'; $)
  $}

  $( 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 syntactic construction introduces a binary non-logical predicate
     symbol ` e. ` (stylized lowercase epsilon) 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 scheme ax-8 (Left Equality for Binary Predicate)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Axiom of Left Equality for Binary Predicate.  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
     an arbitrary binary predicate ` e. ` , which we will use for the set
     membership relation when set theory is introduced.  This axiom scheme is a
     sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose
     general form cannot be represented with our notation.  Also appears as
     Axiom scheme C12' in [Megill] p. 448 (p. 16 of the preprint).
     "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.

     We prove in ~ ax8 that this axiom can be recovered from its weakened
     version ~ ax8v where ` x ` and ` y ` are assumed to be disjoint variables.
     In particular, the only theorem referencing ~ ax-8 should be ~ ax8v .  See
     the comment of ~ ax8v for more details on these matters.  (Contributed by
     NM, 30-Jun-1993.)  (Revised by BJ, 7-Dec-2020.)  Use ~ ax8 instead.
     (New usage is discouraged.) $)
  ax-8 $a |- ( x = y -> ( x e. z -> y e. z ) ) $.

  ${
    $d x y $.
    $( Weakened version of ~ ax-8 , with a disjoint variable condition on
       ` x , y ` .  This should be the only proof referencing ~ ax-8 , and it
       should be referenced only by its two weakened versions ~ ax8v1 and
       ~ ax8v2 , from which ~ ax-8 is then rederived as ~ ax8 , which shows
       that either ~ ax8v or the conjunction of ~ ax8v1 and ~ ax8v2 is
       sufficient.  (Contributed by BJ, 7-Dec-2020.)  Use ~ ax8 instead.
       (New usage is discouraged.) $)
    ax8v $p |- ( x = y -> ( x e. z -> y e. z ) ) $=
      ( ax-8 ) ABCD $.
  $}

  ${
    $d x y $.  $d x z $.
    $( First of two weakened versions of ~ ax8v , with an extra disjoint
       variable condition on ` x , z ` , see comments there.  (Contributed by
       BJ, 7-Dec-2020.) $)
    ax8v1 $p |- ( x = y -> ( x e. z -> y e. z ) ) $=
      ( ax8v ) ABCD $.
  $}

  ${
    $d x y $.  $d y z $.
    $( Second of two weakened versions of ~ ax8v , with an extra disjoint
       variable condition on ` y , z ` see comments there.  (Contributed by BJ,
       7-Dec-2020.) $)
    ax8v2 $p |- ( x = y -> ( x e. z -> y e. z ) ) $=
      ( ax8v ) ABCD $.
  $}

  ${
    $d t x $.  $d t y $.  $d t z $.
    $( Proof of ~ ax-8 from ~ ax8v1 and ~ ax8v2 , proving sufficiency of the
       conjunction of the latter two weakened versions of ~ ax8v , which is
       itself a weakened version of ~ ax-8 .  (Contributed by BJ, 7-Dec-2020.)
       (Proof shortened by Wolf Lammen, 11-Apr-2021.) $)
    ax8 $p |- ( x = y -> ( x e. z -> y e. z ) ) $=
      ( vt weq wa wex wel wi equvinv ax8v2 equcoms ax8v1 sylan9 exlimiv sylbi )
      ABEDAEZDBEZFZDGACHZBCHZIZABDJSUBDQTDCHZRUATUCIADADCKLDBCMNOP $.
  $}

  $( An identity law for the non-logical predicate.  (Contributed by NM,
     30-Jun-1993.) $)
  elequ1 $p |- ( x = y -> ( x e. z <-> y e. z ) ) $=
    ( weq wel ax8 wi equcoms impbid ) ABDACEZBCEZABCFKJGBABACFHI $.

  ${
    $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.)  Reduce axiom usage.  (Revised by Wolf Lammen,
       24-Jul-2023.) $)
    elsb1 $p |- ( [ y / x ] x e. z <-> y e. z ) $=
      ( vw wel elequ1 sbievw2 ) ACEBCEDCEABDADCFDBCFG $.
  $}

  ${
    $d x z $.  $d y z $.
    $( When the class variables in Definition ~ df-clel 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.)
       Revised to use ~ equsexvw in order to remove dependencies on ~ ax-10 ,
       ~ ax-12 , ~ ax-13 .  Note that there is no disjoint variable condition
       on ` x , y ` , that is, on the variables of the left-hand side, as
       should be the case for definitions.  (Revised by BJ, 29-Dec-2020.) $)
    cleljust $p |- ( x e. y <-> E. z ( z = x /\ z e. y ) ) $=
      ( weq wel wa wex elequ1 equsexvw bicomi ) CADCBEZFCGABEZKLCACABHIJ $.
    $( $j usage 'cleljust' avoids 'ax-9' 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Axiom scheme ax-9 (Right Equality for Binary Predicate)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Axiom of Right Equality for Binary Predicate.  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
     an arbitrary binary predicate ` e. ` , which we will use for the set
     membership relation when set theory is introduced.  This axiom scheme is a
     sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose
     general form cannot be represented with our notation.  Also appears as
     Axiom scheme C13' in [Megill] p. 448 (p. 16 of the preprint).

     We prove in ~ ax9 that this axiom can be recovered from its weakened
     version ~ ax9v where ` x ` and ` y ` are assumed to be disjoint variables.
     In particular, the only theorem referencing ~ ax-9 should be ~ ax9v .  See
     the comment of ~ ax9v for more details on these matters.  (Contributed by
     NM, 21-Jun-1993.)  (Revised by BJ, 7-Dec-2020.)  Use ~ ax9 instead.
     (New usage is discouraged.) $)
  ax-9 $a |- ( x = y -> ( z e. x -> z e. y ) ) $.

  ${
    $d x y $.
    $( Weakened version of ~ ax-9 , with a disjoint variable condition on
       ` x , y ` .  This should be the only proof referencing ~ ax-9 , and it
       should be referenced only by its two weakened versions ~ ax9v1 and
       ~ ax9v2 , from which ~ ax-9 is then rederived as ~ ax9 , which shows
       that either ~ ax9v or the conjunction of ~ ax9v1 and ~ ax9v2 is
       sufficient.  (Contributed by BJ, 7-Dec-2020.)  Use ~ ax9 instead.
       (New usage is discouraged.) $)
    ax9v $p |- ( x = y -> ( z e. x -> z e. y ) ) $=
      ( ax-9 ) ABCD $.
  $}

  ${
    $d x y $.  $d x z $.
    $( First of two weakened versions of ~ ax9v , with an extra disjoint
       variable condition on ` x , z ` , see comments there.  (Contributed by
       BJ, 7-Dec-2020.) $)
    ax9v1 $p |- ( x = y -> ( z e. x -> z e. y ) ) $=
      ( ax9v ) ABCD $.
  $}

  ${
    $d x y $.  $d y z $.
    $( Second of two weakened versions of ~ ax9v , with an extra disjoint
       variable condition on ` y , z ` see comments there.  (Contributed by BJ,
       7-Dec-2020.) $)
    ax9v2 $p |- ( x = y -> ( z e. x -> z e. y ) ) $=
      ( ax9v ) ABCD $.
  $}

  ${
    $d t x $.  $d t y $.  $d t z $.
    $( Proof of ~ ax-9 from ~ ax9v1 and ~ ax9v2 , proving sufficiency of the
       conjunction of the latter two weakened versions of ~ ax9v , which is
       itself a weakened version of ~ ax-9 .  (Contributed by BJ, 7-Dec-2020.)
       (Proof shortened by Wolf Lammen, 11-Apr-2021.) $)
    ax9 $p |- ( x = y -> ( z e. x -> z e. y ) ) $=
      ( vt weq wa wex wel wi equvinv ax9v2 equcoms ax9v1 sylan9 exlimiv sylbi )
      ABEDAEZDBEZFZDGCAHZCBHZIZABDJSUBDQTCDHZRUATUCIADADCKLDBCMNOP $.
  $}

  $( An identity law for the non-logical predicate.  (Contributed by NM,
     21-Jun-1993.) $)
  elequ2 $p |- ( x = y -> ( z e. x <-> z e. y ) ) $=
    ( weq wel ax9 wi equcoms impbid ) ABDCAEZCBEZABCFKJGBABACFHI $.

  ${
    $d x z $.  $d y z $.
    $( A form of ~ elequ2 with a universal quantifier.  Its converse is the
       axiom of extensionality ~ ax-ext .  (Contributed by BJ, 3-Oct-2019.) $)
    elequ2g $p |- ( x = y -> A. z ( z e. x <-> z e. y ) ) $=
      ( weq wel wb elequ2 alrimiv ) ABDCAECBEFCABCGH $.
  $}

  ${
    $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.)  Reduce axiom usage.  (Revised by Wolf Lammen,
       24-Jul-2023.) $)
    elsb2 $p |- ( [ y / x ] z e. x <-> z e. y ) $=
      ( vw wel elequ2 sbievw2 ) CAECBECDEABDADCFDBCFG $.
  $}

  $( An identity law for the non-logical predicate, which combines ~ elequ1 and
     ~ elequ2 .  The analogous theorems for class terms are ~ eleq1 , ~ eleq2 ,
     and ~ eleq12 respectively.  (Contributed by BJ, 29-Sep-2019.) $)
  elequ12 $p |- ( ( x = y /\ z = t ) -> ( x e. z <-> y e. t ) ) $=
    ( weq wel elequ1 elequ2 sylan9bb ) ABEACFBCFCDEBDFABCGCDBHI $.

  ${
    $d x y $.
    $( The FOL statement used in the standard proof of Russell's paradox ~ ru .
       (Contributed by NM, 7-Aug-1994.)  Extract from proof of ~ ru and reduce
       axiom usage.  (Revised by BJ, 12-Oct-2019.) $)
    ru0 $p |- -. A. x ( x e. y <-> -. x e. x ) $=
      ( wel wn wb wal pm5.19 weq elequ1 elequ12 anidms notbid bibi12d spvv mto
      ) ABCZAACZDZEZAFBBCZTDZEZTGSUBABABHZPTRUAABBIUCQTUCQTEABABJKLMNO $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  The original axiom schemes of Tarski's predicate calculus are ~ ax-4 ,
  ~ ax-5 , ~ ax6v , ~ ax-7 , ~ ax-8 , and ~ ax-9 , together with rule
  ~ ax-gen .  See ~ mmset.html#compare .  They are given as axiom schemes B4
  through B8 in [KalishMontague] p. 81.  These are shown to be logically
  complete by Theorem 1 of [KalishMontague] p. 85.

  The axiom system of set.mm includes the auxiliary axiom schemes ~ ax-10 ,
  ~ ax-11 , ~ ax-12 , and ~ ax-13 , which are not part of Tarski's axiom
  schemes.  Each object-language instance of them is provable from Tarski's
  axioms, so they are logically redundant.  However, they are conjectured not
  to be provable directly _as schemes_ from Tarski's axiom schemes using only
  Metamath's direct substitution rule.  They are used to make our system
  "scheme complete", i.e., able to prove directly all possible schemes
  with wff and setvar variables, bundled or not, whose object-language
  instances are valid.  ( ~ ax-12 has been proved to be required; see
  ~ https://us.metamath.org/award2003.html#9a .  Metalogical independence of
  the other three are open problems.)

  (There are additional predicate calculus axiom schemes included in set.mm
  such as ~ ax-c5 , but they can all be proved as theorems from the above.)

  Terminology:  Two setvar (individual) metavariables are "bundled" in an axiom
  or theorem scheme when there is no distinct variable constraint ($d) imposed
  on them.  (The term "bundled" is due to Raph Levien.)  For example, the ` x `
  and ` y ` in ~ ax-6 are bundled, but they are not in ~ ax6v . We also say
  that a scheme is bundled when it has at least one pair of bundled setvar
  variables.  If distinct variable conditions are added to all setvar variable
  pairs in a bundled scheme, we call that the "principal" instance
  of the bundled scheme.  For example, ~ ax6v is the principal instance of
  ~ ax-6 . Whenever a common variable is substituted for two or more bundled
  variables in an axiom or theorem scheme, we call the substitution instance
  "degenerate".  For example, the instance ` -. A. x -. x = x ` of ~ ax-6 is
  degenerate.  An advantage of bundling is ease of use since there are fewer
  distinct variable restrictions ($d) to be concerned with, and theorems
  are more general.  There may be some
  economy in being able to prove facts about principal and degenerate instances
  simultaneously.  A disadvantage is that bundling may present difficulties in
  translations to other proof languages, which typically lack the concept (in
  part because their variables often represent the variables of the
  object language rather than metavariables ranging over them).

  Because Tarski's axiom schemes are logically complete, they can be used to
  prove any object-language instance of ~ ax-10 , ~ ax-11 , ~ ax-12 , and
  ~ ax-13 .  "Translating" this to Metamath, it means that Tarski's axioms can
  prove any substitution instance of ~ ax-10 , ~ ax-11 , ~ ax-12 , or ~ ax-13
  in which (1) there are no wff metavariables and (2) all setvar variables
  are mutually distinct i.e. are not bundled.  In effect this is mimicking the
  object language by pretending that each setvar variable is an
  object-language variable.  (There may also be specific instances with wff
  metavariables and/or bundling that are directly provable from Tarski's axiom
  schemes, but it isn't guaranteed.  Whether all of them are possible is part
  of the still open metalogical independence problem for our additional axiom
  schemes.)

  It can be useful to see how this can be done, both to show that our
  additional schemes are valid metatheorems of Tarski's system and to be able
  to translate object-language instances of our proofs into proofs that would
  work with a system using only Tarski's original schemes.  In addition, it may
  (or may not) provide insight into the conjectured metalogical independence of
  our additional schemes.

  The theorem schemes ~ ax10w , ~ ax11w , ~ ax12w , and ~ ax13w are derived
  using only Tarski's axiom schemes, showing that Tarski's schemes can be used
  to derive all substitution instances of ~ ax-10 , ~ ax-11 , ~ ax-12 , and
  ~ ax-13 meeting Conditions (1) and (2).  (The "w" suffix stands for "weak
  version".)  Each hypothesis of ~ ax10w , ~ ax11w , and ~ ax12w is of the form
  ` ( x = y -> ( ph <-> ps ) ) ` where ` ps ` is an auxiliary or "dummy" wff
  metavariable in which ` x ` doesn't occur.  We can show by induction on
  formula length that the hypotheses can be eliminated in all cases meeting
  Conditions (1) and (2).  The example ~ ax12wdemo illustrates the techniques
  (equality theorems and bound variable renaming) used to achieve this.

  We also show the degenerate instances for axioms with bundled variables in
  ~ ax11dgen , ~ ax12dgen , ~ ax13dgen1 , ~ ax13dgen2 , ~ ax13dgen3 , and
  ~ ax13dgen4 . (Their proofs are trivial, but we include them to be thorough.)
  Combining the principal and degenerate cases _outside_ of Metamath, we show
  that the bundled schemes ~ ax-10 , ~ ax-11 , ~ ax-12 , and ~ ax-13 are
  schemes of Tarski's system, meaning that all object-language instances they
  generate are theorems of Tarski's system.

  It is interesting that Tarski used the bundled scheme ~ ax-6 in an older
  system, so it seems the main purpose of his later ~ ax6v was just to show
  that the weaker unbundled form is sufficient rather than an aesthetic
  objection to bundled free and bound variables.  Since we adopt the
  bundled ~ ax-6 as our official axiom, we show that the degenerate
  instance holds in ~ ax6dgen .  (Recall that in set.mm, the only statement
  referencing ~ ax-6 is ~ ax6v .)

  The case of ~ sp is curious:  originally an axiom scheme of Tarski's system,
  it was proved logically redundant by Lemma 9 of [KalishMontague] p. 86.
  However, the proof is by induction on formula length, and the scheme form
  ` A. x ph -> ph ` apparently cannot be proved directly from Tarski's other
  axiom schemes.  The best we can do seems to be ~ spw , again requiring
  substitution instances of ` ph ` that meet Conditions (1) and (2) above.
  Note that our direct proof ~ sp requires ~ ax-12 , which is not part of
  Tarski's system.

$)

  $( Tarski's system uses the weaker ~ ax6v instead of the bundled ~ ax-6 , so
     here we show that the degenerate case of ~ ax-6 can be derived.  Even
     though ~ ax-6 is in the list of axioms used, recall that in set.mm, the
     only statement referencing ~ ax-6 is ~ ax6v .  We later rederive from
     ~ ax6v the bundled form as ~ ax6 with the help of the auxiliary axiom
     schemes.  (Contributed by NM, 23-Apr-2017.) $)
  ax6dgen $p |- -. A. x -. x = x $=
    ( weq wn wal equid notnoti spfalw mt2 ) AABZCZADIAEZJAIKFGH $.

  ${
    $d y ph $.  $d x ps $.  $d x y $.
    ax10w.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Weak version of ~ ax-10 from which we can prove any ~ ax-10 instance not
       involving wff variables or bundling.  Uses only Tarski's FOL axiom
       schemes.  It is an alias of ~ hbn1w introduced for labeling consistency.
       (Contributed by NM, 9-Apr-2017.)  Use ~ hbn1w instead.
       (New usage is discouraged.) $)
    ax10w $p |- ( -. A. x ph -> A. x -. A. x ph ) $=
      ( hbn1w ) ABCDEF $.
  $}

  ${
    $d y z $.  $d x y $.  $d z ph $.  $d y ps $.
    ax11w.1 $e |- ( y = z -> ( ph <-> ps ) ) $.
    $( Weak version of ~ ax-11 from which we can prove any ~ ax-11 instance not
       involving wff variables or bundling.  Uses only Tarski's FOL axiom
       schemes.  Unlike ~ ax-11 , this theorem requires that ` x ` and ` y ` be
       distinct i.e. are not bundled.  It is an alias of ~ alcomimw introduced
       for labeling consistency.  (Contributed by NM, 10-Apr-2017.)  Use
       ~ alcomimw instead.  (New usage is discouraged.) $)
    ax11w $p |- ( A. x A. y ph -> A. y A. x ph ) $=
      ( alcomimw ) ABCDEFG $.
    $( $j usage 'ax11w' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12'
       'ax-13'; $)
  $}

  $( Degenerate instance of ~ ax-11 where bundled variables ` x ` and ` y `
     have a common substitution.  Uses only Tarski's FOL axiom schemes.
     (Contributed by NM, 13-Apr-2017.) $)
  ax11dgen $p |- ( A. x A. x ph -> A. x A. x ph ) $=
    ( wal id ) ABCBCD $.
  $( $j usage 'ax11dgen' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12'
       'ax-13'; $)

  ${
    $d x ps $.
    ax12wlemw.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Lemma for weak version of ~ ax-12 .  Uses only Tarski's FOL axiom
       schemes.  In some cases, this lemma may lead to shorter proofs than
       ~ ax12w .  (Contributed by NM, 10-Apr-2017.) $)
    ax12wlem $p |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) $=
      ( ax-5 ax12i ) ABCDEBCFG $.
  $}

  ${
    $d y z $.  $d x ps $.  $d z ph $.  $d y ch $.
    ax12w.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    ax12w.2 $e |- ( y = z -> ( ph <-> ch ) ) $.
    $( Weak version of ~ ax-12 from which we can prove any ~ ax-12 instance not
       involving wff variables or bundling.  Uses only Tarski's FOL axiom
       schemes.  An instance of the first hypothesis will normally require that
       ` x ` and ` y ` be distinct (unless ` x ` does not occur in ` ph ` ).
       For an example of how the hypotheses can be eliminated when we
       substitute an expression without wff variables for ` ph ` , see
       ~ ax12wdemo .  (Contributed by NM, 10-Apr-2017.) $)
    ax12w $p |- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) ) $=
      ( wal weq wi spw ax12wlem syl5 ) AEIADEJZOAKDIACEFHLABDEGMN $.
    $( $j usage 'ax12w' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12'
       'ax-13'; $)
  $}

  $( Degenerate instance of ~ ax-12 where bundled variables ` x ` and ` y `
     have a common substitution.  Uses only Tarski's FOL axiom schemes.
     (Contributed by NM, 13-Apr-2017.) $)
  ax12dgen $p |- ( x = x -> ( A. x ph -> A. x ( x = x -> ph ) ) ) $=
    ( wal weq wi ala1 a1i ) ABCBBDZAEBCEHAHBFG $.
  $( $j usage 'ax12dgen' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12'
       'ax-13'; $)

  ${
    $d x y z w v $.
    $( Example of an application of ~ ax12w that results in an instance of
       ~ ax-12 for a contrived formula with mixed free and bound variables,
       ` ( x e. y /\ A. x z e. x /\ A. y A. z y e. x ) ` , in place of ` ph ` .
       The proof illustrates bound variable renaming with ~ cbvalvw to obtain
       fresh variables to avoid distinct variable clashes.  Uses only Tarski's
       FOL axiom schemes.  (Contributed by NM, 14-Apr-2017.) $)
    ax12wdemo $p |- ( x = y
              -> ( A. y ( x e. y /\ A. x z e. x /\ A. y A. z y e. x )
     -> A. x ( x = y -> ( x e. y /\ A. x z e. x /\ A. y A. z y e. x ) ) ) ) $=
      ( vw vv wel wal w3a weq elequ1 elequ2 cbvalvw a1i albidv bitrid 3anbi123d
      wb 3anbi13d ax12w ) ABFZCAFZAGZBAFZCGZBGZHBBFZCDFZDGZEBFZCGZEGZHAEFZUBEAF
      ZCGZEGZHABEABIZTUFUBUHUEUKABBJUBUHQUPUAUGADADCKLMUEUOUPUKUDUNBEBEIZUCUMCB
      EAJNLZUPUNUJEUPUMUICABEKNNOPUQTULUEUOUBBEAKUEUOQUQURMRS $.
    $( $j usage 'ax12wdemo' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    $d x y $.  $d x z $.
    $( Weak version (principal instance) of ~ ax-13 .  (Because ` y ` and ` z `
       don't need to be distinct, this actually bundles the principal instance
       and the degenerate instance
       ` ( -. x = y -> ( y = y -> A. x y = y ) ) ` .)  Uses only Tarski's FOL
       axiom schemes.  The proof is trivial but is included to complete the set
       ~ ax10w , ~ ax11w , and ~ ax12w .  (Contributed by NM, 10-Apr-2017.) $)
    ax13w $p |- ( -. x = y -> ( y = z -> A. x y = z ) ) $=
      ( weq wn ax5d ) ABDEBCDAF $.
    $( $j usage 'ax13w' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12'
       'ax-13'; $)
  $}

  $( Degenerate instance of ~ ax-13 where bundled variables ` x ` and ` y `
     have a common substitution.  Uses only Tarski's FOL axiom schemes.
     (Contributed by NM, 13-Apr-2017.) $)
  ax13dgen1 $p |- ( -. x = x -> ( x = z -> A. x x = z ) ) $=
    ( weq wal wi equid pm2.24i ) AACABCZHADEAFG $.
  $( $j usage 'ax13dgen1' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12'
     'ax-13'; $)

  $( Degenerate instance of ~ ax-13 where bundled variables ` x ` and ` z `
     have a common substitution.  Uses only Tarski's FOL axiom schemes.
     (Contributed by NM, 13-Apr-2017.) $)
  ax13dgen2 $p |- ( -. x = y -> ( y = x -> A. x y = x ) ) $=
    ( weq wn wal equcomi pm2.21 syl5 ) BACZABCZJDIAEZBAFJKGH $.
  $( $j usage 'ax13dgen2' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12'
     'ax-13'; $)

  $( Degenerate instance of ~ ax-13 where bundled variables ` y ` and ` z `
     have a common substitution.  Uses only Tarski's FOL axiom schemes.
     (Contributed by NM, 13-Apr-2017.) $)
  ax13dgen3 $p |- ( -. x = y -> ( y = y -> A. x y = y ) ) $=
    ( weq wal wn equid ax-gen 2a1i ) BBCZADABCEIIABFGH $.
  $( $j usage 'ax13dgen3' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12'
     'ax-13'; $)

  $( Degenerate instance of ~ ax-13 where bundled variables ` x ` , ` y ` , and
     ` z ` have a common substitution.  Therefore, also a degenerate instance
     of ~ ax13dgen1 , ~ ax13dgen2 , and ~ ax13dgen3 .  Also an instance of the
     intuitionistic tautology ~ pm2.21 .  Uses only Tarski's FOL axiom schemes.
     (Contributed by NM, 13-Apr-2017.)  Reduce axiom usage.  (Revised by Wolf
     Lammen, 10-Oct-2021.) $)
  ax13dgen4 $p |- ( -. x = x -> ( x = x -> A. x x = x ) ) $=
    ( weq wal pm2.21 ) AABZEACD $.
  $( $j usage 'ax13dgen4' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12'
     'ax-13'; $)


$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
  Predicate calculus with equality:  Auxiliary axiom schemes (4 schemes)
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#

  In this section we introduce four additional schemes ~ ax-10 , ~ ax-11 ,
  ~ ax-12 , and ~ ax-13 that are not part of Tarski's system but can be proved
  (outside of Metamath) as theorem schemes of Tarski's system.  These are
  needed to give our system the property of "scheme completeness", which
  means that we can prove (with Metamath) all possible theorem schemes
  expressible in our language of wff metavariables ranging over object-language
  wffs, and setvar variables ranging over object-language individual variables.

  To show that these schemes are valid metatheorems of Tarski's system S2,
  above we proved from Tarski's system theorems ~ ax10w , ~ ax11w , ~ ax12w ,
  and ~ ax13w , which show that any object-language instance of these schemes
  (emulated by having no wff metavariables and requiring all setvar variables
  to be mutually distinct) can be proved using only the schemes in Tarski's
  system S2.

  An open problem is to show that these four additional schemes are mutually
  _metalogically_ independent and metalogically independent from Tarski's.  So
  far, independence of ~ ax-12 from all others has been shown, and
  independence of Tarski's ~ ax-6 from all others has been shown; see
  items 9a and 11 on ~ https://us.metamath.org/award2003.html .

$)


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Axiom scheme ax-10 (Quantified Negation)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Axiom of Quantified Negation.  Axiom C5-2 of [Monk2] p. 113.  This axiom
     scheme is logically redundant (see ~ ax10w ) but is used as an auxiliary
     axiom scheme to achieve scheme completeness.  It means that ` x ` is not
     free in ` -. A. x ph ` .  (Contributed by NM, 21-May-2008.)  Use its alias
     ~ hbn1 instead if you must use it.  Any theorem in first-order logic (FOL)
     that contains only set variables that are all mutually distinct, and has
     no wff variables, can be proved *without* using ~ ax-10 through ~ ax-13 ,
     by invoking ~ ax10w through ~ ax13w .  We encourage proving theorems
     *without* ~ ax-10 through ~ ax-13 and moving them up to the ~ ax-4 through
     ~ ax-9 section.  (New usage is discouraged.) $)
  ax-10 $a |- ( -. A. x ph -> A. x -. A. x ph ) $.

  $( Alias for ~ ax-10 to be used instead of it.  (Contributed by NM,
     24-Jan-1993.)  (Proof shortened by Wolf Lammen, 18-Aug-2014.) $)
  hbn1 $p |- ( -. A. x ph -> A. x -. A. x ph ) $=
    ( ax-10 ) ABC $.

  $( The setvar ` x ` is not free in ` E. x ph ` .  Corresponds to the axiom
     (5) of modal logic (see also ~ modal5 ).  (Contributed by NM,
     24-Jan-1993.) $)
  hbe1 $p |- ( E. x ph -> A. x E. x ph ) $=
    ( wex wn wal df-ex hbn1 hbxfrbi ) ABCADZBEDBABFIBGH $.

  $( Dual statement of ~ hbe1 .  Modified version of ~ axc7e with a universally
     quantified consequent.  (Contributed by Wolf Lammen, 15-Sep-2021.) $)
  hbe1a $p |- ( E. x A. x ph -> A. x ph ) $=
    ( wal wex wn df-ex hbn1 con1i sylbi ) ABCZBDJEBCZEJJBFJKABGHI $.

  $( One direction of ~ nf5 can be proved with a smaller footprint on axiom
     usage.  (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 nfd ) AABCZDBCZABKABEJBEJAJBFABGHI $.

  ${
    nf5i.1 $e |- ( ph -> A. x ph ) $.
    $( Deduce that ` x ` is not free in ` ph ` from the definition.
       (Contributed by Mario Carneiro, 11-Aug-2016.) $)
    nf5i $p |- F/ x ph $=
      ( wal wi wnf nf5-1 mpg ) AABDEABFBABGCH $.
  $}

  ${
    nf5dh.1 $e |- ( ph -> A. x ph ) $.
    nf5dh.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.) ~ df-nf changed.  (Revised by Wolf Lammen,
       11-Oct-2021.) $)
    nf5dh $p |- ( ph -> F/ x ps ) $=
      ( wal wi wnf alrimih nf5-1 syl ) ABBCFGZCFBCHALCDEIBCJK $.
  $}

  ${
    $d x ph $.
    nf5dv.1 $e |- ( ph -> ( ps -> A. x ps ) ) $.
    $( Apply the definition of not-free in a context.  (Contributed by Mario
       Carneiro, 11-Aug-2016.) ~ df-nf changed.  (Revised by Wolf Lammen,
       18-Sep-2021.)  (Proof shortened by Wolf Lammen, 13-Jul-2022.) $)
    nf5dv $p |- ( ph -> F/ x ps ) $=
      ( ax-5 nf5dh ) ABCACEDF $.
  $}

  ${
    $d x y $.
    $( All variables are effectively bound in a distinct variable specifier.
       Version of ~ nfnae with a disjoint variable condition, which does not
       require ~ ax-13 .  (Contributed by Mario Carneiro, 11-Aug-2016.)  Avoid
       ~ ax-13 .  (Revised by GG, 10-Jan-2024.)  (Proof shortened by Wolf
       Lammen, 25-Sep-2024.) $)
    nfnaew $p |- F/ z -. A. x x = y $=
      ( weq wal wn hbnaev nf5i ) ABDAEFCABCGH $.
  $}

  $( The setvar ` x ` is not free in ` E. x ph ` .  (Contributed by Mario
     Carneiro, 11-Aug-2016.) $)
  nfe1 $p |- F/ x E. x ph $=
    ( wex hbe1 nf5i ) ABCBABDE $.

  $( The setvar ` x ` is not free in ` A. x ph ` .  (Contributed by Mario
     Carneiro, 11-Aug-2016.) ~ df-nf changed.  (Revised by Wolf Lammen,
     11-Sep-2021.)  Remove dependency on ~ ax-12 .  (Revised by Wolf Lammen,
     12-Oct-2021.) $)
  nfa1 $p |- F/ x A. x ph $=
    ( wal wn wex alex nfe1 nfn nfxfr ) ABCADZBEZDBABFKBJBGHI $.

  $( A convenience theorem particularly designed to remove dependencies on
     ~ ax-11 in conjunction with distinctors.  (Contributed by Wolf Lammen,
     2-Sep-2018.) $)
  nfna1 $p |- F/ x -. A. x ph $=
    ( wal nfa1 nfn ) ABCBABDE $.

  $( 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 $.

  $( The setvar ` x ` is not free in ` F/ x ph ` .  (Contributed by Mario
     Carneiro, 11-Aug-2016.)  Remove dependency on ~ ax-12 .  (Revised by Wolf
     Lammen, 12-Oct-2021.) $)
  nfnf1 $p |- F/ x F/ x ph $=
    ( wnf wex wal wi df-nf nfe1 nfa1 nfim nfxfr ) ABCABDZABEZFBABGLMBABHABIJK
    $.

  $( The analogue in our predicate calculus of axiom (5) of modal logic S5.
     See also ~ hbe1 .  (Contributed by NM, 5-Oct-2005.) $)
  modal5 $p |- ( -. A. x -. ph -> A. x -. A. x -. ph ) $=
    ( wn hbn1 ) ACBD $.

  ${
    $d x y $.
    $( The setvar ` x ` is not free in ` [ y / x ] ph ` when ` x ` and ` y `
       are distinct.  (Contributed by Mario Carneiro, 11-Aug-2016.)  Shorten
       ~ nfs1v and ~ hbs1 combined.  (Revised by Wolf Lammen, 28-Jul-2022.) $)
    nfs1v $p |- F/ x [ y / x ] ph $=
      ( wsb weq wi wal sb6 nfa1 nfxfr ) ABCDBCEAFZBGBABCHKBIJ $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Axiom scheme ax-11 (Quantifier Commutation)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Axiom of Quantifier Commutation.  This axiom says universal quantifiers
     can be swapped.  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.  This axiom scheme is logically redundant (see ~ ax11w )
     but is used as an auxiliary axiom scheme to achieve metalogical
     completeness.  Use its weak version ~ alcomimw when it allows to avoid
     dependence on ~ ax-11 .  (Contributed by NM, 12-Mar-1993.) $)
  ax-11 $a |- ( A. x A. y ph -> A. y A. x ph ) $.

  ${
    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-11 syl ) ACFDFADFCFBADCGEH $.
  $}

  $( Theorem 19.5 of [Margaris] p. 89.  Use its weak version ~ alcomw when it
     allows to avoid dependence on ~ ax-11 .  (Contributed by NM,
     30-Jun-1993.) $)
  alcom $p |- ( A. x A. y ph <-> A. y A. x ph ) $=
    ( wal ax-11 impbii ) ACDBDABDCDABCEACBEF $.

  $( 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 four universal quantifiers twice.  (Contributed by NM, 2-Feb-2005.)
     (Proof shortened by Fan Zheng, 6-Jun-2016.) $)
  alrot4 $p |- ( A. x A. y A. z A. w ph <-> A. z A. w A. x A. y ph ) $=
    ( wal alrot3 albii bitri ) AEFDFCFZBFACFZEFDFZBFKBFEFDFJLBACDEGHKBDEGI $.

  $( Theorem 19.11 of [Margaris] p. 89.  (Contributed by NM, 5-Aug-1993.)
     Remove dependencies on ~ ax-5 , ~ ax-6 , ~ ax-7 , ~ ax-10 , ~ ax-12 .
     (Revised by Wolf Lammen, 8-Jan-2018.)  (Proof shortened by Wolf Lammen,
     22-Aug-2020.) $)
  excom $p |- ( E. x E. y ph <-> E. y E. x ph ) $=
    ( wn wal wex alcom notbii 2exnaln 3bitr4i ) ADZCEBEZDKBECEZDACFBFABFCFLMKBC
    GHABCIACBIJ $.
  $( $j usage 'excom' avoids 'ax-5' 'ax-6' 'ax-7' 'ax-10' 'ax-12' 'ax-13'; $)

  $( One direction of Theorem 19.11 of [Margaris] p. 89.  (Contributed by NM,
     5-Aug-1993.)  (Revised by Mario Carneiro, 24-Sep-2016.)  Remove
     dependencies on ~ ax-5 , ~ ax-6 , ~ ax-7 , ~ ax-10 , ~ ax-12 .  (Revised
     by Wolf Lammen, 8-Jan-2018.) $)
  excomim $p |- ( E. x E. y ph -> E. y E. x ph ) $=
    ( wex excom biimpi ) ACDBDABDCDABCEF $.
  $( $j usage 'excomim' avoids 'ax-5' 'ax-6' 'ax-7' 'ax-10' 'ax-12' 'ax-13'; $)

  $( 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 $.

  ${
    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, 12-Mar-1993.) $)
    hbal $p |- ( A. y ph -> A. x A. y ph ) $=
      ( wal alimi ax-11 syl ) ACEZABEZCEIBEAJCDFACBGH $.
  $}

  ${
    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-11 syl6 ) ABDGZBCGZDGKCGABLDEFHBDCIJ $.
  $}

  ${
    $d w x y $.  $d w x z $.  $d w ph $.
    $( Move universal quantifier in and out of substitution.  (Contributed by
       NM, 16-May-1993.)  (Proof shortened by Wolf Lammen, 29-Sep-2018.)
       Reduce dependencies on axioms.  (Revised by Steven Nguyen,
       13-Aug-2023.) $)
    sbal $p |- ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) $=
      ( vw weq wi wal wsb alcom 19.21v albii bitr3i imbi2i df-sb 3bitr4i ) EDFZ
      CEFZAGZCHZGZBHZEHZUAEHZBHABHZCDIZACDIZBHUAEBJQRUEGZCHZGZEHQTBHZGZEHUFUCUJ
      ULEUIUKQUISBHZCHUKUMUHCRABKLSCBJMNLUECEDOUBULEQTBKLPUGUDBACEDOLP $.
  $}

  ${
    $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 x z w $.  $d y z w $.  $d ph w $.
    hbsbw.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.  Version of ~ hbsb with a disjoint
       variable condition, which requires fewer axioms.  (Contributed by NM,
       12-Aug-1993.)  (Revised by GG, 23-May-2024.)  (Proof shortened by Wolf
       Lammen, 14-May-2025.) $)
    hbsbw $p |- ( [ y / x ] ph -> A. z [ y / x ] ph ) $=
      ( wsb wal sbimi sbal sylib ) ABCFZADGZBCFKDGALBCEHADBCIJ $.
    $( $j usage 'hbsbw' avoids 'ax-6' 'ax-7' 'ax-10' 'ax-12' 'ax-13'; $)

    $( Obsolete version of ~ hbsbw as of 14-May-2025.  (Contributed by NM,
       12-Aug-1993.)  (Revised by GG, 23-May-2024.)
       (New usage is discouraged.)  (Proof modification is discouraged.) $)
    hbsbwOLD $p |- ( [ y / x ] ph -> A. z [ y / x ] ph ) $=
      ( vw wsb weq wal df-sb imim2i alimi 19.21v biimpri ax-11 3syl sylibr hbal
      wi hbxfrbi ) ABCGFCHZBFHZASZBIZSZFIDABFCJUEDFUEUAUDDIZSUEDIUDUFUAUDUBADIZ
      SZBIUCDIZBIUFUCUHBAUGUBEKLUHUIBUIUHUBADMNLUCBDOPKUAUDDMQRT $.
    $( $j usage 'hbsbwOLD' avoids 'ax-6' 'ax-7' 'ax-10' 'ax-12' 'ax-13'; $)
  $}

  ${
    $d x z $.  $d u v x w $.  $d u v y z $.  $d u 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 shortened by Wolf Lammen, 23-Dec-2022.) $)
    sbcom2 $p |- ( [ w / z ] [ y / x ] ph <-> [ y / x ] [ w / z ] ph ) $=
      ( vv vu weq wsb wb wi wal 2sb6 alcom ancomst sbequ sbbidv ax6ev exlimiiv
      wa 2albii 3bitri bitr4i bitr3id sylan9bb sylan9bbr bitr3d ex ) FEHZABCIZD
      EIZADEIZBCIZJZFGCHZUIUNKGUOUIUNUOUITADFIZBGIZUKUMUOUQUJDFIZUIUKUQABGIZDFI
      ZUOURUTBGHZDFHZTAKZDLBLZUQUTVBVATAKZBLDLVEDLBLVDADBFGMVEDBNVEVCBDVBVAAOUA
      UBABDGFMUCUOUSUJDFAGCBPQUDUJFEDPUEUIUQULBGIUOUMUIUPULBGAFEDPQULGCBPUFUGUH
      GCRSFERS $.
  $}

  ${
    $d v w ph $.  $d v w x $.  $d v w y $.
    $( Obsolete version of ~ sbco4lem as of 3-Sep-2025.  (Contributed by Jim
       Kingdon, 26-Sep-2018.)  (Proof shortened by Wolf Lammen, 12-Oct-2024.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    sbco4lemOLD $p |- ( [ x / v ] [ y / x ] [ v / y ] ph <->
        [ x / w ] [ y / x ] [ w / y ] ph ) $=
      ( wsb sbcom2 sbbii sbco2vv 2sbbii 3bitr3i ) ACDFZDEFZBCFZEBFLBCFZDEFZEBFA
      CEFZBCFEBFODBFNPEBLDEBCGHMQEBCBACEDIJODBEIK $.
  $}

  ${
    $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 $.
    $( Obsolete version of ~ sbco4 as of 3-Sep-2025.  (Contributed by Jim
       Kingdon, 25-Sep-2018.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    sbco4OLD $p |- ( [ y / u ] [ x / v ] [ u / x ] [ v / y ] ph <->
        [ x / w ] [ y / x ] [ w / y ] ph ) $=
      ( vt wsb sbcom2 sbco2vv sbbii bitr3i sbco4lem 3bitri ) ACEHZBFHZEBHFCHZOB
      CHZEBHZACGHBCHGBHACDHBCHDBHQPFCHZEBHSPFCEBITREBOBCFJKLABCGEMABCDGMN $.
  $}

  $( Lemma 24 of [Monk2] p. 114.  (Contributed by Mario Carneiro, 24-Sep-2016.)
     Remove dependency on ~ ax-12 .  (Revised by Wolf Lammen, 18-Oct-2021.) $)
  nfa2 $p |- F/ x A. y A. x ph $=
    ( wal alcom nfa1 nfxfr ) ABDCDACDZBDBACBEHBFG $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Axiom scheme ax-12 (Substitution)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Axiom of 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.

     The original version of this axiom was ~ ax-c15 and was replaced with this
     shorter ~ ax-12 in Jan. 2007.  The old axiom is proved from this one as
     Theorem ~ axc15 .  Conversely, this axiom is proved from ~ ax-c15 as
     Theorem ~ ax12 .

     Juha Arpiainen proved the metalogical independence of this axiom (in the
     form of the older axiom ~ ax-c15 ) from the others on 19-Jan-2006.  See
     item 9a at ~ https://us.metamath.org/award2003.html .

     See ~ ax12v and ~ ax12v2 for other equivalents of this axiom that (unlike
     this axiom) have distinct variable restrictions.

     This axiom scheme is logically redundant (see ~ ax12w ) but is used as an
     auxiliary axiom scheme to achieve scheme completeness.  (Contributed by
     NM, 22-Jan-2007.)  (New usage is discouraged.) $)
  ax-12 $a |- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) ) $.

  ${
    $d x y $.  $d y ph $.
    $( This is essentially Axiom ~ ax-12 weakened by additional restrictions on
       variables.  Besides ~ axc11r , this theorem should be the only one
       referencing ~ ax-12 directly.

       Both restrictions on variables have their own value.  If for a moment we
       assume ` x ` could be set to ` y ` , then, after elimination of the
       tautology ` y = y ` , immediately we have ` ph -> A. y ph ` for all
       ` ph ` and ` y ` , that is ~ ax-5 , a degenerate result.

       The second restriction is not necessary, but a simplification that makes
       the following interpretation easier to see.  Since ` ph ` textually at
       most depends on ` x ` , we can look at it at some given 'fixed' ` y ` .
       This theorem now states that the truth value of ` ph ` will stay
       constant, as long as we 'vary ` x ` around ` y ` ' only such that
       ` x = y ` still holds.  Or in other words, equality is the finest
       grained logical expression.  If you cannot differ two sets by ` = ` ,
       you won't find a whatever sophisticated expression that does.  One might
       wonder how the described variation of ` x ` is possible at all.  Note
       that Metamath is a text processor that easily sees a difference between
       text chunks ` { x | -. x = x } ` and ` { y | -. y = y } ` .  Our usual
       interpretation is to abstract from textual variations of the same set,
       but we are free to interpret Metamath's formalism differently, and in
       fact let ` x ` run through all textual representations of sets.

       Had we allowed ` ph ` to depend also on ` y ` , this idea is both harder
       to see, and it is less clear that this extra freedom introduces effects
       not covered by other axioms.  (Contributed by Wolf Lammen,
       8-Aug-2020.) $)
    ax12v $p |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) $=
      ( wal weq wi ax-5 ax-12 syl5 ) AACDBCEZJAFBDACGABCHI $.
  $}

  ${
    $d x y z $.  $d z ph $.
    $( It is possible to remove any restriction on ` ph ` in ~ ax12v .  Same as
       Axiom C8 of [Monk2] p. 105.  Use ~ ax12v instead when sufficient.
       (Contributed by NM, 5-Aug-1993.)  Remove dependencies on ~ ax-10 and
       ~ ax-13 .  (Revised by Jim Kingdon, 15-Dec-2017.)  (Proof shortened by
       Wolf Lammen, 8-Dec-2019.) $)
    ax12v2 $p |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) $=
      ( vz weq wi wal equtrr ax12v imim1d alimdv syl9r syld ax6evr exlimiiv ) C
      DEZBCEZAQAFZBGZFZFDPQBDEZTCDBHZUAAUAAFZBGPSABDIPUCRBPQUAAUBJKLMDCNO $.
    $( $j usage 'ax12v2' avoids 'ax-10' 'ax-11' 'ax-13'; $)
  $}

  ${
    $d x y $.
    $( Version of ~ ax12v2 rewritten to use an existential quantifier.  One
       direction of ~ sbalex without the universal quantifier, avoiding
       ~ ax-10 .  (Contributed by SN, 14-Aug-2025.) $)
    ax12ev2 $p |- ( E. x ( x = y /\ ph ) -> ( x = y -> ph ) ) $=
      ( weq wa wex wn wi wal exnalimn ax12v2 con1d biimtrid com12 ) BCDZOAEBFZA
      POAGZHBIZGOAOABJOARQBCKLMN $.
    $( $j usage 'ax12ev2' avoids 'ax-10' 'ax-11' 'ax-13'; $)
  $}

  ${
    $d x y $.  $d y ph $.
    $( If a wff is true, it is true for at least one instance.  Special case of
       Theorem 19.8 of [Margaris] p. 89.  See ~ 19.8v for a version with a
       disjoint variable condition requiring fewer axioms.  (Contributed by NM,
       9-Jan-1993.)  Allow a shortening of ~ sp .  (Revised by Wolf Lammen,
       13-Jan-2018.)  (Proof shortened by Wolf Lammen, 8-Dec-2019.) $)
    19.8a $p |- ( ph -> E. x ph ) $=
      ( vy weq wex wi wal ax12v alequexv syl6 ax6evr exlimiiv ) BCDZAABEZFCMAMA
      FBGNABCHABCIJCBKL $.
  $}

  ${
    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 $.
  $}

  $( Specialization.  A universally quantified wff implies the wff without a
     quantifier.  Axiom scheme B5 of [Tarski] p. 67 (under his system S2,
     defined in the last paragraph on p. 77).  Also appears as Axiom scheme C5'
     in [Megill] p. 448 (p. 16 of the preprint).  This corresponds to the axiom
     (T) of modal logic.

     For the axiom of specialization presented in many logic textbooks, see
     Theorem ~ stdpc4 .

     This theorem shows that our obsolete axiom ~ ax-c5 can be derived from the
     others.  The proof uses ideas from the proof of Lemma 21 of [Monk2]
     p. 114.

     It appears that this scheme cannot be derived directly from Tarski's
     axioms without auxiliary axiom scheme ~ ax-12 .  It is thought the best we
     can do using only Tarski's axioms is ~ spw .  Also see ~ spvw where ` x `
     and ` ph ` are disjoint, using fewer axioms.  (Contributed by NM,
     21-May-2008.)  (Proof shortened by Scott Fenton, 24-Jan-2011.)  (Proof
     shortened by Wolf Lammen, 13-Jan-2018.) $)
  sp $p |- ( A. x ph -> ph ) $=
    ( wal wn wex alex 19.8a con1i sylbi ) ABCADZBEZDAABFAKJBGHI $.

  ${
    spi.1 $e |- A. x ph $.
    $( Inference rule of universal instantiation, or universal specialization.
       Converse of the inference rule of (universal) generalization ~ ax-gen .
       Contrary to the rule of generalization, its closed form is valid, see
       ~ sp .  (Contributed by NM, 5-Aug-1993.) $)
    spi $p |- ph $=
      ( wal sp ax-mp ) ABDACABEF $.
  $}

  ${
    sps.1 $e |- ( ph -> ps ) $.
    $( Generalization of antecedent.  (Contributed by NM, 5-Jan-1993.) $)
    sps $p |- ( A. x ph -> ps ) $=
      ( wal sp syl ) ACEABACFDG $.
  $}

  $( A double specialization (see ~ sp ).  Another double specialization,
     closer to PM*11.1, is ~ 2stdpc4 .  (Contributed by BJ, 15-Sep-2018.) $)
  2sp $p |- ( A. x A. y ph -> ph ) $=
    ( wal sp sps ) ACDABACEF $.

  ${
    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 $.
  $}

  $( Theorem 19.2 of [Margaris] p. 89, generalized to use two setvar variables.
     Use ~ 19.2 when sufficient.  (Contributed by Mel L. O'Cat,
     31-Mar-2008.) $)
  19.2g $p |- ( A. x ph -> E. y ph ) $=
    ( wex 19.8a sps ) AACDBACEF $.

  ${
    19.21bi.1 $e |- ( ph -> A. x ps ) $.
    $( Inference form of ~ 19.21 and also deduction form of ~ sp .
       (Contributed by NM, 26-May-1993.) $)
    19.21bi $p |- ( ph -> ps ) $=
      ( wal sp syl ) ABCEBDBCFG $.
  $}

  ${
    19.21bbi.1 $e |- ( ph -> A. x A. y ps ) $.
    $( Inference removing two universal quantifiers.  Version of ~ 19.21bi with
       two quantifiers.  (Contributed by NM, 20-Apr-1994.) $)
    19.21bbi $p |- ( ph -> ps ) $=
      ( wal 19.21bi ) ABDABDFCEGG $.
  $}

  ${
    19.23bi.1 $e |- ( E. x ph -> ps ) $.
    $( Inference form of Theorem 19.23 of [Margaris] p. 90, see ~ 19.23 .
       (Contributed by NM, 12-Mar-1993.) $)
    19.23bi $p |- ( ph -> ps ) $=
      ( wex 19.8a syl ) AACEBACFDG $.
  $}

  ${
    nexr.1 $e |- -. E. x ph $.
    $( Inference associated with the contrapositive of ~ 19.8a .  (Contributed
       by Jeff Hankins, 26-Jul-2009.) $)
    nexr $p |- -. ph $=
      ( wex 19.8a mto ) AABDCABEF $.
  $}

  $( Quantified excluded middle (see ~ exmid ).  Also known as the drinker
     paradox (if ` ph ( x ) ` is interpreted as " ` x ` drinks", then this
     theorem tells that there exists a person such that, if this person drinks,
     then everyone drinks).  Exercise 9.2a of Boolos, p. 111, _Computability
     and Logic_.  (Contributed by NM, 10-Dec-2000.) $)
  qexmid $p |- E. x ( ph -> A. x ph ) $=
    ( wal 19.8a 19.35ri ) AABCZBFBDE $.

  $( Consequence of the definition of not-free.  (Contributed by Mario
     Carneiro, 26-Sep-2016.) ~ df-nf changed.  (Revised by Wolf Lammen,
     11-Sep-2021.)  (Proof shortened by Wolf Lammen, 23-Nov-2023.) $)
  nf5r $p |- ( F/ x ph -> ( ph -> A. x ph ) ) $=
    ( wex wnf wal 19.8a id nfrd syl5 ) AABCABDZABEABFJABJGHI $.

  ${
    nf5ri.1 $e |- F/ x ph $.
    $( Consequence of the definition of not-free.  (Contributed by Mario
       Carneiro, 11-Aug-2016.)  (Proof shortened by Wolf Lammen,
       15-Mar-2023.) $)
    nf5ri $p |- ( ph -> A. x ph ) $=
      ( wal nfri 19.23bi ) AABDBABCEF $.
  $}

  ${
    nf5rd.1 $e |- ( ph -> F/ x ps ) $.
    $( Consequence of the definition of not-free in a context.  (Contributed by
       Mario Carneiro, 11-Aug-2016.) $)
    nf5rd $p |- ( ph -> ( ps -> A. x ps ) ) $=
      ( wnf wal wi nf5r syl ) ABCEBBCFGDBCHI $.
  $}

  ${
    $d x y $.
    spimedv.1 $e |- ( ch -> F/ x ph ) $.
    spimedv.2 $e |- ( x = y -> ( ph -> ps ) ) $.
    $( Deduction version of ~ spimev .  Version of ~ spimed with a disjoint
       variable condition, which does not require ~ ax-13 .  See ~ spime for a
       non-deduction version.  (Contributed by NM, 14-May-1993.)  (Revised by
       BJ, 31-May-2019.) $)
    spimedv $p |- ( ch -> ( ph -> E. x ps ) ) $=
      ( wal wex nf5rd weq wi ax6ev eximii 19.35i syl6 ) CAADHBDICADFJABDDEKABLD
      DEMGNOP $.
  $}

  ${
    $d x y $.
    spimefv.1 $e |- F/ x ph $.
    spimefv.2 $e |- ( x = y -> ( ph -> ps ) ) $.
    $( Version of ~ spime with a disjoint variable condition, which does not
       require ~ ax-13 .  (Contributed by BJ, 31-May-2019.) $)
    spimefv $p |- ( ph -> E. x ps ) $=
      ( wex wi wtru wnf a1i spimedv mptru ) ABCGHABICDACJIEKFLM $.
  $}

  ${
    nfim1.1 $e |- F/ x ph $.
    nfim1.2 $e |- ( ph -> F/ x ps ) $.
    $( A closed form of ~ nfim .  (Contributed by NM, 2-Jun-1993.)  (Revised by
       Mario Carneiro, 24-Sep-2016.)  (Proof shortened by Wolf Lammen,
       2-Jan-2018.) ~ df-nf changed.  (Revised by Wolf Lammen, 18-Sep-2021.) $)
    nfim1 $p |- F/ x ( ph -> ps ) $=
      ( wal wn wo wi wnf nf3 mpbi nftht sps nfimd pm2.21 alimi syl jaoi ax-mp )
      ACFZAGZCFZHZABIZCJZACJUDDACKLUAUFUCUAABCACMABCJCENOUCUECFUFUBUECABPQUECMR
      ST $.

    $( A closed form of ~ nfan .  (Contributed by Mario Carneiro, 3-Oct-2016.)
       ~ df-nf changed.  (Revised by Wolf Lammen, 18-Sep-2021.)  (Proof
       shortened by Wolf Lammen, 7-Jul-2022.) $)
    nfan1 $p |- F/ x ( ph /\ ps ) $=
      ( wa wn wi df-an nfnd nfim1 nfn nfxfr ) ABFABGZHZGCABIOCANCDABCEJKLM $.
  $}

  $( Closed form of ~ 19.3 and version of ~ 19.9t with a universal quantifier.
     (Contributed by NM, 9-Nov-2020.)  (Proof shortened by BJ, 9-Oct-2022.) $)
  19.3t $p |- ( F/ x ph -> ( A. x ph <-> ph ) ) $=
    ( wnf wal sp nf5r impbid2 ) ABCABDAABEABFG $.

  ${
    19.3.1 $e |- F/ x ph $.
    $( A wff may be quantified with a variable not free in it.  Version of
       ~ 19.9 with a universal quantifier.  Theorem 19.3 of [Margaris] p. 89.
       See ~ 19.3v for a version requiring fewer axioms.  (Contributed by NM,
       12-Mar-1993.)  (Revised by Mario Carneiro, 24-Sep-2016.) $)
    19.3 $p |- ( A. x ph <-> ph ) $=
      ( wal sp nf5ri impbii ) ABDAABEABCFG $.
  $}

  ${
    19.9d.1 $e |- ( ps -> F/ x ph ) $.
    $( A deduction version of one direction of ~ 19.9 .  (Contributed by NM,
       14-May-1993.)  (Revised by Mario Carneiro, 24-Sep-2016.)  Revised to
       shorten other proofs.  (Revised by Wolf Lammen, 14-Jul-2020.) ~ df-nf
       changed.  (Revised by Wolf Lammen, 11-Sep-2021.)  (Proof shortened by
       Wolf Lammen, 8-Jul-2022.) $)
    19.9d $p |- ( ps -> ( E. x ph -> ph ) ) $=
      ( wex wal nfrd sp syl6 ) BACEACFABACDGACHI $.
  $}

  $( Closed form of ~ 19.9 and version of ~ 19.3t with an existential
     quantifier.  (Contributed by NM, 13-May-1993.)  (Revised by Mario
     Carneiro, 24-Sep-2016.)  (Proof shortened by Wolf Lammen, 14-Jul-2020.) $)
  19.9t $p |- ( F/ x ph -> ( E. x ph <-> ph ) ) $=
    ( wnf wex id 19.9d 19.8a impbid1 ) ABCZABDAAIBIEFABGH $.

  ${
    19.9.1 $e |- F/ x ph $.
    $( A wff may be existentially quantified with a variable not free in it.
       Version of ~ 19.3 with an existential quantifier.  Theorem 19.9 of
       [Margaris] p. 89.  See ~ 19.9v for a version requiring fewer axioms.
       (Contributed by FL, 24-Mar-2007.)  (Revised by Mario Carneiro,
       24-Sep-2016.)  (Proof shortened by Wolf Lammen, 30-Dec-2017.)  Revised
       to shorten other proofs.  (Revised by Wolf Lammen, 14-Jul-2020.) $)
    19.9 $p |- ( E. x ph <-> ph ) $=
      ( wnf wex wb 19.9t ax-mp ) ABDABEAFCABGH $.
  $}

  $( Closed form of Theorem 19.21 of [Margaris] p. 90, see ~ 19.21 .
     (Contributed by NM, 27-May-1997.)  (Revised by Mario Carneiro,
     24-Sep-2016.)  (Proof shortened by Wolf Lammen, 3-Jan-2018.) ~ df-nf
     changed.  (Revised by Wolf Lammen, 11-Sep-2021.)  (Proof shortened by BJ,
     3-Nov-2021.) $)
  19.21t $p |- ( F/ x ph -> ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) ) $=
    ( wnf wex wal wi 19.38a 19.9t imbi1d bitr3d ) ACDZACEZBCFZGABGCFANGABCHLMAN
    ACIJK $.

  ${
    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 ` ".  See ~ 19.21v for a version
       requiring fewer axioms.  See also ~ 19.21h .  (Contributed by NM,
       14-May-1993.)  (Revised by Mario Carneiro, 24-Sep-2016.) ~ df-nf
       changed.  (Revised by Wolf Lammen, 18-Sep-2021.) $)
    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.  See ~ stdpc5v for a version requiring fewer axioms.
       (Contributed by NM, 22-Sep-1993.)  (Revised by Mario Carneiro,
       12-Oct-2016.)  (Proof shortened by Wolf Lammen, 1-Jan-2018.)  Remove
       dependency on ~ ax-10 .  (Revised by Wolf Lammen, 4-Jul-2021.)  (Proof
       shortened by Wolf Lammen, 11-Oct-2021.) $)
    stdpc5 $p |- ( A. x ( ph -> ps ) -> ( ph -> A. x ps ) ) $=
      ( wi wal 19.21 biimpi ) ABECFABCFEABCDGH $.
  $}

  ${
    19.21-2.1 $e |- F/ x ph $.
    19.21-2.2 $e |- F/ y ph $.
    $( Version of ~ 19.21 with two 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 $.
  $}

  $( Closed form of Theorem 19.23 of [Margaris] p. 90.  See ~ 19.23 .
     (Contributed by NM, 7-Nov-2005.)  (Proof shortened by Wolf Lammen,
     13-Aug-2020.) ~ df-nf changed.  (Revised by Wolf Lammen, 11-Sep-2021.)
     (Proof shortened by BJ, 8-Oct-2022.) $)
  19.23t $p |- ( F/ x ps -> ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) ) $=
    ( wnf wex wal wi 19.38b 19.3t imbi2d bitr3d ) BCDZACEZBCFZGABGCFMBGABCHLNBM
    BCIJK $.

  ${
    19.23.1 $e |- F/ x ps $.
    $( Theorem 19.23 of [Margaris] p. 90.  See ~ 19.23v for a version requiring
       fewer axioms.  (Contributed by NM, 24-Jan-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 $.
  $}

  ${
    alimd.1 $e |- F/ x ph $.
    alimd.2 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction form of Theorem 19.20 of [Margaris] p. 90, see ~ alim .  See
       ~ alimdh , ~ alimdv for variants requiring fewer axioms.  (Contributed
       by Mario Carneiro, 24-Sep-2016.) $)
    alimd $p |- ( ph -> ( A. x ps -> A. x ch ) ) $=
      ( nf5ri alimdh ) ABCDADEGFH $.
  $}

  ${
    alrimi.1 $e |- F/ x ph $.
    alrimi.2 $e |- ( ph -> ps ) $.
    $( Inference form of Theorem 19.21 of [Margaris] p. 90, see ~ 19.21 .
       (Contributed by Mario Carneiro, 24-Sep-2016.) $)
    alrimi $p |- ( ph -> A. x ps ) $=
      ( nf5ri alrimih ) ABCACDFEG $.
  $}

  ${
    alrimdd.1 $e |- F/ x ph $.
    alrimdd.2 $e |- ( ph -> F/ x ps ) $.
    alrimdd.3 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction form of Theorem 19.21 of [Margaris] p. 90, see ~ 19.21 .
       (Contributed by Mario Carneiro, 24-Sep-2016.) $)
    alrimdd $p |- ( ph -> ( ps -> A. x ch ) ) $=
      ( wal nf5rd alimd syld ) ABBDHCDHABDFIABCDEGJK $.
  $}

  ${
    alrimd.1 $e |- F/ x ph $.
    alrimd.2 $e |- F/ x ps $.
    alrimd.3 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction form of Theorem 19.21 of [Margaris] p. 90, see ~ 19.21 .
       (Contributed by Mario Carneiro, 24-Sep-2016.) $)
    alrimd $p |- ( ph -> ( ps -> A. x ch ) ) $=
      ( wnf a1i alrimdd ) ABCDEBDHAFIGJ $.
  $}

  ${
    eximd.1 $e |- F/ x ph $.
    eximd.2 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction form of Theorem 19.22 of [Margaris] p. 90, see ~ exim .
       (Contributed by NM, 29-Jun-1993.)  (Revised by Mario Carneiro,
       24-Sep-2016.) $)
    eximd $p |- ( ph -> ( E. x ps -> E. x ch ) ) $=
      ( nf5ri eximdh ) ABCDADEGFH $.
  $}

  ${
    exlimi.1 $e |- F/ x ps $.
    exlimi.2 $e |- ( ph -> ps ) $.
    $( Inference associated with ~ 19.23 .  See ~ exlimiv for a version with a
       disjoint variable condition requiring fewer axioms.  (Contributed by NM,
       10-Jan-1993.)  (Revised by Mario Carneiro, 24-Sep-2016.) $)
    exlimi $p |- ( E. x ph -> ps ) $=
      ( wi wex 19.23 mpgbi ) ABFACGBFCABCDHEI $.
  $}

  ${
    exlimd.1 $e |- F/ x ph $.
    exlimd.2 $e |- F/ x ch $.
    exlimd.3 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction form of Theorem 19.9 of [Margaris] p. 89.  (Contributed by NM,
       23-Jan-1993.)  (Revised by Mario Carneiro, 24-Sep-2016.)  (Proof
       shortened by Wolf Lammen, 12-Jan-2018.) $)
    exlimd $p |- ( ph -> ( E. x ps -> ch ) ) $=
      ( wex eximd 19.9 imbitrdi ) ABDHCDHCABCDEGICDFJK $.
  $}

  ${
    exlimdd.1 $e |- F/ x ph $.
    exlimdd.2 $e |- F/ x ch $.
    exlimdd.3 $e |- ( ph -> E. x ps ) $.
    ${
      exlimimdd.4 $e |- ( ph -> ( ps -> ch ) ) $.
      $( Existential elimination rule of natural deduction.  (Contributed by
         ML, 17-Jul-2020.)  Shorten ~ exlimdd .  (Revised by Wolf Lammen,
         3-Sep-2023.) $)
      exlimimdd $p |- ( ph -> ch ) $=
        ( wex exlimd mpd ) ABDICGABCDEFHJK $.
    $}

    ${
      exlimdd.4 $e |- ( ( ph /\ ps ) -> ch ) $.
      $( Existential elimination rule of natural deduction.  (Contributed by
         Mario Carneiro, 9-Feb-2017.)  (Proof shortened by Wolf Lammen,
         3-Sep-2023.) $)
      exlimdd $p |- ( ph -> ch ) $=
        ( ex exlimimdd ) ABCDEFGABCHIJ $.
    $}
  $}

  ${
    nexd.1 $e |- F/ x ph $.
    nexd.2 $e |- ( ph -> -. ps ) $.
    $( Deduction for generalization rule for negated wff.  (Contributed by
       Mario Carneiro, 24-Sep-2016.) $)
    nexd $p |- ( ph -> -. E. x ps ) $=
      ( nf5ri nexdh ) ABCACDFEG $.
  $}

  ${
    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 ) ) $=
      ( nf5ri albidh ) ABCDADEGFH $.

    $( Formula-building rule for existential quantifier (deduction form).
       (Contributed by Mario Carneiro, 24-Sep-2016.) $)
    exbid $p |- ( ph -> ( E. x ps <-> E. x ch ) ) $=
      ( nf5ri exbidh ) ABCDADEGFH $.

    $( An equality theorem for effectively not free.  (Contributed by Mario
       Carneiro, 4-Oct-2016.) ~ df-nf changed.  (Revised by Wolf Lammen,
       18-Sep-2021.) $)
    nfbidf $p |- ( ph -> ( F/ x ps <-> F/ x ch ) ) $=
      ( wex wal wi wnf exbid albid imbi12d df-nf 3bitr4g ) ABDGZBDHZICDGZCDHZIB
      DJCDJAPRQSABCDEFKABCDEFLMBDNCDNO $.
  $}

  ${
    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 $.
  $}

  ${
    19.27.1 $e |- F/ x ps $.
    $( Theorem 19.27 of [Margaris] p. 90.  See ~ 19.27v for a version requiring
       fewer axioms.  (Contributed by NM, 21-Jun-1993.) $)
    19.27 $p |- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ ps ) ) $=
      ( wa wal 19.26 19.3 anbi2i bitri ) ABECFACFZBCFZEKBEABCGLBKBCDHIJ $.
  $}

  ${
    19.28.1 $e |- F/ x ph $.
    $( Theorem 19.28 of [Margaris] p. 90.  See ~ 19.28v for a version requiring
       fewer axioms.  (Contributed by NM, 1-Aug-1993.)  (Proof shortened by
       Wolf Lammen, 7-May-2025.) $)
    19.28 $p |- ( A. x ( ph /\ ps ) <-> ( ph /\ A. x ps ) ) $=
      ( wa wal 19.26 19.3 bianbi ) ABECFACFBCFAABCGACDHI $.
  $}

  ${
    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 $.
  $}

  ${
    19.36.1 $e |- F/ x ps $.
    $( Theorem 19.36 of [Margaris] p. 90.  See ~ 19.36v for a version requiring
       fewer axioms.  (Contributed by NM, 24-Jun-1993.) $)
    19.36 $p |- ( E. x ( ph -> ps ) <-> ( A. x ph -> ps ) ) $=
      ( wi wex wal 19.35 19.9 imbi2i bitri ) ABECFACGZBCFZELBEABCHMBLBCDIJK $.

    19.36i.2 $e |- E. x ( ph -> ps ) $.
    $( Inference associated with ~ 19.36 .  See ~ 19.36iv for a version
       requiring fewer axioms.  (Contributed by NM, 24-Jun-1993.) $)
    19.36i $p |- ( A. x ph -> ps ) $=
      ( wi wex wal 19.36 mpbi ) ABFCGACHBFEABCDIJ $.
  $}

  ${
    19.37.1 $e |- F/ x ph $.
    $( Theorem 19.37 of [Margaris] p. 90.  See ~ 19.37v for a version requiring
       fewer axioms.  (Contributed by NM, 21-Jun-1993.) $)
    19.37 $p |- ( E. x ( ph -> ps ) <-> ( ph -> E. x ps ) ) $=
      ( wi wex wal 19.35 19.3 imbi1i bitri ) ABECFACGZBCFZEAMEABCHLAMACDIJK $.
  $}

  ${
    19.32.1 $e |- F/ x ph $.
    $( Theorem 19.32 of [Margaris] p. 90.  See ~ 19.32v for a version requiring
       fewer axioms.  (Contributed by NM, 14-May-1993.)  (Revised by Mario
       Carneiro, 24-Sep-2016.) $)
    19.32 $p |- ( A. x ( ph \/ ps ) <-> ( ph \/ A. x ps ) ) $=
      ( wn wi wal wo nfn 19.21 df-or albii 3bitr4i ) AEZBFZCGNBCGZFABHZCGAPHNBC
      ACDIJQOCABKLAPKM $.
  $}

  ${
    19.31.1 $e |- F/ x ps $.
    $( Theorem 19.31 of [Margaris] p. 90.  See ~ 19.31v for a version requiring
       fewer axioms.  (Contributed by NM, 14-May-1993.) $)
    19.31 $p |- ( A. x ( ph \/ ps ) <-> ( A. x ph \/ ps ) ) $=
      ( wo wal 19.32 orcom albii 3bitr4i ) BAEZCFBACFZEABEZCFLBEBACDGMKCABHILBH
      J $.
  $}

  ${
    19.41.1 $e |- F/ x ps $.
    $( Theorem 19.41 of [Margaris] p. 90.  See ~ 19.41v for a version requiring
       fewer axioms.  (Contributed by NM, 14-May-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 $.
  $}

  ${
    19.42.1 $e |- F/ x ph $.
    $( Theorem 19.42 of [Margaris] p. 90.  See ~ 19.42v for a version requiring
       fewer axioms.  See ~ exan for an immediate version.  (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
      $.
  $}

  ${
    19.44.1 $e |- F/ x ps $.
    $( Theorem 19.44 of [Margaris] p. 90.  See ~ 19.44v for a version requiring
       fewer axioms.  (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.  See ~ 19.45v for a version requiring
       fewer axioms.  (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 $.
  $}

  ${
    $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, which does not require ~ ax-13 .  See
       ~ spimvw for a version with two disjoint variable conditions, requiring
       fewer axioms, and ~ spimv for another variant.  (Contributed by NM,
       10-Jan-1993.)  (Revised by BJ, 31-May-2019.) $)
    spimfv $p |- ( A. x ph -> ps ) $=
      ( weq wi ax6ev 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, which does not require
       ~ ax-13 .  (Contributed by Raph Levien, 9-Jul-2003.)  (Revised by BJ,
       31-May-2019.) $)
    chvarfv $p |- ps $=
      ( weq biimpd spimfv mpg ) ABCABCDECDHABFIJGK $.
  $}

  ${
    $d x y $.  $d y ph $.
    cbv3v2.nf $e |- F/ x ps $.
    cbv3v2.1 $e |- ( x = y -> ( ph -> ps ) ) $.
    $( Version of ~ cbv3 with two disjoint variable conditions, which does not
       require ~ ax-11 nor ~ ax-13 .  (Contributed by BJ, 24-Jun-2019.)  (Proof
       shortened by Wolf Lammen, 30-Aug-2021.) $)
    cbv3v2 $p |- ( A. x ph -> A. y ps ) $=
      ( wal spimfv alrimiv ) ACGBDABCDEFHI $.
  $}

  ${
    $d x t $.
    $( Equivalence of two ways to express proper substitution of a setvar for
       another setvar disjoint from it in a formula.  This proof of their
       equivalence does not use ~ df-sb .

       That both sides of the biconditional express proper substitution is
       proved by ~ sb5 and ~ sb6 .  The implication "to the left" is ~ equs4v
       and does not require ~ ax-10 nor ~ ax-12 .  It also holds without
       disjoint variable condition if we allow more axioms (see ~ equs4 ).
       Theorem 6.2 of [Quine] p. 40.  Theorem ~ equs5 replaces the disjoint
       variable condition with a distinctor antecedent.  Theorem ~ equs45f
       replaces the disjoint variable condition on ` x , t ` with the
       nonfreeness hypothesis of ` t ` in ` ph ` .  (Contributed by NM,
       14-Apr-2008.)  Revised to use ~ equsexv in place of ~ equsex in order to
       remove dependency on ~ ax-13 .  (Revised by BJ, 20-Dec-2020.)  Revise to
       remove dependency on ~ df-sb .  (Revised by BJ, 21-Sep-2024.)  (Proof
       shortened by SN, 14-Aug-2025.) $)
    sbalex $p |- ( E. x ( x = t /\ ph ) <-> A. x ( x = t -> ph ) ) $=
      ( weq wa wex wi wal nfe1 ax12ev2 alrimi equs4v impbii ) BCDZAEZBFZNAGZBHP
      QBOBIABCJKABCLM $.
    $( $j usage 'sbalex' avoids 'ax-11' 'df-sb' 'ax-13'; $)

    $( Obsolete version of ~ sbalex as of 14-Aug-2025.  (Contributed by NM,
       14-Apr-2008.)  (Revised by BJ, 20-Dec-2020.)  (Revised by BJ,
       21-Sep-2024.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    sbalexOLD $p |- ( E. x ( x = t /\ ph ) <-> A. x ( x = t -> ph ) ) $=
      ( weq wa wex wi wal nfa1 ax12v2 imp exlimi equs4v impbii ) BCDZAEZBFOAGZB
      HZPRBQBIOARABCJKLABCMN $.
  $}

  ${
    $d x t $.
    $( Version of ~ sb4a with a disjoint variable condition, which does not
       require ~ ax-13 .  The distinctor antecedent from ~ sb4b is replaced by
       a disjoint variable condition in this theorem.  (Contributed by NM,
       2-Feb-2007.)  (Revised by BJ, 15-Dec-2023.) $)
    sb4av $p |- ( [ t / x ] A. t ph -> A. x ( x = t -> ph ) ) $=
      ( wal wsb weq wi sp sbimi sb6 sylib ) ACDZBCEABCEBCFAGBDLABCACHIABCJK $.
  $}

  ${
    sbimd.1 $e |- F/ x ph $.
    sbimd.2 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction substituting both sides of an implication.  (Contributed by
       Wolf Lammen, 24-Nov-2022.)  Revise ~ df-sb .  (Revised by Steven Nguyen,
       9-Jul-2023.) $)
    sbimd $p |- ( ph -> ( [ y / x ] ps -> [ y / x ] ch ) ) $=
      ( wi wal wsb alrimi spsbim 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.)  Remove dependency on ~ ax-10 and ~ ax-13 .  (Revised
       by Wolf Lammen, 24-Nov-2022.)  Revise ~ df-sb .  (Revised by Steven
       Nguyen, 11-Jul-2023.) $)
    sbbid $p |- ( ph -> ( [ y / x ] ps <-> [ y / x ] ch ) ) $=
      ( wb wal wsb alrimi spsbbi syl ) ABCHZDIBDEJCDEJHANDFGKBCDELM $.

    2sbbid.1 $e |- F/ y ph $.
    $( Deduction doubly substituting both sides of a biconditional.
       (Contributed by AV, 30-Jul-2023.) $)
    2sbbid $p |- ( ph -> ( [ t / x ] [ u / y ] ps
                       <-> [ t / x ] [ u / y ] ch ) ) $=
      ( wsb sbbid ) ABEFKCEFKDGHABCEFJILL $.
  $}

  ${
    $d y x $.  $d y t $.  $d y ph $.
    $( An equality theorem for substitution.  (Contributed by NM, 16-May-1993.)
       Revise ~ df-sb .  (Revised by BJ, 22-Dec-2020.) $)
    sbequ1 $p |- ( x = t -> ( ph -> [ t / x ] ph ) ) $=
      ( vy weq wi wal wsb equeucl ax12v syl6 com23 alrimdv df-sb imbitrrdi ) BC
      EZADCEZBDEZAFBGZFZDGABCHPATDPQASPQRASFBDCIABDJKLMABDCNO $.
  $}

  ${
    $d y x $.  $d y t $.  $d y ph $.
    $( An equality theorem for substitution.  (Contributed by NM, 16-May-1993.)
       Revise ~ df-sb .  (Revised by BJ, 22-Dec-2020.)  (Proof shortened by
       Wolf Lammen, 3-Feb-2024.) $)
    sbequ2 $p |- ( x = t -> ( [ t / x ] ph -> ph ) ) $=
      ( vy wsb weq wex wi wal wa df-sb equvinva equcomi imim12i impcomd aleximi
      biimpi sp syl2im ax5e syl6com ) ABCEZBCFZADGZAUBDCFZBDFZAHZBIZHZDIZUCUFCD
      FZJZDGUDUBUJABDCKQBCDLUIULADUIUKUFAUKUEUHUGCDMUGBRNOPSADTUA $.
  $}

  $( 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 , x ) ` ".  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,
     14-May-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 $.

  ${
    $d x y $.  $d x ph $.
    $( Elimination of substitution.  Also see ~ sbel2x .  (Contributed by NM,
       5-Aug-1993.)  Avoid ~ ax-13 .  (Revised by Wolf Lammen, 6-Aug-2023.)
       Avoid ~ ax-10 .  (Revised by GG, 20-Aug-2023.) $)
    sbelx $p |- ( ph <-> E. x ( x = y /\ [ x / y ] ph ) ) $=
      ( weq wsb wa wex sbequ12r equsexvw bicomi ) BCDACBEZFBGAKABCABCHIJ $.
    $( $j usage 'sbelx' avoids 'ax-10' 'ax-11' 'ax-13'; $)
  $}

  $( An equality theorem for substitution.  (Contributed by NM, 2-Jun-1993.)
     (Proof shortened by Wolf Lammen, 23-Jun-2019.) $)
  sbequ12a $p |- ( x = y -> ( [ y / x ] ph <-> [ x / y ] ph ) ) $=
    ( weq wsb sbequ12r sbequ12 bitr2d ) BCDACBEAABCEABCFABCGH $.

  $( An identity theorem for substitution.  Remark 9.1 in [Megill] p. 447 (p.
     15 of the preprint).  (Contributed by NM, 26-May-1993.)  (Proof shortened
     by Wolf Lammen, 30-Sep-2018.) $)
  sbid $p |- ( [ x / x ] ph <-> ph ) $=
    ( weq wsb wb equid sbequ12r ax-mp ) BBCABBDAEBFABBGH $.

  ${
    $d x y $.
    $( A composition law for substitution.  Version of ~ sbco with a disjoint
       variable condition using fewer axioms.  (Contributed by NM,
       14-May-1993.)  (Revised by GG, 7-Aug-2023.)  (Proof shortened by SN,
       26-Aug-2025.) $)
    sbcov $p |- ( [ y / x ] [ x / y ] ph <-> [ y / x ] ph ) $=
      ( wsb sbequ12r sbbiiev ) ACBDABCABCEF $.
    $( $j usage 'sbcov' avoids 'ax-10' 'ax-11' 'ax-13'; $)

    $( Obsolete version of ~ sbcov as of 26-Aug-2025.  (Contributed by NM,
       14-May-1993.)  (Revised by GG, 7-Aug-2023.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    sbcovOLD $p |- ( [ y / x ] [ x / y ] ph <-> [ y / x ] ph ) $=
      ( wsb sbcom3vv sbid sbbii bitri ) ACBDBCDACCDZBCDABCDACBCEIABCACFGH $.
    $( $j usage 'sbcov' avoids 'ax-10' 'ax-11' 'ax-13'; $)
  $}

  ${
    $d x y $.
    $( Equivalence for substitution.  (Contributed by NM, 2-Jun-1993.)  (Proof
       shortened by Wolf Lammen, 23-Sep-2018.) $)
    sb6a $p |- ( [ y / x ] ph <-> A. x ( x = y -> [ x / y ] ph ) ) $=
      ( wsb weq wi wal sbcov sb6 bitr3i ) ABCDACBDZBCDBCEKFBGABCHKBCIJ $.
  $}

  ${
    $d t x $.  $d x ph $.
    $( Reverting substitution yields the original expression.  Based on fewer
       axioms than ~ sbid2v , at the expense of an extra distinct variable
       condition.  (Contributed by NM, 14-May-1993.)  (Revised by Wolf Lammen,
       5-Aug-2023.) $)
    sbid2vw $p |- ( [ t / x ] [ x / t ] ph <-> ph ) $=
      ( wsb sbequ12r sbievw ) ACBDABCABCEF $.
  $}

  ${
    $d w x y $.  $d w z $.  $d w ph $.
    $( Generalization of ~ axc16 .  Use the latter when sufficient.  This proof
       only requires, on top of { ~ ax-1 -- ~ ax-7 }, Theorem ~ ax12v .
       (Contributed by NM, 15-May-1993.)  (Proof shortened by Andrew Salmon,
       25-May-2011.)  (Proof shortened by Wolf Lammen, 18-Feb-2018.)  Remove
       dependency on ~ ax-13 , along an idea of BJ. (Revised by Wolf Lammen,
       30-Nov-2019.)  (Revised by BJ, 7-Jul-2021.)  Shorten ~ axc11rv .
       (Revised by Wolf Lammen, 11-Oct-2021.) $)
    axc16g $p |- ( A. x x = y -> ( ph -> A. z ph ) ) $=
      ( vw weq wal wi aevlem ax12v sps pm2.27 al2imi syld syl ) BCFBGDEFZDGZAAD
      GZHBCDEIQAPAHZDGZRPATHDADEJKPSADPALMNO $.
  $}

  ${
    $d x y $.
    $( Proof of older axiom ~ ax-c16 .  (Contributed by NM, 8-Nov-2006.)
       (Revised by NM, 22-Sep-2017.) $)
    axc16 $p |- ( A. x x = y -> ( ph -> A. x ph ) ) $=
      ( axc16g ) ABCBD $.
  $}

  ${
    $d x y $.
    $( Biconditional strengthening of ~ axc16g .  (Contributed by NM,
       15-May-1993.) $)
    axc16gb $p |- ( A. x x = y -> ( ph <-> A. z ph ) ) $=
      ( weq wal axc16g sp impbid1 ) BCEBFAADFABCDGADHI $.
  $}

  ${
    $d x y $.
    $( If ~ dtru is false, then there is only one element in the universe, so
       everything satisfies ` F/ ` .  (Contributed by Mario Carneiro,
       7-Oct-2016.)  Remove dependency on ~ ax-11 .  (Revised by Wolf Lammen,
       9-Sep-2018.)  (Proof shortened by BJ, 14-Jun-2019.)  Remove dependency
       on ~ ax-10 .  (Revised by Wolf Lammen, 12-Oct-2021.) $)
    axc16nf $p |- ( A. x x = y -> F/ z ph ) $=
      ( weq wal wex wn wi axc16g eximal sylibr syld nfd ) BCEBFZADOADGZAADFOAHZ
      QDFIPAIQBCDJAADKLABCDJMN $.
  $}

  ${
    $d x y $.
    $( Version of ~ axc11 with a disjoint variable condition on ` x ` and
       ` y ` , which is provable, on top of { ~ ax-1 -- ~ ax-7 }, from ~ ax12v
       (contrary to ~ axc11 which seems to require the full ~ ax-12 and
       ~ ax-13 ).  (Contributed by NM, 16-May-2008.)  (Revised by BJ,
       6-Jul-2021.)  (Proof shortened by Wolf Lammen, 11-Oct-2021.) $)
    axc11v $p |- ( A. x x = y -> ( A. x ph -> A. y ph ) ) $=
      ( weq wal axc16g spsd ) BCDBEAACEBABCCFG $.

    $( Version of ~ axc11r with a disjoint variable condition on ` x ` and
       ` y ` , which is provable, on top of { ~ ax-1 -- ~ ax-7 }, from ~ ax12v
       (contrary to ~ axc11 which seems to require the full ~ ax-12 and
       ~ ax-13 , and to ~ axc11r which seems to require the full ~ ax-12 ).
       (Contributed by BJ, 6-Jul-2021.)  (Proof shortened by Wolf Lammen,
       11-Oct-2021.) $)
    axc11rv $p |- ( A. x x = y -> ( A. y ph -> A. x ph ) ) $=
      ( weq wal axc16 spsd ) BCDBEAABECABCFG $.
  $}

  $( 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 $.

  ${
    $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, which does not require ~ ax-13 .
       See ~ equsalvw for a version with two disjoint variable conditions
       requiring fewer axioms.  See also the dual form ~ equsexv .
       (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 ax6ev a1bi 3bitr4i ) CDGZBHZCIQCJZBH
      QAHZCIBQBCEKTRCQABFLMSBCDNOP $.

    $( An equivalence related to implicit substitution.  Version of ~ equsex
       with a disjoint variable condition, which does not require ~ ax-13 .
       See ~ equsexvw for a version with two disjoint variable conditions
       requiring fewer axioms.  See also the dual form ~ equsalv .
       (Contributed by NM, 5-Aug-1993.)  (Revised by BJ, 31-May-2019.)  Avoid
       ~ ax-10 .  (Revised by GG, 18-Nov-2024.) $)
    equsexv $p |- ( E. x ( x = y /\ ph ) <-> ps ) $=
      ( weq wa wex biimpa exlimi wi wal equsalv equs4v sylbir impbii ) CDGZAHZC
      IZBSBCERABFJKBRALCMTABCDEFNACDOPQ $.
    $( $j usage 'equsexv' avoids 'ax-10'; $)

    $( Obsolete version of ~ equsexv as of 18-Nov-2024.  (Contributed by NM,
       5-Aug-1993.)  (Revised by BJ, 31-May-2019.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    equsexvOLD $p |- ( E. x ( x = y /\ ph ) <-> ps ) $=
      ( weq wa wex wi wal sbalex equsalv bitri ) CDGZAHCIOAJCKBACDLABCDEFMN $.
  $}

  $( 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 nf5r stdpc4 syl6 impbid ) ABDZABCEZA
    PABFOAABCGABHIOAABJPABKABCLMN $.

  ${
    sbf.1 $e |- F/ x ph $.
    $( Substitution for a variable not free in a wff does not affect it.  For a
       version requiring disjoint variables but fewer axioms, see ~ sbv .
       (Contributed by NM, 14-May-1993.)  (Revised by Mario Carneiro,
       4-Oct-2016.) $)
    sbf $p |- ( [ y / x ] ph <-> ph ) $=
      ( wnf wsb wb sbft ax-mp ) ABEABCFAGDABCHI $.
  $}

  $( 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 $.

  ${
    sbh.1 $e |- ( ph -> A. x ph ) $.
    $( Substitution for a variable not free in a wff does not affect it.
       (Contributed by NM, 14-May-1993.) $)
    sbh $p |- ( [ y / x ] ph <-> ph ) $=
      ( nf5i sbf ) ABCABDEF $.
  $}

  ${
    $d x y $.
    $( The setvar ` x ` is not free in ` [ y / x ] ph ` when ` x ` and ` y `
       are distinct.  (Contributed by NM, 26-May-1993.) $)
    hbs1 $p |- ( [ y / x ] ph -> A. x [ y / x ] ph ) $=
      ( wsb nfs1v nf5ri ) ABCDBABCEF $.
  $}

  ${
    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 sbf nfxfr ) ABCEABABCDFDG $.
  $}

  ${
    $d x y $.
    $( Alternate definition of substitution when variables are disjoint.
       Similar to Theorem 6.1 of [Quine] p. 40.  The implication "to the right"
       is ~ sb1v and even needs no disjoint variable condition, see ~ sb1 .
       Theorem ~ sb5f replaces the disjoint variable condition with a
       nonfreeness hypothesis.  (Contributed by NM, 18-Aug-1993.)  (Revised by
       Wolf Lammen, 4-Sep-2023.) $)
    sb5 $p |- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) ) $=
      ( wsb weq wi wal wa wex sb6 sbalex bitr4i ) ABCDBCEZAFBGMAHBIABCJABCKL $.

    $( A property related to substitution that replaces the distinctor from
       ~ equs5 to a disjoint variable condition.  Version of ~ equs5a with a
       disjoint variable condition, which does not require ~ ax-13 .  See also
       ~ sbalex .  (Contributed by NM, 2-Feb-2007.)  (Revised by GG,
       15-Dec-2023.) $)
    equs5av $p |- ( E. x ( x = y /\ A. y ph ) -> A. x ( x = y -> ph ) ) $=
      ( weq wal wa wi nfa1 ax12v2 spsd imp exlimi ) BCDZACEZFMAGZBEZBOBHMNPMAPC
      ABCIJKL $.
  $}

  ${
    $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 $.
  $}

  ${
    $d y t $.  $d y x $.  $d y ph $.
    $( An alternate definition of proper substitution ~ df-sb .  By introducing
       a dummy variable ` y ` in the definiens, we are able to eliminate any
       distinct variable restrictions among the variables ` t ` , ` x ` , and
       ` ph ` of the definiendum.  No distinct variable conflicts arise because
       ` y ` effectively insulates ` t ` from ` x ` .  To achieve this, we use
       a chain of two substitutions in the form of ~ sb5 , first ` y ` for
       ` x ` then ` t ` for ` y ` .  Compare Definition 2.1'' of [Quine] p. 17,
       which is obtained from this theorem by applying ~ df-clab .  Theorem
       ~ sb7h provides a version where ` ph ` and ` y ` don't have to be
       distinct.  (Contributed by NM, 28-Jan-2004.)  Revise ~ df-sb .  (Revised
       by BJ, 25-Dec-2020.)  (Proof shortened by Wolf Lammen, 3-Sep-2023.) $)
    dfsb7 $p |- ( [ t / x ] ph <-> E. y ( y = t /\ E. x ( x = y /\ ph ) ) ) $=
      ( weq wi wal wa wex wsb sbalex anbi2i exbii df-sb 3bitr4ri ) CDEZBCEZAFBG
      ZHZCIPRFCGPQAHBIZHZCIABDJRCDKUASCTRPABCKLMABCDNO $.
  $}

  ${
    $d y t $.  $d y x $.  $d y ph $.
    $( Negation inside and outside of substitution are equivalent.
       (Contributed by NM, 14-May-1993.)  (Proof shortened by Wolf Lammen,
       30-Apr-2018.)  Revise ~ df-sb .  (Revised by BJ, 25-Dec-2020.) $)
    sbn $p |- ( [ t / x ] -. ph <-> -. [ t / x ] ph ) $=
      ( vy wn wsb weq wi wal wa wex df-sb alinexa imbi2i albii xchbinxr 3bitri
      dfsb7 ) AEZBCFDCGZBDGZSHBIZHZDITUAAJBKZEZHZDIZABCFZESBDCLUCUFDUBUETUAABMN
      OUGTUDJDKUHTUDDMABDCRPQ $.
  $}

  ${
    $d x y $.  $d x z $.
    $( Move existential quantifier in and out of substitution.  (Contributed by
       NM, 27-Sep-2003.) $)
    sbex $p |- ( [ z / y ] E. x ph <-> E. x [ z / y ] ph ) $=
      ( wn wal wsb wex sbn sbalv xchbinx df-ex sbbii 3bitr4i ) AEZBFZEZCDGZACDG
      ZEZBFZEABHZCDGSBHRPCDGUAPCDIOTCDBACDIJKUBQCDABLMSBLN $.
  $}

  $( Alternate definition of ~ df-nf .  (Contributed by Mario Carneiro,
     11-Aug-2016.) ~ df-nf changed.  (Revised by Wolf Lammen, 11-Sep-2021.) $)
  nf5 $p |- ( F/ x ph <-> A. x ( ph -> A. x ph ) ) $=
    ( wnf wex wal wi df-nf nfa1 19.23 bitr4i ) ABCABDABEZFAKFBEABGAKBABHIJ $.

  $( An alternate definition of ~ df-nf .  (Contributed by Mario Carneiro,
     24-Sep-2016.) $)
  nf6 $p |- ( F/ x ph <-> A. x ( E. x ph -> ph ) ) $=
    ( wnf wex wal wi df-nf nfe1 19.21 bitr4i ) ABCABDZABEFKAFBEABGKABABHIJ $.

  ${
    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 $.
  $}

  ${
    nf5di.1 $e |- ( ph -> F/ x ph ) $.
    $( Since the converse holds by ~ a1i , this inference shows that we can
       represent a not-free hypothesis with either ` F/ x ph ` (inference form)
       or ` ( ph -> F/ x ph ) ` (deduction form).  (Contributed by NM,
       17-Aug-2018.)  (Proof shortened by Wolf Lammen, 10-Jul-2019.) $)
    nf5di $p |- F/ x ph $=
      ( wal nf5rd pm2.43i nf5i ) ABAABDAABCEFG $.
  $}

  ${
    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.)
       (Proof shortened by Wolf Lammen, 5-Jan-2018.)  (Proof shortened by Wolf
       Lammen, 14-Jul-2020.) $)
    19.9h $p |- ( E. x ph <-> ph ) $=
      ( nf5i 19.9 ) ABABCDE $.
  $}

  ${
    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 ` ".  See also ~ 19.21 and ~ 19.21v .
       (Contributed by NM, 1-Aug-2017.)  (Proof shortened by Wolf Lammen,
       1-Jan-2018.) $)
    19.21h $p |- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) $=
      ( nf5i 19.21 ) ABCACDEF $.
  $}

  ${
    19.23h.1 $e |- ( ps -> A. x ps ) $.
    $( Theorem 19.23 of [Margaris] p. 90.  See ~ 19.23 .  (Contributed by NM,
       24-Jan-1993.)  (Revised by Mario Carneiro, 24-Sep-2016.)  (Proof
       shortened by Wolf Lammen, 1-Jan-2018.) $)
    19.23h $p |- ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) $=
      ( nf5i 19.23 ) ABCBCDEF $.
  $}

  ${
    exlimih.1 $e |- ( ps -> A. x ps ) $.
    exlimih.2 $e |- ( ph -> ps ) $.
    $( Inference associated with ~ 19.23 .  See ~ exlimiv for a version with a
       disjoint variable condition requiring fewer axioms.  (Contributed by NM,
       10-Jan-1993.)  (Proof shortened by Andrew Salmon, 13-May-2011.)  (Proof
       shortened by Wolf Lammen, 1-Jan-2018.) $)
    exlimih $p |- ( E. x ph -> ps ) $=
      ( nf5i exlimi ) ABCBCDFEG $.
  $}

  ${
    exlimdh.1 $e |- ( ph -> A. x ph ) $.
    exlimdh.2 $e |- ( ch -> A. x ch ) $.
    exlimdh.3 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction form of Theorem 19.9 of [Margaris] p. 89.  (Contributed by NM,
       28-Jan-1997.) $)
    exlimdh $p |- ( ph -> ( E. x ps -> ch ) ) $=
      ( nf5i exlimd ) ABCDADEHCDFHGI $.
  $}

  ${
    $d x y $.
    equsalhw.1 $e |- ( ps -> A. x ps ) $.
    equsalhw.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Version of ~ equsalh with a disjoint variable condition, which does not
       require ~ ax-13 .  (Contributed by NM, 29-Nov-2015.)  (Proof shortened
       by Wolf Lammen, 8-Jul-2022.) $)
    equsalhw $p |- ( A. x ( x = y -> ph ) <-> ps ) $=
      ( nf5i equsalv ) ABCDBCEGFH $.

    $( An equivalence related to implicit substitution.  Version of ~ equsexh
       with a disjoint variable condition, which does not require ~ ax-13 .
       (Contributed by NM, 5-Aug-1993.)  (Revised by BJ, 31-May-2019.) $)
    equsexhv $p |- ( E. x ( x = y /\ ph ) <-> ps ) $=
      ( nf5i equsexv ) ABCDBCEGFH $.
  $}

  $( The setvar ` x ` is not free in ` A. x ph ` .  This corresponds to the
     axiom (4) of modal logic.  Example in Appendix in [Megill] p. 450 (p. 19
     of the preprint).  Also Lemma 22 of [Monk2] p. 114.  (Contributed by NM,
     24-Jan-1993.)  (Proof shortened by Wolf Lammen, 12-Oct-2021.) $)
  hba1 $p |- ( A. x ph -> A. x A. x ph ) $=
    ( wal nfa1 nf5ri ) ABCBABDE $.

  $( Closed theorem version of bound-variable hypothesis builder ~ hbn .
     (Contributed by NM, 10-May-1993.)  (Proof shortened by Wolf Lammen,
     14-Oct-2021.) $)
  hbnt $p |- ( A. x ( ph -> A. x ph ) -> ( -. ph -> A. x -. ph ) ) $=
    ( wal wi wn nf5-1 nfnd nf5rd ) AABCDBCZAEBIABABFGH $.

  ${
    hbn.1 $e |- ( ph -> A. x ph ) $.
    $( If ` x ` is not free in ` ph ` , it is not free in ` -. ph ` .
       (Contributed by NM, 10-Jan-1993.)  (Proof shortened by Wolf Lammen,
       17-Dec-2017.) $)
    hbn $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 $.
  $}

  ${
    hbim1.1 $e |- ( ph -> A. x ph ) $.
    hbim1.2 $e |- ( ph -> ( ps -> A. x ps ) ) $.
    $( A closed form of ~ hbim .  (Contributed by NM, 2-Jun-1993.) $)
    hbim1 $p |- ( ( ph -> ps ) -> A. x ( ph -> ps ) ) $=
      ( wi wal a2i 19.21h sylibr ) ABFZABCGZFKCGABLEHABCDIJ $.
  $}

  ${
    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, 14-May-1993.)  (Proof shortened by Wolf Lammen,
       3-Jan-2018.) $)
    hbimd $p |- ( ph -> ( ( ps -> ch ) -> A. x ( ps -> ch ) ) ) $=
      ( wi nf5dh nfimd nf5rd ) ABCHDABCDABDEFIACDEGIJK $.
  $}

  ${
    hbim.1 $e |- ( ph -> A. x ph ) $.
    hbim.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, 24-Jan-1993.)  (Proof shortened
       by Mel L. O'Cat, 3-Mar-2008.)  (Proof shortened by Wolf Lammen,
       1-Jan-2018.) $)
    hbim $p |- ( ( ph -> ps ) -> A. x ( ph -> ps ) ) $=
      ( wal wi a1i hbim1 ) ABCDBBCFGAEHI $.
  $}

  ${
    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, 14-May-1993.)  (Proof shortened
       by Wolf Lammen, 2-Jan-2018.) $)
    hban $p |- ( ( ph /\ ps ) -> A. x ( ph /\ ps ) ) $=
      ( wa nf5i nfan nf5ri ) ABFCABCACDGBCEGHI $.
    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.)  (Proof
       shortened by Wolf Lammen, 2-Jan-2018.) $)
    hb3an $p |- ( ( ph /\ ps /\ ch ) -> A. x ( ph /\ ps /\ ch ) ) $=
      ( w3a nf5i nf3an nf5ri ) ABCHDABCDADEIBDFICDGIJK $.
  $}

  $( Introduction of implication into substitution.  (Contributed by NM,
     14-May-1993.) $)
  sbi2 $p |- ( ( [ y / x ] ph -> [ y / x ] ps ) -> [ y / x ] ( ph -> ps ) ) $=
    ( wsb wi wn sbn pm2.21 sbimi sylbir ax-1 ja ) ACDEZBCDEABFZCDEZNGAGZCDEPACD
    HQOCDABIJKBOCDBALJM $.

  $( Implication inside and outside of a substitution are equivalent.
     (Contributed by NM, 14-May-1993.) $)
  sbim $p |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) ) $=
    ( wi wsb sbi1 sbi2 impbii ) ABECDFACDFBCDFEABCDGABCDHI $.

  ${
    $d t x $.  $d t y $.  $d ps t $.  $d ph t $.
    sbrim.1 $e |- F/ x ph $.
    $( Substitution in an implication with a variable not free in the
       antecedent affects only the consequent.  (Contributed by NM,
       2-Jun-1993.)  (Revised by Mario Carneiro, 4-Oct-2016.)  Avoid ~ ax-10 .
       (Revised by GG, 20-Nov-2024.) $)
    sbrim $p |- ( [ y / x ] ( ph -> ps ) <-> ( ph -> [ y / x ] ps ) ) $=
      ( vt weq wi wal wsb bi2.04 albii 19.21 bitri imbi2i 19.21v bitr4i 3bitr4i
      df-sb ) FDGZCFGZABHZHZCIZHZFIATUABHZCIZHZHZFIZUBCDJABCDJZHZUEUIFUETAUGHZH
      UIUDUMTUDAUFHZCIUMUCUNCUAABKLAUFCEMNOTAUGKNLUBCFDSULAUHFIZHUJUKUOABCFDSOA
      UHFPQR $.
    $( $j usage 'sbrim' avoids 'ax-10'; $)

    $( Obsolete version of ~ sbrim as of 20-Nov-2024.  (Contributed by NM,
       2-Jun-1993.)  (Revised by Mario Carneiro, 4-Oct-2016.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    sbrimOLD $p |- ( [ y / x ] ( ph -> ps ) <-> ( ph -> [ y / x ] ps ) ) $=
      ( wi wsb sbim sbf imbi1i bitri ) ABFCDGACDGZBCDGZFAMFABCDHLAMACDEIJK $.
  $}

  ${
    sblim.1 $e |- F/ x ps $.
    $( Substitution in an implication with a variable not free in the
       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 $.
  $}

  $( Disjunction inside and outside of a substitution are equivalent.
     (Contributed by NM, 29-Sep-2002.) $)
  sbor $p |- ( [ y / x ] ( ph \/ ps ) <-> ( [ y / x ] ph \/ [ y / x ] ps ) ) $=
    ( wn wi wsb wo sbim sbn imbi1i bitri df-or sbbii 3bitr4i ) AEZBFZCDGZACDGZE
    ZBCDGZFZABHZCDGSUAHRPCDGZUAFUBPBCDIUDTUAACDJKLUCQCDABMNSUAMO $.

  $( Equivalence inside and outside of a substitution are equivalent.
     (Contributed by NM, 14-May-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 |- F/ x ch $.
    sbrbif.2 $e |- ( [ y / x ] ph <-> ps ) $.
    $( Introduce right biconditional inside of a substitution.  (Contributed by
       NM, 18-Aug-1993.)  (Revised by Mario Carneiro, 4-Oct-2016.) $)
    sbrbif $p |- ( [ y / x ] ( ph <-> ch ) <-> ( ps <-> ch ) ) $=
      ( wb wsb sbrbis sbf bibi2i bitri ) ACHDEIBCDEIZHBCHABCDEGJNCBCDEFKLM $.
  $}

  ${
    $d x y $.  $d x z $.
    $( Move nonfree predicate in and out of substitution; see ~ sbal and
       ~ sbex .  (Contributed by BJ, 2-May-2019.)  (Proof shortened by Wolf
       Lammen, 2-May-2025.) $)
    sbnf $p |- ( [ z / y ] F/ x ph <-> F/ x [ z / y ] ph ) $=
      ( wnf wsb wex wal wi df-nf sbbii sbim sbex sbal imbi12i bitr4i 3bitri ) A
      BEZCDFABGZABHZIZCDFSCDFZTCDFZIZACDFZBEZRUACDABJKSTCDLUDUEBGZUEBHZIUFUBUGU
      CUHABCDMABCDNOUEBJPQ $.

    $( Obsolete version of ~ sbnf as of 2-May-2025.  (Contributed by BJ,
       2-May-2019.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    sbnfOLD $p |- ( [ z / y ] F/ x ph <-> F/ x [ z / y ] ph ) $=
      ( wal wi wsb wnf sbim sbal imbi2i bitri albii nf5 sbbii 3bitr4i ) AABEZFZ
      CDGZBEZACDGZUABEZFZBEABHZCDGZUABHSUCBSUAQCDGZFUCAQCDIUFUBUAABCDJKLMUERBEZ
      CDGTUDUGCDABNORBCDJLUABNP $.
  $}

  ${
    $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, not requiring ~ ax-13 .
       See ~ sbievw for a version with a disjoint variable condition requiring
       fewer axioms.  (Contributed by NM, 30-Jun-1994.)  (Revised by Wolf
       Lammen, 18-Jan-2023.)  Remove dependence on ~ ax-10 and shorten proof.
       (Revised by BJ, 18-Jul-2023.)  (Proof shortened by SN, 24-Jul-2025.) $)
    sbiev $p |- ( [ y / x ] ph <-> ps ) $=
      ( wsb sbbiiev sbf bitri ) ACDGBCDGBABCDFHBCDEIJ $.
    $( $j usage 'sbiev' avoids 'ax-10' 'ax-13'; $)

    $( Obsolete version of ~ sbiev as of 24-Aug-2025.  (Contributed by NM,
       30-Jun-1994.)  (Revised by Wolf Lammen, 18-Jan-2023.)  Remove dependence
       on ~ ax-10 and shorten proof.  (Revised by BJ, 18-Jul-2023.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    sbievOLD $p |- ( [ y / x ] ph <-> ps ) $=
      ( wsb weq wi wal sb6 equsalv bitri ) ACDGCDHAICJBACDKABCDEFLM $.
  $}

  ${
    $d x y $.
    sbiedw.1 $e |- F/ x ph $.
    sbiedw.2 $e |- ( ph -> F/ x ch ) $.
    sbiedw.3 $e |- ( ph -> ( x = y -> ( ps <-> ch ) ) ) $.
    $( Conversion of implicit substitution to explicit substitution (deduction
       version of ~ sbiev ).  Version of ~ sbied with a disjoint variable
       condition, requiring fewer axioms.  (Contributed by NM, 30-Jun-1994.)
       Avoid ~ ax-13 .  (Revised by GG, 10-Jan-2024.) $)
    sbiedw $p |- ( ph -> ( [ y / x ] ps <-> ch ) ) $=
      ( wsb wi sbrim nfim1 weq wb com12 pm5.74d sbiev bitr3i pm5.74ri ) ABDEIZC
      ATJABJZDEIACJZABDEFKUAUBDEACDFGLDEMZABCAUCBCNHOPQRS $.
    $( $j usage 'sbiedw' avoids 'ax-10' 'ax-11' 'ax-13'; $)
  $}

  $( Show that the original axiom ~ ax-c7 can be derived from ~ ax-10
     ( ~ hbn1 ), ~ sp and propositional calculus.  See ~ ax10fromc7 for the
     rederivation of ~ ax-10 from ~ ax-c7 .

     Normally, ~ axc7 should be used rather than ~ ax-c7 , except by theorems
     specifically studying the latter's properties.  (Contributed by NM,
     21-May-2008.) $)
  axc7 $p |- ( -. A. x -. A. x ph -> ph ) $=
    ( wal wn sp hbn1 nsyl4 ) ABCZAHDBCABEABFG $.

  $( Abbreviated version of ~ axc7 using the existential quantifier.
     Corresponds to the dual of Axiom (B) of modal logic.  (Contributed by NM,
     5-Aug-1993.)  (Proof shortened by Wolf Lammen, 8-Jul-2022.) $)
  axc7e $p |- ( E. x A. x ph -> ph ) $=
    ( wal wex hbe1a 19.21bi ) ABCBDABABEF $.

  $( The analogue in our predicate calculus of the Brouwer axiom (B) of modal
     logic S5.  (Contributed by NM, 5-Oct-2005.) $)
  modal-b $p |- ( ph -> A. x -. A. x -. ph ) $=
    ( wn wal axc7 con4i ) ACZBDCBDAGBEF $.

  $( A closed version of ~ 19.9h .  (Contributed by NM, 13-May-1993.)  (Proof
     shortened by Wolf Lammen, 3-Mar-2018.) $)
  19.9ht $p |- ( A. x ( ph -> A. x ph ) -> ( E. x ph -> ph ) ) $=
    ( wal wi nf5-1 19.9d ) AAABCDBCBABEF $.

  $( Show that the original axiom ~ ax-c4 can be derived from ~ ax-4
     ( ~ alim ), ~ ax-10 ( ~ hbn1 ), ~ sp and propositional calculus.  See
     ~ ax4fromc4 for the rederivation of ~ ax-4 from ~ ax-c4 .

     Part of the proof is based on the proof of Lemma 22 of [Monk2] p. 114.
     (Contributed by NM, 21-May-2008.)  (Proof modification is discouraged.) $)
  axc4 $p |- ( A. x ( A. x ph -> ps ) -> ( A. x ph -> A. x ps ) ) $=
    ( wal wi wn sp con2i hbn1 con1i alimi 3syl alim syl5 ) ACDZOCDZOBECDBCDOOFZ
    CDZFZSCDPROQCGHQCISOCORACIJKLOBCMN $.

  ${
    axc4i.1 $e |- ( A. x ph -> ps ) $.
    $( Inference version of ~ axc4 .  (Contributed by NM, 3-Jan-1993.) $)
    axc4i $p |- ( A. x ph -> A. x ps ) $=
      ( wal nfa1 alrimi ) ACEBCACFDG $.
  $}

  ${
    nfal.1 $e |- F/ x ph $.
    $( If ` x ` is not free in ` ph ` , then it is not free in ` A. y ph ` .
       (Contributed by Mario Carneiro, 11-Aug-2016.) $)
    nfal $p |- F/ x A. y ph $=
      ( wal nf5ri hbal nf5i ) ACEBABCABDFGH $.
  $}

  ${
    nfex.1 $e |- F/ x ph $.
    $( If ` x ` is not free in ` ph ` , then it is not free in ` E. y ph ` .
       (Contributed by Mario Carneiro, 11-Aug-2016.)  (Proof shortened by Wolf
       Lammen, 30-Dec-2017.)  Reduce symbol count in ~ nfex , ~ hbex .
       (Revised by Wolf Lammen, 16-Oct-2021.) $)
    nfex $p |- F/ x E. y ph $=
      ( wex wn wal df-ex nfn nfal nfxfr ) ACEAFZCGZFBACHMBLBCABDIJIK $.
  $}

  ${
    hbex.1 $e |- ( ph -> A. x ph ) $.
    $( If ` x ` is not free in ` ph ` , then it is not free in ` E. y ph ` .
       (Contributed by NM, 12-Mar-1993.)  Reduce symbol count in ~ nfex ,
       ~ hbex .  (Revised by Wolf Lammen, 16-Oct-2021.) $)
    hbex $p |- ( E. y ph -> A. x E. y ph ) $=
      ( wex nf5i nfex nf5ri ) ACEBABCABDFGH $.
  $}

  ${
    nfnf.1 $e |- F/ x ph $.
    $( If ` x ` is not free in ` ph ` , then 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 wex wal wi df-nf nfex nfal nfim nfxfr ) ACEACFZACGZHBACINOBABCDJABC
      DKLM $.
  $}

  $( Theorem 19.12 of [Margaris] p. 89.  Assuming the converse is a mistake
     sometimes made by beginners!  But sometimes the converse does hold, as in
     ~ 19.12vv and ~ r19.12sn .  (Contributed by NM, 12-Mar-1993.)  (Proof
     shortened by Wolf Lammen, 3-Jan-2018.) $)
  19.12 $p |- ( E. x A. y ph -> A. y E. x ph ) $=
    ( wal wex nfa1 nfex sp eximi alrimi ) ACDZBEABECKCBACFGKABACHIJ $.

  ${
    nfald.1 $e |- F/ y ph $.
    nfald.2 $e |- ( ph -> F/ x ps ) $.
    $( Deduction form of ~ nfal .  (Contributed by Mario Carneiro,
       24-Sep-2016.)  (Proof shortened by Wolf Lammen, 16-Oct-2021.) $)
    nfald $p |- ( ph -> F/ x A. y ps ) $=
      ( wal wex 19.12 nfrd alimd ax-11 syl56 nfd ) ABDGZCOCHBCHZDGABCGZDGOCGBCD
      IAPQDEABCFJKBDCLMN $.

    $( If ` x ` is not free in ` ps ` , then it is not free in ` E. y ps ` .
       (Contributed by Mario Carneiro, 24-Sep-2016.) $)
    nfexd $p |- ( ph -> F/ x E. y ps ) $=
      ( wex wn wal df-ex nfnd nfald nfxfrd ) BDGBHZDIZHACBDJAOCANCDEABCFKLKM $.
  $}

  ${
    $d w x z $.  $d w y z $.  $d w ph $.
    nfsbv.nf $e |- F/ z ph $.
    $( If ` z ` is not free in ` ph ` , then it is not free in ` [ y / x ] ph `
       when ` z ` is disjoint from both ` x ` and ` y ` .  Version of ~ nfsb
       with an additional disjoint variable condition on ` x , z ` but not
       requiring ~ ax-13 .  (Contributed by Mario Carneiro, 11-Aug-2016.)
       (Revised by Wolf Lammen, 7-Feb-2023.)  Remove disjoint variable
       condition on ` x , y ` .  (Revised by Steven Nguyen, 13-Aug-2023.)
       (Proof shortened by Wolf Lammen, 25-Oct-2024.) $)
    nfsbv $p |- F/ z [ y / x ] ph $=
      ( wsb nf5ri hbsbw nf5i ) ABCFDABCDADEGHI $.
  $}

  ${
    $d x z $.  $d y z $.
    sbco2v.1 $e |- F/ z ph $.
    $( A composition law for substitution.  Version of ~ sbco2 with disjoint
       variable conditions but not requiring ~ ax-13 .  (Contributed by NM,
       30-Jun-1994.)  (Revised by Wolf Lammen, 29-Apr-2023.) $)
    sbco2v $p |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph ) $=
      ( wsb nfsbv sbequ sbiev ) ABDFABCFDCABCDEGADCBHI $.
  $}

  ${
    aaan.1 $e |- F/ y ph $.
    aaan.2 $e |- F/ x ps $.
    $( Distribute universal quantifiers.  (Contributed by NM, 12-Aug-1993.)
       Avoid ~ ax-10 .  (Revised by GG, 21-Nov-2024.) $)
    aaan $p |- ( A. x A. y ( ph /\ ps ) <-> ( A. x ph /\ A. y ps ) ) $=
      ( wa wal 19.26-2 19.3 albii alcom bitri anbi12i ) ABGDHCHADHZCHZBDHZCHZGA
      CHZQGABCDIPSRQOACADEJKRBCHZDHQBCDLTBDBCFJKMNM $.
    $( $j usage 'aaan' avoids 'ax-10'; $)

    $( Obsolete version of ~ aaan as of 21-Nov-2024.  (Contributed by NM,
       12-Aug-1993.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    aaanOLD $p |- ( A. x A. y ( ph /\ ps ) <-> ( A. x ph /\ A. y ps ) ) $=
      ( wa wal 19.28 albii nfal 19.27 bitri ) ABGDHZCHABDHZGZCHACHOGNPCABDEIJAO
      CBCDFKLM $.
  $}

  ${
    eeor.1 $e |- F/ y ph $.
    eeor.2 $e |- F/ x ps $.
    $( Distribute existential quantifiers.  (Contributed by NM, 8-Aug-1994.)
       Avoid ~ ax-10 .  (Revised by GG, 21-Nov-2024.) $)
    eeor $p |- ( E. x E. y ( ph \/ ps ) <-> ( E. x ph \/ E. y ps ) ) $=
      ( wo wex 19.43 exbii 19.9 excom bitri orbi12i ) ABGDHZCHADHZBDHZGZCHZACHZ
      QGZORCABDIJSPCHZQCHZGUAPQCIUBTUCQPACADEKJUCBCHZDHQBCDLUDBDBCFKJMNMM $.
    $( $j usage 'eeor' avoids 'ax-10'; $)

    $( Obsolete version of ~ eeor as of 21-Nov-2024.  (Contributed by NM,
       8-Aug-1994.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    eeorOLD $p |- ( E. x E. y ( ph \/ ps ) <-> ( E. x ph \/ E. y ps ) ) $=
      ( wo wex 19.45 exbii nfex 19.44 bitri ) ABGDHZCHABDHZGZCHACHOGNPCABDEIJAO
      CBCDFKLM $.
  $}

  ${
    $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, which does not
       require ~ ax-13 .  (Contributed by NM, 5-Aug-1993.)  (Revised by BJ,
       31-May-2019.) $)
    cbv3v $p |- ( A. x ph -> A. y ps ) $=
      ( wal nf5ri hbal spimfv alrimih ) ACHBDADCADEIJABCDFGKL $.
  $}

  ${
    $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.
       Version of ~ cbv1 with a disjoint variable condition, which does not
       require ~ ax-13 .  (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 $.
  $}

  ${
    $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, which does not
       require ~ ax-13 .  (Contributed by NM, 5-Aug-1993.)  Avoid ~ ax-13 .
       (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 ph $.  $d x ch $.  $d x y $.
    cbvaldw.1 $e |- F/ y ph $.
    cbvaldw.2 $e |- ( ph -> F/ y ps ) $.
    cbvaldw.3 $e |- ( ph -> ( x = y -> ( ps <-> ch ) ) ) $.
    $( Deduction used to change bound variables, using implicit substitution.
       Version of ~ cbvald with a disjoint variable condition, which does not
       require ~ ax-13 .  (Contributed by NM, 2-Jan-2002.)  Avoid ~ ax-13 .
       (Revised by GG, 10-Jan-2024.) $)
    cbvaldw $p |- ( ph -> ( A. x ps <-> A. y ch ) ) $=
      ( nfv nfvd cbv2w ) ABCDEADIFGACDJHK $.

    $( Deduction used to change bound variables, using implicit substitution.
       Version of ~ cbvexd with a disjoint variable condition, which does not
       require ~ ax-13 .  (Contributed by NM, 2-Jan-2002.)  Avoid ~ ax-13 .
       (Revised by GG, 10-Jan-2024.) $)
    cbvexdw $p |- ( ph -> ( E. x ps <-> E. y ch ) ) $=
      ( wex wn wal nfnd weq wb notbi imbitrdi cbvaldw alnex 3bitr3g con4bid ) A
      BDIZCEIZABJZDKCJZEKUAJUBJAUCUDDEFABEGLADEMBCNUCUDNHBCOPQBDRCERST $.
  $}

  ${
    $d x y $.
    cbv3hv.nf1 $e |- ( ph -> A. y ph ) $.
    cbv3hv.nf2 $e |- ( ps -> A. x ps ) $.
    cbv3hv.1 $e |- ( x = y -> ( ph -> ps ) ) $.
    $( Rule used to change bound variables, using implicit substitution.
       Version of ~ cbv3h with a disjoint variable condition on ` x , y ` ,
       which does not require ~ ax-13 .  Was used in a proof of ~ axc11n (but
       of independent interest).  (Contributed by NM, 25-Jul-2015.)  (Proof
       shortened by Wolf Lammen, 29-Nov-2020.)  (Proof shortened by BJ,
       30-Nov-2020.) $)
    cbv3hv $p |- ( A. x ph -> A. y ps ) $=
      ( nf5i cbv3v ) ABCDADEHBCFHGI $.
  $}

  ${
    $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, which does not
       require ~ ax-13 .  See ~ cbvalvw for a version with two more disjoint
       variable conditions, requiring fewer axioms, 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 $.

    $( Rule used to change bound variables, using implicit substitution.
       Version of ~ cbvex with a disjoint variable condition, which does not
       require ~ ax-13 .  See ~ cbvexvw for a version with two disjoint
       variable conditions, requiring fewer axioms, 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 wn wal nfn weq notbid cbvalv1 alnex 3bitr3i con4bii ) ACHZBDHZAIZCJ
      BIZDJRISITUACDADEKBCFKCDLABGMNACOBDOPQ $.
  $}

  ${
    $d x y z w $.
    cbval2v.1 $e |- F/ z ph $.
    cbval2v.2 $e |- F/ w ph $.
    cbval2v.3 $e |- F/ x ps $.
    cbval2v.4 $e |- F/ y ps $.
    cbval2v.5 $e |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) ) $.
    $( Rule used to change bound variables, using implicit substitution.
       Version of ~ cbval2 with a disjoint variable condition, which does not
       require ~ ax-13 .  (Contributed by NM, 22-Dec-2003.)  (Revised by BJ,
       16-Jun-2019.)  (Proof shortened by GG, 10-Jan-2024.) $)
    cbval2v $p |- ( A. x A. y ph <-> A. z A. w ps ) $=
      ( wal nfal weq nfv wnf a1i wb ex cbv2w cbvalv1 ) ADLBFLCEAEDGMBCFIMCENZAB
      DFUBDOUBFOAFPUBHQBDPUBJQUBDFNABRKSTUA $.

    $( Rule used to change bound variables, using implicit substitution.
       Version of ~ cbvex2 with a disjoint variable condition, which does not
       require ~ ax-13 .  (Contributed by NM, 14-Sep-2003.)  (Revised by BJ,
       16-Jun-2019.) $)
    cbvex2v $p |- ( E. x E. y ph <-> E. z E. w ps ) $=
      ( wex wn wal nfn weq wa notbid cbval2v 2nexaln 3bitr4i con4bii ) ADLCLZBF
      LELZAMZDNCNBMZFNENUCMUDMUEUFCDEFAEGOAFHOBCIOBDJOCEPDFPQABKRSACDTBEFTUAUB
      $.
  $}

  ${
    $d x z $.  $d y z $.
    dvelimhw.1 $e |- ( ph -> A. x ph ) $.
    dvelimhw.2 $e |- ( ps -> A. z ps ) $.
    dvelimhw.3 $e |- ( z = y -> ( ph <-> ps ) ) $.
    dvelimhw.4 $e |- ( -. A. x x = y -> ( y = z -> A. x y = z ) ) $.
    $( Proof of ~ dvelimh without using ~ ax-13 but with additional distinct
       variable conditions.  (Contributed by NM, 1-Oct-2002.)  (Revised by
       Andrew Salmon, 21-Jul-2011.)  (Revised by NM, 1-Aug-2017.)  (Proof
       shortened by Wolf Lammen, 23-Dec-2018.) $)
    dvelimhw $p |- ( -. A. x x = y -> ( ps -> A. x ps ) ) $=
      ( weq wal wn wi wnf nfv equcom nfna1 nf5d nfxfrd nf5i a1i nfimd equsalhw
      nfald nfbii sylib nf5rd ) CDJZCKLZBCUIEDJZAMZEKZCNBCNUIUKCEUIEOUIUJACUJDE
      JZUICEDPUIUMCUHCQIRSACNUIACFTUAUBUDULBCABEDGHUCUEUFUG $.
  $}

  ${
    $d ph y $.  $d ps x $.
    $( Theorem *11.53 in [WhiteheadRussell] p. 164.  See ~ pm11.53v for a
       version requiring fewer axioms.  (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 nfv nfal 19.23 bitri ) ABEDFZCFABDFZEZCFACGOENP
      CABDHIAOCBCDBCJKLM $.
  $}

  ${
    $d x ps $.  $d y ph $.
    $( Special case of ~ 19.12 where its converse holds.  See ~ 19.12vvv for a
       version with a disjoint variable condition requiring fewer axioms.
       (Contributed by NM, 18-Jul-2001.)  (Revised by Andrew Salmon,
       11-Jul-2011.) $)
    19.12vv $p |- ( E. x A. y ( ph -> ps ) <-> A. y E. x ( ph -> ps ) ) $=
      ( wi wal wex 19.21v exbii nfv nfal 19.36 19.36v albii 19.21 bitr2i 3bitri
      ) ABEZDFZCGABDFZEZCGACFZTEZRCGZDFZSUACABDHIATCBCDBCJKLUEUBBEZDFUCUDUFDABC
      MNUBBDADCADJKOPQ $.
  $}

  ${
    eean.1 $e |- F/ y ph $.
    eean.2 $e |- F/ x ps $.
    $( Distribute 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 $.
    $( Distribute a pair of existential quantifiers over a conjunction.
       Combination of ~ 19.41v and ~ 19.42v .  For a version requiring fewer
       axioms but with additional disjoint variable conditions, see
       ~ exdistrv .  (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 ps $.  $d z ps $.  $d x ch $.  $d y ch $.
    $( Distribute three existential quantifiers over a conjunction.
       (Contributed by NM, 26-Jul-1995.)  (Proof shortened by Andrew Salmon,
       25-May-2011.)  Reduce distinct variable restrictions.  (Revised by Wolf
       Lammen, 20-Jan-2018.) $)
    eeeanv $p |- ( E. x E. y E. z ( ph /\ ps /\ ch ) <->
                                         ( E. x ph /\ E. y ps /\ E. z ch ) ) $=
      ( wa wex w3a eeanv anbi1i df-3an exbii 19.42v bitri 2exbii nfv nfex 19.41
      3bitri 3bitr4i ) ABGZEHZDHZCFHZGZADHZBEHZGZUEGABCIZFHZEHDHZUGUHUEIUDUIUEA
      BDEJKULUBUEGZEHZDHUCUEGZDHUFUKUMDEUKUBCGZFHUMUJUPFABCLMUBCFNOPUNUODUBUEEC
      EFCEQRSMUCUEDCDFCDQRSTUGUHUELUA $.
  $}

  ${
    $d z ph $.  $d w ph $.  $d x ps $.  $d y ps $.
    $( Distribute two pairs of existential quantifiers over a conjunction.  For
       a version requiring fewer axioms but with additional disjoint variable
       conditions, see ~ 4exdistrv .  (Contributed by NM, 31-Jul-1995.)  Remove
       disjoint variable conditions on ` y , z ` and ` x , w ` .  (Revised by
       Eric Schmidt, 26-Oct-2025.) $)
    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 nfv nfex eean 3bitri ) ABGFHZEHDHZCHQDH
      ZEHZCHADHZBFHZGZEHCHUACHUBEHGRTCQDEIJSUCCEABDFKLUAUBCEAEDAEMNBCFBCMNOP $.

    ${
      $d y z $.  $d w x $.
      $( Obsolete version of ~ ee4anv as of 26-Oct-2025.  (Contributed by NM,
         31-Jul-1995.)  (Proof modification is discouraged.)
         (New usage is discouraged.) $)
      ee4anvOLD $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 ) ABGFHZEHDHZCHNDHZEHZCHADHZBF
        HZGZEHCHRCHSEHGOQCNDEIJPTCEABDFKLRSCEKM $.
    $}
  $}

  ${
    $d ph y $.  $d x y $.
    $( Substitution of variable in universal quantifier.  Version of ~ sb8f
       with a disjoint variable condition replacing the nonfree hypothesis
       ` F/ y ph ` , not requiring ~ ax-12 .  (Contributed by SN,
       5-Dec-2024.) $)
    sb8v $p |- ( A. x ph <-> A. y [ y / x ] ph ) $=
      ( wsb wal weq wi sb6 albii alcom equcom imbi1i equsv bitri 3bitrri ) ABCD
      ZCEBCFZAGZBEZCERCEZBEABEPSCABCHIRCBJTABTCBFZAGZCEARUBCQUAABCKLIACBMNIO $.
    $( $j usage 'sb8v' avoids 'ax-10' 'ax-12'; $)
  $}

  ${
    $d x y $.
    sb8f.nf $e |- F/ y ph $.
    $( Substitution of variable in universal quantifier.  Version of ~ sb8 with
       a disjoint variable condition, not requiring ~ ax-10 or ~ ax-13 .
       (Contributed by NM, 16-May-1993.)  (Revised by Wolf Lammen,
       19-Jan-2023.)  Avoid ~ ax-10 .  (Revised by SN, 5-Dec-2024.) $)
    sb8f $p |- ( A. x ph <-> A. y [ y / x ] ph ) $=
      ( wsb wal weq wi sb6 albii alcom sbf equcom imbi1i 3bitr3ri 3bitrri ) ABC
      EZCFBCGZAHZBFZCFSCFZBFABFQTCABCIJSCBKUAABACBECBGZAHZCFAUAACBIACBDLUCSCUBR
      ACBMNJOJP $.
    $( $j usage 'sb8f' avoids 'ax-10' 'ax-13'; $)

    $( Obsolete version of ~ sb8f as of 5-Dec-2024.  (Contributed by NM,
       16-May-1993.)  (Revised by Wolf Lammen, 19-Jan-2023.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    sb8fOLD $p |- ( A. x ph <-> A. y [ y / x ] ph ) $=
      ( wsb nfs1v sbequ12 cbvalv1 ) AABCEBCDABCFABCGH $.

    $( Substitution of variable in existential quantifier.  Version of ~ sb8e
       with a disjoint variable condition, not requiring ~ ax-13 .
       (Contributed by NM, 12-Aug-1993.)  (Revised by Wolf Lammen,
       19-Jan-2023.) $)
    sb8ef $p |- ( E. x ph <-> E. y [ y / x ] ph ) $=
      ( wsb nfs1v sbequ12 cbvexv1 ) AABCEBCDABCFABCGH $.
  $}

  ${
    $d z x $.  $d z w y $.
    2sb8ef.1 $e |- F/ w ph $.
    2sb8ef.2 $e |- F/ z ph $.
    $( An equivalent expression for double existence.  Version of ~ 2sb8e with
       more disjoint variable conditions, not requiring ~ ax-13 .  (Contributed
       by Wolf Lammen, 28-Jan-2023.) $)
    2sb8ef $p |- ( E. x E. y ph <->
                  E. z E. w [ z / x ] [ w / y ] ph ) $=
      ( wex wsb sb8ef exbii excom bitri nfsbv 3bitri ) ACHZBHZACEIZBHZEHZRBDIZD
      HZEHUAEHDHQREHZBHTPUCBACEFJKRBELMSUBERBDACEDGNJKUAEDLO $.
  $}

  ${
    $d x y $.
    sb6rfv.nf $e |- F/ y ph $.
    $( Reversed substitution.  Version of ~ sb6rf requiring disjoint variables,
       but fewer axioms.  (Contributed by NM, 1-Aug-1993.)  (Revised by Wolf
       Lammen, 7-Feb-2023.) $)
    sb6rfv $p |- ( ph <-> A. y ( y = x -> [ y / x ] ph ) ) $=
      ( weq wsb wi wal sbequ12r equsalv bicomi ) CBEABCFZGCHALACBDACBIJK $.
  $}

  ${
    $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.)  (Proof shortened by Wolf Lammen, 22-Sep-2018.)
       Avoid ~ ax-13 .  (Revised by Wolf Lammen, 30-Jan-2023.) $)
    sbnf2 $p |- ( F/ x ph
       <-> A. y A. z ( [ y / x ] ph <-> [ z / x ] ph ) ) $=
      ( wnf wa wsb wi wal wb wex nfv sb8ef imbi12i df-nf pm11.53v 3bitr4i alcom
      sb8v bitri anbi12i pm4.24 2albiim ) ABEZUDFABCGZABDGZHDICIZUFUEHZDICIZFUD
      UEUFJDICIUDUGUDUIABKZABIZHZUECKZUFDIZHUDUGUJUMUKUNABCACLMABDSNABOZUEUFCDP
      QUDUHCIDIZUIULUFDKZUECIZHUDUPUJUQUKURABDADLMABCSNUOUFUEDCPQUHDCRTUAUDUBUE
      UFCDUCQ $.
  $}

  ${
    $d x y $.  $d y ph $.
    $( An equivalent expression for existence.  One direction ( ~ exsbim )
       needs fewer axioms.  (Contributed by NM, 2-Feb-2005.)  Avoid ~ ax-13 .
       (Revised by Wolf Lammen, 16-Oct-2022.) $)
    exsb $p |- ( E. x ph <-> E. y A. x ( x = y -> ph ) ) $=
      ( weq wi wal nfv nfa1 ax12v sp com12 impbid cbvexv1 ) ABCDZAEZBFZBCACGOBH
      NAPABCIPNAOBJKLM $.
  $}

  ${
    $d x y z $.  $d y w $.  $d z w ph $.
    $( An equivalent expression for double existence.  (Contributed by NM,
       2-Feb-2005.)  (Proof shortened by Wolf Lammen, 30-Sep-2018.) $)
    2exsb $p |- ( E. x E. y ph <->
                  E. z E. w A. x A. y ( ( x = z /\ y = w ) -> ph ) ) $=
      ( wex wsb weq wa wi wal nfv 2sb8ef 2sb6 2exbii bitri ) ACFBFACEGBDGZEFDFB
      DHCEHIAJCKBKZEFDFABCDEAELADLMQRDEABCDENOP $.
  $}

  ${
    $d y x $.
    sbbib.y $e |- F/ y ph $.
    sbbib.x $e |- F/ x ps $.
    $( Reversal of substitution.  (Contributed by AV, 6-Aug-2023.)  (Proof
       shortened by Wolf Lammen, 4-Sep-2023.) $)
    sbbib $p |- ( A. y ( [ y / x ] ph <-> ps )
              <-> A. x ( ph <-> [ x / y ] ps ) ) $=
      ( wsb wb nfs1v nfbi weq sbequ12r sbequ12 bibi12d cbvalv1 ) ACDGZBHABDCGZH
      DCPBCACDIFJAQDEBDCIJDCKPABQADCLBDCMNO $.
  $}

  ${
    $d x ps $.  $d y ph $.  $d x y $.
    $( Reversal of substitution.  (Contributed by AV, 6-Aug-2023.) $)
    sbbibvv $p |- ( A. y ( [ y / x ] ph <-> ps )
              <-> A. x ( ph <-> [ x / y ] ps ) ) $=
      ( nfv sbbib ) ABCDADEBCEF $.
  $}

  ${
    $d x y w $.  $d ph w $.  $d ps w $.  $d w z $.
    cbvsbvf.1 $e |- F/ y ph $.
    cbvsbvf.2 $e |- F/ x ps $.
    cbvsbvf.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Change the bound variable (i.e. the substituted one) in wff's linked by
       implicit substitution.  The proof was part of a former ~ cbvabw version.
       (Contributed by GG and WL, 26-Oct-2024.) $)
    cbvsbvf $p |- ( [ z / x ] ph <-> [ z / y ] ps ) $=
      ( vw weq wi wal wsb nfv nfim equequ1 imbi12d cbvalv1 imbi2i df-sb 3bitr4i
      albii ) IEJZCIJZAKZCLZKZILUCDIJZBKZDLZKZILACEMBDEMUGUKIUFUJUCUEUICDUDADUD
      DNFOUHBCUHCNGOCDJUDUHABCDIPHQRSUBACIETBDIETUA $.
    $( $j usage 'cbvsbvf' avoids 'ax-10' 'ax-13'; $)
  $}

  ${
    $d x z $.  $d y z $.
    $( Alternate proof of ~ cleljust .  It is kept here and should not be
       modified because it is referenced on the Metamath Proof Explorer Home
       Page (mmset.html) as an example of how disjoint variable conditions are
       inherited by substitutions.  (Contributed by NM, 28-Jan-2004.)  (Revised
       by BJ, 29-Dec-2020.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    cleljustALT $p |- ( x e. y <-> E. z ( z = x /\ z e. y ) ) $=
      ( weq wel wa wex ax-5 elequ1 equsexhv bicomi ) CADCBEZFCGABEZLMCAMCHCABIJ
      K $.

    $( Alternate proof of ~ cleljust .  Compared with ~ cleljustALT , it uses
       ~ nfv followed by ~ equsexv instead of ~ ax-5 followed by ~ equsexhv ,
       so it uses the idiom ` F/ x ph ` instead of ` ph -> A. x ph ` to express
       nonfreeness.  This style is generally preferred for later theorems.
       (Contributed by NM, 28-Jan-2004.)  (Revised by Mario Carneiro,
       21-Dec-2016.)  (Revised by BJ, 29-Dec-2020.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    cleljustALT2 $p |- ( x e. y <-> E. z ( z = x /\ z e. y ) ) $=
      ( weq wel wa wex nfv elequ1 equsexv bicomi ) CADCBEZFCGABEZLMCAMCHCABIJK
      $.
  $}

  $( Alternate proof of ~ equs5a .  Uses ~ ax-12 but not ~ ax-13 .
     (Contributed by NM, 2-Feb-2007.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  equs5aALT $p |- ( E. x ( x = y /\ A. y ph ) -> A. x ( x = y -> ph ) ) $=
    ( weq wal wa wi nfa1 ax-12 imp exlimi ) BCDZACEZFLAGZBEZBNBHLMOABCIJK $.

  $( Alternate proof of ~ equs5e .  Uses ~ ax-12 but not ~ ax-13 .
     (Contributed by NM, 2-Feb-2007.)  (Proof shortened by Wolf Lammen,
     15-Jan-2018.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  equs5eALT $p |- ( E. x ( x = y /\ ph ) -> A. x ( x = y -> E. y ph ) ) $=
    ( weq wa wex wi wal nfa1 hbe1 19.23bi ax-12 syl5 imp exlimi ) BCDZAEPACFZGZ
    BHZBRBIPASAQCHZPSATCACJKQBCLMNO $.

  $( Same as ~ axc11 but with reversed antecedent.  Note the use of ~ ax-12
     (and not merely ~ ax12v as in ~ axc11rv ).

     This theorem is mostly used to eliminate conditions requiring set
     variables be distinct (cf. ~ cbvaev and ~ aecom , for example) in proofs.
     In practice, theorems beyond elementary set theory do not really benefit
     from such eliminations.  As of 2024, it is used in conjunction with
     ~ ax-13 only, and like that, it should be applied only in niches where
     indispensable.  (Contributed by NM, 25-Jul-2015.) $)
  axc11r $p |- ( A. y y = x -> ( A. x ph -> A. y ph ) ) $=
    ( weq wal wi ax-12 sps pm2.27 al2imi syld ) CBDZCEABEZLAFZCEZACELMOFCACBGHL
    NACLAIJK $.

  ${
    $d x y $.
    dral1v.1 $e |- ( A. x x = y -> ( ph <-> ps ) ) $.
    $( Formula-building lemma for use with the Distinctor Reduction Theorem.
       Version of ~ dral1 with a disjoint variable condition, which does not
       require ~ ax-13 .  Remark: the corresponding versions for ~ dral2 and
       ~ drex2 are instances of ~ albidv and ~ exbidv respectively.
       (Contributed by NM, 24-Nov-1994.)  (Revised by BJ, 17-Jun-2019.)  Base
       the proof on ~ ax12v .  (Revised by Wolf Lammen, 30-Mar-2024.)  Avoid
       ~ ax-10 .  (Revised by GG, 18-Nov-2024.) $)
    dral1v $p |- ( A. x x = y -> ( A. x ph <-> A. y ps ) ) $=
      ( weq wal hbaev albidh axc11v axc11rv impbid bitrd ) CDFCGZACGBCGZBDGZNAB
      CCDCHEINOPBCDJBCDKLM $.
    $( $j usage 'dral1v' avoids 'ax-10'; $)

    $( Obsolete version of ~ dral1v as of 18-Nov-2024.  (Contributed by NM,
       24-Nov-1994.)  (Revised by BJ, 17-Jun-2019.)  Base the proof on
       ~ ax12v .  (Revised by Wolf Lammen, 30-Mar-2024.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    dral1vOLD $p |- ( A. x x = y -> ( A. x ph <-> A. y ps ) ) $=
      ( weq wal nfa1 albid axc11v axc11rv impbid bitrd ) CDFZCGZACGBCGZBDGZOABC
      NCHEIOPQBCDJBCDKLM $.

    $( Formula-building lemma for use with the Distinctor Reduction Theorem.
       Version of ~ drex1 with a disjoint variable condition, which does not
       require ~ ax-13 .  (Contributed by NM, 27-Feb-2005.)  (Revised by BJ,
       17-Jun-2019.) $)
    drex1v $p |- ( A. x x = y -> ( E. x ph <-> E. y ps ) ) $=
      ( weq wal wn wex notbid dral1v df-ex 3bitr4g ) CDFCGZAHZCGZHBHZDGZHACIBDI
      NPROQCDNABEJKJACLBDLM $.

    $( Formula-building lemma for use with the Distinctor Reduction Theorem.
       Version of ~ drnf1 with a disjoint variable condition, which does not
       require ~ ax-13 .  (Contributed by Mario Carneiro, 4-Oct-2016.)
       (Revised by BJ, 17-Jun-2019.)  Avoid ~ ax-10 .  (Revised by GG,
       18-Nov-2024.) $)
    drnf1v $p |- ( A. x x = y -> ( F/ x ph <-> F/ y ps ) ) $=
      ( weq wal wex wi wnf drex1v dral1v imbi12d df-nf 3bitr4g ) CDFCGZACHZACGZ
      IBDHZBDGZIACJBDJPQSRTABCDEKABCDELMACNBDNO $.
    $( $j usage 'drnf1v' avoids 'ax-10'; $)

    $( Obsolete version of ~ drnf1v as of 18-Nov-2024.  (Contributed by Mario
       Carneiro, 4-Oct-2016.)  (Revised by BJ, 17-Jun-2019.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    drnf1vOLD $p |- ( A. x x = y -> ( F/ x ph <-> F/ y ps ) ) $=
      ( weq wal wi wnf dral1v imbi12d nf5 3bitr4g ) CDFCGZAACGZHZCGBBDGZHZDGACI
      BDIPRCDNABOQEABCDEJKJACLBDLM $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Axiom scheme ax-13 (Quantified Equality)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Axiom of Quantified Equality.  One of the equality and substitution axioms
     of predicate calculus with equality.

     An equivalent way to express this axiom that may be easier to understand
     is ` ( -. x = y -> ( -. x = z -> ( y = z -> A. x y = z ) ) ) ` (see
     ~ ax13b ).  Recall that in the intended interpretation, our variables are
     metavariables ranging over the variables of predicate calculus (the object
     language).  In order for the first antecedent ` -. x = y ` to hold, ` x `
     and ` y ` must have different values and thus cannot be the same
     object-language variable (so they are effectively "distinct variables"
     even though no $d is present).  Similarly, ` x ` and ` z ` cannot be the
     same object-language variable.  Therefore, ` x ` will not occur in the wff
     ` y = z ` when the first two antecedents hold, so analogous to ~ ax-5 ,
     the conclusion ` ( y = z -> A. x y = z ) ` follows.  Note that ~ ax-5
     cannot prove this because its distinct variable ($d) requirement is not
     satisfied directly but only indirectly (outside of Metamath) by the
     argument above.

     The original version of this axiom was ~ ax-c9 and was replaced with this
     shorter ~ ax-13 in December 2015.  The old axiom is proved from this one
     as Theorem ~ axc9 .

     The primary purpose of this axiom is to provide a way to introduce the
     quantifier ` A. x ` on ` y = z ` even when ` x ` and ` y ` are substituted
     with the same variable.  In this case, the first antecedent becomes
     ` -. x = x ` and the axiom still holds.

     This axiom is mostly used to eliminate conditions requiring set variables
     be distinct (cf. ~ ax6ev and ~ ax6e , for example) in proofs.  In
     practice, theorems beyond elementary set theory do not really benefit from
     such eliminations, so direct or indirect application of this axiom is
     discouraged now.  You need to explicitly confirm its use in case you see a
     sensible application in a niche.

     After some assisting contributions by others over the years, it was in
     particular the extensive work of Gino Giotto in 2024 that helped reducing
     dependencies on this axiom on a large scale.

     Although this version is shorter, the original version ~ axc9 may be more
     practical to work with because of the "distinctor" form of its
     antecedents.  A typical application of ~ axc9 is in ~ dvelimh which
     converts a distinct variable pair to the distinctor antecedent
     ` -. A. x x = y ` .  In particular, it is conjectured that it is not
     possible to prove ~ ax6 from ~ ax6v without this axiom.

     This axiom can be weakened if desired by adding distinct variable
     restrictions on pairs ` x , z ` and ` y , z ` .  To show that, we add
     these restrictions to Theorem ~ ax13v and use only ~ ax13v for further
     derivations.  Thus, ~ ax13v should be the only theorem referencing this
     axiom.  Other theorems can reference either ~ ax13v (preferred) or ~ ax13
     (if the stronger form is needed).

     This axiom scheme is logically redundant (see ~ ax13w ) but is used as an
     auxiliary axiom scheme to achieve scheme completeness (i.e. so that all
     possible cases of bundling can be proved; see text linked at
     ~ mmtheorems.html#ax6dgen ).  It is not known whether this axiom can be
     derived from the others.  (Contributed by NM, 21-Dec-2015.)
     (New usage is discouraged.) $)
  ax-13 $a |- ( -. x = y -> ( y = z -> A. x y = z ) ) $.

  ${
    $d x z $.  $d y z $.
    $( A weaker version of ~ ax-13 with distinct variable restrictions on pairs
       ` x , z ` and ` y , z ` .  In order to show (with ~ ax13 ) that this
       weakening is still adequate, this should be the only theorem referencing
       ~ ax-13 directly.

       Had we additionally required ` x ` and ` y ` be distinct, too, this
       theorem would have been a direct consequence of ~ ax-5 .  So essentially
       this theorem states, that a distinct variable condition can be replaced
       with an inequality between set variables.  Preferably, use the version
       ~ ax13w to avoid the propagation of ~ ax-13 .  (Contributed by NM,
       30-Jun-2016.)  (New usage is discouraged.) $)
    ax13v $p |- ( -. x = y -> ( y = z -> A. x y = z ) ) $=
      ( ax-13 ) ABCD $.
  $}

  ${
    $d x z w $.  $d y w $.
    $( A version of ~ ax13v with one distinct variable restriction dropped.
       For convenience, ` y ` is kept on the right side of equations.  The
       proof of ~ ax13 bases on ideas from NM, 24-Dec-2015.  (Contributed by
       Wolf Lammen, 8-Sep-2018.)  (New usage is discouraged.) $)
    ax13lem1 $p |- ( -. x = y -> ( z = y -> A. x z = y ) ) $=
      ( vw weq wa wex wal equvinva ax13v equeucl alimdv syl9 impd exlimdv syl5
      wn ) CBEZCDEZBDEZFZDGABEQZRAHZCBDIUBUAUCDUBSTUCUBTTAHSUCABDJSTRACBDKLMNOP
      $.
  $}

  ${
    $d x w $.  $d z w $.  $d y w $.
    $( Derive ~ ax-13 from ~ ax13v and Tarski's FOL. This shows that the
       weakening in ~ ax13v is still sufficient for a complete system.
       Preferably, use the weaker ~ ax13w to avoid the propagation of ~ ax-13 .
       (Contributed by NM, 21-Dec-2015.)  (Proof shortened by Wolf Lammen,
       31-Jan-2018.)  Reduce axiom usage.  (Revised by Wolf Lammen,
       2-Jun-2021.)  (New usage is discouraged.) $)
    ax13 $p |- ( -. x = y -> ( y = z -> A. x y = z ) ) $=
      ( vw weq wn wal wi wa wex equvinv ax13lem1 imp ax7v1 alanimi an4s exlimdv
      syl2an ex biimtrid ax13b mpbir ) ABEFZBCEZUDAGZHZHUCACEFZUFHHUCUGUFUDDBEZ
      DCEZIZDJUCUGIZUEBCDKUKUJUEDUKUJUEUCUHUGUIUEUCUHIUHAGZUIAGZUEUGUIIUCUHULAB
      DLMUGUIUMACDLMUHUIUDAUHUIUDDBCNMORPSQTSUEABCUAUB $.
  $}

  ${
    $d w x z $.  $d w y $.
    $( Lemma for ~ nfeqf2 .  This lemma is equivalent to ~ ax13v with one
       distinct variable constraint removed.  (Contributed by Wolf Lammen,
       8-Sep-2018.)  Reduce axiom usage.  (Revised by Wolf Lammen,
       18-Oct-2020.)  (New usage is discouraged.) $)
    ax13lem2 $p |- ( -. x = y -> ( E. x z = y -> z = y ) ) $=
      ( vw weq wn wex wi wal ax13lem1 equeucl eximi 19.36v syl9 alrimdv equequ2
      sylib equsalvw imbitrdi ) ABEFZCBEZAGZDBEZCDEZHZDIUATUBUEDTUCUCAIZUBUDABD
      JUBUEAGUFUDHUAUEACDBKLUCUDAMQNOUDUADBDBCPRS $.
  $}

  ${
    $d x z $.
    $( An equation between setvar is free of any other setvar.  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  (Contributed by
       Wolf Lammen, 9-Jun-2019.)  Remove dependency on ~ ax-12 .  (Revised by
       Wolf Lammen, 16-Dec-2022.)  (New usage is discouraged.) $)
    nfeqf2 $p |- ( -. A. x x = y -> F/ x z = y ) $=
      ( weq wal wex wnf exnal hbe1 ax13lem2 ax13lem1 syldc eximdh hbe1a syl6com
      wn nfd sylbir ) ABDZAEPSPZAFZCBDZAGSAHUAUBAUBAFZUAUBAEZAFUDUCTUDAUBAITUCU
      BUDABCJABCKLMUBANOQR $.
  $}

  ${
    $d x z $.
    $( Quantifier introduction when one pair of variables is distinct.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .
       (Contributed by NM, 2-Jan-2002.)  (Revised by NM, 20-Jul-2015.)  Remove
       dependency on ~ ax-11 .  (Revised by Wolf Lammen, 8-Sep-2018.)
       (New usage is discouraged.) $)
    dveeq2 $p |- ( -. A. x x = y -> ( z = y -> A. x z = y ) ) $=
      ( weq wal wn nfeqf2 nf5rd ) ABDAEFCBDAABCGH $.
  $}

  ${
    $d x z $.
    $( An equation between setvar is free of any other setvar.  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  (Contributed by
       Wolf Lammen, 10-Jun-2019.)  (New usage is discouraged.) $)
    nfeqf1 $p |- ( -. A. x x = y -> F/ x y = z ) $=
      ( weq wal wn wnf nfeqf2 equcom nfbii sylib ) ABDAEFCBDZAGBCDZAGABCHLMACBI
      JK $.
  $}

  ${
    $d x z $.
    $( Quantifier introduction when one pair of variables is distinct.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .
       (Contributed by NM, 2-Jan-2002.)  Remove dependency on ~ ax-11 .
       (Revised by Wolf Lammen, 8-Sep-2018.)  (New usage is discouraged.) $)
    dveeq1 $p |- ( -. A. x x = y -> ( y = z -> A. x y = z ) ) $=
      ( weq wal wn nfeqf1 nf5rd ) ABDAEFBCDAABCGH $.
  $}

  ${
    $d x w $.  $d y w $.  $d z w $.
    $( A variable is effectively not free in an equality if it is not either of
       the involved variables. ` F/ ` version of ~ ax-c9 .  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  (Contributed by
       Mario Carneiro, 6-Oct-2016.)  Remove dependency on ~ ax-11 .  (Revised
       by Wolf Lammen, 6-Sep-2018.)  (New usage is discouraged.) $)
    nfeqf $p |- ( ( -. A. z z = x /\ -. A. z z = y ) -> F/ z x = y ) $=
      ( vw weq wal wn wa nfna1 nfan wex equvinva dveeq1 imp equtr2 alanimi an4s
      syl2an ex exlimdv syl5 nf5d ) CAEZCFGZCBEZCFGZHZABEZCUDUFCUCCIUECIJUHADEZ
      BDEZHZDKUGUHCFZABDLUGUKULDUGUKULUDUIUFUJULUDUIHUICFZUJCFZULUFUJHUDUIUMCAD
      MNUFUJUNCBDMNUIUJUHCABDOPRQSTUAUB $.
  $}

  $( Derive set.mm's original ~ ax-c9 from the shorter ~ ax-13 .  Usage is
     discouraged to avoid uninformed ~ ax-13 propagation.  (Contributed by NM,
     29-Nov-2015.)  (Revised by NM, 24-Dec-2015.)  (Proof shortened by Wolf
     Lammen, 29-Apr-2018.)  (New usage is discouraged.) $)
  axc9 $p |- ( -. A. z z = x -> ( -. A. z z = y
              -> ( x = y -> A. z x = y ) ) ) $=
    ( weq wal wn wi wa nfeqf nf5rd ex ) CADCEFZCBDCEFZABDZNCEGLMHNCABCIJK $.

  ${
    $d y w $.  $d x w $.
    $( 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-4 through ~ ax-9 , all axioms other than
       ~ ax-6 are believed to be theorems of free logic, although the system
       without ~ ax-6 is not complete in free logic.

       Usage of this theorem is discouraged because it depends on ~ ax-13 .  It
       is preferred to use ~ ax6ev when it is sufficient.  (Contributed by NM,
       14-May-1993.)  Shortened after ~ ax13lem1 became available.  (Revised by
       Wolf Lammen, 8-Sep-2018.)  (New usage is discouraged.) $)
    ax6e $p |- E. x x = y $=
      ( vw weq wex 19.8a wn wi wal ax13lem1 ax6ev equtr eximii syl6com exlimiiv
      19.35i pm2.61i ) ABDZRAEZRAFCBDZRGZSHCUATTAISABCJTRAACDTRHAACKACBLMPNCBKO
      Q $.
  $}

  $( Theorem showing that ~ ax-6 follows from the weaker version ~ ax6v .
     (Even though this theorem depends on ~ ax-6 , all references of ~ ax-6 are
     made via ~ ax6v .  An earlier version stated ~ ax6v as a separate axiom,
     but having two axioms caused some confusion.)

     This theorem should be referenced in place of ~ ax-6 so that all proofs
     can be traced back to ~ ax6v .  When possible, use the weaker ~ ax6v
     rather than ~ ax6 since the ~ ax6v derivation is much shorter and requires
     fewer axioms.  (Contributed by NM, 12-Nov-2013.)  (Revised by NM,
     25-Jul-2015.)  (Proof shortened by Wolf Lammen, 4-Feb-2018.)  Usage of
     this theorem is discouraged because it depends on ~ ax-13 .  Use ~ ax6v
     instead.  (New usage is discouraged.) $)
  ax6 $p |- -. A. x -. x = y $=
    ( weq wex wn wal ax6e df-ex mpbi ) ABCZADJEAFEABGJAHI $.

  $( Show that the original axiom ~ ax-c10 can be derived from ~ ax6 and ~ axc7
     (on top of propositional calculus, ~ ax-gen , and ~ ax-4 ).  See
     ~ ax6fromc10 for the rederivation of ~ ax6 from ~ ax-c10 .

     Normally, ~ axc10 should be used rather than ~ ax-c10 , except by theorems
     specifically studying the latter's properties.  See ~ bj-axc10v for a
     weaker version requiring fewer axioms.  (Contributed by NM, 5-Aug-1993.)
     (Proof modification is discouraged.)  Usage of this theorem is discouraged
     because it depends on ~ ax-13 .  (New usage is discouraged.) $)
  axc10 $p |- ( A. x ( x = y -> A. x ph ) -> ph ) $=
    ( weq wal wi wn ax6 con3 al2imi mtoi axc7 syl ) BCDZABEZFZBEZOGZBEZGAQSNGZB
    EBCHPRTBNOIJKABLM $.

  $( Closed theorem form of ~ spim .  (Contributed by NM, 15-Jan-2008.)
     (Revised by Mario Carneiro, 17-Oct-2016.)  (Proof shortened by Wolf
     Lammen, 21-Mar-2023.)  Usage of this theorem is discouraged because it
     depends on ~ ax-13 .  (New usage is discouraged.) $)
  spimt $p |- ( ( F/ x ps /\ A. x ( x = y -> ( ph -> ps ) ) ) ->
                                                         ( A. x ph -> ps ) ) $=
    ( weq wi wal wex wnf ax6e exim mpi 19.35 sylib id 19.9d sylan9r ) CDEZABFZF
    CGZACGZBCHZBCIZBTSCHZUAUBFTRCHUDCDJRSCKLABCMNBUCCUCOPQ $.

  ${
    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.  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  See ~ spimw for a version
       requiring fewer axioms.  (Contributed by NM, 10-Jan-1993.)  (Revised by
       Mario Carneiro, 3-Oct-2016.)  (Proof shortened by Wolf Lammen,
       18-Feb-2018.)  (New usage is discouraged.) $)
    spim $p |- ( A. x ph -> ps ) $=
      ( weq wi ax6e eximii 19.36i ) ABCECDGABHCCDIFJK $.
  $}

  ${
    spimed.1 $e |- ( ch -> F/ x ph ) $.
    spimed.2 $e |- ( x = y -> ( ph -> ps ) ) $.
    $( Deduction version of ~ spime .  (Contributed by NM, 14-May-1993.)
       (Revised by Mario Carneiro, 3-Oct-2016.)  (Proof shortened by Wolf
       Lammen, 19-Feb-2018.)  Usage of this theorem is discouraged because it
       depends on ~ ax-13 .  Use ~ spimedv instead.
       (New usage is discouraged.) $)
    spimed $p |- ( ch -> ( ph -> E. x ps ) ) $=
      ( wal wex nf5rd weq wi ax6e eximii 19.35i syl6 ) CAADHBDICADFJABDDEKABLDD
      EMGNOP $.
  $}

  ${
    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.  See ~ spimew and ~ spimevw for weaker versions
       requiring fewer axioms.  (Contributed by NM, 7-Aug-1994.)  (Revised by
       Mario Carneiro, 3-Oct-2016.)  (Proof shortened by Wolf Lammen,
       6-Mar-2018.)  Usage of this theorem is discouraged because it depends on
       ~ ax-13 .  Use ~ spimefv instead.  (New usage is discouraged.) $)
    spime $p |- ( ph -> E. x ps ) $=
      ( wex wi wtru wnf a1i spimed mptru ) ABCGHABICDACJIEKFLM $.
  $}

  ${
    $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.  See ~ spimfv and ~ spimvw for versions
       requiring fewer axioms.  (Contributed by NM, 31-Jul-1993.)  Usage of
       this theorem is discouraged because it depends on ~ ax-13 .  Use
       ~ spimvw instead.  (New usage is discouraged.) $)
    spimv $p |- ( A. x ph -> ps ) $=
      ( nfv spim ) ABCDBCFEG $.

    $( Alternate proof of ~ spimv .  Note that it requires only ~ ax-1 through
       ~ ax-5 together with ~ ax6e .  Currently, proofs derive from ~ ax6v ,
       but if ~ ax-6 could be used instead, this proof would reduce axiom
       usage.  (Contributed by NM, 31-Jul-1993.)  Remove dependency on
       ~ ax-10 .  (Revised by BJ, 29-Nov-2020.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    spimvALT $p |- ( A. x ph -> ps ) $=
      ( weq wi ax6e eximii 19.36iv ) ABCCDFABGCCDHEIJ $.
  $}

  ${
    $d x ph $.
    spimev.1 $e |- ( x = y -> ( ph -> ps ) ) $.
    $( Distinct-variable version of ~ spime .  (Contributed by NM,
       10-Jan-1993.)  Usage of this theorem is discouraged because it depends
       on ~ ax-13 .  Use ~ spimevw instead.  (New usage is discouraged.) $)
    spimev $p |- ( ph -> E. x ps ) $=
      ( nfv spime ) ABCDACFEG $.
  $}

  ${
    $d x ps $.
    spv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Specialization, using implicit substitution.  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  Use the weaker ~ spvv if
       possible.  (Contributed by NM, 30-Aug-1993.)
       (New usage is discouraged.) $)
    spv $p |- ( A. x ph -> ps ) $=
      ( weq biimpd spimv ) ABCDCDFABEGH $.
  $}

  ${
    spei.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    spei.2 $e |- ps $.
    $( Inference from existential specialization, using implicit substitution.
       Remove a distinct variable constraint.  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  Use the weaker ~ speiv if
       possible.  (Contributed by NM, 19-Aug-1993.)  (Proof shortened by Wolf
       Lammen, 12-May-2018.)  (New usage is discouraged.) $)
    spei $p |- E. x ph $=
      ( weq ax6e mpbiri eximii ) CDGZACCDHKABFEIJ $.
  $}

  ${
    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.  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  Use the weaker
       ~ chvarfv if possible.  (Contributed by Raph Levien, 9-Jul-2003.)
       (Revised by Mario Carneiro, 3-Oct-2016.)  (New usage is discouraged.) $)
    chvar $p |- ps $=
      ( weq biimpd spim mpg ) ABCABCDECDHABFIJGK $.
  $}

  ${
    $d x ps $.
    chvarv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    chvarv.2 $e |- ph $.
    $( Implicit substitution of ` y ` for ` x ` into a theorem.  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  Use the weaker
       ~ chvarvv if possible.  (Contributed by NM, 20-Apr-1994.)  (Proof
       shortened by Wolf Lammen, 22-Apr-2018.)  (New usage is discouraged.) $)
    chvarv $p |- ps $=
      ( nfv chvar ) ABCDBCGEFH $.
  $}

  ${
    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, that
       does not use ~ ax-c9 .  Usage of this theorem is discouraged because it
       depends on ~ ax-13 .  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 nf5ri hbal spim alrimih ) ACHBDADCADEIJABCDFGKL $.
  $}

  ${
    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.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .  Check
       out ~ cbvalw , ~ cbvalvw , ~ cbvalv1 for versions requiring fewer
       axioms.  (Contributed by NM, 13-May-1993.)  (Revised by Mario Carneiro,
       3-Oct-2016.)  (New usage is discouraged.) $)
    cbval $p |- ( A. x ph <-> A. y ps ) $=
      ( wal weq biimpd cbv3 wi biimprd equcoms impbii ) ACHBDHABCDEFCDIZABGJKBA
      DCFEBALCDPABGMNKO $.

    $( Rule used to change bound variables, using implicit substitution.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .  Check
       out ~ cbvexvw , ~ cbvexv1 for weaker versions requiring fewer axioms.
       (Contributed by NM, 21-Jun-1993.)  (New usage is discouraged.) $)
    cbvex $p |- ( E. x ph <-> E. y ps ) $=
      ( wex wn wal nfn weq notbid cbval alnex 3bitr3i con4bii ) ACHZBDHZAIZCJBI
      ZDJRISITUACDADEKBCFKCDLABGMNACOBDOPQ $.
  $}

  ${
    $d y ph $.  $d x ps $.
    cbvalv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Rule used to change bound variables, using implicit substitution.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .  See
       ~ cbvalvw for a version requiring fewer axioms, to be preferred when
       sufficient.  (Contributed by NM, 5-Aug-1993.)  Remove dependency on
       ~ ax-10 and shorten proof.  (Revised by Wolf Lammen, 11-Sep-2023.)
       (New usage is discouraged.) $)
    cbvalv $p |- ( A. x ph <-> A. y ps ) $=
      ( nfv cbval ) ABCDADFBCFEG $.

    $( Rule used to change bound variables, using implicit substitution.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .  See
       ~ cbvexvw for a version requiring fewer axioms, to be preferred when
       sufficient.  (Contributed by NM, 21-Jun-1993.)  Remove dependency on
       ~ ax-10 and shorten proof.  (Revised by Wolf Lammen, 11-Sep-2023.)
       (New usage is discouraged.) $)
    cbvexv $p |- ( E. x ph <-> E. y ps ) $=
      ( nfv cbvex ) ABCDADFBCFEG $.
  $}

  ${
    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.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .  See
       ~ cbv1v with disjoint variable conditions, not depending on ~ ax-13 .
       (Contributed by NM, 5-Aug-1993.)  (Revised by Mario Carneiro,
       3-Oct-2016.)  Format hypotheses to common style.  (Revised by Wolf
       Lammen, 13-May-2018.)  (New usage is discouraged.) $)
    cbv1 $p |- ( ph -> ( A. x ps -> A. y ch ) ) $=
      ( wal wi nfim1 weq com12 a2d cbv3 19.21 3imtr3i pm2.86i ) ABDKZCEKZABLZDK
      ACLZEKAUALAUBLUCUDDEABEGHMACDFIMDENZABCAUEBCLJOPQABDFRACEGRST $.
  $}

  ${
    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.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .  See
       ~ cbv2w with disjoint variable conditions, not depending on ~ ax-13 .
       (Contributed by NM, 5-Aug-1993.)  (Revised by Mario Carneiro,
       3-Oct-2016.)  Format hypotheses to common style, avoid ~ ax-10 .
       (Revised by Wolf Lammen, 10-Sep-2023.)  (New usage is discouraged.) $)
    cbv2 $p |- ( ph -> ( A. x ps <-> A. y ch ) ) $=
      ( wal weq wb wi biimp syl6 cbv1 equcomi biimpr syl56 impbid ) ABDKCEKABCD
      EFGHIADELZBCMZBCNJBCOPQACBEDGFIHEDLUBAUCCBNEDRJBCSTQUA $.
    $( $j usage 'cbv2' avoids 'ax-10'; $)
  $}

  ${
    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.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .  Use the
       weaker ~ cbv3hv if possible.  (Contributed by NM, 8-Jun-1993.)  (Proof
       shortened by Andrew Salmon, 25-May-2011.)  (Proof shortened by Wolf
       Lammen, 12-May-2018.)  (New usage is discouraged.) $)
    cbv3h $p |- ( A. x ph -> A. y ps ) $=
      ( nf5i cbv3 ) ABCDADEHBCFHGI $.
  $}

  ${
    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.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .
       (Contributed by NM, 11-May-1993.)  (Proof shortened by Wolf Lammen,
       13-May-2018.)  (New usage is discouraged.) $)
    cbv1h $p |- ( A. x A. y ph -> ( A. x ps -> A. y ch ) ) $=
      ( wal nfa1 nfa2 wi 2sp syl nf5d weq cbv1 ) AEIZDIZBCDERDJZAEDKZSBEUASABBE
      ILADEMZFNOSCDTSACCDILUBGNOSADEPBCLLUBHNQ $.
  $}

  ${
    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.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .
       (Contributed by NM, 11-May-1993.)  (New usage is discouraged.) $)
    cbv2h $p |- ( A. x A. y ph -> ( A. x ps <-> A. y ch ) ) $=
      ( wal weq wb wi biimp syl6 cbv1h equcomi biimpr syl56 alcoms impbid ) AEI
      DIBDIZCEIZABCDEFGADEJZBCKZBCLHBCMNOAUBUALEDACBEDGFEDJUCAUDCBLEDPHBCQROST
      $.
  $}

  ${
    $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 .  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  See ~ cbvaldw
       for a version with ` x , y ` disjoint, not depending on ~ ax-13 .
       (Contributed by NM, 2-Jan-2002.)  (Revised by Mario Carneiro,
       6-Oct-2016.)  (Revised by Wolf Lammen, 13-May-2018.)
       (New usage is discouraged.) $)
    cbvald $p |- ( ph -> ( A. x ps <-> A. y ch ) ) $=
      ( nfv nfvd cbv2 ) ABCDEADIFGACDJHK $.

    $( Deduction used to change bound variables, using implicit substitution,
       particularly useful in conjunction with ~ dvelim .  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  Use the weaker
       ~ cbvexdw if possible.  (Contributed by NM, 2-Jan-2002.)  (Revised by
       Mario Carneiro, 6-Oct-2016.)  (New usage is discouraged.) $)
    cbvexd $p |- ( ph -> ( E. x ps <-> E. y ch ) ) $=
      ( wex wn wal nfnd weq wb notbi imbitrdi cbvald alnex 3bitr3g con4bid ) AB
      DIZCEIZABJZDKCJZEKUAJUBJAUCUDDEFABEGLADEMBCNUCUDNHBCOPQBDRCERST $.
  $}

  ${
    $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.  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  Use the weaker ~ cbvaldvaw
       if possible.  (Contributed by David Moews, 1-May-2017.)
       (New usage is discouraged.) $)
    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.  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  Use the weaker ~ cbvexdvaw
       if possible.  (Contributed by David Moews, 1-May-2017.)
       (New usage is discouraged.) $)
    cbvexdva $p |- ( ph -> ( E. x ps <-> E. y ch ) ) $=
      ( nfv nfvd weq wb ex cbvexd ) ABCDEAEGABEHADEIBCJFKL $.
  $}

  ${
    $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.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .  Use the
       weaker ~ cbval2v if possible.  (Contributed by NM, 22-Dec-2003.)
       (Revised by Mario Carneiro, 6-Oct-2016.)  (Proof shortened by Wolf
       Lammen, 11-Sep-2023.)  (New usage is discouraged.) $)
    cbval2 $p |- ( A. x A. y ph <-> A. z A. w ps ) $=
      ( wal nfal weq nfv wnf a1i wb ex cbv2 cbval ) ADLBFLCEAEDGMBCFIMCENZABDFU
      BDOUBFOAFPUBHQBDPUBJQUBDFNABRKSTUA $.

    $( Rule used to change bound variables, using implicit substitution.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .  Use the
       weaker ~ cbvex2v if possible.  (Contributed by NM, 14-Sep-2003.)
       (Revised by Mario Carneiro, 6-Oct-2016.)  (Proof shortened by Wolf
       Lammen, 16-Jun-2019.)  (New usage is discouraged.) $)
    cbvex2 $p |- ( E. x E. y ph <-> E. z E. w ps ) $=
      ( wex wn wal nfn weq wa notbid cbval2 2nexaln 3bitr4i con4bii ) ADLCLZBFL
      ELZAMZDNCNBMZFNENUCMUDMUEUFCDEFAEGOAFHOBCIOBDJOCEPDFPQABKRSACDTBEFTUAUB
      $.
  $}

  ${
    $d z w ph $.  $d x y ps $.  $d x w $.  $d z y $.
    cbval2vv.1 $e |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) ) $.
    $( Rule used to change bound variables, using implicit substitution.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .  Use the
       weaker ~ cbval2vw if possible.  (Contributed by NM, 4-Feb-2005.)  Remove
       dependency on ~ ax-10 .  (Revised by Wolf Lammen, 18-Jul-2021.)
       (New usage is discouraged.) $)
    cbval2vv $p |- ( A. x A. y ph <-> A. z A. w ps ) $=
      ( wal weq cbvaldva cbvalv ) ADHBFHCECEIABDFGJK $.

    $( Rule used to change bound variables, using implicit substitution.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .  Use the
       weaker ~ cbvex2vw if possible.  (Contributed by NM, 26-Jul-1995.)
       Remove dependency on ~ ax-10 .  (Revised by Wolf Lammen, 18-Jul-2021.)
       (New usage is discouraged.) $)
    cbvex2vv $p |- ( E. x E. y ph <-> E. z E. w ps ) $=
      ( wex weq cbvexdva cbvexv ) 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 substitution.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .  Use the
       weaker ~ cbvex4vw if possible.  (Contributed by NM, 26-Jul-1995.)
       (New usage is discouraged.) $)
    cbvex4v $p |- ( E. x E. y E. z E. w ph <-> E. v E. u E. f E. g ch ) $=
      ( wex weq wa 2exbidv cbvex2vv 2exbii bitri ) AGNFNZENDNBGNFNZINHNCKNJNZIN
      HNUAUBDEHIDHOEIOPABFGLQRUBUCHIBCFGJKMRST $.
  $}

  $( Lemma used in proofs of implicit substitution properties.  The converse
     requires either a disjoint variable condition ( ~ sbalex ) or a
     nonfreeness hypothesis ( ~ equs45f ).  Usage of this theorem is
     discouraged because it depends on ~ ax-13 .  See ~ equs4v for a weaker
     version requiring fewer axioms.  (Contributed by NM, 10-May-1993.)  (Proof
     shortened by Mario Carneiro, 20-May-2014.)  (Proof shortened by Wolf
     Lammen, 5-Feb-2018.)  (New usage is discouraged.) $)
  equs4 $p |- ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) ) $=
    ( weq wi wal wex wa ax6e exintr mpi ) BCDZAEBFLBGLAHBGBCILABJK $.

  ${
    equsal.1 $e |- F/ x ps $.
    equsal.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( An equivalence related to implicit substitution.  Usage of this theorem
       is discouraged because it depends on ~ ax-13 .  See ~ equsalvw and
       ~ equsalv for versions with disjoint variable conditions proved from
       fewer axioms.  See also the dual form ~ equsex .  (Contributed by NM,
       2-Jun-1993.)  (Proof shortened by Andrew Salmon, 12-Aug-2011.)  (Revised
       by Mario Carneiro, 3-Oct-2016.)  (Proof shortened by Wolf Lammen,
       5-Feb-2018.)  (New usage is discouraged.) $)
    equsal $p |- ( A. x ( x = y -> ph ) <-> ps ) $=
      ( weq wi wal wex 19.23 pm5.74i albii ax6e a1bi 3bitr4i ) CDGZBHZCIQCJZBHQ
      AHZCIBQBCEKTRCQABFLMSBCDNOP $.

    $( An equivalence related to implicit substitution.  Usage of this theorem
       is discouraged because it depends on ~ ax-13 .  See ~ equsexvw and
       ~ equsexv for versions with disjoint variable conditions proved from
       fewer axioms.  See also the dual form ~ equsal .  See ~ equsexALT for an
       alternate proof.  (Contributed by NM, 5-Aug-1993.)  (Revised by Mario
       Carneiro, 3-Oct-2016.)  (Proof shortened by Wolf Lammen, 6-Feb-2018.)
       (New usage is discouraged.) $)
    equsex $p |- ( E. x ( x = y /\ ph ) <-> ps ) $=
      ( weq wa wex biimpa exlimi wi wal equsal equs4 sylbir impbii ) CDGZAHZCIZ
      BSBCERABFJKBRALCMTABCDEFNACDOPQ $.

    $( Alternate proof of ~ equsex .  This proves the result directly, instead
       of as a corollary of ~ equsal via ~ equs4 .  Note in particular that
       only existential quantifiers appear in the proof and that the only step
       requiring ~ ax-13 is ~ ax6e .  This proof mimics that of ~ equsal (in
       particular, note that ~ pm5.32i , ~ exbii , ~ 19.41 , ~ mpbiran
       correspond respectively to ~ pm5.74i , ~ albii , ~ 19.23 , ~ a1bi ).
       (Contributed by BJ, 20-Aug-2020.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    equsexALT $p |- ( E. x ( x = y /\ ph ) <-> ps ) $=
      ( weq wa wex pm5.32i exbii ax6e 19.41 mpbiran bitri ) CDGZAHZCIPBHZCIZBQR
      CPABFJKSPCIBCDLPBCEMNO $.
  $}

  ${
    equsalh.1 $e |- ( ps -> A. x ps ) $.
    equsalh.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( An equivalence related to implicit substitution.  Usage of this theorem
       is discouraged because it depends on ~ ax-13 .  See ~ equsalhw for a
       version with a disjoint variable condition requiring fewer axioms.
       (Contributed by NM, 2-Jun-1993.)  (New usage is discouraged.) $)
    equsalh $p |- ( A. x ( x = y -> ph ) <-> ps ) $=
      ( nf5i equsal ) ABCDBCEGFH $.

    $( An equivalence related to implicit substitution.  Usage of this theorem
       is discouraged because it depends on ~ ax-13 .  See ~ equsexhv for a
       version with a disjoint variable condition which does not require
       ~ ax-13 .  (Contributed by NM, 5-Aug-1993.)
       (New usage is discouraged.) $)
    equsexh $p |- ( E. x ( x = y /\ ph ) <-> ps ) $=
      ( nf5i equsex ) ABCDBCEGFH $.
  $}

  ${
    $d x z $.  $d y z $.  $d z ph $.
    $( Derivation of set.mm's original ~ ax-c15 from ~ ax-c11n and the shorter
       ~ ax-12 that has replaced it.

       Theorem ~ ax12 shows the reverse derivation of ~ ax-12 from ~ ax-c15 .

       Normally, ~ axc15 should be used rather than ~ ax-c15 , except by
       theorems specifically studying the latter's properties.  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  (Contributed by
       NM, 2-Feb-2007.)  (Proof shortened by Wolf Lammen, 26-Mar-2023.)
       (New usage is discouraged.) $)
    axc15 $p |- ( -. A. x x = y ->
                 ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) ) $=
      ( vz weq wal wn wex ax6ev dveeq2 ax12v equeuclr sps imim1d al2imi imim12d
      wi imim2d syl6mpi exlimdv mpi ) BCEZBFGZDCEZDHUBAUBAQZBFZQZQZDCIUCUDUHDUC
      UDUDBFZBDEZAUJAQZBFZQZQUHBCDJABDKUIUBUJUMUGUDUBUJQBDBCLZMUIULUFAUDUKUEBUD
      UBUJAUNNORPSTUA $.
  $}

  $( Rederivation of Axiom ~ ax-12 from ~ ax12v (used only via ~ sp ),
     ~ axc11r , and ~ axc15 (on top of Tarski's FOL).  Since this version
     depends on ~ ax-13 , usage of the weaker ~ ax12v , ~ ax12w , ~ ax12i are
     preferred.  (Contributed by NM, 22-Jan-2007.)  Proof uses contemporary
     axioms.  (Revised by Wolf Lammen, 8-Aug-2020.)  (Proof shortened by BJ,
     4-Jul-2021.)  (New usage is discouraged.) $)
  ax12 $p |- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) ) $=
    ( weq wal wi axc11r ala1 syl6 a1d wn sp axc15 syl7 pm2.61i ) BCDZBEZPACEZPA
    FBEZFZFQTPQRABESACBGAPBHIJRAQKPSACLABCMNO $.

  $( A bidirectional version of ~ axc15 .  Usage of this theorem is discouraged
     because it depends on ~ ax-13 .  (Contributed by NM, 30-Jun-2006.)
     (New usage is discouraged.) $)
  ax12b $p |- ( ( -. A. x x = y /\ x = y ) ->
              ( ph <-> A. x ( x = y -> ph ) ) ) $=
    ( weq wal wn wa wi axc15 imp sp com12 adantl impbid ) BCDZBEFZOGAOAHZBEZPOA
    RHABCIJORAHPROAQBKLMN $.

  $( Alternate proof of ~ ax13 from FOL, ~ sp , and ~ axc9 .  (Contributed by
     NM, 21-Dec-2015.)  (Proof shortened by Wolf Lammen, 31-Jan-2018.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  ax13ALT $p |- ( -. x = y -> ( y = z -> A. x y = z ) ) $=
    ( weq wn wal wi sp con3i axc9 syl2im ax13b mpbir ) ABDZEZBCDZPAFZGZGOACDZEZ
    RGGONAFZETSAFZERUANNAHIUBSSAHIBCAJKQABCLM $.

  ${
    $d x z $.  $d y z $.
    $( Derive set.mm's original ~ ax-c11n from others.  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).  If a disjoint variable
       condition is added on ` x ` and ` y ` , then this becomes an instance of
       ~ aevlem .  Use ~ aecom instead when this does not lengthen the proof.
       Usage of this theorem is discouraged because it depends on ~ ax-13 .
       (Contributed by NM, 10-May-1993.)  (Revised by NM, 7-Nov-2015.)  (Proof
       shortened by Wolf Lammen, 6-Mar-2018.)  (Revised by Wolf Lammen,
       30-Nov-2019.)  (Proof shortened by BJ, 29-Mar-2021.)  (Proof shortened
       by Wolf Lammen, 2-Jul-2021.)  (New usage is discouraged.) $)
    axc11n $p |- ( A. x x = y -> A. y y = x ) $=
      ( vz weq wal wn dveeq1 com12 axc11r aev syl6 syl9 ax6evr exlimiiv pm2.18d
      wi ) ABDAEZBADBEZACDZQRFZRPPCSTSBEZQRTSUABACGHQUASAERSBAIACBABJKLCAMNO $.
  $}

  $( Commutation law for identical variable specifiers.  Both sides of the
     biconditional are true when ` x ` and ` y ` are substituted with the same
     variable.  Usage of this theorem is discouraged because it depends on
     ~ ax-13 .  (Contributed by NM, 10-May-1993.)  Change to a biconditional.
     (Revised by BJ, 26-Sep-2019.)  (New usage is discouraged.) $)
  aecom $p |- ( A. x x = y <-> A. y y = x ) $=
    ( weq wal axc11n impbii ) ABCADBACBDABEBAEF $.

  ${
    aecoms.1 $e |- ( A. x x = y -> ph ) $.
    $( A commutation rule for identical variable specifiers.  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  (Contributed by
       NM, 10-May-1993.)  (New usage is discouraged.) $)
    aecoms $p |- ( A. y y = x -> ph ) $=
      ( weq wal aecom sylbi ) CBECFBCEBFACBGDH $.
  $}

  ${
    naecoms.1 $e |- ( -. A. x x = y -> ph ) $.
    $( A commutation rule for distinct variable specifiers.  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  (Contributed by
       NM, 2-Jan-2002.)  (New usage is discouraged.) $)
    naecoms $p |- ( -. A. y y = x -> ph ) $=
      ( weq wal aecom sylnbir ) CBECFBCEBFABCGDH $.
  $}

  $( Show that ~ ax-c11 can be derived from ~ ax-c11n in the form of ~ axc11n .
     Normally, ~ axc11 should be used rather than ~ ax-c11 , except by theorems
     specifically studying the latter's properties.  Usage of this theorem is
     discouraged because it depends on ~ ax-13 .  Use the weaker ~ axc11v when
     possible.  (Contributed by NM, 16-May-2008.)  (Proof shortened by Wolf
     Lammen, 21-Apr-2018.)  (New usage is discouraged.) $)
  axc11 $p |- ( A. x x = y -> ( A. x ph -> A. y ph ) ) $=
    ( wal wi axc11r aecoms ) ABDACDECBABCFG $.

  $( All variables are effectively bound in an identical variable specifier.
     Usage of this theorem is discouraged because it depends on ~ ax-13 .  Use
     the weaker ~ hbaev when possible.  (Contributed by NM, 13-May-1993.)
     (Proof shortened by Wolf Lammen, 21-Apr-2018.)
     (New usage is discouraged.) $)
  hbae $p |- ( A. x x = y -> A. z A. x x = y ) $=
    ( weq wal wi wn sp axc9 syl7 axc11r axc11 pm2.43i syl5 pm2.61ii axc4i ax-11
    syl ) ABDZAEZSCEZAETCESUAACADCEZCBDCEZTUAFTSUBGUCGUASAHABCIJSACKTSBEZUCUATU
    DSABLMSBCKNOPSACQR $.

  $( All variables are effectively bound in a distinct variable specifier.
     Lemma L19 in [Megill] p. 446 (p. 14 of the preprint).  Usage of this
     theorem is discouraged because it depends on ~ ax-13 .  Use the weaker
     ~ hbnaev when possible.  (Contributed by NM, 13-May-1993.)
     (New usage is discouraged.) $)
  hbnae $p |- ( -. A. x x = y -> A. z -. A. x x = y ) $=
    ( weq wal hbae hbn ) ABDAECABCFG $.

  $( All variables are effectively bound in an identical variable specifier.
     Usage of this theorem is discouraged because it depends on ~ ax-13 .
     (Contributed by Mario Carneiro, 11-Aug-2016.)
     (New usage is discouraged.) $)
  nfae $p |- F/ z A. x x = y $=
    ( weq wal hbae nf5i ) ABDAECABCFG $.

  $( All variables are effectively bound in a distinct variable specifier.
     Usage of this theorem is discouraged because it depends on ~ ax-13 .  Use
     the weaker ~ nfnaew when possible.  (Contributed by Mario Carneiro,
     11-Aug-2016.)  (New usage is discouraged.) $)
  nfnae $p |- F/ z -. A. x x = y $=
    ( weq wal nfae nfn ) ABDAECABCFG $.

  ${
    hbnaes.1 $e |- ( A. z -. A. x x = y -> ph ) $.
    $( Rule that applies ~ hbnae to antecedent.  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  (Contributed by NM,
       15-May-1993.)  (New usage is discouraged.) $)
    hbnaes $p |- ( -. A. x x = y -> ph ) $=
      ( weq wal wn hbnae syl ) BCFBGHZKDGABCDIEJ $.
  $}

  ${
    $d x y z $.  $d z ph $.
    axc16i.1 $e |- ( x = z -> ( ph <-> ps ) ) $.
    axc16i.2 $e |- ( ps -> A. x ps ) $.
    $( Inference with ~ axc16 as its conclusion.  (Contributed by NM,
       20-May-2008.)  (Proof modification is discouraged.)  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  Use ~ axc16
       instead.  (New usage is discouraged.) $)
    axc16i $p |- ( A. x x = y -> ( ph -> A. x ph ) ) $=
      ( weq wal wi nfv ax7 cbv3 spimvw equcomi syl syl5com alimdv mpcom alimi
      biimpcd nf5i biimprd syl6com 3syl ) CDHZCIEDHZEIZCEHZEIZAACIZJUFUGCEUFEKU
      GCKCEDLMUHECHZEIZUJUFUHUMUGUFECECDLNUFUGULEUFDCHZUGULCDOUGDEHUNULJEDODECL
      PQRSULUIEECOZTPAUJBEIUKAUIBEUIABFUARBAECBCGUBAEKULUIBAJUOUIABFUCPMUDUE $.
  $}

  ${
    $d x y $.
    $( Alternate proof of ~ axc16nf , shorter but requiring ~ ax-11 and
       ~ ax-13 .  (Contributed by Mario Carneiro, 7-Oct-2016.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    axc16nfALT $p |- ( A. x x = y -> F/ z ph ) $=
      ( weq wal nfae axc16g nf5d ) BCEBFADBCDGABCDHI $.
  $}

  ${
    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).  Usage of
       this theorem is discouraged because it depends on ~ ax-13 .  Usage of
       ~ albidv is preferred, which requires fewer axioms.  (Contributed by NM,
       27-Feb-2005.)  Allow a shortening of ~ dral1 .  (Revised by Wolf Lammen,
       4-Mar-2018.)  (New usage is discouraged.) $)
    dral2 $p |- ( A. x x = y -> ( A. z ph <-> A. z ps ) ) $=
      ( weq wal nfae albid ) CDGCHABECDEIFJ $.

    $( Formula-building lemma for use with the Distinctor Reduction Theorem.
       Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint).  Usage of
       this theorem is discouraged because it depends on ~ ax-13 .  Use the
       weaker ~ dral1v if possible.  (Contributed by NM, 24-Nov-1994.)  Remove
       dependency on ~ ax-11 .  (Revised by Wolf Lammen, 6-Sep-2018.)
       (New usage is discouraged.) $)
    dral1 $p |- ( A. x x = y -> ( A. x ph <-> A. y ps ) ) $=
      ( weq wal nfa1 albid axc11 axc11r impbid bitrd ) CDFZCGZACGBCGZBDGZOABCNC
      HEIOPQBCDJBDCKLM $.

    $( Alternate proof of ~ dral1 , shorter but requiring ~ ax-11 .
       (Contributed by NM, 24-Nov-1994.)  (Proof shortened by Wolf Lammen,
       22-Apr-2018.)  (New usage is discouraged.)
       (Proof modification is discouraged.) $)
    dral1ALT $p |- ( A. x x = y -> ( A. x ph <-> A. y ps ) ) $=
      ( weq wal dral2 axc11 axc11r impbid bitrd ) CDFCGZACGBCGZBDGZABCDCEHMNOBC
      DIBDCJKL $.

    $( Formula-building lemma for use with the Distinctor Reduction Theorem.
       Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint).  Usage of
       this theorem is discouraged because it depends on ~ ax-13 .  Use the
       weaker ~ drex1v if possible.  (Contributed by NM, 27-Feb-2005.)
       (New usage is discouraged.) $)
    drex1 $p |- ( A. x x = y -> ( E. x ph <-> E. y ps ) ) $=
      ( weq wal wn wex notbid dral1 df-ex 3bitr4g ) CDFCGZAHZCGZHBHZDGZHACIBDIN
      PROQCDNABEJKJACLBDLM $.

    $( Formula-building lemma for use with the Distinctor Reduction Theorem.
       Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint).  Usage of
       this theorem is discouraged because it depends on ~ ax-13 .  Usage of
       ~ exbidv is preferred, which requires fewer axioms.  (Contributed by NM,
       27-Feb-2005.)  (New usage is discouraged.) $)
    drex2 $p |- ( A. x x = y -> ( E. z ph <-> E. z ps ) ) $=
      ( weq wal nfae exbid ) CDGCHABECDEIFJ $.

    $( Formula-building lemma for use with the Distinctor Reduction Theorem.
       (Contributed by Mario Carneiro, 4-Oct-2016.)  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  Use ~ drnf1v instead.
       (New usage is discouraged.) $)
    drnf1 $p |- ( A. x x = y -> ( F/ x ph <-> F/ y ps ) ) $=
      ( weq wal wi wnf dral1 imbi12d nf5 3bitr4g ) CDFCGZAACGZHZCGBBDGZHZDGACIB
      DIPRCDNABOQEABCDEJKJACLBDLM $.

    $( Formula-building lemma for use with the Distinctor Reduction Theorem.
       (Contributed by Mario Carneiro, 4-Oct-2016.)  (Proof shortened by Wolf
       Lammen, 5-May-2018.)  Usage of this theorem is discouraged because it
       depends on ~ ax-13 .  Use ~ nfbidv instead.
       (New usage is discouraged.) $)
    drnf2 $p |- ( A. x x = y -> ( F/ z ph <-> F/ z ps ) ) $=
      ( weq wal nfae nfbidf ) CDGCHABECDEIFJ $.
  $}

  ${
    nfald2.1 $e |- F/ y ph $.
    nfald2.2 $e |- ( ( ph /\ -. A. x x = y ) -> F/ x ps ) $.
    $( Variation on ~ nfald which adds the hypothesis that ` x ` and ` y ` are
       distinct in the inner subproof.  (Contributed by Mario Carneiro,
       8-Oct-2016.)  Usage of this theorem is discouraged because it depends on
       ~ ax-13 .  Use ~ nfald instead.  (New usage is discouraged.) $)
    nfald2 $p |- ( ph -> F/ x A. y ps ) $=
      ( weq wal wnf wn wa nfnae nfan nfald ex nfa1 biidd drnf1 mpbiri pm2.61d2
      ) ACDGCHZBDHZCIZAUAJZUCAUDKBCDAUDDECDDLMFNOUAUCUBDIBDPUBUBCDUAUBQRST $.

    $( Variation on ~ nfexd which adds the hypothesis that ` x ` and ` y ` are
       distinct in the inner subproof.  (Contributed by Mario Carneiro,
       8-Oct-2016.)  Usage of this theorem is discouraged because it depends on
       ~ ax-13 .  Use ~ nfexd instead.  (New usage is discouraged.) $)
    nfexd2 $p |- ( ph -> F/ x E. y ps ) $=
      ( wex wn wal df-ex weq wa nfnd nfald2 nfxfrd ) BDGBHZDIZHACBDJAQCAPCDEACD
      KCIHLBCFMNMO $.
  $}

  ${
    exdistrf.1 $e |- ( -. A. x x = y -> 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 Mario Carneiro, 20-Mar-2013.)  (Proof
       shortened by Wolf Lammen, 14-May-2018.)  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  Use ~ exdistr instead.
       (New usage is discouraged.) $)
    exdistrf $p |- ( E. x E. y ( ph /\ ps ) -> E. x ( ph /\ E. y ps ) ) $=
      ( wa wex nfe1 weq wi 19.8a anim2i eximi biidd drex1 imbitrrid 19.40 19.9d
      wal wn anim1d syl56 pm2.61i exlimi ) ABFZDGZABDGZFZCGZCUHCHCDICSZUFUIJUFU
      IUJUHDGUEUHDBUGABDKLMUHUHCDUJUHNOPUFADGZUGFUJTZUHUIABDQULUKAUGAULDERUAUHC
      KUBUCUD $.
  $}

  ${
    dvelimf.1 $e |- F/ x ph $.
    dvelimf.2 $e |- F/ z ps $.
    dvelimf.3 $e |- ( z = y -> ( ph <-> ps ) ) $.
    $( Version of ~ dvelimv without any variable restrictions.  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  (Contributed by
       NM, 1-Oct-2002.)  (Revised by Mario Carneiro, 6-Oct-2016.)  (Proof
       shortened by Wolf Lammen, 11-May-2018.)  (New usage is discouraged.) $)
    dvelimf $p |- ( -. A. x x = y -> F/ x ps ) $=
      ( weq wi wal wn equsal bicomi nfnae wa wnf nfeqf ancoms a1i nfald2 nfxfrd
      nfimd ) BEDIZAJZEKZCDICKLZCUFBABEDGHMNUGUECECDEOUGCEICKLZPZUDACUHUGUDCQED
      CRSACQUIFTUCUAUB $.
  $}

  ${
    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 .  Usage of this theorem is discouraged
       because it depends on ~ ax-13 .  (Contributed by NM, 7-Apr-2004.)
       (Revised by Mario Carneiro, 6-Oct-2016.)  (Proof shortened by Wolf
       Lammen, 11-May-2018.)  (New usage is discouraged.) $)
    dvelimdf $p |- ( ph -> ( -. A. x x = y -> F/ x ch ) ) $=
      ( weq wal wn wi wnf nfim1 wb com12 pm5.74d dvelimf pm5.5 nfbidf imbitrid
      ) DELDMNACOZDPACDPABOUEDEFABDGIQACFHJQFELZABCAUFBCRKSTUAAUECDGACUBUCUD $.
  $}

  ${
    dvelimh.1 $e |- ( ph -> A. x ph ) $.
    dvelimh.2 $e |- ( ps -> A. z ps ) $.
    dvelimh.3 $e |- ( z = y -> ( ph <-> ps ) ) $.
    $( Version of ~ dvelim without any variable restrictions.  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  Check out
       ~ dvelimhw for a version requiring fewer axioms.  (Contributed by NM,
       1-Oct-2002.)  (Proof shortened by Wolf Lammen, 11-May-2018.)
       (New usage is discouraged.) $)
    dvelimh $p |- ( -. A. x x = y -> ( ps -> A. x ps ) ) $=
      ( weq wal wn nf5i dvelimf nf5rd ) CDICJKBCABCDEACFLBEGLHMN $.
  $}

  ${
    $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 do not require that ` x ` and ` y ` be distinct: if
       they are not, 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 ~ dvelimh (with no distinct variable
       restrictions) and ~ dvelimhw (that avoids ~ ax-13 ).  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  Check out
       ~ dvelimhw for a version requiring fewer axioms.  (Contributed by NM,
       23-Nov-1994.)  (New usage is discouraged.) $)
    dvelim $p |- ( -. A. x x = y -> ( ps -> A. x ps ) ) $=
      ( ax-5 dvelimh ) ABCDEFBEHGI $.
  $}

  ${
    $d x ph $.  $d z ps $.
    dvelimv.1 $e |- ( z = y -> ( ph <-> ps ) ) $.
    $( Similar to ~ dvelim with first hypothesis replaced by a distinct
       variable condition.  Usage of this theorem is discouraged because it
       depends on ~ ax-13 .  Check out ~ dvelimhw for a version requiring fewer
       axioms.  (Contributed by NM, 25-Jul-2015.)  (Proof shortened by Wolf
       Lammen, 30-Apr-2018.)  (New usage is discouraged.) $)
    dvelimv $p |- ( -. A. x x = y -> ( ps -> A. x ps ) ) $=
      ( ax-5 dvelim ) ABCDEACGFH $.
  $}

  ${
    $d z ps $.
    dvelimnf.1 $e |- F/ x ph $.
    dvelimnf.2 $e |- ( z = y -> ( ph <-> ps ) ) $.
    $( Version of ~ dvelim using "not free" notation.  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  (Contributed by Mario
       Carneiro, 9-Oct-2016.)  (New usage is discouraged.) $)
    dvelimnf $p |- ( -. A. x x = y -> F/ x ps ) $=
      ( nfv dvelimf ) ABCDEFBEHGI $.
  $}

  ${
    $d w x z $.  $d w y $.
    $( Alternate proof of ~ dveeq2 , shorter but requiring ~ ax-11 .
       (Contributed by NM, 2-Jan-2002.)  (Revised by NM, 20-Jul-2015.)
       (New usage is discouraged.)  (Proof modification is discouraged.) $)
    dveeq2ALT $p |- ( -. A. x x = y -> ( z = y -> A. x z = y ) ) $=
      ( vw weq equequ2 dvelimv ) CDECBEABDDBCFG $.
  $}

  $( 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 ` .
     Usage of this theorem is discouraged because it depends on ~ ax-13 .  See
     ~ equvinv for a shorter proof requiring fewer axioms when ` z ` is
     required to be distinct from ` x ` and ` y ` .  (Contributed by NM,
     10-Jan-1993.)  (Proof shortened by Andrew Salmon, 25-May-2011.)  (Proof
     shortened by Wolf Lammen, 16-Sep-2023.)  (New usage is discouraged.) $)
  equvini $p |- ( x = y -> E. z ( x = z /\ z = y ) ) $=
    ( weq wa wex wi equtr equcomi jctild 19.8a syl6 wal ax13 ax6e eximii 19.35i
    wn pm2.61i ) CADZABDZACDZCBDZEZCFZGTUAUDUETUAUCUBCABHCAIJZUDCKLTRUAUACMUECA
    BNUAUDCTUAUDGCCAOUFPQLS $.

  $( A variable elimination law for equality with no distinct variable
     requirements.  Compare ~ equvini .  Usage of this theorem is discouraged
     because it depends on ~ ax-13 .  Use ~ equvelv when possible.
     (Contributed by NM, 1-Mar-2013.)  (Proof shortened by Mario Carneiro,
     17-Oct-2016.)  (Proof shortened by Wolf Lammen, 15-Jun-2019.)
     (New usage is discouraged.) $)
  equvel $p |- ( A. z ( z = x <-> z = y ) -> x = y ) $=
    ( weq wb wal wex albi wi ax6e biimpr ax7 syli com12 eximii 19.35i spsd a1dd
    sps wn wa nfeqf 19.9d ex bija sylc ) CADZCBDZEZCFUGCFZUHCFZEABDZCGZULUGUHCH
    UIULCUHUIULICCBJUIUHULUHUIUGULUGUHKCABLZMNOPUJUKUMULIZUJUKULUMUGUKULICUGUHU
    LCUNQSRUJTZUKTZUOULUPUQUACABCUBUCUDUEUF $.

  $( A property related to substitution that unlike ~ equs5 does not require a
     distinctor antecedent.  Usage of this theorem is discouraged because it
     depends on ~ ax-13 .  This proof uses ~ ax12 , see ~ equs5aALT for an
     alternative one using ~ ax-12 but not ~ ax13 .  Usage of the weaker
     ~ equs5av is preferred, which uses ~ ax12v2 , but not ~ ax-13 .
     (Contributed by NM, 2-Feb-2007.)  (New usage is discouraged.) $)
  equs5a $p |- ( E. x ( x = y /\ A. y ph ) -> A. x ( x = y -> ph ) ) $=
    ( weq wal wa wi nfa1 ax12 imp exlimi ) BCDZACEZFLAGZBEZBNBHLMOABCIJK $.

  $( A property related to substitution that unlike ~ equs5 does not require a
     distinctor antecedent.  This proof uses ~ ax12 , see ~ equs5eALT for an
     alternative one using ~ ax-12 but not ~ ax13 .  Usage of this theorem is
     discouraged because it depends on ~ ax-13 .  (Contributed by NM,
     2-Feb-2007.)  (Proof shortened by Wolf Lammen, 15-Jan-2018.)
     (New usage is discouraged.) $)
  equs5e $p |- ( E. x ( x = y /\ ph ) -> A. x ( x = y -> E. y ph ) ) $=
    ( weq wa wex wi wal nfa1 ax12 hbe1 19.23bi impel exlimi ) BCDZAEOACFZGZBHZB
    QBIOPCHZRAPBCJASCACKLMN $.

  ${
    equs45f.1 $e |- F/ y ph $.
    $( Two ways of expressing substitution when ` y ` is not free in ` ph ` .
       The implication "to the left" is ~ equs4 and does not require the
       nonfreeness hypothesis.  Theorem ~ sbalex replaces the nonfreeness
       hypothesis with a disjoint variable condition and ~ equs5 replaces it
       with a distinctor antecedent.  (Contributed by NM, 25-Apr-2008.)
       (Revised by Mario Carneiro, 4-Oct-2016.)  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  Use ~ sbalex instead.
       (New usage is discouraged.) $)
    equs45f $p |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) ) $=
      ( weq wa wex wi wal nf5ri anim2i eximi equs5a syl equs4 impbii ) BCEZAFZB
      GZQAHBIZSQACIZFZBGTRUBBAUAQACDJKLABCMNABCOP $.
  $}

  $( Lemma used in proofs of substitution properties.  If there is a disjoint
     variable condition on ` x , y ` , then ~ sbalex can be used instead; if
     ` y ` is not free in ` ph ` , then ~ equs45f can be used.  Usage of this
     theorem is discouraged because it depends on ~ ax-13 .  (Contributed by
     NM, 14-May-1993.)  (Revised by BJ, 1-Oct-2018.)
     (New usage is discouraged.) $)
  equs5 $p |- ( -. A. x x = y ->
             ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) ) ) $=
    ( weq wal wn wa wex wi nfna1 nfa1 axc15 impd exlimd equs4 impbid1 ) BCDZBEF
    ZQAGZBHQAIZBEZRSUABQBJTBKRQAUAABCLMNABCOP $.

  ${
    $d w z x $.  $d w y $.
    $( Quantifier introduction when one pair of variables is disjoint.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .
       (Contributed by NM, 2-Jan-2002.)  (New usage is discouraged.) $)
    dveel1 $p |- ( -. A. x x = y -> ( y e. z -> A. x y e. z ) ) $=
      ( vw wel elequ1 dvelimv ) DCEBCEABDDBCFG $.

    $( Quantifier introduction when one pair of variables is disjoint.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .
       (Contributed by NM, 2-Jan-2002.)  (New usage is discouraged.) $)
    dveel2 $p |- ( -. A. x x = y -> ( z e. y -> A. x z e. y ) ) $=
      ( vw wel elequ2 dvelimv ) CDECBEABDDBCFG $.
  $}

  ${
    $d w y $.  $d w z $.  $d w x $.
    $( Axiom ~ ax-c14 is redundant if we assume ~ ax-5 .  Remark 9.6 in
       [Megill] p. 448 (p. 16 of the preprint), regarding axiom scheme C14'.

       Note that ` w ` is a dummy variable introduced in the proof.  Its
       purpose is to satisfy the distinct variable requirements of ~ dveel2 and
       ~ ax-5 .  By the end of the proof it has vanished, and the final theorem
       has no distinct variable requirements.  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  (Contributed by NM,
       29-Jun-1995.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    axc14 $p |- ( -. A. z z = x -> ( -. A. z z = y ->
              ( x e. y -> A. z x e. y ) ) ) $=
      ( vw weq wal wn wel hbn1 dveel2 hbim1 elequ1 imbi2d dvelim nfa1 nfn 19.21
      wi imbitrdi pm2.86d ) CAECFGZCBEZCFZGZABHZUECFZUAUDUERZUGCFUDUFRUDDBHZRUG
      CADUDUHCUBCICBDJKDAEUHUEUDDABLMNUDUECUCCUBCOPQST $.
  $}

  ${
    sb6x.1 $e |- F/ x ph $.
    $( Equivalence involving substitution for a variable not free.  Usage of
       this theorem is discouraged because it depends on ~ ax-13 .  Usage of
       ~ sb6 is preferred, which requires fewer axioms.  (Contributed by NM,
       2-Jun-1993.)  (Revised by Mario Carneiro, 4-Oct-2016.)
       (New usage is discouraged.) $)
    sb6x $p |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) $=
      ( wsb weq wi wal sbf biidd equsal bitr4i ) ABCEABCFZAGBHABCDIAABCDMAJKL
      $.
  $}

  $( Substitution does not change an identical variable specifier.  Usage of
     this theorem is discouraged because it depends on ~ ax-13 .  (Contributed
     by NM, 15-May-1993.)  (New usage is discouraged.) $)
  sbequ5 $p |- ( [ w / z ] A. x x = y <-> A. x x = y ) $=
    ( weq wal nfae sbf ) ABEAFCDABCGH $.

  $( Substitution does not change a distinctor.  Usage of this theorem is
     discouraged because it depends on ~ ax-13 .  (Contributed by NM,
     5-Aug-1993.)  (New usage is discouraged.) $)
  sbequ6 $p |- ( [ w / z ] -. A. x x = y <-> -. A. x x = y ) $=
    ( weq wal wn nfnae sbf ) ABEAFGCDABCHI $.

  ${
    sb5rf.1 $e |- F/ y ph $.
    $( Reversed substitution.  Usage of this theorem is discouraged because it
       depends on ~ ax-13 .  (Contributed by NM, 3-Feb-2005.)  (Revised by
       Mario Carneiro, 6-Oct-2016.)  (Proof shortened by Wolf Lammen,
       20-Sep-2018.)  (New usage is discouraged.) $)
    sb5rf $p |- ( ph <-> E. y ( y = x /\ [ y / x ] ph ) ) $=
      ( weq wsb wa wex sbequ12r equsex bicomi ) CBEABCFZGCHALACBDACBIJK $.

    $( Reversed substitution.  For a version requiring disjoint variables, but
       fewer axioms, see ~ sb6rfv .  Usage of this theorem is discouraged
       because it depends on ~ ax-13 .  Use the weaker ~ sb6rfv if possible.
       (Contributed by NM, 1-Aug-1993.)  (Revised by Mario Carneiro,
       6-Oct-2016.)  (Proof shortened by Wolf Lammen, 21-Sep-2018.)
       (New usage is discouraged.) $)
    sb6rf $p |- ( ph <-> A. y ( y = x -> [ y / x ] ph ) ) $=
      ( weq wsb wi wal sbequ12r equsal bicomi ) CBEABCFZGCHALACBDACBIJK $.
  $}

  ${
    $d x y $.
    $( Alternate proof of ~ ax12v2 , shorter, but depending on more axioms.
       (Contributed by NM, 5-Aug-1993.)  (New usage is discouraged.)
       (Proof modification is discouraged.) $)
    ax12vALT $p |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) $=
      ( weq wal wi ax-1 axc16 syl5 a1d axc15 pm2.61i ) BCDZBEZMAMAFZBEZFZFNQMAO
      NPAMGOBCHIJABCKL $.
  $}

  $( We can always find values matching ` x ` and ` y ` , as long as they are
     represented by distinct variables.  This theorem merges two ~ ax6e
     instances ` E. z z = x ` and ` E. w w = y ` into a common expression.
     Alan Sare contributed a variant of this theorem with distinct variable
     conditions before, see ~ ax6e2nd .  Usage of this theorem is discouraged
     because it depends on ~ ax-13 .  (Contributed by Wolf Lammen,
     27-Sep-2018.)  (New usage is discouraged.) $)
  2ax6elem $p |- ( -. A. w w = z -> E. z E. w ( z = x /\ w = y ) ) $=
    ( weq wal wn wex ax6e nfnae nfan nfeqf pm3.21 spimed eximd mpi nfae equvini
    wa ex equtrr anim1d aleximi syl5 pm2.61d2 ) DCEDFGZDAEZDFZCAEZDBEZSZDHZCHZU
    FUHGZUMUFUNSZUICHUMCAIUOUIULCUFUNCDCCJDACJKUIUKUODBCADLUJUIMNOPTUHCBEZCHUMC
    BIUHUPULCDACQUPCDEZUJSZDHUHULCBDRUGURUKDUGUQUIUJDACUAUBUCUDOPUE $.

  ${
    $d w z $.
    $( We can always find values matching ` x ` and ` y ` , as long as they are
       represented by distinct variables.  Version of ~ 2ax6elem with a
       distinct variable constraint.  Usage of this theorem is discouraged
       because it depends on ~ ax-13 .  (Contributed by Wolf Lammen,
       28-Sep-2018.)  (Proof shortened by Wolf Lammen, 3-Oct-2023.)
       (New usage is discouraged.) $)
    2ax6e $p |- E. z E. w ( z = x /\ w = y ) $=
      ( weq wal wa wex aeveq jca 19.8ad 2ax6elem pm2.61i ) DCEDFZCAEZDBEZGZDHZC
      HNRCNQDNOPDCCAIDCDBIJKKABCDLM $.
  $}

  ${
    $d w z $.
    2sb5rf.1 $e |- F/ z ph $.
    2sb5rf.2 $e |- F/ w ph $.
    $( Reversed double substitution.  Usage of this theorem is discouraged
       because it depends on ~ ax-13 .  (Contributed by NM, 3-Feb-2005.)
       (Revised by Mario Carneiro, 6-Oct-2016.)  Remove distinct variable
       constraints.  (Revised by Wolf Lammen, 28-Sep-2018.)
       (New usage is discouraged.) $)
    2sb5rf $p |- ( ph <->
                E. z E. w ( ( z = x /\ w = y ) /\ [ z / x ] [ w / y ] ph ) ) $=
      ( weq wa wex 19.41 exbii 2ax6e biantrur 3bitr4ri sbequ12r sylan9bb 2exbii
      wsb pm5.32i bitr4i ) ADBHZECHZIZAIZEJZDJZUDACESZBDSZIZEJDJUDEJZAIZDJUKDJZ
      AIUGAUKADFKUFULDUDAEGKLUMABCDEMNOUJUEDEUDUIAUBUIUHUCAUHDBPAECPQTRUA $.

    $( Reversed double substitution.  Usage of this theorem is discouraged
       because it depends on ~ ax-13 .  (Contributed by NM, 3-Feb-2005.)
       (Revised by Mario Carneiro, 6-Oct-2016.)  Remove variable constraints.
       (Revised by Wolf Lammen, 28-Sep-2018.)  (Proof shortened by Wolf Lammen,
       13-Apr-2023.)  (New usage is discouraged.) $)
    2sb6rf $p |- ( ph <->
                A. z A. w ( ( z = x /\ w = y ) -> [ z / x ] [ w / y ] ph ) ) $=
      ( weq wa wi wal wsb wex 19.23 albii 2ax6e a1bi 3bitr4ri sbequ12r sylan9bb
      pm5.74i 2albii bitr4i ) ADBHZECHZIZAJZEKZDKZUFACELZBDLZJZEKDKUFEMZAJZDKUM
      DMZAJUIAUMADFNUHUNDUFAEGNOUOABCDEPQRULUGDEUFUKAUDUKUJUEAUJDBSAECSTUAUBUC
      $.
  $}

  ${
    $d x y ph $.
    $( Elimination of double substitution.  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  (Contributed by NM,
       5-Aug-1993.)  (Proof shortened by Wolf Lammen, 29-Sep-2018.)
       (New usage is discouraged.) $)
    sbel2x $p |- ( ph <-> E. x E. y ( ( x = z /\ y = w ) /\
                     [ y / w ] [ x / z ] ph ) ) $=
      ( weq wa wsb wex nfv 2sb5rf ancom anbi1i 2exbii excom 3bitri ) ACEFZBDFZG
      ZADBHECHZGZBICIRQGZTGZBICIUCCIBIAEDCBACJABJKUAUCCBSUBTQRLMNUCCBOP $.
  $}

  ${
    $d y x $.  $d y t $.  $d y ph $.
    $( Simplified definition of substitution when variables are distinct.
       Version of ~ sb6 with a distinctor antecedent.  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  (Contributed by NM,
       27-May-1997.)  Revise ~ df-sb .  (Revised by Wolf Lammen, 21-Feb-2024.)
       (New usage is discouraged.) $)
    sb4b $p |- ( -. A. x x = t -> ( [ t / x ] ph
                                              <-> A. x ( x = t -> ph ) ) ) $=
      ( vy weq wal wn wi wa nfna1 nfeqf2 nfan1 wb equequ2 imbi1d albid pm5.74da
      wsb adantl albidv df-sb wex ax6ev a1bi 19.23v bitr4i 3bitr4g ) BCEZBFGZDC
      EZBDEZAHZBFZHZDFUJUHAHZBFZHZDFZABCRUPUIUNUQDUIUJUMUPUIUJIULUOBUIUJBUHBJBC
      DKLUJULUOMUIUJUKUHADCBNOSPQTABDCUAUPUJDUBZUPHURUSUPDCUCUDUJUPDUEUFUG $.
  $}

  $( Simplified definition of substitution when variables are distinct.  This
     is the biconditional strengthening of ~ sb3 .  Usage of this theorem is
     discouraged because it depends on ~ ax-13 .  (Contributed by BJ,
     6-Oct-2018.)  Shorten ~ sb3 .  (Revised by Wolf Lammen, 21-Feb-2021.)
     (New usage is discouraged.) $)
  sb3b $p |- ( -. A. x x = y -> ( [ y / x ] ph <-> E. x ( x = y /\ ph ) ) ) $=
    ( weq wal wn wsb wi wa wex sb4b equs5 bitr4d ) BCDZBEFABCGNAHBENAIBJABCKABC
    LM $.

  $( One direction of a simplified definition of substitution when variables
     are distinct.  Usage of this theorem is discouraged because it depends on
     ~ ax-13 .  (Contributed by NM, 5-Aug-1993.)  (Proof shortened by Wolf
     Lammen, 21-Feb-2024.)  (New usage is discouraged.) $)
  sb3 $p |- ( -. A. x x = y -> ( E. x ( x = y /\ ph ) -> [ y / x ] ph ) ) $=
    ( weq wal wn wsb wa wex sb3b biimprd ) BCDZBEFABCGLAHBIABCJK $.

  $( One direction of a simplified definition of substitution.  The converse
     requires either a disjoint variable condition ( ~ sb5 ) or a nonfreeness
     hypothesis ( ~ sb5f ).  Usage of this theorem is discouraged because it
     depends on ~ ax-13 .  Use the weaker ~ sb1v when possible.  (Contributed
     by NM, 13-May-1993.)  Revise ~ df-sb .  (Revised by Wolf Lammen,
     21-Feb-2024.)  (New usage is discouraged.) $)
  sb1 $p |- ( [ y / x ] ph -> E. x ( x = y /\ ph ) ) $=
    ( weq wal wsb wa wex wi spsbe pm3.2 aleximi syl5 wn sb3b biimpd pm2.61i ) B
    CDZBEZABCFZRAGZBHZITABHSUBABCJRAUABRAKLMSNTUBABCOPQ $.

  $( One direction of a simplified definition of substitution.  The converse
     requires either a disjoint variable condition ( ~ sb6 ) or a nonfreeness
     hypothesis ( ~ sb6f ).  Usage of this theorem is discouraged because it
     depends on ~ ax-13 .  (Contributed by NM, 13-May-1993.)  Revise ~ df-sb .
     (Revised by Wolf Lammen, 26-Jul-2023.)  (New usage is discouraged.) $)
  sb2 $p |- ( A. x ( x = y -> ph ) -> [ y / x ] ph ) $=
    ( weq wal wi wsb pm2.27 al2imi stdpc4 syl6 wn sb4b biimprd pm2.61i ) BCDZBE
    ZPAFZBEZABCGZFQSABETPRABPAHIABCJKQLTSABCMNO $.

  $( A version of one implication of ~ sb4b that does not require a distinctor
     antecedent.  Usage of this theorem is discouraged because it depends on
     ~ ax-13 .  Use the weaker ~ sb4av when possible.  (Contributed by NM,
     2-Feb-2007.)  Revise ~ df-sb .  (Revised by Wolf Lammen, 28-Jul-2023.)
     (New usage is discouraged.) $)
  sb4a $p |- ( [ t / x ] A. t ph -> A. x ( x = t -> ph ) ) $=
    ( weq wal wsb wi sbequ2 sps axc11r ala1 syl6 syld wn sb4b sp alimi biimtrdi
    imim2i pm2.61i ) BCDZBEZACEZBCFZUAAGZBEZGUBUDUCUFUAUDUCGBUCBCHIUBUCABEUFACB
    JAUABKLMUBNUDUAUCGZBEUFUCBCOUGUEBUCAUAACPSQRT $.

  $( Alternate definition of substitution.  Remark 9.1 in [Megill] p. 447 (p.
     15 of the preprint).  This was the original definition before ~ df-sb .
     Note that it does not require dummy variables in its definiens; this is
     done by having ` x ` free in the first conjunct and bound in the second.
     Usage of this theorem is discouraged because it depends on ~ ax-13 .
     (Contributed by BJ, 9-Jul-2023.)  Revise ~ df-sb .  (Revised by Wolf
     Lammen, 29-Jul-2023.)  (New usage is discouraged.) $)
  dfsb1 $p |- ( [ y / x ] ph <->
                ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) ) $=
    ( wsb weq wi wa wex sbequ2 com12 sb1 jca wal sbequ1 embantd sps adantrd sb3
    id wn adantld pm2.61i impbii ) ABCDZBCEZAFZUEAGBHZGZUDUFUGUEUDAABCIJABCKLUE
    BMZUHUDFUIUFUDUGUEUFUDFBUEUEAUDUESABCNOPQUITUGUDUFABCRUAUBUC $.

  $( Bound-variable hypothesis builder for substitution.  Usage of this theorem
     is discouraged because it depends on ~ ax-13 .  (Contributed by NM,
     14-May-1993.)  (New usage is discouraged.) $)
  hbsb2 $p |- ( -. A. x x = y -> ( [ y / x ] ph -> A. x [ y / x ] ph ) ) $=
    ( weq wal wn wsb wi sb4b sb2 axc4i biimtrdi ) BCDZBEFABCGZMAHZBENBEABCIONBA
    BCJKL $.

  $( Bound-variable hypothesis builder for substitution.  Usage of this theorem
     is discouraged because it depends on ~ ax-13 .  (Contributed by Mario
     Carneiro, 4-Oct-2016.)  (New usage is discouraged.) $)
  nfsb2 $p |- ( -. A. x x = y -> F/ x [ y / x ] ph ) $=
    ( weq wal wn wsb nfna1 hbsb2 nf5d ) BCDZBEFABCGBKBHABCIJ $.

  $( Special case of a bound-variable hypothesis builder for substitution.
     Usage of this theorem is discouraged because it depends on ~ ax-13 .
     (Contributed by NM, 2-Feb-2007.)  (New usage is discouraged.) $)
  hbsb2a $p |- ( [ y / x ] A. y ph -> A. x [ y / x ] ph ) $=
    ( wal wsb weq wi sb4a sb2 axc4i syl ) ACDBCEBCFAGZBDABCEZBDABCHLMBABCIJK $.

  $( One direction of a simplified definition of substitution that unlike
     ~ sb4b does not require a distinctor antecedent.  Usage of this theorem is
     discouraged because it depends on ~ ax-13 .  (Contributed by NM,
     2-Feb-2007.)  (New usage is discouraged.) $)
  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.
     Usage of this theorem is discouraged because it depends on ~ ax-13 .
     (Contributed by NM, 2-Feb-2007.)  (New usage is discouraged.) $)
  hbsb2e $p |- ( [ y / x ] ph -> A. x [ y / x ] E. y ph ) $=
    ( wsb weq wex wi wal sb4e sb2 axc4i syl ) ABCDBCEACFZGZBHMBCDZBHABCINOBMBCJ
    KL $.

  ${
    hbsb3.1 $e |- ( ph -> A. y ph ) $.
    $( If ` y ` is not free in ` ph ` , ` x ` is not free in ` [ y / x ] ph ` .
       Usage of this theorem is discouraged because it depends on ~ ax-13 .
       Check out ~ bj-hbsb3v for a weaker version requiring fewer axioms.
       (Contributed by NM, 14-May-1993.)  (New usage is discouraged.) $)
    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 ` .
       Usage of this theorem is discouraged because it depends on ~ ax-13 .
       Check out ~ nfs1v for a version requiring fewer axioms.  (Contributed by
       Mario Carneiro, 11-Aug-2016.)  (New usage is discouraged.) $)
    nfs1 $p |- F/ x [ y / x ] ph $=
      ( wsb nf5ri hbsb3 nf5i ) ABCEBABCACDFGH $.
  $}

  ${
    $d x y z $.  $d z ph $.
    $( Alternate proof of ~ axc16 , shorter but requiring ~ ax-10 , ~ ax-11 ,
       ~ ax-13 and using ~ df-nf and ~ df-sb .  (Contributed by NM,
       17-May-2008.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    axc16ALT $p |- ( A. x x = y -> ( ph -> A. x ph ) ) $=
      ( vz wsb sbequ12 ax-5 hbsb3 axc16i ) AABDEBCDABDFABDADGHI $.
  $}

  ${
    $d x y $.
    $( Alternate proof of ~ axc16g that uses ~ df-sb and requires ~ ax-10 ,
       ~ ax-11 , ~ ax-13 .  (Contributed by NM, 15-May-1993.)  (Proof shortened
       by Andrew Salmon, 25-May-2011.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    axc16gALT $p |- ( A. x x = y -> ( ph -> A. z ph ) ) $=
      ( weq wal aev axc16ALT biidd dral1 biimprd sylsyld ) BCEBFDBEDFZAABFZADFZ
      BCDBDGABCHMONAADBMAIJKL $.
  $}

  $( Substitution applied to an atomic wff.  Usage of this theorem is
     discouraged because it depends on ~ ax-13 .  Use the weaker ~ equsb1v if
     possible.  (Contributed by NM, 10-May-1993.)
     (New usage is discouraged.) $)
  equsb1 $p |- [ y / x ] x = y $=
    ( weq wi wsb sb2 id mpg ) ABCZIDIABEAIABFIGH $.

  $( Substitution applied to an atomic wff.  Usage of this theorem is
     discouraged because it depends on ~ ax-13 .  Check out ~ equsb1v for a
     version requiring fewer axioms.  (Contributed by NM, 10-May-1993.)
     (New usage is discouraged.) $)
  equsb2 $p |- [ y / x ] y = x $=
    ( weq wi wsb sb2 equcomi mpg ) ABCBACZDIABEAIABFABGH $.

  $( An alternate definition of proper substitution that, like ~ dfsb1 , mixes
     free and bound variables to avoid distinct variable requirements.  Usage
     of this theorem is discouraged because it depends on ~ ax-13 .
     (Contributed by NM, 17-Feb-2005.)  (New usage is discouraged.) $)
  dfsb2 $p |- ( [ y / x ] ph <->
              ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) ) $=
    ( wsb weq wa wi wal wo sp sbequ2 sps orc syl6an wn sb4b olc biimtrdi sbequ1
    pm2.61i imp sb2 jaoi impbii ) ABCDZBCEZAFZUFAGBHZIZUFBHZUEUIGUJUFUEAUIUFBJU
    FUEAGBABCKLUGUHMNUJOUEUHUIABCPUHUGQRTUGUEUHUFAUEABCSUAABCUBUCUD $.

  $( An alternate definition of proper substitution ~ df-sb that uses only
     primitive connectives (no defined terms) on the right-hand side.  Usage of
     this theorem is discouraged because it depends on ~ ax-13 .  (Contributed
     by NM, 6-Mar-2007.)  (New usage is discouraged.) $)
  dfsb3 $p |- ( [ y / x ] ph <->
              ( ( x = y -> -. ph ) -> A. x ( x = y -> ph ) ) ) $=
    ( weq wa wi wal wo wn wsb df-or dfsb2 imnan imbi1i 3bitr4i ) BCDZAEZPAFBGZH
    QIZRFABCJPAIFZRFQRKABCLTSRPAMNO $.

  $( Formula-building lemma for use with the Distinctor Reduction Theorem.
     Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint).  Usage of this
     theorem is discouraged because it depends on ~ ax-13 .  (Contributed by
     NM, 2-Jun-1993.)  (New usage is discouraged.) $)
  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
    dfsb1 ) BCEZBFZBDEZAGZUBAHZBIZHCDEZAGZUFAHZCIZHABDJACDJUAUCUGUEUIUAUBUFATUB
    UFKBBCDLMZNUDUHBCUAUBUFAUJOPQABDSACDSR $.

  ${
    $d v y $.
    $( In the case of two successive substitutions for two always equal
       variables, the second substitution has no effect.  Usage of this theorem
       is discouraged because it depends on ~ ax-13 .  (Contributed by BJ and
       WL, 9-Aug-2023.)  (New usage is discouraged.) $)
    sb2ae $p |-
               ( A. x x = y -> ( [ u / x ] [ v / y ] ph <-> [ v / y ] ph ) ) $=
      ( weq wal wsb drsb1 nfs1v sbf bitrdi ) BCFBGACDHZBEHMCEHMMBCEIMCEACDJKL
      $.
  $}

  ${
    sb6f.1 $e |- F/ y ph $.
    $( Equivalence for substitution when ` y ` is not free in ` ph ` .  The
       implication "to the left" is ~ sb2 and does not require the nonfreeness
       hypothesis.  Theorem ~ sb6 replaces the nonfreeness hypothesis with a
       disjoint variable condition on ` x , y ` and requires fewer axioms.
       Usage of this theorem is discouraged because it depends on ~ ax-13 .
       (Contributed by NM, 2-Jun-1993.)  (Revised by Mario Carneiro,
       4-Oct-2016.)  (New usage is discouraged.) $)
    sb6f $p |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) $=
      ( wsb weq wi wal nf5ri sbimi sb4a syl sb2 impbii ) ABCEZBCFAGBHZOACHZBCEP
      AQBCACDIJABCKLABCMN $.

    $( Equivalence for substitution when ` y ` is not free in ` ph ` .  The
       implication "to the right" is ~ sb1 and does not require the nonfreeness
       hypothesis.  Theorem ~ sb5 replaces the nonfreeness hypothesis with a
       disjoint variable condition on ` x , y ` and requires fewer axioms.
       Usage of this theorem is discouraged because it depends on ~ ax-13 .
       (Contributed by NM, 5-Aug-1993.)  (Revised by Mario Carneiro,
       4-Oct-2016.)  (New usage is discouraged.) $)
    sb5f $p |- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) ) $=
      ( wsb weq wi wal wa wex sb6f equs45f bitr4i ) ABCEBCFZAGBHNAIBJABCDKABCDL
      M $.
  $}

  $( A variable not free in a proposition remains so after substitution in that
     proposition with a distinct variable (closed form of ~ nfsb4 ).  Usage of
     this theorem is discouraged because it depends on ~ ax-13 .  (Contributed
     by NM, 7-Apr-2004.)  (Revised by Mario Carneiro, 4-Oct-2016.)  (Proof
     shortened by Wolf Lammen, 11-May-2018.)  (New usage is discouraged.) $)
  nfsb4t $p |- ( A. x F/ z ph ->
                 ( -. A. z z = y -> F/ z [ y / x ] ph ) ) $=
    ( wnf wal weq wn wsb wi wa sbequ12 sps drnf2 biimpd spsd impcom nfnae nfan
    wb a1d nfnf1 nfal nfa1 sp adantr nfsb2 adantl a1i dvelimdf pm2.61dan ) ADEZ
    BFZBCGZBFZDCGDFHZABCIZDEZJUMUOKURUPUOUMURUOULURBUOULURAUQBCDUNAUQTZBABCLZMN
    OPQUAUMUOHZKZAUQDCBUMVADULDBADUBUCBCDRSUMVABULBUDBCBRSUMULVAULBUEUFVAUQBEUM
    ABCUGUHUNUSJVBUTUIUJUK $.

  ${
    nfsb4.1 $e |- F/ z ph $.
    $( A variable not free in a proposition remains so after substitution in
       that proposition with a distinct variable (inference associated with
       ~ nfsb4t ).  Theorem ~ nfsb replaces the distinctor antecedent with a
       disjoint variable condition.  See ~ nfsbv for a weaker version of ~ nfsb
       not requiring ~ ax-13 .  (Contributed by NM, 14-May-1993.)  (Revised by
       Mario Carneiro, 4-Oct-2016.)  Usage of this theorem is discouraged
       because it depends on ~ ax-13 .  Use ~ nfsbv instead.
       (New usage is discouraged.) $)
    nfsb4 $p |- ( -. A. z z = y -> F/ z [ y / x ] ph ) $=
      ( wnf weq wal wn wsb wi nfsb4t mpg ) ADFDCGDHIABCJDFKBABCDLEM $.
  $}

  $( Elimination of equality from antecedent after substitution.  Usage of this
     theorem is discouraged because it depends on ~ ax-13 .  (Contributed by
     NM, 5-Aug-1993.)  Reduce dependencies on axioms.  (Revised by Wolf Lammen,
     28-Jul-2018.)  Revise ~ df-sb .  (Revised by Wolf Lammen, 28-Jul-2023.)
     (New usage is discouraged.) $)
  sbequ8 $p |- ( [ y / x ] ph <-> [ y / x ] ( x = y -> ph ) ) $=
    ( wsb weq wi equsb1 a1bi sbim bitr4i ) ABCDZBCEZBCDZKFLAFBCDMKBCGHLABCIJ $.

  ${
    sbie.1 $e |- F/ x ps $.
    sbie.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Conversion of implicit substitution to explicit substitution.  For
       versions requiring disjoint variables, but fewer axioms, see ~ sbiev and
       ~ sbievw .  Usage of this theorem is discouraged because it depends on
       ~ ax-13 .  (Contributed by NM, 30-Jun-1994.)  (Revised by Mario
       Carneiro, 4-Oct-2016.)  (Proof shortened by Wolf Lammen, 13-Jul-2019.)
       (New usage is discouraged.) $)
    sbie $p |- ( [ y / x ] ph <-> ps ) $=
      ( wb wsb weq equsb1 sbimi ax-mp sbf sblbis mpbi ) ABGZCDHZACDHBGCDIZCDHQC
      DJRPCDFKLBBACDBCDEMNO $.
  $}

  ${
    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 ) Usage of this theorem is discouraged because it
       depends on ~ ax-13 .  See ~ sbiedw , ~ sbiedvw for variants using
       disjoint variables, but requiring fewer axioms.  (Contributed by NM,
       30-Jun-1994.)  (Revised by Mario Carneiro, 4-Oct-2016.)  (Proof
       shortened by Wolf Lammen, 24-Jun-2018.)  (New usage is discouraged.) $)
    sbied $p |- ( ph -> ( [ y / x ] ps <-> ch ) ) $=
      ( wsb wi sbrim nfim1 weq wb com12 pm5.74d sbie bitr3i pm5.74ri ) ABDEIZCA
      TJABJZDEIACJZABDEFKUAUBDEACDFGLDEMZABCAUCBCNHOPQRS $.
  $}

  ${
    $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 ).  Usage of this theorem is discouraged because it
       depends on ~ ax-13 .  Use the weaker ~ sbiedvw when possible.
       (Contributed by NM, 7-Jan-2017.)  (New usage is discouraged.) $)
    sbiedv $p |- ( ph -> ( [ y / x ] ps <-> ch ) ) $=
      ( nfv nfvd weq wb ex sbied ) ABCDEADGACDHADEIBCJFKL $.
  $}

  ${
    $d x y ps $.  $d t y $.
    2sbiev.1 $e |- ( ( x = t /\ y = u ) -> ( ph <-> ps ) ) $.
    $( Conversion of double implicit substitution to explicit substitution.
       Usage of this theorem is discouraged because it depends on ~ ax-13 .
       See ~ 2sbievw for a version with extra disjoint variables, but based on
       fewer axioms.  (Contributed by AV, 29-Jul-2023.)
       (New usage is discouraged.) $)
    2sbiev $p |- ( [ t / x ] [ u / y ] ph <-> ps ) $=
      ( wsb nfv weq sbiedv sbie ) ADEHBCFBCICFJABDEGKL $.
  $}

  $( Substituting ` y ` for ` x ` and then ` z ` for ` y ` is equivalent to
     substituting ` z ` for both ` x ` and ` y ` .  Usage of this theorem is
     discouraged because it depends on ~ ax-13 .  For a version requiring a
     disjoint variable, but fewer axioms, see ~ sbcom3vv .  (Contributed by
     Giovanni Mascellani, 8-Apr-2018.)  Remove dependency on ~ ax-11 .
     (Revised by Wolf Lammen, 16-Sep-2018.)  (Proof shortened by Wolf Lammen,
     16-Sep-2018.)  (New usage is discouraged.) $)
  sbcom3 $p |- ( [ z / y ] [ y / x ] ph <-> [ z / y ] [ z / x ] ph ) $=
    ( weq wal wsb wb nfa1 drsb2 sbbid wn sb4b sbequ pm5.74i albii bitrdi bitr4d
    wi pm2.61i ) CDEZCFZABCGZCDGZABDGZCDGZHUBUCUECDUACIACDBJKUBLZUDUAUESZCFZUFU
    GUDUAUCSZCFUIUCCDMUJUHCUAUCUEACDBNOPQUECDMRT $.

  $( A composition law for substitution.  Usage of this theorem is discouraged
     because it depends on ~ ax-13 .  See ~ sbcov for a version with a disjoint
     variable condition requiring fewer axioms.  (Contributed by NM,
     14-May-1993.)  (Proof shortened by Wolf Lammen, 21-Sep-2018.)
     (New usage is discouraged.) $)
  sbco $p |- ( [ y / x ] [ x / y ] ph <-> [ y / x ] ph ) $=
    ( wsb sbcom3 sbid sbbii bitri ) ACBDBCDACCDZBCDABCDACBCEIABCACFGH $.

  ${
    sbid2.1 $e |- F/ x ph $.
    $( An identity law for substitution.  Usage of this theorem is discouraged
       because it depends on ~ ax-13 .  Check out ~ sbid2vw for a weaker
       version requiring fewer axioms.  (Contributed by NM, 14-May-1993.)
       (Revised by Mario Carneiro, 6-Oct-2016.)  (New usage is discouraged.) $)
    sbid2 $p |- ( [ y / x ] [ x / y ] ph <-> ph ) $=
      ( wsb sbco sbf bitri ) ACBEBCEABCEAABCFABCDGH $.
  $}

  ${
    $d x ph $.
    $( An identity law for substitution.  Used in proof of Theorem 9.7 of
       [Megill] p. 449 (p. 16 of the preprint).  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  See ~ sbid2vw for a version
       with an extra disjoint variable condition requiring fewer axioms.
       (Contributed by NM, 5-Aug-1993.)  (New usage is discouraged.) $)
    sbid2v $p |- ( [ y / x ] [ x / y ] ph <-> ph ) $=
      ( nfv sbid2 ) ABCABDE $.
  $}

  $( An idempotent law for substitution.  Usage of this theorem is discouraged
     because it depends on ~ ax-13 .  (Contributed by NM, 30-Jun-1994.)  (Proof
     shortened by Andrew Salmon, 25-May-2011.)  (Proof shortened by Wolf
     Lammen, 13-Jul-2019.)  (New usage is discouraged.) $)
  sbidm $p |- ( [ y / x ] [ y / x ] ph <-> [ y / x ] ph ) $=
    ( wsb sbcom3 sbid sbbii bitr3i ) ABCDZBCDABBDZBCDIABBCEJABCABFGH $.

  ${
    sbco2.1 $e |- F/ z ph $.
    $( A composition law for substitution.  For versions requiring fewer
       axioms, but more disjoint variable conditions, see ~ sbco2v and
       ~ sbco2vv .  Usage of this theorem is discouraged because it depends on
       ~ ax-13 .  (Contributed by NM, 30-Jun-1994.)  (Revised by Mario
       Carneiro, 6-Oct-2016.)  (Proof shortened by Wolf Lammen, 17-Sep-2018.)
       (New usage is discouraged.) $)
    sbco2 $p |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph ) $=
      ( weq wal wsb wb sbequ12 sbequ bitr3d sps wn nfnae nfsb4 wi sbied pm2.61i
      a1i ) DCFZDGZABDHZDCHZABCHZIZUAUFDUAUCUDUEUCDCJADCBKZLMUBNZUCUEDCDCDOABCD
      EPUAUCUEIQUHUGTRS $.
  $}

  ${
    sbco2d.1 $e |- F/ x ph $.
    sbco2d.2 $e |- F/ z ph $.
    sbco2d.3 $e |- ( ph -> F/ z ps ) $.
    $( A composition law for substitution.  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  (Contributed by NM,
       2-Jun-1993.)  (Revised by Mario Carneiro, 6-Oct-2016.)
       (New usage is discouraged.) $)
    sbco2d $p |- ( ph -> ( [ y / z ] [ z / x ] ps <-> [ y / x ] ps ) ) $=
      ( wsb wi nfim1 sbco2 sbrim sbbii bitri 3bitr3i pm5.74ri ) ABCEIZEDIZBCDIZ
      ABJZCEIZEDIZUACDIASJZATJUACDEABEGHKLUCARJZEDIUDUBUEEDABCEFMNAREDGMOABCDFM
      PQ $.
  $}

  $( A composition law for substitution.  Usage of this theorem is discouraged
     because it depends on ~ ax-13 .  (Contributed by NM, 2-Jun-1993.)  (Proof
     shortened by Wolf Lammen, 18-Sep-2018.)  (New usage is discouraged.) $)
  sbco3 $p |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph ) $=
    ( weq wal wsb wb drsb1 nfae sbequ12a sps sbbid bitr3d wn nfnae nfsb2 sbco2d
    sbco sbbii bitr3di pm2.61i ) BCEZBFZABCGZCDGZACBGZBDGZHUDUEBDGUFUHUEBCDIUDU
    EUGBDBCBJUCUEUGHBABCKLMNUDOZUECBGZBDGUFUHUIUECDBBCCPBCBPABCQRUJUGBDACBSTUAU
    B $.

  $( A commutativity law for substitution.  Usage of this theorem is
     discouraged because it depends on ~ ax-13 .  Check out ~ sbcom3vv for a
     version requiring fewer axioms.  (Contributed by NM, 27-May-1997.)  (Proof
     shortened by Wolf Lammen, 20-Sep-2018.)  (New usage is discouraged.) $)
  sbcom $p |- ( [ y / z ] [ y / x ] ph <-> [ y / x ] [ y / z ] ph ) $=
    ( wsb sbco3 sbcom3 3bitr3i ) ABDEDCEADBEBCEABCEDCEADCEBCEABDCFABDCGADBCGH
    $.

  ${
    sbtrt.nf $e |- F/ y ph $.
    $( Partially closed form of ~ sbtr .  Usage of this theorem is discouraged
       because it depends on ~ ax-13 .  (Contributed by BJ, 4-Jun-2019.)
       (New usage is discouraged.) $)
    sbtrt $p |- ( A. y [ y / x ] ph -> ph ) $=
      ( wsb wal stdpc4 sbid2 sylib ) ABCEZCFJCBEAJCBGACBDHI $.
  $}

  ${
    sbtr.nf $e |- F/ y ph $.
    sbtr.1 $e |- [ y / x ] ph $.
    $( A partial converse to ~ sbt .  If the substitution of a variable for a
       nonfree one in a wff gives a theorem, then the original wff is a
       theorem.  Usage of this theorem is discouraged because it depends on
       ~ ax-13 .  (Contributed by BJ, 15-Sep-2018.)
       (New usage is discouraged.) $)
    sbtr $p |- ph $=
      ( wsb sbtrt mpg ) ABCFACABCDGEH $.
  $}

  ${
    sb8.1 $e |- F/ y ph $.
    $( Substitution of variable in universal quantifier.  Usage of this theorem
       is discouraged because it depends on ~ ax-13 .  For a version requiring
       disjoint variables, but fewer axioms, see ~ sb8f .  (Contributed by NM,
       16-May-1993.)  (Revised by Mario Carneiro, 6-Oct-2016.)  (Proof
       shortened by Jim Kingdon, 15-Jan-2018.)  (New usage is discouraged.) $)
    sb8 $p |- ( A. x ph <-> A. y [ y / x ] ph ) $=
      ( wsb nfs1 sbequ12 cbval ) AABCEBCDABCDFABCGH $.

    $( Substitution of variable in existential quantifier.  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  For a version
       requiring disjoint variables, but fewer axioms, see ~ sb8ef .
       (Contributed by NM, 12-Aug-1993.)  (Revised by Mario Carneiro,
       6-Oct-2016.)  (Proof shortened by Jim Kingdon, 15-Jan-2018.)
       (New usage is discouraged.) $)
    sb8e $p |- ( E. x ph <-> E. y [ y / x ] ph ) $=
      ( wsb nfs1 sbequ12 cbvex ) AABCEBCDABCDFABCGH $.
  $}

  $( Commutation of quantification and substitution variables.  Usage of this
     theorem is discouraged because it depends on ~ ax-13 .  (Contributed by
     NM, 5-Aug-1993.)  Allow a shortening of ~ sb9i .  (Revised by Wolf Lammen,
     15-Jun-2019.)  (New usage is discouraged.) $)
  sb9 $p |- ( A. x [ x / y ] ph <-> A. y [ y / x ] ph ) $=
    ( weq wal wsb wb sbequ12a equcoms sps dral1 wn nfnae wnf nfsb2 naecoms cbv2
    wi a1i pm2.61i ) BCDZBEZACBFZBEABCFZCEGUCUDBCUAUCUDGZBUECBACBHIZJKUBLZUCUDB
    CBCBMBCCMUCCNCBACBOPABCOUAUERUGUFSQT $.

  $( Commutation of quantification and substitution variables.  Usage of this
     theorem is discouraged because it depends on ~ ax-13 .  (Contributed by
     NM, 5-Aug-1993.)  (Proof shortened by Wolf Lammen, 15-Jun-2019.)
     (New usage is discouraged.) $)
  sb9i $p |- ( A. x [ x / y ] ph -> A. y [ y / x ] ph ) $=
    ( wsb wal sb9 biimpi ) ACBDBEABCDCEABCFG $.

  ${
    $d y ph $.
    $( Two ways of expressing " ` x ` is (effectively) not free in ` ph ` ".
       Usage of this theorem is discouraged because it depends on ~ ax-13 .
       (Contributed by NM, 29-May-2009.)  (New usage is discouraged.) $)
    sbhb $p |- ( ( ph -> A. x ph ) <-> A. y ( ph -> [ y / x ] ph ) ) $=
      ( wal wi wsb nfv sb8 imbi2i 19.21v bitr4i ) AABDZEAABCFZCDZEAMECDLNAABCAC
      GHIAMCJK $.
  $}

  ${
    $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.)  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .  Use
       ~ nfsbv instead.  (New usage is discouraged.) $)
    nfsbd $p |- ( ph -> F/ z [ y / x ] ps ) $=
      ( weq wal wsb wnf wn wi alrimi nfsb4t syl axc16nf pm2.61d2 ) AEDHEIZBCDJZ
      EKZABEKZCISLUAMAUBCFGNBCDEOPTEDEQR $.
  $}

  ${
    $d y z $.
    nfsb.1 $e |- F/ z ph $.
    $( If ` z ` is not free in ` ph ` , then it is not free in ` [ y / x ] ph `
       when ` y ` and ` z ` are distinct.  See ~ nfsbv for a version with an
       additional disjoint variable condition on ` x , z ` but not requiring
       ~ ax-13 .  (Contributed by Mario Carneiro, 11-Aug-2016.)  (Proof
       shortened by Wolf Lammen, 25-Feb-2024.)  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  Use ~ nfsbv instead.
       (New usage is discouraged.) $)
    nfsb $p |- F/ z [ y / x ] ph $=
      ( wsb wnf wtru nftru a1i nfsbd mptru ) ABCFDGHABCDBIADGHEJKL $.
  $}

  ${
    $d y z $.
    hbsb.1 $e |- ( ph -> A. z ph ) $.
    $( If ` z ` is not free in ` ph ` , then it is not free in ` [ y / x ] ph `
       when ` y ` and ` z ` are distinct.  (Contributed by NM, 12-Aug-1993.)
       Usage of this theorem is discouraged because it depends on ~ ax-13 .
       Use ~ hbsbw instead.  (New usage is discouraged.) $)
    hbsb $p |- ( [ y / x ] ph -> A. z [ y / x ] ph ) $=
      ( wsb nf5i nfsb nf5ri ) ABCFDABCDADEGHI $.
  $}

  ${
    $d y z $.
    sb7f.1 $e |- F/ 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-5 ,
       i.e., that does not have the concept of a variable not occurring in a
       formula.  (Definition ~ dfsb1 is also suitable, but its mixing of free
       and bound variables is distasteful to some logicians.)  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  (Contributed by
       NM, 26-Jul-2006.)  (Revised by Mario Carneiro, 6-Oct-2016.)
       (New usage is discouraged.) $)
    sb7f $p |- ( [ y / x ] ph <->
               E. z ( z = y /\ E. x ( x = z /\ ph ) ) ) $=
      ( wsb weq wa wex sb5f sbbii sbco2 sb5 3bitr3i ) ABDFZDCFBDGAHBIZDCFABCFDC
      GPHDIOPDCABDEJKABCDELPDCMN $.
  $}

  ${
    $d y z $.
    sb7h.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-5 ,
       i.e., that does not have the concept of a variable not occurring in a
       formula.  (Definition ~ dfsb1 is also suitable, but its mixing of free
       and bound variables is distasteful to some logicians.)  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  (Contributed by
       NM, 26-Jul-2006.)  (Proof shortened by Andrew Salmon, 25-May-2011.)
       (New usage is discouraged.) $)
    sb7h $p |- ( [ y / x ] ph <->
               E. z ( z = y /\ E. x ( x = z /\ ph ) ) ) $=
      ( nf5i sb7f ) ABCDADEFG $.
  $}

  ${
    $d x y $.
    sb10f.1 $e |- F/ 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.  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  (Contributed by
       NM, 9-May-2005.)  (Revised by Mario Carneiro, 6-Oct-2016.)
       (New usage is discouraged.) $)
    sb10f $p |- ( [ y / z ] ph <-> E. x ( x = y /\ [ x / z ] ph ) ) $=
      ( weq wsb wa wex nfsb sbequ equsexv bicomi ) BCFADBGZHBIADCGZNOBCADCBEJAB
      CDKLM $.
  $}

  ${
    $d x y $.
    $( Check out ~ sbal for a version not dependent on ~ ax-13 .  A theorem
       used in elimination of disjoint variable restriction on ` x ` and ` z `
       by replacing it with a distinctor ` -. A. x x = z ` .  (Contributed by
       NM, 15-May-1993.)  (Proof shortened by Wolf Lammen, 3-Oct-2018.)
       (New usage is discouraged.)  (Proof modification is discouraged.) $)
    sbal1 $p |- ( -. A. x x = z ->
             ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) ) $=
      ( weq wal wn wsb wb wa wi sb4b nfnae wnf nfeqf2 19.21t bicomd sbequ12 sps
      albid syl sylan9bbr alcom bitrdi adantl bitr4d ex dral2 bitr3d pm2.61d2 )
      BDEBFGZCDEZCFZABFZCDHZACDHZBFZIZUKUMGZURUKUSJUOULAKZBFZCFZUQUSUOULUNKZCFU
      KVBUNCDLUKVCVACBDCMUKULBNZVCVAIBDCOVDVAVCULABPQUATUBUSUQVBIUKUSUQUTCFZBFV
      BUSUPVEBCDBMACDLTUTBCUCUDUEUFUGUMUNUOUQULUNUOICUNCDRSAUPCDBULAUPICACDRSUH
      UIUJ $.
  $}

  ${
    $d z x $.
    $( Move quantifier in and out of substitution.  (Contributed by NM,
       2-Jan-2002.)  Remove a distinct variable constraint.  (Revised by Wolf
       Lammen, 24-Dec-2022.)  (Proof shortened by Wolf Lammen, 23-Sep-2023.)
       Usage of this theorem is discouraged because it depends on ~ ax-13 .
       Use ~ sbal instead.  (New usage is discouraged.) $)
    sbal2 $p |- ( -. A. x x = y ->
             ( [ z / y ] A. x ph <-> A. x [ z / y ] ph ) ) $=
      ( weq wal wn wsb wb sbequ12 dral2 bitr3d adantl wa sb4b nfnae albid alcom
      sps wi bitrdi wnf nfeqf1 19.21t syl sylan9bbr bitr4d pm2.61dan ) BCEBFGZC
      DEZCFZABFZCDHZACDHZBFZIZUKUPUIUKULUMUOUJULUMICULCDJSAUNCDBUJAUNICACDJSKLM
      UIUKGZNUMUJULTZCFZUOUQUMUSIUIULCDOMUQUOUJATZBFZCFZUIUSUQUOUTCFZBFVBUQUNVC
      BCDBPACDOQUTBCRUAUIVAURCBCCPUIUJBUBVAURIBCDUCUJABUDUEQUFUGUH $.
  $}

  ${
    $d z w ph $.
    $( An equivalent expression for double existence.  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  For a version requiring
       more disjoint variables, but fewer axioms, see ~ 2sb8ef .  (Contributed
       by Wolf Lammen, 2-Nov-2019.)  (New usage is discouraged.) $)
    2sb8e $p |- ( E. x E. y ph <->
                  E. z E. w [ z / x ] [ w / y ] ph ) $=
      ( wex wsb nfv sb8e exbii excom bitri nfsb 3bitri ) ACFZBFZACEGZBFZEFZQBDG
      ZDFZEFTEFDFPQEFZBFSOUBBACEAEHIJQBEKLRUAEQBDACEDADHMIJTEDKN $.
  $}


$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
  Uniqueness and unique existence
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)

  ${
    $d x y $.  $d y ph $.
    $( An elementary proof of ~ moeu in disguise, connecting an expression
       characterizing uniqueness ( ~ df-mo ) to that of existential uniqueness
       ( ~ eu6 ).  No particular order of definition is required, as one can be
       derived from the other.  This is shown here and in ~ dfeumo .
       (Contributed by Wolf Lammen, 27-May-2019.) $)
    dfmoeu $p |- ( ( E. x ph -> E. y A. x ( ph <-> x = y ) ) <->
                      E. y A. x ( ph -> x = y ) ) $=
      ( wex weq wb wal wi wn alnex pm2.21 alimi sylbir 19.8ad biimp eximi nfia1
      ja wa com12 id ax12v embantd ancld albiim imbitrrdi exlimi eximdv impbii
      spsd ) ABDZABCEZFZBGZCDZHAULHZBGZCDZUKUOURUKIZUQCUSAIZBGUQABJUTUPBAULKLMN
      UNUQCUMUPBAULOLPRUKURUOUKUQUNCAUQUNHBUPUMBQAUQUQULAHBGZSUNAUQVAAUPVABAAUL
      VAAUAULAVAABCUBTUCUJUDAULBUEUFUGUHTUI $.

    $( An elementary proof showing the reverse direction of ~ dfmoeu .  Here
       the characterizing expression of existential uniqueness ( ~ eu6 ) is
       derived from that of uniqueness ( ~ df-mo ).  (Contributed by Wolf
       Lammen, 3-Oct-2023.) $)
    dfeumo $p |- ( ( E. x ph /\ E. y A. x ( ph -> x = y ) ) <->
                      E. y A. x ( ph <-> x = y ) ) $=
      ( weq wb wal wex wa ax6ev biimpr aleximi mpi exlimiv pm4.71ri abai dfmoeu
      wi anbi2i 3bitrri ) ABCDZEZBFZCGZABGZUCHUDUDUCQZHUDATQBFCGZHUCUDUBUDCUBTB
      GUDBCIUATABATJKLMNUDUCOUEUFUDABCPRS $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Uniqueness: the at-most-one quantifier
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Token for the at-most-one quantifier. $)
  $c E* $.

  $( Extend wff definition to include the at-most-one quantifier ("there exists
     at most one ` x ` such that ` ph ` "). $)
  wmo $a wff E* x ph $.

  ${
    $d t x y $.  $d t x z $.  $d t y ph $.  $d t z ph $.
    $( Soundness justification theorem for ~ df-mo (note that ` y ` and ` z `
       need not be disjoint, although the weaker theorem with that disjoint
       variable condition added would be enough to justify the soundness of the
       definition).  (Contributed by NM, 11-Mar-2010.)  Added this theorem by
       adapting the proof of ~ eujust .  (Revised by BJ, 30-Sep-2022.) $)
    mojust $p |- ( E. y A. x ( ph -> x = y ) <-> E. z A. x ( ph -> x = z ) ) $=
      ( vt weq wi wal wex equequ2 imbi2d albidv cbvexvw bitri ) ABCFZGZBHZCIABE
      FZGZBHZEIABDFZGZBHZDIQTCECEFZPSBUDORACEBJKLMTUCEDEDFZSUBBUERUAAEDBJKLMN
      $.
  $}

  ${
    $d x y $.  $d ph y $.
    $( Define the at-most-one quantifier.  The expression ` E* x ph ` is read
       "there exists at most one ` x ` such that ` ph ` ".  This is also called
       the "uniqueness quantifier" but that expression is also used for the
       unique existential quantifier ~ df-eu , therefore we avoid that
       ambiguous name.

       Notation of [BellMachover] p. 460, whose definition we show as ~ mo3 .
       For other possible definitions see ~ moeu and ~ mo4 .  (Contributed by
       Wolf Lammen, 27-May-2019.)  Make this the definition (which used to be
       ~ moeu , while this definition was then proved as ~ dfmo ).  (Revised by
       BJ, 30-Sep-2022.) $)
    df-mo $a |- ( E* x ph <-> E. y A. x ( ph -> x = y ) ) $.
  $}

  ${
    $d x y $.  $d ph y $.
    $( Nonexistence implies uniqueness.  (Contributed by BJ, 30-Sep-2022.)
       Avoid ~ ax-11 .  (Revised by Wolf Lammen, 16-Oct-2022.) $)
    nexmo $p |- ( -. E. x ph -> E* x ph ) $=
      ( vy wn wal weq wi wex wmo pm2.21 alimi alrimiv 19.2d alnex df-mo 3imtr4i
      bicomi ) ADZBEZABCFZGZBEZCHABHDZABISUBCSUBCRUABATJKLMSUCABNQABCOP $.
    $( $j usage 'nexmo' avoids 'ax-11'; $)
  $}

  $( Any proposition holds for some ` x ` or holds for at most one ` x ` .
     (Contributed by NM, 8-Mar-1995.)  Shorten proof and avoid ~ df-eu .
     (Revised by BJ, 14-Oct-2022.) $)
  exmo $p |- ( E. x ph \/ E* x ph ) $=
    ( wex wmo nexmo orri ) ABCABDABEF $.

  $( Absorption of existence condition by uniqueness.  (Contributed by NM,
     4-Nov-2002.)  Shorten proof and avoid ~ df-eu .  (Revised by BJ,
     14-Oct-2022.) $)
  moabs $p |- ( E* x ph <-> ( E. x ph -> E* x ph ) ) $=
    ( wmo wex wi ax-1 nexmo id ja impbii ) ABCZABDZKEKLFLKKABGKHIJ $.

  ${
    $d x y $.  $d y ph $.  $d y ps $.
    $( The at-most-one quantifier reverses implication.  (Contributed by NM,
       22-Apr-1995.) $)
    moim $p |- ( A. x ( ph -> ps ) -> ( E* x ps -> E* x ph ) ) $=
      ( vy wi wal weq wex wmo imim1 al2imi eximdv df-mo 3imtr4g ) ABEZCFZBCDGZE
      ZCFZDHAQEZCFZDHBCIACIPSUADORTCABQJKLBCDMACDMN $.
  $}

  ${
    $d x y $.  $d y ph $.  $d y ps $.
    moimi.1 $e |- ( ph -> ps ) $.
    $( The at-most-one quantifier reverses implication.  (Contributed by NM,
       15-Feb-2006.)  Remove use of ~ ax-5 .  (Revised by Steven Nguyen,
       9-May-2023.) $)
    moimi $p |- ( E* x ps -> E* x ph ) $=
      ( vy weq wi wal wex wmo imim1i alimi eximi df-mo 3imtr4i ) BCEFZGZCHZEIAP
      GZCHZEIBCJACJRTEQSCABPDKLMBCENACENO $.
    $( $j usage 'moimi' avoids 'ax-5'; $)
  $}

  ${
    $d x ph $.
    moimdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( The at-most-one quantifier reverses implication, deduction form.
       (Contributed by Thierry Arnoux, 25-Feb-2017.) $)
    moimdv $p |- ( ph -> ( E* x ch -> E* x ps ) ) $=
      ( wi wal wmo alrimiv moim syl ) ABCFZDGCDHBDHFALDEIBCDJK $.
  $}

  $( Equivalence theorem for the at-most-one quantifier.  (Contributed by BJ,
     7-Oct-2022.)  (Proof shortened by Wolf Lammen, 18-Feb-2023.) $)
  mobi $p |- ( A. x ( ph <-> ps ) -> ( E* x ph <-> E* x ps ) ) $=
    ( wb wal wi wa wmo albiim moim impbid21d imp sylbi ) ABDCEABFCEZBAFCEZGACHZ
    BCHZDZABCINORNOPQBACJABCJKLM $.

  ${
    mobii.1 $e |- ( ps <-> ch ) $.
    $( Formula-building rule for the at-most-one quantifier (inference form).
       (Contributed by NM, 9-Mar-1995.)  (Revised by Mario Carneiro,
       17-Oct-2016.)  Avoid ~ ax-5 .  (Revised by Wolf Lammen, 24-Sep-2023.) $)
    mobii $p |- ( E* x ps <-> E* x ch ) $=
      ( wmo biimpri moimi biimpi impbii ) ACEBCEBACABDFGABCABDHGI $.
    $( $j usage 'mobii' avoids 'ax-5'; $)
  $}

  ${
    $d x ph $.
    mobidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for the at-most-one quantifier (deduction form).
       (Contributed by Mario Carneiro, 7-Oct-2016.)  Reduce axiom dependencies
       and shorten proof.  (Revised by BJ, 7-Oct-2022.) $)
    mobidv $p |- ( ph -> ( E* x ps <-> E* x ch ) ) $=
      ( wb wal wmo alrimiv mobi syl ) ABCFZDGBDHCDHFALDEIBCDJK $.
  $}

  ${
    mobid.1 $e |- F/ x ph $.
    mobid.2 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for the at-most-one quantifier (deduction form).
       (Contributed by NM, 8-Mar-1995.)  Remove dependency on ~ ax-10 ,
       ~ ax-11 , ~ ax-13 .  (Revised by BJ, 14-Oct-2022.)  (Proof shortened by
       Wolf Lammen, 18-Feb-2023.) $)
    mobid $p |- ( ph -> ( E* x ps <-> E* x ch ) ) $=
      ( wb wal wmo alrimi mobi syl ) ABCGZDHBDICDIGAMDEFJBCDKL $.
  $}

  $( If an implication holds for at most one value, then its consequent holds
     for at most one value.  See also ~ ala1 and ~ exa1 .  (Contributed by NM,
     28-Jul-1995.)  (Proof shortened by Wolf Lammen, 22-Dec-2018.) $)
  moa1 $p |- ( E* x ( ph -> ps ) -> E* x ps ) $=
    ( wi ax-1 moimi ) BABDCBAEF $.

  $( "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, inference form.
       (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 ph y $.
    $( Bound-variable hypothesis builder for the at-most-one quantifier.
       (Contributed by NM, 8-Mar-1995.)  (Revised by Mario Carneiro,
       7-Oct-2016.)  Adapt to new definition.  (Revised by BJ, 1-Oct-2022.) $)
    nfmo1 $p |- F/ x E* x ph $=
      ( vy wmo weq wi wal wex df-mo nfa1 nfex nfxfr ) ABDABCEFZBGZCHBABCINBCMBJ
      KL $.
  $}

  ${
    $d x z $.  $d y z $.  $d z ph $.  $d z ps $.
    nfmod2.1 $e |- F/ y ph $.
    nfmod2.2 $e |- ( ( ph /\ -. A. x x = y ) -> F/ x ps ) $.
    $( Bound-variable hypothesis builder for the at-most-one quantifier.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .  See
       ~ nfmodv for a version replacing the distinctor with a disjoint variable
       condition, not requiring ~ ax-13 .  (Contributed by Mario Carneiro,
       14-Nov-2016.)  Avoid ~ df-eu .  (Revised by BJ, 14-Oct-2022.)
       (New usage is discouraged.) $)
    nfmod2 $p |- ( ph -> F/ x E* y ps ) $=
      ( vz wmo weq wi wal wex df-mo nfv wn wa wnf nfeqf1 adantl nfimd nfald2
      nfexd nfxfrd ) BDHBDGIZJZDKZGLACBDGMAUFCGAGNAUECDEACDICKOZPBUDCFUGUDCQACD
      GRSTUAUBUC $.
  $}

  ${
    $d x y z $.  $d ph z $.  $d ps z $.
    nfmodv.1 $e |- F/ y ph $.
    nfmodv.2 $e |- ( ph -> F/ x ps ) $.
    $( Bound-variable hypothesis builder for the at-most-one quantifier.  See
       ~ nfmod for a version without disjoint variable conditions but requiring
       ~ ax-13 .  (Contributed by Mario Carneiro, 14-Nov-2016.)  (Revised by
       BJ, 28-Jan-2023.) $)
    nfmodv $p |- ( ph -> F/ x E* y ps ) $=
      ( vz wmo weq wi wal wex df-mo nfv nfvd nfimd nfald nfexd nfxfrd ) BDHBDGI
      ZJZDKZGLACBDGMAUBCGAGNAUACDEABTCFATCOPQRS $.
    $( $j usage 'nfmodv' avoids 'ax-13'; $)
  $}

  ${
    $d x y $.
    nfmov.1 $e |- F/ x ph $.
    $( Bound-variable hypothesis builder for the at-most-one quantifier.  See
       ~ nfmo for a version without disjoint variable conditions but requiring
       ~ ax-13 .  (Contributed by NM, 9-Mar-1995.)  (Revised by Wolf Lammen,
       2-Oct-2023.) $)
    nfmov $p |- F/ x E* y ph $=
      ( wmo wnf wtru nftru a1i nfmodv mptru ) ACEBFGABCCHABFGDIJK $.
    $( $j usage 'nfmov' avoids 'ax-13'; $)
  $}

  ${
    nfmod.1 $e |- F/ y ph $.
    nfmod.2 $e |- ( ph -> F/ x ps ) $.
    $( Bound-variable hypothesis builder for the at-most-one quantifier.
       Deduction version of ~ nfmo .  Usage of this theorem is discouraged
       because it depends on ~ ax-13 .  Use the weaker ~ nfmodv when possible.
       (Contributed by Mario Carneiro, 14-Nov-2016.)
       (New usage is discouraged.) $)
    nfmod $p |- ( ph -> F/ x E* y ps ) $=
      ( wnf weq wal wn adantr nfmod2 ) ABCDEABCGCDHCIJFKL $.
  $}

  ${
    nfmo.1 $e |- F/ x ph $.
    $( Bound-variable hypothesis builder for the at-most-one quantifier.  Note
       that ` x ` and ` y ` need not be disjoint.  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  Use the weaker ~ nfmov when
       possible.  (Contributed by NM, 9-Mar-1995.)
       (New usage is discouraged.) $)
    nfmo $p |- F/ x E* y ph $=
      ( wmo wnf wtru nftru a1i nfmod mptru ) ACEBFGABCCHABFGDIJK $.
  $}

  ${
    $d x y z $.  $d ph z $.
    mof.1 $e |- F/ y ph $.
    $( Version of ~ df-mo with disjoint variable condition replaced by
       nonfreeness hypothesis.  (Contributed by NM, 8-Mar-1995.)  Extract
       ~ dfmo from this proof, and prove ~ mof from it (as of 30-Sep-2022,
       directly from ~ df-mo ).  (Revised by Wolf Lammen, 28-May-2019.)  Avoid
       ~ ax-13 .  (Revised by Wolf Lammen, 16-Oct-2022.) $)
    mof $p |- ( E* x ph <-> E. y A. x ( ph -> x = y ) ) $=
      ( vz wmo weq wi wal wex df-mo nfv nfim nfal equequ2 imbi2d albidv cbvexv1
      bitri ) ABFABEGZHZBIZEJABCGZHZBIZCJABEKUBUEECUACBATCDTCLMNUEELECGZUAUDBUF
      TUCAECBOPQRS $.
  $}

  ${
    $d x y z $.  $d ph z $.
    mo3.nf $e |- F/ y ph $.
    $( Alternate definition of the at-most-one quantifier.  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.)  (Proof shortened by Wolf Lammen, 18-Aug-2019.)
       Remove dependency on ~ ax-13 .  (Revised by BJ and WL, 29-Jan-2023.) $)
    mo3 $p |- ( E* x ph <->
               A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) $=
      ( vz wmo wsb wa weq wi nfmo1 nfmov df-mo sp spsbim equsb3 imbitrdi alrimi
      wal wex anim12d equtr2 syl6 exlimiv sylbi nfs1v pm3.21 imim1d alimd com12
      aleximi sb8ef mof 3imtr4g moabs sylibr alcoms impbii ) ABFZAABCGZHZBCIZJZ
      CSZBSUSVDBABKUSVCCACBDLUSABEIZJZBSZETVCABEMVGVCEVGVAVECEIZHVBVGAVEUTVHVFB
      NVGUTVEBCGVHAVEBCOBCEPQUABCEUBUCUDUERRVCUSCBVCBSZCSZABTZUSJUSVJUTCTAVBJZB
      SZCTVKUSVIUTVMCUTVIVMUTVCVLBABCUFUTAVAVBUTAUGUHUIUJUKABCDULABCDUMUNABUOUP
      UQUR $.
  $}

  ${
    $d x y $.
    mo.nf $e |- F/ y ph $.
    $( Equivalent definitions of "there exists at most one".  (Contributed by
       NM, 7-Aug-1994.)  (Revised by Mario Carneiro, 7-Oct-2016.)  (Proof
       shortened by Wolf Lammen, 2-Dec-2018.) $)
    mo $p |- ( E. y A. x ( ph -> x = y ) <->
               A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) $=
      ( weq wi wal wex wmo wsb wa mof mo3 bitr3i ) ABCEZFBGCHABIAABCJKOFCGBGABC
      DLABCDMN $.
  $}

  ${
    $d x y $.  $d y z ph $.  $d x z ps $.
    mo4.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( At-most-one quantifier expressed using implicit substitution.  This
       theorem is also a direct consequence of ~ mo4f , but this proof is based
       on fewer axioms.

       By the way, swapping ` x ` , ` y ` and ` ph ` , ` ps ` leads to an
       expression for ` E* y ps ` , which is equivalent to ` E* x ph ` (is a
       proof line), so the right hand side is a rare instance of an expression
       where swapping the quantifiers can be done without ~ ax-11 .
       (Contributed by NM, 26-Jul-1995.)  Reduce axiom usage.  (Revised by Wolf
       Lammen, 18-Oct-2023.) $)
    mo4 $p |- ( E* x ph <-> A. x A. y ( ( ph /\ ps ) -> x = y ) ) $=
      ( vz wmo wa weq wi wal df-mo equequ1 imbi12d cbvalvw biimpi pm2.27 alimdv
      wex sylibr adantr adantl anim12d equtr2 syl6com com12 mpcom exlimiv sylbi
      cbvexvw biimpri ax6evr pm3.2 imim1d ax7 syl8 com4r impcom impancom eximdv
      ex mpi expcom aleximi ax5e syl56 exbii 3bitr4i moabs bitri impbii ) ACGZA
      BHZCDIZJZDKZCKZVLACFIZJZCKZFSZVQACFLZVTVQFBDFIZJZDKZVTVQVTWEVSWDCDVNABVRW
      CECDFMNOZPWEVSVPCVSWEVPVSWDVODVSWDVOVMVSWDHVRWCHVNVMVSVRWDWCAVSVRJBAVRQUA
      BWDWCJABWCQUBUCCDFUDUEVARUFRUGUHUIVQBDSZBDGZJZVLWGACSZVQWHCSWHWJWGABCDEUJ
      UKVPAWHCAVPWHAVPHZWEFSZWHWKVRFSWLFCULWKVRWEFAVRVPWEAVRHVOWDDVRAVOWDJAVOBV
      RWCAVOBVNVRWCJABVMVNABUMUNCDFUOUPUQURRUSUTVBBDFLZTVCVDWHCVEVFVLWHWIWAWLVL
      WHVTWEFWFVGWBWMVHBDVIVJTVK $.
  $}

  ${
    $d x y $.  $d y ph $.
    mo4f.1 $e |- F/ x ps $.
    mo4f.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( At-most-one quantifier expressed using implicit substitution.  Note that
       the disjoint variable condition on ` y , ph ` can be replaced by the
       nonfreeness hypothesis ` |- F/ y ph ` with essentially the same proof.
       (Contributed by NM, 10-Apr-2004.)  Remove dependency on ~ ax-13 .
       (Revised by Wolf Lammen, 19-Jan-2023.) $)
    mo4f $p |- ( E* x ph <-> A. x A. y ( ( ph /\ ps ) -> x = y ) ) $=
      ( wmo wsb wa weq wi wal nfv mo3 sbiev anbi2i imbi1i 2albii bitri ) ACGAAC
      DHZIZCDJZKZDLCLABIZUBKZDLCLACDADMNUCUECDUAUDUBTBAABCDEFOPQRS $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Unique existence: the unique existential quantifier
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Token for the unique existential quantifier. $)
  $c E! $.

  $( Extend wff definition to include the unique existential quantifier ("there
     exists a unique ` x ` such that ` ph ` "). $)
  weu $a wff E! x ph $.

  $( Define the existential uniqueness quantifier.  This expresses unique
     existence, or existential uniqueness, which is the conjunction of
     existence ( ~ df-ex ) and uniqueness ( ~ df-mo ).  The expression
     ` E! x ph ` is read "there exists exactly one ` x ` such that ` ph ` " or
     "there exists a unique ` x ` such that ` ph ` ".  This is also called the
     "uniqueness quantifier" but that expression is also used for the
     at-most-one quantifier ~ df-mo , therefore we avoid that ambiguous name.

     Definition 10.1 of [BellMachover] p. 97; also Definition *14.02 of
     [WhiteheadRussell] p. 175.  Other possible definitions are given by
     ~ eu1 , ~ eu2 , ~ eu3v , and ~ eu6 .  As for double unique existence,
     beware that the expression ` E! x E! y ph ` means "there exists a unique
     ` x ` such that there exists a unique ` y ` such that ` ph ` " which is a
     weaker property than "there exists exactly one ` x ` and one ` y ` such
     that ` ph ` " (see ~ 2eu4 ).  (Contributed by NM, 12-Aug-1993.)  Make this
     the definition (which used to be ~ eu6 , while this definition was then
     proved as ~ dfeu ).  (Revised by BJ, 30-Sep-2022.) $)
  df-eu $a |- ( E! x ph <-> ( E. x ph /\ E* x ph ) ) $.

  ${
    $d x y $.  $d ph y $.
    $( An alternate way to express existential uniqueness.  (Contributed by NM,
       8-Jul-1994.)  Replace a nonfreeness hypothesis with a disjoint variable
       condition on ` ph ` , ` y ` to reduce axiom usage.  (Revised by Wolf
       Lammen, 29-May-2019.) $)
    eu3v $p |- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) ) $=
      ( weu wex wmo wa weq wi wal df-eu df-mo anbi2i bitri ) ABDABEZABFZGOABCHI
      BJCEZGABKPQOABCLMN $.
  $}

  ${
    $d w x y $.  $d x z $.  $d y ph $.  $d w z ph $.
    $( Soundness justification theorem for ~ eu6 when this was the definition
       of the unique existential quantifier (note that ` y ` and ` z ` need not
       be disjoint, although the weaker theorem with that disjoint variable
       condition added would be enough to justify the soundness of the
       definition).  See ~ eujustALT for a proof that provides an example of
       how it can be achieved through the use of ~ dvelim .  (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 weq wb wal wex equequ2 bibi2d albidv cbvexvw bitri ) ABCFZGZBHZCIABE
      FZGZBHZEIABDFZGZBHZDIQTCECEFZPSBUDORACEBJKLMTUCEDEDFZSUBBUERUAAEDBJKLMN
      $.

    $( Alternate proof of ~ eujust illustrating the use of ~ dvelim .
       (Contributed by NM, 11-Mar-2010.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    eujustALT $p |- ( E. y A. x ( ph <-> x = y )
                  <-> E. z A. x ( ph <-> x = z ) ) $=
      ( vw weq wal wb wex equequ2 bibi2d albidv sps wn hbnae ax-5 notbid dvelim
      wi df-ex drex1 alrimih naecoms a1i cbv2h syl 3bitr4g pm2.61i ) CDFZCGZABC
      FZHZBGZCIZABDFZHZBGZDIZHUMUQCDUIUMUQHCUIULUPBUIUKUOACDBJKLZMUAUJNZUMNZCGZ
      NUQNZDGZNUNURUTVBVDUTUTDGZCGVBVDHUTVECCDCOCDDOUBUTVAVCCDVAVADGSDCABEFZHZB
      GZNZVADCEVIDPECFZVHUMVJVGULBVJVFUKAECBJKLQRUCVIVCCDEVICPEDFZVHUQVKVGUPBVK
      VFUOAEDBJKLQRUIVAVCHSUTUIUMUQUSQUDUEUFQUMCTUQDTUGUH $.
  $}

  ${
    $d x y z $.  $d y z ph $.
    $( Lemma of ~ eu6im .  A dissection of an idiom characterizing existential
       uniqueness.  (Contributed by NM, 12-Aug-1993.)  This used to be the
       definition of the unique existential quantifier, while ~ df-eu was then
       proved as ~ dfeu .  (Revised by BJ, 30-Sep-2022.)  (Proof shortened by
       Wolf Lammen, 3-Jan-2023.)  Extract common proof lines.  (Revised by Wolf
       Lammen, 3-Mar-2023.) $)
    eu6lem $p |- ( E. y A. x ( ph <-> x = y ) <->
          ( E. y A. x ( x = y -> ph ) /\ E. z A. x ( ph -> x = z ) ) ) $=
      ( weq wb wal wex wi wa 19.42v alsyl equvelv sylib pm4.71i albiim biancomi
      equequ2 imbi2d exbii albidv anbi2d bitrid pm5.32ri bitr4i ax6evr 3bitr4ri
      biantru exdistrv bitri ) ABCEZFBGZCHUKAIBGZABDEZIZBGZJZDHZCHUMCHUPDHJULUR
      CULCDEZJZDHULUSDHZJURULULUSDKUQUTDUQUQUSJUTUQUSUQUKUNIBGUSUKAUNBLCDBMNOUS
      ULUQULUMAUKIZBGZJUSUQULUMVCAUKBPQUSVCUPUMUSVBUOBUSUKUNACDBRSUAUBUCUDUETVA
      ULDCUFUHUGTUMUPCDUIUJ $.

    $( Alternate definition of the unique existential quantifier ~ df-eu not
       using the at-most-one quantifier.  (Contributed by NM, 12-Aug-1993.)
       This used to be the definition of the unique existential quantifier,
       while ~ df-eu was then proved as ~ dfeu .  (Revised by BJ, 30-Sep-2022.)
       (Proof shortened by Wolf Lammen, 3-Jan-2023.)  Remove use of ~ ax-11 .
       (Revised by SN, 21-Sep-2023.) $)
    eu6 $p |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) ) $=
      ( weu wex weq wb wal wa wi dfmoeu anbi2i abai 3bitr4ri ancom biimpr alimi
      eu3v eximi exsbim syl biantru 3bitr4i bitri ) ABDZABEZABCFZGZBHZCEZIZUJUF
      UFUJJZIUFAUGJBHCEZIUKUEULUMUFABCKLUFUJMABCRNUJUFIUJUJUFJZIUKUJUJUFMUFUJOU
      NUJUJUGAJZBHZCEUFUIUPCUHUOBAUGPQSABCTUAUBUCUD $.

    $( One direction of ~ eu6 needs fewer axioms.  (Contributed by Wolf Lammen,
       2-Mar-2023.) $)
    eu6im $p |- ( E. y A. x ( ph <-> x = y ) -> E! x ph ) $=
      ( vz weq wi wal wex wa wb weu exsbim anim1i eu6lem eu3v 3imtr4i ) BCEZAFB
      GCHZABDEFBGDHZIABHZSIAQJBGCHABKRTSABCLMABCDNABDOP $.
  $}

  ${
    $d x y z $.  $d ph z $.
    euf.1 $e |- F/ y ph $.
    $( Version of ~ eu6 with disjoint variable condition replaced by
       nonfreeness hypothesis.  (Contributed by NM, 12-Aug-1993.)  (Proof
       shortened by Wolf Lammen, 30-Oct-2018.)  Avoid ~ ax-13 .  (Revised by
       Wolf Lammen, 16-Oct-2022.) $)
    euf $p |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) ) $=
      ( vz weu weq wb wal wex eu6 nfbi nfal equequ2 bibi2d albidv cbvexv1 bitri
      nfv ) ABFABEGZHZBIZEJABCGZHZBIZCJABEKUBUEECUACBATCDTCSLMUEESECGZUAUDBUFTU
      CAECBNOPQR $.
  $}

  $( Existential uniqueness implies existence.  (Contributed by NM,
     15-Sep-1993.)  (Proof shortened by Andrew Salmon, 9-Jul-2011.)  (Proof
     shortened by Wolf Lammen, 4-Dec-2018.)  (Proof shortened by BJ,
     7-Oct-2022.) $)
  euex $p |- ( E! x ph -> E. x ph ) $=
    ( weu wex wmo df-eu simplbi ) ABCABDABEABFG $.

  $( Existential uniqueness implies uniqueness.  (Contributed by NM,
     23-Mar-1995.) $)
  eumo $p |- ( E! x ph -> E* x ph ) $=
    ( weu wex wmo df-eu simprbi ) ABCABDABEABFG $.

  ${
    eumoi.1 $e |- E! x ph $.
    $( Uniqueness inferred from existential uniqueness.  (Contributed by NM,
       5-Apr-1995.) $)
    eumoi $p |- E* x ph $=
      ( weu wmo eumo ax-mp ) ABDABECABFG $.
  $}

  $( Existence implies that uniqueness is equivalent to unique existence.
     (Contributed by NM, 5-Apr-2004.) $)
  exmoeub $p |- ( E. x ph -> ( E* x ph <-> E! x ph ) ) $=
    ( weu wex wmo df-eu baibr ) ABCABDABEABFG $.

  $( Existence is equivalent to uniqueness implying existential uniqueness.
     (Contributed by NM, 5-Apr-2004.)  (Proof shortened by Wolf Lammen,
     5-Dec-2018.)  (Proof shortened by BJ, 7-Oct-2022.) $)
  exmoeu $p |- ( E. x ph <-> ( E* x ph -> E! x ph ) ) $=
    ( wex wmo weu wi exmoeub biimpd nexmo con1i euex ja impbii ) ABCZABDZABEZFN
    OPABGHOPNNOABIJABKLM $.

  $( Uniqueness implies that existence is equivalent to unique existence.
     (Contributed by BJ, 7-Oct-2022.) $)
  moeuex $p |- ( E* x ph -> ( E. x ph <-> E! x ph ) ) $=
    ( weu wex wmo df-eu rbaibr ) ABCABDABEABFG $.

  $( Uniqueness is equivalent to existence implying unique existence.
     Alternate definition of the at-most-one quantifier, in terms of the
     existential quantifier and the unique existential quantifier.
     (Contributed by NM, 8-Mar-1995.)  This used to be the definition of the
     at-most-one quantifier, while ~ df-mo was then proved as ~ dfmo .
     (Revised by BJ, 30-Sep-2022.) $)
  moeu $p |- ( E* x ph <-> ( E. x ph -> E! x ph ) ) $=
    ( wmo wex wi weu moabs exmoeub pm5.74i bitri ) ABCZABDZKELABFZEABGLKMABHIJ
    $.

  $( Equivalence theorem for the unique existential quantifier.  Theorem
     *14.271 in [WhiteheadRussell] p. 192.  (Contributed by Andrew Salmon,
     11-Jul-2011.)  Reduce dependencies on axioms.  (Revised by BJ,
     7-Oct-2022.) $)
  eubi $p |- ( A. x ( ph <-> ps ) -> ( E! x ph <-> E! x ps ) ) $=
    ( wb wal wex wmo wa weu exbi mobi anbi12d df-eu 3bitr4g ) ABDCEZACFZACGZHBC
    FZBCGZHACIBCIOPRQSABCJABCKLACMBCMN $.

  ${
    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.)  Avoid ~ ax-5 .  (Revised by Wolf Lammen, 27-Sep-2023.) $)
    eubii $p |- ( E! x ph <-> E! x ps ) $=
      ( wex wmo wa weu exbii mobii anbi12i df-eu 3bitr4i ) ACEZACFZGBCEZBCFZGAC
      HBCHNPOQABCDIABCDJKACLBCLM $.
    $( $j usage 'eubii' avoids 'ax-5'; $)
  $}

  ${
    $d x ph $.
    eubidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for unique existential quantifier (deduction
       form).  (Contributed by NM, 9-Jul-1994.)  Reduce axiom dependencies and
       shorten proof.  (Revised by BJ, 7-Oct-2022.) $)
    eubidv $p |- ( ph -> ( E! x ps <-> E! x ch ) ) $=
      ( wb wal weu alrimiv eubi syl ) ABCFZDGBDHCDHFALDEIBCDJK $.
  $}

  ${
    eubid.1 $e |- F/ x ph $.
    eubid.2 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for the unique existential quantifier (deduction
       form).  (Contributed by NM, 9-Jul-1994.)  (Proof shortened by Wolf
       Lammen, 19-Feb-2023.) $)
    eubid $p |- ( ph -> ( E! x ps <-> E! x ch ) ) $=
      ( wb wal weu alrimi eubi syl ) ABCGZDHBDICDIGAMDEFJBCDKL $.
  $}

  ${
    $d x y $.  $d y ph $.
    $( Bound-variable hypothesis builder for uniqueness.  See also ~ nfeu1ALT .
       (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 eu6 nfa1 nfex nfxfr ) ABDABCEFZBGZCHBABCINBCMBJKL
      $.
  $}

  $( Alternate proof of ~ nfeu1 .  This illustrates the systematic way of
     proving nonfreeness in a defined expression: consider the definiens as a
     tree whose nodes are its subformulas, and prove by tree-induction
     nonfreeness of each node, starting from the leaves (generally using ~ nfv
     or nf* theorems for previously defined expressions) and up to the root.
     Here, the definiens is a conjunction of two previously defined
     expressions, which automatically yields the present proof.  (Contributed
     by BJ, 2-Oct-2022.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  nfeu1ALT $p |- F/ x E! x ph $=
    ( weu wex wmo wa df-eu nfe1 nfmo1 nfan nfxfr ) ABCABDZABEZFBABGLMBABHABIJK
    $.

  ${
    nfeud2.1 $e |- F/ y ph $.
    nfeud2.2 $e |- ( ( ph /\ -. A. x x = y ) -> F/ x ps ) $.
    $( Bound-variable hypothesis builder for uniqueness.  (Contributed by Mario
       Carneiro, 14-Nov-2016.)  (Proof shortened by Wolf Lammen, 4-Oct-2018.)
       (Proof shortened by BJ, 14-Oct-2022.)  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  Use ~ nfeudw instead.
       (New usage is discouraged.) $)
    nfeud2 $p |- ( ph -> F/ x E! y ps ) $=
      ( weu wex wmo wa df-eu nfexd2 nfmod2 nfand nfxfrd ) BDGBDHZBDIZJACBDKAPQC
      ABCDEFLABCDEFMNO $.
  $}

  ${
    $d x y $.
    nfeudw.1 $e |- F/ y ph $.
    nfeudw.2 $e |- ( ph -> F/ x ps ) $.
    $( Bound-variable hypothesis builder for the unique existential quantifier.
       Deduction version of ~ nfeu .  Version of ~ nfeud with a disjoint
       variable condition, which does not require ~ ax-13 .  (Contributed by
       NM, 15-Feb-2013.)  Avoid ~ ax-13 .  (Revised by GG, 10-Jan-2024.) $)
    nfeudw $p |- ( ph -> F/ x E! y ps ) $=
      ( weu wex wmo wa df-eu nfexd nfmodv nfand nfxfrd ) BDGBDHZBDIZJACBDKAPQCA
      BCDEFLABCDEFMNO $.
    $( $j usage 'nfeudw' avoids 'ax-13'; $)
  $}

  ${
    nfeud.1 $e |- F/ y ph $.
    nfeud.2 $e |- ( ph -> F/ x ps ) $.
    $( Bound-variable hypothesis builder for the unique existential quantifier.
       Deduction version of ~ nfeu .  Usage of this theorem is discouraged
       because it depends on ~ ax-13 .  Use the weaker ~ nfeudw when possible.
       (Contributed by NM, 15-Feb-2013.)  (Revised by Mario Carneiro,
       7-Oct-2016.)  (New usage is discouraged.) $)
    nfeud $p |- ( ph -> F/ x E! y ps ) $=
      ( wnf weq wal wn adantr nfeud2 ) ABCDEABCGCDHCIJFKL $.
  $}

  ${
    $d x y $.
    nfeuw.1 $e |- F/ x ph $.
    $( Bound-variable hypothesis builder for the unique existential quantifier.
       Version of ~ nfeu with a disjoint variable condition, which does not
       require ~ ax-13 .  (Contributed by NM, 8-Mar-1995.)  Avoid ~ ax-13 .
       (Revised by GG, 10-Jan-2024.) $)
    nfeuw $p |- F/ x E! y ph $=
      ( weu wnf wtru nftru a1i nfeudw mptru ) ACEBFGABCCHABFGDIJK $.
    $( $j usage 'nfeuw' avoids 'ax-13'; $)
  $}

  ${
    nfeu.1 $e |- F/ x ph $.
    $( Bound-variable hypothesis builder for the unique existential quantifier.
       Note that ` x ` and ` y ` need not be disjoint.  Usage of this theorem
       is discouraged because it depends on ~ ax-13 .  Use the weaker ~ nfeuw
       when possible.  (Contributed by NM, 8-Mar-1995.)  (Revised by Mario
       Carneiro, 7-Oct-2016.)  (New usage is discouraged.) $)
    nfeu $p |- F/ x E! y ph $=
      ( weu wnf wtru nftru a1i nfeud mptru ) ACEBFGABCCHABFGDIJK $.
  $}

  $( Rederive ~ df-eu from the old definition ~ eu6 .  (Contributed by NM,
     23-Mar-1995.)  (Proof shortened by Wolf Lammen, 25-May-2019.)  (Proof
     shortened by BJ, 7-Oct-2022.)  (Proof modification is discouraged.)  Use
     ~ df-eu instead.  (New usage is discouraged.) $)
  dfeu $p |- ( E! x ph <-> ( E. x ph /\ E* x ph ) ) $=
    ( wex weu wa wi wmo abai euex pm4.71ri moeu anbi2i 3bitr4i ) ABCZABDZENNOFZ
    EONABGZENOHONABIJQPNABKLM $.

  ${
    $d x y $.  $d y ph $.
    $( Rederive ~ df-mo from the old definition ~ moeu .  (Contributed by Wolf
       Lammen, 27-May-2019.)  (Proof modification is discouraged.)  Use ~ df-mo
       instead.  (New usage is discouraged.) $)
    dfmo $p |- ( E* x ph <-> E. y A. x ( ph -> x = y ) ) $=
      ( wmo wex weu wi weq wb wal moeu eu6 imbi2i dfmoeu 3bitri ) ABDABEZABFZGP
      ABCHZIBJCEZGARGBJCEABKQSPABCLMABCNO $.
  $}

  ${
    $d x y z $.
    $( There exists a unique set equal to a given set.  Special case of ~ eueqi
       proved using only predicate calculus.  The proof needs ` y = z ` be free
       of ` x ` .  This is ensured by having ` x ` and ` y ` be distinct.
       Alternately, a distinctor ` -. A. x x = y ` could have been used
       instead.  See ~ eueq and ~ eueqi for classes.  (Contributed by Stefan
       Allan, 4-Dec-2008.)  (Proof shortened by Wolf Lammen, 8-Sep-2019.)
       Reduce axiom usage.  (Revised by Wolf Lammen, 1-Mar-2023.) $)
    euequ $p |- E! x x = y $=
      ( vz weq weu wex wi wal ax6ev equeuclr alrimiv eximii eu3v mpbir2an ) ABD
      ZAEOAFOACDGZAHZCFABICBDZQCCBIRPACABJKLOACMN $.
  $}

  $( substitution $)

  ${
    $d w y z $.  $d ph z w $.  $d w x z $.
    sb8eulem.nfsb $e |- F/ y [ w / x ] ph $.
    $( Lemma.  Factor out the common proof skeleton of ~ sb8euv and ~ sb8eu .
       Variable substitution in unique existential quantifier.  (Contributed by
       NM, 7-Aug-1994.)  (Revised by Mario Carneiro, 7-Oct-2016.)  (Proof
       shortened by Wolf Lammen, 24-Aug-2019.)  Factor out common proof lines.
       (Revised by Wolf Lammen, 9-Feb-2023.) $)
    sb8eulem $p |- ( E! x ph <-> E! y [ y / x ] ph ) $=
      ( vz weq wb wal wex wsb weu sb8v equsb3 sblbis albii nfv nfbi sbequ eu6
      equequ1 bibi12d cbvalv1 3bitri exbii 3bitr4i ) ABFGZHZBIZFJABCKZCFGZHZCIZ
      FJABLUJCLUIUMFUIUHBDKZDIABDKZDFGZHZDIUMUHBDMUNUQDUGUPABDBDFNOPUQULDCUOUPC
      EUPCQRULDQDCGUOUJUPUKADCBSDCFUAUBUCUDUEABFTUJCFTUF $.
  $}

  ${
    $d w x y $.  $d ph w $.
    sb8euv.nf $e |- F/ y ph $.
    $( Variable substitution in unique existential quantifier.  Version of
       ~ sb8eu requiring more disjoint variables, but fewer axioms.
       (Contributed by NM, 7-Aug-1994.)  (Revised by Wolf Lammen,
       7-Feb-2023.) $)
    sb8euv $p |- ( E! x ph <-> E! y [ y / x ] ph ) $=
      ( vw nfsbv sb8eulem ) ABCEABECDFG $.
    $( $j usage 'sb8euv' avoids 'ax-13'; $)
  $}

  ${
    $d w y $.  $d ph w $.  $d w x $.
    sb8eu.1 $e |- F/ y ph $.
    $( Variable substitution in unique existential quantifier.  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  For a version
       requiring more disjoint variables, but fewer axioms, see ~ sb8euv .
       (Contributed by NM, 7-Aug-1994.)  (Revised by Mario Carneiro,
       7-Oct-2016.)  (Proof shortened by Wolf Lammen, 24-Aug-2019.)
       (New usage is discouraged.) $)
    sb8eu $p |- ( E! x ph <-> E! y [ y / x ] ph ) $=
      ( vw nfsb sb8eulem ) ABCEABECDFG $.

    $( Variable substitution for the at-most-one quantifier.  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  (Contributed by
       Alexander van der Vekens, 17-Jun-2017.)  (New usage is discouraged.) $)
    sb8mo $p |- ( E* x ph <-> E* y [ y / x ] ph ) $=
      ( wex weu wi wsb wmo sb8e sb8eu imbi12i moeu 3bitr4i ) ABEZABFZGABCHZCEZQ
      CFZGABIQCIORPSABCDJABCDKLABMQCMN $.
  $}

  ${
    $d x y $.  $d x z ps $.  $d y z ph $.
    cbvmovw.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Change bound variable.  Uses only Tarski's FOL axiom schemes.  See
       ~ cbvmo and ~ cbvmow for versions with fewer disjoint variable
       conditions but requiring more axioms.  (Contributed by NM, 9-Mar-1995.)
       (Revised by GG, 30-Sep-2024.) $)
    cbvmovw $p |- ( E* x ph <-> E* y ps ) $=
      ( vz weq wi wal wex wmo equequ1 imbi12d cbvalvw exbii df-mo 3bitr4i ) ACF
      GZHZCIZFJBDFGZHZDIZFJACKBDKTUCFSUBCDCDGABRUAECDFLMNOACFPBDFPQ $.
    $( $j usage 'cbvmovw' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    $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, which does not
       require ~ ax-10 , ~ ax-13 .  (Contributed by NM, 9-Mar-1995.)  (Revised
       by GG, 23-May-2024.) $)
    cbvmow $p |- ( E* x ph <-> E* y ps ) $=
      ( vz weq wi wal wex wmo nfv nfim equequ1 imbi12d cbvalv1 exbii df-mo
      3bitr4i ) ACHIZJZCKZHLBDHIZJZDKZHLACMBDMUDUGHUCUFCDAUBDEUBDNOBUECFUECNOCD
      IABUBUEGCDHPQRSACHTBDHTUA $.
    $( $j usage 'cbvmow' avoids 'ax-10' 'ax-13'; $)
  $}

  ${
    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.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .  Use the
       weaker ~ cbvmow , ~ cbvmovw when possible.  (Contributed by NM,
       9-Mar-1995.)  (Revised by Andrew Salmon, 8-Jun-2011.)  (Proof shortened
       by Wolf Lammen, 4-Jan-2023.)  (New usage is discouraged.) $)
    cbvmo $p |- ( E* x ph <-> E* y ps ) $=
      ( wmo wsb sb8mo sbie mobii bitri ) ACHACDIZDHBDHACDEJNBDABCDFGKLM $.
  $}

  ${
    $d x y $.  $d x ps $.  $d y ph $.
    cbveuvw.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Change bound variable.  Uses only Tarski's FOL axiom schemes.  See
       ~ cbveu for a version with fewer disjoint variable conditions but
       requiring more axioms.  (Contributed by NM, 25-Nov-1994.)  (Revised by
       GG, 30-Sep-2024.) $)
    cbveuvw $p |- ( E! x ph <-> E! y ps ) $=
      ( wex wmo wa weu cbvexvw cbvmovw anbi12i df-eu 3bitr4i ) ACFZACGZHBDFZBDG
      ZHACIBDIOQPRABCDEJABCDEKLACMBDMN $.
    $( $j usage 'cbveuvw' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    $d x y $.
    cbveuw.1 $e |- F/ y ph $.
    cbveuw.2 $e |- F/ x ps $.
    cbveuw.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Version of ~ cbveu with a disjoint variable condition, which does not
       require ~ ax-10 , ~ ax-13 .  (Contributed by NM, 25-Nov-1994.)  (Revised
       by GG, 23-May-2024.) $)
    cbveuw $p |- ( E! x ph <-> E! y ps ) $=
      ( wex wmo wa weu cbvexv1 cbvmow anbi12i df-eu 3bitr4i ) ACHZACIZJBDHZBDIZ
      JACKBDKQSRTABCDEFGLABCDEFGMNACOBDOP $.
    $( $j usage 'cbveuw' avoids 'ax-10' 'ax-13'; $)
  $}

  ${
    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.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .  Use the
       weaker ~ cbveuw , ~ cbveuvw when possible.  (Contributed by NM,
       25-Nov-1994.)  (Revised by Mario Carneiro, 7-Oct-2016.)
       (New usage is discouraged.) $)
    cbveu $p |- ( E! x ph <-> E! y ps ) $=
      ( weu wsb sb8eu sbie eubii bitri ) ACHACDIZDHBDHACDEJNBDABCDFGKLM $.

    $( Alternative proof of ~ cbveu .  Since ~ df-eu combines two other
       quantifiers, one can base this theorem on their associated 'change
       bounded variable' kind of theorems as well.  (Contributed by Wolf
       Lammen, 5-Jan-2023.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    cbveuALT $p |- ( E! x ph <-> E! y ps ) $=
      ( wex wmo wa weu cbvex cbvmo anbi12i df-eu 3bitr4i ) ACHZACIZJBDHZBDIZJAC
      KBDKQSRTABCDEFGLABCDEFGMNACOBDOP $.
  $}

  ${
    $d x y $.
    eu2.nf $e |- F/ y ph $.
    $( An alternate way of defining existential uniqueness.  Definition 6.10 of
       [TakeutiZaring] p. 26.  (Contributed by NM, 8-Jul-1994.)  (Proof
       shortened by Wolf Lammen, 2-Dec-2018.) $)
    eu2 $p |- ( E! x ph <->
    ( E. x ph /\ A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) ) $=
      ( weu wex wmo wa wsb weq wi wal df-eu mo3 anbi2i bitri ) ABEABFZABGZHQAAB
      CIHBCJKCLBLZHABMRSQABCDNOP $.
  $}

  ${
    $d x y $.
    eu1.nf $e |- F/ 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.)  (Revised
       by Mario Carneiro, 7-Oct-2016.)  (Proof shortened by Wolf Lammen,
       29-Oct-2018.)  Avoid ~ ax-13 .  (Revised by Wolf Lammen, 7-Feb-2023.) $)
    eu1 $p |- ( E! x ph <->
                E. x ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) ) $=
      ( wsb weu weq wb wal wex wi wa nfs1v sb8euv sb6rfv equcom imbi2i anbi12ci
      euf albii albiim bitr4i exbii 3bitr4i ) ABCEZCFUECBGZHCIZBJABFAUEBCGZKZCI
      ZLZBJUECBABCMSABCDNUKUGBUKUEUFKZCIZUFUEKCIZLUGAUNUJUMABCDOUIULCUHUFUEBCPQ
      TRUEUFCUAUBUCUD $.
    $( $j usage 'eu1' avoids 'ax-13'; $)
  $}

  ${
    euor.nf $e |- F/ x ph $.
    $( Introduce a disjunct into a unique existential quantifier.  For a
       version requiring disjoint variables, but fewer axioms, see ~ euorv .
       (Contributed by NM, 21-Oct-2005.) $)
    euor $p |- ( ( -. ph /\ E! x ps ) -> E! x ( ph \/ ps ) ) $=
      ( wn weu wo nfn biorf eubid biimpa ) AEZBCFABGZCFLBMCACDHABIJK $.
  $}

  ${
    $d x ph $.
    $( Introduce a disjunct into a unique existential quantifier.  Version of
       ~ euor requiring disjoint variables, but fewer axioms.  (Contributed by
       NM, 23-Mar-1995.)  Reduce dependencies on axioms.  (Revised by Wolf
       Lammen, 14-Jan-2023.) $)
    euorv $p |- ( ( -. ph /\ E! x ps ) -> E! x ( ph \/ ps ) ) $=
      ( wn weu wo biorf eubidv biimpa ) ADZBCEABFZCEJBKCABGHI $.
  $}

  $( Introduce or eliminate a disjunct in a unique existential quantifier.
     (Contributed by NM, 21-Oct-2005.)  (Proof shortened by Andrew Salmon,
     9-Jul-2011.)  (Proof shortened by Wolf Lammen, 27-Dec-2018.) $)
  euor2 $p |- ( -. E. x ph -> ( E! x ( ph \/ ps ) <-> E! x ps ) ) $=
    ( wex wn wo nfe1 nfn wb 19.8a biorf bicomd nsyl5 eubid ) ACDZEABFZBCOCACGHA
    OPBIACJAEBPABKLMN $.

  ${
    $d w x z $.  $d w y z $.  $d w ph $.
    $( Substitution into an at-most-one quantifier.  (Contributed by Jeff
       Madsen, 2-Sep-2009.) $)
    sbmo $p |- ( [ y / x ] E* z ph <-> E* z [ y / x ] ph ) $=
      ( vw weq wal wex wsb wmo sbex sblim sbalv exbii bitri df-mo sbbii 3bitr4i
      wi nfv ) ADEFZSZDGZEHZBCIZABCIZUASZDGZEHZADJZBCIUFDJUEUCBCIZEHUIUCEBCKUKU
      HEUBUGBCDAUABCUABTLMNOUJUDBCADEPQUFDEPR $.
  $}

  ${
    $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 df-eu mo4 anbi2i bitri ) ACFACGZACHZIQABICDJK
      DLCLZIACMRSQABCDENOP $.
  $}

  $( Existential uniqueness implies uniqueness through reverse 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 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.)  (Proof shortened by Wolf
     Lammen, 1-Oct-2023.) $)
  euim $p |- ( ( E. x ph /\ A. x ( ph -> ps ) ) -> ( E! x ps -> E! x ph ) ) $=
    ( wi wal weu wmo wex euimmo exmoeub biimpd sylan9r ) ABDCEBCFACGZACHZACFZAB
    CINMOACJKL $.

  ${
    moanimlem.1 $e |- ( ph -> ( E* x ps <-> E* x ( ph /\ ps ) ) ) $.
    moanimlem.2 $e |- ( E. x ( ph /\ ps ) -> ph ) $.
    $( Factor out the common proof skeleton of ~ moanimv and ~ moanim .
       (Contributed by NM, 3-Dec-2001.)  (Proof shortened by Wolf Lammen,
       24-Dec-2018.)  Factor out common proof lines.  (Revised by Wolf Lammen,
       8-Feb-2023.) $)
    moanimlem $p |- ( E* x ( ph /\ ps ) <-> ( ph -> E* x ps ) ) $=
      ( wa wmo wi biimprcd wex nexmo nsyl5 moan ja impbii ) ABFZCGZABCGZHARQDIA
      RQPCJAQEPCKLBACMNO $.
  $}

  ${
    $d x ph $.
    $( Introduction of a conjunct into an at-most-one quantifier.  Version of
       ~ moanim requiring disjoint variables, but fewer axioms.  (Contributed
       by NM, 23-Mar-1995.)  Reduce axiom usage .  (Revised by Wolf Lammen,
       8-Feb-2023.) $)
    moanimv $p |- ( E* x ( ph /\ ps ) <-> ( ph -> E* x ps ) ) $=
      ( wa ibar mobidv simpl exlimiv moanimlem ) ABCABABDZCABEFJACABGHI $.
  $}

  ${
    moanim.1 $e |- F/ x ph $.
    $( Introduction of a conjunct into "at most one" quantifier.  For a version
       requiring disjoint variables, but fewer axioms, see ~ moanimv .
       (Contributed by NM, 3-Dec-2001.)  (Proof shortened by Wolf Lammen,
       24-Dec-2018.) $)
    moanim $p |- ( E* x ( ph /\ ps ) <-> ( ph -> E* x ps ) ) $=
      ( wa ibar mobid simpl exlimi moanimlem ) ABCABABEZCDABFGKACDABHIJ $.

    $( Introduction of a conjunct into unique existential quantifier.
       (Contributed by NM, 19-Feb-2005.)  (Proof shortened by Andrew Salmon,
       9-Jul-2011.)  (Proof shortened by Wolf Lammen, 24-Dec-2018.) $)
    euan $p |- ( E! x ( ph /\ ps ) <-> ( ph /\ E! x ps ) ) $=
      ( wa weu wex euex simpl exlimi syl ibar eubid biimprcd jcai biimpa impbii
      ) ABEZCFZABCFZESATSRCGARCHRACDABIJKATSABRCDABLMZNOATSUAPQ $.
  $}

  $( 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 $.

  $( Nested at-most-one and unique existential quantifiers.  (Contributed by
     NM, 25-Jan-2006.)  (Proof shortened by Wolf Lammen, 27-Dec-2018.) $)
  moaneu $p |- E* x ( ph /\ E! x ph ) $=
    ( wmo wa weu moanmo eumo anim2i moimi ax-mp ) AABCZDZBCAABEZDZBCABFNLBMKAAB
    GHIJ $.

  ${
    $d x ph $.
    $( Introduction of a conjunct into unique existential quantifier.
       (Contributed by NM, 23-Mar-1995.)  Reduce dependencies on axioms.
       (Revised by Wolf Lammen, 14-Jan-2023.) $)
    euanv $p |- ( E! x ( ph /\ ps ) <-> ( ph /\ E! x ps ) ) $=
      ( weu wex euex simpl exlimiv syl ibar eubidv biimprcd jcai biimpa impbii
      wa ) ABPZCDZABCDZPRASRQCEAQCFQACABGHIASRABQCABJKZLMASRTNO $.
  $}

  ${
    $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.)  (Proof shortened by Wolf
       Lammen, 17-Sep-2019.) $)
    mopick $p |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) ) $=
      ( vy wmo wa wex wi weq df-mo sp pm3.45 aleximi ax12ev2 syl6 syl5d exlimiv
      wal sylbi imp ) ACEZABFZCGZABHZUAACDIZHZCRZDGUCUDHZACDJUGUHDUGAUEUCBUFCKU
      GUCUEBFZCGUEBHUFUBUICAUEBLMBCDNOPQST $.
    $( $j usage 'mopick' avoids 'ax-10'; $)
  $}

  ${
    moexexlem.1 $e |- F/ y ph $.
    moexexlem.2 $e |- F/ y E* x ph $.
    moexexlem.3 $e |- F/ x E* y E. x ( ph /\ ps ) $.
    $( Factor out the proof skeleton of ~ moexex and ~ moexexvw .  (Contributed
       by Wolf Lammen, 2-Oct-2023.) $)
    moexexlem $p |- ( ( E* x ph /\ A. x E* y ps ) -> E* y E. x ( ph /\ ps ) )
      $=
      ( wmo wal wa wex wi nfmo1 nfa1 nfim mopick ex com23 alrimd moim spsd syl6
      exlimd wn nfex exsimpl exlimi nexmo nsyl5 a1d pm2.61d1 imp ) ACHZBDHZCIZA
      BJCKZDHZUMACKZUOUQLZUMAUSCACMUOUQCUNCNGOUMAUPBLZDIZUSUMAUTDFEUMUPABUMUPAB
      LABCPQRSVAUNUQCUPBDTUAUBUCURUDUQUOUPDKURUQUPURDADCEUEABCUFUGUPDUHUIUJUKUL
      $.
  $}

  ${
    $d x y $.
    $( Double quantification with "at most one".  (Contributed by NM,
       3-Dec-2001.) $)
    2moexv $p |- ( E* x E. y ph -> A. y E* x ph ) $=
      ( wex wmo nfe1 nfmov 19.8a moimi alrimi ) ACDZBEABECKCBACFGAKBACHIJ $.
    $( $j usage '2moexv' avoids 'ax-13'; $)

    $d y ph $.
    $( "At most one" double quantification.  Version of ~ moexexv with an
       additional disjoint variable condition, which does not require ~ ax-13 .
       (Contributed by NM, 26-Jan-1997.)  (Revised by GG, 22-Aug-2023.)  Factor
       out common proof lines with ~ moexex .  (Revised by Wolf Lammen,
       2-Oct-2023.) $)
    moexexvw $p |- ( ( E* x ph /\ A. x E* y ps ) -> E* y E. x ( ph /\ ps ) ) $=
      ( nfv wmo wa wex nfe1 nfmov moexexlem ) ABCDADEACFDEABGZCHCDLCIJK $.
    $( $j usage 'moexexvw' avoids 'ax-13'; $)
  $}

  ${
    $d x y $.
    $( A condition allowing to swap an existential quantifier and at
       at-most-one quantifier.  Version of ~ 2moswap with a disjoint variable
       condition, which does not require ~ ax-13 .  (Contributed by NM,
       10-Apr-2004.)  (Revised by GG, 22-Aug-2023.)  Factor out common proof
       lines with ~ moexexvw .  (Revised by Wolf Lammen, 2-Oct-2023.) $)
    2moswapv $p |- ( A. x E* y ph -> ( E* x E. y ph -> E* y E. x ph ) ) $=
      ( wmo wal wex nfmov moexexlem expcom 19.8a pm4.71ri exbii mobii imbitrrdi
      wa nfe1 ) ACDBEZACFZBDZRAOZBFZCDZABFZCDSQUBRABCACPZRCBUDGUABCTBPGHIUCUACA
      TBARACJKLMN $.
    $( $j usage '2moswapv' avoids 'ax-13'; $)
  $}

  ${
    $d x y $.
    $( A condition allowing to swap an existential quantifier and a unique
       existential quantifier.  Version of ~ 2euswap with a disjoint variable
       condition, which does not require ~ ax-13 .  (Contributed by NM,
       10-Apr-2004.)  (Revised by GG, 22-Aug-2023.) $)
    2euswapv $p |- ( A. x E* y ph -> ( E! x E. y ph -> E! y E. x ph ) ) $=
      ( wmo wal wex wa weu wi excomim a1i 2moswapv anim12d df-eu 3imtr4g ) ACDB
      EZACFZBFZQBDZGABFZCFZTCDZGQBHTCHPRUASUBRUAIPABCJKABCLMQBNTCNO $.
    $( $j usage '2euswapv' avoids 'ax-13'; $)

    $( Double quantification with existential uniqueness.  Version of ~ 2euex
       with ` x ` and ` y ` distinct, but not requiring ~ ax-13 .  (Contributed
       by NM, 3-Dec-2001.)  (Revised by Wolf Lammen, 2-Oct-2023.) $)
    2euexv $p |- ( E! x E. y ph -> E. y E! x ph ) $=
      ( wex weu wa df-eu excom nfe1 nfmov 19.8a moimi moeu sylib eximd biimtrid
      wmo wi impcom sylbi ) ACDZBEUABDZUABQZFABEZCDZUABGUCUBUEUBABDZCDUCUEABCHU
      CUFUDCUACBACIJUCABQUFUDRAUABACKLABMNOPST $.
    $( $j usage '2euexv' avoids 'ax-13'; $)

    $( Double existential uniqueness implies double unique existential
       quantification.  Version of ~ 2exeu with ` x ` and ` y ` distinct, but
       not requiring ~ ax-13 .  (Contributed by NM, 3-Dec-2001.)  (Revised by
       Wolf Lammen, 2-Oct-2023.) $)
    2exeuv $p |- ( ( E! x E. y ph /\ E! y E. x ph ) -> E! x E! y ph ) $=
      ( wex weu wa wmo eumo euex moimi syl 2euexv anim12ci df-eu sylibr ) ACDZB
      EZABDCEZFACEZBDZSBGZFSBEQUARTQPBGUAPBHSPBACIJKACBLMSBNO $.
    $( $j usage '2exeuv' avoids 'ax-13'; $)
  $}

  $( 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 nfeu1 nfe1 nfan eupick alrimi ) ACDZABEZCFZEABGCMOCACHNCIJA
    BCKL $.

  $( Existential uniqueness "pick" showing wff equivalence.  (Contributed by
     NM, 25-Nov-1994.)  (Proof shortened by Wolf Lammen, 27-Dec-2018.) $)
  eupickb $p |- ( ( E! x ph /\ E! x ps /\ E. x ( ph /\ ps ) ) ->
               ( ph <-> ps ) ) $=
    ( weu wa wex w3a wi eupick 3adant2 exancom sylan2b 3adant1 impbid ) ACDZBCD
    ZABECFZGABOQABHPABCIJPQBAHZOQPBAECFRABCKBACILMN $.

  $( Theorem *14.26 in [WhiteheadRussell] p. 192.  (Contributed by Andrew
     Salmon, 11-Jul-2011.)  (Proof shortened by Wolf Lammen, 27-Dec-2018.) $)
  eupickbi $p |- ( E! x ph -> ( E. x ( ph /\ ps ) <-> A. x ( ph -> ps ) ) ) $=
    ( weu wa wex wi wal eupicka ex euex exintr syl5com impbid ) ACDZABECFZABGCH
    ZOPQABCIJOACFQPACKABCLMN $.

  $( "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 wa wex w3a nfmo1 nfe1 nfan mopick ancld anim1d df-3an imbitrrdi eximd
    3impia ) ADEZABFZDGZACFZDGABCHZDGSUAFZUBUCDSUADADITDJKUDUBTCFUCUDATCUDABABD
    LMNABCOPQR $.

  ${
    moexex.1 $e |- F/ y ph $.
    $( "At most one" double quantification.  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  Use the version ~ moexexvw
       when possible.  (Contributed by NM, 3-Dec-2001.)  (Proof shortened by
       Wolf Lammen, 28-Dec-2018.)  Factor out common proof lines with
       ~ moexexvw .  (Revised by Wolf Lammen, 2-Oct-2023.)
       (New usage is discouraged.) $)
    moexex $p |- ( ( E* x ph /\ A. x E* y ps ) -> E* y E. x ( ph /\ ps ) ) $=
      ( nfmo wa wex nfe1 moexexlem ) ABCDEADCEFABGZCHCDKCIFJ $.
  $}

  ${
    $d y ph $.
    $( "At most one" double quantification.  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  Use the weaker ~ moexexvw
       when possible.  (Contributed by NM, 26-Jan-1997.)
       (New usage is discouraged.) $)
    moexexv $p |- ( ( E* x ph /\ A. x E* y ps ) -> E* y E. x ( ph /\ ps ) ) $=
      ( nfv moexex ) ABCDADEF $.
  $}

  $( Double quantification with "at most one".  Usage of this theorem is
     discouraged because it depends on ~ ax-13 .  Use the weaker ~ 2moexv when
     possible.  (Contributed by NM, 3-Dec-2001.)
     (New usage is discouraged.) $)
  2moex $p |- ( E* x E. y ph -> A. y E* x ph ) $=
    ( wex wmo nfe1 nfmo 19.8a moimi alrimi ) ACDZBEABECKCBACFGAKBACHIJ $.

  $( Double quantification with existential uniqueness.  Usage of this theorem
     is discouraged because it depends on ~ ax-13 .  Use the weaker ~ 2euexv
     when possible.  (Contributed by NM, 3-Dec-2001.)  (Proof shortened by
     Andrew Salmon, 9-Jul-2011.)  (New usage is discouraged.) $)
  2euex $p |- ( E! x E. y ph -> E. y E! x ph ) $=
    ( wex weu wa df-eu excom nfe1 nfmo wi 19.8a moimi moeu sylib eximd biimtrid
    wmo impcom sylbi ) ACDZBEUABDZUABRZFABEZCDZUABGUCUBUEUBABDZCDUCUEABCHUCUFUD
    CUACBACIJUCABRUFUDKAUABACLMABNOPQST $.

  $( Nested unique existential quantifier and at-most-one quantifier.
     (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 to swap an existential quantifier and at at-most-one
     quantifier.  Usage of this theorem is discouraged because it depends on
     ~ ax-13 .  Use the weaker ~ 2moswapv when possible.  (Contributed by NM,
     10-Apr-2004.)  (New usage is discouraged.) $)
  2moswap $p |- ( A. x E* y ph -> ( E* x E. y ph -> E* y E. x ph ) ) $=
    ( wmo wal wex wa nfe1 moexex expcom 19.8a pm4.71ri exbii mobii imbitrrdi )
    ACDBEZACFZBDZQAGZBFZCDZABFZCDRPUAQABCACHIJUBTCASBAQACKLMNO $.

  $( A condition allowing to swap an existential quantifier and a unique
     existential quantifier.  Usage of this theorem is discouraged because it
     depends on ~ ax-13 .  Use the weaker ~ 2euswapv when possible.
     (Contributed by NM, 10-Apr-2004.)  (New usage is discouraged.) $)
  2euswap $p |- ( A. x E* y ph -> ( E! x E. y ph -> E! y E. x ph ) ) $=
    ( wmo wal wex wa weu wi excomim a1i 2moswap anim12d df-eu 3imtr4g ) ACDBEZA
    CFZBFZQBDZGABFZCFZTCDZGQBHTCHPRUASUBRUAIPABCJKABCLMQBNTCNO $.

  $( Double existential uniqueness implies double unique existential
     quantification.  The converse does not hold.  Usage of this theorem is
     discouraged because it depends on ~ ax-13 .  Use the weaker ~ 2exeuv when
     possible.  (Contributed by NM, 3-Dec-2001.)  (Proof shortened by Mario
     Carneiro, 22-Dec-2016.)  (New usage is discouraged.) $)
  2exeu $p |- ( ( E! x E. y ph /\ E! y E. x ph ) -> E! x E! y ph ) $=
    ( wex weu wa wmo eumo euex moimi syl 2euex anim12ci df-eu sylibr ) ACDZBEZA
    BDCEZFACEZBDZSBGZFSBEQUARTQPBGUAPBHSPBACIJKACBLMSBNO $.

  ${
    $d x y z w $.  $d z w ph $.
    $( Two ways of expressing "there exists at most one ordered pair
       ` <. x , y >. ` such that ` ph ( x , y ) ` holds.  Note that this is not
       equivalent to ` E* x E* y ph ` .  See also ~ 2mo .  This is the analogue
       of ~ 2eu4 for existential uniqueness.  (Contributed by Wolf Lammen,
       26-Oct-2019.)  Reduce dependencies on axioms.  (Revised by Wolf Lammen,
       3-Jan-2023.) $)
    2mo2 $p |- ( ( E* x E. y ph /\ E* y E. x ph ) <->
                          E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) ) $=
      ( wex weq wi wal wa wmo exdistrv jcab 2albii 19.26-2 19.23v albii anbi12i
      alcom df-mo bitri 3bitri 2exbii 3bitr4ri ) ACFZBDGZHZBIZABFZCEGZHZCIZJZEF
      DFUHDFZULEFZJAUFUJJHZCIBIZEFDFUEBKZUICKZJUHULDELUQUMDEUQAUFHZAUJHZJZCIBIU
      TCIZBIZVACIBIZJUMUPVBBCAUFUJMNUTVABCOVDUHVEULVCUGBAUFCPQVEVABIZCIULVABCSV
      FUKCAUJBPQUARUBUCURUNUSUOUEBDTUICETRUD $.
  $}

  ${
    $d x y z w $.  $d z w ph $.
    $( Two ways of expressing "there exists at most one ordered pair
       ` <. x , y >. ` such that ` ph ( x , y ) ` holds.  See also ~ 2mo2 .
       (Contributed by NM, 2-Feb-2005.)  (Revised by Mario Carneiro,
       17-Oct-2016.)  (Proof shortened by Wolf Lammen, 2-Nov-2019.) $)
    2mo $p |- ( E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) <->
              A. x A. y A. z A. w ( ( ph /\ [ z / x ] [ w / y ] ph ) ->
                                                      ( x = z /\ y = w ) ) ) $=
      ( weq wa wi wal wex wsb wmo nfmo1 nfe1 nfmov nfan 19.8a spsbe sbimi nfv
      2mo2 biimpi 19.21bbi syl2ani sbcom2 sylbi anim12ii alrimi alrimivv sylbir
      mo3 nfs1v nfsbv pm3.21 imim1d alimd aleximi 2nexaln 2sb8ef xchnxbi pm2.21
      com12 wn 2alimi 2eximi 19.23bi pm2.61d1 impbii alrot4 bitri ) ABDFZCEFZGZ
      HZCIZBIZEJZDJZAACEKZBDKZGZVMHZCIZBIZEIZDIZWBEIDICIBIVRWFVRACJZBLZABJZCLZG
      ZWFABCDEUAWKWDDEWKWCBWHWJBWGBMWIBCABNOPWKWBCWHWJCWGCBACNOWICMPWHWAVKWJVLA
      WHWGWGBDKZVKVTACQVSWGBDACERSWHWGWLGVKHZBDWHWMDIBIWGBDWGDTUKUBUCUDAWJWIWIC
      EKZVLVTABQVTABDKZCEKWNACEBDUEWOWICEABDRSUFWJWIWNGVLHZCEWJWPEICIWICEWIETUK
      UBUCUDUGUHUHUIUJWFVTEJZDJZVRWEWQVQDWDVTVPEVTWDVPVTWCVOBVSBDULVTWBVNCVSBDC
      ACEULUMVTAWAVMVTAUNUOUPUPVBUQUQWRVCAVCZCIBIZVRWGBJWTWRABCURABCDEAETADTUSU
      TWTVREWTEJVRDWTVPDEWSVNBCAVMVAVDVEVFVFUFVGVHWBDEBCVIVJ $.
  $}

  ${
    $d z w ph $.  $d x y ps $.  $d x y z w $.
    2mos.1 $e |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) ) $.
    $( Double "there exists at most one", using implicit substitution.
       (Contributed by NM, 10-Feb-2005.)  (Proof shortened by Wolf Lammen,
       21-May-2025.) $)
    2mos $p |- ( E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) <->
             A. x A. y A. z A. w ( ( ph /\ ps ) -> ( x = z /\ y = w ) ) ) $=
      ( weq wa wi wal wex wsb 2mo 2sbievw anbi2i imbi1i 2albii bitri ) ACEHDFHI
      ZJDKCKFLELAADFMCEMZIZTJZFKEKZDKCKABIZTJZFKEKZDKCKACDEFNUDUGCDUCUFEFUBUETU
      ABAABCDFEGOPQRRS $.

    $( Obsolete version of ~ 2mos as of 21-May-2025.  (Contributed by NM,
       10-Feb-2005.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    2mosOLD $p |- ( E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) <->
             A. x A. y A. z A. w ( ( ph /\ ps ) -> ( x = z /\ y = w ) ) ) $=
      ( weq wa wi wal wex wsb 2mo nfv sbrim wb expcom sbievw 2albii pm5.74ri
      pm5.74d bitr3i anbi2i imbi1i bitri ) ACEHZDFHZIZJDKCKFLELAADFMZCEMZIZUIJZ
      FKEKZDKCKABIZUIJZFKEKZDKCKACDEFNUNUQCDUMUPEFULUOUIUKBAUJBCEUGUJBUGUJJUGAJ
      ZDFMUGBJZUGADFUGDOPURUSDFUHUGABUGUHABQGRUBSUCUASUDUETTUF $.
  $}

  $( Double existential uniqueness.  This theorem shows a condition under which
     a "naive" definition matches the correct one.  Usage of this theorem is
     discouraged because it depends on ~ ax-13 .  Use the weaker ~ 2eu1v when
     possible.  (Contributed by NM, 3-Dec-2001.)  (Proof shortened by Wolf
     Lammen, 23-Apr-2023.)  (New usage is discouraged.) $)
  2eu1 $p |- ( A. x E* y ph ->
        ( E! x E! y ph <-> ( E! x E. y ph /\ E! y E. x ph ) ) ) $=
    ( wmo wal weu wex wa wi 2eu2ex moeu albii euim sylan2b pm2.43b 2euswap syld
    ex syl jcad 2exeu impbid1 ) ACDZBEZACFZBFZACGZBFZABGCFZHUDUFUHUIUDUFUHUFUGB
    GZUDUFUHIZIABCJUJUDUKUDUJUGUEIZBEUKUCULBACKLUGUEBMNRSOZUDUFUHUIUMABCPQTABCU
    AUB $.

  ${
    $d x y $.
    $( Double existential uniqueness.  This theorem shows a condition under
       which a "naive" definition matches the correct one.  Version of ~ 2eu1
       with ` x ` and ` y ` distinct, but not requiring ~ ax-13 .  (Contributed
       by NM, 3-Dec-2001.)  (Revised by Wolf Lammen, 2-Oct-2023.) $)
    2eu1v $p |- ( A. x E* y ph ->
          ( E! x E! y ph <-> ( E! x E. y ph /\ E! y E. x ph ) ) ) $=
      ( wmo wal weu wex wa wi 2eu2ex moeu albii sylan2b ex syl pm2.43b 2euswapv
      euim syld jcad 2exeuv impbid1 ) ACDZBEZACFZBFZACGZBFZABGCFZHUDUFUHUIUDUFU
      HUFUGBGZUDUFUHIZIABCJUJUDUKUDUJUGUEIZBEUKUCULBACKLUGUEBRMNOPZUDUFUHUIUMAB
      CQSTABCUAUB $.
    $( $j usage '2eu1v' avoids 'ax-13'; $)
  $}

  $( Double existential uniqueness.  Usage of this theorem is discouraged
     because it depends on ~ ax-13 .  (Contributed by NM, 3-Dec-2001.)
     (New usage is discouraged.) $)
  2eu2 $p |- ( E! y E. x ph -> ( E! x E! y ph <-> E! x E. y ph ) ) $=
    ( wex weu wmo wi eumo 2moex wa 2eu1 simpl biimtrdi 3syl 2exeu expcom impbid
    wal ) ABDZCEZACEBEZACDBEZTSCFACFBRZUAUBGSCHACBIUCUAUBTJUBABCKUBTLMNUBTUAABC
    OPQ $.

  $( Double existential uniqueness.  Usage of this theorem is discouraged
     because it depends on ~ ax-13 .  (Contributed by NM, 3-Dec-2001.)  (Proof
     shortened by Wolf Lammen, 23-Apr-2023.)  (New usage is discouraged.) $)
  2eu3 $p |- ( A. x A. y ( E* x ph \/ E* y ph ) ->
 ( ( E! x E! y ph /\ E! y E! x ph ) <-> ( E! x E. y ph /\ E! y E. x ph ) ) ) $=
    ( wmo wo wal weu wa wb nfmo1 19.31 albii nfal 19.32 bitri 2eu1 biimpd ancom
    wex 2exeu imbitrdi jaoa ancomsd ancoms jca impbid1 sylbi ) ABDZACDZECFZBFZU
    HCFZUIBFZEZACGBGZABGCGZHZACSBGZABSCGZHZIUKULUIEZBFUNUJVABUHUICACJKLULUIBUHB
    CABJMNOUNUQUTUNUPUOUTULUPUTUMUOULUPUSURHZUTULUPVBACBPQUSURRUAUMUOUTABCPQUBU
    CUTUOUPABCTUSURUPACBTUDUEUFUG $.

  ${
    $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 ~ 2eu1 for a condition under which the naive definition holds and
       ~ 2exeu for a one-way implication.  See ~ 2eu5 and ~ 2eu8 for alternate
       definitions.  (Contributed by NM, 3-Dec-2001.)  (Proof shortened by Wolf
       Lammen, 14-Sep-2019.) $)
    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 wmo weq wi wal df-eu excom bianbi anbi12i anandi 2mo2 anbi2i
      3bitr2i ) ACFZBGZABFZCGZHUABFZUABIZHZUEUCCIZHZHUEUFUHHZHUEABDJCEJHKCLBLEF
      DFZHUBUGUDUIUABMUDUCCFUHUEUCCMACBNOPUEUFUHQUJUKUEABCDERST $.
    $( An alternate definition of double existential uniqueness (see ~ 2eu4 ).
       A mistake sometimes made in the literature is to use ` E! x E! y ` to
       mean "exactly one ` x ` and exactly one ` y ` ".  (For example, see
       Proposition 7.53 of [TakeutiZaring] p. 53.)  It turns out that this is
       actually a weaker assertion, as can be seen by expanding out the formal
       definitions.  This theorem shows that the erroneous definition can be
       repaired by conjoining ` A. x E* y ph ` as an additional condition.  The
       correct definition apparently has never been published.  ( ` E* ` means
       "there exists at most one".)  (Contributed by NM, 26-Oct-2003.)  Avoid
       ~ ax-13 .  (Revised by Wolf Lammen, 2-Oct-2023.) $)
    2eu5 $p |- ( ( E! x E! y ph /\ A. x E* y ph ) <->
      ( E. x E. y ph /\ E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) ) ) $=
      ( weu wmo wal wa wex weq wi 2eu1v pm5.32ri eumo 2moexv syl adantl pm4.71i
      2eu4 3bitr2i ) ACFBFZACGBHZIACJZBFZABJZCFZIZUCIUHUDBJABDKCEKILCHBHEJDJIUC
      UBUHABCMNUHUCUGUCUEUGUFCGUCUFCOACBPQRSABCDETUA $.
  $}

  ${
    $d x y z w $.  $d z w ph $.
    $( Two equivalent expressions for double existential uniqueness.
       (Contributed by NM, 2-Feb-2005.)  (Revised by Mario Carneiro,
       17-Oct-2016.)  (Proof shortened by Wolf Lammen, 2-Oct-2019.) $)
    2eu6 $p |- ( ( E! x E. y ph /\ E! y E. x ph ) <->
               E. z E. w A. x A. y ( ph <-> ( x = z /\ y = w ) ) ) $=
      ( wex weu wa weq wi wal wb 2eu4 imim2i sps exlimd syli wsb 2alimi 2eximi
      nfia1 nfa1 nfv simpl ax12v com12 spsd nfs1v sbequ1 imim2d al2imi sb6 2sb6
      simpr bitr3i imbitrdi sylcom ancld 2albiim imbitrrdi exlimi 2eximdv 2exsb
      imp biimpr sylibr biimp jca impbii bitri ) ACFZBGABFCGHVKBFZABDIZCEIZHZJZ
      CKZBKZEFDFZHZAVOLZCKZBKZEFDFZABCDEMVTWDVLVSWDVLVRWCDEVKVRWCJBVQWBBUAVKVRV
      RVOAJZCKBKZHWCVKVRWFVKVRVMVKJZBKZWFVKVQWHBVQVKWHVKVQVMWHVQAVMCVPCUBZVMCUC
      VPAVMJCVOVMAVMVNUDNOPVKBDUEQUFUGVRWHVMACERZJZBKZWFVQWGWKBVQVKWJVMVQAWJCWI
      ACEUHVPAWJJCAVPVNWJVOVNAVMVNUNNACEUIQOPUJUKWLWJBDRWFWJBDULABCDEUMUOUPUQUR
      AVOBCUSUTVAVBVDWDVLVSWDWFEFDFVLWCWFDEWAWEBCAVOVESTABCDEVCVFWCVRDEWAVPBCAV
      OVGSTVHVIVJ $.
  $}

  $( Two equivalent expressions for double existential uniqueness.  Usage of
     this theorem is discouraged because it depends on ~ ax-13 .  (Contributed
     by NM, 19-Feb-2005.)  (New usage is discouraged.) $)
  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 nfe1 nfeu euan ancom eubii 3bitri 3bitr4ri ) ABDZCEZACDZFZBEOP
    BEZFNPFZCEZBEROFOPBNBCABGHITQBTPNFZCEPOFQSUACNPJKPNCACGIPOJLKROJM $.

  $( Two equivalent expressions for double existential uniqueness.  Curiously,
     we can put ` E! ` on either of the internal conjuncts but not both.  We
     can also commute ` E! x E! y ` using ~ 2eu7 .  Usage of this theorem is
     discouraged because it depends on ~ ax-13 .  (Contributed by NM,
     20-Feb-2005.)  (New usage is discouraged.) $)
  2eu8 $p |- ( E! x E! y ( E. x ph /\ E. y ph ) <->
                E! x E! y ( E! x ph /\ E. y ph ) ) $=
    ( wex wa 2eu2 pm5.32i nfeu1 nfeu euan ancom eubii nfe1 3bitri 3bitr4ri 2eu7
    weu 3bitr3ri ) ACDZBQZABQZCQZEZTABDZCQZEUASEZCQZBQZUDSECQBQTUBUEACBFGUBSEZB
    QUBTEUHUCUBSBUABCABHIJUGUIBUGSUAEZCQSUBEUIUFUJCUASKLSUACACMJSUBKNLTUBKOABCP
    R $.

  ${
    $d x y $.
    $( Two ways to express "exactly one thing exists".  To paraphrase the
       statement and explain the label: there Exists a Unique thing if and only
       if for All ` x ` , ` x ` Equals some given (and disjoint) ` y ` .  Both
       sides are false in set theory, see Theorems ~ neutru and ~ dtru .
       (Contributed by NM, 5-Apr-2004.)  State the theorem using truth constant
       ` T. ` .  (Revised by BJ, 7-Oct-2022.)  Reduce axiom dependencies.
       (Revised by Wolf Lammen, 2-Mar-2023.) $)
    euae $p |- ( E! x T. <-> A. x x = y ) $=
      ( wtru weq wi wal wex wa extru biantrur hbaev 19.8w wn hbnaev alnex sylib
      weu con4i impbii trut albii exbii bitri eu3v 3bitr4ri ) CABDZEZAFZBGZCAGZ
      UIHUFAFZCAQUJUIAIJUKUKBGZUIUKULUKBABBKLUKULUKMZUMBFULMABBNUKBOPRSUKUHBUFU
      GAUFTUAUBUCCABUDUE $.

    $( Two ways to express "exactly one thing exists".  The left-hand side
       requires only one variable to express this.  Both sides are false in set
       theory, see Theorem ~ dtru .  (Contributed by NM, 5-Apr-2004.)  (Proof
       shortened by BJ, 7-Oct-2022.) $)
    exists1 $p |- ( E! x x = x <-> A. x x = y ) $=
      ( weq weu wtru wal equid bitru eubii euae bitri ) AACZADEADABCAFLEALAGHIA
      BJK $.

    $( A condition implying that at least two things exist.  (Contributed by
       NM, 10-Apr-2004.)  (Proof shortened by Andrew Salmon, 9-Jul-2011.)
       Reduce axiom usage.  (Revised by Wolf Lammen, 4-Mar-2023.) $)
    exists2 $p |- ( ( E. x ph /\ E. x -. ph ) -> -. E! x x = x ) $=
      ( vy wex weq weu wal axc16nf nfrd com12 exists1 alex bicomi 3imtr4g con2d
      wn imp ) ABDZAPBDZBBEBFZPRTSRBCEBGZABGZTSPZUARUBUAABABCBHIJBCKUBUCABLMNOQ
      $.
  $}


$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
  Other axiomatizations related to classical predicate calculus
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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).  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 used 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
  approach 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_,
  ~ http://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 form "is", not the plural form "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 more
  specific 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
  Aristotelian 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.  Using such specific
  constructs would also be anti-historical; Aristotle and others who directly
  followed his work focused on relating wholes to their parts, an approach now
  called part-whole theory.  The work of Cantor and Peano (over 2,000 years
  later) led to a sharper distinction between inclusion ( ` C_ ` ) and
  membership ( ` e. ` ); this distinction was not directly made in Aristotle's
  work.

  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 non-existent
  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),
  ~ http://www.iep.utm.edu/aris-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.

  The following twenty-four syllogisms (from ~ barbara to ~ bamalip ) are all
  proven from { ~ ax-mp , ~ ax-1 , ~ ax-2 , ~ ax-3 , ~ ax-gen , ~ ax-4 },
  which corresponds in the usual translation to modal logic (a universal (resp.
  existential) quantifier maps to necessity (resp. possibility)) to the weakest
  normal modal logic (K).  Some proofs could be shortened by using additionally
  ~ spi (inference form of ~ sp , which corresponds to the axiom (T) of modal
  logic), as demonstrated by ~ dariiALT , ~ barbariALT , ~ festinoALT ,
  ~ barocoALT , ~ daraptiALT .

$)

  $( Figure 1.  Aristotelian syllogisms are grouped by "figures", which does
     not 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 ~ http://www.friesian.com/aristotl.htm ,
       ~ http://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 ` .
       Instance of ~ barbara .  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.) $)
    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 .  See ~ dariiALT for a
       shorter proof requiring more axioms.  (Contributed by David A. Wheeler,
       24-Aug-2016.)  Reduce dependencies on axioms.  (Revised by BJ,
       16-Sep-2022.) $)
    darii $p |- E. x ( ch /\ ps ) $=
      ( wa wi wal wex id anim2d alimi ax-mp exim mp2 ) CAGZCBGZHZDIZQDJRDJABHZD
      ITEUASDUAABCUAKLMNFQRDOP $.

    $( Alternate proof of ~ darii , shorter but using more axioms.  This shows
       how the use of ~ spi may shorten some proofs of the Aristotelian
       syllogisms, even though this adds axiom dependencies.  Note that ~ spi
       is the inference associated with ~ sp , which corresponds to the axiom
       (T) of modal logic.  (Contributed by David A. Wheeler, 27-Aug-2016.)
       Added precisions on axiom usage.  (Revised by BJ, 27-Sep-2022.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    dariiALT $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 ` .  Instance of ~ darii .  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.) $)
    ferio $p |- E. x ( ch /\ -. ps ) $=
      ( wn darii ) ABGCDEFH $.
  $}

  ${
    barbarilem.min $e |- E. x ph $.
    barbarilem.maj $e |- A. x ( ph -> ps ) $.
    $( Lemma for ~ barbari and the other Aristotelian syllogisms with
       existential assumption.  (Contributed by BJ, 16-Sep-2022.) $)
    barbarilem $p |- E. x ( ph /\ ps ) $=
      ( wi wal wex wa exintr mp2 ) ABFCGACHABICHEDABCJK $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Barbari", 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.)  Reduce dependencies on axioms.  (Revised by
       BJ, 16-Sep-2022.) $)
    barbari $p |- E. x ( ch /\ ps ) $=
      ( barbara barbarilem ) CBDGABCDEFHI $.

    $( Alternate proof of ~ barbari , shorter but using more axioms.  See
       comment of ~ dariiALT .  (Contributed by David A. Wheeler, 27-Aug-2016.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    barbariALT $p |- E. x ( ch /\ ps ) $=
      ( wa wi barbara spi ancli eximii ) CCBHDGCBCBIDABCDEFJKLM $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Celaront", 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 ` .  Instance of ~ barbari .  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.) $)
    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.)  Reduce
       dependencies on axioms.  (Revised by BJ, 16-Sep-2022.) $)
    cesare $p |- A. x ( ch -> -. ph ) $=
      ( wn wi wal con2 alimi ax-mp celarent ) BACDABGHZDIBAGHZDIENODABJKLFM $.
  $}

  ${
    $( 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.)  Reduce dependencies on axioms.
       (Revised by BJ, 16-Sep-2022.) $)
    camestres $p |- A. x ( ch -> -. ph ) $=
      ( wn wi wal con3 alimi ax-mp celarent ) BGZACDABHZDINAGHZDIEOPDABJKLFM $.
  $}

  ${
    $( 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.)  Reduce dependencies on
       axioms.  (Revised by BJ, 16-Sep-2022.) $)
    festino $p |- E. x ( ch /\ -. ph ) $=
      ( wa wn wi wal wex con2 anim2d alimi ax-mp exim mp2 ) CBGZCAHZGZIZDJZRDKT
      DKABHIZDJUBEUCUADUCBSCABLMNOFRTDPQ $.

    $( Alternate proof of ~ festino , shorter but using more axioms.  See
       comment of ~ dariiALT .  (Contributed by David A. Wheeler, 27-Aug-2016.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    festinoALT $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.)  Reduce dependencies on axioms.
       (Revised by BJ, 16-Sep-2022.) $)
    baroco $p |- E. x ( ch /\ -. ph ) $=
      ( wn wa wi wal wex con3 anim2d alimi ax-mp exim mp2 ) CBGZHZCAGZHZIZDJZSD
      KUADKABIZDJUCEUDUBDUDRTCABLMNOFSUADPQ $.

    $( Alternate proof of ~ festino , shorter but using more axioms.  See
       comment of ~ dariiALT .  (Contributed by David A. Wheeler, 27-Aug-2016.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    barocoALT $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.)  Reduce
       dependencies on axioms.  (Revised by BJ, 16-Sep-2022.) $)
    cesaro $p |- E. x ( ch /\ -. ph ) $=
      ( wn cesare barbarilem ) CAHDGABCDEFIJ $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Camestros" $)
    camestros.maj $e |- A. x ( ph -> ps ) $.
    $( Minor premise for Camestros $)
    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.)  Reduce dependencies on axioms.
       (Revised by BJ, 16-Sep-2022.) $)
    camestros $p |- E. x ( ch /\ -. ph ) $=
      ( wn camestres barbarilem ) CAHDGABCDEFIJ $.
  $}

  $( Figure 3 $)

  ${
    $( Major premise for the Aristotelian syllogism "Datisi" $)
    datisi.maj $e |- A. x ( ph -> ps ) $.
    $( Minor premise for Datisi $)
    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.)  Shorten and reduce
       dependencies on axioms.  (Revised by BJ, 16-Sep-2022.) $)
    datisi $p |- E. x ( ch /\ ps ) $=
      ( wa wex exancom mpbi darii ) ABCDEACGDHCAGDHFACDIJK $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Disamis" $)
    disamis.maj $e |- E. x ( ph /\ ps ) $.
    $( Minor premise for Disamis $)
    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.)  Reduce dependencies on
       axioms.  (Revised by BJ, 16-Sep-2022.) $)
    disamis $p |- E. x ( ch /\ ps ) $=
      ( wa wex datisi exancom mpbi ) BCGDHCBGDHACBDFEIBCDJK $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Ferison" $)
    ferison.maj $e |- A. x ( ph -> -. ps ) $.
    $( Minor premise for Ferison $)
    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 ` .  Instance of ~ datisi .  In Aristotelian notation, EIO-3:  MeP
       and MiS therefore SoP. (Contributed by David A. Wheeler,
       28-Aug-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 $)
    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 ` .  Instance of ~ disamis .  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".  (Contributed by
       David A. Wheeler, 28-Aug-2016.) $)
    bocardo $p |- E. x ( ch /\ -. ps ) $=
      ( wn disamis ) ABGCDEFH $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Darapti" $)
    darapti.maj $e |- A. x ( ph -> ps ) $.
    $( Minor premise for Darapti $)
    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.)  Reduce dependencies on
       axioms.  (Revised by BJ, 16-Sep-2022.) $)
    darapti $p |- E. x ( ch /\ ps ) $=
      ( wa wi wal wex id alanimi mp2an pm3.43 alimi ax-mp exim mp2 ) ACBHZIZDJZ
      ADKTDKACIZABIZHZDJZUBUCDJUDDJUFFEUCUDUEDUELMNUEUADACBOPQGATDRS $.

    $( Alternate proof of ~ darapti , shorter but using more axioms.  See
       comment of ~ dariiALT .  (Contributed by David A. Wheeler, 27-Aug-2016.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    daraptiALT $p |- E. x ( ch /\ ps ) $=
      ( wa wi spi jca eximii ) ACBHDGACBACIDFJABIDEJKL $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Felapton" $)
    felapton.maj $e |- A. x ( ph -> -. ps ) $.
    $( Minor premise for Felapton $)
    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 ` .  Instance of ~ darapti .  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.) $)
    felapton $p |- E. x ( ch /\ -. ps ) $=
      ( wn darapti ) ABHCDEFGI $.
  $}

  $( Figure 4 $)

  ${
    $( Major premise for the Aristotelian syllogism "Calemes" $)
    calemes.maj $e |- A. x ( ph -> ps ) $.
    $( Minor premise for Calemes $)
    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.)  Reduce dependencies on axioms.
       (Revised by BJ, 16-Sep-2022.) $)
    calemes $p |- A. x ( ch -> -. ph ) $=
      ( wn wi wal con2 alimi ax-mp camestres ) ABCDEBCGHZDICBGHZDIFNODBCJKLM $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Dimatis" $)
    dimatis.maj $e |- E. x ( ph /\ ps ) $.
    $( Minor premise for Dimatis $)
    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.)  Shorten
       and reduce dependencies on axioms.  (Revised by BJ, 16-Sep-2022.) $)
    dimatis $p |- E. x ( ch /\ ph ) $=
      ( wa wex darii exancom mpbi ) ACGDHCAGDHBCADFEIACDJK $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Fresison" $)
    fresison.maj $e |- A. x ( ph -> -. ps ) $.
    $( Minor premise for Fresison $)
    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.)  Shorten
       and reduce dependencies on axioms.  (Revised by BJ, 16-Sep-2022.) $)
    fresison $p |- E. x ( ch /\ -. ph ) $=
      ( wa wex exancom mpbi festino ) ABCDEBCGDHCBGDHFBCDIJK $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Calemos" $)
    calemos.maj $e |- A. x ( ph -> ps ) $.
    $( Minor premise for Calemos $)
    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.)
       Shorten and reduce dependencies on axioms.  (Revised by BJ,
       16-Sep-2022.) $)
    calemos $p |- E. x ( ch /\ -. ph ) $=
      ( wn calemes barbarilem ) CAHDGABCDEFIJ $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Fesapo" $)
    fesapo.maj $e |- A. x ( ph -> -. ps ) $.
    $( Minor premise for Fesapo $)
    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.)  Reduce
       dependencies on axioms.  (Revised by BJ, 16-Sep-2022.) $)
    fesapo $p |- E. x ( ch /\ -. ph ) $=
      ( wn wi wal con2 alimi ax-mp felapton ) BACDABHIZDJBAHIZDJEOPDABKLMFGN $.
  $}

  ${
    $( Major premise for the Aristotelian syllogism "Bamalip" $)
    bamalip.maj $e |- A. x ( ph -> ps ) $.
    $( Minor premise for Bamalip $)
    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. Very similar to ~ barbari .  (Contributed by David A. Wheeler,
       28-Aug-2016.)  Shorten and reduce dependencies on axioms.  (Revised by
       BJ, 16-Sep-2022.) $)
    bamalip $p |- E. x ( ch /\ ph ) $=
      ( wa wex barbari exancom mpbi ) ACHDICAHDIBCADFEGJACDKL $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Intuitionistic logic
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  Intuitionistic (constructive) logic is similar to classical logic with the
  notable omission of ~ ax-3 and theorems such as ~ exmid or ~ peirce .  We
  mostly treat intuitionistic logic in a separate file, iset.mm, which is known
  as the Intuitionistic Logic Explorer on the web site.  However, iset.mm has a
  number of additional axioms (mainly to replace definitions like ~ df-or and
  ~ df-ex which are not valid in intuitionistic logic) and we want to prove
  those axioms here to demonstrate that adding those axioms in iset.mm does not
  make iset.mm any less consistent than set.mm.

  The following axioms are unchanged between set.mm and iset.mm: ~ ax-1 ,
  ~ ax-2 , ~ ax-mp , ~ ax-4 , ~ ax-11 , ~ ax-gen , ~ ax-7 , ~ ax-12 , ~ ax-8 ,
  ~ ax-9 , and ~ ax-5 .

  In this list of axioms, the ones that repeat earlier theorems are marked
  "(New usage is discouraged.)" so that the earlier theorems will be used
  consistently in other proofs.

$)

  $( Left 'and' elimination (intuitionistic logic axiom ax-ia1).  (Contributed
     by Jim Kingdon, 21-May-2018.)  (New usage is discouraged.) $)
  axia1 $p |- ( ( ph /\ ps ) -> ph ) $=
    ( simpl ) ABC $.

  $( Right 'and' elimination (intuitionistic logic axiom ax-ia2).  (Contributed
     by Jim Kingdon, 21-May-2018.)  (New usage is discouraged.) $)
  axia2 $p |- ( ( ph /\ ps ) -> ps ) $=
    ( simpr ) ABC $.

  $( 'And' introduction (intuitionistic logic axiom ax-ia3).  (Contributed by
     Jim Kingdon, 21-May-2018.)  (New usage is discouraged.) $)
  axia3 $p |- ( ph -> ( ps -> ( ph /\ ps ) ) ) $=
    ( pm3.2 ) ABC $.

  $( 'Not' introduction (intuitionistic logic axiom ax-in1).  (Contributed by
     Jim Kingdon, 21-May-2018.)  (New usage is discouraged.) $)
  axin1 $p |- ( ( ph -> -. ph ) -> -. ph ) $=
    ( pm2.01 ) AB $.

  $( 'Not' elimination (intuitionistic logic axiom ax-in2).  (Contributed by
     Jim Kingdon, 21-May-2018.)  (New usage is discouraged.) $)
  axin2 $p |- ( -. ph -> ( ph -> ps ) ) $=
    ( pm2.21 ) ABC $.

  $( Definition of 'or' (intuitionistic logic axiom ax-io).  (Contributed by
     Jim Kingdon, 21-May-2018.)  (New usage is discouraged.) $)
  axio $p |- ( ( ( ph \/ ch ) -> ps ) <->
      ( ( ph -> ps ) /\ ( ch -> ps ) ) ) $=
    ( jaob ) ABCD $.

  $( Specialization (intuitionistic logic axiom ax-4).  This is just ~ sp by
     another name.  (Contributed by Jim Kingdon, 31-Dec-2017.)
     (New usage is discouraged.) $)
  axi4 $p |- ( A. x ph -> ph ) $=
    ( sp ) ABC $.

  $( Converse of ~ axc4 (intuitionistic logic axiom ax-i5r).  (Contributed by
     Jim Kingdon, 31-Dec-2017.) $)
  axi5r $p |- ( ( A. x ph -> A. x ps ) -> A. x ( A. x ph -> ps ) ) $=
    ( wal wi hba1 hbim sp imim2i alrimih ) ACDZBCDZEKBECKLCACFBCFGLBKBCHIJ $.

  $( The setvar ` x ` is not free in ` A. x ph ` (intuitionistic logic axiom
     ax-ial).  (Contributed by Jim Kingdon, 31-Dec-2017.)
     (New usage is discouraged.) $)
  axial $p |- ( A. x ph -> A. x A. x ph ) $=
    ( hba1 ) ABC $.

  $( The setvar ` x ` is not free in ` E. x ph ` (intuitionistic logic axiom
     ax-ie1).  (Contributed by Jim Kingdon, 31-Dec-2017.)
     (New usage is discouraged.) $)
  axie1 $p |- ( E. x ph -> A. x E. x ph ) $=
    ( hbe1 ) ABC $.

  $( A key property of existential quantification (intuitionistic logic axiom
     ax-ie2).  (Contributed by Jim Kingdon, 31-Dec-2017.) $)
  axie2 $p |- ( A. x ( ps -> A. x ps ) ->
              ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) ) $=
    ( wal wi wnf wex wb nf5 19.23t sylbir ) BBCDECDBCFABECDACGBEHBCIABCJK $.

  $( Axiom of existence (intuitionistic logic axiom ax-i9).  In classical
     logic, this is equivalent to ~ ax-6 but in intuitionistic logic it needs
     to be stated using the existential quantifier.  (Contributed by Jim
     Kingdon, 31-Dec-2017.)  (New usage is discouraged.) $)
  axi9 $p |- E. x x = y $=
    ( ax6e ) ABC $.

  $( Axiom of Quantifier Substitution (intuitionistic logic axiom ax-10).  This
     is just ~ axc11n by another name.  (Contributed by Jim Kingdon,
     31-Dec-2017.)  (New usage is discouraged.) $)
  axi10 $p |- ( A. x x = y -> A. y y = x ) $=
    ( axc11n ) ABC $.

  $( Axiom of Quantifier Introduction (intuitionistic logic axiom ax-i12).  In
     classical logic, this is mostly a restatement of ~ axc9 (with one
     additional quantifier).  But in intuitionistic logic, changing the
     negations and implications to disjunctions makes it stronger.  Usage of
     this theorem is discouraged because it depends on ~ ax-13 .  (Contributed
     by Jim Kingdon, 31-Dec-2017.)  Avoid ~ ax-11 .  (Revised by Wolf Lammen,
     24-Apr-2023.)  (New usage is discouraged.) $)
  axi12 $p |- ( A. z z = x \/ ( A. z z = y \/
                A. z ( x = y -> A. z x = y ) ) ) $=
    ( weq wal wo wi nfa1 nfor 19.32 wn axc9 orrd orri orass mpbir mpgbi mpbi )
    CADZCEZCBDZCEZFZABDZUDCEGZCEZFZTUBUFFFUCUEFZUGCUCUECTUBCSCHUACHIJUHTUBUEFZF
    TUITKUBUEABCLMNTUBUEOPQTUBUFOR $.

  $( Axiom of Bundling (intuitionistic logic axiom ax-bnd).  In classical
     logic, this and ~ axi12 are fairly straightforward consequences of
     ~ axc9 .  But in intuitionistic logic, it is not easy to add the extra
     ` A. x ` to ~ axi12 and so we treat the two as separate axioms.  Usage of
     this theorem is discouraged because it depends on ~ ax-13 .  (Contributed
     by Jim Kingdon, 22-Mar-2018.)  (Proof shortened by Wolf Lammen,
     24-Apr-2023.)  (New usage is discouraged.) $)
  axbnd $p |- ( A. z z = x \/ ( A. z z = y \/
                 A. x A. z ( x = y -> A. z x = y ) ) ) $=
    ( weq wal wo wi nfae nfor 19.32 orass bitri axi12 mpbir mpgbi ) CADCEZCBDCE
    ZFZABDZSCEGCEZFZPQTAEZFFZAUAAERUBFUCRTAPQACAAHCBAHIJPQUBKLUAPQTFFABCMPQTKNO
    $.

$( End $[ set-pred.mm $] $)


$( Begin $[ set-class.mm $] $)
$( Skip $[ set-pred.mm $] $)
$(
###############################################################################
  ZF (ZERMELO-FRAENKEL) 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.

  A simplistic concept of sets, sometimes called "naive set theory", is
  vulnerable to a paradox called "Russell's Paradox" ( ~ ru ), a discovery that
  revolutionized the foundations of mathematics and logic.  Russell's Paradox
  spawned the development of set theories that countered the paradox, including
  the ZF set theory that is most widely used and is defined here.

  Except for Extensionality, the axioms basically say, "given an arbitrary set
  x (and, in the cases of Replacement and Regularity, provided that an
  antecedent is satisfied), there exists another set y based on or constructed
  from it, with the stated properties".  (The axiom of extensionality can also
  be restated this way as shown by ~ axexte .) The individual axiom links
  provide more detailed descriptions.  We derive the redundant ZF axioms of
  Separation, Null Set, and Pairing from the others as theorems.

$)


$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
  ZF Set Theory - start with the Axiom of Extensionality
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Introduce the Axiom of Extensionality
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x y z $.
    $( Axiom of extensionality.  An axiom of Zermelo-Fraenkel set theory.  It
       states that two sets are identical if they contain the same elements.
       Axiom Ext of [BellMachover] p. 461.  Its converse is a theorem of
       predicate logic, ~ elequ2g .

       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 predicate calculus axioms, we would rewrite all axioms
       involving equality with equality expanded according to the above
       definition.  Some of those axioms may be provable from ax-ext and would
       become redundant, but this hasn't been studied carefully.

       _General remarks_:  Our set theory axioms are presented using defined
       connectives ( ` <-> ` , ` E. ` , etc.) for convenience.  However, it is
       implicitly understood that the actual axioms use only the primitive
       connectives ` -> ` , ` -. ` , ` A. ` , ` = ` , and ` e. ` .  It is
       straightforward to establish the equivalence between the actual axioms
       and the ones we display, and we will not do so.

       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 Replacement ~ ax-rep , which involves a wff
       metavariable).  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, 21-May-1993.) $)
    ax-ext $a |- ( A. z ( z e. x <-> z e. y ) -> x = y ) $.

    $( The axiom of extensionality ( ~ ax-ext ) restated so that it postulates
       the existence of a set ` z ` given two arbitrary sets ` x ` and ` y ` .
       This way to express it follows the general idea of the other ZFC axioms,
       which is to postulate the existence of sets given other sets.
       (Contributed by NM, 28-Sep-2003.) $)
    axexte $p |- E. z ( ( z e. x <-> z e. y ) -> x = y ) $=
      ( wel wb weq wi wex wal ax-ext 19.36v mpbir ) CADCBDEZABFZGCHMCINGABCJMNC
      KL $.
  $}

  ${
    $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.)  Remove dependencies on
       ~ ax-10 , ~ ax-12 , ~ ax-13 .  (Revised by BJ, 12-Jul-2019.)  (Revised
       by Wolf Lammen, 9-Dec-2019.) $)
    axextg $p |- ( A. z ( z e. x <-> z e. y ) -> x = y ) $=
      ( vw wel wb wal weq elequ2 bibi1d albidv equequ1 imbi12d ax-ext chvarvv
      wi ) CDEZCBEZFZCGZDBHZPCAEZRFZCGZABHZPDADAHZTUDUAUEUFSUCCUFQUBRDACIJKDABL
      MDBCNO $.

    $d z x $.  $d z y $.
    $( A bidirectional version of the axiom of extensionality.  Although this
       theorem looks like a definition of equality, it requires the axiom of
       extensionality for its proof under our axiomatization.  See the comments
       for ~ ax-ext and ~ df-cleq .  (Contributed by NM, 14-Nov-2008.) $)
    axextb $p |- ( x = y <-> A. z ( z e. x <-> z e. y ) ) $=
      ( weq wel wb wal elequ2g axextg impbii ) ABDCAECBEFCGABCHABCIJ $.
  $}

  ${
    $d x y z $.  $d ph z $.
    axextmo.1 $e |- F/ x ph $.
    $( There exists at most one set with prescribed elements.  Theorem 1.1 of
       [BellMachover] p. 462.  (Contributed by NM, 30-Jun-1994.)  (Proof
       shortened by Wolf Lammen, 13-Nov-2019.)  Use the at-most-one quantifier.
       (Revised by BJ, 17-Sep-2022.) $)
    axextmo $p |- E* x A. y ( y e. x <-> ph ) $=
      ( vz wel wb wal wmo wa weq biantr alanimi ax-ext syl gen2 nfv nfbi nfal
      wi elequ2 bibi1d albidv mo4f mpbir ) CBFZAGZCHZBIUHCEFZAGZCHZJZBEKZTZEHBH
      UNBEULUFUIGZCHUMUGUJUOCUFAUILMBECNOPUHUKBEUJBCUIABUIBQDRSUMUGUJCUMUFUIABE
      CUAUBUCUDUE $.
  $}

  ${
    $d x y $.
    $( There exists at most one empty set.  With either ~ axnul or ~ axnulALT
       or ~ ax-nul , this proves that there exists a unique empty set.  In
       practice, once the language of classes is available, we use the stronger
       characterization among classes ~ eq0 .  (Contributed by NM,
       22-Dec-2007.)  Use the at-most-one quantifier.  (Revised by BJ,
       17-Sep-2022.)  (Proof shortened by Wolf Lammen, 26-Apr-2023.) $)
    nulmo $p |- E* x A. y -. y e. x $=
      ( wel wn wal wmo wfal wb nfv axextmo nbfal albii mobii mpbir ) BACZDZBEZA
      FOGHZBEZAFGABGAIJQSAPRBOKLMN $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Classes
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare new constants used in class abstractions. $)
  $c { $.  $( Left brace $)
  $c | $.  $( Vertical bar $)
  $c } $.  $( Right brace $)


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Class abstractions
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  $( Introduce the class abstraction (or class builder) notation:
     ` { x | ph } ` is the class of sets ` x ` such that ` ph ( x ) ` is true.
     A setvar variable can be expressed as a class abstraction per Theorem
     ~ cvjust , justifying the substitution of class variables for setvar
     variables via the use of ~ cv . $)
  cab $a class { x | ph } $.

  $( Register class abstraction as a primitive expression (lacking a definition
     in the strict sense). $)
  $( $j primitive 'cab'; $)

  $( Define class abstractions, that is, classes of the form ` { y | ph } ` ,
     which is read "the class of sets ` y ` such that ` ph ( y ) ` ".

     A few remarks are in order:

     1.  The axiomatic statement ~ df-clab does not define the class
     abstraction ` { y | ph } ` itself, that is, it does not have the form
     ` |- { y | ph } = ... ` that a standard definition should have (for a good
     reason: equality itself has not yet been defined or axiomatized for class
     abstractions; it is defined later in ~ df-cleq ).  Instead, ~ df-clab has
     the form ` |- ( x e. { y | ph } <-> ... ) ` , meaning that it only defines
     what it means for a setvar to be a member of a class abstraction.  As a
     consequence, one can say that ~ df-clab defines class abstractions if and
     only if a class abstraction is completely determined by which elements
     belong to it, which is the content of the axiom of extensionality
     ~ ax-ext .  Therefore, ~ df-clab can be considered a definition only in
     systems that can prove ~ ax-ext (and the necessary first-order logic).

     2.  As in all definitions, the definiendum (the left-hand side of the
     biconditional) has no disjoint variable conditions.  In particular, the
     setvar variables ` x ` and ` y ` need not be distinct, and the formula
     ` ph ` may depend on both ` x ` and ` y ` .  This is necessary, as with
     all definitions, since if there was for instance a disjoint variable
     condition on ` x , y ` , then one could not do anything with expressions
     like ` x e. { x | ph } ` which are sometimes useful to shorten proofs
     (because of ~ abid ).  Most often, however, ` x ` does not occur in
     ` { y | ph } ` and ` y ` is free in ` ph ` .

     3.  Remark 1 stresses that ~ df-clab does not have the standard form of a
     definition for a class, but one could be led to think it has the standard
     form of a definition for a formula.  However, it also fails that test
     since the membership predicate ` e. ` has already appeared earlier
     (outside of syntax e.g. in ~ ax-8 ).  Indeed, the definiendum extends, or
     "overloads", the membership predicate ` e. ` from formulas of the form
     "setvar ` e. ` setvar" to formulas of the form "setvar ` e. ` class
     abstraction".  This is possible because of ~ wcel and ~ cab , and it can
     be called an "extension" of the membership predicate because of ~ wel ,
     whose proof uses ~ cv .  An _a posteriori_ justification for ~ cv is given
     by ~ cvjust , stating that every setvar can be written as a class
     abstraction (though conversely not every class abstraction is a set, as
     illustrated by Russell's paradox ~ ru ).

     4.  Proof techniques.  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 done with theorems such as
     ~ vtoclg which is used, for example, to convert ~ elirrv to ~ elirr .

     5.  Definition or axiom?  The question arises with the three axiomatic
     statements introducing classes, ~ df-clab , ~ df-cleq , and ~ df-clel , to
     decide if they qualify as definitions or if they should be called axioms.
     Under the strict definition of "definition" (see ~ conventions ), they are
     not definitions (see Remarks 1 and 3 above, and similarly for ~ df-cleq
     and ~ df-clel ).  One could be less strict and decide to call "definition"
     every axiomatic statement which provides an eliminable and conservative
     extension of the considered axiom system.  But the notion of
     conservativity may be given two different meanings in set.mm, due to the
     difference between the "scheme level" of set.mm and the "object level" of
     classical treatments.  For a proof that these three axiomatic statements
     yield an eliminable and weakly (that is, object-level) conservative
     extension of FOL= plus ~ ax-ext , see Appendix of [Levy] p. 357.

     6.  References and history.  The concept of class abstraction dates back
     to at least Frege, and is used by Whitehead and Russell.  This definition
     is Definition 2.1 of [Quine] p. 16 and Axiom 4.3.1 of [Levy] p. 12.  It is
     called the "axiom of class comprehension" by [Levy] p. 358, who treats the
     theory of classes as an extralogical extension to predicate logic and set
     theory axioms.  He calls the construction ` { y | ph } ` a "class term".
     For a full description of how classes are introduced and how to recover
     the primitive language, see the books of Quine and Levy (and the comment
     of ~ eqabb for a quick overview).  For a general discussion of the theory
     of classes, see ~ mmset.html#class .  (Contributed by NM, 26-May-1993.)
     (Revised by BJ, 19-Aug-2023.) $)
  df-clab $a |- ( x e. { y | ph } <-> [ x / y ] ph ) $.

  $( Extension (in the sense of Remark 3 of the comment of ~ df-clab ) of
     ~ elequ1 from formulas of the form "setvar ` e. ` setvar" to formulas of
     the form "setvar ` e. ` class abstraction".  This extension does not
     require ~ ax-8 contrary to ~ elequ1 , but recall from Remark 3 of the
     comment of ~ df-clab that it can be considered an extension only because
     of ~ cvjust , which does require ~ ax-8 .

     This is an instance of ~ eleq1w where the containing class is a class
     abstraction, and contrary to it, it can be proved without ~ df-clel .  See
     also ~ eleq1 for general classes.

     The straightforward yet important fact that this statement can be proved
     from FOL= plus ~ df-clab (hence without ~ ax-ext , ~ df-cleq or
     ~ df-clel ) was stressed by Mario Carneiro.  (Contributed by BJ,
     17-Aug-2023.) $)
  eleq1ab $p |- ( x = y -> ( x e. { z | ph } <-> y e. { z | ph } ) ) $=
    ( weq wsb cv cab wcel sbequ df-clab 3bitr4g ) BCEADBFADCFBGADHZICGMIABCDJAB
    DKACDKL $.
  $( $j usage 'eleq1ab' avoids 'ax-ext' 'df-cleq' 'df-clel'; $)

  ${
    $d x z $.  $d y z $.  $d ph z $.
    $( Extension of ~ cleljust from formulas of the form "setvar ` e. ` setvar"
       to formulas of the form "setvar ` e. ` class abstraction".  This is an
       instance of ~ dfclel where the containing class is a class abstraction.
       The same remarks as for ~ eleq1ab apply.  (Contributed by BJ,
       8-Nov-2021.)  (Proof shortened by Steven Nguyen, 19-May-2023.) $)
    cleljustab $p |-
                   ( x e. { y | ph } <-> E. z ( z = x /\ z e. { y | ph } ) ) $=
      ( weq cv cab wcel wa wex eleq1ab equsexvw bicomi ) DBEDFACGZHZIDJBFNHZOPD
      BADBCKLM $.
    $( $j usage 'cleljustab' avoids 'ax-ext' 'df-cleq' 'df-clel'; $)
  $}

  $( Simplification of class abstraction notation when the free and bound
     variables are identical.  (Contributed by NM, 26-May-1993.) $)
  abid $p |- ( x e. { x | ph } <-> ph ) $=
    ( cv cab wcel wsb df-clab sbid bitri ) BCABDEABBFAABBGABHI $.

  $( A standard theorem of predicate calculus ( ~ stdpc4 ) expressed using
     class abstractions.  Closed form of ~ vexw .  (Contributed by BJ,
     14-Jun-2019.) $)
  vexwt $p |- ( A. x ph -> y e. { x | ph } ) $=
    ( wal wsb cv cab wcel stdpc4 df-clab sylibr ) ABDABCECFABGHABCIACBJK $.

  ${
    vexw.1 $e |- ph $.
    $( If ` ph ` is a theorem, then any set belongs to the class
       ` { x | ph } ` .  Therefore, ` { x | ph } ` is "a" universal class.

       This is the closest one can get to defining a universal class, or
       proving ~ vex , without using ~ ax-ext .  Note that this theorem has no
       disjoint variable condition and does not use ~ df-clel nor ~ df-cleq
       either: only propositional logic and ~ ax-gen and ~ df-clab .  This is
       ~ sbt expressed using class abstractions.

       Without ~ ax-ext , one cannot define "the" universal class, since one
       could not prove for instance the justification theorem
       ` { x | T. } = { y | T. } ` (see ~ vjust ).  Indeed, in order to prove
       any equality of classes, one needs ~ df-cleq , which has ~ ax-ext as a
       hypothesis.  Therefore, the classes ` { x | T. } ` ,
       ` { y | ( ph -> ph ) } ` , ` { z | ( A. t t = t -> A. t t = t ) } ` and
       countless others are all universal classes whose equality cannot be
       proved without ~ ax-ext .  Once ~ dfcleq is available, we will define
       "the" universal class in ~ df-v .

       Its degenerate instance is also a simple consequence of ~ abid (using
       ~ mpbir ).  (Contributed by BJ, 13-Jun-2019.)  Reduce axiom
       dependencies.  (Revised by Steven Nguyen, 25-Apr-2023.) $)
    vexw $p |- y e. { x | ph } $=
      ( cv cab wcel wsb sbt df-clab mpbir ) CEABFGABCHABCDIACBJK $.
  $}

  $( Every setvar is a member of ` { x | T. } ` , which is therefore "a"
     universal class.  Once class extensionality ~ dfcleq is available, we can
     say "the" universal class (see ~ df-v ).  This is ~ sbtru expressed using
     class abstractions.  (Contributed by BJ, 2-Sep-2023.) $)
  vextru $p |- y e. { x | T. } $=
    ( wtru tru vexw ) CABDE $.

  ${
    $d x y $.
    $( Bound-variable hypothesis builder for a class abstraction.  (Contributed
       by Mario Carneiro, 11-Aug-2016.)  Remove use of ~ ax-12 .  (Revised by
       SN, 20-Sep-2023.) $)
    nfsab1 $p |- F/ x y e. { x | ph } $=
      ( cv cab wcel wsb df-clab nfs1v nfxfr ) CDABEFABCGBACBHABCIJ $.

    $( Bound-variable hypothesis builder for a class abstraction.  (Contributed
       by NM, 26-May-1993.)  (Proof shortened by Wolf Lammen, 25-Oct-2024.) $)
    hbab1 $p |- ( y e. { x | ph } -> A. x y e. { x | ph } ) $=
      ( cv cab wcel nfsab1 nf5ri ) CDABEFBABCGH $.
  $}

  ${
    $d x y $.  $d x z $.
    hbab.1 $e |- ( ph -> A. x ph ) $.
    $( Bound-variable hypothesis builder for a class abstraction.  (Contributed
       by NM, 1-Mar-1995.)  Add disjoint variable condition to avoid ~ ax-13 .
       See ~ hbabg for a less restrictive version requiring more axioms.
       (Revised by GG, 20-Jan-2024.) $)
    hbab $p |- ( z e. { y | ph } -> A. x z e. { y | ph } ) $=
      ( cv cab wcel wsb df-clab hbsbw hbxfrbi ) DFACGHACDIBADCJACDBEKL $.
    $( $j usage 'hbab' avoids 'ax-13'; $)
  $}

  ${
    $d x z $.
    hbabg.1 $e |- ( ph -> A. x ph ) $.
    $( Bound-variable hypothesis builder for a class abstraction.  Usage of
       this theorem is discouraged because it depends on ~ ax-13 .  See ~ hbab
       for a version with more disjoint variable conditions, but not requiring
       ~ ax-13 .  (Contributed by NM, 1-Mar-1995.)
       (New usage is discouraged.) $)
    hbabg $p |- ( z e. { y | ph } -> A. x z e. { y | ph } ) $=
      ( cv cab wcel wsb df-clab hbsb hbxfrbi ) DFACGHACDIBADCJACDBEKL $.
  $}

  ${
    $d x y $.  $d x z $.
    nfsab.1 $e |- F/ x ph $.
    $( Bound-variable hypothesis builder for a class abstraction.  (Contributed
       by Mario Carneiro, 11-Aug-2016.)  Add disjoint variable condition to
       avoid ~ ax-13 .  See ~ nfsabg for a less restrictive version requiring
       more axioms.  (Revised by GG, 20-Jan-2024.) $)
    nfsab $p |- F/ x z e. { y | ph } $=
      ( cv cab wcel nf5ri hbab nf5i ) DFACGHBABCDABEIJK $.
    $( $j usage 'nfsab' avoids 'ax-13'; $)
  $}

  ${
    $d x z $.
    nfsabg.1 $e |- F/ x ph $.
    $( Bound-variable hypothesis builder for a class abstraction.  Usage of
       this theorem is discouraged because it depends on ~ ax-13 .  See ~ nfsab
       for a version with more disjoint variable conditions, but not requiring
       ~ ax-13 .  (Contributed by Mario Carneiro, 11-Aug-2016.)
       (New usage is discouraged.) $)
    nfsabg $p |- F/ x z e. { y | ph } $=
      ( cv cab wcel nf5ri hbabg nf5i ) DFACGHBABCDABEIJK $.
  $}


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Class equality
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-

  This section introduces class equality in ~ df-cleq .

  Note that apart from the local introduction of class variables to state the
  syntax axioms ~ wceq and ~ wcel , this section is the first to use class
  variables.  Therefore, the file set.mm contains declarations of class
  variables at the beginning of this section (not visible on the webpages).

$)

  $( Declare variable symbols that will be used to represent classes.  Note
     that even though any class variable can be chosen to represent a class, it
     is usual to give mnemonic names, like using ` R ` and ` S ` for relations,
     ` F ` and ` G ` for functions, ` .0. ` for an additive identity, etc. $)
  $v ./\ $.
  $v .\/ $.
  $v .<_ $.
  $v .< $.
  $v .+ $.
  $v .- $.
  $v .X. $.
  $v ./ $.
  $v .^ $.
  $v .0. $.
  $v .1. $.
  $v .|| $.
  $v .~ $.
  $v ._|_ $.
  $v .+^ $.
  $v .+b $.
  $v .(+) $.
  $v .* $.
  $v .x. $.
  $v .xb $.
  $v ., $.
  $v .(x) $.
  $v .o. $.
  $v .0b $.

  $v A $.
  $v B $.
  $v C $.
  $v D $.
  $v P $.
  $v Q $.
  $v R $.
  $v S $.
  $v T $.
  $v U $.

  $( 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 .|| $.

  $( 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. $.

  $( Declare variable symbols that will be used to represent setvars. $)
  $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 $.

  ${
    $d x y z t u v A $.  $d x y z t u v B $.
    df-cleq.1 $e |- ( y = z <-> A. u ( u e. y <-> u e. z ) ) $.
    df-cleq.2 $e |- ( t = t <-> A. v ( v e. t <-> v e. t ) ) $.
    $( 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.

       The hypotheses express that all instances of the conclusion where class
       variables are replaced with setvar variables hold.  Therefore, this
       definition merely extends to class variables something that is true for
       setvar variables, hence is conservative.  This is only a proof sketch of
       conservativity; for details see Appendix of [Levy] p. 357.  This is the
       reason why we call this axiomatic statement a "definition", even though
       it does not have the usual form of a definition.  If we required a
       definition to have the usual form, we would call ~ df-cleq an axiom.

       See also comments under ~ df-clab , ~ df-clel , and ~ eqabb .

       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.

       While the three class definitions ~ df-clab , ~ df-cleq , and ~ df-clel
       are eliminable and conservative and thus meet the requirements for sound
       definitions, they are technically axioms in that they do not satisfy the
       requirements for the current definition checker.  The proofs of
       conservativity require external justification that is beyond the scope
       of the definition checker.

       For a general discussion of the theory of classes, see
       ~ mmset.html#class .  (Contributed by NM, 15-Sep-1993.)  (Revised by BJ,
       24-Jun-2019.) $)
    df-cleq $a |- ( A = B <-> A. x ( x e. A <-> x e. B ) ) $.
  $}

  ${
    $d x y z t u v A $.  $d x y z t u v B $.
    $( The defining characterization of class equality.  It is proved, over
       Tarski's FOL, from the axiom of (set) extensionality ( ~ ax-ext ) and
       the definition of class equality ( ~ df-cleq ).  Its forward implication
       is called "class extensionality".  Remark: the proof uses ~ axextb to
       prove also the hypothesis of ~ df-cleq that is a degenerate instance,
       but it could be proved also from minimal propositional calculus and {
       ~ ax-gen , ~ equid }.  (Contributed by NM, 15-Sep-1993.)  (Revised by
       BJ, 24-Jun-2019.) $)
    dfcleq $p |- ( A = B <-> A. x ( x e. A <-> x e. B ) ) $=
      ( vy vz vv vu vt axextb df-cleq ) ADEFGHBCDEGIHHFIJ $.
    $( $j usage 'dfcleq' avoids 'ax-8' 'ax-10' 'ax-11' 'ax-12' 'ax-13'
       'df-clab' 'df-clel'; $)
  $}

  ${
    $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 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."
       See ~ abid1 for the version of ~ cvjust extended to classes.
       (Contributed by NM, 7-Nov-2006.)  Avoid ~ ax-13 .  (Revised by Wolf
       Lammen, 4-May-2023.) $)
    cvjust $p |- x = { y | y e. x } $=
      ( vz cv wel cab wceq wcel wb dfcleq wsb df-clab elsb1 bitr2i mpgbir ) ADZ
      BAEZBFZGCAEZCDRHZICCPRJTQBCKSQCBLBCAMNO $.
    $( $j usage 'cvjust' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13' 'df-clel'; $)
  $}

  ${
    $d t x $.  $d t y $.  $d t z $.
    $( Proof of ~ ax-9 from Tarski's FOL and ~ dfcleq .  For a version not
       using ~ ax-8 either, see ~ eleq2w2 .  This shows that ~ dfcleq is too
       powerful to be used as a definition instead of ~ df-cleq .  Note that
       ~ ax-ext is also a direct consequence of ~ dfcleq (as an instance of its
       forward implication).  (Contributed by BJ, 24-Jun-2019.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    ax9ALT $p |- ( x = y -> ( z e. x -> z e. y ) ) $=
      ( vt weq wel wi wal wb cv dfcleq biimpi biimp sylg equcoms imim12d spimvw
      ax8 syl ) ABEZDAFZDBFZGZDHCAFZCBFZGZTUAUBIZUCDTUGDHDAJBJKLUAUBMNUCUFDCDCE
      UDUAUBUEUDUAGCDCDARODCBRPQS $.
    $( $j usage 'ax9ALT' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13' 'df-clab'
       'df-clel'; $)
  $}

  ${
    $d A x $.  $d B x $.
    $( A weaker version of ~ eleq2 (but stronger than ~ ax-9 and ~ elequ2 )
       that uses ~ ax-12 to avoid ~ ax-8 and ~ df-clel .  Compare ~ eleq2w ,
       whose setvars appear where the class variables are in this theorem, and
       vice versa.  (Contributed by BJ, 24-Jun-2019.)  Strengthen from setvar
       variables to class variables.  (Revised by WL and SN, 23-Aug-2024.) $)
    eleq2w2 $p |- ( A = B -> ( x e. A <-> x e. B ) ) $=
      ( wceq cv wcel wb wal dfcleq biimpi 19.21bi ) BCDZAEZBFMCFGZALNAHABCIJK
      $.
    $( $j usage 'eleq2w2' avoids 'ax-8' 'ax-10' 'ax-11' 'ax-13' 'df-clab'
       'df-clel'; $)
  $}

  ${
    $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, 21-Jun-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.)
       (Proof shortened by Wolf Lammen, 19-Nov-2019.) $)
    eqrdav $p |- ( ph -> A = B ) $=
      ( cv wcel wa biimpd impancom mpd biimprd impbida eqrdv ) ABCDABIZCJZRDJZA
      SKREJZTFAUASTAUAKZSTHLMNATKUASGAUATSUBSTHOMNPQ $.
  $}

  ${
    $d x A $.
    $( Law of identity (reflexivity of class equality).  Theorem 6.4 of [Quine]
       p. 41.

       This is part of Frege's eighth axiom per Proposition 54 of [Frege1879]
       p. 50; see also ~ biid .  An early mention of this law can be found in
       Aristotle, _Metaphysics_, Z.17, 1041a10-20.  (Thanks to Stefan Allan and
       BJ for this information.)  (Contributed by NM, 21-Jun-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 $.  $d x C $.  $d x ph $.
    eqeq1d.1 $e |- ( ph -> A = B ) $.
    $( Deduction from equality to equivalence of equalities.  (Contributed by
       NM, 27-Dec-1993.)  Reduce dependencies on axioms.  (Revised by Wolf
       Lammen, 5-Dec-2019.) $)
    eqeq1d $p |- ( ph -> ( A = C <-> B = C ) ) $=
      ( vx cv wcel wb wal wceq dfcleq biimpi bibi1 alimi albi 4syl 3bitr4g ) AF
      GZBHZSDHZIZFJZSCHZUAIZFJZBDKCDKABCKZTUDIZFJZUBUEIZFJUCUFIEUGUIFBCLMUHUJFT
      UDUANOUBUEFPQFBDLFCDLR $.
    $( $j usage 'eqeq1d' avoids 'ax-8' 'ax-10' 'ax-11' 'ax-12' 'ax-13'
       'df-clab'; $)

    $( Alternate proof of ~ eqeq1d , shorter but requiring ~ ax-12 .
       (Contributed by NM, 27-Dec-1993.)  (Revised by Wolf Lammen,
       19-Nov-2019.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    eqeq1dALT $p |- ( ph -> ( A = C <-> B = C ) ) $=
      ( vx cv wcel wb wal wceq dfcleq sylib 19.21bi bibi1d albidv 3bitr4g ) AFG
      ZBHZRDHZIZFJRCHZTIZFJBDKCDKAUAUCFASUBTASUBIZFABCKUDFJEFBCLMNOPFBDLFCDLQ
      $.
  $}

  $( Equality implies equivalence of equalities.  (Contributed by NM,
     26-May-1993.)  (Proof shortened by Wolf Lammen, 19-Nov-2019.) $)
  eqeq1 $p |- ( A = B -> ( A = C <-> B = C ) ) $=
    ( wceq id eqeq1d ) ABDZABCGEF $.

  ${
    eqeq1i.1 $e |- A = B $.
    $( Inference from equality to equivalence of equalities.  (Contributed by
       NM, 15-Jul-1993.) $)
    eqeq1i $p |- ( A = C <-> B = C ) $=
      ( wceq wb eqeq1 ax-mp ) ABEACEBCEFDABCGH $.
  $}

  ${
    eqcomd.1 $e |- ( ph -> A = B ) $.
    $( Deduction from commutative law for class equality.  (Contributed by NM,
       15-Aug-1994.)  Allow shortening of ~ eqcom .  (Revised by Wolf Lammen,
       19-Nov-2019.) $)
    eqcomd $p |- ( ph -> B = A ) $=
      ( wceq eqid eqeq1d mpbii ) ABBECBEBFABCBDGH $.
  $}

  $( Commutative law for class equality.  Theorem 6.5 of [Quine] p. 41.
     (Contributed by NM, 26-May-1993.)  (Proof shortened by Wolf Lammen,
     19-Nov-2019.) $)
  eqcom $p |- ( A = B <-> B = A ) $=
    ( wceq id eqcomd impbii ) ABCZBACZGABGDEHBAHDEF $.

  ${
    eqcoms.1 $e |- ( A = B -> ph ) $.
    $( Inference applying commutative law for class equality to an antecedent.
       (Contributed by NM, 24-Jun-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,
       26-May-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 $.
  $}

  ${
    eqeq2d.1 $e |- ( ph -> A = B ) $.
    $( Deduction from equality to equivalence of equalities.  (Contributed by
       NM, 27-Dec-1993.)  Allow shortening of ~ eqeq2 .  (Revised by Wolf
       Lammen, 19-Nov-2019.) $)
    eqeq2d $p |- ( ph -> ( C = A <-> C = B ) ) $=
      ( wceq eqeq1d eqcom 3bitr4g ) ABDFCDFDBFDCFABCDEGDBHDCHI $.
  $}

  $( Equality implies equivalence of equalities.  (Contributed by NM,
     26-May-1993.)  (Proof shortened by Wolf Lammen, 19-Nov-2019.) $)
  eqeq2 $p |- ( A = B -> ( C = A <-> C = B ) ) $=
    ( wceq id eqeq2d ) ABDZABCGEF $.

  ${
    eqeq2i.1 $e |- A = B $.
    $( Inference from equality to equivalence of equalities.  (Contributed by
       NM, 26-May-1993.) $)
    eqeq2i $p |- ( C = A <-> C = B ) $=
      ( wceq wb eqeq2 ax-mp ) ABECAECBEFDABCGH $.
  $}

  ${
    eqeqan12d.1 $e |- ( ph -> A = B ) $.
    eqeqan12d.2 $e |- ( ps -> C = D ) $.
    $( A useful inference for substituting definitions into an equality.  See
       also ~ eqeqan12dALT .  (Contributed by NM, 9-Aug-1994.)  (Proof
       shortened by Andrew Salmon, 25-May-2011.)  Shorten other proofs.
       (Revised by Wolf Lammen, 23-Oct-2024.) $)
    eqeqan12d $p |- ( ( ph /\ ps ) -> ( A = C <-> B = D ) ) $=
      ( wceq eqeq1d eqeq2d sylan9bb ) ACEIDEIBDFIACDEGJBEFDHKL $.
  $}

  ${
    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 $.
  $}

  ${
    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.)  (Proof shortened by Wolf Lammen, 23-Oct-2024.) $)
    eqeq12d $p |- ( ph -> ( A = C <-> B = D ) ) $=
      ( wceq wb eqeqan12d anidms ) ABDHCEHIAABCDEFGJK $.
  $}

  $( Equality relationship among four classes.  (Contributed by NM,
     3-Aug-1994.)  (Proof shortened by Wolf Lammen, 23-Oct-2024.) $)
  eqeq12 $p |- ( ( A = B /\ C = D ) -> ( A = C <-> B = D ) ) $=
    ( wceq id eqeqan12d ) ABEZCDEZABCDHFIFG $.

  ${
    eqeq12i.1 $e |- A = B $.
    eqeq12i.2 $e |- C = D $.
    $( A useful inference for substituting definitions into an equality.
       (Contributed by NM, 15-Jul-1993.)  (Proof shortened by Andrew Salmon,
       25-May-2011.)  (Proof shortened by Wolf Lammen, 20-Nov-2019.) $)
    eqeq12i $p |- ( A = C <-> B = D ) $=
      ( wceq eqeq1i eqeq2i bitri ) ACGBCGBDGABCEHCDBFIJ $.

    $( Theorem eqeq12i is the congruence law for equality. $)
    $( $j congruence 'eqeq12i'; $)
  $}

  ${
    eqeqan12dALT.1 $e |- ( ph -> A = B ) $.
    eqeqan12dALT.2 $e |- ( ps -> C = D ) $.
    $( Alternate proof of ~ eqeqan12d .  This proof has one more step but one
       fewer essential step.  (Contributed by NM, 9-Aug-1994.)  (Proof
       shortened by Andrew Salmon, 25-May-2011.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    eqeqan12dALT $p |- ( ( ph /\ ps ) -> ( A = C <-> B = D ) ) $=
      ( wceq wb eqeq12 syl2an ) ACDIEFICEIDFIJBGHCDEFKL $.
  $}

  $( 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.)  (Proof shortened by Wolf
     Lammen, 24-Oct-2024.) $)
  eqtr2 $p |- ( ( A = B /\ A = C ) -> B = C ) $=
    ( wceq eqeq1 biimpa ) ABDACDBCDABCEF $.

  $( A transitive law for class equality.  (Contributed by NM, 20-May-2005.)
     (Proof shortened by Wolf Lammen, 24-Oct-2024.) $)
  eqtr3 $p |- ( ( A = C /\ B = C ) -> A = B ) $=
    ( wceq eqeq2 biimparc ) BCDABDACDBCAEF $.

  ${
    eqtri.1 $e |- A = B $.
    eqtri.2 $e |- B = C $.
    $( An equality transitivity inference.  (Contributed by NM,
       26-May-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,
       26-May-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,
       26-May-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,
       21-Jun-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,
       21-Jun-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,
       21-Jun-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, 21-Jun-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 $.
  $}

  ${
    eq2tri.1 $e |- ( A = C -> D = F ) $.
    eq2tri.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 x z A $.
    $( Lemma for ~ iseqsetv-cleq .  (Contributed by Wolf Lammen, 17-Aug-2025.)
       (Proof modification is discouraged.) $)
    iseqsetvlem $p |- ( E. x x = A <-> E. z z = A ) $=
      ( cv wceq eqeq1 cbvexvw ) ADZCEBDZCEABHICFG $.

    $d y A $.  $d y z $.
    $( Alternate proof of ~ iseqsetv-clel .  The expression ` E. x x = A ` does
       not depend on a particular choice of the set variable.  The proof here
       avoids ~ df-clab , ~ df-clel and ~ ax-8 , but instead is based on
       ~ ax-9 , ~ ax-ext and ~ df-cleq .  In particular it still accepts
       ` x e. A ` being a primitive syntax term, not assuming any specific
       semantics (like elementhood in some form).

       Use it in contexts where you want to avoid ~ df-clab , or you need
       ~ df-cleq anyway.  See the alternative version , not using ~ df-cleq or
       ~ ax-ext or ~ ax-9 .  (Contributed by Wolf Lammen, 6-Aug-2025.)
       (Proof modification is discouraged.) $)
    iseqsetv-cleq $p |- ( E. x x = A <-> E. y y = A ) $=
      ( vz cv wceq wex iseqsetvlem bitr4i ) AECFAGDECFDGBECFBGADCHBDCHI $.
    $( $j usage 'iseqsetv-cleq' avoids 'ax-8' 'ax-10' 'ax-11' 'ax-12' 'ax-13'
       'df-clab'; $)
  $}

  ${
    $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-8' 'ax-10' 'ax-11' 'ax-12' '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.)  Avoid ~ ax-12 , based on an idea of
       Steven Nguyen.  (Revised by Wolf Lammen, 6-May-2023.) $)
    abbidv $p |- ( ph -> { x | ps } = { x | ch } ) $=
      ( wb wal cab wceq alrimiv abbi syl ) ABCFZDGBDHCDHIAMDEJBCDKL $.
  $}

  ${
    abbii.1 $e |- ( ph <-> ps ) $.
    $( Equivalent wff's yield equal class abstractions (inference form).
       (Contributed by NM, 26-May-1993.)  Remove dependency on ~ ax-10 ,
       ~ ax-11 , and ~ ax-12 .  (Revised by Steven Nguyen, 3-May-2023.) $)
    abbii $p |- { x | ph } = { x | ps } $=
      ( wb cab wceq abbi mpg ) 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, with
       nonfreeness hypothesis).  (Contributed by NM, 21-Jun-1993.)  (Revised by
       Mario Carneiro, 7-Oct-2016.)  Avoid ~ ax-10 and ~ ax-11 .  (Revised by
       Wolf Lammen, 6-May-2023.) $)
    abbid $p |- ( ph -> { x | ps } = { x | ch } ) $=
      ( wb wal cab wceq alrimi abbi syl ) 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-8' 'df-clel' ; $)
  $}

  ${
    $d y z ph $.  $d x z ps $.  $d x y $.
    cbvabv.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Rule used to change bound variables, using implicit substitution.
       Version of ~ cbvab with disjoint variable conditions requiring fewer
       axioms.  (Contributed by NM, 26-May-1999.)  Require ` x ` , ` y ` be
       disjoint to avoid ~ ax-11 and ~ ax-13 .  (Revised by Steven Nguyen,
       4-Dec-2022.) $)
    cbvabv $p |- { x | ph } = { y | ps } $=
      ( vz cab wsb cv wcel cbvsbv df-clab 3bitr4i eqriv ) FACGZBDGZACFHBDFHFIZO
      JQPJABCDFEKAFCLBFDLMN $.
    $( $j usage 'cbvabv' avoids 'ax-8' 'ax-10' 'ax-11' 'ax-12' 'ax-13'
       'df-clel' ; $)
  $}

  ${
    $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 ) ) $.
    $( Rule used to change bound variables, using implicit substitution.
       Version of ~ cbvab with a disjoint variable condition, which does not
       require ~ ax-10 , ~ ax-13 .  (Contributed by Andrew Salmon,
       11-Jul-2011.)  (Revised by GG, 23-May-2024.) $)
    cbvabw $p |- { x | ph } = { y | ps } $=
      ( vz cab wsb cv wcel cbvsbvf df-clab 3bitr4i eqriv ) HACIZBDIZACHJBDHJHKZ
      QLSRLABCDHEFGMAHCNBHDNOP $.
    $( $j usage 'cbvabw' avoids 'ax-8' 'ax-10' 'ax-13' 'df-clel'; $)
  $}

  ${
    $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.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .  Usage of
       the weaker ~ cbvabw and ~ cbvabv are preferred.  (Contributed by Andrew
       Salmon, 11-Jul-2011.)  (Proof shortened by Wolf Lammen, 16-Nov-2019.)
       (New usage is discouraged.) $)
    cbvab $p |- { x | ph } = { y | ps } $=
      ( vz cab wsb cv wcel sbco2 sbie sbbii bitr3i df-clab 3bitr4i eqriv ) HACI
      ZBDIZACHJZBDHJZHKZTLUDUALUBACDJZDHJUCACHDEMUEBDHABCDFGNOPAHCQBHDQRS $.
    $( $j usage 'cbvab' avoids 'ax-8' 'df-clel' ; $)
  $}

  ${
    $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, which requires fewer
       axioms.  (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 dfcleq wsb df-clab sbievw bitri bibi2i albii )
      EACGZHDIZEJZUATJZKZDLUBBKZDLDETMUDUEDUCBUBUCACDNBADCOABCDFPQRSQ $.
    $( $j usage 'eqabbw' avoids 'df-clel' 'ax-8' 'ax-10' 'ax-11' 'ax-12'; $)
  $}


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Class membership
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  ${
    $d x y z t u v A $.  $d x y z t u v B $.
    df-clel.1 $e |- ( y e. z <-> E. u ( u = y /\ u e. z ) ) $.
    df-clel.2 $e |- ( t e. t <-> E. v ( v = t /\ v e. t ) ) $.
    $( 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.

       The hypotheses express that all instances of the conclusion where class
       variables are replaced with setvar variables hold.  Therefore, this
       definition merely extends to class variables something that is true for
       setvar variables, hence is conservative.  This is only a proof sketch of
       conservativity; for details see Appendix of [Levy] p. 357.  This is the
       reason why we call this axiomatic statement a "definition", even though
       it does not have the usual form of a definition.  If we required a
       definition to have the usual form, we would call ~ df-clel an axiom.

       See also comments under ~ df-clab , ~ df-cleq , and ~ eqabb .

       Alternate characterizations of ` A e. B ` when either ` A ` or ` B ` is
       a set are given by ~ clel2g , ~ clel3g , and ~ clel4g .

       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.

       While the three class definitions ~ df-clab , ~ df-cleq , and ~ df-clel
       are eliminable and conservative and thus meet the requirements for sound
       definitions, they are technically axioms in that they do not satisfy the
       requirements for the current definition checker.  The proofs of
       conservativity require external justification that is beyond the scope
       of the definition checker.

       For a general discussion of the theory of classes, see
       ~ mmset.html#class .  (Contributed by NM, 26-May-1993.)  (Revised by BJ,
       27-Jun-2019.) $)
    df-clel $a |- ( A e. B <-> E. x ( x = A /\ x e. B ) ) $.
  $}

  ${
    $d x y z t u v A $.  $d x y z t u v B $.
    $( Characterization of the elements of a class.  (Contributed by BJ,
       27-Jun-2019.) $)
    dfclel $p |- ( A e. B <-> E. x ( x = A /\ x e. B ) ) $=
      ( vy vz vv vu vt cleljust df-clel ) ADEFGHBCDEGIHHFIJ $.
    $( $j usage 'dfclel' avoids 'ax-9' 'ax-10' 'ax-11' 'ax-12' 'ax-13'
       'ax-ext' 'df-clab' 'df-cleq'; $)
  $}

  ${
    $d x A $.  $d x B $.
    $( If a class contains another class, then it contains some set.
       (Contributed by Alan Sare, 25-Sep-2011.)  Avoid ~ ax-9 , ~ ax-ext ,
       ~ df-clab .  (Revised by Wolf Lammen, 30-Nov-2024.) $)
    elex2 $p |- ( A e. B -> E. x x e. B ) $=
      ( wcel cv wceq wa wex dfclel exsimpr sylbi ) BCDAEZBFZLCDZGAHNAHABCIMNAJK
      $.
      $( $j usage 'elex2' avoids 'ax-9' 'df-clab' 'ax-ext'; $)
  $}

  ${
    $d A x $.  $d x y $.
    $( Weak version of ~ isset .  (Contributed by BJ, 24-Apr-2024.) $)
    issettru $p |- ( E. x x = A <-> A e. { y | T. } ) $=
      ( cv wceq wex wtru cab wcel wa vextru biantru exbii dfclel bitr4i ) ADZCE
      ZAFQPGBHZIZJZAFCRIQTASQBAKLMACRNO $.
    $( $j usage 'issettru' avoids 'ax-9' 'ax-10' 'ax-11' 'ax-12' 'ax-13'
       'ax-ext' 'df-cleq'; $)
  $}

  ${
    $d x z A $.  $d y z A $.
    $( Alternate proof of ~ iseqsetv-cleq .  The expression ` E. x x = A ` does
       not depend on a particular choice of the set variable.  Use this theorem
       in contexts where ~ df-cleq or ~ ax-ext is not referenced elsewhere in
       your proof.  It is proven from a specific implementation (class builder,
       axiom ~ df-clab ) of the primitive term ` x e. A ` .  (Contributed by
       BJ, 29-Apr-2019.)  (Proof modification is discouraged.) $)
    iseqsetv-clel $p |- ( E. x x = A <-> E. y y = A ) $=
      ( vz cv wceq wex wtru cab wcel issettru bitr4i ) AECFAGCHDIJBECFBGADCKBDC
      KL $.
    $( $j usage 'iseqsetv-clel' avoids 'ax-9' 'ax-10' 'ax-11' 'ax-12' 'ax-13'
       'ax-ext' 'df-cleq'; $)
  $}

  ${
    $d A x $.  $d V x $.
    issetlem.1 $e |- x e. V $.
    $( Lemma for ~ elisset and ~ isset .  (Contributed by NM, 26-May-1993.)
       Extract from the proof of ~ isset .  (Revised by WL, 2-Feb-2025.) $)
    issetlem $p |- ( A e. V <-> E. x x = A ) $=
      ( wcel cv wceq wa wex dfclel biantru exbii bitr4i ) BCEAFZBGZNCEZHZAIOAIA
      BCJOQAPODKLM $.
  $}

  ${
    $d A x $.  $d V x $.
    $( An element of a class exists.  Version of ~ elisset with a disjoint
       variable condition on ` V , x ` , avoiding ~ df-clab .  Prefer its use
       over ~ elisset when sufficient (for instance in usages where ` x ` is a
       dummy variable).  (Contributed by BJ, 14-Sep-2019.) $)
    elissetv $p |- ( A e. V -> E. x x = A ) $=
      ( wcel cv wceq wa wex dfclel exsimpl sylbi ) BCDAEZBFZLCDZGAHMAHABCIMNAJK
      $.
    $( $j usage 'elissetv' avoids 'ax-9' 'ax-10' 'ax-11' 'ax-12' 'ax-13'
       'ax-ext' 'df-clab' 'df-cleq'; $)
  $}

  ${
    $d A x z $.  $d V z $.
    $( An element of a class exists.  Use ~ elissetv instead when sufficient
       (for instance in usages where ` x ` is a dummy variable).  (Contributed
       by NM, 1-May-1995.)  Reduce dependencies on axioms.  (Revised by BJ,
       29-Apr-2019.) $)
    elisset $p |- ( A e. V -> E. x x = A ) $=
      ( vz wcel cv wceq wex elissetv iseqsetv-clel sylib ) BCEDFBGDHAFBGAHDBCID
      ABJK $.
    $( $j usage 'elisset' avoids 'ax-9' 'ax-10' 'ax-11' 'ax-12' 'ax-13'
       'ax-ext' 'df-cleq'; $)
  $}

  ${
    $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 .

       Note that this provides a proof of ~ ax-8 from Tarski's FOL and ~ dfclel
       (simply consider an instance where ` A ` is replaced by a setvar and
       deduce the forward implication by ~ biimpd ), which shows that ~ dfclel
       is too powerful to be used as a definition instead of ~ df-clel .
       (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 dfclel 3bitr4g ) ABEZDAEZDF
      CGZHZDIDBEZQHZDIAFZCGBFZCGORTDOPSQABDJKLDUACMDUBCMN $.
    $( $j usage 'eleq1w' avoids 'ax-9' 'ax-10' 'ax-11' 'ax-12' 'ax-13'
       'ax-ext' 'df-clab' '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 dfclel 3bitr4g ) AEZBEZFZDE
      ZCFZROGZHZDISRPGZHZDICOGCPGQUAUCDQTUBSABDJKLDCOMDCPMN $.
    $( $j usage 'eleq2w' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'
       'ax-ext' 'df-clab' 'df-cleq'; $)
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.  $d x ph $.
    eleq1d.1 $e |- ( ph -> A = B ) $.
    $( Deduction from equality to equivalence of membership.  (Contributed by
       NM, 21-Jun-1993.)  Allow shortening of ~ eleq1 .  (Revised by Wolf
       Lammen, 20-Nov-2019.) $)
    eleq1d $p |- ( ph -> ( A e. C <-> B e. C ) ) $=
      ( vx cv wceq wcel wa wex eqeq2d anbi1d exbidv dfclel 3bitr4g ) AFGZBHZQDI
      ZJZFKQCHZSJZFKBDICDIATUBFARUASABCQELMNFBDOFCDOP $.

    $( Deduction from equality to equivalence of membership.  (Contributed by
       NM, 27-Dec-1993.)  Reduce dependencies on axioms.  (Revised by Wolf
       Lammen, 5-Dec-2019.) $)
    eleq2d $p |- ( ph -> ( C e. A <-> C e. B ) ) $=
      ( vx cv wceq wcel wa wex wb wal dfcleq sylib anbi2 alexbii dfclel 3bitr4g
      syl ) AFGZDHZUABIZJZFKZUBUACIZJZFKZDBIDCIAUCUFLZFMZUEUHLABCHUJEFBCNOUIUDU
      GFUCUFUBPQTFDBRFDCRS $.

    $( Alternate proof of ~ eleq2d , shorter at the expense of requiring
       ~ ax-12 .  (Contributed by NM, 27-Dec-1993.)  (Revised by Wolf Lammen,
       20-Nov-2019.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    eleq2dALT $p |- ( ph -> ( C e. A <-> C e. B ) ) $=
      ( vx cv wceq wcel wa wex wb wal dfcleq sylib 19.21bi anbi2d exbidv dfclel
      3bitr4g ) AFGZDHZUABIZJZFKUBUACIZJZFKDBIDCIAUDUFFAUCUEUBAUCUELZFABCHUGFME
      FBCNOPQRFDBSFDCST $.
  $}

  $( Equality implies equivalence of membership.  (Contributed by NM,
     26-May-1993.)  (Proof shortened by Wolf Lammen, 20-Nov-2019.) $)
  eleq1 $p |- ( A = B -> ( A e. C <-> B e. C ) ) $=
    ( wceq id eleq1d ) ABDZABCGEF $.

  $( Equality implies equivalence of membership.  (Contributed by NM,
     26-May-1993.)  (Proof shortened by Wolf Lammen, 20-Nov-2019.) $)
  eleq2 $p |- ( A = B -> ( C e. A <-> C e. B ) ) $=
    ( wceq id eleq2d ) ABDZABCGEF $.

  $( 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, 21-Jun-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, 26-May-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'; $)
    $}
  $}

  ${
    eleq12d.1 $e |- ( ph -> A = B ) $.
    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 $.

  ${
    eqeltri.1 $e |- A = B $.
    eqeltri.2 $e |- B e. C $.
    $( Substitution of equal classes into membership relation.  (Contributed by
       NM, 21-Jun-1993.) $)
    eqeltri $p |- A e. C $=
      ( wcel eleq1i mpbir ) ACFBCFEABCDGH $.
  $}

  ${
    eqeltrri.1 $e |- A = B $.
    eqeltrri.2 $e |- A e. C $.
    $( Substitution of equal classes into membership relation.  (Contributed by
       NM, 21-Jun-1993.) $)
    eqeltrri $p |- B e. C $=
      ( eqcomi eqeltri ) BACABDFEG $.
  $}

  ${
    eleqtri.1 $e |- A e. B $.
    eleqtri.2 $e |- B = C $.
    $( Substitution of equal classes into membership relation.  (Contributed by
       NM, 15-Jul-1993.) $)
    eleqtri $p |- A e. C $=
      ( wcel eleq2i mpbi ) ABFACFDBCAEGH $.
  $}

  ${
    eleqtrri.1 $e |- A e. B $.
    eleqtrri.2 $e |- C = B $.
    $( Substitution of equal classes into membership relation.  (Contributed by
       NM, 15-Jul-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 $.
  $}

  ${
    eqeltrid.1 $e |- A = B $.
    eqeltrid.2 $e |- ( ph -> B e. C ) $.
    $( A 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 ) $.
    $( A 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 ) $.
    $( A 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 ) $.
    $( A 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 $.
  $}

  ${
    3eltr3i.1 $e |- A e. B $.
    3eltr3i.2 $e |- A = C $.
    3eltr3i.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 $.
  $}

  ${
    3eltr4i.1 $e |- A e. B $.
    3eltr4i.2 $e |- C = A $.
    3eltr4i.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.)  (Proof shortened by Wolf Lammen,
       23-Nov-2019.) $)
    3eltr3g $p |- ( ph -> C e. D ) $=
      ( eqeltrrid eleqtrdi ) ADCEADBCGFIHJ $.
  $}

  ${
    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.)  (Proof shortened by Wolf Lammen,
       23-Nov-2019.) $)
    3eltr4g $p |- ( ph -> C e. D ) $=
      ( eqeltrid eleqtrrdi ) ADCEADBCGFIHJ $.
  $}

  ${
    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 $.
  $}

  ${
    eqneltri.1 $e |- A = B $.
    eqneltri.2 $e |- -. B e. C $.
    $( If a class is not an element of another class, an equal class is also
       not an element.  (Contributed by Glauco Siliprandi, 3-Jan-2021.) $)
    eqneltri $p |- -. A e. C $=
      ( wcel eleq1i mtbir ) ACFBCFEABCDGH $.
  $}

  ${
    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.)  (Proof shortened by Wolf Lammen, 13-Nov-2019.) $)
    eqneltrrd $p |- ( ph -> -. B e. C ) $=
      ( eqcomd eqneltrd ) ACBDABCEGFH $.
  $}

  ${
    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.)  (Proof shortened by Wolf Lammen, 13-Nov-2019.) $)
    neleqtrrd $p |- ( ph -> -. C e. A ) $=
      ( eqcomd neleqtrd ) ACBDEABCFGH $.
  $}

  $( 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 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 eqeq1 sbievw2 ) AEZCFBEZCFDEZCFABDIKCGKJCGH $.
  $}

  ${
    $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 eleq1w sbievw2 ) AECFBECFDECFABDADCGDBCGH $.
  $}

  ${
    $d w x z A $.  $d w y z $.
    $( Substitution for the second argument of the membership predicate in an
       atomic formula (class version of ~ elsb2 ).  (Contributed by Jim
       Kingdon, 22-Nov-2018.)  Reduce dependencies on axioms.  (Revised by Wolf
       Lammen, 24-Nov-2024.) $)
    clelsb2 $p |- ( [ y / x ] A e. x <-> A e. y ) $=
      ( vz cv wcel eleq2w sbievw2 ) CAEFCBEFCDEFABDADCGDBCGH $.
    $( $j usage 'clelsb2' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13' 'ax-ext'; $)

    $( Obsolete version of ~ clelsb2 as of 24-Nov-2024.)  (Contributed by Jim
       Kingdon, 22-Nov-2018.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    clelsb2OLD $p |- ( [ y / x ] A e. x <-> A e. y ) $=
      ( vw cv wcel wsb nfv sbco2 eleq2 sbie sbbii 3bitr3i ) CDEZFZDAGZABGODBGCA
      EZFZABGCBEZFZODBAOAHIPRABORDARDHNQCJKLOTDBTDHNSCJKM $.
  $}

  ${
    $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 as in ~ dfcleq .  See also
       ~ cleqf .  (Contributed by NM, 26-May-1993.)  (Proof shortened by Wolf
       Lammen, 14-Nov-2019.)  Remove dependency on ~ ax-13 .  (Revised by BJ,
       30-Nov-2020.) $)
    cleqh $p |- ( A = B <-> A. x ( x e. A <-> x e. B ) ) $=
      ( wceq cv wcel wal dfcleq nfv nf5i nfbi weq eleq1w bibi12d cbvalv1 bitr4i
      wb ) CDGBHZCIZUADIZTZBJAHZCIZUEDIZTZAJBCDKUHUDABUHBLUBUCAUBAEMUCAFMNABOUF
      UBUGUCABCPABDPQRS $.
  $}

  ${
    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 $.  $d x y $.
    hblem.1 $e |- ( y e. A -> A. x y e. A ) $.
    $( Change the free variable of a hypothesis builder.  (Contributed by NM,
       21-Jun-1993.)  (Revised by Andrew Salmon, 11-Jul-2011.)  Add disjoint
       variable condition to avoid ~ ax-13 .  See ~ hblemg for a less
       restrictive version requiring more axioms.  (Revised by GG,
       20-Jan-2024.) $)
    hblem $p |- ( z e. A -> A. x z e. A ) $=
      ( cv wcel wsb wal hbsbw clelsb1 albii 3imtr3i ) BFDGZBCHZOAICFDGZPAINBCAE
      JBCDKZOPAQLM $.
    $( $j usage 'hblem' avoids 'ax-13'; $)
  $}

  ${
    $d y A $.  $d x z $.
    hblemg.1 $e |- ( y e. A -> A. x y e. A ) $.
    $( Change the free variable of a hypothesis builder.  Usage of this theorem
       is discouraged because it depends on ~ ax-13 .  See ~ hblem for a
       version with more disjoint variable conditions, but not requiring
       ~ ax-13 .  (Contributed by NM, 21-Jun-1993.)  (Revised by Andrew Salmon,
       11-Jul-2011.)  (New usage is discouraged.) $)
    hblemg $p |- ( z e. A -> A. x z e. A ) $=
      ( cv wcel wsb wal hbsb clelsb1 albii 3imtr3i ) BFDGZBCHZOAICFDGZPAINBCAEJ
      BCDKZOPAQLM $.
  $}


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  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.)  Avoid ~ ax-11 .  (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.)  Avoid ~ ax-11 .  (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 $.
    $( A simplification of class abstraction.  Commuted form of ~ abid1 .  See
       comments there.  (Contributed by NM, 26-Dec-1993.) $)
    abid2 $p |- { x | x e. A } = A $=
      ( cv wcel cab abid1 eqcomi ) BACBDAEABFG $.
  $}

  ${
    $d x A y $.  $d ph y $.
    $( One direction of ~ eqabb is provable from fewer axioms.  (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 $.
    $( $j usage 'eqab' avoids 'ax-10' 'ax-11' 'ax-12'; $)

    $( 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 ~ zfauscl to ~ inex1 (look at the instance of
       ~ zfauscl 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.  Examples
       include ~ dfsymdif2 and ~ cp ; the latter derives a formula containing
       wff variables from substitution instances of the class variables in its
       equivalent formulation ~ cplem2 .  For more information on class
       variables, see Quine pp. 15-21 and/or Takeuti and Zaring pp. 10-13.

       Usage of ~ eqabbw is preferred since it requires fewer axioms.
       (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 $.

    $( Obsolete version of ~ eqabb as of 12-Feb-2025.  (Contributed by NM,
       26-May-1993.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    eqabbOLD $p |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) ) $=
      ( vy cab wceq cv wcel wb wal ax-5 hbab1 cleqh abid bibi2i albii bitri ) C
      ABEZFBGZCHZSRHZIZBJTAIZBJBDCRDGCHBKABDLMUBUCBUAATABNOPQ $.
  $}

  ${
    $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 $.
  $}

  ${
    eqabri.1 $e |- A = { x | ph } $.
    $( Equality of a class variable and a class abstraction (inference form).
       (Contributed by NM, 3-Apr-1996.)  (Proof shortened by Wolf Lammen,
       15-Nov-2019.) $)
    eqabri $p |- ( x e. A <-> ph ) $=
      ( cv wcel wb wtru cab wceq a1i eqabrd mptru ) BECFAGHABCCABIJHDKLM $.
  $}

  ${
    eqabcri.1 $e |- { x | ph } = A $.
    $( Equality of a class variable and a class abstraction (inference form).
       (Contributed by NM, 31-Jul-1994.)  (Proof shortened by Wolf Lammen,
       15-Nov-2019.) $)
    eqabcri $p |- ( ph <-> x e. A ) $=
      ( cv wcel cab eqcomi eqabri bicomi ) BECFAABCABGCDHIJ $.
  $}

  ${
    $d x A y $.  $d ph y $.
    $( Membership of a class variable in a class abstraction.  (Contributed by
       NM, 23-Dec-1993.)  (Proof shortened by Wolf Lammen, 16-Nov-2019.)  Avoid
       ~ ax-11 , see ~ sbc5ALT for more details.  (Revised by SN,
       2-Sep-2024.) $)
    clelab $p |- ( A e. { x | ph } <-> E. x ( x = A /\ ph ) ) $=
      ( vy cab wcel cv wex wa elissetv exsimpl iseqsetv-cleq sylib wb eleq1 weq
      wceq wsb df-clab sb5 bitri anbi1d exbidv bitrid bitr3d exlimiv pm5.21nii
      eqeq2 ) CABEZFZDGZCQZDHZBGZCQZAIZBHZDCUIJUQUOBHUMUOABKBDCLMULUJUQNDULUKUI
      FZUJUQUKCUIOURBDPZAIZBHZULUQURABDRVAADBSABDTUAULUTUPBULUSUOAUKCUNUHUBUCUD
      UEUFUG $.
    $( $j usage 'clelab' avoids 'ax-11'; $)
  $}

  ${
    $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 dfclel eqabb anbi2ci exbii bitri ) ABEZD
      FCGZRHZSDFZIZCJUABGSFAKBLZIZCJCRDMUBUDCTUCUAABSNOPQ $.
  $}

  ${
    $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 eqabdv ) ABECFDGZABHCDKABIJ $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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 eleq1w nfbidv cbvalvw ) BEDFZAGCEDFZAGBCBCHLMABCDIJK $.
  $}

  ${
    $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 "for all" or something that expands to one (such
       as "there exists").  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 nf5i nfci ) ABCBECFADGH $.
    $}
  $}

  ${
    $d x y z $.  $d z A $.
    $( Consequence of the not-free predicate.  (Contributed by Mario Carneiro,
       11-Aug-2016.)  Drop ~ ax-12 but use ~ ax-8 , ~ df-clel , and avoid a DV
       condition on ` y ` , ` A ` .  (Revised by SN, 3-Jun-2024.) $)
    nfcr $p |- ( F/_ x A -> F/ x y e. A ) $=
      ( vz wnfc cv wcel wnf wal df-nfc weq eleq1w nfbidv spvv sylbi ) ACEDFCGZA
      HZDIBFCGZAHZADCJQSDBDBKPRADBCLMNO $.
    $( $j usage 'nfcr' avoids 'ax-12'; $)
  $}

  ${
    $d x y $.  $d y A $.
    $( Alternate version of ~ nfcr .  Avoids ~ ax-8 but uses ~ ax-12 .
       (Contributed by Mario Carneiro, 11-Aug-2016.)
       (New usage is discouraged.)  (Proof modification is discouraged.) $)
    nfcrALT $p |- ( F/_ x A -> F/ x y e. A ) $=
      ( wnfc cv wcel wnf wal df-nfc sp sylbi ) ACDBECFAGZBHLABCILBJK $.
    $( $j usage 'nfcrALT' avoids 'ax-8' 'df-clel'; $)
  $}

  ${
    $d x y $.
    nfcri.1 $e |- F/_ x A $.
    $( Consequence of the not-free predicate.  (Contributed by Mario Carneiro,
       11-Aug-2016.)  Avoid ~ ax-10 , ~ ax-11 .  (Revised by GG, 23-May-2024.)
       Avoid ~ ax-12 (adopting Wolf Lammen's 13-May-2023 proof).  (Revised by
       SN, 3-Jun-2024.) $)
    nfcri $p |- F/ x y e. A $=
      ( wnfc cv wcel wnf nfcr ax-mp ) ACEBFCGAHDABCIJ $.
    $( $j usage 'nfcri' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $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 $.
    nfcrd.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 x y $.
    nfcrii.1 $e |- F/_ x A $.
    $( Consequence of the not-free predicate.  (Contributed by Mario Carneiro,
       11-Aug-2016.)  Avoid ~ ax-10 , ~ ax-11 .  (Revised by GG,
       23-May-2024.) $)
    nfcrii $p |- ( y e. A -> A. x y e. A ) $=
      ( cv wcel nfcri nf5ri ) BECFAABCDGH $.
    $( $j usage 'nfcrii' avoids 'ax-10' 'ax-11'; $)
  $}

  ${
    $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.)  Avoid ~ ax-8 and ~ df-clel .  (Revised by WL
       and SN, 23-Aug-2024.) $)
    nfceqdf $p |- ( ph -> ( F/_ x A <-> F/_ x B ) ) $=
      ( vy cv wcel wnf wal wnfc wceq eleq2w2 syl nfbidf albidv df-nfc 3bitr4g
      wb ) AGHZCIZBJZGKUADIZBJZGKBCLBDLAUCUEGAUBUDBEACDMUBUDTFGCDNOPQBGCRBGDRS
      $.
    $( $j usage 'nfceqdf' avoids 'ax-8' 'ax-10' 'ax-11' 'ax-13' 'df-clab'
       'df-clel'; $)
  $}

  ${
    $d x y $.  $d A y $.  $d B y $.
    nfceqi.1 $e |- A = B $.
    $( Equality theorem for class not-free.  (Contributed by Mario Carneiro,
       11-Aug-2016.)  (Proof shortened by Wolf Lammen, 16-Nov-2019.)  Avoid
       ~ ax-12 .  (Revised by Wolf Lammen, 19-Jun-2023.) $)
    nfceqi $p |- ( F/_ x A <-> F/_ x B ) $=
      ( vy cv wcel wnf wal wnfc eleq2i nfbii albii df-nfc 3bitr4i ) EFZBGZAHZEI
      PCGZAHZEIABJACJRTEQSABCPDKLMAEBNAECNO $.
    $( $j usage 'nfceqi' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13' 'df-clab' ; $)
  $}

  ${
    nfcxfr.1 $e |- A = B $.
    ${
      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 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 $.

    $( The setvar ` 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 x y $.
    clelsb1fw.1 $e |- F/_ x A $.
    $( Substitution for the first argument of the membership predicate in an
       atomic formula (class version of ~ elsb1 ).  Version of ~ clelsb1f with
       a disjoint variable condition, which does not require ~ ax-13 .
       (Contributed by Rodolfo Medina, 28-Apr-2010.)  Avoid ~ ax-13 .  (Revised
       by GG, 10-Jan-2024.) $)
    clelsb1fw $p |- ( [ y / x ] x e. A <-> y e. A ) $=
      ( vw cv wcel wsb nfcri sbco2v clelsb1 sbbii 3bitr3i ) EFCGZEAHZABHNEBHAFC
      GZABHBFCGNEBAAECDIJOPABEACKLEBCKM $.
    $( $j usage 'clelsb1fw' avoids 'ax-13'; $)
  $}

  ${
    $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 ).  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  See ~ clelsb1fw not
       requiring ~ ax-13 , but extra disjoint variables.  (Contributed by
       Rodolfo Medina, 28-Apr-2010.)  (Proof shortened by Andrew Salmon,
       14-Jun-2011.)  (Revised by Thierry Arnoux, 13-Mar-2017.)  (Proof
       shortened by Wolf Lammen, 7-May-2023.)  (New usage is discouraged.) $)
    clelsb1f $p |- ( [ y / x ] x e. A <-> y e. A ) $=
      ( vw cv wcel wsb nfcri sbco2 clelsb1 sbbii 3bitr3i ) EFCGZEAHZABHNEBHAFCG
      ZABHBFCGNEBAAECDIJOPABEACKLEBCKM $.
  $}

  ${
    $d x 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.)  Add disjoint variable condition to
       avoid ~ ax-13 .  See ~ nfabg for a less restrictive version requiring
       more axioms.  (Revised by GG, 20-Jan-2024.) $)
    nfab $p |- F/_ x { y | ph } $=
      ( vz cab nfsab nfci ) BEACFABCEDGH $.
    $( $j usage 'nfab' avoids 'ax-13'; $)
  $}

  ${
    $d x z $.  $d y z $.  $d z ph $.
    nfabg.1 $e |- F/ x ph $.
    $( Bound-variable hypothesis builder for a class abstraction.  Usage of
       this theorem is discouraged because it depends on ~ ax-13 .  See ~ nfab
       for a version with more disjoint variable conditions, but not requiring
       ~ ax-13 .  (Contributed by Mario Carneiro, 11-Aug-2016.)
       (New usage is discouraged.) $)
    nfabg $p |- F/_ x { y | ph } $=
      ( vz cab nfsabg nfci ) BEACFABCEDGH $.
  $}

  ${
    $d x y z $.  $d ph z $.
    $( Bound-variable hypothesis builder for a class abstraction.  (Contributed
       by Mario Carneiro, 14-Oct-2016.)  Add disjoint variable condition to
       avoid ~ ax-13 .  See ~ nfaba1g for a less restrictive version requiring
       more axioms.  (Revised by GG, 20-Jan-2024.)  Avoid ~ ax-6 , ~ ax-7 ,
       ~ ax-12 .  (Revised by SN, 14-May-2025.) $)
    nfaba1 $p |- F/_ x { y | A. x ph } $=
      ( vz wal cab cv wcel wsb df-clab sbal nfa1 nfxfr nfci ) BDABEZCFZDGPHOCDI
      ZBODCJQACDIZBEBABCDKRBLMMN $.
    $( $j usage 'nfaba1' avoids 'ax-6' 'ax-7' 'ax-12' 'ax-13'; $)

    $( Obsolete version of ~ nfaba1 as of 14-May-2025.  (Contributed by Mario
       Carneiro, 14-Oct-2016.)  (Revised by GG, 20-Jan-2024.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    nfaba1OLD $p |- F/_ x { y | A. x ph } $=
      ( wal nfa1 nfab ) ABDBCABEF $.
  $}

  $( Bound-variable hypothesis builder for a class abstraction.  Usage of this
     theorem is discouraged because it depends on ~ ax-13 .  See ~ nfaba1 for a
     version with a disjoint variable condition, but not requiring ~ ax-13 .
     (Contributed by Mario Carneiro, 14-Oct-2016.)
     (New usage is discouraged.) $)
  nfaba1g $p |- F/_ x { y | A. x ph } $=
    ( wal nfa1 nfabg ) ABDBCABEF $.

  ${
    $d x y $.  $d y A $.  $d y B $.  $d y ph $.
    nfeqd.1 $e |- ( ph -> F/_ x A ) $.
    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 wnf wnfc df-nfc sylib 19.21bi nfbid
      nfald nfxfrd ) CDHGIZCJZUCDJZKZGLABGCDMAUFBGAGNAUDUEBAUDBOZGABCPUGGLEBGCQ
      RSAUEBOZGABDPUHGLFBGDQRSTUAUB $.
    $( $j usage 'nfeqd' avoids 'ax-8'; $)

    $( 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 dfclel nfv nfcvd nfeqd nfcrd nfand nfexd nfxfrd
      ) CDHGIZCJZUADHZKZGLABGCDMAUDBGAGNAUBUCBABUACABUAOEPABGDFQRST $.
  $}

  ${
    $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.)  Remove dependency on ~ ax-13 .  (Revised by Wolf Lammen,
       10-Dec-2019.) $)
    nfnfc $p |- F/ x F/_ y A $=
      ( vz wnfc cv wcel wnf wal df-nfc mpbi spi nfnf nfal nfxfr ) BCFEGCHZBIZEJ
      ABECKRAEQABQAIZEACFSEJDAECKLMNOP $.
    $( $j usage 'nfnfc' avoids 'ax-8' 'ax-13'; $)

    nfeq.2 $e |- F/_ x B $.
    $( Hypothesis builder for equality.  (Contributed by NM, 21-Jun-1993.)
       (Revised by Mario Carneiro, 11-Aug-2016.)  (Proof shortened by Wolf
       Lammen, 16-Nov-2019.) $)
    nfeq $p |- F/ x A = B $=
      ( wceq wnf wtru wnfc a1i nfeqd mptru ) BCFAGHABCABIHDJACIHEJKL $.

    $( Hypothesis builder for elementhood.  (Contributed by NM, 1-Aug-1993.)
       (Revised by Mario Carneiro, 11-Aug-2016.)  (Proof shortened by Wolf
       Lammen, 16-Nov-2019.) $)
    nfel $p |- F/ x A e. B $=
      ( wcel wnf wtru wnfc a1i nfeld mptru ) BCFAGHABCABIHDJACIHEJKL $.
  $}

  ${
    $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 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.
       Usage of this theorem is discouraged because it depends on ~ ax-13 .
       (Contributed by Mario Carneiro, 8-Oct-2016.)  Avoid ~ ax-8 , ~ ax-11 .
       (Revised by Wolf Lammen, 22-Sep-2024.)  (New usage is discouraged.) $)
    drnfc1 $p |- ( A. x x = y -> ( F/_ x A <-> F/_ y B ) ) $=
      ( vw weq wal cv wcel wnf wnfc wceq wb eleq2w2 drnf1 albidv df-nfc 3bitr4g
      syl ) ABGAHZFIZCJZAKZFHUBDJZBKZFHACLBDLUAUDUFFUCUEABUACDMUCUENEFCDOTPQAFC
      RBFDRS $.
    $( $j usage 'drnfc1' avoids 'ax-8' 'ax-11'; $)

    $( Formula-building lemma for use with the Distinctor Reduction Theorem.
       Proof revision is marked as discouraged because the minimizer replaces
       ~ albidv with ~ dral2 , leading to a one byte longer proof.  However
       feel free to manually edit it according to conventions.  (TODO: ~ dral2
       depends on ~ ax-13 , hence its usage during minimizing is discouraged.
       Check in the long run whether this is a permanent restriction).  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .
       (Contributed by Mario Carneiro, 8-Oct-2016.)  Avoid ~ ax-8 .  (Revised
       by Wolf Lammen, 22-Sep-2024.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    drnfc2 $p |- ( A. x x = y -> ( F/_ z A <-> F/_ z B ) ) $=
      ( vw weq wal cv wcel wnf wnfc wceq wb eleq2w2 syl drnf2 albidv df-nfc
      3bitr4g ) ABHAIZGJZDKZCLZGIUCEKZCLZGICDMCEMUBUEUGGUDUFABCUBDENUDUFOFGDEPQ
      RSCGDTCGETUA $.
  $}

  ${
    $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, which does not require
       ~ ax-13 .  (Contributed by Mario Carneiro, 8-Oct-2016.)  Avoid ~ ax-13 .
       (Revised by GG, 10-Jan-2024.)  (Proof shortened by Wolf Lammen,
       23-Sep-2024.) $)
    nfabdw $p |- ( ph -> F/_ x { y | ps } ) $=
      ( vz cab nfv cv wcel weq wi wal wsb df-clab sb6 bitri nfvd nfimd nfxfrd
      nfald nfcd ) ACGBDHZAGIGJUDKZDGLZBMZDNZACUEBDGOUHBGDPBDGQRAUGCDEAUFBCAUFC
      SFTUBUAUC $.
    $( $j usage 'nfabdw' avoids 'ax-9' 'ax-13' 'ax-ext'; $)
  $}

  ${
    $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.  Usage of
       this theorem is discouraged because it depends on ~ ax-13 .  Use the
       weaker ~ nfabdw when possible.  (Contributed by Mario Carneiro,
       8-Oct-2016.)  Avoid ~ ax-9 and ~ ax-ext .  (Revised by Wolf Lammen,
       23-May-2023.)  (New usage is discouraged.) $)
    nfabd $p |- ( ph -> F/_ x { y | ps } ) $=
      ( vz cab nfv cv wcel wsb df-clab nfsbd nfxfrd nfcd ) ACGBDHZAGIGJQKBDGLAC
      BGDMABDGCEFNOP $.
    $( $j usage 'nfabd' avoids 'ax-9' 'ax-ext'; $)
  $}

  ${
    nfabd2.1 $e |- F/ y ph $.
    nfabd2.2 $e |- ( ( ph /\ -. A. x x = y ) -> F/ x ps ) $.
    $( Bound-variable hypothesis builder for a class abstraction.  Usage of
       this theorem is discouraged because it depends on ~ ax-13 .
       (Contributed by Mario Carneiro, 8-Oct-2016.)  (Proof shortened by Wolf
       Lammen, 10-May-2023.)  (New usage is discouraged.) $)
    nfabd2 $p |- ( ph -> F/_ x { y | ps } ) $=
      ( weq wal cab wnfc wn wa nfnae nfan nfabd ex nfab1 eqidd drnfc1 mpbiri
      pm2.61d2 ) ACDGCHZCBDIZJZAUBKZUDAUELBCDAUEDECDDMNFOPUBUDDUCJBDQCDUCUCUBUC
      RSTUA $.
  $}

  ${
    $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 .  Usage of this theorem is discouraged
       because it depends on ~ ax-13 .  (Contributed by Mario Carneiro,
       8-Oct-2016.)  (New usage is discouraged.) $)
    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.  Usage of this theorem is discouraged
       because it depends on ~ ax-13 .  (Contributed by Mario Carneiro,
       8-Oct-2016.)  (New usage is discouraged.) $)
    dvelimc $p |- ( -. A. x x = y -> F/_ x B ) $=
      ( weq wal wn wnfc wi wtru nftru a1i wceq dvelimdc mptru ) ABIAJKAELMNABCD
      EAOCOADLNFPCELNGPCBIDEQMNHPRS $.
  $}

  ${
    $d x w z $.  $d y w z $.
    $( If ` x ` and ` y ` are distinct, then ` x ` is not free in ` y ` .
       Usage of this theorem is discouraged because it depends on ~ ax-13 .
       See ~ nfcv for a version that replaces the distinctor with a disjoint
       variable condition, requiring fewer axioms.  (Contributed by Mario
       Carneiro, 8-Oct-2016.)  Avoid ~ ax-ext .  (Revised by Wolf Lammen,
       10-May-2023.)  (New usage is discouraged.) $)
    nfcvf $p |- ( -. A. x x = y -> F/_ x y ) $=
      ( vw vz weq wal wn cv nfv wel elequ2 dvelimnf nfcd ) ABEAFGZACBHNCICDJZCB
      JABDOAIDBCKLM $.
    $( $j usage 'nfcvf' avoids 'ax-ext'; $)

    $( If ` x ` and ` y ` are distinct, then ` y ` is not free in ` x ` .
       Usage of this theorem is discouraged because it depends on ~ ax-13 .
       See ~ nfcv for a version that replaces the distinctor with a disjoint
       variable condition, requiring fewer axioms.  (Contributed by Mario
       Carneiro, 5-Dec-2016.)  (New usage is discouraged.) $)
    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 as in ~ dfcleq .  See also
       ~ cleqh .  (Contributed by NM, 26-May-1993.)  (Revised by Mario
       Carneiro, 7-Oct-2016.)  (Proof shortened by Wolf Lammen, 17-Nov-2019.)
       Avoid ~ ax-13 .  (Revised by Wolf Lammen, 10-May-2023.)  Avoid ~ ax-10 .
       (Revised by GG, 20-Aug-2023.) $)
    cleqf $p |- ( A = B <-> A. x ( x e. A <-> x e. B ) ) $=
      ( vy wceq cv wcel wb wal dfcleq nfv nfcri nfbi weq eleq1w bibi12d cbvalv1
      bitr4i ) BCGFHZBIZUACIZJZFKAHZBIZUECIZJZAKFBCLUHUDAFUHFMUBUCAAFBDNAFCENOA
      FPUFUBUGUCAFBQAFCQRST $.
    $( $j usage 'cleqf' avoids 'ax-10' 'ax-13'; $)
  $}

  ${
    eqabf.0 $e |- F/_ x A $.
    $( Equality of a class variable and a class abstraction.  In this version,
       the fact that ` x ` is a nonfree variable in ` A ` is explicitly stated
       as a hypothesis.  (Contributed by Thierry Arnoux, 11-May-2017.)  Avoid
       ~ ax-13 .  (Revised by Wolf Lammen, 13-May-2023.) $)
    eqabf $p |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) ) $=
      ( cab wceq cv wcel wb wal nfab1 cleqf abid bibi2i albii bitri ) CABEZFBGZ
      CHZRQHZIZBJSAIZBJBCQDABKLUAUBBTASABMNOP $.
    $( $j usage 'eqabf' avoids 'ax-13'; $)
  $}

  ${
    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.)  (Proof shortened by Wolf Lammen, 26-Feb-2025.) $)
    abid2f $p |- { x | x e. A } = A $=
      ( cv wcel cab wceq wb eqabf biid mpgbir eqcomi ) BADBEZAFZBNGMMHAMABCIMJK
      L $.

    $( Obsolete version of ~ abid2f as of 26-Feb-2025.  (Contributed by NM,
       5-Sep-2011.)  (Revised by Mario Carneiro, 7-Oct-2016.)  (Proof shortened
       by Wolf Lammen, 17-Nov-2019.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    abid2fOLD $p |- { x | x e. A } = A $=
      ( cv wcel cab wceq wb nfab1 cleqf abid mpgbir ) ADZBEZAFZBGMOENHAAOBNAICJ
      NAKL $.
  $}

  ${
    $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.)  (Proof shortened by Wolf Lammen, 28-Oct-2024.) $)
    sbabel $p |- ( [ y / x ] { z | ph } e. A <-> { z | [ y / x ] ph } e. A ) $=
      ( vv cab wcel wsb cv wel wb wal wa wex clabel sbbii sbex sban nfcri sbalv
      sbf sbv sbrbis anbi12i bitri exbii 3bitri bitr4i ) ADHEIZBCJZGKEIZDGLZABC
      JZMZDNZOZGPZUODHEIULUMUNAMZDNZOZGPZBCJVBBCJZGPUSUKVCBCADGEQRVBGBCSVDURGVD
      UMBCJZVABCJZOURUMVABCTVEUMVFUQUMBCBGEFUAUCUTUPBCDUNUNABCUNBCUDUEUBUFUGUHU
      IUODGEQUJ $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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, 26-May-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 NM, 9-Jun-2006.) $)
  nne $p |- ( -. A =/= B <-> A = B ) $=
    ( wceq wne wn df-ne con2bii bicomi ) ABCZABDZEJIABFGH $.

  ${
    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 $.

  ${
    neqned.1 $e |- ( ph -> -. A = B ) $.
    $( If it is not the case that two classes are equal, then 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.  Inequality is irreflexive.  (Contributed
     by Stefan O'Rear, 1-Jan-2015.) $)
  neirr $p |- -. A =/= A $=
    ( wne wn wceq eqid nne mpbir ) AABCAADAEAAFG $.

  $( Excluded middle with equality and inequality.  (Contributed by NM,
     3-Feb-2012.)  (Proof shortened by Wolf Lammen, 17-Nov-2019.) $)
  exmidne $p |- ( A = B \/ A =/= B ) $=
    ( wceq wne neqne orri ) ABCABDABEF $.

  $( 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 $.

  $( Law of noncontradiction with equality and inequality.  (Contributed by NM,
     3-Feb-2012.)  (Proof shortened by Wolf Lammen, 21-Dec-2019.) $)
  nonconne $p |- -. ( A = B /\ A =/= B ) $=
    ( wceq wne wa wfal fal eqneqall imp mto ) ABCZABDZEFGKLFFABHIJ $.

  ${
    necon3ad.1 $e |- ( ph -> ( ps -> A = B ) ) $.
    $( Contrapositive law deduction for inequality.  (Contributed by NM,
       2-Apr-2007.)  (Proof shortened by Andrew Salmon, 25-May-2011.)  (Proof
       shortened by Wolf Lammen, 23-Nov-2019.) $)
    necon3ad $p |- ( ph -> ( A =/= B -> -. ps ) ) $=
      ( wceq wne neneq nsyli ) ABCDFCDGECDHI $.
  $}

  ${
    necon3bd.1 $e |- ( ph -> ( A = B -> ps ) ) $.
    $( Contrapositive law deduction for inequality.  (Contributed by NM,
       2-Apr-2007.)  (Proof shortened by Andrew Salmon, 25-May-2011.) $)
    necon3bd $p |- ( ph -> ( -. ps -> A =/= B ) ) $=
      ( wne wn wceq nne biimtrid con1d ) ACDFZBLGCDHABCDIEJK $.
  $}

  ${
    necon2ad.1 $e |- ( ph -> ( A = B -> -. ps ) ) $.
    $( Contrapositive inference for inequality.  (Contributed by NM,
       19-Apr-2007.)  (Proof shortened by Andrew Salmon, 25-May-2011.)  (Proof
       shortened by Wolf Lammen, 23-Nov-2019.) $)
    necon2ad $p |- ( ph -> ( ps -> A =/= B ) ) $=
      ( wn wne notnot necon3bd syl5 ) BBFZFACDGBHAKCDEIJ $.
  $}

  ${
    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 $.
  $}

  ${
    necon1ad.1 $e |- ( ph -> ( -. ps -> A = B ) ) $.
    $( Contrapositive deduction for inequality.  (Contributed by NM,
       2-Apr-2007.)  (Proof shortened by Wolf Lammen, 23-Nov-2019.) $)
    necon1ad $p |- ( ph -> ( A =/= B -> ps ) ) $=
      ( wne wn necon3ad notnotr syl6 ) ACDFBGZGBAKCDEHBIJ $.
  $}

  ${
    necon1bd.1 $e |- ( ph -> ( A =/= B -> ps ) ) $.
    $( Contrapositive deduction for inequality.  (Contributed by NM,
       21-Mar-2007.)  (Proof shortened by Andrew Salmon, 25-May-2011.)  (Proof
       shortened by Wolf Lammen, 23-Nov-2019.) $)
    necon1bd $p |- ( ph -> ( -. ps -> A = B ) ) $=
      ( wceq wn wne df-ne biimtrrid con1d ) ACDFZBLGCDHABCDIEJK $.
  $}

  ${
    necon4ad.1 $e |- ( ph -> ( A =/= B -> -. ps ) ) $.
    $( Contrapositive inference for inequality.  (Contributed by NM,
       2-Apr-2007.)  (Proof shortened by Andrew Salmon, 25-May-2011.)  (Proof
       shortened by Wolf Lammen, 23-Nov-2019.) $)
    necon4ad $p |- ( ph -> ( ps -> A = B ) ) $=
      ( wn wceq notnot necon1bd syl5 ) BBFZFACDGBHAKCDEIJ $.
  $}

  ${
    necon4bd.1 $e |- ( ph -> ( -. ps -> A =/= B ) ) $.
    $( Contrapositive inference for inequality.  (Contributed by NM,
       1-Jun-2007.)  (Proof shortened by Andrew Salmon, 25-May-2011.)  (Proof
       shortened by Wolf Lammen, 23-Nov-2019.) $)
    necon4bd $p |- ( ph -> ( A = B -> ps ) ) $=
      ( wceq wn necon2bd notnotr syl6 ) ACDFBGZGBAKCDEHBIJ $.
  $}

  ${
    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 $.
  $}

  ${
    necon1d.1 $e |- ( ph -> ( A =/= B -> C = D ) ) $.
    $( Contrapositive law deduction for inequality.  (Contributed by NM,
       28-Dec-2008.)  (Proof shortened by Andrew Salmon, 25-May-2011.) $)
    necon1d $p |- ( ph -> ( C =/= D -> A = B ) ) $=
      ( wne wceq wn nne imbitrrdi necon4ad ) ADEGZBCABCGDEHMIFDEJKL $.
  $}

  ${
    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 $.
  $}

  ${
    necon4d.1 $e |- ( ph -> ( A =/= B -> C =/= D ) ) $.
    $( Contrapositive inference for inequality.  (Contributed by NM,
       2-Apr-2007.)  (Proof shortened by Andrew Salmon, 25-May-2011.) $)
    necon4d $p |- ( ph -> ( C = D -> A = B ) ) $=
      ( wceq wne wn necon2bd nne imbitrdi ) ADEGBCHZIBCGAMDEFJBCKL $.
  $}

  ${
    necon3ai.1 $e |- ( ph -> A = B ) $.
    $( Contrapositive inference for inequality.  (Contributed by NM,
       23-May-2007.)  (Proof shortened by Andrew Salmon, 25-May-2011.)  (Proof
       shortened by Wolf Lammen, 28-Oct-2024.) $)
    necon3ai $p |- ( A =/= B -> -. ph ) $=
      ( wne wceq neneq nsyl ) BCEBCFABCGDH $.
  $}

  ${
    necon3bi.1 $e |- ( A = B -> ph ) $.
    $( Contrapositive inference for inequality.  (Contributed by NM,
       1-Jun-2007.)  (Proof shortened by Andrew Salmon, 25-May-2011.)  (Proof
       shortened by Wolf Lammen, 22-Nov-2019.) $)
    necon3bi $p |- ( -. ph -> A =/= B ) $=
      ( wn wceq con3i neqned ) AEBCBCFADGH $.
  $}

  ${
    necon1ai.1 $e |- ( -. ph -> A = B ) $.
    $( Contrapositive inference for inequality.  (Contributed by NM,
       12-Feb-2007.)  (Proof shortened by Wolf Lammen, 22-Nov-2019.) $)
    necon1ai $p |- ( A =/= B -> ph ) $=
      ( wne wn necon3ai notnotrd ) BCEAAFBCDGH $.
  $}

  ${
    necon1bi.1 $e |- ( A =/= B -> ph ) $.
    $( Contrapositive inference for inequality.  (Contributed by NM,
       18-Mar-2007.)  (Proof shortened by Andrew Salmon, 25-May-2011.)  (Proof
       shortened by Wolf Lammen, 22-Nov-2019.) $)
    necon1bi $p |- ( -. ph -> A = B ) $=
      ( wceq wn wne df-ne sylbir con1i ) BCEZAKFBCGABCHDIJ $.
  $}

  ${
    necon2ai.1 $e |- ( A = B -> -. ph ) $.
    $( Contrapositive inference for inequality.  (Contributed by NM,
       16-Jan-2007.)  (Proof shortened by Andrew Salmon, 25-May-2011.)  (Proof
       shortened by Wolf Lammen, 22-Nov-2019.) $)
    necon2ai $p |- ( ph -> A =/= B ) $=
      ( wceq con2i neqned ) ABCBCEADFG $.
  $}

  ${
    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 $.
  $}

  ${
    necon4ai.1 $e |- ( A =/= B -> -. ph ) $.
    $( Contrapositive inference for inequality.  (Contributed by NM,
       16-Jan-2007.)  (Proof shortened by Andrew Salmon, 25-May-2011.)  (Proof
       shortened by Wolf Lammen, 22-Nov-2019.) $)
    necon4ai $p |- ( ph -> A = B ) $=
      ( wn wceq notnot necon1bi syl ) AAEZEBCFAGJBCDHI $.
  $}

  ${
    necon3i.1 $e |- ( A = B -> C = D ) $.
    $( Contrapositive inference for inequality.  (Contributed by NM,
       9-Aug-2006.)  (Proof shortened by Wolf Lammen, 22-Nov-2019.) $)
    necon3i $p |- ( C =/= D -> A =/= B ) $=
      ( wne wceq necon3ai neqned ) CDFABABGCDEHI $.
  $}

  ${
    necon1i.1 $e |- ( A =/= B -> C = D ) $.
    $( Contrapositive inference for inequality.  (Contributed by NM,
       18-Mar-2007.) $)
    necon1i $p |- ( C =/= D -> A = B ) $=
      ( wceq wn wne df-ne sylbir necon1ai ) ABFZCDLGABHCDFABIEJK $.
  $}

  ${
    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 $.
  $}

  ${
    necon4i.1 $e |- ( A =/= B -> C =/= D ) $.
    $( Contrapositive inference for inequality.  (Contributed by NM,
       17-Mar-2007.)  (Proof shortened by Andrew Salmon, 25-May-2011.)  (Proof
       shortened by Wolf Lammen, 24-Nov-2019.) $)
    necon4i $p |- ( C = D -> A = B ) $=
      ( wceq wne neneqd necon4ai ) CDFABABGCDEHI $.
  $}

  ${
    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 $.
  $}

  ${
    necon1abid.1 $e |- ( ph -> ( -. ps <-> A = B ) ) $.
    $( Contrapositive deduction for inequality.  (Contributed by NM,
       21-Aug-2007.)  (Proof shortened by Wolf Lammen, 24-Nov-2019.) $)
    necon1abid $p |- ( ph -> ( A =/= B <-> ps ) ) $=
      ( wn wne notnotb necon3bbid bitr2id ) BBFZFACDGBHAKCDEIJ $.
  $}

  ${
    necon1bbid.1 $e |- ( ph -> ( A =/= B <-> ps ) ) $.
    $( Contrapositive inference for inequality.  (Contributed by NM,
       31-Jan-2008.) $)
    necon1bbid $p |- ( ph -> ( -. ps <-> A = B ) ) $=
      ( wceq wn wne df-ne bitr3id con1bid ) ACDFZBLGCDHABCDIEJK $.
  $}

  ${
    necon4abid.1 $e |- ( ph -> ( A =/= B <-> -. ps ) ) $.
    $( Contrapositive law deduction for inequality.  (Contributed by NM,
       11-Jan-2008.)  (Proof shortened by Wolf Lammen, 24-Nov-2019.) $)
    necon4abid $p |- ( ph -> ( A = B <-> ps ) ) $=
      ( wn wceq notnotb necon1bbid bitr2id ) BBFZFACDGBHAKCDEIJ $.
  $}

  ${
    necon4bbid.1 $e |- ( ph -> ( -. ps <-> A =/= B ) ) $.
    $( Contrapositive law deduction for inequality.  (Contributed by NM,
       9-May-2012.) $)
    necon4bbid $p |- ( ph -> ( ps <-> A = B ) ) $=
      ( wceq wn wne bicomd necon4abid ) ACDFBABCDABGCDHEIJI $.
  $}

  ${
    necon2abid.1 $e |- ( ph -> ( A = B <-> -. ps ) ) $.
    $( Contrapositive deduction for inequality.  (Contributed by NM,
       18-Jul-2007.)  (Proof shortened by Wolf Lammen, 24-Nov-2019.) $)
    necon2abid $p |- ( ph -> ( ps <-> A =/= B ) ) $=
      ( wn wne notnotb necon3abid bitr4id ) ABBFZFCDGBHAKCDEIJ $.
  $}

  ${
    necon2bbid.1 $e |- ( ph -> ( ps <-> A =/= B ) ) $.
    $( Contrapositive deduction for inequality.  (Contributed by NM,
       13-Apr-2007.)  (Proof shortened by Wolf Lammen, 24-Nov-2019.) $)
    necon2bbid $p |- ( ph -> ( A = B <-> -. ps ) ) $=
      ( wn wne notnotb bitr3di necon4abid ) ABFZCDABCDGKFEBHIJ $.
  $}

  ${
    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 $.
  $}

  ${
    necon4bid.1 $e |- ( ph -> ( A =/= B <-> C =/= D ) ) $.
    $( Contrapositive law deduction for inequality.  (Contributed by NM,
       29-Jun-2007.) $)
    necon4bid $p |- ( ph -> ( A = B <-> C = D ) ) $=
      ( wceq wne wn necon2bbid nne bitr2di ) ADEGBCHZIBCGAMDEFJBCKL $.
  $}

  ${
    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 $.
  $}

  ${
    necon1abii.1 $e |- ( -. ph <-> A = B ) $.
    $( Contrapositive inference for inequality.  (Contributed by NM,
       17-Mar-2007.)  (Proof shortened by Wolf Lammen, 25-Nov-2019.) $)
    necon1abii $p |- ( A =/= B <-> ph ) $=
      ( wn wne notnotb necon3bbii bitr2i ) AAEZEBCFAGJBCDHI $.
  $}

  ${
    necon1bbii.1 $e |- ( A =/= B <-> ph ) $.
    $( Contrapositive inference for inequality.  (Contributed by NM,
       17-Mar-2007.)  (Proof shortened by Wolf Lammen, 24-Nov-2019.) $)
    necon1bbii $p |- ( -. ph <-> A = B ) $=
      ( wne wceq nne xchnxbi ) BCEBCFABCGDH $.
  $}

  ${
    necon2abii.1 $e |- ( A = B <-> -. ph ) $.
    $( Contrapositive inference for inequality.  (Contributed by NM,
       2-Mar-2007.) $)
    necon2abii $p |- ( ph <-> A =/= B ) $=
      ( wne wceq wn bicomi necon1abii ) BCEAABCBCFAGDHIH $.
  $}

  ${
    necon2bbii.1 $e |- ( ph <-> A =/= B ) $.
    $( Contrapositive inference for inequality.  (Contributed by NM,
       13-Apr-2007.) $)
    necon2bbii $p |- ( A = B <-> -. ph ) $=
      ( wn wceq wne bicomi necon1bbii ) AEBCFABCABCGDHIH $.
  $}

  ${
    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 $.
  $}

  $( 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 $.
  $}

  $( 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.)
       (Proof shortened by Wolf Lammen, 25-Nov-2019.) $)
    nesymi $p |- -. B = A $=
      ( necomi neii ) BAABCDE $.
  $}

  ${
    nesymir.1 $e |- -. A = B $.
    $( Inference associated with ~ nesym .  (Contributed by BJ, 7-Jul-2018.)
       (Proof shortened by Wolf Lammen, 25-Nov-2019.) $)
    nesymir $p |- B =/= A $=
      ( neir necomi ) ABABCDE $.
  $}

  ${
    neeq1d.1 $e |- ( ph -> A = B ) $.
    $( Deduction for inequality.  (Contributed by NM, 25-Oct-1999.)  (Proof
       shortened by Wolf Lammen, 19-Nov-2019.) $)
    neeq1d $p |- ( ph -> ( A =/= C <-> B =/= C ) ) $=
      ( eqeq1d necon3bid ) ABDCDABCDEFG $.

    $( Deduction for inequality.  (Contributed by NM, 25-Oct-1999.)  (Proof
       shortened by Wolf Lammen, 19-Nov-2019.) $)
    neeq2d $p |- ( ph -> ( C =/= A <-> C =/= B ) ) $=
      ( eqeq2d necon3bid ) ADBDCABCDEFG $.

    neeq12d.2 $e |- ( ph -> C = D ) $.
    $( Deduction for inequality.  (Contributed by NM, 24-Jul-2012.)  (Proof
       shortened by Wolf Lammen, 25-Nov-2019.) $)
    neeq12d $p |- ( ph -> ( A =/= C <-> B =/= D ) ) $=
      ( eqeq12d necon3bid ) ABDCEABCDEFGHI $.
  $}

  $( Equality theorem for inequality.  (Contributed by NM, 19-Nov-1994.)
     (Proof shortened by Wolf Lammen, 18-Nov-2019.) $)
  neeq1 $p |- ( A = B -> ( A =/= C <-> B =/= C ) ) $=
    ( wceq id neeq1d ) ABDZABCGEF $.

  $( Equality theorem for inequality.  (Contributed by NM, 19-Nov-1994.)
     (Proof shortened by Wolf Lammen, 18-Nov-2019.) $)
  neeq2 $p |- ( A = B -> ( C =/= A <-> C =/= B ) ) $=
    ( wceq id neeq2d ) ABDZABCGEF $.

  ${
    neeq1i.1 $e |- A = B $.
    $( Inference for inequality.  (Contributed by NM, 29-Apr-2005.)  (Proof
       shortened by Wolf Lammen, 19-Nov-2019.) $)
    neeq1i $p |- ( A =/= C <-> B =/= C ) $=
      ( eqeq1i necon3bii ) ACBCABCDEF $.

    $( Inference for inequality.  (Contributed by NM, 29-Apr-2005.)  (Proof
       shortened by Wolf Lammen, 19-Nov-2019.) $)
    neeq2i $p |- ( C =/= A <-> C =/= B ) $=
      ( eqeq2i necon3bii ) CACBABCDEF $.

    neeq12i.2 $e |- C = D $.
    $( Inference for inequality.  (Contributed by NM, 24-Jul-2012.)  (Proof
       shortened by Wolf Lammen, 25-Nov-2019.) $)
    neeq12i $p |- ( A =/= C <-> B =/= D ) $=
      ( eqeq12i necon3bii ) ACBDABCDEFGH $.
  $}

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  ${
    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 ) $.
    $( A chained equality inference for inequality.  (Contributed by NM,
       6-Jun-2012.)  (Proof shortened by Wolf Lammen, 19-Nov-2019.) $)
    eqnetrrid $p |- ( ph -> A =/= C ) $=
      ( wceq a1i eqnetrrd ) ACBDCBGAEHFI $.
  $}

  ${
    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.)  (Proof shortened by Wolf Lammen, 19-Nov-2019.) $)
    3netr3d $p |- ( ph -> C =/= D ) $=
      ( neeqtrd eqnetrrd ) ABDEGABCEFHIJ $.
  $}

  ${
    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.)  (Proof shortened by Wolf Lammen, 21-Nov-2019.) $)
    3netr4d $p |- ( ph -> C =/= D ) $=
      ( eqnetrd neeqtrrd ) ADCEADBCGFIHJ $.
  $}

  ${
    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 $.
  $}

  $( Contraposition law for inequality.  (Contributed by NM, 28-Dec-2008.) $)
  nebi $p |- ( ( A = B <-> C = D ) <-> ( A =/= B <-> C =/= D ) ) $=
    ( wceq wb wne id necon3bid necon4bid impbii ) ABECDEFZABGCDGFZLABCDLHIMABCD
    MHJK $.

  $( Theorem *13.18 in [WhiteheadRussell] p. 178.  (Contributed by Andrew
     Salmon, 3-Jun-2011.)  (Proof shortened by Wolf Lammen, 29-Oct-2024.) $)
  pm13.18 $p |- ( ( A = B /\ A =/= C ) -> B =/= C ) $=
    ( wceq wne neeq1 biimpa ) ABDACEBCEABCFG $.

  $( Theorem *13.181 in [WhiteheadRussell] p. 178.  (Contributed by Andrew
     Salmon, 3-Jun-2011.)  (Proof shortened by Wolf Lammen, 30-Oct-2024.) $)
  pm13.181 $p |- ( ( A = B /\ B =/= C ) -> A =/= C ) $=
    ( wceq wne neeq1 biimpar ) ABDACEBCEABCFG $.

  ${
    pm2.61ine.1 $e |- ( A = B -> ph ) $.
    pm2.61ine.2 $e |- ( A =/= B -> ph ) $.
    $( Inference eliminating an inequality in an antecedent.  (Contributed by
       NM, 16-Jan-2007.)  (Proof shortened by Andrew Salmon, 25-May-2011.) $)
    pm2.61ine $p |- ph $=
      ( wne wn wceq nne sylbi pm2.61i ) BCFZAELGBCHABCIDJK $.
  $}

  ${
    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 $.
  $}

  ${
    pm2.61ne.1 $e |- ( A = B -> ( ps <-> ch ) ) $.
    pm2.61ne.2 $e |- ( ( ph /\ A =/= B ) -> ps ) $.
    pm2.61ne.3 $e |- ( ph -> ch ) $.
    $( Deduction eliminating an inequality in an antecedent.  (Contributed by
       NM, 24-May-2006.)  (Proof shortened by Andrew Salmon, 25-May-2011.)
       (Proof shortened by Wolf Lammen, 25-Nov-2019.) $)
    pm2.61ne $p |- ( ph -> ps ) $=
      ( wi wceq imbitrrid wne expcom pm2.61ine ) ABIDEABDEJCHFKADELBGMN $.
  $}

  ${
    pm2.61dne.1 $e |- ( ph -> ( A = B -> ps ) ) $.
    pm2.61dne.2 $e |- ( ph -> ( A =/= B -> ps ) ) $.
    $( Deduction eliminating an inequality in an antecedent.  (Contributed by
       NM, 1-Jun-2007.)  (Proof shortened by Andrew Salmon, 25-May-2011.) $)
    pm2.61dne $p |- ( ph -> ps ) $=
      ( wi wceq com12 wne pm2.61ine ) ABGCDACDHBEIACDJBFIK $.
  $}

  ${
    pm2.61dane.1 $e |- ( ( ph /\ A = B ) -> ps ) $.
    pm2.61dane.2 $e |- ( ( ph /\ A =/= B ) -> ps ) $.
    $( Deduction eliminating an inequality in an antecedent.  (Contributed by
       NM, 30-Nov-2011.) $)
    pm2.61dane $p |- ( ph -> ps ) $=
      ( wceq ex wne pm2.61dne ) ABCDACDGBEHACDIBFHJ $.
  $}

  ${
    pm2.61da2ne.1 $e |- ( ( ph /\ A = B ) -> ps ) $.
    pm2.61da2ne.2 $e |- ( ( ph /\ C = D ) -> ps ) $.
    pm2.61da2ne.3 $e |- ( ( ph /\ ( A =/= B /\ C =/= D ) ) -> ps ) $.
    $( Deduction eliminating two inequalities in an antecedent.  (Contributed
       by NM, 29-May-2013.) $)
    pm2.61da2ne $p |- ( ph -> ps ) $=
      ( wne wa wceq adantlr anassrs pm2.61dane ) ABCDGACDJZKBEFAEFLBPHMAPEFJBIN
      OO $.
  $}

  ${
    pm2.61da3ne.1 $e |- ( ( ph /\ A = B ) -> ps ) $.
    pm2.61da3ne.2 $e |- ( ( ph /\ C = D ) -> ps ) $.
    pm2.61da3ne.3 $e |- ( ( ph /\ E = F ) -> ps ) $.
    pm2.61da3ne.4 $e |- ( ( ph /\ ( A =/= B /\ C =/= D /\ E =/= F ) )
          -> ps ) $.
    $( Deduction eliminating three inequalities in an antecedent.  (Contributed
       by NM, 15-Jun-2013.)  (Proof shortened by Wolf Lammen, 25-Nov-2019.) $)
    pm2.61da3ne $p |- ( ph -> ps ) $=
      ( wne wa wi wceq a1d 3exp2 imp4b pm2.61dane imp pm2.61da2ne ) ABEFGHJKAEF
      MZGHMZNZBAUEBOCDACDPNBUEIQACDMZUCUDBAUFUCUDBLRSTUAUB $.
  $}

  ${
    pm2.61iine.1 $e |- ( ( A =/= C /\ B =/= D ) -> ph ) $.
    pm2.61iine.2 $e |- ( A = C -> ph ) $.
    pm2.61iine.3 $e |- ( B = D -> ph ) $.
    $( Equality version of ~ pm2.61ii .  (Contributed by Scott Fenton,
       13-Jun-2013.)  (Proof shortened by Wolf Lammen, 25-Nov-2019.) $)
    pm2.61iine $p |- ph $=
      ( wne wceq adantl pm2.61dane pm2.61ine ) ABDGBDIZACECEJANHKFLM $.
  $}

  ${
    mteqand.1 $e |- ( ph -> C =/= D ) $.
    mteqand.2 $e |- ( ( ph /\ A = B ) -> C = D ) $.
    $( A modus tollens deduction for inequality.  (Contributed by Steven
       Nguyen, 1-Jun-2023.) $)
    mteqand $p |- ( ph -> A =/= B ) $=
      ( wceq neneqd mtand neqned ) ABCABCHDEHADEFIGJK $.
  $}

  $( Logical OR with an equality.  (Contributed by NM, 29-Apr-2007.) $)
  neor $p |- ( ( A = B \/ ps ) <-> ( A =/= B -> ps ) ) $=
    ( wceq wo wn wi wne df-or df-ne imbi1i bitr4i ) BCDZAEMFZAGBCHZAGMAIONABCJK
    L $.

  $( 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.) $)
  ne3anior $p |- ( ( A =/= B /\ C =/= D /\ E =/= F )
        <-> -. ( A = B \/ C = D \/ E = F ) ) $=
    ( wne w3a wn w3o wceq 3anor nne 3orbi123i xchbinx ) ABGZCDGZEFGZHPIZQIZRIZJ
    ABKZCDKZEFKZJPQRLSUBTUCUAUDABMCDMEFMNO $.

  $( A De Morgan's law for inequality.  (Contributed by NM, 18-May-2007.) $)
  neorian $p |- ( ( A =/= B \/ C =/= D ) <-> -. ( A = B /\ C = D ) ) $=
    ( wne wo wceq wn wa df-ne orbi12i ianor bitr4i ) ABEZCDEZFABGZHZCDGZHZFPRIH
    NQOSABJCDJKPRLM $.

  ${
    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 neii mtbir ) ABCFBCDGEH $.
  $}

  $( Two classes are different if they don't contain the same element.
     (Contributed by NM, 3-Feb-2012.)  (Proof shortened by Wolf Lammen,
     14-May-2023.) $)
  nelne1 $p |- ( ( A e. B /\ -. A e. C ) -> B =/= C ) $=
    ( wcel wn wa nelneq2 neqned ) ABDACDEFBCABCGH $.

  $( Two classes are different if they don't belong to the same class.
     (Contributed by NM, 25-Jun-2012.)  (Proof shortened by Wolf Lammen,
     14-May-2023.) $)
  nelne2 $p |- ( ( A e. C /\ -. B e. C ) -> A =/= B ) $=
    ( wcel wn wa nelneq neqned ) ACDBCDEFABABCGH $.

  $( 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 $.

  $( If two classes are different, a third class must be different of at least
     one of them.  (Contributed by Thierry Arnoux, 8-Aug-2020.) $)
  neneor $p |- ( A =/= B -> ( A =/= C \/ B =/= C ) ) $=
    ( wne wceq wa wn wo eqtr3 necon3ai neorian sylibr ) ABDACEBCEFZGACDBCDHMABA
    BCIJACBCKL $.

  ${
    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 $.
  $}

  $( Not equivalent wff's correspond to not equal class abstractions.
     (Contributed by AV, 7-Apr-2019.)  (Proof shortened by Wolf Lammen,
     25-Nov-2019.)  Definitial form.  (Revised by Wolf Lammen, 5-Mar-2025.) $)
  nabbib $p |- ( { x | ph } =/= { x | ps } <-> E. x ( ph <-> -. ps ) ) $=
    ( cab wne wceq wn wb wex df-ne exnal xor3 exbii bitr3i abbib xchnxbir bitri
    wal ) ACDZBCDZESTFZGABGHZCIZSTJABHZCRZUCUAUEGUDGZCIUCUDCKUFUBCABLMNABCOPQ
    $.


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Negated membership
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  $c e/ $.  $( Not an element of (membership symbol 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 $.
  $}

  ${
    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 $.
  $}

  ${
    neleq12d.1 $e |- ( ph -> A = B ) $.
    neleq12d.2 $e |- ( ph -> C = D ) $.
    $( Equality theorem for negated membership.  (Contributed by FL,
       10-Aug-2016.)  (Proof shortened by Wolf Lammen, 25-Nov-2019.) $)
    neleq12d $p |- ( ph -> ( A e/ C <-> B e/ D ) ) $=
      ( wcel wn wnel eleq12d notbid df-nel 3bitr4g ) ABDHZICEHZIBDJCEJAOPABCDEF
      GKLBDMCEMN $.
  $}

  $( Equality theorem for negated membership.  (Contributed by NM,
     20-Nov-1994.)  (Proof shortened by Wolf Lammen, 25-Nov-2019.) $)
  neleq1 $p |- ( A = B -> ( A e/ C <-> B e/ C ) ) $=
    ( wceq id eqidd neleq12d ) ABDZABCCHEHCFG $.

  $( Equality theorem for negated membership.  (Contributed by NM,
     20-Nov-1994.)  (Proof shortened by Wolf Lammen, 25-Nov-2019.) $)
  neleq2 $p |- ( A = B -> ( C e/ A <-> C e/ B ) ) $=
    ( wceq eqidd id neleq12d ) ABDZCCABHCEHFG $.

  ${
    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 $.
  $}

  $( Negation of negated membership, analogous to ~ nne .  (Contributed by
     Alexander van der Vekens, 18-Jan-2018.)  (Proof shortened by Wolf Lammen,
     25-Nov-2019.) $)
  nnel $p |- ( -. A e/ B <-> A e. B ) $=
    ( wcel wnel wn df-nel bicomi con1bii ) ABCZABDZJIEABFGH $.

  $( 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 $.

  $( A contradiction concerning membership implies anything.  (Contributed by
     Alexander van der Vekens, 25-Jan-2018.) $)
  pm2.24nel $p |- ( A e. B -> ( A e/ B -> ph ) ) $=
    ( wnel wcel wn df-nel pm2.24 biimtrid ) BCDBCEZFJABCGJAHI $.

  ${
    pm2.61danel.1 $e |- ( ( ph /\ A e. B ) -> ps ) $.
    pm2.61danel.2 $e |- ( ( ph /\ A e/ B ) -> ps ) $.
    $( Deduction eliminating an elementhood in an antecedent.  (Contributed by
       AV, 5-Dec-2021.) $)
    pm2.61danel $p |- ( ph -> ps ) $=
      ( wcel wn wnel df-nel sylan2br pm2.61dan ) ACDGZBEMHACDIBCDJFKL $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Restricted quantification
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Restricted universal and existential quantification
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  $( Extend wff notation to include restricted universal quantification. $)
  wral $a wff A. x e. A ph $.

  $( Define restricted universal quantification.  Special case of Definition
     4.15(3) of [TakeutiZaring] p. 22.

     Note:  This notation is most often used to express that ` ph ` holds for
     all elements of a given class ` A ` .  For this reading ` F/_ x A ` is
     required, though, for example, asserted when ` x ` and ` A ` are disjoint.

     Should instead ` A ` depend on ` x ` , you rather focus on those ` x `
     that happen to be contained in the corresponding ` A ( x ) ` .  This
     hardly used interpretation could still occur naturally.  For some
     examples, look at ~ ralndv1 or ~ ralndv2 , courtesy of AV.

     So be careful to either keep ` A ` independent of ` x ` , or adjust your
     comments to include such exotic cases.  (Contributed by NM,
     19-Aug-1993.) $)
  df-ral $a |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) ) $.

  $( *** Theorems using axioms up to ax-gen only: *** $)

  ${
    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 $.
  $}

  $( All elements of a class are elements of the class.  (Contributed by AV,
     30-Oct-2020.) $)
  ralel $p |- A. x e. A x e. A $=
    ( cv wcel id rgen ) ACBDZABGEF $.

  ${
    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 $.
  $}

  $( *** Theorems using axioms up to ax-4 only: *** $)

  $( Restricted universally quantified negation expressed as a universally
     quantified negation.  (Contributed by BJ, 16-Jul-2021.) $)
  raln $p |- ( A. x e. A -. ph <-> A. x -. ( x e. A /\ ph ) ) $=
    ( wn wral cv wcel wi wal wa df-ral imnang bitri ) ADZBCEBFCGZNHBIOAJDBINBCK
    OABLM $.

  $( Extend wff notation to include restricted existential quantification. $)
  wrex $a wff E. x e. A ph $.

  $( Define restricted existential quantification.  Special case of Definition
     4.15(4) of [TakeutiZaring] p. 22.

     Note:  This notation is most often used to express that ` ph ` holds for
     at least one element of a given class ` A ` .  For this reading
     ` F/_ x A ` is required, though, for example, asserted when ` x ` and
     ` A ` are disjoint.

     Should instead ` A ` depend on ` x ` , you rather assert at least one
     ` x ` fulfilling ` ph ` happens to be contained in the corresponding
     ` A ( x ) ` .  This interpretation is rarely needed (see also ~ df-ral ).
     (Contributed by NM, 30-Aug-1993.) $)
  df-rex $a |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) ) $.

  $( Relationship between restricted universal and existential quantifiers.
     (Contributed by NM, 21-Jan-1997.)  (Proof shortened by BJ,
     16-Jul-2021.) $)
  ralnex $p |- ( A. x e. A -. ph <-> -. E. x e. A ph ) $=
    ( wn wral cv wcel wa wal wrex raln wex alnex df-rex xchbinxr bitri ) ADBCEB
    FCGAHZDBIZABCJZDABCKRQBLSQBMABCNOP $.

  $( Relationship between restricted universal and existential quantifiers.
     (Contributed by NM, 21-Jan-1997.)  (Proof shortened by Wolf Lammen,
     26-Nov-2019.) $)
  dfrex2 $p |- ( E. x e. A ph <-> -. A. x e. A -. ph ) $=
    ( wn wral wrex ralnex con2bii ) ADBCEABCFABCGH $.

  ${
    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 $.
  $}

  $( 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 ala1 df-ral sylibr ) ABDBECFZAGBDABCHALBIABCJK $.

  $( 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 exsimpr sylbi ) ABCDBECFZAGBHABHABCILABJK $.

  $( Two ways of expressing that a class has at least one element.
     (Contributed by Zhi Wang, 23-Sep-2024.) $)
  rextru $p |- ( E. x x e. A <-> E. x e. A T. ) $=
    ( cv wcel wex wtru wa wrex tru biantru exbii df-rex bitr4i ) ACBDZAENFGZAEF
    ABHNOAFNIJKFABLM $.

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  ${
    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 $.

    $( Inference quantifying both antecedent and consequent.  (Contributed by
       NM, 10-Feb-1997.)  (Proof shortened by Wolf Lammen, 31-Oct-2024.) $)
    reximia $p |- ( E. x e. A ph -> E. x e. A ps ) $=
      ( cv wcel imdistani reximi2 ) 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.  (Contributed by
       NM, 18-Oct-1996.) $)
    reximi $p |- ( E. x e. A ph -> E. x e. A ps ) $=
      ( wi cv wcel a1i reximia ) ABCDABFCGDHEIJ $.
  $}

  ${
    ral2imi.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Inference quantifying antecedent, nested antecedent, and consequent,
       with a strong hypothesis.  (Contributed by NM, 19-Dec-2006.)  Allow
       shortening of ~ ralim .  (Revised by Wolf Lammen, 1-Dec-2019.) $)
    ral2imi $p |- ( A. x e. A ph -> ( A. x e. A ps -> A. x e. A ch ) ) $=
      ( wral cv wcel wi wal df-ral imim3i al2imi 3imtr4g sylbi ) ADEGDHEIZAJZDK
      ZBDEGZCDEGZJADELSQBJZDKQCJZDKTUARUBUCDABCQFMNBDELCDELOP $.
  $}

  $( Distribution of restricted quantification over implication.  (Contributed
     by NM, 9-Feb-1997.)  (Proof shortened by Wolf Lammen, 1-Dec-2019.) $)
  ralim $p |- ( A. x e. A ( ph -> ps ) ->
               ( A. x e. A ph -> A. x e. A ps ) ) $=
    ( wi id ral2imi ) ABEZABCDHFG $.

  $( 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 wrex wn con3 ral2imi ralnex 3imtr3g con4d ) ABEZCDFZBCDGZACDGZOBH
    ZCDFAHZCDFPHQHNRSCDABIJBCDKACDKLM $.

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  ${
    rexbiia.1 $e |- ( x e. A -> ( ph <-> ps ) ) $.
    $( 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 $.
  $}

  ${
    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.)  (Proof shortened by Wolf Lammen,
       4-Dec-2019.) $)
    ralbii $p |- ( A. x e. A ph <-> A. x e. A ps ) $=
      ( wb cv wcel a1i ralbiia ) ABCDABFCGDHEIJ $.
  $}

  ${
    rexbii.1 $e |- ( ph <-> ps ) $.
    $( Inference adding restricted existential quantifier to both sides of an
       equivalence.  (Contributed by NM, 23-Nov-1994.)  (Revised by Mario
       Carneiro, 17-Oct-2016.)  (Proof shortened by Wolf Lammen,
       6-Dec-2019.) $)
    rexbii $p |- ( E. x e. A ph <-> E. x e. A ps ) $=
      ( wb cv wcel a1i rexbiia ) ABCDABFCGDHEIJ $.
  $}

  $( Cancellation law for restricted universal quantification.  (Contributed by
     Peter Mazsa, 30-Dec-2018.)  (Proof shortened by Wolf Lammen,
     29-Jun-2023.) $)
  ralanid $p |- ( A. x e. A ( x e. A /\ ph ) <-> A. x e. A ph ) $=
    ( cv wcel wa ibar bicomd ralbiia ) BDCEZAFZABCJAKJAGHI $.

  $( Cancellation law for restricted existential quantification.  (Contributed
     by Peter Mazsa, 24-May-2018.)  (Proof shortened by Wolf Lammen,
     8-Jul-2023.) $)
  rexanid $p |- ( E. x e. A ( x e. A /\ ph ) <-> E. x e. A ph ) $=
    ( cv wcel wa ibar bicomd rexbiia ) BDCEZAFZABCJAKJAGHI $.

  $( A commutation law for restricted universal quantifiers that swaps the
     domains of the restriction.  (Contributed by NM, 22-Feb-2004.)  (Proof
     shortened by Wolf Lammen, 22-Dec-2024.) $)
  ralcom3 $p |- ( A. x e. A ( x e. B -> ph ) <->
                  A. x e. B ( x e. A -> ph ) ) $=
    ( cv wcel wi bi2.04 ralbii2 ) BEZDFZAGJCFZAGBCDLKAHI $.

  $( Obsolete version of ~ ralcom3 as of 22-Dec-2024.  (Contributed by NM,
     22-Feb-2004.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  ralcom3OLD $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 $.

  $( Relationship between restricted universal and existential quantifiers.
     (Contributed by NM, 21-Jan-1997.)  Allow shortening of ~ rexnal .
     (Revised by Wolf Lammen, 9-Dec-2019.) $)
  dfral2 $p |- ( A. x e. A ph <-> -. E. x e. A -. ph ) $=
    ( wral wn wrex notnotb ralbii ralnex bitri ) ABCDAEZEZBCDKBCFEALBCAGHKBCIJ
    $.

  $( Relationship between restricted universal and existential quantifiers.
     (Contributed by NM, 21-Jan-1997.)  (Proof shortened by Wolf Lammen,
     9-Dec-2019.) $)
  rexnal $p |- ( E. x e. A -. ph <-> -. A. x e. A ph ) $=
    ( wral wn wrex dfral2 con2bii ) ABCDAEBCFABCGH $.

  $( 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 $.

  $( A transformation of restricted quantifiers and logical connectives.
     (Contributed by NM, 4-Sep-2005.)  (Proof shortened by Wolf Lammen,
     27-Dec-2019.) $)
  rexanali $p |- ( E. x e. A ( ph /\ -. ps ) <-> -. A. x e. A ( ph -> ps ) ) $=
    ( wn wa wrex wral wi dfrex2 iman ralbii xchbinxr ) ABEFZCDGNEZCDHABIZCDHNCD
    JPOCDABKLM $.

  $( Distribute a restricted universal quantifier over a biconditional.
     Restricted quantification version of ~ albi .  (Contributed by NM,
     6-Oct-2003.)  Reduce axiom usage.  (Revised by Wolf Lammen,
     17-Jun-2023.) $)
  ralbi $p |- ( A. x e. A ( ph <-> ps ) ->
               ( A. x e. A ph <-> A. x e. A ps ) ) $=
    ( wb wral biimp ral2imi biimpr impbid ) ABEZCDFACDFBCDFKABCDABGHKBACDABIHJ
    $.
  $( $j usage 'ralbi' avoids 'ax-5' 'ax-6' 'ax-7' 'ax-10' 'ax-12' ; $)

  $( Distribute restricted quantification over a biconditional.  (Contributed
     by Scott Fenton, 7-Aug-2024.)  (Proof shortened by Wolf Lammen,
     3-Nov-2024.) $)
  rexbi $p |- ( A. x e. A ( ph <-> ps ) ->
  ( E. x e. A ph <-> E. x e. A ps ) ) $=
    ( wb wral wrex wi biimp ralimi rexim syl biimpr impbid ) ABEZCDFZACDGZBCDGZ
    PABHZCDFQRHOSCDABIJABCDKLPBAHZCDFRQHOTCDABMJBACDKLN $.

  ${
    ralrexbid.1 $e |- ( ph -> ( ps <-> th ) ) $.
    $( Formula-building rule for restricted existential quantifier, using a
       restricted universal quantifier to bind the quantified variable in the
       antecedent.  (Contributed by AV, 21-Oct-2023.)  Reduce axiom usage.
       (Revised by SN, 13-Nov-2023.)  (Proof shortened by Wolf Lammen,
       4-Nov-2024.) $)
    ralrexbid $p |- ( A. x e. A ph -> ( E. x e. A ps <-> E. x e. A th ) ) $=
      ( wral wb wrex ralimi rexbi syl ) ADEGBCHZDEGBDEICDEIHAMDEFJBCDEKL $.
  $}

  $( Restricted quantifier version of ~ 19.35 .  (Contributed by NM,
     20-Sep-2003.)  (Proof shortened by Wolf Lammen, 22-Dec-2024.) $)
  r19.35 $p |- ( E. x e. A ( ph -> ps ) <->
               ( A. x e. A ph -> E. x e. A ps ) ) $=
    ( wrex wral pm5.5 ralrexbid biimpcd rexnal pm2.21 reximi sylbir ax-1 impbii
    wi wn ja ) ABPZCDEZACDFZBCDEZPUATUBASBCDABGHIUAUBTUAQAQZCDETACDJUCSCDABKLMB
    SCDBANLRO $.

  $( Obsolete version of ~ 19.35 as of 22-Dec-2024.  (Contributed by NM,
     20-Sep-2003.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  r19.35OLD $p |- ( E. x e. A ( ph -> ps ) <->
               ( A. x e. A ph -> E. x e. A ps ) ) $=
    ( wi wrex wral rexim pm2.27 ralimi syl11 rexnal pm2.21 reximi sylbir impbii
    wn ax-1 ja ) ABEZCDFZACDGZBCDFZETBEZCDGUAUCUBTBCDHAUDCDABIJKUBUCUAUBQAQZCDF
    UAACDLUETCDABMNOBTCDBARNSP $.

  $( Version of ~ 19.26 and ~ r19.26 with restricted quantifiers ranging over
     different classes.  (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 $.

  $( Restricted quantifier version of ~ 19.26 .  (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 $.

  $( Version of ~ r19.26 with three 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 ) ) $=
    ( wa wral w3a r19.26 bianbi df-3an ralbii 3bitr4i ) ABFZCFZDEGZADEGZBDEGZFZ
    CDEGZFABCHZDEGQRTHPNDEGTSNCDEIABDEIJUAODEABCKLQRTKM $.

  $( Split a biconditional and distribute quantifier.  Restricted quantifier
     version of ~ albiim .  (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 $.

  $( Restricted quantifier version of ~ 19.29 .  See also ~ r19.29r .
     (Contributed by NM, 31-Aug-1999.)  (Proof shortened by Andrew Salmon,
     30-May-2011.)  (Proof shortened by Wolf Lammen, 22-Dec-2024.) $)
  r19.29 $p |- ( ( A. x e. A ph /\ E. x e. A ps ) ->
               E. x e. A ( ph /\ ps ) ) $=
    ( wral wrex wa ibar ralrexbid biimpa ) ACDEBCDFABGZCDFABKCDABHIJ $.

  $( Obsolete version of ~ r19.29 as of 22-Dec-2024.  (Contributed by NM,
     31-Aug-1999.)  (Proof shortened by Andrew Salmon, 30-May-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  r19.29OLD $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 $.

  $( Restricted quantifier version of ~ 19.29r ; variation of ~ r19.29 .
     (Contributed by NM, 31-Aug-1999.)  (Proof shortened by Wolf Lammen,
     29-Jun-2023.) $)
  r19.29r $p |- ( ( E. x e. A ph /\ A. x e. A ps ) ->
                E. x e. A ( ph /\ ps ) ) $=
    ( wral wrex wa iba ralrexbid biimpac ) BCDEACDFABGZCDFBAKCDBAHIJ $.

  $( Obsolete version of ~ r19.29r as of 22-Dec-2024.  (Contributed by NM,
     31-Aug-1999.)  (Proof shortened by Wolf Lammen, 29-Jun-2023.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  r19.29rOLD $p |- ( ( E. x e. A ph /\ A. x e. A ps ) ->
                E. x e. A ( ph /\ ps ) ) $=
    ( wrex wral wa r19.29 ancoms pm3.22 reximi syl ) ACDEZBCDFZGBAGZCDEZABGZCDE
    NMPBACDHIOQCDBAJKL $.

  ${
    r19.29imd.1 $e |- ( ph -> E. x e. A ps ) $.
    r19.29imd.2 $e |- ( ph -> A. x e. A ( ps -> ch ) ) $.
    $( Theorem 19.29 of [Margaris] p. 90 with an implication in the hypothesis
       containing the generalization, deduction version.  (Contributed by AV,
       19-Jan-2019.) $)
    r19.29imd $p |- ( ph -> E. x e. A ( ps /\ ch ) ) $=
      ( wi wa wrex wral r19.29r syl2anc abai rexbii sylibr ) ABBCHZIZDEJZBCIZDE
      JABDEJQDEKSFGBQDELMTRDEBCNOP $.
  $}

  $( 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 $.

  $( Restricted quantifier version of ~ 19.30 .  (Contributed by Scott Fenton,
     25-Feb-2011.)  (Proof shortened by Wolf Lammen, 5-Nov-2024.) $)
  r19.30 $p |- ( A. x e. A ( ph \/ ps ) ->
                 ( A. x e. A ph \/ E. x e. A ps ) ) $=
    ( wo wral wrex wn wi pm2.53 ralimi rexnal biimpri rexim syl2im orrd ) ABEZC
    DFZACDFZBCDGZRAHZBIZCDFSHZUACDGZTQUBCDABJKUDUCACDLMUABCDNOP $.

  $( Restricted quantifier version of ~ 19.43 .  (Contributed by NM,
     27-May-1998.)  (Proof shortened by Andrew Salmon, 30-May-2011.) $)
  r19.43 $p |- ( E. x e. A ( ph \/ ps )
         <-> ( E. x e. A ph \/ E. x e. A ps ) ) $=
    ( wn wi wrex wral wo r19.35 df-or rexbii ralnex imbi1i bitr4i 3bitr4i ) AEZ
    BFZCDGQCDHZBCDGZFZABIZCDGACDGZTIZQBCDJUBRCDABKLUDUCEZTFUAUCTKSUETACDMNOP $.

  ${
    2ralimi.1 $e |- ( ph -> ps ) $.
    $( 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 $.

    $( Inference quantifying both antecedent and consequent three times, with
       strong hypothesis.  (Contributed by Scott Fenton, 5-Mar-2025.) $)
    3ralimi $p |- ( A. x e. A A. y e. B A. z e. C ph ->
    A. x e. A A. y e. B A. z e. C ps ) $=
      ( wral ralimi 2ralimi ) AEHJBEHJCDFGABEHIKL $.

    $( Inference quantifying both antecedent and consequent four times, with
       strong hypothesis.  (Contributed by Scott Fenton, 5-Mar-2025.) $)
    4ralimi $p |- ( A. x e. A A. y e. B A. z e. C A. w e. D ph ->
    A. x e. A A. y e. B A. z e. C A. w e. D ps ) $=
      ( wral ralimi 3ralimi ) AFJLBFJLCDEGHIABFJKMN $.

    $( Inference quantifying both antecedent and consequent five times, with
       strong hypothesis.  (Contributed by Scott Fenton, 5-Mar-2025.) $)
    5ralimi $p |- ( A. x e. A A. y e. B A. z e. C A. w e. D A. t e. E ph ->
    A. x e. A A. y e. B A. z e. C A. w e. D A. t e. E ps ) $=
      ( wral ralimi 4ralimi ) AGLNBGLNCDEFHIJKABGLMOP $.

    $( Inference quantifying both antecedent and consequent six times, with
       strong hypothesis.  (Contributed by Scott Fenton, 5-Mar-2025.) $)
    6ralimi $p |- (
    A. x e. A A. y e. B A. z e. C A. w e. D A. t e. E A. u e. F ph ->
    A. x e. A A. y e. B A. z e. C A. w e. D A. t e. E A. u e. F ps ) $=
      ( wral ralimi 5ralimi ) AGNPBGNPCDEFHIJKLMABGNOQR $.
  $}

  ${
    2ralbii.1 $e |- ( ph <-> ps ) $.
    $( 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 $.
  $}

  ${
    2rexbii.1 $e |- ( ph <-> ps ) $.
    $( 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 $.
  $}


  ${
    3ralbii.1 $e |- ( ph <-> ps ) $.
    $( Inference adding three restricted universal quantifiers to both sides of
       an equivalence.  (Contributed by Peter Mazsa, 25-Jul-2019.) $)
    3ralbii $p |- ( A. x e. A A. y e. B A. z e. C ph <->
                    A. x e. A A. y e. B A. z e. C ps ) $=
      ( wral 2ralbii ralbii ) AEHJDGJBEHJDGJCFABDEGHIKL $.
  $}

  ${
    4ralbii.1 $e |- ( ph <-> ps ) $.
    $( Inference adding four restricted universal quantifiers to both sides of
       an equivalence.  (Contributed by Scott Fenton, 28-Feb-2025.) $)
    4ralbii $p |- ( A. x e. A A. y e. B A. z e. C A. w e. D ph <->
    A. x e. A A. y e. B A. z e. C A. w e. D ps ) $=
      ( wral ralbii 3ralbii ) AFJLBFJLCDEGHIABFJKMN $.
  $}

  $( Split a biconditional and distribute two restricted universal quantifiers,
     analogous to ~ 2albiim and ~ ralbiim .  (Contributed by Alexander van der
     Vekens, 2-Jul-2017.) $)
  2ralbiim $p |- ( A. x e. A A. y e. B ( ph <-> ps ) <->
                   ( A. x e. A A. y e. B ( ph -> ps )
                     /\ A. x e. A A. y e. B ( ps -> ph ) ) ) $=
    ( wb wral wi wa ralbiim ralbii r19.26 bitri ) ABGDFHZCEHABIDFHZBAIDFHZJZCEH
    PCEHQCEHJORCEABDFKLPQCEMN $.

  $( 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
    $.

  $( Relationship between three restricted universal and existential
     quantifiers.  (Contributed by Thierry Arnoux, 12-Jul-2020.)  (Proof
     shortened by Wolf Lammen, 18-May-2023.) $)
  ralnex3 $p |- ( A. x e. A A. y e. B A. z e. C -. ph
    <-> -. E. x e. A E. y e. B E. z e. C ph ) $=
    ( wn wral wrex ralnex 2ralbii ralnex2 bitri ) AHDGIZCFIBEIADGJZHZCFIBEIPCFJ
    BEJHOQBCEFADGKLPBCEFMN $.

  $( Relationship between two restricted universal and existential quantifiers.
     (Contributed by Glauco Siliprandi, 11-Dec-2019.) $)
  rexnal2 $p |- ( E. x e. A E. y e. B -. ph <-> -. A. x e. A A. y e. B ph ) $=
    ( wn wrex wral rexnal rexbii bitri ) AFCEGZBDGACEHZFZBDGMBDHFLNBDACEIJMBDIK
    $.

  $( Relationship between three restricted universal and existential
     quantifiers.  (Contributed by Thierry Arnoux, 12-Jul-2020.) $)
  rexnal3 $p |- ( E. x e. A E. y e. B E. z e. C -. ph
    <-> -. A. x e. A A. y e. B A. z e. C ph ) $=
    ( wn wrex wral rexnal 2rexbii rexnal2 bitri ) AHDGIZCFIBEIADGJZHZCFIBEIPCFJ
    BEJHOQBCEFADGKLPBCEFMN $.

  $( Negation of a complex predicate calculus formula.  (Contributed by FL,
     31-Jul-2009.) $)
  nrexralim $p |- ( -. E. x e. A A. y e. B ( ph -> ps )
     <-> A. x e. A E. y e. B ( ph /\ -. ps ) ) $=
    ( wn wa wrex wral wi rexanali ralbii ralnex bitr2i ) ABGHDFIZCEJABKDFJZGZCE
    JQCEIGPRCEABDFLMQCENO $.

  $( Restricted quantifier version of ~ 19.26-2 .  Version of ~ r19.26 with two
     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 ~ r19.29 with two quantifiers.  (Contributed by Rodolfo Medina,
     25-Sep-2010.) $)
  2r19.29 $p |- ( ( A. x e. A A. y e. B ph /\ E. x e. A E. y e. B ps )
                        -> E. x e. A E. y e. B ( ph /\ ps ) ) $=
    ( wral wrex wa r19.29 reximi syl ) ADFGZCEGBDFHZCEHIMNIZCEHABIDFHZCEHMNCEJO
    PCEABDFJKL $.

  ${
    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.)
       (Proof shortened by Wolf Lammen, 4-Nov-2024.) $)
    r19.29d2r $p |- ( ph -> E. x e. A E. y e. B ( ps /\ ch ) ) $=
      ( wral wrex wa jca 2r19.29 syl ) ABEGJDFJZCEGKDFKZLBCLEGKDFKAPQHIMBCDEFGN
      O $.
  $}

  ${
    r2allem.1 $e |- ( A. y ( x e. A -> ( y e. B -> ph ) )
                      <-> ( x e. A -> A. y ( y e. B -> ph ) ) ) $.
    $( Lemma factoring out common proof steps of ~ r2alf and ~ r2al .
       Introduced to reduce dependencies on axioms.  (Contributed by Wolf
       Lammen, 9-Jan-2020.) $)
    r2allem $p |- ( A. x e. A A. y e. B ph <->
               A. x A. y ( ( x e. A /\ y e. B ) -> ph ) ) $=
      ( wral cv wcel wi wal wa df-ral impexp albii imbi2i 3bitr4i bitr4i ) ACEG
      ZBDGBHDIZSJZBKTCHEIZLAJZCKZBKSBDMUDUABTUBAJZJZCKTUECKZJUDUAFUCUFCTUBANOSU
      GTACEMPQOR $.
  $}

  ${
    r2exlem.1 $e |- ( A. x e. A A. y e. B -. ph <->
               A. x A. y ( ( x e. A /\ y e. B ) -> -. ph ) ) $.
    $( Lemma factoring out common proof steps in ~ r2exf an ~ r2ex .
       Introduced to reduce dependencies on axioms.  (Contributed by Wolf
       Lammen, 10-Jan-2020.) $)
    r2exlem $p |- ( E. x e. A E. y e. B ph <->
                E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) ) $=
      ( cv wcel wa wn wal wex wral wrex exnal xchbinxr exnalimn exbii ralnex2
      wi con2bii 3bitr4ri ) BGDHCGEHIZAJZTCKZJZBLZUDCEMBDMZJUCAICLZBLACENBDNZUG
      UEBKUHUEBOFPUIUFBUCACQRUHUJABCDESUAUB $.
  $}

  ${
    hbralrimi.1 $e |- ( ph -> A. x ph ) $.
    hbralrimi.2 $e |- ( ph -> ( x e. A -> ps ) ) $.
    $( Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier
       version).  This theorem contains the common proof steps for ~ ralrimi
       and ~ ralrimiv .  Its main advantage over these two is its minimal
       references to axioms.  The proof is extracted from NM's previous work.
       (Contributed by Wolf Lammen, 4-Dec-2019.) $)
    hbralrimi $p |- ( ph -> A. x e. A ps ) $=
      ( cv wcel wi wal wral alrimih df-ral sylibr ) ACGDHBIZCJBCDKAOCEFLBCDMN
      $.
  $}

  $( *** Theorems using axioms up to ax-5 only: *** $)

  ${
    $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.)  Reduce
       dependencies on axioms.  (Revised by Wolf Lammen, 4-Dec-2019.) $)
    ralrimiv $p |- ( ph -> A. x e. A ps ) $=
      ( ax-5 hbralrimi ) 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 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.)  Shorten dependent
       theorems.  (Revised by Wolf lammen, 23-Dec-2024.) $)
    rexlimiva $p |- ( E. x e. A ph -> ps ) $=
      ( wrex cv wcel wa wex df-rex exlimiv sylbi ) ACDFCGDHAIZCJBACDKNBCELM $.
  $}

  ${
    $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.)  Reduce
       dependencies on axioms.  (Revised by Wolf Lammen, 14-Jan-2020.) $)
    rexlimiv $p |- ( E. x e. A ph -> ps ) $=
      ( cv wcel imp rexlimiva ) ABCDCFDGABEHI $.
  $}

  ${
    $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.)  (Proof shortened by Wolf Lammen,
       5-Jan-2020.) $)
    nrexdv $p |- ( ph -> -. E. x e. A ps ) $=
      ( wn wral wrex ralrimiva ralnex sylib ) ABFZCDGBCDHFALCDEIBCDJK $.
  $}

  ${
    $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 $.
  $}

  ${
    $d x ps $.
    rexlimivw.1 $e |- ( ph -> ps ) $.
    $( Weaker version of ~ rexlimiv .  (Contributed by FL, 19-Sep-2011.)
       (Proof shortened by Wolf Lammen, 23-Dec-2024.) $)
    rexlimivw $p |- ( E. x e. A ph -> ps ) $=
      ( cv wcel adantl rexlimiva ) ABCDABCFDGEHI $.
  $}

  ${
    $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.)  Reduce
       dependencies on axioms.  (Revised by Wolf Lammen, 28-Dec-2019.) $)
    ralrimdv $p |- ( ph -> ( ps -> A. x e. A ch ) ) $=
      ( wral wa cv wcel wi imp ralrimiv ex ) ABCDEGABHCDEABDIEJCKFLMN $.
  $}

  ${
    $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.)  Reduce dependencies on axioms.  (Revised by Wolf
       Lammen, 14-Jan-2020.) $)
    rexlimdv $p |- ( ph -> ( E. x e. A ps -> ch ) ) $=
      ( wrex wi cv wcel com3l rexlimiv com12 ) BDEGACBACHDEADIEJBCFKLM $.
  $}

  ${
    $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.)  (Proof shortened
       by Wolf Lammen, 28-Dec-2019.) $)
    ralrimdva $p |- ( ph -> ( ps -> A. x e. A ch ) ) $=
      ( cv wcel expimpd expcomd ralrimdv ) ABCDEADGEHZBCALBCFIJK $.
  $}

  ${
    $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 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 $.

    $( A commonly used pattern in the spirit of ~ r19.29 .  (Contributed by
       Thierry Arnoux, 29-Dec-2019.)  (Proof shortened by Wolf Lammen,
       17-Jun-2023.) $)
    r19.29an $p |- ( ( ph /\ E. x e. A ps ) -> ch ) $=
      ( wrex rexlimdva2 imp ) ABDEGCABCDEFHI $.
  $}

  ${
    $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 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 ch x $.  $d ph x $.
    r19.29a.1 $e |- ( ( ( ph /\ x e. A ) /\ ps ) -> ch ) $.
    r19.29a.2 $e |- ( ph -> E. x e. A ps ) $.
    $( A commonly used pattern in the spirit of ~ r19.29 .  (Contributed by
       Thierry Arnoux, 22-Nov-2017.)  Reduce axiom usage.  (Revised by Wolf
       Lammen, 17-Jun-2023.) $)
    r19.29a $p |- ( ph -> ch ) $=
      ( wrex rexlimdva2 mpd ) ABDEHCGABCDEFIJ $.
  $}

  ${
    $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 $.
  $}

  ${
    $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.)  Reduce
       dependencies on axioms.  (Revised by Wolf Lammen, 8-Jan-2020.)  (Proof
       shortened by Wolf Lammen, 4-Nov-2024.) $)
    reximdvai $p |- ( ph -> ( E. x e. A ps -> E. x e. A ch ) ) $=
      ( cv wcel imdistand reximdv2 ) ABCDEEADGEHBCFIJ $.
  $}

  ${
    $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.)  Reduce
       dependencies on axioms.  (Revised by Wolf Lammen, 5-Dec-2019.) $)
    ralimdva $p |- ( ph -> ( A. x e. A ps -> A. x e. A ch ) ) $=
      ( cv wcel wi ex a2d ralimdv2 ) ABCDEEADGEHZBCAMBCIFJKL $.

    $( 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 $.
    ralimdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction quantifying both antecedent and consequent, based on Theorem
       19.20 of [Margaris] p. 90 ( ~ alim ).  (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 $.

    $( 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 $.
    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 ph x $.
    reximddv3.1 $e |- ( ( ( ph /\ x e. A ) /\ ps ) -> ch ) $.
    reximddv3.2 $e |- ( ph -> E. x e. A ps ) $.
    $( Deduction from Theorem 19.22 of [Margaris] p. 90.  (Contributed by
       Glauco Siliprandi, 5-Feb-2022.) $)
    reximddv3 $p |- ( ph -> E. x e. A ch ) $=
      ( cv wcel anasss reximddv ) ABCDEADHEIBCFJGK $.
  $}

  ${
    $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 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 $.
  $}

  ${
    $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.)  Reduce dependencies on axioms.
       (Revised by Wolf Lammen, 29-Dec-2019.) $)
    ralbidva $p |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) ) $=
      ( cv wcel pm5.74da ralbidv2 ) ABCDEEADGEHBCFIJ $.

    $( Formula-building rule for restricted existential quantifier (deduction
       form).  (Contributed by NM, 9-Mar-1997.)  Reduce dependencies on axioms.
       (Revised by Wolf Lammen, 6-Dec-2019.)  (Proof shortened by Wolf Lammen,
       10-Dec-2019.) $)
    rexbidva $p |- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) ) $=
      ( cv wcel pm5.32da rexbidv2 ) ABCDEEADGEHBCFIJ $.
  $}

  ${
    $d x ph $.
    ralbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for restricted universal quantifier (deduction
       form).  (Contributed by NM, 20-Nov-1994.)  Reduce dependencies on
       axioms.  (Revised by Wolf Lammen, 5-Dec-2019.) $)
    ralbidv $p |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) ) $=
      ( wb cv wcel adantr ralbidva ) ABCDEABCGDHEIFJK $.

    $( Formula-building rule for restricted existential quantifier (deduction
       form).  (Contributed by NM, 20-Nov-1994.)  Reduce dependencies on
       axioms.  (Revised by Wolf Lammen, 6-Dec-2019.) $)
    rexbidv $p |- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) ) $=
      ( wb cv wcel adantr rexbidva ) ABCDEABCGDHEIFJK $.
  $}

  ${
    $d x ph $.
    $( Restricted quantifier version of ~ 19.21v .  (Contributed by NM,
       15-Oct-2003.)  (Proof shortened by Andrew Salmon, 30-May-2011.)  Reduce
       dependencies on axioms.  (Revised by Wolf Lammen, 2-Jan-2020.)  (Proof
       shortened by Wolf Lammen, 11-Dec-2024.) $)
    r19.21v $p |- ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) ) $=
      ( wi wral pm2.27 ralimdv com12 wn pm2.21 ralrimivw ax-1 ralimi ja impbii
      ) ABEZCDFZABCDFZEARSAQBCDABGHIASRAJQCDABKLBQCDBAMNOP $.

    $( Obsolete version of ~ r19.21v as of 11-Dec-2024.  (Contributed by NM,
       15-Oct-2003.)  (Proof shortened by Andrew Salmon, 30-May-2011.)  Reduce
       dependencies on axioms.  (Revised by Wolf Lammen, 2-Jan-2020.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    r19.21vOLD $p |- ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) ) $=
      ( cv wcel wi wal wral bi2.04 albii 19.21v bitri df-ral imbi2i 3bitr4i ) C
      EDFZABGZGZCHZAQBGZCHZGZRCDIABCDIZGTAUAGZCHUCSUECQABJKAUACLMRCDNUDUBABCDNO
      P $.

    $( Restricted quantifier version of one direction of ~ 19.37v .  (The other
       direction holds iff ` A ` is nonempty, see ~ r19.37zv .)  (Contributed
       by NM, 2-Apr-2004.)  Reduce axiom usage.  (Revised by Wolf Lammen,
       18-Jun-2023.) $)
    r19.37v $p |- ( E. x e. A ( ph -> ps ) -> ( ph -> E. x e. A ps ) ) $=
      ( wral wi wrex id ralrimivw r19.35 biimpi syl5 ) AACDEZABFCDGZBCDGZAACDAH
      INMOFABCDJKL $.
  $}

  ${
    $d x ps $.
    $( Restricted quantifier version of ~ 19.23v .  Version of ~ r19.23 with a
       disjoint variable condition.  (Contributed by NM, 31-Aug-1999.)  Reduce
       dependencies on axioms.  (Revised by Wolf Lammen, 14-Jan-2020.) $)
    r19.23v $p |- ( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) ) $=
      ( wi wral wn wrex con34b ralbii r19.21v dfrex2 imbi1i con1b bitr2i 3bitri
      ) ABEZCDFBGZAGZEZCDFRSCDFZEZACDHZBEZQTCDABIJRSCDKUDUAGZBEUBUCUEBACDLMUABN
      OP $.

    $( Restricted quantifier version of one direction of ~ 19.36 .  (The other
       direction holds iff ` A ` is nonempty, see ~ r19.36zv .)  (Contributed
       by NM, 22-Oct-2003.) $)
    r19.36v $p |- ( E. x e. A ( ph -> ps ) -> ( A. x e. A ph -> ps ) ) $=
      ( wi wrex wral r19.35 id rexlimivw imim2i sylbi ) ABECDFACDGZBCDFZEMBEABC
      DHNBMBBCDBIJKL $.
  $}

  ${
    $d x ps $.
    rexlimivOLD.1 $e |- ( x e. A -> ( ph -> ps ) ) $.
    $( Obsolete version of ~ rexlimiv as of 19-Dec-2024.)  (Contributed by NM,
       20-Nov-1994.)  Reduce dependencies on axioms.  (Revised by Wolf Lammen,
       14-Jan-2020.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    rexlimivOLD $p |- ( E. x e. A ph -> ps ) $=
      ( wi wral wrex rgen r19.23v mpbi ) ABFZCDGACDHBFLCDEIABCDJK $.
  $}

  ${
    $d x ps $.
    rexlimivaOLD.1 $e |- ( ( x e. A /\ ph ) -> ps ) $.
    $( Obsolete version of ~ rexlimiva as of 23-Dec-2024.  (Contributed by NM,
       18-Dec-2006.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    rexlimivaOLD $p |- ( E. x e. A ph -> ps ) $=
      ( cv wcel ex rexlimiv ) ABCDCFDGABEHI $.
  $}

  ${
    $d ps x $.
    rexlimivwOLD.1 $e |- ( ph -> ps ) $.
    $( Obsolete version of ~ rexlimivw as of 23-Dec-2024.  (Contributed by FL,
       19-Sep-2011.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    rexlimivwOLD $p |- ( E. x e. A ph -> ps ) $=
      ( wi cv wcel a1i rexlimiv ) ABCDABFCGDHEIJ $.
  $}

  ${
    $d x ps $.
    $( Restricted quantitifer version of one direction of ~ 19.27 .  (Assuming
       ` F/_ x A ` , the other direction holds when ` A ` is nonempty, see
       ~ r19.27zv .)  (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 $.

    $( Restricted quantifier version ~ 19.41v .  Version of ~ r19.41 with a
       disjoint variable condition, requiring fewer axioms.  (Contributed by
       NM, 17-Dec-2003.)  Reduce dependencies on axioms.  (Revised by BJ,
       29-Mar-2020.) $)
    r19.41v $p |- ( E. x e. A ( ph /\ ps ) <-> ( E. x e. A ph /\ ps ) ) $=
      ( wa wrex cv wcel wex df-rex anass exbii 19.41v bicomi bianbi 3bitr2i ) A
      BEZCDFCGDHZQEZCIRAEZBEZCIZACDFZBEQCDJUASCRABKLUBTCIZBUCTBCMUCUDACDJNOP $.
  $}

  ${
    $d x ph $.
    $( Restricted quantifier version of one direction of ~ 19.28 .  (Assuming
       ` F/_ x A ` , the other direction holds when ` A ` is nonempty, see
       ~ r19.28zv .)  (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 $.

    $( Restricted quantifier version of ~ 19.42v (see also ~ 19.42 ).
       (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 $.
  $}

  ${
    $d x ph $.
    $( Restricted quantifier version of ~ 19.32v .  (Contributed by NM,
       25-Nov-2003.) $)
    r19.32v $p |- ( A. x e. A ( ph \/ ps ) <-> ( ph \/ A. x e. A ps ) ) $=
      ( wn wi wral wo r19.21v df-or ralbii 3bitr4i ) AEZBFZCDGMBCDGZFABHZCDGAOH
      MBCDIPNCDABJKAOJL $.

    $( Restricted quantifier version of one direction of ~ 19.45 .  The other
       direction holds when ` A ` is nonempty, see ~ r19.45zv .  (Contributed
       by NM, 2-Apr-2004.) $)
    r19.45v $p |- ( E. x e. A ( ph \/ ps ) -> ( ph \/ E. x e. A ps ) ) $=
      ( wo wrex r19.43 id rexlimivw orim1i sylbi ) ABECDFACDFZBCDFZEAMEABCDGLAM
      AACDAHIJK $.
  $}

  ${
    $d x ps $.
    $( One direction of a restricted quantifier version of ~ 19.44 .  The other
       direction holds when ` A ` is nonempty, see ~ r19.44zv .  (Contributed
       by NM, 2-Apr-2004.) $)
    r19.44v $p |- ( E. x e. A ( ph \/ ps ) -> ( E. x e. A ph \/ ps ) ) $=
      ( wo wrex r19.43 id rexlimivw orim2i sylbi ) ABECDFACDFZBCDFZELBEABCDGMBL
      BBCDBHIJK $.
  $}

  ${
    $d x y $.  $d y A $.
    $( Double restricted universal quantification.  (Contributed by NM,
       19-Nov-1995.)  Reduce dependencies on axioms.  (Revised by Wolf Lammen,
       9-Jan-2020.) $)
    r2al $p |- ( A. x e. A A. y e. B ph <->
               A. x A. y ( ( x e. A /\ y e. B ) -> ph ) ) $=
      ( cv wcel wi 19.21v r2allem ) ABCDEBFDGCFEGAHCIJ $.

    $( Double restricted existential quantification.  (Contributed by NM,
       11-Nov-1995.)  Reduce dependencies on axioms.  (Revised by Wolf Lammen,
       10-Jan-2020.) $)
    r2ex $p |- ( E. x e. A E. y e. B ph <->
               E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) ) $=
      ( wn r2al r2exlem ) ABCDEAFBCDEGH $.
  $}

  ${
    $d x y z $.  $d y z A $.  $d z B $.
    $( Triple restricted universal quantification.  (Contributed by NM,
       19-Nov-1995.)  (Proof shortened by Wolf Lammen, 30-Dec-2019.) $)
    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 ) ) $=
      ( wral cv wcel wa wi wal w3a r2al 19.21v df-3an imbi1i impexp bitri albii
      df-ral imbi2i 3bitr4ri 2albii ) ADGHZCFHBEHBIEJZCIFJZKZUFLZCMBMUGUHDIGJZN
      ZALZDMZCMBMUFBCEFOUJUNBCUIUKALZLZDMUIUODMZLUNUJUIUODPUMUPDUMUIUKKZALUPULU
      RAUGUHUKQRUIUKASTUAUFUQUIADGUBUCUDUET $.

    $( Triple existential quantification.  (Contributed by AV, 21-Jul-2025.) $)
    r3ex $p |- ( E. x e. A E. y e. B E. z e. C ph <->
               E. x E. y E. z ( ( x e. A /\ y e. B /\ z e. C ) /\ ph ) ) $=
      ( wrex cv wcel wa wex w3a r2ex df-rex anbi2i 19.42v anass bicomi df-3an
      bianbi exbii 3bitr2i 2exbii bitri ) ADGHZCFHBEHBIEJZCIFJZKZUFKZCLBLUGUHDI
      GJZMZAKZDLZCLBLUFBCEFNUJUNBCUJUIUKAKZDLZKUIUOKZDLUNUFUPUIADGOPUIUODQUQUMD
      UQUIUKKZAULURAKUQUIUKARSULURUGUHUKTSUAUBUCUDUE $.
  $}

  ${
    $d x y $.  $d y A $.
    rgen2.1 $e |- ( ( x e. A /\ y e. B ) -> ph ) $.
    $( Generalization rule for restricted quantification, with two quantifiers.
       This theorem should be used in place of ~ rgen2a since it depends on a
       smaller set of axioms.  (Contributed by NM, 30-May-1999.) $)
    rgen2 $p |- A. x e. A A. y e. B ph $=
      ( wral cv wcel ralrimiva rgen ) ACEGBDBHDIACEFJK $.
    $( $j usage 'rgen2' avoids 'ax-6' 'ax-7' 'ax-8' 'ax-9' 'ax-10' 'ax-11'
       'ax-12' 'ax-13' 'ax-ext'; $)
  $}

  ${
    $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 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 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 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 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, with three
       quantifiers.  (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 $.
  $}

  ${
    $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 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 $.

    $( Deduction doubly quantifying both antecedent and consequent, based on
       Theorem 19.22 of [Margaris] p. 90.  (Contributed by AV, 5-Jan-2022.) $)
    reximdvva $p |- ( ph
                   -> ( E. x e. A E. y e. B ps -> E. x e. A E. y e. B ch ) ) $=
      ( wrex cv wcel wa wi anassrs reximdva ) ABEGICEGIDFADJFKZLBCEGAPEJGKBCMHN
      OO $.
  $}

  ${
    $d y A $.  $d x y ph $.
    ralimdvv.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction doubly quantifying both antecedent and consequent.
       (Contributed by Scott Fenton, 2-Mar-2025.) $)
    ralimdvv $p |- ( ph -> ( A. x e. A A. y e. B ps ->
     A. x e. A A. y e. B ch ) ) $=
      ( wi cv wcel wa adantr ralimdvva ) ABCDEFGABCIDJFKEJGKLHMN $.
  $}

  ${
    $d y z w A $.  $d z w B $.  $d w C $.  $d x y z w ph $.
    ralimd4v.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction quadrupally quantifying both antecedent and consequent.
       (Contributed by Scott Fenton, 2-Mar-2025.) $)
    ralimd4v $p |- ( ph -> ( A. x e. A A. y e. B A. z e. C A. w e. D ps ->
     A. x e. A A. y e. B A. z e. C A. w e. D ch ) ) $=
      ( wral ralimdvv ) ABGKMFJMCGKMFJMDEHIABCFGJKLNN $.
  $}

  ${
    $d y z w A $.  $d z w B $.  $d w C $.  $d q E $.  $d x y z w ph $.
    $d p q ph $.
    ralim6dv.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction sextupally quantifying both antecedent and consequent.
       (Contributed by Scott Fenton, 5-Mar-2025.) $)
    ralimd6v $p |- ( ph -> (
     A. x e. A A. y e. B A. z e. C A. w e. D A. p e. E A. q e. F ps ->
     A. x e. A A. y e. B A. z e. C A. w e. D A. p e. E A. q e. F ch ) ) $=
      ( wral ralimdvv ralimd4v ) ABNMQOLQCNMQOLQDEFGHIJKABCONLMPRS $.
  $}

  ${
    $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 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 $.
  $}

  ${
    $d ph x y z $.  $d ch x y z $.  $d A y z $.  $d B z $.
    rexlimdvvva.1 $e |- ( ( ph /\ ( x e. A /\ y e. B /\ z e. C ) )
                          -> ( ps -> ch ) ) $.
    $( Inference from Theorem 19.23 of [Margaris] p. 90, for three restricted
       quantifiers.  (Contributed by AV, 23-Aug-2025.) $)
    rexlimdvvva $p |- ( ph -> ( E. x e. A E. y e. B E. z e. C ps -> ch ) ) $=
      ( wrex cv wcel wa wi w3a df-3an ex biimtrrid expdimp rexlimdv rexlimdvva
      ) ABFIKCDEGHADLGMZELHMZNZNBCFIAUEFLIMZBCOZUEUFNUCUDUFPZAUGUCUDUFQAUHUGJRS
      TUAUB $.
  $}

  ${
    $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 $.
  $}

  ${
    $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.)  (Proof
       shortened by Wolf Lammen, 4-Nov-2024.) $)
    r19.29vva $p |- ( ph -> ch ) $=
      ( wrex reximddv2 cv wcel wa idd rexlimivv syl ) ACEGJDFJCABCDEFGHIKCCDEFG
      DLFMELGMNCOPQ $.
  $}

  ${
    $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 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.)  Reduce dependencies on axioms.
       (Revised by Wolf Lammen, 9-Dec-2019.) $)
    2ralbidva $p |- ( ph ->
                     ( A. x e. A A. y e. B ps <-> A. x e. A A. y e. B ch ) ) $=
      ( wral cv wcel wa wb anassrs ralbidva ) ABEGICEGIDFADJFKZLBCEGAPEJGKBCMHN
      OO $.

    $( 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 $.
  $}

  ${
    $d ph x $.  $d ph y $.  $d ph z $.
    3ralbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for restricted universal quantifiers (deduction
       form.)  (Contributed by Scott Fenton, 20-Feb-2025.) $)
    3ralbidv $p |- ( ph -> ( A. x e. A A. y e. B A. z e. C ps <->
    A. x e. A A. y e. B A. z e. C ch ) ) $=
      ( wral ralbidv 2ralbidv ) ABFIKCFIKDEGHABCFIJLM $.
  $}

  ${
    $d ph x $.  $d ph y $.  $d ph z $.  $d ph w $.
    4ralbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for restricted universal quantifiers (deduction
       form.)  (Contributed by Scott Fenton, 20-Feb-2025.) $)
    4ralbidv $p |- ( ph -> ( A. x e. A A. y e. B A. z e. C A. w e. D ps <->
    A. x e. A A. y e. B A. z e. C A. w e. D ch ) ) $=
      ( wral ralbidv 3ralbidv ) ABGKMCGKMDEFHIJABCGKLNO $.
  $}

  ${
    $d ph t $.  $d ph u $.  $d ph w $.  $d ph x $.  $d ph y $.  $d ph z $.
    6ralbidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for restricted universal quantifiers (deduction
       form.)  (Contributed by Scott Fenton, 5-Mar-2025.) $)
    6ralbidv $p |- ( ph ->
    ( A. x e. A A. y e. B A. z e. C A. w e. D A. t e. E A. u e. F ps <->
    A. x e. A A. y e. B A. z e. C A. w e. D A. t e. E A. u e. F ch ) ) $=
      ( wral 2ralbidv 4ralbidv ) ABHOQINQCHOQINQDEFGJKLMABCIHNOPRS $.
  $}

  ${
    $d x ps $.  $d y ps $.
    $( Version of ~ r19.41v with two quantifiers.  (Contributed by Thierry
       Arnoux, 25-Jan-2017.) $)
    r19.41vv $p |- ( E. x e. A E. y e. B ( ph /\ ps )
         <-> ( E. x e. A E. y e. B ph /\ ps ) ) $=
      ( wa wrex r19.41v rexbii bitri ) ABGDFHZCEHADFHZBGZCEHMCEHBGLNCEABDFIJMBC
      EIK $.
  $}

  ${
    $d y A $.  $d x y $.
    reeanlem.1 $e |- ( E. x E. y ( ( x e. A /\ ph ) /\ ( y e. B /\ ps ) )
                    <-> ( E. x ( x e. A /\ ph ) /\ E. y ( y e. B /\ ps ) ) ) $.
    $( Lemma factoring out common proof steps of ~ reeanv and ~ reean .
       (Contributed by Wolf Lammen, 20-Aug-2023.) $)
    reeanlem $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 bitri r2ex df-rex anbi12i 3bitr4i ) CHEI
      ZDHFIZJABJZJZDKCKZTAJZCKZUABJZDKZJZUBDFLCELACELZBDFLZJUDUEUGJZDKCKUIUCULC
      DTUAABMNGOUBCDEFPUJUFUKUHACEQBDFQRS $.
  $}

  ${
    $d y ph $.  $d x ps $.  $d x y $.  $d y A $.  $d x B $.
    $( Rearrange restricted 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 ) ) $=
      ( cv wcel wa exdistrv reeanlem ) ABCDEFCGEHAIDGFHBICDJK $.
  $}

  ${
    $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 restricted 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 bianbi df-3an 2rexbii bitri rexbii 3bitr4i )
      ABJZEHKZCFIKZJZDGKZADGKZBEHKZJZUCJABCLZFIKEHKZDGKUFUGUCLUEUBDGKUCUHUBUCDG
      MABDEGHNOUJUDDGUJUACJZFIKEHKUDUIUKEFHIABCPQUACEFHINRSUFUGUCPT $.
  $}

  ${
    $d ph y $.  $d ps x $.  $d B x $.  $d x y $.
    $( Distribute restricted universal quantification over "or".  (Contributed
       by Jeff Madsen, 19-Jun-2010.)  Shorten, reduce dv conditions.  (Revised
       by Wolf Lammen, 20-Nov-2024.) $)
    2ralor $p |- ( A. x e. A A. y e. B ( ph \/ ps ) <->
                  ( A. x e. A ph \/ A. y e. B ps ) ) $=
      ( wo wral r19.32v orcom bitri ralbii 3bitri ) ABGDFHZCEHBDFHZAGZCEHOACEHZ
      GQOGNPCENAOGPABDFIAOJKLOACEIOQJM $.
  $}

  ${
    $d ph y $.  $d ps x $.  $d A y $.  $d B x $.  $d x y $.
    $( Obsolete version of ~ 2ralor as of 20-Nov-2024.  (Contributed by Jeff
       Madsen, 19-Jun-2010.)  (New usage is discouraged.)
       (Proof modification is discouraged.) $)
    2ralorOLD $p |- ( A. x e. A A. y e. B ( ph \/ ps ) <->
                  ( A. x e. A ph \/ A. y e. B ps ) ) $=
      ( wo wral wn wrex wa rexnal anbi12i rexbii bitr3i reeanv 3bitr3ri 3bitr4i
      ioran con4bii ) ABGZDFHZCEHZACEHZBDFHZGZAIZCEJZBIZDFJZKZUDIZUEIZKUCIZUFIU
      HULUJUMACELBDFLMUGUIKZDFJZCEJUBIZCEJUKUNUPUQCEUPUAIZDFJUQURUODFABSNUADFLO
      NUGUICDEFPUBCELQUDUESRT $.
  $}

  $( *** Theorems using axioms up to ax-8 only: *** $)

  ${
    $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 dfclel 3bitr4ri ) ADZCEZNBFZGAH
      POGAHPACIBCEOPAJPACKABCLM $.

    $( A definition of ` -. A e. B ` .  (Contributed by Thierry Arnoux,
       20-Nov-2023.)  (Proof shortened by SN, 23-Jan-2024.)  (Proof shortened
       by Wolf Lammen, 3-Nov-2024.) $)
    nelb $p |- ( -. A e. B <-> A. x e. B x =/= A ) $=
      ( cv wceq wrex wne wral wcel df-ne ralbii ralnex bitr2i risset xchnxbir
      wn ) ADZBEZACFZQBGZACHZBCIUARPZACHSPTUBACQBJKRACLMABCNO $.
  $}

  ${
    $d x y A $.  $d x ps $.  $d y ph $.
    rspw.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Restricted specialization.  Weak version of ~ rsp , requiring ~ ax-8 ,
       but not ~ ax-12 .  (Contributed by GG, 3-Oct-2024.) $)
    rspw $p |- ( A. x e. A ph -> ( x e. A -> ph ) ) $=
      ( wral cv wcel wi wal df-ral weq eleq1w imbi12d spw sylbi ) ACEGCHEIZAJZC
      KSACELSDHEIZBJCDCDMRTABCDENFOPQ $.
    $( $j usage 'rspw' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $d x y A $.  $d y ph $.  $d x ps $.
    cbvralvw.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Change the bound variable of a restricted universal quantifier using
       implicit substitution.  Version of ~ cbvralv with a disjoint variable
       condition, which does not require ~ ax-10 , ~ ax-11 , ~ ax-12 ,
       ~ ax-13 .  (Contributed by NM, 28-Jan-1997.)  Avoid ~ ax-13 .  (Revised
       by GG, 10-Jan-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-10' 'ax-11' 'ax-12' 'ax-13'; $)

    $( Change the bound variable of a restricted existential quantifier using
       implicit substitution.  Version of ~ cbvrexv with a disjoint variable
       condition, which does not require ~ ax-10 , ~ ax-11 , ~ ax-12 ,
       ~ ax-13 .  (Contributed by NM, 2-Jun-1998.)  Avoid ~ ax-13 .  (Revised
       by GG, 10-Jan-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 'cbvrexvw' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    $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.)  Avoid ~ ax-9 , ~ ax-ext .  (Revised by Wolf
       Lammen, 9-Mar-2025.) $)
    cbvraldva $p |- ( ph -> ( A. x e. A ps <-> A. y e. A ch ) ) $=
      ( wral wi weq wb ancoms pm5.74da cbvralvw r19.21v 3bitr3i pm5.74ri ) ABDF
      HZCEFHZABIZDFHACIZEFHARIASITUADEFDEJZABCAUBBCKGLMNABDFOACEFOPQ $.
    $( $j usage 'cbvraldva' avoids 'ax-9' 'ax-ext'; $)

    $( Rule used to change the bound variable in a restricted existential
       quantifier with implicit substitution.  Deduction form.  (Contributed by
       David Moews, 1-May-2017.)  Avoid ~ ax-9 , ~ ax-ext .  (Revised by Wolf
       Lammen, 9-Mar-2025.) $)
    cbvrexdva $p |- ( ph -> ( E. x e. A ps <-> E. y e. A ch ) ) $=
      ( wrex wn wral weq wa notbid cbvraldva ralnex 3bitr3g con4bid ) ABDFHZCEF
      HZABIZDFJCIZEFJRISIATUADEFADEKLBCGMNBDFOCEFOPQ $.
    $( $j usage 'cbvrexdva' avoids 'ax-9' 'ax-ext'; $)
  $}

  ${
    $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.)  Avoid ~ ax-13 .  (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-13'; $)
  $}

  ${
    $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.)  Avoid ~ ax-13 .  (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-13'; $)
  $}

  ${
    $d w ph $.  $d z ps $.  $d x ch $.  $d v ch $.  $d y th $.  $d u th $.
    $d x A $.  $d u z $.  $d w A $.  $d x y B $.  $d w y B $.  $d v B $.
    $d x y z C $.  $d w x $.  $d w y z C $.  $d v z C $.  $d z y C $.
    $d z C $.  $d u C $.  $d v y $.
    cbvral3vw.1 $e |- ( x = w -> ( ph <-> ch ) ) $.
    cbvral3vw.2 $e |- ( y = v -> ( ch <-> th ) ) $.
    cbvral3vw.3 $e |- ( z = u -> ( th <-> ps ) ) $.
    $( Change bound variables of triple restricted universal quantification,
       using implicit substitution.  Version of ~ cbvral3v with a disjoint
       variable condition, which does not require ~ ax-13 .  (Contributed by
       NM, 10-May-2005.)  Avoid ~ ax-13 .  (Revised by GG, 10-Jan-2024.) $)
    cbvral3vw $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 cbvralvw cbvral2vw ralbii bitri ) AGMQFLQZEKQCGMQFLQZ
      HKQBJMQILQZHKQUDUEEHKEHRACFGLMNSTUEUFHKCBDFGIJLMOPUAUBUC $.
    $( $j usage 'cbvral3vw' avoids 'ax-13'; $)
  $}

  ${
    $d a x A $.  $d a x y B $.  $d b y B $.  $d a x C $.  $d b y C $.
    $d c z C $.  $d a x y z w D $.  $d b y z w D $.  $d c z w D $.  $d d w D $.
    $d a ph $.  $d b ch $.  $d c th $.  $d x ch $.  $d d ta $.  $d w ps $.
    $d z ta $.  $d y th $.
    cbvral4vw.1 $e |- ( x = a -> ( ph <-> ch ) ) $.
    cbvral4vw.2 $e |- ( y = b -> ( ch <-> th ) ) $.
    cbvral4vw.3 $e |- ( z = c -> ( th <-> ta ) ) $.
    cbvral4vw.4 $e |- ( w = d -> ( ta <-> ps ) ) $.
    $( Change bound variables of quadruple restricted universal quantification,
       using implicit substitution.  (Contributed by Scott Fenton,
       2-Mar-2025.) $)
    cbvral4vw $p |- ( A. x e. A A. y e. B A. z e. C A. w e. D ph <->
       A. a e. A A. b e. B A. c e. C A. d e. D ps ) $=
      ( wral weq ralbidv cbvral3vw cbvralvw 3ralbii bitri ) AIMUBZHLUBGKUBFJUBE
      IMUBZPLUBOKUBNJUBBQMUBZPLUBOKUBNJUBUIUJCIMUBDIMUBFGHNOPJKLFNUCACIMRUDGOUC
      CDIMSUDHPUCDEIMTUDUEUJUKNOPJKLEBIQMUAUFUGUH $.
     $( $j usage 'cbvral4vw' avoids 'ax-13'; $)
  $}

  ${
    $d A a x $.  $d et w $.  $d th y $.  $d ps q $.  $d c th $.  $d d ta $.
    $d p ze $.  $d ch x $.  $d e et $.  $d b ch $.  $d a ph $.  $d f ze $.
    $d E c $.  $d E e $.  $d E d $.  $d E b z $.  $d D c $.  $d D a x $.
    $d D b y z $.  $d B a x $.  $d B b y $.  $d C a x $.  $d F e p q $.
    $d F a q x z $.  $d F b p q w y $.  $d F c p q w z $.  $d F d p q $.
    $d ta z $.  $d C c $.  $d E a p w x y $.  $d F f q $.  $d C b y z $.
    $d D d w $.
    cbvral6vw.1 $e |- ( x = a -> ( ph <-> ch ) ) $.
    cbvral6vw.2 $e |- ( y = b -> ( ch <-> th ) ) $.
    cbvral6vw.3 $e |- ( z = c -> ( th <-> ta ) ) $.
    cbvral6vw.4 $e |- ( w = d -> ( ta <-> et ) ) $.
    cbvral6vw.5 $e |- ( p = e -> ( et <-> ze ) ) $.
    cbvral6vw.6 $e |- ( q = f -> ( ze <-> ps ) ) $.
    $( Change bound variables of sextuple restricted universal quantification,
       using implicit substitution.  (Contributed by Scott Fenton,
       5-Mar-2025.) $)
    cbvral6vw $p |-
      ( A. x e. A A. y e. B A. z e. C A. w e. D A. p e. E A. q e. F ph <->
        A. a e. A A. b e. B A. c e. C A. d e. D A. e e. E A. f e. F ps ) $=
      ( wral weq 2ralbidv cbvral4vw cbvral2vw 4ralbii bitri ) ATSULUARULZKOULJN
      ULIMULHLULFTSULUARULZUEOULUDNULUCMULUBLULBQSULPRULZUEOULUDNULUCMULUBLULUS
      UTCTSULUARULDTSULUARULETSULUARULHIJKLMNOUBUCUDUEHUBUMACUATRSUFUNIUCUMCDUA
      TRSUGUNJUDUMDEUATRSUHUNKUEUMEFUATRSUIUNUOUTVAUBUCUDUELMNOFBGUATPQRSUJUKUP
      UQUR $.
    $( $j usage 'cbvral6vw' avoids 'ax-13'; $)
  $}

  ${
    $d A a x $.  $d F a q w x y z $.  $d f ze $.  $d r rh $.  $d h rh $.
    $d G g r $.  $d q si $.  $d G d $.  $d et w $.  $d e et $.  $d th y $.
    $d H c s $.  $d H h s $.  $d G e $.  $d ta z $.  $d ps s $.  $d d ta $.
    $d G c r $.  $d F c q $.  $d c th $.  $d F d $.  $d F b $.  $d G b w z $.
    $d F e p $.  $d a ph $.  $d G f $.  $d F f $.  $d p ze $.  $d b ch $.
    $d B b $.  $d C c $.  $d C b $.  $d E a x $.  $d E c p $.  $d E b y z $.
    $d E e $.  $d H b p q r s z $.  $d g si $.  $d H d q r s $.
    $d G a p q x y z $.  $d B a x y $.  $d C a x y z $.  $d D a x $.
    $d D b y $.  $d H f q r s $.  $d E d p w $.  $d D c w z $.
    $d H a r s w x y $.  $d D d $.  $d H e q r s $.  $d H g s $.  $d ch x $.
    cbvral8vw.1 $e |- ( x = a -> ( ph <-> ch ) ) $.
    cbvral8vw.2 $e |- ( y = b -> ( ch <-> th ) ) $.
    cbvral8vw.3 $e |- ( z = c -> ( th <-> ta ) ) $.
    cbvral8vw.4 $e |- ( w = d -> ( ta <-> et ) ) $.
    cbvral8vw.5 $e |- ( p = e -> ( et <-> ze ) ) $.
    cbvral8vw.6 $e |- ( q = f -> ( ze <-> si ) ) $.
    cbvral8vw.7 $e |- ( r = g -> ( si <-> rh ) ) $.
    cbvral8vw.8 $e |- ( s = h -> ( rh <-> ps ) ) $.
    $( Change bound variables of octuple restricted universal quantification,
       using implicit substitution.  (Contributed by Scott Fenton,
       2-Mar-2025.) $)
    cbvral8vw $p |- ( A. x e. A A. y e. B A. z e. C A. w e. D A. p e. E
       A. q e. F A. r e. G A. s e. H ph <->
       A. a e. A A. b e. B A. c e. C A. d e. D A. e e. E A. f e. F
       A. g e. G A. h e. H ps ) $=
      ( wral weq 4ralbidv cbvral4vw 4ralbii bitri ) AUFUEVBUGUDVBUHUCVBUIUBVBZM
      QVBLPVBKOVBJNVBFUFUEVBUGUDVBUHUCVBUIUBVBZUMQVBULPVBUKOVBUJNVBBUAUEVBTUDVB
      SUCVBRUBVBZUMQVBULPVBUKOVBUJNVBVHVICUFUEVBUGUDVBUHUCVBUIUBVBDUFUEVBUGUDVB
      UHUCVBUIUBVBEUFUEVBUGUDVBUHUCVBUIUBVBJKLMNOPQUJUKULUMJUJVCACUIUHUGUFUBUCU
      DUEUNVDKUKVCCDUIUHUGUFUBUCUDUEUOVDLULVCDEUIUHUGUFUBUCUDUEUPVDMUMVCEFUIUHU
      GUFUBUCUDUEUQVDVEVIVJUJUKULUMNOPQFBGHIUIUHUGUFUBUCUDUERSTUAURUSUTVAVEVFVG
      $.
     $( $j usage 'cbvral8vw' avoids 'ax-13'; $)
  $}

  $( *** Theorems using axioms up to ax-8 and ax-12 only: *** $)

  $( 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 $.

  ${
    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 $.
  $}

  ${
    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.)  (Proof
       shortened by Wolf Lammen, 11-Jun-2023.) $)
    r19.21bi $p |- ( ( ph /\ x e. A ) -> ps ) $=
      ( wral cv wcel rspa sylan ) ABCDFCGDHBEBCDIJ $.
  $}

  ${
    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 $.
  $}

  $( Restricted quantifier version of ~ 19.21t ; closed form of ~ r19.21 .
     (Contributed by NM, 1-Mar-2008.)  (Proof shortened by Wolf Lammen,
     2-Jan-2020.) $)
  r19.21t $p |- ( F/ x ph ->
           ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) ) ) $=
    ( wnf cv wcel wi wal wral 19.21t df-ral bi2.04 albii bitri imbi2i 3bitr4g )
    ACEACFDGZBHZHZCIZASCIZHABHZCDJZABCDJZHASCKUDRUCHZCIUAUCCDLUFTCRABMNOUEUBABC
    DLPQ $.

  ${
    r19.21.1 $e |- F/ x ph $.
    $( Restricted quantifier version of ~ 19.21 .  (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 $.
  $}

  $( 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 $.
    $( Restricted quantifier version of ~ 19.23 .  See ~ r19.23v for a version
       requiring fewer axioms.  (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 $.
  $}

  ${
    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).  For a version based on fewer axioms see ~ ralrimiv .
       (Contributed by NM, 10-Oct-1999.)  Shortened after introduction of
       ~ hbralrimi .  (Revised by Wolf Lammen, 4-Dec-2019.) $)
    ralrimi $p |- ( ph -> A. x e. A ps ) $=
      ( nf5ri hbralrimi ) ABCDACEGFH $.
  $}

  ${
    ralrimia.1 $e |- F/ x ph $.
    ralrimia.2 $e |- ( ( ph /\ x e. A ) -> ps ) $.
    $( Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier
       version).  (Contributed by Glauco Siliprandi, 23-Oct-2021.) $)
    ralrimia $p |- ( ph -> A. x e. A ps ) $=
      ( cv wcel ex ralrimi ) ABCDEACGDHBFIJ $.
  $}

  ${
    rexlimi.1 $e |- F/ x ps $.
    rexlimi.2 $e |- ( x e. A -> ( ph -> ps ) ) $.
    $( Restricted quantifier version of ~ exlimi .  For a version based on
       fewer axioms see ~ rexlimiv .  (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 $.
  $}

  ${
    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.)  (Proof
       shortened by Wolf Lammen, 29-Dec-2019.) $)
    ralimdaa $p |- ( ph -> ( A. x e. A ps -> A. x e. A ch ) ) $=
      ( wi wral cv wcel ex ralrimi ralim syl ) ABCHZDEIBDEICDEIHAPDEFADJEKPGLMB
      CDENO $.
  $}

  ${
    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 $.
  $}

  ${
    r19.37.1 $e |- F/ x ph $.
    $( Restricted quantifier version of one direction of ~ 19.37 .  (The other
       direction does not hold when ` A ` is empty.)  (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 ) ) $=
      ( wi wrex wral r19.35 cv wcel ax-1 ralrimi imim1i sylbi ) ABFCDGACDHZBCDG
      ZFAQFABCDIAPQAACDEACJDKLMNO $.
  $}

  ${
    r19.41.1 $e |- F/ x ps $.
    $( Restricted quantifier version of ~ 19.41 .  See ~ r19.41v for a version
       with a disjoint variable condition, requiring fewer axioms.
       (Contributed by NM, 1-Nov-2010.) $)
    r19.41 $p |- ( E. x e. A ( ph /\ ps ) <-> ( E. x e. A ph /\ ps ) ) $=
      ( wa wrex cv wcel wex df-rex anass exbii 19.41 bicomi bianbi 3bitr2i ) AB
      FZCDGCHDIZRFZCJSAFZBFZCJZACDGZBFRCDKUBTCSABLMUCUACJZBUDUABCENUDUEACDKOPQ
      $.
  $}

  ${
    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.)  For a version based on fewer axioms see
       ~ ralrimdv .  (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 $.
  $}

  ${
    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 $.
  $}

  ${
    rexlimd.1 $e |- F/ x ph $.
    rexlimd.2 $e |- F/ x ch $.
    rexlimd.3 $e |- ( ph -> ( x e. A -> ( ps -> ch ) ) ) $.
    $( Deduction form of ~ rexlimd .  For a version based on fewer axioms see
       ~ rexlimdv .  (Contributed by NM, 27-May-1998.)  (Proof shortened by
       Andrew Salmon, 30-May-2011.)  (Proof shortened by Wolf Lammen,
       14-Jan-2020.) $)
    rexlimd $p |- ( ph -> ( E. x e. A ps -> ch ) ) $=
      ( wnf a1i rexlimd2 ) ABCDEFCDIAGJHK $.
  $}

  ${
    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.)  (Proof shortened by OpenAI, 25-Mar-2020.) $)
    r19.29af2 $p |- ( ph -> ch ) $=
      ( wrex cv wcel exp31 rexlimd mpd ) ABDEJCIABCDEFGADKELBCHMNO $.
  $}

  ${
    $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 .  See ~ r19.29a , ~ r19.29an
       for a variant when ` x ` is disjoint from ` ph ` .  (Contributed by
       Thierry Arnoux, 29-Nov-2017.) $)
    r19.29af $p |- ( ph -> ch ) $=
      ( nfv r19.29af2 ) ABCDEFCDIGHJ $.
  $}

  ${
    reximd2a.1 $e |- F/ x ph $.
    reximd2a.2 $e |- ( ( ( ph /\ x e. A ) /\ ps ) -> x e. B ) $.
    reximd2a.3 $e |- ( ( ( ph /\ x e. A ) /\ ps ) -> ch ) $.
    reximd2a.4 $e |- ( ph -> E. x e. A ps ) $.
    $( Deduction quantifying both antecedent and consequent, based on Theorem
       19.22 of [Margaris] p. 90.  (Contributed by Thierry Arnoux,
       27-Jan-2020.) $)
    reximd2a $p |- ( ph -> E. x e. B ch ) $=
      ( wrex cv wcel wa wex jca expl eximd df-rex 3imtr4g mpd ) ABDEKZCDFKZJADL
      ZEMZBNZDOUDFMZCNZDOUBUCAUFUHDGAUEBUHAUENBNUGCHIPQRBDESCDFSTUA $.
  $}

  ${
    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.)  (Proof shortened by Wolf
       Lammen, 31-Oct-2024.) $)
    ralbida $p |- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) ) $=
      ( wral cv wcel wa biimpd ralimdaa biimprd impbid ) ABDEHCDEHABCDEFADIEJKZ
      BCGLMACBDEFPBCGNMO $.
  $}

  ${
    rexbida.1 $e |- F/ x ph $.
    rexbida.2 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
    $( 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 $.
  $}

  ${
    ralbid.1 $e |- F/ x ph $.
    ralbid.2 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for restricted universal quantifier (deduction
       form).  For a version based on fewer axioms see ~ ralbidv .
       (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 $.
  $}

  ${
    rexbid.1 $e |- F/ x ph $.
    rexbid.2 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for restricted existential quantifier (deduction
       form).  For a version based on fewer axioms see ~ rexbidv .
       (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 $.
    rexbidvALT.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Alternate proof of ~ rexbidv , shorter but requires more axioms.
       (Contributed by NM, 20-Nov-1994.)  (New usage is discouraged.)
       (Proof modification is discouraged.) $)
    rexbidvALT $p |- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) ) $=
      ( nfv rexbid ) ABCDEADGFH $.
  $}

  ${
    $d x ph $.
    rexbidvaALT.1 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
    $( Alternate proof of ~ rexbidva , shorter but requires more axioms.
       (Contributed by NM, 9-Mar-1997.)  (New usage is discouraged.)
       (Proof modification is discouraged.) $)
    rexbidvaALT $p |- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) ) $=
      ( nfv rexbida ) ABCDEADGFH $.
  $}

  $( Restricted specialization, with two quantifiers.  (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.)  (Proof
     shortened by Wolf Lammen, 7-Jan-2020.) $)
  rsp2e $p |- ( ( x e. A /\ y e. B /\ ph ) -> E. x e. A E. y e. B ph ) $=
    ( cv wcel wrex wa rspe sylan2 3impb ) BFDGZCFEGZAACEHZBDHZNAIMOPACEJOBDJKL
    $.

  ${
    rspec2.1 $e |- A. x e. A A. y e. B ph $.
    $( Specialization rule for restricted quantification, with two quantifiers.
       (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, with three
       quantifiers.  (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 $.
  $}

  ${
    $d x y $.
    r2alf.1 $e |- F/_ y A $.
    $( Double restricted universal quantification.  For a version based on
       fewer axioms see ~ r2al .  (Contributed by Mario Carneiro, 14-Oct-2016.)
       Use ~ r2allem .  (Revised by Wolf Lammen, 9-Jan-2020.) $)
    r2alf $p |- ( A. x e. A A. y e. B ph <->
               A. x A. y ( ( x e. A /\ y e. B ) -> ph ) ) $=
      ( cv wcel wi nfcri 19.21 r2allem ) ABCDEBGDHCGEHAICCBDFJKL $.
  $}

  ${
    $d x y $.
    r2exf.1 $e |- F/_ y A $.
    $( Double restricted existential quantification.  For a version based on
       fewer axioms see ~ r2ex .  (Contributed by Mario Carneiro, 14-Oct-2016.)
       Use ~ r2exlem .  (Revised by Wolf Lammen, 10-Jan-2020.) $)
    r2exf $p |- ( E. x e. A E. y e. B ph <->
               E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) ) $=
      ( wn r2alf r2exlem ) ABCDEAGBCDEFHI $.
  $}

  ${
    $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 $.
  $}

  $( *** Theorems using axioms up to ax-8 and ax-10 only: *** $)

  $( The setvar ` 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 $.

  $( The setvar ` 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 $.

  $( *** Theorems using axioms up to ax-8 and ax-11 only: *** $)

  ${
    $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.)  Reduce axiom dependencies.  (Revised by BJ, 13-Jun-2019.)
       (Proof shortened by Wolf Lammen, 31-Oct-2024.) $)
    ralcom4 $p |- ( A. x e. A A. y ph <-> A. y A. x e. A ph ) $=
      ( wal wral cv wcel wi 19.21v albii alcom df-ral 3bitr4ri bitr4i ) ACEZBDF
      ZBGDHZAIZBEZCEZABDFZCESCEZBERPIZBEUAQUCUDBRACJKSCBLPBDMNUBTCABDMKO $.

    $( Commutation of restricted and unrestricted existential quantifiers.
       (Contributed by NM, 12-Apr-2004.)  (Proof shortened by Andrew Salmon,
       8-Jun-2011.)  Reduce axiom dependencies.  (Revised by BJ,
       13-Jun-2019.) $)
    rexcom4 $p |- ( E. x e. A E. y ph <-> E. y E. x e. A ph ) $=
      ( cv wcel wa wex wrex exdistr df-rex exbii excom bitri 3bitr4ri ) BEDFZAG
      ZCHBHZPACHZGBHABDIZCHZSBDIPABCJUAQBHZCHRTUBCABDKLQCBMNSBDKO $.

    $d x B $.
    $( Commutation of restricted universal quantifiers.  See ~ ralcom2 for a
       version without disjoint variable condition on ` x , y ` .  This theorem
       should be used in place of ~ ralcom2 since it depends on a smaller set
       of axioms.  (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 ) $=
      ( cv wcel wa wi wal wral ancomst 2albii alcom bitri r2al 3bitr4i ) BFDGZC
      FEGZHAIZCJBJZSRHAIZBJCJZACEKBDKABDKCEKUAUBCJBJUCTUBBCRSALMUBBCNOABCDEPACB
      EDPQ $.
    $( $j usage 'ralcom' avoids 'ax-6' 'ax-7' 'ax-8' 'ax-9' 'ax-10' 'ax-12'
       'ax-13' 'ax-ext'; $)

    $( Commutation of restricted existential quantifiers.  (Contributed by NM,
       19-Nov-1995.)  (Revised by Mario Carneiro, 14-Oct-2016.)  (Proof
       shortened by BJ, 26-Aug-2023.)  (Proof shortened by Wolf Lammen,
       8-Dec-2024.) $)
    rexcom $p |- ( E. x e. A E. y e. B ph <-> E. y e. B E. x e. A ph ) $=
      ( wrex wn wral ralcom ralnex2 3bitr3i con4bii ) ACEFBDFZABDFCEFZAGZCEHBDH
      OBDHCEHMGNGOBCDEIABCDEJACBEDJKL $.
    $( $j usage 'rexcom' avoids 'ax-6' 'ax-7' 'ax-8' 'ax-9' 'ax-10' 'ax-12'
       'ax-13' 'ax-ext'; $)

    $( Obsolete version of ~ rexcom as of 8-Dec-2024.  (Contributed by NM,
       19-Nov-1995.)  (Revised by Mario Carneiro, 14-Oct-2016.)  (Proof
       shortened by BJ, 26-Aug-2023.)  (New usage is discouraged.)
       (Proof modification is discouraged.) $)
    rexcomOLD $p |- ( E. x e. A E. y e. B ph <-> E. y e. B E. x e. A ph ) $=
      ( wrex cv wcel wa wex df-rex rexbii rexcom4 r19.42v exbii bitr4i 3bitri )
      ACEFZBDFCGEHZAIZCJZBDFTBDFZCJZABDFZCEFZRUABDACEKLTBCDMUCSUDIZCJUEUBUFCSAB
      DNOUDCEKPQ $.
    $( $j usage 'rexcomOLD' avoids 'ax-6' 'ax-7' 'ax-8' 'ax-9' 'ax-10' 'ax-12'
       'ax-13' 'ax-ext'; $)
  $}

  ${
    $d x A $.  $d x y $.  $d x ph $.
    $( 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 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 first and third restricted universal quantifiers.  (Contributed by
       AV, 3-Dec-2021.)  (Proof shortened by Wolf Lammen, 2-Jan-2025.) $)
    ralcom13 $p |- ( A. x e. A A. y e. B A. z e. C ph
         <-> A. z e. C A. y e. B A. x e. A ph ) $=
      ( wral ralrot3 ralcom ralbii bitri ) ADGHCFHBEHACFHBEHZDGHABEHCFHZDGHABCD
      EFGIMNDGABCEFJKL $.

    $( Obsolete version of ~ ralcom13 as of 2-Jan-2025.  (Contributed by AV,
       3-Dec-2021.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    ralcom13OLD $p |- ( A. x e. A A. y e. B A. z e. C ph
         <-> A. z e. C A. y e. B A. x e. A ph ) $=
      ( wral ralcom ralbii 3bitri ) ADGHZCFHBEHLBEHZCFHABEHZDGHZCFHNCFHDGHLBCEF
      IMOCFABDEGIJNCDFGIK $.

    $( Swap first and third 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 C $.  $d w y C $.  $d x z D $.
    $d y z D $.
    $( Rotate four restricted existential 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 $.
  $}

  ${
    $d A x $.  $d A y $.  $d B x $.  $d B y $.  $d w x $.  $d w y $.  $d x z $.
    $d y z $.
    $( Rotate two existential quantifiers and two restricted existential
       quantifiers.  (Contributed by AV, 9-Nov-2023.) $)
    2ex2rexrot $p |- ( E. x E. y E. z e. A E. w e. B ph
                       <-> E. z e. A E. w e. B E. x E. y ph ) $=
      ( wex wrex rexcom4 rexbii bitri exbii 3bitrri ) ACHZBHEGIZDFIOEGIZBHZDFIQ
      DFIZBHAEGIZDFICHZBHPRDFOEBGJKQDBFJSUABSTCHZDFIUAQUBDFAECGJKTDCFJLMN $.
  $}

  $( *** Theorems using axioms up to ax-8 and ax-10, ax-11 only: *** $)

  ${
    $d A y $.  $d x y $.
    $( Similar to Lemma 24 of [Monk2] p. 114, except that quantification is
       restricted.  Once derived from ~ hbra2VD .  Version of ~ nfra2 with a
       disjoint variable condition not requiring ~ ax-13 .  (Contributed by
       Alan Sare, 31-Dec-2011.)  Reduce axiom usage.  (Revised by GG,
       24-Sep-2024.)  (Proof shortened by Wolf Lammen, 3-Jan-2025.) $)
    nfra2w $p |- F/ y A. x e. A A. y e. B ph $=
      ( wral cv wcel wa wi wal r2al nfa2 nfxfr ) ACEFBDFBGDHCGEHIAJZCKBKCABCDEL
      OCBMN $.
    $( $j usage 'nfra2w' avoids 'ax-6' 'ax-7' 'ax-8' 'df-clel' 'ax-9' 'df-cleq'
       'ax-12' 'ax-13' 'ax-ext'; $)
  $}

  $( *** Theorems using axioms up to ax-8 and ax-10, ax-12 only: *** $)

  $( The setvar ` x ` is not free in ` A. x e. A ph ` .  (Contributed by NM,
     18-Oct-1996.)  (Proof shortened by Wolf Lammen, 7-Dec-2019.) $)
  hbra1 $p |- ( A. x e. A ph -> A. x A. x e. A ph ) $=
    ( wral nfra1 nf5ri ) ABCDBABCEF $.

  $( *** Theorems using axioms up to ax-8 and ax-11, ax-12 only: *** $)

  ${
    $d x y $.
    ralcomf.1 $e |- F/_ y A $.
    ralcomf.2 $e |- F/_ x B $.
    $( Commutation of restricted universal quantifiers.  For a version based on
       fewer axioms see ~ ralcom .  (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 ancomst 2albii alcom bitri r2alf 3bitr4i ) BHDIZ
      CHEIZJAKZCLBLZUATJAKZBLCLZACEMBDMABDMCEMUCUDCLBLUEUBUDBCTUAANOUDBCPQABCDE
      FRACBEDGRS $.

    $( Commutation of restricted existential quantifiers.  For a version based
       on fewer axioms see ~ rexcom .  (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 $.
    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, which does not
       require ~ ax-10 , ~ ax-13 .  For a version not dependent on ~ ax-11 and
       ax-12, see ~ cbvralvw .  (Contributed by NM, 7-Mar-2004.)  Avoid
       ~ ax-10 , ~ ax-13 .  (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 $.
    $( $j usage 'cbvralfw' avoids 'ax-10' 'ax-13'; $)
  $}

  ${
    $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, which does not
       require ~ ax-13 .  For a version not dependent on ~ ax-11 and ax-12, see
       ~ cbvrexvw .  (Contributed by FL, 27-Apr-2008.)  Avoid ~ ax-10 ,
       ~ ax-13 .  (Revised by GG, 10-Jan-2024.) $)
    cbvrexfw $p |- ( E. x e. A ph <-> E. y e. A ps ) $=
      ( wrex wn wral nfn weq notbid cbvralfw ralnex 3bitr3i con4bii ) ACEKZBDEK
      ZALZCEMBLZDEMUALUBLUCUDCDEFGADHNBCINCDOABJPQACERBDERST $.
    $( $j usage 'cbvrexfw' avoids 'ax-13'; $)
  $}

  ${
    $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 ~ cbvralfw with more disjoint variable conditions.
       (Contributed by NM, 31-Jul-2003.)  Avoid ~ ax-13 .  (Revised by GG,
       10-Jan-2024.) $)
    cbvralw $p |- ( A. x e. A ph <-> A. y e. A ps ) $=
      ( nfcv cbvralfw ) ABCDECEIDEIFGHJ $.
    $( $j usage 'cbvralw' avoids 'ax-13'; $)

    $( Rule used to change bound variables, using implicit substitution.
       Version of ~ cbvrexfw with more disjoint variable conditions.
       (Contributed by NM, 31-Jul-2003.)  Avoid ~ ax-13 .  (Revised by GG,
       10-Jan-2024.) $)
    cbvrexw $p |- ( E. x e. A ph <-> E. y e. A ps ) $=
      ( nfcv cbvrexfw ) ABCDECEIDEIFGHJ $.
    $( $j usage 'cbvrexw' avoids 'ax-13'; $)
  $}

  $( *** Theorems using axioms up to ax-12 only: *** $)

  ${
    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 $.
  $}

  ${
    $d x y $.
    nfraldw.1 $e |- F/ y ph $.
    nfraldw.2 $e |- ( ph -> F/_ x A ) $.
    nfraldw.3 $e |- ( ph -> F/ x ps ) $.
    $( Deduction version of ~ nfralw .  Version of ~ nfrald with a disjoint
       variable condition, which does not require ~ ax-13 .  (Contributed by
       NM, 15-Feb-2013.)  Avoid ~ ax-9 , ~ ax-ext .  (Revised by GG,
       24-Sep-2024.) $)
    nfraldw $p |- ( ph -> F/ x A. y e. A ps ) $=
      ( wral cv wcel wi wal df-ral nfcrd nfimd nfald nfxfrd ) BDEIDJEKZBLZDMACB
      DENATCDFASBCACDEGOHPQR $.
    $( $j usage 'nfraldw' avoids 'ax-13' 'ax-9' 'df-cleq' 'ax-ext'; $)

    $( Deduction version of ~ nfrexw .  (Contributed by Mario Carneiro,
       14-Oct-2016.)  Add disjoint variable condition to avoid ~ ax-13 .  See
       ~ nfrexd for a less restrictive version requiring more axioms.  (Revised
       by GG, 20-Jan-2024.) $)
    nfrexdw $p |- ( ph -> F/ x E. y e. A ps ) $=
      ( wrex wn wral dfrex2 nfnd nfraldw nfxfrd ) BDEIBJZDEKZJACBDELAQCAPCDEFGA
      BCHMNMO $.
    $( $j usage 'nfrexdw' avoids 'ax-13'; $)
  $}

  ${
    $d x y $.
    nfralw.1 $e |- F/_ x A $.
    nfralw.2 $e |- F/ x ph $.
    $( Bound-variable hypothesis builder for restricted quantification.
       Version of ~ nfral with a disjoint variable condition, which does not
       require ~ ax-13 .  (Contributed by NM, 1-Sep-1999.)  Avoid ~ ax-13 .
       (Revised by GG, 10-Jan-2024.)  (Proof shortened by Wolf Lammen,
       13-Dec-2024.) $)
    nfralw $p |- F/ x A. y e. A ph $=
      ( wral cv wcel nfcri nf5ri hbral nf5i ) ACDGBABCDCHDIBBCDEJKABFKLM $.
    $( $j usage 'nfralw' avoids 'ax-13'; $)

    $( Obsolete version of ~ nfralw as of 13-Dec-2024.  (Contributed by NM,
       1-Sep-1999.)  (Revised by GG, 10-Jan-2024.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    nfralwOLD $p |- F/ x A. y e. A ph $=
      ( wral wnf wtru nftru wnfc a1i nfraldw mptru ) ACDGBHIABCDCJBDKIELABHIFLM
      N $.

    $( Bound-variable hypothesis builder for restricted quantification.
       (Contributed by NM, 1-Sep-1999.)  (Revised by Mario Carneiro,
       7-Oct-2016.)  (Proof shortened by Wolf Lammen, 30-Dec-2019.)  Add
       disjoint variable condition to avoid ~ ax-13 .  See ~ nfrex for a less
       restrictive version requiring more axioms.  (Revised by GG,
       20-Jan-2024.) $)
    nfrexw $p |- F/ x E. y e. A ph $=
      ( wrex wnf wtru nftru wnfc a1i nfrexdw mptru ) ACDGBHIABCDCJBDKIELABHIFLM
      N $.
    $( $j usage 'nfrexw' avoids 'ax-13'; $)
  $}

  ${
    $d x y $.  $d y A $.  $d x B $.
    $( Restricted quantifier version of ~ 19.12 .  (Contributed by NM,
       15-Oct-2003.)  (Proof shortened by Andrew Salmon, 30-May-2011.)  Avoid
       ~ ax-13 , ~ ax-ext .  (Revised by Wolf Lammen, 17-Jun-2023.)  (Proof
       shortened by Wolf Lammen, 4-Nov-2024.) $)
    r19.12 $p |- ( E. x e. A A. y e. B ph -> A. y e. B E. x e. A ph ) $=
      ( wral wrex cv wcel wa wex df-rex nfv nfra1 nfan nfex nfxfr com12 reximdv
      rsp ralrimi ) ACEFZBDGZABDGZCEUCBHDIZUBJZBKCUBBDLUFCBUEUBCUECMACENOPQCHEI
      ZUCUDUGUBABDUBUGAACETRSRUA $.
    $( $j usage 'r19.12' avoids 'ax-13' 'ax-ext'; $)
  $}

  ${
    $d y A $.  $d x B $.  $d x y $.
    reean.1 $e |- F/ y ph $.
    reean.2 $e |- F/ x ps $.
    $( Rearrange restricted existential quantifiers.  For a version based on
       fewer axioms see ~ reeanv .  (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 nfv nfan eean reeanlem ) ABCDEFCIEJZAKDIFJZBKCDPADPDLGMQBCQC
      LHMNO $.
  $}

  ${
    $d x y A $.  $d y ph $.
    $( Change bound variable by using a substitution.  Version of ~ cbvralsv
       with a disjoint variable condition, which does not require ~ ax-13 .
       (Contributed by NM, 20-Nov-2005.)  Avoid ~ ax-13 .  (Revised by GG,
       10-Jan-2024.)  (Proof shortened by Wolf Lammen, 8-Mar-2025.)  Avoid
       ~ ax-10 , ~ ax-12 .  (Revised by SN, 21-Aug-2025.) $)
    cbvralsvw $p |- ( A. x e. A ph <-> A. y e. A [ y / x ] ph ) $=
      ( cv wcel wi wal wsb wral df-ral weq eleq1w imbi1d sbbiiev sbrimvw bitr2i
      sb8v albii bitri 3bitr4i ) BEDFZAGZBHUCBCIZCHZABDJABCIZCDJZUCBCRABDKUGCED
      FZUFGZCHUEUFCDKUIUDCUDUHAGZBCIUIUCUJBCBCLUBUHABCDMNOUHABCPQSTUA $.
    $( $j usage 'cbvralsvw' avoids 'ax-10' 'ax-12' 'ax-13'; $)

    $( Change bound variable by using a substitution.  Version of ~ cbvrexsv
       with a disjoint variable condition, which does not require ~ ax-13 .
       (Contributed by NM, 2-Mar-2008.)  Avoid ~ ax-13 .  (Revised by GG,
       10-Jan-2024.)  (Proof shortened by Wolf Lammen, 8-Mar-2025.) $)
    cbvrexsvw $p |- ( E. x e. A ph <-> E. y e. A [ y / x ] ph ) $=
      ( wsb nfv nfs1v sbequ12 cbvrexw ) AABCEBCDACFABCGABCHI $.
    $( $j usage 'cbvrexsvw' avoids 'ax-13'; $)

    $( Obsolete version of ~ cbvralsvw as of 21-Aug-2025.  (Contributed by NM,
       20-Nov-2005.)  Avoid ~ ax-13 .  (Revised by GG, 10-Jan-2024.)  (Proof
       shortened by Wolf Lammen, 8-Mar-2025.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    cbvralsvwOLD $p |- ( A. x e. A ph <-> A. y e. A [ y / x ] ph ) $=
      ( wsb nfv nfs1v sbequ12 cbvralw ) AABCEBCDACFABCGABCHI $.
  $}

  ${
    $d z x A $.  $d y A $.  $d z y ph $.  $d x y $.
    $( Obsolete version of ~ cbvralsvw as of 8-Mar-2025.  (Contributed by NM,
       20-Nov-2005.)  Avoid ~ ax-13 .  (Revised by GG, 10-Jan-2024.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    cbvralsvwOLDOLD $p |- ( A. x e. A ph <-> A. y e. A [ y / x ] ph ) $=
      ( vz wral wsb nfv nfs1v sbequ12 cbvralw sbequ bitri ) ABDFABEGZEDFABCGZCD
      FANBEDAEHABEIABEJKNOECDNCHOEHAECBLKM $.

    $( Obsolete version of ~ cbvrexsvw as of 8-Mar-2025.  (Contributed by NM,
       20-Nov-2005.)  Avoid ~ ax-13 .  (Revised by GG, 10-Jan-2024.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    cbvrexsvwOLD $p |- ( E. x e. A ph <-> E. y e. A [ y / x ] ph ) $=
      ( vz wrex wsb nfv nfs1v sbequ12 cbvrexw sbequ bitri ) ABDFABEGZEDFABCGZCD
      FANBEDAEHABEIABEJKNOECDNCHOEHAECBLKM $.
  $}

  $( *** Theorems using axioms up to ax-9 and ax-ext only: *** $)

  ${
    $d x A $.  $d x B $.
    $( Equality theorem for restricted existential quantifier.  (Contributed by
       NM, 29-Oct-1995.)  Remove usage of ~ ax-10 , ~ ax-11 , and ~ ax-12 .
       (Revised by Steven Nguyen, 30-Apr-2023.)  Shorten other proofs.
       (Revised by Wolf Lammen, 8-Mar-2025.) $)
    rexeq $p |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) ) $=
      ( wceq cv wcel wa wex wrex wal dfcleq anbi1 alexbii sylbi df-rex 3bitr4g
      wb ) CDEZBFZCGZAHZBIZTDGZAHZBIZABCJABDJSUAUDRZBKUCUFRBCDLUGUBUEBUAUDAMNOA
      BCPABDPQ $.

    $( Equality theorem for restricted universal quantifier.  (Contributed by
       NM, 16-Nov-1995.)  Remove usage of ~ ax-10 , ~ ax-11 , and ~ ax-12 .
       (Revised by Steven Nguyen, 30-Apr-2023.)  Shorten other proofs.
       (Revised by Wolf Lammen, 8-Mar-2025.) $)
    raleq $p |- ( A = B -> ( A. x e. A ph <-> A. x e. B ph ) ) $=
      ( wceq wral wn wrex rexeq rexnal 3bitr3g con4bid ) CDEZABCFZABDFZMAGZBCHP
      BDHNGOGPBCDIABCJABDJKL $.
  $}

  ${
    $d A x $.  $d B x $.
    raleq1i.1 $e |- A = B $.
    $( Equality inference for restricted universal quantifier.  (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 quantifier.  (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 $.
    raleqdv.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 $.  $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 $.
  $}

  ${
    $d ph x $.  $d A x $.  $d B x $.
    raleqbidvv.1 $e |- ( ph -> A = B ) $.
    raleqbidvv.2 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Version of ~ raleqbidv with additional disjoint variable conditions, not
       requiring ~ ax-8 nor ~ df-clel .  (Contributed by BJ, 22-Sep-2024.) $)
    raleqbidvv $p |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) ) $=
      ( wb cv wcel adantr raleqbidva ) ABCDEFGABCIDJEKHLM $.
    $( $j usage 'raleqbidvv' avoids 'ax-8' 'ax-10' 'ax-11' 'ax-12'
          'df-clel'; $)

    $( Obsolete version of ~ raleqbidvv as of 9-Mar-2025.  (Contributed by BJ,
       22-Sep-2024.)  (New usage is discouraged.)
       (Proof modification is discouraged.) $)
    raleqbidvvOLD $p |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) ) $=
      ( cv wcel wi wal wral wb wa alrimiv wceq dfcleq sylib df-ral 19.26 imbi12
      sylanbrc impcom sylg albi syl 3bitr4g ) ADIZEJZBKZDLZUIFJZCKZDLZBDEMCDFMA
      UKUNNZDLULUONABCNZUJUMNZOZUPDAUQDLURDLZUSDLAUQDHPAEFQUTGDEFRSUQURDUAUCURU
      QUPUJUMBCUBUDUEUKUNDUFUGBDETCDFTUH $.

    $( Version of ~ rexeqbidv with additional disjoint variable conditions, not
       requiring ~ ax-8 nor ~ df-clel .  (Contributed by Wolf Lammen,
       25-Sep-2024.) $)
    rexeqbidvv $p |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) ) $=
      ( wb cv wcel adantr rexeqbidva ) ABCDEFGABCIDJEKHLM $.
    $( $j usage 'rexeqbidvv' avoids 'ax-8' 'ax-10' 'ax-11' 'ax-12'
          'df-clel'; $)

    $( Obsolete version of ~ rexeqbidvv as of 9-Mar-2025.  (Contributed by Wolf
       Lammen, 25-Sep-2024.)  (New usage is discouraged.)
       (Proof modification is discouraged.) $)
    rexeqbidvvOLD $p |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) ) $=
      ( wrex wn wral notbid raleqbidvv ralnex 3bitr3g con4bid ) ABDEIZCDFIZABJZ
      DEKCJZDFKQJRJASTDEFGABCHLMBDENCDFNOP $.
  $}

  ${
    $d x A $.  $d x B $.
    raleqbi1dv.1 $e |- ( A = B -> ( ph <-> ps ) ) $.
    $( Equality deduction for restricted universal quantifier.  (Contributed by
       NM, 16-Nov-1995.)  (Proof shortened by Steven Nguyen, 5-May-2023.) $)
    raleqbi1dv $p |- ( A = B -> ( A. x e. A ph <-> A. x e. B ps ) ) $=
      ( wceq id raleqbidvv ) DEGZABCDEJHFI $.
    $( $j usage 'raleqbi1dv' avoids 'ax-8' 'ax-12'; $)

    $( Equality deduction for restricted existential quantifier.  (Contributed
       by NM, 18-Mar-1997.)  (Proof shortened by Steven Nguyen, 5-May-2023.) $)
    rexeqbi1dv $p |- ( A = B -> ( E. x e. A ph <-> E. x e. B ps ) ) $=
      ( wceq id rexeqbidvv ) DEGZABCDEJHFI $.
    $( $j usage 'rexeqbi1dv' avoids 'ax-8' 'ax-12'; $)
  $}

  ${
    $d x A $.  $d x B $.
    $( Obsolete version of ~ raleq as of 9-Mar-2025.  (Contributed by NM,
       16-Nov-1995.)  Remove usage of ~ ax-10 , ~ ax-11 , and ~ ax-12 .
       (Revised by Steven Nguyen, 30-Apr-2023.)  (New usage is discouraged.)
       (Proof modification is discouraged.) $)
    raleqOLD $p |- ( A = B -> ( A. x e. A ph <-> A. x e. B ph ) ) $=
      ( wceq biidd raleqbi1dv ) AABCDCDEAFG $.

    $( Obsolete version of ~ raleq as of 9-Mar-2025.  (Contributed by NM,
       29-Oct-1995.)  Remove usage of ~ ax-10 , ~ ax-11 , and ~ ax-12 .
       (Revised by Steven Nguyen, 30-Apr-2023.)  (New usage is discouraged.)
       (Proof modification is discouraged.) $)
    rexeqOLD $p |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) ) $=
      ( wceq biidd rexeqbi1dv ) AABCDCDEAFG $.
  $}

  ${
    $d A x $.  $d B x $.
    $( All elements of a class are elements of a class equal to this class.
       (Contributed by AV, 30-Oct-2020.)  (Proof shortened by Wolf Lammen,
       18-Jul-2025.) $)
    raleleq $p |- ( A = B -> A. x e. A x e. B ) $=
      ( wceq cv wcel wral ralel raleq mpbiri ) BCDAECFZABGKACGACHKABCIJ $.
    $( $j usage 'raleleq' avoids 'ax-8' 'df-clel'; $)

    $( Obsolete version of ~ raleleq as of 18-Jul-2025.  (Contributed by AV,
       30-Oct-2020.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    raleleqOLD $p |- ( A = B -> A. x e. A x e. B ) $=
      ( wceq cv wcel wral ralel id raleqdv mpbiri ) BCDZAECFZABGMACGACHLMABCLIJ
      K $.
    $( $j usage 'raleleqOLD' avoids 'ax-8' 'df-clel'; $)
  $}

  ${
    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 $.
  $}

  ${
    rexeqbii.1 $e |- A = B $.
    rexeqbii.2 $e |- ( ps <-> ch ) $.
    $( 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 $.
  $}

  ${
    $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.)  Remove usage of ~ ax-10 , ~ ax-11 , and ~ ax-12 and
       reduce distinct variable conditions.  (Revised by Steven Nguyen,
       30-Apr-2023.) $)
    raleqbidv $p |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) ) $=
      ( cv wcel eleq2d imbi12d ralbidv2 ) ABCDEFADIZEJNFJBCAEFNGKHLM $.

    $( Equality deduction for restricted universal quantifier.  (Contributed by
       NM, 6-Nov-2007.)  Remove usage of ~ ax-10 , ~ ax-11 , and ~ ax-12 and
       reduce distinct variable conditions.  (Revised by Steven Nguyen,
       30-Apr-2023.) $)
    rexeqbidv $p |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) ) $=
      ( cv wcel eleq2d anbi12d rexbidv2 ) ABCDEFADIZEJNFJBCAEFNGKHLM $.
  $}

  ${
    $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 wb wa simpr 19.21v df-ral eleq12d pm5.74ri
      imbi12d expcom pm5.74d cbvalvw 3bitr3i 3bitr4g ) ADJZFKZBLZDMZEJZGKZCLZEM
      ZBDFNCEGNAULUPAUKLZDMAUOLZEMAULLAUPLUQURDEDEOZAUKUOAUSUKUOPAUSQZUJUNBCUTU
      IUMFGAUSRIUAHUCUDUEUFAUKDSAUOESUGUBBDFTCEGTUH $.

    $( 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.)
       (Proof shortened by Wolf Lammen, 8-Jan-2025.) $)
    cbvrexdva2 $p |- ( ph -> ( E. x e. A ps <-> E. y e. B ch ) ) $=
      ( wrex wn wral weq wa notbid cbvraldva2 ralnex 3bitr3g con4bid ) ABDFJZCE
      GJZABKZDFLCKZEGLTKUAKAUBUCDEFGADEMNBCHOIPBDFQCEGQRS $.

    $( Obsolete version of ~ cbvrexdva2 as of 8-Jan-2025.  (Contributed by
       David Moews, 1-May-2017.)  (Proof shortened by Wolf Lammen,
       12-Aug-2023.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    cbvrexdva2OLD $p |- ( ph -> ( E. x e. A ps <-> E. y e. B ch ) ) $=
      ( cv wcel wa wex wrex wb wi weq simpr 19.42v df-rex eleq12d ancoms pm5.32
      anbi12d pm5.32da cbvexvw 3bitr3i mpbir 3bitr4g ) ADJZFKZBLZDMZEJZGKZCLZEM
      ZBDFNCEGNAUMUQOPAUMLZAUQLZOAULLZDMAUPLZEMURUSUTVADEDEQZAULUPAVBULUPOAVBLZ
      UKUOBCVCUJUNFGAVBRIUAHUDUBUEUFAULDSAUPESUGAUMUQUCUHBDFTCEGTUI $.
  $}

  ${
    $d ps y $.  $d ch x $.  $d A x y $.  $d x ph y $.
    cbvrexdva.1 $e |- ( ( ph /\ x = y ) -> ( ps <-> ch ) ) $.
    $( Obsolete version of ~ cbvraldva as of 9-Mar-2025.  (Contributed by David
       Moews, 1-May-2017.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    cbvraldvaOLD $p |- ( ph -> ( A. x e. A ps <-> A. y e. A ch ) ) $=
      ( weq wa eqidd cbvraldva2 ) ABCDEFFGADEHIFJK $.

    $( Obsolete version of ~ cbvrexdva as of 9-Mar-2025.  (Contributed by David
       Moews, 1-May-2017.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    cbvrexdvaOLD $p |- ( ph -> ( E. x e. A ps <-> E. y e. A ch ) ) $=
      ( weq wa eqidd cbvrexdva2 ) ABCDEFFGADEHIFJK $.
  $}

  $( *** Theorems using axioms up to ax-12 and ax-ext only: *** $)

  ${
    raleqf.1 $e |- F/_ x A $.
    raleqf.2 $e |- F/_ x B $.
    $( Equality theorem for restricted universal quantifier, with
       bound-variable hypotheses instead of distinct variable restrictions.
       See ~ raleq for a version based on fewer axioms.  (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.
       See ~ rexeq for a version based on fewer axioms.  (Contributed by NM,
       9-Oct-2003.)  (Revised by Andrew Salmon, 11-Jul-2011.)  (Proof shortened
       by Wolf Lammen, 9-Mar-2025.) $)
    rexeqf $p |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) ) $=
      ( wceq wrex wn wral raleqf ralnex 3bitr3g con4bid ) CDGZABCHZABDHZOAIZBCJ
      RBDJPIQIRBCDEFKABCLABDLMN $.

    $( Obsolete version of ~ rexeqf as of 9-Mar-2025.  (Contributed by NM,
       9-Oct-2003.)  (Revised by Andrew Salmon, 11-Jul-2011.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    rexeqfOLD $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 $.
  $}

  ${
    raleqbid.0 $e |- F/ x ph $.
    raleqbid.1 $e |- F/_ x A $.
    raleqbid.2 $e |- F/_ x B $.
    raleqbid.3 $e |- ( ph -> A = B ) $.
    raleqbid.4 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Equality deduction for restricted universal quantifier.  See ~ raleqbidv
       for a version based on fewer axioms.  (Contributed by Thierry Arnoux,
       8-Mar-2017.) $)
    raleqbid $p |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) ) $=
      ( wral wceq wb raleqf syl ralbid bitrd ) ABDELZBDFLZCDFLAEFMSTNJBDEFHIOPA
      BCDFGKQR $.

    $( Equality deduction for restricted existential quantifier.  See
       ~ rexeqbidv for a version based on fewer axioms.  (Contributed by
       Thierry Arnoux, 8-Mar-2017.) $)
    rexeqbid $p |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) ) $=
      ( wrex wceq wb rexeqf syl rexbid bitrd ) ABDELZBDFLZCDFLAEFMSTNJBDEFHIOPA
      BCDFGKQR $.
  $}

  ${
    $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 rectrictedly
       universally quantified expression.  (Contributed by NM, 5-Sep-2004.)
       Avoid ~ ax-ext , ~ df-cleq , ~ df-clel .  (Revised by Wolf Lammen,
       10-Mar-2025.) $)
    sbralie $p |- ( A. x e. y ph <-> [ y / x ] A. y e. x ps ) $=
      ( vz cv wral wel wal wsb df-ral nfv sblim elsb2 imbi1i bitri albii sbbii
      wi weq elequ1 imbi12d cbvalvw sbal 3bitrri sbco2vv 3bitr3i 3bitr2i ) ACDG
      ZHCDIZATZCJCFIZATZFDKZCJZBDCGZHZCDKZACUJLUOULCUOUMFDKZATULUMAFDAFMNUTUKAF
      DCOPQRUNCJZFDKURCFKZFDKUPUSVAVBFDVADFIZBTZDJZVBUNVDCDCDUAUMVCABCDFUBEUCUD
      VBDCIZBTZDJZCFKVGCFKZDJVEURVHCFBDUQLSVGDCFUEVIVDDVIVFCFKZBTVDVFBCFBCMNVJV
      CBCFDOPQRUFQSUNCFDUEURCDFUGUHUI $.
    $( $j usage 'sbralie' avoids 'ax-ext' 'df-cleq' 'df-clel'; $)

    $( Alternative shorter proof of ~ sbralie dependent on ~ ax-ext ,
       ~ df-cleq , ~ df-clel .  (Contributed by NM, 5-Sep-2004.)
       (New usage is discouraged.)  (Proof modification is discouraged.) $)
    sbralieALT $p |- ( A. x e. y ph <-> [ y / x ] A. y e. x ps ) $=
      ( vz cv wral wsb cbvralsvw sbbii raleq sbievw sbco2vv wb weq bicomd bitri
      equcoms ralbii 3bitrri ) BDCGZHZCDIBDFIZFUBHZCDIUDFDGZHZACUFHZUCUECDBDFUB
      JKUEUGCDUDFUBUFLMUGUDFCIZCUFHUHUDFCUFJUIACUFUIBDCIABDCFNBADCBAOCDCDPABEQS
      MRTRUA $.
  $}

  $( *** Theorems using axioms ax-13 or axc11r: *** $)

  ${
    $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.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .  Use the
       weaker ~ cbvralfw when possible.  (Contributed by NM, 7-Mar-2004.)
       (Revised by Mario Carneiro, 9-Oct-2016.)  (New usage is discouraged.) $)
    cbvralf $p |- ( A. x e. A ph <-> A. y e. A ps ) $=
      ( vz cv wcel wi wal wral wsb nfv nfcri nfim nfs1v sbequ12 imbi12d cbvalv1
      weq eleq1w nfsb sbequ sbie bitrdi bitri df-ral 3bitr4i ) CLEMZANZCOZDLEMZ
      BNZDOZACEPBDEPUPKLEMZACKQZNZKOUSUOVBCKUOKRUTVACCKEFSACKUATCKUEUNUTAVACKEU
      FACKUBUCUDVBURKDUTVADDKEGSACKDHUGTURKRKDUEZUTUQVABKDEUFVCVAACDQBAKDCUHABC
      DIJUIUJUCUDUKACEULBDEULUM $.

    $( Rule used to change bound variables, using implicit substitution.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .  Use the
       weaker ~ cbvrexfw when possible.  (Contributed by FL, 27-Apr-2008.)
       (Revised by Mario Carneiro, 9-Oct-2016.)  (New usage is discouraged.) $)
    cbvrexf $p |- ( E. x e. A ph <-> E. y e. A ps ) $=
      ( wn wral wrex nfn weq notbid cbvralf notbii dfrex2 3bitr4i ) AKZCELZKBKZ
      DELZKACEMBDEMUBUDUAUCCDEFGADHNBCINCDOABJPQRACESBDEST $.
  $}

  ${
    $d x A $.  $d y A $.
    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.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .  Use the
       weaker ~ cbvralw when possible.  (Contributed by NM, 31-Jul-2003.)
       (New usage is discouraged.) $)
    cbvral $p |- ( A. x e. A ph <-> A. y e. A ps ) $=
      ( nfcv cbvralf ) ABCDECEIDEIFGHJ $.

    $( Rule used to change bound variables, using implicit substitution.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .  Use the
       weaker ~ cbvrexw when possible.  (Contributed by NM, 31-Jul-2003.)
       (Proof shortened by Andrew Salmon, 8-Jun-2011.)
       (New usage is discouraged.) $)
    cbvrex $p |- ( E. x e. A ph <-> E. y e. A ps ) $=
      ( nfcv cbvrexf ) ABCDECEIDEIFGHJ $.
  $}

  ${
    $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.  See ~ cbvralvw based on fewer axioms , but extra
       disjoint variables.  Usage of this theorem is discouraged because it
       depends on ~ ax-13 .  Use the weaker ~ cbvralvw when possible.
       (Contributed by NM, 28-Jan-1997.)  (New usage is discouraged.) $)
    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.  See ~ cbvrexvw based on fewer axioms , but extra
       disjoint variables.  Usage of this theorem is discouraged because it
       depends on ~ ax-13 .  Use the weaker ~ cbvrexvw when possible.
       (Contributed by NM, 2-Jun-1998.)  (New usage is discouraged.) $)
    cbvrexv $p |- ( E. x e. A ph <-> E. y e. A ps ) $=
      ( nfv cbvrex ) ABCDEADGBCGFH $.
  $}

  ${
    $d z x A $.  $d y A $.  $d z y ph $.
    $( Change bound variable by using a substitution.  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  Use the weaker ~ cbvralsvw
       when possible.  (Contributed by NM, 20-Nov-2005.)  (Revised by Andrew
       Salmon, 11-Jul-2011.)  (New usage is discouraged.) $)
    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 $.

    $( Change bound variable by using a substitution.  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  Use the weaker ~ cbvrexsvw
       when possible.  (Contributed by NM, 2-Mar-2008.)  (Revised by Andrew
       Salmon, 11-Jul-2011.)  (New usage is discouraged.) $)
    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 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.  Usage of this theorem is discouraged
       because it depends on ~ ax-13 .  Use the weaker ~ cbvral2vw when
       possible.  (Contributed by NM, 10-Aug-2004.)
       (New usage is discouraged.) $)
    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 $.

    $( Change bound variables of double restricted universal quantification,
       using implicit substitution.  Usage of this theorem is discouraged
       because it depends on ~ ax-13 .  Use the weaker ~ cbvrex2vw when
       possible.  (Contributed by FL, 2-Jul-2012.)
       (New usage is discouraged.) $)
    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 th $.  $d 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.  Usage of this theorem is discouraged
       because it depends on ~ ax-13 .  Use the weaker ~ cbvral3vw when
       possible.  (Contributed by NM, 10-May-2005.)
       (New usage is discouraged.) $)
    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 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 .  This theorem relies on the full set of axioms up to
       ~ ax-ext and it should no longer be used.  Usage of ~ rgen2 is highly
       encouraged.  (Contributed by NM, 23-Nov-1994.)  (Proof shortened by
       Andrew Salmon, 25-May-2011.)  (Proof shortened by Wolf Lammen,
       1-Jan-2020.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    rgen2a $p |- A. x e. A A. y e. A ph $=
      ( vz wral cv wcel wi wal weq wn eleq1 dvelimv alimi syl6com biimpd syli
      ex pm2.61d2 df-ral sylibr rgen ) ACDGZBDBHZDIZCHZDIZAJZCKZUEUGCBLZCKZUKUM
      MUGUGCKUKFHZDIUGCBFUNUFDNOUGUJCUGUIAETZPQULUJCUIULUGAULUIUGUHUFDNRUOSPUAA
      CDUBUCUD $.
  $}

  ${
    nfrald.1 $e |- F/ y ph $.
    nfrald.2 $e |- ( ph -> F/_ x A ) $.
    nfrald.3 $e |- ( ph -> F/ x ps ) $.
    $( Deduction version of ~ nfral .  Usage of this theorem is discouraged
       because it depends on ~ ax-13 .  Use the weaker ~ nfraldw when possible.
       (Contributed by NM, 15-Feb-2013.)  (Revised by Mario Carneiro,
       7-Oct-2016.)  (New usage is discouraged.) $)
    nfrald $p |- ( ph -> F/ x A. y e. A ps ) $=
      ( wral cv wcel wi wal df-ral weq wn wa wnfc nfcvf adantr adantl nfeld wnf
      nfimd nfald2 nfxfrd ) BDEIDJZEKZBLZDMACBDENAUICDFACDOCMPZQZUHBCUKCUGEUJCU
      GRACDSUAACERUJGTUBABCUCUJHTUDUEUF $.

    $( Deduction version of ~ nfrex .  Usage of this theorem is discouraged
       because it depends on ~ ax-13 .  See ~ nfrexdw for a version with a
       disjoint variable condition, but not requiring ~ ax-13 .  (Contributed
       by Mario Carneiro, 14-Oct-2016.)  (New usage is discouraged.) $)
    nfrexd $p |- ( ph -> F/ x E. y e. A ps ) $=
      ( wrex wn wral dfrex2 nfnd nfrald nfxfrd ) BDEIBJZDEKZJACBDELAQCAPCDEFGAB
      CHMNMO $.
  $}

  ${
    nfral.1 $e |- F/_ x A $.
    nfral.2 $e |- F/ x ph $.
    $( Bound-variable hypothesis builder for restricted quantification.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .  Use the
       weaker ~ nfralw when possible.  (Contributed by NM, 1-Sep-1999.)
       (Revised by Mario Carneiro, 7-Oct-2016.)  (New usage is discouraged.) $)
    nfral $p |- F/ x A. y e. A ph $=
      ( wral wnf wtru nftru wnfc a1i nfrald mptru ) ACDGBHIABCDCJBDKIELABHIFLMN
      $.

    $( Bound-variable hypothesis builder for restricted quantification.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .  See
       ~ nfrexw for a version with a disjoint variable condition, but not
       requiring ~ ax-13 .  (Contributed by NM, 1-Sep-1999.)  (Revised by Mario
       Carneiro, 7-Oct-2016.)  (Proof shortened by Wolf Lammen, 30-Dec-2019.)
       (New usage is discouraged.) $)
    nfrex $p |- F/ x E. y e. A ph $=
      ( wrex wnf wtru nftru wnfc a1i nfrexd mptru ) ACDGBHIABCDCJBDKIELABHIFLMN
      $.
  $}

  ${
    $d A y $.
    $( Similar to Lemma 24 of [Monk2] p. 114, except the quantification of the
       antecedent is restricted.  Derived automatically from ~ hbra2VD .  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .  Use the
       weaker ~ nfra2w when possible.  (Contributed by Alan Sare, 31-Dec-2011.)
       (New usage is discouraged.) $)
    nfra2 $p |- F/ y A. x e. A A. y e. B ph $=
      ( wral nfcv nfra1 nfral ) ACEFCBDCDGACEHI $.
  $}

  ${
    $d y A $.  $d x A $.
    $( Commutation of restricted universal quantifiers.  Note that ` x ` and
       ` y ` need not be disjoint (this makes the proof longer).  This theorem
       relies on the full set of axioms up to ~ ax-ext and it should no longer
       be used.  Usage of ~ ralcom is highly encouraged.  (Contributed by NM,
       24-Nov-1994.)  (Proof shortened by Mario Carneiro, 17-Oct-2016.)
       (New usage is discouraged.) $)
    ralcom2 $p |- ( A. x e. A A. y e. A ph -> A. y e. A A. x e. A ph ) $=
      ( weq wal wral wi cv wcel wb eleq1w dral1 df-ral 3bitr4g wa nfnae ralrimi
      nfan ex sps imbi1d bicomd imbi12d biimpd wn nfra2 nfra1 wnfc nfcvf adantr
      nfcvd nfeld nfan1 rsp2 ancomsd expdimp adantll pm2.61i ) BCEZBFZACDGZBDGZ
      ABDGZCDGZHVAVCVEVABIDJZVBHZBFCIZDJZVDHZCFVCVEVGVJBCVAVFVIVBVDUTVFVIKBBCDL
      UAZVAVIAHZCFZVFAHZBFZVBVDVAVOVMVNVLBCVAVFVIAVKUBMUCACDNABDNOUDMVBBDNVDCDN
      OUEVAUFZVCVEVPVCPZVDCDVPVCCBCCQABCDDUGSVQVIVDVQVIPABDVQVIBVPVCBBCBQVBBDUH
      SVQBVHDVPBVHUIVCBCUJUKVQBDULUMUNVCVIVNVPVCVIVFAVCVFVIAABCDDUOUPUQURRTRTUS
      $.
  $}


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Restricted existential uniqueness and at-most-one quantifier
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  $( 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 $.

  $( Define restricted "at most one".  Note:  This notation is most often used
     to express that ` ph ` holds for at most one element of a given class
     ` A ` .  For this reading ` F/_ x A ` is required, though, for example,
     asserted when ` x ` and ` A ` are disjoint.

     Should instead ` A ` depend on ` x ` , you rather assert at most one ` x `
     fulfilling ` ph ` happens to be contained in the corresponding
     ` A ( x ) ` .  This interpretation is rarely needed (see also ~ df-ral ).
     (Contributed by NM, 16-Jun-2017.) $)
  df-rmo $a |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) ) $.

  $( Define restricted existential uniqueness.

     Note:  This notation is most often used to express that ` ph ` holds for
     exactly one element of a given class ` A ` .  For this reading ` F/_ x A `
     is required, though, for example, asserted when ` x ` and ` A ` are
     disjoint.

     Should instead ` A ` depend on ` x ` , you rather assert exactly one ` x `
     fulfilling ` ph ` happens to be contained in the corresponding
     ` A ( x ) ` .  This interpretation is rarely needed (see also ~ df-ral ).
     (Contributed by NM, 22-Nov-1994.) $)
  df-reu $a |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) ) $.

  $( *** Theorems using axioms up to ax-4 only: *** $)

  $( 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 wrmo df-eu df-reu df-rex anbi12i 3bitr4i
    df-rmo ) BDCEAFZBGSBHZSBIZFABCJABCKZABCLZFSBMABCNUBTUCUAABCOABCRPQ $.

  $( 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 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 $.

  $( 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 $.

  ${
    rmobiia.1 $e |- ( x e. A -> ( ph <-> ps ) ) $.
    $( Formula-building rule for restricted at-most-one 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 $.

    $( Formula-building rule for restricted existential uniqueness 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 $.
  $}

  ${
    rmobii.1 $e |- ( ph <-> ps ) $.
    $( Formula-building rule for restricted at-most-one 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 $.

    $( Formula-building rule for restricted existential uniqueness 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 $.
  $}

  $( Cancellation law for restricted at-most-one quantification.  (Contributed
     by Peter Mazsa, 24-May-2018.)  (Proof shortened by Wolf Lammen,
     12-Jan-2025.) $)
  rmoanid $p |- ( E* x e. A ( x e. A /\ ph ) <-> E* x e. A ph ) $=
    ( cv wcel wa ibar bicomd rmobiia ) BDCEZAFZABCJAKJAGHI $.

  $( Cancellation law for restricted unique existential quantification.
     (Contributed by Peter Mazsa, 12-Feb-2018.)  (Proof shortened by Wolf
     Lammen, 12-Jan-2025.) $)
  reuanid $p |- ( E! x e. A ( x e. A /\ ph ) <-> E! x e. A ph ) $=
    ( cv wcel wa ibar bicomd reubiia ) BDCEZAFZABCJAKJAGHI $.

  $( Obsolete version of ~ rmoanid as of 12-Jan-2025.  (Contributed by Peter
     Mazsa, 24-May-2018.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  rmoanidOLD $p |- ( E* x e. A ( x e. A /\ ph ) <-> E* x e. A ph ) $=
    ( cv wcel wa wmo wrmo anabs5 mobii df-rmo 3bitr4i ) BDCEZMAFZFZBGNBGNBCHABC
    HONBMAIJNBCKABCKL $.

  $( Obsolete version of ~ reuanid as of 12-Jan-2025.  (Contributed by Peter
     Mazsa, 12-Feb-2018.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  reuanidOLD $p |- ( E! x e. A ( x e. A /\ ph ) <-> E! x e. A ph ) $=
    ( cv wcel wa weu wreu anabs5 eubii df-reu 3bitr4i ) BDCEZMAFZFZBGNBGNBCHABC
    HONBMAIJNBCKABCKL $.

  $( Double restricted existential uniqueness, analogous to ~ 2eu2ex .
     (Contributed by Alexander van der Vekens, 25-Jun-2017.) $)
  2reu2rex $p |- ( E! x e. A E! y e. B ph -> E. x e. A E. y e. B ph ) $=
    ( wreu wrex reurex reximi syl ) ACEFZBDFKBDGACEGZBDGKBDHKLBDACEHIJ $.

  $( *** Theorems using axioms up to ax-5 only: *** $)

  ${
    $d x ph $.
    rmobidva.1 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
    $( Formula-building rule for restricted at-most-one quantifier (deduction
       form).  (Contributed by NM, 16-Jun-2017.)  Avoid ~ ax-6 , ~ ax-7 ,
       ~ ax-12 .  (Revised by Wolf Lammen, 23-Nov-2024.) $)
    rmobidva $p |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) ) $=
      ( cv wcel wa wmo wrmo pm5.32da mobidv df-rmo 3bitr4g ) ADGEHZBIZDJPCIZDJB
      DEKCDEKAQRDAPBCFLMBDENCDENO $.
    $( $j usage 'rmobidva' avoids 'ax-6' 'ax-7' 'ax-12'; $)

    $( Formula-building rule for restricted existential uniqueness quantifier
       (deduction form).  (Contributed by NM, 13-Nov-2004.)  Reduce axiom
       usage.  (Revised by Wolf Lammen, 14-Jan-2023.) $)
    reubidva $p |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) ) $=
      ( cv wcel wa weu wreu pm5.32da eubidv df-reu 3bitr4g ) ADGEHZBIZDJPCIZDJB
      DEKCDEKAQRDAPBCFLMBDENCDENO $.
  $}

  ${
    $d x ph $.
    rmobidv.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for restricted at-most-one 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 $.

    $( Formula-building rule for restricted existential uniqueness 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 $.
  $}

  ${
    $d ph x $.
    reueubd.1 $e |- ( ( ph /\ ps ) -> x e. V ) $.
    $( Restricted existential uniqueness is equivalent to existential
       uniqueness if the unique element is in the restricting class.
       (Contributed by AV, 4-Jan-2021.) $)
    reueubd $p |- ( ph -> ( E! x e. V ps <-> E! x ps ) ) $=
      ( wreu cv wcel wa weu df-reu ex pm4.71rd eubidv bitr4id ) ABCDFCGDHZBIZCJ
      BCJBCDKABQCABPABPELMNO $.
  $}

  $( *** Theorems using axioms up to ax-8 only: *** $)

  $( 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 wi wrmo wrex wreu df-rmo df-rex df-reu imbi12i 3bitr4i
    weu moeu ) BDCEAFZBGTBHZTBRZIABCJABCKZABCLZITBSABCMUCUAUDUBABCNABCOPQ $.

  $( 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 A $.
    $( "At most one" element in a set.  (Contributed by Thierry Arnoux,
       26-Jul-2018.)  Avoid ~ ax-11 .  (Revised by Wolf Lammen,
       23-Nov-2024.) $)
    moel $p |- ( E* x x e. A <-> A. x e. A A. y e. A x = y ) $=
      ( cv wcel wmo wa weq wi wal wral eleq1w mo4 r2al bitr4i ) ADCEZAFPBDCEZGA
      BHZIBJAJRBCKACKPQABABCLMRABCCNO $.
    $( $j usage 'rmobidva' avoids 'ax-11'; $)
  $}

  ${
    $d x y A $.  $d y ph $.  $d x ps $.
    cbvrmovw.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Change the bound variable of a restricted at-most-one quantifier using
       implicit substitution.  Version of ~ cbvrmov with a disjoint variable
       condition, which requires fewer axioms.  (Contributed by NM,
       16-Jun-2017.)  (Revised by GG, 30-Sep-2024.) $)
    cbvrmovw $p |- ( E* x e. A ph <-> E* y e. A ps ) $=
      ( cv wcel wa wmo wrmo weq eleq1w anbi12d cbvmovw df-rmo 3bitr4i ) CGEHZAI
      ZCJDGEHZBIZDJACEKBDEKSUACDCDLRTABCDEMFNOACEPBDEPQ $.
    $( $j usage 'cbvrmovw' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)

    $( Change the bound variable of a restricted unique existential quantifier
       using implicit substitution.  Version of ~ cbvreuv with a disjoint
       variable condition, which requires fewer axioms.  (Contributed by NM,
       5-Apr-2004.)  (Revised by GG, 30-Sep-2024.) $)
    cbvreuvw $p |- ( E! x e. A ph <-> E! y e. A ps ) $=
      ( cv wcel wa weu wreu weq eleq1w anbi12d cbveuvw df-reu 3bitr4i ) CGEHZAI
      ZCJDGEHZBIZDJACEKBDEKSUACDCDLRTABCDEMFNOACEPBDEPQ $.
    $( $j usage 'cbvreuvw' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  $( *** Theorems using axioms up to ax-12 only: *** $)

  ${
    $d x y A $.
    $( Obsolete version of ~ moel as of 23-Nov-2024.  (Contributed by Thierry
       Arnoux, 26-Jul-2018.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    moelOLD $p |- ( E* x x e. A <-> A. x e. A A. y e. A x = y ) $=
      ( cv wcel weq wi wal wmo ralcom4 df-ral ralbii wa alcom eleq1w mo4 impexp
      wral albii bitr4i 3bitr4i 3bitr4ri ) BDCEZABFZGZBHZACRUEACRZBHZUDBCRZACRA
      DCEZAIZUEABCJUIUFACUDBCKLUJUCMUDGZBHAHULAHZBHUKUHULABNUJUCABABCOPUGUMBUGU
      JUEGZAHUMUEACKULUNAUJUCUDQSTSUAUB $.
  $}

  ${
    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 $.

    $( 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 ph $.
    rmobidvaOLD.1 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
    $( Obsolete version of ~ rmobidv as of 23-Nov-2024.  (Contributed by NM,
       16-Jun-2017.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    rmobidvaOLD $p |- ( ph -> ( E* x e. A ps <-> E* x e. A ch ) ) $=
      ( nfv rmobida ) ABCDEADGFH $.
  $}

  ${
    $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, which does not require ~ ax-10 , ~ ax-13 .  (Contributed by
       NM, 16-Jun-2017.)  Avoid ~ ax-10 , ~ ax-13 .  (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 $.
    $( $j usage 'cbvrmow' avoids 'ax-10' 'ax-13'; $)
  $}

  ${
    $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, which does not require ~ ax-13 .  (Contributed by
       Mario Carneiro, 15-Oct-2016.)  Avoid ~ ax-13 .  (Revised by GG,
       10-Jan-2024.)  Avoid ~ ax-10 .  (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 $.
    $( $j usage 'cbvreuw' avoids 'ax-10' 'ax-13'; $)
  $}

  $( The setvar ` 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 $.

  $( The setvar ` 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 $.

  ${
    $d x y $.
    nfrmow.1 $e |- F/_ x A $.
    nfrmow.2 $e |- F/ x ph $.
    $( Bound-variable hypothesis builder for restricted uniqueness.  Version of
       ~ nfrmo with a disjoint variable condition, which does not require
       ~ ax-13 .  (Contributed by NM, 16-Jun-2017.)  Avoid ~ ax-13 .  (Revised
       by GG, 10-Jan-2024.)  Avoid ~ ax-9 , ~ ax-ext .  (Revised by Wolf
       Lammen, 21-Nov-2024.) $)
    nfrmow $p |- F/ x E* y e. A ph $=
      ( wrmo cv wcel wa wmo df-rmo nfcri nfan nfmov nfxfr ) ACDGCHDIZAJZCKBACDL
      RBCQABBCDEMFNOP $.
    $( $j usage 'nfrmow' avoids 'ax-9' 'ax-13' 'ax-ext'; $)

    $( Bound-variable hypothesis builder for restricted unique existence.
       Version of ~ nfreu with a disjoint variable condition, which does not
       require ~ ax-13 .  (Contributed by NM, 30-Oct-2010.)  Avoid ~ ax-13 .
       (Revised by GG, 10-Jan-2024.)  Avoid ~ ax-9 , ~ ax-ext .  (Revised by
       Wolf Lammen, 21-Nov-2024.) $)
    nfreuw $p |- F/ x E! y e. A ph $=
      ( wreu cv wcel wa weu df-reu nfcri nfan nfeuw nfxfr ) ACDGCHDIZAJZCKBACDL
      RBCQABBCDEMFNOP $.
    $( $j usage 'nfreuw' avoids 'ax-9' 'ax-13' 'ax-ext'; $)
  $}

  ${
    $d A x y z $.  $d ph z $.  $d ps z $.
    cbvreuwOLD.1 $e |- F/ y ph $.
    cbvreuwOLD.2 $e |- F/ x ps $.
    cbvreuwOLD.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Obsolete version of ~ cbvreuw as of 10-Dec-2024.  (Contributed by Mario
       Carneiro, 15-Oct-2016.)  (Revised by GG, 10-Jan-2024.)
       (New usage is discouraged.)  (Proof modification is discouraged.) $)
    cbvreuwOLD $p |- ( E! x e. A ph <-> E! y e. A ps ) $=
      ( vz cv wcel wa weu wreu wsb nfv sb8euv sban eubii df-reu anbi1i nfan weq
      clelsb1 nfsbv eleq1w sbequ sbiev bitrdi anbi12d cbveuw 3bitri 3bitr4i
      bitri ) CJEKZALZCMZDJEKZBLZDMZACENBDENUQUPCIOZIMUOCIOZACIOZLZIMZUTUPCIUPI
      PQVAVDIUOACIRSVEIJEKZVCLZIMUTVDVGIVBVFVCCIEUDUASVGUSIDVFVCDVFDPACIDFUEUBU
      SIPIDUCZVFURVCBIDEUFVHVCACDOBAIDCUGABCDGHUHUIUJUKUNULACETBDETUM $.
    $( $j usage 'cbvreuwOLD' avoids 'ax-13'; $)
  $}

  $( *** Theorems using axioms up to ax-12 and ax-ext only: *** $)

  ${
    $d z x A $.  $d z x B $.  $d ph z $.
    $( Equality theorem for restricted at-most-one quantifier.  (Contributed by
       Alexander van der Vekens, 17-Jun-2017.)  Remove usage of ~ ax-10 ,
       ~ ax-11 , and ~ ax-12 .  (Revised by Steven Nguyen, 30-Apr-2023.)  Avoid
       ~ ax-8 .  (Revised by Wolf Lammen, 12-Mar-2025.) $)
    rmoeq1 $p |- ( A = B -> ( E* x e. A ph <-> E* x e. B ph ) ) $=
      ( vz wceq cv wcel wa wmo wrmo weq wi wal wex dfcleq df-mo 3bitr4g df-rmo
      wb biimpi anbi1 imbi1d alimi albi 3syl exbidv ) CDFZBGZCHZAIZBJZUIDHZAIZB
      JZABCKABDKUHUKBELZMZBNZEOUNUPMZBNZEOULUOUHURUTEUHUJUMTZBNZUQUSTZBNURUTTUH
      VBBCDPUAVAVCBVAUKUNUPUJUMAUBUCUDUQUSBUEUFUGUKBEQUNBEQRABCSABDSR $.
    $( $j usage 'rmoeq1' avoids 'ax-8' 'df-clel'; $)
  $}

  ${
    $d x A $.  $d x B $.
    $( Equality theorem for restricted unique existential quantifier.
       (Contributed by NM, 5-Apr-2004.)  Remove usage of ~ ax-10 , ~ ax-11 ,
       and ~ ax-12 .  (Revised by Steven Nguyen, 30-Apr-2023.)  Avoid ~ ax-8 .
       (Revised by Wolf Lammen, 12-Mar-2025.) $)
    reueq1 $p |- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) ) $=
      ( wceq wrex wrmo wa wreu rexeq rmoeq1 anbi12d reu5 3bitr4g ) CDEZABCFZABC
      GZHABDFZABDGZHABCIABDIOPRQSABCDJABCDKLABCMABDMN $.
    $( $j usage 'reueq1' avoids 'ax-8' 'df-clel'; $)

    $( Obsolete version of ~ rmoeq1 as of 12-Mar-2025.  (Contributed by
       Alexander van der Vekens, 17-Jun-2017.)  Remove usage of ~ ax-10 ,
       ~ ax-11 , and ~ ax-12 .  (Revised by Steven Nguyen, 30-Apr-2023.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    rmoeq1OLD $p |- ( A = B -> ( E* x e. A ph <-> E* x e. B ph ) ) $=
      ( wceq cv wcel wa wmo wrmo eleq2 anbi1d mobidv df-rmo 3bitr4g ) CDEZBFZCG
      ZAHZBIQDGZAHZBIABCJABDJPSUABPRTACDQKLMABCNABDNO $.

    $( Obsolete version of ~ reueq1 as of 12-Mar-2025.  (Contributed by NM,
       5-Apr-2004.)  Remove usage of ~ ax-10 , ~ ax-11 , and ~ ax-12 .
       (Revised by Steven Nguyen, 30-Apr-2023.)  (New usage is discouraged.)
       (Proof modification is discouraged.) $)
    reueq1OLD $p |- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) ) $=
      ( wceq cv wcel wa weu wreu eleq2 anbi1d eubidv df-reu 3bitr4g ) CDEZBFZCG
      ZAHZBIQDGZAHZBIABCJABDJPSUABPRTACDQKLMABCNABDNO $.
  $}

  ${
    $d x A $.  $d x B $.
    rmoeqd.1 $e |- ( A = B -> ( ph <-> ps ) ) $.
    $( 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 $.

    $( 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 $.
  $}

  ${
    $d A x $.  $d B x $.
    reueqdv.1 $e |- ( ph -> A = B ) $.
    $( Formula-building rule for restricted existential uniqueness quantifier.
       Deduction form.  (Contributed by GG, 1-Sep-2025.) $)
    reueqdv $p |- ( ph -> ( E! x e. A ps <-> E! x e. B ps ) ) $=
      ( wceq wreu wb reueq1 syl ) ADEGBCDHBCEHIFBCDEJK $.
    $( $j usage 'reueqdv' avoids 'ax-8' 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    $d ph x $.
    reueqbidv.1 $e |- ( ph -> A = B ) $.
    reueqbidv.2 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Formula-building rule for restricted existential uniqueness quantifier.
       Deduction form.  General version of ~ reubidv .  (Contributed by GG,
       1-Sep-2025.) $)
    reueqbidv $p |- ( ph -> ( E! x e. A ps <-> E! x e. B ch ) ) $=
      ( cv wcel wa weu wreu eleq2d anbi12d eubidv df-reu 3bitr4g ) ADIZEJZBKZDL
      SFJZCKZDLBDEMCDFMAUAUCDATUBBCAEFSGNHOPBDEQCDFQR $.
    $( $j usage 'reueqbidv' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    rmoeq1f.1 $e |- F/_ x A $.
    rmoeq1f.2 $e |- F/_ x B $.
    $( 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 $.

    $( 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 wrex wrmo wa wreu rexeqf rmoeq1f anbi12d reu5 3bitr4g ) CDGZABCHZA
      BCIZJABDHZABDIZJABCKABDKQRTSUAABCDEFLABCDEFMNABCOABDOP $.
  $}

  ${
    $d x y $.
    nfreuwOLD.1 $e |- F/_ x A $.
    nfreuwOLD.2 $e |- F/ x ph $.
    $( Obsolete version of ~ nfreuw as of 21-Nov-2024.  (Contributed by NM,
       30-Oct-2010.)  (Revised by GG, 10-Jan-2024.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    nfreuwOLD $p |- F/ x E! y e. A ph $=
      ( wreu wnf cv wcel wa weu wtru df-reu nftru nfcvd wnfc a1i nfeld nfand
      nfeudw nfxfrd mptru ) ACDGZBHUDCIZDJZAKZCLMBACDNMUGBCCOMUFABMBUEDMBUEPBDQ
      MERSABHMFRTUAUBUC $.
    $( $j usage 'nfreuwOLD' avoids 'ax-13'; $)

    $( Obsolete version of ~ nfrmow as of 21-Nov-2024.  (Contributed by NM,
       16-Jun-2017.)  (Revised by GG, 10-Jan-2024.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    nfrmowOLD $p |- F/ x E* y e. A ph $=
      ( wrmo cv wcel wa wmo df-rmo wnf wtru nftru nfcvd wnfc a1i nfeld nfand
      nfmodv mptru nfxfr ) ACDGCHZDIZAJZCKZBACDLUGBMNUFBCCONUEABNBUDDNBUDPBDQNE
      RSABMNFRTUAUBUC $.
    $( $j usage 'nfrmowOLD' avoids 'ax-13'; $)
  $}

  $( *** Theorems using axioms ax-13 or axc11r: *** $)

  ${
    $d x z A $.  $d y z A $.  $d z ph $.  $d z ps $.
    cbvrmo.1 $e |- F/ y ph $.
    cbvrmo.2 $e |- F/ x ps $.
    cbvrmo.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Change the bound variable of a restricted unique existential quantifier
       using implicit substitution.  Usage of this theorem is discouraged
       because it depends on ~ ax-13 .  Use the weaker ~ cbvreuw when possible.
       (Contributed by Mario Carneiro, 15-Oct-2016.)
       (New usage is discouraged.) $)
    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 eleq1w sbequ sbie bitrdi anbi12d cbveu bitri 3bitri 3bitr4i )
      CJEKZALZCMZDJEKZBLZDMZACENBDENUQUPCIOZIMUOCIOZACIOZLZIMZUTUPCIUPIPQVAVDIU
      OACIRSVEIJEKZVCLZIMUTVDVGIVBVFVCCIEUDUASVGUSIDVFVCDVFDPACIDFUBUCUSIPIDUEZ
      VFURVCBIDEUFVHVCACDOBAIDCUGABCDGHUHUIUJUKULUMACETBDETUN $.

    $( Change the bound variable of a restricted at-most-one quantifier using
       implicit substitution.  Usage of this theorem is discouraged because it
       depends on ~ ax-13 .  Use the weaker ~ cbvrmow , ~ cbvrmovw when
       possible.  (Contributed by NM, 16-Jun-2017.)
       (New usage is discouraged.) $)
    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 $.
    cbvrmov.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Change the bound variable of a restricted at-most-one quantifier using
       implicit substitution.  Usage of this theorem is discouraged because it
       depends on ~ ax-13 .  (Contributed by Alexander van der Vekens,
       17-Jun-2017.)  (New usage is discouraged.) $)
    cbvrmov $p |- ( E* x e. A ph <-> E* y e. A ps ) $=
      ( nfv cbvrmo ) ABCDEADGBCGFH $.

    $( Change the bound variable of a restricted unique existential quantifier
       using implicit substitution.  See ~ cbvreuvw for a version without
       ~ ax-13 , but extra disjoint variables.  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  Use the weaker ~ cbvreuvw
       when possible.  (Contributed by NM, 5-Apr-2004.)  (Revised by Mario
       Carneiro, 15-Oct-2016.)  (New usage is discouraged.) $)
    cbvreuv $p |- ( E! x e. A ph <-> E! y e. A ps ) $=
      ( nfv cbvreu ) ABCDEADGBCGFH $.
  $}

  ${
    nfrmod.1 $e |- F/ y ph $.
    nfrmod.2 $e |- ( ph -> F/_ x A ) $.
    nfrmod.3 $e |- ( ph -> F/ x ps ) $.
    $( Deduction version of ~ nfrmo .  Usage of this theorem is discouraged
       because it depends on ~ ax-13 .  (Contributed by NM, 17-Jun-2017.)
       (New usage is discouraged.) $)
    nfrmod $p |- ( ph -> F/ x E* y e. A ps ) $=
      ( wrmo cv wcel wa wmo df-rmo weq wal wn wnfc nfcvf adantr nfeld wnf nfand
      adantl nfmod2 nfxfrd ) BDEIDJZEKZBLZDMACBDENAUICDFACDOCPQZLZUHBCUKCUGEUJC
      UGRACDSUDACERUJGTUAABCUBUJHTUCUEUF $.

    $( Deduction version of ~ nfreu .  Usage of this theorem is discouraged
       because it depends on ~ ax-13 .  (Contributed by NM, 15-Feb-2013.)
       (Revised by Mario Carneiro, 8-Oct-2016.)  (New usage is discouraged.) $)
    nfreud $p |- ( ph -> F/ x E! y e. A ps ) $=
      ( wreu cv wcel wa weu df-reu weq wal wn wnfc nfcvf adantr nfeld wnf nfand
      adantl nfeud2 nfxfrd ) BDEIDJZEKZBLZDMACBDENAUICDFACDOCPQZLZUHBCUKCUGEUJC
      UGRACDSUDACERUJGTUAABCUBUJHTUCUEUF $.
  $}

  ${
    nfrmo.1 $e |- F/_ x A $.
    nfrmo.2 $e |- F/ x ph $.
    $( Bound-variable hypothesis builder for restricted uniqueness.  Usage of
       this theorem is discouraged because it depends on ~ ax-13 .  Use the
       weaker ~ nfrmow when possible.  (Contributed by NM, 16-Jun-2017.)
       (New usage is discouraged.) $)
    nfrmo $p |- F/ x E* y e. A ph $=
      ( wrmo cv wcel wa wmo df-rmo wnf wtru nftru weq wal wn nfcvf a1i adantl
      wnfc nfeld nfand nfmod2 mptru nfxfr ) ACDGCHZDIZAJZCKZBACDLUKBMNUJBCCOBCP
      BQRZUJBMNULUIABULBUHDBCSBDUBULETUCABMULFTUDUAUEUFUG $.

    $( Bound-variable hypothesis builder for restricted unique existence.
       Usage of this theorem is discouraged because it depends on ~ ax-13 .
       Use the weaker ~ nfreuw when possible.  (Contributed by NM,
       30-Oct-2010.)  (Revised by Mario Carneiro, 8-Oct-2016.)
       (New usage is discouraged.) $)
    nfreu $p |- F/ x E! y e. A ph $=
      ( wreu wnf wtru nftru wnfc a1i nfreud mptru ) ACDGBHIABCDCJBDKIELABHIFLMN
      $.
  $}


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Restricted class abstraction
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  $( Extend class notation to include the restricted class abstraction (class
     builder). $)
  crab $a class { x e. A | ph } $.

  $( Define a restricted class abstraction (class builder): ` { x e. A | ph } `
     is the class of all sets ` x ` in ` A ` such that ` ph ( x ) ` is true.
     Definition of [TakeutiZaring] p. 20.

     For the interpretation given in the previous paragraph to be correct, we
     need to assume ` F/_ x A ` , which is the case as soon as ` x ` and ` A `
     are disjoint, which is generally the case.  If ` A ` were to depend on
     ` x ` , then the interpretation would be less obvious (think of the two
     extreme cases ` A = { x } ` and ` A = x ` , for instance).  See also
     ~ df-ral .  (Contributed by NM, 22-Nov-1994.) $)
  df-rab $a |- { x e. A | ph } = { x | ( x e. A /\ ph ) } $.

  $( *** Theorems using axioms up to ax-9 and ax-ext only: *** $)

  ${
    $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 formulas yield equal restricted class abstractions (inference
       form).  (Contributed by NM, 22-May-1999.)  (Proof shortened by Wolf
       Lammen, 12-Jan-2025.) $)
    rabbiia $p |- { x e. A | ph } = { x e. A | ps } $=
      ( cv wcel pm5.32i rabbia2 ) ABCDDCFDGABEHI $.

    $( Obsolete version of ~ rabbiia as of 12-Jan-2025.  (Contributed by NM,
       22-May-1999.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    rabbiiaOLD $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.)  (Proof shortened by SN,
       3-Dec-2023.) $)
    rabbidva $p |- ( ph -> { x e. A | ps } = { x e. A | ch } ) $=
      ( cv wcel pm5.32da rabbidva2 ) ABCDEEADGEHBCFIJ $.
    $( $j usage 'rabbidva' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    $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 $.
  $}

  ${
    rabbieq.1 $e |- B = { x e. A | ph } $.
    rabbieq.2 $e |- ( ph <-> ps ) $.
    $( Equivalent wff's correspond to restricted class abstractions which are
       equal with the same class.  (Contributed by Peter Mazsa, 8-Jul-2019.) $)
    rabbieq $p |- B = { x e. A | ps } $=
      ( crab rabbii eqtri ) EACDHBCDHFABCDGIJ $.
  $}

  $( 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 ancom rabbia2 ) ADZCEZHBEZABCJIFG $.

  ${
    $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.)  Require
       ` x ` , ` y ` be disjoint to avoid ~ ax-11 and ~ ax-13 .  (Revised by
       Steven Nguyen, 4-Dec-2022.) $)
    cbvrabv $p |- { x e. A | ph } = { y e. A | ps } $=
      ( cv wcel wa cab crab weq eleq1w anbi12d cbvabv df-rab 3eqtr4i ) CGEHZAIZ
      CJDGEHZBIZDJACEKBDEKSUACDCDLRTABCDEMFNOACEPBDEPQ $.
  $}

  ${
    $d x A $.  $d x ph $.
    rabeqcda.1 $e |- ( ( ph /\ x e. A ) -> ps ) $.
    $( When ` ps ` is always true in a context, a restricted class abstraction
       is equal to the restricting class.  Deduction form of ~ rabeqc .
       (Contributed by Steven Nguyen, 7-Jun-2023.) $)
    rabeqcda $p |- ( ph -> { x e. A | ps } = A ) $=
      ( crab cv wcel wa cab df-rab ex pm4.71d eqabdv eqtr4id ) ABCDFCGDHZBIZCJD
      BCDKAQCDAPBAPBELMNO $.
  $}

  ${
    $d A x $.
    rabeqc.1 $e |- ( x e. A -> ph ) $.
    $( A restricted class abstraction equals the restricting class if its
       condition follows from the membership of the free setvar variable in the
       restricting class.  (Contributed by AV, 20-Apr-2022.)  (Proof shortened
       by SN, 15-Jan-2025.) $)
    rabeqc $p |- { x e. A | ph } = A $=
      ( crab wceq wtru cv wcel adantl rabeqcda mptru ) ABCECFGABCBHCIAGDJKL $.
  $}

  ${
    rabeqi.1 $e |- A = B $.
    $( Equality theorem for restricted class abstractions.  Inference form of
       ~ rabeqf .  (Contributed by Glauco Siliprandi, 26-Jun-2021.)  Avoid
       ~ ax-10 , ~ ax-11 , ~ ax-12 .  (Revised by GG, 3-Jun-2024.) $)
    rabeqi $p |- { x e. A | ph } = { x e. B | ph } $=
      ( cv wcel eleq2i anbi1i rabbia2 ) AABCDBFZCGKDGACDKEHIJ $.
    $( $j usage 'rabeqi' avoids 'ax-10' 'ax-11' 'ax-12'; $)

  $}

  ${
    $d x A $.  $d x B $.
    $( Equality theorem for restricted class abstractions.  (Contributed by NM,
       15-Oct-2003.)  Avoid ~ ax-10 , ~ ax-11 , ~ ax-12 .  (Revised by GG,
       20-Aug-2023.) $)
    rabeq $p |- ( A = B -> { x e. A | ph } = { x e. B | ph } ) $=
      ( wceq cv wcel eleq2 anbi1d rabbidva2 ) CDEZAABCDKBFZCGLDGACDLHIJ $.
    $( $j usage 'rabeq' avoids 'ax-10' 'ax-11' 'ax-12'; $)

    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 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.)  Remove DV conditions.  (Revised by GG,
       1-Sep-2025.) $)
    rabeqbidva $p |- ( ph -> { x e. A | ps } = { x e. B | ch } ) $=
      ( crab rabbidva cv wcel eleq2d anbi1d rabbidva2 eqtrd ) ABDEICDEICDFIABCD
      EHJACCDEFADKZELQFLCAEFQGMNOP $.
    $( $j usage 'rabeqbidva' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    $d A x $.  $d B x $.  $d ph x $.
    rabeqbidvaOLD.1 $e |- ( ph -> A = B ) $.
    rabeqbidvaOLD.2 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
    $( Obsolete version of ~ rabeqbidva as of 1-Sep-2025.  (Contributed by
       Mario Carneiro, 26-Jan-2017.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    rabeqbidvaOLD $p |- ( ph -> { x e. A | ps } = { x e. B | ch } ) $=
      ( crab rabbidva rabeqdv eqtrd ) ABDEICDEICDFIABCDEHJACDEFGKL $.
  $}

  ${
    $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 } ) $=
      ( wb cv wcel adantr rabeqbidva ) ABCDEFGABCIDJEKHLM $.
  $}

  ${
    $d x y $.  $d A y $.  $d ch y $.
    rabrabi.1 $e |- ( x = y -> ( ch <-> ph ) ) $.
    $( Abstract builder restricted to another restricted abstract builder with
       implicit substitution.  (Contributed by AV, 2-Aug-2022.)  Avoid
       ~ ax-10 , ~ ax-11 and ~ ax-12 .  (Revised by GG, 12-Oct-2024.) $)
    rabrabi $p |- { x e. { y e. A | ph } | ps } = { x e. A | ( ch /\ ps ) } $=
      ( wa crab cv wcel cab wsb df-rab eleq2i df-clab weq eleq1w wb bicomd
      equcoms anbi12d sbievw 3bitri anbi1i anass bitri rabbia2 ) BCBHZDAEFIZFDJ
      ZUJKZBHUKFKZCHZBHUMUIHULUNBULUKEJFKZAHZELZKUPEDMUNUJUQUKAEFNOUPDEPUPUNEDE
      DQUOUMACEDFRACSDEDEQCAGTUAUBUCUDUEUMCBUFUGUH $.
    $( $j usage 'rabrabi' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  $( *** Theorems using axioms up to ax-9 , ax-10, ax-ext only: *** $)

  $( 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 $.

  $( *** Theorems using axioms up to ax-9 , ax-12, ax-ext only: *** $)

  $( 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 eqabri ) BDCEAFBABCGABCHI $.

  $( Membership in a restricted abstraction, implication.  (Contributed by
     Glauco Siliprandi, 26-Jun-2021.) $)
  rabidim1 $p |- ( x e. { x e. A | ph } -> x e. A ) $=
    ( cv crab wcel rabid simplbi ) BDZABCEFICFAABCGH $.

  ${
    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 $.
  $}

  $( Abstract builder restricted to another restricted abstract builder.
     (Contributed by Thierry Arnoux, 30-Aug-2017.) $)
  rabrab $p |- { x e. { x e. A | ph } | ps } = { x e. A | ( ph /\ ps ) } $=
    ( cv crab wcel wa cab rabid anbi1i anass bitri abbii df-rab 3eqtr4i ) CEZAC
    DFZGZBHZCIQDGZABHZHZCIBCRFUBCDFTUCCTUAAHZBHUCSUDBACDJKUAABLMNBCROUBCDOP $.

  $( *** Theorems using axioms up to ax-12, ax-ext only: *** $)

  ${
    rabbida4.nf $e |- F/ x ph $.
    rabbida4.1 $e |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) ) ) $.
    $( Version of ~ rabbidva2 with disjoint variable condition replaced by
       nonfreeness hypothesis.  (Contributed by BJ, 27-Apr-2019.) $)
    rabbida4 $p |- ( ph -> { x e. A | ps } = { x e. B | ch } ) $=
      ( cv wcel wa cab crab abbid df-rab 3eqtr4g ) ADIZEJBKZDLQFJCKZDLBDEMCDFMA
      RSDGHNBDEOCDFOP $.
  $}

  ${
    rabbida.n $e |- F/ x ph $.
    rabbida.1 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
    $( Equivalent wff's yield equal restricted class abstractions (deduction
       form).  Version of ~ rabbidva with disjoint variable condition replaced
       by nonfreeness hypothesis.  (Contributed by BJ, 27-Apr-2019.)  Avoid
       ~ ax-10 , ~ ax-11 .  (Revised by Wolf Lammen, 14-Mar-2025.) $)
    rabbida $p |- ( ph -> { x e. A | ps } = { x e. A | ch } ) $=
      ( cv wcel pm5.32da rabbida4 ) ABCDEEFADHEIBCGJK $.
    $( $j usage 'rabbida' avoids 'ax-10' 'ax-11'; $)
  $}

  ${
    rabbid.n $e |- F/ x ph $.
    rabbid.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    $( Version of ~ rabbidv with disjoint variable condition replaced by
       nonfreeness hypothesis.  (Contributed by BJ, 27-Apr-2019.) $)
    rabbid $p |- ( ph -> { x e. A | ps } = { x e. A | ch } ) $=
      ( wb cv wcel adantr rabbida ) ABCDEFABCHDIEJGKL $.
  $}

  ${
    rabeqd.nf $e |- F/ x ph $.
    rabeqd.1 $e |- ( ph -> A = B ) $.
    $( Deduction form of ~ rabeq .  Note that contrary to ~ rabeq it has no
       disjoint variable condition.  (Contributed by BJ, 27-Apr-2019.) $)
    rabeqd $p |- ( ph -> { x e. A | ps } = { x e. B | ps } ) $=
      ( wceq cv wcel wa wb eleq2 anbi1d syl rabbida4 ) ABBCDEFADEHZCIZDJZBKREJZ
      BKLGQSTBDERMNOP $.
  $}

  ${
    rabeqbida.nf $e |- F/ x ph $.
    rabeqbida.1 $e |- ( ph -> A = B ) $.
    rabeqbida.2 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
    $( Version of ~ rabeqbidva with two disjoint variable conditions removed
       and the third replaced by a nonfreeness hypothesis.  (Contributed by BJ,
       27-Apr-2019.) $)
    rabeqbida $p |- ( ph -> { x e. A | ps } = { x e. B | ch } ) $=
      ( crab rabbida rabeqd eqtrd ) ABDEJCDEJCDFJABCDEGIKACDEFGHLM $.
  $}

  $( Equivalent wff's correspond to equal restricted class abstractions.
     Closed theorem form of ~ rabbii .  (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 cab wceq wb wal crab wral abbib df-rab eqeq12i wi df-ral albii
    pm5.32 bitri 3bitr4ri ) CEDFZAGZCHZUCBGZCHZIUDUFJZCKZACDLZBCDLZIABJZCDMZUDU
    FCNUJUEUKUGACDOBCDOPUMUCULQZCKUIULCDRUNUHCUCABTSUAUB $.

  ${
    rabid2f.1 $e |- F/_ x A $.
    $( An "identity" law for restricted class abstraction.  (Contributed by NM,
       9-Oct-2003.)  (Proof shortened by Andrew Salmon, 30-May-2011.)  (Revised
       by Thierry Arnoux, 13-Mar-2017.) $)
    rabid2f $p |- ( A = { x e. A | ph } <-> A. x e. A ph ) $=
      ( cv wcel wa cab wceq wi wal crab eqabf pm4.71 albii bitr4i df-rab eqeq2i
      wral wb df-ral 3bitr4i ) CBECFZAGZBHZIZUCAJZBKZCABCLZIABCSUFUCUDTZBKUHUDB
      CDMUGUJBUCANOPUIUECABCQRABCUAUB $.
  $}

  ${
    $d x A $.
    $( One direction of ~ rabid2 is based on fewer axioms.  (Contributed by
       Wolf Lammen, 26-May-2025.) $)
    rabid2im $p |- ( A. x e. A ph -> A = { x e. A | ph } ) $=
      ( cv wcel wi wal wa cab wceq wral crab wb pm4.71 albii eqab df-ral df-rab
      sylbi eqeq2i 3imtr4i ) BDCEZAFZBGZCUBAHZBIZJZABCKCABCLZJUDUBUEMZBGUGUCUIB
      UBANOUEBCPSABCQUHUFCABCRTUA $.

    $( An "identity" law for restricted class abstraction.  Prefer ~ rabid2im
       if one direction is sufficient.  (Contributed by NM, 9-Oct-2003.)
       (Proof shortened by Andrew Salmon, 30-May-2011.)  (Proof shortened by
       Wolf Lammen, 24-Nov-2024.) $)
    rabid2 $p |- ( A = { x e. A | ph } <-> A. x e. A ph ) $=
      ( nfcv rabid2f ) ABCBCDE $.

    $( Obsolete version of ~ rabid2 as of 24-Nov-2024.  (Contributed by NM,
       9-Oct-2003.)  (Proof shortened by Andrew Salmon, 30-May-2011.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    rabid2OLD $p |- ( A = { x e. A | ph } <-> A. x e. A ph ) $=
      ( cv wcel wa cab wceq wi wal crab eqabb pm4.71 albii bitr4i df-rab eqeq2i
      wral wb df-ral 3bitr4i ) CBDCEZAFZBGZHZUBAIZBJZCABCKZHABCRUEUBUCSZBJUGUCB
      CLUFUIBUBAMNOUHUDCABCPQABCTUA $.
  $}

  ${
    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 $.
  $}

  ${
    $d x y $.
    cbvrabw.1 $e |- F/_ x A $.
    cbvrabw.2 $e |- F/_ y A $.
    cbvrabw.3 $e |- F/ y ph $.
    cbvrabw.4 $e |- F/ x ps $.
    cbvrabw.5 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Rule to change the bound variable in a restricted class abstraction,
       using implicit substitution.  Version of ~ cbvrab with a disjoint
       variable condition, which does not require ~ ax-13 .  (Contributed by
       Andrew Salmon, 11-Jul-2011.)  Avoid ~ ax-13 .  (Revised by GG,
       10-Jan-2024.)  Avoid ~ ax-10 .  (Revised by Wolf Lammen,
       19-Jul-2025.) $)
    cbvrabw $p |- { x e. A | ph } = { y e. A | ps } $=
      ( cv wcel wa cab crab nfcri nfan weq eleq1w df-rab anbi12d cbvabw 3eqtr4i
      ) CKELZAMZCNDKELZBMZDNACEOBDEOUEUGCDUDADDCEGPHQUFBCCDEFPIQCDRUDUFABCDESJU
      AUBACETBDETUC $.
    $( $j usage 'cbvrabw' avoids 'ax-10' 'ax-13'; $)

    $d x z $.  $d y z $.  $d A z $.  $d ph z $.  $d ps z $.
    $( Obsolete version of ~ cbvrabw as of 19-Jul-2025.  (Contributed by Andrew
       Salmon, 11-Jul-2011.)  Avoid ~ ax-13 .  (Revised by GG, 10-Jan-2024.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    cbvrabwOLD $p |- { x e. A | ph } = { y e. A | ps } $=
      ( vz cv wcel wa cab crab wsb nfv nfcri nfan eleq1w sbequ12 anbi12d cbvabw
      nfs1v weq nfsbv sbequ sbiev bitrdi eqtri df-rab 3eqtr4i ) CLEMZANZCOZDLEM
      ZBNZDOZACEPBDEPUPKLEMZACKQZNZKOUSUOVBCKUOKRUTVACCKEFSACKUETCKUFUNUTAVACKE
      UAACKUBUCUDVBURKDUTVADDKEGSACKDHUGTURKRKDUFZUTUQVABKDEUAVCVAACDQBAKDCUHAB
      CDIJUIUJUCUDUKACEULBDEULUM $.
    $( $j usage 'cbvrabw' avoids 'ax-13'; $)
  $}

  ${
    $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.  Version of ~ nfrab with a disjoint variable condition,
       which does not require ~ ax-13 .  (Contributed by NM, 13-Oct-2003.)
       Avoid ~ ax-13 .  (Revised by GG, 10-Jan-2024.)  (Proof shortened by Wolf
       Lammen, 23-Nov-2024.) $)
    nfrabw $p |- F/_ x { y e. A | ph } $=
      ( crab cv wcel wa cab df-rab wnfc wtru nftru wnf nfcri nfan a1i nfabdw
      mptru nfcxfr ) BACDGCHDIZAJZCKZACDLBUEMNUDBCCOUDBPNUCABBCDFQERSTUAUB $.
    $( $j usage 'nfrabw' avoids 'ax-13'; $)

    $( Obsolete version of ~ nfrabw as of 23-Nov-2024.  (Contributed by NM,
       13-Oct-2003.)  (Revised by GG, 10-Jan-2024.)
       (New usage is discouraged.)  (Proof modification is discouraged.) $)
    nfrabwOLD $p |- F/_ x { y e. A | ph } $=
      ( crab cv wcel wa cab df-rab wnfc wtru nftru wnf nfcri a1i nfand nfabdw
      mptru nfcxfr ) BACDGCHDIZAJZCKZACDLBUEMNUDBCCONUCABUCBPNBCDFQRABPNERSTUAU
      B $.
    $( $j usage 'nfrabwOLD' avoids 'ax-13'; $)
  $}

  ${
    rabbidaOLD.n $e |- F/ x ph $.
    rabbidaOLD.1 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $.
    $( Obsolete version of ~ rabbida as of 14-Mar-2025.  (Contributed by BJ,
       27-Apr-2019.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    rabbidaOLD $p |- ( ph -> { x e. A | ps } = { x e. A | ch } ) $=
      ( wb wral crab wceq cv wcel ex ralrimi rabbi sylib ) ABCHZDEIBDEJCDEJKARD
      EFADLEMRGNOBCDEPQ $.
  $}

  $( *** Theorems using axioms ax-13 or axc11r: *** $)

  ${
    $d x z $.  $d y z $.  $d z A $.
    nfrab.1 $e |- F/ x ph $.
    nfrab.2 $e |- F/_ x A $.
    $( A variable not free in a wff remains so in a restricted class
       abstraction.  Usage of this theorem is discouraged because it depends on
       ~ ax-13 .  Use the weaker ~ nfrabw when possible.  (Contributed by NM,
       13-Oct-2003.)  (Revised by Mario Carneiro, 9-Oct-2016.)
       (New usage is discouraged.) $)
    nfrab $p |- F/_ x { y e. A | ph } $=
      ( vz crab cv wcel wa cab df-rab wnfc wtru nftru weq wal wn wnf eleq1w a1i
      nfcri dvelimnf nfand adantl nfabd2 mptru nfcxfr ) BACDHCIDJZAKZCLZACDMBUL
      NOUKBCCPBCQBRSZUKBTOUMUJABGIDJUJBCGBGDFUCGCDUAUDABTUMEUBUEUFUGUHUI $.
  $}

  ${
    $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.  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  Use the weaker ~ cbvrabw
       when possible.  (Contributed by Andrew Salmon, 11-Jul-2011.)  (Revised
       by Mario Carneiro, 9-Oct-2016.)  (New usage is discouraged.) $)
    cbvrab $p |- { x e. A | ph } = { y e. A | ps } $=
      ( vz cv wcel wa cab crab wsb nfv nfcri nfan nfs1v weq eleq1w sbequ12 nfsb
      anbi12d cbvab sbequ sbie bitrdi eqtri df-rab 3eqtr4i ) CLEMZANZCOZDLEMZBN
      ZDOZACEPBDEPUPKLEMZACKQZNZKOUSUOVBCKUOKRUTVACCKEFSACKUATCKUBUNUTAVACKEUCA
      CKUDUFUGVBURKDUTVADDKEGSACKDHUETURKRKDUBZUTUQVABKDEUCVCVAACDQBAKDCUHABCDI
      JUIUJUFUGUKACEULBDEULUM $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  The universal class
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare the symbol for the universal class. $)
  $c _V $.  $( Upright uppercase 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 $.
    $( Justification theorem for ~ df-v .  (Contributed by Rodolfo Medina,
       27-Apr-2010.) $)
    vjust $p |- { x | x = x } = { y | y = y } $=
      ( vz weq cab cv wcel equid vexw 2th eqriv ) CAADZAEZBBDZBEZCFZMGPOGLACAHI
      NBCBHIJK $.
  $}

  $( Define the universal class.  Definition 5.20 of [TakeutiZaring] p. 21.
     Also Definition 2.9 of [Quine] p. 19.  The class ` _V ` can be described
     as the "class of all sets"; ~ vprc proves that ` _V ` is not itself a set
     in ZF. We will frequently use the expression ` A e. _V ` as a short way to
     say " ` A ` is a set", and ~ isset proves that this expression has the
     same meaning as ` E. x x = A ` .

     In well-founded set theories without urelements, like ZF, the class ` _V `
     is equal to the von Neumann universe.  However, the letter "V" does not
     stand for "von Neumann".  The letter "V" was used earlier by Peano in 1889
     for the universe of sets, where the letter V is derived from the Latin
     word "Verum", referring to the true truth constant ` T ` .  Peano's
     notation ` _V ` was adopted by Whitehead and Russell in Principia
     Mathematica for the class of all sets in 1910.

     The class constant ` _V ` is the first class constant introduced in this
     database.  As a constant, as opposed to a variable, it cannot be
     substituted with anything, and in particular it is not part of any
     disjoint variable condition.

     For a general discussion of the theory of classes, see
     ~ mmset.html#class .  See ~ dfv2 for an alternate definition.
     (Contributed by NM, 26-May-1993.) $)
  df-v $a |- _V = { x | x = x } $.

  $( Alternate definition of the universal class (see ~ df-v ).  (Contributed
     by BJ, 30-Nov-2019.) $)
  dfv2 $p |- _V = { x | T. } $=
    ( cvv weq cab wtru df-v equid bitru abbii eqtri ) BAACZADEADAFKEAKAGHIJ $.

  $( All setvar variables are sets (see ~ isset ).  Theorem 6.8 of [Quine]
     p. 43.  A shorter proof is possible from ~ eleq2i but it uses more axioms.
     (Contributed by NM, 26-May-1993.)  Remove use of ~ ax-12 .  (Revised by
     SN, 28-Aug-2023.)  (Proof shortened by BJ, 4-Sep-2024.) $)
  vex $p |- x e. _V $=
    ( cv wtru cab cvv vextru dfv2 eleqtrri ) ABCADEAAFAGH $.

  ${
    elv.1 $e |- ( x e. _V -> ph ) $.
    $( 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 ) $.
    $( If a proposition is implied by ` x e. _V ` (which is true, see ~ vex )
       and another antecedent, then it is implied by that other antecedent.
       Deduction associated with ~ elv .  (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 $.
  $}

  ${
    el3v.1 $e |- ( ( x e. _V /\ y e. _V /\ z e. _V ) -> ph ) $.
    $( If a proposition is implied by ` x e. _V ` , ` y e. _V ` and ` z e. _V `
       (which is true, see ~ vex ), then it is true.  Inference forms (with
       ` |- A e. _V ` , ` |- B e. _V ` and ` |- C e. _V ` hypotheses) of the
       general theorems (proving ` |- ( ( A e. V /\ B e. W /\ C e. X ) -> `
       assertions) may be superfluous.  (Contributed by Peter Mazsa,
       13-Oct-2018.) $)
    el3v $p |- ph $=
      ( cv cvv wcel vex mp3an ) BFGHCFGHDFGHABICIDIEJ $.
  $}

  ${
    el3v3.1 $e |- ( ( ph /\ ps /\ z e. _V ) -> th ) $.
    $( If a proposition is implied by ` z e. _V ` (which is true, see ~ vex )
       and two other antecedents, then it is implied by these other
       antecedents.  (Contributed by Peter Mazsa, 16-Oct-2020.) $)
    el3v3 $p |- ( ( ph /\ ps ) -> th ) $=
      ( cv cvv wcel vex mp3an3 ) ABDFGHCDIEJ $.
  $}

  ${
    $d x A $.
    $( The universe contains every set.  (Contributed by NM, 11-Sep-2006.)
       Remove dependency on ~ ax-10 , ~ ax-11 , ~ ax-13 .  (Revised by BJ,
       10-Aug-2022.) $)
    eqv $p |- ( A = _V <-> A. x x e. A ) $=
      ( cvv wceq cv wcel wb wal dfcleq vex tbt albii bitr4i ) BCDAEZBFZNCFZGZAH
      OAHABCIOQAPOAJKLM $.
    $( $j usage 'eqv' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    eqvf.1 $e |- F/_ x A $.
    $( The universe contains every set.  (Contributed by BJ, 15-Jul-2021.) $)
    eqvf $p |- ( A = _V <-> A. x x e. A ) $=
      ( cvv wceq cv wcel wb wal nfcv cleqf vex tbt albii bitr4i ) BDEAFZBGZPDGZ
      HZAIQAIABDCADJKQSARQALMNO $.
  $}

  ${
    $d y x $.  $d y ph $.
    $( The class of sets verifying a property is the universal class if and
       only if that property is a tautology.  The reverse implication
       ( ~ bj-abv ) requires fewer axioms.  (Contributed by BJ, 19-Mar-2021.)
       Avoid ~ df-clel , ~ ax-8 .  (Revised by GG, 30-Aug-2024.)  (Proof
       shortened by BJ, 30-Aug-2024.) $)
    abv $p |- ( { x | ph } = _V <-> A. x ph ) $=
      ( vy cab wtru wceq wsb wal cvv cv wcel wb dfcleq vextru tbt df-clab albii
      bitr3i bitri dfv2 eqeq2i sb8v 3bitr4i ) ABDZEBDZFZABCGZCHZUDIFABHUFCJZUDK
      ZUIUEKZLZCHUHCUDUEMULUGCULUJUGUKUJBCNOACBPRQSIUEUDBTUAABCUBUC $.
    $( $j usage 'abv' avoids 'df-clel' 'ax-8' 'ax-13'; $)
  $}

  ${
    $d y x $.  $d y ph $.
    $( Alternate proof of ~ abv , shorter but using more axioms.  (Contributed
       by BJ, 19-Mar-2021.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    abvALT $p |- ( { x | ph } = _V <-> A. x ph ) $=
      ( vy cv cab wcel wal wsb cvv wceq df-clab albii eqv sb8v 3bitr4i ) CDABEZ
      FZCGABCHZCGPIJABGQRCACBKLCPMABCNO $.
  $}

  ${
    $d x A $.
    $( Two ways to express that " ` 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.

       A class ` A ` which is not a set is called a _proper class_.

       _Conventions_:  We will often use the expression " ` A e. _V ` " to
       mean " ` A ` is a set", for example in ~ uniex .  To make some theorems
       more readily applicable, we will also use the more general expression
       ` A e. V ` instead of ` A e. _V ` to mean " ` A ` is a set", typically
       in an antecedent, or in a hypothesis for theorems in deduction form (see
       for instance ~ uniexg compared with ~ uniex ).  That this is more
       general is seen either by substitution (when the variable ` V ` has no
       other occurrences), or by ~ elex .  (Contributed by NM, 26-May-1993.) $)
    isset $p |- ( A e. _V <-> E. x x = A ) $=
      ( cvv vex issetlem ) ABCADE $.
      $( $j usage 'isset' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    $d A y $.  $d x y $.
    $( The expression ` E. x x = A ` means " ` A ` is a set" even if ` A `
       contains ` x ` as a bound variable.  This lemma helps minimizing axiom
       or ~ df-clab usage in some cases.  Extracted from the proof of
       ~ issetft .  (Contributed by Wolf Lammen, 30-Jul-2025.) $)
    cbvexeqsetf $p |- ( F/_ x A -> ( E. x x = A <-> E. y y = A ) ) $=
      ( wnfc cv wceq wex wn wal nfnfc1 nfv nfvd nfcvd id nfeqd nfnd wb wi alnex
      weq eqeq1 notbid a1i cbv2w 3bitr3g con4bid ) ACDZAEZCFZAGZBEZCFZBGZUGUIHZ
      AIULHZBIUJHUMHUGUNUOABACJUGBKUGUNBLUGULAUGAUKCUGAUKMUGNOPABTZUNUOQRUGUPUI
      ULUHUKCUAUBUCUDUIASULBSUEUF $.
      $( $j usage 'cbvexeqsetf' avoids 'ax-8' 'df-clab'; $)
  $}

  ${
    $d A y $.  $d x y $.
    $( Closed theorem form of ~ isset that does not require ` x ` and ` A ` to
       be distinct.  Extracted from the proof of ~ vtoclgft .  (Contributed by
       Wolf Lammen, 9-Apr-2025.) $)
    issetft $p |- ( F/_ x A -> ( A e. _V <-> E. x x = A ) ) $=
      ( vy wnfc cvv wcel cv wceq wex isset cbvexeqsetf bitr4id ) ABDBEFCGBHCIAG
      BHAICBJACBKL $.
  $}

  ${
    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 ) $=
      ( wnfc cvv wcel cv wceq wex wb issetft ax-mp ) ABDBEFAGBHAIJCABKL $.
  $}

  ${
    $d x A $.
    isseti.1 $e |- A e. _V $.
    $( A way to say " ` A ` is a set" (inference form).  (Contributed by NM,
       24-Jun-1993.)  Remove dependencies on axioms.  (Revised by BJ,
       13-Jul-2019.) $)
    isseti $p |- E. x x = A $=
      ( cvv wcel cv wceq wex elissetv ax-mp ) BDEAFBGAHCABDIJ $.
      $( $j usage 'isseti' avoids 'ax-9' 'ax-10' 'ax-11' 'ax-12' 'ax-13'
       'ax-ext' 'df-clab' 'df-cleq'; $)
  $}

  ${
    $d x A $.
    issetri.1 $e |- E. x x = A $.
    $( A way to say " ` A ` is a set" (inference form).  (Contributed by NM,
       21-Jun-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, 26-May-1993.)  (Proof shortened
       by Andrew Salmon, 8-Jun-2011.)  (Proof shortened by Wolf Lammen,
       28-May-2025.) $)
    elex $p |- ( A e. B -> A e. _V ) $=
      ( vx wcel cv wceq wex cvv elissetv isset sylibr ) ABDCEAFCGAHDCABICAJK $.

    $( Obsolete version of ~ elex as of 28-May-2025.  (Contributed by NM,
       26-May-1993.)  (Proof shortened by Andrew Salmon, 8-Jun-2011.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    elexOLD $p |- ( A e. B -> A e. _V ) $=
      ( vx cv wceq wcel wa wex cvv exsimpl dfclel isset 3imtr4i ) CDZAEZNBFZGCH
      OCHABFAIFOPCJCABKCALM $.
  $}

  ${
    elexi.1 $e |- A e. B $.
    $( If a class is a member of another class, then it is a set.  Inference
       associated with ~ elex .  (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, then it is a set.  Deduction
       associated with ~ elex .  (Contributed by Glauco Siliprandi,
       11-Oct-2020.) $)
    elexd $p |- ( ph -> A e. _V ) $=
      ( wcel cvv elex syl ) ABCEBFEDBCGH $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Obsolete version of ~ elex2 as of 30-Nov-2024.  (Contributed by Alan
       Sare, 25-Sep-2011.)  (New usage is discouraged.)
       (Proof modification is discouraged.) $)
    elex2OLD $p |- ( A e. B -> E. x x e. B ) $=
      ( wcel cv wceq wi wal wex eleq1a alrimiv elisset exim sylc ) BCDZAEZBFZPC
      DZGZAHQAIRAIOSABCPJKABCLQRAMN $.
  $}

  ${
    $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 wex eleq1a anim12ii alrimiv elissetv adantr exim
      sylc ) BCEZBDEZFZAGZBHZUBCEZUBDEZFZIZAJUCAKZUFAKUAUGASUCUDTUEBCUBLBDUBLMN
      SUHTABCOPUCUFAQR $.
  $}

  $( A proper class doesn't belong to any class.  (Contributed by Glauco
     Siliprandi, 17-Aug-2020.)  (Proof shortened by AV, 14-Nov-2020.) $)
  prcnel $p |- ( -. A e. _V -> -. A e. V ) $=
    ( wcel cvv elex con3i ) ABCADCABEF $.

  $( 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 A $.  $d x B $.  $d x y $.  $d x ph $.
    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 $.
  $}

  ${
    $( One direction of ~ ceqsalt is based on fewer assumptions and fewer
       axioms.  It is at the same time the reverse direction of ~ vtoclgft .
       Extracted from a proof of ~ ceqsalt .  (Contributed by Wolf Lammen,
       25-Mar-2025.) $)
    ceqsal1t $p |- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) )
         -> ( ps -> A. x ( x = A -> ph ) ) ) $=
      ( wnf cv wceq wb wi wal biimpr imim2i com23 alimi 19.21t imbitrid imp ) B
      CEZCFDGZABHZIZCJZBSAIZCJIZUBBUCIZCJRUDUAUECUASBATBAISABKLMNBUCCOPQ $.
  $}

  ${
    $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 biimp imim3i al2imi wex elisset 19.23t
      wa biimpd syl7 sylan9r com23 3impia ceqsal1t 3adant3 impbid ) BCFZCGDHZAB
      IZJZCKZDELZMUJAJZCKZBUIUMUNUPBJUIUMTUPUNBUMUPUJBJZCKZUIUNBJULUOUQCUKABUJA
      BNOPUNUJCQZUIURBCDERUIURUSBJUJBCSUAUBUCUDUEUIUMBUPJUNABCDUFUGUH $.
  $}

  ${
    $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 biimt df-ral eleq1 pm5.32ri impexp
      wa imbi1i 3bitr3i albii 19.21v 3bitrri bitrdi 3ad2ant3 ceqsalt bitr3d ) B
      CFZCGZDHZABIJCKZDELZMULAJZCKZUOCENZBUNUJUPUQIUMUNUPUNUPJZUQUNUPOUQUKELZUO
      JZCKUNUOJZCKURUOCEPUTVACUSULTZAJUNULTZAJUTVAVBVCAULUSUNUKDEQRUAUSULASUNUL
      ASUBUCUNUOCUDUEUFUGABCDEUHUI $.
  $}

  ${
    $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.  For an alternate
       proof, see ~ ceqsalgALT .  (Contributed by NM, 29-Oct-2003.)  (Proof
       shortened by BJ, 29-Sep-2019.) $)
    ceqsalg $p |- ( A e. V -> ( A. x ( x = A -> ph ) <-> ps ) ) $=
      ( wnf cv wceq wb wi wal wcel ax-gen ceqsalt mp3an12 ) BCHCIDJZABKLZCMDENR
      ALCMBKFSCGOABCDEPQ $.

    $( Alternate proof of ~ ceqsalg , not using ~ ceqsalt .  (Contributed by
       NM, 29-Oct-2003.)  (Proof shortened by Andrew Salmon, 8-Jun-2011.)
       (Revised by BJ, 29-Sep-2019.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    ceqsalgALT $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.)  Avoid ~ df-clab .  (Revised by Wolf Lammen,
       23-Jan-2025.) $)
    ceqsal $p |- ( A. x ( x = A -> ph ) <-> ps ) $=
      ( cv wceq wi wal wex 19.23 pm5.74i albii isseti a1bi 3bitr4i ) CHDIZBJZCK
      SCLZBJSAJZCKBSBCEMUBTCSABGNOUABCDFPQR $.
    $( $j usage 'ceqsal' avoids 'df-clab'; $)

    $( A representation of explicit substitution of a class for a variable,
       inferred from an implicit substitution hypothesis.  Shorter proof uses
       ~ df-clab .  (Contributed by NM, 18-Aug-1993.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    ceqsalALT $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.)  Avoid ~ ax-12 .  (Revised by SN, 8-Sep-2024.) $)
    ceqsalv $p |- ( A. x ( x = A -> ph ) <-> ps ) $=
      ( cv wceq wi wal wex 19.23v pm5.74i albii isseti a1bi 3bitr4i ) CGDHZBIZC
      JRCKZBIRAIZCJBRBCLUASCRABFMNTBCDEOPQ $.
    $( $j usage 'ceqsalv' avoids 'ax-12'; $)
  $}

  ${
    $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.)  Avoid ~ ax-9 , ~ ax-12 , ~ ax-ext .  (Revised by SN,
       8-Sep-2024.) $)
    ceqsralv $p |- ( A e. B -> ( A. x e. B ( x = A -> ph ) <-> ps ) ) $=
      ( cv wceq wi wral wcel pm5.74i ralbii wrex r19.23v wb risset pm5.5 bitrid
      sylbi ) CGDHZAIZCEJUABIZCEJZDEKZBUBUCCEUAABFLMUDUACENZBIZUEBUABCEOUEUFUGB
      PCDEQUFBRTSS $.
    $( $j usage 'ceqsralv' avoids 'ax-9' 'ax-12' 'ax-ext'; $)
  $}

  ${
    $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 exdistrv sylibr syl
      2eximi biimprcd ancld 2eximdv syl5com impbid2 ) FHLZGILZMZCAMZENDNZBUMBDE
      CABKOPULCENDNZBUNULDQFRZEQGRZMZENDNZUOULUPDNZUQENZMUSUJUTUKVADFHSEGISTUPU
      QDEUAUBURCDEJUDUCBCUMDEBCACABKUEUFUGUHUI $.
  $}

  ${
    $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.)  Avoid ~ ax-10 , ~ ax-11 .  (Revised by GG,
       28-Jun-2024.)  Avoid ~ ax-9 , ~ ax-ext .  (Revised by Wolf Lammen,
       21-Mar-2025.) $)
    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 elisset anim12i 19.42vv 2exbii 3bitri sylibr
      exlimivv 19.41vv exdistrv anbi12i 2eximi biimprcd 2eximdv syl5com impbid2
      syl ancld ) HLPZIMPZQZJLPZKMPZQZQZCAQZGRFRZERDRZBVGBDEVFBFGCABOUAUHUHVECG
      RFRZERDRZBVHVEDSHTZESITZQZFSJTZGSKTZQZQZGRFRZERDRZVJVEVKDRZVLERZQZVNFRZVO
      GRZQZQZVSVAWBVDWEUSVTUTWADHLUBEIMUBUCVBWCVCWDFJLUBGKMUBUCUCVSVMVPGRFRZQZE
      RDRVMERDRZWGQWFVRWHDEVMVPFGUDUEVMWGDEUIWIWBWGWEVKVLDEUJVNVOFGUJUKUFUGVRVI
      DEVQCFGNULULUQBVIVGDEBCVFFGBCACABOUMURUNUNUOUP $.
    $( $j usage 'cgsex4g' avoids 'ax-9' 'ax-10' 'ax-11' 'ax-12' 'ax-ext'; $)
  $}

  ${
    $d x y z w v A $.  $d x y z w v B $.  $d x y z w v C $.  $d x y z w v D $.
    $d x y z w ps $.
    cgsex4gOLD.1 $e |- ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) )
                    -> ch ) $.
    cgsex4gOLD.2 $e |- ( ch -> ( ph <-> ps ) ) $.
    $( Obsolete version of ~ cgsex4g as of 21-Mar-2025.  (Contributed by NM,
       5-Aug-1995.)  Avoid ~ ax-10 , ~ ax-11 .  (Revised by GG, 28-Jun-2024.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    cgsex4gOLD $p |- ( ( ( A e. R /\ B e. S ) /\ ( C e. R /\ D e. S ) ) ->
                    ( E. x E. y E. z E. w ( ch /\ ph ) <-> ps ) ) $=
      ( vv wa wex cv wceq wcel biimpa exlimivv elisset anim12i exdistrv wal weq
      sylibr wn eqeq1 anbi2d anbi1d exbidv notbid alcomw notbii 2exnaln 3bitr4i
      exbii 4exdistrv bitri 2eximi syl biimprcd ancld 2eximdv syl5com impbid2 )
      HLUAZIMUAZQZJLUAZKMUAZQZQZCAQZGRFRZERDRZBVRBDEVQBFGCABOUBUCUCVPCGRFRZERDR
      ZBVSVPDSHTZESZITZQZFSZJTZGSKTZQZQZGRZFRZERZDRZWAVPWEERDRZWIGRFRZQZWNVLWOV
      OWPVLWBDRZWDERZQWOVJWRVKWSDHLUDEIMUDUEWBWDDEUFUIVOWGFRZWHGRZQWPVMWTVNXAFJ
      LUDGKMUDUEWGWHFGUFUIUEWNWKERFRZDRWQWMXBDWKUJZFUGEUGZUJXCEUGFUGZUJWMXBXDXE
      XCWBPSZITZQZWIQZGRZUJWEXFJTZWHQZQZGRZUJEFPPEPUHZWKXJXOWJXIGXOWEXHWIXOWDXG
      WBWCXFIUKULUMUNUOFPUHZWKXNXPWJXMGXPWIXLWEXPWGXKWHWFXFJUKUMULUNUOUPUQWKEFU
      RWKFEURUSUTWEWIDEFGVAVBUIWLVTDEWJCFGNVCVCVDBVTVRDEBCVQFGBCACABOVEVFVGVGVH
      VI $.
    $( $j usage 'cgsex4gOLD' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $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.)  (Proof shortened by Wolf Lammen, 22-Jan-2025.) $)
    ceqsex $p |- ( E. x ( x = A /\ ph ) <-> ps ) $=
      ( cv wceq wa wex wn wi wal alinexa nfn notbid ceqsal bitr3i con4bii ) CHD
      IZAJCKZBUBLUAALZMCNBLZUAACOUCUDCDBCEPFUAABGQRST $.
    $( $j usage 'ceqsex' avoids 'df-clab'; $)

    $( Obsolete version of ~ ceqsex as of 22-Jan-2025.  (Contributed by NM,
       2-Mar-1995.)  (Revised by Mario Carneiro, 10-Oct-2016.)
       (New usage is discouraged.)  (Proof modification is discouraged.) $)
    ceqsexOLD $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.)  Avoid ~ ax-12 .  (Revised by GG,
       12-Oct-2024.)  (Proof shortened by Wolf Lammen, 22-Jan-2025.) $)
    ceqsexv $p |- ( E. x ( x = A /\ ph ) <-> ps ) $=
      ( cv wceq wa wex wn wi wal alinexa notbid ceqsalv bitr3i con4bii ) CGDHZA
      ICJZBTKSAKZLCMBKZSACNUAUBCDESABFOPQR $.
    $( $j usage 'ceqsexv' avoids 'ax-10' 'ax-11' 'ax-12' 'df-clab'; $)
  $}

  ${
    $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 A $.  $d x ps $.
    ceqsexv2dOLD.1 $e |- A e. _V $.
    ceqsexv2dOLD.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
    ceqsexv2dOLD.3 $e |- ps $.
    $( Obsolete version of ~ ceqsexv2d as of 5-Jun-2025.  (Contributed by
       Thierry Arnoux, 10-Sep-2016.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    ceqsexv2dOLD $p |- E. x ph $=
      ( cv wceq wa wex ceqsexv biimpri exsimpr mp2b ) BCHDIZAJCKZACKGQBABCDEFLM
      PACNO $.
  $}

  ${
    $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.)  Avoid ~ ax-10 and ~ ax-11 .
       (Revised by GG, 20-Aug-2023.) $)
    ceqsex2v $p |- ( E. x E. y ( x = A /\ y = B /\ ph ) <-> ch ) $=
      ( cv wceq w3a wex wa 3anass exbii 19.42v ceqsexv anbi2d exbidv 3bitri
      bitri ) DLFMZELGMZANZEOZDOUEUFAPZEOZPZDOUFBPZEOZCUHUKDUHUEUIPZEOUKUGUNEUE
      UFAQRUEUIESUDRUJUMDFHUEUIULEUEABUFJUAUBTBCEGIKTUC $.
    $( $j usage 'ceqsex2v' avoids 'ax-10' 'ax-11'; $)
  $}

  ${
    $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 3bitri ) EQHRZFQIRZGQJRZ
      UIZASZGTFTZETUQURUSAUIZGTFTZSZETURUSBUIZGTFTZDVBVEEVBUQVCSZGTFTVEVAVHFGUQ
      URUSSZSZASUQVIASZSVAVHUQVIAUAUTVJAUQURUSUBUCVCVKUQURUSAUDUEUFUGUQVCFGUHUJ
      UKVDVGEHKUQVCVFFGUQABURUSNULUMUNBCDFGIJLMOPUOUP $.
  $}

  ${
    $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
      3bitri ) HULNUMZIULOUMZJULPUMZUNZKULQUMLULRUMMULSUMUNZAUNZMUOLUOKUOZJUOIU
      OHUOVHVIAUPZMUOLUOKUOZUPZJUOIUOHUOVIDUPZMUOLUOKUOZGVKVNHIJVKVHVLUPZMUOLUO
      KUOVNVJVQKLMVHVIAUQURVHVLKLMUSUTURVMVIBUPZMUOLUOKUOVICUPZMUOLUOKUOVPHIJNO
      PTUAUBVEVLVRKLMVEABVIUFVAVBVFVRVSKLMVFBCVIUGVAVBVGVSVOKLMVGCDVIUHVAVBVCDE
      FGKLMQRSUCUDUEUIUJUKVCVD $.
  $}

  ${
    $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 19.42vv 2exbii bitri 3anass df-3an anbi2i bitr4i 3bitr4i
      wex 3anbi3d 4exbidv ceqsex4v 3bitri ) JVBQVCZKVBRVCZVDZLVBSVCZMVBTVCZVDZV
      DZNVBUAVCOVBUBVCVDZPVBUCVCUEVBUDVCVDZVDZAVEZUEVNPVNZOVNNVNZMVNLVNZKVNJVNW
      AWDWFWGAVEZUEVNPVNZOVNNVNZVEZMVNLVNZKVNJVNWFWGEVEZUEVNPVNOVNNVNZIWLWQJKWK
      WPLMWEWMVDZUEVNPVNZOVNNVNZWEWOVDZWKWPXBWEWNVDZOVNNVNXCXAXDNOWEWMPUEVFVGWE
      WNNOVFVHWJXANOWIWTPUEWIWEWHAVDZVDWTWEWHAVIWMXEWEWFWGAVJVKVLVGVGWAWDWOVJVM
      VGVGWOWFWGBVEZUEVNPVNOVNNVNWFWGCVEZUEVNPVNOVNNVNWFWGDVEZUEVNPVNOVNNVNWSJK
      LMQRSTUFUGUHUIVSWMXFNOPUEVSABWFWGUNVOVPVTXFXGNOPUEVTBCWFWGUOVOVPWBXGXHNOP
      UEWBCDWFWGUPVOVPWCXHWRNOPUEWCDEWFWGUQVOVPVQEFGHINOPUEUAUBUCUDUJUKULUMURUS
      UTVAVQVR $.
  $}

  ${
    $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.) $)
    gencbval $p |- ( A. x ( ch -> ph ) <-> A. y ( th -> ps ) ) $=
      ( wi wal wn wa wex cv wceq notbid exanali gencbvex 3bitr3i con4bii ) CALE
      MZDBLFMZCANZOEPDBNZOFPUDNUENUFUGCDEFGHGFQRABISJKUACAETDBFTUBUC $.
  $}

  ${
    $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.)  (Proof shortened by Wolf Lammen, 25-Jan-2025.) $)
    sbhypf $p |- ( y = A -> ( [ y / x ] ph <-> ps ) ) $=
      ( cv wceq wsb wb sbimi eqsb1 sbf sblbis 3imtr3i ) CHEIZCDJABKZCDJDHEIACDJ
      BKQRCDGLCDEMBBACDBCDFNOP $.

    $( Obsolete version of ~ sbhypf as of 25-Jan-2025.  (Contributed by Raph
       Levien, 10-Apr-2004.)  (New usage is discouraged.)
       (Proof modification is discouraged.) $)
    sbhypfOLD $p |- ( y = A -> ( [ y / x ] ph <-> ps ) ) $=
      ( cv wceq wa wex wsb wb eqeq1 equsexvw nfs1v nfbi sbequ12 bicomd sylan9bb
      exlimi sylbir ) DHZEIZCHZUCIZUEEIZJZCKACDLZBMZUGUDCDUEUCENOUHUJCUIBCACDPF
      QUFUIAUGBUFAUIACDRSGTUAUB $.
  $}

  ${
    $d A y $.  $d V y $.  $d x y $.  $d ph y $.  $d ps y $.
    $( Closed theorem form of ~ spcimgf .  (Contributed by Wolf Lammen,
       28-Jul-2025.) $)
    spcimgft $p |- ( ( ( F/_ x A /\ F/ x ps )
                  /\ A. x ( x = A -> ( ph -> ps ) ) )
                  -> ( A e. V -> ( A. x ph -> ps ) ) ) $=
      ( vy wnfc wnf cv wceq wi wal wex wa elissetv cbvexeqsetf imbitrrid pm2.04
      wcel al2imi 19.23t biimpd sylan9r com23 sylan9 anassrs ) CDGZBCHZCIDJZABK
      KZCLZDESZACLZBKZKUGULUICMZUHUKNZUNULUOUGFIDJFMFDEOCFDPQUPUMUOBUKUMUIBKZCL
      ZUHUOBKZUJAUQCUIABRTUHURUSUIBCUAUBUCUDUEUF $.
    $( $j usage 'spcimgft' avoids 'df-clab'; $)
  $}

  ${
    spcimgfi1.1 $e |- F/ x ps $.
    spcimgfi1.2 $e |- F/_ x A $.
    $( A closed version of ~ spcimgf .  (Contributed by Mario Carneiro,
       4-Jan-2017.)  (Proof shortened by Wolf Lammen, 27-Jul-2025.) $)
    spcimgfi1 $p |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. B ->
                      ( A. x ph -> ps ) ) ) $=
      ( wnfc wnf cv wceq wi wal wcel wa spcimgft ex mp2an ) CDHZBCIZCJDKABLLCMZ
      DENACMBLLZLGFSTOUAUBABCDEPQR $.

    $( Obsolete version of ~ spcimgfi1 as of 27-Jul-2025.  (Contributed by
       Mario Carneiro, 4-Jan-2017.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    spcimgfi1OLD $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 imbitrdi
      syl5 ) DEHDIHZCJDKZABLZLCMZACMBLZDENUEUBUDCOZUFUBUCCOUEUGCDGPUCUDCQRABCFS
      TUA $.

    $( 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 spcimgfi1 syl ) CHDIZABJZKZCL
      SABKZKZCLDEMACLBKKUAUCCTUBSABNOPABCDEFGQR $.
  $}

  ${
    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 spcimgfi1 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 ) ) $=
      ( wcel wn wal wex nfn cv wceq con3d spcimgf con2d df-ex imbitrrdi ) DEIZB
      AJZCKZJACLUAUCBUBBJCDEFBCGMCNDOBAHPQRACST $.
  $}

  ${
    $( Closed theorem form of ~ vtoclgf .  The reverse implication is proven in
       ~ ceqsal1t .  See ~ ceqsalt for a version with ` x ` and ` A ` disjoint.
       (Contributed by NM, 17-Feb-2013.)  (Revised by Mario Carneiro,
       12-Oct-2016.)  (Proof shortened by JJ, 11-Aug-2021.)  Avoid ~ ax-13 .
       (Revised by GG, 6-Oct-2023.) $)
    vtoclgft $p |- ( ( ( F/_ x A /\ F/ x ps )
                  /\ ( A. x ( x = A -> ( ph <-> ps ) )
                     /\ A. x ph ) /\ A e. V ) -> ps ) $=
      ( wnfc wnf wa cv wceq wb wi wcel biimp imim2i alimi spcimgft sylan2 com23
      wal impr 3impia ) CDFBCGHZCIDJZABKZLZCTZACTZHDEMZBUCUGUHUIBLUCUGHUIUHBUGU
      CUDABLZLZCTUIUHBLLUFUKCUEUJUDABNOPABCDEQRSUAUB $.
    $( $j usage 'vtoclgft' avoids 'ax-13' 'df-clab'; $)
  $}

  ${
    $d x A $.  $d x ph $.
    vtocleg.1 $e |- ( x = A -> ph ) $.
    $( Implicit substitution of a class for a setvar variable.  (Contributed by
       NM, 21-Jun-1993.) $)
    vtocleg $p |- ( A e. V -> ph ) $=
      ( wcel cv wceq wex elisset exlimiv syl ) CDFBGCHZBIABCDJMABEKL $.
  $}

  ${
    $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.)  Avoid ~ ax-12 .  (Revised by SN,
       20-Apr-2024.)  (Proof shortened by Wolf Lammen, 26-Jan-2025.) $)
    vtoclg $p |- ( A e. V -> ps ) $=
      ( cv wceq mpbii vtocleg ) BCDECHDIABGFJK $.
  $}

  ${
    $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.)  Avoid ~ df-clab .  (Revised by Wolf Lammen,
       31-May-2025.) $)
    vtocle $p |- ph $=
      ( cv wceq isseti exlimiiv ) BFCGABEBCDHI $.
    $( $j usage 'vtocle' avoids 'df-clab' 'df-cleq'; $)

    $( Obsolete version of ~ vtocle as of 31-May-2025.  (Contributed by NM,
       9-Sep-1993.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    vtocleOLD $p |- ph $=
      ( cvv wcel vtocleg ax-mp ) CFGADABCFEHI $.
  $}

  ${
    $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 $.
  $}

  ${
    $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.)  Remove dependency on ~ ax-10 .  (Revised by BJ,
       29-Nov-2020.)  (Proof shortened by SN, 20-Apr-2024.)  (Proof shortened
       by Wolf Lammen, 20-Jun-2025.) $)
    vtocl $p |- ps $=
      ( cv wceq mpbii vtocle ) BCDECHDIABGFJK $.
    $( $j usage 'vtocl' avoids 'ax-10' 'df-clab' 'df-cleq'; $)

    $( Obsolete version of ~ vtocl as of 20-Jun-2025.  (Contributed by NM,
       30-Aug-1993.)  Remove dependency on ~ ax-10 .  (Revised by BJ,
       29-Nov-2020.)  (Proof shortened by SN, 20-Apr-2024.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    vtoclOLD $p |- ps $=
      ( cv wceq mpbii isseti exlimiiv ) CHDIZBCMABGFJCDEKL $.
    $( $j usage 'vtoclOLD' avoids 'df-clab' 'df-cleq'; $)
  $}

  ${
    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.)  Avoid ~ ax-10 , ~ ax-11 , ~ ax-12 .
       (Revised by SN, 2-Sep-2024.) $)
    vtocld $p |- ( ph -> ch ) $=
      ( cv wceq wcel wex elisset syl wa adantr mpbid exlimddv ) ADJEKZCDAEFLTDM
      GDEFNOATPBCABTIQHRS $.
    $( $j usage 'vtocld' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $d A x $.  $d B x y $.  $d ch x y $.  $d ph x y $.
    vtocl2d.a $e |- ( ph -> A e. V ) $.
    vtocl2d.b $e |- ( ph -> B e. W ) $.
    vtocl2d.1 $e |- ( ( x = A /\ y = B ) -> ( ps <-> ch ) ) $.
    vtocl2d.3 $e |- ( ph -> ps ) $.
    $( Implicit substitution of two classes for two setvar variables.
       (Contributed by Thierry Arnoux, 25-Aug-2020.)  (Revised by BTernaryTau,
       19-Oct-2023.) $)
    vtocl2d $p |- ( ph -> ch ) $=
      ( cv wceq wa adantr wi wb vtocld adantll pm5.74da a1d imp 2thd ) ABCEGIKA
      ENGOZPBCABUFMQAUFCAUFBRUFCRDFHJADNFOZPUFBCUGUFBCSALUAUBABUFMUCTUDUEMT $.
  $}

  ${
    $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 $.
    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.)  (Proof
       shortened by Wolf Lammen, 26-Jan-2025.) $)
    vtoclf $p |- ps $=
      ( cv wceq mpbii vtoclef ) BCDEFCIDJABHGKL $.

    $( Obsolete version of ~ vtoclf as of 26-Jan-2025.  (Contributed by NM,
       30-Aug-1993.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    vtoclfOLD $p |- ps $=
      ( cv wceq wi isseti biimpd eximii 19.36i mpg ) ABCABCECIDJZABKCCDFLQABGMN
      OHP $.
  $}

  ${
    $d x 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 $=
      ( cv wceq a1i wi pm5.74da vtocl 2thd ) ABDFHDKFLZABARJMZRANRBNCEGCKELRABI
      OSPQJP $.
    $( $j usage 'vtocl2' avoids 'df-clab'; $)
  $}

  ${
    $d x A $.  $d x y 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.)  Avoid
       ~ ax-10 and ~ ax-11 .  (Revised by GG, 20-Aug-2023.)  (Proof shortened
       by Wolf Lammen, 23-Aug-2023.) $)
    vtocl3 $p |- ps $=
      ( cv wceq a1i wi wa wb 3expa pm5.74da vtocl2 2thd vtocl ) ABEHKENHOZABAUE
      MPZUEAQUEBQCDFGIJCNFOZDNGOZRUEABUGUHUEABSLTUAUFUBUCMUD $.
    $( $j usage 'vtocl3' avoids 'ax-10' 'ax-11' 'ax-12' 'df-clab'; $)
  $}

  ${
    $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 $.
    $( $j usage 'vtoclb' avoids 'df-clab' 'df-cleq'; $)
  $}

  ${
    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 disjoint 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-10 and
       ~ ax-11 .  (Contributed by BJ, 1-May-2019.) $)
    vtoclg1f $p |- ( A e. V -> ps ) $=
      ( wcel cv wceq wex elisset mpbii exlimi syl ) DEICJDKZCLBCDEMQBCFQABHGNOP
      $.
  $}

  ${
    $d x A $.  $d x ps $.
    vtoclgOLD.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    vtoclgOLD.2 $e |- ph $.
    $( Obsolete version of ~ vtoclg as of 26-Jan-2025.  (Contributed by NM,
       17-Apr-1995.)  Avoid ~ ax-12 .  (Revised by SN, 20-Apr-2024.)
       (New usage is discouraged.)  (Proof modification is discouraged.) $)
    vtoclgOLD $p |- ( A e. V -> ps ) $=
      ( wcel cv wceq wex elisset mpbii exlimiv syl ) DEHCIDJZCKBCDELPBCPABGFMNO
      $.
  $}

  ${
    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.)  Remove dependency on ~ ax-10 , ~ ax-11 , and
       ~ ax-13 .  (Revised by Steven Nguyen, 29-Nov-2022.) $)
    vtocl2g $p |- ( ( A e. V /\ B e. W ) -> ch ) $=
      ( wcel cvv elex wi cv wceq imbi2d vtoclg mpan9 ) FHMFNMZGIMCFHOUBBPUBCPEG
      IEQGRBCUBKSABDFNJLTTUA $.
  $}

  ${
    $d x A $.  $d y A $.  $d z A $.  $d y B $.  $d z B $.  $d z C $.
    $d x ps $.  $d y ch $.  $d z th $.
    vtocl3g.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    vtocl3g.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
    vtocl3g.3 $e |- ( z = C -> ( ch <-> th ) ) $.
    vtocl3g.4 $e |- ph $.
    $( Implicit substitution of a class for a setvar variable.  Version of
       ~ vtocl3gf with disjoint variable conditions instead of nonfreeness
       hypotheses, requiring fewer axioms.  (Contributed by GG, 3-Oct-2024.) $)
    vtocl3g $p |- ( ( A e. V /\ B e. W /\ C e. X ) -> th ) $=
      ( wcel cvv wi wa elex cv wceq imbi2d vtoclg vtocl2g mpan9 3impb ) HKRZILR
      ZJMRZDUJHSRZUKULUADHKUBUMBTUMCTUMDTFGIJLMFUCIUDBCUMOUEGUCJUDCDUMPUEABEHSN
      QUFUGUHUI $.
    $( $j usage 'vtocl3g' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $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.)  Avoid ~ ax-10 and ~ ax-11 .  (Revised by GG,
       20-Aug-2023.) $)
    vtoclga $p |- ( A e. B -> ps ) $=
      ( wcel cv wi wceq eleq1 imbi12d vtoclg pm2.43i ) DEHZBCIZEHZAJPBJCDEQDKRP
      ABQDELFMGNO $.
    $( $j usage 'vtoclga' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $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.)  Avoid ~ ax-10 and ~ ax-11 .  (Revised by GG,
       20-Aug-2023.)  (Proof shortened by Wolf Lammen, 23-Aug-2023.) $)
    vtocl2ga $p |- ( ( A e. C /\ B e. D ) -> ch ) $=
      ( wcel wi cv wceq imbi2d ex vtoclga com12 impcom ) GIMFHMZCUBBNUBCNEGIEOZ
      GPBCUBKQUBUCIMZBUDANUDBNDFHDOZFPABUDJQUEHMUDALRSTSUA $.
    $( $j usage 'vtocl2ga' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $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.)  (Proof shortened by Wolf Lammen, 31-May-2025.) $)
    vtocl2gaf $p |- ( ( A e. C /\ B e. D ) -> ch ) $=
      ( wcel wi nfim nfel1 cv wceq imbi2d nfv ex vtoclgaf com12 impcom ) GIRFHR
      ZCUJBSUJCSEGILUJCEEFHKUANTEUBZGUCBCUJPUDUJUKIRZBULASULBSDFHJULBDULDUEMTDU
      BZFUCABULOUDUMHRULAQUFUGUHUGUI $.

    $( Obsolete version of ~ vtocl2gaf as of 31-May-2025.  (Contributed by NM,
       10-Aug-2013.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    vtocl2gafOLD $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 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.)  (Proof
       shortened by Wolf Lammen, 31-May-2025.) $)
    vtocl3gaf $p |- ( ( A e. R /\ B e. S /\ C e. T ) -> th ) $=
      ( wcel wa wi nfel1 nfan nfim cv wceq imbi2d nfv 3expia vtocl2gaf vtoclgaf
      com12 impcom 3impa ) HKUGZILUGZJMUGZDVEVCVDUHZDVFCUIVFDUIGJMSVFDGVCVDGGHK
      PUJGILRUJUKUBULGUMZJUNCDVFUEUOVFVGMUGZCVHAUIVHBUIVHCUIEFHIKLNOQVHBEVHEUPT
      ULVHCFVHFUPUAULEUMZHUNABVHUCUOFUMZIUNBCVHUDUOVIKUGVJLUGVHAUFUQURUTUSVAVB
      $.

    $( Obsolete version of ~ vtocl3gaf as of 31-May-2025.  (Contributed by NM,
       10-Aug-2013.)  (Revised by Mario Carneiro, 11-Oct-2016.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    vtocl3gafOLD $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.)  Reduce axiom usage.  (Revised by GG, 3-Oct-2024.)
       (Proof shortened by Wolf Lammen, 31-May-2025.) $)
    vtocl3ga $p |- ( ( A e. D /\ B e. R /\ C e. S ) -> th ) $=
      ( wcel wi cv wa wceq imbi2d 3expia vtocl2ga com12 vtoclga impcom 3impa )
      HKRZILRZJMRZDULUJUKUAZDUMCSUMDSGJMGTZJUBCDUMPUCUMUNMRZCUOASUOBSUOCSEFHIKL
      ETZHUBABUONUCFTZIUBBCUOOUCUPKRUQLRUOAQUDUEUFUGUHUI $.
    $( $j usage 'vtocl3ga' avoids 'ax-10' 'ax-11' 'ax-12'; $)

    $( Obsolete version of ~ vtocl3ga as of 31-May-2025.  (Contributed by NM,
       20-Aug-1995.)  Reduce axiom usage.  (Revised by GG, 3-Oct-2024.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    vtocl3gaOLD $p |- ( ( A e. D /\ B e. R /\ C e. S ) -> th ) $=
      ( wcel w3a wi wceq eleq1 3anbi1d imbi12d 3anbi2d 3anbi3d vtocl3g pm2.43i
      cv ) HKRZILRZJMRZSZDEUIZKRZFUIZLRZGUIZMRZSZATUJUQUSSZBTUJUKUSSZCTUMDTEFGH
      IJKLMUNHUAZUTVAABVCUOUJUQUSUNHKUBUCNUDUPIUAZVAVBBCVDUQUKUJUSUPILUBUEOUDUR
      JUAZVBUMCDVEUSULUJUKURJMUBUFPUDQUGUH $.
  $}

  ${
    $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
      $.

    $( Obsolete version of ~ vtocl4ga as of 31-May-2025.  (Contributed by AV,
       22-Jan-2019.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    vtocl4gaOLD $p |- ( ( ( A e. Q /\ B e. R )
                       /\ ( C e. S /\ D e. T ) ) -> th ) $=
      ( wcel wa cv wi wceq eleq1 anbi1d imbi12d anbi2d vtocl4g pm2.43i ) JNUCZK
      OUCZUDZLPUCZMQUCZUDZUDZDFUEZNUCZGUEZOUCZUDZHUEZPUCZIUEZQUCZUDZUDZAUFUNVDU
      DZVJUDZBUFUPVJUDZCUFUTDUFUPUQVIUDZUDZEUFFGHIJKLMNOPQVAJUGZVKVMABVQVEVLVJV
      QVBUNVDVAJNUHUIUIRUJVCKUGZVMVNBCVRVLUPVJVRVDUOUNVCKOUHUKUISUJVFLUGZVNVPCE
      VSVJVOUPVSVGUQVIVFLPUHUIUKTUJVHMUGZVPUTEDVTVOUSUPVTVIURUQVHMQUHUKUKUAUJUB
      ULUM $.
  $}

  ${
    $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.)  (Proof shortened by Wolf Lammen,
       26-Jan-2025.) $)
    vtoclegft $p |- ( ( A e. B /\ F/ x ph /\
                   A. x ( x = A -> ph ) ) -> ph ) $=
      ( wcel wnf cv wceq wi wal wa wb biidd ax-gen ceqsalt mp3an2 ancoms biimpd
      3impia ) CDEZABFZBGCHZAIBJZATUAKUCAUATUCALZUAUBAALIZBJTUDUEBUBAMNAABCDOPQ
      RS $.

    $( Obsolete version of ~ vtoclegft as of 26-Jan-2025.  (Contributed by NM,
       7-Nov-2005.)  (Revised by Mario Carneiro, 11-Oct-2016.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    vtoclegftOLD $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 $.  $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 $.
  $}

  ${
    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 ) ) $=
      ( wcel wn wal wex nfn cv wceq notbid spcgf con2d df-ex imbitrrdi ) DEIZBA
      JZCKZJACLUAUCBUBBJCDEFBCGMCNDOABHPQRACST $.
  $}

  ${
    $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.)  Avoid ~ ax-10 and ~ ax-11 .  (Revised
         by GG, 20-Aug-2023.) $)
      spcimdv $p |- ( ph -> ( A. x ps -> ch ) ) $=
        ( wi wex wal cv wceq wcel elisset syl ex eximdv mpd 19.36v sylib ) ABCI
        ZDJZBDKCIADLEMZDJZUCAEFNUEGDEFOPAUDUBDAUDUBHQRSBCDTUA $.
      $( $j usage 'spcimdv' avoids 'ax-10' 'ax-11'; $)
    $}

    ${
      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 ) ) $=
      ( wn wal wex cv wceq wa con3d spcimdv con2d df-ex imbitrrdi ) ACBIZDJZIBD
      KAUACATCIDEFGADLEMNCBHOPQBDRS $.
  $}

  ${
    $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.)  Avoid
       ~ ax-10 , ~ ax-11 .  (Revised by Wolf Lammen, 25-Aug-2023.) $)
    spcgv $p |- ( A e. V -> ( A. x ph -> ps ) ) $=
      ( wcel cvv wal wi elex id cv wceq wb adantl spcdv syl ) DEGDHGZACIBJDEKSA
      BCDHSLCMDNABOSFPQR $.

    $( Existential specialization, using implicit substitution.  (Contributed
       by NM, 14-Aug-1994.)  Avoid ~ ax-10 , ~ ax-11 .  (Revised by Wolf
       Lammen, 25-Aug-2023.) $)
    spcegv $p |- ( A e. V -> ( ps -> E. x ph ) ) $=
      ( wcel cv wceq wex elisset biimprcd eximdv syl5com ) DEGCHDIZCJBACJCDEKBO
      ACOABFLMN $.
  $}

  ${
    $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.)  (Revised by AV,
       16-Aug-2024.) $)
    spcedv $p |- ( ph -> E. x ps ) $=
      ( wcel wex spcegv sylc ) AFEJCBDKGHBCDFEILM $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y ps $.
    spc2egv.1 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
    $( Existential specialization with two 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 exdistrv sylibr biimprcd 2eximdv
      syl5com ) EGJZFHJZKZCLEMZDLFMZKZDNCNZBADNCNUDUECNZUFDNZKUHUBUIUCUJCEGODFH
      OPUEUFCDQRBUGACDUGABISTUA $.

    $( Specialization with two 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 wal wn wex cv wceq notbid spc2egv 2nalexn imbitrrdi con4d ) EGJ
      FHJKZBADLCLZUBBMZAMZDNCNUCMUEUDCDEFGHCOEPDOFPKABIQRACDSTUA $.
  $}

  ${
    $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 exdistrv sylibr anass ancom anbi1i
      wi anim12i bitr3i biimparc sylbir ex eximd impancom sylan2 ) FHMZGIMZNZAD
      OFPZEOGPZNZEQZDQZCBEQZDQZUFUQURDQZUSEQZNVBUOVEUPVFDFHREGIRUGURUSDEUAUBACV
      BVDACNZVAVCDACDADSJTVGUTBEACEAESKTVGUTBVGUTNZCAUTNZNZBVJCANZUTNVHCAUTUCVK
      VGUTCAUDUEUHVIBCLUIUJUKULULUMUN $.

    $( Specialization with 2 quantifiers, using implicit substitution.
       (Contributed by Thierry Arnoux, 23-Aug-2017.) $)
    spc2d $p |- ( ( ph /\ ( A e. V /\ B e. W ) ) -> ( A. x A. y ps -> ch ) ) $=
      ( wal wn wex wcel wa nfn cv wceq 2nalexn con1bii notbid spc2ed biimtrrid
      con1d ) BEMDMZBNZEODOZNAFHPGIPQQZCUGUIBDEUAUBUJCUIAUHCNDEFGHICDJRCEKRADSF
      TESGTQQBCLUCUDUFUE $.
  $}

  ${
    $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 three quantifiers, using implicit
       substitution.  (Contributed by NM, 12-May-2008.)  Avoid ~ ax-10 and
       ~ ax-11 .  (Revised by GG, 20-Aug-2023.)  (Proof shortened by Wolf
       Lammen, 25-Aug-2023.) $)
    spc3egv $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
               ( ps -> E. x E. y E. z ph ) ) $=
      ( wcel cvv wex wi elex cv wceq wa w3a simp1 wb 3coml 3expa spc2egv 19.37v
      pm5.74da exbii bitri imbitrdi pm2.86d 3adant1 imp spcimedv syl3an ) FIMFN
      MZGJMGNMZHKMHNMZBAEOZDOZCOPFIQGJQHKQUQURUSUAZVABCFNUQURUSUBVBCRFSZBVAPZUR
      USVCVDPUQURUSTZVCBVAVEVCBPZVCAPZEOZDOZVCVAPZVGVFDEGHNNDRGSZERHSZTVCABVKVL
      VCABUCZVCVKVLVMLUDUEUHUFVIVCUTPZDOVJVHVNDVCAEUGUIVCUTDUGUJUKULUMUNUOUP $.
    $( $j usage 'spc3egv' avoids 'ax-10' 'ax-11'; $)

    $( Specialization with three 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 wal wn wex cv wceq exnal notbid spc3egv exbii bitr2i imbitrrdi
      bitri con4d ) FIMGJMHKMNZBAEOZDOZCOZUHBPZAPZEQZDQZCQZUKPZUMULCDEFGHIJKCRF
      SDRGSERHSNABLUAUBUPUJPZCQUQUOURCUOUIPZDQURUNUSDAETUCUIDTUFUCUJCTUDUEUG $.
  $}

  ${
    $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 $.
    rspcdf.1 $e |- F/ x ph $.
    rspcdf.2 $e |- F/ x ch $.
    rspcdf.3 $e |- ( ph -> A e. B ) $.
    rspcdf.4 $e |- ( ( ph /\ x = A ) -> ( ps <-> ch ) ) $.
    $( Restricted specialization, using implicit substitution.  (Contributed by
       Emmett Weisz, 16-Jan-2020.) $)
    rspcdf $p |- ( ph -> ( A. x e. B ps -> ch ) ) $=
      ( cv wceq wb wi wal wcel wral ex alrimi rspct sylc ) ADKELZBCMZNZDOEFPBDF
      QCNAUDDGAUBUCJRSIBCDEFHTUA $.
  $}

  ${
    $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 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 ) ) $=
      ( wn wral wrex cv wceq wa con3d rspcimdv con2d dfrex2 imbitrrdi ) ACBIZDF
      JZIBDFKAUACATCIDEFGADLEMNCBHOPQBDFRS $.
  $}

  ${
    $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 $.

    rspcebdv.1 $e |- ( ( ph /\ ps ) -> x = A ) $.
    $( Restricted existential specialization, using implicit substitution in
       both directions.  (Contributed by AV, 8-Jan-2022.) $)
    rspcebdv $p |- ( ph -> ( E. x e. B ps <-> ch ) ) $=
      ( wrex wi wa cv wceq wb syldan biimpd expcom pm2.43b rexlimdvw rspcedv
      impbid ) ABDFJCABCDFABCABBCKABLBCABDMENBCOIHPQRSTABCDEFGHUAUB $.
  $}

  ${
    $d A x $.  $d B x $.  $d ch x $.  $d ph x $.
    rspcdv2.1 $e |- ( ( ph /\ x = A ) -> ( ps <-> ch ) ) $.
    rspcdv2.2 $e |- ( ph -> A e. B ) $.
    rspcdv2.3 $e |- ( ph -> A. x e. B ps ) $.
    $( Restricted specialization, using implicit substitution.  (Contributed by
       Stanislas Polu, 9-Mar-2020.) $)
    rspcdv2 $p |- ( ph -> ch ) $=
      ( wral rspcdv mpd ) ABDFJCIABCDEFHGKL $.
  $}

  ${
    $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.)  Drop ~ ax-10 , ~ ax-11 , ~ ax-12 .  (Revised by SN,
       12-Dec-2023.) $)
    rspcv $p |- ( A e. B -> ( A. x e. B ph -> ps ) ) $=
      ( wcel id cv wceq wb adantl rspcdv ) DEGZABCDENHCIDJABKNFLM $.

    $( 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.)  Drop ~ ax-10 , ~ ax-11 , ~ ax-12 .
       (Revised by SN, 12-Dec-2023.) $)
    rspcev $p |- ( ( A e. B /\ ps ) -> E. x e. B ph ) $=
      ( wcel wrex id cv wceq wb adantl rspcedv imp ) DEGZBACEHPABCDEPICJDKABLPF
      MNO $.
  $}

  ${
    $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 A x $.  $d B x $.  $d ch x $.  $d ph x $.  $d th x $.
    rspceb2dv.1 $e |- ( ( ph /\ x e. B ) -> ( ps -> ch ) ) $.
    rspceb2dv.2 $e |- ( ( ph /\ ch ) -> A e. B ) $.
    rspceb2dv.3 $e |- ( ( ph /\ ch ) -> th ) $.
    rspceb2dv.4 $e |- ( x = A -> ( ps <-> th ) ) $.
    $( Restricted existential specialization, using implicit substitution in
       both directions.  (Contributed by Zhi Wang, 28-Sep-2024.) $)
    rspceb2dv $p |- ( ph -> ( E. x e. B ps <-> ch ) ) $=
      ( wrex rexlimdva wa wcel rspcev syl2anc ex impbid ) ABEGLZCABCEGHMACTACNF
      GODTIJBDEFGKPQRS $.
  $}

  ${
    $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 ) ) $.
    $( Restricted specialization with two quantifiers, 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 nfralw cv wceq rspc ralbidv sylan9 ) FHNAEIOZDHOCEIOZGIN
      BUCUDDFHCDEIDIPJQDRFSACEILUATCBEGIKMTUB $.
  $}

  ${
    $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 ) ) $=
      ( wcel wral cv wceq ralbidv rspcv sylan9 ) FHLAEIMZDHMCEIMZGILBSTDFHDNFOA
      CEIJPQCBEGIKQR $.

    $( 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 A x y $.  $d B y $.  $d X x $.  $d Y x y $.  $d ch x $.  $d th y $.
    2rspcedvdw.1 $e |- ( x = A -> ( ps <-> ch ) ) $.
    2rspcedvdw.2 $e |- ( y = B -> ( ch <-> th ) ) $.
    2rspcedvdw.a $e |- ( ph -> A e. X ) $.
    2rspcedvdw.b $e |- ( ph -> B e. Y ) $.
    2rspcedvdw.3 $e |- ( ph -> th ) $.
    $( Double application of ~ rspcedvdw .  (Contributed by SN,
       24-Aug-2024.) $)
    2rspcedvdw $p |- ( ph -> E. x e. X E. y e. Y ps ) $=
      ( wcel wrex rspc2ev syl3anc ) AGIPHJPDBFJQEIQMNOBDCEFGHIJKLRS $.
  $}

  ${
    $d x y A $.  $d y B $.  $d x C $.  $d x y D $.  $d y ch $.  $d x th $.
    rspc2dv.1 $e |- ( x = A -> ( ps <-> th ) ) $.
    rspc2dv.2 $e |- ( y = B -> ( th <-> ch ) ) $.
    rspc2dv.3 $e |- ( ph -> A. x e. C A. y e. D ps ) $.
    rspc2dv.4 $e |- ( ph -> A e. C ) $.
    rspc2dv.5 $e |- ( ph -> B e. D ) $.
    $( 2-variable restricted specialization, using implicit substitution.
       (Contributed by Scott Fenton, 6-Mar-2025.) $)
    rspc2dv $p |- ( ph -> ch ) $=
      ( wcel wral rspc2va syl21anc ) AGIPHJPBFJQEIQCNOMBCDEFGHIJKLRS $.
  $}

  ${
    $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 existential 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 ch x $.  $d th y $.  $d ta z $.  $d A x y z $.  $d B y z $.  $d C z $.
    $d X x $.  $d Y x y $.  $d Z x y z $.
    3rspcedvdw.1 $e |- ( x = A -> ( ps <-> ch ) ) $.
    3rspcedvdw.2 $e |- ( y = B -> ( ch <-> th ) ) $.
    3rspcedvdw.3 $e |- ( z = C -> ( th <-> ta ) ) $.
    3rspcedvdw.a $e |- ( ph -> A e. X ) $.
    3rspcedvdw.b $e |- ( ph -> B e. Y ) $.
    3rspcedvdw.c $e |- ( ph -> C e. Z ) $.
    3rspcedvdw.4 $e |- ( ph -> ta ) $.
    $( Triple application of ~ rspcedvdw .  (Contributed by SN,
       20-Aug-2024.) $)
    3rspcedvdw $p |- ( ph -> E. x e. X E. y e. Y E. z e. Z ps ) $=
      ( wcel wrex rspc3ev syl31anc ) AILUBJMUBKNUBEBHNUCGMUCFLUCRSTUABECDFGHIJK
      LMNOPQUDUE $.
  $}

  ${
    $d x y z A $.  $d y z B $.  $d z C $.  $d x D $.  $d x y E $.
    $d x y z F $.  $d z ch $.  $d y ta $.  $d x th $.
    rspc3dv.1 $e |- ( x = A -> ( ps <-> th ) ) $.
    rspc3dv.2 $e |- ( y = B -> ( th <-> ta ) ) $.
    rspc3dv.3 $e |- ( z = C -> ( ta <-> ch ) ) $.
    rspc3dv.4 $e |- ( ph -> A. x e. D A. y e. E A. z e. F ps ) $.
    rspc3dv.5 $e |- ( ph -> A e. D ) $.
    rspc3dv.6 $e |- ( ph -> B e. E ) $.
    rspc3dv.7 $e |- ( ph -> C e. F ) $.
    $( 3-variable restricted specialization, using implicit substitution.
       (Contributed by Scott Fenton, 10-Mar-2025.) $)
    rspc3dv $p |- ( ph -> ch ) $=
      ( wcel w3a wral 3jca rspc3v sylc ) AILUBZJMUBZKNUBZUCBHNUDGMUDFLUDCAUHUIU
      JSTUAUERBCDEFGHIJKLMNOPQUFUG $.
  $}

  ${
    $d A w x y z $.  $d B w y z $.  $d C w z $.  $d D w $.  $d R x $.
    $d S x y $.  $d T x y z $.  $d U w x y z $.  $d ch x $.  $d ps w $.
    $d ta z $.  $d th y $.
    rspc4v.1 $e |- ( x = A -> ( ph <-> ch ) ) $.
    rspc4v.2 $e |- ( y = B -> ( ch <-> th ) ) $.
    rspc4v.3 $e |- ( z = C -> ( th <-> ta ) ) $.
    rspc4v.4 $e |- ( w = D -> ( ta <-> ps ) ) $.
    $( 4-variable restricted specialization, using implicit substitution.
       (Contributed by Scott Fenton, 7-Feb-2025.) $)
    rspc4v $p |- ( ( ( A e. R /\ B e. S ) /\ ( C e. T /\ D e. U ) ) ->
     ( A. x e. R A. y e. S A. z e. T A. w e. U ph -> ps ) ) $=
      ( wcel wa wral wi w3a df-3an cv wceq ralbidv rspc3v sylan9 sylanbr anasss
      rspcv ) JNUBZKOUBZUCZLPUBZMQUBZAIQUDZHPUDGOUDFNUDZBUEZURUSUCUPUQUSUFZUTVC
      UPUQUSUGVDVBEIQUDZUTBVAVECIQUDDIQUDFGHJKLNOPFUHJUIACIQRUJGUHKUICDIQSUJHUH
      LUIDEIQTUJUKEBIMQUAUOULUMUN $.
  $}

  ${
    $d A x y z w p q $.  $d B y z w p q $.  $d C z w p q $.  $d D w p q $.
    $d E p q $.  $d F q $.  $d R x $.  $d S x y $.  $d T x y z $.
    $d U x y z w $.  $d V x y z w p $.  $d W x y z w p q $.  $d ch x $.
    $d th y $.  $d ta z $.  $d et w $.  $d ze p $.  $d ps q $.
    rspc6v.1 $e |- ( x = A -> ( ph <-> ch ) ) $.
    rspc6v.2 $e |- ( y = B -> ( ch <-> th ) ) $.
    rspc6v.3 $e |- ( z = C -> ( th <-> ta ) ) $.
    rspc6v.4 $e |- ( w = D -> ( ta <-> et ) ) $.
    rspc6v.5 $e |- ( p = E -> ( et <-> ze ) ) $.
    rspc6v.6 $e |- ( q = F -> ( ze <-> ps ) ) $.
    $( 6-variable restricted specialization, using implicit substitution.
       (Contributed by Scott Fenton, 20-Feb-2025.) $)
    rspc6v $p |- ( ( ( A e. R /\ B e. S ) /\ ( C e. T /\ D e. U ) /\
    ( E e. V /\ F e. W ) ) ->
     ( A. x e. R A. y e. S A. z e. T A. w e. U
       A. p e. V A. q e. W ph -> ps ) ) $=
      ( wcel wa wral wi cv wceq 2ralbidv rspc4v rspc2v syl9 3impia ) LPULMQULUM
      ZNRULOSULUMZTUBULUAUCULUMZAUDUCUNUEUBUNZKSUNJRUNIQUNHPUNZBUOVCVDUMVGFUDUC
      UNUEUBUNZVEBVFVHCUDUCUNUEUBUNDUDUCUNUEUBUNEUDUCUNUEUBUNHIJKLMNOPQRSHUPLUQ
      ACUEUDUBUCUFURIUPMUQCDUEUDUBUCUGURJUPNUQDEUEUDUBUCUHURKUPOUQEFUEUDUBUCUIU
      RUSFBGUEUDTUAUBUCUJUKUTVAVB $.
  $}

  ${
    $d A p q r s w x y z $.  $d B p q r s w y z $.  $d C p q r s w z $.
    $d D p q r s w $.  $d E p q r s $.  $d F q r s $.  $d G r s $.  $d H s $.
    $d R x $.  $d S x y $.  $d T x y z $.  $d U w x y z $.  $d V p w x y z $.
    $d W p q w x y z $.  $d X p q r w x y z $.  $d Y p q r s w x y z $.
    $d ch x $.  $d th y $.  $d ta z $.  $d et w $.  $d ze p $.  $d si q $.
    $d rh r $.  $d ps s $.
    rspc8v.1 $e |- ( x = A -> ( ph <-> ch ) ) $.
    rspc8v.2 $e |- ( y = B -> ( ch <-> th ) ) $.
    rspc8v.3 $e |- ( z = C -> ( th <-> ta ) ) $.
    rspc8v.4 $e |- ( w = D -> ( ta <-> et ) ) $.
    rspc8v.5 $e |- ( p = E -> ( et <-> ze ) ) $.
    rspc8v.6 $e |- ( q = F -> ( ze <-> si ) ) $.
    rspc8v.7 $e |- ( r = G -> ( si <-> rh ) ) $.
    rspc8v.8 $e |- ( s = H -> ( rh <-> ps ) ) $.
    $( 8-variable restricted specialization, using implicit substitution.
       (Contributed by Scott Fenton, 20-Feb-2025.) $)
    rspc8v $p |- ( ( ( ( A e. R /\ B e. S ) /\ ( C e. T /\ D e. U ) ) /\
    ( ( E e. V /\ F e. W ) /\ ( G e. X /\ H e. Y ) ) ) ->
     ( A. x e. R A. y e. S A. z e. T A. w e. U
       A. p e. V A. q e. W A. r e. X A. s e. Y ph -> ps ) ) $=
      ( wcel wa wral cv wceq 4ralbidv rspc4v sylan9 ) NRVBOSVBVCPTVBQUAVBVCVCAU
      JUIVDUKUHVDULUGVDUMUFVDZMUAVDLTVDKSVDJRVDFUJUIVDUKUHVDULUGVDUMUFVDZUBUFVB
      UCUGVBVCUDUHVBUEUIVBVCVCBVJVKCUJUIVDUKUHVDULUGVDUMUFVDDUJUIVDUKUHVDULUGVD
      UMUFVDEUJUIVDUKUHVDULUGVDUMUFVDJKLMNOPQRSTUAJVENVFACUMULUKUJUFUGUHUIUNVGK
      VEOVFCDUMULUKUJUFUGUHUIUOVGLVEPVFDEUMULUKUJUFUGUHUIUPVGMVEQVFEFUMULUKUJUF
      UGUHUIUQVGVHFBGHIUMULUKUJUBUCUDUEUFUGUHUIURUSUTVAVHVI $.
  $}

  ${
    $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 ph x z $.  $d ph x y $.  $d ps y $.  $d ps z $.  $d A x $.  $d C x $.
    $d D x $.  $d ch x $.
    ralxpxfr2d.a $e |- A e. _V $.
    ralxpxfr2d.b $e |- ( ph -> ( x e. B <-> E. y e. C E. z e. D x = A ) ) $.
    ralxpxfr2d.c $e |- ( ( ph /\ x = A ) -> ( ps <-> ch ) ) $.
    $( Transfer a universal quantifier between one variable with pair-like
       semantics and two.  (Contributed by Stefan O'Rear, 27-Feb-2015.) $)
    ralxpxfr2d $p |- ( ph -> ( A. x e. B ps <-> A. y e. C A. z e. D ch ) ) $=
      ( wral wi wal wrex albidv ralcom4 ralbii wceq df-ral imbi1d bitrid bitr2i
      wcel r19.23v albii 3bitr4ri bitrdi pm5.74da biidd ceqsalv 2ralbidv bitrd
      cv ) ABDHNZDUPZGUAZBOZDPZFJNZEINZCFJNEINAUQUSFJQZEIQZBOZDPZVCUQURHUFZBOZD
      PAVGBDHUBAVIVFDAVHVEBLUCRUDUTFJNZDPZEINVJEINZDPVCVGVJEDISVBVKEIUTFDJSTVFV
      LDVLVDBOZEINVFVJVMEIUSBFJUGTVDBEIUGUEUHUIUJAVACEFIJAVAUSCOZDPCAUTVNDAUSBC
      MUKRCCDGKUSCULUMUJUNUO $.
  $}

  ${
    $d A x z $.  $d Y x z $.  $d ph x $.  $d ps z $.  $d th z $.
    rexraleqim.1 $e |- ( x = z -> ( ps <-> ph ) ) $.
    rexraleqim.2 $e |- ( z = Y -> ( ph <-> th ) ) $.
    $( Statement following from existence and generalization with equality.
       (Contributed by AV, 9-Feb-2019.) $)
    rexraleqim $p |- ( ( E. z e. A ph /\ A. x e. A ( ps -> x = Y ) ) -> th ) $=
      ( wrex cv wceq wi wral wcel wa weq eqeq1 imbi12d rspcva syli impancom imp
      biimpd rexlimiva ) AEFJBDKZGLZMZDFNZCAUICMEFEKZFOZUIACAUKUIPUJGLZCUHAULMD
      UJFDEQBAUGULHUFUJGRSTULACIUDUAUBUEUC $.
  $}

  ${
    $d x A $.  $d x B $.
    $( A variable introduction law for class equality, closed form.
       (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 eximi pm3.43 3syl sylib
      19.37v eqtr2 exlimiv impbid1 ) BDEZBCFZAGZBFZUFCFZHZAIZUDUEUIJZAIZUEUJJUD
      UGAIUEUGJZUEUHJZHZAIULABDKUGUOAUGUMUNUGUELUGUEUHUFBCMNOPUOUKAUEUGUHQPRUEU
      IATSUIUEAUFBCUAUBUC $.

    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.)  (Proof
       shortened by Thierry Arnoux, 23-Jan-2022.) $)
    eqvinc $p |- ( A = B <-> E. x ( x = A /\ x = B ) ) $=
      ( cvv wcel wceq cv wa wex wb eqvincg ax-mp ) BEFBCGAHZBGNCGIAJKDABCELM $.
  $}

  ${
    $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 cbvexv1 bitri ) B
      CHGIZBHZTCHZJZGKAIZBHZUDCHZJZAKGBCFLUCUGGAUAUBAATBDMATCEMNUGGOTUDHUAUEUBU
      FTUDBPTUDCPQRS $.
  $}

  ${
    $d x y A $.  $d y ph $.
    $( Two ways to express substitution of ` A ` for ` x ` in ` ph ` .  This is
       the analogue for classes of ~ sbalex .  (Contributed by NM, 2-Mar-1995.)
       (Revised by BJ, 27-Apr-2019.) $)
    alexeqg $p |- ( A e. V -> ( A. x ( x = A -> ph )
                                                <-> E. x ( x = A /\ ph ) ) ) $=
      ( vy wcel cv wceq wa wex wi wal eqeq2 anbi1d exbidv imbi1d albidv vtoclbg
      sbalex bicomd ) CDFBGZCHZAIZBJZUBAKZBLZUAEGZHZAIZBJUHAKZBLUDUFECDUGCHZUIU
      CBUKUHUBAUGCUAMZNOUKUJUEBUKUHUBAULPQABESRT $.
  $}

  ${
    $d x A $.
    $( Equality implies equivalence with substitution.  (Contributed by NM,
       2-Mar-1995.)  (Proof shortened by BJ, 1-May-2019.) $)
    ceqex $p |- ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) ) $=
      ( cv wceq wa wex 19.8a ex wi wal cvv wcel wb eqvisset alexeqg syl sylbird
      sp com12 impbid ) BDCEZAUBAFZBGZUBAUDUCBHIUBUDUBAJZBKZAUBCLMUFUDNBCOABCLP
      QUFUBAUEBSTRUA $.
  $}

  ${
    $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 nfe1 nfbi ceqex bibi12d biid vtoclg1f ) AAHCIDJZAKZCL
      ZBHCDEUABCTCMFNSAUAABACDOGPAQR $.
  $}

  ${
    $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.)  Drop ~ ax-10 and ~ ax-12 .  (Revised
       by GG, 1-Dec-2023.) $)
    ceqsexgv $p |- ( A e. V -> ( E. x ( x = A /\ ph ) <-> ps ) ) $=
      ( cv wceq id cgsexg ) ABCGDHZCDEKIFJ $.
    $( $j usage 'ceqsexgv' avoids 'ax-10' 'ax-12'; $)
  $}

  ${
    $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 $.

    $( Elimination of a restricted universal quantifier, using implicit
       substitution.  (Contributed by Scott Fenton, 7-Dec-2020.) $)
    ceqsralbv $p |- ( A. x e. B ( x = A -> ph ) <-> ( A e. B -> ps ) ) $=
      ( cv wceq wi wral wcel wn wa wrex notbid ceqsrexbv rexanali annim 3bitr3i
      con4bii ) CGDHZAICEJZDEKZBIZUAALZMCENUCBLZMUBLUDLUEUFCDEUAABFOPUAACEQUCBR
      ST $.
  $}

  ${
    $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 $.
    $( Alternate definition of membership when the member is a set.
       (Contributed by NM, 18-Aug-1993.)  Strengthen from sethood hypothesis to
       sethood antecedent.  (Revised by BJ, 12-Feb-2022.)  Avoid ~ ax-12 .
       (Revised by BJ, 1-Sep-2024.) $)
    clel2g $p |- ( A e. V -> ( A e. B <-> A. x ( x = A -> x e. B ) ) ) $=
      ( wcel cv wceq wex wi wal wb elisset biimt syl 19.23v eleq1 pm5.74i albii
      bicomd bitr3i bitrdi ) BDEZBCEZAFZBGZAHZUCIZUEUDCEZIZAJZUBUFUCUGKABDLUFUC
      MNUGUEUCIZAJUJUEUCAOUKUIAUEUCUHUEUHUCUDBCPSQRTUA $.
  $}

  ${
    $d x A $.  $d x B $.
    clel2.1 $e |- A e. _V $.
    $( Alternate definition of membership when the member is a set.
       (Contributed by NM, 18-Aug-1993.) $)
    clel2 $p |- ( A e. B <-> A. x ( x = A -> x e. B ) ) $=
      ( cvv wcel cv wceq wi wal wb clel2g ax-mp ) BEFBCFAGZBHNCFIAJKDABCELM $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Alternate definition of membership in 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 $.
    $( Alternate definition of membership in 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 A x $.  $d B x $.
    $( Alternate definition of membership in a set.  (Contributed by NM,
       18-Aug-1993.)  Strengthen from sethood hypothesis to sethood antecedent
       and avoid ~ ax-12 .  (Revised by BJ, 1-Sep-2024.) $)
    clel4g $p |- ( B e. V -> ( A e. B <-> A. x ( x = B -> A e. x ) ) ) $=
      ( wcel cv wceq wi wal wex elisset biimt syl 19.23v bitr4di bicomd pm5.74i
      wb eleq2 albii bitrdi ) CDEZBCEZAFZCGZUCHZAIZUEBUDEZHZAIUBUCUEAJZUCHZUGUB
      UJUCUKRACDKUJUCLMUEUCANOUFUIAUEUCUHUEUHUCUDCBSPQTUA $.
  $}

  ${
    $d x A $.  $d x B $.
    clel4.1 $e |- B e. _V $.
    $( Alternate definition of membership in a set.  (Contributed by NM,
       18-Aug-1993.) $)
    clel4 $p |- ( A e. B <-> A. x ( x = B -> A e. x ) ) $=
      ( cvv wcel cv wceq wi wal wb clel4g ax-mp ) CEFBCFAGZCHBNFIAJKDABCELM $.
  $}

  ${
    $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.)  Remove use of ~ ax-10 , ~ ax-11 , and
       ~ ax-12 .  (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.)
       Avoid ~ ax-13 .  (Revised by Wolf Lammen, 29-Apr-2023.) $)
    pm13.183 $p |- ( A e. V -> ( A = B <-> A. z ( z = A <-> z = B ) ) ) $=
      ( vy cv wceq weq wb wal eqeq1 eqeq2 bibi1d albidv alrimiv wsb stdpc4 sbbi
      equsb1v tbt bicom bitri eqsb1 3bitr2i sylib impbii vtoclbg ) EFZCGZAEHZAF
      ZCGZIZAJZBCGUKBGZULIZAJEBDUHBCKUHBGZUMUPAUQUJUOULUHBUKLMNUIUNUIUMAUHCUKLO
      UNUMAEPZUIUMAEQURUJAEPZULAEPZIZUTUIUJULAERUTUTUSIVAUSUTAESTUTUSUAUBAECUCU
      DUEUFUG $.
  $}

  ${
    $d y A $.  $d x y $.  $d y ph $.
    $( Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89.  We
       don't need the nonempty class condition of ~ r19.3rzv when there is an
       outer quantifier.  (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.  We
       don't need the nonempty class condition of ~ r19.28zv when there is an
       outer quantifier.  (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 simpr ralimia r19.28v
      jca2 impbii ) ABFZDEGZCEGABDEGZFZCEGUBUDCECHZEIZUBAUCUBADEGUFAUAADEABJKAA
      DUEEDCLAMNOUABDEABPKSQUDUBCEABDERKT $.
  $}

  ${
    $d A x y $.  $d ph y $.
    $( Membership in a class abstraction.  Class version of ~ sb6 .
       (Contributed by SN, 5-Oct-2024.) $)
    elab6g $p |- ( A e. V -> ( A e. { x | ph } <-> A. x ( x = A -> ph ) ) ) $=
      ( vy cv cab wcel weq wal wceq eleq1 eqeq2 imbi1d albidv wsb df-clab bitri
      wi sb6 vtoclbg ) EFZABGZHZBEIZASZBJZCUCHBFZCKZASZBJECDUBCUCLUBCKZUFUJBUKU
      EUIAUBCUHMNOUDABEPUGAEBQABETRUA $.
  $}

  ${
    $d x ph $.  $d x ch $.  $d x A $.
    elabd2.ex $e |- ( ph -> A e. V ) $.
    elabd2.eq $e |- ( ph -> B = { x | ps } ) $.
    elabd2.is $e |- ( ( ph /\ x = A ) -> ( ps <-> ch ) ) $.
    $( Membership in a class abstraction, using implicit substitution.
       Deduction version of ~ elab .  (Contributed by GG, 12-Oct-2024.)
       (Revised by BJ, 16-Oct-2024.) $)
    elabd2 $p |- ( ph -> ( A e. B <-> ch ) ) $=
      ( wcel wb wa cv wceq wi wal cab eleq2d sylan9bb elab6g wex elisset albidv
      pm5.74da 19.23v bitrdi pm5.5 sylan2 bitrd mpdan ) AEGKZEFKZCLHAULMUMDNEOZ
      BPZDQZCAUMEBDRZKULUPAFUQEISBDEGUATULAUNDUBZUPCLDEGUCAUPURCPZURCAUPUNCPZDQ
      USAUOUTDAUNBCJUEUDUNCDUFUGURCUHTUIUJUK $.
    $( $j usage 'elabd2' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $d x ph $.  $d x ch $.  $d x A $.
    elabd3.ex $e |- ( ph -> A e. V ) $.
    elabd3.is $e |- ( ( ph /\ x = A ) -> ( ps <-> ch ) ) $.
    $( Membership in a class abstraction, using implicit substitution.
       Deduction version of ~ elab .  (Contributed by GG, 12-Oct-2024.) $)
    elabd3 $p |- ( ph -> ( A e. { x | ps } <-> ch ) ) $=
      ( cab eqidd elabd2 ) ABCDEBDIZFGALJHK $.
    $( $j usage 'elabd3' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $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.)  Reduce axiom usage.  (Revised
       by GG, 12-Oct-2024.)  (Proof shortend by Wolf Lammen, 11-May-2025.) $)
    elabgt $p |- ( ( A e. B /\ A. x ( x = A -> ( ph <-> ps ) ) ) ->
                 ( A e. { x | ph } <-> ps ) ) $=
      ( wcel cv wceq wb wi wal wa cab elab6g adantr elisset biimp imim3i al2imi
      wex 19.23v imbitrdi syl7 com3r imp biimpr imim2i com23 alimi 19.21v sylib
      adantl impbid bitrd ) DEFZCGDHZABIZJZCKZLZDACMFZUPAJZCKZBUOVAVCIUSACDENOU
      TVCBUOUSVCBJUSVCUOBUOUPCTZUSVCBCDEPUSVCUPBJZCKVDBJURVBVECUQABUPABQRSUPBCU
      AUBUCUDUEUSBVCJZUOUSBVBJZCKVFURVGCURUPBAUQBAJUPABUFUGUHUIBVBCUJUKULUMUN
      $.
    $( $j usage 'elabgt' avoids 'ax-10' 'ax-11' 'ax-12'; $)

    $( Obsolete version of ~ elabgt as of 11-May-2025.  (Contributed by NM,
       7-Nov-2005.)  (Proof shortened by Andrew Salmon, 8-Jun-2011.)  Reduce
       axiom usage.  (Revised by GG, 12-Oct-2024.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    elabgtOLD $p |- ( ( A e. B /\ A. x ( x = A -> ( ph <-> ps ) ) ) ->
                 ( A e. { x | ph } <-> ps ) ) $=
      ( wcel cv wceq wb wi wal cab elab6g adantr wex elisset biimp imim3i syl
      wa al2imi 19.23v imbitrdi com3r biimpr imim2i alimi bi2.04 19.21v impbidd
      albii sylbb a1i imp bitrd ) DEFZCGDHZABIZJZCKZTDACLFZUQAJZCKZBUPVAVCIUTAC
      DEMNUPUTVCBIZUPUQCOZUTVDJCDEPVEUTVCBUTVCVEBUTVCUQBJZCKVEBJUSVBVFCURABUQAB
      QRUAUQBCUBUCUDUTBVCJZJVEUTUQBAJZJZCKZVGUSVICURVHUQABUEUFUGVJBVBJZCKVGVIVK
      CUQBAUHUKBVBCUIULSUMUJSUNUO $.
    $( $j usage 'elabgtOLD' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    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 $.
    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.)  Avoid
       ~ ax-13 .  (Revised by SN, 23-Nov-2022.)  Avoid ~ ax-10 , ~ ax-11 ,
       ~ ax-12 .  (Revised by SN, 5-Oct-2024.) $)
    elabg $p |- ( A e. V -> ( A e. { x | ph } <-> ps ) ) $=
      ( wcel cab cv wceq wi wal elab6g wex pm5.74i albii 19.23v bitri elisset
      wb pm5.5 syl bitrid bitrd ) DEGZDACHGCIDJZAKZCLZBACDEMUHUFCNZBKZUEBUHUFBK
      ZCLUJUGUKCUFABFOPUFBCQRUEUIUJBTCDESUIBUAUBUCUD $.
    $( $j usage 'elabg' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    $d A y $.  $d x y $.  $d ph y $.  $d ch y $.  $d ps x $.
    elabgw.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    elabgw.2 $e |- ( y = A -> ( ps <-> ch ) ) $.
    $( Membership in a class abstraction, using two substitution hypotheses to
       avoid a disjoint variable condition on ` x ` and ` A ` .  This is to
       ~ elabg what ~ sbievw2 is to ~ sbievw .  (Contributed by SN,
       20-Apr-2024.) $)
    elabgw $p |- ( A e. V -> ( A e. { x | ph } <-> ch ) ) $=
      ( cv cab wcel eleq1 wsb df-clab sbievw bitri vtoclbg ) EJZADKZLZBFTLCEFGS
      FTMIUAADENBAEDOABDEHPQR $.
    $( $j usage 'elabgw' avoids 'ax-10' 'ax-11' 'ax-12'; $)

    elab2gw.3 $e |- B = { x | ph } $.
    $( Membership in a class abstraction, using two substitution hypotheses to
       avoid a disjoint variable condition on ` x ` and ` A ` , which is not
       usually significant since ` B ` is usually a constant.  (Contributed by
       SN, 16-May-2024.) $)
    elab2gw $p |- ( A e. V -> ( A e. B <-> ch ) ) $=
      ( wcel cab eleq2i elabgw bitrid ) FGLFADMZLFHLCGQFKNABCDEFHIJOP $.
    $( $j usage 'elab2gw' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $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.)  Avoid
       ~ ax-10 , ~ ax-11 , ~ ax-12 .  (Revised by SN, 5-Oct-2024.) $)
    elab $p |- ( A e. { x | ph } <-> ps ) $=
      ( cvv wcel cab wb elabg ax-mp ) DGHDACIHBJEABCDGFKL $.
    $( $j usage 'elab' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $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 A x $.  $d ch x $.
    elabd.1 $e |- ( ph -> A e. V ) $.
    elabd.2 $e |- ( ph -> ch ) $.
    elabd.3 $e |- ( x = A -> ( ps <-> ch ) ) $.
    $( Explicit demonstration the class ` { x | ps } ` is not empty by the
       example ` A ` .  (Contributed by RP, 12-Aug-2020.)  (Revised by AV,
       23-Mar-2024.) $)
    elabd $p |- ( ph -> A e. { x | ps } ) $=
      ( cab wcel wb elabg syl mpbird ) AEBDJKZCHAEFKPCLGBCDEFIMNO $.
  $}

  ${
    $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 biadanii ) 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 cab wb wn elabgf ibi pm2.21 impbid2 ja ) BDEIDACJZIZBKBLSBSBABCDRF
      GHMNBSOPABCDEFGHMQ $.
  $}

  ${
    $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 ) ) $=
      ( wcel cab wb wn elabg ibi pm2.21 impbid2 ja ) BDEGDACHZGZBIBJQBQBABCDPFK
      LBQMNABCDEFKO $.
    $( $j usage 'elab3g' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $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.)  (Revised by AV, 16-Aug-2024.) $)
    elab3 $p |- ( A e. { x | ph } <-> ps ) $=
      ( wcel wi cab wb elab3g ax-mp ) BDEHIDACJHBKFABCDEGLM $.
  $}

  ${
    $d A y $.  $d V x y $.  $d ph y $.
    $( Implication for the membership in a restricted class abstraction.
       (Contributed by Alexander van der Vekens, 31-Dec-2017.)  Remove disjoint
       variable condition on ` A , x ` and avoid ~ ax-10 , ~ ax-11 , ~ ax-12 .
       (Revised by SN, 5-Aug-2024.) $)
    elrabi $p |- ( A e. { x e. V | ph } -> A e. V ) $=
      ( vy wcel cv wa cab crab wex dfclel wsb df-clab simpl sbimi clelsb1 sylib
      wceq sylbi eleq1 biimpa sylan2 exlimiv df-rab eleq2s ) CDFZCBGDFZAHZBIZAB
      DJCUJFEGZCSZUKUJFZHZEKUGECUJLUNUGEUMULUKDFZUGUMUIBEMZUOUIEBNUPUHBEMUOUIUH
      BEUHAOPBEDQRTULUOUGUKCDUAUBUCUDTABDUEUF $.
    $( $j usage 'elrabi' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    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 y $.  $d A y $.
    rabtru.1 $e |- F/_ x A $.
    $( Abstract builder using the constant wff ` T. ` .  (Contributed by
       Thierry Arnoux, 4-May-2020.) $)
    rabtru $p |- { x e. A | T. } = A $=
      ( vy wtru crab cv wcel tru nfcv nftru weq biidd elrabf mpbiran2 eqriv ) D
      EABFZBDGZQHRBHEIEEARBARJCAKADLEMNOP $.
  $}

  ${
    $d A x $.
    rabeqcOLD.1 $e |- ( x e. A -> ph ) $.
    $( Obsolete version of ~ rabeqc as of 15-Jan-2025.  (Contributed by AV,
       20-Apr-2022.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    rabeqcOLD $p |- { x e. A | ph } = A $=
      ( crab cv wcel wa cab df-rab wceq wb eqabcb pm4.71i bicomi mpgbir eqtri )
      ABCEBFCGZAHZBIZCABCJTCKSRLBSBCMRSRADNOPQ $.
  $}

  ${
    $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 ) ) $=
      ( crab wcel cv wa cab wceq wb wi wal df-rab eleq2i id nfa1 nfv nfan eleq1
      sp biimparc biantrurd bibi1d pm5.74da syl5ibcom imp alrimi elabgt syl2an2
      bitrid ) DACEFZGDCHZEGZAIZCJZGZUNDKZABLZMZCNZDEGZIZBUMUQDACEOPVCVCVBUSUPB
      LZMZCNURBLVCQVDVFCVBVCCVACRVCCSTVBVCVFVBVAVCVFVACUBVCUSUTVEVCUSIZAUPBVGUO
      AUSUOVCUNDEUAUCUDUEUFUGUHUIUPBCDEUJUKUL $.
  $}

  ${
    $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.)  Remove dependency on
       ~ ax-13 .  (Revised by Steven Nguyen, 23-Nov-2022.) $)
    elrab $p |- ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) ) $=
      ( crab wcel cvv wa elex adantr wceq eleq1 anbi12d df-rab elab2g pm5.21nii
      cv ) DACEGZHDIHZDEHZBJZDTKUBUABDEKLCSZEHZAJUCCDTIUDDMUEUBABUDDENFOACEPQR
      $.

    $( 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 crab elrab sylanbrc ) AEFJCEBDFKJHIBCDEFGLM $.
  $}

  ${
    $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 restricted 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 A y $.  $d B x y $.  $d ph y $.  $d ch y $.  $d ps x $.
    elrab2w.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    elrab2w.2 $e |- ( y = A -> ( ps <-> ch ) ) $.
    elrab2w.3 $e |- C = { x e. B | ph } $.
    $( Membership in a restricted class abstraction.  This is to ~ elrab2 what
       ~ elab2gw is to ~ elab2g .  (Contributed by SN, 2-Sep-2024.) $)
    elrab2w $p |- ( A e. C <-> ( A e. B /\ ch ) ) $=
      ( wcel cvv wa elex adantr cv weq eleq1w anbi12d wceq eleq1 crab cab eqtri
      df-rab elab2gw pm5.21nii ) FHLFMLZFGLZCNZFHOUJUICFGOPDQGLZANZEQZGLZBNUKDE
      FHMDERULUOABDEGSITUNFUAUOUJBCUNFGUBJTHADGUCUMDUDKADGUFUEUGUH $.
    $( $j usage 'elrab2w' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $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.)  Reduce axiom usage.  (Revised by GG,
       2-Nov-2024.) $)
    ralab $p |- ( A. x e. { y | ph } ch <-> A. x ( ps -> ch ) ) $=
      ( cab wral cv wcel wal df-ral wsb df-clab sbievw bitri imbi1i biid albii
      wi ) CDAEGZHDIUAJZCTZDKBCTZDKCDUALUCUDDUCUDUDUBBCUBAEDMBADENABEDFOPQUDRPS
      P $.
    $( $j usage 'ralab' avoids 'ax-8' 'ax-9' 'df-clel' 'df-cleq' 'ax-ext'; $)

    $( 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.)
       Reduce axiom usage.  (Revised by GG, 2-Nov-2024.) $)
    rexab $p |- ( E. x e. { y | ph } ch <-> E. x ( ps /\ ch ) ) $=
      ( cab wrex wa wn wal wex wi wral dfrex2 ralab xchbinx imnang df-ex bitr4i
      ) CDAEGZHZBCIZJDKZJUCDLUBBCJZMDKZUDUBUEDUANUFCDUAOABUEDEFPQBCDRQUCDST $.
    $( $j usage 'rexab' avoids 'ax-8' 'ax-9' 'df-clel' 'df-cleq' 'ax-ext'; $)

    $( 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.)  Drop ~ ax-8 .  (Revised by GG,
       1-Dec-2023.) $)
    ralab2 $p |- ( A. x e. { y | ph } ps <-> A. y ( ph -> ch ) ) $=
      ( cab wral cv wcel wi wal df-ral nfsab1 nfv nfim weq eleq1ab abid bitrdi
      imbi12d cbvalv1 bitri ) BDAEGZHDIUDJZBKZDLACKZELBDUDMUFUGDEUEBEAEDNBEOPUG
      DODEQZUEABCUHUEEIUDJAADEERAESTFUAUBUC $.
    $( $j usage 'ralab2' avoids 'ax-8'; $)

    $( 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 $.
    $( $j usage 'ralrab2' avoids 'ax-8'; $)

    $( Existential quantification over a class abstraction.  (Contributed by
       Mario Carneiro, 3-Sep-2015.)  Drop ~ ax-8 .  (Revised by GG,
       1-Dec-2023.) $)
    rexab2 $p |- ( E. x e. { y | ph } ps <-> E. y ( ph /\ ch ) ) $=
      ( cab wrex cv wcel wa wex df-rex nfsab1 nfv nfan weq eleq1ab abid bitrdi
      anbi12d cbvexv1 bitri ) BDAEGZHDIUDJZBKZDLACKZELBDUDMUFUGDEUEBEAEDNBEOPUG
      DODEQZUEABCUHUEEIUDJAADEERAESTFUAUBUC $.
    $( $j usage 'rexab2' avoids 'ax-8'; $)

    $( 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 A y $.  $d ph y $.  $d x y $.
    reurab.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Restricted existential uniqueness of a restricted abstraction.
       (Contributed by Scott Fenton, 8-Aug-2024.) $)
    reurab $p |- ( E! x e. { y e. A | ps } ch <-> E! x e. A ( ph /\ ch ) ) $=
      ( cv crab wcel wa weu wreu wb weq bicomd equcoms elrab anbi1i df-reu
      anass bitri eubii 3bitr4i ) DHZBEFIZJZCKZDLUEFJZACKZKZDLCDUFMUJDFMUHUKDUH
      UIAKZCKUKUGULCBAEUEFBANDEDEOABGPQRSUIACUAUBUCCDUFTUJDFTUD $.
  $}

  ${
    $d x z $.  $d A z $.
    $( 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 nf5rd impbid2 eqabcdv ) ACDZBECFZAGZBCMONNAHMN
      AABCIJKL $.
  $}

  ${
    $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 x A $.
    $( Writing a set as a class abstraction.  This theorem looks artificial but
       was added to characterize the class abstraction whose existence is
       proved in ~ class2set .  (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 wral crab wceq elex cv ax-1 ralrimiv rabid2im eqcomd 3syl ) BC
      DBEDZPABFZPABGZBHBCIPPABPAJBDKLQBRPABMNO $.
    $( $j usage 'class2seteq' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $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 $.
  $}

  ${
    $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 weq weu spcegv imp 3adant3 eqeq2 3adant2
      imbi2d albidv eu3v sylanbrc ) DEHZBACIZDJZKZCLZMACNZACGOZKZCLZGNZACPUFBUK
      UJUFBUKABCDEFQRSUFUJUOBUFUJUOUNUJGDEGIZDJZUMUICUQULUHAUPDUGTUBUCQRUAACGUD
      UE $.
  $}

  ${
    $d x y A $.
    $( There exists at most one set equal to a given class.  (Contributed by
       NM, 8-Mar-1995.)  Shorten combined proofs of ~ moeq and ~ eueq .  (Proof
       shortened by BJ, 24-Sep-2022.) $)
    moeq $p |- E* x x = A $=
      ( vy cv wceq wmo wa weq wi wal eqtr3 gen2 eqeq1 mo4 mpbir ) ADZBEZAFQCDZB
      EZGACHIZCJAJTACPRBKLQSACPRBMNO $.
  $}

  ${
    $d x A $.
    $( A class is a set if and only if there exists a unique set equal to it.
       (Contributed by NM, 25-Nov-1994.)  Shorten combined proofs of ~ moeq and
       ~ eueq .  (Proof shortened by BJ, 24-Sep-2022.) $)
    eueq $p |- ( A e. _V <-> E! x x = A ) $=
      ( cv wceq wex wmo wa cvv wcel weu moeq biantru isset df-eu 3bitr4i ) ACBD
      ZAEZQPAFZGBHIPAJRQABKLABMPANO $.
  $}

  ${
    $d x A $.
    eueqi.1 $e |- A e. _V $.
    $( There exists a unique set equal to a given set.  Inference associated
       with ~ euequ .  See ~ euequ in the case of a setvar.  (Contributed by
       NM, 5-Apr-1995.) $)
    eueqi $p |- E! x x = A $=
      ( cvv wcel cv wceq weu eueq mpbi ) BDEAFBGAHCABIJ $.
  $}

  ${
    $d x ph $.  $d x A $.  $d x B $.
    eueq2.1 $e |- A e. _V $.
    eueq2.2 $e |- B e. _V $.
    $( Equality has existential uniqueness (split into 2 cases).  (Contributed
       by NM, 5-Apr-1995.) $)
    eueq2 $p |- E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) $=
      ( cv wceq wa wn weu eueqi euanv biimpri mpan2 euorv bianfd eubidv mpbid
      wo notnot syl2anc orcom orbi2d bitrid mpdan id orbi1d pm2.61i ) AABGZCHZI
      ZAJZUJDHZIZTZBKZAUMULTZBKZUQAUMJULBKZUSAUAZAUKBKZUTBCELUTAVBIAUKBMNOUMULB
      PUBAURUPBURULUMTAUPUMULUCAUMUOULAUMUNVAQUDUERSUMAUOTZBKZUQUMUOBKZVDUMUNBK
      ZVEBDFLVEUMVFIUMUNBMNOAUOBPUFUMVCUPBUMAULUOUMAUKUMUGQUHRSUI $.
  $}

  ${
    $d x ph $.  $d x ps $.  $d x A $.  $d x B $.  $d x C $.
    eueq3.1 $e |- A e. _V $.
    eueq3.2 $e |- B e. _V $.
    eueq3.3 $e |- C e. _V $.
    eueq3.4 $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.) $)
    eueq3 $p |- E! x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B )
                \/ ( ps /\ x = C ) ) $=
      ( wceq wa wo wn w3o weu eueqi ibar wb con2i cv pm2.45 imnani jaoi orbi12d
      bianfd mtbid biorf bitrd 3orrot df-3or bitri bitr4di eubidv adantr pm2.46
      syl mpbii simpl orim12i nsyl5 3orcomb ecase3 ) ABACUAZDKZLZABMZNZVDEKZLZB
      VDFKZLZOZCPZAVECPVNCDGQAVEVMCAVEVJVLMZVFMZVMAVEVFVPAVERAVONVFVPSAVHBMZVOV
      QAVHANBABUBZABABJUCZTUDTAVHVJBVLAVHVIVHAVRTUFABVKVSUFUEUGVOVFUHUQUIVMVJVL
      VFOVPVFVJVLUJVJVLVFUKULUMUNURBVKCPVNCFIQBVKVMCBVKVFVJMZVLMZVMBVKVLWABVKRB
      VTNVLWASVTBVFBNZVJAWBVEVSUOVHWBVIABUPUOUDTVTVLUHUQUIVFVJVLUKUMUNURVHVICPV
      NCEHQVHVIVMCVHVIVFVLMZVJMZVMVHVIVJWDVHVIRWCVGVJWDSVFAVLBAVEUSBVKUSUTWCVJU
      HVAUIVMVFVLVJOWDVFVJVLVBVFVLVJUKULUMUNURVC $.
  $}

  ${
    $d x y ph $.  $d x y ps $.  $d x y A $.  $d x y B $.  $d x y C $.
    moeq3.1 $e |- B e. _V $.
    moeq3.2 $e |- C e. _V $.
    moeq3.3 $e |- -. ( ph /\ ps ) $.
    $( "At most one" property of equality (split into 3 cases).  (The first two
       hypotheses could be eliminated with longer proof.)  (Contributed by NM,
       23-Apr-1995.) $)
    moeq3 $p |- E* x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B )
                \/ ( ps /\ x = C ) ) $=
      ( vy cvv cv wceq wa wo wn w3o weu biidd 3orass wmo eqeq2 anbi2d 3orbi123d
      wcel eubidv vex vtoclg eumo syl wi wal eqvisset pm2.21 syl5 anim2d orim1d
      eueq3 3imtr4g alrimiv euimmo mpisyl pm2.61i ) DKUEZACLZDMZNZABOPVEEMNZBVE
      FMNZQZCUAZVDVJCRZVKAVEJLZMZNZVHVIQZCRZVLJDKVMDMZVPVJCVRVOVGVHVHVIVIVRVNVF
      AVMDVEUBUCVRVHSVRVISUDUFABCVMEFJUGGHIURZUHVJCUIUJVDPZVJVPUKZCULVQVKVTWACV
      TVGVHVIOZOVOWBOVJVPVTVGVOWBVTVFVNAVFVDVTVNCDUMVDVNUNUOUPUQVGVHVITVOVHVITU
      SUTVSVJVPCVAVBVC $.
  $}

  ${
    $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 ) $=
      ( cv wceq wmo wal wa wex moeq ax-gen moexexvw mp2an ) CFDGZCHABHZCIPAJCKB
      HCDLQCEMPACBNO $.
  $}

  ${
    $d x y z A $.  $d y z ph $.
    $( Theorem for inferring "at most one".  (Contributed by NM,
       17-Oct-1996.) $)
    mo2icl $p |- ( A. x ( ph -> x = A ) -> E* x ph ) $=
      ( vy vz cvv wcel cv wi wal wmo weq eqeq2 imbi2d albidv imbi1d wex equequ2
      wceq wn 19.8aw df-mo sylibr vtoclg imim2i con3rr3 alimdv alnex nexmo syl6
      eqvisset sylbi pm2.61i ) CFGZABHZCSZIZBJZABKZIZABDLZIZBJZUSIUTDCFDHZCSZVC
      URUSVEVBUQBVEVAUPAVDCUOMNOPVCVCDQUSVCABELZIZBJDEDELZVBVGBVHVAVFADEBRNOUAA
      BDUBUCUDUNTZURATZBJZUSVIUQVJBUQAUNUPUNABCUKUEUFUGVKABQTUSABUHABUIULUJUM
      $.
  $}

  ${
    $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 nfv sbhypf anbi2d
      eqeq2 imbi12d spcgv nfs1v sbequ12 mo4f sp sylbi impel expd 3impia impbid
      ) DEHZACIZAJZCKZDLZBUPAURBUNUOAMFNUNUOABUROUNUOPABURUNAACGQZPZUQGKZLZOZGR
      ZABPZUROZUOVCVFGDEVADLZUTVEVBURVGUSBAABCGDBCSFTUAVADUQUBUCUDUOVDCRVDAUSCG
      ACGUEACGUFUGVDCUHUIUJUKULUM $.

    $( 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 nfmo1 nf3an nfim 3anbi3d bibi1d
      cv eqeq1 imbi12d mob2 vtoclg1f com12 3expib syl com3r imp 3impib ) EGKZFH
      KZLADMZBEFNZCOZUQURUSBLZVAPURVBUQVAURFQKZVBUQVAPZPFHRVCUSBVDUQVCUSBSZVAVC
      USASZDUFZFNZCOZPVEVAPDEGVEVADVCUSBDVCDTADUABDTUBVADTUCVGENZVFVEVIVAVJABVC
      USIUDVJVHUTCVGEFUGUEUHACDFQJUIUJUKULUMUNUOUP $.

    $( 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 $.  $d x y $.
    euxfr2w.1 $e |- A e. _V $.
    euxfr2w.2 $e |- E* y x = A $.
    $( Transfer existential uniqueness from a variable ` x ` to another
       variable ` y ` contained in expression ` A ` .  Version of ~ euxfr2 with
       a disjoint variable condition, which does not require ~ ax-13 .
       (Contributed by NM, 14-Nov-2004.)  Avoid ~ ax-13 .  (Revised by GG,
       10-Jan-2024.) $)
    euxfr2w $p |- ( E! x E. y ( x = A /\ ph ) <-> E! y ph ) $=
      ( cv wceq wa wex weu wmo wi 2euswapv moani ancom mobii mpbi mpg moeq
      impbii biidd ceqsexv eubii bitri ) BGDHZAIZCJBKZUGBJZCKZACKUHUJUGCLZUHUJM
      BUGBCNAUFIZCLUKUFACFOULUGCAUFPZQRSUGBLZUJUHMCUGCBNULBLUNUFABBDTOULUGBUMQR
      SUAUIACAABDEUFAUBUCUDUE $.
    $( $j usage 'euxfr2w' avoids 'ax-13'; $)
  $}

  ${
    $d x ps $.  $d y ph $.  $d x A $.  $d x y $.
    euxfrw.1 $e |- A e. _V $.
    euxfrw.2 $e |- E! y x = A $.
    euxfrw.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Transfer existential uniqueness from a variable ` x ` to another
       variable ` y ` contained in expression ` A ` .  Version of ~ euxfr with
       a disjoint variable condition, which does not require ~ ax-13 .
       (Contributed by NM, 14-Nov-2004.)  Avoid ~ ax-13 .  (Revised by GG,
       10-Jan-2024.) $)
    euxfrw $p |- ( E! x ph <-> E! y ps ) $=
      ( weu cv wceq wa wex euex ax-mp biantrur 19.41v pm5.32i exbii 3bitr2i
      eubii eumoi euxfr2w bitri ) ACICJEKZBLZDMZCIBDIAUGCAUEDMZALUEALZDMUGUHAUE
      DIUHGUEDNOPUEADQUIUFDUEABHRSTUABCDEFUEDGUBUCUD $.
    $( $j usage 'euxfrw' avoids 'ax-13'; $)
  $}

  ${
    $d x ph $.  $d x A $.
    euxfr2.1 $e |- A e. _V $.
    euxfr2.2 $e |- E* y x = A $.
    $( Transfer existential uniqueness from a variable ` x ` to another
       variable ` y ` contained in expression ` A ` .  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  Use the weaker ~ euxfr2w
       when possible.  (Contributed by NM, 14-Nov-2004.)
       (New usage is discouraged.) $)
    euxfr2 $p |- ( E! x E. y ( x = A /\ ph ) <-> E! y ph ) $=
      ( cv wceq wa wex weu wmo wi 2euswap moani ancom mobii mpbi mpg moeq biidd
      impbii ceqsexv eubii bitri ) BGDHZAIZCJBKZUGBJZCKZACKUHUJUGCLZUHUJMBUGBCN
      AUFIZCLUKUFACFOULUGCAUFPZQRSUGBLZUJUHMCUGCBNULBLUNUFABBDTOULUGBUMQRSUBUIA
      CAABDEUFAUAUCUDUE $.
  $}

  ${
    $d x ps $.  $d y ph $.  $d x A $.
    euxfr.1 $e |- A e. _V $.
    euxfr.2 $e |- E! y x = A $.
    euxfr.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Transfer existential uniqueness from a variable ` x ` to another
       variable ` y ` contained in expression ` A ` .  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  Use the weaker ~ euxfrw
       when possible.  (Contributed by NM, 14-Nov-2004.)
       (New usage is discouraged.) $)
    euxfr $p |- ( E! x ph <-> E! y ps ) $=
      ( weu cv wceq wa wex euex ax-mp biantrur 19.41v pm5.32i exbii 3bitr2i
      eubii eumoi euxfr2 bitri ) ACICJEKZBLZDMZCIBDIAUGCAUEDMZALUEALZDMUGUHAUED
      IUHGUEDNOPUEADQUIUFDUEABHRSTUABCDEFUEDGUBUCUD $.
  $}

  ${
    $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 ) ) $.
    $( 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 sylib cbvexvw isseti 19.41v
      weu biantrur excom 3bitr2i wb eqeq2 biimpr an31 imbi1i impexp 3bitr3i syl
      2alimi 19.23v albii 19.21v eximdv biimtrid imp pm4.24 biimpi anim12 eqtr3
      syl56 alanimi com12 alrimivv adantl eqeq1 imbi2d albidv eu4 sylanbrc ) AB
      KZFGLZMZDNCNZACOZKAEPZFLZMZCNZEOZWEAJPZFLZMZCNZKZWBWGLZMZJNENZWEEUDVTWAWF
      WAWBGLZBKZDOZEOZVTWFWABDOZWRABCDIUAWSWOEOZBKZDOWPEOZDOWRBXADWTBEGHUBUEQXB
      XADWOBEUCQWPDEUFUGRVTWQWEEVTWPWDMZDNZCNZWQWEMZVSXCCDVSVQWCWOUHZMZXCVRXGVQ
      FGWBUISXHVQWOWCMZMZXCXGXIVQWCWOUJSVQWOKZWCMWPAKZWCMXJXCXKXLWCABWOUKULVQWO
      WCUMWPAWCUMUNTUOUPXEWQWDMZCNXFXDXMCWPWDDUQURWQWDCUSRTUTVAVBWAWNVTWAWMEJWK
      WAWLWKAWLMZCNWAWLMWDWIXNCAAAKZWDWIKWCWHKWLAXOAVCVDAWCAWHVEWBWGFVFVGVHAWLC
      UQTVIVJVKWEWJEJWLWDWICWLWCWHAWBWGFVLVMVNVOVP $.
  $}

  ${
    $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 eleq1w sbequ12 anbi12d sbiev
      anbi2i an4 bitri imbi1i 3bitri imbi2i bitr4i anbi12i ) BEDFZAGZBHVABIZVAV
      ABCJZGZBCKZLZCMZBMZGABDNABDUAZAABCJZGZVELZCDOZBDOZGVABCVACTUBABDUCVIVBVNV
      HABDUDVNUTVMLZBMVHVMBDPVGVOBUTCEDFZVLLZLZCMUTVQCMZLVGVOUTVQCUEVFVRCVFUTVP
      GZVKGZVELVTVLLVRVDWAVEVDVAVPVJGZGWAVCWBVAVAWBBCVPVJBVPBTABCUFUGVEUTVPAVJB
      CDUHABCUIUJUKULUTAVPVJUMUNUOVTVKVEQUTVPVLQUPRVMVSUTVLCDPUQSRURUSS $.

    $( 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 ) ) $=
      ( wreu cv wcel wa weu weq wb wral wrex df-reu wal wex wi biimprcd 3bitr4i
      19.28v eleq1w sbequ12 anbi12d equequ1 bibi12d equid simpl sylbir biimtrdi
      wsb tbt spimvw ibar bibi1d sps jca axc4i imim2i impd adantl adantr simplr
      biimp simpr biimpr syl6ci jcai ex impbid alimi impbii df-ral anbi2i exbii
      imp eu6 df-rex bitri ) ABDEBFDGZAHZBIZABCJZKZBDLZCDMZABDNVTWBKZBOZCPCFDGZ
      WDHZCPWAWEWGWICWHVSWCQZHZBOZWHWJBOZHWGWIWHWJBTWGWLWFWKBWGWHWJWFWHBCWBWFWH
      ABCUJZHZCCJZKZWHWBVTWOWBWPWBVSWHAWNBCDUAZABCUBUCBCCUDUEWQWOWHWPWOCUFUKWHW
      NUGUHUIULWFWJBVSWCWFVSAVTWBVSAUMUNRUOUPUQWKWFBWKVTWBWJVTWBQWHWJVSAWBWCAWB
      QVSAWBVCURUSUTWKWBVTWKWBHZVSAWKWBVSWHWBVSQWJWBVSWHWRRVAVOWSVSWCWBAWHWJWBV
      BWKWBVDAWBVEVFVGVHVIVJVKWDWMWHWCBDVLVMSVNVTBCVPWDCDVQSVR $.

    $( 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 wcel weu eu3v df-reu df-rex df-ral impexp albii bitr4i
      cv exbii anbi12i 3bitr4i sylibr impbii ) ABDEZABDFZABCGZHZBDIZCDFZJZURUSV
      CABDKURAUTLZBDIZCDFVCABCDMVFVBCDVEVABDAUTNOPTQVDUSVBCRZJZURVCVGUSVBCDUAUB
      BULDUCZAJZBUDVJBRZVJUTHZBSZCRZJURVHVJBCUEABDUFUSVKVGVNABDUGVBVMCVBVIVAHZB
      SVMVABDUHVLVOBVIAUTUIUJUKUMUNUOUPUQ $.

    $( 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 an4 impexp albii df-ral mo4
      ancom bianbi imbi1i 3bitri r19.21v 3bitr2i eleq1w anbi12d 3bitr4i bitri )
      ACEGCHEIZAJZCKZABJZCDLZMZDENZCENZACEOUMDHEIZBJZJZUPMZDPZCPULURMZCPUNUSVDV
      ECVDUTULUQMZMZDPVFDENVEVCVGDVCUTULJZUOJZUPMVHUQMVGVBVIUPVBULUTJUOVHULAUTB
      QULUTUBUCUDVHUOUPRUTULUQRUESVFDETULUQDEUFUGSUMVACDUPULUTABCDEUHFUIUAURCET
      UJUK $.

    $( 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 reu3 equequ1 equcom bitrdi imbi12d cbvralvw
      bitri rexbii imbi2d ralbidv cbvrexvw anbi2i ) ACEHACEIZACGJZKZCELZGEIZMUF
      BCDJZKZDELZCEIZMACGENUJUNUFUJBGDJZKZDELZGEIUNUIUQGEUHUPCDEUKABUGUOFUKUGDG
      JUOCDGODGPQRSUAUQUMGCEGCJZUPULDEURUOUKBGCDOUBUCUDTUET $.

    $( 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 cbvreuvw reu6 cv wcel ralbii wal bitrid a1i
      dfbi2 r19.26 ancom equcom imbi2i biimt df-ral bi2.04 albii eleq1w imbi12d
      bicomd equcoms equsalvw 3bitrri bitrdi anbi12d bitr4id rexbiia 3bitri ) A
      CEGBDEGBDCHZIZDEJZCEKABCDHZLZDEJZMZCEKABCDEFNBDCEOVDVHCEVDBVBLZVBBLZMZDEJ
      ZCPEQZVHVCVKDEBVBUBRVMVLVIDEJZVJDEJZMZVHVIVJDEUCVHVGAMVMVPAVGUDVMVGVNAVOV
      GVNIVMVFVIDEVEVBBCDUEUFRUAVMAVMALZVOVMAUGVODPEQZVJLZDSVBVRBLZLZDSVQVJDEUH
      VSWADVRVBBUIUJVTVQDCVTVQICDVEVQVTVEVMVRABCDEUKFULUMUNUOUPUQURTUSTUTVA $.
  $}

  ${
    $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 ) ) $=
      ( wrmo cv wcel wa wmo wsb wi wral wal 3bitri impexp albii df-ral weq sban
      df-rmo clelsb1fw bianbi anbi2i an4 ancom anbi1i imbi1i nfcri 3bitr2i nfan
      r19.21 mo3 3bitr4i bitri ) ABDHBIDJZAKZBLZAABCMZKZBCUAZNZCDOZBDOZABDUCUSU
      SBCMZKZVCNZCPZBPURVENZBPUTVFVJVKBVJCIDJZURVDNZNZCPVMCDOVKVIVNCVIVLURKZVBK
      ZVCNVOVDNVNVHVPVCVHUSVLVAKZKURVLKZVBKVPVGVQUSVGURBCMVAVLURABCUBBCDEUDUEUF
      URAVLVAUGVRVOVBURVLUHUIQUJVOVBVCRVLURVDRQSVMCDTURVDCDCBDFUKZUNULSUSBCURAC
      VSGUMUOVEBDTUPUQ $.
  $}

  ${
    $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 sbiev anbi2i imbi1i 2ralbii bitri ) A
      CEJAACDKZLZCDMZNZDEOCEOABLZUENZDEOCEOACDEFGADPQUFUHCDEEUDUGUEUCBAABCDHIRS
      TUAUB $.
  $}

  ${
    $d A x y $.  $d B x y $.  $d C x y $.  $d ch x y $.  $d ps y $.
    $d th x y $.
    reu2eqd.1 $e |- ( x = B -> ( ps <-> ch ) ) $.
    reu2eqd.2 $e |- ( x = C -> ( ps <-> th ) ) $.
    reu2eqd.3 $e |- ( ph -> E! x e. A ps ) $.
    reu2eqd.4 $e |- ( ph -> B e. A ) $.
    reu2eqd.5 $e |- ( ph -> C e. A ) $.
    reu2eqd.6 $e |- ( ph -> ch ) $.
    reu2eqd.7 $e |- ( ph -> th ) $.
    $( Deduce equality from restricted uniqueness, deduction version.
       (Contributed by Thierry Arnoux, 27-Nov-2019.) $)
    reu2eqd $p |- ( ph -> B = C ) $=
      ( vy wceq wa wi nfv wsb cv wral wrex wreu reu2 sylib wcel nfs1v nfan nfim
      simprd anbi1d eqeq1 imbi12d sbhypf anbi2d eqeq2 rspc2 syl2anc mpd mp2and
      ) ACDGHQZNOABBEPUAZRZEUBZPUBZQZSZPFUCEFUCZCDRZVCSZABEFUDZVJABEFUEVMVJRKBE
      PFUFUGULAGFUHHFUHVJVLSLMVIVLCVDRZGVGQZSEPGHFFVNVOECVDECETBEPUIUJVOETUKVLP
      TVFGQZVEVNVHVOVPBCVDIUMVFGVGUNUOVGHQZVNVKVOVCVQVDDCBDEPHDETJUPUQVGHGURUOU
      SUTVAVB $.
  $}

  ${
    $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 $.
  $}

  ${
    $d x A $.
    $( Equality's restricted existential "at most one" property.  (Contributed
       by Thierry Arnoux, 30-Mar-2018.)  (Revised by AV, 27-Oct-2020.)  (Proof
       shortened by NM, 29-Oct-2020.) $)
    rmoeq $p |- E* x e. B x = A $=
      ( cv wceq wrmo wcel wa wmo moeq moani df-rmo mpbir ) ADZBEZACFNCGZOHAIOPA
      ABJKOACLM $.
  $}

  $( 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 $.
  $}

  ${
    rmoimi.1 $e |- ( ph -> ps ) $.
    $( Restricted "at most one" is preserved through implication (note wff
       reversal).  (Contributed by Alexander van der Vekens, 17-Jun-2017.) $)
    rmoimi $p |- ( E* x e. A ps -> E* x e. A ph ) $=
      ( wi cv wcel a1i rmoimia ) ABCDABFCGDHEIJ $.
  $}

  ${
    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 $.
  $}

  $( Double restricted existential uniqueness in terms of restricted existence
     and restricted "at most one".  (Contributed by Alexander van der Vekens,
     17-Jun-2017.) $)
  2reu5a $p |- ( E! x e. A E! y e. B ph <->
                ( E. x e. A ( E. y e. B ph /\ E* y e. B ph )
                  /\ E* x e. A ( E. y e. B ph /\ E* y e. B ph ) ) ) $=
    ( wreu wrex wrmo wa reu5 rexbii rmobii anbi12i bitri ) ACEFZBDFOBDGZOBDHZIA
    CEGACEHIZBDGZRBDHZIOBDJPSQTORBDACEJZKORBDUALMN $.

  $( Restricted uniqueness implies restricted "at most one" through
     implication, analogous to ~ euimmo .  (Contributed by Alexander van der
     Vekens, 25-Jun-2017.) $)
  reuimrmo $p |- ( A. x e. A ( ph -> ps )
                   -> ( E! x e. A ps -> E* x e. A ph ) ) $=
    ( wreu wrmo wi wral reurmo rmoim syl5 ) BCDEBCDFABGCDHACDFBCDIABCDJK $.

  ${
    $d x y $.  $d 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.) $)
    2reuswap $p |- ( 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 wrex wreu wal wex weu df-reu r19.42v df-rex bitri
      wi wrmo df-rmo ralbii df-ral moanimv albii bitr4i bitr3i an12 exbii eubii
      2euswapv 3imtr4g sylbi ) ACEUAZBDFCGEHZAIZCJZBDFZACEKZBDLZABDKZCELZTZUOUR
      BDACEUBUCUSBGDHZUQIZCJZBMZVDUSVEURTZBMVHURBDUDVGVIBVEUQCUEUFUGVHVFCNZBOZV
      FBNZCOZVAVCVFBCULVAVEUTIZBOVKUTBDPVNVJBVNUPVEAIZIZCNZVJVNVOCEKVQVEACEQVOC
      ERUHVPVFCUPVEAUIUJSUKSVCUPVBIZCOVMVBCEPVRVLCVRUQBDKVLUPABDQUQBDRUHUKSUMUN
      UN $.
  $}

  ${
    $d x y $.  $d y A $.  $d x B $.
    $( A condition allowing swap of uniqueness and existential quantifiers.
       (Contributed by Thierry Arnoux, 7-Apr-2017.) $)
    2reuswap2 $p |- ( A. x e. A E* y ( y e. B /\ ph ) ->
                   ( E! x e. A E. y e. B ph -> E! y e. B E. x e. A ph ) ) $=
      ( cv wcel wa wmo wal wrex wreu wex weu df-reu r19.42v df-rex bitr3i bitri
      wi wral df-ral moanimv albii bitr4i 2euswapv exbii eubii 3imtr4g sylbi
      an12 ) CFEGZAHZCIZBDUAZBFDGZUMHZCIZBJZACEKZBDLZABDKZCELZTUOUPUNTZBJUSUNBD
      UBURVDBUPUMCUCUDUEUSUQCMZBNZUQBMZCNZVAVCUQBCUFVAUPUTHZBNVFUTBDOVIVEBVIULU
      PAHZHZCMZVEVIVJCEKVLUPACEPVJCEQRVKUQCULUPAUKUGSUHSVCULVBHZCNVHVBCEOVMVGCV
      MUMBDKVGULABDPUMBDQRUHSUIUJ $.
  $}

  ${
    $d x y ph $.  $d x ps $.  $d x A $.  $d x y B $.  $d x y C $.
    reuxfrd.1 $e |- ( ( ph /\ y e. C ) -> A e. B ) $.
    reuxfrd.2 $e |- ( ( ph /\ x e. B ) -> E* y e. C x = A ) $.
    $( Transfer existential uniqueness from a variable ` x ` to another
       variable ` y ` contained in expression ` A ` .  (Contributed by NM,
       16-Jan-2012.)  Separate variables ` B ` and ` C ` .  (Revised by Thierry
       Arnoux, 8-Oct-2017.) $)
    reuxfrd $p |- ( ph
        -> ( E! x e. B E. y e. C ( x = A /\ ps ) <-> E! y e. C ps ) ) $=
      ( cv wceq wa wrex wreu wrmo wi wcel syl ancom wmo wral rmoan rmobii sylib
      ralrimiva 2reuswap 2reuswap2 moeq moani an12 bitri mobii mpbi a1i impbid1
      mprg wb biidd ceqsrexv reubidva bitrd ) ACJZEKZBLZDGMCFNZVDCFMZDGNZBDGNAV
      EVGAVDDGOZCFUAVEVGPAVHCFAVBFQZLZBVCLZDGOZVHVJVCDGOVLIVCBDGUBRVKVDDGBVCSUC
      UDUEVDCDFGUFRVIVDLZCTZVGVEPDGVDDCGFUGVNDJGQZVIBLZVCLZCTVNVCVPCCEUHUIVQVMC
      VQVCVPLVMVPVCSVCVIBUJUKULUMUNUPUOAVFBDGAVOLEFQVFBUQHBBCEFVCBURUSRUTVA $.
  $}

  ${
    $d x ph $.  $d x A $.  $d x y B $.  $d x y C $.
    reuxfr.1 $e |- ( y e. C -> A e. B ) $.
    reuxfr.2 $e |- ( x e. B -> E* y e. C x = A ) $.
    $( Transfer existential uniqueness from a variable ` x ` to another
       variable ` y ` contained in expression ` A ` .  (Contributed by NM,
       14-Nov-2004.)  (Revised by NM, 16-Jun-2017.) $)
    reuxfr $p |- ( E! x e. B E. y e. C ( x = A /\ ph ) <-> E! y e. C ph ) $=
      ( cv wceq wa wrex wreu wb wtru wcel adantl wrmo reuxfrd mptru ) BIZDJZAKC
      FLBEMACFMNOABCDEFCIFPDEPOGQUAEPUBCFROHQST $.
  $}

  ${
    $d x y ph $.  $d y ps $.  $d x ch $.  $d x A $.  $d x y B $.  $d x y C $.
    reuxfr1d.1 $e |- ( ( ph /\ y e. C ) -> A e. B ) $.
    reuxfr1d.2 $e |- ( ( ph /\ x e. B ) -> E! y e. C x = A ) $.
    reuxfr1d.3 $e |- ( ( ph /\ x = A ) -> ( ps <-> ch ) ) $.
    $( Transfer existential uniqueness from a variable ` x ` to another
       variable ` y ` contained in expression ` A ` .  Cf. ~ reuxfr1ds .
       (Contributed by Thierry Arnoux, 7-Apr-2017.) $)
    reuxfr1d $p |- ( ph -> ( E! x e. B ps <-> E! y e. C ch ) ) $=
      ( wreu cv wceq wa wrex wcel reurex syl bitrd biantrurd wb r19.41v rexbidv
      pm5.32da bitr3id adantr reubidva wrmo reurmo reuxfrd ) ABDGLDMZFNZCOZEHPZ
      DGLCEHLABUODGAULGQZOZBUMEHPZBOZUOUQURBUQUMEHLZURJUMEHRSUAAUSUOUBUPUSUMBOZ
      EHPAUOUMBEHUCAVAUNEHAUMBCKUEUDUFUGTUHACDEFGHIUQUTUMEHUIJUMEHUJSUKT $.
  $}

  ${
    $d x y ph $.  $d y ps $.  $d x ch $.  $d x A $.  $d x y B $.  $d x y C $.
    reuxfr1ds.1 $e |- ( ( ph /\ y e. C ) -> A e. B ) $.
    reuxfr1ds.2 $e |- ( ( ph /\ x e. B ) -> E! y e. C x = A ) $.
    reuxfr1ds.3 $e |- ( x = A -> ( ps <-> ch ) ) $.
    $( Transfer existential uniqueness from a variable ` x ` to another
       variable ` y ` contained in expression ` A ` .  Use ~ reuhypd to
       eliminate the second hypothesis.  (Contributed by NM, 16-Jan-2012.) $)
    reuxfr1ds $p |- ( ph -> ( E! x e. B ps <-> E! y e. C ch ) ) $=
      ( cv wceq wb adantl reuxfr1d ) ABCDEFGHIJDLFMBCNAKOP $.
  $}

  ${
    $d x ps $.  $d y ph $.  $d x A $.  $d x y B $.  $d x y C $.
    reuxfr1.1 $e |- ( y e. C -> A e. B ) $.
    reuxfr1.2 $e |- ( x e. B -> E! y e. C x = A ) $.
    reuxfr1.3 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Transfer existential uniqueness from a variable ` x ` to another
       variable ` y ` contained in expression ` A ` .  Use ~ reuhyp to
       eliminate the second hypothesis.  (Contributed by NM, 14-Nov-2004.) $)
    reuxfr1 $p |- ( E! x e. B ph <-> E! y e. C ps ) $=
      ( wreu wb wtru cv wcel adantl wceq reuxfr1ds mptru ) ACFKBDGKLMABCDEFGDNG
      OEFOMHPCNZFOTEQDGKMIPJRS $.
  $}

  ${
    $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 eleq1d anbi12d cbvexvw r19.41v
      wral wreu exbii rexcom4 risset anbi1i 3bitr4ri eqeq2 imim2i biimpr imbi1i
      wb an31 impexp 3bitr3i sylib syl 2alimi 19.23v an12 eleq1 adantr pm5.32ri
      bitr4i 19.42v albii 19.21v expd reximdvai biimtrid imp pm4.24 eqtr3 syl56
      biimpi anim12 alanimi com12 a1d ralrimivv adantl eqeq1 imbi2d albidv reu4
      sylanbrc ) FHLZAMZGHLZBMZMZFGNZOZDPCPZWLCQZMWLERZFNZOZCPZEHSZXCWLKRZFNZOZ
      CPZMZWTXENZOZKHUEEHUEZXCEHUFWRWSXDWSWTGNZBMZDQZEHSZWRXDWSWNDQZXPWLWNCDCRD
      RNZWKWMABXRFGHJUAIUBUCXNEHSZDQXMEHSZBMZDQXPXQXSYADXMBEHUDUGXNEDHUHWNYADWM
      XTBEGHUIUJUGUKTWRXOXCEHWRWTHLZXOXCWRXMWNMZXBOZDPZCPZYBXOMZXCOZWQYDCDWQWOX
      AXMUPZOZYDWPYIWOFGWTULUMYJWOXMXAOZOZYDYIYKWOXAXMUNUMWOXMMZXAOYCWLMZXAOYLY
      DYMYNXAWLWNXMUQUOWOXMXAURYCWLXAURUSUTVAVBYFYGXBOZCPYHYEYOCYEYCDQZXBOYOYCX
      BDVCYPYGXBYPYBXNMZDQYGYCYQDYCWMXNMYQXMWMBVDXNYBWMXMYBWMUPBWTGHVEVFVGVHUGY
      BXNDVITUOTVJYGXBCVKTUTVLVMVNVOWSXLWRWSXKEKHHWSXKYBXEHLMXIWSXJXIWLXJOZCPWS
      XJOXBXGYRCWLWLWLMZXBXGMXAXFMXJWLYSWLVPVSWLXAWLXFVTWTXEFVQVRWAWLXJCVCUTWBW
      CWDWEXCXHEKHXJXBXGCXJXAXFWLWTXEFWFWGWHWIWJ $.
  $}

  ${
    $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 nfcv nfre1 nfrmow wi wral cv wcel rmoim rspe ex syl11 ralrimi
      ralrimivw ) ACEFZBDGZABDGZCEUACBDCDHACEIJAUAKZBDLUBUCCMENZAUABDOUEUDBDUEA
      UAACEPQTRS $.
  $}

  ${
    $d y A $.  $d x y $.
    $( Lemma for ~ 2reu5 .  Note that ` E! x e. A E! y e. B ph ` does not mean
       "there is exactly one ` x ` in ` A ` and exactly one ` y ` in ` B ` such
       that ` ph ` holds"; see comment for ~ 2eu5 .  (Contributed by Alexander
       van der Vekens, 17-Jun-2017.) $)
    2reu5lem1 $p |- ( E! x e. A E! y e. B ph <->
                   E! x E! y ( x e. A /\ y e. B /\ ph ) ) $=
      ( wreu cv wcel wa weu w3a df-reu reubii euanv bicomi 3anass eubii bitri )
      ACEFZBDFCGEHZAIZCJZBDFZBGDHZTAKZCJZBJZSUBBDACELMUCUDUBIZBJUGUBBDLUHUFBUHU
      DUAIZCJZUFUJUHUDUACNOUIUECUEUIUDTAPOQRQRR $.

    $( Lemma for ~ 2reu5 .  (Contributed by Alexander van der Vekens,
       17-Jun-2017.) $)
    2reu5lem2 $p |- ( A. x e. A E* y e. B ph <->
                   A. x E* y ( x e. A /\ y e. B /\ ph ) ) $=
      ( wrmo wral cv wcel wa wmo w3a wal df-rmo ralbii wi df-ral moanimv bicomi
      bitri 3anass mobii albii ) ACEFZBDGCHEIZAJZCKZBDGZBHDIZUEALZCKZBMZUDUGBDA
      CENOUHUIUGPZBMULUGBDQUMUKBUMUIUFJZCKZUKUOUMUIUFCRSUNUJCUJUNUIUEAUASUBTUCT
      T $.
  $}

  ${
    $d w y z A $.  $d w x z B $.  $d x y $.  $d ph w $.  $d ph z $.
    $( Lemma for ~ 2reu5 .  This lemma is interesting in its own right, showing
       that existential restriction in the last conjunct (the "at most one"
       part) is optional; compare ~ rmo2 .  (Contributed by Alexander van der
       Vekens, 17-Jun-2017.) $)
    2reu5lem3 $p |- ( ( E! x e. A E! y e. B ph /\ A. x e. A E* y e. B ph )
                   <-> ( E. x e. A E. y e. B ph
                         /\ E. z E. w A. x e. A A. y e. B
                            ( ph -> ( x = z /\ y = w ) ) ) ) $=
      ( wreu wral wa cv wcel weu wal wex weq wi wrex exbii 3bitri w3a 2reu5lem1
      wrmo wmo 2reu5lem2 anbi12i 2eu5 3anass 19.42v df-rex bicomi anbi2i bitr4i
      3anan12 imbi1i impexp imbi2i albii df-ral r19.21v 3bitr2i ) ACGHBFHZACGUC
      BFIZJBKFLZCKGLZAUAZCMBMZVFCUDBNZJVFCOZBOZVFBDPCEPJZQZCNZBNZEOZDOZJACGRZBF
      RZAVKQZCGIZBFIZEOZDOZJVBVGVCVHABCFGUBABCFGUEUFVFBCDEUGVJVRVPWCVJVDVQJZBOV
      RVIWDBVIVDVEAJZJZCOVDWECOZJWDVFWFCVDVEAUHSVDWECUIWGVQVDVQWGACGUJUKULTSVQB
      FUJUMVOWBDVNWAEVNVDVTQZBNWAVMWHBVMVEVDVSQZQZCNWICGIWHVLWJCVLVEVDAJZJZVKQV
      EWKVKQZQWJVFWLVKVDVEAUNUOVEWKVKUPWMWIVEVDAVKUPUQTURWICGUSVDVSCGUTVAURVTBF
      USUMSSUFT $.

    $d x A $.  $d y B $.
    $( Double restricted existential uniqueness in terms of restricted
       existential quantification and restricted universal quantification,
       analogous to ~ 2eu5 and ~ reu3 .  (Contributed by Alexander van der
       Vekens, 17-Jun-2017.) $)
    2reu5 $p |- ( ( E! x e. A E! y e. B ph /\ A. x e. A E* y e. B ph )
                  <-> ( E. x e. A E. y e. B ph
                        /\ E. z e. A E. w e. B A. x e. A A. y e. B
                           ( ph -> ( x = z /\ y = w ) ) ) ) $=
      ( wrex weq wa wral wex cv wcel wreu r19.29r reximi eleq1w ex df-rex anass
      wi wrmo pm3.35 bi2anan9 biimpac ancomd rexlimivv pm4.71rd 2exbidv pm5.32i
      4syl bitrdi 2reu5lem3 r19.42v bitr3i exbii bitri anbi2i 3bitr4i ) ACGHZBF
      HZABDIZCEIZJZUBZCGKZBFKZELDLZJVBEMGNZDMFNZVHJZJZELZDLZJACGOBFOACGUCBFKJVB
      VHEGHZDFHZJVBVIVOVBVHVMDEVBVHVJVKJZVHJVMVBVHVRVBVHVRVBVHJVAVGJZBFHAVFJZCG
      HZBFHVECGHZBFHVRVAVGBFPVSWABFAVFCGPQWAWBBFVTVECGAVEUDQQVEVRBCFGBMFNZCMGNZ
      JZVEVRWEVEJVKVJVEWEVKVJJVCWCVKVDWDVJBDFRCEGRUEUFUGSUHULSUIVJVKVHUAUMUJUKA
      BCDEFGUNVQVOVBVQVKVPJZDLVOVPDFTWFVNDWFVLEGHVNVKVHEGUOVLEGTUPUQURUSUT $.
  $}

  $( Double restricted quantification with restricted existential uniqueness
     and restricted "at most one", analogous to ~ 2eumo .  (Contributed by
     Alexander van der Vekens, 24-Jun-2017.) $)
  2reurmo $p |- ( E! x e. A E* y e. B ph -> E* x e. A E! y e. B ph ) $=
    ( wreu wrmo wi reuimrmo cv wcel reurmo a1i mprg ) ACEFZACEGZHZPBDFOBDGHBDOP
    BDIQBJDKACELMN $.

  ${
    $d y A $.  $d x y $.  $d x B $.
    $( Double restricted quantification with existential uniqueness, analogous
       to ~ 2euex .  (Contributed by Alexander van der Vekens, 24-Jun-2017.) $)
    2reurex $p |- ( E! x e. A E. y e. B ph -> E. y e. B E! x e. A ph ) $=
      ( wrex wreu wrmo wa reu5 rexcom nfcv nfre1 nfrmow cv wcel wi wral impcom
      ex rspe ralrimivw rmoim syl rmo5 sylib reximdai biimtrid sylbi ) ACEFZBDG
      UJBDFZUJBDHZIABDGZCEFZUJBDJULUKUNUKABDFZCEFULUNABCDEKULUOUMCEUJCBDCDLACEM
      NULCOEPZUOUMQZULUPIABDHZUQUPULURUPAUJQZBDRULURQUPUSBDUPAUJACEUATUBAUJBDUC
      UDSABDUEUFTUGUHSUI $.

    $( A condition allowing to swap restricted "at most one" and restricted
       existential quantifiers, analogous to ~ 2moswap .  (Contributed by
       Alexander van der Vekens, 25-Jun-2017.) $)
    2rmoswap $p |- ( 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 ) ) $=
      ( wrmo wral cv wcel wa wmo wrex wi df-rmo wal r19.42v df-rex bitri bitr3i
      wex ralbii df-ral moanimv albii bitr4i 2moswapv exbii mobii 3imtr4g sylbi
      an12 ) ACEFZBDGCHEIZAJZCKZBDGZACELZBDFZABDLZCEFZMZULUOBDACENUAUPBHDIZUNJZ
      CKZBOZVAUPVBUOMZBOVEUOBDUBVDVFBVBUNCUCUDUEVEVCCTZBKZVCBTZCKZURUTVCBCUFURV
      BUQJZBKVHUQBDNVKVGBVKVBAJZCELZVGVBACEPVMUMVLJZCTVGVLCEQVNVCCUMVBAUKUGRSUH
      RUTUMUSJZCKVJUSCENVOVICVOUNBDLVIUMABDPUNBDQSUHRUIUJUJ $.

    $( Double restricted existential uniqueness implies double restricted
       unique existential quantification, analogous to ~ 2exeu .  (Contributed
       by Alexander van der Vekens, 25-Jun-2017.) $)
    2rexreu $p |- ( ( E! x e. A E. y e. B ph /\ E! y e. B E. x e. A ph )
                  -> E! x e. A E! y e. B ph ) $=
      ( wrex wreu wa wrmo reurmo reurex rmoimi syl 2reurex anim12ci reu5 sylibr
      ) ACEFZBDGZABDFCEGZHACEGZBDFZUABDIZHUABDGSUCTUBSRBDIUCRBDJUARBDACEKLMACBE
      DNOUABDPQ $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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 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 assumption: 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 setvar
  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 universal
  quantifier 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 ) $.

  $( Indicate that the variables "x" and "y" are free in wcdeq. $)
  $( $j free_var 'wcdeq' with 'x' 'y'; $)

  $( 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.  Usage of this
         theorem is discouraged because it depends on ~ ax-13 .  (Contributed
         by Mario Carneiro, 11-Aug-2016.)  (New usage is discouraged.) $)
      cdeqal1 $p |- CondEq ( x = y -> ( A. x ph <-> A. y ps ) ) $=
        ( wal wb cdeqri cbvalv cdeqth ) ACFBDFGCDABCDABGCDEHIJ $.

      $( Distribute conditional equality over abstraction.  Usage of this
         theorem is discouraged because it depends on ~ ax-13 .  (Contributed
         by Mario Carneiro, 11-Aug-2016.)  (New usage is discouraged.) $)
      cdeqab1 $p |- CondEq ( x = y -> { x | ph } = { y | ps } ) $=
        ( cab wceq nfv wb cdeqri cbvab cdeqth ) ACFBDFGCDABCDADHBCHABICDEJKL $.
    $}

    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 $.
    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.  Usage of this theorem is discouraged because it depends on
       ~ ax-13 .  (Contributed by Mario Carneiro, 11-Aug-2016.)
       (New usage is discouraged.) $)
    nfcdeq $p |- ( ph <-> ps ) $=
      ( wsb sbf nfv wb cdeqri sbie bitr3i ) AACDGBACDEHABCDBCIABJCDFKLM $.
  $}

  ${
    $d x z B $.  $d y z $.  $d z A $.
    nfccdeq.1 $e |- F/_ x A $.
    nfccdeq.2 $e |- CondEq ( x = y -> A = B ) $.
    $( Variation of ~ nfcdeq for classes.  Usage of this theorem is discouraged
       because it depends on ~ ax-13 .  (Contributed by Mario Carneiro,
       11-Aug-2016.)  Avoid ~ ax-11 .  (Revised by GG, 19-May-2023.)
       (New usage is discouraged.) $)
    nfccdeq $p |- A = B $=
      ( vz cv wcel nfcri weq eqid cdeqth cdeqel nfcdeq eqriv ) GCDGHZCIQDIABAGC
      EJABQQCDGGKABQLMFNOP $.
    $( $j usage 'nfccdeq' avoids 'ax-11'; $)
  $}

  $( 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 A $.
    $( Relative version of Russell's paradox ~ ru (which corresponds to the
       case ` A = _V ` ).

       Originally a subproof in ~ pwnss .  (Contributed by Stefan O'Rear,
       22-Feb-2015.)  Avoid ~ df-nel .  (Revised by Steven Nguyen,
       23-Nov-2022.)  Reduce axiom usage.  (Revised by GG, 30-Aug-2024.) $)
    rru $p |- -. { x e. A | -. x e. x } e. A $=
      ( vy wel wn crab wcel wa wb cv eleq12 anidms notbid cbvrabv elrab2 pclem6
      wceq weq ax-mp ) AADZEZABFZUBGZUBBGZUCEZHIUDECCDZEZUECUBBUBCJZUBQZUFUCUIU
      FUCIUHUBUHUBKLMUAUGACBACRZTUFUJTUFIAJZUHUKUHKLMNOUCUDPS $.
    $( $j usage 'rru' avoids 'df-nel' 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $d x y z $.
    $( 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, which
       Frege acknowledged in the second edition of his _Grundgesetze der
       Arithmetik_.

       In 1908, Zermelo rectified this fatal flaw by replacing Comprehension
       with a weaker Subset (or Separation) Axiom ~ ssex asserting that ` A `
       is a set only when it is smaller than some other set ` B ` .  However,
       Zermelo was then faced with a "chicken and egg" problem of how to show
       ` B ` is a set, leading him to introduce the set-building axioms of Null
       Set ~ 0ex , Pairing ~ prex , Union ~ uniex , Power Set ~ pwex , and
       Infinity ~ omex to give him some starting sets to work with (all of
       which, before Russell's Paradox, were immediate consequences of Frege's
       Comprehension).  In 1922 Fraenkel strengthened the Subset Axiom with our
       present Replacement Axiom ~ funimaex (whose modern formalization is due
       to Skolem, also in 1922).  Thus, in a very real sense Russell's Paradox
       spawned the invention of ZF set theory and completely revised the
       foundations of mathematics!

       Another mainstream formalization of set theory, devised by von Neumann,
       Bernays, and Goedel, uses class variables rather than setvar variables
       as its primitives.  The axiom system NBG in [Mendelson] p. 225 is
       suitable for a Metamath encoding.  NBG is a conservative extension of ZF
       in that it proves exactly the same theorems as ZF that are expressible
       in the language of ZF. An advantage of NBG is that it is finitely
       axiomatizable - the Axiom of Replacement can be broken down into a
       finite set of formulas that eliminate its wff metavariable.  Finite
       axiomatizability is required by some proof languages (although not by
       Metamath).  There is a stronger version of NBG called Morse-Kelley
       (axiom system MK in [Mendelson] p. 287).

       Russell himself continued in a different direction, avoiding the paradox
       with his "theory of types".  Quine extended Russell's ideas to formulate
       his New Foundations set theory (axiom system NF of [Quine] p. 331).  In
       NF, the collection of all sets is a set, contrarily to ZF and NBG set
       theories.  Russell's paradox has other consequences: when classes are
       too large (beyond the size of those used in standard mathematics), the
       axiom of choice ~ ac4 and Cantor's theorem ~ canth are provably false.
       (See ~ ncanth for some intuition behind the latter.)  Recent results (as
       of 2014) seem to show that NF is equiconsistent to Z (ZF in which
       ~ ax-sep replaces ~ ax-rep ) with ~ ax-sep restricted to only bounded
       quantifiers.  NF is finitely axiomatizable and can be encoded in
       Metamath using the axioms from T. Hailperin, "A set of axioms for
       logic", _J. Symb.  Logic_ 9:1-19 (1944).

       Under our ZF set theory, every set is a member of the Russell class by
       ~ elirrv (derived from the Axiom of Regularity), so for us the Russell
       class equals the universe ` _V ` (Theorem ~ ruv ).  See ~ ruALT for an
       alternate proof of ~ ru derived from that fact.  (Contributed by NM,
       7-Aug-1994.)  Remove use of ~ ax-13 .  (Revised by BJ, 12-Oct-2019.)
       Remove use of ~ ax-10 , ~ ax-11 , and ~ ax-12 .  (Revised by
       BTernaryTau, 20-Jun-2025.)  (Proof modification is discouraged.) $)
    ru $p |- { x | x e/ x } e/ _V $=
      ( vz vy cv wnel cab cvv wcel wex wel wn wb wal ru0 weq id neleq12d df-nel
      wceq mtbir bitrdi eqabbw nex isset nelir ) ADZUFEZAFZGUHGHBDZUHSZBIUJBUJC
      BJCCJKZLCMCBNUGUKACUIACOZUGCDZUMEUKULUFUMUFUMULPZUNQUMUMRUAUBTUCBUHUDTUE
      $.
    $( $j usage 'ru' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13' 'ax-reg'; $)
  $}

  ${
    $d x y $.
    $( Obsolete version of ~ ru as of 20-Jun-2025.  (Contributed by NM,
       7-Aug-1994.)  Remove use of ~ ax-13 .  (Revised by BJ, 12-Oct-2019.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    ruOLD $p |- { x | x e/ x } e/ _V $=
      ( vy cv wnel cab cvv wcel wceq wex wel wb wn pm5.19 eleq1w df-nel eleq12d
      wal id notbid mtbir bitrid bibi12d spvv mto eqabb nex isset nelir ) ACZUI
      DZAEZFUKFGBCZUKHZBIUMBUMABJZUJKZAQZUPBBJZUQLZKZUQMUOUSABUIULHZUNUQUJURABU
      LNUJAAJZLUTURUIUIOUTVAUQUTUIULUIULUTRZVBPSUAUBUCUDUJAULUETUFBUKUGTUH $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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 $.

  $( Indicate that "x" is not bound in "A" in the definition of wsbc.
     This is similar to $d x A $. $)
  $( $j free_var_in 'wsbc' with 'A' 'x'; $)

  $( Define the proper substitution of a class for a set.

     When ` A ` is a proper class, our definition evaluates to false (see
     ~ sbcex ).  This is somewhat arbitrary: we could have, instead, chosen the
     conclusion of ~ sbc6 for our definition, whose right-hand side 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).  For example, if ` A ` is a proper
     class, Quine's substitution of ` A ` for ` y ` in ` 0 e. y ` evaluates to
     ` 0 e. A ` rather than our falsehood.  (This can be seen by substituting
     ` A ` , ` y ` , and ` 0 ` for alpha, beta, and gamma in Subcase 1 of
     Quine's discussion on p. 42.)  Unfortunately, Quine's definition requires
     a recursive syntactic 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.

     Theorem ~ sbc2or shows the apparently "strongest" statement we can make
     regarding behavior at proper classes if we start from ~ dfsbcq .

     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 } ) $.

  $( Proper substitution of a class for a set in a wff given equal classes.
     This is the essence of the sixth axiom of Frege, specifically Proposition
     52 of [Frege1879] p. 50.

     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 after Theorem ~ sbc2or
     below.  Proper substitution 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 $.
  $}

  ${
    $d x y A $.  $d y ph $.
    $( The disjunction of two equivalences for class substitution does not
       require a class existence hypothesis.  This theorem tells us that there
       are only 2 possibilities for ` [ A / x ] ph ` behavior at proper
       classes, matching the ~ sbc5 (false) and ~ sbc6 (true) conclusions.
       This is interesting since ~ dfsbcq and ~ dfsbcq2 (from which it is
       derived) do not appear to say anything obvious about proper class
       behavior.  Note that this theorem does not tell us that it is always one
       or the other at proper classes; it could "flip" between false (the first
       disjunct) and true (the second disjunct) as a function of some other
       variable ` y ` that ` ph ` or ` A ` may contain.  (Contributed by NM,
       11-Oct-2004.)  (Proof modification is discouraged.) $)
    sbc2or $p |- ( ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) ) \/
                  ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) ) $=
      ( vy cvv wcel wsbc cv wceq wa wex wb wi wal wo wsb weq wn mpbii con3i sb5
      dfsbcq2 eqeq2 anbi1d exbidv vtoclbg pm5.15 vex eleq1 adantr nexdv pm2.21d
      orcd alrimiv 2thd bibi2d orbi2d pm2.61i ) CEFZABCGZBHZCIZAJZBKZLZUTVBAMZB
      NZLZOZUSVEVHABDPBDQZAJZBKUTVDDCEABDCUBDHZCIZVKVCBVMVJVBAVLCVAUCUDUEABDUAU
      FUMUSRZVEUTVDRZLZOVIUTVDUGVNVPVHVEVNVOVGUTVNVOVGVNVCBVCUSVBUSAVBVAEFUSBUH
      VACEUISZUJTUKVNVFBVNVBAVBUSVQTULUNUOUPUQSUR $.
  $}

  $( Direct use of ~ df-sbc is allowed from here on. -NM $)

  $( 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.  This
       is Frege's ninth axiom per Proposition 58 of [Frege1879] p. 51.  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 NM, 5-Aug-1993.)  (Revised 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 $.
  $}

  ${
    $d x y $.
    nfsbcdw.1 $e |- F/ y ph $.
    nfsbcdw.2 $e |- ( ph -> F/_ x A ) $.
    nfsbcdw.3 $e |- ( ph -> F/ x ps ) $.
    $( Deduction version of ~ nfsbcw .  Version of ~ nfsbcd with a disjoint
       variable condition, which does not require ~ ax-13 .  (Contributed by
       NM, 23-Nov-2005.)  Avoid ~ ax-13 .  (Revised by GG, 10-Jan-2024.) $)
    nfsbcdw $p |- ( ph -> F/ x [. A / y ]. ps ) $=
      ( wsbc cab wcel df-sbc nfabdw nfeld nfxfrd ) BDEIEBDJZKACBDELACEPGABCDFHM
      NO $.
    $( $j usage 'nfsbcdw' avoids 'ax-13'; $)
  $}

  ${
    $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 does not require
       ~ ax-13 .  (Contributed by NM, 7-Sep-2014.)  Avoid ~ ax-13 .  (Revised
       by GG, 10-Jan-2024.) $)
    nfsbcw $p |- F/ x [. A / y ]. ph $=
      ( wsbc wnf wtru nftru wnfc a1i nfsbcdw mptru ) ACDGBHIABCDCJBDKIELABHIFLM
      N $.
    $( $j usage 'nfsbcw' avoids 'ax-13'; $)
  $}

  ${
    $d x z $.  $d z A $.  $d y z ph $.  $d x y $.
    $( A composition law for class substitution.  Version of ~ sbcco with a
       disjoint variable condition, which requires fewer axioms.  (Contributed
       by NM, 26-Sep-2003.)  Avoid ~ ax-13 .  (Revised by GG, 10-Jan-2024.) $)
    sbccow $p |- ( [. A / y ]. [. y / x ]. ph <-> [. A / x ]. ph ) $=
      ( vz cv wsbc cvv wcel sbcex dfsbcq wsb sbsbc sbbii sbco2vv 3bitr3ri bitri
      vtoclbg pm5.21nii ) ABCFGZCDGZDHIABDGZTCDJABDJTCEFZGZABUCGZUAUBEDHTCUCDKA
      BUCDKUDABELZUEABCLZCELTCELUFUDUGTCEABCMNABECOTCEMPABEMQRS $.
    $( $j usage 'sbccow' avoids 'ax-10' 'ax-11' 'ax-13'; $)
  $}

  ${
    nfsbcd.1 $e |- F/ y ph $.
    nfsbcd.2 $e |- ( ph -> F/_ x A ) $.
    nfsbcd.3 $e |- ( ph -> F/ x ps ) $.
    $( Deduction version of ~ nfsbc .  Usage of this theorem is discouraged
       because it depends on ~ ax-13 .  Use the weaker ~ nfsbcdw when possible.
       (Contributed by NM, 23-Nov-2005.)  (Revised by Mario Carneiro,
       12-Oct-2016.)  (New usage is discouraged.) $)
    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.  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  Use the weaker
       ~ nfsbcw when possible.  (Contributed by NM, 7-Sep-2014.)  (Revised by
       Mario Carneiro, 12-Oct-2016.)  (New usage is discouraged.) $)
    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.  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  Use the weaker ~ sbccow
       when possible.  (Contributed by NM, 26-Sep-2003.)  (Revised by Mario
       Carneiro, 13-Oct-2016.)  (New usage is discouraged.) $)
    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 weq wceq wb equcoms dfsbcq bicomd syl sbievw bitr3i )
      ABEGZCBHGTCBIABDGZTCBJTUACBCBKDELZTUAMUBBCFNUBUATABDEOPQRS $.
  $}

  ${
    $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.)  (Proof
       shortened by SN, 2-Sep-2024.) $)
    sbc5 $p |- ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) ) $=
      ( wsbc cab wcel cv wceq wa wex df-sbc clelab bitri ) ABCDCABEFBGCHAIBJABC
      KABCLM $.

    $( Alternate proof of ~ sbc5 .  This proof helps show how ~ clelab works,
       since it is equivalent but shorter thanks to now-available library
       theorems like ~ vtoclbg and ~ isset .  (Contributed by NM, 23-Aug-1993.)
       (Revised by Mario Carneiro, 12-Oct-2016.)  (New usage is discouraged.)
       (Proof modification is discouraged.) $)
    sbc5ALT $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.)  (Proof
       shortened by SN, 5-Oct-2024.) $)
    sbc6g $p |- ( A e. V -> ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) ) $=
      ( wsbc cab wcel cv wceq wi wal df-sbc elab6g bitrid ) ABCECABFGCDGBHCIAJB
      KABCLABCDMN $.
  $}

  ${
    $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 sbccow sbc5 bitr3i ) ABDEABCFZEZCDEMDGNHCIABCDJNCDK
      L $.
  $}

  ${
    $d x y $.
    cbvsbcw.1 $e |- F/ y ph $.
    cbvsbcw.2 $e |- F/ x ps $.
    cbvsbcw.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Change bound variables in a wff substitution.  Version of ~ cbvsbc with
       a disjoint variable condition, which does not require ~ ax-13 .
       (Contributed by Jeff Hankins, 19-Sep-2009.)  Avoid ~ ax-13 .  (Revised
       by GG, 10-Jan-2024.) $)
    cbvsbcw $p |- ( [. A / x ]. ph <-> [. A / y ]. ps ) $=
      ( cab wcel wsbc cbvabw eleq2i df-sbc 3bitr4i ) EACIZJEBDIZJACEKBDEKPQEABC
      DFGHLMACENBDENO $.
    $( $j usage 'cbvsbcw' avoids 'ax-13'; $)
  $}

  ${
    $d y ph $.  $d x ps $.  $d x y $.
    cbvsbcvw.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Change the bound variable of a class substitution using implicit
       substitution.  Version of ~ cbvsbcv with a disjoint variable condition,
       which does not require ~ ax-13 .  (Contributed by NM, 30-Sep-2008.)
       Avoid ~ ax-13 .  (Revised by GG, 10-Jan-2024.) $)
    cbvsbcvw $p |- ( [. A / x ]. ph <-> [. A / y ]. ps ) $=
      ( cab wcel wsbc cbvabv eleq2i df-sbc 3bitr4i ) EACGZHEBDGZHACEIBDEINOEABC
      DFJKACELBDELM $.
    $( $j usage 'cbvsbcvw' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    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.  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  Use the weaker ~ cbvsbcw
       when possible.  (Contributed by Jeff Hankins, 19-Sep-2009.)  (Proof
       shortened by Andrew Salmon, 8-Jun-2011.)  (New usage is discouraged.) $)
    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.  Usage of this theorem is discouraged because it depends
       on ~ ax-13 .  Use the weaker ~ cbvsbcvw when possible.  (Contributed by
       NM, 30-Sep-2008.)  (Revised by Mario Carneiro, 13-Oct-2016.)
       (New usage is discouraged.) $)
    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.)  (Proof
       shortened by SN, 14-May-2025.) $)
    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 sbc6g 3ad2ant1 ceqsalt 3comr bitrd
      ) DEFZBCGZCHDIZABJKCLZMACDNZUBAKCLZBTUAUDUEJUCACDEOPUAUCTUEBJABCDEQRS $.
    $( $j usage 'sbciegft' avoids 'ax-10'; $)

    $( Obsolete version of ~ sbciegft as of 14-May-2025.  (Contributed by NM,
       10-Nov-2005.)  (Revised by Mario Carneiro, 13-Oct-2016.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    sbciegftOLD $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.)  Avoid ~ ax-10 , ~ ax-12 .  (Revised
       by GG, 12-Oct-2024.) $)
    sbcieg $p |- ( A e. V -> ( [. A / x ]. ph <-> ps ) ) $=
      ( wsbc cab wcel df-sbc elabg bitrid ) ACDGDACHIDEIBACDJABCDEFKL $.
    $( $j usage 'sbcieg' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $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 , 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 sbievw bitr3i vtoclbg ) ADEJZKZBADFKCEFGADRFLI
      SADEMBADENABDEHOPQ $.
  $}

  ${
    $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.)  Avoid ~ ax-10 ,
       ~ ax-12 .  (Revised by GG, 12-Oct-2024.) $)
    sbcied $p |- ( ph -> ( [. A / x ]. ps <-> ch ) ) $=
      ( wsbc cab wcel df-sbc elabd3 bitrid ) BDEIEBDJKACBDELABCDEFGHMN $.
    $( $j usage 'sbcied' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $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 cbvrabw elrab2 ) ABFGZH
      ZABCHFCDABDIABQCJARBFDEFDKAFLABQMABQNOP $.
  $}

  ${
    $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.)  Avoid ~ ax-13 .
       (Revised by Wolf Lammen, 29-Apr-2023.) $)
    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.)  (Revised by NM, 17-Aug-2018.) $)
    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 disjunction.  (Contributed by
       NM, 31-Dec-2016.)  (Revised by NM, 17-Aug-2018.) $)
    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 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 $.

  ${
    $d ph y $.  $d ps y $.  $d x y $.  $d A y $.
    $( Distribution of class substitution over implication.  One direction of
       ~ sbcimg that holds for proper classes.  (Contributed by NM,
       17-Aug-2018.)  Avoid ~ ax-10 , ~ ax-12 .  (Revised by SN,
       26-Oct-2024.) $)
    sbcim1 $p |- ( [. A / x ]. ( ph -> ps )
              -> ( [. A / x ]. ph -> [. A / x ]. ps ) ) $=
      ( vy cvv wcel wi wsbc sbcex wsb cv wceq dfsbcq2 imbi12d sbi1 vtoclg mpcom
      ) DFGABHZCDIZACDIZBCDIZHZSCDJSCEKZACEKZBCEKZHZHTUCHEDFELDMZUDTUGUCSCEDNUH
      UEUAUFUBACEDNBCEDNOOABCEPQR $.
    $( $j usage 'sbcim1' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    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.)  Drop ~ ax-12 .  (Revised by GG, 1-Dec-2023.) $)
    sbcbidv $p |- ( ph -> ( [. A / x ]. ps <-> [. A / x ]. ch ) ) $=
      ( eqidd sbceqbid ) ABCDEEAEGFH $.
    $( $j usage 'sbcbidv' avoids 'ax-12'; $)
  $}

  ${
    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 $.
  $}

  $( 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.)  (Proof shortened by Wolf Lammen,
     4-May-2023.)  Avoid ~ ax-10 , ~ ax-12 .  (Revised by Steven Nguyen,
     5-May-2024.) $)
  sbcbi2 $p |- ( A. x ( ph <-> ps ) ->
       ( [. A / x ]. ph <-> [. A / x ]. ps ) ) $=
    ( wb wal cab wcel wsbc wceq abbi eleq2 syl df-sbc 3bitr4g ) ABECFZDACGZHZDB
    CGZHZACDIBCDIPQSJRTEABCKQSDLMACDNBCDNO $.

  ${
    $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.)  (Revised by NM, 18-Aug-2018.) $)
    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 existential quantifier in and out of class substitution.
       (Contributed by NM, 21-May-2004.)  (Revised by NM, 18-Aug-2018.) $)
    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 $.
  $}

  ${
    $d x B $.  $d x A $.
    $( Class version of one implication of ~ equvelv .  (Contributed by Andrew
       Salmon, 28-Jun-2011.)  (Proof shortened by SN, 26-Oct-2024.) $)
    sbceqal $p |- ( A e. V -> ( A. x ( x = A -> x = B ) -> A = B ) ) $=
      ( cv wceq wi eqeq1 imbi12d eqid a1bi bitr4di spcgv ) AEZBFZNCFZGZBCFZABDO
      QBBFZRGROOSPRNBBHNBCHISRBJKLM $.
    $( $j usage 'sbceqal' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $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 $.
  $}

  ${
    $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 ) ) $=
    ( wa wsbc w3a sbcan bianbi df-3an sbcbii 3bitr4i ) ABFZCFZDEGZADEGZBDEGZFZC
    DEGZFABCHZDEGQRTHPNDEGTSNCDEIABDEIJUAODEABCKLQRTKM $.

  ${
    $d y A $.  $d x y B $.
    $( Class substitution into a membership relation.  (Contributed by NM,
       17-Aug-2018.)  Avoid ~ ax-13 .  (Revised by Wolf Lammen,
       30-Apr-2023.) $)
    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 y ph $.  $d A y $.  $d y ps $.  $d y ch $.
    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.)  Reduce axiom usage.  (Revised by GG,
       12-Oct-2024.) $)
    sbcimdv $p |- ( ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) $=
      ( vy wsbc cv wsb wex cab wcel df-sbc dfclel df-clab anbi2i exbii biimpi
      wa wceq 3bitri sbimdv anim2d eximdv 3bitrri syl56 ) BDEHZGIZEUAZBDGJZTZGK
      ZAUJCDGJZTZGKZCDEHZUHUMUHEBDLZMUJUIURMZTZGKUMBDENGEUROUTULGUSUKUJBGDPQRUB
      SAULUOGAUKUNUJABCDGFUCUDUEUPUQUQECDLZMUJUIVAMZTZGKUPCDENGEVAOVCUOGVBUNUJC
      GDPQRUFSUG $.
    $( $j usage 'sbcimdv' avoids 'ax-9' 'df-cleq' 'ax-10' 'ax-11' 'ax-12'
       'ax-ext'; $)
  $}

  ${
    $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 y ph $.  $d A y $.  $d V y $.
    $( 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.)  Reduce axiom usage.  (Revised by GG, 12-Oct-2024.) $)
    sbcg $p |- ( A e. V -> ( [. A / x ]. ph <-> ph ) ) $=
      ( vy wsbc cv wceq wa wex wcel cab df-sbc dfclel wsb df-clab sbv biimpi wi
      syl bitri anbi2i exbii 3bitrri wal simpr ax-gen 19.23v mp1i 2a1 imp ancrd
      wb eximi 19.37imv impbid bitr3id ) ABCFZEGZCHZAIZEJZCDKZAURCABLZKUTUSVDKZ
      IZEJVBABCMECVDNVFVAEVEAUTVEABEOAAEBPABEQUAUBUCUDVCUTUSDKZIZEJZVBAUMVCVIEC
      DNRVIVBAVAASZEUEZVBASZVIVJEUTAUFUGVKVLVAAEUHRUIVIAVASZEJAVBSVHVMEVHAUTUTV
      GAUTSUTVGAUJUKULUNAVAEUOTUPTUQ $.
    $( $j usage 'sbcg' avoids 'ax-9' 'df-cleq' 'ax-10' 'ax-11' 'ax-12'
       'ax-ext'; $)
  $}

  ${
    sbcgfi.1 $e |- A e. _V $.
    sbcgfi.2 $e |- F/ x ph $.
    $( Substitution for a variable not free in a wff does not affect it, in
       inference form.  (Contributed by Giovanni Mascellani, 1-Jun-2019.) $)
    sbcgfi $p |- ( [. A / x ]. ph <-> ph ) $=
      ( cvv wcel wsbc wb sbcgf ax-mp ) CFGABCHAIDABCFEJK $.
  $}

  ${
    $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.)  (Proof shortened by GG, 12-Oct-2024.) $)
    sbc2ie $p |- ( [. A / x ]. [. B / y ]. ph <-> ps ) $=
      ( wsbc cv wceq cvv wcel a1i sbcied sbcie ) ADFJBCEGCKELZABDFMFMNRHOIPQ $.
    $( $j usage 'sbc2ie' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $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.)  Avoid ~ ax-10 , ~ ax-12 .  (Revised by
       SN, 20-Aug-2025.) $)
    sbccomlem $p |- ( [. A / x ]. [. B / y ]. ph
         <-> [. B / y ]. [. A / x ]. ph ) $=
      ( cvv wcel wsbc wb sbcex cv wceq wi wal sbc6g isset exim biimtrid exlimiv
      wex syl6com sylbid mpcom pm5.21ni bi2.04 2albii alcom bitri albii 3bitr3i
      wa 19.21v a1i imbi2d albidv adantl adantr 3bitr4d ecase ) DFGZEFGZACEHZBD
      HZABDHZCEHZIVCUTVEVBBDJZVAVEUTVDCEJZVAVECKELZVDMZCNZUTVDCEFOZVJVAVDCTZUTV
      AVHCTVJVLCEPVHVDCQRVDUTCABDJSUAUBUCUDVCVAVEUTVCVAVFUTVCBKDLZVBMZBNZVAVBBD
      FOZVOUTVBBTZVAUTVMBTVOVQBDPVMVBBQRVBVABACEJSUAUBUCVGUDUTVAUKZVOVJVCVEVRVM
      VHAMZCNZMZBNZVHVMAMZBNZMZCNZVOVJWBWFIVRVMVSMZCNZBNZVHWCMZBNZCNZWBWFWIWJCN
      BNWLWGWJBCVMVHAUEUFWJBCUGUHWHWABVMVSCULUIWKWECVHWCBULUIUJUMVAVOWBIUTVAVNW
      ABVAVBVTVMACEFOUNUOUPUTVJWFIVAUTVIWECUTVDWDVHABDFOUNUOUQURUTVCVOIVAVPUQVA
      VEVJIUTVKUPURUS $.
    $( $j usage 'sbccomlem' avoids 'ax-10' 'ax-12'; $)

    $( Obsolete version of ~ sbccomlem as of 20-Aug-2025.  (Contributed by NM,
       14-Nov-2005.)  (Revised by Mario Carneiro, 18-Oct-2016.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    sbccomlemOLD $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 sbccow ) ACFHZIZFEIZBDIZAB
      GHZIZGDIZCEIZACEIZBDIABDIZCEIQBSIZGDIZUACOIZFEIZRUBTCOIZFEIZGDIUIGDIZFEIU
      FUHUIGFDEJUJUEGDUJPBSIZFEIUEUIULFEACBOSJKPFBESJLKUKUGFETGCDOJKMQBGDNUACFE
      NMQUCBDACFENKUAUDCEABGDNKM $.
  $}

  ${
    $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 sbccow simpl wsb wceq sbsbc nfcv wb nfralw
      nfs1v weq sbequ12 ralbidv sbiev bitr3i nfnfc1 nfcvd id nfeqd nfan1 adantl
      dfsbcq2 ralbid adantll bitrid sbcied bitr3id ) ACEHZBDIUTBGJZIZGDIDFKZCDL
      ZMZABDIZCEHZUTBGDNVEVBVGGDFVCVDOVBABGPZCEHZVEVADQZMVGVBUTBGPVIUTBGRUTVIBG
      VHBCEBESABGUBUABGUCAVHCEABGUDUEUFUGVDVJVIVGTVCVDVJMVHVFCEVDVJCCDUHVDCVADV
      DCVAUIVDUJUKULVJVHVFTVDABGDUNUMUOUPUQURUS $.

    $( Interchange class substitution and restricted existential quantifier.
       (Contributed by NM, 1-Mar-2008.)  (Proof shortened by Mario Carneiro,
       13-Oct-2016.)  (Revised by NM, 18-Aug-2018.)  (Proof shortened by JJ,
       7-Jul-2021.) $)
    sbcrext $p |- ( F/_ y A ->
          ( [. A / x ]. E. y e. B ph <-> E. y e. B [. A / x ]. ph ) ) $=
      ( wnfc cvv wcel wrex wsbc wi sbcex a1i wb wral sbcng adantl bitrd dfrex2
      wn nfnfc1 id nfcvd nfeld cv 2a1i rexlimd2 wa sbcralt ancoms ralbid notbid
      nfan1 sbcbii 3bitr4g ex pm5.21ndd ) CDFZDGHZACEIZBDJZABDJZCEIZVAUSKURUTBD
      LMURVBUSCECDUAZURCDGURUBURCGUCUDZVBUSKURCUEEHABDLUFUGURUSVAVCNURUSUHZATZC
      EOZTZBDJZVBTZCEOZTZVAVCVFVJVHBDJZTZVMUSVJVONURVHBDGPQVFVNVLVFVNVGBDJZCEOZ
      VLUSURVNVQNVGBCDEGUIUJVFVPVKCEURUSCVDVEUMUSVPVKNURABDGPQUKRULRUTVIBDACESU
      NVBCESUOUPUQ $.
  $}

  ${
    $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 ) ) $=
      ( wcel wnfc wral wsbc wb nfcv sbcralt mpan2 ) DFGCDHACEIBDJABDJCEIKCDLABC
      DEFMN $.

    $( 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.)  (Revised by NM,
       18-Aug-2018.) $)
    sbcreu $p |- ( [. A / x ]. E! y e. B ph <-> E! y e. B [. A / x ]. ph ) $=
      ( vz wreu wsbc cvv wcel sbcex reurex rexlimivw syl wsb dfsbcq2 cv reubidv
      wrex wceq nfcv nfs1v nfreuw weq sbequ12 sbiev vtoclbg pm5.21nii ) ACEGZBD
      HZDIJZABDHZCEGZUIBDKUMULCESUKULCELULUKCEABDKMNUIBFOABFOZCEGZUJUMFDIUIBFDP
      FQDTUNULCEABFDPRUIUOBFUNBCEBEUAABFUBUCBFUDAUNCEABFUERUFUGUH $.
  $}

  ${
    $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 bitrid eqabb elex sbcan sbcg
      sbcex2 sbcal sbcbig bibi1d bitrd albidv sbcbii 3bitr4g nfcri sbcgf exbidv
      anbi12d dfclel syl ) DFIDJIZACKZEIZBDLZABDLZCKZEIZMDFUAURHNZUSOZVEEIZPZHQ
      ZBDLZVEVCOZVGPZHQZVAVDVJVHBDLZHQURVMVHHBDUDURVNVLHVNVFBDLZVGBDLZPURVLVFVG
      BDUBURVOVKVPVGURCNVEIZAMZCRZBDLZVQVBMZCRZVOVKVTVRBDLZCRURWBVRCBDUEURWCWAC
      URWCVQBDLZVBMWAVQABDJUFURWDVQVBVQBDJUCUGUHUISVFVSBDACVETUJVBCVETUKVGBDJBH
      EGULUMUOSUNSUTVIBDHUSEUPUJHVCEUPUKUQ $.
  $}

  ${
    $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 cbvralsvw dfsbcq2 rspcv biimtrid ) ABDFABEGZEDFCD
      HABCIZABEDJNOECDABECKLM $.

    $( 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 cbvrexsvw sylibr ) CDFABCGZHABE
      IZEDJABDJPOECDABECKLABEDMN $.

    $( 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 $.
  $}

  ${
    $d x ph $.
    $( Version of ~ ra4 with a disjoint variable condition, requiring fewer
       axioms.  This is ~ stdpc5v for a restricted domain.  (Contributed by BJ,
       27-Mar-2020.) $)
    ra4v $p |- ( A. x e. A ( ph -> ps ) -> ( ph -> A. x e. A ps ) ) $=
      ( wi wral r19.21v biimpi ) ABECDFABCDFEABCDGH $.
  $}

  ${
    ra4.1 $e |- F/ x ph $.
    $( Restricted quantifier version of Axiom 5 of [Mendelson] p. 69.  This is
       the axiom ~ stdpc5 of standard predicate calculus for a restricted
       domain.  See ~ ra4v for a version requiring fewer axioms.  (Contributed
       by NM, 16-Jan-2004.)  (Proof shortened by BJ, 27-Mar-2020.) $)
    ra4 $p |- ( A. x e. A ( ph -> ps ) -> ( ph -> A. x e. A ps ) ) $=
      ( wi wral r19.21 biimpi ) ABFCDGABCDGFABCDEHI $.
  $}

  ${
    $d x y $.  $d y A $.
    rmo2.1 $e |- F/ y ph $.
    $( Alternate definition of restricted "at most one".  Note that
       ` E* x e. A ph ` is not equivalent to
       ` E. y e. A A. x e. A ( ph -> x = y ) ` (in analogy to ~ reu6 ); to see
       this, let ` A ` be the empty set.  However, one direction of this
       pattern holds; see ~ rmo2i .  (Contributed by NM, 17-Jun-2017.) $)
    rmo2 $p |- ( E* x e. A ph <-> E. y A. x e. A ( ph -> x = y ) ) $=
      ( wrmo cv wcel wa wmo weq wi wal wex wral df-rmo nfv nfan mof impexp
      albii df-ral bitr4i exbii 3bitri ) ABDFBGDHZAIZBJUGBCKZLZBMZCNAUHLZBDOZCN
      ABDPUGBCUFACUFCQERSUJULCUJUFUKLZBMULUIUMBUFAUHTUAUKBDUBUCUDUE $.

    $( Condition implying restricted "at most one".  (Contributed by NM,
       17-Jun-2017.) $)
    rmo2i $p |- ( E. y e. A A. x e. A ( ph -> x = y ) -> E* x e. A ph ) $=
      ( weq wi wral wrex wex wrmo rexex rmo2 sylibr ) ABCFGBDHZCDIOCJABDKOCDLAB
      CDEMN $.

    $d x A $.
    $( Restricted "at most one" using explicit substitution.  (Contributed by
       NM, 4-Nov-2012.)  (Revised by NM, 16-Jun-2017.)  Avoid ~ ax-13 .
       (Revised by Wolf Lammen, 30-Apr-2023.) $)
    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 weq wral df-rmo wal 3bitri impexp albii df-ral
      wi sban clelsb1 bianbi anbi2i an4 ancom anbi1i imbi1i r19.21v 3bitr2i nfv
      nfan mo3 3bitr4i bitri ) ABDFBGDHZAIZBJZAABCKZIZBCLZTZCDMZBDMZABDNUQUQBCK
      ZIZVATZCOZBOUPVCTZBOURVDVHVIBVHCGDHZUPVBTZTZCOVKCDMVIVGVLCVGVJUPIZUTIZVAT
      VMVBTVLVFVNVAVFUQVJUSIZIUPVJIZUTIVNVEVOUQVEUPBCKUSVJUPABCUABCDUBUCUDUPAVJ
      USUEVPVMUTUPVJUFUGPUHVMUTVAQVJUPVBQPRVKCDSUPVBCDUIUJRUQBCUPACUPCUKEULUMVC
      BDSUNUO $.
  $}

  ${
    $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 anim1i simpll simplr mob syl3anc ex pm5.21ndd sylanb ) ADEJDKZELZA
      MZDNZFELZBMZFGOZGELZCMZPZADEQUPURMZUTUSVAVCUQUSUTUPUQBRZFGESUAVAUTUCVCUTC
      UBUDVCUTVBVCUTMUQUTMUPURVBVCUQUTVDUEUPURUTUFUPURUTUGUOURVADFGEEUMFOUNUQAB
      UMFESHTUMGOUNUTACUMGESITUHUIUJUKUL $.

    $( 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 $.
  $}

  ${
    $d A x $.  $d B x $.  $d ch x $.
    rmoi2.1 $e |- ( x = B -> ( ps <-> ch ) ) $.
    rmoi2.2 $e |- ( ph -> B e. A ) $.
    rmoi2.3 $e |- ( ph -> E* x e. A ps ) $.
    rmoi2.4 $e |- ( ph -> x e. A ) $.
    rmoi2.5 $e |- ( ph -> ps ) $.
    $( Consequence of "restricted at most one".  (Contributed by Thierry
       Arnoux, 9-Dec-2019.) $)
    rmob2 $p |- ( ph -> ( x = B <-> ch ) ) $=
      ( cv wceq wcel wa wmo wb wrmo df-rmo sylib eleq1 mob2 syl112anc mpbirand
      anbi12d ) ADLZFMZFENZCHAUHUFENZBOZDPZUIBUGUHCOZQHABDERUKIBDESTJKUJULDFEUG
      UIUHBCUFFEUAGUEUBUCUD $.

    rmoi2.6 $e |- ( ph -> ch ) $.
    $( Consequence of "restricted at most one".  (Contributed by Thierry
       Arnoux, 9-Dec-2019.) $)
    rmoi2 $p |- ( ph -> x = B ) $=
      ( cv wceq rmob2 mpbird ) ADMFNCLABCDEFGHIJKOP $.
  $}

  ${
    $d x y $.  $d y A $.  $d y ph $.  $d y ps $.
    rmoanim.1 $e |- F/ x ph $.
    $( Introduction of a conjunct into restricted "at most one" quantifier,
       analogous to ~ moanim .  (Contributed by Alexander van der Vekens,
       25-Jun-2017.)  Avoid ~ ax-10 and ~ ax-11 .  (Revised by GG,
       24-Aug-2023.) $)
    rmoanim $p |- ( E* x e. A ( ph /\ ps ) <-> ( ph -> E* x e. A ps ) ) $=
      ( vy wa wi wral wex impexp exbii wmo wal df-rmo df-mo albii df-ral bitr4i
      wrmo weq ralbii r19.21 bitri cv wcel 3bitri imbi2i 19.37v 3bitr4i ) ABGZC
      FUAZHZCDIZFJZABULHZCDIZHZFJZUKCDTZABCDTZHZUNURFUNAUPHZCDIURUMVCCDABULKUBA
      UPCDEUCUDLUTCUEDUFZUKGZCMVEULHZCNZFJUOUKCDOVECFPVGUNFVGVDUMHZCNUNVFVHCVDU
      KULKQUMCDRSLUGVBAUQFJZHUSVAVIAVAVDBGZCMVJULHZCNZFJVIBCDOVJCFPVLUQFVLVDUPH
      ZCNUQVKVMCVDBULKQUPCDRSLUGUHAUQFUISUJ $.
    $( $j usage 'rmoanim' avoids 'ax-10' 'ax-11'; $)

    $( Alternate proof of ~ rmoanim , shorter but requiring ~ ax-10 and
       ~ ax-11 .  (Contributed by Alexander van der Vekens, 25-Jun-2017.)
       (New usage is discouraged.)  (Proof modification is discouraged.) $)
    rmoanimALT $p |- ( E* x e. A ( ph /\ ps ) <-> ( ph -> E* x e. A ps ) ) $=
      ( vy wa weq wi wral wex wrmo impexp ralbii r19.21 bitri exbii rmo2 imbi2i
      nfv 19.37v bitr4i 3bitr4i ) ABGZCFHZIZCDJZFKABUEIZCDJZIZFKZUDCDLABCDLZIZU
      GUJFUGAUHIZCDJUJUFUNCDABUEMNAUHCDEOPQUDCFDUDFTRUMAUIFKZIUKULUOABCFDBFTRSA
      UIFUAUBUC $.

    $( Introduction of a conjunct into restricted unique existential
       quantifier, analogous to ~ euan .  (Contributed by Alexander van der
       Vekens, 2-Jul-2017.) $)
    reuan $p |- ( E! x e. A ( ph /\ ps ) <-> ( ph /\ E! x e. A ps ) ) $=
      ( wa wreu wrex wrmo wi cv wcel simpl a1i rexlimi adantr simpr reximi reu5
      biimpa nfre1 ancrd impbid2 rmobida jca32 anbi2i 3imtr4i wb reubida impbii
      a1d ibar ) ABFZCDGZABCDGZFZUMCDHZUMCDIZFZABCDHZBCDIZFZFUNUPUSAUTVAUQAURUM
      ACDEUMAJCKDLZABMNOZPUQUTURUMBCDABQZRPUQURVAUQUMBCDUMCDUAUQVCFZUMBVEVFBAVF
      ABUQAVCVDPUKUBUCUDTUEUMCDSUOVBABCDSUFUGAUOUNABUMCDEABUMUHVCABULPUITUJ $.
  $}

  ${
    $d x y $.  $d y A $.  $d x B $.
    $( Double restricted existential uniqueness.  This theorem shows a
       condition under which a "naive" definition matches the correct one,
       analogous to ~ 2eu1 .  (Contributed by Alexander van der Vekens,
       25-Jun-2017.) $)
    2reu1 $p |- ( A. x e. A E* y e. B ph -> ( E! 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 ) ) ) $=
      ( wrmo wral wreu wrex wa wi 2reu5a wcel simprr rsp jca sylib com12 reu5
      cv adantr impcom rmoimia nfra1 rmoanim ancrd 2rmoswap imdistani simplbiim
      syl6 2reu2rex rexcom jctild anbi12i an4 bitri imbitrrdi 2rexreu impbid1
      ex ) ACEFZBDGZACEHBDHZACEIZBDHZABDIZCEHZJZVCVBVHVCVBVDBDIZVFCEIZJZVDBDFZV
      FCEFZJZJZVHVCVBVNVKVCVDVAJZBDIVPBDFZVBVNKABCDELVQVBVLVBJVNVQVBVLVQVBVDJZB
      DFVBVLKVRVPBDBTDMZVRVPVSVRJVDVAVSVBVDNVRVSVAVBVSVAKVDVABDOUAUBPUTUCVBVDBD
      VABDUDUEQUFVLVBVMVBVLVMABCDEUGRUHUJUIVCVIVJABCDEUKZVCVIVJVTABCDEULQPUMVHV
      IVLJZVJVMJZJVOVEWAVGWBVDBDSVFCESUNVIVLVJVMUOUPUQRABCDEURUS $.

    $( Double restricted existential uniqueness, analogous to ~ 2eu2 .
       (Contributed by Alexander van der Vekens, 29-Jun-2017.) $)
    2reu2 $p |- ( E! y e. B E. x e. A ph
                  -> ( E! x e. A E! y e. B ph <-> E! x e. A E. y e. B ph ) ) $=
      ( wrex wreu wrmo wral wi reurmo 2rmorex 2reu1 simpl biimtrdi 3syl 2rexreu
      wa expcom impbid ) ABDFZCEGZACEGBDGZACEFBDGZUBUACEHACEHBDIZUCUDJUACEKACBE
      DLUEUCUDUBRUDABCDEMUDUBNOPUDUBUCABCDEQST $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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 $.

  $( Indicate that "x" is not bound in "A" in the definition of csb.
     This is similar to $d x A $. $)
  $( $j free_var_in 'csb' with 'A' 'x'; $)

  ${
    $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 did not use distinguished brackets.
       When ` A ` is a proper class, this evaluates to the empty set (see
       ~ csbprc ).  Theorem ~ sbccsb recovers 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 $.
    $( Analogue 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 $.
  $}

  ${
    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 $.  $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 y ph $.  $d A y $.  $d B y $.  $d C y $.
    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 ) $=
      ( vy cv wcel wsbc cab csb eleq2d sbcbidv abbidv df-csb 3eqtr4g ) AGHZDIZB
      CJZGKREIZBCJZGKBCDLBCELATUBGASUABCADERFMNOBGCDPBGCEPQ $.
    $( $j usage 'csbeq2dv' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    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 x ph $.
    csbeq12dv.1 $e |- ( ph -> A = C ) $.
    csbeq12dv.2 $e |- ( ph -> B = D ) $.
    $( Formula-building inference for class substitution.  (Contributed by SN,
       3-Nov-2023.) $)
    csbeq12dv $p |- ( ph -> [_ A / x ]_ B = [_ C / x ]_ D ) $=
      ( csb csbeq1d csbeq2dv eqtrd ) ABCDIBEDIBEFIABCEDGJABEDFHKL $.
  $}

  ${
    $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 ) $.
    $( Change bound variables in a class substitution.  Interestingly, this
       does not require any bound variable conditions on ` A ` .  Version of
       ~ cbvcsb with a disjoint variable condition, which does not require
       ~ ax-13 .  (Contributed by Jeff Hankins, 13-Sep-2009.)  Avoid ~ ax-13 .
       (Revised 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 $.
    $( $j usage 'cbvcsbw' avoids 'ax-13'; $)
  $}

  ${
    $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 ` .  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  Use the weaker
       ~ cbvcsbw when possible.  (Contributed by Jeff Hankins, 13-Sep-2009.)
       (Revised by Mario Carneiro, 11-Dec-2016.)
       (New usage is discouraged.) $)
    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 z $.  $d y z B $.  $d x z C $.  $d A z $.
    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 $=
      ( vz cv wcel wsbc cab csb weq eleq2d cbvsbcvw abbii df-csb 3eqtr4i ) GHZD
      IZACJZGKSEIZBCJZGKACDLBCELUAUCGTUBABCABMDESFNOPAGCDQBGCEQR $.
    $( $j usage 'cbvcsbv' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    $d x y $.  $d y A $.
    $( Analogue 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 x y z $.  $d y z B $.
    $( Composition law for chained substitutions into a class.  Version of
       ~ csbco with a disjoint variable condition, which does not require
       ~ ax-13 .  (Contributed by NM, 10-Nov-2005.)  Avoid ~ ax-13 .  (Revised
       by GG, 10-Jan-2024.) $)
    csbcow $p |- [_ A / y ]_ [_ y / x ]_ B = [_ A / x ]_ B $=
      ( vz cv csb wcel wsbc cab df-csb eqabri sbcbii sbccow bitri abbii 3eqtr4i
      ) EFZABFZDGZHZBCIZEJRDHZACIZEJBCTGACDGUBUDEUBUCASIZBCIUDUAUEBCUEETAESDKLM
      UCABCNOPBECTKAECDKQ $.
    $( $j usage 'csbcow' avoids 'ax-13'; $)
  $}

  ${
    $d z A $.  $d y z B $.  $d x z $.
    $( Composition law for chained substitutions into a class.  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  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 eqabri sbcbii sbcco bitri abbii 3eqtr4i
      ) EFZABFZDGZHZBCIZEJRDHZACIZEJBCTGACDGUBUDEUBUCASIZBCIUDUAUEBCUEETAESDKLM
      UCABCNOPBECTKAECDKQ $.
  $}

  ${
    $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 eqabcdv
      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 y z $.  $d A y $.
    $( Substitution doesn't affect a constant ` B ` (in which ` x ` does not
       occur). ~ csbconstgf with distinct variable requirement.  (Contributed
       by Alan Sare, 22-Jul-2012.)  Avoid ~ ax-12 .  (Revised by GG,
       15-Oct-2024.) $)
    csbconstg $p |- ( A e. V -> [_ A / x ]_ B = B ) $=
      ( vy vz cv csb wceq csbeq1 eqeq1d wcel wsbc cab df-csb cvv sbcg elv abbii
      wb abid2 3eqtri vtoclg ) AEGZCHZCIABCHZCIEBDUDBIUEUFCAUDBCJKUEFGCLZAUDMZF
      NUGFNCAFUDCOUHUGFUHUGTEUGAUDPQRSFCUAUBUC $.
    $( $j usage 'csbconstg' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $d x y $.  $d y A $.  $d y B $.
    csbgfi.1 $e |- A e. _V $.
    csbgfi.2 $e |- F/_ x B $.
    $( Substitution for a variable not free in a class does not affect it, in
       inference form.  (Contributed by Giovanni Mascellani, 4-Jun-2019.) $)
    csbgfi $p |- [_ A / x ]_ B = B $=
      ( vy csb cv wcel wsbc df-csb eqabri nfcri sbcgfi bitri eqriv ) FABCGZCFHZ
      QIRCIZABJZSTFQAFBCKLSABDAFCEMNOP $.
  $}

  ${
    $d x y $.
    csbconstgi.1 $e |- A e. _V $.
    $( The proper substitution of a class for a variable in another variable
       does not modify it, in inference form.  (Contributed by Giovanni
       Mascellani, 30-May-2019.) $)
    csbconstgi $p |- [_ A / x ]_ y = y $=
      ( cvv wcel cv csb wceq csbconstg ax-mp ) CEFACBGZHLIDACLEJK $.
  $}

  ${
    $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 nfabdw nfcxfrd ) ABBCDGFHDIZ
      BCJZFKBFCDLARBFAFMAQBCENOP $.
  $}

  ${
    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 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 .  Usage of this theorem is discouraged
       because it depends on ~ ax-13 .  (Contributed by NM, 21-Nov-2005.)
       (Revised by Mario Carneiro, 12-Oct-2016.)
       (New usage is discouraged.) $)
    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, which does not
       require ~ ax-13 .  (Contributed by Mario Carneiro, 12-Oct-2016.)  Avoid
       ~ ax-13 .  (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 $.
    $( $j usage 'nfcsbw' avoids 'ax-13'; $)
  $}

  ${
    nfcsb.1 $e |- F/_ x A $.
    nfcsb.2 $e |- F/_ x B $.
    $( Bound-variable hypothesis builder for substitution into a class.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .  Use the
       weaker ~ nfcsbw when possible.  (Contributed by Mario Carneiro,
       12-Oct-2016.)  (New usage is discouraged.) $)
    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
      chvarfv ) AIZCJZDEJZKBIZCJZAUEDLZEJZKABUFUHAAUECFMAUGEAUEDNGOPUBUEJZUCUFU
      DUHUBUECQUIDUGEAUEDRSTHUA $.
  $}

  ${
    $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 $.
    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 y A $.  $d x y C $.  $d B y $.
    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.)  Reduce axiom usage.  (Revised
       by GG, 15-Oct-2024.) $)
    csbie $p |- [_ A / x ]_ B = C $=
      ( vy csb cv wcel wsbc cab df-csb wceq eleq2d sbcie abbii abid2 3eqtri ) A
      BCHGIZCJZABKZGLTDJZGLDAGBCMUBUCGUAUCABEAIBNCDTFOPQGDRS $.
    $( $j usage 'csbie' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $d x y z A $.  $d B y z $.  $d x y z C $.  $d x z 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.)  Reduce axiom usage.  (Revised by GG,
       15-Oct-2024.) $)
    csbied $p |- ( ph -> [_ A / x ]_ B = C ) $=
      ( vy vz csb cv wcel wsbc cab df-csb wb wal wceq wa eleq2d alrimiv df-clab
      sbcied wsb eleq1w sbcbidv sbievw bitr2i bibi1i biimpi sylg dfcleq sylibr
      weq eqtrid ) ABCDKILDMZBCNZIOZEBICDPAJLZUSMZUTEMZQZJRUSESAUTDMZBCNZVBQZVC
      JAVFJAVDVBBCFGABLCSTDEUTHUAUDUBVFVCVEVAVBVAURIJUEVEURJIUCURVEIJIJUOUQVDBC
      IJDUFUGUHUIUJUKULJUSEUMUNUP $.
    $( $j usage 'csbied' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $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 2sp
      impl ) AICJZBIDJZKEFJZLZBMZAMZACBDENFOUJAPUKAFQCORUKGSUKUFKZBDEFOUKUFBUIB
      AUAUFBUBUCULBFQDORULHSUKUFUGUHUIABUDUETT $.

    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 ~ csbie avoids a disjointness condition on ` x , A ` and
       ` x , D ` 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 eqabcdv eqtrid ) CGK
      ZACDLJMZDKZACNZJOFAJCDPUBUEJFUDUCEKUCFKABCGAMBMZQDEUCHRUFCQEFUCIRSTUA $.
  $}

  ${
    $d x y z $.  $d A z $.  $d B z $.  $d ph z $.  $d ps z $.
    cbvrabcsfw.1 $e |- F/_ y A $.
    cbvrabcsfw.2 $e |- F/_ x B $.
    cbvrabcsfw.3 $e |- F/ y ph $.
    cbvrabcsfw.4 $e |- F/ x ps $.
    cbvrabcsfw.5 $e |- ( x = y -> A = B ) $.
    cbvrabcsfw.6 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Version of ~ cbvrabcsf with a disjoint variable condition, which does
       not require ~ ax-13 .  (Contributed by Andrew Salmon, 13-Jul-2011.)
       (Revised by GG, 26-Jan-2024.) $)
    cbvrabcsfw $p |- { x e. A | ph } = { y e. B | ps } $=
      ( vz cv wcel wa cab crab csb nfv wsb nfcsb1v nfcri nfs1v nfan weq csbeq1a
      eleq12d sbequ12 anbi12d cbvabw nfcv nfcsbw nfsbv csbeq1 vex csbief eqtrdi
      id sbhypf eqtri df-rab 3eqtr4i ) CNZEOZAPZCQZDNZFOZBPZDQZACERBDFRVGMNZCVL
      ESZOZACMUAZPZMQVKVFVPCMVFMTVNVOCCMVMCVLEUBUCACMUDUECMUFZVEVNAVOVQVDVLEVMV
      QUSCVLEUGUHACMUIUJUKVPVJMDVNVODDMVMDCVLEDVLULGUMUCACMDIUNUEVJMTMDUFZVNVIV
      OBVRVLVHVMFVRUSVRVMCVHESFCVLVHEUOCVHEFDUPHKUQURUHABCMVHJLUTUJUKVAACEVBBDF
      VBVC $.
    $( $j usage 'cbvrabcsfw' avoids 'ax-13'; $)
  $}

  ${
    $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.  Usage of this theorem is discouraged because it
       depends on ~ ax-13 .  (Contributed by Andrew Salmon, 13-Jul-2011.)
       (New usage is discouraged.) $)
    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 cbvalv1 nfcv nfcsb csbeq1 cab df-csb wsb
      nfsbc eleq2d sbsbc bitr3i eqabi eqtr4i eqtrdi dfsbcq bitrdi bitri 3bitr4i
      sbie df-ral ) COZEPZAQZCRZDOZFPZBQZDRZACEUABDFUAVPMOZCWAEUBZPZACWASZQZMRV
      TVOWECMVOMUCWCWDCCMWBCWAEUDTACWAUEUHCMUIZVNWCAWDWFVMWAEWBWFUFCWAEUGUJACWA
      UKULUMWEVSMDWCWDDDMWBDCWAEDWAUNZGUOTADCWAWGIUTUHVSMUCMDUIZWCVRWDBWHWAVQWB
      FWHUFWHWBCVQEUBZFCWAVQEUPWINOZEPZCVQSZNUQFCNVQEURWLNFWJFPZWKCDUSWLWKWMCDC
      NFHTCDUIEFWJKVAVKWKCDVBVCVDVEVFUJWHWDACVQSZBACWAVQVGWNACDUSBACDVBABCDJLVK
      VCVHULUMVIACEVLBDFVLVJ $.

    $( A more general version of ~ cbvrexf that has no distinct variable
       restrictions.  Changes bound variables using implicit substitution.
       Usage of this theorem is discouraged because it depends on ~ ax-13 .
       (Contributed by Andrew Salmon, 13-Jul-2011.)  (Proof shortened by Mario
       Carneiro, 7-Dec-2014.)  (New usage is discouraged.) $)
    cbvrexcsf $p |- ( E. x e. A ph <-> E. y e. B ps ) $=
      ( wn wral wrex nfn weq notbid cbvralcsf dfrex2 notbii 3bitr4i ) AMZCENZMB
      MZDFNZMACEOBDFOUDUFUCUECDEFGHADIPBCJPKCDQABLRSUAACETBDFTUB $.

    $( A more general version of ~ cbvreuv that has no distinct variable
       restrictions.  Changes bound variables using implicit substitution.
       Usage of this theorem is discouraged because it depends on ~ ax-13 .
       (Contributed by Andrew Salmon, 13-Jul-2011.)
       (New usage is discouraged.) $)
    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 eqabi df-csb 3eqtr4ri eqtrdi sbequ bitrdi df-reu
      bitri 3bitr4i ) COZEPZAQZCRZDOZFPZBQZDRZACEUABDFUAVQMOZCWBEUBZPZACMSZQZMR
      WAVPWFCMVPMUCWDWECCMWCCWBEUDTACMUIUECMUFZVOWDAWEWGVNWBEWCWGUGCWBEUHUJACMU
      KULUMWFVTMDWDWEDDMWCDCWBEDWBUNGUOTACMDIUPUEVTMUCMDUFZWDVSWEBWHWBVRWCFWHUG
      WHWCCVREUBZFCWBVREUQNOZEPZCDSZNURWKCVRUSZNURFWIWLWMNWKCDUTVAWLNFWLWJFPZWK
      WNCDCNFHTCDUFEFWJKVBVCVDVECNVREVFVGVHUJWHWEACDSBAMDCVIABCDJLVCVJULUMVLACE
      VKBDFVKVM $.

    $( A more general version of ~ cbvrab with no distinct variable
       restrictions.  Usage of this theorem is discouraged because it depends
       on ~ ax-13 .  (Contributed by Andrew Salmon, 13-Jul-2011.)
       (New usage is discouraged.) $)
    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 eqabi 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.  Usage of this theorem is discouraged because it depends on
       ~ ax-13 .  (Contributed by David Moews, 1-May-2017.)
       (New usage is discouraged.) $)
    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.  Usage of this theorem is discouraged because it depends on
       ~ ax-13 .  (Contributed by David Moews, 1-May-2017.)
       (New usage is discouraged.) $)
    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 $)
  $c C. $.  $( Proper 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 $.

  $( Extend wff notation with proper subclass relation. $)
  wpss $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 eleq1w notbid anbi12d cbvabv eqtri ) AFZCGZQDG
      ZHZIZAJEFZCGZUBDGZHZIZEJBFZCGZUGDGZHZIZBJUAUFAEAEKZRUCTUEAECLULSUDAEDLMNO
      UFUKEBEBKZUCUHUEUJEBCLUMUDUIEBDLMNOP $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Define class difference, also called relative complement.  Definition
       5.12 of [TakeutiZaring] p. 20.  For example,
       ` ( { 1 , 3 } \ { 1 , 8 } ) = { 3 } ` ( ~ ex-dif ).  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 eleq1w orbi12d cbvabv eqtri ) AFZCGZODGZHZAIEFZCG
      ZSDGZHZEIBFZCGZUCDGZHZBIRUBAEAEJPTQUAAECKAEDKLMUBUFEBEBJTUDUAUEEBCKEBDKLM
      N $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Define the union of two classes.  Definition 5.6 of [TakeutiZaring]
       p. 16.  For example, ` ( { 1 , 3 } u. { 1 , 8 } ) = { 1 , 3 , 8 } `
       ( ~ ex-un ).  Contrast this operation with difference ` ( A \ B ) `
       ( ~ df-dif ) and intersection ` ( A i^i B ) ` ( ~ df-in ).  For an
       alternate definition in terms of class difference, requiring no dummy
       variables, see ~ dfun2 .  For union defined in terms of intersection,
       see ~ dfun3 .  (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 eleq1w anbi12d cbvabv eqtri ) AFZCGZODGZHZAIEFZCG
      ZSDGZHZEIBFZCGZUCDGZHZBIRUBAEAEJPTQUAAECKAEDKLMUBUFEBEBJTUDUAUEEBCKEBDKLM
      N $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Define the intersection of two classes.  Definition 5.6 of
       [TakeutiZaring] p. 16.  For example,
       ` ( { 1 , 3 } i^i { 1 , 8 } ) = { 1 } ` ( ~ ex-in ).  Contrast this
       operation with union ` ( A u. B ) ` ( ~ df-un ) and difference
       ` ( A \ B ) ` ( ~ df-dif ).  For alternate definitions in terms of class
       difference, requiring no dummy variables, see ~ dfin2 and ~ dfin4 .  For
       intersection defined in terms of union, see ~ dfin3 .  (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 A x $.  $d B x $.  $d C x $.
    $( 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 $.
  $}

  $( The elements of a set difference and the minuend are not equal.
     (Contributed by AV, 21-Oct-2023.) $)
  elneeldif $p |- ( ( X e. A /\ Y e. ( B \ A ) ) -> X =/= Y ) $=
    ( wcel cdif wne wn wa eldif nelne2 ex adantld biimtrid imp ) CAEZDBAFEZCDGZ
    QDBEZDAEHZIPRDBAJPTRSPTRCDAKLMNO $.

  $( Characterization of setvar elements of the complement of a class.
     (Contributed by Andrew Salmon, 15-Jul-2011.) $)
  velcomp $p |- ( x e. ( _V \ A ) <-> -. x e. A ) $=
    ( cv cvv cdif wcel wn vex eldif mpbiran ) ACZDBEFKDFKBFGAHKDBIJ $.

  ${
    $d A x $.  $d B x $.  $d C x $.
    $( 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 $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Subclasses and subsets
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d A x $.  $d B x $.
    $( Define the subclass relationship.  Definition 5.9 of [TakeutiZaring]
       p. 17.  For example, ` { 1 , 2 } C_ { 1 , 2 , 3 } ` ( ~ ex-ss ).  Note
       that ` A C_ A ` (proved in ~ ssid ).  Contrast this relationship with
       the relationship ` A C. B ` (as will be defined in ~ df-pss ).  For an
       alternative definition, not requiring a dummy variable, see ~ dfss2 .
       Other possible definitions are given by ~ dfss3 , ~ dfss4 , ~ sspss ,
       ~ ssequn1 , ~ ssequn2 , ~ sseqin2 , and ~ ssdif0 .

       We prefer the label "ss" ("subset") for ` C_ ` , despite the fact that
       it applies to classes.  It is much more common to refer to this as the
       subset relation than subclass, especially since most of the time the
       arguments are in fact sets (and for pragmatic reasons we don't want to
       need to use different operations for sets).  The way set.mm is set up,
       many things are technically classes despite morally (and provably) being
       sets, like ` 1 ` (cf. ~ df-1 and ~ 1ex ) or ` RR ` ( cf. ~ df-r and
       ~ reex ).  This has to do with the fact that there are no "set
       expressions": classes are expressions but there are only set variables
       in set.mm (cf.
       ~ https://us.metamath.org/downloads/grammar-ambiguity.txt ).  This is
       why we use ` C_ ` both for subclass relations and for subset relations
       and call it "subset".  (Contributed by NM, 8-Jan-2002.)  Revised from
       the original definition ~ dfss2 .  (Revised by GG, 15-May-2025.) $)
    df-ss $a |- ( A C_ B <-> A. x ( x e. A -> x e. B ) ) $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( Alternate definition of the subclass relationship between two classes.
       Exercise 9 of [TakeutiZaring] p. 18.  This was the original definition
       before ~ df-ss .  (Contributed by NM, 27-Apr-1994.)  Revise ~ df-ss .
       (Revised by GG, 15-May-2025.) $)
    dfss2 $p |- ( A C_ B <-> ( A i^i B ) = A ) $=
      ( vy vx cv wcel wa cab wb wal cin wss dfcleq df-in eqeq1i wi df-ss eleq1w
      wceq bitri simp2 3expib impbid dfbi2 pm2.21 pm3.4 ja simplbiim impbii wsb
      ancl df-clab weq anbi12d sbievw bitr2i bibi1i albii 3bitr4ri ) CEZAFZUTBF
      ZGZCHZASDEZVDFZVEAFZIZDJZABKZASABLZDVDAMVJVDACABNOVKVGVEBFZPZDJVIDABQVMVH
      DVMVGVLGZVGIZVHVMVOVMVNVGVMVGVLVGVMVGVLUAUBVGVLUKUCVOVNVGPVGVNPVMVNVGUDVG
      VNVMVGVLUEVGVLUFUGUHUIVNVFVGVFVCCDUJVNVCDCULVCVNCDCDUMVAVGVBVLCDARCDBRUNU
      OUPUQTURTUS $.
    $( $j usage 'dfss2' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  $( Variant of subclass definition ~ dfss2 .  (Contributed by NM,
     21-Jun-1993.) $)
  dfss $p |- ( A C_ B <-> A = ( A i^i B ) ) $=
    ( wss cin wceq dfss2 eqcom bitri ) ABCABDZAEAIEABFIAGH $.

  $( Define proper subclass (or strict subclass) relationship between two
     classes.  Definition 5.9 of [TakeutiZaring] p. 17.  For example,
     ` { 1 , 2 } C. { 1 , 2 , 3 } ` ( ~ ex-pss ).  Note that ` -. A C. A `
     (proved in ~ pssirr ).  Contrast this relationship with the relationship
     ` A C_ B ` (as defined in ~ df-ss ).  Other possible definitions are given
     by ~ dfpss2 and ~ dfpss3 .  (Contributed by NM, 7-Feb-1996.) $)
  df-pss $a |- ( A C. B <-> ( A C_ B /\ A =/= B ) ) $.

  ${
    $d x A $.  $d x B $.
    $( 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 df-ss df-ral bitr4i ) BCDAEZBFMCFZGAHNABIABCJNA
      BKL $.

    $( Alternate definition of subclass relationship.  (Contributed by RP,
       16-Apr-2020.) $)
    dfss6 $p |- ( A C_ B <-> -. E. x ( x e. A /\ -. x e. B ) ) $=
      ( wss cv wcel wi wal wn wa wex df-ss notnotb bitri exanali xchbinxr ) BCD
      ZAEZBFZRCFZGAHZIZSTIJAKQUAUBIABCLUAMNSTAOP $.
  $}

  ${
    $d z A $.  $d z B $.  $d x z $.
    dfssf.1 $e |- F/_ x A $.
    dfssf.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.)  Avoid ~ ax-13 .
       (Revised by GG, 19-May-2023.) $)
    dfssf $p |- ( A C_ B <-> A. x ( x e. A -> x e. B ) ) $=
      ( vz wss cv wcel wi wal df-ss nfcri nfim nfv eleq1w imbi12d cbvalv1 bitri
      weq ) BCGFHZBIZUACIZJZFKAHZBIZUECIZJZAKFBCLUDUHFAUBUCAAFBDMAFCEMNUHFOFATU
      BUFUCUGFABPFACPQRS $.
    $( $j usage 'dfssf' avoids 'ax-13'; $)

    $( 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 dfssf df-ral bitr4i ) BCFAGZBHOCHZIAJPABKABCDEL
      PABMN $.

    $( 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.)  Avoid ~ ax-12 .  (Revised by SN,
       27-May-2024.) $)
    ssel $p |- ( A C_ B -> ( C e. A -> C e. B ) ) $=
      ( vx wss cv wcel wi wal df-ss wceq wa wex id anim2d aleximi 3imtr4g sylbi
      dfclel ) ABEDFZAGZTBGZHZDIZCAGZCBGZHDABJUDTCKZUALZDMUGUBLZDMUEUFUCUHUIDUC
      UAUBUGUCNOPDCASDCBSQR $.
    $( $j usage 'ssel' avoids 'ax-12'; $)
  $}

  $( 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 implication 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,
       21-Jun-1993.) $)
    ssriv $p |- A C_ B $=
      ( wss cv wcel wi df-ss 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 dfssf 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 df-ss sylibr ) ABFZCGNDGHZBICDJAOBEKBCDLM $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Transitivity of subclass relationship.  Exercise 5 of [TakeutiZaring]
       p. 17.  (Contributed by NM, 24-Jun-1993.)  (Proof shortened by Andrew
       Salmon, 14-Jun-2011.)  Avoid axioms.  (Revised by GG, 19-May-2025.) $)
    sstr2 $p |- ( A C_ B -> ( B C_ C -> A C_ C ) ) $=
      ( vx cv wcel wi wal wss imim1 al2imi df-ss imbi12i 3imtr4i ) DEZAFZOBFZGZ
      DHQOCFZGZDHZPSGZDHZGABIBCIZACIZGRTUBDPQSJKDABLUDUAUEUCDBCLDACLMN $.
    $( $j usage 'sstr2' avoids 'ax-5' 'ax-6' 'ax-7' 'ax-8' 'ax-9' 'ax-10'
       'ax-11' 'ax-12' 'ax-ext' 'df-clel' 'df-cleq'; $)

    $( Obsolete version of ~ sstr2 as of 19-May-2025.  (Contributed by NM,
       24-Jun-1993.)  (Proof shortened by Andrew Salmon, 14-Jun-2011.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    sstr2OLD $p |- ( A C_ B -> ( B C_ C -> A C_ C ) ) $=
      ( vx wss cv wcel wi wal ssel imim1d alimdv df-ss 3imtr4g ) ABEZDFZBGZPCGZ
      HZDIPAGZRHZDIBCEACEOSUADOTQRABPJKLDBCMDACMN $.
  $}

  $( Transitivity of subclass relationship.  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, 21-May-1993.) $)
    eqss $p |- ( A = B <-> ( A C_ B /\ B C_ A ) ) $=
      ( vx cv wcel wb wal wi wa wceq wss albiim dfcleq df-ss 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 $.
  $}

  $( If a class is a subclass of another class, then the classes are equal if
     and only if the other class is a subclass of the first class.
     (Contributed by AV, 23-Dec-2020.) $)
  sssseq $p |- ( B C_ A -> ( A C_ B <-> A = B ) ) $=
    ( wceq wss eqss rbaibr ) ABCABDBADABEF $.

  ${
    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.)  (Proof shortened by BJ,
       1-Dec-2021.) $)
    eqrd $p |- ( ph -> A = B ) $=
      ( cv wcel wb wal wceq alrimi cleqf sylibr ) ABIZCJQDJKZBLCDMARBEHNBCDFGOP
      $.
  $}

  ${
    eqri.1 $e |- F/_ x A $.
    eqri.2 $e |- F/_ x B $.
    eqri.3 $e |- ( x e. A <-> x e. B ) $.
    $( Infer equality of classes from equivalence of membership.  (Contributed
       by Thierry Arnoux, 7-Oct-2017.) $)
    eqri $p |- A = B $=
      ( wceq wtru nftru cv wcel wb a1i eqrd mptru ) BCGHABCAIDEAJZBKPCKLHFMNO
      $.
  $}

  ${
    $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, 21-Jun-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.  (Contributed by NM,
       31-Oct-1995.) $)
    ssv $p |- A C_ _V $=
      ( vx cvv cv elex ssriv ) BACBDAEF $.
  $}

  $( Equality theorem for subclasses.  (Contributed by NM, 24-Jun-1993.)
     (Proof shortened by Andrew Salmon, 21-Jun-2011.) $)
  sseq1 $p |- ( A = B -> ( A C_ C <-> B C_ C ) ) $=
    ( wceq wss wa wb eqss sstr2 anbiim ancoms sylbi ) ABDABEZBAEZFACEZBCEZGZABH
    NMQNMOPBACIABCIJKL $.

  $( Equality theorem for the subclass relationship.  (Contributed by NM,
     25-Jun-1998.) $)
  sseq2 $p |- ( A = B -> ( C C_ A <-> C C_ B ) ) $=
    ( wceq wss wa wb eqss sstr2 com12 anbiim sylbi ) ABDABEZBAEZFCAEZCBEZGABHMN
    OPOMPCABIJPNOCBAIJKL $.

  $( 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 $.
    $}
  $}

  ${
    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 $.
  $}

  ${
    eqsstrid.1 $e |- A = B $.
    eqsstrid.2 $e |- ( ph -> B C_ C ) $.
    $( A 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 ) $.
    $( A 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 ) $=
      ( wss a1i sseqtrd ) ACBDCBGAEHFI $.
  $}

  ${
    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 ) $=
      ( eqcomd sseqtrid ) ABCDEADBFGH $.
  $}

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  ${
    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 $=
      ( eqsstrri sseqtri ) CBDCABFEHGI $.
  $}

  ${
    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 $=
      ( eqsstri sseqtrri ) CBDCABFEHGI $.
  $}

  ${
    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 ) $=
      ( eqsstrrid sseqtrdi ) ADCEADBCGFIHJ $.
  $}

  ${
    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 ) $=
      ( eqsstrid sseqtrrdi ) ADCEADBCGFIHJ $.
  $}

  ${
    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 ) $=
      ( eqsstrrd sseqtrd ) ADCEADBCGFIHJ $.
  $}

  ${
    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 ) $=
      ( eqsstrd sseqtrrd ) ADCEADBCGFIHJ $.
  $}

  ${
    eqimssd.1 $e |- ( ph -> A = B ) $.
    $( Equality implies inclusion, deduction version.  (Contributed by SN,
       6-Nov-2024.) $)
    eqimssd $p |- ( ph -> A C_ B ) $=
      ( ssid eqsstrdi ) ABCCDCEF $.

    $( Equality implies inclusion, deduction version.  (Contributed by SN,
       15-Feb-2025.) $)
    eqimsscd $p |- ( ph -> B C_ A ) $=
      ( ssid eqsstrrdi ) ACBBDBEF $.
  $}

  $( Equality implies inclusion.  (Contributed by NM, 21-Jun-1993.)  (Proof
     shortened by Andrew Salmon, 21-Jun-2011.) $)
  eqimss $p |- ( A = B -> A C_ B ) $=
    ( wceq id eqimssd ) ABCZABFDE $.

  $( Equality implies inclusion.  (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.  Exercise 13 of [TakeutiZaring]
       p. 18.  (Contributed by NM, 25-Feb-1996.)  (Proof shortened by Andrew
       Salmon, 21-Jun-2011.) $)
    nss $p |- ( -. A C_ B <-> E. x ( x e. A /\ -. x e. B ) ) $=
      ( cv wcel wn wa wex wss wi wal exanali df-ss xchbinxr bicomi ) ADZBEZPCEZ
      FGAHZBCIZFSQRJAKTQRALABCMNO $.
  $}

  $( 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 dfssf 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.)  Avoid axioms.  (Revised by GG, 19-May-2025.) $)
    ssralv $p |- ( A C_ B -> ( A. x e. B ph -> A. x e. A ph ) ) $=
      ( wss cv wcel wi wal wral df-ss imim1 al2imi df-ral 3imtr4g sylbi ) CDEBF
      ZCGZQDGZHZBIZABDJZABCJZHBCDKUASAHZBIRAHZBIUBUCTUDUEBRSALMABDNABCNOP $.
    $( $j usage 'ssralv' avoids 'ax-5' 'ax-6' 'ax-7' 'ax-8' 'ax-9' 'ax-ext'
       'df-clel' 'df-cleq'; $)

    $( Existential quantification restricted to a subclass.  (Contributed by
       NM, 11-Jan-2007.)  Avoid axioms.  (Revised by GG, 19-May-2025.) $)
    ssrexv $p |- ( A C_ B -> ( E. x e. A ph -> E. x e. B ph ) ) $=
      ( wss cv wcel wi wal wrex df-ss wex pm3.45 aleximi df-rex 3imtr4g sylbi
      wa ) CDEBFZCGZSDGZHZBIZABCJZABDJZHBCDKUCTARZBLUAARZBLUDUEUBUFUGBTUAAMNABC
      OABDOPQ $.
    $( $j usage 'ssrexv' avoids 'ax-5' 'ax-6' 'ax-7' 'ax-8' 'ax-9' 'ax-ext'
       'df-clel' 'df-cleq'; $)

    $d y A $.  $d y B $.
    $( Two quantifications restricted to a subclass.  (Contributed by AV,
       11-Mar-2023.) $)
    ss2ralv $p |- ( A C_ B -> ( A. x e. B A. y e. B ph
                                -> A. x e. A A. y e. A ph ) ) $=
      ( wss wral ssralv ralimdv syld ) DEFZACEGZBEGACDGZBEGMBDGKLMBEACDEHIMBDEH
      J $.

    $( Two existential quantifications restricted to a subclass.  (Contributed
       by AV, 11-Mar-2023.) $)
    ss2rexv $p |- ( A C_ B -> ( E. x e. A E. y e. A ph
                                -> E. x e. B E. y e. B ph ) ) $=
      ( wss wrex ssrexv reximdv syld ) DEFZACDGZBDGACEGZBDGMBEGKLMBDACDEHIMBDEH
      J $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Obsolete version of ~ ssralv as of 19-May-2025.  (Contributed by NM,
       11-Mar-2006.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    ssralvOLD $p |- ( A C_ B -> ( A. x e. B ph -> A. x e. A ph ) ) $=
      ( wss cv wcel ssel imim1d ralimdv2 ) CDEZAABDCKBFZCGLDGACDLHIJ $.

    $( Obsolete version of ~ ssrexv as of 19-May-2025.  (Contributed by NM,
       11-Jan-2007.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    ssrexvOLD $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.)  Avoid axioms.  (Revised by
       SN, 14-Oct-2025.) $)
    ralss $p |- ( A C_ B -> ( A. x e. A ph <->
                              A. x e. B ( x e. A -> ph ) ) ) $=
      ( wss cv wcel wi wal wral wb df-ss wa pm4.71rd imbi1d impexp bitrdi alimi
      id df-ral sylbi albi syl 3bitr4g ) CDEZBFZCGZAHZBIZUFDGZUHHZBIZABCJUHBDJU
      EUHUKKZBIZUIULKUEUGUJHZBIUNBCDLUOUMBUOUHUJUGMZAHUKUOUGUPAUOUGUJUOSNOUJUGA
      PQRUAUHUKBUBUCABCTUHBDTUD $.
    $( $j usage 'ralss' avoids 'ax-5' 'ax-6' 'ax-7' 'ax-8'; $)

    $( Restricted existential quantification on a subset in terms of superset.
       (Contributed by Stefan O'Rear, 3-Apr-2015.)  Avoid axioms.  (Revised by
       SN, 14-Oct-2025.) $)
    rexss $p |- ( A C_ B -> ( E. x e. A ph <->
                              E. x e. B ( x e. A /\ ph ) ) ) $=
      ( wss cv wcel wa wex wi wal wb df-ss pm3.41 pm4.71rd alexbii sylbi df-rex
      wrex 3bitr4g ) CDEZBFZCGZAHZBIZUBDGZUDHZBIZABCSUDBDSUAUCUFJZBKUEUHLBCDMUI
      UDUGBUIUDUFUCAUFNOPQABCRUDBDRT $.
    $( $j usage 'rexss' avoids 'ax-5' 'ax-6' 'ax-7' 'ax-8'; $)

    $( Obsolete version of ~ ralss as of 14-Oct-2025.  (Contributed by Stefan
       O'Rear, 3-Apr-2015.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    ralssOLD $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 $.

    $( Obsolete version of ~ rexss as of 14-Oct-2025.  (Contributed by Stefan
       O'Rear, 3-Apr-2015.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    rexssOLD $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 dfssf abid imbi12i albii bitri ) ACDZBCDZECF
    ZPGZRQGZHZCIABHZCICPQACJBCJKUAUBCSATBACLBCLMNO $.

  ${
    $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 $.
  $}

  ${
    $d x y ph $.  $d y ch $.  $d y ps $.
    ss2abdv.1 $e |- ( ph -> ( ps -> ch ) ) $.
    $( Deduction of abstraction subclass from implication.  (Contributed by NM,
       29-Jul-2011.)  (Revised by Steven Nguyen, 28-Jun-2024.) $)
    ss2abdv $p |- ( ph -> { x | ps } C_ { x | ch } ) $=
      ( vy cab wsb cv wcel sbimdv df-clab 3imtr4g ssrdv ) AFBDGZCDGZABDFHCDFHFI
      ZOJQPJABCDFEKBFDLCFDLMN $.
    $( $j usage 'ss2abdv' avoids 'ax-6' 'ax-7' 'ax-8' 'ax-9' 'ax-10' 'ax-11'
       'ax-12' 'df-clel' 'df-cleq' 'ax-ext' ; $)
  $}

  ${
    ss2abi.1 $e |- ( ph -> ps ) $.
    $( Inference of abstraction subclass from implication.  (Contributed by NM,
       31-Mar-1995.)  Avoid ~ ax-8 , ~ ax-10 , ~ ax-11 , ~ ax-12 .  (Revised by
       GG, 28-Jun-2024.) $)
    ss2abi $p |- { x | ph } C_ { x | ps } $=
      ( wtru cab wss tru wi a1i ss2abdv ax-mp ) EACFBCFGHEABCABIEDJKL $.
    $( $j usage 'ss2abi' avoids 'ax-8' 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $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.)  (Proof shortened by SN, 22-Dec-2024.) $)
    abssdv $p |- ( ph -> { x | ps } C_ A ) $=
      ( cab cv wcel ss2abdv abid1 sseqtrrdi ) ABCFCGDHZCFDABLCEICDJK $.
    $( $j usage 'abssdv' avoids 'ax-10' 'ax-11' 'ax-12'; $)

    $( Obsolete version of ~ abssdv as of 12-Dec-2024.  (Contributed by NM,
       20-Jan-2006.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    abssdvOLD $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 df-ss ss2ab 3imtr4i 3sstr4g
      df-rab ) CDEZBFZCGZAHZBIZUADGZAHZBIZABCJABDJUBUEKZBLUCUFKZBLTUDUGEUHUIBUB
      UEAMNBCDOUCUFBPQABCSABDSR $.

    $( Subclass relation for the restriction of a class abstraction.
       (Contributed by NM, 31-Mar-1995.) $)
    ssab2 $p |- { x | ( x e. A /\ ph ) } C_ A $=
      ( cv wcel wa simpl abssi ) BDCEZAFBCIAGH $.

    $d A x y $.  $d ph y $.
    $( Subclass relation for a restricted class.  (Contributed by NM,
       19-Mar-1997.)  (Proof shortened by BJ and SN, 8-Aug-2024.) $)
    ssrab2 $p |- { x e. A | ph } C_ A $=
      ( vy crab cv elrabi ssriv ) DABCECABDFCGH $.
  $}

  ${
    $d x A $.  $d x B $.  $d x ph $.
    rabss3d.1 $e |- ( ( ph /\ ( x e. A /\ ps ) ) -> x e. B ) $.
    $( Subclass law for restricted abstraction.  (Contributed by Thierry
       Arnoux, 25-Sep-2017.) $)
    rabss3d $p |- ( ph -> { x e. A | ps } C_ { x e. B | ps } ) $=
      ( crab nfv nfrab1 cv wcel wa simprr jca ex rabid 3imtr4g ssrd ) ACBCDGZBC
      EGZACHBCDIBCEIACJZDKZBLZUAEKZBLZUASKUATKAUCUEAUCLUDBFAUBBMNOBCDPBCEPQR $.
  $}

  ${
    $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 B x $.  $d ph x $.
    eqrrabd.1 $e |- ( ph -> B C_ A ) $.
    eqrrabd.2 $e |- ( ( ph /\ x e. A ) -> ( x e. B <-> ps ) ) $.
    $( Deduce equality with a restricted abstraction.  (Contributed by Thierry
       Arnoux, 11-Apr-2024.) $)
    eqrrabd $p |- ( ph -> B = { x e. A | ps } ) $=
      ( crab nfcv nfrab1 cv wcel wa sseld pm4.71rd pm5.32da bitrd rabid bitr4di
      nfv eqrd ) ACEBCDHZACTCEIBCDJACKZELZUCDLZBMZUCUBLAUDUEUDMUFAUDUEAEDUCFNOA
      UEUDBGPQBCDRSUA $.
  $}

  ${
    $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 $.
  $}

  $( Alternate definition of proper subclass.  (Contributed by NM,
     7-Feb-1996.) $)
  dfpss2 $p |- ( A C. B <-> ( A C_ B /\ -. A = B ) ) $=
    ( wpss wss wne wa wceq wn df-pss df-ne anbi2i bitri ) ABCABDZABEZFMABGHZFAB
    INOMABJKL $.

  $( Alternate definition of proper subclass.  (Contributed by NM, 7-Feb-1996.)
     (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
  dfpss3 $p |- ( A C. B <-> ( A C_ B /\ -. B C_ A ) ) $=
    ( wpss wss wceq wn wa dfpss2 eqss baib notbid pm5.32i bitri ) ABCABDZABEZFZ
    GNBADZFZGABHNPRNOQONQABIJKLM $.

  $( Equality theorem for proper subclass.  (Contributed by NM, 7-Feb-1996.) $)
  psseq1 $p |- ( A = B -> ( A C. C <-> B C. C ) ) $=
    ( wceq wss wne wa wpss sseq1 neeq1 anbi12d df-pss 3bitr4g ) ABDZACEZACFZGBC
    EZBCFZGACHBCHNOQPRABCIABCJKACLBCLM $.

  $( Equality theorem for proper subclass.  (Contributed by NM, 7-Feb-1996.) $)
  psseq2 $p |- ( A = B -> ( C C. A <-> C C. B ) ) $=
    ( wceq wss wne wa wpss sseq2 neeq2 anbi12d df-pss 3bitr4g ) ABDZCAEZCAFZGCB
    EZCBFZGCAHCBHNOQPRABCIABCJKCALCBLM $.

  ${
    psseq1i.1 $e |- A = B $.
    $( An equality inference for the proper subclass relationship.
       (Contributed by NM, 9-Jun-2004.) $)
    psseq1i $p |- ( A C. C <-> B C. C ) $=
      ( wceq wpss wb psseq1 ax-mp ) ABEACFBCFGDABCHI $.

    $( An equality inference for the proper subclass relationship.
       (Contributed by NM, 9-Jun-2004.) $)
    psseq2i $p |- ( C C. A <-> C C. B ) $=
      ( wceq wpss wb psseq2 ax-mp ) ABECAFCBFGDABCHI $.

    ${
      psseq12i.2 $e |- C = D $.
      $( An equality inference for the proper subclass relationship.
         (Contributed by NM, 9-Jun-2004.) $)
      psseq12i $p |- ( A C. C <-> B C. D ) $=
        ( wpss psseq1i psseq2i bitri ) ACGBCGBDGABCEHCDBFIJ $.
    $}
  $}

  ${
    psseq1d.1 $e |- ( ph -> A = B ) $.
    $( An equality deduction for the proper subclass relationship.
       (Contributed by NM, 9-Jun-2004.) $)
    psseq1d $p |- ( ph -> ( A C. C <-> B C. C ) ) $=
      ( wceq wpss wb psseq1 syl ) ABCFBDGCDGHEBCDIJ $.

    $( An equality deduction for the proper subclass relationship.
       (Contributed by NM, 9-Jun-2004.) $)
    psseq2d $p |- ( ph -> ( C C. A <-> C C. B ) ) $=
      ( wceq wpss wb psseq2 syl ) ABCFDBGDCGHEBCDIJ $.

    ${
      psseq12d.2 $e |- ( ph -> C = D ) $.
      $( An equality deduction for the proper subclass relationship.
         (Contributed by NM, 9-Jun-2004.) $)
      psseq12d $p |- ( ph -> ( A C. C <-> B C. D ) ) $=
        ( wpss psseq1d psseq2d bitrd ) ABDHCDHCEHABCDFIADECGJK $.
    $}
  $}

  $( A proper subclass is a subclass.  Theorem 10 of [Suppes] p. 23.
     (Contributed by NM, 7-Feb-1996.) $)
  pssss $p |- ( A C. B -> A C_ B ) $=
    ( wpss wss wne df-pss simplbi ) ABCABDABEABFG $.

  $( Two classes in a proper subclass relationship are not equal.  (Contributed
     by NM, 16-Feb-2015.) $)
  pssne $p |- ( A C. B -> A =/= B ) $=
    ( wpss wss wne df-pss simprbi ) ABCABDABEABFG $.

  ${
    pssssd.1 $e |- ( ph -> A C. B ) $.
    $( Deduce subclass from proper subclass.  (Contributed by NM,
       29-Feb-1996.) $)
    pssssd $p |- ( ph -> A C_ B ) $=
      ( wpss wss pssss syl ) ABCEBCFDBCGH $.

    $( Proper subclasses are unequal.  Deduction form of ~ pssne .
       (Contributed by David Moews, 1-May-2017.) $)
    pssned $p |- ( ph -> A =/= B ) $=
      ( wpss wne pssne syl ) ABCEBCFDBCGH $.
  $}

  $( Subclass in terms of proper subclass.  (Contributed by NM,
     25-Feb-1996.) $)
  sspss $p |- ( A C_ B <-> ( A C. B \/ A = B ) ) $=
    ( wss wpss wceq wo wn dfpss2 simplbi2 con1d orrd pssss eqimss jaoi impbii )
    ABCZABDZABEZFPQRPRQQPRGABHIJKQPRABLABMNO $.

  $( Proper subclass is irreflexive.  Theorem 7 of [Suppes] p. 23.
     (Contributed by NM, 7-Feb-1996.) $)
  pssirr $p |- -. A C. A $=
    ( wpss wss wn wa pm3.24 dfpss3 mtbir ) AABAACZIDEIFAAGH $.

  $( Proper subclass has no 2-cycle loops.  Compare Theorem 8 of [Suppes]
     p. 23.  (Contributed by NM, 7-Feb-1996.)  (Proof shortened by Andrew
     Salmon, 26-Jun-2011.) $)
  pssn2lp $p |- -. ( A C. B /\ B C. A ) $=
    ( wpss wn wi wa wss dfpss3 simprbi pssss nsyl imnan mpbi ) ABCZBACZDENOFDNB
    AGZONABGPDABHIBAJKNOLM $.

  $( Two ways of stating trichotomy with respect to inclusion.  (Contributed by
     NM, 12-Aug-2004.) $)
  sspsstri $p |- ( ( A C_ B \/ B C_ A ) <-> ( A C. B \/ A = B \/ B C. A ) ) $=
    ( wpss wo wceq wss w3o or32 sspss eqcom orbi2i bitri orbi12i orordir bitr4i
    df-3or 3bitr4i ) ABCZBACZDABEZDZRTDZSDABFZBAFZDZRTSGRSTHUEUBSTDZDUAUCUBUDUF
    ABIUDSBAEZDUFBAIUGTSBAJKLMRSTNORTSPQ $.

  $( Partial trichotomy law for subclasses.  (Contributed by NM, 16-May-1996.)
     (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
  ssnpss $p |- ( A C_ B -> -. B C. A ) $=
    ( wpss wss wn dfpss3 simprbi con2i ) BACZABDZIBADJEBAFGH $.

  $( Transitive law for proper subclass.  Theorem 9 of [Suppes] p. 23.
     (Contributed by NM, 7-Feb-1996.) $)
  psstr $p |- ( ( A C. B /\ B C. C ) -> A C. C ) $=
    ( wpss wa wss wceq pssss sylan9ss pssn2lp psseq1 anbi1d mtbiri con2i dfpss2
    wn sylanbrc ) ABDZBCDZEZACFACGZPACDRSABCABHBCHIUATUATCBDZSECBJUARUBSACBKLMN
    ACOQ $.

  $( Transitive law for subclass and proper subclass.  (Contributed by NM,
     3-Apr-1996.) $)
  sspsstr $p |- ( ( A C_ B /\ B C. C ) -> A C. C ) $=
    ( wss wpss wceq wo sspss wi psstr ex psseq1 biimprd jaoi imp sylanb ) ABDAB
    EZABFZGZBCEZACEZABHSTUAQTUAIRQTUAABCJKRUATABCLMNOP $.

  $( Transitive law for subclass and proper subclass.  (Contributed by NM,
     3-Apr-1996.) $)
  psssstr $p |- ( ( A C. B /\ B C_ C ) -> A C. C ) $=
    ( wss wpss wceq wo sspss psstr ex psseq2 biimpcd jaod imp sylan2b ) BCDABEZ
    BCEZBCFZGZACEZBCHPSTPQTRPQTABCIJRPTBCAKLMNO $.

  ${
    psstrd.1 $e |- ( ph -> A C. B ) $.
    psstrd.2 $e |- ( ph -> B C. C ) $.
    $( Proper subclass inclusion is transitive.  Deduction form of ~ psstr .
       (Contributed by David Moews, 1-May-2017.) $)
    psstrd $p |- ( ph -> A C. C ) $=
      ( wpss psstr syl2anc ) ABCGCDGBDGEFBCDHI $.
  $}

  ${
    sspsstrd.1 $e |- ( ph -> A C_ B ) $.
    sspsstrd.2 $e |- ( ph -> B C. C ) $.
    $( Transitivity involving subclass and proper subclass inclusion.
       Deduction form of ~ sspsstr .  (Contributed by David Moews,
       1-May-2017.) $)
    sspsstrd $p |- ( ph -> A C. C ) $=
      ( wss wpss sspsstr syl2anc ) ABCGCDHBDHEFBCDIJ $.
  $}

  ${
    psssstrd.1 $e |- ( ph -> A C. B ) $.
    psssstrd.2 $e |- ( ph -> B C_ C ) $.
    $( Transitivity involving subclass and proper subclass inclusion.
       Deduction form of ~ psssstr .  (Contributed by David Moews,
       1-May-2017.) $)
    psssstrd $p |- ( ph -> A C. C ) $=
      ( wpss wss psssstr syl2anc ) ABCGCDHBDGEFBCDIJ $.
  $}

  $( A class is not a proper subclass of another iff it satisfies a
     one-directional form of ~ eqss .  (Contributed by Mario Carneiro,
     15-May-2015.) $)
  npss $p |- ( -. A C. B <-> ( A C_ B -> A = B ) ) $=
    ( wss wceq wi wpss wn wa pm4.61 dfpss2 bitr4i con1bii ) ABCZABDZEZABFZOGMNG
    HPMNIABJKL $.

  $( A subclass missing a member is a proper subclass.  (Contributed by NM,
     12-Jan-2002.) $)
  ssnelpss $p |- ( A C_ B -> ( ( C e. B /\ -. C e. A ) -> A C. B ) ) $=
    ( wcel wn wa wceq wss wpss nelneq2 eqcom sylnib dfpss2 baibr imbitrid ) CBD
    CADEFZABGZEZABHZABIZPBAGQCBAJBAKLTSRABMNO $.

  ${
    ssnelpssd.1 $e |- ( ph -> A C_ B ) $.
    ssnelpssd.2 $e |- ( ph -> C e. B ) $.
    ssnelpssd.3 $e |- ( ph -> -. C e. A ) $.
    $( Subclass inclusion with one element of the superclass missing is proper
       subclass inclusion.  Deduction form of ~ ssnelpss .  (Contributed by
       David Moews, 1-May-2017.) $)
    ssnelpssd $p |- ( ph -> A C. B ) $=
      ( wcel wn wpss wss wa wi ssnelpss syl mp2and ) ADCHZDBHIZBCJZFGABCKQRLSME
      BCDNOP $.
  $}

  ${
    $d A x $.  $d B x $.
    $( If there is an element of a class which is not contained in a subclass,
       the subclass is a proper subclass.  (Contributed by AV, 29-Jan-2020.) $)
    ssexnelpss $p |- ( ( A C_ B /\ E. x e. B x e/ A ) -> A C. B ) $=
      ( cv wnel wrex wpss wcel wn wa df-nel ssnelpss expdimp biimtrid rexlimdva
      wss imp ) BCPZADZBEZACFBCGZRTUAACTSBHIZRSCHZJUASBKRUCUBUABCSLMNOQ $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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.  (Contributed by BJ and Jim
       Kingdon, 16-Jun-2022.)  (Proof shortened by SN, 15-Aug-2025.) $)
    dfdif3 $p |- ( A \ B ) = { x e. A | A. y e. B x =/= y } $=
      ( cv wcel wn wne wral cdif dfdif2 nelb necom ralbii bitri rabbieq ) AEZDF
      GZQBEZHZBDIZACCDJACDKRSQHZBDIUABQDLUBTBDSQMNOP $.
    $( $j usage 'dfdif3' avoids 'ax-10' 'ax-11' 'ax-12'; $)

    $( Obsolete version of ~ dfdif3 as of 15-Aug-2025.  (Contributed by BJ and
       Jim Kingdon, 16-Jun-2022.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    dfdif3OLD $p |- ( A \ B ) = { x e. A | A. y e. B x =/= y } $=
      ( cdif cv wcel wn crab wne wral dfdif2 wi wal weq wex ax6ev bitr4i 3bitri
      wa biantrur 19.41v sbalex equcom imbi1i eleq1w notbid pm5.74i con2b df-ne
      bicomi imbi2i bitri albii df-ral rabbii eqtri ) CDEAFZDGZHZACIURBFZJZBDKZ
      ACIACDLUTVCACUTVADGZVBMZBNZVCUTBAOZUTTBPZVGUTMZBNVFUTVGBPZUTTVHVJUTBAQUAV
      GUTBUBRUTBAUCVIVEBVIABOZUTMZVEVGVKUTBAUDUEVLVKVDHZMVDVKHZMVEVKUTVMVKUSVDA
      BDUFUGUHVKVDUIVNVBVDVBVNURVAUJUKULSUMUNSVBBDUORUPUQ $.
  $}

  ${
    $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.)  Avoid
       ~ ax-10 , ~ ax-11 , ~ ax-12 .  (Revised by SN, 14-May-2025.) $)
    nfdif $p |- F/_ x ( A \ B ) $=
      ( vy cdif cv wcel wn wa eldif nfcri nfn nfan nfxfr nfci ) AFBCGZFHZRISBIZ
      SCIZJZKASBCLTUBAAFBDMUAAAFCEMNOPQ $.
    $( $j usage 'nfdif' avoids 'ax-10' 'ax-11' 'ax-12'; $)

    $( Obsolete version of ~ nfdif as of 14-May-2025.  (Contributed by NM,
       3-Dec-2003.)  (Revised by Mario Carneiro, 13-Oct-2016.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    nfdifOLD $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 $.

  $( Implication of membership in a class difference.  (Contributed by NM,
     28-Jun-1994.) $)
  neldif $p |- ( ( A e. B /\ -. A e. ( B \ C ) ) -> A e. C ) $=
    ( wcel cdif wn eldif simplbi2 con1d imp ) ABDZABCEDZFACDZKMLLKMFABCGHIJ $.

  ${
    $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 wi wa wn pm4.45im eldif xchbinxr anbi2i bitr2i difeqri
      iman ) CABADZACEZAFZSRBFZSGZHSRQFZIZHSTJUAUCSUATSIHUBTSPRBAKLMNO $.

    $( 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 $.
    $( Double complement under universal class.  Exercise 4.10(s) of
       [Mendelson] p. 231.  (Contributed by NM, 8-Jan-2002.) $)
    ddif $p |- ( _V \ ( _V \ A ) ) = A $=
      ( vx cvv cdif cv wcel wn wa eldif mpbiran con2bii biantrur bitr2i difeqri
      vex ) BCCADZABEZAFZQPFZGZQCFZTHSRSUARGBOZQCAIJKUATUBLMN $.
  $}

  ${
    $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 3bitr4g eldif df-ss
      imbi2i syl2an con2b a1i anbi12d albidv ) ACEZBCEZFZDGZAHZUICBIZHZJZDKUIBH
      ZUICAIZHZJZDKAUKEBUOEUHUMUQDUHUJUICHZUNLZFZJZUNURUJLZFZJZUMUQUHUJURJZUJUS
      JZFUNURJZUNVBJZFVAVDUHVEVGVFVHUFVEVGVEVGMUGACUINBCUINVEVGOUAVFVHMUHUJUNUB
      UCUDUJURUSPUNURVBPQULUTUJUICBRTUPVCUNUICARTQUEDAUKSDBUOSQ $.

    $( 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 df-nel anbi2i bitr4i imbi1i bitr3i ralbii2
    wn eldif ) BEZDFZAGZABCCDHZTCIZUBGUDUAJZAGTUCIZAGUDUAAKUEUFAUEUDTDIRZJUFUAU
    GUDTDLMTCDSNOPQ $.

  $( Restricted existential quantification over a difference.  (Contributed by
     AV, 25-Oct-2023.) $)
  rexdifi $p |- ( ( E. x e. A ph /\ A. x e. B -. ph )
                    -> E. x e. ( A \ B ) ph ) $=
    ( wrex wn wral wa cv cdif wcel wex wi df-rex df-ral nfa1 simprl con2 impcom
    wal sps com12 adantl eldifd simprr jca ex eximd syl2anb sylibr ) ABCEZAFZBD
    GZHBIZCDJZKZAHZBLZABUOEUKUNCKZAHZBLZUNDKZULMZBTZURUMABCNULBDOVDVAURVDUTUQBV
    CBPVDUTUQVDUTHZUPAVEUNCDVDUSAQUTVDVBFZAVDVFMUSVDAVFVCAVFMBVBARUAUBUCSUDVDUS
    AUEUFUGUHSUIABUONUJ $.

  $( Complementation reverses inclusion.  (Contributed by Andrew Salmon,
     15-Jul-2011.)  (Proof shortened by BJ, 19-Mar-2021.) $)
  complss $p |- ( A C_ B <-> ( _V \ B ) C_ ( _V \ A ) ) $=
    ( wss cvv cdif sscon ddif 3sstr3g impbii ) ABCDBEZDAEZCZABDFLDKEDJEABJKDFAG
    BGHI $.

  $( Two classes are equal if and only if their complements are equal.
     (Contributed by BJ, 19-Mar-2021.) $)
  compleq $p |- ( A = B <-> ( _V \ A ) = ( _V \ B ) ) $=
    ( wss wa cvv cdif wceq complss anbi12ci eqss 3bitr4i ) ABCZBACZDEAFZEBFZCZO
    NCZDABGNOGLQMPABHBAHIABJNOJK $.


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  The union of two classes
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  ${
    $d A x $.  $d B x $.  $d C x $.
    $( 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 $.
  $}

  $( A member of a union that is not a member of the first class, is a member
     of the second class.  (Contributed by Glauco Siliprandi, 11-Dec-2019.) $)
  elunnel1 $p |- ( ( A e. ( B u. C ) /\ -. A e. B ) -> A e. C ) $=
    ( cun wcel wo elun biimpi orcanai ) ABCDEZABEZACEZJKLFABCGHI $.

  $( A member of a union that is not a member of the second class, is a member
     of the first class.  (Contributed by Glauco Siliprandi, 11-Dec-2019.) $)
  elunnel2 $p |- ( ( A e. ( B u. C ) /\ -. A e. C ) -> A e. B ) $=
    ( cun wcel wo elun biimpi orcomd orcanai ) ABCDEZACEZABEZKMLKMLFABCGHIJ $.

  ${
    $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,
       21-Jun-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, 21-Jun-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. ~ equncom was automatically derived from
     ~ equncomVD using the tools program translate__without__overwriting.cmd
     and minimizing.  (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 . ~ equncomi was automatically derived from
       ~ equncomiVD using the tools program translate__without__overwriting.cmd
       and minimizing.  (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 the union of two classes.  (Contributed by NM,
       15-Jul-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 the 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 the 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 the 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.)  Avoid ~ ax-10 , ~ ax-11 , ~ ax-12 .  (Revised by SN,
       14-May-2025.) $)
    nfun $p |- F/_ x ( A u. B ) $=
      ( vy cun cv wcel wo elun nfcri nfor nfxfr nfci ) AFBCGZFHZPIQBIZQCIZJAQBC
      KRSAAFBDLAFCELMNO $.
    $( $j usage 'nfun' avoids 'ax-10' 'ax-11' 'ax-12'; $)

    $( Obsolete version of ~ nfun as of 14-May-2025.  (Contributed by NM,
       15-Sep-2003.)  (Revised by Mario Carneiro, 14-Oct-2016.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    nfunOLD $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 $.

  $( A statement is true for every element of the union of a pair of classes if
     and only if it is true for every element of the first class and for every
     element of the second class.  (Contributed by BTernaryTau,
     27-Sep-2023.) $)
  elunant $p |- ( ( C e. ( A u. B ) -> ph ) <->
    ( ( C e. A -> ph ) /\ ( C e. B -> ph ) ) ) $=
    ( cun wcel wi wo wa elun imbi1i jaob bitri ) DBCEFZAGDBFZDCFZHZAGOAGPAGINQA
    DBCJKOAPLM $.

  ${
    $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 df-ss dfcleq ) CDZAEZUABEZFZCGUAABHZEZUCIZCGABRUEBJUDUGCUCUBUCKZIUHUC
      IUDUGUCUHLUBUCMUFUHUCUAABNOPQCABSCUEBTP $.
  $}

  $( 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 wal wa df-ss 19.26 elunant anbi12i 3bitr4i bitr2i
      albii ) ABEZCFDGZSHTCHZIZDJZACFZBCFZKZDSCLTAHUAIZTBHUAIZKZDJUGDJZUHDJZKUC
      UFUGUHDMUBUIDUAABTNRUDUJUEUKDACLDBCLOPQ $.
  $}

  ${
    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 wral elunant albii 19.26 bitri df-ral anbi12i 3bitr4i
    wa ) BEZCDFZGAHZBIZSCGAHZBIZSDGAHZBIZRZABTJABCJZABDJZRUBUCUERZBIUGUAUJBACDS
    KLUCUEBMNABTOUHUDUIUFABCOABDOPQ $.

  $( 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
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  ${
    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 $.
  $}

  $( Membership in an intersection implies membership in the first set.
     (Contributed by Glauco Siliprandi, 2-Jan-2022.) $)
  nel1nelin $p |- ( -. A e. B -> -. A e. ( B i^i C ) ) $=
    ( cin wcel elinel1 con3i ) ABCDEABEABCFG $.

  $( Membership in an intersection implies membership in the second set.
     (Contributed by Glauco Siliprandi, 2-Jan-2022.) $)
  nel2nelin $p |- ( -. A e. C -> -. A e. ( B i^i C ) ) $=
    ( cin wcel elinel2 con3i ) ABCDEACEABCFG $.

  ${
    $d A x $.  $d B x $.
    $( Commutative law for intersection of classes.  Exercise 7 of
       [TakeutiZaring] p. 17.  (Contributed by NM, 21-Jun-1993.)  (Proof
       shortened by SN, 12-Dec-2023.) $)
    incom $p |- ( A i^i B ) = ( B i^i A ) $=
      ( vx cv wcel crab cin rabswap dfin5 3eqtr4i ) CDZBECAFKAECBFABGBAGCABHCAB
      ICBAIJ $.
    $( $j usage 'incom' avoids 'ax-8' 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  $( 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,
       21-Jun-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.)  (Proof shortened by SN, 20-Sep-2023.) $)
    ineq1 $p |- ( A = B -> ( A i^i C ) = ( B i^i C ) ) $=
      ( vx wceq cv wcel crab cin rabeq dfin5 3eqtr4g ) ABEDFCGZDAHMDBHACIBCIMDA
      BJDACKDBCKL $.
  $}

  $( 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 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 ) $=
    ( wss cin wceq dfss2 incom eqeq1i bitri ) ABCABDZAEBADZAEABFJKAABGHI $.

  ${
    $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.)  Avoid ~ ax-10 , ~ ax-11 , ~ ax-12 .  (Revised by SN,
       14-May-2025.) $)
    nfin $p |- F/_ x ( A i^i B ) $=
      ( vy cin cv wcel wa elin nfcri nfan nfxfr nfci ) AFBCGZFHZPIQBIZQCIZJAQBC
      KRSAAFBDLAFCELMNO $.
    $( $j usage 'nfin' avoids 'ax-10' 'ax-11' 'ax-12'; $)

    $( Obsolete version of ~ nfin as of 14-May-2025.  (Contributed by NM,
       15-Sep-2003.)  (Revised by Mario Carneiro, 14-Oct-2016.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    nfinOLD $p |- F/_ x ( A i^i B ) $=
      ( vy cin cv wcel crab dfin5 nfcri nfrabw nfcxfr ) ABCGFHCIZFBJFBCKOAFBAFC
      ELDMN $.
  $}

  ${
    $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 $.

  ${
    $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 elinel1 ssriv ) CABDACEABFG $.
  $}

  $( 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 wa cin wss elin imbi2i albii jcab 19.26 3bitrri df-ss
      anbi12i 3bitr4i ) DEZAFZUABFZGZDHZUBUACFZGZDHZIZUBUABCJZFZGZDHZABKZACKZIA
      UJKUMUBUCUFIZGZDHUDUGIZDHUIULUQDUKUPUBUABCLMNUQURDUBUCUFONUDUGDPQUNUEUOUH
      DABRDACRSDAUJRT $.
  $}

  ${
    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 wa cin jca ssin sylib ) ABCGZBDGZHBCDIGAMNEFJBCDKL $.
  $}

  ${
    $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 $.

  ${
    ssinss1d.1 $e |- ( ph -> A C_ C ) $.
    $( Intersection preserves subclass relationship.  (Contributed by Glauco
       Siliprandi, 17-Aug-2020.) $)
    ssinss1d $p |- ( ph -> ( A i^i B ) C_ C ) $=
      ( wss cin ssinss1 syl ) ABDFBCGDFEBCDHI $.
  $}

  $( 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 $.

  $( Restricted universal quantification over intersection.  (Contributed by
     Peter Mazsa, 8-Sep-2023.) $)
  ralin $p |- ( A. x e. ( A i^i B ) ph <->
                A. x e. A ( x e. B -> ph ) ) $=
    ( cv wcel wi cin wa elin imbi1i impexp bitri ralbii2 ) ABEZDFZAGZBCDHZCORFZ
    AGOCFZPIZAGTQGSUAAOCDJKTPALMN $.

  $( Restricted existential quantification over intersection.  (Contributed by
     Peter Mazsa, 17-Dec-2018.) $)
  rexin $p |- ( E. x e. ( A i^i B ) ph <-> E. x e. A ( x e. B /\ ph ) ) $=
    ( cv wcel wa cin elin anbi1i anass bitri rexbii2 ) ABEZDFZAGZBCDHZCNQFZAGNC
    FZOGZAGSPGRTANCDIJSOAKLM $.

  ${
    $d x A $.  $d x B $.
    $( Alternate definition of subclass relationship.  (Contributed by AV,
       1-Aug-2022.) $)
    dfss7 $p |- ( B C_ A <-> { x e. A | x e. B } = B ) $=
      ( wss cin wceq cv wcel crab dfss2 incom dfin5 eqtri eqeq1i bitri ) CBDCBE
      ZCFAGCHABIZCFCBJPQCPBCEQCBKABCLMNO $.
  $}


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  The symmetric difference of two classes
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  $c /_\ $.  $( Symmetric difference $)

  $( Declare the syntax for symmetric difference. $)
  csymdif $a class ( A /_\ B ) $.

  $( Define the symmetric difference of two classes.  Alternate definitions are
     ~ dfsymdif2 , ~ dfsymdif3 and ~ dfsymdif4 .  (Contributed by Scott Fenton,
     31-Mar-2012.) $)
  df-symdif $a |- ( A /_\ B ) = ( ( A \ B ) u. ( B \ A ) ) $.

  $( Symmetric difference commutes.  (Contributed by Scott Fenton,
     24-Apr-2012.) $)
  symdifcom $p |- ( A /_\ B ) = ( B /_\ A ) $=
    ( cdif cun csymdif uncom df-symdif 3eqtr4i ) ABCZBACZDJIDABEBAEIJFABGBAGH
    $.

  $( Equality theorem for symmetric difference.  (Contributed by Scott Fenton,
     24-Apr-2012.) $)
  symdifeq1 $p |- ( A = B -> ( A /_\ C ) = ( B /_\ C ) ) $=
    ( wceq cdif cun csymdif difeq1 difeq2 uneq12d df-symdif 3eqtr4g ) ABDZACEZC
    AEZFBCEZCBEZFACGBCGMNPOQABCHABCIJACKBCKL $.

  $( Equality theorem for symmetric difference.  (Contributed by Scott Fenton,
     24-Apr-2012.) $)
  symdifeq2 $p |- ( A = B -> ( C /_\ A ) = ( C /_\ B ) ) $=
    ( wceq csymdif symdifeq1 symdifcom 3eqtr4g ) ABDACEBCECAECBEABCFCAGCBGH $.

  ${
    nfsymdif.1 $e |- F/_ x A $.
    nfsymdif.2 $e |- F/_ x B $.
    $( Hypothesis builder for symmetric difference.  (Contributed by Scott
       Fenton, 19-Feb-2013.)  (Revised by Mario Carneiro, 11-Dec-2016.) $)
    nfsymdif $p |- F/_ x ( A /_\ B ) $=
      ( csymdif cdif cun df-symdif nfdif nfun nfcxfr ) ABCFBCGZCBGZHBCIAMNABCDE
      JACBEDJKL $.
  $}

  $( Membership in a symmetric difference.  (Contributed by Scott Fenton,
     31-Mar-2012.) $)
  elsymdif $p |- ( A e. ( B /_\ C ) <-> -. ( A e. B <-> A e. C ) ) $=
    ( cdif cun wcel wn wa wo csymdif wb elun eldif orbi12i df-symdif eleq2i xor
    bitri 3bitr4i ) ABCDZCBDZEZFZABFZACFZGHZUEUDGHZIZABCJZFUDUEKGUCATFZAUAFZIUH
    ATUALUJUFUKUGABCMACBMNRUIUBABCOPUDUEQS $.

  ${
    $d x A $.  $d x B $.
    $( Alternate definition of the symmetric difference.  (Contributed by NM,
       17-Aug-2004.)  (Revised by AV, 17-Aug-2022.) $)
    dfsymdif4 $p |- ( A /_\ B ) = { x | -. ( x e. A <-> x e. B ) } $=
      ( cv wcel wb wn csymdif elsymdif eqabi ) ADZBEKCEFGABCHKBCIJ $.
  $}

  $( Membership in a symmetric difference is an exclusive-or relationship.
     (Contributed by David A. Wheeler, 26-Apr-2020.)  (Proof shortened by BJ,
     13-Aug-2022.) $)
  elsymdifxor $p |- ( A e. ( B /_\ C ) <-> ( A e. B \/_ A e. C ) ) $=
    ( csymdif wcel wb wn wxo elsymdif df-xor bitr4i ) ABCDEABEZACEZFGLMHABCILMJ
    K $.

  ${
    $d x A $.  $d x B $.
    $( Alternate definition of the symmetric difference.  (Contributed by BJ,
       30-Apr-2020.) $)
    dfsymdif2 $p |- ( A /_\ B ) = { x | ( x e. A \/_ x e. B ) } $=
      ( cv wcel wxo csymdif elsymdifxor eqabi ) ADZBEJCEFABCGJBCHI $.
  $}

  ${
    $d A x $.  $d B x $.  $d C x $.
    $( Symmetric difference is associative.  (Contributed by Scott Fenton,
       24-Apr-2012.)  (Proof shortened by BJ, 7-Sep-2022.) $)
    symdifass $p |- ( ( A /_\ B ) /_\ C ) = ( A /_\ ( B /_\ C ) ) $=
      ( vx csymdif cv wcel elsymdifxor biid xorbi12i xorass bicomi 3bitri eqriv
      wxo ) DABEZCEZABCEZEZDFZQGTPGZTCGZOZTAGZTRGZOZTSGZTPCHUCUDTBGZOZUBOUDUHUB
      OZOUFUAUIUBUBTABHUBIJUDUHUBKUDUDUJUEUDIUEUJTBCHLJMUGUFTARHLMN $.
  $}

  $( The symmetric difference contains one of the differences.  (Proposed by
     BJ, 18-Aug-2022.)  (Contributed by AV, 19-Aug-2022.) $)
  difsssymdif $p |- ( A \ B ) C_ ( A /_\ B ) $=
    ( cdif cun csymdif ssun1 df-symdif sseqtrri ) ABCZIBACZDABEIJFABGH $.

  ${
    difsymssdifssd.1 $e |- ( ph -> ( A /_\ B ) C_ C ) $.
    $( If the symmetric difference is contained in ` C ` , so is one of the
       differences.  (Contributed by AV, 17-Aug-2022.) $)
    difsymssdifssd $p |- ( ph -> ( A \ B ) C_ C ) $=
      ( cdif csymdif difsssymdif sstrid ) ABCFBCGDBCHEI $.
  $}


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  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 dfss2 mpbi ) AABCZDAJEAFABGAJHI $.

  $( Negation of subclass expressed in terms of intersection and proper
     subclass.  (Contributed by NM, 30-Jun-2004.)  (Proof shortened by Andrew
     Salmon, 26-Jun-2011.) $)
  nssinpss $p |- ( -. A C_ B <-> ( A i^i B ) C. A ) $=
    ( cin wne wss wa wn wpss inss1 biantrur dfss2 necon3bbii df-pss 3bitr4i ) A
    BCZADZOAEZPFABEZGOAHQPABIJROAABKLOAMN $.

  $( Negation of subclass expressed in terms of proper subclass and union.
     (Contributed by NM, 15-Sep-2004.) $)
  nsspssun $p |- ( -. A C_ B <-> B C. ( A u. B ) ) $=
    ( wss wn cun wa wpss ssun2 biantrur ssid biantru unss bitri xchnxbir dfpss3
    bitr4i ) ABCZDBABEZCZRBCZDZFZBRGTUBQSUABAHIQQBBCZFTUCQBJKABBLMNBROP $.

  ${
    $d x A $.  $d x B $.
    $( Subclass defined in terms of class difference.  See comments under
       ~ dfun2 .  (Contributed by NM, 22-Mar-1998.)  (Proof shortened by Andrew
       Salmon, 26-Jun-2011.) $)
    dfss4 $p |- ( A C_ B <-> ( B \ ( B \ A ) ) = A ) $=
      ( vx wss wceq cdif sseqin2 cv wcel wn wa eldif notbii anbi2i wi elin abai
      cin iman bitr4i 3bitri difeqri eqeq1i ) ABDBARZAEBBAFZFZAEABGUFUDACBUEUDC
      HZBIZUGUEIZJZKUHUHUGAIZJKZJZKZUGUDIZUJUMUHUIULUGBALMNUOUHUKKUHUHUKOZKUNUG
      BAPUHUKQUPUMUHUHUKSNUATUBUCT $.

    $( An alternate definition of the union of two classes in terms of class
       difference, requiring no dummy variables.  Along with ~ dfin2 and
       ~ dfss4 it shows we can express union, intersection, and subset directly
       in terms of the single "primitive" operation ` \ ` (class difference).
       (Contributed by NM, 10-Jun-2004.) $)
    dfun2 $p |- ( A u. B ) = ( _V \ ( ( _V \ A ) \ B ) ) $=
      ( vx cvv cdif cv wcel wo wn wa eldif mpbiran anbi1i ioran 3bitr4i con2bii
      vex bitr4i uneqri ) CABDDAEZBEZEZCFZAGZUCBGZHZUCUAGZIZUCUBGZUGUFUCTGZUEIZ
      JUDIZUKJUGUFIUJULUKUJUCDGZULCQZUCDAKLMUCTBKUDUENOPUIUMUHUNUCDUAKLRS $.

    $( An alternate definition of the intersection of two classes in terms of
       class difference, requiring no dummy variables.  See comments under
       ~ dfun2 .  Another version is given by ~ dfin4 .  (Contributed by NM,
       10-Jun-2004.) $)
    dfin2 $p |- ( A i^i B ) = ( A \ ( _V \ B ) ) $=
      ( vx cvv cdif cv wcel wa vex eldif mpbiran con2bii anbi2i bitr4i ineqri
      wn ) CABADBEZEZCFZAGZSBGZHTSQGZPZHSRGUAUCTUBUAUBSDGUAPCISDBJKLMSAQJNO $.

    $( 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 wi wn pm4.61 anclb elin imbi2i iman 3bitr2i con2bii
      wa eldif 3bitr4i difeqri ) CAABDZABEZCFZAGZUCBGZHZIUDUEIQUDUCUAGZIQZUCUBG
      UDUEJUFUHUFUDUDUEQZHUDUGHUHIUDUEKUGUIUDUCABLMUDUGNOPUCABRST $.
  $}

  $( Implication of a class difference with a subclass.  (Contributed by AV,
     3-Jan-2022.) $)
  ssdifim $p |- ( ( A C_ V /\ B = ( V \ A ) ) -> A = ( V \ B ) ) $=
    ( wss cdif wceq dfss4 eqcom sylbb difeq2 eqcomd sylan9eq ) ACDZBCAEZFZACNEZ
    CBEZMPAFAPFACGPAHIOQPBNCJKL $.

  $( Symmetric class differences for subclasses.  (Contributed by AV,
     3-Jan-2022.) $)
  ssdifsym $p |- ( ( A C_ V /\ B C_ V )
                   -> ( B = ( V \ A ) <-> A = ( V \ B ) ) ) $=
    ( wss wa cdif wceq wi ssdifim ex adantr adantl impbid ) ACDZBCDZEBCAFGZACBF
    GZNPQHONPQABCIJKOQPHNOQPBACIJLM $.

  ${
    $d A x $.  $d B x y $.
    $( Alternate definition of subclass relationship: a class ` A ` is a
       subclass of another class ` B ` iff each element of ` A ` is equal to an
       element of ` B ` .  (Contributed by AV, 13-Nov-2020.) $)
    dfss5 $p |- ( A C_ B <-> A. x e. A E. y e. B x = y ) $=
      ( wss cv wcel wral weq wrex dfss3 clel5 ralbii bitri ) CDEAFZDGZACHABIBDJ
      ZACHACDKPQACBDOLMN $.
  $}

  $( Union defined in terms of intersection (De Morgan's law).  Definition of
     union in [Mendelson] p. 231.  (Contributed by NM, 8-Jan-2002.) $)
  dfun3 $p |- ( A u. B ) = ( _V \ ( ( _V \ A ) i^i ( _V \ B ) ) ) $=
    ( cun cvv cdif cin dfun2 dfin2 ddif difeq2i eqtr2i eqtri ) ABCDDAEZBEZEDMDB
    EZFZEABGNPDPMDOEZENMOHQBMBIJKJL $.

  $( Intersection defined in terms of union (De Morgan's law).  Similar to
     Exercise 4.10(n) of [Mendelson] p. 231.  (Contributed by NM,
     8-Jan-2002.) $)
  dfin3 $p |- ( A i^i B ) = ( _V \ ( ( _V \ A ) u. ( _V \ B ) ) ) $=
    ( cvv cdif cun cin ddif dfun2 difeq1i difeq2i eqtri dfin2 3eqtr4ri ) CCACBD
    ZDZDZDOCCADZNEZDABFOGRPCRCCQDZNDZDPQNHTOCSANAGIJKJABLM $.

  $( Alternate definition of the intersection of two classes.  Exercise 4.10(q)
     of [Mendelson] p. 231.  (Contributed by NM, 25-Nov-2003.) $)
  dfin4 $p |- ( A i^i B ) = ( A \ ( A \ B ) ) $=
    ( cin cdif wss wceq inss1 dfss4 mpbi difin difeq2i eqtr3i ) AAABCZDZDZMAABD
    ZDMAEOMFABGMAHINPAABJKL $.

  $( Intersection with universal complement.  Remark in [Stoll] p. 20.
     (Contributed by NM, 17-Aug-2004.) $)
  invdif $p |- ( A i^i ( _V \ B ) ) = ( A \ B ) $=
    ( cvv cdif cin dfin2 ddif difeq2i eqtri ) ACBDZEACJDZDABDAJFKBABGHI $.

  $( 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 ) $=
    ( cdif cin dfin4 difeq2i difin 3eqtr2i ) AABCZDAAICZCAABDZCIAIEKJAABEFABGH
    $.

  $( 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 $.

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Infer equality from equalities of union and intersection.  Exercise 20
       of [Enderton] p. 32 and its converse.  (Contributed by NM,
       10-Aug-2004.) $)
    unineq $p |- ( ( ( A u. C ) = ( B u. C ) /\ ( A i^i C ) = ( B i^i C ) )
                 <-> A = B ) $=
      ( vx cun wceq cin wa wcel wb eleq2 3bitr3g iba bibi12d imbitrrid wo uncom
      elin elun biorf cv wi adantld eqeq12i sylbi adantrd eqrdv uneq1 ineq1 jca
      wn pm2.61i impbii ) ACEZBCEZFZACGZBCGZFZHZABFZUTDABDUAZCIZUTVBAIZVBBIZJZU
      BVCUSVFUPUSVFVCVDVCHZVEVCHZJUSVBUQIVBURIVGVHUQURVBKVBACRVBBCRLVCVDVGVEVHV
      CVDMVCVEMNOUCVCUKZUPVFUSUPVFVIVCVDPZVCVEPZJUPVBCAEZIZVBCBEZIZVJVKUPVLVNFV
      MVOJUNVLUOVNACQBCQUDVLVNVBKUEVBCASVBCBSLVIVDVJVEVKVCVDTVCVETNOUFULUGVAUPU
      SABCUHABCUIUJUM $.
  $}

  $( Equality of union and intersection implies equality of their 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 $.

  $( 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 ) ) $=
    ( cun cdif cvv cin dfun3 difeq2i inindi dfin2 invdif ineq12i 3eqtr3i eqtri
    ) ABCDZEAFFBEZFCEZGZEZEZABEZACEZGZPTABCHIASGAQGZARGZGUAUDAQRJASKUEUBUFUCABL
    ACLMNO $.

  $( 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 $.

  $( Distributive law for class difference.  Theorem 40 of [Suppes] p. 29.
     (Contributed by NM, 17-Aug-2004.) $)
  difindi $p |- ( A \ ( B i^i C ) ) = ( ( A \ B ) u. ( A \ C ) ) $=
    ( cin cdif cvv cun dfin3 difeq2i indi dfin2 invdif uneq12i 3eqtr3i eqtri )
    ABCDZEAFFBEZFCEZGZEZEZABEZACEZGZPTABCHIASDAQDZARDZGUAUDAQRJASKUEUBUFUCABLAC
    LMNO $.

  $( 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 BTernaryTau,
       14-Aug-2024.) $)
    indifdi $p |- ( A i^i ( B \ C ) ) = ( ( A i^i B ) \ ( A i^i C ) ) $=
      ( vx cdif cin cv wcel wa wn elin eldif anbi2i wi abai an12 imnan xchbinxr
      bicomi anbi12i an21 3bitr2i 3bitr4i 3bitri eqriv ) DABCEZFZABFZACFZEZDGZU
      GHUKAHZUKUFHZIULUKBHZUKCHZJZIZIZUKUJHZUKAUFKUMUQULUKBCLMUNULUPIZIUNULULUP
      NZIZIZURUSUTVBUNULUPOMULUNUPPUSUKUHHZUKUIHZJZIULUNIZVAIVCUKUHUILVGVDVAVFV
      DVGUKABKSVAULUOIVEULUOQUKACKRTULUNVAUAUBUCUDUE $.
  $}

  $( Distribute intersection over difference.  (Contributed by Scott Fenton,
     14-Apr-2011.)  (Revised by BTernaryTau, 14-Aug-2024.) $)
  indifdir $p |- ( ( A \ B ) i^i C ) = ( ( A i^i C ) \ ( B i^i C ) ) $=
    ( cdif cin indifdi incom difeq12i 3eqtr4i ) CABDZECAEZCBEZDJCEACEZBCEZDCABF
    JCGMKNLACGBCGHI $.

  $( Class difference by a class difference.  (Contributed by Thierry Arnoux,
     18-Dec-2017.) $)
  difdif2 $p |- ( A \ ( B \ C ) ) = ( ( A \ B ) u. ( A i^i C ) ) $=
    ( cvv cdif cin cun difindi invdif eqcomi difeq2i dfin2 uneq2i 3eqtr4i ) ABD
    CEZFZEABEZAOEZGABCEZEQACFZGABOHSPAPSBCIJKTRQACLMN $.

  $( 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.  Theorem 5.2(13') of [Stoll] p. 19.
     (Contributed by NM, 18-Aug-2004.) $)
  indm $p |- ( _V \ ( A i^i B ) ) = ( ( _V \ A ) u. ( _V \ B ) ) $=
    ( cvv difindi ) 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 $.
    $( An equality involving class union and class difference.  The first
       equality of Exercise 13 of [TakeutiZaring] p. 22.  (Contributed by Alan
       Sare, 17-Apr-2012.)  (Proof shortened by JJ, 13-Jul-2021.) $)
    undif3 $p |- ( A u. ( B \ C ) ) = ( ( A u. B ) \ ( C \ A ) ) $=
      ( vx cdif cv wcel wn wa wo elun pm4.53 eldif xchnxbir anbi12i orbi2i ordi
      cun orcom anbi2i bitri 3bitri 3bitr4ri eqriv ) DABCEZRZABRZCAEZEZDFZUGGZU
      JUHGZHZIUJAGZUJBGZJZUJCGZHZUNJZIZUJUIGUJUFGZUKUPUMUSUJABKUQUNHIUSULUQUNLU
      JCAMNOUJUGUHMVAUNUJUEGZJUNUOURIZJZUTUJAUEKVBVCUNUJBCMPVDUPUNURJZIUTUNUOUR
      QVEUSUPUNURSTUAUBUCUD $.

    $( Represent a class 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 ancom bianbi anass
      wa elin 3bitr4i 3bitr4g eqrdv ) ACEZDABFZCBFZAGZUCDHZAIZUGBIJZRUHUGCIZRZU
      IRZUGUDIUGUFIZUCUHUKUIUCUHUJACUGKLMUGABNUJUIRZUHRUHUNRUMULUNUHOUMUGUEIUHU
      NUGUEASUGCBNPUHUJUIQTUAUB $.
  $}

  $( 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 $.

  $( Relative complementation reverses inclusion of subclasses.  Relativized
     version of ~ complss .  (Contributed by RP, 3-Jun-2021.) $)
  sscon34b $p |- ( ( A C_ C /\ B C_ C ) -> ( A C_ B
                                         <-> ( C \ B ) C_ ( C \ A ) ) ) $=
    ( wss cdif sscon wceq dfss4 biimpi adantr adantl sseq12d imbitrid impbid2
    wa ) ACDZBCDZOZABDZCBEZCAEZDZABCFUBCUAEZCTEZDRSTUACFRUCAUDBPUCAGZQPUEACHIJQ
    UDBGZPQUFBCHIKLMN $.

  $( Two subclasses are equal if and only if their relative complements are
     equal.  Relativized version of ~ compleq .  (Contributed by RP,
     10-Jun-2021.) $)
  rcompleq $p |- ( ( A C_ C /\ B C_ C ) -> ( A = B
                                         <-> ( C \ A ) = ( C \ B ) ) ) $=
    ( wss wa cdif wceq ancom wb sscon34b ancoms anbi12d bitrid eqss 3bitr4g ) A
    CDZBCDZEZABDZBADZEZCAFZCBFZDZUCUBDZEZABGUBUCGUATSERUFSTHRTUDSUEQPTUDIBACJKA
    BCJLMABNUBUCNO $.

  $( Alternate definition of the symmetric difference, given in Example 4.1 of
     [Stoll] p. 262 (the original definition corresponds to [Stoll] p. 13).
     (Contributed by NM, 17-Aug-2004.)  (Revised by BJ, 30-Apr-2020.) $)
  dfsymdif3 $p |- ( A /_\ B ) = ( ( A u. B ) \ ( A i^i B ) ) $=
    ( cin cdif cun csymdif difin incom difeq2i eqtri uneq12i difundir df-symdif
    3eqtr4ri ) AABCZDZBODZEABDZBADZEABEODABFPRQSABGQBBACZDSOTBABHIBAGJKABOLABMN
    $.


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Class abstractions with difference, union, and intersection of two classes
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  ${
    $d x y $.  $d ph y $.  $d ps y $.  $d ch x $.  $d th x $.
    unabw.1 $e |- ( x = y -> ( ph <-> ch ) ) $.
    unabw.2 $e |- ( x = y -> ( ps <-> th ) ) $.
    $( Union of two class abstractions.  Version of ~ unab using implicit
       substitution, which does not require ~ ax-8 , ~ ax-10 , ~ ax-12 .
       (Contributed by GG, 15-Oct-2024.) $)
    unabw $p |- ( { x | ph } u. { x | ps } ) = { y | ( ch \/ th ) } $=
      ( cab cun cv wcel wo df-un wsb df-clab sbievw bitri orbi12i abbii eqtri )
      AEIZBEIZJFKZUBLZUDUCLZMZFICDMZFIFUBUCNUGUHFUECUFDUEAEFOCAFEPACEFGQRUFBEFO
      DBFEPBDEFHQRSTUA $.
    $( $j usage 'unabw' avoids 'df-clel' 'ax-8' 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $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 $.
  $}

  ${
    $d x y $.  $d y ph $.  $d x ps $.
    notabw.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( A class abstraction defined by a negation.  Version of ~ notab using
       implicit substitution, which does not require ~ ax-10 , ~ ax-12 .
       (Contributed by GG, 15-Oct-2024.) $)
    notabw $p |- { x | -. ph } = ( _V \ { y | ps } ) $=
      ( wn cab cv cvv wcel wa cdif vex biantrur wsb df-clab weq bicomd equcoms
      wb sbievw bitri xchnxbi abbii df-dif eqtr4i ) AFZCGCHZIJZUHBDGZJZFZKZCGIU
      JLUGUMCUKUMAUIULCMNUKBDCOABCDPBADCBATCDCDQABERSUAUBUCUDCIUJUEUF $.
    $( $j usage 'notabw' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  $( A class abstraction 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 wa cab df-rab abid1 ineq12i inab elin an32 bitri abbii
      anbi1i eqtr4i ) ABCEZDFBGZCHZAIZBJZUBDHZBJZFZABCDFZEZUAUEDUGABCKBDLMUJUBU
      IHZAIZBJZUHABUIKUHUDUFIZBJUMUDUFBNULUNBULUCUFIZAIUNUKUOAUBCDOSUCUFAPQRTTT
      $.
  $}

  $( 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
       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 $.

    $( Alternate definition of restricted class abstraction.  (Contributed by
       NM, 20-Sep-2003.)  (Proof shortened by BJ, 22-Apr-2019.) $)
    dfrab2 $p |- { x e. A | ph } = ( { x | ph } i^i A ) $=
      ( crab cab cin dfrab3 incom eqtri ) ABCDCABEZFJCFABCGCJHI $.

    $( Move difference in and out of a restricted class abstraction.
       (Contributed by Steven Nguyen, 6-Jun-2023.) $)
    rabdif $p |- ( { x e. A | ph } \ B ) = { x e. ( A \ B ) | ph } $=
      ( cab cdif cin crab indif2 dfrab2 difeq1i 3eqtr4ri ) ABEZCDFZGMCGZDFABNHA
      BCHZDFMCDIABNJPODABCJKL $.

    $( 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 dfss2 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
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  $( Transfer uniqueness to a smaller or larger class.  (Contributed by NM,
     21-Oct-2005.)  (Proof shortened by Wolf Lammen, 15-May-2025.) $)
  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 wb df-rex euor2 sylnbi df-reu elun
    anbi1i andir orcom 3bitri eubii bitri 3bitr4g ) ABDEZFBGZDHZAIZUICHZAIZJZBK
    ZUMBKZABCDLZMZABCMUHUKBNUOUPOABDPUKUMBQRURUIUQHZAIZBKUOABUQSUTUNBUTULUJJZAI
    UMUKJUNUSVAAUICDTUAULUJAUBUMUKUCUDUEUFABCSUG $.

  ${
    $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 pm3.2 imim2d syl6 a2d imp4a alimdv imp euimmo simplbi2
      syl df-eu syl9 imp32 sylibr ) ACDFZBCEGZHDEIZABJZCDKZHZCTZDLZAHZCMZVDELZB
      HZCNZHZACDGZURVGUSVJACDOBCEPQVCVKHVFCNZVLVCVGVJVMVCVJVFCUAZVGVMVCVFVIJZCR
      ZVJVNJVBUTVEVAJZCRZVPVACDUBUTVRVPUTVQVOCUTVQVEAVIUTVEVAAVIJZUTVEVHVAVSJDE
      VDUCVHBVIAVHBUDUEUFUGUHUIUJSVFVICUKUMVMVGVNVFCUNULUOUPACDPUQS $.

    $( 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 id rgenw reuss2 mpanl2 3impb ) CDEZABCFZABDGZA
      BCGZPAAHZBCIQRJSTBCAKLAABCDMNO $.

    $( 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 $.

    $( 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 $.
  $}

  ${
    $d A x $.  $d B x $.
    $( Transfer uniqueness of an element to a smaller subclass.  (Contributed
       by AV, 14-Apr-2020.) $)
    euelss $p |- ( ( A C_ B /\ E. x x e. A /\ E! x x e. B ) -> E! x x e. A ) $=
      ( wss cv wcel wex weu w3a wtru wa wreu id ancom truan bitri sylbbr df-reu
      wrex eubii df-rex exbii reuss syl3an sylib bitr3i sylibr ) BCDZAEZBFZAGZU
      ICFZAHZIZUJJKZAHZUJAHUNJABLZUPUHUHUKJABSZUMJACLZUQUHMURUOAGUKJABUAUOUJAUO
      JUJKZUJUJJNUJOZPUBQUSULJKZAHUMJACRVBULAVBJULKULULJNULOPTQJABCUCUDJABRUEUJ
      UOAUJUTUOVAJUJNUFTUG $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  The empty set
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare the symbol for the empty set or null set. $)
  $c (/) $.  $( empty set $)

  $( Extend class notation to include the empty set. $)
  c0 $a class (/) $.

  $( Define the empty set.  More precisely, we should write "empty class".  It
     will be posited in ~ ax-nul that an empty set exists.  Then, by uniqueness
     among classes ( ~ eq0 , as opposed to the weaker uniqueness among sets,
     ~ nulmo ), it will follow that ` (/) ` is indeed a set ( ~ 0ex ).  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, 17-Jun-1993.)  Clarify that at this point, it is not established
     that it is a set.  (Revised by BJ, 22-Sep-2022.) $)
  df-nul $a |- (/) = ( _V \ _V ) $.

  $( Alternate definition of the empty class/set.  (Contributed by BJ,
     30-Nov-2019.)  Avoid ~ ax-8 , ~ 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-8'; $)

  $( Alternate definition of the empty set.  Definition 5.14 of [TakeutiZaring]
     p. 20.  (Contributed by NM, 26-Dec-1996.)  Remove dependency on ~ ax-10 ,
     ~ ax-11 , and ~ ax-12 .  (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 $.

  $( 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 $.

  ${
    $d A x y $.
    $( The empty set has no elements.  Theorem 6.14 of [Quine] p. 44.
       (Contributed by NM, 21-Jun-1993.)  (Proof shortened by Mario Carneiro,
       1-Sep-2015.)  Remove dependency on ~ ax-10 , ~ ax-11 , and ~ ax-12 .
       (Revised by Steven Nguyen, 3-May-2023.)  (Proof shortened by BJ,
       23-Sep-2024.) $)
    noel $p |- -. A e. (/) $=
      ( vx vy c0 wcel cv wceq wa wex wfal wsb wn nsb fal mpg cab dfnul4 df-clab
      eleq2i mtbir bitri intnan nex dfclel ) ADEBFZAGZUEDEZHZBIUHBUGUFUGJCBKZJL
      UILCJCBMNOUGUEJCPZEUIDUJUECQSJBCRUATUBUCBADUDT $.
  $}

  $( 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 class has elements, then it is not empty.  (Contributed by NM,
     31-Dec-1993.) $)
  n0i $p |- ( B e. A -> -. A = (/) ) $=
    ( c0 wceq wcel noel eleq2 mtbiri con2i ) ACDZBAEZJKBCEBFACBGHI $.

  $( If a class has elements, then it is nonempty.  (Contributed by NM,
     31-Dec-1993.) $)
  ne0i $p |- ( B e. A -> A =/= (/) ) $=
    ( wcel c0 n0i neqned ) 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 $.
  $}

  ${
    $d x y $.
    $( The universal class is not equal to the empty set.  (Contributed by NM,
       11-Sep-2008.)  Avoid ~ ax-8 , ~ df-clel .  (Revised by GG,
       6-Sep-2024.) $)
    vn0 $p |- _V =/= (/) $=
      ( vy vx cvv c0 cv wtru cab wcel wfal wb wal wceq wex mtbi wsb df-clab sbv
      wn bitr2i eqcomi pm5.501 mptru exgen exnal mpbi bibi12i albii dfcleq dfv2
      fal dfnul4 eqeq12i bitr3i neir ) CDAEZFBGZHZUOIBGZHZJZAKZCDLZFIJZAKZVAVCR
      ZAMVDRVEAIVCUJIVCJFIUAUBNUCVCAUDUEVCUTAFUQIUSUQFBAOFFABPFBAQSUSIBAOIIABPI
      BAQSUFUGNVAUPURLVBAUPURUHUPCURDCUPBUITDURBUKTULUMNUN $.
    $( $j usage 'vn0' avoids 'df-clel' 'ax-8'; $)
  $}

  $( Alternate proof of ~ vn0 .  Shorter, but requiring ~ df-clel , ~ ax-8 .
     (Contributed by NM, 11-Sep-2008.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  vn0ALT $p |- _V =/= (/) $=
    ( vx cv cvv vex ne0ii ) ABCADE $.

  ${
    eq0f.1 $e |- F/_ x A $.
    $( A class is equal to the empty set if and only if it has no elements.
       Theorem 2 of [Suppes] p. 22.  (Contributed by BJ, 15-Jul-2021.) $)
    eq0f $p |- ( A = (/) <-> A. x -. x e. A ) $=
      ( c0 wceq cv wcel wb wal wn nfcv cleqf noel nbn albii bitr4i ) BDEAFZBGZQ
      DGZHZAIRJZAIABDCADKLUATASRQMNOP $.

    $( A class is not empty if and only if it has at least one element.
       Proposition 5.17(1) of [TakeutiZaring] p. 20.  This version of ~ neq0
       requires only that ` x ` not be free in, rather than not occur in,
       ` A ` .  (Contributed by BJ, 15-Jul-2021.) $)
    neq0f $p |- ( -. A = (/) <-> E. x x e. A ) $=
      ( c0 wceq wn cv wcel wal wex eq0f notbii df-ex bitr4i ) BDEZFAGBHZFAIZFPA
      JOQABCKLPAMN $.

    $( A class is nonempty if and only if it has at least one element.
       Proposition 5.17(1) of [TakeutiZaring] p. 20.  This version of ~ n0
       requires only that ` x ` not be free in, rather than not occur in,
       ` A ` .  (Contributed by NM, 17-Oct-2003.) $)
    n0f $p |- ( A =/= (/) <-> E. x x e. A ) $=
      ( c0 wne wceq wn cv wcel wex df-ne neq0f bitri ) BDEBDFGAHBIAJBDKABCLM $.
  $}

  ${
    $d x A $.  $d x y $.
    $( A class is equal to the empty set if and only if it has no elements.
       Theorem 2 of [Suppes] p. 22.  (Contributed by NM, 29-Aug-1993.)  Avoid
       ~ ax-11 , ~ ax-12 .  (Revised by GG and Steven Nguyen, 28-Jun-2024.)
       Avoid ~ ax-8 , ~ df-clel .  (Revised by GG, 6-Sep-2024.) $)
    eq0 $p |- ( A = (/) <-> A. x -. x e. A ) $=
      ( vy c0 wceq wfal cab cv wcel wn wal dfnul4 eqeq2i dfcleq wsb df-clab sbv
      wb bitri bibi2i nbfal bitr4i albii ) BDEBFCGZEZAHZBIZJZAKZDUDBCLMUEUGUFUD
      IZRZAKUIABUDNUKUHAUKUGFRUHUJFUGUJFCAOFFACPFCAQSTUGUAUBUCSS $.
    $( $j usage 'eq0' avoids 'df-clel' 'ax-8' 'ax-10' 'ax-11' 'ax-12'; $)

    $( Alternate proof of ~ eq0 .  Shorter, but requiring ~ df-clel , ~ ax-8 .
       (Contributed by NM, 29-Aug-1993.)  Avoid ~ ax-11 , ~ ax-12 .  (Revised
       by GG and Steven Nguyen, 28-Jun-2024.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    eq0ALT $p |- ( A = (/) <-> A. x -. x e. A ) $=
      ( c0 wceq cv wcel wb wal wn dfcleq noel nbn albii bitr4i ) BCDAEZBFZOCFZG
      ZAHPIZAHABCJSRAQPOKLMN $.
    $( $j usage 'eq0ALT' avoids 'ax-10' 'ax-11' 'ax-12'; $)

    $( A class is not empty if and only if it has at least one element.
       Proposition 5.17(1) of [TakeutiZaring] p. 20.  (Contributed by NM,
       21-Jun-1993.)  Avoid ~ ax-11 , ~ ax-12 .  (Revised by GG,
       28-Jun-2024.) $)
    neq0 $p |- ( -. A = (/) <-> E. x x e. A ) $=
      ( cv wcel wex c0 wceq wn wal df-ex eq0 xchbinxr bicomi ) ACBDZAEZBFGZHONH
      AIPNAJABKLM $.
    $( $j usage 'neq0' avoids 'ax-10' 'ax-11' 'ax-12'; $)

    $( A class is nonempty if and only if it has at least one element.
       Proposition 5.17(1) of [TakeutiZaring] p. 20.  (Contributed by NM,
       29-Sep-2006.)  Avoid ~ ax-11 , ~ ax-12 .  (Revised by GG,
       28-Jun-2024.) $)
    n0 $p |- ( A =/= (/) <-> E. x x e. A ) $=
      ( c0 wne wceq wn cv wcel wex df-ne neq0 bitri ) BCDBCEFAGBHAIBCJABKL $.
    $( $j usage 'n0' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $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 ph $.
    reximdva0.1 $e |- ( ( ph /\ x e. A ) -> ps ) $.
    $( Restricted existence deduced from nonempty class.  (Contributed by NM,
       1-Feb-2012.) $)
    reximdva0 $p |- ( ( ph /\ A =/= (/) ) -> E. x e. A ps ) $=
      ( c0 wne wa cv wcel wex wrex n0 ex ancld eximdv imp sylan2b df-rex sylibr
      ) ADFGZHCIDJZBHZCKZBCDLUAAUBCKZUDCDMAUEUDAUBUCCAUBBAUBBENOPQRBCDST $.
  $}

  ${
    $d A x $.  $d ph x $.
    $( Specialization for restricted generalization with a nonempty class.
       (Contributed by Alexander van der Vekens, 6-Sep-2018.)  Avoid ~ ax-10 ,
       ~ ax-12 .  (Revised by GG, 28-Jun-2024.) $)
    rspn0 $p |- ( A =/= (/) -> ( A. x e. A ph -> ph ) ) $=
      ( c0 wne cv wcel wex wral wi wal df-ral exim ax5e syl6com biimtrid sylbi
      n0 ) CDEBFCGZBHZABCIZAJBCRUASAJBKZTAABCLUBTABHASABMABNOPQ $.
    $( $j usage 'rspn0' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $d A x $.
    $( There is an element in a nonempty class which is an element of the
       class.  (Contributed by AV, 17-Dec-2020.) $)
    n0rex $p |- ( A =/= (/) -> E. x e. A x e. A ) $=
      ( cv wcel wex wa c0 wne wrex id ancli eximi n0 df-rex 3imtr4i ) ACBDZAEPP
      FZAEBGHPABIPQAPPPJKLABMPABNO $.

    $d B x $.
    $( There is an element in a class with a nonempty subclass which is an
       element of the subclass.  (Contributed by AV, 17-Dec-2020.) $)
    ssn0rex $p |- ( ( A C_ B /\ A =/= (/) ) -> E. x e. B x e. A ) $=
      ( wss cv wcel wrex c0 wne ssrexv n0rex impel ) BCDAEBFZABGMACGBHIMABCJABK
      L $.
  $}

  ${
    $d A x $.
    $( A case of equivalence of "at most one" and "only one".  (Contributed by
       FL, 6-Dec-2010.) $)
    n0moeu $p |- ( A =/= (/) -> ( E* x x e. A <-> E! x x e. A ) ) $=
      ( c0 wne cv wcel wmo wex wa weu n0 biimpi biantrurd df-eu bitr4di ) BCDZA
      EBFZAGZQAHZRIQAJPSRPSABKLMQANO $.
  $}

  $( Vacuous restricted 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 $.

  $( Vacuous restricted uniqueness is always false.  (Contributed by AV,
     3-Apr-2023.) $)
  reu0 $p |- -. E! x e. (/) ph $=
    ( c0 wreu wrex rex0 reurex mto ) ABCDABCEABFABCGH $.

  $( Vacuous restricted at-most-one quantifier is always true.  (Contributed by
     AV, 3-Apr-2023.) $)
  rmo0 $p |- E* x e. (/) ph $=
    ( c0 wrmo wrex wreu wi rex0 pm2.21i rmo5 mpbir ) ABCDABCEZABCFZGLMABHIABCJK
    $.

  ${
    $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 A $.  $d x u $.
    $( Negated membership of the empty set in another class.  (Contributed by
       Rodolfo Medina, 25-Sep-2010.) $)
    n0el $p |- ( -. (/) e. A <-> A. x e. A E. u u e. x ) $=
      ( cv wcel wn wal wral wi wex c0 df-ral df-ex ralbii alnex imnang wrex 0el
      wa 3bitr4ri df-rex bitri notbii ) BDADZEZFBGZFZACHUDCEZUGIAGZUEBJZACHKCEZ
      FZUGACLUJUGACUEBMNUHUFSZFAGUMAJZFUIULUMAOUHUFAPUKUNUKUFACQUNABCRUFACUAUBU
      CTT $.
  $}

  ${
    $d x y A $.
    $( A condition which implies the existence of a unique element of a class.
       (Contributed by AV, 4-Jan-2022.) $)
    eqeuel $p |- ( ( A =/= (/) /\ A. x A. y ( ( x e. A /\ y e. A ) -> x = y ) )
                   -> E! x x e. A ) $=
      ( c0 wne cv wcel wa weq wi wal wex weu n0 biimpi anim1i eleq1w eu4 sylibr
      ) CDEZAFCGZBFCGZHABIJBKAKZHUAALZUCHUAAMTUDUCTUDACNOPUAUBABABCQRS $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Subclass expressed in terms of difference.  Exercise 7 of
       [TakeutiZaring] p. 22.  (Contributed by NM, 29-Apr-1994.) $)
    ssdif0 $p |- ( A C_ B <-> ( A \ B ) = (/) ) $=
      ( vx cv wcel wi wal cdif wn wss c0 wceq wa eldif xchbinxr albii df-ss eq0
      iman 3bitr4i ) CDZAEZUABEZFZCGUAABHZEZIZCGABJUEKLUDUGCUDUBUCIMUFUBUCSUAAB
      NOPCABQCUERT $.
  $}

  $( If the difference of two sets is not empty, then the sets are not equal.
     (Contributed by Thierry Arnoux, 28-Feb-2017.) $)
  difn0 $p |- ( ( A \ B ) =/= (/) -> A =/= B ) $=
    ( cdif c0 wceq wss eqimss ssdif0 sylib necon3i ) ABABCZDABEABFKDEABGABHIJ
    $.

  $( A proper subclass has a nonempty difference.  (Contributed by NM,
     3-May-1994.) $)
  pssdifn0 $p |- ( ( A C_ B /\ A =/= B ) -> ( B \ A ) =/= (/) ) $=
    ( wss wne cdif c0 wceq ssdif0 eqss simplbi2 biimtrrid necon3d imp ) ABCZABD
    BAEZFDNOFABOFGBACZNABGZBAHQNPABIJKLM $.

  $( A proper subclass has a nonempty difference.  (Contributed by Mario
     Carneiro, 27-Apr-2016.) $)
  pssdif $p |- ( A C. B -> ( B \ A ) =/= (/) ) $=
    ( wpss wss wne wa cdif c0 df-pss pssdifn0 sylbi ) ABCABDABEFBAGHEABIABJK $.

  ${
    $d x A $.  $d x B $.
    $( Express that an intersection is not empty.  (Contributed by RP,
       16-Apr-2020.) $)
    ndisj $p |- ( ( A i^i B ) =/= (/) <-> E. x ( x e. A /\ x e. B ) ) $=
      ( cin c0 wne cv wcel wex wa n0 elin exbii bitri ) BCDZEFAGZOHZAIPBHPCHJZA
      IAOKQRAPBCLMN $.
  $}

  ${
    inn0f.1 $e |- F/_ x A $.
    inn0f.2 $e |- F/_ x B $.
    $( A nonempty intersection.  (Contributed by Glauco Siliprandi,
       24-Dec-2020.) $)
    inn0f $p |- ( ( A i^i B ) =/= (/) <-> E. x e. A x e. B ) $=
      ( cv cin wcel wex wa c0 wne wrex elin exbii nfin n0f df-rex 3bitr4i ) AFZ
      BCGZHZAITBHTCHZJZAIUAKLUCABMUBUDATBCNOAUAABCDEPQUCABRS $.
  $}

  ${
    $d A x $.  $d B x $.
    $( A nonempty intersection.  (Contributed by Glauco Siliprandi,
       24-Dec-2020.) $)
    inn0 $p |- ( ( A i^i B ) =/= (/) <-> E. x e. A x e. B ) $=
      ( nfcv inn0f ) ABCABDACDE $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Difference, intersection, and subclass relationship.  (Contributed by
       NM, 30-Apr-1994.)  (Proof shortened by Wolf Lammen, 30-Sep-2014.) $)
    difin0ss $p |- ( ( ( A \ B ) i^i C ) = (/) -> ( C C_ A -> C C_ B ) ) $=
      ( vx cdif cin c0 wceq cv wcel wn wal wss wi eq0 wa iman elin eldif df-ss
      anbi2ci annim anbi2i 3bitri xchbinxr ax-2 sylbir al2imi 3imtr4g sylbi ) A
      BEZCFZGHDIZULJZKZDLZCAMZCBMZNDULOUPUMCJZUMAJZNZDLUSUMBJZNZDLUQURUOVAVCDUO
      USUTVBNZNZVAVCNVEUSVDKZPZUNUSVDQUNUMUKJZUSPUSUTVBKPZPVGUMUKCRVHVIUSUMABSU
      AVIVFUSUTVBUBUCUDUEUSUTVBUFUGUHDCATDCBTUIUJ $.

    $( Intersection, subclass, and difference relationship.  (Contributed by
       NM, 27-Oct-1996.)  (Proof shortened by Andrew Salmon, 26-Jun-2011.)
       (Proof shortened by Wolf Lammen, 30-Sep-2014.) $)
    inssdif0 $p |- ( ( A i^i B ) C_ C <-> ( A i^i ( B \ C ) ) = (/) ) $=
      ( vx cv cin wcel wi wal cdif wn wss c0 wceq elin imbi1i iman bitri eldif
      wa anbi2i anass 3bitr4ri xchbinx albii df-ss eq0 3bitr4i ) DEZABFZGZUICGZ
      HZDIUIABCJZFZGZKZDIUJCLUOMNUMUQDUMUIAGZUIBGZTZULKZTZUPUMUTULHVBKUKUTULUIA
      BOPUTULQRURUIUNGZTURUSVATZTUPVBVCVDURUIBCSUAUIAUNOURUSVAUBUCUDUEDUJCUFDUO
      UGUH $.
  $}

  $( The intersection and class difference of a class with another class are
     disjoint.  With ~ inundif , this shows that such intersection and class
     difference partition the class ` A ` .  (Contributed by Thierry Arnoux,
     13-Sep-2017.) $)
  inindif $p |- ( ( A i^i C ) i^i ( A \ C ) ) = (/) $=
    ( cin wss cdif c0 wceq inss2 ssinss1 ax-mp inssdif0 mpbi ) ABCZACBDZMABECFG
    MBDNABHMABIJMABKL $.

  ${
    $d A x $.
    $( 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.)  (Revised by David Abernethy,
       17-Jun-2012.) $)
    difid $p |- ( A \ A ) = (/) $=
      ( vx cdif cv wcel wn crab c0 dfdif2 dfnul3 eqtr4i ) AACBDAEFBAGHBAAIBAJK
      $.
    $( $j usage 'difid' avoids 'ax-8' 'df-clel'; $)
  $}

  $( Alternate proof of ~ difid .  Shorter, but requiring ~ ax-8 , ~ df-clel .
     (Contributed by NM, 22-Apr-2004.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  difidALT $p |- ( A \ A ) = (/) $=
    ( wss cdif c0 wceq ssid ssdif0 mpbi ) AABAACDEAFAAGH $.

  $( 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 $.

  ${
    $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.  Version of ~ ab0 using implicit
       substitution, which requires fewer axioms.  (Contributed by GG,
       3-Oct-2024.) $)
    ab0w $p |- ( { x | ph } = (/) <-> A. y -. ps ) $=
      ( cab c0 wceq wfal wn wal dfnul4 wcel wb df-clab bitri bibi1i albii falim
      wsb eqeq2i cv dfcleq sbievw bicom sbv wi idd bija id pm5.21ni impbii ) AC
      FZGHUMICFZHZBJZDKZGUNUMCLUAUODUBZUNMZBNZDKZUQUOURUMMZUSNZDKVADUMUNUCVCUTD
      VCBUSNUTVBBUSVBACDTBADCOABCDEUDPQBUSUEPRPVAIBNZDKUQUTVDDUSIBUSICDTIIDCOIC
      DUFPQRVDUPDVDUPIBUPBUPUGSIJUPUHUIIBBBSBUJUKULRPPP $.
    $( $j usage 'ab0w' avoids 'df-clel' 'ax-8' 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  $( The class of sets verifying a property is the empty class if and only if
     that property is a contradiction.  See also ~ abn0 (from which it could be
     proved using as many essential proof steps but one fewer syntactic step,
     at the cost of depending on ~ df-ne ).  (Contributed by BJ, 19-Mar-2021.)
     Avoid ~ df-clel , ~ ax-8 .  (Revised by GG, 30-Aug-2024.)  (Proof
     shortened by SN, 8-Sep-2024.) $)
  ab0 $p |- ( { x | ph } = (/) <-> A. x -. ph ) $=
    ( cab wfal wceq wb wal c0 wn abbib dfnul4 eqeq2i nbfal albii 3bitr4i ) ABCZ
    DBCZEADFZBGPHEAIZBGADBJHQPBKLSRBAMNO $.
  $( $j usage 'ab0' avoids 'df-clel' 'ax-8'; $)

  $( Alternate proof of ~ ab0 , shorter but using more axioms.  (Contributed by
     BJ, 19-Mar-2021.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  ab0ALT $p |- ( { x | ph } = (/) <-> A. x -. ph ) $=
    ( cab c0 wceq cv wcel wn wal nfab1 eq0f abid notbii albii bitri ) ABCZDEBFP
    GZHZBIAHZBIBPABJKRSBQAABLMNO $.

  $( Characterization of nonfreeness in a formula in terms of its extension.
     (Contributed by BJ, 19-Mar-2021.) $)
  dfnf5 $p |- ( F/ x ph <-> ( { x | ph } = _V \/ { x | ph } = (/) ) ) $=
    ( wnf wal wn wo cab cvv wceq c0 nf3 abv ab0 orbi12i bitr4i ) ABCABDZAEBDZFA
    BGZHIZRJIZFABKSPTQABLABMNO $.

  ${
    $d x y ph $.
    $( The class abstraction defined by a formula not containing the
       abstraction variable is either the empty set or the universal class.
       (Contributed by Mario Carneiro, 29-Aug-2013.)  (Revised by BJ,
       22-Mar-2020.)  Reduce axiom usage.  (Revised by GG, 30-Aug-2024.) $)
    ab0orv $p |- ( { x | ph } = _V \/ { x | ph } = (/) ) $=
      ( vy wal wn wo cab cvv wceq mpbi wcel wtru wsb df-clab bitr2i 2th bibi12i
      c0 wb weq wnf nfv nf3 tbtru sbv tru vextru bitri albii dfcleq dfv2 eqcomi
      cv eqeq2i 3bitr2i equid nbn3 notbii a1i sbievw dfnul2 orbi12i ) ACDZAEZCD
      ZFZABGZHIZVGRIZFACUAVFACUBACUCJVCVHVEVIVCCUMZVGKZVJLBGZKZSZCDVGVLIVHAVNCA
      ALSVNAUDAVKLVMVKABCMAACBNABCUEOZLVMUFBCUGPQUHUICVGVLUJVLHVGHVLBUKULUNUOVE
      VKVJBBTZEZBGZKZSZCDVGVRIVIVDVTCVDACCTZEZSVTWAACUPZUQAVKWBVSVOVSVQBCMWBVQC
      BNVQWBBCVQWBSBCTVPWAVPWABUPWCPURUSUTOQUHUICVGVRUJVRRVGRVRBVAULUNUOVBJ $.
    $( $j usage 'ab0orv' avoids 'df-clel' 'ax-8' 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $d x ph $.
    $( Alternate proof of ~ ab0orv , shorter but using more axioms.
       (Contributed by Mario Carneiro, 29-Aug-2013.)  (Revised by BJ,
       22-Mar-2020.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    ab0orvALT $p |- ( { x | ph } = _V \/ { x | ph } = (/) ) $=
      ( wnf cab cvv wceq c0 wo nfv dfnf5 mpbi ) ABCABDZEFLGFHABIABJK $.
  $}

  $( Nonempty class abstraction.  See also ~ ab0 .  (Contributed by NM,
     26-Dec-1996.)  (Proof shortened by Mario Carneiro, 11-Nov-2016.)  Avoid
     ~ df-clel , ~ ax-8 .  (Revised by GG, 30-Aug-2024.) $)
  abn0 $p |- ( { x | ph } =/= (/) <-> E. x ph ) $=
    ( cab c0 wceq wn wal wne wex ab0 notbii df-ne df-ex 3bitr4i ) ABCZDEZFAFBGZ
    FODHABIPQABJKODLABMN $.
  $( $j usage 'abn0' avoids 'df-clel' 'ax-8'; $)

  $( 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.)  (Proof shortened by JJ, 14-Jul-2021.) $)
  rab0 $p |- { x e. (/) | ph } = (/) $=
    ( c0 crab cv wcel wa cab df-rab wceq wn ab0 noel intnanr mpgbir eqtri ) ABC
    DBEZCFZAGZBHZCABCITCJSKBSBLRAQMNOP $.

  ${
    $d x y A $.  $d x ps $.  $d y ph $.
    rabeq0w.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Condition for a restricted class abstraction to be empty.  Version of
       ~ rabeq0 using implicit substitution, which does not require ~ ax-10 ,
       ~ ax-11 , ~ ax-12 , but requires ~ ax-8 .  (Contributed by GG,
       30-Sep-2024.) $)
    rabeq0w $p |- ( { x e. A | ph } = (/) <-> A. y e. A -. ps ) $=
      ( cv wcel wa cab c0 wceq wn wal crab wral weq eleq1w anbi12d ab0w 3bitr4i
      df-rab eqeq1i raln ) CGEHZAIZCJZKLDGEHZBIZMDNACEOZKLBMDEPUFUICDCDQUEUHABC
      DERFSTUJUGKACEUBUCBDEUDUA $.
    $( $j usage 'rabeq0w' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  $( Condition for a restricted class abstraction to be empty.  (Contributed by
     Jeff Madsen, 7-Jun-2010.)  (Revised by BJ, 16-Jul-2021.) $)
  rabeq0 $p |- ( { x e. A | ph } = (/) <-> A. x e. A -. ph ) $=
    ( cv wcel wa cab c0 wceq wn wal crab wral ab0 df-rab eqeq1i raln 3bitr4i )
    BDCEAFZBGZHISJBKABCLZHIAJBCMSBNUATHABCOPABCQR $.

  $( Nonempty restricted class abstraction.  (Contributed by NM, 29-Aug-1999.)
     (Revised by BJ, 16-Jul-2021.) $)
  rabn0 $p |- ( { x e. A | ph } =/= (/) <-> E. x e. A ph ) $=
    ( crab c0 wne wn wral wrex rabeq0 necon3abii dfrex2 bitr4i ) ABCDZEFAGBCHZG
    ABCIONEABCJKABCLM $.

  ${
    $d A x $.
    $( Law of excluded middle, in terms of restricted class abstractions.
       (Contributed by Jeff Madsen, 20-Jun-2011.) $)
    rabxm $p |- A = ( { x e. A | ph } u. { x e. A | -. ph } ) $=
      ( wn wo crab cun wceq rabid2im cv wcel exmidd mprg unrab eqtr4i ) CAADZEZ
      BCFZABCFPBCFGQCRHBCQBCIBJCKALMAPBCNO $.
    $( $j usage 'rabxm' avoids 'ax-11'; $)
  $}

  $( 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 wral pm3.24 rgenw rabeq0 mpbir eqtri ) ABCDA
    EZBCDFAQGZBCDZHAQBCISHJREZBCKTBCALMRBCNOP $.

  ${
    $d A s $.
    elneldisj.e $e |- E = { s e. A | B e. C } $.
    elneldisj.n $e |- N = { s e. A | B e/ C } $.
    $( The set of elements ` s ` determining classes ` C ` (which may depend on
       ` s ` ) containing a special element and the set of elements ` s `
       determining classes ` C ` not containing the special element are
       disjoint.  (Contributed by Alexander van der Vekens, 11-Jan-2018.)
       (Revised by AV, 9-Nov-2020.)  (Revised by AV, 17-Dec-2021.) $)
    elneldisj $p |- ( E i^i N ) = (/) $=
      ( cin wcel crab wn c0 wnel df-nel rabbii eqtri ineq12i rabnc ) DEIBCJZFAK
      ZTLZFAKZIMDUAEUCGEBCNZFAKUCHUDUBFABCOPQRTFASQ $.

    $( The union of the set of elements ` s ` determining classes ` C ` (which
       may depend on ` s ` ) containing a special element and the set of
       elements ` s ` determining classes ` C ` not containing the special
       element yields the original set.  (Contributed by Alexander van der
       Vekens, 11-Jan-2018.)  (Revised by AV, 9-Nov-2020.)  (Revised by AV,
       17-Dec-2021.) $)
    elnelun $p |- ( E u. N ) = A $=
      ( cun wcel crab wn wnel df-nel rabbii eqtri uneq12i rabxm eqtr4i ) DEIBCJ
      ZFAKZTLZFAKZIADUAEUCGEBCMZFAKUCHUDUBFABCNOPQTFARS $.
  $}

  ${
    $d x A $.
    $( The union of a class with the empty set is itself.  Theorem 24 of
       [Suppes] p. 27.  (Contributed by NM, 15-Jul-1993.) $)
    un0 $p |- ( A u. (/) ) = A $=
      ( vx c0 cv wcel wo noel biorfri bicomi uneqri ) BACABDZAEZLKCEZFMLKGHIJ
      $.

    $( The intersection of a class with the empty set is the empty set.
       Theorem 16 of [Suppes] p. 26.  (Contributed by NM, 21-Jun-1993.) $)
    in0 $p |- ( A i^i (/) ) = (/) $=
      ( vx c0 cv wcel wa noel bianfi bicomi ineqri ) BACCBDZCEZKAEZLFLMKGHIJ $.
  $}

  $( The union of the empty set with a class is itself.  (Contributed by Glauco
     Siliprandi, 17-Aug-2020.) $)
  0un $p |- ( (/) u. A ) = A $=
    ( c0 cun uncom un0 eqtri ) BACABCABADAEF $.

  $( The intersection of the empty set with a class is the empty set.
     (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.  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.
     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 null set is a subset of any class.  Part of Exercise 1 of
       [TakeutiZaring] p. 22.  (Contributed by NM, 21-Jun-1993.) $)
    0ss $p |- (/) C_ A $=
      ( vx c0 cv wcel noel pm2.21i ssriv ) BCABDZCEIAEIFGH $.
  $}

  $( Any subset of the empty set is empty.  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 $.

  $( 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 sseq2 ss0 biimtrdi impcom ) BCDZABEZACDZJKACELBCAFAGHI $.

  $( 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 $.

  $( 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 $.

  ${
    abf.1 $e |- -. ph $.
    $( A class abstraction determined by a false formula is empty.
       (Contributed by NM, 20-Jan-2012.)  Avoid ~ ax-8 , ~ ax-10 , ~ ax-11 ,
       ~ ax-12 .  (Revised by GG, 30-Jun-2024.) $)
    abf $p |- { x | ph } = (/) $=
      ( cab wfal c0 bifal abbii dfnul4 eqtr4i ) ABDEBDFAEBACGHBIJ $.
    $( $j usage 'abf' avoids 'ax-8' 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $d x A $.  $d x ph $.  $d x y $.
    eq0rdv.1 $e |- ( ph -> -. x e. A ) $.
    $( Deduction for equality to the empty set.  (Contributed by NM,
       11-Jul-2014.)  Avoid ~ ax-8 , ~ df-clel .  (Revised by GG,
       6-Sep-2024.) $)
    eq0rdv $p |- ( ph -> A = (/) ) $=
      ( vy cv wcel wn wal c0 wceq alrimiv wfal cab wb dfnul4 eqeq2i bitri albii
      dfcleq wsb df-clab sbv bibi2i nbfal bicomi 3bitrri sylib ) ABFZCGZHZBIZCJ
      KZAUKBDLUMCMENZKUJUIUNGZOZBIZULJUNCEPQBCUNTUQUJMOZBIULUPURBUOMUJUOMEBUAMM
      BEUBMEBUCRUDSURUKBUKURUJUEUFSRUGUH $.
    $( $j usage 'eq0rdv' avoids 'df-clel' 'ax-8'; $)
  $}

  ${
    $d x A $.  $d x ph $.
    eq0rdvALT.1 $e |- ( ph -> -. x e. A ) $.
    $( Alternate proof of ~ eq0rdv .  Shorter, but requiring ~ df-clel ,
       ~ ax-8 .  (Contributed by NM, 11-Jul-2014.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    eq0rdvALT $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.)  (Proof shortened
       by JJ, 27-Aug-2021.) $)
    csbprc $p |- ( -. A e. _V -> [_ A / x ]_ B = (/) ) $=
      ( vy cvv wcel wn cv wsbc cab wfal csb c0 sbcex pm5.21ni abbidv df-csb fal
      falim abf eqcomi 3eqtr4g ) BEFZGZDHCFZABIZDJKDJZABCLMUDUFKDUFUCKUEABNUCSO
      PADBCQUGMKDRTUAUB $.
  $}

  $( The proper substitution of a class into the empty set is the empty set.
     (Contributed by NM, 18-Aug-2018.) $)
  csb0 $p |- [_ A / x ]_ (/) = (/) $=
    ( cvv wcel c0 csb wceq csbconstg csbprc pm2.61i ) BCDABEFEGABECHABEIJ $.
  $( $j usage 'csb0' avoids 'ax-10' 'ax-11' 'ax-12'; $)

  ${
    $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.)  (Revised by NM, 18-Aug-2018.) $)
    sbcel12 $p |- ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C ) $=
      ( vy vz cvv wcel wsbc csb cv cab dfsbcq2 abbidv eleq12d nfs1v nfab df-csb
      wsb sbab wb wceq nfel weq sbiev vtoclbg eleq12i bitr4di wn sbcex con3i c0
      noel csbprc eleq2d mtbiri 2falsed pm2.61i ) BGHZCDHZABIZABCJZABDJZHZUAUSV
      AEKZCHZABIZELZVEDHZABIZELZHZVDUTAFSVFAFSZELZVIAFSZELZHZVAVLFBGUTAFBMFKBUB
      ZVNVHVPVKVRVMVGEVFAFBMNVRVOVJEVIAFBMNOUTVQAFAVNVPVMAEVFAFPQVOAEVIAFPQUCAF
      UDCVNDVPAFECTAFEDTOUEUFVBVHVCVKAEBCRAEBDRUGUHUSUIZVAVDVAUSUTABUJUKVSVDVBU
      LHVBUMVSVCULVBABDUNUOUPUQUR $.

    $( 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 sbiev vtoclbg df-csb eqeq12i bitr4di ) BEHCDIZABJZFTZCHZABJZF
      KZUJDHZABJZFKZIZABCLZABDLZIUHAGMUKAGMZFKZUNAGMZFKZIZUIUQGBEUHAGBNGTBIZVAU
      MVCUPVEUTULFUKAGBNOVEVBUOFUNAGBNOPUHVDAGAVAVCUTAFUKAGQRVBAFUNAGQRUAAGUBCV
      ADVCAGFCSAGFDSPUCUDURUMUSUPAFBCUEAFBDUEUFUG $.
  $}

  ${
    sbceqi.1 $e |- A e. _V $.
    sbceqi.2 $e |- [_ A / x ]_ B = D $.
    sbceqi.3 $e |- [_ A / x ]_ C = E $.
    $( Distribution of class substitution over equality, in inference form.
       (Contributed by Giovanni Mascellani, 27-May-2019.) $)
    sbceqi $p |- ( [. A / x ]. B = C <-> D = E ) $=
      ( wceq wsbc csb cvv wcel wb sbceqg ax-mp eqeq12i bitri ) CDJABKZABCLZABDL
      ZJZEFJBMNTUCOGABCDMPQUAEUBFHIRS $.
  $}

  $( 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 wn wsbc wnel csb sbcng df-nel sbcbii sbcel12 xchbinxr 3bitr4g ) BEFC
    DFZGZABHQABHZGCDIZABHABCJZABDJZIZQABEKTRABCDLMUCUAUBFSUAUBLABCDNOP $.

  $( Distribute proper substitution through an inequality.  (Contributed by
     Andrew Salmon, 18-Jun-2011.)  (Revised by NM, 18-Aug-2018.) $)
  sbcne12 $p |- ( [. A / x ]. B =/= C <-> [_ A / x ]_ B =/= [_ A / x ]_ C ) $=
    ( cvv wcel wne wsbc csb wb wceq nne sbcbii a1i sbcng sbceqg bitr4di 3bitr3d
    wn csbprc con4bid sbcex con3i c0 eqtr4d sylibr 2falsed pm2.61i ) BEFZCDGZAB
    HZABCIZABDIZGZJUIUKUNUIUJSZABHZCDKZABHZUKSUNSZUPURJUIUOUQABCDLMNUJABEOUIURU
    LUMKZUSABCDEPULUMLZQRUAUISZUKUNUKUIUJABUBUCVBUTUSVBULUDUMABCTABDTUEVAUFUGUH
    $.

  ${
    $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 sbcel12 csbconstg eleq2d bitrid ) CDFABGABCHZABDHZFBEFZMD
      FABCDIONDMABDEJKL $.

    $( 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.)  (Revised by NM, 18-Aug-2018.) $)
    sbcel2 $p |- ( [. A / x ]. B e. C <-> B e. [_ A / x ]_ C ) $=
      ( cvv wcel wsbc wb sbcel12 csbconstg eleq1d bitrid wn sbcex con3i c0 noel
      csb csbprc eleq2d mtbiri 2falsed pm2.61i ) BEFZCDFZABGZCABDRZFZHUFABCRZUG
      FUDUHABCDIUDUICUGABCEJKLUDMZUFUHUFUDUEABNOUJUHCPFCQUJUGPCABDSTUAUBUC $.

    $( 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.)  (Revised by NM, 18-Aug-2018.) $)
    csbcom $p |- [_ A / x ]_ [_ B / y ]_ C = [_ B / y ]_ [_ A / x ]_ C $=
      ( vz csb cv wcel wsbc sbccom sbcel2 sbcbii 3bitr3i eqriv ) FACBDEGZGZBDAC
      EGZGZFHZPIZACJZTRIZBDJZTQITSITEIZBDJZACJUEACJZBDJUBUDUEABCDKUFUAACBDTELMU
      GUCBDACTELMNACTPLBDTRLNO $.
  $}

  ${
    $d x y z $.  $d z A $.  $d z B $.  $d ph z $.
    $( Nest the composition of two substitutions.  Version of ~ sbcnestgf with
       a disjoint variable condition, which does not require ~ ax-13 .
       (Contributed by Mario Carneiro, 11-Nov-2016.)  Avoid ~ ax-13 .  (Revised
       by GG, 26-Jan-2024.) $)
    sbcnestgfw $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 sbceq1d cvv a1i vex nfnf1
      csbeq1 bibi12d imbi2d csbeq1a adantl nfal nfa1 nfcsb1v sp nfsbcdw sbciedf
      wnfc vtoclg imp ) DFHABIZCJZACEKZBDKZACBDELZKZMZURUSBGNZKZACBVDELZKZMZOUR
      VCOGDFVDDPZVHVCURVIVEUTVGVBUSBVDDQVIACVFVABVDDEUCRUDUEURUSVGBVDSVDSHURGUA
      TBNVDPZUSVGMURVJACEVFBVDEUFRUGUQBCABUBUHURABCVFUQCUIBVFUNURBVDEUJTUQCUKUL
      UMUOUP $.
    $( $j usage 'sbcnestgfw' avoids 'ax-13'; $)
  $}

  ${
    $d x y z $.  $d z A $.  $d z B $.  $d z C $.
    $( Nest the composition of two substitutions.  Version of ~ csbnestgf with
       a disjoint variable condition, which does not require ~ ax-13 .
       (Contributed by NM, 23-Nov-2005.)  Avoid ~ ax-13 .  (Revised by GG,
       26-Jan-2024.) $)
    csbnestgfw $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 eqabri wb nfcr
      sbcbii wnf alimi sbcnestgfw sylan2 bitrid abbidv sylan 3eqtr4g ) CFHZAEIZ
      BJZKGLZBDEMZHZACNZGOZUOEHZBACDMZNZGOZACUPMBVAEMULCPHZUNUSVCQCFRVDUNKZURVB
      GURUTBDNZACNZVEVBUQVFACVFGUPBGDESTUCUNVDUTAUDZBJVGVBUAUMVHBAGEUBUEUTABCDP
      UFUGUHUIUJAGCUPSBGVAESUK $.
    $( $j usage 'csbnestgfw' avoids 'ax-13'; $)
  $}

  ${
    $d x y $.  $d x ph $.
    $( Nest the composition of two substitutions.  Version of ~ sbcnestg with a
       disjoint variable condition, which does not require ~ ax-13 .
       (Contributed by NM, 27-Nov-2005.)  Avoid ~ ax-13 .  (Revised by GG,
       26-Jan-2024.) $)
    sbcnestgw $p |- ( A e. V ->
          ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) ) $=
      ( wcel wnf wal wsbc csb wb nfv ax-gen sbcnestgfw mpan2 ) DFGABHZCIACEJBDJ
      ACBDEKJLQCABMNABCDEFOP $.
    $( $j usage 'sbcnestgw' avoids 'ax-13'; $)
  $}

  ${
    $d x y $.  $d x C $.
    $( Nest the composition of two substitutions.  Version of ~ csbnestg with a
       disjoint variable condition, which does not require ~ ax-13 .
       (Contributed by NM, 23-Nov-2005.)  Avoid ~ ax-13 .  (Revised by GG,
       26-Jan-2024.) $)
    csbnestgw $p |- ( A e. V ->
          [_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C ) $=
      ( wcel wnfc wal csb wceq nfcv ax-gen csbnestgfw mpan2 ) CFGAEHZBIACBDEJJB
      ACDJEJKPBAELMABCDEFNO $.
    $( $j usage 'csbnestgw' avoids 'ax-13'; $)
  $}

  ${
    $d x A $.  $d x ph $.  $d x C $.  $d x y $.
    sbcco3gw.1 $e |- ( x = A -> B = C ) $.
    $( Composition of two substitutions.  Version of ~ sbcco3g with a disjoint
       variable condition, which does not require ~ ax-13 .  (Contributed by
       NM, 27-Nov-2005.)  Avoid ~ ax-13 .  (Revised by GG, 26-Jan-2024.) $)
    sbcco3gw $p |- ( A e. V ->
         ( [. A / x ]. [. B / y ]. ph <-> [. C / y ]. ph ) ) $=
      ( wcel wsbc csb sbcnestgw cvv wceq wb elex nfcvd csbiegf dfsbcq 3syl
      bitrd ) DGIZACEJBDJACBDEKZJZACFJZABCDEGLUBDMIZUCFNUDUEODGPBDEFMUFBFQHRACU
      CFSTUA $.
    $( $j usage 'sbcco3gw' avoids 'ax-13'; $)
  $}

  ${
    $d x z $.  $d y z $.  $d z A $.  $d z B $.  $d z C $.  $d z ph $.
    $( Nest the composition of two substitutions.  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  Use the weaker ~ sbcnestgfw
       when possible.  (Contributed by Mario Carneiro, 11-Nov-2016.)
       (New usage is discouraged.) $)
    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 sbceq1d cvv a1i vex nfnf1
      csbeq1 bibi12d imbi2d csbeq1a adantl nfal nfa1 wnfc nfcsb1v nfsbcd vtoclg
      sp sbciedf imp ) DFHABIZCJZACEKZBDKZACBDELZKZMZURUSBGNZKZACBVDELZKZMZOURV
      COGDFVDDPZVHVCURVIVEUTVGVBUSBVDDQVIACVFVABVDDEUCRUDUEURUSVGBVDSVDSHURGUAT
      BNVDPZUSVGMURVJACEVFBVDEUFRUGUQBCABUBUHURABCVFUQCUIBVFUJURBVDEUKTUQCUNULU
      OUMUP $.

    $( Nest the composition of two substitutions.  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  Use the weaker ~ csbnestgfw
       when possible.  (Contributed by NM, 23-Nov-2005.)  (Proof shortened by
       Mario Carneiro, 10-Nov-2016.)  (New usage is discouraged.) $)
    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 eqabri wb nfcr
      sbcbii wnf alimi sbcnestgf sylan2 bitrid abbidv sylan 3eqtr4g ) CFHZAEIZB
      JZKGLZBDEMZHZACNZGOZUOEHZBACDMZNZGOZACUPMBVAEMULCPHZUNUSVCQCFRVDUNKZURVBG
      URUTBDNZACNZVEVBUQVFACVFGUPBGDESTUCUNVDUTAUDZBJVGVBUAUMVHBAGEUBUEUTABCDPU
      FUGUHUIUJAGCUPSBGVAESUK $.

    $d x ph $.
    $( Nest the composition of two substitutions.  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  Use the weaker ~ sbcnestgw
       when possible.  (Contributed by NM, 27-Nov-2005.)  (Proof shortened by
       Mario Carneiro, 11-Nov-2016.)  (New usage is discouraged.) $)
    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.  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  Use the weaker ~ csbnestgw
       when possible.  (Contributed by NM, 23-Nov-2005.)  (Proof shortened by
       Mario Carneiro, 10-Nov-2016.)  (New usage is discouraged.) $)
    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 A $.  $d x ph $.  $d x C $.  $d x D $.
    sbcco3g.1 $e |- ( x = A -> B = C ) $.
    $( Composition of two substitutions.  Usage of this theorem is discouraged
       because it depends on ~ ax-13 .  Use the weaker ~ sbcco3gw when
       possible.  (Contributed by NM, 27-Nov-2005.)  (Revised by Mario
       Carneiro, 11-Nov-2016.)  (New usage is discouraged.) $)
    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.  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  (Contributed by NM,
       27-Nov-2005.)  (Revised by Mario Carneiro, 11-Nov-2016.)
       (New usage is discouraged.) $)
    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 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 csbnestgfw mpan2 csbcow csbeq2i
      wal 3eqtr3g ) BEGZABFCAFHZDIZIZIZFABCIZUBIZABACDIZIAUEDITAUBJZFRUDUFKUHFA
      UADLMAFBCUBENOABUCUGAFCDPQAFUEDPS $.
  $}

  ${
    $d x A $.
    $( Idempotent law for class substitutions.  (Contributed by NM,
       1-Mar-2008.)  (Revised by NM, 18-Aug-2018.) $)
    csbidm $p |- [_ A / x ]_ [_ A / x ]_ B = [_ A / x ]_ B $=
      ( cvv wcel csb wceq csbnest1g csbconstg csbeq1d eqtrd wn c0 csbprc eqtr4d
      pm2.61i ) BDEZABABCFZFZRGQSAABBFZCFRABBCDHQATBCABBDIJKQLSMRABRNABCNOP $.
  $}

  ${
    $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 eqabcdv eqtrid
      elv csbeq2i csbcow 3eqtr3i syl ) BCFBGFZABAHZIZBJBCKUCUEDHZEHZFEBLZDMZBEB
      AUGUDIZIEBUGIUEUIEBUJUGUJUGJEUGGFZUJUFUDFAUGLZDMUGADUGUDNUKULDUGAUFUGGOPQ
      RSAEBUDTEDBUGNUAUCUHDBEUFBGOPQUB $.
  $}

  ${
    csbvargi.1 $e |- A e. _V $.
    $( The proper substitution of a class for a setvar variable results in the
       class (if the class exists), in inference form of ~ csbvarg .
       (Contributed by Giovanni Mascellani, 30-May-2019.) $)
    csbvargi $p |- [_ A / x ]_ x = A $=
      ( cvv wcel cv csb wceq csbvarg ax-mp ) BDEABAFGBHCABDIJ $.
  $}

  ${
    $d x y $.
    $( Substitution into a wff expressed in terms of substitution into a class.
       (Contributed by NM, 15-Aug-2007.)  (Revised by NM, 18-Aug-2018.) $)
    sbccsb $p |- ( [. A / x ]. ph <-> y e. [_ A / x ]_ { y | ph } ) $=
      ( wsbc cv cab wcel csb abid sbcbii sbcel2 bitr3i ) ABDECFZACGZHZBDENBDOIH
      PABDACJKBDNOLM $.
  $}

  $( Substitution into a wff expressed in using substitution into a class.
     (Contributed by NM, 27-Nov-2005.)  (Revised by NM, 18-Aug-2018.) $)
  sbccsb2 $p |- ( [. A / x ]. ph <-> A e. [_ A / x ]_ { x | ph } ) $=
    ( wsbc cvv wcel cab sbcex elex cv abid sbcbii sbcel12 csbvarg eleq1d bitrid
    csb bitr3id pm5.21nii ) ABCDZCEFZCBCABGZQZFZABCHCUCITBJZUBFZBCDZUAUDUFABCAB
    KLUGBCUEQZUCFUAUDBCUEUBMUAUHCUCBCENOPRS $.

  ${
    $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 wnfc cv csb wceq wal cvv wcel vex csbtt mpan wsbc sbsbc sbcel2 bitri
      wsb eqtr4d alrimivv nfv wnf eleq2 3bitr4g 2alimi sbnf2 sylibr nfcd impbii
      wb ) ADFZABGZDHZACGZDHZIZCJBJZUMURBCUMUODUQUNKLUMUODIBMAUNDKNOUPKLUMUQDIC
      MAUPDKNOUAUBUSAEDUSEUCUSEGZDLZABTZVAACTZULZCJBJVAAUDURVDBCURUTUOLZUTUQLZV
      BVCUOUQUTUEVBVAAUNPVEVAABQAUNUTDRSVCVAAUPPVFVAACQAUPUTDRSUFUGVAABCUHUIUJU
      K $.
  $}

  ${
    $d y z A $.  $d z ph $.  $d x y z $.
    $( Move substitution into a class abstraction.  (Contributed by NM,
       13-Dec-2005.)  (Revised by NM, 19-Aug-2018.) $)
    csbab $p |- [_ A / x ]_ { y | ph } = { y | [. A / x ]. ph } $=
      ( vz cab csb wsbc cv wcel df-clab sbsbc bitri sbccom sbcbii bitr4i sbcel2
      wsb 3bitrri eqriv ) EBDACFZGZABDHZCFZEIZUDJZUCCUEHZUEUAJZBDHZUEUBJUFUCCER
      UGUCECKUCCELMUGACUEHZBDHUIACBUEDNUHUJBDUHACERUJAECKACELMOPBDUEUAQST $.
  $}

  ${
    $d A y $.  $d B y $.  $d C y $.  $d x y $.
    $( Distribution of class substitution over union of two classes.
       (Contributed by Drahflow, 23-Sep-2015.)  (Revised by Mario Carneiro,
       11-Dec-2016.)  (Revised by NM, 13-Sep-2018.) $)
    csbun $p |- [_ A / x ]_ ( B u. C ) = ( [_ A / x ]_ B u. [_ A / x ]_ C ) $=
      ( vy cvv wcel cun csb wceq cv csbeq1 uneq12d eqeq12d nfcsb1v nfun csbeq1a
      vex c0 csbprc weq csbief vtoclg wn un0 a1i 3eqtr4rd pm2.61i ) BFGZABCDHZI
      ZABCIZABDIZHZJZAEKZUJIZAUPCIZAUPDIZHZJUOEBFUPBJZUQUKUTUNAUPBUJLVAURULUSUM
      AUPBCLAUPBDLMNAUPUJUTERAURUSAUPCOAUPDOPAEUACURDUSAUPCQAUPDQMUBUCUIUDZSSHZ
      SUNUKVCSJVBSUEUFVBULSUMSABCTABDTMABUJTUGUH $.
  $}

  ${
    $d A y $.  $d B y $.  $d C y $.  $d x y $.
    $( Distribute proper substitution into a class through an intersection
       relation.  (Contributed by Alan Sare, 22-Jul-2012.)  (Revised by NM,
       18-Aug-2018.) $)
    csbin $p |- [_ A / x ]_ ( B i^i C )
                 = ( [_ A / x ]_ B i^i [_ A / x ]_ C ) $=
      ( vy cvv wcel cin csb wceq cv csbeq1 ineq12d eqeq12d nfcsb1v nfin csbeq1a
      vex c0 csbprc weq csbief vtoclg wn in0 eqtr2di eqtrd pm2.61i ) BFGZABCDHZ
      IZABCIZABDIZHZJZAEKZUJIZAUPCIZAUPDIZHZJUOEBFUPBJZUQUKUTUNAUPBUJLVAURULUSU
      MAUPBCLAUPBDLMNAUPUJUTERAURUSAUPCOAUPDOPAEUACURDUSAUPCQAUPDQMUBUCUIUDZUKS
      UNABUJTVBUNSSHSVBULSUMSABCTABDTMSUEUFUGUH $.
  $}

  ${
    $d x y $.  $d A y $.  $d B y $.  $d D y $.  $d ph y $.
    csbie2df.p $e |- F/ x ph $.
    csbie2df.c $e |- ( ph -> F/_ x C ) $.
    csbie2df.d $e |- ( ph -> F/_ x D ) $.
    csbie2df.a $e |- ( ph -> A e. V ) $.
    csbie2df.1 $e |- ( ( ph /\ x = y ) -> B = C ) $.
    csbie2df.2 $e |- ( ( ph /\ y = A ) -> C = D ) $.
    $( Conversion of implicit substitution to explicit class substitution.
       This version of ~ csbiedf avoids a disjointness condition on ` x , A `
       and ` x , D ` by substituting twice.  Deduction form of ~ csbie2 .
       (Contributed by AV, 29-Mar-2024.) $)
    csbie2df $p |- ( ph -> [_ A / x ]_ B = D ) $=
      ( csb wceq wi wsbc wb wa wcel eqidd cv dfsbcq sbceqg adantr sylan2 eqeq2d
      wnfc csbtt bitrd mpancom sylan9bb pm5.74da eqeq1d pm5.74d wsb sbsbc nfeqd
      expcom weq ex sbiedw bitr3id pm5.74i vtoclbg mpbiri mpcom ) DHUAZABDEOZGP
      ZLVIAVKQZAGGPZQZAGUBAEGPZBCUCZRZQAFGPZQVLVNCDHVPDPZAVQVKVSVQVOBDRZAVKVOBV
      PDUDVIAVTVKSLVIATZVTVJBDGOZPZVKVIVTWCSABDEGHUEUFWAWBGVJAVIBGUIWBGPKBDGHUJ
      UGUHUKULUMUNVSAVRVMAVSVRVMSAVSTFGGNUOUTUPAVQVRVQVOBCUQAVRVOBCURAVOVRBCIAB
      FGJKUSABCVAZVOVRSAWDTEFGMUOVBVCVDVEVFVGVH $.
  $}

  ${
    $d A x y $.  $d B x y $.  $d X x y $.  $d ph y $.  $d ch x y $.
    $d ps x y $.
    2nreu.a $e |- ( x = A -> ( ph <-> ps ) ) $.
    2nreu.b $e |- ( x = B -> ( ph <-> ch ) ) $.
    $( If there are two different sets fulfilling a wff (by implicit
       substitution), then there is no unique set fulfilling the wff.
       (Contributed by AV, 20-Jun-2023.) $)
    2nreu $p |- ( ( A e. X /\ B e. X /\ A =/= B )
                  -> ( ( ps /\ ch ) -> -. E! x e. X ph ) ) $=
      ( vy wne wa wn wrex wsbc wb sbcan anbi12d bitrid csb wcel w3a wreu wsb wi
      weq wo cv simpl1 simpl2 simprl sbcieg 3ad2ant2 biimprd adantld imp simpl3
      jca simp1 simp2 simp3 nfs1v sbcgf sbcne12 csbvarg csbconstg 3ad2ant1 sbcg
      neeq12d sbcbidv sbsbc sbcbii sbccow mpbird rspesbca syl2anc sbcrex sylibr
      a1i bitrd syl112anc pm4.61 df-ne bicomi anbi2i bitri 2rexbii olcd rexnal2
      wral ianor orbi2i reu2 xchnxbir ex ) EGUAZFGUAZEFKZUBZBCLZADGUCZMZWSWTLZA
      DGNZMZAADJUDZLZDJUFZUEZMZJGNDGNZUGZXBXCXKXEXCXGDUHZJUHZKZLZJGNZDGNZXKXCWP
      WQBADFOZLZWRXRWPWQWRWTUIWPWQWRWTUJXCBXSWSBCUKWSWTXSWSCXSBWSXSCWQWPXSCPWRA
      CDFGIULUMUNUOUPURWPWQWRWTUQWPWQXTWRLZUBZWPXQDEOZXRWPWQYAUSYBXPDEOZJGNZYCY
      BWQYDJFOZYEWPWQYAUTYBYFYAWPWQYAVAYBYFBXFLZEXNKZLZJFOZYAYBYDYIJFWPWQYDYIPY
      AYDXGDEOZXODEOZLWPYIXGXODEQWPYKYGYLYHYKADEOZXFDEOZLWPYGAXFDEQWPYMBYNXFABD
      EGHULXFDEGADJVBVCRSYLDEXMTZDEXNTZKWPYHDEXMXNVDWPYOEYPXNDEGVEDEXNGVFVISRSV
      GVJYJYGJFOZYHJFOZLYBYAYGYHJFQYBYQXTYRWRYQBJFOZXFJFOZLZYBXTBXFJFQWQWPUUAXT
      PYAWQYSBYTXSBJFGVHYTADXNOZJFOZWQXSXFUUBJFADJVKVLUUCXSPWQADJFVMVSSRUMSYRJF
      ETZJFXNTZKZYBWRJFEXNVDWQWPUUFWRPYAWQUUDEUUEFJFEGVFJFGVEVIUMSRSVTVNYDJFGVO
      VPXPDJEGVQVRXQDEGVOVPWAXJXPDJGGXJXGXHMZLXPXGXHWBUUGXOXGXOUUGXMXNWCWDWEWFW
      GVRWHXDXIJGWJDGWJZLZXLXAUUIMXEUUHMZUGXLXDUUHWKUUJXKXEXKUUJXIDJGGWIWDWLWFA
      DJGWMWNVRWO $.
  $}

  $( Two classes are empty iff their union is empty.  (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.
     (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 $.

  $( The null set is a proper subset of any nonempty set.  (Contributed by NM,
     27-Feb-1996.) $)
  0pss $p |- ( (/) C. A <-> A =/= (/) ) $=
    ( c0 wpss wne wss 0ss df-pss mpbiran necom bitri ) BACZBADZABDKBAELAFBAGHBA
    IJ $.

  $( No set is a proper subset of the empty set.  (Contributed by NM,
     17-Jun-1998.)  (Proof shortened by Andrew Salmon, 26-Jun-2011.)  (Proof
     shortened by JJ, 14-Jul-2021.) $)
  npss0 $p |- -. A C. (/) $=
    ( c0 wss wpss wn 0ss ssnpss ax-mp ) BACABDEAFBAGH $.

  $( Any non-universal class is a proper subclass of the universal class.
     (Contributed by NM, 17-May-1998.) $)
  pssv $p |- ( A C. _V <-> -. A = _V ) $=
    ( cvv wpss wss wceq wn ssv dfpss2 mpbiran ) ABCABDABEFAGABHI $.

  ${
    $d x y z A $.  $d x y z B $.  $d x y C $.
    $( Two ways of saying that two classes are disjoint (have no members in
       common).  (Contributed by NM, 17-Feb-2004.)  Avoid ~ ax-10 , ~ ax-11 ,
       ~ ax-12 .  (Revised by GG, 28-Jun-2024.) $)
    disj $p |- ( ( A i^i B ) = (/) <-> A. x e. A -. x e. B ) $=
      ( vz cin c0 wceq cv wcel wn wi wal wral wa wb eleq1w biimpd eleq1a 3bitri
      albii cab df-in eqeq1i dfcleq wsb weq df-clab id anim12d embantd anim12ii
      sb6 spimvw alrimiv impbii bibi2i bitri eqcom bicom 3bitr4i imnan noel nbn
      bitr2i df-ral bitr4i ) BCEZFGZAHZBIZVICIZJZKZALZVLABMVHDHZBIZVOCIZNZDUAZF
      GZVJVKNZVIFIZOZALZVNVGVSFDBCUBUCFVSGZWBWAOZALZVTWDWEWBVIVSIZOZALWGAFVSUDW
      IWFAWHWAWBWHVRDAUEDAUFZVRKZDLZWAVRADUGVRDAULWLWAWKWADAWJWJVRWAWJUHWJVPVJV
      QVKWJVPVJDABPQWJVQVKDACPQUIUJUMWAWKDVJWJVPVKVQVIBVORVICVORUKUNUOSUPTUQVSF
      URWCWFAWAWBUSTUTWCVMAVMWAJWCVJVKVAWBWAVIVBVCVDTSVLABVEVF $.
    $( $j usage 'disj' avoids 'ax-10' 'ax-11' 'ax-12'; $)

    $( 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.)  Avoid
       ~ ax-12 .  (Revised by GG, 28-Jun-2024.) $)
    reldisj $p |- ( A C_ C -> ( ( A i^i B ) = (/) <-> A C_ ( C \ B ) ) ) $=
      ( vx vy wss cv wcel wn wi wal cdif cin c0 wceq df-ss weq eleq1w imbi12d
      wb spw wa pm5.44 eldif imbi2i bitr4di syl sylbi albidv disj1 3bitr4g ) AC
      FZDGZAHZUMBHIZJZDKUNUMCBLZHZJZDKABMNOAUQFULUPUSDULUNUMCHZJZDKZUPUSTZDACPV
      BVAVCVAEGZAHZVDCHZJDEDEQUNVEUTVFDEARDECRSUAVAUPUNUTUOUBZJUSUNUTUOUCURVGUN
      UMCBUDUEUFUGUHUIDABUJDAUQPUK $.
    $( $j usage 'reldisj' avoids 'ax-10' 'ax-11' 'ax-12'; $)

    $( 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 $.
  $}

  $( Two disjoint sets are equal iff both are empty.  (Contributed by AV,
     19-Jun-2022.) $)
  disjeq0 $p |- ( ( A i^i B ) = (/)
                  -> ( A = B <-> ( A = (/) /\ B = (/) ) ) ) $=
    ( cin c0 wceq wa ineq1 inidm eqtrdi eqeq1d eqtr simpr ex sylbid com12 eqtr3
    jca impbid1 ) ABCZDEZABEZADEZBDEZFZUATUDUATUCUDUASBDUASBBCBABBGBHIJUAUCUDUA
    UCFUBUCABDKUAUCLQMNOABDPR $.

  $( 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 $.

  $( Two ways of saying that two classes are disjoint.  (Contributed by NM,
     21-Mar-2004.) $)
  disj4 $p |- ( ( A i^i B ) = (/) <-> -. ( A \ B ) C. A ) $=
    ( cin c0 wceq cdif wpss disj3 eqcom wss difss dfpss2 mpbiran con2bii 3bitri
    wn ) ABCDEAABFZEQAEZQAGZPABHAQISRSQAJRPABKQALMNO $.

  $( Intersection with a subclass of a disjoint class.  (Contributed by FL,
     24-Jan-2007.)  (Proof shortened by JJ, 14-Jul-2021.) $)
  ssdisj $p |- ( ( A C_ B /\ ( B i^i C ) = (/) ) -> ( A i^i C ) = (/) ) $=
    ( wss cin c0 wceq wa ssrin eqimss sylan9ss ss0 syl ) ABDZBCEZFGZHACEZFDQFGN
    PQOFABCIOFJKQLM $.

  $( A class is a proper subset of its union with a disjoint nonempty class.
     (Contributed by NM, 15-Sep-2004.) $)
  disjpss $p |- ( ( ( A i^i B ) = (/) /\ B =/= (/) ) -> A C. ( A u. B ) ) $=
    ( cin c0 wceq wne wa wss wn cun wpss ssid biantru ssin bitri sseq2 biimtrdi
    bitrid ss0 necon3ad imp nsspssun uncom psseq2i sylib ) ABCZDEZBDFZGBAHZIZAA
    BJZKZUGUHUJUGUIBDUGUIBDHZBDEUIBUFHZUGUMUIUIBBHZGUNUOUIBLMBABNOUFDBPRBSQTUAU
    JABAJZKULBAUBUPUKABAUCUDOUE $.

  $( 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 expressed in terms of intersection with difference from the
     universal class.  (Contributed by NM, 17-Sep-2003.) $)
  ssindif0 $p |- ( A C_ B <-> ( A i^i ( _V \ B ) ) = (/) ) $=
    ( cvv cdif cin c0 wceq wss disj2 ddif sseq2i bitr2i ) ACBDZEFGACMDZHABHAMIN
    BABJKL $.

  $( 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.)  (Proof shortened by JJ,
     14-Jul-2021.) $)
  minel $p |- ( ( A e. B /\ ( C i^i B ) = (/) ) -> -. A e. C ) $=
    ( wcel cin c0 wceq wn wne inelcm expcom necon2bd imp ) ABDZCBEZFGACDZHNPOFP
    NOFIACBJKLM $.

  ${
    $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 uneq2 indi equncomi un0 eqcomi 3eqtr4g eqeq1d dfss2
      3bitr4g ) ABDZEFZABCGZDZAFACDZAFATHACHSUAUBASUBRGUBEGZUAUBREUBIUARUBABCJK
      UCUBUBLMNOATPACPQ $.
  $}

  $( Universal class equality in terms of empty difference.  (Contributed by
     NM, 17-Sep-2003.) $)
  vdif0 $p |- ( A = _V <-> ( _V \ A ) = (/) ) $=
    ( cvv wceq wss cdif c0 vss ssdif0 bitr3i ) 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 Alexander van der Vekens, 31-Dec-2017.) $)
    difrab0eq $p |- ( ( V \ { x e. V | ph } ) = (/)
                      <-> V = { x e. V | ph } ) $=
      ( crab cdif c0 wceq wss ssdif0 ssrabeq bitr3i ) CABCDZEFGCLHCLGCLIABCJK
      $.
  $}

  ${
    $d x A $.  $d x B $.
    $( A proper subclass has a member in one argument that's not in both.
       (Contributed by NM, 29-Feb-1996.) $)
    pssnel $p |- ( A C. B -> E. x ( x e. B /\ -. x e. A ) ) $=
      ( wpss cv cdif wcel wex wn wa c0 wne pssdif n0 sylib eldif exbii ) BCDZAE
      ZCBFZGZAHZSCGSBGIJZAHRTKLUBBCMATNOUAUCASCBPQO $.
  $}

  $( 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 ) ) = (/) $=
    ( cin wss cdif c0 wceq inss1 inssdif0 mpbi ) ABCADABAECFGABHABAIJ $.

  $( A class and its relative complement are disjoint.  (Contributed by Thierry
     Arnoux, 29-Nov-2023.) $)
  disjdifr $p |- ( ( B \ A ) i^i A ) = (/) $=
    ( cdif cin c0 incom disjdif eqtr3i ) ABACZDIADEAIFABGH $.

  $( 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 ssdif0 mpbi ) ABCZBDKBEFGABHKBIJ $.

  $( The union of a class and its complement is the universe.  Theorem 5.1(5)
     of [Stoll] p. 17.  (Contributed by NM, 17-Aug-2004.) $)
  unvdif $p |- ( A u. ( _V \ A ) ) = _V $=
    ( cvv cdif cun cin c0 dfun3 disjdif difeq2i dif0 3eqtri ) ABACZDBLBLCEZCBFC
    BALGMFBLBHIBJK $.

  $( Absorption of difference by union.  This decomposes a union into two
     disjoint classes (see ~ disjdif ).  Theorem 35 of [Suppes] p. 29.
     (Contributed by NM, 19-May-1998.) $)
  undif1 $p |- ( ( A \ B ) u. B ) = ( A u. B ) $=
    ( cvv cdif cin undir invdif uneq1i uncom unvdif eqtri ineq2i inv1 3eqtr3i
    cun ) ACBDZEZBOABOZPBOZEZABDZBORAPBFQUABABGHTRCERSCRSBPOCPBIBJKLRMKN $.

  $( Absorption of difference by union.  This decomposes a union into two
     disjoint classes (see ~ disjdif ).  Part of proof of Corollary 6K of
     [Enderton] p. 144.  (Contributed by NM, 19-May-1998.) $)
  undif2 $p |- ( A u. ( B \ A ) ) = ( A u. B ) $=
    ( cdif cun uncom undif1 3eqtri ) ABACZDHADBADABDAHEBAFBAEG $.

  $( Absorption of difference by union.  (Contributed by NM, 18-Aug-2013.) $)
  undifabs $p |- ( A u. ( A \ B ) ) = A $=
    ( cdif cun undif3 unidm difeq1i difdif 3eqtri ) AABCDAADZBACZCAKCAAABEJAKAF
    GABHI $.

  ${
    $d x A $.  $d x B $.
    $( The intersection and class difference of a class with another class
       unite to give the original class.  (Contributed by Paul Chapman,
       5-Jun-2009.)  (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
    inundif $p |- ( ( A i^i B ) u. ( A \ B ) ) = A $=
      ( vx cin cdif cv wcel wo wa wn elin eldif orbi12i pm4.42 bitr4i uneqri )
      CABDZABEZACFZQGZSRGZHSAGZSBGZIZUBUCJIZHUBTUDUAUESABKSABLMUBUCNOP $.
  $}

  $( 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 NM,
     22-Mar-1998.) $)
  undif $p |- ( A C_ B <-> ( A u. ( B \ A ) ) = B ) $=
    ( wss cun wceq cdif ssequn1 undif2 eqeq1i bitr4i ) ABCABDZBEABAFDZBEABGLKBA
    BHIJ $.

  $( Union of complementary parts into whole.  (Contributed by Thierry Arnoux,
     21-Nov-2023.)  (Proof shortened by SN, 11-Mar-2025.) $)
  undifr $p |- ( A C_ B <-> ( ( B \ A ) u. A ) = B ) $=
    ( wss cun wceq cdif ssequn2 undif1 eqeq1i bitr4i ) ABCBADZBEBAFADZBEABGLKBB
    AHIJ $.

  $( Obsolete version of ~ undifr as of 11-Mar-2025.  (Contributed by Thierry
     Arnoux, 21-Nov-2023.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  undifrOLD $p |- ( A C_ B <-> ( ( B \ A ) u. A ) = B ) $=
    ( wss cdif cun wceq undif uncom eqeq1i bitri ) ABCABADZEZBFKAEZBFABGLMBAKHI
    J $.

  $( An equality involving class union and class difference.  (Contributed by
     Thierry Arnoux, 26-Jun-2024.) $)
  undif5 $p |- ( ( A i^i B ) = (/) -> ( ( A u. B ) \ B ) = A ) $=
    ( cin c0 wceq cun cdif difun2 disjdif2 eqtrid ) ABCDEABFBGABGAABHABIJ $.

  $( 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 disjdifr sseq0 sylancl ) ABCDZEACFZMCFZEOGHNGH
    AMCICBJNOKL $.

  $( 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 condition equivalent to inclusion in the union of two classes.
       (Contributed by NM, 26-Mar-2007.) $)
    ssundif $p |- ( A C_ ( B u. C ) <-> ( A \ B ) C_ C ) $=
      ( vx cv wcel cun wi wal cdif wss wn wa wo pm5.6 eldif imbi1i imbi2i df-ss
      elun 3bitr4ri albii 3bitr4i ) DEZAFZUDBCGZFZHZDIUDABJZFZUDCFZHZDIAUFKUICK
      UHULDUEUDBFZLMZUKHUEUMUKNZHULUHUEUMUKOUJUNUKUDABPQUGUOUEUDBCTRUAUBDAUFSDU
      ICSUC $.
  $}

  $( Swap the arguments of a class difference.  (Contributed by NM,
     29-Mar-2007.) $)
  difcom $p |- ( ( A \ B ) C_ C <-> ( A \ C ) C_ B ) $=
    ( cun wss cdif uncom sseq2i ssundif 3bitr3i ) ABCDZEACBDZEABFCEACFBEKLABCGH
    ABCIACBIJ $.

  $( Two ways to express overlapping subsets.  (Contributed by Stefan O'Rear,
     31-Oct-2014.) $)
  pssdifcom1 $p |- ( ( A C_ C /\ B C_ C ) ->
    ( ( C \ A ) C. B <-> ( C \ B ) C. A ) ) $=
    ( wss wa cdif wn wpss wb difcom ssconb ancoms notbid anbi12d dfpss3 3bitr4g
    a1i ) ACDZBCDZEZCAFZBDZBUADZGZECBFZADZAUEDZGZEUABHUEAHTUBUFUDUHUBUFITCABJQT
    UCUGSRUCUGIBACKLMNUABOUEAOP $.

  $( Two ways to express non-covering pairs of subsets.  (Contributed by Stefan
     O'Rear, 31-Oct-2014.) $)
  pssdifcom2 $p |- ( ( A C_ C /\ B C_ C ) ->
    ( B C. ( C \ A ) <-> A C. ( C \ B ) ) ) $=
    ( wss wa cdif wn wpss wb ssconb ancoms difcom notbii anbi12d dfpss3 3bitr4g
    a1i ) ACDZBCDZEZBCAFZDZUABDZGZEACBFZDZUEADZGZEBUAHAUEHTUBUFUDUHSRUBUFIBACJK
    UDUHITUCUGCABLMQNBUAOAUEOP $.

  $( Distributive law for class difference.  Exercise 4.8 of [Stoll] p. 16.
     (Contributed by NM, 18-Aug-2004.) $)
  difdifdir $p |- ( ( A \ B ) \ C ) = ( ( A \ C ) \ ( B \ C ) ) $=
    ( cdif cvv cun cin dif32 invdif eqtr4i un0 indi disjdif incom eqtr3i uneq2i
    c0 ddif indm difeq2i ineq2i 3eqtri ) ABDCDZACDZEBDZCFZGZUDEBCDZDZGUDUHDUCUD
    UEGZQFZUGUCUJUKUCUDBDUJABCHUDBIJUJKJUGUJUDCGZFUKUDUECLQULUJCUDGQULCAMCUDNOP
    JJUFUIUDUEEECDZDZFZUFUIUNCUECRPEBUMGZDUOUIBUMSUPUHEBCITOOUAUDUHIUB $.

  $( Two ways to say that ` A ` and ` B ` partition ` C ` (when ` A ` and ` B `
     don't overlap and ` A ` is a part of ` C ` ).  (Contributed by FL,
     17-Nov-2008.)  (Proof shortened by JJ, 14-Jul-2021.) $)
  uneqdifeq $p |- ( ( A C_ C /\ ( A i^i B ) = (/) )
     -> ( ( A u. B ) = C <-> ( C \ A ) = B ) ) $=
    ( wss cin c0 wceq wa cun cdif uncom eqcomd difeq1 difun2 incom eqeq1i disj3
    wi eqtr adantl bitri expcom eqcoms sylbi syl5com sylancl com12 simpl difssd
    syl mpan sseq1 mpbid unssd eqimss ssundif sylibr eqssd ex adantr impbid ) A
    CDZABEZFGZHABIZCGZCAJZBGZVDVFVHRVBVFVDVHBAIZVEGZVFVDVHRZBAKVJVFHZCVIGZVKVLV
    ICVIVECSLVMVGVIAJZGZVNBAJZGZVKCVIAMBANVOVQHVGVPGZVDVHVGVNVPSVDBVPGZVRVHRZVD
    BAEZFGVSVCWAFABOPBAQUAVTVPBVRVPBGVHVGVPBSUBUCUDUEUFUJUKUGTVBVHVFRVDVBVHVFVB
    VHHZVECWBABCVBVHUHVHBCDZVBVHVGCDWCVHCAUIVGBCULUMTUNVHCVEDZVBVHVGBDWDVGBUOCA
    BUPUQTURUSUTVA $.

  ${
    $d A x $.  $d B x $.
    raldifeq.1 $e |- ( ph -> A C_ B ) $.
    raldifeq.2 $e |- ( ph -> A. x e. ( B \ A ) ps ) $.
    $( Equality theorem for restricted universal quantifier.  (Contributed by
       Thierry Arnoux, 6-Jul-2019.) $)
    raldifeq $p |- ( ph -> ( A. x e. A ps <-> A. x e. B ps ) ) $=
      ( wral cdif cun wa biantrud ralunb bitr4di wceq undif sylib raleqdv bitrd
      wss ) ABCDHZBCDEDIZJZHZBCEHAUAUABCUBHZKUDAUEUAGLBCDUBMNABCUCEADETUCEOFDEP
      QRS $.
  $}

  ${
    $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 not empty.  (Contributed by NM, 15-Nov-2003.) $)
    r19.2z $p |- ( ( A =/= (/) /\ A. x e. A ph ) -> E. x e. A ph ) $=
      ( wral c0 wne wrex cv wex wa wi wal df-ral exintr sylbi n0 df-rex 3imtr4g
      wcel impcom ) ABCDZCEFZABCGZUABHCSZBIZUDAJBIZUBUCUAUDAKBLUEUFKABCMUDABNOB
      CPABCQRT $.

    $( A response to the notion that the condition ` A =/= (/) ` can be removed
       in ~ r19.2z .  Interestingly enough, ` ph ` does not figure in the
       left-hand side.  (Contributed by Jeff Hankins, 24-Aug-2009.) $)
    r19.2zb $p |- ( A =/= (/) <-> ( A. x e. A ph -> E. x e. A ph ) ) $=
      ( c0 wral wrex wi r19.2z ex wceq cv wcel noel pm2.21i rgen raleq necon3bi
      wne mpbiri wex wa exsimpl df-rex n0 3imtr4i ja impbii ) CDRZABCEZABCFZGUH
      UIUJABCHIUIUJUHUICDCDJUIABDEABDBKZDLAUKMNOABCDPSQUKCLZAUABTULBTUJUHULABUB
      ABCUCBCUDUEUFUG $.
  $}

  ${
    $d x A $.
    r19.3rz.1 $e |- F/ x ph $.
    $( Restricted quantification of wff not containing quantified variable.
       (Contributed by FL, 3-Jan-2008.) $)
    r19.3rz $p |- ( A =/= (/) -> ( ph <-> A. x e. A ph ) ) $=
      ( c0 wne cv wcel wex wi wral wb n0 biimt sylbi df-ral 19.23 bitri bitr4di
      wal ) CEFZABGCHZBIZAJZABCKZUAUCAUDLBCMUCANOUEUBAJBTUDABCPUBABDQRS $.

    $( Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90.  It
       is valid only when the domain of quantification is not empty.
       (Contributed by NM, 26-Oct-2010.) $)
    r19.28z $p |- ( A =/= (/) ->
                   ( A. x e. A ( ph /\ ps ) <-> ( ph /\ A. x e. A ps ) ) ) $=
      ( c0 wne wa wral r19.26 r19.3rz anbi1d bitr4id ) DFGZABHCDIACDIZBCDIZHAPH
      ABCDJNAOPACDEKLM $.
  $}

  ${
    $d x A $.  $d x ph $.
    $( Restricted quantification of wff not containing quantified variable.
       (Contributed by NM, 10-Mar-1997.) $)
    r19.3rzv $p |- ( A =/= (/) -> ( ph <-> A. x e. A ph ) ) $=
      ( nfv r19.3rz ) ABCABDE $.

    $( Restricted quantification of wff not containing quantified variable.
       (Contributed by NM, 27-May-1998.) $)
    r19.9rzv $p |- ( A =/= (/) -> ( ph <-> E. x e. A ph ) ) $=
      ( wrex wn wral c0 wne dfrex2 r19.3rzv con1bid bitr2id ) ABCDAEZBCFZECGHZA
      ABCIOANMBCJKL $.

    $( Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90.  It
       is valid only when the domain of quantification is not empty.
       (Contributed by NM, 19-Aug-2004.) $)
    r19.28zv $p |- ( A =/= (/) ->
                   ( A. x e. A ( ph /\ ps ) <-> ( ph /\ A. x e. A ps ) ) ) $=
      ( nfv r19.28z ) ABCDACEF $.

    $( Restricted quantifier version of Theorem 19.37 of [Margaris] p. 90.  It
       is valid only when the domain of quantification is not empty.
       (Contributed by Paul Chapman, 8-Oct-2007.) $)
    r19.37zv $p |- ( A =/= (/) ->
                   ( E. x e. A ( ph -> ps ) <-> ( ph -> E. x e. A ps ) ) ) $=
      ( c0 wne wi wrex wral r19.35 r19.3rzv imbi1d bitr4id ) DEFZABGCDHACDIZBCD
      HZGAPGABCDJNAOPACDKLM $.

    $( Restricted version of Theorem 19.45 of [Margaris] p. 90.  (Contributed
       by NM, 27-May-1998.) $)
    r19.45zv $p |- ( A =/= (/) ->
                   ( E. x e. A ( ph \/ ps ) <-> ( ph \/ E. x e. A ps ) ) ) $=
      ( c0 wne wo wrex r19.43 r19.9rzv orbi1d bitr4id ) DEFZABGCDHACDHZBCDHZGAO
      GABCDIMANOACDJKL $.
  $}

  ${
    $d x A $.  $d x ps $.
    $( Restricted version of Theorem 19.44 of [Margaris] p. 90.  (Contributed
       by NM, 27-May-1998.) $)
    r19.44zv $p |- ( A =/= (/) ->
                   ( E. x e. A ( ph \/ ps ) <-> ( E. x e. A ph \/ ps ) ) ) $=
      ( c0 wne wo wrex r19.43 r19.9rzv orbi2d bitr4id ) DEFZABGCDHACDHZBCDHZGNB
      GABCDIMBONBCDJKL $.
  $}

  ${
    $d x A $.
    r19.27z.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 not empty.
       (Contributed by NM, 26-Oct-2010.) $)
    r19.27z $p |- ( A =/= (/) ->
                   ( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ ps ) ) ) $=
      ( c0 wne wa wral r19.26 r19.3rz anbi2d bitr4id ) DFGZABHCDIACDIZBCDIZHOBH
      ABCDJNBPOBCDEKLM $.
  $}

  ${
    $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 not empty.
       (Contributed by NM, 19-Aug-2004.) $)
    r19.27zv $p |- ( A =/= (/) ->
                   ( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ ps ) ) ) $=
      ( nfv r19.27z ) ABCDBCEF $.

    $( Restricted quantifier version of Theorem 19.36 of [Margaris] p. 90.  It
       is valid only when the domain of quantification is not empty.
       (Contributed by NM, 20-Sep-2003.) $)
    r19.36zv $p |- ( A =/= (/) ->
                   ( E. x e. A ( ph -> ps ) <-> ( A. x e. A ph -> ps ) ) ) $=
      ( c0 wne wi wrex wral r19.35 r19.9rzv imbi2d bitr4id ) DEFZABGCDHACDIZBCD
      HZGOBGABCDJNBPOBCDKLM $.
  $}

  ${
    $d x y A $.  $d ph y $.  $d ps x $.
    ralidmw.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Idempotent law for restricted quantifier.  Weak version of ~ ralidm ,
       which does not require ~ ax-10 , ~ ax-12 , but requires ~ ax-8 .
       (Contributed by GG, 30-Sep-2024.) $)
    ralidmw $p |- ( A. x e. A A. x e. A ph <-> A. x e. A ph ) $=
      ( cv wcel wral wi wal df-ral imbi2i albii pm2.21 weq eleq1w imbi12d spw
      ja alimi hba1w ax-1 alrimih impbii bitri 3bitr4i ) CGEHZACEIZJZCKZUHAJZCK
      ZUICEIUIUKUHUMJZCKZUMUJUNCUIUMUHACELZMNUOUMUNULCUHUMULUHAOULDGEHZBJZCDCDP
      UHUQABCDEQFRZSTUAUMUNCULURCDUSUBUMUHUCUDUEUFUICELUPUG $.
    $( $j usage 'ralidmw' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $d x A $.  $d x y $.
    $( Vacuous quantification is always true.  (Contributed by NM,
       11-Mar-1997.)  (Proof shortened by Andrew Salmon, 26-Jun-2011.)  Avoid
       ~ df-clel , ~ ax-8 .  (Revised by GG, 2-Sep-2024.) $)
    rzal $p |- ( A = (/) -> A. x e. A ph ) $=
      ( vy wfal cab wceq cv wcel wi wal c0 wral dfcleq biimpi wsb df-clab bitri
      wb sbv bibi2i nbfal pm2.21 sylbir sylbi sylg dfnul4 eqeq2i df-ral 3imtr4i
      wn ) CEDFZGZBHZCIZAJZBKCLGABCMUMUOUNULIZSZUPBUMURBKBCULNOURUOESZUPUQEUOUQ
      EDBPEEBDQEDBTRUAUSUOUKUPUOUBUOAUCUDUEUFLULCDUGUHABCUIUJ $.
    $( $j usage 'rzal' avoids 'df-clel' 'ax-8'; $)

    $( Alternate proof of ~ rzal .  Shorter, but requiring ~ df-clel , ~ ax-8 .
       (Contributed by NM, 11-Mar-1997.)  (Proof shortened by Andrew Salmon,
       26-Jun-2011.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    rzalALT $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.  (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)  Avoid
       ~ df-clel , ~ ax-8 .  (Revised by GG, 2-Sep-2024.) $)
    rexn0 $p |- ( E. x e. A ph -> A =/= (/) ) $=
      ( wrex c0 wn wral wceq dfrex2 rzal con3i sylbi neqned ) ABCDZCENAFZBCGZFC
      EHZFABCIQPOBCJKLM $.
    $( $j usage 'rexn0' avoids 'df-clel' 'ax-8'; $)
  $}

  $( Idempotent law for restricted quantifier.  (Contributed by NM,
     28-Mar-1997.)  Reduce axiom usage.  (Revised by GG, 2-Sep-2024.) $)
  ralidm $p |- ( A. x e. A A. x e. A ph <-> A. x e. A ph ) $=
    ( wral cv wcel wi df-ral ax-1 axc4i pm2.21 sp ja alimi impbii bicomi imbi2i
    wal albii 3bitrri bitri ) ABCDZBCDBECFZUBGZBRZUBUBBCHUBUCAGZBRZUCUGGZBRZUEA
    BCHZUGUIUFUHBUGUCIJUHUFBUCUGUFUCAKUFBLMNOUHUDBUGUBUCUBUGUJPQSTUA $.
  $( $j usage 'ralidm' avoids 'df-clel' 'df-cleq' 'ax-8' 'ax-9' 'ax-ext'; $)

  $( Vacuous universal quantification is always true.  (Contributed by NM,
     20-Oct-2005.)  Avoid ~ df-clel , ~ ax-8 .  (Revised by GG, 2-Sep-2024.) $)
  ral0 $p |- A. x e. (/) ph $=
    ( c0 wceq wral eqid rzal ax-mp ) CCDABCECFABCGH $.
  $( $j usage 'rzal' avoids 'df-clel' 'ax-8'; $)

  ${
    $d x A $.  $d x y $.
    ralf0.1 $e |- -. ph $.
    $( The quantification of a falsehood is vacuous when true.  (Contributed by
       NM, 26-Nov-2005.)  (Proof shortened by JJ, 14-Jul-2021.)  Avoid
       ~ df-clel , ~ ax-8 .  (Revised by GG, 2-Sep-2024.) $)
    ralf0 $p |- ( A. x e. A ph <-> A = (/) ) $=
      ( vy wral c0 wceq wn wi wa r19.26 wfal cv wal pm2.24 impcom ralimi df-ral
      wcel dfnot bicomi albii sylbb cab wb falim pm5.21ni wsb df-clab sbv bitri
      id bitr4di alimi dfcleq sylibr dfnul4 eqtr4di 3syl sylbir a1i mprg impbii
      ex rzal ) ABCFZCGHZAIZVGVHJBCVIBCFZVGVHVJVGKVIAKZBCFZVHVIABCLVLMBCFZBNZCT
      ZIZBOZVHVKMBCAVIMAMPQRVMVOMJZBOVQMBCSVRVPBVPVRVOUAUBUCUDVQCMEUEZGVQVOVNVS
      TZUFZBOCVSHVPWABVPVOMVTVOVOMVOUMVOUGUHVTMEBUIMMBEUJMEBUKULUNUOBCVSUPUQEUR
      USUTVAVEVIVODVBVCABCVFVD $.
    $( $j usage 'ralf0' avoids 'df-clel' 'ax-8'; $)
  $}

  ${
    $d A x $.
    $( A contradiction concerning restricted generalization for a nonempty set
       implies anything.  (Contributed by Alexander van der Vekens,
       4-Sep-2018.) $)
    ralnralall $p |- ( A =/= (/)
                       -> ( ( A. x e. A ph /\ A. x e. A -. ph ) -> ps ) ) $=
      ( wral wn wa wne r19.26 wfal pm3.24 bifal ralbii r19.3rzv falim biimtrrdi
      c0 biimtrid biimtrrid ) ACDEAFZCDEGATGZCDEZDQHZBATCDIUBJCDEZUCBUAJCDUAAKL
      MUCUDJBJCDNBOPRS $.

    $( A false statement can only be true for elements of an empty set.
       (Contributed by AV, 30-Oct-2020.) $)
    falseral0 $p |- ( ( A. x -. ph /\ A. x e. A ph ) -> A = (/) ) $=
      ( wral wn wal cv wcel wi c0 wceq df-ral 19.26 wex con3 impcom alimi alnex
      wa sylib notnotb neq0 xchbinx sylibr sylbir sylan2b ) ABCDAEZBFZBGCHZAIZB
      FZCJKZABCLUHUKSUGUJSZBFZULUGUJBMUNUIBNZEZULUNUIEZBFUPUMUQBUJUGUQUIAOPQUIB
      RTULULEUOULUABCUBUCUDUEUF $.
  $}

  ${
    $d x y A $.
    raaan.1 $e |- F/ y ph $.
    raaan.2 $e |- F/ x ps $.
    $( 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 ) ) $=
      ( wa wral wb c0 wceq rzal pm5.1 syl12anc wne r19.28z ralbidv nfcv nfralw
      r19.27z bitrd pm2.61ine ) ABHDEIZCEIZACEIZBDEIZHZJZEKEKLUEUFUGUIUDCEMACEM
      BDEMUEUHNOEKPZUEAUGHZCEIUHUJUDUKCEABDEFQRAUGCEBCDECESGTUAUBUC $.
  $}

  ${
    $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 ) ) $=
      ( wa wral wb wceq rzal pm5.1 syl12anc wne r19.28zv ralbidv r19.27zv bitrd
      c0 pm2.61ine ) ABFDEGZCEGZACEGZBDEGZFZHZERERIUAUBUCUETCEJACEJBDEJUAUDKLER
      MZUAAUCFZCEGUDUFTUGCEABDENOAUCCEPQS $.
  $}

  ${
    $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 sseq1 sbievw2 ) AEZCFBEZCFDEZCFABDIKCGKJCGH $.
  $}

  ${
    $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 wi wal wsbc csb wss sbcal sbcimg sbcel2 imbi12i bitrdi df-ss
      albidv bitrid sbcbii 3bitr4g ) BEGZFHZCGZUEDGZIZFJZABKZUEABCLZGZUEABDLZGZ
      IZFJZCDMZABKUKUMMUJUHABKZFJUDUPUHFABNUDURUOFUDURUFABKZUGABKZIUOUFUGABEOUS
      ULUTUNABUECPABUEDPQRTUAUQUIABFCDSUBFUKUMSUC $.
  $}

  ${
    $d x y $.  $d x A $.  $d x y B $.
    raaan2.1 $e |- F/ y ph $.
    raaan2.2 $e |- F/ x ps $.
    $( Rearrange restricted quantifiers with two different restricting classes,
       analogous to ~ raaan .  It is necessary that either both restricting
       classes are empty or both are not empty.  (Contributed by Alexander van
       der Vekens, 29-Jun-2017.) $)
    raaan2 $p |- ( ( A = (/) <-> B = (/) )
                   -> ( A. x e. A A. y e. B ( ph /\ ps ) <->
                        ( A. x e. A ph /\ A. y e. B ps ) ) ) $=
      ( c0 wceq wb wa wn wo wral rzal adantr wne df-ne sylbir dfbi3 adantl nfcv
      pm5.1 syl12anc r19.28z ralbidv nfralw r19.27z sylan9bbr jaoi sylbi ) EIJZ
      FIJZKUMUNLZUMMZUNMZLZNABLDFOZCEOZACEOZBDFOZLZKZUMUNUAUOVDURUOUTVAVBVDUMUT
      UNUSCEPQUMVAUNACEPQUNVBUMBDFPUBUTVCUDUEUQUTAVBLZCEOZUPVCUQFIRZUTVFKFISVGU
      SVECEABDFGUFUGTUPEIRVFVCKEISAVBCEBCDFCFUCHUHUITUJUKUL $.
  $}

  ${
    $d z w ph $.  $d w x y z A $.  $d w x y z B $.
    $( Lemma for ~ 2reu4 .  (Contributed by Alexander van der Vekens,
       1-Jul-2017.) $)
    2reu4lem $p |- ( ( A =/= (/) /\ B =/= (/) )
                -> ( ( E! x e. A E. y e. B ph /\ E! y e. B E. x e. A ph ) <->
                    ( E. x e. A E. y e. B ph /\ E. z e. A E. w e. B
                      A. x e. A A. y e. B ( ph -> ( x = z /\ y = w ) ) ) ) ) $=
      ( c0 wne wa wrex wreu weq wi wral wb a1i bitri r19.26 adantr reu3 anbi12i
      an4 rexcom anbi2i anidm cv wcel nfra1 r19.3rz bicomd anbi2d ralbii bitr4d
      jcab bitr2id ad2antlr ralcom anbi12d bitrid ralbidv r19.23v 2ralbii wn wo
      wceq neneq anim12i olcd dfbi3 sylibr nfv nfim raaan2 syl 3bitrd 2rexbidva
      nfre1 reeanv bitr2di ) FHIZGHIZJZACGKZBFLZABFKZCGLZJZWDBFKZWDBDMZNZBFOZDF
      KZJZWFCGKZWFCEMZNZCGOZEGKZJZJZWIWOJZWMWSJZJZWIAWJWPJNZCGOZBFOZEGKDFKZJWHX
      APWCWEWNWGWTWDBDFUAWFCEGUAUBQXAXDPWCWIWMWOWSUCQWCXBWIXCXHXBWIPWCXBWIWIJWI
      WOWIWIACBGFUDUEWIUFRQWCXHWLWRJZEGKDFKXCWCXGXIDEFGWCDUGFUHEUGGUHJZJZXGAWJN
      ZCGOZAWPNZBFOZJZCGOZBFOZWKWQJZCGOBFOZXIXKXGXMXNCGOZBFOZJZBFOZXRYDXMBFOZYB
      BFOZJZXKXGXMYBBFSXKYGYEYBJZXGXKYFYBYEWCYFYBPZXJWAYIWBWAYBYFYBBFYABFUIUJUK
      TTULXGYHPXKXGXMYAJZBFOYHXFYJBFXFXLXNJZCGOYJXEYKCGAWJWPUOUMXLXNCGSRUMXMYAB
      FSRQUNUPXKXQYCBFXQXMCGOZXOCGOZJXKYCXMXOCGSXKYLXMYMYBXKXMYLWBXMYLPWAXJXMCG
      XLCGUIUJUQUKYMYBPXKXNCBGFURQUSUTVAUNXRXTPXKXPXSBCFGXMWKXOWQAWJCGVBAWPBFVB
      UBVCQWCXTXIPZXJWCFHVFZGHVFZPZYNWCYOYPJZYOVDZYPVDZJZVEYQWCUUAYRWAYSWBYTFHV
      GGHVGVHVIYOYPVJVKWKWQBCFGWDWJCACGVRWJCVLVMWFWPBABFVRWPBVLVMVNVOTVPVQWLWRD
      EFGVSVTUSVP $.

    $( Definition of double restricted existential uniqueness ("exactly one
       ` x ` and exactly one ` y ` "), analogous to ~ 2eu4 .  (Contributed by
       Alexander van der Vekens, 1-Jul-2017.) $)
    2reu4 $p |- ( ( E! x e. A E. y e. B ph /\ E! y e. B E. x e. A ph ) <->
                    ( E. x e. A E. y e. B ph /\ E. z e. A E. w e. B
                      A. x e. A A. y e. B ( ph -> ( x = z /\ y = w ) ) ) ) $=
      ( wrex wreu wa c0 wne weq wral reurex rexn0 syl anim12i cv wcel rexlimivv
      wi ne0i a1d adantr 2reu4lem pm5.21nii ) ACGHZBFIZABFHZCGIZJFKLZGKLZJZUHBF
      HZABDMCEMJUBCGNBFNEGHDFHZJUIULUKUMUIUOULUHBFOUHBFPQUKUJCGHUMUJCGOUJCGPQRU
      OUNUPAUNBCFGBSZFTZCSZGTZJUNAURULUTUMFUQUCGUSUCRUDUAUEABCDEFGUFUG $.
  $}

  ${
    $d A y $.  $d B y $.  $d C y $.  $d x y $.
    $( Distribution of class substitution over difference of two classes.
       (Contributed by ML, 14-Jul-2020.) $)
    csbdif $p |- [_ A / x ]_ ( B \ C ) = ( [_ A / x ]_ B \ [_ A / x ]_ C ) $=
      ( vy cvv wcel cdif csb wceq csbeq1 difeq12d eqeq12d nfcsb1v nfdif csbeq1a
      cv vex c0 csbprc csbief vtoclg wn dif0 a1i 3eqtr4rd pm2.61i ) BFGZABCDHZI
      ZABCIZABDIZHZJZAEQZUIIZAUOCIZAUODIZHZJUNEBFUOBJZUPUJUSUMAUOBUIKUTUQUKURUL
      AUOBCKAUOBDKLMAUOUIUSERAUQURAUOCNAUODNOAQUOJCUQDURAUOCPAUODPLUAUBUHUCZSSH
      ZSUMUJVBSJVASUDUEVAUKSULSABCTABDTLABUITUFUG $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  The conditional operator for classes
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  This subsection introduces the conditional operator for classes, denoted
  by ` if ( ph , A , B ) ` (see ~ df-if ).  It is the analogue for classes
   of the conditional operator for propositions, denoted by
  ` if- ( ph , ps , ch ) ` (see ~ df-ifp ).

$)

  $( Symbol for the conditional operator for classes. $)
  $c if $.

  $( Extend class notation to include the conditional operator for classes. $)
  cif $a class if ( ph , A , B ) $.

  ${
    $d x ph $.  $d x A $.  $d x B $.
    $( Definition of the conditional operator for classes.  The expression
       ` if ( ph , A , B ) ` is read "if ` ph ` then ` A ` else ` B ` ".  See
       ~ iftrue and ~ iffalse for its values.  In the 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".

       An important use for us is in conjunction with the weak deduction
       theorem, which is described in the next section, beginning at ~ dedth .
       (Contributed by NM, 15-May-1999.) $)
    df-if $a |- if ( ph , A , B ) =
                 { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) } $.
  $}

  ${
    $d x y ph $.  $d x y A $.  $d x y B $.  $d x C $.
    $( An alternate definition of the conditional operator ~ df-if with one
       fewer connectives (but probably less intuitive to understand).
       (Contributed by NM, 30-Jan-2006.) $)
    dfif2 $p |- if ( ph , A , B ) =
                 { x | ( ( x e. B -> ph ) -> ( x e. A /\ ph ) ) } $=
      ( cif cv wcel wa wn wo cab wi df-if df-or orcom iman imbi1i 3bitr4i abbii
      eqtri ) ACDEBFZCGAHZUADGZAIHZJZBKUCALZUBLZBKABCDMUEUGBUDUBJUDIZUBLUEUGUDU
      BNUBUDOUFUHUBUCAPQRST $.

    $( 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 } ) $=
      ( vy cv wcel wa cab wn cun wo crab cif eleq1w anbi1d unabw df-rab uneq12i
      weq df-if 3eqtr4ri ) BFZCGZAHZBIZUCDGZAJZHZBIZKEFZCGZAHZUKDGZUHHZLEIABCMZ
      UHBDMZKACDNUEUIUMUOBEBETZUDULABECOPURUGUNUHBEDOPQUPUFUQUJABCRUHBDRSAECDUA
      UB $.
    $( $j usage 'dfif6' avoids 'ax-10' 'ax-11' 'ax-12'; $)

    $( 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 wi wa cab dfif2 dedlem0a eqabdv eqtr4id ) AABCEDFZCGZAHO
      BGZAIHZDJBADBCKARDBAQPLMN $.
  $}

  ${
    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 eqabdv eqtr4id ) AEZABCFDGZBH
      ZAIQCHZPIJZDKCADBCLPTDCARSMNO $.
  $}

  ${
    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.  It happens, e.g., in ~ oevn0 .  (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 $.

  $( When the branches are not equal, an "if" condition results in the first
     branch if and only if its condition is true.  (Contributed by SN,
     16-Oct-2025.) $)
  iftrueb $p |- ( A =/= B -> ( if ( ph , A , B ) = A <-> ph ) ) $=
    ( wne cif wn necom biimpi iffalse neeq1d syl5ibrcom necon4bd iftrue impbid1
    wceq ) BCDZABCEZBOAPAQBPQBDAFZCBDZPSBCGHRQCBABCIJKLABCMN $.

  ${
    ifsb.1 $e |- ( if ( ph , A , B ) = A -> C = D ) $.
    ifsb.2 $e |- ( if ( ph , A , B ) = B -> C = E ) $.
    $( Distribute a function over an if-clause.  (Contributed by Mario
       Carneiro, 14-Aug-2013.) $)
    ifsb $p |- C = if ( ph , D , E ) $=
      ( cif wceq iftrue syl eqtr4d wn iffalse pm2.61i ) ADAEFIZJADEQAABCIZBJDEJ
      ABCKGLAEFKMANZDFQSRCJDFJABCOHLAEFOMP $.
  $}

  ${
    $d y A $.  $d y B $.  $d x y z 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 vz crab cun cin cvv cdif cab weq biidd cbvabv eqtri ineq2i eqtr4i wn
      cif dfif6 dfrab3 notabw difeq2i eqtr2i uneq12i ) ACDUBAGCIZAUAZGDIZJCEKZD
      LEMZKZJAGCDUCULUIUNUKULCAGNZKUIEUOCEABNZUOFAABGBGOAPQRSAGCUDTUKDUJGNZKUNU
      JGDUDUQUMDUQLAHNZMUMAAGHGHOAPUEEURLEUPURFAABHBHOAPQRUFTSUGUHT $.
    $( $j usage 'dfif3' avoids 'ax-10' 'ax-11' 'ax-12'; $)

    $( 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.) $)
    dfif4 $p |- if ( ph , A , B )
        = ( ( A u. B ) i^i ( ( A u. ( _V \ C ) ) i^i ( B u. C ) ) ) $=
      ( cif cin cvv cdif cun dfif3 undir undi uncom unvdif ineq12i 3eqtri inass
      inv1 eqtri ) ACDGCEHDIEJZHZKCUCKZEUCKZHZCDKZCUBKZDEKZHHZABCDEFLCEUCMUFUGU
      HHZUIHUJUDUKUEUICDUBNUEEDKZEUBKZHUIIHUIEDUBNULUIUMIEDOEPQUITRQUGUHUISUAR
      $.

    $( Alternate definition of the conditional operator ~ df-if .  Note that
       ` ph ` is independent of ` x ` i.e. a constant true or false (see also
       ~ ab0orv ).  (Contributed by G&eacute;rard Lang, 18-Aug-2013.) $)
    dfif5 $p |- if ( ph , A , B ) = ( ( A i^i B )
          u. ( ( ( A \ B ) i^i C ) u. ( ( B \ A ) i^i ( _V \ C ) ) ) ) $=
      ( cun cdif undir unidm unass undi 3eqtr3ri undifabs ineq1i undif2 3eqtr4i
      cin inabs eqtr4i cvv cif inindi dfif4 uneq1i 3eqtri uneq12i uncom 3eqtrri
      unundi uneq2i ineq2i ineq12i ) CDGZCUAEHZGZDEGZRRUNUPRZUNUQRZRZACDUBCDRCD
      HZERZDCHZUORZGZGZUNUPUQUCABCDEFUDVFCVEGZDVEGZRUTCDVEIURVGUSVHURCVBGZCVDGZ
      GZVGURCCDUORZGZGZVKCCGZVLGVMVNURVOCVLCJUECCVLKCDUOLZMVICVJVMVICVAGZCEGZRC
      VRRCCVAELVQCVRCDNOCESUFCVCGZUPRURVJVMVSUNUPCDPOCVCUOLVPQUGTCVBVDUJTCERZDG
      ZDVBGZDVDGZGZUSVHWADGVTDDGZGWDWAVTDDKWAWBDWCWADVAGZUQRZWBDVTGDCGZUQRWAWGD
      CELVTDUHWFWHUQDCPOQDVAELTWCDVCGZDUOGZRDWJRDDVCUOLWIDWJDCNODUOSUIUGWEDVTDJ
      UKMUSUNEDGZRWAUQWKUNDEUHULCEDITDVBVDUJQUMTQ $.
  $}

  ${
    $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 cun cvv cab cdif cin eqid dfif4 inss1 eqsstri ) ABCEBCFZBGADHZIF
      CPFJZJOADBCPPKLOQMN $.
  $}

  $( 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 $.
  $}

  $( Equivalence theorem for conditional operators.  (Contributed by Raph
     Levien, 15-Jan-2004.) $)
  ifbi $p |- ( ( ph <-> ps ) -> if ( ph , A , B ) = if ( ps , A , B ) ) $=
    ( wb wa wn wo cif wceq dfbi3 iftrue eqcomd sylan9eq iffalse jaoi sylbi ) AB
    EABFZAGZBGZFZHACDIZBCDIZJZABKRUDUAABUBCUCACDLBUCCBCDLMNSTUBDUCACDOTUCDBCDOM
    NPQ $.

  ${
    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 form 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 wi wa cab dfif2 nfv nfcrd nfimd nfand nfabdw nfcxfrd ) A
      CBDEJIKZELZBMZUCDLZBNZMZIOBIDEPAUHCIAIQAUEUGCAUDBCACIEHRFSAUFBCACIDGRFTSU
      AUB $.
  $}

  ${
    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 $.
  $}

  ${
    ifeq1da.1 $e |- ( ( ph /\ ps ) -> A = B ) $.
    $( Conditional equality.  (Contributed by Jeff Madsen, 2-Sep-2009.) $)
    ifeq1da $p |- ( ph -> if ( ps , A , C ) = if ( ps , B , C ) ) $=
      ( cif wceq wa ifeq1d wn iffalse eqtr4d adantl pm2.61dan ) ABBCEGZBDEGZHZA
      BIBCDEFJBKZRASPEQBCELBDELMNO $.
  $}

  ${
    ifeq2da.1 $e |- ( ( ph /\ -. ps ) -> A = B ) $.
    $( Conditional equality.  (Contributed by Jeff Madsen, 2-Sep-2009.) $)
    ifeq2da $p |- ( ph -> if ( ps , C , A ) = if ( ps , C , B ) ) $=
      ( cif wceq iftrue eqtr4d adantl wn wa ifeq2d pm2.61dan ) ABBECGZBEDGZHZBR
      ABPEQBECIBEDIJKABLMBCDEFNO $.
  $}

  ${
    ifeq12da.1 $e |- ( ( ph /\ ps ) -> A = C ) $.
    ifeq12da.2 $e |- ( ( ph /\ -. ps ) -> B = D ) $.
    $( Equivalence deduction for conditional operators.  (Contributed by Wolf
       Lammen, 24-Jun-2021.) $)
    ifeq12da $p |- ( ph -> if ( ps , A , B ) = if ( ps , C , D ) ) $=
      ( cif wceq ifeq1da iftrue eqtr4d sylan9eq wn ifeq2da iffalse pm2.61dan )
      ABBCDIZBEFIZJABSBEDIZTABCEDGKBUAETBEDLBEFLMNABOZSBCFIZTABDFCHPUBUCFTBCFQB
      EFQMNR $.
  $}

  ${
    ifbieq12d2.1 $e |- ( ph -> ( ps <-> ch ) ) $.
    ifbieq12d2.2 $e |- ( ( ph /\ ps ) -> A = C ) $.
    ifbieq12d2.3 $e |- ( ( ph /\ -. ps ) -> B = D ) $.
    $( Equivalence deduction for conditional operators.  (Contributed by
       Thierry Arnoux, 14-Feb-2017.)  (Proof shortened by Wolf Lammen,
       24-Jun-2021.) $)
    ifbieq12d2 $p |- ( ph -> if ( ps , A , B ) = if ( ch , C , D ) ) $=
      ( cif ifeq12da ifbid eqtrd ) ABDEKBFGKCFGKABDEFGIJLABCFGHMN $.
  $}

  ${
    ifclda.1 $e |- ( ( ph /\ ps ) -> A e. C ) $.
    ifclda.2 $e |- ( ( ph /\ -. ps ) -> B e. C ) $.
    $( Conditional closure.  (Contributed by Jeff Madsen, 2-Sep-2009.) $)
    ifclda $p |- ( ph -> if ( ps , A , B ) e. C ) $=
      ( cif wcel wa wceq iftrue adantl eqeltrd wn iffalse pm2.61dan ) ABBCDHZEI
      ABJRCEBRCKABCDLMFNABOZJRDESRDKABCDPMGNQ $.
  $}

  ${
    ifeqda.1 $e |- ( ( ph /\ ps ) -> A = C ) $.
    ifeqda.2 $e |- ( ( ph /\ -. ps ) -> B = C ) $.
    $( Separation of the values of the conditional operator.  (Contributed by
       Alexander van der Vekens, 13-Apr-2018.) $)
    ifeqda $p |- ( ph -> if ( ps , A , B ) = C ) $=
      ( cif wceq wa iftrue adantl eqtrd wn iffalse pm2.61dan ) ABBCDHZEIABJQCEB
      QCIABCDKLFMABNZJQDERQDIABCDOLGMP $.
  $}

  ${
    elimif.1 $e |- ( if ( ph , A , B ) = A -> ( ps <-> ch ) ) $.
    elimif.2 $e |- ( if ( ph , A , B ) = B -> ( ps <-> th ) ) $.
    $( Elimination of a conditional operator contained in a wff ` ps ` .
       (Contributed by NM, 15-Feb-2005.)  (Proof shortened by NM,
       25-Apr-2019.) $)
    elimif $p |- ( ps <-> ( ( ph /\ ch ) \/ ( -. ph /\ th ) ) ) $=
      ( cif wceq wb iftrue syl wn iffalse cases ) ABCDAAEFIZEJBCKAEFLGMANQFJBDK
      AEFOHMP $.
  $}

  ${
    ifboth.1 $e |- ( A = if ( ph , A , B ) -> ( ps <-> th ) ) $.
    ifboth.2 $e |- ( B = if ( ph , A , B ) -> ( ch <-> th ) ) $.
    ${
      ifbothda.3 $e |- ( ( et /\ ph ) -> ps ) $.
      ifbothda.4 $e |- ( ( et /\ -. ph ) -> ch ) $.
      $( A wff ` th ` containing a conditional operator is true when both of
         its cases are true.  (Contributed by NM, 15-Feb-2015.) $)
      ifbothda $p |- ( et -> th ) $=
        ( wa wb cif wceq iftrue eqcomd syl adantl mpbid wn iffalse pm2.61dan )
        EADEALBDJABDMZEAFAFGNZOUDAUEFAFGPQHRSTEAUAZLCDKUFCDMZEUFGUEOUGUFUEGAFGU
        BQIRSTUC $.
    $}

    $( A wff ` th ` containing a conditional operator is true when both of its
       cases are true.  (Contributed by NM, 3-Sep-2006.)  (Revised by Mario
       Carneiro, 15-Feb-2015.) $)
    ifboth $p |- ( ( ps /\ ch ) -> th ) $=
      ( wa simpll wn simplr ifbothda ) ABCDBCIEFGHBCAJBCAKLM $.
  $}

  $( Identical true and false arguments in the conditional operator.
     (Contributed by NM, 18-Apr-2005.) $)
  ifid $p |- if ( ph , A , A ) = A $=
    ( cif wceq iftrue iffalse pm2.61i ) AABBCBDABBEABBFG $.

  $( Expansion of an equality with a conditional operator.  (Contributed by NM,
     14-Feb-2005.) $)
  eqif $p |- ( A = if ( ph , B , C ) <->
             ( ( ph /\ A = B ) \/ ( -. ph /\ A = C ) ) ) $=
    ( cif wceq eqeq2 elimif ) ABACDEZFBCFBDFCDICBGIDBGH $.

  $( Another expression of the value of the ` if ` predicate, analogous to
     ~ eqif .  See also the more specialized ~ iftrue and ~ iffalse .
     (Contributed by BJ, 6-Apr-2019.) $)
  ifval $p |- ( A = if ( ph , B , C )
                             <-> ( ( ph -> A = B ) /\ ( -. ph -> A = C ) ) ) $=
    ( cif wceq wa wn wo wi eqif cases2 bitri ) BACDEFABCFZGAHZBDFZGIANJOPJGABCD
    KANPLM $.

  $( 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 ) ) ) $=
    ( cif wcel eleq2 elimif ) ABACDEZFBCFBDFCDICBGIDBGH $.

  $( Membership of a conditional operator.  (Contributed by NM,
     10-Sep-2005.) $)
  ifel $p |- ( if ( ph , A , B ) e. C <->
             ( ( ph /\ A e. C ) \/ ( -. ph /\ B e. C ) ) ) $=
    ( cif wcel eleq1 elimif ) AABCEZDFBDFCDFBCIBDGICDGH $.

  $( Membership (closure) of a conditional operator.  (Contributed by NM,
     4-Apr-2005.) $)
  ifcl $p |- ( ( A e. C /\ B e. C ) -> if ( ph , A , B ) e. C ) $=
    ( wcel cif eleq1 ifboth ) ABDECDEABCFZDEBCBIDGCIDGH $.

  ${
    ifcld.a $e |- ( ph -> A e. C ) $.
    ifcld.b $e |- ( ph -> B e. C ) $.
    $( Membership (closure) of a conditional operator, deduction form.
       (Contributed by SO, 16-Jul-2018.) $)
    ifcld $p |- ( ph -> if ( ps , A , B ) e. C ) $=
      ( wcel cif ifcl syl2anc ) ACEHDEHBCDIEHFGBCDEJK $.
  $}

  ${
    ifcli.1 $e |- A e. C $.
    ifcli.2 $e |- B e. C $.
    $( Inference associated with ~ ifcl .  Membership (closure) of a
       conditional operator.  Also usable to keep a membership hypothesis for
       the weak deduction theorem ~ dedth when the special case ` B e. C ` is
       provable.  (Contributed by NM, 14-Aug-1999.)  (Proof shortened by BJ,
       1-Sep-2022.) $)
    ifcli $p |- if ( ph , A , B ) e. C $=
      ( wcel cif ifcl mp2an ) BDGCDGABCHDGEFABCDIJ $.
  $}

  ${
    ifexd.1 $e |- ( ph -> A e. V ) $.
    ifexd.2 $e |- ( ph -> B e. W ) $.
    $( Existence of the conditional operator (deduction form).  (Contributed by
       SN, 26-Jul-2024.) $)
    ifexd $p |- ( ph -> if ( ps , A , B ) e. _V ) $=
      ( cvv elexd ifcld ) ABCDIACEGJADFHJK $.
  $}

  $( 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 ifcli ) ABCFDEG $.
  $}

  $( The possible values of a conditional operator.  (Contributed by NM,
     17-Jun-2007.)  (Proof shortened by Andrew Salmon, 26-Jun-2011.) $)
  ifeqor $p |- ( if ( ph , A , B ) = A \/ if ( ph , A , B ) = B ) $=
    ( cif wceq wn iftrue con3i iffalsed orri ) ABCDZBEZKCELFABCALABCGHIJ $.

  $( Negating the first argument swaps the last two arguments of a conditional
     operator.  (Contributed by NM, 21-Jun-2007.) $)
  ifnot $p |- if ( -. ph , A , B ) = if ( ph , B , A ) $=
    ( wn cif wceq notnot iffalsed iftrue eqtr4d iffalse pm2.61i ) AADZBCEZACBEZ
    FANCOAMBCAGHACBIJMNBOMBCIACBKJL $.

  $( Rewrite a conjunction in a conditional as two nested conditionals.
     (Contributed by Mario Carneiro, 28-Jul-2014.) $)
  ifan $p |- if ( ( ph /\ ps ) , A , B ) = if ( ph , if ( ps , A , B ) , B ) $=
    ( wa cif wceq iftrue ibar ifbid eqtr2d simpl con3i iffalsed iffalse pm2.61i
    wn eqtr4d ) AABEZCDFZABCDFZDFZGAUBUATAUADHABSCDABIJKAQZTDUBUCSCDSAABLMNAUAD
    ORP $.

  $( Rewrite a disjunction in a conditional as two nested conditionals.
     (Contributed by Mario Carneiro, 28-Jul-2014.) $)
  ifor $p |- if ( ( ph \/ ps ) , A , B ) = if ( ph , A , if ( ps , A , B ) ) $=
    ( wo cif wceq iftrue orcs eqtr4d wn iffalse biorf ifbid eqtr2d pm2.61i ) AA
    BEZCDFZACBCDFZFZGARCTABRCGQCDHIACSHJAKZTSRACSLUABQCDABMNOP $.

  ${
    2if2.1 $e |- ( ( ph /\ ps ) -> D = A ) $.
    2if2.2 $e |- ( ( ph /\ -. ps /\ th ) -> D = B ) $.
    2if2.3 $e |- ( ( ph /\ -. ps /\ -. th ) -> D = C ) $.
    $( Resolve two nested conditionals.  (Contributed by Alexander van der
       Vekens, 27-Mar-2018.) $)
    2if2 $p |- ( ph -> D = if ( ps , A , if ( th , B , C ) ) ) $=
      ( cif wceq wa iftrue adantl eqtr4d wn 3expa iffalse pm2.61dan eqcomd
      eqtrd ) ABGBDCEFKZKZLABMGDUDHBUDDLABDUCNOPABQZMZGUCUDUFCGUCLUFCMGEUCAUECG
      ELIRCUCELUFCEFNOPUFCQZMGFUCAUEUGGFLJRUGFUCLUFUGUCFCEFSUAOUBTUEUDUCLABDUCS
      OPT $.
  $}

  $( Commute the conditions in two nested conditionals if both conditions are
     not simultaneously true.  (Contributed by SO, 15-Jul-2018.) $)
  ifcomnan $p |- ( -. ( ph /\ ps ) -> if ( ph , A , if ( ps , B , C ) ) =
      if ( ps , B , if ( ph , A , C ) ) ) $=
    ( wa wn wo cif wceq pm3.13 iffalse ifeq2d eqtr4d jaoi syl ) ABFGAGZBGZHACBD
    EIZIZBDACEIZIZJZABKQUCRQTSUBACSLQBUAEDACELMNRTUAUBRASECBDELMBDUALNOP $.

  ${
    $d y A $.  $d y B $.  $d y C $.  $d y ph $.  $d x y $.
    $( Distribute proper substitution through the conditional operator.
       (Contributed by NM, 24-Feb-2013.)  (Revised by NM, 19-Aug-2018.) $)
    csbif $p |- [_ A / x ]_ if ( ph , B , C )
          = if ( [. A / x ]. ph , [_ A / x ]_ B , [_ A / x ]_ C ) $=
      ( vy cvv wcel cif csb wsbc cv wsb csbeq1 ifbieq12d nfcsb1v csbeq1a csbprc
      wceq c0 dfsbcq2 eqeq12d vex nfs1v nfif sbequ12 csbief vtoclg ifeq12d ifid
      weq wn eqtr2di eqtrd pm2.61i ) CGHZBCADEIZJZABCKZBCDJZBCEJZIZSZBFLZUQJZAB
      FMZBVDDJZBVDEJZIZSVCFCGVDCSZVEURVIVBBVDCUQNVJVFUSVGVHUTVAABFCUABVDCDNBVDC
      ENOUBBVDUQVIFUCVFBVGVHABFUDBVDDPBVDEPUEBFUKAVFDEVGVHABFUFBVDDQBVDEQOUGUHU
      PULZURTVBBCUQRVKVBUSTTITVKUSUTTVATBCDRBCERUIUSTUJUMUNUO $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  The weak deduction theorem for set theory
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  This subsection contains a few results related to the weak deduction theorem
  in set theory.  For the weak deduction theorem in propositional calculus, see
  the section beginning with ~ elimh .  For more information on the weak
  deduction theorem, see the Weak Deduction Theorem page ~ mmdeduction.html .

  In a Hilbert system of logic (which consists of a set of axioms, modus
  ponens, and the generalization rule), converting a deduction to a proof using
  the Deduction Theorem (taught in introductory logic books) involves an
  exponential increase of the number of steps as hypotheses are successively
  eliminated.  Here is a trick that is not as general as the Deduction Theorem
  but requires only a linear increase in the number of steps.

  The general problem:  We want to convert a deduction
    P |- Q
  into a proof of the theorem
    |- P -> Q
  i.e., we want to eliminate the hypothesis P.  Normally this is done using the
  Deduction (meta)Theorem, which looks at the microscopic steps of the
  deduction and usually doubles or triples the number of these microscopic
  steps for each hypothesis that is eliminated.  We will look at a special case
  of this problem, without appealing to the Deduction Theorem.

  We assume ZF with class notation.  A and B are arbitrary (possibly proper)
  classes.  P, Q, R, S and T are wffs.

  We define the conditional operator, if(P, A, B), as follows:
    if(P, A, B) =def= { x | (x \in A & P) v (x \in B & -. P) }
  (where x does not occur in A, B, or P).

  Lemma 1.
    A = if(P, A, B) -> (P <-> R), B = if(P, A, B) -> (S <-> R), S |- R
  Proof:  Logic and Axiom of Extensionality.

  Lemma 2.
    A = if(P, A, B) -> (Q <-> T), T |- P -> Q
  Proof:  Logic and Axiom of Extensionality.

  Here is a simple example that illustrates how it works.  Suppose we have a
  deduction
    Ord A |- Tr A
  which means, "Assume A is an ordinal class.  Then A is a transitive class."
  Note that A is a class variable that may be substituted with any class
  expression, so this is really a deduction scheme.

  We want to convert this to a proof of the theorem (scheme)
    |- Ord A -> Tr A.

  The catch is that we must be able to prove "Ord A" for at least one object A
  (and this is what makes it weaker than the ordinary Deduction Theorem).
  However, it is easy to prove |- Ord 0 (the empty set is ordinal).  (For a
  typical textbook "theorem", i.e., deduction, there is usually at least one
  object satisfying each hypothesis, otherwise the theorem would not be very
  useful.  We can always go back to the standard Deduction Theorem for those
  hypotheses where this is not the case.)  Continuing with the example:

  Equality axioms (and Extensionality) yield
    |- A = if(Ord A, A, 0) -> (Ord A <-> Ord if(Ord A, A, 0))    (1)
    |- 0 = if(Ord A, A, 0) -> (Ord 0 <-> Ord if(Ord A, A, 0))    (2)
  From (1), (2) and |- Ord 0, Lemma 1 yields
    |- Ord if(Ord A, A, 0)                                       (3)
  From (3) and substituting if(Ord A, A, 0) for
  A in the original deduction,
    |- Tr if(Ord A, A, 0)                                        (4)
  Equality axioms (and Extensionality) yield
    |- A = if(Ord A, A, 0) -> (Tr A <-> Tr if(Ord A, A, 0))      (5)
  From (4) and (5), Lemma 2 yields
    |- Ord A -> Tr A                                             (Q.E.D.)

$)

  ${
    dedth.1 $e |- ( A = if ( ph , A , B ) -> ( ps <-> ch ) ) $.
    dedth.2 $e |- ch $.
    $( Weak deduction theorem that eliminates a hypothesis ` ph ` , making it
       become an antecedent.  We assume that a proof exists for ` ph ` when the
       class variable ` A ` is replaced with a specific class ` B ` .  The
       hypothesis ` ch ` should be assigned to the inference, and the inference
       hypothesis eliminated with ~ elimhyp .  If the inference has other
       hypotheses with class variable ` A ` , these can be kept by assigning
       ~ keephyp to them.  For more information, see the Weak Deduction Theorem
       page ~ mmdeduction.html .  (Contributed by NM, 15-May-1999.) $)
    dedth $p |- ( ph -> ps ) $=
      ( cif wceq wb iftrue eqcomd syl mpbiri ) ABCGADADEHZIBCJAODADEKLFMN $.
  $}

  ${
    dedth2h.1 $e |- ( A = if ( ph , A , C ) -> ( ch <-> th ) ) $.
    dedth2h.2 $e |- ( B = if ( ps , B , D ) -> ( th <-> ta ) ) $.
    dedth2h.3 $e |- ta $.
    $( Weak deduction theorem eliminating two hypotheses.  This theorem is
       simpler to use than ~ dedth2v but requires that each hypothesis have
       exactly one class variable.  See also comments in ~ dedth .
       (Contributed by NM, 15-May-1999.) $)
    dedth2h $p |- ( ( ph /\ ps ) -> ch ) $=
      ( wi cif wceq imbi2d dedth imp ) ABCABCMBDMFHFAFHNOCDBJPBDEGIKLQQR $.
  $}

  ${
    dedth3h.1 $e |- ( A = if ( ph , A , D ) -> ( th <-> ta ) ) $.
    dedth3h.2 $e |- ( B = if ( ps , B , R ) -> ( ta <-> et ) ) $.
    dedth3h.3 $e |- ( C = if ( ch , C , S ) -> ( et <-> ze ) ) $.
    dedth3h.4 $e |- ze $.
    $( Weak deduction theorem eliminating three hypotheses.  See comments in
       ~ dedth2h .  (Contributed by NM, 15-May-1999.) $)
    dedth3h $p |- ( ( ph /\ ps /\ ch ) -> th ) $=
      ( wa wi cif wceq imbi2d dedth2h dedth 3impib ) ABCDABCRZDSUFESHKHAHKTUADE
      UFNUBBCEFGIJLMOPQUCUDUE $.
  $}

  ${
    dedth4h.1 $e |- ( A = if ( ph , A , R ) -> ( ta <-> et ) ) $.
    dedth4h.2 $e |- ( B = if ( ps , B , S ) -> ( et <-> ze ) ) $.
    dedth4h.3 $e |- ( C = if ( ch , C , F ) -> ( ze <-> si ) ) $.
    dedth4h.4 $e |- ( D = if ( th , D , G ) -> ( si <-> rh ) ) $.
    dedth4h.5 $e |- rh $.
    $( Weak deduction theorem eliminating four hypotheses.  See comments in
       ~ dedth2h .  (Contributed by NM, 16-May-1999.) $)
    dedth4h $p |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) -> ta ) $=
      ( wa wi cif wceq imbi2d dedth2h imp ) ABUCCDUCZEABUJEUDUJFUDUJGUDJKNOJAJN
      UEUFEFUJRUGKBKOUEUFFGUJSUGCDGHILMPQTUAUBUHUHUI $.
  $}

  ${
    dedth2v.1 $e |- ( A = if ( ph , A , C ) -> ( ps <-> ch ) ) $.
    dedth2v.2 $e |- ( B = if ( ph , B , D ) -> ( ch <-> th ) ) $.
    dedth2v.3 $e |- th $.
    $( Weak deduction theorem for eliminating a hypothesis with 2 class
       variables.  Note: if the hypothesis can be separated into two
       hypotheses, each with one class variable, then ~ dedth2h is simpler to
       use.  See also comments in ~ dedth .  (Contributed by NM, 13-Aug-1999.)
       (Proof shortened by Eric Schmidt, 28-Jul-2009.) $)
    dedth2v $p |- ( ph -> ps ) $=
      ( dedth2h anidms ) ABAABCDEFGHIJKLM $.
  $}

  ${
    dedth3v.1 $e |- ( A = if ( ph , A , D ) -> ( ps <-> ch ) ) $.
    dedth3v.2 $e |- ( B = if ( ph , B , R ) -> ( ch <-> th ) ) $.
    dedth3v.3 $e |- ( C = if ( ph , C , S ) -> ( th <-> ta ) ) $.
    dedth3v.4 $e |- ta $.
    $( Weak deduction theorem for eliminating a hypothesis with 3 class
       variables.  See comments in ~ dedth2v .  (Contributed by NM,
       13-Aug-1999.)  (Proof shortened by Eric Schmidt, 28-Jul-2009.) $)
    dedth3v $p |- ( ph -> ps ) $=
      ( dedth3h 3anidm12 anidms ) ABAABAAABCDEFGHIJKLMNOPQR $.
  $}

  ${
    dedth4v.1 $e |- ( A = if ( ph , A , R ) -> ( ps <-> ch ) ) $.
    dedth4v.2 $e |- ( B = if ( ph , B , S ) -> ( ch <-> th ) ) $.
    dedth4v.3 $e |- ( C = if ( ph , C , T ) -> ( th <-> ta ) ) $.
    dedth4v.4 $e |- ( D = if ( ph , D , U ) -> ( ta <-> et ) ) $.
    dedth4v.5 $e |- et $.
    $( Weak deduction theorem for eliminating a hypothesis with 4 class
       variables.  See comments in ~ dedth2v .  (Contributed by NM,
       21-Apr-2007.)  (Proof shortened by Eric Schmidt, 28-Jul-2009.) $)
    dedth4v $p |- ( ph -> ps ) $=
      ( anidms wa dedth4h ) ABAAUABAAAABCDEFGHIJKLMNOPQRSUBTT $.
  $}

  ${
    elimhyp.1 $e |- ( A = if ( ph , A , B ) -> ( ph <-> ps ) ) $.
    elimhyp.2 $e |- ( B = if ( ph , A , B ) -> ( ch <-> ps ) ) $.
    elimhyp.3 $e |- ch $.
    $( Eliminate a hypothesis containing class variable ` A ` when it is known
       for a specific class ` B ` .  For more information, see comments in
       ~ dedth .  (Contributed by NM, 15-May-1999.) $)
    elimhyp $p |- ps $=
      ( cif wceq wb iftrue eqcomd syl ibi wn iffalse mpbii pm2.61i ) ABABADADEI
      ZJABKATDADELMFNOAPZCBHUAETJCBKUATEADEQMGNRS $.
  $}

  ${
    elimhyp2v.1 $e |- ( A = if ( ph , A , C ) -> ( ph <-> ch ) ) $.
    elimhyp2v.2 $e |- ( B = if ( ph , B , D ) -> ( ch <-> th ) ) $.
    elimhyp2v.3 $e |- ( C = if ( ph , A , C ) -> ( ta <-> et ) ) $.
    elimhyp2v.4 $e |- ( D = if ( ph , B , D ) -> ( et <-> th ) ) $.
    elimhyp2v.5 $e |- ta $.
    $( Eliminate a hypothesis containing 2 class variables.  (Contributed by
       NM, 14-Aug-1999.) $)
    elimhyp2v $p |- th $=
      ( cif wceq wb iftrue eqcomd syl bitrd ibi wn iffalse mpbii pm2.61i ) ACAC
      AABCAFAFHOZPABQAUGFAFHRSJTAGAGIOZPBCQAUHGAGIRSKTUAUBAUCZDCNUIDECUIHUGPDEQ
      UIUGHAFHUDSLTUIIUHPECQUIUHIAGIUDSMTUAUEUF $.
  $}

  ${
    elimhyp3v.1 $e |- ( A = if ( ph , A , D ) -> ( ph <-> ch ) ) $.
    elimhyp3v.2 $e |- ( B = if ( ph , B , R ) -> ( ch <-> th ) ) $.
    elimhyp3v.3 $e |- ( C = if ( ph , C , S ) -> ( th <-> ta ) ) $.
    elimhyp3v.4 $e |- ( D = if ( ph , A , D ) -> ( et <-> ze ) ) $.
    elimhyp3v.5 $e |- ( R = if ( ph , B , R ) -> ( ze <-> si ) ) $.
    elimhyp3v.6 $e |- ( S = if ( ph , C , S ) -> ( si <-> ta ) ) $.
    elimhyp3v.7 $e |- et $.
    $( Eliminate a hypothesis containing 3 class variables.  (Contributed by
       NM, 14-Aug-1999.) $)
    elimhyp3v $p |- ta $=
      ( cif wceq wb iftrue eqcomd syl 3bitrd ibi wn iffalse mpbii pm2.61i ) ADA
      DAABCDAHAHKUAZUBABUCAUMHAHKUDUENUFAIAILUAZUBBCUCAUNIAILUDUEOUFAJAJMUAZUBC
      DUCAUOJAJMUDUEPUFUGUHAUIZEDTUPEFGDUPKUMUBEFUCUPUMKAHKUJUEQUFUPLUNUBFGUCUP
      UNLAILUJUERUFUPMUOUBGDUCUPUOMAJMUJUESUFUGUKUL $.
  $}

  ${
    elimhyp4v.1 $e |- ( A = if ( ph , A , D ) -> ( ph <-> ch ) ) $.
    elimhyp4v.2 $e |- ( B = if ( ph , B , R ) -> ( ch <-> th ) ) $.
    elimhyp4v.3 $e |- ( C = if ( ph , C , S ) -> ( th <-> ta ) ) $.
    elimhyp4v.4 $e |- ( F = if ( ph , F , G ) -> ( ta <-> ps ) ) $.
    elimhyp4v.5 $e |- ( D = if ( ph , A , D ) -> ( et <-> ze ) ) $.
    elimhyp4v.6 $e |- ( R = if ( ph , B , R ) -> ( ze <-> si ) ) $.
    elimhyp4v.7 $e |- ( S = if ( ph , C , S ) -> ( si <-> rh ) ) $.
    elimhyp4v.8 $e |- ( G = if ( ph , F , G ) -> ( rh <-> ps ) ) $.
    elimhyp4v.9 $e |- et $.
    $( Eliminate a hypothesis containing 4 class variables (for use with the
       weak deduction theorem ~ dedth ).  (Contributed by NM, 16-Apr-2005.) $)
    elimhyp4v $p |- ps $=
      ( cif wceq wb iftrue eqcomd syl bitrd 3bitrd ibi wn iffalse mpbii pm2.61i
      ) ABABAADEBAACDAJAJMUGZUHACUIAUTJAJMUJUKRULAKAKNUGZUHCDUIAVAKAKNUJUKSULUM
      ALALOUGZUHDEUIAVBLALOUJUKTULAPAPQUGZUHEBUIAVCPAPQUJUKUAULUNUOAUPZFBUFVDFH
      IBVDFGHVDMUTUHFGUIVDUTMAJMUQUKUBULVDNVAUHGHUIVDVANAKNUQUKUCULUMVDOVBUHHIU
      IVDVBOALOUQUKUDULVDQVCUHIBUIVDVCQAPQUQUKUEULUNURUS $.
  $}

  ${
    elimel.1 $e |- B e. C $.
    $( Eliminate a membership hypothesis for weak deduction theorem, when
       special case ` B e. C ` is provable.  (Contributed by NM,
       15-May-1999.) $)
    elimel $p |- if ( A e. C , A , B ) e. C $=
      ( wcel cif eleq1 elimhyp ) ACEZIABFZCEBCEABAJCGBJCGDH $.
  $}

  ${
    elimdhyp.1 $e |- ( ph -> ps ) $.
    elimdhyp.2 $e |- ( A = if ( ph , A , B ) -> ( ps <-> ch ) ) $.
    elimdhyp.3 $e |- ( B = if ( ph , A , B ) -> ( th <-> ch ) ) $.
    elimdhyp.4 $e |- th $.
    $( Version of ~ elimhyp where the hypothesis is deduced from the final
       antecedent.  See ~ divalg for an example of its use.  (Contributed by
       Paul Chapman, 25-Mar-2008.) $)
    elimdhyp $p |- ch $=
      ( cif wceq wb iftrue eqcomd syl mpbid wn iffalse mpbii pm2.61i ) ACABCGAE
      AEFKZLBCMAUBEAEFNOHPQARZDCJUCFUBLDCMUCUBFAEFSOIPTUA $.
  $}

  ${
    keephyp.1 $e |- ( A = if ( ph , A , B ) -> ( ps <-> th ) ) $.
    keephyp.2 $e |- ( B = if ( ph , A , B ) -> ( ch <-> th ) ) $.
    keephyp.3 $e |- ps $.
    keephyp.4 $e |- ch $.
    $( Transform a hypothesis ` ps ` that we want to keep (but contains the
       same class variable ` A ` used in the eliminated hypothesis) for use
       with the weak deduction theorem.  (Contributed by NM, 15-May-1999.) $)
    keephyp $p |- th $=
      ( ifboth mp2an ) BCDIJABCDEFGHKL $.
  $}

  ${
    keephyp2v.1 $e |- ( A = if ( ph , A , C ) -> ( ps <-> ch ) ) $.
    keephyp2v.2 $e |- ( B = if ( ph , B , D ) -> ( ch <-> th ) ) $.
    keephyp2v.3 $e |- ( C = if ( ph , A , C ) -> ( ta <-> et ) ) $.
    keephyp2v.4 $e |- ( D = if ( ph , B , D ) -> ( et <-> th ) ) $.
    keephyp2v.5 $e |- ps $.
    keephyp2v.6 $e |- ta $.
    $( Keep a hypothesis containing 2 class variables (for use with the weak
       deduction theorem ~ dedth ).  (Contributed by NM, 16-Apr-2005.) $)
    keephyp2v $p |- th $=
      ( wceq wb eqcomd syl cif iftrue bitrd mpbii wn iffalse pm2.61i ) ADABDOAB
      CDAGAGIUAZQBCRAUHGAGIUBSKTAHAHJUAZQCDRAUIHAHJUBSLTUCUDAUEZEDPUJEFDUJIUHQE
      FRUJUHIAGIUFSMTUJJUIQFDRUJUIJAHJUFSNTUCUDUG $.
  $}

  ${
    keephyp3v.1 $e |- ( A = if ( ph , A , D ) -> ( rh <-> ch ) ) $.
    keephyp3v.2 $e |- ( B = if ( ph , B , R ) -> ( ch <-> th ) ) $.
    keephyp3v.3 $e |- ( C = if ( ph , C , S ) -> ( th <-> ta ) ) $.
    keephyp3v.4 $e |- ( D = if ( ph , A , D ) -> ( et <-> ze ) ) $.
    keephyp3v.5 $e |- ( R = if ( ph , B , R ) -> ( ze <-> si ) ) $.
    keephyp3v.6 $e |- ( S = if ( ph , C , S ) -> ( si <-> ta ) ) $.
    keephyp3v.7 $e |- rh $.
    keephyp3v.8 $e |- et $.
    $( Keep a hypothesis containing 3 class variables.  (Contributed by NM,
       27-Sep-1999.) $)
    keephyp3v $p |- ta $=
      ( cif wceq wb iftrue eqcomd syl 3bitrd mpbii wn iffalse pm2.61i ) ADAHDUA
      AHBCDAIAILUCZUDHBUEAUNIAILUFUGOUHAJAJMUCZUDBCUEAUOJAJMUFUGPUHAKAKNUCZUDCD
      UEAUPKAKNUFUGQUHUIUJAUKZEDUBUQEFGDUQLUNUDEFUEUQUNLAILULUGRUHUQMUOUDFGUEUQ
      UOMAJMULUGSUHUQNUPUDGDUEUQUPNAKNULUGTUHUIUJUM $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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 x $.  $d z y $.  $d z A $.
    $( 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 = { 3 , 5 , 7 } ` , then
       ` ~P A = { (/) , { 3 } , { 5 } , { 7 } , { 3 , 5 } , `
       ` { 3 , 7 } , { 5 , 7 } , { 3 , 5 , 7 } } ` ( ~ ex-pw ).  We will later
       introduce the Axiom of Power Sets ~ ax-pow , which can be expressed in
       class notation per ~ pwexg .  Still later we will prove, in ~ hashpw ,
       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, 24-Jun-1993.) $)
    df-pw $a |- ~P A = { x | x C_ A } $.
  $}

  ${
    $d A x $.  $d B x $.
    $( Membership in a power class.  Theorem 86 of [Suppes] p. 47.  See also
       ~ elpw2g .  (Contributed by NM, 6-Aug-2000.)  (Proof shortened by BJ,
       31-Dec-2023.) $)
    elpwg $p |- ( A e. V -> ( A e. ~P B <-> A C_ B ) ) $=
      ( vx cv wss cpw sseq1 df-pw elab2g ) DEZBFABFDABGCKABHDBIJ $.
  $}

  ${
    elpw.1 $e |- A e. _V $.
    $( Membership in a power class.  Theorem 86 of [Suppes] p. 47.
       (Contributed by NM, 31-Dec-1993.)  (Proof shortened by BJ,
       31-Dec-2023.) $)
    elpw $p |- ( A e. ~P B <-> A C_ B ) $=
      ( cvv wcel cpw wss wb elpwg ax-mp ) ADEABFEABGHCABDIJ $.
  $}

  $( Setvar variable membership in a power class.  (Contributed by David A.
     Wheeler, 8-Dec-2018.) $)
  velpw $p |- ( x e. ~P A <-> x C_ A ) $=
    ( cv vex elpw ) ACBADE $.

  ${
    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 biadanii ) 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 $.
  $}

  $( Equality theorem for power class.  (Contributed by NM, 21-Jun-1993.)
     (Proof shortened by BJ, 13-Apr-2024.) $)
  pweq $p |- ( A = B -> ~P A = ~P B ) $=
    ( wceq cpw eqimss sspwd eqimss2 eqssd ) ABCZADBDIABABEFIBABAGFH $.

  ${
    $d x A $.  $d x B $.
    $( Alternate proof of ~ pweq directly from the definition.  (Contributed by
       NM, 21-Jun-1993.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    pweqALT $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 $.
  $}

  $( 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.)  Remove use of ~ ax-sep , ~ ax-nul , ~ ax-pr and shorten
     proof.  (Revised by BJ, 13-Apr-2024.) $)
  pwunss $p |- ( ~P A u. ~P B ) C_ ~P ( A u. B ) $=
    ( cpw cun ssun1 sspwi ssun2 unssi ) ACBCABDZCAIABEFBIBAGFH $.

  ${
    $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 $.
  $}

  $( A set is an element of its 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 $.

  $( A class is an element of its powerclass if and only if it is a set.
     (Contributed by BJ, 31-Dec-2023.) $)
  pwidb $p |- ( A e. _V <-> A e. ~P A ) $=
    ( cvv wcel cpw pwidg elex impbii ) ABCAADZCABEAHFG $.

  ${
    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 df-ss velpw imbi1i albii bitri ) BDZCEAFZOGZPCGZ
      HZAIPBEZRHZAIAOCJSUAAQTRABKLMN $.
  $}

  $( Break up the power class of a union into a union of smaller classes.
     (Contributed by NM, 25-Mar-2007.)  (Proof shortened by Thierry Arnoux,
     20-Dec-2016.)  Remove use of ~ ax-sep , ~ ax-nul , ~ ax-pr and shorten
     proof.  (Revised by BJ, 14-Apr-2024.) $)
  pwundif $p |- ~P ( A u. B ) = ( ( ~P ( A u. B ) \ ~P A ) u. ~P A ) $=
    ( cpw cun cdif wss wceq ssun1 sspwi undif mpbi uncom eqtr3i ) ACZABDZCZNEZD
    ZPQNDNPFRPGAOABHINPJKNQLM $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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') $)

  ${
    $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 $.
  $}

  $( Extend class notation to include singleton. $)
  csn $a class { A } $.

  ${
    $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 ` , see ~ snprc .  For an alternate definition see
       ~ dfsn2 .  (Contributed by NM, 21-Jun-1993.) $)
    df-sn $a |- { A } = { x | x = A } $.
  $}

  $( Extend class notation to include unordered pair. $)
  cpr $a class { A , B } $.

  $( Define unordered pair of classes.  Definition 7.1 of [Quine] p. 48.  For
     example, ` A e. { 1 , -u 1 } -> ( A ^ 2 ) = 1 ` ( ~ ex-pr ).  They are
     unordered, so ` { A , B } = { B , A } ` as proven by ~ prcom .  For a more
     traditional definition, but requiring a dummy variable, see ~ dfpr2 .
     ` { A , A } ` is also an unordered pair, but also a singleton because of
     ` { A } = { A , A } ` (see ~ dfsn2 ).  Therefore, ` { A , B } ` is called
     a _proper_ (unordered) pair iff ` A =/= B ` and ` A ` and ` B ` are sets.

     Note: ordered pairs are a completely different object defined below in
     ~ df-op .  When the term "pair" is used without qualifier, it generally
     means "unordered pair", and the context makes it clear which version is
     meant.  (Contributed by NM, 21-Jun-1993.) $)
  df-pr $a |- { A , B } = ( { A } u. { B } ) $.

  $( Extend class notation to include unordered triple (sometimes called
     "unordered triplet"). $)
  ctp $a class { A , B , C } $.

  $( Define unordered triple of classes.  Definition of [Enderton] p. 19.

     Note: ordered triples are a completely different object defined below in
     ~ df-ot .  As with all tuples, when the term "triple" is used without
     qualifier, it means "ordered triple".  (Contributed by NM, 9-Apr-1994.) $)
  df-tp $a |- { A , B , C } = ( { A , B } u. { C } ) $.

  $( Extend class notation to include ordered pair. $)
  cop $a class <. A , B >. $.

  ${
    $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 , ~ opprc2 , and
       ~ 0nelop ).  For Kuratowski's actual definition when the arguments are
       sets, see ~ dfop .  For the justifying theorem (for sets) see ~ opth .
       See ~ dfopif for an equivalent formulation using the ` if ` operation.

       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 } } } ` , justified by
       ~ opthwiener .  This was simplified by Kazimierz Kuratowski in 1921 to
       our present definition.  An even simpler definition ` <. A , B >. `_3
       ` = { A , { A , B } } ` is justified by ~ opthreg , but it requires the
       Axiom of Regularity for its justification and is not commonly used.  A
       definition that also works for proper classes is ` <. A , B >. `_4
       ` = ( ( A X. { (/) } ) u. ( B X. { { (/) } } ) ) ` , justified by
       ~ opthprc .  Nearly at the same time as Norbert Wiener, Felix Hausdorff
       proposed the following definition in "Grundz&uuml;ge der Mengenlehre"
       ("Basics of Set Theory"), p. 32, in 1914: ` <. A , B >. `_5
       ` = { { A , O } , { B , T } } ` .  Hausdorff used ` 1 ` and ` 2 `
       instead of ` O ` and ` T ` , but actually any two different fixed sets
       will do (e.g., ` O = (/) ` and ` T = { (/) } ` , see ~ 0nep0 ).
       Furthermore, Hausdorff demanded that ` O ` and ` T ` are both different
       from ` A ` as well as ` B ` , which is actually not necessary (at least
       not in full extent), see ~ opthhausdorff0 and ~ opthhausdorff .  If we
       restrict our sets to nonnegative integers, an ordered pair definition
       that involves only elementary arithmetic is provided by ~ nn0opthi .  An
       ordered pair of real numbers can also be represented by a complex number
       as shown by ~ cru .  Kuratowski's ordered pair definition is standard
       for ZFC set theory, but it is very inconvenient to use in New
       Foundations theory because it is not type-level; a common alternate
       definition in New Foundations is the definition from [Rosser] p. 281.

       Since there are other ways to define ordered pairs, we discourage direct
       use of this definition so that most theorems won't depend on this
       particular construction; theorems will instead rely on ~ dfopif .
       (Contributed by NM, 28-May-1995.)  (Revised by Mario Carneiro,
       26-Apr-2015.)  (Avoid depending on this detail.) $)
    df-op $a |- <. A , B >. =
                { x | ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) } $.
  $}

  $( Extend class notation to include ordered triple. $)
  cotp $a class <. A , B , C >. $.

  $( 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, 21-Jun-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 at most 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 N x $.  $d V x $.
    $( Equality of class abstractions restricted to a singleton.  (Contributed
       by AV, 17-May-2025.) $)
    rabsneq $p |- ( N e. V -> { x e. { N } | ps }
                              = { x e. V | ( x = N /\ ps ) } ) $=
      ( wcel cv csn wa cab wceq crab velsn eleq1a pm4.71rd bitrid anbi1d bitrdi
      anass abbidv df-rab 3eqtr4g ) CDEZBFZCGZEZAHZBIUCDEZUCCJZAHZHZBIABUDKUIBD
      KUBUFUJBUBUFUGUHHZAHUJUBUEUKAUEUHUBUKBCLUBUHUGCDUCMNOPUGUHARQSABUDTUIBDTU
      A $.
  $}

  ${
    $d Y x $.
    $( Condition for a class abstraction to be a singleton.  Formerly part of
       proof of ~ dfiota2 .  (Contributed by Andrew Salmon, 30-Jun-2011.)
       (Revised by AV, 24-Aug-2022.) $)
    absn $p |- ( { x | ph } = { Y } <-> A. x ( ph <-> x = Y ) ) $=
      ( cab csn wceq cv wb wal df-sn eqeq2i abbib bitri ) ABDZCEZFNBGCFZBDZFAPH
      BIOQNBCJKAPBLM $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Alternate definition of a 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 cab df-pr wcel elun velsn orbi12i bitri eqabi eqtri
      wo ) BCDBEZCEZFZAGZBHZUBCHZRZAIBCJUEAUAUBUAKUBSKZUBTKZRUEUBSTLUFUCUGUDABM
      ACMNOPQ $.
  $}

  ${
    $d x A $.
    $( Alternate definition of singleton, based on the (alternate) definition
       of pair.  Definition 5.1 of [TakeutiZaring] p. 15.  (Contributed by AV,
       12-Jun-2022.)  (New usage is discouraged.)
       (Proof modification is discouraged.) $)
    dfsn2ALT $p |- { A } = { A , A } $=
      ( vx cv wceq wo cab cpr csn oridm abbii dfpr2 df-sn 3eqtr4ri ) BCADZNEZBF
      NBFAAGAHONBNIJBAAKBALM $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( A member of a pair of classes is one or the other of them, and
       conversely as soon as it is a set.  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 $.
  $}

  $( 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 $.

  ${
    elpr.1 $e |- A e. _V $.
    $( A member of a pair of classes is one or the other of them, and
       conversely as soon as it is a set.  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 $.
  $}

  ${
    $( A member of a pair of sets is one or the other of them, and conversely.
       Exercise 1 of [TakeutiZaring] p. 15.  (Contributed by NM, 14-Oct-2005.)
       Generalize from sethood hypothesis to sethood antecedent.  (Revised by
       BJ, 25-May-2024.) $)
    elpr2g $p |-
       ( ( B e. V /\ C e. W ) -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) ) $=
      ( wcel wa cvv cpr wceq wo wi elex a1i eleq1a syl jaao wb elprg pm5.21ndd
      ) BDFZCEFZGZAHFZABCIZFZABJZACJZKZUFUDLUCAUEMNUAUGUDUBUHUABHFUGUDLBDMBHAOP
      UBCHFUHUDLCEMCHAOPQUDUFUIRLUCABCHSNT $.
  $}

  ${
    elpr2.1 $e |- B e. _V $.
    elpr2.2 $e |- C e. _V $.
    $( A member of a pair of sets is one or the other of them, and conversely.
       Exercise 1 of [TakeutiZaring] p. 15.  (Contributed by NM, 14-Oct-2005.)
       (Proof shortened by JJ, 23-Jul-2021.) $)
    elpr2 $p |- ( A e. { B , C } <-> ( A = B \/ A = C ) ) $=
      ( cvv wcel cpr wceq wo wb elpr2g mp2an ) BFGCFGABCHGABIACIJKDEABCFFLM $.
  $}

  ${
    nelpr2.a $e |- ( ph -> A e. V ) $.
    nelpr2.n $e |- ( ph -> -. A e. { B , C } ) $.
    $( If a class is not an element of an unordered pair, it is not the second
       listed element.  (Contributed by Glauco Siliprandi, 3-Mar-2021.) $)
    nelpr2 $p |- ( ph -> A =/= C ) $=
      ( wceq cpr wcel wa wo animorr wb elprg syl adantr mpbird mtand neqned ) A
      BDABDHZBCDIJZGAUAKUBBCHZUALZAUAUCMAUBUDNZUAABEJUEFBCDEOPQRST $.
  $}

  ${
    nelpr1.a $e |- ( ph -> A e. V ) $.
    nelpr1.n $e |- ( ph -> -. A e. { B , C } ) $.
    $( If a class is not an element of an unordered pair, it is not the first
       listed element.  (Contributed by Glauco Siliprandi, 3-Mar-2021.) $)
    nelpr1 $p |- ( ph -> A =/= B ) $=
      ( wceq cpr wcel wa wo animorrl wb elprg syl adantr mpbird mtand neqned )
      ABCABCHZBCDIJZGAUAKUBUABDHZLZAUAUCMAUBUDNZUAABEJUEFBCDEOPQRST $.
  $}

  ${
    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.)  (Proof
     shortened by Wolf Lammen, 15-May-2025.) $)
  rexdifpr $p |- ( E. x e. ( A \ { B , C } ) ph
      <-> E. x e. A ( x =/= B /\ x =/= C /\ ph ) ) $=
    ( cv wne w3a cpr cdif wcel anass eldifpr 3anass bitri anbi1i df-3an 3bitr4i
    wa anbi2i rexbii2 ) ABFZDGZUBEGZAHZBCDEIJZCUBCKZUCUDSZSZASUGUHASZSUBUFKZASU
    GUESUGUHALUKUIAUKUGUCUDHUIUBCDEMUGUCUDNOPUEUJUGUCUDAQTRUA $.

  $( 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.  (Contributed by David A.
     Wheeler, 8-Dec-2018.) $)
  vsnid $p |- x e. { x } $=
    ( cv vex snid ) ABACD $.

  $( There is exactly 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 exactly 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 x $.
    $( Conditions for a restricted class abstraction to be a singleton.
       (Contributed by AV, 18-Apr-2019.)  (Proof shortened by AV,
       26-Aug-2022.) $)
    rabeqsn $p |- ( { x e. V | ph } = { X }
                    <-> A. x ( ( x e. V /\ ph ) <-> x = X ) ) $=
      ( crab csn wceq cv wcel wa cab wb wal df-rab eqeq1i absn bitri ) ABCEZDFZ
      GBHZCIAJZBKZSGUATDGLBMRUBSABCNOUABDPQ $.

    $( Conditions for a restricted class abstraction to be a subset of a
       singleton, i.e. to be a singleton or the empty set.  (Contributed by AV,
       18-Apr-2019.) $)
    rabsssn $p |- ( { x e. V | ph } C_ { X } <-> A. x e. V ( ph -> x = X ) ) $=
      ( crab csn wss cv wcel wa cab wceq wi wal wral df-rab df-sn sseq12i ss2ab
      impexp albii df-ral bitr4i 3bitri ) ABCEZDFZGBHZCIZAJZBKZUGDLZBKZGUIUKMZB
      NZAUKMZBCOZUEUJUFULABCPBDQRUIUKBSUNUHUOMZBNUPUMUQBUHAUKTUAUOBCUBUCUD $.
  $}

  ${
    $d B x $.  $d ph x $.
    rabeqsnd.0 $e |- ( x = B -> ( ps <-> ch ) ) $.
    rabeqsnd.1 $e |- ( ph -> B e. A ) $.
    rabeqsnd.2 $e |- ( ph -> ch ) $.
    rabeqsnd.3 $e |- ( ( ( ph /\ x e. A ) /\ ps ) -> x = B ) $.
    $( Conditions for a restricted class abstraction to be a singleton, in
       deduction form.  (Contributed by Thierry Arnoux, 2-Dec-2021.) $)
    rabeqsnd $p |- ( ph -> { x e. A | ps } = { B } ) $=
      ( cv wcel wa wceq wb wal wi alrimiv jca sylibr crab csn a1d eleq1 anbi12d
      expl pm5.74i albii albiim rabeqsn ) ADKZELZBMZUKFNZODPZBDEUAFUBNAUMUNQZDP
      ZUNUMQZDPZMUOAUQUSAUPDAULBUNJUFRAUNFELZCMZQZDPUSAVBDAVAUNAUTCHISUCRURVBDU
      NUMVAUNULUTBCUKFEUDGUEUGUHTSUMUNDUITBDEFUJT $.
  $}

  ${
    $d A 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 $.

    $( 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 $.
  $}

  ${
    $d A x $.
    rexsngf.1 $e |- F/ x ps $.
    rexsngf.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Restricted existential quantification over a singleton.  (Contributed by
       NM, 29-Jan-2012.)  (Revised by Glauco Siliprandi, 17-Aug-2020.) $)
    rexsngf $p |- ( A e. V -> ( E. x e. { A } ph <-> ps ) ) $=
      ( csn wrex wsbc wcel rexsns sbciegf bitrid ) ACDHIACDJDEKBACDLABCDEFGMN
      $.

    $( Restricted universal quantification over a singleton.  (Contributed by
       NM, 14-Dec-2005.)  (Revised by AV, 3-Apr-2023.) $)
    ralsngf $p |- ( A e. V -> ( A. x e. { A } ph <-> ps ) ) $=
      ( wcel csn wral wsbc ralsnsg sbciegf bitrd ) DEHACDIJACDKBACDELABCDEFGMN
      $.

    $d c w x A $.  $d c w ph $.
    $( Restricted existential uniqueness over a singleton.  (Contributed by AV,
       3-Apr-2023.) $)
    reusngf $p |- ( A e. V -> ( E! x e. { A } ph <-> ps ) ) $=
      ( vc vw csn wreu cv wsbc wi wral wa nfsbc1v dfsbcq wceq nfv weq wrex wcel
      sbceq1a reu8nf nfcv nfim nfralw nfan eqeq1 imbi2d ralbidv anbi12d rexsngf
      eqeq2 imbi12d ralsngf anbi2d eqidd biantru bitr4di bitrd bitrid ) ACDJZKA
      ACHLZMZCHUAZNZHVDOZPZCVDUBZDEUCZBAVFACILZMCHIVDACVEQZACVMQACVMUDACVMVERUE
      VLVKBVFDVESZNZHVDOZPZBVJVRCDEBVQCFVPCHVDCVDUFVFVOCVNVOCTUGUHUICLZDSZABVIV
      QGVTVHVPHVDVTVGVOVFVSDVEUJUKULUMUNVLVRBACDMZDDSZNZPBVLVQWCBVPWCHDEWCHTVED
      SVFWAVOWBACVEDRVEDDUOUPUQURWCBWADUSUTVAVBVC $.
  $}

  ${
    $d A x $.  $d ps x $.
    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.)  Avoid ~ ax-10 , ~ ax-12 .  (Revised by GG,
       30-Sep-2024.) $)
    ralsng $p |- ( A e. V -> ( A. x e. { A } ph <-> ps ) ) $=
      ( csn wral cv wceq wi wal wcel df-ral velsn imbi1i albii bitri wb a1i wex
      elisset pm5.74i 19.23v pm5.5 3bitrd syl bitrid ) ACDGZHZCIZDJZAKZCLZDEMZB
      UJUKUIMZAKZCLUNACUINUQUMCUPULACDOPQRUOULCUAZUNBSCDEUBURUNULBKZCLZURBKZBUN
      UTSURUMUSCULABFUCQTUTVASURULBCUDTURBUEUFUGUH $.
    $( $j usage 'ralsng' avoids 'ax-10' 'ax-11' 'ax-12'; $)

    $( Restricted existential quantification over a singleton.  (Contributed by
       NM, 29-Jan-2012.)  Avoid ~ ax-10 , ~ ax-12 .  (Revised by GG,
       30-Sep-2024.) $)
    rexsng $p |- ( A e. V -> ( E. x e. { A } ph <-> ps ) ) $=
      ( wcel wn wral wb wrex cv wceq notbid ralsng dfrex2 bicom1 con1bid bitrid
      csn syl ) DEGAHZCDTZIZBHZJZACUCKZBJUBUECDECLDMABFNOUGUDHUFBACUCPUFBUDUDUE
      QRSUA $.
    $( $j usage 'rexsng' avoids 'ax-10' 'ax-11' 'ax-12'; $)

    $( Restricted existential uniqueness over a singleton.  (Contributed by AV,
       3-Apr-2023.) $)
    reusng $p |- ( A e. V -> ( E! x e. { A } ph <-> ps ) ) $=
      ( nfv reusngf ) ABCDEBCGFH $.

    $d A y $.  $d B x y $.  $d ch y $.
    2ralsng.1 $e |- ( y = B -> ( ps <-> ch ) ) $.
    $( Substitution expressed in terms of two quantifications over singletons.
       (Contributed by AV, 22-Dec-2019.) $)
    2ralsng $p |- ( ( A e. V /\ B e. W )
                    -> ( A. x e. { A } A. y e. { B } ph <-> ch ) ) $=
      ( wcel csn wral cv wceq ralbidv ralsng sylan9bb ) FHLAEGMZNZDFMNBETNZGILC
      UAUBDFHDOFPABETJQRBCEGIKRS $.
  $}

  ${
    $d A x y $.  $d ph y $.
    $( Restricted existential uniqueness over a singleton is equivalent to a
       restricted existential quantification over a singleton.  (Contributed by
       AV, 3-Apr-2023.) $)
    rexreusng $p |- ( A e. V -> ( E. x e. { A } ph <-> E! x e. { A } ph ) ) $=
      ( vy wcel csn wa wi wral wsbc cv wceq nfsbc1v nfv sbceq1a imbi12d ralsngf
      nfan nfim wrex wsb weq wreu eqidd dfsbcq2 anbi12d eqeq2 mpbiri nfcv nfs1v
      nfralw anbi1d eqeq1 ralbidv mpbird biantrud reu2 bitr4di ) CDFZABCGZUAZVB
      AABEUBZHZBEUCZIZEVAJZBVAJZHABVAUDUTVHVBUTVHABCKZVCHZCELZMZIZEVAJZUTVNVIEC
      KZVIHZCCMZIZVPCUEVMVRECDVPVQEVOVIEVIECNVIEOSVQEOTVKCMZVJVPVLVQVSVIVOVCVIV
      IECPABECUFUGVKCCUHQRUIVGVNBCDVMBEVABVAUJVJVLBVIVCBABCNABEUKSVLBOTULBLZCMZ
      VFVMEVAWAVDVJVEVLWAAVIVCABCPUMVTCVKUNQUORUPUQABEVAURUS $.
  $}

  $( 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 wrex vsnid eleq2 mpbiri pm4.71ri exbii df-rex bitr4i
    wa ) BACZDZEZAFQBGZSPZAFSABHSUAASTSTQRGAIBRQJKLMSABNO $.

  ${
    $d A x $.  $d ps x $.
    ralsn.1 $e |- A e. _V $.
    ralsn.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Convert a universal quantification restricted to 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 $.

    $( Convert an existential quantification restricted to a singleton to a
       substitution.  (Contributed by Jeff Madsen, 5-Jan-2011.) $)
    rexsn $p |- ( E. x e. { A } ph <-> ps ) $=
      ( cvv wcel csn wrex wb rexsng ax-mp ) DGHACDIJBKEABCDGFLM $.
  $}

  $( Elementhood in a union with a singleton.  (Contributed by Thierry Arnoux,
     14-Dec-2023.) $)
  elunsn $p |- ( A e. V -> ( A e. ( B u. { C } ) <-> ( A e. B \/ A = C ) ) )
      $=
    ( csn cun wcel wo wceq elun elsng orbi2d bitrid ) ABCEZFGABGZANGZHADGZOACIZ
    HABNJQPROACDKLM $.

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Membership in an extension of a power class.  (Contributed by NM,
       26-Mar-2007.) $)
    elpwunsn $p |- ( A e. ( ~P ( B u. { C } ) \ ~P B ) -> C e. A ) $=
      ( vx csn cun cpw cdif wcel wn wa eldif cv wrex wral wss elpwg dfss3 sylbi
      wi bitrdi notbid biimpa rexnal sylibr wceq elpwi ssel wo elsni orim2i ord
      elun imim2i impd 3syl eleq1 biimpd syl6 com4r pm2.43b rexlimdv imp syldan
      expd ) ABCEZFZGZBGZHIAVHIZAVIIZJZKZCAIZAVHVILVJVLDMZBIZJZDANZVNVMVPDAOZJZ
      VRVJVLVTVJVKVSVJVKABPVSABVHQDABRUAUBUCVPDAUDUEVJVRVNVJVQVNDAVJVOAIZVQVNTV
      JWAVQWAVNVJWAVQWAVNTZVJWAVQKZVOCUFZWBVJAVGPWAVOVGIZTZWCWDTAVGUGAVGVOUHWFW
      AVQWDWEVQWDTZWAWEVPVOVFIZUIZWGVOBVFUMWIVPWDWHWDVPVOCUJUKULSUNUOUPWDWAVNVO
      CAUQURUSVEUTVAVBVCVDS $.
  $}

  $( An element of a set is either equal to another element of the set or a
     member of the difference of the set and the singleton containing the other
     element.  (Contributed by AV, 25-Aug-2020.)  (Proof shortened by JJ,
     23-Jul-2021.) $)
  eqoreldif $p |- ( B e. C
                    -> ( A e. C <-> ( A = B \/ A e. ( C \ { B } ) ) ) ) $=
    ( wcel wceq cdif wo wn wa simpl elsni con3i adantl eldifd ex orrd eleq1a wi
    csn eldifi a1i jaod impbid2 ) BCDZACDZABEZACBSZFDZGUEUFUHUEUFHZUHUEUIIACUGU
    EUIJUIAUGDZHUEUJUFABKLMNOPUDUFUEUHBCAQUHUERUDACUGTUAUBUC $.

  $( 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 $.
  $}

  $( Members of a set with seven elements.  Lemma for ~ usgrexmpl2nb0 etc.
     (Contributed by AV, 9-Aug-2025.) $)
  el7g $p |- ( X e. V -> ( X e. ( { A } u. ( { B , C , D } u. { E , F , G } ) )
                        <-> ( X = A \/ ( ( X = B \/ X = C \/ X = D )
                                      \/ ( X = E \/ X = F \/ X = G ) ) ) ) ) $=
    ( csn ctp cun wcel wo wceq w3o elun eltpg orbi12d bitrid elsng ) IAJZBCDKZE
    FGKZLZLMIUBMZIUEMZNIHMZIAOZIBOICOIDOPZIEOIFOIGOPZNZNIUBUEQUHUFUIUGULIAHUAUG
    IUCMZIUDMZNUHULIUCUDQUHUMUJUNUKIBCDHRIEFGHRSTST $.

  ${
    $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 eqabi ) 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 $.
  $}

  $( Membership of a conditional operator in an unordered pair.  (Contributed
     by NM, 17-Jun-2007.) $)
  ifpr $p |- ( ( A e. C /\ B e. D ) -> if ( ph , A , B ) e. { A , B } ) $=
    ( wcel cvv cif cpr elex wa ifcl wceq wo ifeqor elprg mpbiri syl syl2an ) BD
    FBGFZCGFZABCHZBCIFZCEFBDJCEJTUAKUBGFZUCABCGLUDUCUBBMUBCMNABCOUBBCGPQRS $.

  ${
    $d x A $.  $d x B $.
    ralprgf.1 $e |- F/ x ps $.
    ralprgf.2 $e |- F/ x ch $.
    ralprgf.a $e |- ( x = A -> ( ph <-> ps ) ) $.
    ralprgf.b $e |- ( x = B -> ( ph <-> ch ) ) $.
    $( Convert a restricted universal quantification over a pair to a
       conjunction, using bound-variable hypotheses instead of distinct
       variable conditions.  (Contributed by NM, 17-Sep-2011.)  (Revised by AV,
       8-Apr-2023.) $)
    ralprgf $p |- ( ( A e. V /\ B e. W )
                    -> ( A. x e. { A , B } ph <-> ( ps /\ ch ) ) ) $=
      ( cpr wral csn wa wcel cun df-pr ralsngf raleqi ralunb bi2anan9 bitrid
      bitri ) ADEFMZNZADEOZNZADFOZNZPZEGQZFHQZPBCPUGADUHUJRZNULADUFUOEFSUAADUHU
      JUBUEUMUIBUNUKCABDEGIKTACDFHJLTUCUD $.

    $( Convert a restricted existential quantification over a pair to a
       disjunction, using bound-variable hypotheses instead of distinct
       variable conditions.  (Contributed by NM, 17-Sep-2011.)  (Revised by AV,
       2-Apr-2023.) $)
    rexprgf $p |- ( ( A e. V /\ B e. W )
                    -> ( E. x e. { A , B } ph <-> ( ps \/ ch ) ) ) $=
      ( cpr wrex csn wo wcel wa cun rexsngf df-pr rexeqi orbi1d orbi2d sylan9bb
      rexun bitri bitrid ) ADEFMZNZADEOZNZADFOZNZPZEGQZFHQZRBCPZUJADUKUMSZNUOAD
      UIUSEFUAUBADUKUMUFUGUPUOBUNPUQURUPULBUNABDEGIKTUCUQUNCBACDFHJLTUDUEUH $.
  $}

  ${
    $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 restricted universal quantification over a pair to a
       conjunction.  (Contributed by NM, 17-Sep-2011.)  (Revised by Mario
       Carneiro, 23-Apr-2015.)  Avoid ~ ax-10 , ~ ax-12 .  (Revised by GG,
       30-Sep-2024.) $)
    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 $.
    $( $j usage 'ralprg' avoids 'ax-10' 'ax-11' 'ax-12'; $)

    $( Convert a restricted existential quantification over a pair to a
       disjunction.  (Contributed by NM, 17-Sep-2011.)  (Revised by Mario
       Carneiro, 23-Apr-2015.)  Avoid ~ ax-10 , ~ ax-12 .  (Revised by GG,
       30-Sep-2024.) $)
    rexprg $p |- ( ( A e. V /\ B e. W ) ->
      ( E. x e. { A , B } ph <-> ( ps \/ ch ) ) ) $=
      ( wcel wa wn cpr wral wb wrex wo wceq notbid ralprg ralnex pm4.56 bibi12i
      cv notbi sylbb2 syl ) EGKFHKLAMZDEFNZOZBMZCMZLZPZADUJQZBCRZPZUIULUMDEFGHD
      UEZESABITUSFSACJTUAUOUPMZUQMZPURUKUTUNVAADUJUBBCUCUDUPUQUFUGUH $.
    $( $j usage 'rexprg' avoids 'ax-10' 'ax-11' 'ax-12'; $)

    raltpg.3 $e |- ( x = C -> ( ph <-> th ) ) $.
    $( Convert a restricted universal 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 restricted existential 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 restricted universal 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 a restricted 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 c w x A $.  $d c w x B $.  $d x ps $.  $d c w ph $.  $d x ch $.
    reuprg.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    reuprg.2 $e |- ( x = B -> ( ph <-> ch ) ) $.
    $( Convert a restricted existential uniqueness over a pair to a disjunction
       of conjunctions.  (Contributed by AV, 2-Apr-2023.) $)
    reuprg0 $p |- ( ( A e. V /\ B e. W )
             -> ( E! x e. { A , B } ph <-> ( ( ps /\ ( ch -> A = B ) )
                                          \/ ( ch /\ ( ps -> A = B ) ) ) ) ) $=
      ( vc cv wsbc wceq wi wral wa nfv eqeq2 imbi12d cpr wreu wrex wcel nfsbc1v
      vw wo sbceq1a dfsbcq reu8nf nfcv nfim nfralw eqeq1 imbi2d ralbidv anbi12d
      nfan rexprgf ralprg eqidd biantrur wb sbcieg adantl imbi1d bitr3id anbi2d
      bitrd biantru adantr eqcom imbi2i anbi2i bitrdi orbi12d bitrid ) ADEFUAZU
      BAADKLZMZDLZVSNZOZKVRPZQZDVRUCZEGUDZFHUDZQZBCEFNZOZQZCBWJOZQZUGZAVTADUFLZ
      MDKUFVRADVSUEZADWPUEADWPUHADWPVSUIUJWIWFBVTEVSNZOZKVRPZQZCVTFVSNZOZKVRPZQ
      ZUGWOWEXAXEDEFGHBWTDBDRWSDKVRDVRUKZVTWRDWQWRDRULUMURCXDDCDRXCDKVRXFVTXBDW
      QXBDRULUMURWAENZABWDWTIXGWCWSKVRXGWBWRVTWAEVSUNUOUPUQWAFNZACWDXDJXHWCXCKV
      RXHWBXBVTWAFVSUNUOUPUQUSWIXAWLXEWNWIWTWKBWIWTADEMZEENZOZADFMZWJOZQZWKWSXK
      XMKEFGHVSENZVTXIWRXJADVSEUIZVSEESTVSFNZVTXLWRWJADVSFUIZVSFESTUTXNXMWIWKXK
      XMXIEVAVBWIXLCWJWHXLCVCWGACDFHJVDVEVFVGVIVHWIXECBFENZOZQWNWIXDXTCWIXDXIXS
      OZXLFFNZOZQZXTXCYAYCKEFGHXOVTXIXBXSXPVSEFSTXQVTXLXBYBXRVSFFSTUTYDYAWIXTYC
      YAXLFVAVJWIXIBXSWGXIBVCWHABDEGIVDVKVFVGVIVHXTWMCXSWJBFEVLVMVNVOVPVIVQ $.

    $( Convert a restricted existential uniqueness over a pair to a disjunction
       and an implication .  (Contributed by AV, 2-Apr-2023.) $)
    reuprg $p |- ( ( A e. V /\ B e. W ) -> ( E! x e. { A , B } ph
                       <-> ( ( ps \/ ch ) /\ ( ( ch /\ ps ) -> A = B ) ) ) ) $=
      ( wcel wa cpr wreu wceq wi wo curryax bicomi bitri reuprg0 orddi biantrur
      biantru orcom mpbir pm4.79 anbi12i bitrdi ) EGKFHKLADEFMNBCEFOZPZLCBUJPZL
      QZBCQZCBLUJPZLZABCDEFGHIJUAUMUNBULQZLZUKCQZUKULQZLZLUPBUKCULUBURUNVAUOUNU
      RUQUNBUJRUDSVAUTUOUTVAUSUTUSCUKQCUJRUKCUEUFUCSUJCBUGTUHTUI $.

    $( Convert a restricted existential uniqueness over a pair to a restricted
       existential quantification and an implication .  (Contributed by AV,
       3-Apr-2023.) $)
    reurexprg $p |- ( ( A e. V /\ B e. W ) -> ( E! x e. { A , B } ph
               <-> ( E. x e. { A , B } ph /\ ( ( ch /\ ps ) -> A = B ) ) ) ) $=
      ( wcel wa cpr wreu wo wceq wi wrex reuprg rexprg bicomd anbi1d bitrd ) EG
      KFHKLZADEFMZNBCOZCBLEFPQZLADUERZUGLABCDEFGHIJSUDUFUHUGUDUHUFABCDEFGHIJTUA
      UBUC $.
  $}

  ${
    $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 universal quantification over an unordered 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 an existential quantification over an unordered 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 $.
  $}

  ${
    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.
       ~ csbsng is derived from the virtual deduction proof ~ csbsngVD .
       (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 csbab sbceq2g abbidv eqtrid df-sn csbeq2i
      cv 3eqtr4g ) BDFZABERZCGZEHZIZUAABCIZGZEHZABCJZIUEJTUDUBABKZEHUGUBAEBLTUI
      UFEABUACDMNOABUHUCECPQEUEPS $.
  $}

  $( Distribute proper substitution through a pair of classes.  (Contributed by
     Alexander van der Vekens, 4-Sep-2018.) $)
  csbprg $p |- ( C e. V
            -> [_ C / x ]_ { A , B } = { [_ C / x ]_ A , [_ C / x ]_ B } ) $=
    ( wcel csn cun csb cpr csbun csbsng uneq12d eqtrid df-pr csbeq2i 3eqtr4g )
    DEFZADBGZCGZHZIZADBIZGZADCIZGZHZADBCJZIUCUEJRUBADSIZADTIZHUGADSTKRUIUDUJUFA
    DBELADCELMNADUHUABCOPUCUEOQ $.

  $( If the intersection of two classes is a (proper) singleton, then the
     singleton element is a member of both classes.  (Contributed by AV,
     30-Dec-2021.) $)
  elinsn $p |- ( ( A e. V /\ ( B i^i C ) = { A } ) -> ( A e. B /\ A e. C ) ) $=
    ( wcel csn cin wceq wa snidg eleq2 elin biimpi biimtrrdi mpan9 ) ADEAAFZEZB
    CGZPHZABEACEIZADJSQAREZTRPAKUATABCLMNO $.

  ${
    $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 dfclel xchbinxr ) ABDZEFGCHZAIZUFUEIZJKZCLUFBGZUGMZ
      JZCLZBAIZJCAUENUIULCUIUHUGJZKUJUOKULUGUHOUHUJUOCBPQUJUGSRTUMUKCUAUNUKCUBC
      BAUCUDR $.
  $}

  $( Two distinct singletons are 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 $.

  $( Two completely distinct unordered pairs are disjoint.  (Contributed by
     Alexander van der Vekens, 11-Nov-2017.)  (Proof shortened by JJ,
     23-Jul-2021.) $)
  disjpr2 $p |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) )
                  -> ( { A , B } i^i { C , D } ) = (/) ) $=
    ( wne wa cpr cin csn cun df-pr eqtri wceq ineq1i indir disjsn2 anim12i un00
    c0 eqtrid ineq2i indi sylib adantr adantl uneq12d un0 eqtrdi ) ACEZBCEZFZAD
    EZBDEZFZFZABGZCDGZHZUPCIZHZUPDIZHZJZSURUPUSVAJZHVCUQVDUPCDKUAUPUSVAUBLUOVCS
    SJSUOUTSVBSUKUTSMUNUKUTAIZUSHZBIZUSHZJZSUTVEVGJZUSHVIUPVJUSABKZNVEVGUSOLUKV
    FSMZVHSMZFVISMUIVLUJVMACPBCPQVFVHRUCTUDUNVBSMUKUNVBVEVAHZVGVAHZJZSVBVJVAHVP
    UPVJVAVKNVEVGVAOLUNVNSMZVOSMZFVPSMULVQUMVRADPBDPQVNVORUCTUEUFSUGUHT $.

  $( The disjoint intersection of an unordered pair and a singleton.
     (Contributed by AV, 23-Jan-2021.) $)
  disjprsn $p |- ( ( A =/= C /\ B =/= C ) -> ( { A , B } i^i { C } ) = (/) ) $=
    ( wne wa cpr csn cin c0 dfsn2 ineq2i wceq disjpr2 anidms eqtrid ) ACDBCDEZA
    BFZCGZHQCCFZHZIRSQCJKPTILABCCMNO $.

  $( The disjoint intersection of an unordered triple and a singleton.
     (Contributed by AV, 14-Nov-2021.) $)
  disjtpsn $p |- ( ( A =/= D /\ B =/= D /\ C =/= D )
                   -> ( { A , B , C } i^i { D } ) = (/) ) $=
    ( wne w3a ctp csn cin cpr cun c0 df-tp ineq1i wceq disjprsn 3adant3 disjsn2
    wa 3ad2ant3 jca undisj1 sylib eqtrid ) ADEZBDEZCDEZFZABCGZDHZIABJZCHZKZUJIZ
    LUIUMUJABCMNUHUKUJILOZULUJILOZSUNLOUHUOUPUEUFUOUGABDPQUGUEUPUFCDRTUAUKULUJU
    BUCUD $.

  $( Two completely distinct unordered triples are disjoint.  (Contributed by
     AV, 14-Nov-2021.) $)
  disjtp2 $p |- ( ( ( A =/= D /\ B =/= D /\ C =/= D )
                    /\ ( A =/= E /\ B =/= E /\ C =/= E )
                    /\ ( A =/= F /\ B =/= F /\ C =/= F ) )
                  -> ( { A , B , C } i^i { D , E , F } ) = (/) ) $=
    ( wne w3a ctp cin cpr csn cun c0 df-tp wceq wa 3simpa 3ad2ant3 eqtrid incom
    ineq2i ineq1i disjpr2 syl2an 3adant3 necom biimpi disjprsn undisj1 disjtpsn
    jca sylib undisj2 ) ADGZBDGZCDGZHZAEGZBEGZCEGZHZAFGBFGCFGHZHZABCIZDEFIZJVED
    EKZFLZMZJZNVFVIVEDEFOUBVDVEVGJZNPZVEVHJNPZQVJNPVDVLVMVDVKABKZCLZMZVGJZNVEVP
    VGABCOUCVDVNVGJNPZVOVGJZNPZQVQNPVDVRVTURVBVRVCURUOUPQUSUTQVRVBUOUPUQRUSUTVA
    RABDEUDUEUFVDVSVGVOJZNVOVGUAURVBWANPZVCURDCGZECGZWBVBUQUOWCUPUQWCCDUGUHSVAU
    SWDUTVAWDCEUGUHSDECUIUEUFTULVNVOVGUJUMTVCURVMVBABCFUKSULVEVGVHUNUMT $.

  ${
    $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,
       21-Jun-1993.) $)
    snprc $p |- ( -. A e. _V <-> { A } = (/) ) $=
      ( vx csn c0 wceq cvv wcel cv wex velsn exbii neq0 isset 3bitr4i con1bii
      wn ) ACZDEZAFGZBHZQGZBITAEZBIRPSUAUBBBAJKBQLBAMNO $.
  $}

  $( A singleton is nonempty iff its argument is a set.  (Contributed by Scott
     Fenton, 8-May-2018.) $)
  snnzb $p |- ( A e. _V <-> { A } =/= (/) ) $=
    ( cvv wcel csn c0 wne wn wceq snprc df-ne con2bii bitri con4bii ) ABCZADZEF
    ZNGOEHZPGAIPQOEJKLM $.

  ${
    $d A x $.
    $( A restricted at-most-one quantifier over a singleton is always true.
       (Contributed by AV, 3-Apr-2023.) $)
    rmosn $p |- E* x e. { A } ph $=
      ( cvv wcel csn wrmo wrex wreu wi wsbc idd nfsbc1v sbceq1a rexsngf reusngf
      3imtr4d rmo5 sylibr c0 wn rmo0 wceq wb snprc rmoeq1 sylbi mpbiri pm2.61i
      ) CDEZABCFZGZUJABUKHZABUKIZJULUJABCKZUOUMUNUJUOLAUOBCDABCMZABCNZOAUOBCDUP
      UQPQABUKRSUJUAZULABTGZABUBURUKTUCULUSUDCUEABUKTUFUGUHUI $.
  $}

  ${
    $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, 18-Mar-2020.) $)
    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.)  (Proof shortened by AV,
       26-Aug-2022.) $)
    rabsn $p |- ( B e. A -> { x e. A | x = B } = { B } ) $=
      ( wcel cv wceq wa wal crab csn eleq1 pm5.32ri baib alrimiv rabeqsn sylibr
      wb ) CBDZAEZBDZSCFZGZUAQZAHUAABICJFRUCAUBRUAUATRSCBKLMNUAABCOP $.
  $}

  ${
    $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 x A $.  $d x ps $.
    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 } , (/) ) $=
      ( cvv wcel csn crab cif wceq wsbc rabsnifsb sbcieg ifbid eqtrid rab0 ifid
      c0 wn eqtr4i snprc biimpi rabeqdv ifeq1d 3eqtr4a pm2.61i ) DFGZACDHZIZBUI
      SJZKUHUJACDLZUISJUKACDMUHULBUISABCDFENOPUHTZACSIZBSSJZUJUKUNSUOACQBSRUAUM
      ACUISUMUISKDUBUCZUDUMBUISSUPUEUFUG $.
  $}

  ${
    $d x A $.
    $( A restricted class abstraction restricted to a singleton is either the
       empty set or the singleton itself.  (Contributed by Alexander van der
       Vekens, 22-Dec-2017.)  (Proof shortened by AV, 21-Jul-2019.) $)
    rabrsn $p |- ( M = { x e. { A } | ph } -> ( M = (/) \/ M = { A } ) ) $=
      ( csn crab wceq wsbc c0 cif wo rabsnifsb eqeq2i ifeqor orcom mpbi orbi12d
      eqeq1 mpbiri sylbi ) DABCEZFZGDABCHZUAIJZGZDIGZDUAGZKZUBUDDABCLMUEUHUDIGZ
      UDUAGZKZUJUIKUKUCUAINUJUIOPUEUFUIUGUJDUDIRDUDUARQST $.
  $}

  ${
    $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 eu6 absn exbii bitr4i ) ABDABECEZFGBHZCI
      ABJPKFZCIABCLRQCABPMNO $.

    $( 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 nfv nfab1 sneq eqeq2d cbvexv1 bitr4i
      nfeq1 ) ABDABEZCFZGZHZCIRBFZGZHZBIABCJUDUABCUDCKBRTABLQUBSHUCTRUBSMNOP $.

    $( 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, 15-Jul-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,
     15-Jul-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 unordered 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 unordered 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, 24-Jun-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.  (Note: the proof from ~ prid2g and ~ ax-mp has one fewer
       essential step but one more total step.)  (Contributed by NM,
       5-Aug-1993.) $)
    prid2 $p |- B e. { A , B } $=
      ( cpr prid1 prcom eleqtri ) BBADABDBACEBAFG $.
  $}

  $( 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.) $)
  ifpprsnss $p |- ( P = { A , B }
                    -> if- ( A = B , P = { A } , { A , B } C_ P ) ) $=
    ( cpr wceq csn wss preq2 dfsn2 eqtr4di eqcoms eqeq2d biimpac eqimss2 adantr
    wn ifpimpda ) CABDZEZABEZCAFZEZRCGZTSUBTRUACRUAEBABAERAADUABAAHAIJKLMSUCTPR
    CNOQ $.

  $( A proper class vanishes in an unordered pair.  (Contributed by NM,
     15-Jul-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 $.

  $( Closed theorem form of ~ tpid3 .  (Contributed by Alan Sare, 24-Oct-2011.)
     (Proof shortened by JJ, 30-Apr-2021.) $)
  tpid3g $p |- ( A e. B -> A e. { C , D , A } ) $=
    ( wcel ctp wceq w3o eqid 3mix3i eltpg mpbiri ) ABEACDAFEACGZADGZAAGZHOMNAIJ
    ACDABKL $.

  ${
    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.)  (Proof
       shortened by JJ, 30-Apr-2021.) $)
    tpid3 $p |- C e. { A , B , C } $=
      ( cvv wcel ctp tpid3g ax-mp ) CEFCABCGFDCEABHI $.
  $}

  $( The singleton of a set is not empty.  (Contributed by NM, 14-Dec-2008.) $)
  snnzg $p |- ( A e. V -> { A } =/= (/) ) $=
    ( wcel csn snidg ne0d ) ABCADAABEF $.

  ${
    snn0d.1 $e |- ( ph -> A e. V ) $.
    $( The singleton of a set is not empty.  (Contributed by Glauco Siliprandi,
       3-Mar-2021.) $)
    snn0d $p |- ( ph -> { A } =/= (/) ) $=
      ( wcel csn c0 wne snnzg syl ) ABCEBFGHDBCIJ $.
  $}

  ${
    snnz.1 $e |- A e. _V $.
    $( The singleton of a set is not empty.  (Contributed by NM,
       10-Apr-1994.) $)
    snnz $p |- { A } =/= (/) $=
      ( cvv wcel csn c0 wne snnzg ax-mp ) ACDAEFGBACHI $.
  $}

  ${
    prnz.1 $e |- A e. _V $.
    $( A pair containing a set is not empty.  (Contributed by NM,
       9-Apr-1994.) $)
    prnz $p |- { A , B } =/= (/) $=
      ( cpr prid1 ne0ii ) AABDABCEF $.
  $}

  $( A pair containing a set is not empty.  (Contributed by FL, 19-Sep-2011.)
     (Proof shortened by JJ, 23-Jul-2021.) $)
  prnzg $p |- ( A e. V -> { A , B } =/= (/) ) $=
    ( wcel cpr prid1g ne0d ) ACDABEAABCFG $.

  ${
    tpnz.1 $e |- A e. _V $.
    $( An unordered triple containing a set is not empty.  (Contributed by NM,
       10-Apr-1994.) $)
    tpnz $p |- { A , B , C } =/= (/) $=
      ( ctp tpid1 ne0ii ) AABCEABCDFG $.
  $}

  ${
    tpnzd.1 $e |- ( ph -> A e. V ) $.
    $( An unordered triple containing a set is not empty.  (Contributed by
       Thierry Arnoux, 8-Apr-2019.) $)
    tpnzd $p |- ( ph -> { A , B , C } =/= (/) ) $=
      ( wcel ctp c0 wne tpid1g ne0i 3syl ) ABEGBBCDHZGNIJFBECDKNBLM $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.  $d x ph $.  $d x ch $.  $d x th $.
    $d x ta $.
    ralprd.1 $e |- ( ( ph /\ x = A ) -> ( ps <-> ch ) ) $.
    ralprd.2 $e |- ( ( ph /\ x = B ) -> ( ps <-> th ) ) $.
    raltpd.3 $e |- ( ( ph /\ x = C ) -> ( ps <-> ta ) ) $.
    ralprd.a $e |- ( ph -> A e. V ) $.
    ralprd.b $e |- ( ph -> B e. W ) $.
    raltpd.c $e |- ( ph -> C e. X ) $.
    $( Convert a universal quantification over an unordered triple to a
       conjunction.  (Contributed by Thierry Arnoux, 8-Apr-2019.) $)
    raltpd $p |- ( ph -> ( A. x e. { A , B , C } ps <-> ( ch /\ th /\ ta ) ) )
      $=
      ( wa wb ctp wral w3a an3andi a1i wcel cv expcom pm5.32d raltpg syl3anc c0
      wceq wne tpnzd r19.28zv syl 3bitr2d bianabs bicomd ) ABFGHIUAZUBZCDEUCZAA
      VCSZVBAVDVBAVDACSZADSZAESZUCZABSZFVAUBZAVBSZVDVHTAACDEUDUEAGJUFHKUFILUFVJ
      VHTPQRVIVEVFVGFGHIJKLFUGZGUMZABCAVMBCTMUHUIVLHUMZABDAVNBDTNUHUIVLIUMZABEA
      VOBETOUHUIUJUKAVAULUNVJVKTAGHIJPUOABFVAUPUQURUSUTUS $.
  $}

  ${
    $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 wi wal wceq cvv df-ss velsn imbi1i albii wex pm5.74i
      eleq1 19.23v 3bitri isset bicomi ) ADZBECFZUCGZUDBGZHZCIUDAJZUFHZCIZAKGZA
      BGZHZCUCBLUGUICUEUHUFCAMNOUJUHULHZCIUHCPZULHUMUIUNCUHUFULUDABRQOUHULCSUOU
      KULUKUOCAUAUBNTT $.
  $}

  $( 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 $.

  ${
    $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 wcel cv csn wi wal wceq wss wb velsn imbi1i albii a1i df-ss 3bitr4rd
      clel2g ) ACEZDFZAGZEZUABEZHZDIZUAAJZUDHZDIZUBBKZABEUFUILTUEUHDUCUGUDDAMNO
      PUJUFLTDUBBQPDABCSR $.
  $}

  ${
    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.) $)
    snss $p |- ( A e. B <-> { A } C_ B ) $=
      ( cvv wcel csn wss wb snssg ax-mp ) ADEABEAFBGHCABDIJ $.
  $}

  $( 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 $.

  ${
    eldifsnd.1 $e |- ( ph -> A e. B ) $.
    eldifsnd.2 $e |- ( ph -> A =/= C ) $.
    $( Membership in a set with an element removed : deduction version.
       (Contributed by Thierry Arnoux, 4-May-2025.) $)
    eldifsnd $p |- ( ph -> A e. ( B \ { C } ) ) $=
      ( wcel wne csn cdif eldifsn sylanbrc ) ABCGBDHBCDIJGEFBCDKL $.
  $}

  $( 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 wa wcel wn difss2 reldisj bicomd biadanii disjsn
    anbi2i bitri ) ABCDZEFZABFZATGHIZJUBCAKLZJUAUBUCABTMUBUCUAATBNOPUCUDUBACQRS
    $.

  ${
    $d A x $.  $d S x $.  $d V x $.  $d W x $.
    $( A subset of a set is an element of the power set of the difference of
       the set with a singleton if the subset does not contain the singleton
       element.  (Contributed by AV, 10-Jan-2020.) $)
    elpwdifsn $p |- ( ( S e. W /\ S C_ V /\ A e/ S )
                      -> S e. ~P ( V \ { A } ) ) $=
      ( vx wcel wss wnel w3a csn cdif cpw cv wa wne simp2 sselda df-nel biimpi
      wn 3ad2ant3 anim1ci nelne2 syl sylanbrc ex ssrdv wb elpwg 3ad2ant1 mpbird
      eldifsn ) BDFZBCGZABHZIZBCAJKZLFZBUQGZUPEBUQUPEMZBFZUTUQFZUPVANZUTCFUTAOZ
      VBUPBCUTUMUNUOPQVCVAABFTZNVDUPVEVAUOUMVEUNUOVEABRSUAUBUTABUCUDUTCAULUEUFU
      GUMUNURUSUHUOBUQDUIUJUK $.
  $}

  $( 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 $.

  $( An element of a difference with a singleton is not equal to the element of
     that singleton.  Note that ` ( -. A e. { C } -> -. A = C ) ` need not hold
     if ` A ` is a proper class.  (Contributed by BJ, 18-Mar-2023.)  (Proof
     shortened by Steven Nguyen, 1-Jun-2023.) $)
  eldifsnneq $p |- ( A e. ( B \ { C } ) -> -. A = C ) $=
    ( csn cdif wcel eldifsni neneqd ) ABCDEFACABCGH $.

  $( The class ` 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 $.

  $( The class ` 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 $.

  $( Rearrangement of a property of a singleton difference.  (Contributed by
     Stefan O'Rear, 27-Feb-2015.) $)
  raldifsni $p |- ( A. x e. ( A \ { B } ) -. ph <->
      A. x e. A ( ph -> x = B ) ) $=
    ( wn cv wceq wi csn cdif wcel wne eldifsn imbi1i impexp df-ne con34b bitr4i
    wa imbi2i 3bitri ralbii2 ) AEZABFZDGZHZBCDIJZCUDUGKZUCHUDCKZUDDLZSZUCHUIUJU
    CHZHUIUFHUHUKUCUDCDMNUIUJUCOULUFUIULUEEZUCHUFUJUMUCUDDPNAUEQRTUAUB $.

  $( Restricted universal quantification on a class difference with a singleton
     in terms of an implication.  (Contributed by Alexander van der Vekens,
     26-Jan-2018.) $)
  raldifsnb $p |- ( A. x e. A ( x =/= Y -> ph )
                    <-> A. x e. ( A \ { Y } ) ph ) $=
    ( cv wne wi wral csn wnel cdif wcel wceq wn velsn nnel nne 3bitr4ri con4bii
    imbi1i ralbii raldifb bitri ) BEZDFZAGZBCHUDDIZJZAGZBCHABCUGKHUFUIBCUEUHAUE
    UHUDUGLUDDMUHNUENBDOUDUGPUDDQRSTUAABCUGUBUC $.

  $( 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 A x $.  $d B x $.
    $( 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 nelelne ancld impbid2 eqrdv
      bitrid ) ABDEZCBAFGZBCHZSDTBDZTAIZJZRUATBAKRUCUAUAUBLRUAUBABTMNOQP $.

    $( 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.)  (Proof shortened by JJ,
     23-Jul-2021.) $)
  diftpsn3 $p |- ( ( A =/= C /\ B =/= C )
                    -> ( { A , B , C } \ { C } ) = { A , B } ) $=
    ( wne wa cpr csn cdif cun c0 ctp cin wceq disjprsn disj3 sylib eqcomd difid
    a1i uneq12d df-tp difeq1i difundir eqtr2i un0 3eqtr3g ) ACDBCDEZABFZCGZHZUI
    UIHZIZUHJIABCKZUIHZUHUGUJUHUKJUGUHUJUGUHUILJMUHUJMABCNUHUIOPQUKJMUGUIRSTUNU
    HUIIZUIHULUMUOUIABCUAUBUHUIUIUCUDUHUEUF $.

  $( 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 $.

  $( An unordered triple is an unordered pair if one of its elements is a
     proper class or is identical with another element.  (Contributed by
     Alexander van der Vekens, 6-Oct-2017.) $)
  tpprceq3 $p |- ( -. ( C e. _V /\ C =/= B ) -> { A , B , C } = { A , B } ) $=
    ( cvv wcel wne wa wn wo ctp cpr wceq ianor csn cun prprc2 tprot eqtri sylbi
    uneq1d df-tp prcom df-pr 3eqtr4g nne tppreq3 eqcoms jaoi ) CDEZCBFZGHUIHZUJ
    HZIABCJZABKZLZUIUJMUKUOULUKBCKZANZOZBNZUQOZUMUNUKUPUSUQBCPTUMBCAJURABCQBCAU
    ARUNBAKUTABUBBAUCRUDULCBLUOCBUEUOBCABCUFUGSUHS $.

  $( An unordered triple is an unordered pair if and only if one of its
     elements is a proper class or is identical with one of the another
     elements.  (Contributed by Alexander van der Vekens, 15-Jan-2018.) $)
  tppreqb $p |- ( -. ( C e. _V /\ C =/= A /\ C =/= B )
                     <-> { A , B , C } = { A , B } ) $=
    ( cvv wcel wne w3a wn ctp cpr wceq wo w3o wa ianor tpprceq3 sylbir jaoi nne
    sylbi 3ianor df-3or bitri orass tpcoma prcom 3eqtr3g orcom bitr4i csn df-tp
    orcs cun eqeq1i wss ssequn2 snssg elpri 3mix2 3mix3 syl biimtrrdi 3mix1 a1d
    wi pm2.61i sylibr impbii ) CDEZCAFZCBFZGHZABCIZABJZKZVLVIHZVJHZLZVKHZLZVOVL
    VPVQVSMZVTVIVJVKUAZVPVQVSUBUCVTVPVOVTVPLVRVSVPLZLVOVRVSVPUDVRVOWCVRBACIZBAJ
    ZVMVNVRVIVJNHWDWEKVIVJOBACPQBACUEBAUFUGWCVIVKNHZVOWCVPVSLWFVSVPUHVIVKOUIABC
    PTRTULTVOVNCUJZUMZVNKZVLVMWHVNABCUKUNWIWGVNUOZVLWGVNUPWJWAVLVIWJWAVEVIWJCVN
    EZWACVNDUQWKCAKZCBKZLWACABURWLWAWMWLVQWACASVQVPVSUSQWMVSWACBSVSVPVQUTQRVAVB
    VPWAWJVPVQVSVCVDVFWBVGQTVH $.

  $( ` ( 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 $.

  $( ` ( B \ { A } ) ` is a proper subclass of ` B ` if and only if ` A ` is a
     member of ` B ` .  (Contributed by David Moews, 1-May-2017.) $)
  difsnpss $p |- ( A e. B <-> ( B \ { A } ) C. B ) $=
    ( wcel wn csn cdif wpss notnotb wne wss wa difss biantrur difsnb necon3bbii
    df-pss 3bitr4i bitri ) ABCZSDZDZBAEZFZBGZSHUCBIZUCBJZUEKUAUDUFUEBUBLMTUCBAB
    NOUCBPQR $.

  $( 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 snssi syl ) ABCEBFCGDBCHI $.
  $}

  $( If we remove a single element from a class then put it back in, we end up
     with the original class.  (Contributed by NM, 2-Oct-2006.) $)
  difsnid $p |- ( B e. A -> ( ( A \ { B } ) u. { B } ) = A ) $=
    ( wcel csn wss cdif cun wceq snssi undifr sylib ) BACBDZAEALFLGAHBAILAJK $.

  $( An element of a difference set is an element of the difference with a
     singleton.  (Contributed by AV, 2-Jan-2022.) $)
  eldifeldifsn $p |- ( ( X e. A /\ Y e. ( B \ A ) ) -> Y e. ( B \ { X } ) ) $=
    ( wcel cdif csn snssi sscond sselda ) CAEZBAFBCGZFDKLABCAHIJ $.

  $( 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 $.

  ${
    $d x y $.
    $( 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.
       Although this theorem is a special case of ~ pwsn , we have chosen to
       show a direct elementary proof.  (Contributed by NM, 7-Aug-1994.) $)
    pwpw0 $p |- ~P { (/) } = { (/) , { (/) } } $=
      ( vx vy cv c0 csn wss cab wceq wo cpw cpr wn wa wcel wi df-ss velsn sylbi
      wal wex imbi2i albii bitri neq0 exintr biimtrid exancom dfclel snssi syl6
      bitr4i anc2li eqss imbitrrdi orrd 0ss sseq1 mpbiri jaoi abbii df-pw dfpr2
      eqimss impbii 3eqtr4i ) ACZDEZFZAGVFDHZVFVGHZIZAGVGJDVGKVHVKAVHVKVHVIVJVH
      VILZVHVGVFFZMVJVHVLVMVHBCZVFNZVNDHZOZBSZVLVMOVHVOVNVGNZOZBSVRBVFVGPVTVQBV
      SVPVOBDQUAUBUCVRVLVOVPMBTZVMVLVOBTVRWABVFUDVOVPBUEUFWADVFNZVMWAVPVOMBTWBV
      OVPBUGBDVFUHUKDVFUIRUJRULVFVGUMUNUOVIVHVJVIVHDVGFVGUPVFDVGUQURVFVGVCUSVDU
      TAVGVAADVGVBVE $.
  $}

  $( 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 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 $.

  ${
    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.)  (Proof shortened by JJ, 23-Jul-2021.) $)
    prss $p |- ( ( A e. C /\ B e. C ) <-> { A , B } C_ C ) $=
      ( cvv wcel wa cpr wss wb prssg mp2an ) AFGBFGACGBCGHABICJKDEABCFFLM $.
  $}

  $( 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 $.
    $( The subsets of a singleton.  (Contributed by NM, 24-Apr-2004.) $)
    sssn $p |- ( A C_ { B } <-> ( A = (/) \/ A = { B } ) ) $=
      ( vx csn wss c0 wceq wo wn wa wcel cv wex neq0 ssel elsni syl6 eleq1 ibd
      wb biimtrid snssi anc2li eqss imbitrrdi orrd 0ss sseq1 mpbiri eqimss jaoi
      exlimdv impbii ) ABDZEZAFGZAUNGZHUOUPUQUOUPIZUOUNAEZJUQUOURUSUOURBAKZUSUR
      CLZAKZCMUOUTCANUOVBUTCUOVBUTUOVBVABGZVBUTTUOVBVAUNKVCAUNVAOVABPQVABARQSUL
      UABAUBQUCAUNUDUEUFUPUOUQUPUOFUNEUNUGAFUNUHUIAUNUJUKUM $.

    $( The property of being sandwiched between two sets naturally splits under
       union with a singleton.  This is the induction hypothesis for the
       determination of large powersets such as ~ pwtp .  (Contributed by Mario
       Carneiro, 2-Jul-2016.) $)
    ssunsn2 $p |- ( ( B C_ A /\ A C_ ( C u. { D } ) )
                    <-> ( ( B C_ A /\ A C_ C ) \/
                      ( ( B u. { D } ) C_ A /\ A C_ ( C u. { D } ) ) ) ) $=
      ( wcel wss csn cun wa wo wb syl wi anim12d pm4.72 sylib bitrd bitr3i wceq
      a1i snssi unss bicomi rbaibr anbi1d biimpi expcom ssun3 cdif uncom sseq2i
      wn ssundif cin c0 disjsn disj3 sseq1 bitr4id anbi2d simplbi biimpd bitrdi
      sylbi orcom pm2.61i ) DAEZBAFZACDGZHZFZIZVHACFZIZBVIHAFZVKIZJZKVGVLVPVQVG
      VHVOVKVGVIAFZVHVOKDAUAZVOVHVRVHVRIZVOBVIAUBZUCZUDLUEVGVNVPMVPVQKVGVHVOVMV
      KVGVRVHVOMVSVHVRVOVTVOWAUFUGLVMVKMVGACVIUHTNVNVPOPQVGULZVLVNVQWCVKVMVHWCV
      KAVIUIZCFZVMVKAVICHZFWEWFVJAVICUJUKAVICUMRWCAWDSZVMWEKWCAVIUNUOSWGADUPAVI
      UQRAWDCURVDUSZUTWCVNVPVNJZVQWCVPVNMVNWIKWCVOVHVKVMVOVHMWCVOVHVRWBVATWCVKV
      MWHVBNVPVNOPVPVNVEVCQVF $.

    $( Possible values for a set sandwiched between another set and it plus a
       singleton.  (Contributed by Mario Carneiro, 2-Jul-2016.) $)
    ssunsn $p |- ( ( B C_ A /\ A C_ ( B u. { C } ) )
        <-> ( A = B \/ A = ( B u. { C } ) ) ) $=
      ( wss csn cun wa wo wceq ssunsn2 ancom eqss bitr4i orbi12i bitri ) BADZAB
      CEFZDZGPABDZGZQADZRGZHABIZAQIZHABBCJTUCUBUDTSPGUCPSKABLMUBRUAGUDUARKAQLMN
      O $.

    $( Two ways to express that a nonempty set equals a singleton.
       (Contributed by NM, 15-Dec-2007.)  (Proof shortened by JJ,
       23-Jul-2021.) $)
    eqsn $p |- ( A =/= (/) -> ( A = { B } <-> A. x e. A x = B ) ) $=
      ( c0 wne csn wceq wo cv wral wn wb df-ne biorf sylbi wss wcel dfss3 velsn
      sssn ralbii 3bitr3i bitrdi ) BDEZBCFZGZBDGZUFHZAIZCGZABJZUDUGKUFUHLBDMUGU
      FNOBUEPUIUEQZABJUHUKABUERBCTULUJABACSUAUBUC $.

    $d ph x $.
    eqsnd.1 $e |- ( ( ph /\ x e. A ) -> x = B ) $.
    eqsnd.2 $e |- ( ph -> B e. A ) $.
    $( Deduce that a set is a singleton.  (Contributed by Thierry Arnoux,
       10-May-2023.)  (Proof shortened by SN, 3-Jul-2025.) $)
    eqsnd $p |- ( ph -> A = { B } ) $=
      ( csn wceq cv wral ralrimiva c0 wne wb ne0d eqsn syl mpbird ) ACDGHZBIDHZ
      BCJZATBCEKACLMSUANACDFOBCDPQR $.

    $( Obsolete version of ~ eqsnd as of 3-Jul-2025.  (Contributed by Thierry
       Arnoux, 10-May-2023.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    eqsndOLD $p |- ( ph -> A = { B } ) $=
      ( csn cv wcel wceq wa simpr adantr eqeltrd impbida velsn bitr4di eqrdv )
      ABCDGZABHZCIZTDJZTSIAUAUBEAUBKTDCAUBLADCIUBFMNOBDPQR $.
  $}

  ${
    $d A x y $.  $d A x z $.
    $( A sufficient condition for a (nonempty) set to be a singleton.
       (Contributed by AV, 20-Sep-2020.) $)
    issn $p |- ( E. x e. A A. y e. A x = y -> E. z A = { z } ) $=
      ( weq wral cv csn wceq wex wcel wb equcom a1i ralbidv wne ne0i eqsn syl
      c0 bitr4d sneq eqeq2d spcegv sylbid rexlimiv ) ABEZBDFZDCGZHZIZCJZADAGZDK
      ZUHDUMHZIZULUNUHBAEZBDFZUPUNUGUQBDUGUQLUNABMNOUNDTPUPURLDUMQBDUMRSUAUKUPC
      UMDCAEUJUODUIUMUBUCUDUEUF $.
  $}

  ${
    $d A w y x $.  $d A w y $.  $d A w z $.
    $( A nonempty set is either a singleton or contains at least two different
       elements.  (Contributed by AV, 20-Sep-2020.) $)
    n0snor2el $p |- ( A =/= (/)
                      -> ( E. x e. A E. y e. A x =/= y \/ E. z A = { z } ) ) $=
      ( vw weq wral wrex c0 wne cv csn wceq wex wo wi issn olcd wn bitri rexbii
      a1d df-ne rexnal ralbii ralnex wa neeq1 rexbidv rspccva reximdva0 orcd ex
      sylbir pm2.61i ) EBFZBDGZEDHZDIJZAKZBKZJZBDHZADHZDCKLMCNZOZPZURVFUSURVEVD
      EBCDQRUBURSZEKZVAJZBDHZEDGZVGVLUQSZEDGVHVKVMEDVKUPSZBDHVMVJVNBDVIVAUCUAUP
      BDUDTUEUQEDUFTVLUSVFVLUSUGVDVEVLVCADVKVCEUTDEAFVJVBBDVIUTVAUHUIUJUKULUMUN
      UO $.
  $}

  $( Possible values for a set sandwiched between another set and it plus a
     singleton.  (Contributed by Mario Carneiro, 2-Jul-2016.) $)
  ssunpr $p |- ( ( B C_ A /\ A C_ ( B u. { C , D } ) )
      <-> ( ( A = B \/ A = ( B u. { C } ) ) \/
        ( A = ( B u. { D } ) \/ A = ( B u. { C , D } ) ) ) ) $=
    ( wss cpr cun wa csn wo wceq df-pr uneq2i unass eqtr4i sseq2i anbi2i ssunsn
    ssunsn2 3bitri un23 eqtr2i eqeq2i orbi2i orbi12i ) BAEZABCDFZGZEZHUFABCIZGZ
    DIZGZEZHUFAUKEHZBULGZAEZUNHZJABKAUKKJZAUPKZAUHKZJZJUIUNUFUHUMAUHBUJULGZGUMU
    GVCBCDLMBUJULNOZPQABUKDSUOUSURVBABCRURUQAUPUJGZEZHUTAVEKZJVBUNVFUQUMVEABUJU
    LUAZPQAUPCRVGVAUTVEUHAUHUMVEVDVHUBUCUDTUET $.

  $( The subsets of a pair.  (Contributed by NM, 16-Mar-2006.)  (Proof
     shortened by Mario Carneiro, 2-Jul-2016.) $)
  sspr $p |- ( A C_ { B , C }
        <-> ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) ) $=
    ( cpr wss c0 cun wa wceq csn wo uncom un0 sseq2i 0ss biantrur bitr3i eqeq2i
    eqtri orbi12i ssunpr orbi2i 3bitri ) ABCDZEZFAEZAFUDGZEZHZAFIZAFBJZGZIZKZAF
    CJZGZIZAUGIZKZKUJAUKIZKZAUOIZAUDIZKZKUEUHUIUGUDAUGUDFGUDFUDLUDMSZNUFUHAOPQA
    FBCUAUNVAUSVDUMUTUJULUKAULUKFGUKFUKLUKMSRUBUQVBURVCUPUOAUPUOFGUOFUOLUOMSRUG
    UDAVERTTUC $.

  $( The subsets of an unordered triple.  (Contributed by Mario Carneiro,
     2-Jul-2016.) $)
  sstp $p |- ( A C_ { B , C , D }
        <-> ( ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) \/
          ( ( A = { D } \/ A = { B , D } ) \/
            ( A = { C , D } \/ A = { B , C , D } ) ) ) ) $=
    ( wss cpr csn cun c0 wa wceq wo sseq2i biantrur uncom eqtr4i eqeq2i orbi12i
    df-pr 3bitri ctp df-tp 0ss ssunsn2 sspr bitr3i sseq1i anbi12i ssunpr orbi2i
    un0 eqtri eqtr2i bitri ) ABCDUAZEABCFZDGZHZEZIAEZUSJZAIKABGZKLACGZKAUPKLLZA
    UQKZABDFZKZLZACDFZKZAUOKZLZLZLZUOURABCDUBZMUTUSAUCZNVAUTAUPEZJZIUQHZAEZUSJZ
    LVNAIUPDUDVRVDWAVMVRVQVDUTVQVPNABCUEUFWAUQAEZAUQUPHZEZJVEAUQVBHZKZLZAUQVCHZ
    KZAWCKZLZLVMVTWBUSWDVSUQAVSUQIHUQIUQOUQUKULUGURWCAUPUQOZMUHAUQBCUIWGVHWKVLW
    FVGVEWEVFAWEVBUQHVFUQVBOBDSPQUJWIVJWJVKWHVIAWHVCUQHVIUQVCOCDSPQWCUOAUOURWCV
    OWLUMQRRTRUNT $.

  ${
    tpss.1 $e |- A e. _V $.
    tpss.2 $e |- B e. _V $.
    tpss.3 $e |- C e. _V $.
    $( An unordered triple 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 $.
  $}

  $( An unordered 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 $.

  $( Closed form of ~ sneqr .  (Contributed by Scott Fenton, 1-Apr-2011.)
     (Proof shortened by JJ, 23-Jul-2021.) $)
  sneqrg $p |- ( A e. V -> ( { A } = { B } -> A = B ) ) $=
    ( wcel csn wceq snidg eleq2 syl5ibcom elsng sylibd ) ACDZAEZBEZFZANDZABFLAM
    DOPACGMNAHIABCJK $.

  ${
    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 ) $=
      ( cvv wcel csn wceq wi sneqrg ax-mp ) ADEAFBFGABGHCABDIJ $.

    $( 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 ) $=
      ( csn wss c0 wceq wo sssn snnz neii pm2.21i sneqr jaoi sylbi ) ADZBDZEPFG
      ZPQGZHABGZPBIRTSRTPFACJKLABCMNO $.
  $}

  ${
    $d x y A $.
    $( There exists at most one set whose singleton is equal to a given class.
       See also ~ moeq .  (Contributed by BJ, 24-Sep-2022.) $)
    mosneq $p |- E* x { x } = A $=
      ( vy cv csn wceq wmo wa weq wi wal eqtr3 vex sneqr gen2 sneq eqeq1d mpbir
      syl mo4 ) ADZEZBFZAGUCCDZEZBFZHZACIZJZCKAKUIACUGUBUEFUHUBUEBLUAUDAMNSOUCU
      FACUHUBUEBUAUDPQTR $.
  $}

  $( 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, 21-Jun-1993.) $)
    snsspw $p |- { A } C_ ~P A $=
      ( vx csn cpw cv wceq wss wcel eqimss velsn velpw 3imtr4i ssriv ) BACZADZB
      EZAFPAGPNHPOHPAIBAJBAKLM $.
  $}

  ${
    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.)  (Proof shortened by OpenAI,
       25-Mar-2020.) $)
    prsspw $p |- ( { A , B } C_ ~P C <-> ( A C_ C /\ B C_ C ) ) $=
      ( cvv wcel cpr cpw wss wa wb prsspwg mp2an ) AFGBFGABHCIJACJBCJKLDEABCFFM
      N $.
  $}

  ${
    preq1b.a $e |- ( ph -> A e. V ) $.
    preq1b.b $e |- ( ph -> B e. W ) $.
    $( Biconditional equality lemma for unordered pairs, deduction form.  Two
       unordered pairs have the same second element iff the first elements are
       equal.  (Contributed by AV, 18-Dec-2020.) $)
    preq1b $p |- ( ph -> ( { A , C } = { B , C } <-> A = B ) ) $=
      ( cpr wceq wa wo wcel prid1g syl eleq2 wb elprg sylibd imp eqcom eqeq2 ex
      syl5ibcom syl5ibrcom oplem1 preq1 impbid1 ) ABDIZCDIZJZBCJZAUKULAUKKULBDJ
      ZCBJZCDJZAUKULUMLZAUKBUJMZUPABUIMZUKUQABEMZURGBDENOUIUJBPUDAUSUQUPQGBCDER
      OSTAUKUNUOLZAUKCUIMZUTAVAUKCUJMZACFMZVBHCDFNOUIUJCPUEAVCVAUTQHCBDFROSTBCU
      ABDCUBUFUCBCDUGUH $.

    $( Biconditional equality lemma for unordered pairs, deduction form.  Two
       unordered pairs have the same first element iff the second elements are
       equal.  (Contributed by AV, 18-Dec-2020.) $)
    preq2b $p |- ( ph -> ( { C , A } = { C , B } <-> A = B ) ) $=
      ( cpr wceq prcom eqeq12i preq1b bitrid ) DBIZDCIZJBDIZCDIZJABCJOQPRDBKDCK
      LABCDEFGHMN $.
  $}

  ${
    preqr1.a $e |- A e. _V $.
    preqr1.b $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 cvv wcel wb id a1i preq1b ax-mp biimpi ) ACFBCFGZABGZAHIZPQJDR
      ABCHHRKBHIRELMNO $.

    $( Reverse equality lemma for unordered pairs.  If two unordered pairs have
       the same first element, the second elements are equal.  (Contributed by
       NM, 15-Jul-1993.) $)
    preqr2 $p |- ( { C , A } = { C , B } -> A = B ) $=
      ( cpr wceq prcom eqeq12i preqr1 sylbi ) CAFZCBFZGACFZBCFZGABGLNMOCAHCBHIA
      BCDEJK $.

    preq12b.c $e |- C e. _V $.
    preq12b.d $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 sylbi orim12d preq12 eqtrdi jaoi impbii
      mpd ) ABIZCDIZJZACJZBDJZKZADJZBCJZKZLZUQURVALZVDUQAUPMZVEUQAUOMVFABEUBUOU
      PAUCUDACDEUAUEUQURUTVAVCUQURUSURUQUSURUQCBIZUPJUSURUOVGUPACBNOBDCFHPQRSUQ
      VAVBUQUODCIZJZVAVBUFUPVHUOCDTUGVAVIVBVAVIDBIZVHJVBVAUOVJVHADBNOBCDFGPQRUH
      SUIUNUTUQVCABCDUJVCUOVHUPABDCUJDCTUKULUM $.

    $( An unordered pair has the ordered pair property (compare ~ opth ) under
       certain conditions.  (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 $.
  $}

  $( 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 AV, 29-Jan-2021.)  (Revised by AV,
     18-Sep-2021.) $)
  preqr1g $p |- ( ( A e. V /\ B e. W )
                  -> ( { A , C } = { B , C } -> A = B ) ) $=
    ( wcel wa cpr wceq simpl simpr preq1b biimpd ) ADFZBEFZGZACHBCHIABIPABCDENO
    JNOKLM $.

  ${
    $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 ianor nne orbi12i bitr2i imbitrdi orddi simpll
    preq12bg pm1.4 ad2antll jca sylbi biimtrdi anbi12i pm4.56 necon2ad ) AEIBFI
    JCGIDHIJJZACKZADKZJZBCKZBDKZJZLZABMZCDMZULUTVANZUOOZUROZJZUSOULVBACNZADNZLZ
    BCNZBDNZLZJZVEULVBVFVJJVGVIJLZVLABCDEFGHUCVMVHVFVILZJZVJVGLZVJVILZJZJZVLVFV
    JVGVIUAVSVHVKVHVNVRUBVQVKVOVPVJVIUDUEUFUGUHVHVCVKVDVCUMOZUNOZLVHUMUNPVTVFWA
    VGACQADQRSVDUPOZUQOZLVKUPUQPWBVIWCVJBCQBDQRSUITUOURUJTUK $.

  $( Two pairs are not equal if their counterparts are not equal.  (Contributed
     by AV, 5-Sep-2025.) $)
  prneimg2 $p |- ( ( ( A e. U /\ B e. V ) /\ ( C e. X /\ D e. Y ) )
           -> ( { A , B } =/= { C , D }
                <-> ( ( A =/= C \/ B =/= D ) /\ ( A =/= D \/ B =/= C ) ) ) ) $=
    ( wcel wa cpr wne wceq wo wn preq12bg ianor df-ne orbi12i bitr4i necon3abid
    ioran anbi12i bitri bitrdi ) AEIBFIJCGIDHIJJZABKZCDKZLACMZBDMZJZADMZBCMZJZN
    ZOZACLZBDLZNZADLZBCLZNZJZUFUOUGUHABCDEFGHPUAUPUKOZUNOZJVCUKUNUBVDUSVEVBVDUI
    OZUJOZNUSUIUJQUQVFURVGACRBDRSTVEULOZUMOZNVBULUMQUTVHVAVIADRBCRSTUCUDUE $.

  $( A (proper) pair is not equal to another (maybe improper) pair if and only
     if an element of the first pair is not contained in the second pair.
     (Contributed by Alexander van der Vekens, 16-Jan-2018.) $)
  prnebg $p |- ( ( ( A e. U /\ B e. V ) /\ ( C e. X /\ D e. Y ) /\ A =/= B )
                  -> ( ( ( A =/= C /\ A =/= D ) \/ ( B =/= C /\ B =/= D ) )
                       <-> { A , B } =/= { C , D } ) ) $=
    ( wcel wa wne wo cpr wi wn wceq ianor nne bitri jaoi prneimg ioran eqneqall
    w3a 3adant3 orbi12i anbi12i anddi eqtr3 syl preq12 a1d prcom com12 3ad2ant3
    eqtrdi biimtrid necon1ad impbid ) AEIBFIJZCGIDHIJZABKZUDZACKZADKZJZBCKZBDKZ
    JZLZABMZCDMZKZUTVAVJVMNVBABCDEFGHUAUEVCVJVKVLVJOZACPZADPZLZBCPZBDPZLZJZVCVK
    VLPZVNVFOZVIOZJWAVFVIUBWCVQWDVTWCVDOZVEOZLVQVDVEQWEVOWFVPACRADRUFSWDVGOZVHO
    ZLVTVGVHQWGVRWHVSBCRBDRUFSUGSWAVOVRJZVOVSJZLZVPVRJZVPVSJZLZLZVCWBVOVPVRVSUH
    VBUTWOWBNVAWOVBWBWKVBWBNZWNWIWPWJWIABPZWPABCUIWBABUCZUJWJWBVBABCDUKULTWLWPW
    MWLWBVBWLVKDCMVLABDCUKDCUMUPULWMWQWPABDUIWRUJTTUNUOUQUQURUS $.

  $( A (proper) pair is equal to another (maybe improper) pair containing one
     element of the first pair if and only if the other element of the first
     pair is contained in the second pair.  (Contributed by Alexander van der
     Vekens, 26-Jan-2018.) $)
  pr1eqbg $p |- ( ( ( A e. U /\ B e. V /\ C e. X ) /\ A =/= B )
                  -> ( A = C <-> { A , B } = { B , C } ) ) $=
    ( wcel w3a wne wa wceq wo cpr wb eqid biantru orbi2i a1i wn syl neneq biorf
    adantl intnanrd 3simpa 3simpc jca adantr preq12bg 3bitr4d ) ADGZBEGZCFGZHZA
    BIZJZABKZBCKZJZACKZLZUSUTBBKZJZLZUTABMBCMKZVAVDNUPUTVCUSVBUTBOPQRUPUSSUTVAN
    UPUQURUOUQSUNABUAUCUDUSUTUBTUPUKULJZULUMJZJZVEVDNUNVHUOUNVFVGUKULUMUEUKULUM
    UFUGUHABBCDEEFUITUJ $.

  $( A (proper) pair is not equal to another (maybe improper) pair containing
     one element of the first pair if and only if the other element of the
     first pair is not contained in the second pair.  (Contributed by Alexander
     van der Vekens, 26-Jan-2018.) $)
  pr1nebg $p |- ( ( ( A e. U /\ B e. V /\ C e. X ) /\ A =/= B )
                  -> ( A =/= C <-> { A , B } =/= { B , C } ) ) $=
    ( wcel w3a wne wa cpr pr1eqbg necon3bid ) ADGBEGCFGHABIJACABKBCKABCDEFLM $.

  ${
    preqsnd.1 $e |- ( ph -> A e. V ) $.
    preqsnd.2 $e |- ( ph -> B e. W ) $.
    $( Equivalence for a pair equal to a singleton, deduction form.
       (Contributed by Thierry Arnoux, 27-Dec-2016.)  (Revised by AV,
       13-Jun-2022.)  (Revised by AV, 16-Aug-2024.) $)
    preqsnd $p |- ( ph -> ( { A , B } = { C } <-> ( A = C /\ B = C ) ) ) $=
      ( cvv wcel cpr csn wceq wa wb adantl wn c0 wi syl simpl dfsn2 wo preq12bg
      eqeq2i oridm bitrdi bitrid syl22anc snprc biimpi adantr wne prnzg syl5com
      eqeq2d eqneqall sylbid eleq1 eqcoms notbid pm2.24 elex syl11 com13 impcom
      biimtrdi impd impbid pm2.61ian ) DIJZABCKZDLZMZBDMZCDMZNZOZVKANBEJZCFJZVK
      VKVRAVSVKGPAVTVKHPVKAUAZWAVNVLDDKZMZVSVTNVKVKNNZVQVMWBVLDUBUEWDWCVQVQUCVQ
      BCDDEFIIUDVQUFUGUHUIVKQZANZVNVQWFVNVLRMZVQWFVMRVLWEVMRMZAWEWHDUJUKULUPAWG
      VQSZWEAVSWIGVSVLRUMWGVQBCEUNVQVLRUQUOTPURWFVOVPVNAWEVOVPVNSZSZAVSWEWKSGVO
      WEVSWJVOWEBIJZQZVSWJSVOVKWLVKWLODBDBIUSUTVAWLWMWJVSWLWJVBBEVCVDVGVETVFVHV
      IVJ $.
  $}

  $( A proper unordered pair is not a (proper or improper) singleton.
     (Contributed by AV, 13-Jun-2022.) $)
  prnesn $p |- ( ( A e. V /\ B e. W /\ A =/= B ) -> { A , B } =/= { C } ) $=
    ( wcel wne w3a cpr csn wa wn eqtr3 necon3ai 3ad2ant3 simp1 simp2 necon3abid
    wceq preqsnd mpbird ) ADFZBEFZABGZHZABIZCJZGACSBCSKZLZUDUBUIUCUHABABCMNOUEU
    HUFUGUEABCDEUBUCUDPUBUCUDQTRUA $.

  $( A proper unordered pair is not an improper unordered pair.  (Contributed
     by AV, 13-Jun-2022.) $)
  prneprprc $p |- ( ( ( A e. V /\ B e. W /\ A =/= B ) /\ -. C e. _V )
                    -> { A , B } =/= { C , D } ) $=
    ( wcel wne w3a cvv wn wa cpr csn prnesn adantr wb prprc1 neeq2d adantl
    mpbird ) AEGBFGABHIZCJGKZLABMZCDMZHZUDDNZHZUBUHUCABDEFOPUCUFUHQUBUCUEUGUDCD
    RSTUA $.

  ${
    preqsn.1 $e |- A e. _V $.
    preqsn.2 $e |- B e. _V $.
    $( Equivalence for a pair equal to a singleton.  (Contributed by NM,
       3-Jun-2008.)  (Revised by AV, 12-Jun-2022.) $)
    preqsn $p |- ( { A , B } = { C } <-> ( A = B /\ B = C ) ) $=
      ( cpr csn wceq wa cvv wcel wb id a1i preqsnd ax-mp eqeq2 pm5.32ri bitr4i
      ) ABFCGHZACHZBCHZIZABHZUBIAJKZTUCLDUEABCJJUEMBJKUEENOPUBUDUABCAQRS $.
  $}

  $( Equality relationship for two proper unordered pairs.  (Contributed by AV,
     12-Jun-2022.) $)
  preq12nebg $p |- ( ( A e. V /\ B e. W /\ A =/= B )
                   -> ( { A , B } = { C , D }
                        <-> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) ) ) $=
    ( cvv wcel wa wne cpr wceq wo wi ancoms ex wn prneprprc eqneqall syl5com wb
    w3a 3simpa anim1i preq12bg syl ianor prcom eqeq2i jaoian preq12 eqtrdi jaoi
    sylbi impbid1 pm2.61i ) CGHZDGHZIZAEHZBFHZABJZUBZABKZCDKZLZACLBDLIZADLBCLIZ
    MZUAZNZUSVCVJUSVCIUTVAIZUSIZVJVCUSVMVCVLUSUTVAVBUCUDOABCDEFGGUEUFPUSQUQQZUR
    QZMZVKUQURUGVPVCVJVPVCIVFVIVNVCVFVINVOVNVCIVDVEJZVFVIVCVNVQABCDEFROVIVDVEST
    VOVCIVDDCKZJZVFVIVCVOVSABDCEFROVFVDVRLVSVINVEVRVDCDUHUIVIVDVRSUNTUJVGVFVHAB
    CDUKVHVDVRVEABDCUKDCUHULUMUOPUNUP $.

  $( Equality of two unordered pairs.  (Contributed by NM, 17-Oct-1996.)
     (Revised by AV, 9-Dec-2018.)  (Revised by AV, 12-Jun-2022.) $)
  prel12g $p |- ( ( A e. V /\ B e. W /\ A =/= B )
                  -> ( { A , B } = { C , D }
                       <-> ( A e. { C , D } /\ B e. { C , D } ) ) ) $=
    ( wcel cpr wceq wa wo prid1g 3ad2ant1 preq1 prid2g 3ad2ant2 preq2 syl5ibcom
    eleq2d anim12d wne w3a preq12nebg adantr adantl eleqtrd ex wb elprg anbi12d
    jaod wi eqtr3 eqneqall syl olc a1d ccase com12 3ad2ant3 sylbid impbid bitrd
    orc ) AEGZBFGZABUAZUBZABHCDHZIACIZBDIZJZADIZBCIZJZKZAVIGZBVIGZJZABCDEFUCVHV
    PVSVHVLVSVOVHVJVQVKVRVHVJVQVHVJJAADHZVIVHAVTGZVJVEVFWAVGADELMUDVJVTVIIVHACD
    NUEUFUGVHBCBHZGZVKVRVFVEWCVGCBFOPVKWBVIBBDCQSRTVHVMVQVNVRVHACAHZGZVMVQVEVFW
    EVGCAEOMVMWDVIAADCQSRVHBBDHZGZVNVRVFVEWGVGBDFLPVNWFVIBBCDNSRTUKVHVSVJVMKZVN
    VKKZJZVPVHVQWHVRWIVEVFVQWHUHVGACDEUIMVFVEVRWIUHVGBCDFUIPUJVGVEWJVPULVFWJVGV
    PVJVNVMVKVGVPULZVJVNJABIZWKABCUMVPABUNZUOVOVPVGVOVLUPUQVLVPVGVLVOVDUQVMVKJW
    LWKABDUMWMUOURUSUTVAVBVC $.

  $( An unordered pair has the ordered pair property (compare ~ opth ) under
     certain conditions.  Variant of ~ opthpr in closed form.  (Contributed by
     AV, 13-Jun-2022.) $)
  opthprneg $p |- ( ( ( A e. V /\ B e. W ) /\ ( A =/= B /\ A =/= D ) )
                    -> ( { A , B } = { C , D } <-> ( A = C /\ B = D ) ) ) $=
    ( cvv wcel wa wne cpr wceq wb wi wo wn sylbi impbid1 prneprprc sylan pm2.21
    preq12bg adantlr idd df-ne impd jaod orc adantl ad2antlr bitrd expcom ianor
    w3a simpl anim2i df-3an sylibr ancoms eqneqall syl5com eqeq2i jaoian preq12
    prcom ex pm2.61i ) CGHZDGHZIZAEHZBFHZIZABJZADJZIZIZABKZCDKZLZACLBDLIZMZNZVQ
    VJWBVQVJIVTWAADLZBCLZIZOZWAVMVJVTWGMVPABCDEFGGUBUCVPWGWAMZVMVJVOWHVNVOWGWAV
    OWAWAWFVOWAUDVOWDWEWAVOWDPWDWEWANZNADUEWDWIUAQUFUGWAWFUHRUIUJUKULVJPVHPZVIP
    ZOZWCVHVIUMWLVQWBWLVQIVTWAWJVQVTWANWKWJVQIVRVSJZVTWAVQWJWMVQVKVLVNUNZWJWMVQ
    VMVNIWNVPVNVMVNVOUOUPVKVLVNUQURZABCDEFSTUSWAVRVSUTVAWKVQIVRDCKZJZVTWAVQWKWQ
    VQWNWKWQWOABDCEFSTUSVTVRWPLWQWANVSWPVRCDVEVBWAVRWPUTQVAVCABCDVDRVFQVG $.

  ${
    $d A x $.  $d B x $.  $d C x $.
    $( Lemma for ~ elpreqpr .  (Contributed by Scott Fenton, 7-Dec-2020.)
       (Revised by AV, 9-Dec-2020.) $)
    elpreqprlem $p |- ( B e. V -> E. x { B , C } = { B , x } ) $=
      ( cvv wcel cpr cv wceq wex wi eqid preq2 eqeq2d spcegv mpi a1d csn dfsn2
      wn prprc2 eqeq1d exbidv imbitrrid pm2.61i ) CEFZBDFZBCGZBAHZGZIZAJZKUFULU
      GUFUHUHIZULUHLUKUMACEUICIUJUHUHUICBMNOPQUGULUFTZBRZUJIZAJZUGUOBBGZIZUQBSU
      PUSABDUIBIUJURUOUIBBMNOPUNUKUPAUNUHUOUJBCUAUBUCUDUE $.

    $( Equality and membership rule for pairs.  (Contributed by Scott Fenton,
       7-Dec-2020.) $)
    elpreqpr $p |- ( A e. { B , C } -> E. x { B , C } = { A , x } ) $=
      ( cpr wcel wo cvv cv wex wi elpreqprlem eleq1 preq1 eqeq2d exbidv imbi12d
      wceq mpbiri imp elpri elex prcom eqeq1i exbii sylib jaoian syl2anc ) BCDE
      ZFBCRZBDRZGBHFZUIBAIZEZRZAJZBCDUABUIUBUJULUPUKUJULUPUJULUPKZCHFZUICUMEZRZ
      AJZKACDHLUJULURUPVABCHMUJUOUTAUJUNUSUIBCUMNOPQSTUKULUPUKUQDHFZUIDUMEZRZAJ
      ZKVBDCEZVCRZAJVEADCHLVGVDAVFUIVCDCUCUDUEUFUKULVBUPVEBDHMUKUOVDAUKUNVCUIBD
      UMNOPQSTUGUH $.

    $d V x $.
    $( A set is an element of an unordered pair iff there is another (maybe the
       same) set which is an element of the unordered pair.  (Proposed by BJ,
       8-Dec-2020.)  (Contributed by AV, 9-Dec-2020.) $)
    elpreqprb $p |- ( A e. V
                      -> ( A e. { B , C } <-> E. x { B , C } = { A , x } ) ) $=
      ( wcel cpr cv wceq wex elpreqpr prid1g eleq2 syl5ibrcom exlimdv impbid2 )
      BEFZBCDGZFZRBAHZGZIZAJABCDKQUBSAQSUBBUAFBTELRUABMNOP $.
  $}

  ${
    $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 A x $.  $d B x $.
    $( Rewrite ~ df-op using ` if ` .  When both arguments are sets, it reduces
       to the standard Kuratowski definition; otherwise, it is defined to be
       the empty set.  Avoid directly depending on this detail so that theorems
       will not depend on the Kuratowski construction.  (Contributed by Mario
       Carneiro, 26-Apr-2015.)  (Avoid depending on this detail.) $)
    dfopif $p |- <. A , B >. =
      if ( ( A e. _V /\ B e. _V ) , { { A } , { A , B } } , (/) ) $=
      ( vx cop cvv wcel cv csn cpr w3a cab wa c0 df-op df-3an abbii wceq iftrue
      cif ibar eqabdv eqtr2d wn wss pm2.21 adantrd abssdv ss0 syl eqtr4d 3eqtri
      iffalse pm2.61i ) ABDAEFZBEFZCGZAHABIIZFZJZCKUNUOLZURLZCKZUTUQMSZCABNUSVA
      CUNUOUROPUTVBVCQUTVCUQVBUTUQMRUTVACUQUTURTUAUBUTUCZVBMVCVDVBMUDVBMQVDVACM
      VDUTUPMFZURUTVEUEUFUGVBUHUIUTUQMULUJUMUK $.
    $( $j usage 'dfopif' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  $( Value of the ordered pair when the arguments are sets.  (Contributed by
     Mario Carneiro, 26-Apr-2015.)  (Avoid depending on this detail.) $)
  dfopg $p |- ( ( A e. V /\ B e. W ) ->
    <. A , B >. = { { A } , { A , B } } ) $=
    ( wcel cvv cop csn cpr wceq elex wa c0 cif dfopif iftrue eqtrid syl2an ) AC
    EAFEZBFEZABGZAHABIIZJBDEACKBDKSTLZUAUCUBMNUBABOUCUBMPQR $.

  ${
    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.)  (Avoid depending on this detail.) $)
    dfop $p |- <. A , B >. = { { A } , { A , B } } $=
      ( cvv wcel cop csn cpr wceq dfopg mp2an ) AEFBEFABGAHABIIJCDABEEKL $.
  $}

  $( 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 >. ) $=
    ( wceq cvv wcel wa csn cpr cif cop eleq1 anbi1d sneq preq1 preq12d ifbieq1d
    c0 dfopif 3eqtr4g ) ABDZAEFZCEFZGZAHZACIZIZRJBEFZUCGZBHZBCIZIZRJACKBCKUAUDU
    IUGULRUAUBUHUCABELMUAUEUJUFUKABNABCOPQACSBCST $.
  $( $j usage 'opeq1' avoids 'ax-10' 'ax-11' 'ax-12'; $)

  $( 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 >. ) $=
    ( wceq cvv wcel wa csn cpr c0 cif eleq1 anbi2d preq2 preq2d ifbieq1d dfopif
    cop 3eqtr4g ) ABDZCEFZAEFZGZCHZCAIZIZJKUABEFZGZUDCBIZIZJKCARCBRTUCUHUFUJJTU
    BUGUAABELMTUEUIUDABCNOPCAQCBQS $.
  $( $j usage 'opeq2' avoids 'ax-10' 'ax-11' 'ax-12'; $)

  $( 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 $.
  $}

  ${
    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 >. $=
      ( cop cvv wcel wa csn cpr c0 cif dfopif nfel1 nfan nfsn nfpr nfcv nfif
      nfcxfr ) ABCFBGHZCGHZIZBJZBCKZKZLMBCNUDAUGLUBUCAABGDOACGEOPAUEUFABDQABCDE
      RRALSTUA $.
  $}

  ${
    $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 $.
  $}

  $( Distribution of class substitution over ordered pairs.  (Contributed by
     Drahflow, 25-Sep-2015.)  (Revised by Mario Carneiro, 29-Oct-2015.)
     (Revised by ML, 25-Oct-2020.) $)
  csbopg $p |- ( A e. V -> [_ A / x ]_ <. C , D >. =
                         <. [_ A / x ]_ C , [_ A / x ]_ D >. ) $=
    ( wcel cvv wa csn cpr c0 cif csb cop wsbc csbif sbcan sbcel1g csbprg dfopif
    anbi12d bitrid csbsng preq12d csbconstg ifbieq12d eqtrid csbeq2i 3eqtr4g
    eqtrd ) BEFZABCGFZDGFZHZCIZCDJZJZKLZMZABCMZGFZABDMZGFZHZUTIZUTVBJZJZKLZABCD
    NZMUTVBNUKUSUNABOZABUQMZABKMZLVHUNABUQKPUKVJVDVKVLVGKVJULABOZUMABOZHUKVDULU
    MABQUKVMVAVNVCABCGERABDGERUAUBUKVKABUOMZABUPMZJVGAUOUPBESUKVOVEVPVFABCEUCAC
    DBESUDUJABKEUEUFUGABVIURCDTUHUTVBTUI $.

  $( The ordered pair ` <. A , A >. ` in Kuratowski's representation.  Closed
     form of ~ opid .  (Contributed by Peter Mazsa, 22-Jul-2019.)  (Avoid
     depending on this detail.) $)
  opidg $p |- ( A e. V -> <. A , A >. = { { A } } ) $=
    ( wcel cop csn cpr wceq dfopg anidms dfsn2 eqcomi preq2i eqtr4i eqtrdi ) AB
    CZAADZAEZAAFZFZQEZOPSGAABBHISQQFTRQQQRAJKLQJMN $.

  ${
    opid.1 $e |- A e. _V $.
    $( The ordered pair ` <. A , A >. ` in Kuratowski's representation.
       Inference form of ~ opidg .  (Contributed by FL, 28-Dec-2011.)  (Proof
       shortened by AV, 16-Feb-2022.)  (Avoid depending on this detail.) $)
    opid $p |- <. A , A >. = { { A } } $=
      ( cvv wcel cop csn wceq opidg ax-mp ) ACDAAEAFFGBACHI $.
  $}

  ${
    $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
      $.
  $}

  $( 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 >. = (/) ) $=
    ( cvv wcel wa wn cop csn cpr c0 cif dfopif iffalse eqtrid ) ACDBCDEZFABGOAH
    ABIIZJKJABLOPJMN $.

  $( 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 wa cop c0 wceq simpl opprc nsyl5 ) ACDZBCDZELABFGHLMIABJK $.

  $( 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 wa cop c0 wceq simpr opprc nsyl5 ) ACDZBCDZEMABFGHLMIABJK $.

  $( 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 ) ) $=
    ( cop wcel c0 wceq cvv wa n0i opprc nsyl2 ) CABDZEMFGAHEBHEIMCJABKL $.

  ${
    $d A x $.
    $( The power set of a singleton.  (Contributed by NM, 5-Jun-2006.) $)
    pwsn $p |- ~P { A } = { (/) , { A } } $=
      ( vx cv csn wss cab c0 wceq wo cpw cpr sssn abbii df-pw dfpr2 3eqtr4i ) B
      CZADZEZBFQGHQRHIZBFRJGRKSTBQALMBRNBGROP $.
  $}

  ${
    $d A x $.  $d B x $.
    $( The power set of an unordered pair.  (Contributed by NM, 1-May-2009.) $)
    pwpr $p |- ~P { A , B } = ( { (/) , { A } } u. { { B } , { A , B } } ) $=
      ( vx cpr cpw c0 csn cun cv wss wcel wo wceq sspr vex orbi12i bitr4i velpw
      elpr elun 3bitr4i eqriv ) CABDZEZFAGZDZBGZUCDZHZCIZUCJZUJUFKZUJUHKZLZUJUD
      KUJUIKUKUJFMUJUEMLZUJUGMUJUCMLZLUNUJABNULUOUMUPUJFUECOZSUJUGUCUQSPQCUCRUJ
      UFUHTUAUB $.
  $}

  ${
    $d A x $.  $d B x $.  $d C x $.
    $( The power set of an unordered triple.  (Contributed by Mario Carneiro,
       2-Jul-2016.) $)
    pwtp $p |- ~P { A , B , C } =
      ( ( { (/) , { A } } u. { { B } , { A , B } } ) u.
        ( { { C } , { A , C } } u. { { B , C } , { A , B , C } } ) ) $=
      ( vx ctp cpw c0 csn cpr cun cv wcel velpw wo wceq elun elpr orbi12i bitri
      wss vex sstp 3bitr4ri eqriv ) DABCEZFZGAHZIZBHZABIZIZJZCHZACIZIZBCIZUEIZJ
      ZJZDKZUFLUTUETZUTUSLZDUEMUTULLZUTURLZNUTGOUTUGONZUTUIOUTUJONZNZUTUMOUTUNO
      NZUTUPOUTUEONZNZNVBVAVCVGVDVJVCUTUHLZUTUKLZNVGUTUHUKPVKVEVLVFUTGUGDUAZQUT
      UIUJVMQRSVDUTUOLZUTUQLZNVJUTUOUQPVNVHVOVIUTUMUNVMQUTUPUEVMQRSRUTULURPUTAB
      CUBUCSUD $.
  $}

  $( Compute the power set of the power set of the power set of the empty set.
     (See also ~ pw0 and ~ pwpw0 .)  (Contributed by NM, 2-May-2009.) $)
  pwpwpw0 $p |- ~P { (/) , { (/) } } =
                ( { (/) , { (/) } } u. { { { (/) } } , { (/) , { (/) } } } ) $=
    ( c0 csn pwpr ) AABC $.

  $( The power class of the universe is the universe.  Exercise 4.12(d) of
     [Mendelson] p. 235.

     The collection of all classes is of course larger than ` _V ` , which is
     the collection of all sets.  But ` ~P _V ` , being a class, cannot contain
     proper classes, so ` ~P _V ` is actually no larger than ` _V ` .  This
     fact is exploited in ~ ncanth .  (Contributed by NM, 14-Sep-2003.) $)
  pwv $p |- ~P _V = _V $=
    ( vx cvv cpw cv wcel wss ssv velpw mpbir vex 2th eqriv ) ABCZBADZMEZNBEONBF
    NGABHIAJKL $.

  ${
    $d A v $.  $d B v $.  $d C v $.  $d V v $.
    $( For an element of a proper unordered pair of elements of a class ` V ` ,
       there is another (different) element of the class ` V ` which is an
       element of the proper pair.  (Contributed by AV, 18-Dec-2021.) $)
    prproe $p |- ( ( C e. { A , B } /\ A =/= B /\ ( A e. V /\ B e. V ) )
                   -> E. v e. ( V \ { C } ) v e. { A , B } ) $=
      ( wcel wne wa wceq wi eleq1 neeq2 eqcoms biimpcd adantr eldifsnd ad2antll
      wb impcom rspcedvdw cpr cv csn cdif wrex elpri simprrr necom sylbi prid2g
      wo adantl ex simprrl prid1g jaoi syl 3impib ) DBCUAZFZBCGZBEFZCEFZHZAUBZU
      SFZAEDUCUDZUEZUTDBIZDCIZUKVAVDHZVHJZDBCUFVIVLVJVIVKVHVIVKHZVFCUSFZACVGVEC
      USKVMCEDVIVAVBVCUGVKVICDGZVAVIVOJZVDVACBGZVPBCUHVIVQVOVQVORBDBDCLMNUIOSPV
      DVNVIVAVCVNVBBCEUJULQTUMVJVKVHVJVKHZVFBUSFZABVGVEBUSKVRBEDVJVAVBVCUNVKVJB
      DGZVAVJVTJVDVJVAVTVAVTRCDCDBLMNOSPVDVSVJVAVBVSVCBCEUOOQTUMUPUQUR $.
  $}

  $( If there are three elements in a proper unordered pair, and two of them
     are different from the third one, the two must be equal.  (Contributed by
     AV, 19-Dec-2021.) $)
  3elpr2eq $p |- ( ( ( X e. { A , B } /\ Y e. { A , B } /\ Z e. { A , B } )
                     /\ ( Y =/= X /\ Z =/= X ) ) -> Y = Z ) $=
    ( wcel wne wa wceq wo wi elpri eqtr3 eqneqall syl adantld ex a1d impd jaoi
    cpr w3a 2a1d com12 com13 3imp syl3an imp ) CABUAZFZDUIFZEUIFZUBDCGZECGZHZDE
    IZUJCAIZCBIZJZUKDAIZDBIZJZULEAIZEBIZJZUOUPKZCABLDABLEABLUSVBVEVFUQVBVEVFKKU
    RVEVBUQVFVCVBUQVFKZKVDVCVGVBVCUQVFVCUQHZUNUPUMVHECIZUNUPKZECAMUPECNZOPQRVBV
    DVGUTVDVGKVAUTVGVDUTUQVFUTUQHZUMUNUPVLDCIZUMVJKZDCAMVJDCNZOSQRVAVDVGVAVDHUP
    UQUODEBMUCQTUDTUEVEVBURVFVCVBURVFKZKVDVBVCVPUTVCVPKVAUTVCVPUTVCHUPURUODEAMU
    CQVAVPVCVAURVFVAURHZUMUNUPVQVMVNDCBMVOOSQRTUDVDVPVBVDURVFVDURHZUNUPUMVRVIVJ
    ECBMVKOPQRTUETUFUGUH $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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 } ` ( ~ ex-uni ).
       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 y A $.  $d x y B $.
    $( 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 $.  $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 nfrexdw nfabdw nfcxfrd ) ABCGEF
      HZFCIZEJEFCKARBEAELAQBFCAFLDAQBMNOP $.
  $}

  ${
    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 $=
      ( wnfc cuni id nfunid ax-mp ) ABDZABEDCIABIFGH $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( 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 $.
  $}

  ${
    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 $.
  $}

  $( 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.)  (Proof shortened by BJ, 13-Apr-2024.) $)
  unieq $p |- ( A = B -> U. A = U. B ) $=
    ( wceq cuni eqimss unissd eqimss2 eqssd ) ABCZADBDIABABEFIBABAGFH $.

  ${
    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 eleq2w eleq1w abid
      bitrdi anbi12d cbvexv1 bitri ) CABEZFGCDHZGZUDUCGZIZDJCBHZGZAIZBJDCUCKUGU
      JDBUEUFBUEBLABDMNUJDLDBOZUEUIUFADBCPUKUFUHUCGADBUCQABRSTUAUB $.

    $( 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 $.  $d V x $.  $d W x $.
    $( The union of a pair is the union of its members.  Proposition 5.7 of
       [TakeutiZaring] p. 16.  (Contributed by NM, 25-Aug-2006.)  Avoid using
       ~ unipr to prove it from ~ uniprg .  (Revised by BJ, 1-Sep-2024.) $)
    uniprg $p |- ( ( A e. V /\ B e. W ) -> U. { A , B } = ( A u. B ) ) $=
      ( vx vy wcel wa wel cv cpr wex cab wo cuni wceq bitri wb clel3g bicomd
      cun vex elpr anbi2i ancom andir exbii adantr adantl orbi12d bitrid abbidv
      19.43 df-uni df-un 3eqtr4g ) ACGZBDGZHZEFIZFJZABKZGZHZFLZEMEJZAGZVFBGZNZE
      MVBOABUAUSVEVIEVEVAAPZUTHZFLZVABPZUTHZFLZNZUSVIVEVKVNNZFLVPVDVQFVDUTVJVMN
      ZHZVQVCVRUTVAABFUBUCUDVSVRUTHVQUTVRUEVJVMUTUFQQUGVKVNFUMQUSVLVGVOVHUQVLVG
      RURUQVGVLFVFACSTUHURVOVHRUQURVHVOFVFBDSTUIUJUKULEFVBUNEABUOUP $.
  $}

  ${
    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.)  (Proof
       shortened by BJ, 1-Sep-2024.) $)
    unipr $p |- U. { A , B } = ( A u. B ) $=
      ( cvv wcel cpr cuni cun wceq uniprg mp2an ) AEFBEFABGHABIJCDABEEKL $.
  $}

  $( 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 ) $=
    ( wcel csn cuni cpr cun wceq dfsn2 unieqi a1i uniprg anidms unidm 3eqtrd )
    ABCZADZEZAAFZEZAAGZARTHPQSAIJKPTUAHAABBLMUAAHPANKO $.

  ${
    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 $=
      ( cvv wcel csn cuni wceq unisng ax-mp ) ACDAEFAGBACHI $.
  $}

  $( A set equals the union of its singleton (setvar case).  (Contributed by
     NM, 30-Aug-1993.) $)
  unisnv $p |- U. { x } = x $=
    ( cv vex unisn ) ABACD $.

  ${
    $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 x y $.  $d y A $.
    $( An alternative statement of the effective freeness of a class ` A ` ,
       when it is a set.  (Contributed by Mario Carneiro, 14-Oct-2016.)  (Proof
       shortened by JJ, 26-Jul-2021.) $)
    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 df-nfc velsn nfbii
      cuni albii sylbbr nfunid nfa1 unisng sps nfceqdf imbitrid impbid2 ) CDEZA
      FZACGZBHZCIZAJZBFZUKUNBUKAULCUKAULKUKLMNUOACOZSZGUJUKUOAUPAUPGULUPEZAJZBF
      UOABUPPUSUNBURUMABCQRTUAUBUJAUQCUIAUCUIUQCIACDUDUEUFUGUH $.
  $}

  ${
    $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.  See ~ uniinqs for a condition where equality holds.
       (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 A $.  $d x B $.  $d x C $.
    $( Subclass relationship for class union.  (Contributed by NM,
       24-May-1994.)  (Proof shortened by Andrew Salmon, 29-Jun-2011.)  (Proof
       shortened by JJ, 26-Jul-2021.) $)
    ssuni $p |- ( ( A C_ B /\ B e. C ) -> A C_ U. C ) $=
      ( vx wcel wss cuni cv wi elunii expcom imim2d alimdv df-ss 3imtr4g impcom
      wal ) BCEZABFZACGZFZRDHZAEZUBBEZIZDQUCUBTEZIZDQSUARUEUGDRUDUFUCUDRUFUBBCJ
      KLMDABNDATNOP $.
  $}

  ${
    $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 csn wcel wral wceq wss cuni velsn ralbii dfss3 wn wrex neq0
      wex rexcom4 3bitr4ri rexbii eluni2 exbii rexnal 3bitri con4bii ) BDZEFZGZ
      BAHUGEIZBAHZAUHJAKZEIZUIUJBABELMBAUHNUMUKUMOCDZULGZCRZUJOZBAPZUKOCULQUNUG
      GZCRZBAPUSBAPZCRURUPUSBCASUQUTBACUGQUAUOVACBUNAUBUCTUJBAUDUEUFT $.

    $( 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.  (Contributed by NM, 16-Sep-1993.)  Remove use of ~ ax-nul .
     (Revised by Eric Schmidt, 4-Apr-2007.) $)
  uni0 $p |- U. (/) = (/) $=
    ( c0 cuni wceq csn wss 0ss uni0b mpbir ) ABACAADZEIFAGH $.

  ${
    $d A y z $.  $d B y z $.  $d x y z $.
    $( Distribute proper substitution through the union of a class.
       (Contributed by Alan Sare, 10-Nov-2012.)  (Revised by NM,
       22-Aug-2018.) $)
    csbuni $p |- [_ A / x ]_ U. B = U. [_ A / x ]_ B $=
      ( vz vy cvv wcel cuni csb wceq wel cv wa wex wsbc bitrid df-uni c0 csbprc
      cab csbab sbcex2 anbi1d sbcel2 anbi2i bitrdi exbidv abbidv eqtrid csbeq2i
      sbcan sbcg 3eqtr4g wn unieqd uni0 eqtr2di eqtrd pm2.61i ) BFGZABCHZIZABCI
      ZHZJUTABDEKZELZCGZMZENZDTZIZVEVFVCGZMZENZDTZVBVDUTVKVIABOZDTVOVIADBUAUTVP
      VNDVPVHABOZENUTVNVHEABUBUTVQVMEVQVEABOZVGABOZMZUTVMVEVGABUKUTVTVEVSMVMUTV
      RVEVSVEABFULUCVSVLVEABVFCUDUEUFPUGPUHUIABVAVJDECQUJDEVCQUMUTUNZVBRVDABVAS
      WAVDRHRWAVCRABCSUOUPUQURUS $.
  $}

  $( 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 A x y z $.  $d B x y z $.
    $( Relationship involving membership, subset, and union.  Exercise 5 of
       [Enderton] p. 26 and its converse.  (Contributed by NM, 20-Sep-2003.)
       Avoid ~ ax-11 .  (Revised by BTernaryTau, 28-Dec-2024.) $)
    unissb $p |- ( U. A C_ B <-> A. x e. A x C_ B ) $=
      ( vy vz cv cuni wcel wi wal wss wral wel wa albii weq bitri df-ss 3bitr4i
      eleq1w wex eluni imbi1i 19.23v bitr4i elequ1 anbi1d imbi12d elequ2 imbi1d
      anbi12d alcomw 19.21v impexp bi2.04 imbi2i df-ral ) DFZBGZHZURCHZIZDJZAFZ
      BHZVDCKZIZAJZUSCKVFABLVCDAMZVENZVAIZAJZDJZVHVBVLDVBVJAUAZVAIVLUTVNVAAURBU
      BUCVJVAAUDUEOVMVKDJZAJVHVKEAMZVENZEFZCHZIDEMZVRBHZNZVAIDAEEDEPZVJVQVAVSWC
      VIVPVEDEAUFUGDECTUHAEPZVJWBVAWDVIVTVEWAAEDUIAEBTUKUJULVOVGAVEVIVAIZIZDJVE
      WEDJZIVOVGVEWEDUMVKWFDVKVIVEVAIIWFVIVEVAUNVIVEVAUOQOVFWGVEDVDCRUPSOQQDUSC
      RVFABUQS $.
    $( $j usage 'unissb' avoids 'ax-11'; $)
  $}

  ${
    $d x y A $.  $d x y B $.
    $( Obsolete version of ~ unissb as of 28-Dec-2024.  (Contributed by NM,
       20-Sep-2003.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    unissbOLD $p |- ( U. A C_ B <-> A. x e. A x C_ B ) $=
      ( vy cv cuni wcel wi wal wss wa wex eluni imbi1i 19.23v albii bitri df-ss
      wral 3bitr4i bitr4i alcom 19.21v impexp bi2.04 imbi2i df-ral ) DEZBFZGZUH
      CGZHZDIZAEZBGZUNCJZHZAIZUICJUPABSUMUHUNGZUOKZUKHZAIZDIZURULVBDULUTALZUKHV
      BUJVDUKAUHBMNUTUKAOUAPVCVADIZAIURVADAUBVEUQAUOUSUKHZHZDIUOVFDIZHVEUQUOVFD
      UCVAVGDVAUSUOUKHHVGUSUOUKUDUSUOUKUEQPUPVHUODUNCRUFTPQQDUICRUPABUGT $.
  $}

  ${
    $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.  See ~ iunss2 for a generalization to indexed unions.
       (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 $.
  $}

  ${
    $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 velpw sylibr ssriv ) BAACZDZBEZAFNLGNMF
      NAHBLIJK $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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 A x y $.  $d B x 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 cvv wcel cint cv wi wal wb wel wceq eleq1 imbi2d albidv df-int ax-mp
      elab2g ) BFGBCHZGAIZCGZBUBGZJZAKZLDUCEAMZJZAKUFEBUAFEIZBNZUHUEAUJUGUDUCUI
      BUBOPQEACRTS $.
    $( $j usage 'elint' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $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.)  (Proof shortened
       by JJ, 26-Jul-2021.) $)
    elintg $p |- ( A e. V -> ( A e. |^| B <-> A. x e. B A e. x ) ) $=
      ( vy wel wral cv wcel cint wceq eleq1 ralbidv dfint2 elab2g ) EAFZACGBAHZ
      IZACGEBCJDEHZBKPRACSBQLMEACNO $.
  $}

  ${
    $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 nfralw nfab nfcxfr ) ABFDEGZEBHZDIDE
      BJPADOAEBCOAKLMN $.
  $}

  ${
    $d y ph $.  $d x y A $.
    $( Two ways of saying a set is an element of the intersection of a class.
       (Contributed by NM, 30-Aug-1993.)  Put in closed form.  (Revised by RP,
       13-Aug-2020.) $)
    elintabg $p |-
             ( A e. V -> ( A e. |^| { x | ph } <-> A. x ( ph -> A e. x ) ) ) $=
      ( vy wcel cab cint cv wral wi wal elintg eleq2w ralab2 bitrdi ) CDFCABGZH
      FCEIFZEQJACBIFZKBLECQDMARSEBEBCNOP $.
  $}

  ${
    $d A x y $.  $d ph y $.
    elintab.ex $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 ) ) $=
      ( cvv wcel cab cint cv wi wal wb elintabg ax-mp ) CEFCABGHFACBIFJBKLDABCE
      MN $.

    $( Obsolete version of ~ elintab as of 17-Jan-2025.  (Contributed by NM,
       30-Aug-1993.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    elintabOLD $p |- ( A e. |^| { x | ph } <-> A. x ( ph -> A e. x ) ) $=
      ( vy cab cint wcel cv wi wal elint nfsab1 nfv nfim weq eleq1w abid bitrdi
      eleq2w imbi12d cbvalv1 bitri ) CABFZGHEIZUDHZCUEHZJZEKACBIZHZJZBKECUDDLUH
      UKEBUFUGBABEMUGBNOUKENEBPZUFAUGUJULUFUIUDHAEBUDQABRSEBCTUAUBUC $.

    $( 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 $.
  $}

  ${
    $d x y $.
    $( The intersection of the empty set is the universal class.  Exercise 2 of
       [TakeutiZaring] p. 44.  (Contributed by NM, 18-Aug-1993.)  (Proof
       shortened by JJ, 26-Jul-2021.) $)
    int0 $p |- |^| (/) = _V $=
      ( vy vx c0 cint cvv cv wcel wel wral ral0 vex elint2 mpbir 2th eqriv ) AC
      DZEAFZPGZQEGRABHZBCISBJBQCAKZLMTNO $.
  $}

  ${
    $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.)  (Proof
       shortened by OpenAI, 25-Mar-2020.) $)
    intss $p |- ( A C_ B -> |^| B C_ |^| A ) $=
      ( vy vx wss wel wral cab cint ssralv ss2abdv dfint2 3sstr4g ) ABEZCDFZDBG
      ZCHODAGZCHBIAINPQCODABJKCDBLCDALM $.

    $( The intersection of a nonempty set is a subclass of its union.
       (Contributed by NM, 29-Jul-2006.) $)
    intssuni $p |- ( A =/= (/) -> |^| A C_ U. A ) $=
      ( vx vy c0 wne cint cuni cv wcel wral r19.2z ex vex elint2 eluni2 3imtr4g
      wrex ssrdv ) ADEZBAFZAGZSBHZCHIZCAJZUCCAQZUBTIUBUAISUDUEUCCAKLCUBABMNCUBA
      OPR $.
  $}

  ${
    $d x A $.
    $( Subclass of the intersection of a restricted class abstraction.
       (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 $.
  $}

  $( If the union of a class is included in its intersection, the class is
     either the empty set or a singleton ( ~ uniintsn ).  (Contributed by NM,
     30-Oct-2010.)  (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
  unissint $p |- ( U. A C_ |^| A <-> ( A = (/) \/ U. A = |^| A ) ) $=
    ( cuni cint wss c0 wo wn wa simpl wne df-ne intssuni sylbir adantl eqssd ex
    wceq orrd cvv ssv int0 sseqtrri inteq sseqtrrid eqimss jaoi impbii ) ABZACZ
    DZAEQZUHUIQZFUJUKULUJUKGZULUJUMHUHUIUJUMIUMUIUHDZUJUMAEJUNAEKALMNOPRUKUJULU
    KECZUHUIUHSUOUHTUAUBAEUCUDUHUIUEUFUG $.

  $( Subclass relationship for intersection and union.  (Contributed by NM,
     29-Jul-2006.) $)
  intssuni2 $p |- ( ( A C_ B /\ A =/= (/) ) -> |^| A C_ U. B ) $=
    ( c0 wne wss cint cuni intssuni uniss sylan9ssr ) ACDABEAFAGBGAHABIJ $.

  ${
    $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 ) ` .  Typically, ~ abrexex2 or ~ abexssex can be used to
       satisfy the second hypothesis.  (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 19.26 elunant albii vex elint anbi12i
      wa 3bitr4i elin eqriv ) CABEZFZAFZBFZGZDHZUBICHZUGIZJZDKZUHUDIZUHUEIZRZUH
      UCIUHUFIUGAIUIJZUGBIUIJZRZDKUODKZUPDKZRUKUNUOUPDLUJUQDUIABUGMNULURUMUSDUH
      ACOZPDUHBUTPQSDUHUBUTPUHUDUETSUA $.
  $}

  ${
    $d A x y $.  $d B x y $.  $d V x $.  $d W x $.
    $( 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.)  (Proof shortened by BJ, 1-Sep-2024.) $)
    intprg $p |- ( ( A e. V /\ B e. W ) -> |^| { A , B } = ( A i^i B ) ) $=
      ( vx vy wcel wa cv cpr cint cin wb wal wceq wel wi vex elint clel4g bitri
      wo elpr imbi1i jaob albii 19.26 3bitri elin bitr2id bitrid alrimiv dfcleq
      bi2anan9 sylibr ) ACGZBDGZHZEIZABJZKZGZUSABLZGZMZENVAVCOURVEEVBFIZAOZEFPZ
      QZFNZVFBOZVHQZFNZHZURVDVBVFUTGZVHQZFNVIVLHZFNVNFUSUTERSVPVQFVPVGVKUBZVHQV
      QVOVRVHVFABFRUCUDVGVHVKUEUAUFVIVLFUGUHVDUSAGZUSBGZHURVNUSABUIUPVSVJUQVTVM
      FUSACTFUSBDTUNUJUKULEVAVCUMUO $.
  $}

  ${
    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.)  Prove from
       ~ intprg .  (Revised by BJ, 1-Sep-2024.) $)
    intpr $p |- |^| { A , B } = ( A i^i B ) $=
      ( cvv wcel cpr cint cin wceq intprg mp2an ) AEFBEFABGHABIJCDABEEKL $.
  $}

  $( 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 $.
    $( Two ways to express " ` A ` is a singleton".  See also ~ en1 , ~ en1b ,
       ~ card1 , and ~ eusn .  (Contributed by NM, 2-Aug-2010.) $)
    uniintsn $p |- ( U. A = |^| A <-> E. x A = { x } ) $=
      ( vy cuni cint wceq cv wex wa wal c0 wne cvv inteq eqtrdi adantl unieq ex
      wcel wss csn wi vn0 int0 uni0 eqeq1 imbitrid imp eqtr3d necon3d mpi sylib
      n0 cpr vex prss cun cin uniss simpl sseqtrd intss sstrd unipr intpr inss1
      3sstr3g ssun1 sstri eqss uneqin bitr3i sylanblc biimtrid alrimivv jca weu
      cab euabsn eleq1w abid2 eqeq1i exbii 3bitr3i unisnv intsn 3eqtr4a exlimiv
      eu4 impbii ) BDZBEZFZBAGZUAZFZAHZWMWNBSZAHZWRCGZBSZIZWNWTFZUBZCJAJZIZWQWM
      WSXEWMBKLZWSWMMKLXGUCWMBKMKWMBKFZMKFWMXHIWLMKXHWLMFWMXHWLKEMBKNUDOPWMXHWL
      KFZXHWKKFWMXIXHWKKDKBKQUEOWKWLKUFUGUHUIRUJUKABUMULWMXDACXBWNWTUNZBTZWMXCW
      NWTBAUOZCUOZUPWMXKXCWMXKIZWNWTUQZWNWTURZTZXPXOTZXCXNXJDZXJEZXOXPXNXSWLXTX
      NXSWKWLXKXSWKTWMXJBUSPWMXKUTVAXKWLXTTWMXJBVBPVCWNWTXLXMVDWNWTXLXMVEVGXPWN
      XOWNWTVFWNWTVHVIXQXRIXOXPFXCXOXPVJWNWTVKVLVMRVNVOVPWRAVQWRAVRZWOFZAHXFWQW
      RAVSWRXAACACBVTWIYBWPAYABWOABWAWBWCWDULWPWMAWPWODWNWKWLAWEBWOQWPWLWOEWNBW
      ONWNXLWFOWGWHWJ $.

    $( The union and the intersection of a class abstraction are equal exactly
       when there is a unique satisfying value of ` ph ( x ) ` .  (Contributed
       by Mario Carneiro, 24-Dec-2016.) $)
    uniintab $p |- ( E! x ph <-> U. { x | ph } = |^| { x | ph } ) $=
      ( vy weu cab cv csn wceq wex cuni cint euabsn2 uniintsn bitr4i ) 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 a 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 distinct variable group (meaning ` A ` cannot depend on ` x ` ) and
       that ` B ` and ` x ` do not share a distinct 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.  Theorems
       ~ fniunfv and ~ funiunfv are useful when ` B ` is a function.
       (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 A x y $.  $d B y $.  $d C y $.
    $( 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 x A $.  $d x C $.  $d x D $.  $d x E $.
    eliuni.1 $e |- ( x = A -> B = C ) $.
    $( Membership in an indexed union, one way.  (Contributed by JJ,
       27-Jul-2021.) $)
    eliuni $p |- ( ( A e. D /\ E e. C ) -> E e. U_ x e. D B ) $=
      ( wcel wa wrex ciun cv wceq eleq2d rspcev eliun sylibr ) BEHFDHZIFCHZAEJF
      AECKHSRABEALBMCDFGNOAFECPQ $.
  $}

  ${
    $d A x $.
    eliund.1 $e |- ( ph -> E. x e. B A e. C ) $.
    $( Membership in indexed union.  (Contributed by Glauco Siliprandi,
       15-Feb-2025.) $)
    eliund $p |- ( ph -> A e. U_ x e. B C ) $=
      ( wcel wrex ciun eliun sylibr ) ACEGBDHCBDEIGFBCDEJK $.
  $}

  ${
    $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 NM, 5-Sep-2004.)  (Proof shortened by Andrew
       Salmon, 25-Jul-2011.) $)
    iunconst $p |- ( A =/= (/) -> U_ x e. A B = B ) $=
      ( vy c0 wne ciun cv wcel wrex eliun r19.9rzv bitr4id eqrdv ) BEFZDABCGZCO
      DHZPIQCIZABJRAQBCKRABLMN $.

    $( Indexed intersection of a constant class, i.e. where ` B ` does not
       depend on ` x ` .  (Contributed by Mario Carneiro, 6-Feb-2015.) $)
    iinconst $p |- ( A =/= (/) -> |^|_ x e. A B = B ) $=
      ( vy c0 wne ciin cv wcel wral wb cvv eliin elv r19.3rzv bitr4id eqrdv ) B
      EFZDABCGZCRDHZSIZTCIZABJZUBUAUCKDATBCLMNUBABOPQ $.
  $}

  ${
    $d A x y $.  $d B y $.  $d C x y $.  $d X x y $.
    iuneqconst.p $e |- ( x = X -> B = C ) $.
    $( Indexed union of identical classes.  (Contributed by AV, 5-Mar-2024.) $)
    iuneqconst $p |- ( ( X e. A /\ A. x e. A B = C ) -> U_ x e. A B = C ) $=
      ( vy wcel wceq wral wa ciun cv wrex eliun eleq2d rspcev adantlr nfv wi ex
      nfra1 nfan rsp eleq2 biimpd syl6 adantl rexlimd impbid bitr4id eqrdv ) EB
      HZCDIZABJZKZGABCLZDUPGMZUQHURCHZABNZURDHZAURBCOUPVAUTUPVAUTUMVAUTUOUSVAAE
      BAMZEICDURFPQRUAUPUSVAABUMUOAUMASUNABUBUCVAASUOVBBHZUSVATZTUMUOVCUNVDUNAB
      UDUNUSVACDURUEUFUGUHUIUJUKUL $.
  $}

  ${
    $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 wb cvv eliin rexbii eliun 3imtr4i
      elv ralbii ssriv ) FACBDEGZHZBDACEHZGZFIZUCJZACKZUGUEJZBDLZUGUDJUGUFJZUGE
      JZBDLZACKUMACKZBDLUIUKUMABCDMUHUNACUHUNNFBUGDEOPTQUJUOBDAUGCERUASAUGCUCRU
      LUKNFBUGDUEOPTSUB $.
  $}

  ${
    $d A x y $.  $d B y $.
    $( An indexed intersection is a subset of the corresponding indexed union.
       (Contributed by Thierry Arnoux, 31-Dec-2021.) $)
    iinssiun $p |- ( A =/= (/) -> |^|_ x e. A B C_ U_ x e. A B ) $=
      ( vy c0 wne ciin ciun cv wcel wral wrex r19.2z ex cvv eliin eliun 3imtr4g
      wb elv ssrdv ) BEFZDABCGZABCHZUBDIZCJZABKZUFABLZUEUCJZUEUDJUBUGUHUFABMNUI
      UGSDAUEBCOPTAUEBCQRUA $.
  $}

  ${
    $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 intersection.  (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 wb cvv eliin elv 3imtr4g ssrdv ) BCFZEA
      CDGZABDGZREHZDIZACJZUBABJZUASIZUATIZUBABCKUEUCLEAUACDMNOUFUDLEAUABDMNOPQ
      $.

    $( 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 indexed intersection.  (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 $.
    $( $j usage 'iuneq2dv' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)

    $( Equality deduction for indexed intersection.  (Contributed by NM,
       3-Aug-2004.) $)
    iineq2dv $p |- ( ph -> |^|_ x e. A B = |^|_ x e. A C ) $=
      ( wceq wral ciin ralrimiva iineq2 syl ) ADEGZBCHBCDIBCEIGAMBCFJBCDEKL $.
    $( $j usage 'iineq2dv' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    $d x y $.  $d y A $.  $d y B $.  $d y C $.  $d y D $.  $d y ph $.
    iuneq12df.1 $e |- F/ x ph $.
    iuneq12df.2 $e |- F/_ x A $.
    iuneq12df.3 $e |- F/_ x B $.
    iuneq12df.4 $e |- ( ph -> A = B ) $.
    iuneq12df.5 $e |- ( ph -> C = D ) $.
    $( Equality deduction for indexed union, deduction version.  (Contributed
       by Thierry Arnoux, 31-Dec-2016.) $)
    iuneq12df $p |- ( ph -> U_ x e. A C = U_ x e. B D ) $=
      ( vy cv wcel wrex wb wal ciun cab df-iun wceq eleq2d alrimiv abbi 3eqtr4g
      rexeqbid syl ) ALMZENZBCOZUHFNZBDOZPZLQZBCERZBDFRZUAAUMLAUIUKBCDGHIJAEFUH
      KUBUFUCUNUJLSULLSUOUPUJULLUDBLCETBLDFTUEUG $.
  $}

  ${
    $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 $.
      iuneq12dOLD.2 $e |- ( ph -> C = D ) $.
      $( Obsolete version of ~ iuneq12d as of 1-Sep-2025.  (Contributed by
         Drahflow, 22-Oct-2015.)  (Proof modification is discouraged.)
         (New usage is discouraged.) $)
      iuneq12dOLD $p |- ( ph -> U_ x e. A C = U_ x e. B D ) $=
        ( ciun iuneq1d wceq cv wcel adantr iuneq2dv eqtrd ) ABCEIBDEIBDFIABCDEG
        JABDEFAEFKBLDMHNOP $.
    $}
  $}

  ${
    $d x t ph $.  $d A t $.  $d B t $.  $d C t $.
    iuneq12d.1 $e |- ( ph -> A = B ) $.
    iuneq12d.2 $e |- ( ph -> C = D ) $.
    $( Equality deduction for indexed union, deduction version.  (Contributed
       by Drahflow, 22-Oct-2015.)  Remove DV conditions (Revised by GG,
       1-Sep-2025.) $)
    iuneq12d $p |- ( ph -> U_ x e. A C = U_ x e. B D ) $=
      ( vt ciun cv wcel wrex cab eleq2d anbi1d rexbidv2 abbidv df-iun 3eqtr4g
      wceq adantr iuneq2dv eqtrd ) ABCEJZBDEJZBDFJAIKELZBCMZINUGBDMZINUEUFAUHUI
      IAUGUGBCDABKZCLUJDLZUGACDUJGOPQRBICESBIDESTABDEFAEFUAUKHUBUCUD $.
    $( $j usage 'iuneq12d' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    $d x ph $.
    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 z $.  $d z A $.  $d z B $.
    nfiun.1 $e |- F/_ y A $.
    nfiun.2 $e |- F/_ y B $.
    $( Bound-variable hypothesis builder for indexed union.  (Contributed by
       Mario Carneiro, 25-Jan-2014.)  Add disjoint variable condition to avoid
       ~ ax-13 .  See ~ nfiung for a less restrictive version requiring more
       axioms.  (Revised by GG, 20-Jan-2024.) $)
    nfiun $p |- F/_ y U_ x e. A B $=
      ( vz ciun cv wcel wrex cab df-iun nfcri nfrexw nfab nfcxfr ) BACDHGIDJZAC
      KZGLAGCDMSBGRBACEBGDFNOPQ $.
    $( $j usage 'nfiun' avoids 'ax-13'; $)

    $( Bound-variable hypothesis builder for indexed intersection.
       (Contributed by Mario Carneiro, 25-Jan-2014.)  Add disjoint variable
       condition to avoid ~ ax-13 .  See ~ nfiing for a less restrictive
       version requiring more axioms.  (Revised by GG, 20-Jan-2024.) $)
    nfiin $p |- F/_ y |^|_ x e. A B $=
      ( vz ciin cv wcel wral cab df-iin nfcri nfralw nfab nfcxfr ) BACDHGIDJZAC
      KZGLAGCDMSBGRBACEBGDFNOPQ $.
    $( $j usage 'nfiin' avoids 'ax-13'; $)
  $}

  ${
    $d z A $.  $d z B $.  $d x z $.  $d y z $.
    nfiung.1 $e |- F/_ y A $.
    nfiung.2 $e |- F/_ y B $.
    $( Bound-variable hypothesis builder for indexed union.  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  See ~ nfiun for
       a version with more disjoint variable conditions, but not requiring
       ~ ax-13 .  (Contributed by Mario Carneiro, 25-Jan-2014.)
       (New usage is discouraged.) $)
    nfiung $p |- F/_ y U_ x e. A B $=
      ( vz ciun cv wcel wrex cab df-iun nfcri nfrex nfabg nfcxfr ) BACDHGIDJZAC
      KZGLAGCDMSBGRBACEBGDFNOPQ $.

    $( Bound-variable hypothesis builder for indexed intersection.  Usage of
       this theorem is discouraged because it depends on ~ ax-13 .  See ~ nfiin
       for a version with more disjoint variable conditions, but not requiring
       ~ ax-13 .  (Contributed by Mario Carneiro, 25-Jan-2014.)
       (New usage is discouraged.) $)
    nfiing $p |- F/_ y |^|_ x e. A B $=
      ( vz ciin cv wcel wral cab df-iin nfcri nfral nfabg nfcxfr ) BACDHGIDJZAC
      KZGLAGCDMSBGRBACEBGDFNOPQ $.
  $}

  ${
    $d y A $.  $d y B $.  $d x y $.
    $( Bound-variable hypothesis builder for indexed union.  (Contributed by
       NM, 12-Oct-2003.)  Avoid ~ ax-11 , ~ ax-12 .  (Revised by SN,
       14-May-2025.) $)
    nfiu1 $p |- F/_ x U_ x e. A B $=
      ( vy ciun cv wcel wrex eliun nfre1 nfxfr nfci ) ADABCEZDFZMGNCGZABHAANBCI
      OABJKL $.
    $( $j usage 'nfiu1' avoids 'ax-11' 'ax-12'; $)

    $( Obsolete version of ~ nfiu1 as of 14-May-2025.  (Contributed by NM,
       12-Oct-2003.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    nfiu1OLD $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 A w y z $.  $d B w y z $.  $d C w $.  $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.)  (Proof shortened by Rohan
       Ridenour, 11-Aug-2023.)  Avoid ~ ax-10 , ~ ax-12 .  (Revised by SN,
       11-Dec-2024.) $)
    dfiun2g $p |- ( A. x e. A B e. C ->
                  U_ x e. A B = U. { y | E. x e. A y = B } ) $=
      ( vz vw wcel wral cv wrex cab wceq wa wex wi exbii bitri vex rexbidv ciun
      cuni df-iun wel elisset eleq2 pm5.32ri simplbi2 eximdv syl5com ralimi syl
      rexim rexcom4 r19.42v imbitrdi biimpac reximi sylbir exlimiv impbid1 elab
      eleq1w eluni eqeq1 anbi2i 3bitr4g eqrdv eqtrid ) DEHZACIZACDUAFJZDHZACKZF
      LZBJZDMZACKZBLZUBZAFCDUCVKGVOVTVKGJZDHZACKZGFUDZVLDMZACKZNZFOZWAVOHWAVTHZ
      VKWCWHVKWCWDWENZFOZACKZWHVKWBWKPZACIWCWLPVJWMACVJWEFOWBWKFDEUEWBWEWJFWJWB
      WEWEWDWBVLDWAUFZUGUHUIUJUKWBWKACUMULWLWJACKZFOWHWJAFCUNWOWGFWDWEACUOZQRUP
      WGWCFWGWOWCWPWJWBACWEWDWBWNUQURUSUTVAVNWCFWAGSVLWAMVMWBACFGDVCTVBWIWDVLVS
      HZNZFOWHFWAVSVDWRWGFWQWFWDVRWFBVLFSVPVLMVQWEACVPVLDVETVBVFQRVGVHVI $.
    $( $j usage 'dfiun2g' avoids 'ax-10' 'ax-12'; $)
  $}

  ${
    $d y z w A $.  $d y z w B $.  $d w C z $.  $d w x y z $.
    $( Obsolete version of ~ dfiun2g as of 11-Dec-2024.  (Contributed by NM,
       23-Mar-2006.)  (Proof shortened by Andrew Salmon, 25-Jul-2011.)  (Proof
       shortened by Rohan Ridenour, 11-Aug-2023.)  (New usage is discouraged.)
       (Proof modification is discouraged.) $)
    dfiun2gOLD $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 rspa bitrdi syl
      clel3g rexbida rexcom4 r19.41v exbii exancom bitri eliun eluniab 3bitr4g
      eqrdv ) DEGZACHZFACDIZBJZDKZACLZBMNZUNFJZDGZACLZUTUPGZUROBPZUTUOGUTUSGUNV
      BUQVCOZACLZBPZVDUNVBVEBPZACLVGUNVAVHACUMACQUNAJCGOUMVAVHRUMACSBUTDEUBUAUC
      VEABCUDTVGURVCOZBPVDVFVIBUQVCACUEUFURVCBUGUHTAUTCDUIURBUTUJUKUL $.

    $( 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 eleq1w rexbidv
      elab rexbii abbii 3eqtri ) ADBEFHZHADGIFJZBEKZGLZHCIZUHJZADKZCLUIFJZBEKZA
      DKZCLADUEUHUEUHMAIDJBGEFNOPACDUHNUKUNCUJUMADUGUMGUICQGCRUFULBEGCFSTUAUBUC
      UD $.
  $}

  ${
    $d x y z 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.)  Add
       disjoint variable condition to avoid ~ ax-13 .  See ~ cbviung for a less
       restrictive version requiring more axioms.  (Revised by GG,
       20-Jan-2024.) $)
    cbviun $p |- U_ x e. A B = U_ y e. A C $=
      ( vz cv wcel wrex cab ciun nfcri weq eleq2d cbvrexw abbii df-iun 3eqtr4i
      ) IJZDKZACLZIMUBEKZBCLZIMACDNBCENUDUFIUCUEABCBIDFOAIEGOABPDEUBHQRSAICDTBI
      CETUA $.
    $( $j usage 'cbviun' avoids 'ax-13'; $)

    $( Change bound variables in an indexed intersection.  (Contributed by Jeff
       Hankins, 26-Aug-2009.)  (Revised by Mario Carneiro, 14-Oct-2016.)  Add
       disjoint variable condition to avoid ~ ax-13 .  See ~ cbviing for a less
       restrictive version requiring more axioms.  (Revised by GG,
       20-Jan-2024.) $)
    cbviin $p |- |^|_ x e. A B = |^|_ y e. A C $=
      ( vz cv wcel wral cab ciin nfcri weq eleq2d cbvralw abbii df-iin 3eqtr4i
      ) IJZDKZACLZIMUBEKZBCLZIMACDNBCENUDUFIUCUEABCBIDFOAIEGOABPDEUBHQRSAICDTBI
      CETUA $.
    $( $j usage 'cbviin' avoids 'ax-13'; $)
  $}

  ${
    $d z y A $.  $d z x A $.  $d z B $.  $d z C $.
    cbviung.1 $e |- F/_ y B $.
    cbviung.2 $e |- F/_ x C $.
    cbviung.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.  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  See ~ cbviun for
       a version with more disjoint variable conditions, but not requiring
       ~ ax-13 .  (Contributed by NM, 26-Mar-2006.)  (Revised by Andrew Salmon,
       25-Jul-2011.)  (New usage is discouraged.) $)
    cbviung $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.  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  See ~ cbviin for
       a version with more disjoint variable conditions, but not requiring
       ~ ax-13 .  (Contributed by Jeff Hankins, 26-Aug-2009.)  (Revised by
       Mario Carneiro, 14-Oct-2016.)  (New usage is discouraged.) $)
    cbviing $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 y z A $.  $d y z B $.  $d x z 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.)  Add disjoint variable condition to avoid ~ ax-13 .
       See ~ cbviunvg for a less restrictive version requiring more axioms.
       (Revised by GG, 14-Aug-2025.) $)
    cbviunv $p |- U_ x e. A B = U_ y e. A C $=
      ( vz cv wcel wrex cab ciun weq eleq2d cbvrexvw abbii df-iun 3eqtr4i ) GHZ
      DIZACJZGKSEIZBCJZGKACDLBCELUAUCGTUBABCABMDESFNOPAGCDQBGCEQR $.
    $( $j usage 'cbviunv' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)

    $( Change bound variables in an indexed intersection.  (Contributed by Jeff
       Hankins, 26-Aug-2009.)  Add disjoint variable condition to avoid
       ~ ax-13 .  See ~ cbviinvg for a less restrictive version requiring more
       axioms.  (Revised by GG, 14-Aug-2025.) $)
    cbviinv $p |- |^|_ x e. A B = |^|_ y e. A C $=
      ( vz cv wcel wral cab ciin weq eleq2d cbvralvw abbii df-iin 3eqtr4i ) GHZ
      DIZACJZGKSEIZBCJZGKACDLBCELUAUCGTUBABCABMDESFNOPAGCDQBGCEQR $.
    $( $j usage 'cbviinv' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    $d x A $.  $d y A $.  $d y B $.  $d x C $.
    cbviunvg.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.  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  Usage of the
       weaker ~ cbviunv is preferred.  (Contributed by NM, 15-Sep-2003.)
       (New usage is discouraged.) $)
    cbviunvg $p |- U_ x e. A B = U_ y e. A C $=
      ( nfcv cbviung ) ABCDEBDGAEGFH $.

    $( Change bound variables in an indexed intersection.  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  Usage of the
       weaker ~ cbviinv is preferred.  (Contributed by Jeff Hankins,
       26-Aug-2009.)  (New usage is discouraged.) $)
    cbviinvg $p |- |^|_ x e. A B = |^|_ y e. A C $=
      ( nfcv cbviing ) ABCDEBDGAEGFH $.
  $}

  ${
    $d A y $.  $d B y $.  $d C y $.  $d x y $.
    iunssf.1 $e |- F/_ x C $.
    $( Subset theorem for an indexed union.  (Contributed by Glauco Siliprandi,
       3-Mar-2021.) $)
    iunssf $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 df-ss ralbii
      wi ralcom4 nfcri r19.23 albii 3bitrri 3bitri ) ABCGZDHFIZCJZABKZFLZDHUJUH
      DJZTZFMZCDHZABNZUGUKDAFBCOPUJFDQUPUIULTZFMZABNUQABNZFMUNUOURABFCDRSUQAFBU
      AUSUMFUIULABAFDEUBUCUDUEUF $.
  $}

  ${
    $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 ralbii ralcom4
      wi df-ss r19.23v albii 3bitrri 3bitri ) ABCFZDGEHZCIZABJZEKZDGUHUFDIZSZEL
      ZCDGZABMZUEUIDAEBCNOUHEDPUNUGUJSZELZABMUOABMZELULUMUPABECDTQUOAEBRUQUKEUG
      UJABUAUBUCUD $.
  $}

  ${
    $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 r19.37v 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 wb nfcv nfab1 nfiun cleqf abid eliun 3bitr4i
      rexbii mpgbir ) BDACEZFZABDGZCEZHCIZUBJZUEUDJZKCCUBUDBCDUACDLACMNUCCMOUEU
      AJZBDGUCUFUGUHABDACPSBUEDUAQUCCPRT $.

    $( 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 eliin elv ralbii nfcv ralcomf bitri dfss3
      wb 3bitr4i ) FGZABCHZIZFDJZUBCIZFDJZABJZDUCKDCKZABJUEUFABJZFDJUHUDUJFDUDU
      JTFAUBBCLMNOUFFADBEFBPQRFDUCSUIUGABFDCSOUA $.
  $}

  ${
    $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 wb cvv eliin elv ssel reximi r19.36v syl
      wi biimtrid ssrdv ) CDFZABGZEABCHZDEIZUEJZUFCJZABKZUDUFDJZUGUILEAUFBCMNOU
      DUHUJTZABGUIUJTUCUKABCDUFPQUHUJABRSUAUB $.
  $}

  ${
    $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 wb cvv eliin elv rsp com12 biimtrid ssrdv ) AEBFZD
      ABCGZCDEZRFZSCFZABHZQUATUBIDASBCJKLUBQUAUAABMNOP $.
  $}

  ${
    $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.)  (Proof shortened by SN, 15-Jan-2025.) $)
    iunid $p |- U_ x e. A { x } = A $=
      ( vy cv csn ciun wcel wrex cab df-iun weq clel5 velsn rexbii bitr4i eqabi
      eqtr4i ) ABADZEZFCDZSGZABHZCIBACBSJUBCBTBGCAKZABHUBABTLUAUCABCRMNOPQ $.
    $( $j usage 'iunid' avoids 'ax-10' 'ax-11' 'ax-12'; $)

    $( Obsolete version of ~ iunid as of 15-Jan-2025.  (Contributed by NM,
       6-Sep-2005.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    iunidOLD $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 nel0 ) CABDEZCFZPG
      QDGZABHRABRIAFBGQJKLAQBDMNO $.

    $( 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 nel0 ) CADBEZCFZMGNBGZADHOAIAN
      DBJKL $.

    $( 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 eqabi eqtr4i ) ADBE
      CFZBGZADHZCIJACDBKSCJQJGSCLRAMNOP $.

    $( Indexed intersection with a universal index class.  When ` A ` doesn't
       depend on ` x ` , this evaluates to ` A ` by ~ 19.3 and ~ abid2 .  When
       ` A = x ` , this evaluates to ` (/) ` by ~ intiin and ~ intv .
       (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 $.  $d y A $.  $d y B $.
    $( Indexed union of a singleton.  Compare ~ dfiun2 and ~ rnmpt .
       (Contributed by Steven Nguyen, 7-Jun-2023.) $)
    iunsn $p |- U_ x e. A { B } = { y | E. x e. A y = B } $=
      ( csn ciun cv wcel wrex cab wceq df-iun velsn rexbii abbii eqtri ) ACDEZF
      BGZQHZACIZBJRDKZACIZBJABCQLTUBBSUAACBDMNOP $.
  $}

  ${
    $d x y A $.  $d y B $.
    $( There is a nonempty class in an indexed collection ` B ( x ) ` iff the
       indexed union of them is nonempty.  (Contributed by NM, 15-Oct-2003.)
       (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
    iunn0 $p |- ( E. x e. A B =/= (/) <-> U_ x e. A B =/= (/) ) $=
      ( vy cv wcel wex wrex c0 wne rexcom4 eliun exbii bitr4i n0 rexbii 3bitr4i
      ciun ) DEZCFZDGZABHZSABCRZFZDGZCIJZABHUCIJUBTABHZDGUETADBKUDUGDASBCLMNUFU
      AABDCOPDUCOQ $.
  $}

  ${
    $d y A $.  $d x y $.
    $( Indexed intersection of a class abstraction.  (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 nfiin cleqf abid ralbii cvv eliin
      wb elv 3bitr4i mpgbir ) BDACEZFZABDGZCEZHCIZUDJZUGUFJZSCCUDUFBCDUCCDKACLM
      UECLNUGUCJZBDGZUEUHUIUJABDACOPUHUKSCBUGDUCQRTUECOUAUB $.

    $d x A $.  $d x B $.
    $( Indexed intersection of a restricted class abstraction.  (Contributed by
       NM, 6-Dec-2011.) $)
    iinrab $p |- ( A =/= (/)
          -> |^|_ x e. A { y e. B | ph } = { y e. B | A. x e. A ph } ) $=
      ( c0 wne cv wcel wa wral cab crab ciin r19.28zv abbidv df-rab a1i iineq2i
      wceq iinab eqtri 3eqtr4g ) DFGZCHEIZAJZBDKZCLZUEABDKZJZCLBDACEMZNZUICEMUD
      UGUJCUEABDOPULBDUFCLZNUHBDUKUMUKUMTBHDIACEQRSUFBCDUAUBUICEQUC $.

    $d y B $.
    $( Indexed intersection of a restricted class abstraction.  (Contributed by
       NM, 6-Dec-2011.) $)
    iinrab2 $p |- ( |^|_ x e. A { y e. B | ph } i^i B )
                      = { y e. B | A. x e. A ph } $=
      ( crab ciin cin wral wceq c0 cvv iineq1 0iin eqtrdi incom inv1 eqtri rzal
      ineq1d rabid2 ralcom bitr2i sylib wne iinrab wss ssrab2 dfss mpbi eqtr4di
      eqtrd pm2.61ine ) BDACEFZGZEHZABDIZCEFZJDKDKJZUPEURUSUPLEHZEUSUOLEUSUOBKU
      NGLBDKUNMBUNNOTUTELHELEPEQROUSACEIZBDIZEURJZVABDSVCUQCEIVBUQCEUAACBEDUBUC
      UDULDKUEZUPUREHZURVDUOUREABCDEUFTUREUGURVEJUQCEUHUREUIUJUKUM $.
  $}

  ${
    $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 intersection of union.  Generalization of half of theorem
       "Distributive laws" in [Enderton] p. 30.  Use ~ intiin to recover
       Enderton's theorem.  (Contributed by NM, 19-Aug-2004.) $)
    iinun2 $p |- |^|_ x e. A ( B u. C ) = ( B u. |^|_ x e. A C ) $=
      ( vy cun ciin cv wcel wral wo r19.32v elun ralbii wb cvv eliin elv orbi2i
      3bitr4i eqriv ) EABCDFZGZCABDGZFZEHZUBIZABJZUFCIZUFUDIZKZUFUCIZUFUEIUIUFD
      IZKZABJUIUMABJZKUHUKUIUMABLUGUNABUFCDMNUJUOUIUJUOOEAUFBDPQRSTULUHOEAUFBUB
      PQRUFCUDMTUA $.

    $( Indexed union of class difference.  Generalization of half of theorem
       "De Morgan's laws" in [Enderton] p. 31.  Use ~ intiin to recover
       Enderton's theorem.  (Contributed by NM, 19-Aug-2004.) $)
    iundif2 $p |- U_ x e. A ( B \ C ) = ( B \ |^|_ x e. A C ) $=
      ( vy cdif ciun ciin cv wcel wrex wn wa eldif rexbii r19.42v rexnal wb cvv
      wral eliin elv xchbinxr anbi2i 3bitri eliun 3bitr4i eqriv ) EABCDFZGZCABD
      HZFZEIZUIJZABKZUMCJZUMUKJZLZMZUMUJJUMULJUOUPUMDJZLZMZABKUPVAABKZMUSUNVBAB
      UMCDNOUPVAABPVCURUPVCUTABTZUQUTABQUQVDREAUMBDSUAUBUCUDUEAUMBUIUFUMCUKNUGU
      H $.
  $}

  ${
    $d A x y $.  $d B y $.  $d C x y $.
    $( Indexed intersection of class difference with the subtrahend held
       constant.  (Contributed by Thierry Arnoux, 21-Aug-2023.) $)
    iindif1 $p |- ( A =/= (/) -> |^|_ x e. A ( B \ C ) = ( |^|_ x e. A B \ C )
       ) $=
      ( vy c0 wne cdif ciin cv wcel wral wn wa eldif wb cvv eliin elv 3bitr4g
      r19.27zv ralbii anbi1i eqrdv ) BFGZEABCDHZIZABCIZDHZUEEJZUFKZABLZUJUHKZUJ
      DKMZNZUJUGKZUJUIKUEUJCKZUNNZABLUQABLZUNNULUOUQUNABUAUKURABUJCDOUBUMUSUNUM
      USPEAUJBCQRSUCTUPULPEAUJBUFQRSUJUHDOTUD $.
  $}

  ${
    $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.  Generalization of half of
       theorem "De Morgan's laws" in [Enderton] p. 31.  Use ~ uniiun to recover
       Enderton's theorem.  (Contributed by NM, 5-Oct-2006.) $)
    iindif2 $p |- ( A =/= (/) ->
                 |^|_ x e. A ( B \ C ) = ( B \ U_ x e. A C ) ) $=
      ( vy c0 wne cdif ciin ciun cv wcel wral wn wa r19.28zv bicomi ralbii wrex
      eldif ralnex eliun xchbinxr anbi2i 3bitr3g wb cvv eliin elv 3bitr4g eqrdv
      ) BFGZEABCDHZIZCABDJZHZULEKZUMLZABMZUQCLZUQUOLZNZOZUQUNLZUQUPLULUTUQDLZNZ
      OZABMUTVFABMZOUSVCUTVFABPVGURABURVGUQCDTQRVHVBUTVHVEABSVAVEABUAAUQBDUBUCU
      DUEVDUSUFEAUQBUMUGUHUIUQCUOTUJUK $.

    $( Indexed intersection of intersection.  Generalization of half of theorem
       "Distributive laws" in [Enderton] p. 30.  Use ~ intiin to recover
       Enderton's theorem.  (Contributed by Mario Carneiro, 19-Mar-2015.) $)
    iinin2 $p |- ( A =/= (/) ->
      |^|_ x e. A ( B i^i C ) = ( B i^i |^|_ x e. A C ) ) $=
      ( vy c0 wne cin ciin cv wcel wral wa r19.28zv elin cvv eliin elv 3bitr4g
      wb ralbii anbi2i eqrdv ) BFGZEABCDHZIZCABDIZHZUDEJZUEKZABLZUICKZUIUGKZMZU
      IUFKZUIUHKUDULUIDKZMZABLULUPABLZMUKUNULUPABNUJUQABUICDOUAUMURULUMURTEAUIB
      DPQRUBSUOUKTEAUIBUEPQRUICUGOSUC $.

    $( Indexed intersection of intersection.  Generalization of half of theorem
       "Distributive laws" in [Enderton] p. 30.  Use ~ intiin to recover
       Enderton's theorem.  (Contributed by Mario Carneiro, 19-Mar-2015.) $)
    iinin1 $p |- ( A =/= (/) ->
      |^|_ x e. A ( C i^i B ) = ( |^|_ x e. A C i^i B ) ) $=
      ( c0 wne cin ciin iinin2 wceq cv wcel incom a1i iineq2i 3eqtr4g ) BEFABCD
      GZHCABDHZGABDCGZHRCGABCDIABSQSQJAKBLDCMNORCMP $.
  $}

  ${
    $d x A $.
    $( The indexed intersection of a complement.  (Contributed by G&eacute;rard
       Lang, 5-Aug-2018.) $)
    iinvdif $p |- |^|_ x e. A ( _V \ B ) = ( _V \ U_ x e. A B ) $=
      ( cvv cdif ciin ciun wceq c0 dif0 0iun difeq2i 0iin iineq1 iuneq1 difeq2d
      3eqtr4ri 3eqtr4a iindif2 pm2.61ine ) ABDCEZFZDABCGZEZHBIBIHZAIUAFZDAICGZE
      ZUBUDDIEDUHUFDJUGIDACKLAUAMQABIUANUEUCUGDABICOPRABDCST $.
  $}

  ${
    $d A x y $.  $d X x y $.  $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 a nonempty family.  (Contributed by Stefan
       O'Rear, 3-Apr-2015.) $)
    riinn0 $p |- ( ( A. x e. X S C_ A /\ X =/= (/) ) ->
        ( A i^i |^|_ x e. X S ) = |^|_ x e. X S ) $=
      ( wss wral c0 wne wa ciin incom wceq wrex r19.2z ancoms iinss dfss2 sylib
      cin syl eqtrid ) CBEZADFZDGHZIZBADCJZSUFBSZUFBUFKUEUFBEZUGUFLUEUBADMZUHUD
      UCUIUBADNOADCBPTUFBQRUA $.

    $( Relative intersection of a relative abstraction.  (Contributed by Stefan
       O'Rear, 3-Apr-2015.) $)
    riinrab $p |- ( A i^i |^|_ x e. X { y e. A | ph } ) =
        { y e. A | A. x e. X ph } $=
      ( crab ciin cin wral wceq c0 riin0 rzal ralrimivw rabid2 sylibr eqtrd wne
      wss ssrab2 rgenw riinn0 mpan iinrab pm2.61ine ) DBEACDFZGZHZABEIZCDFZJEKE
      KJZUHDUJBDUFELUKUICDIDUJJUKUICDABEMNUICDOPQEKRZUHUGUJUFDSZBEIULUHUGJUMBEA
      CDTUABDUFEUBUCABCEDUDQUE $.
  $}

  $( Symmetric difference with the empty class.  The empty class is the
     identity element for symmetric difference.  (Contributed by Scott Fenton,
     24-Apr-2012.) $)
  symdif0 $p |- ( A /_\ (/) ) = A $=
    ( c0 csymdif cdif cun df-symdif dif0 0dif uneq12i un0 3eqtri ) ABCABDZBADZE
    ABEAABFLAMBAGAHIAJK $.

  $( The symmetric difference with the universal class is the complement.
     (Contributed by Scott Fenton, 24-Apr-2012.) $)
  symdifv $p |- ( A /_\ _V ) = ( _V \ A ) $=
    ( cvv csymdif cdif cun df-symdif c0 wss wceq ssdif0 mpbi uneq1i uncom eqtri
    ssv un0 ) ABCABDZBADZEZRABFSGREZRQGRABHQGIAOABJKLTRGERGRMRPNNN $.

  $( The symmetric difference of a class with itself is the empty class.
     (Contributed by Scott Fenton, 25-Apr-2012.) $)
  symdifid $p |- ( A /_\ A ) = (/) $=
    ( csymdif cdif cun c0 df-symdif difid uneq12i un0 3eqtri ) AABAACZKDEEDEAAF
    KEKEAGZLHEIJ $.

  ${
    $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 eqabcdv 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.)  Avoid ~ ax-13 .  (Revised by GG, 19-May-2023.) $)
    iunxsngf $p |- ( A e. V -> U_ x e. { A } B = C ) $=
      ( vy wcel csn ciun cv wrex eliun nfcri wceq eleq2d rexsngf bitrid eqrdv )
      BEIZHABJZCKZDHLZUCIUDCIZAUBMUAUDDIZAUDUBCNUEUFABEAHDFOALBPCDUDGQRST $.
    $( $j usage 'iunxsngf' avoids 'ax-13'; $)
  $}

  ${
    $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 A x y $.  $d B y $.  $d E y $.
    iunxdif3.1 $e |- F/_ x E $.
    $( An indexed union where some terms are the empty set.  See ~ iunxdif2 .
       (Contributed by Thierry Arnoux, 4-May-2020.) $)
    iunxdif3 $p |- ( A. x e. E B = (/) -> U_ x e. ( A \ E ) B = U_ x e. A B )
      $=
      ( vy c0 wceq wral cin ciun cdif cun wss inss2 wcel wrex eliun ax-mp a1i
      nfcv nfin ssrexf 3imtr4g ssrdv iuneq2 iun0 eqtrdi sseqtrid ss0 syl uneq1d
      cv iunxun inundif nfth nfdif id eqidd iuneq12df eqtr3i uncom un0 3eqtr3rd
      nfun eqtri ) CGHADIZABDJZCKZABDLZCKZMZGVKMZABCKZVKVGVIGVKVGVIGNVIGHVGADCK
      ZVIGVHDNZVIVONBDOVPFVIVOVPFUMZCPZAVHQVRADQVQVIPVQVOPVRAVHDABDABUAZEUBZEUC
      AVQVHCRAVQDCRUDUESVGVOADGKGADCGUFADUGUHUIVIUJUKULVLVNHVGAVHVJMZCKZVLVNAVH
      VJCUNWABHZWBVNHBDUOZWCAWABCCWCAWDUPAVHVJVTABDVSEUQVEVSWCURWCCUSUTSVATVMVK
      HVGVMVKGMVKGVKVBVKVCVFTVD $.
  $}

  ${
    $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.  (Contributed by NM, 25-Nov-2003.)  (Proof shortened
       by Mario Carneiro, 17-Nov-2016.) $)
    iinuni $p |- ( A u. |^| B ) = |^|_ x e. B ( A u. x ) $=
      ( vy cv wcel cint cab cun wral ciin r19.32v elun ralbii vex elint2 orbi2i
      wo 3bitr4ri abbii df-un df-iin 3eqtr4i ) DEZBFZUDCGZFZRZDHUDBAEZIZFZACJZD
      HBUFIACUJKUHULDUEUDUIFZRZACJUEUMACJZRULUHUEUMACLUKUNACUDBUIMNUGUOUEAUDCDO
      PQSTDBUFUAADCUJUBUC $.

    $( A relationship involving union and indexed union.  Exercise 25 of
       [Enderton] p. 33.  (Contributed by NM, 25-Nov-2003.)  (Proof shortened
       by Mario Carneiro, 17-Nov-2016.) $)
    iununi $p |- ( ( B = (/) -> A = (/) ) <->
                ( A u. U. B ) = U_ x e. B ( A u. x ) ) $=
      ( c0 wceq wi cuni cun cv ciun wn wne iunconst sylbir iun0 iuneq2d 3eqtr4a
      df-ne id eqtrdi ja eqcomd uneq1d uniiun uneq2i iunun 3eqtr4g unieq uneq2d
      uni0 un0 iuneq1 0iun eqeq12d biimpcd impbii ) CDEZBDEZFZBCGZHZACBAIZHZJZE
      ZUSBACVBJZHACBJZVFHVAVDUSBVGVFUSVGBUQURVGBEZUQKCDLVHCDRACBMNURACDJDVGBACO
      URACBDURSZPVIQUAUBUCUTVFBACUDUEACBVBUFUGUQVEURUQVABVDDUQVABDHBUQUTDBUQUTD
      GDCDUHUJTUIBUKTUQVDADVCJDACDVCULAVCUMTUNUOUP $.
  $}

  ${
    $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 velpw ralbii dfss3 unissb 3bitr4i ) CDZBEZ
      FZCAGOBHZCAGAPHAIBHQRCACBJKCAPLCABMN $.

    $( 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 eqabi ) ACZDBEZABFFZMOGMHGN
      AIMBJKL $.
  $}

  ${
    $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 ssint velpw ralbii bitr4i wb cvv eliin elv
      wral 3bitr4i eqriv ) CBDZEZABAFZEZGZCFZUAHZUFUDIZABRZUFUBIUFUEIZUGUFUCHZA
      BRUIAUFBJUHUKABCUCKLMCUAKUJUINCAUFBUDOPQST $.

    $( 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 cv cpw ciun cuni wss wrex wcel ssiun eliun velpw rexbii bitri uniiun
      sseq2i 3imtr4i ssriv ) CABADZEZFZBGZEZCDZTHZABIZUEABTFZHZUEUBJZUEUDJZABTU
      EKUJUEUAJZABIUGAUEBUALULUFABCTMNOUKUEUCHUICUCMUCUHUEABPQORS $.
  $}

  $( A nonempty intersection of a family of subsets of a class is included in
     that class.  (Contributed by BJ, 7-Dec-2021.) $)
  intss2 $p |- ( A C_ ~P X -> ( A =/= (/) -> |^| A C_ X ) ) $=
    ( cpw wss cuni c0 wne cint sspwuni biimpi intssuni sstr expcom syl2im ) ABC
    DZAEZBDZAFGAHZPDZRBDZOQABIJAKSQTRPBLMN $.

  $( Relative intersection of a nonempty set.  (Contributed by Stefan O'Rear,
     3-Apr-2015.)  (Revised by Mario Carneiro, 5-Jun-2015.) $)
  rintn0 $p |- ( ( X C_ ~P A /\ X =/= (/) ) -> ( A i^i |^| X ) = |^| X ) $=
    ( cpw wss wne cint cin wceq cuni intssuni2 ssid sspwuni mpbi sstrdi sseqin2
    c0 wa sylib ) BACZDBPEQZBFZADAUAGUAHTUASIZABSJSSDUBADSKSALMNUAAOR $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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 wmo wal wdisj ssel anim1d moimdv alimdv dfdisj2 3imtr4g
      wa ) BCFZAGZCHZEGDHZRZAIZEJTBHZUBRZAIZEJACDKABDKSUDUGESUFUCASUEUAUBBCTLMN
      OAECDPAEBDPQ $.

    $( 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 cbvrmow albii df-disj
      3bitr4i ) IJZDKZACLZIMUBEKZBCLZIMACDNBCENUDUFIUCUEABCBIDFOAIEGOABPDEUBHQR
      SAICDTBICETUA $.
  $}

  ${
    $d x y z A $.  $d y z B $.  $d x z 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 ) $=
      ( vz cv wcel wrmo wal wdisj weq eleq2d cbvrmovw albii df-disj 3bitr4i ) G
      HZDIZACJZGKSEIZBCJZGKACDLBCELUAUCGTUBABCABMDESFNOPAGCDQBGCEQR $.
    $( $j usage 'cbvdisjv' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    $d z A $.  $d z B $.  $d x y z $.
    nfdisjw.1 $e |- F/_ y A $.
    nfdisjw.2 $e |- F/_ y B $.
    $( Bound-variable hypothesis builder for disjoint collection.  Version of
       ~ nfdisj with a disjoint variable condition, which does not require
       ~ ax-13 .  (Contributed by Mario Carneiro, 14-Nov-2016.)  Avoid
       ~ ax-13 .  (Revised by GG, 26-Jan-2024.) $)
    nfdisjw $p |- F/ y Disj_ x e. A B $=
      ( vz wdisj cv wcel wa wmo wal dfdisj2 wnf wtru nftru nfcvd wnfc a1i nfeld
      nfcri nfand nfmodv mptru nfal nfxfr ) ACDHAIZCJZGIDJZKZALZGMBAGCDNULBGULB
      OPUKBAAQPUIUJBPBUHCPBUHRBCSPETUAUJBOPBGDFUBTUCUDUEUFUG $.
    $( $j usage 'nfdisjw' avoids 'ax-13'; $)
  $}

  ${
    $d z A $.  $d z B $.  $d x z $.  $d y z $.
    nfdisj.1 $e |- F/_ y A $.
    nfdisj.2 $e |- F/_ y B $.
    $( Bound-variable hypothesis builder for disjoint collection.  Usage of
       this theorem is discouraged because it depends on ~ ax-13 .  Use the
       weaker ~ nfdisjw when possible.  (Contributed by Mario Carneiro,
       14-Nov-2016.)  (New usage is discouraged.) $)
    nfdisj $p |- F/ y Disj_ x e. A B $=
      ( vz wdisj cv wcel wa wmo wal dfdisj2 wnf wtru nftru weq wn a1i wnfc nfal
      nfcvf nfeld nfcri nfand adantl nfmod2 mptru nfxfr ) ACDHAIZCJZGIDJZKZALZG
      MBAGCDNUOBGUOBOPUNBAAQBARBMSZUNBOPUPULUMBUPBUKCBAUCBCUAUPETUDUMBOUPBGDFUE
      TUFUGUHUIUBUJ $.
  $}

  ${
    $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 $.
    disjor.1 $e |- ( i = j -> B = C ) $.
    $( Two ways to say that a collection ` B ( i ) ` for ` i e. A ` is
       disjoint.  (Contributed by Mario Carneiro, 26-Mar-2015.)  (Revised by
       Mario Carneiro, 14-Nov-2016.) $)
    disjor $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 wo wral wi ralcom4 wex bitri bitr4i ralbii c0
      weq cin wceq df-disj wa orcom df-or neq0 elin imbi1i 19.23v 3bitri eleq2d
      wn exbii rmo4 albii 3bitr4i ) DABHGIZBJZDAKZGLZDEUBZBCUCZUAUDZMZEANZDANZD
      GABUEVAUTCJZUFZVDOZEANZGLZDANVMDANZGLVIVCVMDGAPVHVNDAVHVLGLZEANVNVGVPEAVG
      VFVDMVFUOZVDOZVPVDVFUGVFVDUHVRVKGQZVDOVPVQVSVDVQUTVEJZGQVSGVEUIVTVKGUTBCU
      JUPRUKVKVDGULSUMTVLEGAPRTVBVOGVAVJDEAVDBCUTFUNUQURUSS $.
  $}

  ${
    $d i j x A $.  $d i j B $.
    $( Two ways to say that a collection ` B ( i ) ` for ` i e. A ` is
       disjoint.  (Contributed by Mario Carneiro, 14-Nov-2016.) $)
    disjors $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 wceq c0 wo wral nfcv nfcsb1v csbeq1a cbvdisj csbeq1 disjor
      cin bitri ) ABCFDBADGZCHZFUAEGZIUBAUCCHZSJIKEBLDBLADBCUBDCMAUACNAUACOPBUB
      UDDEAUAUCCQRT $.
  $}

  ${
    $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 wa cin c0 wceq wo cv csb nfcv wdisj df-ne wral disjors eqeq1
      wne csbhypf ineq1d eqeq1d orbi12d eqeq2 ineq2d rspc2v biimtrid impcom ord
      wn 3impia ) ABCUAZFBLGBLMZFGUFZDENZOPZVAFGPZUQUSUTMZVCFGUBVEVDVCUTUSVDVCQ
      ZUSJRZKRZPZAVGCSZAVHCSZNZOPZQZKBUCJBUCUTVFABCJKUDVNVFFVHPZDVKNZOPZQJKFGBB
      VGFPZVIVOVMVQVGFVHUEVRVLVPOVRVJDVKAJFCDAFTADTHUGUHUIUJVHGPZVOVDVQVCVHGFUK
      VSVPVBOVSVKEDAKGCEAGTAETIUGULUIUJUMUNUOUPUNUR $.

    $( Property of a disjoint collection: if ` B ( X ) = C ` and
       ` B ( Y ) = D ` have a common element ` Z ` , then ` X = Y ` .
       (Contributed by Mario Carneiro, 14-Nov-2016.) $)
    disji $p |- ( ( Disj_ x e. A B /\ ( X e. A /\ Y e. A ) /\
      ( Z e. C /\ Z e. D ) ) -> X = Y ) $=
      ( wcel wa wdisj cin c0 wne wceq inelcm disji2 3expia necon1d syl3an3
      3impia ) HDKHEKLABCMZFBKGBKLZDENZOPZFGQZHDERUDUEUGUHUDUELFGUFOUDUEFGPUFOQ
      ABCDEFGIJSTUAUCUB $.
  $}

  ${
    $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 nfra2w wi df-ral rsp imbitrdi imim2i
      eqcom impd alimi sylbi mo2icl syl alrimi dfdisj2 sylibr ) EAFZGZBDHZACHZU
      ICIZBFDIZJZAKZBLACDMULUPBUJABCDNULUOUIEGZOZALZUPULUMUKOZALUSUKACPUTURAUTU
      MUNUQUKUNUQOUMUKUNUJUQUJBDQEUITRSUAUBUCUOAEUDUEUFABCDUGUH $.
  $}

  ${
    $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 $.
    $( $j usage 'invdisjrab' avoids 'ax-13'; $)
  $}

  ${
    $d x y z A $.  $d y z B $.  $d x y z C $.  $d x y z D $.
    $( 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 vz wss cin c0 wceq wa cv wcel wi wrex wral impcom syl biimtrid wdisj
      w3a ciun wn wal wrmo df-disj wne elin eliun anbi12i bitri weq wex nfv an4
      rmo2 ssralv r19.29 id imp eleq1d biimpcd rexlimiv ex expimpd anim12d syl6
      inelcm exlimiv expd sylbi necon2bd impancom 3impa alimdv eq0 sylibr ) ABC
      UAZDBHZEBHZDEIZJKZUBZLFMZADCUCZAECUCZIZNZUDZFUEZWHJKWDVSWKVSWECNZABUFZFUE
      WDWKAFBCUGWDWMWJFVTWAWCWMWJOVTWALZWMWCWJWNWMLZWIWBJWIWLADPZWLAEPZLZWOWBJU
      HZWIWEWFNZWEWGNZLWRWEWFWGUIWTWPXAWQAWEDCUJAWEECUJUKULWMWNWRWSOZWMWLAGUMZO
      ZABQZGUNZWNXBOWLAGBWLGUOUQXFWNWRWSWNWRLVTWPLZWAWQLZLZXFWSVTWAWPWQUPXEXIWS
      OGXEXIGMZDNZXJENZLWSXEXGXKXHXLXEVTWPXKXEVTLXDADQZWPXKOVTXEXMXDADBURRXMWPX
      KXMWPLXDWLLZADPXKXDWLADUSXNXKADXNAMZDNXKXNXOXJDXDWLXCXDUTVAZVBVCVDSVESVFX
      EWAWQXLXEWALXDAEQZWQXLOWAXEXQXDAEBURRXQWQXLXQWQLXNAEPXLXDWLAEUSXNXLAEXNXO
      ENXLXNXOXJEXPVBVCVDSVESVFVGXJDEVIVHVJTVKVLRTVMVNVOVPTRFWHVQVR $.
  $}

  ${
    $d A b x $.  $d B a x $.  $d V a b x $.  $d ph a b x $.
    disjord.1 $e |- ( a = b -> A = B ) $.
    disjord.2 $e |- ( ( ph /\ x e. A /\ x e. B ) -> a = b ) $.
    $( Conditions for a collection of sets ` A ( a ) ` for ` a e. V ` to be
       disjoint.  (Contributed by AV, 9-Jan-2022.) $)
    disjord $p |- ( ph -> Disj_ a e. V A ) $=
      ( weq cin c0 wceq wo wral cv wcel wa wn sylibr wdisj orc a1d 3expia con3d
      wi impancom ralrimiv disj olcd expcom pm2.61i adantr ralrimivva disjor )
      AFGJZCDKLMZNZGEOFEOFECUAAURFGEEAURFPEQGPEQRUPAURUFUPURAUPUQUBUCAUPSZURAUS
      RZUQUPUTBPZDQZSZBCOUQUTVCBCAVACQZUSVCAVDRVBUPAVDVBUPIUDUEUGUHBCDUITUJUKUL
      UMUNECDFGHUOT $.
  $}

  ${
    $d A i j $.  $d B j x $.  $d C j $.  $d E i $.  $d D i x $.  $d i j x $.
    disjiunb.1 $e |- ( i = j -> B = D ) $.
    disjiunb.2 $e |- ( i = j -> C = E ) $.
    $( Two ways to say that a collection of index unions ` C ( i , x ) ` for
       ` i e. A ` and ` x e. B ` is disjoint.  (Contributed by AV,
       9-Jan-2022.) $)
    disjiunb $p |- ( Disj_ i e. A U_ x e. B C <-> A. i e. A A. j e. A
                        ( i = j \/ ( U_ x e. B C i^i U_ x e. D E ) = (/) ) ) $=
      ( ciun weq iuneq12d disjor ) BACDKAEHKFGFGLACEDHIJMN $.
  $}

  ${
    $d A c d x $.  $d C a d x $.  $d D b $.  $d V a c $.  $d W b c d x $.
    $d X a b d x $.  $d ph a b c d x $.
    disjiund.1 $e |- ( a = c -> A = C ) $.
    disjiund.2 $e |- ( b = d -> C = D ) $.
    disjiund.3 $e |- ( a = c -> W = X ) $.
    disjiund.4 $e |- ( ( ph /\ x e. A /\ x e. D ) -> a = c ) $.
    $( Conditions for a collection of index unions of sets ` A ( a , b ) ` for
       ` a e. V ` and ` b e. W ` to be disjoint.  (Contributed by AV,
       9-Jan-2022.) $)
    disjiund $p |- ( ph -> Disj_ a e. V U_ b e. W A ) $=
      ( wral cv wcel wa weq ciun c0 wceq wo wdisj wn wrex eliun eleq2d cbvrexvw
      cin wi 3exp rexlimdvw imp biimtrid con3d impancom ralrimiv disj sylibr ex
      orrd a1d ralrimivv disjiunb ) AIKUAZJGCUBZJHDUBZULUCUDZUEZKFQIFQIFVIUFAVL
      IKFFAVLIRFSKRFSTAVHVKAVHUGZVKAVMTZBRZVJSZUGZBVIQVKVNVQBVIVOVISVOCSZJGUHZV
      NVQJVOGCUIAVSVMVQAVSTZVPVHVPVODSZJHUHZVTVHJVOHDUIWBVOESZLHUHVTVHWAWCJLHJL
      UADEVONUJUKVTWCVHLHAVSWCVHUMZAVRWDJGAVRWCVHPUNUOUPUOUQUQURUSUQUTBVIVJVAVB
      VCVDVEVFJFGCHIKDOMVGVB $.
  $}

  ${
    $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 weq simpr velsn sylib equcomd moimi
      moeq ax-mp mpgbir ) ABADZEZFTBGZCDZUAGZHZAIZCACBUAJACKZAIUFAUCQUEUGAUECAU
      EUDCAKUBUDLCTMNOPRS $.
  $}

  $( 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 $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d y z C $.  $d x y z D $.  $d x y z E $.
    disjprg.1 $e |- ( x = A -> C = D ) $.
    disjprg.2 $e |- ( x = B -> C = E ) $.
    $( A pair collection is disjoint iff the two sets in the family have empty
       intersection.  (Contributed by Mario Carneiro, 14-Nov-2016.) $)
    disjprg $p |- ( ( A e. V /\ B e. V /\ A =/= B ) ->
      ( Disj_ x e. { A , B } C <-> ( D i^i E ) = (/) ) ) $=
      ( vy vz wceq cin c0 wo wral wa wb nfcv wtru wcel wne w3a cv csb cpr wdisj
      eqeq1 csbhypf ineq1d eqeq1d orbi12d ralbidv ralprg 3adant3 id eqcomd orcd
      trud 2thd eqeq2 ineq2d wn simp3 neneqd biorf biantrur bitrdi bitr4d eqcom
      syl tru incom eqtrdi biantru anbi12d bitrd disjors pm4.24 3bitr4g ) BGUAZ
      CGUAZBCUBZUCZJUDZKUDZLZAWEDUEZAWFDUEZMZNLZOZKBCUFZPZJWMPZEFMZNLZWQQZAWMDU
      GWQWDWOBWFLZEWIMZNLZOZKWMPZCWFLZFWIMZNLZOZKWMPZQZWRWAWBWOXIRWCWNXCXHJBCGG
      WEBLZWLXBKWMXJWGWSWKXAWEBWFUHXJWJWTNXJWHEWIAJBDEABSZAESZHUIUJUKULUMWECLZW
      LXGKWMXMWGXDWKXFWECWFUHXMWJXENXMWHFWIAJCDFACSZAFSZIUIUJUKULUMUNUOWDXCWQXH
      WQWDXCTBCLZWQOZQZWQWAWBXCXRRWCXBTXQKBCGGWFBLZXBTXSWSXAXSWFBXSUPUQURXSUSUT
      WFCLZWSXPXAWQWFCBVAXTWTWPNXTWIFEAKCDFXNXOIUIVBUKULUNUOWDWQXQXRWDXPVCWQXQR
      WDBCWAWBWCVDVEXPWQVFVKZTXQVLVGVHVIWDXHXQTQZWQWAWBXHYBRWCXGXQTKBCGGXSXDXPX
      FWQXSXDCBLXPWFBCVACBVJVHXSXEWPNXSXEFEMWPXSWIEFAKBDEXKXLHUIVBFEVMVNUKULXTX
      GTXTXDXFXTWFCXTUPUQURXTUSUTUNUOWDWQXQYBYATXQVLVOVHVIVPVQAWMDJKVRWQVSVT $.
    $( $j usage 'disjprg' avoids 'ax-13'; $)
  $}

  ${
    $d r s u v x y z A $.  $d r s u v x z B $.  $d r s u v y z C $.
    $( An indexed union of a disjoint collection of disjoint collections is
       disjoint if each component is disjoint, and the disjoint unions in the
       collection are also disjoint.  Note that ` B ( y ) ` and ` C ( x ) ` may
       have the displayed free variables.  (Contributed by Mario Carneiro,
       14-Nov-2016.)  (Proof shortened by JJ, 27-May-2021.) $)
    disjxiun $p |- ( Disj_ y e. A B -> ( Disj_ x e. U_ y e. A B C <->
      ( A. y e. A Disj_ x e. B C /\ Disj_ y e. A U_ x e. B C ) ) ) $=
      ( vu vv vr vs vz wdisj ciun wral wa nfcv wss cv wcel csb weq nfiu1 ssiun2
      wi nfdisjw disjss1 syl11 ralrimi a1i c0 wceq wo wn simplr nfcsb1v csbeq1a
      cin cbviun sseqtrrdi adantr ad2antrl csbeq1 sseq1d vtoclga adantl cbvdisj
      sylbb rsp2 syl imp ord impr adantlr disjiun syl13anc expr orrd ralrimivva
      disjor iuneq1d sylibr nfiun ex jcad wrex eleq2i eliun bitri reeanv bitr4i
      anbi12i simplrr disjeq1d rspc disjors sylib ad2ant2r simplrl eleqtrrd jca
      impcom syl2an2r adantlrr ss2in syl2an biimpi ad3antlr id disji2 syl2an3an
      mpd wne sseq0 pm2.61dane rexlimdvva biimtrid ralrimivv impbid1 ) BCDKZABC
      DLZEKZADEKZBCMZBCADELZKZNZXRXTYBYDXTYBUCXRXTYABCABXSEBCDUABEOZUDDXSPXTYAB
      QCRADXSEUEBCDUBUFUGUHXRXTYDXRXTNZFCABFQZDSZELZKZYDYGFGTZYJABGQZDSZELZUPZU
      IUJZUKZGCMFCMYKYGYRFGCCYGYHCRZYMCRZNZNYLYQYGUUAYLULZYQYGUUAUUBNZNXTYIXSPZ
      YNXSPZYIYNUPUIUJZYQXRXTUUCUMUUAUUDYGUUBYSUUDYTYSYIFCYILZXSFCYIUBBFCDYIFDO
      ZBYHDUNZBYHDUOZUQZURZUSUTUUAUUEYGUUBYTUUEYSUUDUUEFYMCYLYIYNXSBYHYMDVAZVBU
      ULVCVDUTXRUUCUUFXTXRUUAUUBUUFXRUUANYLUUFXRUUAYLUUFUKZXRUUNGCMFCMZUUAUUNUC
      XRFCYIKUUOBFCDYIUUHUUIUUJVECYIYNFGUUMVRVFUUNFGCCVGVHVIVJVKVLAXSEYIYNVMVNV
      OVPVQCYJYOFGYLAYIYNEUUMVSVRVTBFCYCYJFYCOABYIEUUIYFWABFTZADYIEUUJVSVEVTWBW
      CYEHITZAHQZESZAIQZESZUPZUIUJZUKZIXSMHXSMXTYEUVDHIXSXSUURXSRZUUTXSRZNZUURY
      IRZUUTYNRZNZGCWDFCWDZYEUVDUVGUVHFCWDZUVIGCWDZNUVKUVEUVLUVFUVMUVEUURUUGRUV
      LXSUUGUURUUKWEFUURCYIWFWGUVFUUTGCYNLZRUVMXSUVNUUTBGCDYNGDOBYMDUNBYMDUOUQW
      EGUUTCYNWFWGWJUVHUVIFGCCWHWIYEUVJUVDFGCCYEUUANZUVJUVDUVOUVJNZUUQUVCUVOUVJ
      UUQULZUVCUVOUVJUVQNZNZUVCYHYMUVSYLNZUVQUVCUVOUVJUVQYLWKUVTUUQUVCUVOUVJYLU
      VDUVQUVPUVDIYIMHYIMZYLUVHUUTYIRZNZUVDUVOUWAUVJYBYSUWAYDYTYBYSNAYIEKZUWAYS
      YBUWDYAUWDBYHCABYIEUUIYFUDUUPADYIEUUJWLWMWTAYIEHIWNWOWPUSUVPYLNZUVHUWBUVO
      UVHUVIYLWQUWEUUTYNYIUVOUVHUVIYLWKYLYIYNUJUVPUUMVDWRWSUWAUWCUVDUVDHIYIYIVG
      VIXAXBVJXJUVSUVBYPPZYHYMXKZYQUVCUVJUWFUVOUVQUVHUUSYJPUVAYOPUWFUVIUVHUUSHY
      IUUSLYJHYIUUSUBAHYIEUUSHEOAUUREUNAUUREUOUQURUVIUVAIYNUVALYOIYNUVAUBAIYNEU
      VAIEOAUUTEUNAUUTEUOUQURUUSYJUVAYOXCXDUTUVSJCABJQZDSZELZKZUUAUWGUWGYQYDUWK
      YBUUAUVRYDUWKBJCYCUWJJYCOABUWIEBUWHDUNYFWABJTADUWIEBUWHDUOVSVEXEXFYEUUAUV
      RUMUWGXGJCUWJYJYOYHYMJFTAUWIYIEBUWHYHDVAVSJGTAUWIYNEBUWHYMDVAVSXHXIUVBYPX
      LXAXMVOVPWBXNXOXPAXSEHIWNVTXQ $.
  $}

  ${
    $d w x y z A $.  $d w x y z B $.  $d w y z C $.  $d w x z D $.
    disjxun.1 $e |- ( x = y -> C = D ) $.
    $( The union of two disjoint collections.  (Contributed by Mario Carneiro,
       14-Nov-2016.) $)
    disjxun $p |- ( ( A i^i B ) = (/) -> ( Disj_ x e. ( A u. B ) C <->
      ( Disj_ x e. A C /\ Disj_ x e. B C /\
        A. x e. A A. y e. B ( C i^i D ) = (/) ) ) ) $=
      ( vz vw cin c0 wceq weq cv wo wral wa eqeq1d orbi12d bitri csb wdisj wcel
      cun w3a wn wb disjel eleq1w notbid syl5ibcom con2d biorf bicomd 2ralbidva
      impr syl anbi2d ralunb ralbii nfcv nfcsb1v nfin nfeq1 nfor nfralw equequ2
      nfv csbhypf ineq2d cbvralvw equequ1 csbeq1a ineq1d ralbidv bitr3id r19.26
      cbvralw 3bitr3i disjor anbi1i 3bitr4g equcom bitrdi eqtrdi ralcom disjors
      incom bitrid anbi1d anbi2ci anbi12d df-3an anandir ) CDJKLZHIMZAHNZEUAZAI
      NZEUAZJZKLZOZICDUDZPZHCPZXEHDPZQZACEUBZEFJZKLZBDPACPZQZADEUBZXLQZQZAXDEUB
      ZXIXNXLUEZWOXFXMXGXOWOABMZXKOZBCPZACPZXTBDPZACPZQZYBXLQXFXMWOYDXLYBWOXTXK
      ABCDWOANZCUCZBNZDUCZQQZXKXTYJXSUFZXKXTUGWOYGYIYKWOYGQZXSYIYLYFDUCZUFXSYIU
      FCDYFUHXSYMYIABDUIUJUKULUPXSXKUMUQUNUOZURXTBXDPZACPYAYCQZACPXFYEYOYPACXTB
      CDUSUTYOXEAHCYOHVHXCAIXDAXDVAWPXBAWPAVHAXAKAWRWTAWQEVBAWSEVBVCVDVEZVFYOAI
      MZEWTJZKLZOZIXDPAHMZXEUUAXTIBXDIBMZYRXSYTXKIBAVGUUCYSXJKUUCWTFEAIYHEFAYHV
      AZAFVAZGVIVJRSVKUUBUUAXCIXDUUBYRWPYTXBAHIVLUUBYSXAKUUBEWRWTAWQEVMVNRSVOVP
      VRYAYCACVQVSXIYBXLCEFABGVTWAWBWOXCICPZHDPZXCIDPZHDPZQZXLUUIQXGXOWOUUGXLUU
      IUUGYDWOXLUUGXTACPZBDPYDUUFUUKHBDUUFHAMZWREJZKLZOZACPHBMZUUKUUOXCAICUUOIV
      HYQYRUULWPUUNXBAIHVGYRUUMXAKYREWTWRAWSEVMVJRSVRUUPUUOXTACUUPUULXSUUNXKUUP
      UULBAMXSHBAVLBAWCWDUUPUUMXJKUUPUUMFEJXJUUPWRFEAHYHEFUUDUUEGVIVNFEWHWERSVO
      VPVKXTBADCWFTYNWIWJXGUUFUUHQZHDPUUJXEUUQHDXCICDUSUTUUFUUHHDVQTXNUUIXLADEH
      IWGWKWBWLXQXEHXDPXHAXDEHIWGXEHCDUSTXRXIXNQXLQXPXIXNXLWMXIXNXLWNTWB $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d y C $.
    $( Expand a disjoint collection with any number of empty sets.
       (Contributed by Mario Carneiro, 15-Nov-2016.) $)
    disjss3 $p |- ( ( A C_ B /\ A. x e. ( B \ A ) C = (/) ) ->
      ( Disj_ x e. A C <-> Disj_ x e. B C ) ) $=
      ( vy wss c0 wceq wa wdisj wi cv wcel wmo wal wn syl simpl adantl dfdisj2
      cdif wral df-ral simprr n0i eldif biimtrrid mpand mt3d jca ex alimi sylbi
      moim alimdv 3imtr4g disjss1 adantr impbid ) BCFZDGHZACBUAZUBZIABDJZACDJZV
      CVDVEKUTVCALZBMZELZDMZIZANZEOVFCMZVIIZANZEOVDVEVCVKVNEVCVMVJKZAOZVKVNKVCV
      FVBMZVAKZAOVPVAAVBUCVRVOAVRVMVJVRVMIZVGVIVSVGVAVSVIVAPVRVLVIUDZDVHUEQVSVL
      VGPZVAVMVLVRVLVIRSVLWAIVQVSVAVFCBUFVRVMRUGUHUIVTUJUKULUMVMVJAUNQUOAEBDTAE
      CDTUPSUTVEVDKVCABCDUQURUS $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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.  (For an example of how
     syntax could become ambiguous if we are not careful, see the comment in
     ~ cneg .) $)
  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.  Class ` R ` often denotes a relation
     such as " ` < ` " that compares two classes ` A ` and ` B ` , which might
     be numbers such as ` 1 ` and ` 2 ` (see ~ df-ltxr for the specific
     definition of ` < ` ).  As a wff, relations are true or false.  For
     example, ` ( R = { <. 2 , 6 >. , <. 3 , 9 >. } -> 3 R 9 ) ` ( ~ ex-br ).
     Often class ` R ` meets the ` Rel ` criteria to be defined in ~ df-rel ,
     and in particular ` R ` may be a function (see ~ df-fun ).  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, 1-Aug-1999.) $)
    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, 1-Aug-1999.) $)
    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, 1-Aug-1999.) $)
    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, 1-Aug-1999.) $)
    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 ) $.
    $( A 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 ) $.
    $( A 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 ) $.
    $( A 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 ) $.
    $( A 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 ) $=
      ( wss wbr wi ssbr ax-mp ) ABFCDAGCDBGHEABCDIJ $.
  $}

  ${
    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 wb cvv breq1 sbcie2g elv df-sbc bitr3i ) AFZCDG
      ZBFZCDGZBRHZRUABIJUBSKAUAEFZCDGSBERLTUCCDMUCRCDMNOUABRPQ $.
  $}

  $( 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.
     (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 $.

  $( 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 R $.  $d x y $.
    $( Move substitution in and out of a binary relation.  (Contributed by NM,
       13-Dec-2005.)  (Revised by NM, 22-Aug-2018.) $)
    sbcbr123 $p |- ( [. A / x ]. B R C
           <-> [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C ) $=
      ( vy wbr wsbc cvv wcel csb sbcex wn c0 br0 csbprc csbeq1 breq123d nfcsb1v
      csbeq1a breqd mtbiri con4i wsb cv dfsbcq2 wceq nfbr weq vtoclbg pm5.21nii
      sbiev ) CDEGZABHZBIJZABCKZABDKZABEKZGZUMABLUOUSUOMZUSUPUQNGUPUQOUTURNUPUQ
      ABEPUAUBUCUMAFUDAFUEZCKZAVADKZAVAEKZGZUNUSFBIUMAFBUFVABUGVBUPVCUQVDURAVAB
      CQAVABEQAVABDQRUMVEAFAVBVCVDAVACSAVAESAVADSUHAFUICVBDVCEVDAVACTAVAETAVADT
      RULUJUK $.
  $}

  ${
    $d x B $.  $d x C $.
    $( Move substitution in and out of a binary relation.  (Contributed by NM,
       23-Aug-2018.) $)
    sbcbr $p |- ( [. A / x ]. B R C <-> B [_ A / x ]_ R C ) $=
      ( wbr wsbc csb sbcbr123 cvv wcel wb csbconstg breq12d wn br0 csbprc breqd
      c0 mtbiri 2falsed pm2.61i bitri ) CDEFABGABCHZABDHZABEHZFZCDUFFZABCDEIBJK
      ZUGUHLUIUDCUEDUFABCJMABDJMNUIOZUGUHUJUGUDUESFUDUEPUJUFSUDUEABEQZRTUJUHCDS
      FCDPUJUFSCDUKRTUAUBUC $.
  $}

  ${
    $d x R $.
    $( Move substitution in and out of a binary relation.  (Contributed by NM,
       13-Dec-2005.) $)
    sbcbr12g $p |- ( A e. V ->
                 ( [. A / x ]. B R C <-> [_ A / x ]_ B R [_ A / x ]_ C ) ) $=
      ( wbr wsbc csb wcel sbcbr123 csbconstg breqd bitrid ) CDEGABHABCIZABDIZAB
      EIZGBFJZOPEGABCDEKRQEOPABEFLMN $.
  $}

  ${
    $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. V -> ( [. A / x ]. B R C <-> [_ A / x ]_ B R C ) ) $=
      ( wcel wbr wsbc csb sbcbr12g csbconstg breq2d bitrd ) BFGZCDEHABIABCJZABD
      JZEHPDEHABCDEFKOQDPEABDFLMN $.
  $}

  ${
    $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. V -> ( [. A / x ]. B R C <-> B R [_ A / x ]_ C ) ) $=
      ( wcel wbr wsbc csb sbcbr12g csbconstg breq1d bitrd ) BFGZCDEHABIABCJZABD
      JZEHCQEHABCDEFKOPCQEABCFLMN $.
  $}

  $( Characterization of the symmetric difference of two binary relations.
     (Contributed by Scott Fenton, 11-Apr-2012.) $)
  brsymdif $p |- ( A ( R /_\ S ) B <-> -. ( A R B <-> A S B ) ) $=
    ( csymdif wbr cop wcel wb wn df-br elsymdif bibi12i xchbinxr bitri ) ABCDEZ
    FABGZPHZABCFZABDFZIZJABPKRQCHZQDHZIUAQCDLSUBTUCABCKABDKMNO $.

  ${
    $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)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Extend class notation to include ordered-pair class abstraction (class
     builder). $)
  copab $a class { <. x , y >. | ph } $.

  ${
    $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 does not require it (see ~ dfid2 for a case
       where they are not distinct).  The brace notation is called "class
       abstraction" by Quine; it is also called "class builder" in the
       literature.  An alternate definition using no existential quantifiers is
       shown by ~ dfopab2 .  An example is given by ~ ex-opab .  (Contributed
       by NM, 4-Jul-1994.) $)
    df-opab $a |- { <. x , y >. | ph } =
                                 { z | E. x E. y ( z = <. x , y >. /\ ph ) } $.
  $}

  ${
    $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 ph x z $.  $d ph y z $.  $d ps z $.  $d ch z $.
    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 } ) $=
      ( vz cv cop wceq wa wex cab copab anbi2d 2exbidv abbidv df-opab 3eqtr4g )
      AGHDHEHIJZBKZELDLZGMTCKZELDLZGMBDENCDENAUBUDGAUAUCDEABCTFOPQBDEGRCDEGRS
      $.
    $( $j usage 'opabbidv' avoids 'ax-12'; $)
  $}

  ${
    $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 $.  $d ps w $.
    nfopabd.1 $e |- F/ x ph $.
    nfopabd.2 $e |- F/ y ph $.
    nfopabd.4 $e |- ( ph -> F/ z ps ) $.
    $( Bound-variable hypothesis builder for class abstraction.  Deduction
       form.  (Contributed by Scott Fenton, 26-Oct-2024.) $)
    nfopabd $p |- ( ph -> F/_ z { <. x , y >. | ps } ) $=
      ( vw copab cv cop wceq wa wex cab df-opab nfv nfvd nfexd nfabdw nfcxfrd
      nfand ) AEBCDJIKCKDKLMZBNZDOZCOZIPBCDIQAUGEIAIRAUFECFAUEEDGAUDBEAUDESHUCT
      TUAUB $.
  $}

  ${
    $d x z $.  $d y z $.
    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.)  (Revised by Scott Fenton, 26-Oct-2024.) $)
    nfopab $p |- F/_ z { <. x , y >. | ph } $=
      ( copab wnfc wtru nftru wnf a1i nfopabd mptru ) DABCFGHABCDBICIADJHEKLM
      $.
  $}

  ${
    $d x z $.  $d y z $.  $d z ph $.
    $( The first abstraction variable in an ordered-pair class abstraction 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 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 copab opeq12 anbi12d cbvex2v df-opab
      eqeq2d abbii 3eqtr4i ) LMZCMZDMZNZOZAPZDQCQZLRUIEMZFMZNZOZBPZFQEQZLRACDUA
      BEFUAUOVALUNUTCDEFUMAEUMESGTUMAFUMFSHTUSBCUSCSITUSBDUSDSJTUJUPOUKUQOPZUMU
      SABVBULURUIUJUKUPUQUBUFKUCUDUGACDLUEBEFLUEUH $.
  $}

  ${
    $d x y z w $.  $d z w v ph $.  $d x y v 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.)  Reduce axiom usage.  (Revised by GG, 15-Oct-2024.) $)
    cbvopabv $p |- { <. x , y >. | ph } = { <. z , w >. | ps } $=
      ( vv cv cop wceq wa wex cab copab weq opeq12 eqeq2d anbi12d df-opab abbii
      cbvex2vw 3eqtr4i ) HIZCIZDIZJZKZALZDMCMZHNUDEIZFIZJZKZBLZFMEMZHNACDOBEFOU
      JUPHUIUOCDEFCEPDFPLZUHUNABUQUGUMUDUEUFUKULQRGSUBUAACDHTBEFHTUC $.
    $( $j usage 'cbvopabv' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $d v w x 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.)  Add disjoint variable condition to avoid
       ~ ax-13 .  See ~ cbvopab1g for a less restrictive version requiring more
       axioms.  (Revised by GG, 17-Jan-2024.) $)
    cbvopab1 $p |- { <. x , y >. | ph } = { <. z , y >. | ps } $=
      ( vw vv cv cop wceq wa wex cab copab nfv nfan nfex wsb nfs1v opeq1 eqeq2d
      sbequ12 anbi12d exbidv cbvexv1 nfsbv sbhypf bitri abbii df-opab 3eqtr4i )
      IKZCKZDKZLZMZANZDOZCOZIPUOEKZUQLZMZBNZDOZEOZIPACDQBEDQVBVHIVBUOJKZUQLZMZA
      CJUAZNZDOZJOVHVAVNCJVAJRVMCDVKVLCVKCRACJUBSTUPVIMZUTVMDVOUSVKAVLVOURVJUOU
      PVIUQUCUDACJUEUFUGUHVNVGJEVMEDVKVLEVKERACJEFUISTVGJRVIVCMZVMVFDVPVKVEVLBV
      PVJVDUOVIVCUQUCUDABCJVCGHUJUFUGUHUKULACDIUMBEDIUMUN $.
    $( $j usage 'cbvopab1' avoids 'ax-13'; $)
  $}

  ${
    $d v w x y $.  $d v w y z $.  $d v w ph $.  $d v w ps $.
    cbvopab1g.1 $e |- F/ z ph $.
    cbvopab1g.2 $e |- F/ x ps $.
    cbvopab1g.3 $e |- ( x = z -> ( ph <-> ps ) ) $.
    $( Change first bound variable in an ordered-pair class abstraction, using
       explicit substitution.  Usage of this theorem is discouraged because it
       depends on ~ ax-13 .  See ~ cbvopab1 for a version with more disjoint
       variable conditions, but not requiring ~ ax-13 .  (Contributed by NM,
       6-Oct-2004.)  (Revised by Mario Carneiro, 14-Oct-2016.)
       (New usage is discouraged.) $)
    cbvopab1g $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 cbvexv1 nfsb sbequ sbie bitrdi bitri abbii df-opab
      3eqtr4i ) IKZCKZDKZLZMZANZDOZCOZIPUQEKZUSLZMZBNZDOZEOZIPACDQBEDQVDVJIVDUQ
      JKZUSLZMZACJRZNZDOZJOVJVCVPCJVCJSVOCDVMVNCVMCSACJUATUBURVKMZVBVODVQVAVMAV
      NVQUTVLUQURVKUSUCUDACJUEUFUGUHVPVIJEVOEDVMVNEVMESACJEFUITUBVIJSVKVEMZVOVH
      DVRVMVGVNBVRVLVFUQVKVEUSUCUDVRVNACERBAJECUJABCEGHUKULUFUGUHUMUNACDIUOBEDI
      UOUP $.
  $}

  ${
    $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 cbvexv1
      eqeq2d exbii abbii 3eqtr4i ) IJZCJZDJZKZLZAMZDNZCNZIOUGUHEJZKZLZBMZENZCNZ
      IOACDPBCEPUNUTIUMUSCULURDEUKAEUKEQFRUQBDUQDQGRUIUOLZUKUQABVAUJUPUGUIUOUHS
      UCHUAUBUDUEACDITBCEITUF $.
  $}

  ${
    $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 cbvexv1 abbii 3eqtr4i ) EFZBFZCFZGZHZAIZCJZBJ
      ZEKUGDFZUIGZHZABDLZIZCJZDJZEKABCMURDCMUNVAEUMUTBDUMDNUSBCUQURBUQBNABDOPTU
      HUOHZULUSCVBUKUQAURVBUJUPUGUHUOUIQRABDUAUBUCUDUEABCESURDCESUF $.
  $}

  ${
    $d x y z w $.  $d z w ph $.  $d x w 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.)  Reduce
       axiom usage.  (Revised by GG, 17-Nov-2024.) $)
    cbvopab1v $p |- { <. x , y >. | ph } = { <. z , y >. | ps } $=
      ( vw cv cop wceq wa wex cab copab weq opeq1 eqeq2d anbi12d exbidv df-opab
      cbvexvw abbii 3eqtr4i ) GHZCHZDHZIZJZAKZDLZCLZGMUDEHZUFIZJZBKZDLZELZGMACD
      NBEDNUKUQGUJUPCECEOZUIUODURUHUNABURUGUMUDUEULUFPQFRSUAUBACDGTBEDGTUC $.
    $( $j usage 'cbvopab1v' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    $d x y z $.  $d z ph $.  $d x ps $.
    cbvopab1vOLD.1 $e |- ( x = z -> ( ph <-> ps ) ) $.
    $( Obsolete version of ~ cbvopab1v as of 17-Nov-2024.  (Contributed by NM,
       31-Jul-2003.)  (Proof shortened by Eric Schmidt, 4-Apr-2007.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    cbvopab1vOLD $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 cbvexvw exbii df-opab
      wa abbii 3eqtr4i ) GHZCHZDHZIZJZATZDKZCKZGLUCUDEHZIZJZBTZEKZCKZGLACDMBCEM
      UJUPGUIUOCUHUNDEUEUKJZUGUMABUQUFULUCUEUKUDNOFPQRUAACDGSBCEGSUB $.
  $}

  ${
    $d x z w $.  $d y z w $.  $d ph z w $.  $d ps z w $.
    $( 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 vw cv cop wa wex cab cun wo copab anbi1d 2exbidv 19.43 exbii df-opab
      wceq weq eqeq1 unabw andi bitr2i bitr3i abbii eqtri uneq12i 3eqtr4i ) EGZ
      CGDGHZTZAIZDJCJZEKZUMBIZDJCJZEKZLZFGZULTZABMZIZDJZCJZFKZACDNZBCDNZLVCCDNU
      TVBAIZDJZCJZVBBIZDJZCJZMZFKVGUOURVLVOEFEFUAZUNVJCDVQUMVBAUKVAULUBZOPVQUQV
      MCDVQUMVBBVROPUCVPVFFVPVKVNMZCJVFVKVNCQVSVECVEVJVMMZDJVSVDVTDVBABUDRVJVMD
      QUERUFUGUHVHUPVIUSACDESBCDESUIVCCDFSUJ $.
    $( $j usage 'unopab' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Functions in maps-to notation
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c |-> $.  $( Maps-to symbol $)

  $( 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 y $.  $d y A $.  $d y B $.
    $( Define maps-to notation for defining a function via a rule.  Read as
       "the function which maps ` x ` (in ` A ` ) to ` B ( x ) ` ".  The class
       expression ` B ` is the value of the function at ` x ` and normally
       contains the variable ` x ` .  An example is the square function for
       complex numbers, ` ( x e. CC |-> ( x ^ 2 ) ) ` .  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 A y $.  $d B y $.  $d C y $.  $d D y $.  $d ph y $.  $d x y $.
    mpteq12da.1 $e |- F/ x ph $.
    mpteq12da.2 $e |- ( ph -> A = C ) $.
    mpteq12da.3 $e |- ( ( ph /\ x e. A ) -> B = D ) $.
    $( An equality inference for the maps-to notation.  (Contributed by Glauco
       Siliprandi, 23-Oct-2021.)  Remove dependency on ~ ax-10 .  (Revised by
       SN, 11-Nov-2024.) $)
    mpteq12da $p |- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) ) $=
      ( vy cv wcel wceq wa copab cmpt nfv eqeq2d pm5.32da df-mpt eleq2d opabbid
      anbi1d bitrd 3eqtr4g ) ABKZCLZJKZDMZNZBJOUFELZUHFMZNZBJOBCDPBEFPAUJUMBJGA
      JQAUJUGULNUMAUGUIULAUGNDFUHIRSAUGUKULACEUFHUAUCUDUBBJCDTBJEFTUE $.
  $}

  ${
    mpteq12df.1 $e |- F/ x ph $.
    mpteq12df.2 $e |- ( ph -> A = C ) $.
    mpteq12df.3 $e |- ( ph -> B = D ) $.
    $( An equality inference for the maps-to notation.  Compare ~ mpteq12dv .
       (Contributed by Scott Fenton, 8-Aug-2013.)  (Revised by Mario Carneiro,
       11-Dec-2016.)  (Proof shortened by SN, 11-Nov-2024.) $)
    mpteq12df $p |- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) ) $=
      ( wceq cv wcel adantr mpteq12da ) ABCDEFGHADFJBKCLIMN $.
  $}

  ${
    $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 rspa df-mpt
      eqeq2d pm5.32da sp eleq2d anbi1d sylan9bbr opabbid 3eqtr4g ) BDGZAHZCEGZA
      BIZJZAKZBLZFKZCGZJZAFMUNDLZUPEGZJZAFMABCNADENUMURVAAFUJULAUIAOUKABPQUMFRU
      LURUOUTJUJVAULUOUQUTULUOJCEUPUKABSUAUBUJUOUSUTUJBDUNUIAUCUDUEUFUGAFBCTAFD
      ETUH $.

    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.)  Remove dependency on ~ ax-10 , ~ ax-12 .
         (Revised by SN, 11-Nov-2024.) $)
      mpteq12dva $p |- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) ) $=
        ( vy cv wcel wceq wa copab cmpt eqeq2d pm5.32da eleq2d anbi1d df-mpt
        bitrd opabbidv 3eqtr4g ) ABJZCKZIJZDLZMZBINUDEKZUFFLZMZBINBCDOBEFOAUHUK
        BIAUHUEUJMUKAUEUGUJAUEMDFUFHPQAUEUIUJACEUDGRSUAUBBICDTBIEFTUC $.
      $( $j usage 'mpteq12dva' avoids 'ax-10' 'ax-12'; $)
    $}

    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.)  Remove
       dependency on ~ ax-10 , ~ ax-12 .  (Revised by SN and GG,
       1-Dec-2023.) $)
    mpteq12dv $p |- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) ) $=
      ( wceq cv wcel adantr mpteq12dva ) ABCDEFGADFIBJCKHLM $.
    $( $j usage 'mpteq12dv' avoids 'ax-10' 'ax-12'; $)
  $}

  ${
    $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-5 mpteq12f sylan ) BDFZMAGCEFABHABCIADEIFMAJABCDE
      KL $.
  $}

  ${
    $d x A $.  $d x B $.
    $( An equality theorem for the maps-to notation.  (Contributed by Mario
       Carneiro, 16-Dec-2013.)  (Proof shortened by SN, 11-Nov-2024.) $)
    mpteq1 $p |- ( A = B -> ( x e. A |-> C ) = ( x e. B |-> C ) ) $=
      ( wceq id eqidd mpteq12dv ) BCEZABDCDIFIDGH $.
    $( $j usage 'mpteq1' avoids 'ax-10' 'ax-11' 'ax-12'; $)

    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 $.
  $}

  ${
    mpteq1i.1 $e |- A = B $.
    $( An equality theorem for the maps-to notation.  (Contributed by Glauco
       Siliprandi, 17-Aug-2020.)  Remove all disjoint variable conditions.
       (Revised by SN, 11-Nov-2024.) $)
    mpteq1i $p |- ( x e. A |-> C ) = ( x e. B |-> C ) $=
      ( cmpt wceq wtru a1i eqidd mpteq12dv mptru ) ABDFACDFGHABDCDBCGHEIHDJKL
      $.

    $d A x $.  $d B x $.
    $( Obsolete version of ~ mpteq1i as of 15-Nov-2024.  (Contributed by Glauco
       Siliprandi, 17-Aug-2020.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    mpteq1iOLD $p |- ( x e. A |-> C ) = ( x e. B |-> C ) $=
      ( wceq cmpt mpteq1 ax-mp ) BCFABDGACDGFEABCDHI $.
  $}

  ${
    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.)  (Proof shortened by SN, 11-Nov-2024.) $)
    mpteq2da $p |- ( ph -> ( x e. A |-> B ) = ( x e. A |-> C ) ) $=
      ( eqidd mpteq12da ) ABCDCEFACHGI $.
  $}

  ${
    $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.)  Remove dependency on
       ~ ax-10 .  (Revised by SN, 11-Nov-2024.) $)
    mpteq2dva $p |- ( ph -> ( x e. A |-> B ) = ( x e. A |-> C ) ) $=
      ( eqidd mpteq12dva ) ABCDCEACGFH $.
    $( $j usage 'mpteq2dva' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $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 $.
  $}

  ${
    mpteq2ia.1 $e |- ( x e. A -> B = C ) $.
    $( An equality inference for the maps-to notation.  (Contributed by Mario
       Carneiro, 16-Dec-2013.)  (Proof shortened by SN, 11-Nov-2024.) $)
    mpteq2ia $p |- ( x e. A |-> B ) = ( x e. A |-> C ) $=
      ( cmpt wceq wtru cv wcel adantl mpteq2dva mptru ) ABCFABDFGHABCDAIBJCDGHE
      KLM $.
    $( $j usage 'mpteq2ia' avoids 'ax-10' 'ax-12'; $)
  $}

  ${
    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 $.
  $}

  ${
    $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 x y z w $.  $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 NM, 11-Sep-2011.)
       (Revised by Thierry Arnoux, 9-Mar-2017.)  Add disjoint variable
       condition to avoid ~ ax-13 .  See ~ cbvmptfg for a less restrictive
       version requiring more axioms.  (Revised by GG, 17-Jan-2024.) $)
    cbvmptf $p |- ( x e. A |-> B ) = ( y e. A |-> C ) $=
      ( vz vw cv wcel wceq wa copab cmpt wsb weq nfv nfcri nfs1v eleq1w sbequ12
      nfan anbi12d cbvopab1 nfeq2 nfsbv sbequ eqeq2d sbiev bitrdi eqtri 3eqtr4i
      df-mpt ) AMCNZKMZDOZPZAKQZBMCNZUSEOZPZBKQZACDRBCERVBLMCNZUTALSZPZLKQVFVAV
      IAKLVALUAVGVHAALCFUBUTALUCUFALTURVGUTVHALCUDUTALUEUGUHVIVELKBVGVHBBLCGUBU
      TALBBUSDHUIUJUFVELUALBTZVGVCVHVDLBCUDVJVHUTABSVDUTLBAUKUTVDABAUSEIUIABTDE
      USJULUMUNUGUHUOAKCDUQBKCEUQUP $.
    $( $j usage 'cbvmptf' avoids 'ax-13'; $)
  $}

  ${
    $d w z x $.  $d w z y $.  $d w z A $.  $d w z B $.  $d w z C $.
    cbvmptfg.1 $e |- F/_ x A $.
    cbvmptfg.2 $e |- F/_ y A $.
    cbvmptfg.3 $e |- F/_ y B $.
    cbvmptfg.4 $e |- F/_ x C $.
    cbvmptfg.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.  Usage of this theorem is discouraged
       because it depends on ~ ax-13 .  See ~ cbvmptf for a version with more
       disjoint variable conditions, but not requiring ~ ax-13 .  (Contributed
       by NM, 11-Sep-2011.)  (Revised by Thierry Arnoux, 9-Mar-2017.)
       (New usage is discouraged.) $)
    cbvmptfg $p |- ( x e. A |-> B ) = ( y e. A |-> C ) $=
      ( vz vw cv wcel wceq wa copab cmpt wsb weq nfv nfcri nfs1v eleq1w sbequ12
      nfan anbi12d cbvopab1g nfeq2 nfsb sbequ eqeq2d sbie bitrdi df-mpt 3eqtr4i
      eqtri ) AMCNZKMZDOZPZAKQZBMCNZUSEOZPZBKQZACDRBCERVBLMCNZUTALSZPZLKQVFVAVI
      AKLVALUAVGVHAALCFUBUTALUCUFALTURVGUTVHALCUDUTALUEUGUHVIVELKBVGVHBBLCGUBUT
      ALBBUSDHUIUJUFVELUALBTZVGVCVHVDLBCUDVJVHUTABSVDUTLBAUKUTVDABAUSEIUIABTDEU
      SJULUMUNUGUHUQAKCDUOBKCEUOUP $.
  $}

  ${
    $d x y A $.
    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.)  Add
       disjoint variable condition to avoid ~ ax-13 .  See ~ cbvmptg for a less
       restrictive version requiring more axioms.  (Revised by GG,
       17-Jan-2024.) $)
    cbvmpt $p |- ( x e. A |-> B ) = ( y e. A |-> C ) $=
      ( nfcv cbvmptf ) ABCDEACIBCIFGHJ $.
    $( $j usage 'cbvmpt' avoids 'ax-13'; $)
  $}

  ${
    $d x A $.  $d y A $.
    cbvmptg.1 $e |- F/_ y B $.
    cbvmptg.2 $e |- F/_ x C $.
    cbvmptg.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.  Usage of this theorem is discouraged
       because it depends on ~ ax-13 .  See ~ cbvmpt for a version with more
       disjoint variable conditions, but not requiring ~ ax-13 .  (Contributed
       by NM, 11-Sep-2011.)  (New usage is discouraged.) $)
    cbvmptg $p |- ( x e. A |-> B ) = ( y e. A |-> C ) $=
      ( nfcv cbvmptfg ) ABCDEACIBCIFGHJ $.
  $}

  ${
    $d A x y z $.  $d B y z $.  $d C x z $.
    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.)  Add
       disjoint variable condition to avoid auxiliary axioms .  See ~ cbvmptvg
       for a less restrictive version requiring more axioms.  (Revised by GG,
       17-Nov-2024.) $)
    cbvmptv $p |- ( x e. A |-> B ) = ( y e. A |-> C ) $=
      ( vz cv wcel wceq wa copab eleq1w eqeq2d anbi12d cbvopab1v df-mpt 3eqtr4i
      cmpt weq ) AHCIZGHZDJZKZAGLBHCIZUBEJZKZBGLACDSBCESUDUGAGBABTZUAUEUCUFABCM
      UHDEUBFNOPAGCDQBGCEQR $.
    $( $j usage 'cbvmptv' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    $d A x y $.  $d B y $.  $d C x $.
    cbvmptvOLD.1 $e |- ( x = y -> B = C ) $.
    $( Obsolete version of ~ cbvmptv as of 17-Nov-2024.  (Contributed by Mario
       Carneiro, 19-Feb-2013.)  Add disjoint variable condition to avoid
       ~ ax-13 .  See ~ cbvmptvg for a less restrictive version requiring more
       axioms.  (Revised by GG, 17-Jan-2024.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    cbvmptvOLD $p |- ( x e. A |-> B ) = ( y e. A |-> C ) $=
      ( nfcv cbvmpt ) ABCDEBDGAEGFH $.
    $( $j usage 'cbvmptv' avoids 'ax-13'; $)
  $}

  ${
    $d A x $.  $d A y $.  $d B y $.  $d C x $.
    cbvmptvg.1 $e |- ( x = y -> B = C ) $.
    $( Rule to change the bound variable in a maps-to function, using implicit
       substitution.  Usage of this theorem is discouraged because it depends
       on ~ ax-13 .  See ~ cbvmptv for a version with more disjoint variable
       conditions, but not requiring ~ ax-13 .  (Contributed by Mario Carneiro,
       19-Feb-2013.)  (New usage is discouraged.) $)
    cbvmptvg $p |- ( x e. A |-> B ) = ( y e. A |-> C ) $=
      ( nfcv cbvmptg ) 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.  Not to be confused with a
     transitive relation (see ~ cotr ).  Definition of [Enderton] p. 71
     extended to arbitrary classes.  For alternate definitions, see ~ dftr2
     (which is suggestive of the word "transitive"), ~ dftr2c , ~ dftr3 ,
     ~ dftr4 , ~ dftr5 , and (when ` A ` is a set) ~ unisuc .  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.  Using ~ dftr2c instead may avoid dependences on
       ~ ax-11 .  (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 wa df-ss df-tr 19.23v eluni imbi1i bitr4i albii
      wtr wex 3bitr4i ) CDZCEAFZUAGZUBCGZHZAICRUBBFZGUFCGJZUDHBIZAIAUACKCLUHUEA
      UHUGBSZUDHUEUGUDBMUCUIUDBUBCNOPQT $.
  $}

  ${
    $d A x y z $.
    $( Variant of ~ dftr2 with commuted quantifiers, useful for shortening
       proofs and avoiding ~ ax-11 .  (Contributed by BTernaryTau,
       28-Dec-2024.) $)
    dftr2c $p |- ( Tr A <-> A. y A. x ( ( x e. y /\ y e. A ) -> x e. A ) ) $=
      ( vz wtr wel cv wcel wa wal dftr2 weq elequ1 anbi1d eleq1w imbi12d elequ2
      wi anbi12d imbi1d alcomw bitri ) CEABFZBGCHZIZAGCHZRZBJAJUGAJBJABCKUGDBFZ
      UDIZDGCHZRADFZUJIZUFRABDDADLZUEUIUFUJUMUCUHUDADBMNADCOPBDLZUEULUFUNUCUKUD
      UJBDAQBDCOSTUAUB $.
    $( $j usage 'dftr2c' avoids 'ax-11'; $)
  $}

  ${
    $d A x y $.
    $( An alternate way of defining a transitive class.  Definition 1.1 of
       [Schloeder] p. 1.  (Contributed by NM, 20-Mar-2004.)  Avoid ~ ax-11 .
       (Revised by BTernaryTau, 28-Dec-2024.) $)
    dftr5 $p |- ( Tr A <-> A. x e. A A. y e. x y e. A ) $=
      ( wel cv wcel wa wal wral wtr impexp albii df-ral r19.21v 3bitr2i 3bitr4i
      wi dftr2c ) BADZAEZCFZGBECFZQZBHZAHUAUBBTIZQZAHCJUEACIUDUFAUDSUAUBQZQZBHU
      GBTIUFUCUHBSUAUBKLUGBTMUAUBBTNOLBACRUEACMP $.
    $( $j usage 'dftr5' avoids 'ax-11'; $)

    $( Obsolete version of ~ dftr5 as of 28-Dec-2024.  (Contributed by NM,
       20-Mar-2004.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    dftr5OLD $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.)  (Proof shortened by JJ, 26-Jul-2021.) $)
    trss $p |- ( Tr A -> ( B e. A -> B C_ A ) ) $=
      ( vx wtr cv wss wral wcel wi dftr3 sseq1 rspccv sylbi ) ADCEZAFZCAGBAHBAF
      ZICAJOPCBANBAKLM $.
  $}

  ${
    $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 y x $.  $d y A $.  $d y B $.
    $( An 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 syl2imc rexlimi syl sylan2b ralrimiva dftr3 sylibr ) CEZA
      BFZDGZABCHZIZDULFULEUJUMDULUKULJUJUKCJZABKZUMAUKBCLUJUOSUIUNSZABKUMUIUNAB
      MUPUMABAUKULAUKNABCOPUPUKCIZAGBJCULIUMUIUNUQCUKQRABCTUKCULUAUBUCUDUEUFDUL
      UGUH $.
  $}

  ${
    $d A x $.
    $( 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 $.
  $}

  ${
    $d y x $.  $d y A $.  $d y B $.
    $( An indexed intersection of a class of transitive sets is transitive.
       (Contributed by BJ, 3-Oct-2022.) $)
    triin $p |- ( A. x e. A Tr B -> Tr |^|_ x e. A B ) $=
      ( vy wtr wral cv ciin wss wcel wb cvv eliin elv wa r19.26 trss imp ralimi
      sylibr sylbir ssiin sylan2b ralrimiva dftr3 ) CEZABFZDGZABCHZIZDUIFUIEUGU
      JDUIUHUIJZUGUHCJZABFZUJUKUMKDAUHBCLMNUGUMOZUHCIZABFZUJUNUFULOZABFUPUFULAB
      PUQUOABUFULUOCUHQRSUAABCUHUBTUCUDDUIUET $.
  $}

  ${
    $d A x $.
    $( The intersection of a class of transitive sets is transitive.  Exercise
       5(b) of [Enderton] p. 73.  (Contributed by Scott Fenton, 25-Feb-2011.)
       (Proof shortened by BJ, 3-Oct-2022.) $)
    trint $p |- ( A. x e. A Tr x -> Tr |^| A ) $=
      ( cv wtr wral ciin cint triin wceq wb intiin treq ax-mp sylibr ) ACZDABEA
      BOFZDZBGZDZABOHRPISQJABKRPLMN $.
  $}

  ${
    $d x A $.
    $( Any nonempty transitive class includes its intersection.  Exercise 3 in
       [TakeutiZaring] p. 44 (which mistakenly does not include the
       nonemptiness hypothesis).  (Contributed by Scott Fenton, 3-Mar-2011.)
       (Proof shortened by Andrew Salmon, 14-Nov-2011.) $)
    trintss $p |- ( ( Tr A /\ A =/= (/) ) -> |^| A C_ A ) $=
      ( vx c0 wne wtr cint wss cv wcel wex wi intss1 trss com12 sylsyld exlimiv
      n0 sstr2 sylbi impcom ) ACDZAEZAFZAGZUABHZAIZBJUBUDKZBAQUFUGBUFUCUEGUBUEA
      GZUDUEALUBUFUHAUEMNUCUEAROPST $.
  $}

$( End $[ set-class.mm $] $)


$( Begin $[ set-zf.mm $] $)
$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
  ZF Set Theory - add the Axiom of Replacement
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Introduce the Axiom of Replacement
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x y z w $.
    $( Axiom of Replacement.  An axiom scheme of Zermelo-Fraenkel set theory.
       Axiom 5 of [TakeutiZaring] p. 19.  It tells us that the image of any set
       under a function is also a set (see the variant ~ funimaex ).  Although
       ` ph ` may be any wff whatsoever, this axiom is useful (i.e. its
       antecedent is satisfied) when we are given some function and ` ph `
       encodes the predicate "the value of the function at ` w ` is ` z ` ".
       Thus, ` ph ` will ordinarily have free variables ` w ` and ` z ` - think
       of it informally as ` ph ( w , z ) ` .  We prefix ` ph ` with the
       quantifier ` A. y ` in order to "protect" the axiom from any ` ph `
       containing ` y ` , thus allowing us to eliminate any restrictions on
       ` ph ` .  Another common variant is derived as ~ axrep5 , where you can
       find some further remarks.  A slightly more compact version is shown as
       ~ axrep2 .  A quite different variant is ~ zfrep6 , which if used in
       place of ~ ax-rep would also require that the Separation Scheme ~ axsep
       be stated as a separate axiom.

       There is a very strong generalization of Replacement that doesn't demand
       function-like behavior of ` ph ` .  Two versions of this generalization
       are called the Collection Principle ~ cp and the Boundedness Axiom
       ~ bnd .

       Many developments of set theory distinguish the uses of Replacement from
       uses of the weaker axioms of Separation ~ axsep , Null Set ~ axnul , and
       Pairing ~ axpr , all of which we derive from Replacement.  In order to
       make it easier to identify the uses of those redundant axioms, we
       restate them as Axioms ~ ax-sep , ~ ax-nul , and ~ ax-pr below the
       theorems that prove them.  (Contributed by NM, 23-Dec-1993.) $)
    ax-rep $a |- ( A. w E. y A. z ( A. y ph -> z = y ) ->
                     E. y A. z ( z e. y <-> E. w ( w e. x /\ A. y ph ) ) ) $.
  $}

  ${
    $d w y ph $.  $d w x y z $.
    $( The version of the Axiom of Replacement used in the Metamath Solitaire
       applet ~ https://us.metamath.org/mmsolitaire/mms.html .  Equivalence is
       shown via the path ~ ax-rep ` -> ` ~ axrep1 ` -> ` ~ axrep2 ` -> `
       ~ axrepnd ` -> ` ~ zfcndrep = ~ ax-rep .  (Contributed by NM,
       19-Nov-2005.)  (Proof shortened by Mario Carneiro, 17-Nov-2016.)  Remove
       dependency on ~ ax-13 .  (Revised by BJ, 31-May-2019.) $)
    axrep1 $p |- E. x ( E. y A. z ( ph -> z = y ) ->
                 A. z ( z e. x <-> E. x ( x e. y /\ ph ) ) ) $=
      ( vw weq wi wal wex wel wa elequ2 anbi1d exbidv bibi2d albidv albii exbii
      wb nfv imbi2d ax-rep imbi1i nfe1 nfbi nfal anbi2i bibi12d cbvexv1 3imtr3i
      19.3v a1i chvarvv 19.35ri ) ADCFZGZDHZCIZDBJZBCJZAKZBIZSZDHZBURBHZUSBEJZA
      KZBIZSZDHZBIZGVEVDBIZGECECFZVKVLVEVMVJVDBVMVIVCDVMVHVBUSVMVGVABVMVFUTAECB
      LMNOPNUAACHZUOGZDHZCIZBHDCJZVFVNKZBIZSZDHZCIVEVKAECDBUBVQURBVPUQCVOUPDVNA
      UOACUKZUCQRQWBVJCBWABDVRVTBVRBTVSBUDUEUFVJCTCBFZWAVIDWDVRUSVTVHCBDLVTVHSW
      DVSVGBVNAVFWCUGRULUHPUIUJUMUN $.
  $}

  ${
    $d u x $.  $d u y $.  $d v x $.  $d v y $.  $d w x $.  $d w y $.
    $( Lemma for ~ axrep2 and ~ axrep3 .  (Contributed by BJ, 6-Aug-2022.) $)
    axreplem $p |- ( x = y ->
                  ( E. u ( ph -> A. v ( ps <-> E. w ( z e. x /\ ch ) ) ) <->
                    E. u ( ph -> A. v ( ps <-> E. w ( z e. y /\ ch ) ) ) ) ) $=
      ( weq wel wa wex wb wal wi elequ2 anbi1d exbidv bibi2d albidv imbi2d ) DE
      JZABFDKZCLZGMZNZHOZPABFEKZCLZGMZNZHOZPIUCUHUMAUCUGULHUCUFUKBUCUEUJGUCUDUI
      CDEFQRSTUAUBS $.
  $}

  ${
    $d w ph $.  $d w x y z $.
    $( Axiom of Replacement expressed with the fewest number of different
       variables and without any restrictions on ` ph ` .  (Contributed by NM,
       15-Aug-2003.)  Remove dependency on ~ ax-13 .  (Revised by BJ,
       31-May-2019.) $)
    axrep2 $p |- E. x ( E. y A. z ( ph -> z = y ) ->
                        A. z ( z e. x <-> E. x ( x e. y /\ A. y ph ) ) ) $=
      ( vw wal weq wi wex wel nfe1 nfv nfim nfex axreplem axrep1 chvarfv imim1i
      wa wb sp alimi eximi nfa1 nfal equequ2 imbi2d albidv cbvexv1 sylib eximii
      ) ACFZDEGZHZDFZEIZDBJZBCJULSBITDFZHZADCGZHZDFZCIZURHBUPUQBEJULSBITDFHBIUS
      BIECUSEBUPUREUOEKURELMNUPUQULECBBDBOULBEDPQVCUPURVCULUTHZDFZCIUPVBVECVAVD
      DULAUTACUARUBUCVEUOCEVEELUNCDULUMCACUDUMCLMUECEGZVDUNDVFUTUMULCEDUFUGUHUI
      UJRUK $.
  $}

  ${
    $d w x y z $.
    $( Axiom of Replacement slightly strengthened from ~ axrep2 ; ` w ` may
       occur free in ` ph ` .  (Contributed by NM, 2-Jan-1997.)  Remove
       dependency on ~ ax-13 .  (Revised by BJ, 31-May-2019.) $)
    axrep3 $p |- E. x ( E. y A. z ( ph -> z = y ) ->
                 A. z ( z e. x <-> E. x ( x e. w /\ A. y ph ) ) ) $=
      ( weq wi wal wex wel wa wb nfe1 nfv nfa1 nfan nfex nfbi nfal nfim chvarfv
      axreplem axrep2 ) ADCFGDHZCIZDBJZBCJACHZKBILDHGBIUEUFBEJZUGKZBIZLZDHZGZBI
      CEUMCBUEULCUDCMUKCDUFUJCUFCNUICBUHUGCUHCNACOPQRSTQUEUFUGCEBBDBUBABCDUCUA
      $.
  $}

  ${
    $d w x y z $.  $d z ph $.
    $( Version of ~ axrep4 with a disjoint variable condition, requiring fewer
       axioms.  (Contributed by Matthew House, 18-Sep-2025.) $)
    axrep4v $p |- ( A. x E. z A. y ( ph -> y = z ) ->
                E. z A. y ( y e. z <-> E. x ( x e. w /\ ph ) ) ) $=
      ( wal weq wi wex wel wa wb ax-rep 19.3v imbi1i albii exbii anbi2i 3imtr3i
      bibi2i ) ADFZCDGZHZCFZDIZBFCDJZBEJZUAKZBIZLZCFZDIAUBHZCFZDIZBFUFUGAKZBIZL
      ZCFZDIAEDCBMUEUNBUDUMDUCULCUAAUBADNZOPQPUKURDUJUQCUIUPUFUHUOBUAAUGUSRQTPQ
      S $.
    $( $j usage 'axrep4v' avoids 'ax-12'; $)
  $}

  ${
    $d w x y z $.
    axrep4.1 $e |- F/ z ph $.
    $( A more traditional version of the Axiom of Replacement.  (Contributed by
       NM, 14-Aug-1994.)  (Proof shortened by Matthew House, 18-Sep-2025.) $)
    axrep4 $p |- ( A. x E. z A. y ( ph -> y = z ) ->
                E. z A. y ( y e. z <-> E. x ( x e. w /\ ph ) ) ) $=
      ( wal weq wi wex wel wa wb ax-rep 19.3 imbi1i albii exbii anbi2i bibi2i
      3imtr3i ) ADGZCDHZIZCGZDJZBGCDKZBEKZUBLZBJZMZCGZDJAUCIZCGZDJZBGUGUHALZBJZ
      MZCGZDJAEDCBNUFUOBUEUNDUDUMCUBAUCADFOZPQRQULUSDUKURCUJUQUGUIUPBUBAUHUTSRT
      QRUA $.
  $}

  ${
    $d x y z w $.
    axrep4OLD.1 $e |- F/ z ph $.
    $( Obsolete version of ~ axrep4 as of 18-Sep-2025.  (Contributed by NM,
       14-Aug-1994.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    axrep4OLD $p |- ( A. x E. z A. y ( ph -> y = z ) ->
                E. z A. y ( y e. z <-> E. x ( x e. w /\ ph ) ) ) $=
      ( weq wi wal wex wel wa wb axrep3 19.35i nfv nfa1 nfan nfbi nfal nfex a1i
      nfe1 elequ2 19.3 anbi2i exbii bibi12d albidv cbvexv1 sylib ) ACDGHCIDJZBI
      CBKZBEKZADIZLZBJZMZCIZBJCDKZUNALZBJZMZCIZDJULUSBABDCENOUSVDBDURDCUMUQDUMD
      PUPDBUNUODUNDPADQRUASTVCBCUTVBBUTBPVABUCSTBDGZURVCCVEUMUTUQVBBDCUDUQVBMVE
      UPVABUOAUNADFUEUFUGUBUHUIUJUK $.
  $}

  ${
    $d x y z w $.
    axrep5.1 $e |- F/ z ph $.
    $( Axiom of Replacement (similar to Axiom Rep of [BellMachover] p. 463).
       The antecedent tells us ` ph ` is analogous to a "function" from ` x `
       to ` y ` (although it is not really a function since it is a wff and not
       a class).  In the consequent we postulate the existence of a set ` z `
       that corresponds to the "image" of ` ph ` restricted to some other set
       ` w ` .  The hypothesis says ` z ` must not be free in ` ph ` .
       (Contributed by NM, 26-Nov-1995.)  (Revised by Mario Carneiro,
       14-Oct-2016.) $)
    axrep5 $p |- ( A. x ( x e. w -> E. z A. y ( ph -> y = z ) ) ->
                E. z A. y ( y e. z <-> E. x ( x e. w /\ ph ) ) ) $=
      ( wel weq wi wal wex wa wb 19.37v impexp albii 19.21v bitr2i exbii bitr3i
      nfv nfan axrep4 sylbi anabs5 bibi2i sylib ) BEGZACDHZIZCJZDKIZBJZCDGZUHUH
      ALZLZBKZMZCJZDKZUNUOBKZMZCJZDKUMUOUIIZCJZDKZBJUTULVFBULUHUKIZDKVFUHUKDNVG
      VEDVEUHUJIZCJVGVDVHCUHAUIOPUHUJCQRSTPUOBCDEUHADUHDUAFUBUCUDUSVCDURVBCUQVA
      UNUPUOBUHAUESUFPSUG $.
  $}

  ${
    $d w x y z $.  $d y ph $.
    $( A condensed form of ~ ax-rep .  (Contributed by SN, 18-Sep-2023.)
       (Proof shortened by Matthew House, 18-Sep-2025.) $)
    axrep6 $p |- ( A. w E* z ph -> E. y A. z ( z e. y <-> E. w e. x ph ) ) $=
      ( weq wi wal wex wel wa wb wmo cv wrex axrep4v df-mo albii df-rex bibi2i
      exbii 3imtr4i ) ADCFGDHCIZEHDCJZEBJAKEIZLZDHZCIADMZEHUDAEBNZOZLZDHZCIAEDC
      BPUHUCEADCQRULUGCUKUFDUJUEUDAEUISTRUAUB $.
  $}

  ${
    $d w x y z $.  $d y ph $.
    $( Obsolete version of ~ axrep6 as of 18-Sep-2025.  (Contributed by SN,
       18-Sep-2023.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    axrep6OLD $p |- ( A. w E* z ph -> E. y A. z ( z e. y <-> E. w e. x ph ) )
      $=
      ( wal weq wi wex wel wa wb wmo cv wrex ax-rep df-mo 19.3v albii exbii
      imbi1i bitr4i rexbii df-rex bitr3i bibi2i 3imtr4i ) ACFZDCGZHZDFZCIZEFDCJ
      ZEBJUHKEIZLZDFZCIADMZEFUMAEBNZOZLZDFZCIABCDEPUQULEUQAUIHZDFZCIULADCQUKVCC
      UJVBDUHAUIACRZUASTUBSVAUPCUTUODUSUNUMUSUHEUROUNUHAEURVDUCUHEURUDUEUFSTUG
      $.
  $}

  ${
    $d ps w z $.  $d w x y z $.  $d A x y z $.
    $( ~ axrep6 in class notation.  It is equivalent to both ~ ax-rep and
       ~ abrexexg , providing a direct link between the two.  (Contributed by
       SN, 11-Dec-2024.) $)
    axrep6g $p |-
      ( ( A e. V /\ A. x E* y ps ) -> { y | E. x e. A ps } e. _V ) $=
      ( vz vw wcel wmo wal wrex cab cvv cv wi wceq rexeq abbidv eleq1d imbi2d
      wel wb wex axrep6 abbi abid2 vex eqeltri eqeltrrdi exlimiv syl vtoclg imp
      ) DEHACIBJZABDKZCLZMHZUNABFNZKZCLZMHZOUNUQOFDEURDPZVAUQUNVBUTUPMVBUSUOCAB
      URDQRSTUNCGUAZUSUBCJZGUCVAAFGCBUDVDVAGVDUTVCCLZMVCUSCUEVEGNZMCVFUFGUGUHUI
      UJUKULUM $.
  $}

  ${
    $d y z A v $.  $d z ph v $.  $d x y z v $.
    zfrepclf.1 $e |- F/_ x A $.
    zfrepclf.2 $e |- A e. _V $.
    zfrepclf.3 $e |- ( x e. A -> E. z A. y ( ph -> y = z ) ) $.
    $( An inference based on the Axiom of Replacement.  Typically, ` ph `
       defines a function from ` x ` to ` y ` .  (Contributed by NM,
       26-Nov-1995.) $)
    zfrepclf $p |- E. z A. y ( y e. z <-> E. x ( x e. A /\ ph ) ) $=
      ( vv wel cv wcel wa wex wb wal wceq wi nfeq2 eleq2 biimtrdi alrimi axrep5
      nfv syl anbi1d exbid bibi2d albidv exbidv mpbid vtocle ) CDJZBKZELZAMZBNZ
      OZCPZDNZIEGIKZEQZUMBIJZAMZBNZOZCPZDNZUTVBVCACKDKQRCPDNZRZBPVHVBVJBBVAEFSZ
      VBVCUOVIVAEUNTZHUAUBABCDIADUDUCUEVBVGUSDVBVFURCVBVEUQUMVBVDUPBVKVBVCUOAVL
      UFUGUHUIUJUKUL $.
  $}

  ${
    $d x y z A $.  $d z ph $.
    zfrep3cl.1 $e |- A e. _V $.
    zfrep3cl.2 $e |- ( x e. A -> E. z A. y ( ph -> y = z ) ) $.
    $( An inference based on the Axiom of Replacement.  Typically, ` ph `
       defines a function from ` x ` to ` y ` .  (Contributed by NM,
       26-Nov-1995.) $)
    zfrep3cl $p |- E. z A. y ( y e. z <-> E. x ( x e. A /\ ph ) ) $=
      ( nfcv zfrepclf ) ABCDEBEHFGI $.
  $}

  ${
    $d ph y z $.  $d ps z $.  $d x y z $.
    zfrep4.1 $e |- { x | ph } e. _V $.
    zfrep4.2 $e |- ( ph -> E. z A. y ( ps -> y = z ) ) $.
    $( A version of Replacement using class abstractions.  (Contributed by NM,
       26-Nov-1995.) $)
    zfrep4 $p |- { y | E. x ( ph /\ ps ) } e. _V $=
      ( cv cab wcel wa wex cvv abid anbi1i exbii abbii wceq wb wal nfab1 sylbi
      wi zfrepclf eqabb mpbir issetri eqeltrri ) CHACIZJZBKZCLZDIZABKZCLZDIMULU
      ODUKUNCUJABACNZOPQEUMEHZUMRZELDHZUQJULSDTZELBCDEUIACUAFUJABUSUQRUCDTELUPG
      UBUDURUTEULDUQUEPUFUGUH $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Derive the Axiom of Separation
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x y z w $.  $d y ph w $.
    $( A more general version ~ axsepg of the axiom scheme of separation
       ~ ax-sep derived from the axiom scheme of replacement ~ ax-rep (and
       first-order logic).  The extra generality consists in the absence of a
       disjoint variable condition on ` z , ph ` (that is, variable ` z ` may
       occur in formula ` ph ` ).  See linked statements for more information.
       (Contributed by NM, 11-Sep-2006.)  Remove dependencies on ~ ax-9 to
       ~ ax-13 .  (Revised by SN, 25-Sep-2023.)  Use ~ ax-sep instead (or
       ~ axsepg if the extra generality is needed).
       (New usage is discouraged.) $)
    axsepgfromrep $p |- E. y A. x ( x e. y <-> ( x e. z /\ ph ) ) $=
      ( vw wel weq wa cv wb wal wex wmo axrep6 euequ eumoi equcomi adantr exbii
      wrex moimi ax-mp mpg df-rex an12 elequ1 anbi1d equsexvw bibi2i albii mpbi
      3bitr2i ) BCFZEBGZAHZEDIZTZJZBKZCLZUMBDFZAHZJZBKZCLUOBMZUTEUODCBENBEGZBMV
      EVFBBEOPUOVFBUNVFAEBQRUAUBUCUSVDCURVCBUQVBUMUQEDFZUOHZELUNVGAHZHZELVBUOEU
      PUDVJVHEUNVGAUESVIVBEBUNVGVAAEBDUFUGUHULUIUJSUK $.
  $}

  ${
    $d x y z $.  $d y z ph $.
    $( Axiom scheme of separation ~ ax-sep derived from the axiom scheme of
       replacement ~ ax-rep .  The statement is identical to that of ~ ax-sep ,
       and therefore shows that ~ ax-sep is redundant when ~ ax-rep is allowed.
       See ~ ax-sep for more information.  (Contributed by NM, 11-Sep-2006.)
       Use ~ ax-sep instead.  (New usage is discouraged.) $)
    axsep $p |- E. y A. x ( x e. y <-> ( x e. z /\ ph ) ) $=
      ( axsepgfromrep ) ABCDE $.

    $( Axiom scheme of separation.  This is an axiom scheme of Zermelo and
       Zermelo-Fraenkel set theories.

       It was derived as ~ axsep above and is therefore redundant in ZF set
       theory, which contains ~ ax-rep as an axiom (contrary to Zermelo set
       theory).  We state it as a separate axiom here so that some of its uses
       can be identified more easily.  Some textbooks present the axiom scheme
       of separation as a separate axiom scheme in order to show that much of
       set theory can be derived without the stronger axiom scheme of
       replacement (which is not part of Zermelo set theory).

       The axiom scheme of separation is a weak form of Frege's axiom scheme of
       (unrestricted) comprehension, in that it conditions it with the
       condition ` 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" (German for "separation") or "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 ~ zfauscl , 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, as ~ notzfaus shows (showing the necessity of
       that condition in ~ zfauscl ).

       Scheme Sep of [BellMachover] p. 463.  (Contributed by NM,
       11-Sep-2006.) $)
    ax-sep $a |- E. y A. x ( x e. y <-> ( x e. z /\ ph ) ) $.
    $( $j restatement 'ax-sep' of 'axsep'; $)
  $}

  ${
    $d x y z w $.  $d y ph w $.
    $( A more general version of the axiom scheme of separation ~ ax-sep ,
       where variable ` z ` can also occur (in addition to ` x ` ) in formula
       ` ph ` , which can therefore be thought of as ` ph ( x , z ) ` .  This
       version is derived from the more restrictive ~ ax-sep with no additional
       set theory axioms.  Note that it was also derived from ~ ax-rep but
       without ~ ax-sep as ~ axsepgfromrep .  (Contributed by NM, 10-Dec-2006.)
       (Proof shortened by Mario Carneiro, 17-Nov-2016.)  Remove dependency on
       ~ ax-12 and ~ ax-13 and shorten proof.  (Revised by BJ, 6-Oct-2019.) $)
    axsepg $p |- E. y A. x ( x e. y <-> ( x e. z /\ ph ) ) $=
      ( vw wel wa wal wex weq elequ2 anbi1d bibi2d albidv exbidv ax-sep chvarvv
      wb ) BCFZBEFZAGZRZBHZCISBDFZAGZRZBHZCIEDEDJZUCUGCUHUBUFBUHUAUESUHTUDAEDBK
      LMNOABCEPQ $.
  $}

  ${
    $d x y A z $.  $d y ph z $.
    zfauscl.1 $e |- A e. _V $.
    $( Separation Scheme (Aussonderung) using a class variable.  To derive this
       from ~ ax-sep , we invoke the Axiom of Extensionality (indirectly via
       ~ vtocl ), which is needed for the justification of class variable
       notation.

       If we omit the requirement that ` y ` not occur in ` ph ` , we can
       derive a contradiction, as ~ notzfaus shows.  (Contributed by NM,
       21-Jun-1993.) $)
    zfauscl $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 y z $.  $d y z ph $.
    $( Lemma for ~ sepex .  Use ~ sepex instead.  (Contributed by Matthew
       House, 19-Sep-2025.)  (New usage is discouraged.) $)
    sepexlem $p |- ( E. y A. x ( ph -> x e. y ) ->
        E. z A. x ( x e. z <-> ph ) ) $=
      ( wel wi wal wb wex wa ax-sep bimsc1 ex al2imi eximdv mpi exlimiv ) ABCEZ
      FZBGZBDEZAHZBGZDIZCTUARAJHZBGZDIUDABDCKTUFUCDSUEUBBSUEUBARUALMNOPQ $.
  $}

  ${
    $d x y w $.  $d x z w $.  $d y w ph $.  $d z w ph $.
    $( Convert implication to equivalence within an existence statement using
       the Separation Scheme (Aussonderung) ~ ax-sep .  Similar to Theorem
       1.3(ii) of [BellMachover] p. 463.  (Contributed by Matthew House,
       19-Sep-2025.) $)
    sepex $p |- ( E. y A. x ( ph -> x e. y ) ->
        E. z A. x ( x e. z <-> ph ) ) $=
      ( vw wel wi wal wex wb sepexlem biimpr alimi eximi 3syl ) ABCFGBHCIBEFZAJ
      ZBHZEIAPGZBHZEIBDFAJBHDIABCEKRTEQSBPALMNABEDKO $.
    $( $j usage 'sepex' avoids 'ax-9'; $)
  $}

  ${
    $d x y $.  $d x z $.  $d y ph $.  $d z ph $.
    sepexi.1 $e |- E. y A. x ( ph -> x e. y ) $.
    $( Convert implication to equivalence within an existence statement using
       the Separation Scheme (Aussonderung) ~ ax-sep .  Inference associated
       with ~ sepex .  (Contributed by NM, 21-Jun-1993.)  Generalize
       conclusion, extract closed form, and avoid ~ ax-9 .  (Revised by Matthew
       House, 19-Sep-2025.) $)
    sepexi $p |- E. z A. x ( x e. z <-> ph ) $=
      ( wel wi wal wex wb sepex ax-mp ) ABCFGBHCIBDFAJBHDIEABCDKL $.
    $( $j usage 'sepexi' avoids 'ax-9'; $)
  $}

  ${
    $d x ph z $.  $d x y z $.
    bm1.3iiOLD.1 $e |- E. x A. y ( ph -> y e. x ) $.
    $( Obsolete version of ~ sepexi as of 18-Sep-2025.  (Contributed by NM,
       21-Jun-1993.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    bm1.3iiOLD $p |- E. x A. y ( y e. x <-> ph ) $=
      ( vz wel wi wal wa wb wex 19.42v bimsc1 eximi sylbir elequ2 imbi2d albidv
      alanimi weq cbvexvw mpbi ax-sep exan exlimiiv ) ACEFZGZCHZCBFZUFAIJZCHZBK
      ZIZUIAJZCHZBKZEUMUHUKIZBKUPUHUKBLUQUOBUGUJUNCAUFUIMSNOUHULEAUIGZCHZBKUHEK
      DUSUHBEBETZURUGCUTUIUFABECPQRUAUBACBEUCUDUE $.
  $}

  ${
    $d x y z $.
    $( Derive ~ ax6v (a weakened version of ~ ax-6 where ` x ` and ` y ` must
       be distinct), from Separation ~ ax-sep and Extensionality ~ ax-ext .
       See ~ ax6 for the derivation of ~ ax-6 from ~ ax6v .  (Contributed by
       NM, 12-Nov-2013.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    ax6vsep $p |- -. A. x -. x = y $=
      ( vz cv wceq wex wn wal wcel wi wa wb ax-sep biantru bibi2i biimpri alimi
      id ax-ext syl eximi ax-mp df-ex mpbi ) ADZBDZEZAFZUGGAHGCDZUEIZUIUFIZUIUI
      EZULJZKZLZCHZAFUHUMCABMUPUGAUPUJUKLZCHUGUOUQCUQUOUKUNUJUMUKULRNOPQABCSTUA
      UBUGAUCUD $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Derive the Null Set Axiom
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x y z w $.
    $( Alternate proof of ~ axnul , proved from propositional calculus,
       ~ ax-gen , ~ ax-4 , ~ sp , and ~ ax-rep .  To check this, replace ~ sp
       with the obsolete axiom ~ ax-c5 in the proof of ~ axnulALT and type the
       Metamath program "MM> SHOW TRACE__BACK axnulALT / AXIOMS" command.
       (Contributed by Jeff Hoffman, 3-Feb-2008.)  (Proof shortened by Mario
       Carneiro, 17-Nov-2016.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    axnulALT $p |- E. x A. y -. y e. x $=
      ( vw vz cv wcel wn wal wex wfal wa wb wi ax-rep sp con2i df-ex sylibr mpg
      wceq fal mto pm2.21i intnan nex nbn albii exbii mpbir ) BEZAEZFZGZBHZAIUL
      CEDEFZJAHZKZCIZLZBHZAIZUPUJUKTZMZBHZAIZVACJDABCNVCVEBVDVDGZAHZGVEVGVDVFAO
      PVDAQRUPVBUPJUAJAOUBZUCSSUNUTAUMUSBURULUQCUPUOVHUDUEUFUGUHUI $.

    $( 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 ~ nulmo ).

       This proof, suggested by Jeff Hoffman, uses only ~ ax-4 and ~ ax-gen
       from predicate calculus, which are valid in "free logic" i.e. logic
       holding in an empty domain (see Axiom A5 and Rule R2 of [LeBlanc]
       p. 277).  Thus, our ~ ax-sep implies the existence of at least one set.
       Note that Kunen's version of ~ ax-sep (Axiom 3 of [Kunen] p. 11) does
       not imply the existence of a set because his is universally closed,
       i.e., prefixed with universal quantifiers to eliminate all free
       variables.  His existence is provided by a separate axiom stating
       ` E. x x = x ` (Axiom 0 of [Kunen] p. 10).

       See ~ axnulALT for a proof directly from ~ ax-rep .

       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 wel wfal wa wb wal wn ax-sep fal intnan id mtbiri alimi eximii ) BAD
      ZBCDZEFZGZBHQIZBHAEBACJTUABTQSERKLTMNOP $.

    $( The Null Set Axiom of ZF 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.  (Contributed by NM,
       7-Aug-2003.) $)
    ax-nul $a |- E. x A. y -. y e. x $.
    $( $j restatement 'ax-nul' of 'axnul'; $)

    $( 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, 21-Jun-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 X y $.
    $( The empty set is the unique class which is a subclass of any set.
       (Contributed by AV, 24-Aug-2022.) $)
    al0ssb $p |- ( A. y X C_ y <-> X = (/) ) $=
      ( cv wss wal c0 wceq 0ex sseq2 ss0b bitrdi 0ss ax-gen sseq1 albidv mpbiri
      spcv impbii ) BACZDZAEZBFGZTUBAFHSFGTBFDUBSFBIBJKQUBUAFSDZAEUCASLMUBTUCAB
      FSNOPR $.
  $}

  ${
    sseliALT.1 $e |- A C_ B $.
    $( Alternate proof of ~ sseli illustrating the use of the weak deduction
       theorem to prove it from the inference ~ sselii .  (Contributed by NM,
       24-Aug-2018.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    sseliALT $p |- ( C e. A -> C e. B ) $=
      ( wcel csn cif wceq biidd eleq2 eleq1 wss sseq1 sseq2 ssid keephyp3v snid
      c0 0ex elimhyp3v sselii dedth3v ) CAEZCBEZUDCUCBRFZGZEUCCRGZUFEABCUEUERAU
      CAUEGZHUDIBUFCJCUGUFKUHUFUGUCUHBLUHUFLZUIUEUELUHUELUIABLABCUEUERAUHBMBUFU
      HNCUGHUIIUEUHUEMUEUFUHNRUGHUIIDUEOPUCCUHEZUJUGUHERUEERUHEZUKABCUEUERAUHCJ
      BUFHUJICUGUHKUEUHRJUEUFHUKIRUGUHKRSQTUAUB $.
  $}

  ${
    $d y A $.  $d y B $.  $d x y $.
    $( The existence of proper substitution into a class.  (Contributed by NM,
       10-Nov-2005.)  (Revised by NM, 17-Aug-2018.) $)
    csbexg $p |- ( A. x B e. W -> [_ A / x ]_ B e. _V ) $=
      ( vy cvv wcel wal csb wa cv wsbc cab abid2 elex eqeltrid alimi spsbc syl5
      df-csb nfcv sbcabel sylibd imp wn c0 csbprc 0ex eqeltrdi adantr pm2.61ian
      ) BFGZCDGZAHZABCIZFGZULUNJUOEKCGZABLEMZFAEBCTULUNURFGZULUNUQEMZFGZABLZUSU
      NVAAHULVBUMVAAUMUTCFECNCDOPQVAABFRSUQAEBFFAFUAUBUCUDPULUEZUPUNVCUOUFFABCU
      GUHUIUJUK $.
  $}

  ${
    csbex.1 $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.)  (Revised
       by NM, 17-Aug-2018.) $)
    csbex $p |- [_ A / x ]_ B e. _V $=
      ( cvv wcel csb csbexg mpg ) CEFABCGEFAABCEHDI $.
  $}

  $( A version of ~ unisn without the ` A e. _V ` hypothesis.  (Contributed by
     Stefan Allan, 14-Mar-2006.) $)
  unisn2 $p |- U. { A } e. { (/) , A } $=
    ( cvv wcel csn cuni c0 cpr unisng prid2g eqeltrd wn wceq biimpi unieqd uni0
    snprc 0ex prid1 eqeltri eqeltrdi pm2.61i ) ABCZADZEZFAGZCUBUDAUEABHFABIJUBK
    ZUDFEZUEUFUCFUFUCFLAPMNUGFUEOFAQRSTUA $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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.)  Remove use of ~ ax-12 and ~ ax-13 .  (Revised by
       BJ, 31-May-2019.) $)
    nalset $p |- -. E. x A. y y e. x $=
      ( vz wel wn wex wal alexn wa wb ax-sep elequ1 elequ2 bitrd notbid anbi12d
      weq bibi12d spvv pclem6 syl eximii mpgbi ) BADZEZBFUDBGAFEAUDABHCBDZCADZC
      CDZEZIZJZCGZUEBUICBAKULBBDZUDUMEZIZJZUEUKUPCBCBQZUFUMUJUOCBBLUQUGUDUIUNCB
      ALUQUHUMUQUHBCDUMCBCLCBBMNOPRSUMUDTUAUBUC $.
    $( $j usage 'nalset' avoids 'ax-12' 'ax-13'; $)
  $}

  ${
    $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, 21-Jun-1993.) $)
    inex1 $p |- ( A i^i B ) e. _V $=
      ( vx vy cin cv wceq wex wcel wa wb zfauscl dfcleq elin bibi2i albii bitri
      wal exbii mpbir issetri ) DABFZDGZUCHZDIEGZUDJZUFAJUFBJZKZLZESZDIUHEDACMU
      EUKDUEUGUFUCJZLZESUKEUDUCNUMUJEULUIUGUFABOPQRTUAUB $.
  $}

  ${
    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 $.
  $}

  $( Sufficient condition for an intersection to be a set.  Commuted form of
     ~ inex1g .  (Contributed by Peter Mazsa, 19-Dec-2018.) $)
  inex2g $p |- ( A e. V -> ( B i^i A ) e. _V ) $=
    ( wcel cin cvv incom inex1g eqeltrid ) ACDBAEABEFBAGABCHI $.

  ${
    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 dfss2 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 $.
  $}

  ${
    $d A x $.  $d ph x $.
    abexd.1 $e |- ( ( ph /\ ps ) -> x e. A ) $.
    abexd.2 $e |- ( ph -> A e. V ) $.
    $( Conditions for a class abstraction to be a set, deduction form.
       (Contributed by AV, 19-Apr-2025.) $)
    abexd $p |- ( ph -> { x | ps } e. _V ) $=
      ( cab cv wcel wi wal wss ex alrimiv abss sylibr ssexd ) ABCHZDEGABCIDJZKZ
      CLSDMAUACABTFNOBCDPQR $.
  $}

  ${
    $d A x $.
    abex.1 $e |- ( ph -> x e. A ) $.
    abex.2 $e |- A e. _V $.
    $( Conditions for a class abstraction to be a set.  Remark:  This proof is
       shorter than a proof using ~ abexd .  (Contributed by AV,
       19-Apr-2025.) $)
    abex $p |- { x | ph } e. _V $=
      ( cab wss cv wcel wi abss mpgbir ssexi ) ABFZCENCGABHCIJBABCKDLM $.
  $}

  $( 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
    $.

  ${
    sselpwd.1 $e |- ( ph -> B e. V ) $.
    sselpwd.2 $e |- ( ph -> A C_ B ) $.
    $( Elementhood to a power set.  (Contributed by Thierry Arnoux,
       18-May-2020.) $)
    sselpwd $p |- ( ph -> A e. ~P B ) $=
      ( cvv ssexd elpwd ) ABCGABCDEFHFI $.
  $}

  $( 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 $.
    zfausab.1 $e |- A e. _V $.
    $( Separation Scheme (Aussonderung) in terms of a class abstraction.
       (Contributed by NM, 8-Jun-1994.) $)
    zfausab $p |- { x | ( x e. A /\ ph ) } e. _V $=
      ( cv wcel wa cab ssab2 ssexi ) BECFAGBHCDABCIJ $.
  $}

  $( 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 $.
  $}

  ${
    $d A x $.
    $( A restricted class abstraction is an element of the power set of its
       restricting set.  (Contributed by AV, 9-Oct-2023.) $)
    rabelpw $p |- ( A e. V -> { x e. A | ph } e. ~P A ) $=
      ( wcel crab cpw wss ssrab2 elpw2g mpbiri ) CDEABCFZCGELCHABCILCDJK $.
  $}

  ${
    $d x A $.
    $( Separation Scheme in terms of a restricted class abstraction.
       (Contributed by NM, 23-Oct-1999.)  (Proof shortened by BJ,
       24-Jul-2025.) $)
    rabexg $p |- ( A e. V -> { x e. A | ph } e. _V ) $=
      ( wcel crab cpw rabelpw elexd ) CDEABCFCGABCDHI $.
  $}

  ${
    $d x A $.
    $( Obsolete version of ~ rabexg as of 24-Jul-2025).  (Contributed by NM,
       23-Oct-1999.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    rabexgOLD $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 a nonempty class exists.  Exercise 5 of
       [TakeutiZaring] p. 44 and its converse.  (Contributed by NM,
       13-Aug-2002.) $)
    intex $p |- ( A =/= (/) <-> |^| A e. _V ) $=
      ( vx c0 wne cint cvv wcel cv wex n0 wss intss1 vex ssex syl exlimiv sylbi
      wceq vprc inteq int0 eqtrdi eleq1d mtbiri necon2ai impbii ) ACDZAEZFGZUGB
      HZAGZBIUIBAJUKUIBUKUHUJKUIUJALUHUJBMNOPQUIACACRZUIFFGSULUHFFULUHCEFACTUAU
      BUCUDUEUF $.
  $}

  $( If a class intersection is not a set, it must be the universe.
     (Contributed by NM, 3-Jul-2005.) $)
  intnex $p |- ( -. |^| A e. _V <-> |^| A = _V ) $=
    ( cint cvv wcel wn wceq intex necon1bbii inteq int0 eqtrdi sylbi vprc eleq1
    c0 mtbiri impbii ) ABZCDZEZRCFZTAOFZUASAOAGHUBROBCAOIJKLUASCCDMRCCNPQ $.

  $( The intersection of a nonempty class abstraction exists.  (Contributed by
     NM, 21-Oct-2003.) $)
  intexab $p |- ( E. x ph <-> |^| { x | ph } e. _V ) $=
    ( wex cab c0 wne cint cvv wcel abn0 intex bitr3i ) ABCABDZEFMGHIABJMKL $.

  $( The intersection of a nonempty restricted class abstraction exists.
     (Contributed by NM, 21-Oct-2003.) $)
  intexrab $p |- ( E. x e. A ph <-> |^| { x e. A | ph } e. _V ) $=
    ( cv wcel wa wex cab cint cvv wrex crab intexab df-rex df-rab inteqi eleq1i
    3bitr4i ) BDCEAFZBGSBHZIZJEABCKABCLZIZJESBMABCNUCUAJUBTABCOPQR $.

  ${
    $d A x y $.  $d B y $.
    $( The existence of a class intersection. ` x ` is normally a free-variable
       parameter in ` B ` , which should be read ` B ( x ) ` .  (Contributed by
       FL, 19-Sep-2011.) $)
    iinexg $p |- ( ( A =/= (/) /\ A. x e. A B e. C )
    -> |^|_ x e. A B e. _V ) $=
      ( vy c0 wne wcel wral wa ciin cv wceq wrex cab cint cvv wex wi sylib abn0
      dfiin2g adantl elisset rgenw r19.2z mpan2 r19.35 imp rexcom4 sylibr intex
      eqeltrd ) BFGZCDHZABIZJZABCKZELCMZABNZEOZPZQUPURVBMUNAEBCDUBUCUQVAFGZVBQH
      UQUTERZVCUQUSERZABNZVDUNUPVFUNUOVESZABNZUPVFSUNVGABIVHVGABECDUDUEVGABUFUG
      UOVEABUHTUIUSAEBUJTUTEUAUKVAULTUM $.
  $}

  ${
    $d x y $.  $d x A $.  $d y ph $.  $d x ps $.  $d x ch $.
    intabs.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    intabs.2 $e |- ( x = |^| { y | ps } -> ( ph <-> ch ) ) $.
    intabs.3 $e |- ( |^| { y | ps } C_ A /\ ch ) $.
    $( Absorption of a redundant conjunct in the intersection of a class
       abstraction.  (Contributed by NM, 3-Jul-2005.) $)
    intabs $p |- |^| { x | ( x C_ A /\ ph ) } = |^| { x | ph } $=
      ( cv wss wa cab cint cvv wcel wceq sseq1 anbi12d intmin3 intnex ssv sseq2
      wn mpbiri sylbi pm2.61i cbvabv inteqi sseqtrri simpr ss2abi intss ax-mp
      eqssi ) DJZFKZALZDMZNZADMZNZUTBEMZNZVBVDOPZUTVDKZURVDFKZCLDVDOUPVDQUQVGAC
      UPVDFRHSITVEUDVDOQZVFVCUAVHVFUTOKUTUBVDOUTUCUEUFUGVAVCABDEGUHUIUJUSVAKVBU
      TKURADUQAUKULUSVAUMUNUO $.
  $}

  ${
    $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.)  (Proof shortened by Wolf Lammen,
       15-May-2025.) $)
    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 wel wa wex wcel ancom r19.41v bitr4i exbii
      elin eluniab eluni2 anbi1i 3bitr4i vex inex1 eleq2 ceqsexv rexbii rexcom4
      bitri 3bitr2i 3bitr4ri eqriv ) ECFZDGZAHZBHZDGZIZBCJZAKFZEALZVAMZANUTVCMZ
      BCJZANZEHZVBOVHUPOZVDVFAVDVAVCMVFVCVAPUTVCBCQRSVAAVHUAVIEBLZVHDOZMZBCJZVE
      ANZBCJVGVHUOOZVKMVJBCJZVKMVIVMVOVPVKBVHCUBUCVHUODTVJVKBCQUDVNVLBCVNVHUSOZ
      VLVCVQAUSURDBUEUFUQUSVHUGUHVHURDTUKUIVEBACUJULUMUN $.
  $}

  ${
    $d A x y $.  $d A y z $.
    $( Two equivalent ways to express the Power Set Axiom.  Note that ~ ax-pow
       is not used by the proof.  When ~ ax-pow is assumed and ` A ` is a set,
       both sides of the biconditional hold.  In ZF, both sides hold if and
       only if ` A ` is a set (see ~ pwexr ).  (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 wal pwidg wceq pweq eleq2d spcegv mpd wss bitri
      wi elex exlimiv impbii vex elpw2 pwss df-ss imbi1i albii exbii ) DEZFGZUK
      AHZEZGZAIZCBJCHDGTCKZBAJZTZBKZAIULUPULUKUKEZGZUPUKFLUOVBAUKFUMUKMUNVAUKUM
      UKNOPQUOULAUKUNUAUBUCUOUTAUOUKUMRZUTUKUMAUDUEVCBHZDRZURTZBKUTBDUMUFVFUSBV
      EUQURCVDDUGUHUISSUJS $.
  $}

  ${
    $d A x $.
    $( The power set of a set is never a subset.  (Contributed by Stefan
       O'Rear, 22-Feb-2015.)  (Proof shortened by BJ, 24-Jul-2025.) $)
    pwnss $p |- ( A e. V -> -. ~P A C_ A ) $=
      ( vx cpw wss wel wn crab wcel rru ssel mtoi rabelpw nsyl3 ) ADZAEZCCFGZCA
      HZOIZABIPSRAICAJOARKLQCABMN $.
  $}

  $( 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
    $.

  $( A difference is an element of the power set of its minuend.  (Contributed
     by AV, 9-Oct-2023.) $)
  difelpw $p |- ( A e. V -> ( A \ B ) e. ~P A ) $=
    ( wcel cdif cpw wss difss elpw2g mpbiri ) ACDABEZAFDKAGABHKACIJ $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Theorems requiring empty set existence
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x A $.
    $( The class of elements of ` A ` "such that ` A ` is a set" is a set.
       That class is equal to ` A ` when ` A ` is a set (see ~ class2seteq )
       and to the empty set when ` A ` is a proper class.  (Contributed by NM,
       16-Oct-2003.) $)
    class2set $p |- { x e. A | A e. _V } e. _V $=
      ( wcel crab rabexg wn c0 wrex wceq cv simpl nrexdv rabn0 necon1bbii sylib
      cvv 0ex eqeltrdi pm2.61i ) BPCZTABDZPCTABPETFZUAGPUBTABHZFUAGIUBTABUBAJBC
      KLUCUAGTABMNOQRS $.
  $}

  $( 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 $.

  $( A power class is never empty.  (Contributed by NM, 3-Sep-2018.) $)
  pwne0 $p |- ~P A =/= (/) $=
    ( c0 cpw 0elpw ne0ii ) BACADE $.

  $( 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 null set and the power set of the
     null set.  (Contributed by NM, 21-Jun-1993.) $)
  0inp0 $p |- ( A = (/) -> -. A = { (/) } ) $=
    ( c0 wceq csn wne 0nep0 neeq1 mpbiri neneqd ) ABCZABDZJAKEBKEFABKGHI $.

  $( 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 $=
    ( c0 csn cdif cuni cun uniun undif1 uncom eqtr2i unieqi 0ex uneq2i 3eqtr4ri
    unisn un0 uneq1i 3eqtri ) ABCZDZEZBAEZFZUBBFUBUASAFZEZSEZUBFUCTSFZEUAUFFZUE
    UATSGUDUGUGASFUDASHASIJKUHUABFUAUFBUABLOZMUAPJNSAGUFBUBUIQRBUBIUBPR $.

  $( If a class is equal to the singleton of its union, then its union exists.
     (Contributed by BTernaryTau, 24-Sep-2024.) $)
  eqsnuniex $p |- ( A = { U. A } -> U. A e. _V ) $=
    ( cuni csn wceq c0 wn wcel wa unieq uni0 eqtrdi sylan9eq sneqd 0inp0 adantl
    cvv pm2.65da snprc bicomi con2bii sylibr ) AABZCZDZUCEDZFUBPGZUDUEUCECDZUDU
    EHUBEUDUEUBUCBZEAUCIUEUHEBEUCEIJKLMUEUGFUDUCNOQUEUFUFFUEUBRSTUA $.

  ${
    $d x A $.
    $( An indexed intersection of the empty set, with a nonempty index set, is
       empty.  (Contributed by NM, 20-Oct-2005.) $)
    iin0 $p |- ( A =/= (/) <-> |^|_ x e. A (/) = (/) ) $=
      ( c0 wne ciin wceq iinconst cvv 0ex n0ii 0iin eqeq1i iineq1 eqeq1d mtbiri
      mtbir necon2ai impbii ) BCDABCEZCFZABCGTBCBCFZTACCEZCFZUCHCFCHIJUBHCACKLP
      UASUBCABCCMNOQR $.
  $}

  ${
    $d x A $.
    notzfaus.1 $e |- A = { (/) } $.
    notzfaus.2 $e |- ( ph <-> -. x e. y ) $.
    $( In the Separation Scheme ~ zfauscl , we require that ` y ` not occur in
       ` ph ` (which can be generalized to "not be free in").  Here we show
       special cases of ` A ` and ` ph ` that result in a contradiction if that
       requirement is not met.  (Contributed by NM, 8-Feb-2006.)  (Proof
       shortened by BJ, 18-Nov-2023.) $)
    notzfaus $p |- -. E. y A. x ( x e. y <-> ( x e. A /\ ph ) ) $=
      ( cv wcel wa wb wal wn wex c0 wne csn 0ex snnz eqnetri mpbi pm5.19 bibi2d
      n0 ibar bitr3di mtbiri eximii exnal nex ) BGZCGHZUJDHZAIZJZBKZCUNLZBMUOLU
      LUPBDNOULBMDNPNENQRSBDUCTULUNUKUKLZJUKUAULUMUQUKULAUMUQULAUDFUEUBUFUGUNBU
      HTUI $.
  $}

  $( 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 $.


$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
  ZF Set Theory - add the Axiom of Power Sets
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Introduce the Axiom of Power Sets
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x y z w $.
    $( Axiom of Power Sets.  An axiom of 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 ` .  The variant ~ axpow2
       uses explicit subset notation.  A version using class notation is
       ~ pwex .  (Contributed by NM, 21-Jun-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 wi wal wex ax-pow elequ1 imbi12d cbvalvw imbi1i albii exbii mpbi
      weq ) DBEZDCEZFZDGZBAEZFZBGZAHABEZACEZFZAGZUBFZBGZAHCABDIUDUJAUCUIBUAUHUB
      TUGDADAQRUESUFDABJDACJKLMNOP $.

    $( 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 df-ss imbi1i albii exbii mpbir ) CEZAEZ
      FZCBGZHZCIZBJDCGDAGHDIZTHZCIZBJABCDKUBUEBUAUDCSUCTDQRLMNOP $.

    $( 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 ) $=
      ( wel cv wss wb wal axpow2 sepexi bicom1 alimi eximii ) CBDZCEAEFZGZCHONG
      ZCHBOCBBABCIJPQCNOKLM $.
  $}

  ${
    $d x y z $.
    $( Alternate proof of ~ el using ~ ax-9 and ~ ax-pow instead of ~ ax-pr .
       (Contributed by NM, 4-Jan-2002.)  (Proof shortened by Andrew Salmon,
       25-Jul-2011.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    elALT2 $p |- E. y x e. y $=
      ( vz wel wi wal zfpow weq ax9 alrimiv ax8 embantd spimvw eximii ) BCDBADE
      ZBFZCBDZEZCFABDZBBCAGRSCACAHZPQSTOBCABIJCABKLMN $.
    $( $j usage 'elALT2' avoids 'ax-ext' 'ax-sep' 'ax-nul' 'ax-pr'; $)
  $}

  ${
    $d w x y z $.
    $( Alternate proof of ~ dtru using ~ ax-pow instead of ~ ax-pr .  See
       ~ dtruALT for another proof using ~ ax-pow instead of ~ ax-pr .
       (Contributed by NM, 7-Nov-2006.)  Avoid ~ ax-13 .  (Revised by BJ,
       31-May-2019.)  Avoid ~ ax-12 .  (Revised by Rohan Ridenour, 9-Oct-2024.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    dtruALT2 $p |- -. A. x x = y $=
      ( vw vz cv wceq wn wex wal wel wcel elALT2 ax-nul elequ1 notbid spw ax7v1
      wa con3d spimevw eximii exdistrv mpbir2an ax9v2 com12 con3dimp 2eximi weq
      wi equequ2 biimtrdi a1d pm2.61i exlimivv mp2b exnal mpbi ) AEZBEZFZGZAHZU
      TAIGACJZURDEZKZGZRZDHCHZCEZVDFZGZDHCHVBVHVCCHVFDHACLVFAIVFDDAMVFVIVDKZGAC
      URVIFZVEVLACDNOPUAVCVFCDUBUCVGVKCDVCVJVEVJVCVECDAUDUEUFUGVKVBCDDBUHZVKVBU
      IVNVKVIUSFZGZVBVNVJVODBCUJOVPVAACVMUTVOACBQSTUKVNGZVBVKVQVAADURVDFUTVNADB
      QSTULUMUNUOUTAUPUQ $.
    $( $j usage 'dtruALT2' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13' 'ax-ext'
       'ax-sep' 'ax-pr'; $)
  $}

  ${
    $d x y $.
    dtrucor.1 $e |- x = y $.
    $( Corollary of ~ dtru .  This example illustrates the danger of blindly
       trusting the standard Deduction Theorem without accounting for free
       variables: the theorem form of this deduction is not valid, as shown by
       ~ dtrucor2 .  (Contributed by NM, 27-Jun-2002.) $)
    dtrucor $p |- x =/= y $=
      ( weq cv wne wal dtruALT2 pm2.21i mpg ) ABDZAEBEFZAKAGLABHICJ $.
  $}

  ${
    dtrucor2.1 $e |- ( x = y -> x =/= y ) $.
    $( The theorem form of the deduction ~ dtrucor leads to a contradiction, as
       mentioned in the "Wrong!" example at ~ mmdeduction.html#bad .  Usage of
       this theorem is discouraged because it depends on ~ ax-13 .
       (Contributed by NM, 20-Oct-2007.)  (New usage is discouraged.) $)
    dtrucor2 $p |- ( ph /\ -. ph ) $=
      ( weq wex wn wa ax6e wi cv necon2bi pm2.01 ax-mp nex pm2.24ii ) BCEZBFAAG
      HBCIQBQQGZJRQBKCKDLQMNOP $.
  $}

  ${
    $d x y $.
    $( Demonstration of a theorem that requires the setvar variables ` x ` and
       ` y ` to be disjoint (but without any other disjointness conditions, and
       in particular, none on ` z ` ).

       That theorem bundles the theorems ( ` |- E. x ( x = y -> z e. x ) ` with
       ` x , y , z ` disjoint), often called its "principal instance", and the
       two "degenerate instances" ( ` |- E. x ( x = y -> x e. x ) ` with
       ` x , y ` disjoint) and ( ` |- E. x ( x = y -> y e. x ) ` with ` x , y `
       disjoint).

       Compare with ~ dvdemo2 , which has the same principal instance and one
       common degenerate instance but crucially differs in the other degenerate
       instance.

       See ~ https://us.metamath.org/mpeuni/mmset.html#distinct for details on
       the "disjoint variable" mechanism.  (The verb "bundle" to express this
       phenomenon was introduced by Raph Levien.)

       Note that ~ dvdemo1 is partially bundled, in that the pairs of setvar
       variables ` x , z ` and ` y , z ` need not be disjoint, and in spite of
       that, its proof does not require ~ ax-11 nor ~ ax-13 .  (Contributed by
       NM, 1-Dec-2006.)  (Revised by BJ, 13-Jan-2024.) $)
    dvdemo1 $p |- E. x ( x = y -> z e. x ) $=
      ( weq wn wel wi wex wal dtruALT2 exnal mpbir pm2.21 eximii ) ABDZEZOCAFZG
      APAHOAIEABJOAKLOQMN $.
  $}

  ${
    $d x z $.
    $( Demonstration of a theorem that requires the setvar variables ` x ` and
       ` z ` to be disjoint (but without any other disjointness conditions, and
       in particular, none on ` y ` ).

       That theorem bundles the theorems ( ` |- E. x ( x = y -> z e. x ) ` with
       ` x , y , z ` disjoint), often called its "principal instance", and the
       two "degenerate instances" ( ` |- E. x ( x = x -> z e. x ) ` with
       ` x , z ` disjoint) and ( ` |- E. x ( x = z -> z e. x ) ` with ` x , z `
       disjoint).

       Compare with ~ dvdemo1 , which has the same principal instance and one
       common degenerate instance but crucially differs in the other degenerate
       instance.

       See ~ https://us.metamath.org/mpeuni/mmset.html#distinct for details on
       the "disjoint variable" mechanism.

       Note that ~ dvdemo2 is partially bundled, in that the pairs of setvar
       variables ` x , y ` and ` y , z ` need not be disjoint, and in spite of
       that, its proof does not require any of the auxiliary axioms ~ ax-10 ,
       ~ ax-11 , ~ ax-12 , ~ ax-13 .  (Contributed by NM, 1-Dec-2006.)  Avoid
       ~ ax-13 .  (Revised by BJ, 13-Jan-2024.) $)
    dvdemo2 $p |- E. x ( x = y -> z e. x ) $=
      ( wel weq wi elALT2 ax-1 eximii ) CADZABEZJFACAGJKHI $.
  $}

  ${
    $d w x y z $.
    $( A setvar variable is not free from itself.  This theorem is not true in
       a one-element domain, as illustrated by the use of ~ dtruALT2 in its
       proof.  (Contributed by Mario Carneiro, 8-Oct-2016.) $)
    nfnid $p |- -. F/_ x x $=
      ( vy vz vw cv wel wb wal weq dtruALT2 ax-ext sps alimi mto wnf df-nfc wsb
      wnfc sbnf2 elsb2 bibi12i 2albii bitri albii alrot3 3bitri mtbir ) AAEZRZB
      CFZBDFZGZBHZDHZCHZUOCDIZCHCDJUNUPCUMUPDCDBKLMNUIBAFZAOZBHULDHCHZBHUOABUHP
      URUSBURUQACQZUQADQZGZDHCHUSUQACDSVBULCDUTUJVAUKACBTADBTUAUBUCUDULBCDUEUFU
      G $.
  $}

  $( The "distinctor" expression ` -. A. x x = y ` , stating that ` x ` and
     ` y ` are not the same variable, can be written in terms of ` F/ ` in the
     obvious way.  This theorem is not true in a one-element domain, because
     then ` F/_ x y ` and ` A. x x = y ` will both be true.  (Contributed by
     Mario Carneiro, 8-Oct-2016.)  Usage of this theorem is discouraged because
     it depends on ~ ax-13 .  (New usage is discouraged.) $)
  nfcvb $p |- ( F/_ x y <-> -. A. x x = y ) $=
    ( cv wnfc weq wal wn nfnid eqidd drnfc1 mtbiri con2i nfcvf impbii ) ABCZDZA
    BEAFZGQPQPBODBHABOOQOIJKLABMN $.

  ${
    $d x y z w $.
    $( 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 $=
      ( vw vy vz cv cpw wss cab cvv df-pw wceq wex wel wal axpow2 sepexi eqabbw
      wb sseq1 exbii mpbir issetri eqeltri ) AEZFBEZUDGZBHZIBUDJCUGCEZUGKZCLDCM
      DEZUDGZRDNZCLUKDCCACDOPUIULCUFUKBDUHUEUJUDSQTUAUBUC $.
    $( $j usage 'vpwex' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $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 $.
  $}

  ${
    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 $.
  $}

  $( Quantitative version of ~ pwexg : the powerset of an element of a class is
     an element of the double powerclass of the union of that class.  Exercise
     10 of [Enderton] p. 26.  (Contributed by NM, 13-Jan-2007.)  Remove use of
     ~ ax-nul and ~ ax-pr and shorten proof.  (Revised by BJ, 13-Apr-2024.) $)
  pwel $p |- ( A e. B -> ~P A e. ~P ~P U. B ) $=
    ( wcel cpw cuni cvv pwexg elssuni sspwd elpwd ) ABCZADBEZDFABGKALABHIJ $.

  ${
    $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 $.
  $}

  $( Alternate proof of ~ snex using Power Set ( ~ ax-pow ) instead of Pairing
     ( ~ ax-pr ).  Unlike in the proof of ~ zfpair , Replacement ( ~ ax-rep )
     is not needed.  (Contributed by NM, 7-Aug-1994.)  (Proof shortened by
     Andrew Salmon, 25-Jul-2011.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  snexALT $p |- { A } e. _V $=
    ( cpw cvv wcel csn wss snsspw ssexg mpan wn pwexg con3i c0 snprc biimpi 0ex
    wceq eqeltrdi syl pm2.61i ) ABZCDZAEZCDZUCUAFUBUDAGUCUACHIUBJACDZJZUDUEUBAC
    KLUFUCMCUFUCMQANOPRST $.

  $( The power set of the empty set (the ordinal 1) is a set.  See also
     ~ p0exALT .  (Contributed by NM, 23-Dec-1993.) $)
  p0ex $p |- { (/) } e. _V $=
    ( c0 cpw csn cvv pw0 0ex pwex eqeltrri ) ABACDEAFGH $.

  $( Alternate proof of ~ p0ex which is quite different and longer if ~ snexALT
     is expanded.  (Contributed by NM, 23-Dec-1993.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  p0exALT $p |- { (/) } e. _V $=
    ( c0 snexALT ) AB $.

  $( The power set of the power set of the empty set (the ordinal 2) is a set.
     (Contributed by NM, 24-Jun-1993.) $)
  pp0ex $p |- { (/) , { (/) } } e. _V $=
    ( c0 csn cpw cpr cvv pwpw0 p0ex pwex eqeltrri ) ABZCAJDEFJGHI $.

  $( The ordinal number 3 is a set, proved without the Axiom of Union ~ ax-un .
     (Contributed by NM, 2-May-2009.) $)
  ord3ex $p |- { (/) , { (/) } , { (/) , { (/) } } } e. _V $=
    ( c0 csn cpr ctp cun cvv df-tp cpw pwpr pp0ex pwex eqeltrri wss unss2 ax-mp
    snsspr2 ssexi eqeltri ) AABZASCZDTTBZEZFASTGUBTSBZTCZEZTHUEFASITJKLUAUDMUBU
    EMUCTPUAUDTNOQR $.

  ${
    $d x y $.
    $( Alternate proof of ~ dtru which requires more axioms but is shorter and
       may be easier to understand.  Like ~ dtruALT2 , it uses ~ ax-pow rather
       than ~ ax-pr .

       Assuming that ZF set theory is consistent, we cannot prove this theorem
       unless we specify that ` x ` and ` y ` be distinct.  Specifically,
       Theorem ~ spcev requires that ` x ` must not occur in the subexpression
       ` -. y = { (/) } ` in step 4 nor in the subexpression ` -. y = (/) ` in
       step 9.  The proof verifier will require that ` x ` and ` y ` be in a
       distinct variable group to ensure this.  You can check this by deleting
       the $d statement in set.mm and rerunning the verifier, which will print
       a detailed explanation of the distinct variable violation.  (Contributed
       by NM, 15-Jul-1994.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    dtruALT $p |- -. A. x x = y $=
      ( cv wceq wn wex wal c0 csn 0inp0 p0ex eqeq2 notbid spcev syl 0ex pm2.61i
      exnal eqcom albii xchbinx mpbi ) BCZACZDZEZAFZUDUCDZAGZEUCHDZUGUJUCHIZDZE
      ZUGUCJUFUMAUKKUDUKDUEULUDUKUCLMNOUFUJEAHPUDHDUEUJUDHUCLMNQUGUEAGUIUEARUEU
      HAUCUDSTUAUB $.
      $( $j usage 'dtruALT' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13' 'ax-pr'; $)
  $}

  ${
    $d x y $.
    $( This theorem shows that Axiom ~ ax-c16 is redundant in the presence of
       Theorem ~ dtruALT2 , which states simply that at least two things exist.
       This justifies the remark at ~ mmzfcnd.html#twoness (which links to this
       theorem).  (Proof modification is discouraged.)
       (New usage is discouraged.)  (Contributed by NM, 7-Nov-2006.) $)
    axc16b $p |- ( A. x x = y -> ( ph -> A. x ph ) ) $=
      ( weq wal wi dtruALT2 pm2.21i ) BCDBEAABEFBCGH $.
  $}

  ${
    $d x y $.  $d y ph $.
    $( Existential uniqueness implies there is a value for which the wff
       argument is false.  (Contributed by NM, 24-Oct-2010.)  (Proof shortened
       by BJ, 2-Jan-2023.) $)
    eunex $p |- ( E! x ph -> E. x -. ph ) $=
      ( vy weq wb wal wex wn weu dtruALT2 albi mtbiri exlimiv eu6 exnal 3imtr4i
      ) ABCDZEBFZCGABFZHZABIAHBGRTCRSQBFBCJAQBKLMABCNABOP $.
  $}

  ${
    $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 weu csb wi cvv nfcsb1v nfeq2 weq csbeq1a eqeq2d spcgf
      nfcv elv wex wnfc euex eqtr3d alrimivv sbnfc2 sylibr exlimiv syl ) BFZCGZ
      AHZBIULBUAACUBZULBUCULUMBULADFZCJZAEFZCJZGZEHDHUMULURDEULUJUOUQULUJUOGZKD
      UKUSAUNLAUNSAUJUOAUNCMNADOCUOUJAUNCPQRTULUJUQGZKEUKUTAUPLAUPSAUJUQAUPCMNA
      EOCUQUJAUPCPQRTUDUEADECUFUGUHUI $.

    $( 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 ) ) $=
      ( wceq wal weu wnfc cvv wcel eusvnf wex euex eqvisset sps exlimiv syl jca
      cv wa isset nfcvd id nfeqd nf5rd eximdv biimtrid imp eusv1 sylibr impbii
      ) BRZCDZAEZBFZACGZCHIZSZUNUOUPABCJUNUMBKZUPUMBLUMUPBULUPABCMNOPQUQURUNUOU
      PURUPULBKUOURBCTUOULUMBUOULAUOAUKCUOAUKUAUOUBUCUDUEUFUGABCUHUIUJ $.
  $}

  ${
    $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 ) $=
      ( wceq wal weu wex nfeu1 nfcvd eusvnf nfeqd nfrd 19.2 impbid1 eubid ibir
      cv ) BQZCDZAEZBFZSAGZBFUAUBTBTBHUAUBTUASAUAARCUAARIABCJKLSAMNOP $.
  $}

  ${
    $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 nfeuw wa isseti alrimi sylibr cvv
      nfe1 19.8a ancri eximii eupick mpan2 nf6 wcel dfnfc2 mpg eusvnfb mpbiran2
      wb eusv2i sylbir impbii ) BECFZAGZBHZACIZURUPAJZBKZUSURUTBUQBLURUQUPMZAKU
      TURVBAUQABUPATNURUQUPOZBGVBUPVCBBCDPUPUQUPAUAUBUCUQUPBUDUEQUPAUFRQCSUGZUS
      VAULAABCSUHDUIRUSUPAKBHZURVEUSVDDABCUJUKABCUMUNUO $.

    $( 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.)  (Proof shortened by JJ, 7-Aug-2021.) $)
    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 nfmov wcel wal rsp alrimdv
      com3l mo2icl syl6 rexlimi mormo reu5 rbaib 3syl ) ACEGABHFIZJZCEKZBLZULBD
      MZULBDNZULBDGZOAUMCEULCBUKCEPQCHERZAULUJJZBSUMUQAURBULUQAUJUKCETUBUAULBFU
      CUDUEULBDUFUOUPUNULBDUGUHUI $.
  $}

  ${
    $d x y z A $.  $d x z B $.  $d x z C $.  $d x z ph $.
    $( Lemma for ~ reusv2 .  (Contributed by NM, 22-Oct-2010.)  (Proof
       shortened by Mario Carneiro, 19-Nov-2016.) $)
    reusv2lem1 $p |- ( A =/= (/) -> ( E! x A. y e. A x = B
                     <-> E. x A. y e. A x = B ) ) $=
      ( c0 wne cv wceq wral wmo weu wex wb wcel n0 nfra1 nfmov wi wal syl com12
      rsp alrimiv mo2icl exlimi sylbi df-eu rbaib ) CEFZAGDHZBCIZAJZUKAKZUKALZM
      UIBGCNZBLULBCOUOULBUKBAUJBCPQUOUKUJRZASULUOUPAUKUOUJUJBCUBUAUCUKADUDTUEUF
      UMUNULUKAUGUHT $.

    $( Lemma for ~ reusv2 .  (Contributed by NM, 27-Oct-2010.)  (Proof
       shortened by Mario Carneiro, 19-Nov-2016.)  (Proof shortened by JJ,
       7-Aug-2021.) $)
    reusv2lem2 $p |- ( E! x A. y e. A x = B -> E! x E. y e. A x = B ) $=
      ( vz cv wceq wral weu wrex wi c0 wn wex sylib wa simpr nfra1 adantr ex wb
      wal eunex exnal rzal alrimiv nsyl3 pm2.21d wcel rspa eqtr4d eqeq1 ralbidv
      wne biimprcd ad2antrr mpd exp31 rexlimd adantl r19.2z impbid exlimdv euex
      eubidv cbvexvw impel mpbird pm2.61ine ) AFZDGZBCHZAIZVKBCJZAIZKCLCLGZVMVO
      VMVLAUBZVPVMVLMANVQMVLAUCVLAUDOVPVLAVKBCUEUFUGUHCLUNZVMVOVRVMPVOVMVRVMQVR
      EFZDGZBCHZENZVOVMUAZVMVRWAWCEVRWAWCVRWAPZVNVLAWDVNVLWAVNVLKVRWAVKVLBCVTBC
      RVKBCRWABFCUIZVKVLWAWEPZVKPZVJVSGZVLWGVJDVSWFVKQWFVTVKVTBCUJSUKWAWHVLKWEV
      KWHVLWAWHVKVTBCVJVSDULUMZUOUPUQURUSUTVRVLVNKWAVRVLVNVKBCVATSVBVETVCVMVLAN
      WBVLAVDVLWAAEWIVFOVGVHTVI $.

    $( Lemma for ~ reusv2 .  (Contributed by NM, 14-Dec-2012.)  (Proof
       shortened by Mario Carneiro, 19-Nov-2016.) $)
    reusv2lem3 $p |- ( A. y e. A B e. _V ->
          ( E! x E. y e. A x = B <-> E! x A. y e. A x = B ) ) $=
      ( cvv wcel wral cv wceq wrex weu wa simpr nfv nfeu1 nfan wi wex c0 ex wne
      euex rexn0 exlimiv r19.2z adantl nfra1 nfre1 nfeuw rsp impcom isset sylib
      3syl adantrr rspe ancrd eximdv syldan eupick syl2an2 com3l ralrimd impbid
      imp eubid mpbird reusv2lem2 impbid1 ) DEFZBCGZAHDIZBCJZAKZVLBCGZAKZVKVNVP
      VKVNLZVPVNVKVNMZVQVOVMAVKVNAVKANVMAOPVQVOVMVNVOVMQZVKVNVMARCSUAZVSVMAUBVM
      VTAVLBCUCUDVTVOVMVLBCUETUNUFVQVMVLBCVKVNBVJBCUGVMBAVLBCUHZUIPWABHCFZVQVMV
      LWBVQVMVLQZVQVNWBVMVLLZARZWCVRWBVQVLARZWEWBVKWFVNWBVKLVJWFVKWBVJVJBCUJUKA
      DULUMUOWBWFWEWBVLWDAWBVLVMWBVLVMVLBCUPTUQURVEUSVMVLAUTVATVBVCVDVFVGTABCDV
      HVI $.

    $( Lemma for ~ reusv2 .  (Contributed by NM, 13-Dec-2012.) $)
    reusv2lem4 $p |- ( E! x e. A E. y e. B ( ph /\ x = C )
            <-> E! x A. y e. B ( ( C e. A /\ ph ) -> x = C ) ) $=
      ( vz cv wceq wa wrex wcel weu wi wral anass eleq1 nfv cvv wal wreu df-reu
      crab anbi1i anbi1d pm5.32ri bitr3i anbi2i 3bitr4ri rexbii2 r19.42v nfrab1
      csb rabid nfcsb1v nfeq2 csbeq1a eqeq2d cbvrexfw 3bitr3i eubii wb ad2antrl
      nfcv elex sylbi rgen nfel1 eleq1d mpbi reusv2lem3 ax-mp df-ral nfcri nfim
      cbvralfw imbi12d cbvalv1 imbi1i impexp bitri albii bitr4i 3bitr2i 3bitri
      ) ABHZFIZJZCEKZBDUAWFDLZWIJZBMWFCGHZFUMZIZGFDLZAJZCEUCZKZBMZWPWGNZCEOZBMZ
      WIBDUBWKWRBWJWHJZCEKWGCWQKWKWRXCWGCEWQCHZELZWPJZWGJXEWPWGJZJXDWQLZWGJXEXC
      JXEWPWGPXHXFWGWPCEUNZUDXCXGXEXCWJAJZWGJXGWJAWGPWGXJWPWGWJWOAWFFDQUEUFUGUH
      UIUJWJWHCEUKWGWNCGWQWPCEULZGWQVDZWGGRCWFWMCWLFUOZUPZXDWLIZFWMWFCWLFUQZURZ
      USUTVAWSWNGWQOZBMZXBWMSLZGWQOZWSXSVBFSLZCWQOYAYBCWQXHXFYBXIWOYBXEAFDVEVCV
      FVGYBXTCGWQXKXLYBGRCWMSXMVHXOFWMSXPVIVPVJBGWQWMVKVLXRXABXRWLWQLZWNNZGTXHW
      GNZCTZXAWNGWQVMYEYDCGYEGRYCWNCCGWQXKVNXNVOXOXHYCWGWNXDWLWQQXQVQVRYFXEWTNZ
      CTXAYEYGCYEXFWGNYGXHXFWGXIVSXEWPWGVTWAWBWTCEVMWCWDVAWAWE $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x C $.
    $( Lemma for ~ reusv2 .  (Contributed by NM, 4-Jan-2013.)  (Proof shortened
       by Mario Carneiro, 19-Nov-2016.) $)
    reusv2lem5 $p |- ( ( A. y e. B C e. A /\ B =/= (/) )
          -> ( E! x e. A E. y e. B x = C <-> E! x e. A A. y e. B x = C ) ) $=
      ( wcel wral c0 wne wa wtru cv wceq wi weu wrex wreu wb tru eubidv bitr4di
      biimt mpan2 ibar bitr3d eleq1 pm5.32ri ralimi ralbi syl r19.28zv sylan9bb
      biantrur rexbii reubii reusv2lem4 bitri df-reu 3bitr4g ) ECFZBDGZDHIZJUTK
      JZALZEMZNZBDGZAOZVDCFZVEBDGZJZAOZVEBDPZACQZVJACQVAVHVIVEJZBDGZAOVBVLVAVGV
      PAVAVFVORZBDGVGVPRUTVQBDUTVFUTVEJZVOUTVEVFVRUTKVEVFRSVCVEUBUCUTVEUDUEVEVI
      UTVDECUFUGUAUHVFVOBDUIUJTVBVPVKAVIVEBDUKTULVNKVEJZBDPZACQVHVMVTACVEVSBDKV
      ESUMUNUOKABCDEUPUQVJACURUS $.
  $}

  ${
    $d x y z A $.  $d x z B $.  $d x z C $.  $d x z ph $.
    $( Two ways to express single-valuedness of a class expression ` C ( y ) `
       that is constant for those ` y e. B ` such that ` ph ` .  The first
       antecedent ensures that the constant value belongs to the existential
       uniqueness domain ` A ` , and the second ensures that ` C ( y ) ` is
       evaluated for at least one ` y ` .  (Contributed by NM, 4-Jan-2013.)
       (Proof shortened by Mario Carneiro, 19-Nov-2016.) $)
    reusv2 $p |- ( ( A. y e. B ( ph -> C e. A ) /\ E. y e. B ph )
          -> ( E! x e. A E. y e. B ( ph /\ x = C )
                   <-> E! x e. A A. y e. B ( ph -> x = C ) ) ) $=
      ( vz wcel wi wral cv wceq wa wrex wreu nfv cbvralfw imbi1i bitri bitr3i
      csb crab c0 wne wb nfrab1 nfcv nfcsb1v nfel1 csbeq1a eleq1d rabid ralbii2
      impexp rabn0 reusv2lem5 nfeq2 eqeq2d cbvrexfw anbi1i anass rexbii2 reubii
      3bitr3g syl2anbr ) AFDHZIZCEJZCGKZFUAZDHZGACEUBZJZVLUCUDZABKZFLZMZCENZBDO
      ZAVPIZCEJZBDOZUEACENVMVFCVLJVHVFVKCGVLACEUFZGVLUGZVFGPCVJDCVIFUHZUICKZVIL
      ZFVJDCVIFUJZUKQVFVGCVLEWFVLHZVFIWFEHZAMZVFIWJVGIWIWKVFACEULZRWJAVFUNSUMTA
      CEUOVMVNMVOVJLZGVLNZBDOWMGVLJZBDOVSWBBGDVLVJUPWNVRBDWNVPCVLNVRVPWMCGVLWCW
      DVPGPZCVOVJWEUQZWGFVJVOWHURZUSVPVQCVLEWIVPMWKVPMWJVQMWIWKVPWLUTWJAVPVASVB
      TVCWOWABDWOVPCVLJWAVPWMCGVLWCWDWPWQWRQVPVTCVLEWIVPIWKVPIWJVTIWIWKVPWLRWJA
      VPUNSUMTVCVDVE $.
  $}

  ${
    $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 cbvralvw biimpi raaanv anim12 2ralimi
      eqtr2 syl6 sylbir mpdan rexlimivw ) ACLZHMZNZDGOZABPZHIMZNZEGODGOZCFULBUI
      IMZNZEGOZUPULUSUKURDEGDLELMZABUJUQJUTHIUIKQRSTULUSPUKURPZEGODGOUPUKURDEGU
      AVAUODEGGVAUMUJUQPUNAUJBUQUBUIHIUDUEUCUFUGUH $.

    $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 cbvrexvw nfra2w risset
      anbi12d 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
      FLZMZEGNVOVTPZVKWBDEGDRERZOZABVJWAJWEHIFKUAUEUBWBWCEGVOVTEVNDEGGUCVTEUFUG
      WDGLZBWAWCWAVPIOZCFNZWFBMZWCCIFUDWIVOWHVTWIVOWHVTPWIVOMZWGVSCFWJVSWGAVMPZ
      DGQZWFBVOWLVOWFBWLVOBWLPZEGQZWFWMPVOVNDGQZEGQWNVNDEGGUHWOWMEGWOBWKPZDGQWM
      VNWPDGVNABVMPPWPABVMUIABVMUJSTBWKDGUKSTSWMEGULUMUNUOWGVRWKDGWGVQVMAWGVQIH
      OVMVPIHUPIHUQURVGUSUTVAVBVCVDVEVFUMABCDEFGHIJKVHVI $.
  $}

  ${
    $d x y A $.  $d x B $.
    eusv4.1 $e |- B e. _V $.
    $( Two ways to express single-valuedness of a class expression
       ` B ( y ) ` .  (Contributed by NM, 27-Oct-2010.) $)
    eusv4 $p |- ( E! x E. y e. A x = B <-> E! x A. y e. A x = B ) $=
      ( cvv wcel cv wceq wrex weu wral wb reusv2lem3 a1i mprg ) DFGZAHDIZBCJAKR
      BCLAKMBCABCDNQBHCGEOP $.
  $}

  ${
    $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.)  (Proof
       shortened by JJ, 7-Aug-2021.) $)
    ralxfrd $p |- ( ph -> ( A. x e. B ps <-> A. y e. C ch ) ) $=
      ( wral cv wcel wa wceq wb adantlr ralrimdva wrex rspcdv wi r19.29 impcomd
      exbiri rexlimdvw syl5 adantr mpan2d impbid ) ABDGLZCEHLZAUKCEHAEMHNZOBCDF
      GIADMZFPZBCQUMKRUASAULBDGAUNGNZOULUOEHTZBJAULUQOZBUBUPURCUOOZEHTABCUOEHUC
      AUSBEHAUOCBAUOBCKUEUDUFUGUHUISUJ $.

    $( Transfer existential 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 ) ) $=
      ( wn wral wrex cv wceq wa notbid ralxfrd dfrex2 3bitr4g ) ABLZDGMZLCLZEHM
      ZLBDGNCEHNAUCUEAUBUDDEFGHIJADOFPQBCKRSRBDGTCEHTUA $.
  $}

  ${
    $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 existential 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 ) ) $=
      ( wn wral wrex cv wceq wa notbid dfrex2 ralxfr2d 3bitr4g ) ABMZDGNZMCMZEH
      NZMBDGOCEHOAUDUFAUCUEDEFGHIJKADPFQRBCLSUASBDGTCEHTUB $.
  $}

  ${
    $d x A $.  $d x y B $.  $d x C $.  $d x ch $.  $d x y ph $.  $d y ps $.
    ralxfrd2.1 $e |- ( ( ph /\ y e. C ) -> A e. B ) $.
    ralxfrd2.2 $e |- ( ( ph /\ x e. B ) -> E. y e. C x = A ) $.
    ralxfrd2.3 $e |- ( ( ph /\ y e. C /\ x = A ) -> ( ps <-> ch ) ) $.
    $( Transfer universal quantification from a variable ` x ` to another
       variable ` y ` contained in expression ` A ` .  Variant of ~ ralxfrd .
       (Contributed by Alexander van der Vekens, 25-Apr-2018.) $)
    ralxfrd2 $p |- ( ph -> ( A. x e. B ps <-> A. y e. C ch ) ) $=
      ( wral cv wcel wa wceq wb 3expa ralrimdva wrex rspcdv ad4ant134 rexlimdva
      r19.29 exbiri impcomd syl5 mpan2d impbid ) ABDGLZCEHLZAUJCEHAEMHNZOBCDFGI
      AULDMZFPZBCQZKRUASAUKBDGAUMGNZOZUKUNEHTZBJUKUROCUNOZEHTUQBCUNEHUDUQUSBEHU
      QULOZUNCBUTUNBCAULUNUOUPKUBUEUFUCUGUHSUI $.

    $( Transfer existence from a variable ` x ` to another variable ` y `
       contained in expression ` A ` .  Variant of ~ rexxfrd .  (Contributed by
       Alexander van der Vekens, 25-Apr-2018.) $)
    rexxfrd2 $p |- ( ph -> ( E. x e. B ps <-> E. y e. C ch ) ) $=
      ( wn wral wrex cv wcel wceq w3a notbid dfrex2 ralxfrd2 3bitr4g ) ABLZDGMZ
      LCLZEHMZLBDGNCEHNAUDUFAUCUEDEFGHIJAEOHPDOFQRBCKSUASBDGTCEHTUB $.
  $}

  ${
    $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 $.

    $( Alternate proof of ~ ralxfr which 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 wn wral dfrex2 cv wceq notbid ralxfr xchbinxr bitr4i ) ACFKALZCFMZ
      LBDGKZACFNUCBLZDGMUBBDGNUAUDCDEFGHICOEPABJQRST $.
  $}

  ${
    $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 ) $.
    $( Membership in a restricted class abstraction after substituting an
       expression ` A ` (containing ` y ` ) for ` x ` in the formula defining
       the class abstraction.  (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 ) $.
    $( Membership in a restricted class abstraction after substituting an
       expression ` A ` (containing ` y ` ) for ` x ` in the formula defining
       the class abstraction.  (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 the restricted existential uniqueness
       hypotheses in ~ riotaxfrd .  (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 elexd eueq sylib eleq1 syl5ibrcom pm4.71rd
      wb 3expa pm5.32da bitr4d eubidv mpbid df-reu sylibr ) ABIZFJZKZCIZFJZUJDL
      ZKZCMZUOCFNULUMELZCMZUQULEOJUSULEFGPCEQRULURUPCULURUNURKUPULURUNULUNUREFJ
      GUMEFSTUAULUNUOURAUKUNUOURUBHUCUDUEUFUGUOCFUHUI $.
  $}

  ${
    $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 the restricted existential uniqueness
       hypotheses in ~ reuxfr1 .  (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 $.
  $}

  ${
    $d x z w v $.  $d y z w v $.
    $( The Axiom of Pairing of Zermelo-Fraenkel set theory.  Axiom 2 of
       [TakeutiZaring] p. 15.  In some textbooks this is stated as a separate
       axiom; here we show it is redundant since it can be derived from the
       other axioms.

       This theorem should not be referenced by any proof other than
       ~ axprALT .  Instead, use ~ zfpair2 below so that the uses of the Axiom
       of Pairing can be more easily identified.  (Contributed by NM,
       18-Oct-1995.)  (New usage is discouraged.) $)
    zfpair $p |- { x , y } e. _V $=
      ( vw vz vv cv cpr weq wo cab cvv dfpr2 c0 wceq wex isseti mpbiran equequ2
      wa 19.41v csn 19.43 prlem2 exbii 0ex p0ex orbi12i 3bitr3ri abbii eqeltrri
      pp0ex wi wal 0inp0 prlem1 alrimdv spimevw orcom con2i syl7bi jaoi eqeltri
      zfrep4 ) AFZBFZGCAHZCBHZIZCJZKCVDVELVIDFZMNZVJMUAZNZIZVKVFSZVMVGSZIZSZDOZ
      CJKVHVSCVQDOVODOZVPDOZIVSVHVOVPDUBVQVRDVKVFVMVGUCUDVTVFWAVGVTVKDOVFDMUEPV
      KVFDTQWAVMDOVGDVLUFPVMVGDTQUGUHUIVNVQDCEMVLGVNDJKDMVLLUKUJVKVQCEHZULZCUMZ
      EOVMVKWDEAEAHZVKWCCWEVKVFVMVGWBEACRVJUNZUOUPUQVMWDEBEBHZVMWCCVQVPVOIWGVMW
      BVOVPURWGVMVGVKVFWBEBCRVKVMWFUSUOUTUPUQVAVCVBVB $.

    $( Alternate proof of ~ axpr .  (Contributed by NM, 14-Nov-2006.)
       (New usage is discouraged.)  (Proof modification is discouraged.) $)
    axprALT $p |- E. z A. w ( ( w = x \/ w = y ) -> w e. z ) $=
      ( cv cpr wceq wo wcel wi wal zfpair isseti dfcleq vex bibi2i biimpr sylbi
      wb elpr alimi eximii ) CEZAEZBEZFZGZDEZUDGUHUEGHZUHUCIZJZDKZCCUFABLMUGUJU
      HUFIZSZDKULDUCUFNUNUKDUNUJUISUKUMUIUJUHUDUEDOTPUJUIQRUARUB $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Derive the Axiom of Pairing
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x y z w $.
    $( Lemma for ~ axpr .  There exists a set to which all empty sets belong.
       (Contributed by Rohan Ridenour, 10-Aug-2023.)  (Revised by BJ,
       13-Aug-2023.) $)
    axprlem1 $p |- E. x A. y ( A. z -. z e. y -> y e. x ) $=
      ( vw wel wn wal wex ax-pow pm2.21 alimi a1i imim1d alimdv eximdv exlimiiv
      wi mpi ax-nul ) CDEZFCGZCBEZFZCGZBAEZQZBGZAHZDUAUBTQZCGZUEQZBGZAHUHDABCIU
      AULUGAUAUKUFBUAUDUJUEUDUJQUAUCUICUBTJKLMNORDCSP $.
    $( $j usage 'axprlem1' avoids 'ax-8' 'ax-9' 'ax-12' 'ax-ext' 'ax-sep'; $)
  $}

  ${
    $d x y z w v $.
    $( Lemma for ~ axpr .  There exists a set to which all sets whose only
       members are empty sets belong.  (Contributed by Rohan Ridenour,
       9-Aug-2023.)  (Revised by BJ, 13-Aug-2023.) $)
    axprlem2 $p |- E. x A. y ( A. z e. y A. w -. w e. z -> y e. x ) $=
      ( vv wel wn wal wi cv wral wex ax-pow df-ral imim2 al2imi biimtrid imim1d
      alimdv eximdv mpi axprlem1 exlimiiv ) DCFGDHZCEFZIZCHZUDCBJZKZBAFZIZBHZAL
      ZEUGCBFZUEIZCHZUJIZBHZALUMEABCMUGURULAUGUQUKBUGUIUPUJUIUNUDIZCHUGUPUDCUHN
      UFUSUOCUDUEUNOPQRSTUAECDUBUC $.
    $( $j usage 'axprlem2' avoids 'ax-8' 'ax-9' 'ax-12' 'ax-ext' 'ax-sep'; $)
  $}

  ${
    $d x z w $.  $d y z w $.  $d z w n $.  $d z w s p $.
    $( Lemma for ~ axpr .  Eliminate the antecedent of the relevant replacement
       instance.  (Contributed by Rohan Ridenour, 10-Aug-2023.)  (Proof
       shortened by Matthew House, 18-Sep-2025.) $)
    axprlem3 $p |- E. z A. w ( w e. z <->
                    E. s ( s e. p /\ if- ( E. n n e. s , w = x , w = y ) ) ) $=
      ( wel wex weq wif wi wal wa wb biimpd equeuclr syl9r alrimdv spimevw mpg
      axrep4v ifptru wn ifpfal pm2.61i ) EFHEIZDAJZDBJZKZDCJZLZDMZCIZDCHFGHUJNF
      IODMCIFUJFDCGUBUGUNUGUMCACAJZUGULDUGUJUHUOUKUGUJUHUGUHUIUCPCDAQRSTUGUDZUM
      CBCBJZUPULDUPUJUIUQUKUPUJUIUGUHUIUEPCDBQRSTUFUA $.
    $( $j usage 'axprlem3' avoids 'ax-8' 'ax-9' 'ax-12' 'ax-ext'; $)
  $}

  ${
    $d w s $.  $d v s $.
    axprlem4.1 $e |- E. s A. n ph $.
    axprlem4.2 $e |- ( ph -> ( n e. s -> A. t -. t e. n ) ) $.
    axprlem4.3 $e |- ( A. n ph ->
      ( if- ( E. n n e. s , w = x , w = y ) <-> w = v ) ) $.
    $( Lemma for ~ axpr .  If an existing set of empty sets corresponds to one
       element of the pair, then the element is included in any superset of the
       set whose existence is asserted by the axiom of replacement.
       (Contributed by Rohan Ridenour, 10-Aug-2023.)  (Revised by BJ,
       13-Aug-2023.)  (Revised by Matthew House, 18-Sep-2025.) $)
    axprlem4 $p |- ( A. s ( A. n e. s A. t -. t e. n -> s e. p ) -> ( w = v ->
      E. s ( s e. p /\ if- ( E. n n e. s , w = x , w = y ) ) ) ) $=
      ( wel wn wal cv wi wa wex weq wral wif alimi df-ral sylibr imim1i aleximi
      ancrd mpi biimprcd anim2d eximdv syl5com ) FGMNFOZGHPZUAZHIMZQZHOZUQAGOZR
      ZHSZDETZUQGHMZGSDBTDCTUBZRZHSUSUTHSVBJURUTVAHURUTUQUTUPUQUTVDUNQZGOUPAVGG
      KUCUNGUOUDUEUFUHUGUIVCVAVFHVCUTVEUQUTVEVCLUJUKULUM $.
    $( $j usage 'axprlem4' avoids 'ax-8' 'ax-9' 'ax-12' 'ax-ext'; $)
  $}

  ${
    $d x z w s p $.  $d y z w s p $.  $d z w t n s p $.
    $( Unabbreviated version of the Axiom of Pairing of ZF set theory, derived
       as a theorem from the other axioms.

       This theorem should not be referenced by any proof.  Instead, use
       ~ ax-pr below so that the uses of the Axiom of Pairing can be more
       easily identified.

       For a shorter proof using ~ ax-ext , see ~ axprALT .  (Contributed by
       NM, 14-Nov-2006.)  Remove dependency on ~ ax-ext .  (Revised by Rohan
       Ridenour, 10-Aug-2023.)  (Proof shortened by BJ, 13-Aug-2023.)  (Proof
       shortened by Matthew House, 18-Sep-2025.)  Use ~ ax-pr instead.
       (New usage is discouraged.) $)
    axpr $p |- E. z A. w ( ( w = x \/ w = y ) -> w e. z ) $=
      ( vt vn vs vp wel wn wal cv wral wi weq wo wex wb ax-nul axprlem4 wa exbi
      wif axprlem3 axprlem1 sepexi biimp mpbiri ifptru pm2.21 alnex ifpfal jaod
      syl sylbi imbi2 syl5ibrcom alimdv eximdv mpi axprlem2 exlimiiv ) EFIJEKZF
      GLMGHIZNGKZDAOZDBOZPZDCIZNZDKZCQZHVEVIVDFGIZFQZVFVGUCZUAGQZRZDKZCQVLABCDF
      GHUDVEVRVKCVEVQVJDVEVJVQVHVPNVEVFVPVGVMVCRZABDAEFGHVCFGGGFEUEUFVMVCUGVSFK
      ZVNVOVFRVTVNVCFQFESVMVCFUBUHVNVFVGUIUNTVMJZABDBEFGHGFSVMVCUJWAFKVNJVOVGRV
      MFUKVNVFVGULUOTUMVIVPVHUPUQURUSUTHGFEVAVB $.
    $( $j usage 'axpr' avoids 'ax-8' 'ax-9' 'ax-10' 'ax-11' 'ax-12'
       'ax-ext'; $)
  $}

  ${
    $d x z w $.  $d y z w $.  $d z w n $.  $d z w s p $.
    $( Obsolete version of ~ axprlem3 as of 18-Sep-2025.  (Contributed by Rohan
       Ridenour, 10-Aug-2023.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    axprlem3OLD $p |- E. z A. w ( w e. z <->
                    E. s ( s e. p /\ if- ( E. n n e. s , w = x , w = y ) ) ) $=
      ( cv wcel wex weq wi wal wa ax6evr biimpd alrimiv expcom eximdv mpi wb wn
      wif axrep4 ifptru equtrr sylan9r ifpfal adantl simpl equtr syl6ci pm2.61i
      nfv mpg ) EHFHZIEJZDAKZDBKZUCZDCKZLZDMZCJZDHCHIUPGHIUTNFJUADMCJFUTFDCGUTC
      UNUDUQVDUQACKZCJVDCAOUQVEVCCVEUQVCVEUQNVBDUQUTURVEVAUQUTURUQURUSUEPACDUFU
      GQRSTUQUBZBCKZCJVDCBOVFVGVCCVGVFVCVGVFNZVBDVHUTUSVGVAVFUTUSLVGVFUTUSUQURU
      SUHPUIVGVFUJDBCUKULQRSTUMUO $.
    $( $j usage 'axprlem3OLD' avoids 'ax-ext'; $)
  $}

  ${
    $d x s $.  $d w s $.  $d t n s $.
    $( Obsolete version of ~ axprlem4 as of 18-Sep-2025.  (Contributed by Rohan
       Ridenour, 10-Aug-2023.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    axprlem4OLD $p |- ( ( A. s ( A. n e. s A. t -. t e. n -> s e. p )
      /\ w = x ) -> E. s ( s e. p /\ if- ( E. n n e. s , w = x , w = y ) ) ) $=
      ( wel wn wal cv wral wi weq wa wex nfa1 sp eximd mpi wb wif axprlem1 nfan
      bm1.3iiOLD biimp alimi df-ral sylibr adantrr ax-nul biimprd simprr ifptru
      nfv mpan9 biimpar syl2an2r jca expcom ) DEHIDJZEFKZLZFGHZMZFJZCANZOZEFHZV
      AUAZEJZFPVDVIEPZVGCBNZUBZOZFPVAFEFEDUCUEVHVKVOFVFVGFVEFQVGFUOUDVKVHVOVKVH
      OVDVNVKVFVDVGVKVCVFVDVKVIVAMZEJVCVJVPEVIVAUFUGVAEVBUHUIVEFRUPUJVKVLVHVGVN
      VKVAEPVLEDUKVKVAVIEVJEQVKVIVAVJERULSTVKVFVGUMVLVNVGVLVGVMUNUQURUSUTST $.
    $( $j usage 'axprlem4OLD' avoids 'ax-ext'; $)
  $}

  ${
    $d y s $.  $d w s $.  $d n s $.
    $( Obsolete version of ~ axprlem4 as of 18-Sep-2025.  (Contributed by Rohan
       Ridenour, 10-Aug-2023.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    axprlem5OLD $p |- ( ( A. s ( A. n e. s A. t -. t e. n -> s e. p )
      /\ w = y ) -> E. s ( s e. p /\ if- ( E. n n e. s , w = x , w = y ) ) ) $=
      ( wel wn wal cv wral wi weq wa wex wif ax-nul nfa1 nfv nfan pm2.21 adantr
      alimi df-ral sylibr ad2antrl mpd simpl alnex sylib simprr biimpar syl2anc
      sp ifpfal jca expcom eximd mpi ) DEHIDJZEFKZLZFGHZMZFJZCBNZOZEFHZIZEJZFPV
      DVIEPZCANZVGQZOZFPFERVHVKVOFVFVGFVEFSVGFTUAVKVHVOVKVHOZVDVNVPVCVDVPVIVAMZ
      EJZVCVKVRVHVJVQEVIVAUBUDUCVAEVBUEUFVFVEVKVGVEFUOUGUHVPVLIZVGVNVPVKVSVKVHU
      IVIEUJUKVKVFVGULVSVNVGVLVMVGUPUMUNUQURUSUT $.
    $( $j usage 'axprlem5OLD' avoids 'ax-ext'; $)
  $}

  ${
    $d x z w s p $.  $d y z w s p $.  $d z w t n s p $.
    $( Obsolete version of ~ axpr as of 18-Sep-2025.  (Contributed by NM,
       14-Nov-2006.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    axprOLD $p |- E. z A. w ( ( w = x \/ w = y ) -> w e. z ) $=
      ( vt vn vs vp wel wn wal cv wral wi weq wo wex wif wa wb biimpr eximii ex
      axprlem3OLD alimi axprlem4OLD axprlem5OLD jaodan imim1d alimdv eximdv mpi
      axprlem2 exlimiiv ) EFIJEKFGLMGHIZNGKZDAOZDBOZPZDCIZNZDKZCQZHUPUOFGIFQUQU
      RRSGQZUTNZDKZCQVCUTVDTZDKVFCABCDFGHUDVGVEDUTVDUAUEUBUPVFVBCUPVEVADUPUSVDU
      TUPUSVDUPUQVDURABDEFGHUFABDEFGHUGUHUCUIUJUKULHGFEUMUN $.
    $( $j usage 'axprOLD' avoids 'ax-ext'; $)
  $}

  ${
    $d x z w $.  $d y z w $.
    $( The Axiom of Pairing of ZF set theory.  It was derived as Theorem ~ axpr
       above and is therefore redundant, but we state it as a separate axiom
       here so that its uses can be identified more easily.  (Contributed by
       NM, 14-Nov-2006.) $)
    ax-pr $a |- E. z A. w ( ( w = x \/ w = y ) -> w e. z ) $.
    $( $j restatement 'ax-pr' of 'axpr'; $)

    $( Derive the abbreviated version of the Axiom of Pairing from ~ ax-pr .
       See ~ zfpair for its derivation from the other axioms.  (Contributed by
       NM, 14-Nov-2006.) $)
    zfpair2 $p |- { x , y } e. _V $=
      ( vz vw cv cpr wceq wex wcel wo wb wal ax-pr sepexi dfcleq vex elpr albii
      bibi2i bitri exbii mpbir issetri ) CAEZBEZFZCEZUFGZCHDEZUGIZUIUDGUIUEGJZK
      ZDLZCHUKDCCABCDMNUHUMCUHUJUIUFIZKZDLUMDUGUFOUOULDUNUKUJUIUDUEDPQSRTUAUBUC
      $.
  $}

  $( 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 $.
    $( A singleton built on a set is a set.  Special case of ~ snex which does
       not require ~ ax-nul and is intuitionistically valid.  (Contributed by
       NM, 7-Aug-1994.)  (Revised by Mario Carneiro, 19-May-2013.)  Extract
       from ~ snex and shorten proof.  (Revised by BJ, 15-Jan-2025.) $)
    snexg $p |- ( A e. V -> { A } e. _V ) $=
      ( vx csn cvv wcel cv wceq sneq vsnex eqeltrrdi vtocleg ) ADZEFCABCGZAHMND
      ENAICJKL $.
  $}

  $( A singleton is a set.  Theorem 7.12 of [Quine] p. 51, proved using
     Extensionality, Separation, Null Set, and Pairing.  See also ~ snexALT .
     (Contributed by NM, 7-Aug-1994.)  (Revised by Mario Carneiro,
     19-May-2013.) $)
  snex $p |- { A } e. _V $=
    ( cvv wcel csn snexg wn c0 wceq snprc biimpi 0ex eqeltrdi pm2.61i ) ABCZADZ
    BCABENFZOGBPOGHAIJKLM $.

  ${
    $d x A $.  $d x y B $.
    $( The Axiom of Pairing using class variables.  Theorem 7.13 of [Quine]
       p. 51.  By virtue of its definition, an unordered pair remains a set
       (even though no longer a pair) even when its components are proper
       classes (see ~ prprc ), so we can dispense with hypotheses requiring
       them to be sets.  (Contributed by NM, 15-Jul-1993.) $)
    prex $p |- { A , B } e. _V $=
      ( vx vy cvv wcel wi cv wceq preq2 eleq1d zfpair2 vtoclg preq1 imbitrid wn
      cpr csn snex eqeltrdi vtocleg prprc1 prprc2 pm2.61nii ) AEFZBEFZABQZEFZUF
      UHGCAEUFCHZBQZEFZUIAIZUHUIDHZQZEFUKDBEUMBIUNUJEUMBUIJKCDLMULUJUGEUIABNKOU
      AUEPUGBREABUBBSTUFPUGAREABUCASTUD $.
  $}

  ${
    $d x y z $.
    $( There exist two sets, one a member of the other.

       This theorem looks similar to ~ el , but its meaning is different.  It
       only depends on the axioms ~ ax-mp to ~ ax-4 , ~ ax-6 , and ~ ax-pr .
       This theorem does not exclude that these two sets could actually be one
       single set containing itself.  That two different sets exist is proved
       by ~ exexneq .  (Contributed by SN, 23-Dec-2024.) $)
    exel $p |- E. y E. x x e. y $=
      ( vz weq wo wel wi wal wex ax-pr ax6ev pm2.07 eximii exim mpi ) ACDZPEZAB
      FZGAHZRAIZBCCBAJSQAITPQAACKPLMQRANOM $.
    $( $j usage 'exel' avoids 'ax-5' 'ax-7' 'ax-8' 'ax-9' 'ax-10' 'ax-11'
       'ax-12' 'ax-13' 'ax-ext' 'ax-sep' 'ax-nul' 'ax-pow'; $)
  $}

  ${
    $d x y z $.
    $( There exist two different sets.  (Contributed by NM, 7-Nov-2006.)  Avoid
       ~ ax-13 .  (Revised by BJ, 31-May-2019.)  Avoid ~ ax-8 .  (Revised by
       SN, 21-Sep-2023.)  Avoid ~ ax-12 .  (Revised by Rohan Ridenour,
       9-Oct-2024.)  Use ~ ax-pr instead of ~ ax-pow .  (Revised by
       BTernaryTau, 3-Dec-2024.)  Extract this result from the proof of
       ~ dtru .  (Revised by BJ, 2-Jan-2025.) $)
    exexneq $p |- E. x E. y -. x = y $=
      ( vz wel wex wn wal wa weq ax-nul exdistrv mpbir2an ax9v1 eximdv imbitrdi
      exel df-ex com12 con2d imp 2eximi ax-mp ) CADZCEZCBDZFCGZHZBEAEZABIZFZBEA
      EUHUDAEUFBECAPBCJUDUFABKLUGUJABUDUFUJUDUIUFUIUDUFFZUIUDUECEUKUIUCUECABCMN
      UECQORSTUAUB $.
    $( $j usage 'exexneq' avoids 'ax-7' 'ax-8' 'ax-10' 'ax-11' 'ax-12' 'ax-13'
       'ax-ext' 'ax-sep' 'ax-pow'; $)
  $}

  ${
    $d w x y z $.
    $( Given any set (the " ` y ` " in the statement), there exists a set not
       equal to it.

       The same statement without disjoint variable condition is false, since
       we do not have ` E. x -. x = x ` .  This theorem is proved directly from
       set theory axioms (no class definitions) and does not depend on
       ~ ax-ext , ~ ax-sep , or ~ ax-pow nor auxiliary logical axiom schemes
       ~ ax-10 to ~ ax-13 .  See ~ dtruALT for a shorter proof using more
       axioms, and ~ dtruALT2 for a proof using ~ ax-pow instead of ~ ax-pr .
       (Contributed by NM, 7-Nov-2006.)  Avoid ~ ax-13 .  (Revised by BJ,
       31-May-2019.)  Avoid ~ ax-8 .  (Revised by SN, 21-Sep-2023.)  Avoid
       ~ ax-12 .  (Revised by Rohan Ridenour, 9-Oct-2024.)  Use ~ ax-pr instead
       of ~ ax-pow .  (Revised by BTernaryTau, 3-Dec-2024.)  Extract this
       result from the proof of ~ dtru .  (Revised by BJ, 2-Jan-2025.) $)
    exneq $p |- E. x -. x = y $=
      ( vz vw weq wn wex exexneq equeuclr con3d ax7v1 spimevw syl6 a1d exlimivv
      wi pm2.61i ax-mp ) CDEZFZDGCGABEZFZAGZCDHTUCCDDBEZTUCPUDTCBEZFZUCUDUESDCB
      IJUFUBACACEUAUEACBKJLMUDFZUCTUGUBADADEUAUDADBKJLNQOR $.
    $( $j usage 'exneq' avoids 'ax-8' 'ax-10' 'ax-11' 'ax-12' 'ax-13' 'ax-ext'
       'ax-sep' 'ax-pow' 'df-clab' 'df-cleq' 'df-clel'; $)
  $}

  ${
    $d x y $.
    $( Given any set (the " ` y ` " in the statement), not all sets are equal
       to it.  The same statement without disjoint variable condition is false
       since it contradicts ~ stdpc6 .  The same comments and revision history
       concerning axiom usage as in ~ exneq apply.  See ~ dtruALT and
       ~ dtruALT2 for alternate proofs avoiding ~ ax-pr .  (Contributed by NM,
       7-Nov-2006.)  Extract ~ exneq as an intermediate result.  (Revised by
       BJ, 2-Jan-2025.) $)
    dtru $p |- -. A. x x = y $=
      ( weq wn wex wal exneq exnal mpbi ) ABCZDAEJAFDABGJAHI $.
    $( $j usage 'dtru' avoids 'ax-8' 'ax-10' 'ax-11' 'ax-12' 'ax-13' 'ax-ext'
       'ax-sep' 'ax-pow' 'df-clab' 'df-cleq' 'df-clel'; $)
  $}

  ${
    $d x y z $.
    $( Any set is an element of some other set.  See ~ elALT for a shorter
       proof using more axioms, and see ~ elALT2 for a proof that uses ~ ax-9
       and ~ ax-pow instead of ~ ax-pr .  (Contributed by NM, 4-Jan-2002.)
       (Proof shortened by Andrew Salmon, 25-Jul-2011.)  Use ~ ax-pr instead of
       ~ ax-9 and ~ ax-pow .  (Revised by BTernaryTau, 2-Dec-2024.) $)
    el $p |- E. y x e. y $=
      ( vz weq wo wel wi ax-pr pm4.25 imbi1i albii elequ1 equsalvw bitr3i exbii
      wal wex mpbi ) CADZSEZCBFZGZCPZBQABFZBQAABCHUCUDBUCSUAGZCPUDUEUBCSTUASIJK
      UAUDCACABLMNOR $.
    $( $j usage 'el' avoids 'ax-9' 'ax-ext' 'ax-sep' 'ax-nul' 'ax-pow'; $)
  $}

  ${
    $d x y A $.
    $( If a class is a set, then it is a member of a set.  (Contributed by NM,
       4-Jan-2002.)  Generalize from the proof of ~ elALT .  (Revised by BJ,
       3-Apr-2019.)  Avoid ~ ax-sep , ~ ax-nul , ~ ax-pow .  (Revised by
       BTernaryTau, 15-Jan-2025.) $)
    sels $p |- ( A e. V -> E. x A e. x ) $=
      ( vy wel wex cv wcel wceq eleq1 exbidv el vtoclg ) DAEZAFBAGZHZAFDBCDGZBI
      NPAQBOJKDALM $.
    $( $j usage 'sels' avoids 'ax-sep' 'ax-nul' 'ax-pow'; $)
  $}

  ${
    $d x A $.
    $( Alternate proof of ~ sels , requiring ~ ax-sep but not using ~ el (which
       is proved from it as ~ elALT ).  (especially when the proof of ~ el is
       inlined in ~ sels ).  (Contributed by NM, 4-Jan-2002.)  Generalize from
       the proof of ~ elALT .  (Revised by BJ, 3-Apr-2019.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    selsALT $p |- ( A e. V -> E. x A e. x ) $=
      ( wcel csn cv wex snidg cvv snexg eleq2 spcedv syl ) BCDBBEZDZBAFZDZAGBCH
      OQOAINBNJBNHPNBKLM $.
    $( $j usage 'sels' avoids 'ax-nul' 'ax-pow'; $)
  $}

  ${
    $d x y $.
    $( Alternate proof of ~ el , shorter but requiring ~ ax-sep .  (Contributed
       by NM, 4-Jan-2002.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    elALT $p |- E. y x e. y $=
      ( cv cvv wcel wel wex vex selsALT ax-mp ) ACZDEABFBGAHBKDIJ $.
    $( $j usage 'sels' avoids 'ax-nul' 'ax-pow'; $)
  $}

  ${
    $d w x y z $.
    $( Obsolete version of ~ dtru as of 1-Jan-2025.  (Contributed by NM,
       7-Nov-2006.)  Avoid ~ ax-13 .  (Revised by BJ, 31-May-2019.)  Avoid
       ~ ax-12 .  (Revised by Rohan Ridenour, 9-Oct-2024.)  Use ~ ax-pr instead
       of ~ ax-pow .  (Revised by BTernaryTau, 3-Dec-2024.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    dtruOLD $p |- -. A. x x = y $=
      ( vw vz weq wn wex wal wel wa el ax-nul elequ1 notbid spw eximii exdistrv
      ax7v1 con3d spimevw mpbir2an ax9v2 com12 con3dimp 2eximi equequ2 biimtrdi
      wi a1d pm2.61i exlimivv mp2b exnal mpbi ) ABEZFZAGZUOAHFACIZADIZFZJZDGCGZ
      CDEZFZDGCGUQVBURCGUTDGACKUTAHUTDDALUTCDIZFACACEZUSVEACDMNOPURUTCDQUAVAVDC
      DURVCUSVCURUSCDAUBUCUDUEVDUQCDDBEZVDUQUHVGVDCBEZFZUQVGVCVHDBCUFNVIUPACVFU
      OVHACBRSTUGVGFZUQVDVJUPADADEUOVGADBRSTUIUJUKULUOAUMUN $.
  $}

  $( 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 $.

  $( If a set is a member of a class, then the singleton of that set is a
     member of the powerclass of that class.  (Contributed by Alan Sare,
     25-Aug-2011.) $)
  snelpwi $p |- ( A e. B -> { A } e. ~P B ) $=
    ( wcel csn cpw snelpwg ibi ) ABCADBECABBFG $.

  $( Obsolete version of ~ snelpwi as of 17-Jan-2025.  (Contributed by NM,
     28-May-1995.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  snelpwiOLD $p |- ( A e. B -> { A } e. ~P B ) $=
    ( wcel csn wss cpw snssi snex elpw sylibr ) ABCADZBEKBFCABGKBAHIJ $.

  ${
    snelpw.ex $e |- A e. _V $.
    $( 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.) $)
    snelpw $p |- ( A e. B <-> { A } e. ~P B ) $=
      ( cvv wcel csn cpw wb snelpwg ax-mp ) ADEABEAFBGEHCABDIJ $.
  $}

  $( 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 prex elpw bitr4di ) ADFBEFGACFBCFGABHZCIOCJFABC
    DEKOCABLMN $.

  $( If two sets are members of a class, then the unordered pair of those two
     sets is a member of the powerclass of that class.  (Contributed by Thierry
     Arnoux, 10-Mar-2017.)  (Proof shortened by AV, 23-Oct-2021.) $)
  prelpwi $p |- ( ( A e. C /\ B e. C ) -> { A , B } e. ~P C ) $=
    ( wcel wa cpr cpw prelpw ibi ) ACDBCDEABFCGDABCCCHI $.

  ${
    $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 wal csn wceq vsnid vsnex eleq2 imbi12d spcv velsn equcomi sylbi
      wi mpi syl ) ADZCDZEZBDZUAEZQZCFZUCTGZEZTUCHZUFTUGEZUHAIUEUJUHQCUGAJUAUGH
      UBUJUDUHUAUGTKUAUGUCKLMRUHUCTHUIBTNBAOPS $.
  $}

  ${
    $d A x $.  $d B x $.
    $( The powerclass construction preserves and reflects inclusion.  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 sspw csn wcel ssel vsnex elpw vex snss bitr4i 3imtr3g impbii
      cv ssrdv ) ABDAEZBEZDZABFUACABUACQZGZSHZUCTHZUBAHZUBBHZSTUCIUDUCADUFUCACJ
      ZKUBACLZMNUEUCBDUGUCBUHKUBBUIMNORP $.
  $}

  ${
    $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 wa wex eluni elelpwi exlimiv sylbi vsnid snelpwi
      csn elunii sylancr impbii eqriv ) BADZEZABFZUBGZUCAGZUDUCCFZGUFUAGHZCIUEC
      UCUAJUGUECUCUFAKLMUEUCUCPZGUHUAGUDBNUCAOUCUHUAQRST $.
  $}

  $( 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 $.

  $( 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 df-ss velpw imbi12i albii 3bitri ) BCDBEZC
      EZDAFZPGZRQGZHZAIRBDZRCDZHZAIBCJAPQKUAUDASUBTUCABLACLMNO $.

    $( 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 ~ nss .  (Contributed by NM,
       30-Jun-2004.)  (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
    nssss $p |- ( -. A C_ B <-> E. x ( x C_ A /\ -. x C_ B ) ) $=
      ( cv wss wn wa wex wi wal exanali ssextss xchbinxr bicomi ) ADZBEZOCEZFGA
      HZBCEZFRPQIAJSPQAKABCLMN $.
  $}

  $( 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 $.
    $( The intersection of all sets to which a set belongs is the singleton of
       that set.  (Contributed by NM, 5-Jun-2009.)  Put in closed form and
       avoid ~ ax-nul .  (Revised by BJ, 17-Jan-2025.) $)
    intidg $p |- ( A e. V -> |^| { x | A e. x } = { A } ) $=
      ( wcel cab cint csn wss cvv snexg snidg eleq2 elabd intss1 syl wal ax-gen
      cv wi id elintabg mpbiri snssd eqssd ) BCDZBARZDZAEZFZBGZUEUJUHDUIUJHUEUG
      BUJDAUJIBCJBCKUFUJBLMUJUHNOUEBUIUEBUIDUGUGSZAPUKAUGTQUGABCUAUBUCUD $.
  $}

  ${
    $d x A $.
    intid.1 $e |- A e. _V $.
    $( Obsolete version of ~ intidg as of 18-Jan-2025.  (Contributed by NM,
       5-Jun-2009.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    intidOLD $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 ) BADZEZAFGZBHZUEIEUDUEJBKUCBUEEAUEIUBUEBLBCRMNBUDEZU
      EUDJUFUCUCOAUCABCPUCSQBUDTNUA $.
  $}

  ${
    $d x y $.  $d y ph $.
    $( "At most one" existence implies a class abstraction exists.
       (Contributed by NM, 30-Dec-1996.) $)
    moabex $p |- ( E* x ph -> { x | ph } e. _V ) $=
      ( vy wmo weq wi wal wex cab cvv wcel df-mo cv csn abss velsn imbi2i albii
      wss bitri vsnex ssex sylbir exlimiv sylbi ) ABDABCEZFZBGZCHABIZJKZABCLUHU
      JCUHUICMZNZSZUJUMABMULKZFZBGUHABULOUOUGBUNUFABUKPQRTUIULCUAUBUCUDUE $.
  $}

  $( Restricted "at most one" existence implies a restricted class abstraction
     exists.  (Contributed by NM, 17-Jun-2017.) $)
  rmorabex $p |- ( E* x e. A ph -> { x e. A | ph } e. _V ) $=
    ( cv wcel wa wmo cab cvv wrmo crab moabex df-rmo df-rab eleq1i 3imtr4i ) BD
    CEAFZBGQBHZIEABCJABCKZIEQBLABCMSRIABCNOP $.

  $( The abstraction of a wff with existential uniqueness exists.  (Contributed
     by NM, 25-Nov-1994.) $)
  euabex $p |- ( E! x ph -> { x | ph } e. _V ) $=
    ( weu wmo cab cvv wcel eumo moabex syl ) ABCABDABEFGABHABIJ $.

  ${
    $d x y A $.
    $( A nonempty class (even if proper) has a nonempty subset.  (Contributed
       by NM, 23-Aug-2003.) $)
    nnullss $p |- ( A =/= (/) -> E. x ( x C_ A /\ x =/= (/) ) ) $=
      ( vy c0 wne cv wcel wex wss wa csn snss snnz vsnex wceq sseq1 neeq1 sylbi
      n0 vex anbi12d spcev mpan2 exlimiv ) BDECFZBGZCHAFZBIZUGDEZJZAHZCBSUFUKCU
      FUEKZBIZUKUEBCTZLUMULDEZUKUEUNMUJUMUOJAULCNUGULOUHUMUIUOUGULBPUGULDQUAUBU
      CRUDR $.
  $}

  ${
    $d x y z A $.  $d y z ph $.
    $( 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 wrex cv wcel wa cab wex wss crab wne df-rab rabn0 sylbi wsb df-clab
      c0 neeq1i n0 3bitr3i csn vex snss ssab2 sstr2 mpi simpr weq equsb1v velsn
      sbbii mpbir jctil sban bitri eleq2i 3bitri 3imtr4i ne0d sylib vsnex sseq1
      wceq rexeq anbi12d spcev syl2anc exlimiv ) ABDFZEGZBGZDHZAIZBJZHZEKZCGZDL
      ZABVTFZIZCKZABDMZTNVQTNVLVSWEVQTABDOUAABDPEVQUBUCVRWDEVRVMUDZDLZABWFFZWDV
      RWFVQLZWGVMVQEUEUFWIVQDLWGABDUGWFVQDUHUIQVRABWFMZTNWHVRWJVMVOBERZABERZIZV
      NWFHZBERZWLIZVRVMWJHZWMWLWOWKWLUJWOBEUKZBERBEULWNWRBEBVMUMUNUOUPVRVPBERWM
      VPEBSVOABEUQURWQVMWNAIZBJZHWSBERWPWJWTVMABWFOUSWSEBSWNABEUQUTVAVBABWFPVCW
      CWGWHICWFEVDVTWFVFWAWGWBWHVTWFDVEABVTWFVGVHVIVJVKQ $.
  $}

  $( An ordered pair of classes is a set.  Exercise 7 of [TakeutiZaring] p. 16.
     (Contributed by NM, 18-Aug-1993.)  (Revised by Mario Carneiro,
     26-Apr-2015.) $)
  opex $p |- <. A , B >. e. _V $=
    ( cop cvv wcel wa csn cpr c0 cif dfopif prex 0ex ifex eqeltri ) ABCADEBDEFZ
    AGZABHZHZIJDABKPSIQRLMNO $.

  $( An ordered triple of classes is a set.  (Contributed by NM,
     3-Apr-2015.) $)
  otex $p |- <. A , B , C >. e. _V $=
    ( cotp cop cvv df-ot opex eqeltri ) ABCDABEZCEFABCGJCHI $.

  $( Characterization of the elements of an ordered pair.  Closed form of
     ~ elop .  (Contributed by BJ, 22-Jun-2019.)  (Avoid depending on this
     detail.) $)
  elopg $p |- ( ( A e. V /\ B e. W ) ->
                   ( C e. <. A , B >. <-> ( C = { A } \/ C = { A , B } ) ) ) $=
    ( wcel wa cop csn cpr wceq wo wb dfopg eleq2 snex prex elpr2 bitrdi syl ) A
    DFBEFGABHZAIZABJZJZKZCUAFZCUBKCUCKLZMABDENUEUFCUDFUGUAUDCOCUBUCAPABQRST $.

  ${
    elop.1 $e |- B e. _V $.
    elop.2 $e |- C e. _V $.
    $( Characterization of the elements of an ordered pair.  Exercise 3 of
       [TakeutiZaring] p. 15.  (Contributed by NM, 15-Jul-1993.)  (Revised by
       Mario Carneiro, 26-Apr-2015.)  Remove an extraneous hypothesis.
       (Revised by BJ, 25-Dec-2020.)  (Avoid depending on this detail.) $)
    elop $p |- ( A e. <. B , C >. <-> ( A = { B } \/ A = { B , C } ) ) $=
      ( cvv wcel cop csn wceq cpr wo wb elopg mp2an ) BFGCFGABCHGABIJABCKJLMDEB
      CAFFNO $.
  $}

  ${
    opi1.1 $e |- A e. _V $.
    opi1.2 $e |- B e. _V $.
    $( One of the two elements in an ordered pair.  (Contributed by NM,
       15-Jul-1993.)  (Revised by Mario Carneiro, 26-Apr-2015.)  (Avoid
       depending on this detail.) $)
    opi1 $p |- { A } e. <. A , B >. $=
      ( csn cpr cop snex prid1 dfop eleqtrri ) AEZLABFZFABGLMAHIABCDJK $.

    $( One of the two elements of an ordered pair.  (Contributed by NM,
       5-Aug-1993.)  (Revised by Mario Carneiro, 26-Apr-2015.)  (Avoid
       depending on this detail.) $)
    opi2 $p |- { A , B } e. <. A , B >. $=
      ( cpr csn cop prex prid2 dfop eleqtrri ) ABEZAFZLEABGMLABHIABCDJK $.
  $}

  ${
    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 $.
  $}

  ${
    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.  (See ~ op2ndb to extract the second member, ~ op1sta for an
       alternate version, and ~ op1st for the preferred version.)  (Contributed
       by NM, 25-Nov-2003.) $)
    op1stb $p |- |^| |^| <. A , B >. = A $=
      ( cop cint csn cpr dfop inteqi cin snex prex intpr wss wceq snsspr1 dfss2
      mpbi eqtri intsn ) ABEZFZFAGZFAUCUDUCUDABHZHZFZUDUBUFABCDIJUGUDUEKZUDUDUE
      ALABMNUDUEOUHUDPABQUDUERSTTJACUAT $.
  $}

  $( Two classes are always in relation by ` _V ` .  This is simply equivalent
     to ` <. A , B >. e. _V ` , and does not imply that ` _V ` is a relation:
     see ~ nrelv .  (Contributed by Scott Fenton, 11-Apr-2012.) $)
  brv $p |- A _V B $=
    ( cvv wbr cop wcel opex df-br mpbir ) ABCDABECFABGABCHI $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Ordered pair theorem
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( An ordered pair is nonempty iff the arguments are sets.  (Contributed by
     NM, 24-Jan-2004.)  (Revised by Mario Carneiro, 26-Apr-2015.) $)
  opnz $p |- ( <. A , B >. =/= (/) <-> ( A e. _V /\ B e. _V ) ) $=
    ( cop c0 wne cvv wcel wa opprc necon1ai csn cpr dfopg snex prnz a1i eqnetrd
    impbii ) ABCZDEAFGBFGHZTSDABIJTSAKZABLZLZDABFFMUCDETUAUBANOPQR $.

  ${
    opth1.1 $e |- A e. _V $.
    opth1.2 $e |- B e. _V $.
    $( An ordered pair is nonempty if the arguments are sets.  (Contributed by
       Mario Carneiro, 26-Apr-2015.) $)
    opnzi $p |- <. A , B >. =/= (/) $=
      ( cop c0 wne cvv wcel opnz mpbir2an ) ABEFGAHIBHICDABJK $.

    $( 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 wcel opi1 id eleqtrid cpr wi sneqr a1i cvv oprcl syl eleq2
      simpld prid1g syl5ibrcom elsni eqcomd syl6 wo dfopg eleqtrd elpri mpjaod
      wa ) ABGZCDGZHZAIZUOJZACHZUPUQUNUOABEFKUPLMURUQCIZHZUSUQCDNZHZVAUSOURACEP
      QURVCCUQJZUSURVDVCCVBJZURCRJZVEURVFDRJZCDUQSZUBCDRUCTUQVBCUAUDVDCACAUEUFU
      GURUQUTVBNZJVAVCUHURUQUOVIURLURVFVGUMUOVIHVHCDRRUITUJUQUTVBUKTULT $.

    $d x B $.  $d x C $.  $d x D $.
    $( 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 prex preqr2 wi opi1 id
      opth1 eleqtrid oprcl simprd opeq1d simpld sylancl cv preq2 eqeq2d imbi12d
      eqeq2 vex vtoclg sylc jca opeq12 impbii ) ABHZCDHZIZACIZBDIZJVCVDVEABCDEF
      UCZVCDKLZCBMZCDMZIZVEVCCKLZVGVCANZVBLVKVGJZVCVLVAVBABEFUAVCUBZUDCDVLUEOZU
      FVCCNZVHMZVPVIMZIVJVCVBVQVRVCCBHZVBVQVCVAVSVBVCACBVFUGVNPVCVKBKLVSVQIVCVK
      VGVOUHFCBKKQUIPVCVMVBVRIVOCDKKQOPVHVIVPCBRCDRSOVHCGUJZMZIZBVTIZTVJVETGDKV
      TDIZWBVJWCVEWDWAVIVHVTDCUKULVTDBUNUMBVTCFGUOSUPUQURABCDUSUT $.
    $( $j usage 'opth' avoids 'ax-reg'; $)
  $}

  ${
    $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 $.
  $}

  $( Equality of the first members of equal ordered pairs.  Closed form of
     ~ opth1 .  (Contributed by AV, 14-Oct-2018.) $)
  opth1g $p |- ( ( A e. V /\ B e. W )
                 -> ( <. A , B >. = <. C , D >. -> A = C ) ) $=
    ( wcel wa cop wceq opthg simpl biimtrdi ) AEGBFGHABICDIJACJZBDJZHNABCDEFKNO
    LM $.

  $( 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
      $.
  $}

  $( Two ordered pairs are not equal iff their first components or their second
     components are not equal.  (Contributed by AV, 13-Dec-2018.) $)
  opthneg $p |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. =/= <. C , D >.
                                            <-> ( A =/= C \/ B =/= D ) ) ) $=
    ( cop wne wceq wn wcel wa wo df-ne opthg notbid ianor orbi12i bitr4i bitrdi
    bitrid ) ABGZCDGZHUBUCIZJZAEKBFKLZACHZBDHZMZUBUCNUFUEACIZBDIZLZJZUIUFUDULAB
    CDEFOPUMUJJZUKJZMUIUJUKQUGUNUHUOACNBDNRSTUA $.

  ${
    opthne.1 $e |- A e. _V $.
    opthne.2 $e |- B e. _V $.
    $( Two ordered pairs are not equal iff their first components or their
       second components are not equal.  (Contributed by AV, 13-Dec-2018.) $)
    opthne $p |- ( <. A , B >. =/= <. C , D >. <-> ( A =/= C \/ B =/= D ) ) $=
      ( cvv wcel cop wne wo wb opthneg mp2an ) AGHBGHABICDIJACJBDJKLEFABCDGGMN
      $.
  $}

  ${
    otth.1 $e |- A e. _V $.
    otth.2 $e |- B e. _V $.
    otth.3 $e |- R e. _V $.
    $( Ordered triple theorem, with triple expressed 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
      KZLZUBLSEJTFJKUCUDUBMUAUEUBABCDGHNOSETFABPINUCUDUBQR $.

    $( 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 $.
  $}

  $( Ordered triple theorem, closed form.  (Contributed by Alexander van der
     Vekens, 10-Mar-2018.) $)
  otthg $p |- ( ( A e. U /\ B e. V /\ C e. W )
                  -> ( <. A , B , C >. = <. D , E , F >.
                       <-> ( A = D /\ B = E /\ C = F ) ) ) $=
    ( cotp wceq cop wcel w3a df-ot eqeq12i wb wa cvv opthg anbi1d bitr4di 3impa
    opex mpan df-3an sylan9bbr bitrid ) ABCJZDFGJZKABLZCLZDFLZGLZKZAEMZBHMZCIMZ
    NADKZBFKZCGKZNZUIULUJUNABCODFGOPUPUQURUOVBQURUOUKUMKZVARZUPUQRZVBUKSMURUOVD
    QABUDUKCUMGSITUEVEVDUSUTRZVARVBVEVCVFVAABDFEHTUAUSUTVAUFUBUGUCUH $.

  ${
    otthne.1 $e |- A e. _V $.
    otthne.2 $e |- B e. _V $.
    otthne.3 $e |- C e. _V $.
    $( Contrapositive of the ordered triple theorem.  (Contributed by Scott
       Fenton, 31-Jan-2025.) $)
    otthne $p |- ( <. A , B , C >. =/= <. D , E , F >. <->
     ( A =/= D \/ B =/= E \/ C =/= F ) ) $=
      ( cotp wceq wn w3o wne w3a otth notbii 3ianor bitri df-ne 3orbi123i
      3bitr4i ) ABCJZDEFJZKZLZADKZLZBEKZLZCFKZLZMZUCUDNADNZBENZCFNZMUFUGUIUKOZL
      UMUEUQABDECFGHIPQUGUIUKRSUCUDTUNUHUOUJUPULADTBETCFTUAUB $.
  $}

  ${
    $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.  Analogue 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 a x y z $.  $d ph x y $.  $d ps z $.
    sbcop.z $e |- ( z = <. x , y >. -> ( ph <-> ps ) ) $.
    $( The proper substitution of an ordered pair for a setvar variable
       corresponds to a proper substitution of its first component.
       (Contributed by AV, 8-Apr-2023.) $)
    sbcop1 $p |- ( [. a / x ]. ps <-> [. <. a , y >. / z ]. ph ) $=
      ( cv wsbc cop wceq wi wal wa wex sbc5 biimtrdi imp exlimiv sylbi weq opex
      opeq1 equcoms eqeq2d biimprd com23 alrimiv sylibr biimpd com3l vex impbii
      sbc6 ) BCFHZIZAEUODHZJZIZUPEHZURKZALZEMUSUPVBEUPCFUAZBNZCOVBBCUOPVDVBCVCB
      VBVCVABAVCVAUTCHZUQJZKZBALVCURVFUTURVFKFCUOVEUQUCUDUEZVGABGUFQUGRSTUHAEUR
      UOUQUBUNUIUSVAANZEOUPAEURPVIUPEVIVCBLZCMUPVIVJCVAAVJVCVAABVCVAVGABLVHVGAB
      GUJQUKRUHBCUOFULUNUISTUM $.

    $d b x y z $.
    $( The proper substitution of an ordered pair for a setvar variable
       corresponds to a proper substitution of each of its components.
       (Contributed by AV, 8-Apr-2023.) $)
    sbcop $p |- ( [. b / y ]. [. a / x ]. ps <-> [. <. a , b >. / z ]. ph ) $=
      ( cv wsbc cop csb sbcop1 sbcbii wb cvv sbcnestgw elv wceq csbopg csbvargi
      vex csbconstgi opeq12i eqtri dfsbcq ax-mp 3bitri ) BCFIZJZDGIZJAEUIDIZKZJ
      ZDUKJZAEDUKUMLZJZAEUIUKKZJZUJUNDUKABCDEFHMNUOUQOGADEUKUMPQRUPURSUQUSOUPDU
      KUILZDUKULLZKZURUPVBSGDUKUIULPTRUTUIVAUKDFUKGUBZUCDUKVCUAUDUEAEUPURUFUGUH
      $.
  $}

  ${
    $d x y z w A $.  $d z w ph $.
    $( Version of ~ copsexg with a disjoint variable condition, which does not
       require ~ ax-13 .  (Contributed by GG, 26-Jan-2024.) $)
    copsexgw $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 syl5 biimtrid weu euequ ex
      eqvinop 19.8ad opth anbi1i 2exbii wal anim2i anassrs eximi biidd imbitrid
      nfe1 drex1v anass exbii 19.40 nfvd 19.9d anim1d syl6 pm2.61i exlimi eubii
      wn equcom mpbi eupick mpan com12 sylan9 sylbi impbid eqeq1 anbi1d 2exbidv
      bibi2d imbi12d mpbiri adantr exlimivv pm2.43i ) DBGZCGZHZIZAWFAJZCKBKZLZW
      FDEGZFGZHZIZWLWEIZJZFKEKWFWIMZEFDWCWDBNCNUBWOWPEFWMWPWNWMWPWNAWNAJZCKZBKZ
      LZMWNAWSWNAWSWQWRBWQCOUCUAWNEBPZFCPZJZWSAMWJWKWCWDENFNUDZWSXCAJZCKZBKZXCA
      WQXEBCWNXCAXDUEUFXGXAXBAJZCKZJZBKZXCAXFXKBXJBUMCBPCUGZXFXKMXFXJCKXLXKXEXJ
      CXAXBAXJXHXIXAXHCOUHUIUJXJXJCBXLXJUKUNULXLVEZXFXJXKXFXAXHJZCKZXMXJXEXNCXA
      XBAUOUPXOXACKZXIJXMXJXAXHCUQXMXPXAXIXAXMCXMXACURUSUTQRXJBOVAVBVCXAXKXIXBA
      XKXAXIXABSZXKXAXIMBEPZBSXQBETXRXABBEVFVDVGXAXIBVHVIVJXIXBAXBCSZXIXBAMCFPZ
      CSXSCFTXTXBCCFVFVDVGXBACVHVIVJVKQRVLVMWMWFWNWIWTDWLWEVNZWMWHWSAWMWGWQBCWM
      WFWNAYAVOVPVQVRVSVTWAVLWB $.
    $( $j usage 'copsexgw' avoids 'ax-13'; $)
  $}

  ${
    $d x z w A $.  $d y z w A $.  $d z w ph $.
    $( Substitution of class ` A ` for ordered pair ` <. x , y >. ` .  Usage of
       this theorem is discouraged because it depends on ~ ax-13 .  Use the
       weaker ~ copsexgw when possible.  (Contributed by NM, 27-Dec-1996.)
       (Revised by Andrew Salmon, 11-Jul-2011.)  (Proof shortened by Wolf
       Lammen, 25-Aug-2019.)  (New usage is discouraged.) $)
    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 syl5 biimtrid weu euequ ex
      eqvinop 19.23bi opth anbi1i 2exbii nfe1 wal anim2i anassrs eximi imbitrid
      biidd drex1 wn anass exbii 19.40 nfeqf2 anim1d syl6 pm2.61i exlimi equcom
      19.9d eubii eupick com12 sylan9 sylbi impbid eqeq1 anbi1d 2exbidv imbi12d
      mpbi mpan bibi2d mpbiri adantr exlimivv pm2.43i ) DBGZCGZHZIZAWFAJZCKBKZL
      ZWFDEGZFGZHZIZWLWEIZJZFKEKWFWIMZEFDWCWDBNCNUBWOWPEFWMWPWNWMWPWNAWNAJZCKZB
      KZLZMWNAWSWNAWSWQWSCWRBOUCUAWNEBPZFCPZJZWSAMWJWKWCWDENFNUDZWSXCAJZCKZBKZX
      CAWQXEBCWNXCAXDUEUFXGXAXBAJZCKZJZBKZXCAXFXKBXJBUGCBPCUHZXFXKMXFXJCKXLXKXE
      XJCXAXBAXJXHXIXAXHCOUIUJUKXJXJCBXLXJUMUNULXLUOZXFXJXKXFXAXHJZCKZXMXJXEXNC
      XAXBAUPUQXOXACKZXIJXMXJXAXHCURXMXPXAXIXAXMCCBEUSVEUTQRXJBOVAVBVCXAXKXIXBA
      XKXAXIXABSZXKXAXIMBEPZBSXQBETXRXABBEVDVFVPXAXIBVGVQVHXIXBAXBCSZXIXBAMCFPZ
      CSXSCFTXTXBCCFVDVFVPXBACVGVQVHVIQRVJVKWMWFWNWIWTDWLWEVLZWMWHWSAWMWGWQBCWM
      WFWNAYAVMVNVRVOVSVTWAVJWB $.
  $}

  ${
    $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 ) ) $=
      ( cv wceq wa wb wal wex cop wcel nfe1 nfv nfbi exlimd wi nfa1 nfa2 opeq12
      nfex copsexgw eqcoms syl adantl 2sp imp bitr3d ex elisset exdistrv sylibr
      anim12i impel ) CIZEJZDIZFJZKZABLZUAZDMZCMZVCDNZCNZEFOZUSVAOZJAKZDNZCNZBL
      ZEGPZFHPZKZVGVHVOCVFCUBVNBCVMCQBCRSVGVCVODVEDCUCVNBDVMDCVLDQUEBDRSVGVCVOV
      GVCKAVNBVCAVNLZVGVCVKVJJVSUSVAEFUDVSVJVKACDVJUFUGUHUIVGVCVDVECDUJUKULUMTT
      VRUTCNZVBDNZKVIVPVTVQWACEGUNDFHUNUQUTVBCDUOUPUR $.
  $}

  ${
    $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.)  Use a similar proof to ~ copsex4g to reduce axiom usage.
       (Revised by SN, 1-Sep-2024.) $)
    copsex2g $p |- ( ( A e. V /\ B e. W ) ->
                  ( E. x E. y ( <. A , B >. = <. x , y >. /\ ph ) <-> ps ) ) $=
      ( cop cv wceq wa wex wcel eqcom vex opth bitri anbi1i 2exbii id cgsex2g
      bitrid ) EFJZCKZDKZJZLZAMZDNCNUFELUGFLMZAMZDNCNEGOFHOMBUJULCDUIUKAUIUHUEL
      UKUEUHPUFUGEFCQDQRSTUAABUKCDEFGHUKUBIUCUD $.
    $( $j usage 'copsex2g' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $d A x y $.  $d B x y $.  $d ch x y $.  $d ph x y $.
    copsex2dv.a $e |- ( ph -> A e. U ) $.
    copsex2dv.b $e |- ( ph -> B e. V ) $.
    copsex2dv.1 $e |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ps <-> ch ) ) $.
    $( Implicit substitution deduction for ordered pairs.  (Contributed by
       Thierry Arnoux, 4-May-2025.) $)
    copsex2dv $p |- ( ph -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ ps )
                            <-> ch ) ) $=
      ( cv wceq wa wb wal wcel cop wex wi ex alrimivv copsex2t syl12anc ) ADMZF
      NEMZGNOZBCPZUAZEQDQFHRGIRFGSUFUGSNBOETDTCPAUJDEAUHUILUBUCJKBCDEFGHIUDUE
      $.
  $}

  ${
    $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 snn0d prnzg jca neanior sylib pm2.65i ) CABDZEZCAFZSCABGZSHZ
    UICUJUKGZEULUICUHUMUIIUIAJEZBJEZKUHUMSABCLZABJJMNOCUJUKPNUICUJQZCUKQZKULTUI
    UQURUIUJCUIAJUIUNUOUPUAZUBRUIUKCUIUNUKCQUSABJUCNRUDCUJCUKUEUFUG $.

  $( 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 $.

  $( 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 ) ) ) $=
    ( cop wceq c0 wne cvv wcel wa neeq1 opnz 3bitr3g ) ABEZCDEZFOGHPGHAIJBIJKCI
    JDIJKOPGLABMCDMN $.

  $( Equivalence of existence implied by equality of ordered triples.
     (Contributed by NM, 26-Apr-2015.) $)
  oteqex2 $p |- ( <. <. A , B >. , C >. = <. <. R , S >. , T >. ->
    ( C e. _V <-> T e. _V ) ) $=
    ( cop wceq cvv wcel wa opeqex opex biantrur 3bitr4g ) ABGZCGDEGZFGHPIJZCIJZ
    KQIJZFIJZKSUAPCQFLRSABMNTUADEMNO $.

  $( Equivalence of existence implied by equality of ordered triples.
     (Contributed by NM, 28-May-2008.)  (Revised by Mario Carneiro,
     26-Apr-2015.) $)
  oteqex $p |- ( <. <. A , B >. , C >. = <. <. R , S >. , T >. ->
    ( ( A e. _V /\ B e. _V /\ C e. _V ) <->
      ( R e. _V /\ S e. _V /\ T e. _V ) ) ) $=
    ( cop wceq cvv wcel w3a wi simp3 a1i oteqex2 imbitrrid wb wa opex df-3an
    opthg mpan simprbda opeqex syl adantl anbi12d 3bitr4g expcom pm5.21ndd ) AB
    GZCGDEGZFGHZCIJZAIJZBIJZUNKZDIJZEIJZFIJZKZUQUNLUMUOUPUNMNVAUNUMUTURUSUTMABC
    DEFOZPUNUMUQVAQUNUMRZUOUPRZUNRURUSRZUTRUQVAVCVDVEUNUTVCUKULHZVDVEQUNUMVFCFH
    ZUKIJUNUMVFVGRQABSUKCULFIIUAUBUCABDEUDUEUMUNUTQUNVBUFUGUOUPUNTURUSUTTUHUIUJ
    $.

  ${
    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 $.
  $}

  $( Equivalence for an ordered pair equal to a singleton.  (Contributed by NM,
     3-Jun-2008.)  (Revised by AV, 15-Jul-2022.)  (Avoid depending on this
     detail.) $)
  opeqsng $p |- ( ( A e. V /\ B e. W )
                  -> ( <. A , B >. = { C } <-> ( A = B /\ C = { A } ) ) ) $=
    ( wcel wa cop csn wceq cpr dfopg eqeq1d wb snex prex a1i eqcom simpr bitrdi
    preqsn simpl preqsnd eqid impbii bitri bitrid anbi1d wi dfsn2 preq2 eqtr2id
    jctl pm5.32d bitrd 3bitrd ) ADFZBEFZGZABHZCIZJAIZABKZKZVAJZVBVCJZVCCJZGZABJ
    ZCVBJZGZUSUTVDVAABDELMVEVHNUSVBVCCAOABPUAQUSVHVIVGGVKUSVFVIVGVFVCVBJZUSVIVB
    VCRUSVLAAJZBAJZGZVIUSABADEUQURUBUQURSUCVOVNVIVOVNVMVNSVNVMAUDUMUEBARUFTUGUH
    USVIVGVJVIVGVJNUIUSVIVGVBCJVJVIVCVBCVIVBAAKVCAUJABAUKULMVBCRTQUNUOUP $.

  ${
    opeqsn.1 $e |- A e. _V $.
    opeqsn.2 $e |- B e. _V $.
    $( Equivalence for an ordered pair equal to a singleton.  (Contributed by
       NM, 3-Jun-2008.)  (Revised by AV, 15-Jul-2022.)  (Avoid depending on
       this detail.) $)
    opeqsn $p |- ( <. A , B >. = { C } <-> ( A = B /\ C = { A } ) ) $=
      ( cvv wcel cop csn wceq wa wb opeqsng mp2an ) AFGBFGABHCIJABJCAIJKLDEABCF
      FMN $.
  $}

  ${
    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.)  (Avoid depending on this detail.) $)
    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 prex preq12b 3bitri ) ABI
      ZCDJZKUCUBKUCALZABJZJZKCUDKDUEKMCUEKDUDKMNUBUCOUBUFUCABEFPQCDUDUEGHARABST
      UA $.
  $}

  ${
    snopeqop.a $e |- A e. _V $.
    snopeqop.b $e |- B e. _V $.
    $( Equivalence for an ordered pair equal to a singleton of an ordered pair.
       (Contributed by AV, 18-Sep-2020.)  (Revised by AV, 15-Jul-2022.)  (Avoid
       depending on this detail.) $)
    snopeqop $p |- ( { <. A , B >. } = <. C , D >.
                     <-> ( A = B /\ C = D /\ C = { A } ) ) $=
      ( cvv wcel cop csn wceq wb wa a1i wn wi c0 eqeq2d biimtrdi eleq1 w3a opex
      opeqsng ancoms bitrid opeqsn anbi2d 3anan12 bicomi 3bitrd opprc2 wne snnz
      eqcom eqneqall mpi adantr notbid eqcoms pm2.21 impd 3ad2ant2 com12 impbid
      pm2.61ian opprc1 snex pm2.24i 3ad2ant3 pm2.61i ) CGHZABIZJZCDIZKZABKZCDKZ
      CAJZKZUAZLZDGHZVKWAWBVKMZVOVQVLCJKZMZVQVPVSMZMZVTVOVNVMKZWCWEVMVNUNVKWBWH
      WELCDVLGGUCUDUEWCWDWFVQWDWFLWCABCEFUFNUGWGVTLWCVTWGVPVQVSUHUINUJWBOZVKMZV
      OVTWIVOVTPVKWIVOVMQKZVTWIVNQVMCDUKRWKVMQULVTVLABUBUMVTVMQUOUPZSUQVTWJVOVQ
      VPWJVOPVSVQWIVKVOVQWIVKOZVKVOPWIWMLDCDCKWBVKDCGTURUSVKVOUTSVAVBVCVDVEWMVO
      VTWMVOWKVTWMVNQVMCDVFRWLSVTWMVOVSVPWMVOPVQVSWMVRGHZOVOVSVKWNCVRGTURWNVOAV
      GVHSVIVCVDVJ $.

    propeqop.c $e |- C e. _V $.
    propeqop.d $e |- D e. _V $.
    propeqop.e $e |- E e. _V $.
    propeqop.f $e |- F e. _V $.
    $( Equivalence for an ordered pair equal to a pair of ordered pairs.
       (Contributed by AV, 18-Sep-2020.)  (Proof shortened by AV, 16-Jun-2022.)
       (Avoid depending on this detail.) $)
    propeqop $p |- ( { <. A , B >. , <. C , D >. } = <. E , F >.
                     <-> ( ( A = C /\ E = { A } ) /\
                           ( ( A = B /\ F = { A , D } )
                             \/ ( A = D /\ F = { A , B } ) ) ) ) $=
      ( wceq wa wi adantr imp jca ex impcom cop csn cpr wo opeqsn anbi12i eqcom
      opeqpr orbi12i opex bitri simpl anim12i sneq eqeq2d biimpa adantl biimpcd
      simpr preq1 orcd ancoms eqeq1d biimpd a1dd olcd jaoi eqeq1 sneqr biimtrdi
      eqcomd preqsn eqtr sylbi com12 preq1d impancom biimpi com23 com13 orim12d
      sylanbrc syl a1i13 com14 biimtrid sylbid com24 dfsn2 eqtrid anim1d orcoms
      preq2 impbii 3bitr4i ) ABUAZEUBZMZCDUAZEFUCZMZNZWPWTMZWSWQMZNZUDZABMZEAUB
      ZMZNZECUBZMZFCDUCZMZNZEXMMZFXKMZNZUDZNZXIFABUCZMZNZEYAMZFXHMZNZUDZCDMZXLN
      ZNZUDZWPWSUCZEFUAZMZACMZXINZXGFADUCZMZNZADMZYBNZUDZNZXBXTXEYJWRXJXAXSABEG
      HUECDEFIJKLUHUFXCYGXDYIABEFGHKLUHCDEIJUEUFUIYNYMYLMXFYLYMUGEFWPWSKLABUJCD
      UJUHUKUUCYKUUBYPYKYSYPYKOUUAYSYPYKYSYPNZXTYJUUDXJXSYSXGYPXIXGYRULYOXIUSZU
      MUUDXOXRUUDXLXNYPXLYSYOXIXLYOXHXKEACUNUOZUPUQYSYPXNYRYPXNOXGYPYRXNYPYQXMF
      YOYQXMMXIACDUTPUOURUQQRVARVASUUAYPYKUUAYPNZYJXTUUGYGYIUUGYCYFYPUUAYCYPXIU
      UAYBUUEYTYBUSUMVBVAUUGYHXLUUAYPYHYTYPYHOYBYPYTYHYPACDYOXIULVCURPQYPUUAXLY
      OXIUUAXLOYOXIXLUUAYOXIXLUUFVDVEQTRRVFSVGTXTUUCYJXSXJUUCXSXJNZYPUUBUUHYOXI
      XJXSYOXIXSYOOXGXSXIYOXOXIYOOZXRXLUUIXNXLXIXKXHMZYOEXKXHVHUUJCACAIVIVKVJZP
      XPUUIXQXPXIXMXHMZYOEXMXHVHZUULYHDAMZNZYOCDAIJVLZUUOCACDAVMVKZVNVJPVGVOUQT
      XJXIXSXGXIUSUQRXJXSUUBXJXOYSXRUUAXJXOYSXJXONXGYRXJXGXOXGXIULPXOXJYRXLXJXN
      YRXLXJNZXNYRUURXMYQFUURCADXJXLCAMZXIXLUUSOXGXIXLXHXKMZUUSEXHXKVHUUTACACGV
      IVKVJZUQTVPUOVDVQTRSXGXIXRUUAOXRXIXGUUAXPXQXIXGUUAOZOXPXIXQUVBXPXIUULXQUV
      BOZUUMUULUUOUVCUUPUUOXGXQUUAUUOXGXQUUAOUUOXGNZXQUUAUVDXQNYTYBUVDYTXQUUOYT
      XGUUNYTYHUUNYTDAUGVRUQPPUVDXQYBUVDXKYAFUVDYAXKUVDXGBCMZYAXKMZUUOXGUSUVDCB
      UUOXGCBMUUOACBUUQVCUPVKABCGHVLZWBVKUOUPRSSVSVNVJVSQVTQWATRVBYGYIUUCYFYCYI
      UUCOYIYFYCUDZUUCYIUVHUUCYIUVHNYPUUBUVHYIYPYFYIYPOZYCYDYEUVIYIYEYDYPXLYHYE
      YDYPOOXLYDYEYHYPXLYDXKYAMZYEYHYPOOZEXKYAVHUVJXGUVENZXLUVKUVJUVFUVLXKYAUGU
      VGUKYHUVLYEXLYPYHUVLYEXLYPOUVLXLYPUVLXLNYOXIUVLYOXLABCVMZPUVLXLXIUVLXKXHE
      UVLUUSUUJUVLACUVMVKCAUNWCUOUPRSWDWEWFWGWHTVTQYCYIYPYCYINYOXIYCYIYOXIYIYOO
      YBYIXIYOXLUUIYHUUKUQVOPQYCXIYIXIYBULPRSVGTYIUVHUUBYIYFYSYCUUAYHXLYFYSOYFX
      LYHYSYDYEXLYHYSOZOYDXLYEUVNYDXLUVFYEUVNOZEYAXKVHUVFUVLUVOUVGUVLYHYEYSUVLY
      HYEYSOUVLYHNZYEYSUVPYENXGYRUVPXGYEUVLXGYHXGUVEULPPUVPYEYRUVPXHYQFUVLYHXHY
      QMZUVLYOYHUVQOUVMYOYHUVQYOYHNYTUVQACDVMYTXHAAUCYQAWIADAWMWJWCSWCQUOUPRSSV
      SVNVJVSQVTQYIXIYTYBYHXLXIYTOXIXLYHYTXIXLYHYTOXIXLNZYHYTUVRCADXIXLUUSUVAQV
      CVDSVTQWKWAQRSVOWLQVGWNWO $.

    $( If a pair of ordered pairs is a subset of an ordered pair, their first
       components are equal.  (Contributed by AV, 20-Sep-2020.)  (Proof
       shortened by AV, 16-Jun-2022.)  (Avoid depending on this detail.) $)
    propssopi $p |- ( { <. A , B >. , <. C , D >. } C_ <. E , F >.
                      -> A = C ) $=
      ( cop cpr csn wceq c0 wo wa sylbi wss dfop sseq2i sspr opex prnz eqneqall
      wne mpi preqsn opth simpl adantr jaoi wi a1d eqcomi eqeq2i propeqop bitri
      imp simpll ) ABMZCDMZNZEFMZUAVEEOZEFNZNZUAZACPZVFVIVEEFKLUBZUCVJVEQPZVEVG
      OPZRZVEVHOPZVEVIPZRZRVKVEVGVHUDVOVKVRVMVKVNVMVEQUHVKVCVDABUEZUFVKVEQUGUIV
      NVCVDPZVDVGPZSVKVCVDVGVSCDUEZUJVTVKWAVTVKBDPZSZVKABCDGHUKZVKWCULZTUMTUNVP
      VKVQVPVTVDVHPZSVKVCVDVHVSWBUJVTWGVKVTWDWGVKUOWEWDVKWGWFUPTVATVQVKEAOPZSAB
      PFADNPSADPFABNPSRZSZVKVQVEVFPWJVIVFVEVFVIVLUQURABCDEFGHIJKLUSUTVKWHWIVBTU
      NUNTT $.
  $}

  ${
    snopeqopsnid.a $e |- A e. _V $.
    $( Equivalence for an ordered pair of two identical singletons equal to a
       singleton of an ordered pair.  (Contributed by AV, 24-Sep-2020.)
       (Revised by AV, 15-Jul-2022.)  (Avoid depending on this detail.) $)
    snopeqopsnid $p |- { <. A , A >. } = <. { A } , { A } >. $=
      ( cop csn wceq eqid snopeqop mpbir3an ) AACDADZICEAAEIIEZJAFIFZKAAIIBBGH
      $.
  $}

  ${
    $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 nfa1 nfe1 nfmov wi nfex copsexgw sps exlimd
      mobidv biimpcd simpl 2eximi exlimiv nexmo nsyl5 pm2.61d1 ) ABFZDGZCGZECHD
      HIJZDKZCKZULALZDKZCKZBFZUKUMURCUJCMUQCBUPCNOUJUMURPCUJULURDUIDMUQDBUPDCUO
      DNQOUIULURPDULUIURULAUQBACDERUAUBSTSTUQBKUNURUQUNBUOULCDULAUCUDUEUQBUFUGU
      H $.
  $}

  ${
    $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 $.
  $}

  ${
    $d x ph $.  $d x A $.  $d x y $.
    euop2.1 $e |- A e. _V $.
    $( Transfer existential uniqueness to second member of an ordered pair.
       (Contributed by NM, 10-Apr-2004.) $)
    euop2 $p |- ( E! x E. y ( x = <. A , y >. /\ ph ) <-> E! y ph ) $=
      ( cv cop opex moop2 euxfr2w ) ABCDCFZGDKHCBFDEIJ $.
  $}

  ${
    $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. U ) $.
    euotd.2 $e |- ( ph -> B e. V ) $.
    euotd.3 $e |- ( ph -> C e. W ) $.
    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 ) ) $=
      ( wceq wa wex wtru vy cv cotp wal weu w3a biimpa vex sylibr eqeq2d biimpd
      wb otth impancom expimpd exlimdv exlimdvv wcel wsbc adantr ad2antrr simpr
      tru eqcomd biimpar jca trud 2thd 3anassrs sbcied mpbiri spesbcd nfcv nfex
      nfsbc1v sbceq1a exbidv spcegf sylc 2exbidv excom13 sylib eqeq1 syl5ibrcom
      anbi1d 3exbidv impbid alrimiv otex eqeq2 bibi2d albidv spcev syl eu6 ) AC
      UBZJUBZKUBZLUBZUCZQZBRZLSZKSJSZWPUAUBZQZULZCUDZUASZXDCUEAXDWPDEFUCZQZULZC
      UDZXIAXLCAXDXKAXCXKJKAXBXKLAXABXKABXAXKABRZXAXKXNWTXJWPXNWQDQZWREQZWSFQZU
      FZWTXJQZABXRPUGWQWRDEWSFJUHKUHLUHUMZUIUJUKUNUOUPUQAXDXKXJWTQZBRZLSKSJSZAY
      BJSKSZLSZYCAFIURZYBLFUSZJSZKSZYEOAEHURZYGKEUSZJSZYINAYKJDAYKJDUSTVCAYKTJD
      GMAXORZYGTKEHAYJXONUTYMXPRYBTLFIAYFXOXPOVAAXOXPXQYBTULAXRRZYBTYNYABYNWTXJ
      YNXRXSAXRVBXTUIVDABXRPVEVFYNVGVHVIVJVJVJVKVLYHYLKEHKEVMYKKJYGKEVOVNXPYGYK
      JYGKEVPVQVRVSYDYILFILFVMYHLKYGLJYBLFVOVNVNXQYBYGKJYBLFVPVTVRVSYBLKJWAWBXK
      XBYBJKLXKXAYABWPXJWTWCWEWFWDWGWHXHXMUAXJDEFWIXEXJQZXGXLCYOXFXKXDXEXJWPWJW
      KWLWMWNXDCUAWOUI $.
  $}

  ${
    opthw.1 $e |- A e. _V $.
    opthw.2 $e |- B e. _V $.
    $( Justification theorem for the ordered pair definition in Norbert Wiener,
       _A simplification of the logic of relations_, Proceedings of the
       Cambridge Philosophical Society, 1914, vol. 17, pp.387-390.  It is also
       shown as a definition in [Enderton] p. 36 and as Exercise 4.8(b) of
       [Mendelson] p. 230.  It is meaningful only for classes that exist as
       sets (i.e., are not proper classes).  See ~ df-op for other ordered pair
       definitions.  (Contributed by NM, 28-Sep-2003.) $)
    opthwiener $p |- ( { { { A } , (/) } , { { B } } } =
                    { { { C } , (/) } , { { D } } } <-> ( A = C /\ B = D ) ) $=
      ( csn c0 cpr wceq wcel snex prid2 wn 0ex eqcom preq2d syl sneqr sneq elpr
      wa id wo eleq2 mpbii sylib wb snnz elsn bitri nemtbir nelneq2 mp2an biorf
      mtbi ax-mp sylibr eqtr4d prex preqr1 jca preq1d sylan9eq impbii ) AGZHIZB
      GZGZIZCGZHIZDGZGZIZJZACJZBDJZUBVPVQVRVPVFVKJZVQVPVGVLJZVSVPVJVLVIIZJVTVPV
      JVOWAVPUCVPVIVNVLVPVIVLJZVIVNJZUDZWCVPVIVOKZWDVPVIVJKWEVGVIVHLZMVJVOVIUEU
      FVIVLVNWFUAUGWBNWCWDUHVLVIJZWBHVLKHVIKZNWGNVKHOMWHVHHBFUIWHHVHJVHHJHVHOUJ
      HVHPUKULHVLVIUMUNVLVIPUPWBWCUOUQURZQUSVGVLVIVFHUTVKHUTVARVFVKHALCLVARACES
      RVPVHVMJZVRVPWCWJWIVHVMBLSRBDFSRVBVQVRVJWAVOVQVGVLVIVQVFVKHACTVCVCVRVIVNV
      LVRWJWCBDTVHVMTRQVDVE $.

    $( 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 snex prex unipr wss wceq snsspr1 ssequn1 mpbi
      unieqi 3eqtri ) ABEZFAGZABHZHZFUBUCIZUCUAUDABCDJSUBUCAKABLMUBUCNUEUCOABPU
      BUCQRT $.

    $( 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 $.
  $}

  ${
    opthhausdorff.a $e |- A e. _V $.
    opthhausdorff.b $e |- B e. _V $.
    opthhausdorff.o $e |- A =/= O $.
    opthhausdorff.n $e |- B =/= O $.
    opthhausdorff.t $e |- B =/= T $.
    opthhausdorff.1 $e |- O e. _V $.
    opthhausdorff.2 $e |- T e. _V $.
    opthhausdorff.3 $e |- O =/= T $.
    $( Justification theorem for the ordered pair definition of Felix Hausdorff
       in "Grundz&uuml;ge der Mengenlehre" ("Basics of Set Theory"), 1914,
       p. 32: ` <. A , B >. `_H ` = { { A , O } , { B , T } } ` .  Hausdorff
       used ` 1 ` and ` 2 ` instead of ` O ` and ` T ` , but actually, any two
       different fixed sets will do (e.g., ` O = (/) ` and ` T = { (/) } ` ,
       see ~ 0nep0 ).  Furthermore, Hausdorff demanded that ` O ` and ` T ` are
       both different from ` A ` as well as ` B ` , which is actually not
       necessary in full extent ( ` A =/= T ` is not required).  This
       definition is meaningful only for classes ` A ` and ` B ` that exist as
       sets (i.e., are not proper classes):  If ` A ` and ` C ` were different
       proper classes ( ` A =/= C ` ), then
       ` { { A , O } , { B , T } } = { { C , O } , { D , T } `
       ` <-> { { O } , { B , T } } = { { O } , { D , T } ` is true if
       ` B = D ` , but ` ( A = C /\ B = D ) ` would be false.  See ~ df-op for
       other ordered pair definitions.  (Contributed by AV, 14-Jun-2022.) $)
    opthhausdorff $p |- ( { { A , O } , { B , T } } = { { C , O } , { D , T } }
                                                    <-> ( A = C /\ B = D ) ) $=
      ( cpr wceq wa wo cvv wne wcel wb prex pm3.2i necomi olci preq12nebg mp3an
      prneimg mp2 anim12i eqneqall mpi adantr ccase2 syl2anb wi adantl a1d jaoi
      simpl com12 imp sylbi preq1 preq12d impbii ) AFOZBEOZOCFOZDEOZOPZACPZBDPZ
      QZVLVHVJPZVIVKPZQZVHVKPZVIVJPZQZRZVOVHSUAVISUAVHVITZVLWBUBAFUCBEUCASUAZFS
      UAZQZBSUAZESUAZQZQABTAETQZFBTZFETZQZRWCWFWIWDWEGLUDWGWHHMUDUDWMWJWKWLBFJU
      ENUDUFAFBESSSSUIUJVHVIVJVKSSUGUHVRVOWAVPVMFFPZQZAFPZFCPZQZRZVNEEPZQZBEPZE
      DPZQZRZVOVQWDWEAFTZVPWSUBGLIAFCFSSUGUHWGWHBETZVQXEUBHMKBEDESSUGUHWOXAWRXD
      VOWOVMXAVNVMWNVAVNWTVAUKWPVOWQWPXFVOIVOAFULUMUNXBVOXCXBXGVOKVOBEULUMUNUOU
      PVSADPZFEPZQZAEPFDPQZRZBCPZEFPZQZBFPZECPZQZRZVOVTWDWEXFVSXLUBGLIAFDESSUGU
      HWGWHXGVTXSUBHMKBECFSSUGUHXLXSVOXJXSVOUQXKXJVOXSXIVOXHXIWLVONVOFEULUMURUS
      XSXKVOXOXKVOUQXRXOVOXKXNVOXMXNEFTVOFENUEVOEFULUMURUSXRVOXKXPVOXQXPBFTVOJV
      OBFULUMUNUSUTVBUTVCUPUTVDVOVHVJVIVKVMVPVNACFVEUNVNVQVMBDEVEURVFVG $.
  $}

  ${
    opthhausdorff0.a $e |- A e. _V $.
    opthhausdorff0.b $e |- B e. _V $.
    opthhausdorff0.c $e |- C e. _V $.
    opthhausdorff0.d $e |- D e. _V $.
    opthhausdorff0.1 $e |- O e. _V $.
    opthhausdorff0.2 $e |- T e. _V $.
    opthhausdorff0.3 $e |- O =/= T $.
    $( Justification theorem for the ordered pair definition of Felix Hausdorff
       in "Grundz&uuml;ge der Mengenlehre" ("Basics of Set Theory"), 1914,
       p. 32: ` <. A , B >. `_H ` = { { A , O } , { B , T } } ` .  Hausdorff
       used ` 1 ` and ` 2 ` instead of ` O ` and ` T ` , but actually, any two
       different fixed sets will do (e.g., ` O = (/) ` and ` T = { (/) } ` ,
       see ~ 0nep0 ).  Furthermore, Hausdorff demanded that ` O ` and ` T ` are
       both different from ` A ` as well as ` B ` , which is actually not
       necessary if all involved classes exist as sets (i.e. are not proper
       classes), in contrast to ~ opthhausdorff .  See ~ df-op for other
       ordered pair definitions.  (Contributed by AV, 12-Jun-2022.) $)
    opthhausdorff0 $p |- ( { { A , O } , { B , T } }
                           = { { C , O } , { D , T } }
                           <-> ( A = C /\ B = D ) ) $=
      ( cpr wceq wa wo prex preq12b adantl preqr1 anim12i wi wne eqneqall simpl
      mpi eqcoms simpr sylan9eqr sylan9eq jca jaoi sylbi com12 imp preq1 adantr
      ex preq12d impbii ) AFNZBENZNCFNZDENZNOZACOZBDOZPZVFVBVDOZVCVEOZPZVBVEOZV
      CVDOZPZQVIVBVCVDVEAFRBERCFRDERSVLVIVOVJVGVKVHACFGIUABDEHJUAUBVMVNVIVMADOZ
      FEOZPZAEOZFDOZPZQVNVIUCZAFDEGKJLSVRWBWAVQWBVPVQFEUDZWBMWBFEUEUGTVNWAVIVNB
      COZEFOZPZBFOZECOZPZQWAVIUCZBECFHLIKSWFWJWIWEWJWDWJFEVQWCWJMWJFEUEUGUHTWIW
      AVIWIWAPVGVHWAWIAECVSVTUFWGWHUIUJWIWABFDWGWHUFVSVTUIUKULUSUMUNUOUMUNUPUMU
      NVIVBVDVCVEVGVJVHACFUQURVHVKVGBDEUQTUTVA $.
  $}

  ${
    $d A c d $.  $d B c d $.  $d V c d $.  $d X c d $.  $d Y c d $.
    $( The singletons consisting of ordered triples which have distinct third
       components are disjoint.  (Contributed by Alexander van der Vekens,
       10-Mar-2018.) $)
    otsndisj $p |- ( ( A e. X /\ B e. Y )
                   -> Disj_ c e. V { <. A , B , c >. } ) $=
      ( vd wcel wa weq cv cotp csn cin c0 wceq wo wral wdisj wn wne otthg 3expa
      w3a simp3 biimtrdi con3rr3 imp neqned disjsn2 syl orrd adantrr ralrimivva
      wb expcom oteq3 sneqd disjor sylibr ) ADHZBEHZIZFGJZABFKZLZMZABGKZLZMZNOP
      ZQZGCRFCRFCVGSVCVLFGCCVCVECHZVLVHCHVCVMIZVDVKVDTZVNVKVOVNIZVFVIUAVKVPVFVI
      VOVNVFVIPZTVNVQVDVNVQAAPZBBPZVDUDZVDVAVBVMVQVTUOABVEADBVHECUBUCVRVSVDUEUF
      UGUHUIVFVIUJUKUPULUMUNCVGVJFGVDVFVIVEVHABUQURUSUT $.
  $}

  ${
    $d B a c d e s $.  $d V a c d e s $.  $d W a c d e s $.  $d X a c d e s $.
    $( The union of singletons consisting of ordered triples which have
       distinct first and third components are disjoint.  (Contributed by
       Alexander van der Vekens, 10-Mar-2018.) $)
    otiunsndisj $p |- ( B e. X
                 -> Disj_ a e. V U_ c e. ( W \ { a } ) { <. a , B , c >. } ) $=
      ( vd vs ve wcel weq cv csn cotp ciun wceq wral wa wn sylibr cdif c0 wdisj
      cin wo wrex eliun wi w3a otthg simp1 biimtrdi con3d impcom com3r imp31 wb
      3exp velsn eqeq1 notbid sylbi syl5ibrcom sylnibr adantr nrexdv rexlimdva2
      imp biimtrid ralrimiv oteq3 cbviunv eleq2i notbii ralbii disj expcom orrd
      sneqd adantrr ralrimivva sneq difeq2d oteq1 disjiunb ) ADJZEGKZFCELZMZUAZ
      WHAFLZNZMZOZFCGLZMZUAZWOAWKNZMZOZUDUBPZUEZGBQEBQEBWNUCWFXBEGBBWFWHBJZXBWO
      BJWFXCRZWGXAWGSZXDXAXEXDRZHLZWTJZSZHWNQZXAXFXGIWQWOAILZNZMZOZJZSZHWNQXJXF
      XPHWNXGWNJXGWMJZFWJUFXFXPFXGWJWMUGXFXQXPFWJXFWKWJJZRZXQRZXGXMJZIWQUFXOXTY
      AIWQXTYASXKWQJXTXGXLPZYAXSXQYBSZXSYCXQWLXLPZSZXEXDXRYEXDXRXEYEXCWFXRXEYEU
      HZUHXCWFXRYFXCWFXRUIZYDWGYGYDWGAAPZFIKZUIWGWHAWKWOBAXKDWJUJWGYHYIUKULUMUR
      UNUOUPXQXGWLPZYCYEUQHWLUSYJYBYDXGWLXLUTVAVBVCVHHXLUSVDVEVFIXGWQXMUGVDVGVI
      VJXIXPHWNXHXOWTXNXGFIWQWSXMYIWRXLWKXKWOAVKVSVLVMVNVOTHWNWTVPTVQVRVTWAFBWJ
      WMWQEGWSWGWIWPCWHWOWBWCWGWLWRWHWOAWKWDVSWET $.
  $}

  ${
    $d A x y z $.  $d B y $.  $d C x y $.  $d D x y $.
    iunopeqop.b $e |- B e. _V $.
    iunopeqop.c $e |- C e. _V $.
    iunopeqop.d $e |- D e. _V $.
    $( Implication of an ordered pair being equal to an indexed union of
       singletons of ordered pairs.  (Contributed by AV, 20-Sep-2020.)  (Avoid
       depending on this detail.) $)
    iunopeqop $p |- ( A =/= (/) -> ( U_ x e. A { <. x , B >. } = <. C , D >.
                                     -> E. z A = { z } ) ) $=
      ( vy wne cv wrex csn wceq cop wi wcel wa wss c0 wex n0snor2el nfiu1 nfeq1
      wo ciun nfv nfim ssiun2 nfcv nfcsb1v nfop nfsn nfss csbeq1a opeq12d sneqd
      csb id sseq1d vtoclgaf anim12i cun unss sseq2 cpr df-pr eqcomi sseq1i vex
      csbex propssopi eqneqall syl sylbi biimtrdi com14 biimtrid rexlimdva ax-1
      mpd rexlimi jaoi ) CUAKALZJLZKZJCMZACMZCBLNOBUBZUFACWEDPZNZUGZEFPZOZWJQZA
      JBCUCWIWPWJWHWPACWOWJAAWMWNACWLUDZUEWJAUHUIWECRZWGWPJCWRWFCRZSZWLWMTZWFAW
      FDUSZPZNZWMTZSZWGWPQZWRXAWSXEACWLUJZXAXEAWFCAWFUKZAXDWMAXCAWFXBXIAWFDULUM
      UNWQUOWEWFOZWLXDWMXJWKXCXJWEWFDXBXJUTAWFDUPUQURVAXHVBVCXFWLXDVDZWMTZWTXGW
      LXDWMVEWOXLWGWTWJWOXLXKWNTZWGWTWJQZQZWMWNXKVFXMWKXCVGZWNTZXOXKXPWNXPXKWKX
      CVHVIVJXQXJXOWEDWFXBEFAVKGJVKAWFDGVLHIVMXNWEWFVNVOVPVQVRVSWBVTWCWJWOWAWDV
      O $.
  $}

  $( Binary relation for an ordered pair singleton.  (Contributed by Thierry
     Arnoux, 23-Sep-2023.) $)
  brsnop $p |- ( ( A e. V /\ B e. W )
                   -> ( X { <. A , B >. } Y <-> ( X = A /\ Y = B ) ) ) $=
    ( cop csn wbr wcel wa wceq df-br opex elsn opthg2 bitrid ) EFABGZHZIEFGZSJZ
    ACJBDJKZEALFBLKZEFSMUATRLUBUCTREFNOEFABCDPQQ $.

  ${
    brtp.1 $e |- X e. _V $.
    brtp.2 $e |- Y e. _V $.
    $( A necessary and sufficient condition for two sets to be related under a
       binary relation which is an unordered triple.  (Contributed by Scott
       Fenton, 8-Jun-2011.) $)
    brtp $p |- ( X { <. A , B >. , <. C , D >. , <. E , F >. } Y <->
                 ( ( X = A /\ Y = B ) \/
                   ( X = C /\ Y = D ) \/
                   ( X = E /\ Y = F ) ) ) $=
      ( cop ctp wbr wcel wceq w3o wa df-br opex opth eltp 3orbi123i 3bitri ) GH
      ABKZCDKZEFKZLZMGHKZUGNUHUDOZUHUEOZUHUFOZPGAOHBOQZGCOHDOQZGEOHFOQZPGHUGRUH
      UDUEUFGHSUAUIULUJUMUKUNGHABIJTGHCDIJTGHEFIJTUBUC $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Ordered-pair class abstractions (cont.)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $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, which does not
       require ~ ax-13 .  (Contributed by NM, 14-Apr-1995.)  Avoid ~ ax-13 .
       (Revised by GG, 26-Jan-2024.) $)
    opabidw $p |- ( <. x , y >. e. { <. x , y >. | ph } <-> ph ) $=
      ( vz cv cop wceq wa wex copab opex copsexgw bicomd df-opab elab2 ) DEZBEZ
      CEZFZGZAHCIBIZADSABCJQRKTAUAABCPLMABCDNO $.
    $( $j usage 'opabidw' avoids 'ax-13'; $)
  $}

  ${
    $d x z $.  $d y z $.  $d ph z $.
    $( The law of concretion.  Special case of Theorem 9.5 of [Quine] p. 61.
       Usage of this theorem is discouraged because it depends on ~ ax-13 .
       Use the weaker ~ opabidw when possible.  (Contributed by NM,
       14-Apr-1995.)  (Proof shortened by Andrew Salmon, 25-Jul-2011.)
       (New usage is discouraged.) $)
    opabid $p |- ( <. x , y >. e. { <. x , y >. | ph } <-> ph ) $=
      ( vz cv cop wceq wa wex copab opex copsexg bicomd df-opab elab2 ) DEZBEZC
      EZFZGZAHCIBIZADSABCJQRKTAUAABCPLMABCDNO $.
  $}

  ${
    $d A x z $.  $d A y z $.  $d ph z $.
    $( Membership in a class abstraction of ordered pairs.  Weaker version of
       ~ elopab with a sethood antecedent, avoiding ~ ax-sep , ~ ax-nul , and
       ~ ax-pr .  Originally a subproof of ~ elopab .  (Contributed by SN,
       11-Dec-2024.) $)
    elopabw $p |- ( A e. V ->
     ( A e. { <. x , y >. | ph } <-> E. x E. y ( A = <. x , y >. /\ ph ) ) ) $=
      ( vz cv cop wceq wa wex copab eqeq1 anbi1d 2exbidv df-opab elab2g ) FGZBG
      CGHZIZAJZCKBKDSIZAJZCKBKFDABCLERDIZUAUCBCUDTUBARDSMNOABCFPQ $.
    $( $j usage 'elopabw' avoids 'ax-sep' 'ax-nul' 'ax-pr'; $)

    $( 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 ) ) $=
      ( copab wcel cvv cv cop wceq wa wex elex opex eleq1 mpbiri adantr elopabw
      exlimivv pm5.21nii ) DABCEZFDGFZDBHZCHZIZJZAKZCLBLDUAMUGUBBCUFUBAUFUBUEGF
      UCUDNDUEGOPQSABCDGRT $.
  $}

  ${
    $d O o $.  $d o x y $.  $d ph o $.  $d ps x y $.  $d ch o $.
    rexopabb.o $e |- O = { <. x , y >. | ph } $.
    rexopabb.p $e |- ( o = <. x , y >. -> ( ps <-> ch ) ) $.
    $( Restricted existential quantification over an ordered-pair class
       abstraction.  (Contributed by AV, 8-Nov-2023.) $)
    rexopabb $p |- ( E. o e. O ps <-> E. x E. y ( ph /\ ch ) ) $=
      ( wrex copab wa wex rexeqi cv wcel impcom nfv nfrexw exlimi elopab simprr
      cop wceq wi biimpd adantr jca ex 2eximdv sylanb rexlimiva nfopab1 nfopab2
      wsbc opabidw opex sbcie rspesbca syl2anbr impbii bitri ) BFGJBFADEKZJZACL
      ZEMZDMZBFGVCHNVDVGBVGFVCFOZVCPVHDOZEOZUCZUDZALZEMDMZBVGADEVHUABVNVGBVMVED
      EBVMVEBVMLACBVLAUBVMBCVLBCUEAVLBCIUFUGQUHUIUJQUKULVFVDDBDFVCADEUMBDRSVEVD
      EBEFVCADEUNBERSAVKVCPBFVKUOVDCADEUPBCFVKVIVJUQIURBFVKVCUSUTTTVAVB $.
  $}

  ${
    $d x y z $.  $d x y w $.
    $( The law of concretion in terms of substitutions.  Version of
       ~ opelopabsb with set variables.  (Contributed by NM, 30-Sep-2002.)
       (Proof shortened by Andrew Salmon, 25-Jul-2011.)  Remove unnecessary
       commutation.  (Revised by SN, 1-Sep-2024.) $)
    vopelopabsb $p |- ( <. z , w >. e. { <. x , y >. | ph }
                 <-> [ z / x ] [ w / y ] ph ) $=
      ( cv cop wceq wex weq copab wcel wsb eqcom vex opth bitri anbi1i 2exbii
      wa elopab 2sb5 3bitr4i ) DFZEFZGZBFZCFZGZHZATZCIBIBDJCEJTZATZCIBIUFABCKLA
      CEMBDMUKUMBCUJULAUJUIUFHULUFUINUGUHUDUEBOCOPQRSABCUFUAABCDEUBUC $.
  $}

  ${
    $d x y z w $.  $d w z A $.  $d w x B $.  $d 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 ) $=
      ( vz vw cop wcel cvv wa wsbc c0 wne wceq cv wex vex nex wb copab wn opnzi
      simpl eqcomd necon3ai ax-mp elopab mtbir eleq1 mtbiri necon2ai opnz sylib
      sbcex spesbc exlimiv syl jca wsb opeq1 eleq1d dfsbcq2 bibi12d vopelopabsb
      opeq2 sbcbidv vtocl2g pm5.21nii ) DEHZABCUAZIZDJIZEJIZKZACELZBDLZVLVJMNVO
      VLVJMVJMOVLMVKIZVRMBPZCPZHZOZAKZCQZBQWDBWCCWAMNWCUBVSVTBRCRUCWCWAMWCMWAWB
      AUDUEUFUGSSABCMUHUIVJMVKUJUKULDEUMUNVQVMVNVPBDUOVQVPBQVNVPBDUPVPVNBACEUOU
      QURUSFPZGPZHZVKIZACGUTZBFUTZTDWFHZVKIZWIBDLZTVLVQTFGDEJJWEDOZWHWLWJWMWNWG
      WKVKWEDWFVAVBWIBFDVCVDWFEOZWLVLWMVQWOWKVJVKWFEDVFVBWOWIVPBDACGEVCVGVDABCF
      GVEVHVI $.

    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 stoic3 bitrid ) FGJZADEKLUKDMZEMZJNAOEPDPZULFN
      ZABQZRZESDSZUMGNZBCQZRZESDSZFHLGILOZTCADEUKUAURVBUOUSOZACQZRZESDSZVCUNCQU
      RVBOUQVAOZESDSVGUQVADEUEVHVFDEVHVDUPUTOVEUOUPUSUTUBABCUFUGUCUDACDEFGHIUHU
      IUJ $.
  $}

  ${
    $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 nfsbcw cv wceq
      sbcbidv 3bitri ) FGNADEUAOAEGPZDFPZBEGPZCADEFGUBFQOUJUKRJUIUKDFQBDEGDGUCH
      UDDUEFUFABEGLUGSTGQOUKCRKBCEGQIMSTUH $.
  $}

  ${
    $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 anim2d aleximi ss2abdv df-opab 3sstr4g
      copab id ) ABFZDGZCGZEHCHDHIJZAKZDLZCLZEMUDBKZDLZCLZEMACDSBCDSUCUGUJEUBUF
      UICUAUEUHDUAABUDUATNOOPACDEQBCDEQR $.
  $}

  ${
    $d x y $.
    $( Equivalence of ordered pair abstraction subclass and implication.
       Version of ~ ssopab2b with a disjoint variable condition, which does not
       require ~ ax-13 .  (Contributed by NM, 27-Dec-1996.)  Avoid ~ ax-13 .
       (Revised by GG, 26-Jan-2024.) $)
    ssopab2bw $p |- ( { <. x , y >. | ph } C_ { <. x , y >. | ps } <->
                 A. x A. y ( ph -> ps ) ) $=
      ( copab wss wal nfopab1 nfss nfopab2 cop wcel ssel opabidw 3imtr3g alrimi
      wi cv ssopab2 impbii ) ACDEZBCDEZFZABQZDGZCGUCUECCUAUBACDHBCDHIUCUDDDUAUB
      ACDJBCDJIUCCRDRKZUALUFUBLABUAUBUFMACDNBCDNOPPABCDST $.
    $( $j usage 'ssopab2bw' avoids 'ax-13'; $)

    $( Equivalence of ordered pair abstraction equality and biconditional.
       Version of ~ eqopab2b with a disjoint variable condition, which does not
       require ~ ax-13 .  (Contributed by Mario Carneiro, 4-Jan-2017.)  Avoid
       ~ ax-13 .  (Revised by GG, 26-Jan-2024.) $)
    eqopab2bw $p |- ( { <. x , y >. | ph } = { <. x , y >. | ps }
                  <-> A. x A. y ( ph <-> ps ) ) $=
      ( copab wss wa wi wal wceq wb ssopab2bw anbi12i eqss 2albiim 3bitr4i ) AC
      DEZBCDEZFZRQFZGABHDICIZBAHDICIZGQRJABKDICISUATUBABCDLBACDLMQRNABCDOP $.
    $( $j usage 'eqopab2bw' avoids 'ax-13'; $)
  $}

  $( Equivalence of ordered pair abstraction subclass and implication.  Usage
     of this theorem is discouraged because it depends on ~ ax-13 .  Use the
     weaker ~ ssopab2bw when possible.  (Contributed by NM, 27-Dec-1996.)
     (Proof shortened by Mario Carneiro, 18-Nov-2016.)
     (New usage is discouraged.) $)
  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.  Usage
     of this theorem is discouraged because it depends on ~ ax-13 .  Use the
     weaker ~ eqopab2bw when possible.  (Contributed by Mario Carneiro,
     4-Jan-2017.)  (New usage is discouraged.) $)
  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 $.
    $( Nonempty ordered pair class abstraction.  (Contributed by NM,
       10-Oct-2007.) $)
    opabn0 $p |- ( { <. x , y >. | ph } =/= (/) <-> E. x E. y ph ) $=
      ( vz copab c0 wne cv wcel wex n0 cop wceq elopab exbii exrot3 opex isseti
      wa bitri 19.41v mpbiran 2exbii ) ABCEZFGDHZUDIZDJZACJBJZDUDKUGUEBHZCHZLZM
      ZASZCJBJZDJZUHUFUNDABCUENOUOUMDJZCJBJUHUMDBCPUPABCUPULDJADUKUIUJQRULADUAU
      BUCTTT $.
  $}

  $( Empty ordered pair class abstraction.  (Contributed by AV,
     29-Oct-2021.) $)
  opab0 $p |- ( { <. x , y >. | ph } = (/) <-> A. x A. y -. ph ) $=
    ( copab c0 wceq wn wal wne wex opabn0 df-ne 2exnaln 3bitr3i con4bii ) ABCDZ
    EFZAGCHBHZPEIACJBJQGRGABCKPELABCMNO $.

  ${
    $d w y z A $.  $d w ph $.  $d w x y z $.
    $( Move substitution into a class abstraction.  Version of ~ csbopabgALT
       without a sethood antecedent but depending on more axioms.  (Contributed
       by NM, 6-Aug-2007.)  (Revised by NM, 23-Aug-2018.) $)
    csbopab $p |- [_ A / x ]_ { <. y , z >. | ph }
              = { <. y , z >. | [. A / x ]. ph } $=
      ( vw cvv wcel copab csb wsbc wceq cv wsb csbeq1 opabbidv wn c0 wex nexdv
      dfsbcq2 eqeq12d vex nfs1v nfopab sbequ12 csbief vtoclg csbprc sbcex con3i
      weq opabn0 necon1bbii sylib eqtr4d pm2.61i ) EGHZBEACDIZJZABEKZCDIZLZBFMZ
      USJZABFNZCDIZLVCFEGVDELZVEUTVGVBBVDEUSOVHVFVACDABFEUAPUBBVDUSVGFUCVFCDBAB
      FUDUEBFULAVFCDABFUFPUGUHURQZUTRVBBEUSUIVIVADSZCSZQVBRLVIVJCVIVADVAURABEUJ
      UKTTVKVBRVACDUMUNUOUPUQ $.

    $( Move substitution into a class abstraction.  Version of ~ csbopab with a
       sethood antecedent but depending on fewer axioms.  (Contributed by NM,
       6-Aug-2007.)  (Proof shortened by Mario Carneiro, 17-Nov-2016.)
       (New usage is discouraged.)  (Proof modification is discouraged.) $)
    csbopabgALT $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 A y z $.  $d V y z $.  $d Y y z $.  $d Z z $.  $d x y z $.
    $( Move substitution into a maps-to notation.  (Contributed by AV,
       26-Sep-2019.) $)
    csbmpt12 $p |- ( A e. V -> [_ A / x ]_ ( y e. Y |-> Z )
                             = ( y e. [_ A / x ]_ Y |-> [_ A / x ]_ Z ) ) $=
      ( vz wcel cv wceq copab csb cmpt wsbc csbopab sbcan sbcel12 bitrid df-mpt
      wa csbconstg eleq1d sbceq2g anbi12d opabbidv eqtrid csbeq2i 3eqtr4g ) CDH
      ZACBIZEHZGIZFJZTZBGKZLZUJACELZHZULACFLZJZTZBGKZACBEFMZLBUQUSMUIUPUNACNZBG
      KVBUNABGCOUIVDVABGVDUKACNZUMACNZTUIVAUKUMACPUIVEURVFUTVEACUJLZUQHUIURACUJ
      EQUIVGUJUQACUJDUAUBRACULFDUCUDRUEUFACVCUOBGEFSUGBGUQUSSUH $.

    $d Y x $.
    $( Move substitution into the second part of a maps-to notation.
       (Contributed by AV, 26-Sep-2019.) $)
    csbmpt2 $p |- ( A e. V -> [_ A / x ]_ ( y e. Y |-> Z )
                              = ( y e. Y |-> [_ A / x ]_ Z ) ) $=
      ( wcel cmpt csb csbmpt12 csbconstg mpteq1d eqtrd ) CDGZACBEFHIBACEIZACFIZ
      HBEPHABCDEFJNBOEPACEDKLM $.
  $}

  ${
    $d ph w $.  $d A w x $.  $d A y $.  $d w y z $.  $d x z $.
    $( Move indexed union inside an ordered-pair class abstraction.
       (Contributed by Stefan O'Rear, 20-Feb-2015.)  Avoid ~ ax-sep ,
       ~ ax-nul , ~ ax-pr .  (Revised by SN, 11-Nov-2024.) $)
    iunopab $p |- U_ z e. A { <. x , y >. | ph } =
        { <. x , y >. | E. z e. A ph } $=
      ( vw cv copab wcel wrex cab cop wceq wa wex ciun wb rexcom4 exbii bitri
      cvv elopabw elv rexbii r19.42v abbii df-iun df-opab 3eqtr4i ) FGZABCHZIZD
      EJZFKUJBGCGLMZADEJZNZCOZBOZFKDEUKPUOBCHUMURFUMUNANZCOZBOZDEJZURULVADEULVA
      QFABCUJUAUBUCUDVBUTDEJZBOURUTDBERVCUQBVCUSDEJZCOUQUSDCERVDUPCUNADEUESTSTT
      UFDFEUKUGUOBCFUHUI $.
    $( $j usage 'iunopab' avoids 'ax-sep' 'ax-nul' 'ax-pr'; $)

    $( Obsolete version of ~ iunopab as of 11-Dec-2024.  (Contributed by Stefan
       O'Rear, 20-Feb-2015.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    iunopabOLD $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 $.
  $}

  ${
    $d A x y $.  $d R x y $.
    $( Obsolete version of ~ elopabr as of 11-Dec-2024.  (Contributed by AV,
       16-Feb-2021.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    elopabrOLD $p |- ( A e. { <. x , y >. | x R y } -> A e. R ) $=
      ( cv wbr copab wcel cop wceq wa wex elopab df-br eleq1 imbitrrid exlimivv
      biimpi imp sylbi ) CAEZBEZDFZABGHCUAUBIZJZUCKZBLALCDHZUCABCMUFUGABUEUCUGU
      CUGUEUDDHZUCUHUAUBDNRCUDDOPSQT $.
  $}

  ${
    $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 $.
  $}

  ${
    $d F f p $.  $d P f p $.  $d W f p $.  $d ch f p $.  $d th f p $.
    2rbropap.1 $e |- ( ph -> M = { <. f , p >. | ( f W p /\ ps /\ ta ) } ) $.
    2rbropap.2 $e |- ( ( f = F /\ p = P ) -> ( ps <-> ch ) ) $.
    2rbropap.3 $e |- ( ( f = F /\ p = P ) -> ( ta <-> th ) ) $.
    $( 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 conditions ` ps ` and ` ta ` .  (Contributed by AV,
       31-Jan-2021.) $)
    2rbropap $p |- ( ( ph /\ F e. X /\ P e. Y )
                    -> ( F M P <-> ( F W P /\ ch /\ th ) ) ) $=
      ( wcel w3a wbr wa cv copab 3anass opabbii eqtrdi anbi12d rbropap bitr4di
      wceq ) AHKQFLQRHFISHFJSZCDTZTUJCDRABETZUKFGHIJKLMAIGUAZMUAZJSZBERZGMUBUOU
      LTZGMUBNUPUQGMUOBEUCUDUEUMHUIUNFUITBCEDOPUFUGUJCDUCUH $.
  $}

  $( The empty set is never an element in an ordered-pair class abstraction.
     (Contributed by Alexander van der Vekens, 5-Nov-2017.)  Reduce axiom usage
     and shorten proof.  (Revised by GG, 3-Oct-2024.) $)
  0nelopab $p |- -. (/) e. { <. x , y >. | ph } $=
    ( c0 copab wcel cv cop wceq wex vex opnzi nesymi intnanr nex elopab mtbir
    wa ) DABCEFDBGZCGZHZIZARZCJZBJUDBUCCUBAUADSTBKCKLMNOOABCDPQ $.
  $( $j usage '0nelopab' avoids 'ax-10' 'ax-11' 'ax-12'; $)

  $( 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 wbr cop wcel cvv wa df-br wn wceq opprc 0nelopab eleq1 mtbiri syl
    c0 con4i sylbi ) DEABCFZGDEHZUCIZDJIEJIKZDEUCLUFUEUFMUDTNZUEMDEOUGUETUCIABC
    PUDTUCQRSUAUB $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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 velpw anbi12i 3bitr4i ineqri eqcomi ) AD
      ZBDZEABEZDZCPQSCFZAGZTBGZHTRGTPIZTQIZHTSITABJUCUAUDUBCAKCBKLCRKMNO $.
  $}

  ${
    $d A x y $.  $d B x y $.
    $( The power class of the union of two classes is a subset of the union of
       their power classes, iff one class is a subclass of the other.  Exercise
       4.12(l) of [Mendelson] p. 235.  (Contributed by NM, 23-Nov-2003.) $)
    pwssun $p |- ( ( A C_ B \/ B C_ A ) <->
               ~P ( A u. B ) C_ ( ~P A u. ~P B ) ) $=
      ( vy vx wss wo cun cpw wceq pweq eqimss syl sylbi orim12i wn wa wcel elpw
      cv wi ssequn2 ssequn1 orcoms ssun cpr csn vex snss unss12 syl2anb zfpair2
      df-pr sseq1i bitr2i sylib ssel syl5 expcomd imp31 elun bitr4i simprbi ord
      prss simplbi impancom ssrdv exp31 con1b imbitrdi com23 imp ex orrd impbii
      ) ABEZBAEZFZABGZHZAHZBHZGZEZVRVTWAEZVTWBEZFZWDVQVPWGVQWEVPWFVQVSAIZWEBAUA
      WHVTWAIWEVSAJVTWAKLMVPVSBIZWFABUBWIVTWBIWFVSBJVTWBKLMNUCVTWAWBUDLWDVPVQWD
      VPOZVQWDWJPCBAWDWJCSZBQZWKAQZTWDWLWJWMWDWLWMOZVPTWJWMTWDWLWNVPWDWLPZWNPDA
      BWODSZAQZWNWPBQZWOWQPZWMWRWSWPWKUEZWAQZWTWBQZFZWMWRFWSWTWCQZXCWDWLWQXDWDW
      QWLXDWQWLPZWTVTQZWDXDXEWPUFZWKUFZGZVSEZXFWQXGAEXHBEXJWLWPADUGZUHWKBCUGZUH
      XGAXHBUIUJXFWTVSEXJWTVSDCUKZRWTXIVSWPWKULUMUNUOVTWCWTUPUQURUSWTWAWBUTUOXA
      WMXBWRXAWQWMXAWTAEWQWMPWTAXMRWPWKAXKXLVDVAVBXBWRWLXBWTBEWRWLPWTBXMRWPWKBX
      KXLVDVAVENLVCVFVGVHWMVPVIVJVKVLVGVMVNVO $.
  $}

  $( 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 NM, 23-Nov-2003.) $)
  pwun $p |- ( ( A C_ B \/ B C_ A ) <-> ~P ( A u. B ) = ( ~P A u. ~P B ) ) $=
    ( cun cpw wss wa wo wceq pwunss biantru pwssun eqss 3bitr4i ) ABCDZADBDCZEZ
    PONEZFABEBAEGNOHQPABIJABKNOLM $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  The identity relation
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c _I $.  $( Letter I for the identity relation. $)

  $( Extend the definition of a class to include the identity relation. $)
  cid $a class _I $.

  ${
    $d x y $.
    $( Define the identity relation.  Definition 9.15 of [Quine] p. 64.  For
       example, ` 5 _I 5 ` and ` -. 4 _I 5 ` ( ~ ex-id ).  (Contributed by NM,
       13-Aug-1995.) $)
    df-id $a |- _I = { <. x , y >. | x = y } $.
  $}

  ${
    $d x y $.
    $( The identity function expressed using maps-to notation.  (Contributed by
       Scott Fenton, 15-Dec-2017.) $)
    dfid4 $p |- _I = ( x e. _V |-> x ) $=
      ( vy weq copab cv cvv wcel wa cid equcom vex biantrur bitri opabbii df-id
      cmpt df-mpt 3eqtr4i ) ABCZABDAEZFGZBACZHZABDIAFTPSUCABSUBUCABJUAUBAKLMNAB
      OABFTQR $.
  $}

  ${
    $d x y z u $.
    $( Alternate definition of the identity relation.  Instance of ~ dfid3 not
       requiring auxiliary axioms.  (Contributed by NM, 15-Mar-2007.)  Reduce
       axiom usage.  (Revised by GG, 4-Nov-2024.)  (Proof shortened by BJ,
       5-Nov-2024.)

       Use ~ df-id instead to make the semantics of the constructor ~ df-opab
       clearer.  (New usage is discouraged.) $)
    dfid2 $p |- _I = { <. x , x >. | x = x } $=
      ( vy vz vu cid weq copab df-id cv cop wceq wa wex cab opeq2d eqeq2d exbii
      equcomi bitri df-opab pm5.32ri ax6evr mpbiran2 pm4.71i id opeq12d eqeq12d
      19.42v eqidd anbi12d exexw abbii 3eqtr4i eqtri ) EABFZABGZAAFZAAGZABHCIZA
      IZBIZJZKZUOLZBMZAMZCNUSUTUTJZKZUQLZAMZAMZCNUPURVFVKCVFVJVKVEVIAVEVHVIVEVH
      UOLZBMZVHVDVLBUOVCVHUOVBVGUSUOVAUTUTABROPUAQVMVHUOBMBAUBVHUOBUHUCSVHUQVHU
      TUIUDSQVIUSDIZVNJZKZDDFZLADADFZVHVPUQVQVRVGVOUSVRUTVNUTVNVRUEZVSUFPVRUTVN
      UTVNVSVSUGUJUKSULUOABCTUQAACTUMUN $.
    $( $j usage 'dfid2' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    $d w z x $.  $d w z y $.
    $( A stronger version of ~ df-id that does not require ` x ` and ` y ` to
       be disjoint.  This is not the definition since, in order to pass our
       definition soundness test, a definition has to have disjoint dummy
       variables, see ~ conventions .  The proof can be instructive in showing
       how disjoint variable conditions may be eliminated, a task that is not
       necessarily obvious.  (Contributed by NM, 5-Feb-2008.)  (Revised by
       Mario Carneiro, 18-Nov-2016.)

       Use ~ df-id instead to make the semantics of the constructor ~ df-opab
       clearer (in usages, ` x , y ` will typically be dummy variables, so can
       be assumed disjoint).  (New usage is discouraged.) $)
    dfid3 $p |- _I = { <. x , y >. | x = y } $=
      ( vz vw weq copab cv cop wa wex cab wb exbii opeq2 eqeq2d equequ2 anbi12d
      wceq nfnae nfcvd cid df-id wal equcom anbi1ci equsexvw equid biantru nfe1
      3bitri 19.9 bitr4i sps drex1 drex2 bitrid wn nfcvf2 nfopd nfeqd nfand a1i
      wi cbvexd exbid pm2.61i abbii df-opab 3eqtr4i eqtri ) UAACEZACFZABEZABFZA
      CUBDGZAGZCGZHZRZVKIZCJZAJZDKVOVPBGZHZRZVMIZBJZAJZDKVLVNWBWHDVMAUCZWBWHLWB
      VOVPVPHZRZAAEZIZAJZAJZWIWHWBWNWOWAWMAWACAEZVSIZCJWKWMVTWQCVKWPVSACUDUEMVS
      WKCAWPVRWJVOVQVPVPNOUFWLWKAUGUHUJMWNAWMAUIUKULWNWGABAWMWFABVMWMWFLAVMWKWE
      WLVMVMWJWDVOVPWCVPNOABAPQUMUNUOUPWIUQZWAWGAABASWRVTWFCBABBSWRVSVKBWRBVOVR
      WRBVOTWRBVPVQABURZWRBVQTZUSUTWRBVPVQWSWTUTVACBEZVTWFLVCWRXAVSWEVKVMXAVRWD
      VOVQWCVPNOCBAPQVBVDVEVFVGVKACDVHVMABDVHVIVJ $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  The membership relation (or epsilon relation)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c _E $.  $( Upright uppercase letter E for the membership relation. $)

  $( Extend class notation to include the membership relation. $)
  cep $a class _E $.

  ${
    $d x y $.
    $( Define the membership relation (also called "epsilon relation" since it
       is sometimes denoted by the lowercase Greek letter "epsilon").  Similar
       to Definition 6.22 of [TakeutiZaring] p. 30.  The membership relation
       and the membership predicate agree, that is, ` ( A _E B <-> A e. B ) ` ,
       when ` B ` is a set (see ~ epelg ).  Thus, ` |- 5 _E { 1 , 5 } `
       ( ~ ex-eprel ).  (Contributed by NM, 13-Aug-1995.) $)
    df-eprel $a |- _E = { <. x , y >. | x e. y } $.
  $}

  ${
    $d A x y $.  $d B x y $.
    $( The membership relation and the membership predicate agree when the
       "containing" class is a set.  General version of ~ epel and closed form
       of ~ epeli .  Definition 1.6 of [Schloeder] p. 1.  (Contributed by Scott
       Fenton, 27-Mar-2011.)  (Revised by Mario Carneiro, 28-Apr-2015.)  (Proof
       shortened by BJ, 14-Jul-2023.) $)
    epelg $p |- ( B e. V -> ( A _E B <-> A e. B ) ) $=
      ( vx vy wcel cvv cep wbr wi cop df-br c0 wceq cv 0nelopab df-eprel eqcomi
      copab a1i eleq2i mtbi eleq1 mtbiri con2i opprc1 nsyl2 sylbi eleq12 brabga
      elex wb expcom pm5.21ndd ) BCFZAGFZABHIZABFZUQUPJUOUQABKZHFZUPABHLUTUSMNZ
      UPVAUTVAUTMHFZMDOZEOZFZDESZFVBVEDEPVFHMHVFDEQZRUAUBUSMHUCUDUEABUFUGUHTURU
      PJUOABUKTUPUOUQURULVEURDEABHGCVCAVDBUIVGUJUMUN $.
  $}

  ${
    epeli.1 $e |- B e. _V $.
    $( The membership relation and the membership predicate agree when the
       "containing" class is a set.  Inference associated with ~ epelg .
       (Contributed by Scott Fenton, 11-Apr-2012.) $)
    epeli $p |- ( A _E B <-> A e. B ) $=
      ( cvv wcel cep wbr wb epelg ax-mp ) BDEABFGABEHCABDIJ $.
  $}

  $( The membership relation and the membership predicate agree when the
     "containing" class is a setvar.  Definition 1.6 of [Schloeder] p. 1.
     (Contributed by NM, 13-Aug-1995.)  Replace the first setvar variable with
     a class variable.  (Revised by BJ, 13-Sep-2022.) $)
  epel $p |- ( A _E x <-> A e. x ) $=
    ( cv vex epeli ) BACADE $.

  $( An example for the membership relation.  (Contributed by AV,
     19-Jun-2022.) $)
  0sn0ep $p |- (/) _E { (/) } $=
    ( c0 csn cep wbr wcel 0ex snid snex epeli mpbir ) AABZCDAKEAFGAKAHIJ $.

  $( The membership relation is nonempty.  (Contributed by AV, 19-Jun-2022.) $)
  epn0 $p |- _E =/= (/) $=
    ( c0 csn cep wbr wne 0sn0ep brne0 ax-mp ) AABZCDCAEFAICGH $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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 total 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 ` .  For example, ` < Po RR ` is true, while ` <_ Po RR ` is false
       ( ~ ex-po ).  (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 complete (linear) order predicate.  The expression
       ` R Or A ` is true if relationship ` R ` orders ` A ` .  For example,
       ` < Or RR ` is true ( ~ ltso ).  Equivalent to Definition 6.19(1) of
       [TakeutiZaring] p. 29.  (Contributed by NM, 21-Jan-1996.) $)
    df-so $a |- ( R Or A <-> ( R Po A /\ A. x e. A A. y e. A
                ( x R y \/ x = y \/ y R x ) ) ) $.
  $}

  ${
    $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 ss2ralv ralimdv syld df-po
      3imtr4g ) ABGZDHZUBCIJUBEHZCIUCFHZCIKUBUDCILKZFBMEBMZDBMZUEFAMEAMZDAMZBCN
      ACNUAUGUFDAMUIUFDABOUAUFUHDAUEEFABPQRDEFBCSDEFACST $.
  $}

  ${
    $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 $.

  ${
    poeq12d.1 $e |- ( ph -> R = S ) $.
    poeq12d.2 $e |- ( ph -> A = B ) $.
    $( Equality deduction for partial orderings.  (Contributed by Matthew
       House, 10-Sep-2025.) $)
    poeq12d $p |- ( ph -> ( R Po A <-> S Po B ) ) $=
      ( wceq wpo wb poeq1 poeq2 sylan9bb syl2anc ) ADEHZBCHZBDIZCEIZJFGOQBEIPRB
      DEKBCELMN $.
  $}

  ${
    $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 nfralw nfn nfxfr
      nfim ) BCIFJZUDCKZLZUDGJZCKZUGHJZCKZMZUDUICKZNZMZHBOZGBOZFBOAFGHBCPUPAFBE
      UOAGBEUNAHBEUFUMAUEAAUDUDCAUDQZDUQRUAUKULAUHUJAAUDUGCUQDAUGQZRAUGUICURDAU
      IQZRSAUDUICUQDUSRUCSTTTUB $.

    $( Bound-variable hypothesis builder for total orders.  (Contributed by
       Stefan O'Rear, 20-Jan-2015.) $)
    nfso $p |- F/ x R Or A $=
      ( va vb wor wpo cv wbr weq w3o wral wa df-so nfpo nfcv nfbr nfralw nf3or
      nfv nfan nfxfr ) BCHBCIZFJZGJZCKZFGLZUGUFCKZMZGBNZFBNZOAFGBCPUEUMAABCDEQU
      LAFBEUKAGBEUHUIUJAAUFUGCAUFRZDAUGRZSUIAUBAUGUFCUODUNSUATTUCUD $.
  $}

  ${
    $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 $.
    $( Characteristic properties of a partial order in class notation.
       (Contributed by NM, 27-Mar-1997.)  Reduce axiom usage and shorten proof.
       (Revised by GG, 3-Oct-2024.) $)
    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 cv wbr wn wa wi wral wcel wceq breq1 imbi12d breq2 anbi2d wpo
      w3a df-po biimpi id breq12d notbid anbi1d anbi12d imbi1d rspc3v syl5com )
      AEUAZFIZUNEJZKZUNGIZEJZUQHIZEJZLZUNUSEJZMZLZHANGANFANZBAOCAODAOUBBBEJZKZB
      CEJZCDEJZLZBDEJZMZLZUMVEFGHAEUCUDVDVMVGBUQEJZUTLZBUSEJZMZLVGVHCUSEJZLZVPM
      ZLFGHBCDAAAUNBPZUPVGVCVQWAUOVFWAUNBUNBEWAUEZWBUFUGWAVAVOVBVPWAURVNUTUNBUQ
      EQUHUNBUSEQRUIUQCPZVQVTVGWCVOVSVPWCVNVHUTVRUQCBESUQCUSEQUIUJTUSDPZVTVLVGW
      DVSVJVPVKWDVRVIVHUSDCESTUSDBESRTUKUL $.
    $( $j usage 'pocl' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $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 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 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 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 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 $.

  $( Two sets related by a partial order are not equal.  (Contributed by AV,
     13-Mar-2023.) $)
  po2ne $p |- ( ( R Po V /\ ( A e. V /\ B e. V ) /\ A R B ) -> A =/= B ) $=
    ( wpo wcel wa wbr w3a wi wceq breq1 wn poirr adantrl pm2.21d com13 biimtrdi
    wne ex com24 3impd ax-1 pm2.61ine ) DCEZADFZBDFZGZABCHZIZABSZJABABKZUEUHUIU
    KULUIUHUEUKULUIBBCHZUHUEUKJJABBCLUEUHUMUKUEUHUMUKJUEUHGUMUKUEUGUMMUFDBCNOPT
    QRUAUBUKUJUCUD $.

  ${
    $d x y z R $.
    $( Any relation is a partial order on 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 ) $.
    $( The inverse image of a partial order is a partial order.  (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 csbeq1
      csbie eqtr3id breq2d opelopabf 3bitri sylnibr sylan2 anassrs w3a wi com12
      3anim123d imp adantll potr anbi12i 3imtr4g adantlr syldan ispod ) DEUAZGD
      NZACUGZOZKLMCFVTWBKPZCNZWDWDFQZUBZWBWEOVTAWDGRZDNZWGWEWBWIWAWIAWDCAWHDAWD
      GUCZUDAKSZGWHDAWDGUEZUFUHZUIVTWIOWHWHEQZWFDWHEUJWFWDWDTZFNWOGHEQZABUKZNWN
      WDWDFULFWQWOIUMWPWHHEQZWNABWDWDAWHHEWJAEUNZAHUNZUOZWNBUPKUQZXBWKGWHHEWLUR
      ZBKSZHWHWHEXDHABPZGRZWHAXEGHBUQJUTZAXEWDGUSVAVBVCVDVEVFVGWCWELPZCNZMPZCNZ
      VHZWIAXHGRZDNZAXJGRZDNZVHZWDXHFQZXHXJFQZOZWDXJFQZVIZWBXLXQVTWBXLXQWBWEWIX
      IXNXKXPWEWBWIWMVJXIWBXNWAXNAXHCAXMDAXHGUCZUDALSZGXMDAXHGUEZUFUHVJXKWBXPWA
      XPAXJCAXODAXJGUCUDAMSGXODAXJGUEUFUHVJVKVLVMVTXQYBWBVTXQOWHXMEQZXMXOEQZOWH
      XOEQZXTYADWHXMXOEVNXRYFXSYGXRWDXHTZFNYIWQNYFWDXHFULFWQYIIUMWPWRYFABWDXHXA
      YFBUPXBLUQZXCBLSZHXMWHEYKHXFXMXGAXEXHGUSVAVBVCVDXSXHXJTZFNYLWQNYGXHXJFULF
      WQYLIUMWPXMHEQYGABXHXJAXMHEYCWSWTUOYGBUPYJMUQZYDGXMHEYEURBMSZHXOXMEYNHXFX
      OXGAXEXJGUSVAZVBVCVDVOYAWDXJTZFNYPWQNYHWDXJFULFWQYPIUMWPWRYHABWDXJXAYHBUP
      XBYMXCYNHXOWHEYOVBVCVDVPVQVRVS $.
  $}

  ${
    $d x y R $.  $d x y 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 wor wpo cv wbr weq w3o wral df-so simplbi ) ABEABFCGZDGZBHCDIONBH
      JDAKCAKCDABLM $.
  $}

  ${
    $d x y R $.  $d x y A $.  $d x y 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 wss wpo cv wbr weq w3o wral wa poss ss2ralv anim12d df-so 3imtr4g
      wor ) ABFZBCGZDHZEHZCIDEJUCUBCIKZEBLDBLZMACGZUDEALDALZMBCSACSTUAUFUEUGABC
      NUDDEABOPDEBCQDEACQR $.
  $}

  ${
    $d x y R $.  $d x y S $.  $d x y 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 wceq wpo cv wbr w3o wral wa wor poeq1 breq biidd 2ralbidv anbi12d
      3orbi123d df-so 3bitr4g ) BCFZABGZDHZEHZBIZUDUEFZUEUDBIZJZEAKDAKZLACGZUDU
      ECIZUGUEUDCIZJZEAKDAKZLABMACMUBUCUKUJUOABCNUBUIUNDEAAUBUFULUGUGUHUMUDUEBC
      OUBUGPUEUDBCOSQRDEABTDEACTUA $.
  $}

  $( 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 $.

  ${
    soeq12d.1 $e |- ( ph -> R = S ) $.
    soeq12d.2 $e |- ( ph -> A = B ) $.
    $( Equality deduction for total orderings.  (Contributed by Stefan O'Rear,
       19-Jan-2015.)  (Proof shortened by Matthew House, 10-Sep-2025.) $)
    soeq12d $p |- ( ph -> ( R Or A <-> S Or B ) ) $=
      ( wceq wor wb soeq1 soeq2 sylan9bb syl2anc ) ADEHZBCHZBDIZCEIZJFGOQBEIPRB
      DEKBCELMN $.
  $}

  $( 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 $.

  ${
    sotrd.1 $e |- ( ph -> R Or A ) $.
    sotrd.2 $e |- ( ph -> X e. A ) $.
    sotrd.3 $e |- ( ph -> Y e. A ) $.
    sotrd.4 $e |- ( ph -> Z e. A ) $.
    sotrd.5 $e |- ( ph -> X R Y ) $.
    sotrd.6 $e |- ( ph -> Y R Z ) $.
    $( Transitivity law for strict orderings, deduction form.  (Contributed by
       Scott Fenton, 24-Nov-2021.) $)
    sotrd $p |- ( ph -> X R Z ) $=
      ( wbr wor wcel wa wi sotr syl13anc mp2and ) ADECMZEFCMZDFCMZKLABCNDBOEBOF
      BOUAUBPUCQGHIJBDEFCRST $.
  $}

  ${
    $d x y z A $.  $d x y B $.  $d x y C $.  $d x y z R $.
    $( A strict order relation is linear (satisfies trichotomy).  (Contributed
       by NM, 21-Jan-1996.) $)
    solin $p |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) ->
              ( B R C \/ B = C \/ C R B ) ) $=
      ( vx vy vz wcel wa wbr wceq w3o cv weq breq1 breq2 3orbi123d imbi2d wral
      wi wor eqeq1 eqeq2 df-so equequ1 ralbidv rspw equequ2 syl6 impd simplbiim
      wpo com12 vtocl2ga impcom ) BAHCAHIADUAZBCDJZBCKZCBDJZLZUPEMZFMZDJZEFNZVB
      VADJZLZTUPBVBDJZBVBKZVBBDJZLZTUPUTTEFBCAAVABKZVFVJUPVKVCVGVDVHVEVIVABVBDO
      VABVBUBVABVBDPQRVBCKZVJUTUPVLVGUQVHURVIUSVBCBDPVBCBUCVBCBDOQRUPVAAHZVBAHZ
      IZVFUPADULVFFASZEASZVOVFTEFADUDVQVMVNVFVQVMVPVNVFTVPGMZVBDJZGFNZVBVRDJZLZ
      FASEGAEGNZVFWBFAWCVCVSVDVTVEWAVAVRVBDOEGFUEVAVRVBDPQUFUGVFVAVRDJZWCVRVADJ
      ZLFGAFGNVCWDVDWCVEWEVBVRVADPFGEUHVBVRVADOQUGUIUJUKUMUNUO $.
  $}

  $( 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 $.

  $( A strict order relation satisfies strict trichotomy.  (Contributed by NM,
     19-Feb-1996.) $)
  sotric $p |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) ->
              ( B R C <-> -. ( B = C \/ C R B ) ) ) $=
    ( wor wcel wa wceq wo wn wi sonr breq2 notbid syl5ibcom adantrr so2nr imnan
    wbr sylibr con2d jaod w3o solin 3orass sylib ord impbid con2bid ) ADEZBAFZC
    AFZGGZBCHZCBDSZIZBCDSZUMUPUQJZUMUNURUOUJUKUNURKULUJUKGBBDSZJUNURABDLUNUSUQB
    CBDMNOPUMUQUOUMUQUOGJUQUOJKABCDQUQUORTUAUBUMUQUPUMUQUNUOUCUQUPIABCDUDUQUNUO
    UEUFUGUHUI $.

  $( Trichotomy law for strict order relation.  (Contributed by NM,
     9-Apr-1996.)  (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
  sotrieq $p |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) ->
              ( B = C <-> -. ( B R C \/ C R B ) ) ) $=
    ( wor wcel wa wo wceq wn sonr adantrr pm1.2 nsyl breq2 breq1 orbi12d notbid
    wbr sylib syl5ibcom con2d w3o solin 3orass or12 ord impbid con2bid ) ADEZBA
    FZCAFZGGZBCDSZCBDSZHZBCIZUMUPUQJUMUQUPUMBBDSZURHZJUQUPJUMURUSUJUKURJULABDKL
    URMNUQUSUPUQURUNURUOBCBDOBCBDPQRUAUBUMUQUPUMUNUQUOHHZUQUPHUMUNUQUOUCUTABCDU
    DUNUQUOUETUNUQUOUFTUGUHUI $.

  $( Trichotomy law for strict order relation.  (Contributed by NM,
     5-May-1999.) $)
  sotrieq2 $p |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) ->
              ( B = C <-> ( -. B R C /\ -. C R B ) ) ) $=
    ( wor wcel wa wceq wbr wo wn sotrieq ioran bitrdi ) ADEBAFCAFGGBCHBCDIZCBDI
    ZJKOKPKGABCDLOPMN $.

  $( Asymmetry law for strict orderings.  (Contributed by Scott Fenton,
     24-Nov-2021.) $)
  soasym $p |- ( ( R Or A /\ ( X e. A /\ Y e. A ) ) ->
  ( X R Y -> -. Y R X ) ) $=
    ( wor wcel wa wbr wceq wo wn sotric pm2.46 biimtrdi ) ABECAFDAFGGCDBHCDIZDC
    BHZJKPKACDBLOPMN $.

  $( A transitivity relation.  (Read ` B <_ C ` and ` C < D ` implies
     ` B < D ` .)  (Contributed by Mario Carneiro, 10-May-2013.) $)
  sotr2 $p |- ( ( R Or A /\ ( B e. A /\ C e. A /\ D e. A ) ) ->
                ( ( -. C R B /\ C R D ) -> B R D ) ) $=
    ( wor wcel w3a wa wbr wn wceq wo wi sotric ancom2s 3adantr3 con2bid breq1
    wb biimpd a1i sotr expd jaod sylbird impd ) AEFZBAGZCAGZDAGZHIZCBEJZKZCDEJZ
    BDEJZULUNCBLZBCEJZMZUOUPNZULUMUSUHUIUJUMUSKTZUKUHUJUIVAACBEOPQRULUQUTURUQUT
    NULUQUOUPCBDESUAUBULURUOUPABCDEUCUDUEUFUG $.

  ${
    $d x y R $.  $d x y A $.  $d x y 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, linear relation is a strict ordering.
       (Contributed by NM, 21-Jan-1996.)  (Revised by Mario Carneiro,
       9-Jul-2014.) $)
    issod $p |- ( ph -> R Or A ) $=
      ( wpo cv wbr weq w3o wral wor ralrimivva df-so sylanbrc ) ADEHBIZCIZEJBCK
      SREJLZCDMBDMDENFATBCDDGOBCDEPQ $.
  $}

  ${
    $d x y z R $.  $d x y z A $.
    issoi.1 $e |- ( x e. A -> -. x R x ) $.
    issoi.2 $e |- ( ( x e. A /\ y e. A /\ z e. A ) ->
                  ( ( x R y /\ y R z ) -> x R z ) ) $.
    issoi.3 $e |- ( ( x e. A /\ y e. A ) -> ( x R y \/ x = y \/ y R x ) ) $.
    $( An irreflexive, transitive, linear relation is a strict ordering.
       (Contributed by NM, 21-Jan-1996.)  (Revised by Mario Carneiro,
       9-Jul-2014.) $)
    issoi $p |- R Or A $=
      ( wor wtru cv wcel wbr wn adantl w3a wa wi ispod weq w3o issod mptru ) DE
      IJABDEJABCDEAKZDLZUDUDEMNJFOUEBKZDLZCKZDLPUDUFEMZUFUHEMQUDUHEMRJGOSUEUGQU
      IABTUFUDEMUAJHOUBUC $.
  $}

  ${
    $d x y z R $.  $d x y z A $.
    isso2i.1 $e |- ( ( x e. A /\ y e. A ) ->
                     ( x R y <-> -. ( x = y \/ y R x ) ) ) $.
    isso2i.2 $e |- ( ( x e. A /\ y e. A /\ z e. A ) ->
                     ( ( x R y /\ y R z ) -> x R z ) ) $.
    $( Deduce strict ordering from its properties.  (Contributed by NM,
       29-Jan-1996.)  (Revised by Mario Carneiro, 9-Jul-2014.) $)
    isso2i $p |- R Or A $=
      ( cv wcel wbr wn wa weq wo equid orci wb wi nfv eleq1w anbi2d breq1 breq2
      equequ2 orbi12d notbid bibi12d imbi12d con2bid chvarfv anidms w3o biimprd
      mpbii orrd 3orass sylibr issoi ) ABCDEAHZDIZUSUSEJZKZUTUTLZAAMZVANZVBVDVA
      AOPUTBHZDIZLZABMZVFUSEJZNZUSVFEJZKZQZRVCVEVBQZRZBAVPBSBAMZVHVCVNVOVQVGUTU
      TBADTUAVQVKVEVMVBVQVIVDVJVABAAUDVFUSUSEUBUEVQVLVAVFUSUSEUCUFUGUHVHVLVKFUI
      ZUJUNUKGVHVLVKNVLVIVJULVHVLVKVHVKVMVRUMUOVLVIVJUPUQUR $.
  $}

  ${
    $d x y 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 c0 wor wpo cv wbr weq w3o wral po0 ral0 df-so mpbir2an ) DAEDAFBG
      ZCGZAHBCIQPAHJCDKZBDKALRBMBCDANO $.
  $}

  ${
    $d x y z A $.  $d x y z R $.
    $( A totally ordered set has at most one minimal element.  (Contributed by
       Mario Carneiro, 24-Jun-2015.)  (Revised by NM, 16-Jun-2017.) $)
    somo $p |- ( R Or A -> E* x e. A A. y e. A -. y R x ) $=
      ( vz wor cv wbr wn wral wa weq wi wrmo wcel breq1 notbid rspcv wo sylib
      im2anan9 ancomsd imp ioran w3o solin df-3or or32 ord biimtrrid syl5 exp4b
      pm2.43d ralrimivv breq2 ralbidv rmo4 sylibr ) CDFZBGZAGZDHZIZBCJZUTEGZDHZ
      IZBCJZKZAELZMZECJACJVDACNUSVKAECCUSVACOZVECOZKZVKUSVNVNVIVJVNVIKVAVEDHZIZ
      VEVADHZIZKZUSVNKZVJVNVIVSVNVHVDVSVLVHVPVMVDVRVGVPBVACBALVFVOUTVAVEDPQRVCV
      RBVECBELVBVQUTVEVADPQRUAUBUCVSVOVQSZIVTVJVOVQUDVTWAVJVTVOVJSVQSZWAVJSVTVO
      VJVQUEWBCVAVEDUFVOVJVQUGTVOVJVQUHTUIUJUKULUMUNVDVHAECVJVCVGBCVJVBVFVAVEUT
      DUOQUPUQUR $.
  $}

  $( Trichotomy law for strict orderings.  (Contributed by Scott Fenton,
     8-Dec-2021.) $)
  sotrine $p |- ( ( R Or A /\ ( B e. A /\ C e. A ) ) ->
    ( B =/= C <-> ( B R C \/ C R B ) ) ) $=
    ( wor wcel wa wbr wo wceq wn sotrieq bicomd necon1abid ) ADEBAFCAFGGZBCDHCB
    DHIZBCOBCJPKABCDLMN $.

  $( Transitivity law for strict orderings.  (Contributed by Scott Fenton,
     24-Nov-2021.) $)
  sotr3 $p |- ( ( R Or A /\ ( X e. A /\ Y e. A /\ Z e. A ) ) ->
  ( ( X R Y /\ -. Z R Y ) -> X R Z ) ) $=
    ( wor wcel w3a wa wbr wn wceq wo wb simp3 simp2 jca sotric sylan2 con2bid
    adantr wi breq2 biimprcd adantl sotr expdimp jaod sylbird expimpd ) ABFZCAG
    ZDAGZEAGZHZIZCDBJZEDBJZKZCEBJZUPUQIZUSEDLZDEBJZMZUTUPVDUSNUQUPURVDUOUKUNUMI
    URVDKNUOUNUMULUMUNOULUMUNPQAEDBRSTUAVAVBUTVCUQVBUTUBUPVBUTUQEDCBUCUDUEUPUQV
    CUTACDEBUFUGUHUIUJ $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Founded and well-ordering relations
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare new constant symbols. $)
  $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.  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 z R $.  $d x y z A $.
    $( Define the well-founded relation predicate.  Definition 6.24(1) of
       [TakeutiZaring] p. 30.  For alternate definitions, see ~ dffr2 and
       ~ dffr3 .  A class is called well-founded when the membership relation
       ` _E ` (see ~ df-eprel ) is well-founded on it, that is, ` A ` is
       well-founded if ` _E Fr A ` (some sources request that the membership
       relation be well-founded on its transitive closure).  (Contributed by
       NM, 3-Apr-1994.) $)
    df-fr $a |- ( R Fr A <-> A. x ( ( x C_ A /\ x =/= (/) ) ->
                                            E. y e. x A. z e. x -. z R y ) ) $.

    $( 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 ) $.
  $}

  $( Define the well-ordering predicate.  For an alternate definition, see
     ~ dfwe2 .  (Contributed by NM, 3-Apr-1994.) $)
  df-we $a |- ( R We A <-> ( R Fr A /\ R Or A ) ) $.

  ${
    $d A x y z $.  $d R x y z $.
    $( Alternate definition of ~ df-fr .  See ~ dffr5 for a definition without
       dummy variables (but note that their equivalence uses ~ ax-sep ).
       (Contributed by BJ, 16-Nov-2024.) $)
    dffr6 $p |- ( R Fr A <->
                   A. x e. ( ~P A \ { (/) } ) E. y e. x A. z e. x -. z R y ) $=
      ( cv wss c0 wne wa wbr wn wral wrex wi wal cpw csn wcel bicomi cdif velpw
      wfr wceq velsn necon3abii anbi12i eldif bitr4i imbi1i albii df-fr 3bitr4i
      df-ral ) AFZDGZUOHIZJZCFBFEKLCUOMBUONZOZAPUODQZHRZUAZSZUSOZAPDEUCUSAVCMUT
      VEAURVDUSURUOVASZUOVBSZLZJVDUPVFUQVHVFUPADUBTVGUOHVGUOHUDAHUETUFUGUOVAVBU
      HUIUJUKABCDEULUSAVCUNUM $.
  $}

  ${
    $d ph x y z $.  $d A x y z $.  $d B x y z $.  $d R x y z $.
    frd.fr $e |- ( ph -> R Fr A ) $.
    frd.ss $e |- ( ph -> B C_ A ) $.
    frd.ex $e |- ( ph -> B e. V ) $.
    frd.n0 $e |- ( ph -> B =/= (/) ) $.
    $( A nonempty subset of an ` R ` -well-founded class has an ` R ` -minimal
       element (deduction form).  (Contributed by BJ, 16-Nov-2024.) $)
    frd $p |- ( ph -> E. x e. B A. y e. B -. y R x ) $=
      ( vz cv wbr wn wral wrex cpw c0 csn cdif wceq simpr biidd raleqbidv elpwd
      wa rexeqbidv wne wcel nelsn syl eldifd wfr dffr6 sylib rspcdv2 ) ACMBMFNO
      ZCLMZPZBUSQZURCEPZBEQLEDRZSTZUAZAUSEUBZUGZUTVBBUSEAVFUCZVGURURCUSEVHVGURU
      DUEUHAEVCVDAEDGJIUFAESUIEVDUJOKESUKULUMADFUNVALVEPHLBCDFUOUPUQ $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.  $d x y R $.
    $( A nonempty subset of an ` R ` -well-founded class has an ` R ` -minimal
       element (inference form).  (Contributed by BJ, 16-Nov-2024.)  (Proof
       shortened by BJ, 19-Nov-2024.) $)
    fri $p |- ( ( ( B e. C /\ R Fr A ) /\ ( B C_ A /\ B =/= (/) ) ) ->
                E. x e. B A. y e. B -. y R x ) $=
      ( wcel wfr wa wss c0 wne simplr simprl simpll simprr frd ) DEGZCFHZIZDCJZ
      DKLZIZIABCDFERSUCMTUAUBNRSUCOTUAUBPQ $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x y z R $.  $d x y V $.
    $( Obsolete version of ~ fri as of 16-Nov-2024.  (Contributed by NM,
       18-Mar-1997.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    friOLD $p |- ( ( ( B e. C /\ R Fr A ) /\ ( B C_ A /\ B =/= (/) ) ) ->
                E. x e. B A. y e. B -. y R x ) $=
      ( vz wcel wfr wss c0 wne wa cv wbr wn wral wrex wi wal df-fr wceq anbi12d
      sseq1 neeq1 raleq rexeqbi1dv imbi12d spcgv biimtrid imp31 ) DEHZCFIZDCJZD
      KLZMZBNANFOPZBDQZADRZUMGNZCJZUTKLZMZUQBUTQZAUTRZSZGTULUPUSSZGABCFUAVFVGGD
      EUTDUBZVCUPVEUSVHVAUNVBUOUTDCUDUTDKUEUCVDURAUTDUQBUTDUFUGUHUIUJUK $.

    $( 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 z w A $.  $d x y z w R $.
    $( Alternate definition of well-founded relation.  Similar to Definition
       6.21 of [TakeutiZaring] p. 30.  (Contributed by NM, 17-Feb-2004.)
       (Proof shortened by Andrew Salmon, 27-Aug-2011.)  (Proof shortened by
       Mario Carneiro, 23-Jun-2015.)  Avoid ~ ax-10 , ~ ax-11 , ~ ax-12 , but
       use ~ ax-8 .  (Revised by GG, 3-Oct-2024.) $)
    dffr2 $p |- ( R Fr A <-> A. x ( ( x C_ A /\ x =/= (/) ) ->
                E. y e. x { z e. x | z R y } = (/) ) ) $=
      ( vw wfr cv wss c0 wne wa wbr wn wral wrex wi wal crab wceq df-fr rabeq0w
      breq1 rexbii imbi2i albii bitr4i ) DEGAHZDIUHJKLZFHZBHZEMZNFUHOZBUHPZQZAR
      UICHZUKEMZCUHSJTZBUHPZQZARABFDEUAUTUOAUSUNUIURUMBUHUQULCFUHUPUJUKEUCUBUDU
      EUFUG $.
    $( $j usage 'dffr2' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $d x y z A $.  $d x y z R $.
    $( Alternate proof of ~ dffr2 , which avoids ~ ax-8 but requires ~ ax-10 ,
       ~ ax-11 , ~ ax-12 .  (Contributed by NM, 17-Feb-2004.)  (Proof shortened
       by Andrew Salmon, 27-Aug-2011.)  (Proof shortened by Mario Carneiro,
       23-Jun-2015.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    dffr2ALT $p |- ( R Fr A <-> A. x ( ( x C_ A /\ x =/= (/) ) ->
                E. y e. x { z e. x | z R y } = (/) ) ) $=
      ( wfr cv wss c0 wne wa wbr wn wral wrex wi wal crab wceq df-fr rabeq0
      rexbii imbi2i albii bitr4i ) DEFAGZDHUFIJKZCGBGELZMCUFNZBUFOZPZAQUGUHCUFR
      ISZBUFOZPZAQABCDETUNUKAUMUJUGULUIBUFUHCUFUAUBUCUDUE $.
    $( $j usage 'dffr2ALT' avoids 'ax-8'; $)
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x y z R $.
    frc.1 $e |- B e. _V $.
    $( Property of well-founded relation (one direction of definition using
       class variables).  (Contributed by NM, 17-Feb-2004.)  (Revised by Mario
       Carneiro, 19-Nov-2014.) $)
    frc $p |- ( ( R Fr A /\ B C_ A /\ B =/= (/) ) ->
                E. x e. B { y e. B | y R x } = (/) ) $=
      ( vz wfr wss c0 wne w3a cv wbr wn wral wrex crab wceq cvv wcel fri mpanl1
      wa 3impb breq1 rabeq0w rexbii sylibr ) CEHZDCIZDJKZLGMZAMZENZOGDPZADQZBMZ
      UNENZBDRJSZADQUJUKULUQDTUAUJUKULUDUQFAGCDTEUBUCUEUTUPADUSUOBGDURUMUNEUFUG
      UHUI $.
    $( $j usage 'frc' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x y z R $.  $d x y S $.
    $( Subset theorem for the well-founded predicate.  Exercise 1 of
       [TakeutiZaring] p. 31.  (Contributed by NM, 3-Apr-1994.)  (Proof
       shortened by Andrew Salmon, 25-Jul-2011.) $)
    frss $p |- ( A C_ B -> ( R Fr B -> R Fr A ) ) $=
      ( vx vz vy wss cv c0 wne wa wbr wn wral wrex wi wal wfr sstr2 df-fr com12
      anim1d imim1d alimdv 3imtr4g ) ABGZDHZBGZUGIJZKZEHFHCLMEUGNFUGOZPZDQUGAGZ
      UIKZUKPZDQBCRACRUFULUODUFUNUJUKUFUMUHUIUMUFUHUGABSUAUBUCUDDFEBCTDFEACTUE
      $.

    $( 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 $.
  $}

  ${
    $d x y z R $.  $d x y z S $.  $d x y z A $.
    $( Equality theorem for the well-founded predicate.  (Contributed by NM,
       9-Mar-1997.) $)
    freq1 $p |- ( R = S -> ( R Fr A <-> S Fr A ) ) $=
      ( vx vz vy wceq cv wss c0 wne wa wbr wn wral wrex wi wal wfr df-fr notbid
      breq rexralbidv imbi2d albidv 3bitr4g ) BCGZDHZAIUHJKLZEHZFHZBMZNZEUHOFUH
      PZQZDRUIUJUKCMZNZEUHOFUHPZQZDRABSACSUGUOUSDUGUNURUIUGUMUQFEUHUHUGULUPUJUK
      BCUBUAUCUDUEDFEABTDFEACTUF $.
  $}

  $( Equality theorem for the well-founded predicate.  (Contributed by NM,
     3-Apr-1994.) $)
  freq2 $p |- ( A = B -> ( R Fr A <-> R Fr B ) ) $=
    ( wceq wfr wss wi eqimss2 frss syl eqimss impbid ) ABDZACEZBCEZMBAFNOGBAHBA
    CIJMABFONGABKABCIJL $.

  ${
    freq12d.1 $e |- ( ph -> R = S ) $.
    freq12d.2 $e |- ( ph -> A = B ) $.
    $( Equality deduction for well-founded relations.  (Contributed by Stefan
       O'Rear, 19-Jan-2015.)  (Proof shortened by Matthew House,
       10-Sep-2025.) $)
    freq12d $p |- ( ph -> ( R Fr A <-> S Fr B ) ) $=
      ( wceq wfr wb freq1 freq2 sylan9bb syl2anc ) ADEHZBCHZBDIZCEIZJFGOQBEIPRB
      DEKBCELMN $.
  $}

  $( 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 $.

  ${
    seeq12d.1 $e |- ( ph -> R = S ) $.
    seeq12d.2 $e |- ( ph -> A = B ) $.
    $( Equality deduction for the set-like predicate.  (Contributed by Matthew
       House, 10-Sep-2025.) $)
    seeq12d $p |- ( ph -> ( R Se A <-> S Se B ) ) $=
      ( wceq wse wb seeq1 seeq2 sylan9bb syl2anc ) ADEHZBCHZBDIZCEIZJFGOQBEIPRB
      DEKBCELMN $.
  $}

  ${
    $d R a b c $.  $d A a b c $.  $d x a b c $.
    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 $=
      ( va vc vb wfr cv wss c0 wne wa wbr wn wral wrex wi nfcv wal nfss nfv nfn
      df-fr nfan nfbr nfralw nfrexw nfim nfal nfxfr ) BCIFJZBKZUMLMZNZGJZHJZCOZ
      PZGUMQZHUMRZSZFUAAFHGBCUEVCAFUPVBAUNUOAAUMBAUMTZEUBUOAUCUFVAAHUMVDUTAGUMV
      DUSAAUQURCAUQTDAURTUGUDUHUIUJUKUL $.

    $( 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 cv wbr crab cvv wcel wral df-se nfcv nfbr nfrabw nfel1 nfralw
      nfxfr ) BCHFIZGIZCJZFBKZLMZGBNAGFBCOUFAGBEAUELUDAFBAUBUCCAUBPDAUCPQERSTUA
      $.

    $( 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 $=
      ( wwe wfr wor wa df-we nffr nfso nfan nfxfr ) BCFBCGZBCHZIABCJOPAABCDEKAB
      CDELMN $.
  $}

  ${
    $d x y A $.  $d x y z B $.  $d x y z R $.
    $( A well-founded relation is irreflexive.  Special case of Proposition
       6.23 of [TakeutiZaring] p. 30.  (Contributed by NM, 2-Jan-1994.)
       (Revised by Mario Carneiro, 22-Jun-2015.) $)
    frirr $p |- ( ( R Fr A /\ B e. A ) -> -. B R B ) $=
      ( vx vy vz wfr wcel wa cv wbr csn crab c0 wceq adantl wral breq1 notbid
      wn wrex wss wne simpl snssi snnzg snex frc syl3anc rabeq0w ralbidv bitrid
      wb breq2 rexsng ralsng bitrd mpbid ) ACGZBAHZIZDJZEJZCKZDBLZMNOZEVEUAZBBC
      KZTZVAUSVEAUBZVENUCZVGUSUTUDUTVJUSBAUEPUTVKUSBAUFPEDAVECBUGUHUIUTVGVIUMUS
      UTVGFJZBCKZTZFVEQZVIVFVOEBAVFVLVCCKZTZFVEQVCBOZVOVDVPDFVEVBVLVCCRUJVRVQVN
      FVEVRVPVMVCBVLCUNSUKULUOVNVIFBAVLBOVMVHVLBBCRSUPUQPUR $.
    $( $j usage 'frirr' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.  $d x y R $.
    $( A well-founded relation has no 2-cycle loops.  Special case of
       Proposition 6.23 of [TakeutiZaring] p. 30.  (Contributed by NM,
       30-May-1994.)  (Revised by Mario Carneiro, 22-Jun-2015.) $)
    fr2nr $p |- ( ( R Fr A /\ ( B e. A /\ C e. A ) ) ->
                -. ( B R C /\ C R B ) ) $=
      ( vx vy wcel wa wbr wn wo cv wral cvv adantl ad2antrl wceq notbid ralbidv
      breq2 wfr cpr wrex wss wne prex a1i simpl prssi prnzg fri syl22anc rexprg
      c0 wb mpbid wi prid2g ad2antll breq1 syl prid1g orim12d mpd orcomd sylibr
      rspcv ianor ) ADUAZBAGZCAGZHZHZBCDIZJZCBDIZJZKVNVPHJVMVQVOVMELZBDIZJZEBCU
      BZMZVRCDIZJZEWAMZKZVQVOKVMVRFLZDIZJZEWAMZFWAUCZWFVMWANGZVIWAAUDZWAUNUEZWK
      WLVMBCUFUGVIVLUHVLWMVIBCAUIOVJWNVIVKBCAUJPFEAWANDUKULVLWKWFUOVIWJWBWEFBCA
      AWGBQZWIVTEWAWOWHVSWGBVRDTRSWGCQZWIWDEWAWPWHWCWGCVRDTRSUMOUPVMWBVQWEVOVMC
      WAGZWBVQUQVKWQVIVJBCAURUSVTVQECWAVRCQVSVPVRCBDUTRVGVAVMBWAGZWEVOUQVJWRVIV
      KBCAVBPWDVOEBWAVRBQWCVNVRBCDUTRVGVAVCVDVEVNVPVHVF $.
  $}

  ${
    $d x y z R $.
    $( Any relation is well-founded on the empty set.  (Contributed by NM,
       17-Sep-1993.) $)
    fr0 $p |- R Fr (/) $=
      ( vx vz vy c0 wfr cv wss wne wa wbr crab wceq wrex dffr2 ss0 a1d necon1ad
      wi wn imp mpgbir ) EAFBGZEHZUCEIZJCGDGAKCUCLEMDUCNZSBBDCEAOUDUEUFUDUFUCEU
      DUCEMUFTUCPQRUAUB $.
  $}

  ${
    $d A x y z $.  $d R x y z $.  $d ph y z $.  $d ps x z $.
    frminex.1 $e |- A e. _V $.
    frminex.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( If an element of a well-founded set satisfies a property ` ph ` , then
       there is a minimal element that satisfies ` ph ` .  (Contributed by Jeff
       Madsen, 18-Jun-2010.)  (Proof shortened by Mario Carneiro,
       18-Nov-2016.) $)
    frminex $p |- ( R Fr A -> ( E. x e. A ph ->
                      E. x e. A ( ph /\ A. y e. A ( ps -> -. y R x ) ) ) ) $=
      ( vz wrex crab c0 wne cv wbr wn wi wral wa cvv wfr rabn0 wss rabex ssrab2
      wcel fri ralrab rexbii weq breq2 notbid imbi2d ralbidv rexrab2 bitri an4s
      sylib mpanl12 ex biimtrrid ) ACEJACEKZLMZEFUAZABDNZCNZFOZPZQZDERZSCEJZACE
      UBVDVCVKVBTUFZVBEUCZVDVCSVKACEGUDACEUEVLVDVMVCVKVLVDSVMVCSSVEINZFOZPZDVBR
      ZIVBJZVKIDEVBTFUGVRBVPQZDERZIVBJVKVQVTIVBABVPDCEHUHUIAVTVJICEICUJZVSVIDEW
      AVPVHBWAVOVGVNVFVEFUKULUMUNUOUPURUQUSUTVA $.
  $}

  $( A well-founded class does not belong to 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 frirr wb epelg adantl mtbid pm2.01da ) ABCZAADZMNEAAB
    FZNAABGNONHMAAAIJKL $.

  $( A well-founded class contains no 2-cycle loops.  (Contributed by NM,
     19-Apr-1994.) $)
  efrn2lp $p |- ( ( _E Fr A /\ ( B e. A /\ C e. A ) ) ->
                -. ( B e. C /\ C e. B ) ) $=
    ( cep wfr wcel wa wbr fr2nr wb epelg bi2anan9r adantl mtbid ) ADEZBAFZCAFZG
    ZGBCDHZCBDHZGZBCFZCBFZGZABCDIRUAUDJOQSUBPTUCBCAKCBAKLMN $.

  ${
    $d x y A $.
    $( The membership relation is set-like on any class.  (This is the origin
       of the term "set-like": a set-like relation "acts like" the membership
       relation of sets and their elements.)  (Contributed by Mario Carneiro,
       22-Jun-2015.) $)
    epse $p |- _E Se A $=
      ( vy vx cep wse wbr crab cvv wcel wral cab epel bicomi eqabi vex eqeltrri
      cv rabssab ssexi rgenw df-se mpbir ) ADEBQZCQZDFZBAGZHIZCAJUGCAUFUEBKZUDU
      HHUEBUDUEUCUDICUCLMNCOPUEBARSTCBADUAUB $.
  $}

  $( Similar to Theorem 7.2 of [TakeutiZaring] p. 35, except that the Axiom of
     Regularity is not required due to the 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 $.
  $( $j usage 'tz7.2' avoids 'ax-reg'; $)

  ${
    $d x y z A $.
    $( An alternate way of saying that the membership relation is well-founded.
       (Contributed by NM, 17-Feb-2004.)  (Revised by Mario Carneiro,
       23-Jun-2015.) $)
    dfepfr $p |- ( _E Fr A <-> A. x ( ( x C_ A /\ x =/= (/) ) ->
                  E. y e. x ( x i^i y ) = (/) ) ) $=
      ( vz cep wfr cv wss c0 wne wa wbr crab wceq wrex wi wal cin dffr2 wel
      epel rabbii dfin5 eqtr4i eqeq1i rexbii imbi2i albii bitri ) CEFAGZCHUJIJK
      ZDGZBGZELZDUJMZINZBUJOZPZAQUKUJUMRZINZBUJOZPZAQABDCESURVBAUQVAUKUPUTBUJUO
      USIUODBTZDUJMUSUNVCDUJBULUAUBDUJUMUCUDUEUFUGUHUI $.
  $}

  ${
    $d x y A $.  $d x y B $.
    epfrc.1 $e |- B e. _V $.
    $( A subset of a well-founded class has a minimal element.  (Contributed by
       NM, 17-Feb-2004.)  (Revised by David Abernethy, 22-Feb-2011.) $)
    epfrc $p |- ( ( _E Fr A /\ B C_ A /\ B =/= (/) ) ->
                  E. x e. B ( B i^i x ) = (/) ) $=
      ( vy cep wfr wss c0 wne w3a cv wbr crab wceq wrex cin frc wel dfin5 epel
      rabbii eqtr4i eqeq1i rexbii sylibr ) BFGCBHCIJKELZALZFMZECNZIOZACPCUHQZIO
      ZACPAEBCFDRUMUKACULUJIULEASZECNUJECUHTUIUNECAUGUAUBUCUDUEUF $.
  $}

  $( Subset theorem for the well-ordering predicate.  Exercise 4 of
     [TakeutiZaring] p. 31.  (Contributed by NM, 19-Apr-1994.) $)
  wess $p |- ( A C_ B -> ( R We B -> R We A ) ) $=
    ( wss wfr wor wa wwe frss soss anim12d df-we 3imtr4g ) ABDZBCEZBCFZGACEZACF
    ZGBCHACHNOQPRABCIABCJKBCLACLM $.

  $( Equality theorem for the well-ordering predicate.  (Contributed by NM,
     9-Mar-1997.) $)
  weeq1 $p |- ( R = S -> ( R We A <-> S We A ) ) $=
    ( wceq wfr wor wa wwe freq1 soeq1 anbi12d df-we 3bitr4g ) BCDZABEZABFZGACEZ
    ACFZGABHACHNOQPRABCIABCJKABLACLM $.

  $( Equality theorem for the well-ordering predicate.  (Contributed by NM,
     3-Apr-1994.) $)
  weeq2 $p |- ( A = B -> ( R We A <-> R We B ) ) $=
    ( wceq wfr wor wa wwe freq2 soeq2 anbi12d df-we 3bitr4g ) ABDZACEZACFZGBCEZ
    BCFZGACHBCHNOQPRABCIABCJKACLBCLM $.

  ${
    weeq12d.1 $e |- ( ph -> R = S ) $.
    weeq12d.2 $e |- ( ph -> A = B ) $.
    $( Equality deduction for well-orderings.  (Contributed by Stefan O'Rear,
       19-Jan-2015.)  (Proof shortened by Matthew House, 10-Sep-2025.) $)
    weeq12d $p |- ( ph -> ( R We A <-> S We B ) ) $=
      ( wceq wwe wb weeq1 weeq2 sylan9bb syl2anc ) ADEHZBCHZBDIZCEIZJFGOQBEIPRB
      DEKBCELMN $.
  $}

  $( A well-ordering is well-founded.  (Contributed by NM, 22-Apr-1994.) $)
  wefr $p |- ( R We A -> R Fr A ) $=
    ( wwe wfr wor df-we simplbi ) ABCABDABEABFG $.

  $( A well-ordering is a strict ordering.  (Contributed by NM,
     16-Mar-1997.) $)
  weso $p |- ( R We A -> R Or A ) $=
    ( wwe wfr wor df-we simprbi ) ABCABDABEABFG $.

  $( The elements of a class well-ordered by membership are comparable.
     (Contributed by NM, 17-May-1994.) $)
  wecmpep $p |- ( ( _E We A /\ ( x e. A /\ y e. A ) ) ->
                 ( x e. y \/ x = y \/ y e. x ) ) $=
    ( cep wwe wor cv wcel wa weq w3o weso solin epel biid 3orbi123i sylib sylan
    wbr ) CDECDFZAGZCHBGZCHIZUAUBHZABJZUBUAHZKZCDLTUCIUAUBDSZUEUBUADSZKUGCUAUBD
    MUHUDUEUEUIUFBUANUEOAUBNPQR $.

  $( On a class well-ordered by membership, the membership predicate is
     transitive.  (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 wor weso sotr sylan epel anbi12i 3imtr3g
    wi ) DEFZAGZDHBGZDHCGZDHIZJUBUCEKZUCUDEKZJZUBUDEKZABLZBCLZJACLUADEMUEUHUITD
    ENDUBUCUDEOPUFUJUGUKBUBQCUCQRCUBQS $.

  ${
    $d y z A $.  $d x y z B $.
    $( A nonempty subclass of a class well-ordered by membership has a minimal
       element.  Special case of Proposition 6.26 of [TakeutiZaring] p. 31.
       (Contributed by NM, 17-Feb-2004.) $)
    wefrc $p |- ( ( _E We A /\ B C_ A /\ B =/= (/) ) ->
               E. x e. B ( B i^i x ) = (/) ) $=
      ( vy vz cep wwe wss c0 wne cv cin wceq wrex wi wcel wa eqeq1d ex wel wess
      wex ineq2 rspcev adantl inss1 wfr wefr vex inex2 epfrc syl3an1 3exp rexin
      n0 mpi imbitrdi adantr wral elin df-3an 3anrot bitr3i wetrep expd sylan2b
      w3a exp44 com34 impd biimtrid imp4a com23 ralrimdv dfss3 imbitrrdi eqeq2i
      imp dfss in32 sylbb biimprd syl6 reximdvai syld pm2.61dne exlimdv syl6com
      3imp ) BFGZCBHZCIJZCAKZLZIMZACNZWKWJCFGZWLWPOCBFUAWLDKZCPZDUBWQWPDCUOWQWS
      WPDWQWSWPWQWSQZWPCWRLZIWSXAIMZWPOWQWSXBWPWOXBAWRCWMWRMWNXAIWMWRCUCRUDSUEW
      TXAIJZADTZXAWMLZIMZQZACNZWPWQXCXHOWSWQXCXFAXANZXHWQXACHZXCXIOCWRUFWQXJXCX
      IWQCFUGXJXCXICFUHACXAWRCDUIUJUKULUMUPXFACWRUNUQURWTXGWOACWTWMCPZXDXFWOWTX
      KXDXFWOOZWTXKXDQZWNWRHZXLWTXMEDTZEWNUSXNWTXMXOEWNWTEKZWNPZXMXOWTXQXKXDXOX
      QXPCPZEATZQWTXKXDXOOZOZXPCWMUTWTXRXSYAWTXRXKXSXTWQWSXRXKXSXTOZOOWQWSXRXKY
      BWSXRQXKQZWQXRXKWSVGZYBYCWSXRXKVGYDWSXRXKVAWSXRXKVBVCWQYDQXSXDXOEADCVDVEV
      FVHVRVIVJVKVLVMVNEWNWRVOVPXNWOXFXNWNXEIXNWNWNWRLZMWNXEMWNWRVSYEXEWNCWMWRV
      TVQWARWBWCVEVLWDWEWFSWGVKWHWI $.
  $}

  $( Any relation is a well-ordering of the empty set.  (Contributed by NM,
     16-Mar-1997.) $)
  we0 $p |- R We (/) $=
    ( c0 wwe wfr wor fr0 so0 df-we mpbir2an ) BACBADBAEAFAGBAHI $.

  ${
    $d x y z A $.  $d w x y z B $.  $d w x y z R $.  $d x y V $.
    $( A nonempty subset of an ` R ` -well-ordered class has a unique ` R `
       -minimal element.  (Contributed by NM, 18-Mar-1997.)  (Revised by Mario
       Carneiro, 28-Apr-2015.) $)
    wereu $p |- ( ( R We A /\ ( B e. V /\ B C_ A /\ B =/= (/) ) ) ->
                E! x e. B A. y e. B -. y R x ) $=
      ( wwe wcel wss c0 wne w3a wa cv wbr wn wral wrex wi wor wrmo wreu wfr fri
      wefr exp32 expcom 3imp2 sylan weso soss mpan9 somo syl 3ad2antr2 sylanbrc
      reu5 ) CEGZDFHZDCIZDJKZLZMBNANEOPBDQZADRZVCADUAZVCADUBURCEUCZVBVDCEUEVFUS
      UTVAVDUSVFUTVAVDSSUSVFMUTVAVDABCDFEUDUFUGUHUIURUSUTVEVAURUTMDETZVEURCETUT
      VGCEUJDCEUKULABDEUMUNUOVCADUQUP $.

    $( A nonempty subclass of an ` R ` -well-ordered and ` R ` -setlike class
       has a unique ` R ` -minimal element.  Proposition 6.26 of
       [TakeutiZaring] p. 31.  (Contributed by Scott Fenton, 29-Jan-2011.)
       (Revised by Mario Carneiro, 24-Jun-2015.) $)
    wereu2 $p |- ( ( ( R We A /\ R Se A ) /\ ( B C_ A /\ B =/= (/) ) ) ->
                E! x e. B A. y e. B -. y R x ) $=
      ( vz vw wa c0 cv wbr wn wral wrex wcel wi breq1 biimtrid sylc ad2antrr n0
      wwe wse wss wne wrmo wreu wex crab wceq rabeq0 weq cbvralvw breq2 ralbidv
      notbid bitrid rspcev ex ad2antll cvv wfr simprl simplr sess2 seex syl2anc
      simprr wefr ssrab2 sstrid fri expr syl21anc rexrab ralrab weso soss simpr
      wor sotr syl13anc ancomsd expdimp an32s idd jad ralimdva expimpd reximdva
      con3d syld pm2.61dne exlimdv impr somo syl reu5 sylanbrc ) CEUBZCEUCZHZDC
      UDZDIUEZHZHZBJZAJZEKZLZBDMZADNZXKADUFZXKADUGXBXCXDXLXDFJZDOZFUHXBXCHZXLFD
      UAXPXOXLFXBXCXOXLXBXCXOHZHZXLGJZXNEKZGDUIZIYAIUJXTLZGDMZXRXLXTGDUKXOYCXLP
      XBXCXOYCXLXKYCAXNDXKXSXHEKZLZGDMAFULZYCXJYEBGDBGULXIYDXGXSXHEQUPUMYFYEYBG
      DYFYDXTXHXNXSEUNUPUOUQURUSUTRXRYAIUEZXJBYAMZAYANZXLXRYAVAOZCEVBZYACUDZYGY
      IPXRDEUCZXOYJXRXCXAYMXBXCXOVCZWTXAXQVDDCEVESXBXCXOVHZGDXNEVFVGWTYKXAXQCEV
      ITXRYADCXTGDVJYNVKYJYKHYLYGYIABCYAVAEVLVMVNYIXHXNEKZYHHZADNXRXLXTYPYHAGDX
      SXHXNEQVOXRYQXKADXRXHDOZHZYPYHXKYHXGXNEKZXJPZBDMYSYPHZXKXTYTXJBGDXSXGXNEQ
      VPUUBUUAXJBDUUBXGDOZHZYTXJXJUUDXIYTYSUUCYPXIYTPYSUUCHZYPXIYTUUEXIYPYTUUED
      EVTZUUCYRXOXIYPHYTPXRUUFYRUUCXRXCCEVTZUUFYNWTUUGXAXQCEVQZTDCEVRZSTYSUUCVS
      XRYRUUCVDXRXOYRUUCYOTDXGXHXNEWAWBWCWDWEWKUUDXJWFWGWHRWIWJRWLWMVMWNRWOXFUU
      FXMXFXCUUGUUFXBXCXDVCWTUUGXAXEUUHTUUISABDEWPWQXKADWRWS $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Relations
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Introduce new constant symbols. $)
  $c X. $.  $( Times symbol (Cartesian product symbol) $)
  $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 Cartesian 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 >. } ) `
       ( ~ ex-xp ).  Another example is that the set of rational numbers is
       defined in ~ df-q using the Cartesian product ` ( 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 >. } `
       ( ~ ex-cnv ).

       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.  For example, ` ( ( exp o. cos ) `` 0 ) = _e `
       ( ~ ex-co ) because ` ( cos `` 0 ) = 1 ` (see ~ cos0 ) and
       ` ( exp `` 1 ) = _e ` (see ~ df-e ).  Note that Definition 7 of [Suppes]
       p. 63 reverses ` A ` and ` B ` , uses ` /. ` 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 } `
       ( ~ ex-dm ).  Another example is the domain of the complex arctangent,
       ` ( A e. dom arctan <-> ( A e. CC /\ A =/= -u _i /\ A =/= _i ) ) ` (for
       proof see ~ atandm ).  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 } ` ( ~ ex-rn ).
       Contrast with domain (defined in ~ df-dm ).  For alternate definitions,
       see ~ dfrn2 , ~ dfrn3 , and ~ dfrn4 .  The notation " ` ran ` " is used
       by Enderton.  The range of a function is often also called "the image of
       the function" (see definition in [Lang] p. ix), which can be justified
       by ~ imadmrn .  Not to be confused with "codomain" (see ~ df-f ), which
       may be a superset/superclass of the range (see ~ frn ).  (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, the expression ` ( exp |`` RR ) ` (used in
       ~ reeff1 ) means "the exponential function e to the x, but the exponent
       x must be in the reals" ( ~ df-ef defines the exponential function,
       which normally allows the exponent to be a complex number).  Another
       example is ` ( F = { <. 2 , 6 >. , <. 3 , 9 >. } /\ B = { 1 , 2 } ) -> `
       ` ( F |`` B ) = { <. 2 , 6 >. } ` ( ~ ex-res ).  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 (see ~ cnvrescnv ).  (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 } ` ( ~ ex-ima ).  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 Cartesian 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 $.

    ${
      $d x y A $.  $d x y B $.  $d x y C $.  $d x y D $.
      $( Subset theorem for Cartesian 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
        ) ABGZCDGZHZEIZAJZFIZCJZHZEFKUABJZUCDJZHZEFKACLBDLTUEUHEFRUBUFSUDUGABUA
        MCDUCMNOEFACPEFBDPQ $.
    $}

    $( A Cartesian 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 $.

    $( Intersection with a Cartesian product is a subclass of restriction.
       (Contributed by Peter Mazsa, 19-Jul-2019.) $)
    inxpssres $p |- ( R i^i ( A X. B ) ) C_ ( R |` A ) $=
      ( cxp cin cvv cres wss ssid ssv xpss12 mp2an sslin ax-mp df-res sseqtrri
      ) CABDZEZCAFDZEZCAGQSHZRTHAAHBFHUAAIBJAABFKLQSCMNCAOP $.

    $( A Cartesian 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 Cartesian 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 Cartesian 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 $.

    $( Equality theorem for Cartesian 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 Cartesian 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 cxp wcel cv cop wceq wa wex eqeq1 anbi1d 2exbidv copab df-xp df-opab
      cab eqtri elab2g ibi ) CDEGZHCAIZBIZJZKZUEDHUFEHLZLZBMAMZFIZUGKZUILZBMAMZ
      UKFCUDUDULCKZUNUJABUPUMUHUIULCUGNOPUDUIABQUOFTABDERUIABFSUAUBUC $.

    $( Membership in a Cartesian 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 Cartesian product.  (Contributed by NM, 23-Feb-2004.)
       (Proof shortened by JJ, 13-Aug-2021.) $)
    elxp2 $p |- ( A e. ( B X. C ) <-> E. x e. B E. y e. C A = <. x , y >. ) $=
      ( cv cop wceq wcel wa wex cxp wrex ancom 2exbii elxp r2ex 3bitr4i ) CAFZB
      FZGHZSDITEIJZJZBKAKUBUAJZBKAKCDELIUABEMADMUCUDABUAUBNOABCDEPUAABDEQR $.
  $}

  $( Equality theorem for Cartesian 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 Cartesian product.  (Contributed by NM,
       21-Dec-2008.) $)
    xpeq1i $p |- ( A X. C ) = ( B X. C ) $=
      ( wceq cxp xpeq1 ax-mp ) ABEACFBCFEDABCGH $.

    $( Equality inference for Cartesian 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 Cartesian 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 Cartesian 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 Cartesian 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 Cartesian 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 Cartesian product.  (Contributed by
       NM, 2-May-1996.)  (Revised by Mario Carneiro, 26-Apr-2015.)  (Proof
       shortened by JJ, 13-Aug-2021.) $)
    0nelxp $p |- -. (/) e. ( A X. B ) $=
      ( vx vy c0 cxp wcel cv cop wceq wa wex vex opnzi nesymi intnanr nex mtbir
      elxp ) EABFGECHZDHZIZJZTAGUABGKZKZDLZCLUFCUEDUCUDUBETUACMDMNOPQQCDEABSR
      $.

    $( A member of a Cartesian 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 eleq2 mtbiri adantr
      exlimivv sylbi ) CABFGCDHZEHZIZJZUBAGUCBGKZKZELDLMCGZNZDECABOUGUIDEUEUIUF
      UEUHMUDGUBUCPCUDMQRSTUA $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.  $d x y D $.
    $( Ordered pair membership in a Cartesian 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 $.
  $}

  $( Ordered pair membership in a Cartesian 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 $.

  ${
    opelxpii.1 $e |- A e. C $.
    opelxpii.2 $e |- B e. D $.
    $( Ordered pair membership in a Cartesian product (implication), induction
       form.  (Contributed by Steven Nguyen, 17-Jul-2022.) $)
    opelxpii $p |- <. A , B >. e. ( C X. D ) $=
      ( wcel cop cxp opelxpi mp2an ) ACGBDGABHCDIGEFABCDJK $.
  $}

  ${
    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 $.
  $}

  ${
    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 $.
  $}

  $( 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 $.

  $( The first member of an ordered pair of classes in a Cartesian product
     belongs to first Cartesian 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 Cartesian product
     belongs to second Cartesian 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 $.

  $( Ordered triple membership in a triple Cartesian product.  (Contributed by
     Scott Fenton, 31-Jan-2025.) $)
  otelxp $p |- ( <. A , B , C >. e. ( ( D X. E ) X. F ) <->
  ( A e. D /\ B e. E /\ C e. F ) ) $=
    ( cop cxp wcel wa cotp w3a opelxp bianbi df-ot eleq1i df-3an 3bitr4i ) ABGZ
    CGZDEHZFHZIZADIZBEIZJZCFIZJABCKZUBIUDUEUGLUCSUAIUGUFSCUAFMABDEMNUHTUBABCOPU
    DUEUGQR $.

  $( The first member of an ordered triple of classes in a Cartesian product
     belongs to first Cartesian 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 $.

  $( An ordered triple is an element of a doubled Cartesian product.
     (Contributed by Alexander van der Vekens, 26-Feb-2018.) $)
  otel3xp $p |- ( ( T = <. A , B , C >. /\ ( A e. X /\ B e. Y /\ C e. Z ) )
                         -> T e. ( ( X X. Y ) X. Z ) ) $=
    ( cotp wceq wcel w3a cxp cop df-ot wa 3simpa opelxp sylibr simp3 opelxpd
    eqeltrid eleq1 imbitrrid imp ) DABCHZIZAEJZBFJZCGJZKZDEFLZGLZJZUJUMUFUEULJU
    JUEABMZCMULABCNUJUNCUKGUJUGUHOUNUKJUGUHUIPABEFQRUGUHUISTUADUEULUBUCUD $.

  ${
    $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 ) ) $.
    $( Class abstraction restricted to a Cartesian product as an ordered-pair
       class abstraction.  (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 $.
  $}

  $( Binary relation on a Cartesian 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 $.

  $( A set is a binary relation if and only if it belongs to the powerclass of
     the cartesian square of the universal class.  (Contributed by Peter Mazsa,
     14-Jun-2018.)  (Revised by BJ, 16-Dec-2023.) $)
  pwvrel $p |- ( A e. V -> ( A e. ~P ( _V X. _V ) <-> Rel A ) ) $=
    ( wcel cvv cxp cpw wss wrel elpwg df-rel bitr4di ) ABCADDEZFCALGAHALBIAJK
    $.

  $( The powerclass of the cartesian square of the universal class is the class
     of all sets which are binary relations.  (Contributed by BJ,
     21-Dec-2023.) $)
  pwvabrel $p |- ~P ( _V X. _V ) = { x | Rel x } $=
    ( cv wrel cvv cxp cpw wcel wb pwvrel elv eqabi ) ABZCZADDEFZLNGMHALDIJK $.

  $( Two classes related by a binary relation are sets.  (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 $.

  $( If two classes are related by a binary relation, then the first class is a
     set.  (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 $.

  $( If two classes are related by a binary relation, then the second class is
     a set.  (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.  Inference
       form.  (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 $.
  $}

  ${
    nprrel12.1 $e |- Rel R $.
    $( Proper classes are not related via any relation.  (Contributed by AV,
       29-Oct-2021.) $)
    nprrel12 $p |- ( -. ( A e. _V /\ B e. _V ) -> -. A R B ) $=
      ( wbr cvv wcel wa brrelex12i con3i ) ABCEAFGBFGHABCDIJ $.

    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.)  (Revised by BJ, 14-Jul-2023.) $)
  0nelrel0 $p |- ( Rel R -> -. (/) e. R ) $=
    ( wrel cvv cxp c0 wss df-rel biimpi wcel wn 0nelxp a1i ssneldd ) ABZACCDZEN
    AOFAGHEOIJNCCKLM $.

  $( A binary relation does not contain the empty set.  (Contributed by AV,
     15-Nov-2021.) $)
  0nelrel $p |- ( Rel R -> (/) e/ R ) $=
    ( wrel c0 wcel wn wnel 0nelrel0 df-nel sylibr ) ABCADECAFAGCAHI $.

  ${
    $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 cv wceq breq1 imbi12d imbi2d breq2 brrelex12i anbi1d
      brrelex2i anbi12d imbi1d anbi2d vtoclg vtocl2g syl2im imp pm2.43i ) DEGJZ
      EFGJZKZDFGJZULUMUNUOLZULDMNEMNKUMFMNZUPDEGHUAEFGHUCUQAOZBOZGJZUSFGJZKZURF
      GJZLZLUQDUSGJZVAKZUOLZLUQUPLABDEMMURDPZVDVGUQVHVBVFVCUOVHUTVEVAURDUSGQUBU
      RDFGQRSUSEPZVGUPUQVIVFUNUOVIVEULVAUMUSEDGTUSEFGQUDUESUTUSCOZGJZKZURVJGJZL
      VDCFMVJFPZVLVBVMVCVNVKVAUTVJFUSGTUFVJFURGTRIUGUHUIUJUK $.
  $}

  ${
    $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 cop eleq2 0ex snid 0nep0 elsn nemtbir biorfri 3bitr4g eqrdv
      cv snex eqcom bitri biorfi jca xpeq1 uneq12 syl2an impbii ) AFGZHZBVAGZHZ
      IZCVAHZDVCHZIZJZACJZBDJZKVIVJVKVIEACVIEUKZFUAZVELZVMVHLZVLALZVLCLZVEVHVMU
      BVMVBLZVMVDLZMVPFVCLZMVNVPVRVPVSVTVRVPFVALZFUCUDZVLFAVANOVSVLBLZVTKVTVLFB
      VCNVTWCVTFVAUEFVAUCUFUGZPQRVMVBVDSVTVPWDUHTVMVFLZVMVGLZMVQVTMVOVQWEVQWFVT
      WEVQWAWBVLFCVANOWFVLDLZVTKVTVLFDVCNVTWGWDPQRVMVFVGSVTVQWDUHTUIUJVIEBDVIVL
      VAUAZVELZWHVHLZWCWGVEVHWHUBWHVBLZWHVDLZMVAVALZWCMWIWCWKWMWLWCWKVPWMKWMVLV
      AAVANWMVPWMFVAUEWMVAFJFVAJVAFFULZUFVAFUMUNUGZPQWLWCVAVCLZVAWNUDZVLVABVCNO
      RWHVBVDSWMWCWOUOTWHVFLZWHVGLZMWMWGMWJWGWRWMWSWGWRVQWMKWMVLVACVANWMVQWOPQW
      SWGWPWQVLVADVCNORWHVFVGSWMWGWOUOTUIUJUPVJVBVFJVDVGJVIVKACVAUQBDVCUQVBVFVD
      VGURUSUT $.
  $}

  ${
    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 $.
    $( Membership in a Cartesian 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 Cartesian 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 ) ) $=
      ( vy vz cv cop csn cxp ciun wcel wrex wa wceq eleq2d anbi12d bitri 3bitri
      wex cab wsb csb df-iun eleq2i opex nfv nfs1v nfcv nfcsb1v nfxp nfcri nfan
      df-rex sbequ12 csbeq1a xpeq12d cbvexv1 anbi2d exbidv bitrid opelxp anbi2i
      sneq eleq1 elab an12 velsn equcom anbi1i sbequ12r equcoms eqcomd equsexvw
      exbii ) AGZDHZABVPIZCJZKZLVQEGZVSLZABMZEUAZLVPBLZAFUBZVQFGZIZAWGCUCZJZLZN
      ZFTZWEDCLZNZVTWDVQAEBVSUDUEWCWMEVQVPDUFWCWFWAWJLZNZFTZWAVQOZWMWCWEWBNZATW
      RWBABUNWTWQAFWTFUGWFWPAWEAFUHAEWJAWHWIAWHUIAWGCUJUKULUMVPWGOZWEWFWBWPWEAF
      UOXAVSWJWAXAVRWHCWIVPWGVDAWGCUPZUQPQURRWSWQWLFWSWPWKWFWAVQWJVEUSUTVAVFWMW
      GVPOZWFDWILZNZNZFTWOWLXFFWLWFVPWHLZXDNZNXGXENXFWKXHWFVPDWHWIVBVCWFXGXDVGX
      GXCXEXGXAXCAWGVHAFVIRVJSVOXEWOFAXCWFWEXDWNWEFAVKXCWICDXCCWICWIOAFXBVLVMPQ
      VNRS $.
  $}

  ${
    $d x z A $.  $d x z B $.  $d x z C $.  $d x y z $.
    $( Membership of an ordered pair in a union of Cartesian products over its
       second component, analogous to ~ opeliunxp .  (Contributed by AV,
       30-Mar-2019.) $)
    opeliun2xp $p |- ( <. C , y >. e. U_ y e. B ( A X. { y } ) <->
                     ( y e. B /\ C e. A ) ) $=
      ( vx vz cv cop csn cxp wcel wa wex weq eleq2d anbi12d bitri anbi2i 3bitri
      wceq ciun wrex cab wsb csb df-iun eleq2i opex nfv nfs1v nfcsb1v nfcv nfxp
      df-rex nfcri nfan sbequ12 csbeq1a sneq xpeq12d eleq1 anbi2d exbidv bitrid
      cbvexv1 elab opelxp an13 ancom velsn equcom anbi1i exbii sbequ12r equcoms
      eqcomd equsexvw ) DAGZHZACBVRIZJZUAZKVSEGZWAKZACUBZEUCZKVRCKZAFUDZVSAFGZB
      UEZWIIZJZKZLZFMZWGDBKZLZWBWFVSAECWAUFUGWEWOEVSDVRUHWEWHWCWLKZLZFMZWCVSTZW
      OWEWGWDLZAMWTWDACUNXBWSAFXBFUIWHWRAWGAFUJAEWLAWJWKAWIBUKAWKULUMUOUPAFNZWG
      WHWDWRWGAFUQXCWAWLWCXCBWJVTWKAWIBURZVRWIUSUTOPVEQXAWSWNFXAWRWMWHWCVSWLVAV
      BVCVDVFWOFANZWHDWJKZLZLZFMWQWNXHFWNWHXFVRWKKZLZLZXIXGLZXHWMXJWHDVRWJWKVGR
      XKXIXFWHLZLXLWHXFXIVHXMXGXIXFWHVIRQXIXEXGXIXCXEAWIVJAFVKQVLSVMXGWQFAXEWHW
      GXFWPWGFAVNXEWJBDXEBWJBWJTAFXDVOVPOPVQQS $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.
    $( Distributive law for Cartesian 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 Cartesian 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 Cartesian 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 Cartesian 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 Cartesian 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 Cartesian 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 ancom 2exbii 19.42vv elvv bitri
      3bitr2i anbi2i vex biantru anass 3bitrri exrot4 excom opex eqeq2d ceqsexv
      opeq1 exbii ) DFFGZFGHDEIZCIZJZKZUNUMHZUOFHZLLZCMEMZDAIZBIZJZUOJZKZCMZBMA
      MZECDUMFNVAUNVDKZUQLZBMAMZCMEMVJCMEMZBMAMVHUTVKECVKUQVILZBMAMZUQURLZUSLZU
      TVJVMABVIUQOPVNUQVIBMAMZLVOVPUQVIABQURVQUQABUNRUAUSVOCUBUCTUQURUSUDUEPVJA
      BECUFVLVGABVLVJEMZCMVGVJECUGVRVFCUQVFEVDVBVCUHVIUPVEDUNVDUOUKUIUJULSPTS
      $.

    $( 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 $.
  $}

  $( Intersection of binary relation with Cartesian product.  (Contributed by
     NM, 3-Mar-2007.)  (Revised by Mario Carneiro, 26-Apr-2015.)  Group
     conjuncts and avoid ~ df-3an .  (Revised by Peter Mazsa, 18-Sep-2022.) $)
  brinxp2 $p |- ( C ( R i^i ( A X. B ) ) D <->
                ( ( C e. A /\ D e. B ) /\ C R D ) ) $=
    ( cxp cin wbr wa wcel brin ancom brxp anbi1i 3bitri ) CDEABFZGHCDEHZCDPHZIR
    QICAJDBJIZQICDEPKQRLRSQCDABMNO $.

  $( 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 brinxp2 baibr ) ABECDFGHACIBDIJABEHCDABEKL $.

  $( Ordered pair element in an intersection with Cartesian product.
     (Contributed by Peter Mazsa, 21-Jul-2019.) $)
  opelinxp $p |- ( <. C , D >. e. ( R i^i ( A X. B ) ) <->
                   ( ( C e. A /\ D e. B ) /\ <. C , D >. e. R ) ) $=
    ( cxp cin wbr wcel wa cop brinxp2 df-br anbi2i 3bitr3i ) CDEABFGZHCAIDBIJZC
    DEHZJCDKZPIQSEIZJABCDELCDPMRTQCDEMNO $.

  ${
    $d x y z A $.  $d x y z R $.
    $( Intersection of partial order with Cartesian 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 anidms ad2antrr notbid adantll adantlr imbi12d ralbiia 3bitr4i
      adantr ) CFZUJBGZHZUJDFZBGZUMEFZBGZIZUJUOBGZJZIZEAKZDAKZCAKUJUJBAALMZGZHZ
      UJUMVCGZUMUOVCGZIZUJUOVCGZJZIZEAKZDAKZCAKABNAVCNVBVMCAUJAOZVAVLDAVNUMAOZI
      ZUTVKEAVPUOAOZIZULVEUSVJVRUKVDVNUKVDPZVOVQVNVSUJUJAABQUAUBUCVRUQVHURVIVRU
      NVFUPVGVPUNVFPVQUJUMAABQUIVOVQUPVGPVNUMUOAABQUDTVNVQURVIPVOUJUOAABQUEUFTR
      RUGCDEABSCDEAVCSUH $.

    $( Intersection of total order with Cartesian 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 wpo cv wbr weq w3o wral wa cxp cin poinxp wcel brinxp biidd df-so
      wor wb ancoms 3orbi123d ralbidva ralbiia anbi12i 3bitr4i ) ABEZCFZDFZBGZC
      DHZUIUHBGZIZDAJZCAJZKABAALMZEZUHUIUPGZUKUIUHUPGZIZDAJZCAJZKABSAUPSUGUQUOV
      BABNUNVACAUHAOZUMUTDAVCUIAOZKZUJURUKUKULUSUHUIAABPVEUKQVDVCULUSTUIUHAABPU
      AUBUCUDUECDABRCDAUPRUF $.

    $( Intersection of well-founded relation with Cartesian product of its
       field.  (Contributed by Mario Carneiro, 10-Jul-2014.) $)
    frinxp $p |- ( R Fr A <-> ( R i^i ( A X. A ) ) Fr A ) $=
      ( vz vy vx cv wss c0 wa wbr wn wral wrex wi wal wfr wb wcel ssel df-fr
      wne cxp cin anim12d brinxp ancoms notbid ralbidva rexbidva adantr pm5.74i
      syl6 impl albii 3bitr4i ) CFZAGZUPHUAZIZDFZEFZBJZKZDUPLZEUPMZNZCOUSUTVABA
      AUBUCZJZKZDUPLZEUPMZNZCOABPAVGPVFVLCUSVEVKUQVEVKQURUQVDVJEUPUQVAUPRZIZVCV
      IDUPVNUTUPRZIVBVHUQVMVOVBVHQZUQVMVOIVAARZUTARZIVPUQVMVQVOVRUPAVASUPAUTSUD
      VRVQVPUTVAAABUEUFULUMUGUHUIUJUKUNCEDABTCEDAVGTUO $.

    $( Intersection of set-like relation with Cartesian 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 $.

    $( Intersection of well-ordering with Cartesian product of its field.
       (Contributed by NM, 9-Mar-1997.)  (Revised by Mario Carneiro,
       10-Jul-2014.) $)
    weinxp $p |- ( R We A <-> ( R i^i ( A X. A ) ) We A ) $=
      ( wfr wor wa cxp cin wwe frinxp soinxp anbi12i df-we 3bitr4i ) ABCZABDZEA
      BAAFGZCZAPDZEABHAPHNQORABIABJKABLAPLM $.
  $}

  ${
    $d x y z A $.  $d x y z R $.
    $( Partial ordering of a singleton.  (Contributed by NM, 27-Apr-2009.)
       (Revised by Mario Carneiro, 23-Apr-2015.) $)
    posn $p |- ( Rel R -> ( R Po { A } <-> -. A R A ) ) $=
      ( vx vy vz cvv wpo wbr wn wb wa c0 wceq cv wi breq2 anbi2d ralsng ralbidv
      wral wrel wcel csn po0 snprc poeq2 mpbiri adantl brrelex1 stoic1a 2thd ex
      sylbi df-po imbi12d simpl imbitrid biantrud bicomd breq12 anidms pm2.61d2
      bitrd notbid bitrid ) BUAZAFUBZAUCZBGZAABHZIZJZVFVGIZVLVFVMKVIVKVMVIVFVMV
      ILBGZBUDVMVHLMVIVNJAUEVHLBUFUMUGUHVFVJVGAABUIUJUKULVICNZVOBHZIZVODNZBHZVR
      ENZBHZKZVOVTBHZOZKZEVHTZDVHTZCVHTZVGVKCDEVHBUNVGWHVQCVHTVKVGWGVQCVHVGWGVQ
      VSVRABHZKZVOABHZOZKZDVHTVQVGWFWMDVHWEWMEAFVTAMZWDWLVQWNWBWJWCWKWNWAWIVSVT
      AVRBPQVTAVOBPUOQRSWMVQDAFVRAMZVQWMWOWLVQWJVSWOWKVSWIUPVRAVOBPUQURUSRVCSVQ
      VKCAFVOAMZVPVJWPVPVJJVOAVOABUTVAVDRVCVEVB $.

    $( Strict ordering on a singleton.  (Contributed by Mario Carneiro,
       28-Dec-2014.) $)
    sosn $p |- ( Rel R -> ( R Or { A } <-> -. A R A ) ) $=
      ( vx vy csn wor wpo wrel wbr wn cv weq wral wcel wa elsni eqcomd sylan9eq
      w3o 3mix2d rgen2 df-so mpbiran2 posn bitrid ) AEZBFZUFBGZBHAABIJUGUHCKZDK
      ZBIZCDLZUJUIBIZSZDUFMCUFMUNCDUFUFUIUFNZUJUFNZOULUKUMUOUPUIAUJUIAPUPUJAUJA
      PQRTUACDUFBUBUCABUDUE $.

    $( Founded relation on a singleton.  (Contributed by Mario Carneiro,
       28-Dec-2014.)  (Revised by Mario Carneiro, 23-Apr-2015.) $)
    frsn $p |- ( Rel R -> ( R Fr { A } <-> -. A R A ) ) $=
      ( vz vy vx cvv wfr wbr wn wa c0 wceq adantl cv wral wrex wne wi wal imp
      wrel wcel csn wb snprc fr0 freq2 mpbiri sylbi brrelex1 stoic1a 2thd df-fr
      ex wss wo sssn neor sylbb eqimss snnzg neeq1 syl5ibrcom jca imbi1d albidv
      impbida snex raleq rexeqbi1dv ceqsalv bitrdi bitrid notbid ralbidv rexsng
      breq2 breq1 ralsng 3bitrd pm2.61d2 ) BUAZAFUBZAUCZBGZAABHZIZUDZWBWCIZWHWB
      WIJWEWGWIWEWBWIWDKLZWEAUEWJWEKBGBUFWDKBUGUHUIMWBWFWCAABUJUKULUNWCWECNZDNZ
      BHZIZCWDOZDWDPZWKABHZIZCWDOZWGWEENZWDUOZWTKQZJZWNCWTOZDWTPZRZESZWCWPEDCWD
      BUMWCXGWTWDLZXERZESWPWCXFXIEWCXCXHXEWCXCXHXCXHWCXAXBXHXAWTKLXHUPXBXHRWTAU
      QXHWTKURUSTMWCXHJXAXBXHXAWCWTWDUTMWCXHXBWCXBXHWDKQAFVAWTWDKVBVCTVDVGVEVFX
      EWPEWDAVHXDWODWTWDWNCWTWDVIVJVKVLVMWOWSDAFWLALZWNWRCWDXJWMWQWLAWKBVQVNVOV
      PWRWGCAFWKALWQWFWKAABVRVNVSVTWA $.

    $( Well-ordering of a singleton.  (Contributed by Mario Carneiro,
       28-Dec-2014.) $)
    wesn $p |- ( Rel R -> ( R We { A } <-> -. A R A ) ) $=
      ( wrel csn wfr wor wa wbr wn wwe frsn sosn anbi12d df-we pm4.24 3bitr4g )
      BCZADZBEZRBFZGAABHIZUAGRBJUAQSUATUAABKABLMRBNUAOP $.
  $}

  ${
    $d x y $.
    $( Membership in an ordered-pair class abstraction implies membership in a
       Cartesian product.  (Contributed by Alexander van der Vekens,
       23-Jun-2018.)  Avoid ~ ax-sep , ~ ax-nul , ~ ax-pr .  (Revised by SN,
       11-Dec-2024.) $)
    elopaelxp $p |- ( A e. { <. x , y >. | ps } -> A e. ( _V X. _V ) ) $=
      ( copab cvv cxp cv wcel wa vex pm3.2i a1i ssopab2i df-xp sseqtrri sseli )
      ABCEZFFGZDRBHFIZCHFIZJZBCESAUBBCUBATUABKCKLMNBCFFOPQ $.
    $( $j usage 'elopaelxp' avoids 'ax-sep' 'ax-nul' 'ax-pr'; $)

    $d A x y $.
    $( Obsolete version of ~ elopaelxp as of 11-Dec-2024.  (Contributed by
       Alexander van der Vekens, 23-Jun-2018.)  (New usage is discouraged.)
       (Proof modification is discouraged.) $)
    elopaelxpOLD $p |- ( A e. { <. x , y >. | ps } -> A e. ( _V X. _V ) ) $=
      ( cv cop wceq wa wex copab wcel cvv cxp simpl 2eximi elopab elvv 3imtr4i
      ) DBECEFGZAHZCIBISCIBIDABCJKDLLMKTSBCSANOABCDPBCDQR $.
  $}

  ${
    $d A x y $.  $d B x y $.
    bropaex12.1 $e |- R = { <. x , y >. | ps } $.
    $( Two classes related by an ordered-pair class abstraction are sets.
       (Contributed by AV, 21-Jan-2020.) $)
    bropaex12 $p |- ( A R B -> ( A e. _V /\ B e. _V ) ) $=
      ( wbr cop cvv cxp wcel wa copab df-br eleq2i bitri elopaelxp sylbi opelxp
      sylib ) DEFHZDEIZJJKLZDJLEJLMUBUCABCNZLZUDUBUCFLUFDEFOFUEUCGPQABCUCRSDEJJ
      TUA $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( An abstraction relation is a subset of a related Cartesian 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 $.
    brab2a.1 $e |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) ) $.
    brab2a.2 $e |- R = { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) } $.
    $( The law of concretion for a binary relation.  Ordered pair membership in
       an ordered pair class abstraction.  (Contributed by Mario Carneiro,
       28-Apr-2015.) $)
    brab2a $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 biadanii ) EFILZEGMFHMNZBEFGHIICOGMDOHMNANCDPZGHQKACDGHRS
      TUHEFUDZUJMZUIBUHUKIMULEFIUEIUJUKKUAUFABCDEFGHJUBUCUG $.
  $}

  ${
    $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 wbr cxp opelxpi anim12i wceq opex eleq1 anbi1d eqeq1
      4exbidv anbi12d anbi2d brab copsex4g bitrid mpbirand ) INQJNQRZKNQLNQRZRZ
      IJUAZKLUAZMUBZVANNUCZQZVBVDQZRZBURVEUSVFIJNNUDKLNNUDUEVCVGVAESFSUAZUFZVBG
      SHSUAZUFZRZARZHTGTFTETZRZUTVGBRCSZVDQZDSZVDQZRZVPVHUFZVRVJUFZRZARZHTGTFTE
      TZRVEVSRZVIWBRZARZHTGTFTETZRVOCDVAVBMIJUGKLUGVPVAUFZVTWFWEWIWJVQVEVSVPVAV
      DUHUIWJWDWHEFGHWJWCWGAWJWAVIWBVPVAVHUJUIUIUKULVRVBUFZWFVGWIVNWKVSVFVEVRVB
      VDUHUMWKWHVMEFGHWKWGVLAWKWBVKVIVRVBVJUJUMUIUKULPUNUTVNBVGABEFGHIJKLNNOUOU
      MUPUQ $.
  $}

  ${
    $d x y z A $.
    $( The Cartesian 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 noel simprl mto elxp mtbir nel0
      nex ) BEAFZBGZTHUACGZDGZIJZUBEHZUCAHZKKZDLZCLUHCUGDUGUEUBMUDUEUFNOSSCDUAE
      APQR $.
  $}

  ${
    $d A w y z $.  $d B w y z $.  $d C w y z $.  $d w x y z $.
    $( Distribute proper substitution through the Cartesian product of two
       classes.  (Contributed by Alan Sare, 10-Nov-2012.)  (Revised by NM,
       23-Aug-2018.) $)
    csbxp $p |- [_ A / x ]_ ( B X. C ) = ( [_ A / x ]_ B X. [_ A / x ]_ C ) $=
      ( vz vw vy cv wcel wex cab csb cxp wsbc sbcex2 sbcan bitri intnand eqtri
      wa cop wceq csbab cvv wb sbcg sbcel2 anbi12i a1i anbi12d sbcex con3i noel
      wn c0 neleqtrrd 2falsed pm2.61i exbii abbii copab df-opab csbeq2i 3eqtr4i
      csbprc df-xp ) ABEHFHZGHZUAUBZVGCIZVHDIZTZTZGJZFJZEKZLZVIVGABCLZIZVHABDLZ
      IZTZTZGJZFJZEKZABCDMZLVRVTMZVQVOABNZEKWFVOAEBUCWIWEEWIVNABNZFJWEVNFABOWJW
      DFWJVMABNZGJWDVMGABOWKWCGWKVIABNZVLABNZTZWCVIVLABPBUDIZWNWCUEWOWLVIWMWBVI
      ABUDUFWMWBUEWOWMVJABNZVKABNZTWBVJVKABPWPVSWQWAABVGCUGABVHDUGUHQUIUJWOUNZW
      NWCWRWMWLWMWOVLABUKULRWRWBVIWRWAVSWRVTUOVHVHUOIUNWRVHUMUIABDVEUPRRUQURQUS
      QUSQUTSABWGVPWGVLFGVAVPFGCDVFVLFGEVBSVCWHWBFGVAWFFGVRVTVFWBFGEVBSVD $.
  $}

  $( 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.)  Remove dependency on
       ~ ax-sep , ~ ax-nul , ~ ax-pr .  (Revised by KP, 25-Oct-2021.)  Remove
       dependency on ~ ax-12 .  (Revised by SN, 11-Dec-2024.) $)
    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 cvv df-ss eleq1
      wa cxp df-rel sylbb copab wb elopabw elv simpl 2eximi sylbi eleq2s imim2i
      df-xp sylg imbi12d biimprcd 2alimi 19.23vv sylib com23 a2d syl5 imbitrrdi
      alimdv com12 impbid2 ) CFZCDGZAHZBHZIZCJZVKDJZKZBLALZVHVNABCDVKMNVOVGVHVO
      VGEHZCJZVPDJZKZELZVHVGVQVPVKOZBPAPZKZELVOVTVGVQVPQQUAZJZKZWCEVGCWDGWFELCU
      BECWDRUCWEWBVQWBVPVIQJVJQJTZABUDZWDVPWHJZWAWGTZBPAPZWBWIWKUEEWGABVPQUFUGW
      JWAABWAWGUHUIUJABQQUMUKULUNVOWCVSEVOVQWBVRVOWBVQVRVOWAVSKZBLALWBVSKVNWLAB
      WAVSVNWAVQVLVRVMVPVKCSVPVKDSUOUPUQWAVSABURUSUTVAVDVBECDRVCVEVF $.
    $( $j usage 'ssrel' avoids 'ax-12' 'ax-sep' 'ax-nul' 'ax-pr'; $)

    $( Obsolete version of ~ ssrel as of 11-Dec-2024.  (Contributed by NM,
       2-Aug-1994.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.)  Remove
       dependency on ~ ax-sep , ~ ax-nul , ~ ax-pr .  (Revised by KP,
       25-Oct-2021.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    ssrelOLD $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 cvv df-ss eleq1
      wa cxp df-rel sylbb copab cab df-xp df-opab eqtri eqabri simpl sylbi sylg
      2eximi imim2i imbi12d biimprcd 2alimi 19.23vv sylib com23 a2d alimdv syl5
      imbitrrdi com12 impbid2 ) CFZCDGZAHZBHZIZCJZVKDJZKZBLALZVHVNABCDVKMNVOVGV
      HVOVGEHZCJZVPDJZKZELZVHVGVQVPVKOZBPAPZKZELVOVTVGVQVPQQUAZJZKZWCEVGCWDGWFE
      LCUBECWDRUCWEWBVQWEWAVIQJVJQJTZTZBPAPZWBWIEWDWDWGABUDWIEUEABQQUFWGABEUGUH
      UIWHWAABWAWGUJUMUKUNULVOWCVSEVOVQWBVRVOWBVQVRVOWAVSKZBLALWBVSKVNWJABWAVSV
      NWAVQVLVRVMVPVKCSVPVKDSUOUPUQWAVSABURUSUTVAVBVCECDRVDVEVF $.

    $( 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 cop wcel wi wral wrex wal eleq1 r19.23v bitri df-ss sylib
      wa ssel ralrimivv wceq imbi12d biimprcd 2ralimi ralbii com23 alimdv elxp2
      a1d a2d imbi2i albii 3imtr4g com12 impbid2 ) ECDHZIZEFIZAJZBJZKZELZVEFLZM
      ZBDNACNZVBVHABCDVBVHVCCLVDDLUBEFVEUCUMUDVIVAVBVIGJZELZVJVEUEZBDOZACOZMZGP
      ZVKVJFLZMZGPVAVBVIVOVRGVIVKVNVQVIVNVKVQVIVLVRMZBDNZACNZVNVRMZVHVSABCDVLVR
      VHVLVKVFVQVGVJVEEQVJVEFQUFUGUHWAVMVRMZACNWBVTWCACVLVRBDRUIVMVRACRSUAUJUNU
      KVAVKVJUTLZMZGPVPGEUTTWEVOGWDVNVKABVJCDULUOUPSGEFTUQURUS $.
  $}

  ${
    $d A x y $.  $d B x y $.
    $( Subclass relation in another form when the subclass is a relation.
       (Contributed by Peter Mazsa, 16-Feb-2019.) $)
    ssrel3 $p |- ( Rel A -> ( A C_ B <-> A. x A. y ( x A y -> x B y ) ) ) $=
      ( wrel wss cv cop wcel wi wal wbr ssrel df-br imbi12i 2albii bitr4di ) CE
      CDFAGZBGZHZCIZTDIZJZBKAKRSCLZRSDLZJZBKAKABCDMUFUCABUDUAUEUBRSCNRSDNOPQ $.
  $}

  ${
    $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 eqrel imbitrrid imp ) DGEGHZAD
      EIZATSBJCJKZDLUAELMZCNBNAUBBCFOBCDEPQR $.
  $}

  ${
    $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 df-ss
      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 (used by 2-place operations
       ~ df-oprab ), 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 $.
  $}

  $( 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 $.

  $( The universal class is not a relation.  (Contributed by Thierry Arnoux,
     23-Jan-2022.) $)
  nrelv $p |- -. Rel _V $=
    ( cvv wrel cxp wss c0 wcel wn 0ex 0nelxp nelss mp2an df-rel mtbir ) ABAAACZ
    DZEAFENFGOGHAAIEANJKALM $.

  $( A singleton is a relation iff it is a singleton on 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
    $.

  $( An at-most-singleton is a relation iff it is empty (because it is a
     "singleton on a proper class") or it is a singleton of an ordered pair.
     (Contributed by BJ, 26-Feb-2023.) $)
  relsnb $p |- ( Rel { A } <-> ( -. A e. _V \/ A e. ( _V X. _V ) ) ) $=
    ( csn wrel cvv wcel wn cxp wo wi relsng biimpcd imor sylib wceq snprc releq
    c0 rel0 mpbiri sylbi ibir jaoi impbii ) ABZCZADEZFZADDGZEZHZUEUFUIIUJUFUEUI
    ADJKUFUILMUGUEUIUGUDQNZUEAOUKUEQCRUDQPSTUIUEAUHJUAUBUC $.

  $( 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 opex relsng mp1i mpbird ) ACEBDEF
    ZABGZHIZSJJKEZABCDLSJETUAMRABNSJOPQ $.

  ${
    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 ) ) $=
      ( cvv wcel csn wrel cxp wb relsng ax-mp ) ACDAEFACCGDHBACIJ $.

    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 >. } $=
      ( cvv wcel cop csn wrel relsnopg mp2an ) AEFBEFABGHICDABEEJK $.
  $}

  ${
    $d x y A $.  $d x y ph $.
    copsex2ga.1 $e |- ( A = <. x , y >. -> ( ph <-> ps ) ) $.
    $( Implicit substitution inference for ordered pairs.  Compare
       ~ copsex2ga .  (Contributed by NM, 12-Mar-2014.) $)
    copsex2gb $p |- ( E. x E. y ( A = <. x , y >. /\ ps )
         <-> ( A e. ( _V X. _V ) /\ ph ) ) $=
      ( cvv cxp wcel wa cv cop wceq elvv anbi1i 19.41vv pm5.32i 2exbii 3bitr2ri
      wex ) EGGHIZAJECKDKLMZDTCTZAJUBAJZDTCTUBBJZDTCTUAUCACDENOUBACDPUDUECDUBAB
      FQRS $.

    $( Implicit substitution inference for ordered pairs.  Compare ~ copsex2g .
       (Contributed by NM, 26-Feb-2014.)  (Proof shortened by Mario Carneiro,
       31-Aug-2015.) $)
    copsex2ga $p |- ( A e. ( V X. W )
        -> ( ph <-> E. x E. y ( A = <. x , y >. /\ ps ) ) ) $=
      ( cxp wcel cvv cv cop wceq wa wex wb xpss sseli copsex2gb baibr syl ) EFG
      IZJEKKIZJZAECLDLMNBODPCPZQUCUDEFGRSUFUEAABCDEHTUAUB $.

    $( Membership in an ordered-pair class abstraction.  (Contributed by NM,
       25-Feb-2014.)  (Revised by Mario Carneiro, 31-Aug-2015.) $)
    elopaba $p |- ( A e. { <. x , y >. | ps } <->
            ( A e. ( _V X. _V ) /\ ph ) ) $=
      ( copab wcel cv cop wceq wa wex cvv cxp elopab copsex2gb bitri ) EBCDGHEC
      IDIJKBLDMCMENNOHALBCDEPABCDEFQR $.
  $}

  ${
    $d A x y $.  $d B x y $.
    $( A Cartesian 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 ) $=
      ( vx vy cxp cun cpw relxp cv wcel csn cpr wss snssi syl elpw sylibr sylib
      wa vex cop opelxp ssun3 vsnex adantr df-pr ssun4 anim12i eqsstrid zfpair2
      unss jca prex dfop eleq1i prss 3bitr4ri sylbi relssi ) CDABEZABFZGZGZABHC
      IZDIZUAZUTJVDAJZVEBJZSZVFVCJZVDVEABUBVIVDKZVBJZVDVELZVBJZSZVJVIVLVNVGVLVH
      VGVKVAMZVLVGVKAMVPVDANVKABUCOZVKVACUDZPQUEVIVMVAMVNVIVMVKVEKZFZVAVDVEUFVI
      VPVSVAMZSVTVAMVGVPVHWAVQVHVSBMWAVEBNVSBAUGOUHVKVSVAUKRUIVMVACDUJZPQULVKVM
      LZVCJWCVBMVJVOWCVBVKVMUMPVFWCVCVDVECTDTUNUOVKVMVBVRWBUPUQRURUS $.
  $}

  $( The double class union of a Cartesian 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 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 $.

  $( Intersection with a Cartesian product is a relation.  (Contributed by
     Peter Mazsa, 4-Mar-2019.) $)
  relinxp $p |- Rel ( R i^i ( A X. B ) ) $=
    ( cxp wrel cin relxp relin2 ax-mp ) ABDZECJFEABGCJHI $.

  $( 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 df-ss bitri ralbii ralcom4 r19.23v albii bitr4i ) ABCEZFDG
      ZCHZABIZDJZFUMKKLZMZCFZABNZUIUMADBCOPUMQUOULUJUNHZSZDRZUQULDUNUAUQUKURSZD
      RZABNVAABNZDRUTUPVBABUPCUNMVBCQDCUNUBUCUDVAADBUEVCUSDUKURABUFUGTUHT $.
  $}

  $( 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 $.
  $}

  ${
    $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, but a longer proof using ~ ax-11 and
       ~ ax-12 , 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, but using ~ ax-11 and ~ ax-12 , see
       ~ relopab .  (Contributed by SN, 8-Sep-2024.) $)
    relopabv $p |- Rel { <. x , y >. | ph } $=
      ( copab eqid relopabiv ) ABCABCDZGEF $.
  $}

  ${
    $d ph z $.  $d A z $.  $d u x z $.  $d u y z $.
    relopabi.1 $e |- A = { <. x , y >. | ph } $.
    $( A class of ordered pairs is a relation.  (Contributed by Mario Carneiro,
       21-Dec-2013.)  Remove dependency on ~ ax-sep , ~ ax-nul , ~ ax-pr .
       (Revised by KP, 25-Oct-2021.) $)
    relopabi $p |- Rel A $=
      ( vz vu cvv cv wcel cop wceq wex wa copab cab df-opab eqtri eqabri 2eximi
      wrel cxp wss simpl sylbi ax6evr pm3.21 eximdv mpi opeq2 eqtr2 sylan eximi
      eqcomd syl eqcoms excomim pm3.2i jctr df-xp sylibr ax5e 4syl ssriv df-rel
      vex mpbir ) DUADHHUBZUCFDVHFIZDJZVIBIZCIZKZLZCMBMZVIVKGIZKZLZGMZBMZCMZVIV
      HJZCMWBVJVNANZCMBMZVOWDFDDABCOWDFPEABCFQRSWCVNBCVNAUDTUEVOVSCMBMWAVNVSBCV
      SVMVIVMVILZVLVPLZWENZGMZVSWEWFGMWHGCUFWEWFWGGWEWFUGUHUIWGVRGWFVMVQLZWEVRV
      LVPVKUJWIWENVQVIVMVQVIUKUNULUMUOUPTVSBCUQUOVTWBCVTVRVKHJZVPHJZNZNZGMBMZWB
      VRWMBGVRWLWJWKBVFGVFURUSTWNFVHVHWLBGOWNFPBGHHUTWLBGFQRSVAUMWBCVBVCVDDVEVG
      $.
    $( $j usage 'relopabi' avoids 'ax-sep' 'ax-nul' 'ax-pr'; $)

    $( Alternate proof of ~ relopabi (shorter but uses more axioms).
       (Contributed by Mario Carneiro, 21-Dec-2013.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    relopabiALT $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.)
     Remove disjoint variable conditions.  (Revised by Alan Sare, 9-Jul-2013.)
     (Proof shortened by Mario Carneiro, 21-Dec-2013.) $)
  relopab $p |- Rel { <. x , y >. | ph } $=
    ( copab eqid relopabi ) ABCABCDZGEF $.

  ${
    $d A y $.  $d B y $.  $d x y $.
    $( The maps-to notation always describes a binary relation.  (Contributed
       by Scott Fenton, 16-Apr-2012.) $)
    mptrel $p |- Rel ( x e. A |-> B ) $=
      ( vy cv wcel wceq wa cmpt df-mpt relopabiv ) 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 weq cid df-id relopabiv ) ABCABDABEF $.

    $( 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 relopabiv ) 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 relopabv eqrel mpan mpbiri ) CFZAGZBGZHZCIZABJZCKZDGZEGZHZUIIUMCIZL
      ZEMDMZUODEUHUKUFHZCIUNABUKULDNENUEUKKUGUQCUEUKUFOPUFULKUQUMCUFULUKSPQRUIF
      UDUJUPLUHABTDEUICUAUBUC $.

    $( 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 relopabv relin1 ax-mp cop wcel sban vopelopabsb
      cv wsb sbbii anbi12i 3bitr4ri elin 3bitr4i eqrelriiv ) EFACDGZBCDGZHZABIZ
      CDGZUFJUHJACDKUFUGLMUICDKERFRNZUFOZUKUGOZIZUIDFSZCESZUKUHOUKUJOADFSZBDFSZ
      IZCESUQCESZURCESZIUPUNUQURCEPUOUSCEABDFPTULUTUMVAACDEFQBCDEFQUAUBUKUFUGUC
      UICDEFQUDUE $.

    $( Difference of two ordered-pair class abstractions.  (Contributed by
       Stefan O'Rear, 17-Jan-2015.)  (Proof shortened by SN, 19-Dec-2024.) $)
    difopab $p |- ( { <. x , y >. | ph } \ { <. x , y >. | ps } ) =
        { <. x , y >. | ( ph /\ -. ps ) } $=
      ( vz vw copab cdif wn wa wrel relopabv cv wcel wsb sban sbbii vopelopabsb
      sbn 3bitr4ri reldif ax-mp cop notbii anbi12i eldif 3bitr4i eqrelriiv ) EF
      ACDGZBCDGZHZABIZJZCDGZUIKUKKACDLUIUJUAUBUMCDLEMFMUCZUINZUOUJNZIZJZUMDFOZC
      EOZUOUKNUOUNNADFOZULDFOZJZCEOVBCEOZVCCEOZJVAUSVBVCCEPUTVDCEAULDFPQUPVEURV
      FACDEFRBDFOZIZCEOVGCEOZIVFURVGCESVCVHCEBDFSQUQVIBCDEFRUDTUETUOUIUJUFUMCDE
      FRUGUH $.

    $( Obsolete version of ~ difopab as of 19-Dec-2024.  (Contributed by Stefan
       O'Rear, 17-Jan-2015.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    difopabOLD $p |- ( { <. x , y >. | ph } \ { <. x , y >. | ps } ) =
        { <. x , y >. | ( ph /\ -. ps ) } $=
      ( vz vw copab wn wa wrel relopabv cv wcel wsbc sbcan sbcbii opelopabsb wb
      cvv sbcng cdif reldif ax-mp cop elv notbii 3bitr4ri anbi12i eldif 3bitr4i
      eqrelriiv ) EFACDGZBCDGZUAZABHZIZCDGZULJUNJACDKULUMUBUCUPCDKELZFLZUDZULMZ
      UTUMMZHZIZUPDUSNZCURNZUTUNMUTUQMADUSNZUODUSNZIZCURNVGCURNZVHCURNZIVFVDVGV
      HCUROVEVICURAUODUSOPVAVJVCVKACDURUSQBDUSNZHZCURNZVLCURNZHZVKVCVNVPREVLCUR
      STUEVHVMCURVHVMRFBDUSSTUEPVBVOBCDURUSQUFUGUHUGUTULUMUIUPCDURUSQUJUK $.

    $( Intersection of two Cartesian products.  Exercise 9 of [TakeutiZaring]
       p. 25.  (Contributed by NM, 3-Aug-1994.)  (Proof shortened by Andrew
       Salmon, 27-Aug-2011.)  Avoid ~ ax-10 , ~ ax-12 .  (Revised by SN,
       5-May-2025.) $)
    inxp $p |- ( ( A X. B ) i^i ( C X. D ) ) =
                   ( ( A i^i C ) X. ( B i^i D ) ) $=
      ( vx vy cxp cin relinxp relxp cv cop wcel an4 opelxp anbi12i elin 3bitr4i
      wa eqrelriiv ) EFABGZCDGZHZACHZBDHZGZCDUAIUDUEJEKZFKZLZUAMZUIUBMZSZUGUDMZ
      UHUEMZSZUIUCMUIUFMUGAMZUHBMZSZUGCMZUHDMZSZSUPUSSZUQUTSZSULUOUPUQUSUTNUJUR
      UKVAUGUHABOUGUHCDOPUMVBUNVCUGACQUHBDQPRUIUAUBQUGUHUDUEORT $.
    $( $j usage 'inxp' avoids 'ax-10' 'ax-11' 'ax-12'; $)

    $( Obsolete version of ~ inxp as of 5-May-2025.  (Contributed by NM,
       3-Aug-1994.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    inxpOLD $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 Cartesian 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 Cartesian 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 A $.  $d x y z C $.  $d y z B $.
    $( Distributive law for Cartesian product over indexed intersection.
       (Contributed by Mario Carneiro, 21-Mar-2015.) $)
    xpiindi $p |- ( A =/= (/) ->
      ( C X. |^|_ x e. A B ) = |^|_ x e. A ( C X. B ) ) $=
      ( vy vz ciin cxp wrel wa c0 wral relxp cv wcel cvv eliin opelxp 3bitr4g
      wb wne wceq wrex rgenw r19.2z mpan2 reliin syl cop r19.28zv bicomd anbi2i
      jctil elv ralbii opex ax-mp eqrelrdv2 mpancom ) DABCGZHZIZABDCHZGZIZJBKUA
      ZVAVDUBVFVEVBVFVCIZABUCZVEVFVGABLVHVGABDCMUDVGABUEUFABVCUGUHDUTMUMVFEFVAV
      DVFENZDOZFNZUTOZJZVIVKUIZVCOZABLZVNVAOVNVDOZVFVJVKCOZABLZJZVJVRJZABLZVMVP
      VFWBVTVJVRABUJUKVLVSVJVLVSTFAVKBCPQUNULVOWAABVIVKDCRUOSVIVKDUTRVNPOVQVPTV
      IVKUPAVNBVCPQUQSURUS $.

    $( Distributive law for Cartesian product over relativized indexed
       intersection.  (Contributed by Mario Carneiro, 21-Mar-2015.) $)
    xpriindi $p |- ( C X. ( D i^i |^|_ x e. A B ) ) =
      ( ( C X. D ) i^i |^|_ x e. A ( C X. B ) ) $=
      ( ciin cin cxp wceq c0 cvv iineq1 0iin eqtrdi ineq2d xpeq2d eqtr4d xpindi
      inv1 wne xpiindi eqtrid pm2.61ine ) DEABCFZGZHZDEHZABDCHZFZGZIBJBJIZUFUGU
      JUKUEEDUKUEEKGEUKUDKEUKUDAJCFKABJCLACMNOESNPUKUJUGKGUGUKUIKUGUKUIAJUHFKAB
      JUHLAUHMNOUGSNQBJTZUFUGDUDHZGUJDEUDRULUMUIUGABCDUAOUBUC $.
  $}

  ${
    $d y A $.  $d y B $.  $d x y C $.  $d x D $.  $d x E $.
    $( Membership in a union of Cartesian 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 Cartesian 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 nfiu1 nfel2 nfbi opeq1 eleq1d eleq2d
      nfv wceq eleq1 anbi12d bibi12d opeliunxp vtoclg1f pm5.21nii ) DEHZABAIZJZ
      CKZLZMZDNMZDBMZEFMZOZVHDEVGPVIDEVGQDEVGVGRVFRZABSVMABVECUCUDABVFUAUEUBUFV
      JVIVKDBUGUHVDEHZVGMZVDBMZECMZOZTVHVLTADNVHVLAAVCVGABVFUIUJVLAUOUKVDDUPZVO
      VHVRVLVSVNVCVGVDDEULUMVSVPVJVQVKVDDBUQVSCFEGUNURUSABCEUTVAVB $.
  $}

  ${
    $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 ciun wcel wi wal wa wral wex albii bitri wceq eliunxp imbi1i
      cop 19.23vv bitr4i alrot3 impexp opex imbi2d ceqsalv 2albii r2al 3bitr4i
      df-ral ) CIZDFDIZJGKLZMZANZCOZUQFMEIZGMPZBNZEODOZACURQBEGQDFQVAUPUQVBUDZU
      AZVCPZANZEODOZCOZVEUTVJCUTVHERDRZANVJUSVLADEFGUPUBUCVHADEUEUFSVKVICOZEODO
      VEVICDEUGVMVDDEVMVGVCANZNZCOVDVIVOCVGVCAUHSVNVDCVFUQVBUIVGABVCHUJUKTULTTA
      CURUOBDEFGUMUN $.

    $( 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 ) $=
      ( wn cv csn cxp ciun wral wrex cop wceq notbid raliunxp dfrex2 3bitr4i
      ralnex ralbii bitri notbii ) AIZCDFDJZKGLMZNZIBEGOZIZDFNZIACUHOUJDFOUIULU
      IBIZEGNZDFNULUFUMCDEFGCJUGEJPQABHRSUNUKDFBEGUBUCUDUEACUHTUJDFTUA $.

    $d y B $.
    $( Universal quantification restricted to a Cartesian 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 Cartesian 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 y z ph $.  $d x ps $.  $d x y z $.
    exopxfr.1 $e |- ( x = <. y , z >. -> ( ph <-> ps ) ) $.
    $( Transfer ordered-pair existence from/to single variable existence.
       (Contributed by NM, 26-Feb-2014.)  (Proof shortened by Mario Carneiro,
       31-Aug-2015.) $)
    exopxfr $p |- ( E. x e. ( _V X. _V ) ph <-> E. y E. z ps ) $=
      ( cvv cxp wrex wex rexxp rexv exbii 3bitri ) ACGGHIBEGIZDGIODJBEJZDJABCDE
      GGFKODLOPDBELMN $.
  $}

  ${
    $d x y z A $.  $d y z ph $.  $d x ps $.
    exopxfr2.1 $e |- ( x = <. y , z >. -> ( ph <-> ps ) ) $.
    $( Transfer ordered-pair existence from/to single variable existence.
       (Contributed by NM, 26-Feb-2014.) $)
    exopxfr2 $p |- ( Rel A -> ( E. x e. A ph
              <-> E. y E. z ( <. y , z >. e. A /\ ps ) ) ) $=
      ( wrel wrex cv wcel wa cvv cxp cop wex wss df-rel biimpi sseld wceq eleq1
      adantrd pm4.71rd rexbidv2 anbi12d exopxfr bitrdi ) FHZACFICJZFKZALZCMMNZI
      DJEJOZFKZBLZEPDPUIAULCFUMUIULUJUMKZUIUKUQAUIFUMUJUIFUMQFRSTUCUDUEULUPCDEU
      JUNUAUKUOABUJUNFUBGUFUGUH $.
  $}

  ${
    $d x A $.
    $( Disjoint union is a subset of a Cartesian 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 ) $=
      ( vw vu vv wral wsb cv wceq vex nfsbv cxp cbvralsvw ralbii nfv nfcv nfs1v
      nfralw sbequ12 ralbidv cbvralw cop wa wex wb eqvinop nfbi sbhypf sylan9bb
      opth sylbi exlimi ralxp 3bitr4ri bitri ) ACFGUAZOACLPZLVEOZBEGOZDFOZACLVE
      UBBDMPZEGOZMFOVJENPZNGOZMFOVIVGVKVMMFVJENGUBUCVHVKDMFVHMUDVJDEGDGUEBDMUFZ
      UGDQZMQZRZBVJEGBDMUHZUIUJVFVLLMNFGLQZVPNQZUKZRVSVOEQZUKZRZWCWARZULZEUMZDU
      MVFVLUNZDEVSVPVTMSNSUOWGWHDVFVLDACLDHTVJENDVNTUPWFWHEVFVLEACLEITVJENUFUPW
      DVFBWEVLABCLWCJKUQWEVQWBVTRZULBVLUNVOWBVPVTDSESUSVQBVJWIVLVRVJENUHURUTURV
      AVAUTVBVCVD $.

    $( 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 ) $=
      ( wn cxp wral wrex nfn cv cop wceq dfrex2 notbid ralxpf ralnex 3bitr4i
      ralbii bitri notbii ) ALZCFGMZNZLBEGOZLZDFNZLACUIOUKDFOUJUMUJBLZEGNZDFNUM
      UHUNCDEFGADHPAEIPBCJPCQDQEQRSABKUAUBUOULDFBEGUCUEUFUGACUITUKDFTUD $.
  $}

  ${
    $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 Cartesian product 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 abstraction.
       Compare ~ eqabdv .  (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.)  A relation is a class of
       ordered pairs, so the fact that an ordered pair may sometimes be itself
       a relation is an "accident" depending on the specific encoding of
       ordered pairs as classes (in set.mm, the Kuratowski encoding).  A more
       meaningful statement is ~ relsnopg , as ~ funsng is to ~ funop .
       (New usage is discouraged.) $)
    relop $p |-
            ( Rel <. A , B >. <-> E. x E. y ( A = { x } /\ B = { x , y } ) ) $=
      ( vz vw vv cop cv wceq cpr wa wex wi wal bitri weq eqeq2d cvv cxp wss csn
      wrel df-rel wcel df-ss elop elvv imbi12i jaob albii 19.26 eqeq1 eqcom vex
      snex opeqsn bitrdi 2exbidv imbi12d spcv sneq cbvexvw ax6evr mpbiran exbii
      wo 19.41v eqid a1bi 3bitr2ri sylib prex mpi opeqpr idd preqsn simplbi syl
      eqtr2 dfsn2 preq2 eqtr2id eqtrid biimpd expd com12 adantr mpd expcom impd
      anbi12d jaod biimtrid 2eximdv exlimiv imp syl2an sylbi simpr equid sylibr
      jctl eqtr4d opeq12 spc2ev adantlr preq12 biimpa eqtr4di jaodan ex 3imtr4g
      dfop ssrdv exlimivv impbii ) CDJZUEXTUAUAUBZUCZCAKZUDZLZDYCBKZMZLZNZBOAOZ
      XTUFYBYJYBGKZCUDZLZYKYCYFJZLZBOAOZPZGQZYKCDMZLZYPPZGQZNZYJYBYKXTUGZYKYAUG
      ZPZGQZUUCGXTYAUHUUGYQUUANZGQUUCUUFUUHGUUFYMYTVIZYPPUUHUUDUUIUUEYPYKCDEFUI
      ZABYKUJUKYMYPYTULRUMYQUUAGUNRRYRCHKZUDZLZHOZYSYNLZBOAOZYJUUBYRYLYLLZABSZY
      ENZBOZAOZPZUUNYQUVBGYLCURYMYMUUQYPUVAYKYLYLUOYMYOUUSABYMYOYLYNLZUUSYKYLYN
      UOUVCYNYLLUUSYLYNUPYCYFCAUQZBUQZUSRUTVAVBVCUUNYEAOUVAUVBUUMYEHAHASZUULYDC
      UUKYCVDTVEUUTYEAUUTUURBOYEBAVFUURYEBVJVGVHUUQUVAYLVKVLVMVNUUBYSYSLZUUPYSV
      KUUAUVGUUPPGYSCDVOYTYTUVGYPUUPYKYSYSUOYTYOUUOABYKYSYNUOVAVBVCVPUUNUUPYJUU
      MUUPYJPHUUMUUOYIABUUOYICYGLZDYDLZNZVIZUUMYIUUOYNYSLUVKYSYNUPYCYFCDUVDUVEE
      FVQRUUMYIYIUVJUUMYIVRUUMUVHUVIYIUVHUUMUVIYIPZUVHUUMNZUURUVLUVMYGUULLZUURC
      YGUULWBUVNUURBHSYCYFUUKUVDUVEVSVTWAUVHUURUVLPUUMUURUVHUVLUURUVHUVIYIUURUV
      JYIUURUVHYEUVIYHUURYGYDCUURYDYCYCMZYGYCWCZYCYFYCWDZWETUURYDYGDUURYDUVOYGU
      VPUVQWFTWNWGWHWIWJWKWLWMWOWPWQWRWSWTXAYIYBABYIGXTYAYIUUIYKUUKIKZJZLZIOHOZ
      UUDUUEYIUUIUWAYIYMUWAYTYEYMUWAYHYEYMNZYKYCYCJZLZUWAUWBYKYLUWCYEYMXBYEUWCY
      LLZYMYEAASZYENUWEYEUWFAXCXEYCYCCUVDUVDUSXDWJXFUVTUWDHIYCYCUVDUVDUVFIASNUV
      SUWCYKUUKUVRYCYCXGTXHWAXIYIYTNZYOUWAUWGYKYDYGMZYNYIYTYKUWHLYIYSUWHYKCDYDY
      GXJTXKYCYFUVDUVEXPXLUVTYOHIYCYFUVDUVEUVFIBSNUVSYNYKUUKUVRYCYFXGTXHWAXMXNU
      UJHIYKUJXOXQXRXSR $.
  $}

  ${
    $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 wa reli brrelex1i anim12ci eleq1 biimparc elexd
      cvv id simpl cv jca eqeq1 eqeq2 df-id brabg pm5.21nd ) BCFZABGHZABIZAQFZU
      GJUGUGUHUJUGRABGKLMUGUIJZUJUGUKACUIACFUGABCNOPUGUISUADTZETZIAUMIUIDEABQCG
      ULAUMUBUMBAUCDEUDUEUF $.
  $}

  ${
    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 wex copab ccom ssbr anim2d eximdv ssopab2dv 3sstr4g
      wa df-co ) ABGZDHEHZCIZUAFHZAIZRZEJZDFKUBUAUCBIZRZEJZDFKACLBCLTUFUIDFTUEU
      HETUDUGUBABUAUCMNOPDFEACSDFEBCSQ $.

    $( 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 wex copab ccom ssbr anim1d eximdv ssopab2dv 3sstr4g
      wa df-co ) ABGZDHZEHZAIZUBFHCIZRZEJZDFKUAUBBIZUDRZEJZDFKCALCBLTUFUIDFTUEU
      HETUCUGUDABUAUBMNOPDFECASDFECBSQ $.
  $}

  $( 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 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 wa wbr cv wex ccom 3simpa wceq breq2 breq1 spcegv 3ad2antl3
      anbi12d imp brcog biimpar syl2an2r ) AEJZBFJZGHJZKUHUILZAGDMZGBCMZLZAINZD
      MZUOBCMZLZIOZABCDPMZUHUIUJQUJUHUNUSUIUJUNUSURUNIGHUOGRUPULUQUMUOGADSUOGBC
      TUCUAUDUBUKUTUSIABCDEFUEUFUG $.
  $}

  ${
    $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 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.)  (Proof
       shortened by Kyle Wyonch, 27-Apr-2021.) $)
    cnvss $p |- ( A C_ B -> `' A C_ `' B ) $=
      ( vy vx wss cv wbr copab ccnv ssbr ssopab2dv df-cnv 3sstr4g ) ABEZCFZDFZA
      GZDCHOPBGZDCHAIBINQRDCABOPJKDCALDCBLM $.
  $}

  $( Equality theorem for converse relation.  (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 relation.  (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 relation.  (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 relation.  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 relation.  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 relation.  (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 $.
    $( 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 ) ) $=
      ( vy vx cv wbr ccnv breq2 breq1 df-cnv brabg ) FHZGHZEIOAEIBAEIGFABCDEJPA
      OEKOBAELGFEMN $.
  $}

  $( Ordered-pair membership in converse relation.  (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 ) ) $=
    ( wcel wa ccnv wbr cop brcnvg df-br 3bitr3g ) ACFBDFGABEHZIBAEIABJNFBAJEFAB
    CDEKABNLBAELM $.

  ${
    opelcnv.1 $e |- A e. _V $.
    opelcnv.2 $e |- B e. _V $.
    $( Ordered-pair membership in converse relation.  (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 relation.
       Version of ~ csbcnvgALT without a sethood antecedent but depending on
       more axioms.  (Contributed by Thierry Arnoux, 8-Feb-2017.)  (Revised by
       NM, 23-Aug-2018.) $)
    csbcnv $p |- `' [_ A / x ]_ F = [_ A / x ]_ `' F $=
      ( vz vy csb ccnv cv wbr copab wsbc sbcbr opabbii csbopab 3eqtr4ri csbeq2i
      df-cnv eqtr4i ) ABCFZGZABDHZEHZCIZEDJZFZABCGZFUCABKZEDJUAUBSIZEDJUETUGUHE
      DABUAUBCLMUCAEDBNEDSQOABUFUDEDCQPR $.

    $( Move class substitution in and out of the converse of a relation.
       Version of ~ csbcnv with a sethood antecedent but depending on fewer
       axioms.  (Contributed by Thierry Arnoux, 8-Feb-2017.)
       (New usage is discouraged.)  (Proof modification is discouraged.) $)
    csbcnvgALT $p |- ( A e. V -> `' [_ A / x ]_ F = [_ A / x ]_ `' F ) $=
      ( vz vy wcel csb ccnv cv wbr copab wsbc sbcbr123 csbconstg breq12d bitrid
      opabbidv csbopabgALT df-cnv wceq a1i 3eqtr4rd csbeq2i eqtr4di ) BDGZABCHZ
      IZABEJZFJZCKZFELZHZABCIZHUFUKABMZFELUIUJUGKZFELZUMUHUFUOUPFEUOABUIHZABUJH
      ZUGKUFUPABUIUJCNUFURUIUSUJUGABUIDOABUJDOPQRUKAFEBDSUHUQUAUFFEUGTUBUCABUNU
      LFECTUDUE $.
  $}

  ${
    $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 x $.  $d B x $.
    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 ) $=
      ( cvv wcel crn cv cop wex wb elrn2g ax-mp ) BEFBCGFAHBICFAJKDABCELM $.

    $( Membership in a range.  (Contributed by NM, 2-Apr-2004.) $)
    elrn $p |- ( A e. ran B <-> E. x x B A ) $=
      ( cvv wcel crn cv wbr wex wb elrng ax-mp ) BEFBCGFAHBCIAJKDABCELM $.
  $}

  ${
    $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 cbvexv1 abbii nfex
      weq breq1 exbidv cbvabw 3eqtri ) CHFIZGIZCJZGKZFLUFBIZCJZBKZFLAIZUJCJZBKZ
      ALFGCMUIULFUHUKGBBUFUGCBUFNEBUGNOUKGPUGUJUFCQRSULUOFAUKABAUFUJCAUFNDAUJNO
      TUOFPFAUAUKUNBUFUMUJCUBUCUDUE $.
  $}

  ${
    $d A w y $.  $d B w y $.  $d x w y $.
    $( Distribute proper substitution through the domain of a class.
       (Contributed by Alexander van der Vekens, 23-Jul-2017.)  (Revised by NM,
       24-Aug-2018.) $)
    csbdm $p |- [_ A / x ]_ dom B = dom [_ A / x ]_ B $=
      ( vy vw cv cop wcel wex cab csb cdm csbab sbcex2 sbcel2 exbii bitri abbii
      wsbc dfdm3 eqtri csbeq2i 3eqtr4i ) ABDFEFGZCHZEIZDJZKZUDABCKZHZEIZDJZABCL
      ZKUILUHUFABSZDJULUFADBMUNUKDUNUEABSZEIUKUEEABNUOUJEABUDCOPQRUAABUMUGDECTU
      BDEUITUC $.
  $}

  ${
    $d A x y $.  $d B x y $.
    $( 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 $.
    opeldmd.1 $e |- ( ph -> A e. V ) $.
    opeldmd.2 $e |- ( ph -> B e. W ) $.
    $( Membership of first of an ordered pair in a domain.  Deduction version
       of ~ opeldm .  (Contributed by AV, 11-Mar-2021.) $)
    opeldmd $p |- ( ph -> ( <. A , B >. e. C -> A e. dom C ) ) $=
      ( vy cop wcel cv wex cdm wi wceq opeq2 eleq1d spcegv syl eldm2g sylibrd
      wb ) ABCJZDKZBILZJZDKZIMZBDNKZACFKUEUIOHUHUEICFUFCPUGUDDUFCBQRSTABEKUJUIU
      CGIBDEUATUB $.
  $}

  ${
    $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 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 wcel wbr cdm wa cv wex breq2 spcegv imp eldmg imbitrrid 3impib ) ACG
      ZBDGZABEHZAEIGZTUAJUBSAFKZEHZFLZTUAUEUDUAFBDUCBAEMNOFAECPQR $.

    $( 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 is included in the intersection of the
       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 $.
  $}

  ${
    breldmd.1 $e |- ( ph -> A e. C ) $.
    breldmd.2 $e |- ( ph -> B e. D ) $.
    breldmd.3 $e |- ( ph -> A R B ) $.
    $( Membership of first of a binary relation in a domain.  (Contributed by
       Glauco Siliprandi, 23-Apr-2023.) $)
    breldmd $p |- ( ph -> A e. dom R ) $=
      ( wcel wbr cdm breldmg syl3anc ) ABDJCEJBCFKBFLJGHIBCDEFMN $.
  $}

  ${
    $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 cv wbr wex cab nfopab1 nfopab2 dfdmf wcel df-br opabidw bitri
      cop exbii abbii eqtri ) ABCDZEBFZCFZUAGZCHZBIACHZBIBCUAABCJABCKLUEUFBUDAC
      UDUBUCQUAMAUBUCUANABCOPRST $.
  $}

  ${
    $d X x y $.  $d ps x $.
    dmopabel.d $e |- ( x = X -> ( ph <-> ps ) ) $.
    $( A set is an element of the domain of a ordered pair class abstraction
       iff there is a second set so that both sets fulfil the wff of the class
       abstraction.  (Contributed by AV, 19-Oct-2023.) $)
    dmopabelb $p |- ( X e. V -> ( X e. dom { <. x , y >. | ph }
                                  <-> E. y ps ) ) $=
      ( copab cdm wcel wex cab dmopab eleq2i cv wceq exbidv eqid elab2g bitrid
      ) FACDHIZJFADKZCLZJFEJBDKZUAUCFACDMNUBUDCFUCECOFPABDGQUCRST $.
  $}

  ${
    $d A x y z $.  $d B i x y z $.  $d C x y z $.  $d D x y z $.  $d I x y z $.
    $d U i x y z $.  $d V i x y z $.  $d W z $.  $d X i z $.  $d i u x y z $.
    $d i v x y z $.
    $( The domain of an ordered pair class abstraction with two nested
       restricted existential quantifiers.  (Contributed by AV,
       23-Oct-2023.) $)
    dmopab2rex $p |- ( A. u e. U ( A. v e. V B e. X /\ A. i e. I D e. W )
                       -> dom { <. x , y >. | E. u e. U ( E. v e. V
                         ( x = A /\ y = B ) \/ E. i e. I ( x = C /\ y = D ) ) }
                = { x | E. u e. U ( E. v e. V x = A \/ E. i e. I x = C ) } ) $=
      ( wa wceq wrex wo wex rexbidv vz wcel wral cv copab cdm cab rexcom4 19.43
      orbi12i bitr4i rexbii bitri exlimiv elisset ibar bicomd exbidv syl5ibrcom
      wb simpl impbid2 ralrexbid adantr adantl orbi12d bitr3id cvv eqeq1 anbi1d
      dmopabelb elv vex elab 3bitr4g eqrdv ) FNUBZCLUCZHMUBZJKUCZOZDIUCZUAAUDZE
      PZBUDZFPZOZCLQZWCGPZWEHPZOZJKQZRZDIQZABUEUFZWDCLQZWIJKQZRZDIQZAUGZWBUAUDZ
      EPZWFOZCLQZXAGPZWJOZJKQZRZDIQZBSZXBCLQZXEJKQZRZDIQZXAWOUBZXAWTUBXJXCBSZCL
      QZXFBSZJKQZRZDIQZWBXNYAXHBSZDIQXJXTYBDIXTXDBSZXGBSZRYBXQYCXSYDXCCBLUHXFJB
      KUHUJXDXGBUIUKULXHDBIUHUMWAXTXMDIWAXQXKXSXLVRXQXKUTVTVQXPXBCLVQXPXBXCXBBX
      BWFVAUNVQXPXBWFBSBFNUOXBXCWFBXBWFXCXBWFUPUQURUSVBVCVDVTXSXLUTVRVSXRXEJKVS
      XRXEXFXEBXEWJVAUNVSXRXEWJBSBHMUOXEXFWJBXEWJXFXEWJUPUQURUSVBVCVEVFVCVGXOXJ
      UTUAWNXIABVHXAWCXAPZWMXHDIYEWHXDWLXGYEWGXCCLYEWDXBWFWCXAEVIZVJTYEWKXFJKYE
      WIXEWJWCXAGVIZVJTVFTVKVLWSXNAXAUAVMYEWRXMDIYEWPXKWQXLYEWDXBCLYFTYEWIXEJKY
      GTVFTVNVOVP $.
  $}

  ${
    $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 eqabb 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 cv wcel cop wex noel nex vex eldm2 mtbir nel0 ) ACDZAEZOFP
      BEGZCFZBHRBQIJBPCAKLMN $.

    $( 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 ax6ev vex equcom bitri exbii
      ideq mpbir eldm mpgbir ) CDZEFAGZUAHZAAUAIUCUBBGZCJZBKZUFUDUBFZBKBALUEUGB
      UEUBUDFUGUBUDBMQABNOPRBUBCAMSRT $.

    $( 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 $.
    $( The domain of the membership relation is the universal class.
       (Contributed by Scott Fenton, 27-Oct-2010.)  (Proof shortened by BJ,
       26-Dec-2023.) $)
    dmep $p |- dom _E = _V $=
      ( vx vy cep cdm cvv wceq cv wcel eqv wbr wex wel el epel exbii mpbir eldm
      vex mpgbir ) CDZEFAGZTHZAATIUBUABGCJZBKZUDABLZBKABMUCUEBBUANOPBUACARQPS
      $.
  $}

  ${
    $d x y A $.
    $( An empty domain is equivalent to an empty range.  (Contributed by NM,
       21-May-1998.) $)
    dm0rn0 $p |- ( dom A = (/) <-> ran A = (/) ) $=
      ( vx vy cv wbr wex cab c0 wceq wcel wb wal wn alnex noel nbn albii eqabcb
      3bitr4i eqeq1i cdm crn excom xchbinx bitr4i 3bitr3i df-dm dfrn2 ) BDZCDZA
      EZCFZBGZHIZUKBFZCGZHIZAUAZHIAUBZHIULUIHJZKZBLZUOUJHJZKZCLZUNUQULMZBLZUOMZ
      CLZVBVEVGUOCFZMVIVGULBFVJULBNUKBCUCUDUOCNUEVFVABUTULUIOPQVHVDCVCUOUJOPQUF
      ULBHRUOCHRSURUMHBCAUGTUSUPHBCAUHTS $.
  $}

  $( 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 x y $.
    $( The range of the membership relation is the universal class minus the
       empty set.  (Contributed by BJ, 26-Dec-2023.) $)
    rnep $p |- ran _E = ( _V \ { (/) } ) $=
      ( vy vx cep crn cv wbr wex cab cvv c0 csn cdif dfrn2 nfab1 nfcv wcel abid
      wn bicomi 3bitri wel wceq epel exbii neq0 velsn notbii velcomp eqri eqtri
      ) CDAEZBEZCFZAGZBHZIJKZLZABCMBUOUQUNBNBUQOULUOPUNULUPPZRZULUQPZUNBQUNABUA
      ZAGZULJUBZRZUSUMVAABUKUCUDVDVBAULUESVCURURVCBJUFSUGTUTUSBUPUHSTUIUJ $.
  $}

  ${
    $d x y A $.
    $( A relation is empty iff its domain is empty.  (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 x y A $.  $d x y B $.
    $( The domain of a Cartesian 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.)  Avoid ~ ax-10 , ~ ax-11 , ~ ax-12 .  (Revised by
       SN, 12-Aug-2025.) $)
    dmxp $p |- ( B =/= (/) -> dom ( A X. B ) = A ) $=
      ( vx vy c0 wne cxp cdm cv wcel wex wa wbr vex eldm exbii 19.42v 3bitri n0
      brxp biimpi biantrud bitr4id eqrdv ) BEFZCABGZHZAUECIZUGJZUHAJZDIZBJZDKZL
      ZUJUIUHUKUFMZDKUJULLZDKUNDUHUFCNOUOUPDUHUKABTPUJULDQRUEUMUJUEUMDBSUAUBUCU
      D $.
    $( $j usage 'dmxp' avoids 'ax-10' 'ax-11' 'ax-12'; $)

    $( Obsolete version of ~ dmxp as of 19-Dec-2024.  (Contributed by NM,
       28-Jul-1995.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    dmxpOLD $p |- ( B =/= (/) -> dom ( A X. B ) = A ) $=
      ( vy vx c0 wne cxp cdm cv wcel wa copab df-xp dmeqi wex wral n0 ralrimivw
      wceq biimpi dmopab3 sylib eqtrid ) BEFZABGZHCIAJDIBJZKCDLZHZAUEUGCDABMNUD
      UFDOZCAPUHASUDUICAUDUIDBQTRUFCDAUAUBUC $.
  $}

  $( The domain of a Cartesian square.  (Contributed by NM, 28-Jul-1995.) $)
  dmxpid $p |- dom ( A X. A ) = A $=
    ( cxp cdm wceq c0 dm0 xpeq1 0xp eqtrdi dmeqd id 3eqtr4a dmxp pm2.61ine ) AA
    BZCZADAEAEDZECEPAFQOEQOEABEAEAGAHIJQKLAAMN $.

  $( The domain of the intersection of two Cartesian squares.  Unlike in
     ~ 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 square is a one-to-one construction.  (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 is equal to its domain (even
     when that class in not a relation, in which case ~ dfrel2 gives another
     proof).  (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 is equal to its range (even
     when that class in not a relation).  (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 (see ~ op1stb ).  (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 $.

  ${
    rnssi.1 $e |- A C_ B $.
    $( Subclass inference for range.  (Contributed by Peter Mazsa,
       24-Sep-2022.) $)
    rnssi $p |- ran A C_ ran B $=
      ( wss crn rnss ax-mp ) ABDAEBEDCABFG $.
  $}

  $( 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 $.

  ${
    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.  Note that
     ` A R B ` does not imply ` Rel R ` : see for example ~ nrelv and ~ brv .
     (Contributed by NM, 2-Jul-2008.) $)
  releldm $p |- ( ( Rel R /\ A R B ) -> A e. dom R ) $=
    ( wrel wbr wa cvv wcel cdm brrelex1 brrelex2 simpr breldmg syl3anc ) CDZABC
    EZFAGHBGHPACIHABCJABCKOPLABGGCMN $.

  $( 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 brrelex1 brrelex2 simpr brelrng syl3anc ) CDZABC
    EZFAGHBGHPBCIHABCJABCKOPLABCGGMN $.

  ${
    $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 cbvexv1 abbii nfex
      weq breq2 exbidv cbvabw 3eqtri ) CHFIZGIZCJZFKZGLAIZUGCJZAKZGLUJBIZCJZAKZ
      BLFGCMUIULGUHUKFAAUFUGCAUFNDAUGNOUKFPUFUJUGCQRSULUOGBUKBABUJUGCBUJNEBUGNO
      TUOGPGBUAUKUNAUGUMUJCUBUCUDUE $.
  $}

  ${
    $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 cv wbr wex cab nfopab1 nfopab2 dfrnf wcel df-br opabidw bitri
      cop exbii abbii eqtri ) ABCDZEBFZCFZUAGZBHZCIABHZCIBCUAABCJABCKLUEUFCUDAB
      UDUBUCQUAMAUBUCUANABCOPRST $.
  $}

  ${
    $d x y A $.
    $( Upper bound for the range of a restricted class of ordered pairs.
       (Contributed by Eric Schmidt, 16-Sep-2025.) $)
    rnopabss $p |- ran { <. x , y >. | ( y e. A /\ ph ) } C_ A $=
      ( cv wcel wa copab crn wex cab rnopab 19.42v abbii ssab2 eqsstri ) CEDFZA
      GZBCHIRBJZCKZDRBCLTQABJZGZCKDSUBCQABMNUACDOPP $.
  $}

  ${
    $d x y A $.
    $( The range of a restricted class of ordered pairs.  (Contributed by Eric
       Schmidt, 16-Sep-2025.) $)
    rnopab3 $p |- ( A. y e. A E. x ph <->
               ran { <. x , y >. | ( y e. A /\ ph ) } = A ) $=
      ( wex wral cv wcel wi wal wa wb copab crn wceq df-ral pm4.71 albii rnopab
      cab 19.42v abbii eqtri eqeq1i eqcom eqabb 3bitr2ri 3bitri ) ABEZCDFCGDHZU
      IIZCJUJUJUIKZLZCJZUJAKZBCMNZDOZUICDPUKUMCUJUIQRUQULCTZDODUROUNUPURDUPUOBE
      ZCTURUOBCSUSULCUJABUAUBUCUDDURUEULCDUFUGUH $.
  $}

  ${
    $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 rspceeqv mpan2 elrnmpt biimparc sylan ) EBJZD
        CKABLZDGJZDFMJZTDDKUADNAEBCDDIOPUBUCUAABCDFGHQRS $.
    $}

    $( 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
      cbvexv1 bitri eqeq1 anbi2d exbidv bitrid rnmpt elab2g imbitrrid impcom )
      CEIZAJZBIZCDKZIZVFVHVDGJZAVIBLZIZCAVICLZMZNZGOZVNVFGVEAPVNVFQAGVEVIMZVFVK
      VNVPVEVIBVJVPRAVIBSUCZVPVMVKAVICSZUDUAUEUFHJZCMZABUBZVOHCVGEWAVKVSVLMZNZG
      OZVTVOWAVFVTNZAOWDVTABUGWEWCAGWEGUHVKWBAAGVJAVIBTUIAVSVLAVICTUJUKVPVFVKVT
      WBVQVPCVLVSVRULUMUNUOVTWCVNGVTWBVMVKVSCVLUPUQURUSAHBCDFUTVAVBVC $.

    $( 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 wb rexbidv
      r19.29 eleq1 biimparc elexd rexlimivw ex eqeq1 elab3g bitrid ) DEIZJDHKZC
      LZABMZHNZJZCFJZABOZDCLZABMZUMUQDAHBCEGPQUTVBDRJZUAURVBUBUTVBVCUTVBSUSVASZ
      ABMVCUSVAABUDVDVCABVDDFVADFJUSDCFUEUFUGUHTUIUPVBHDRUNDLUOVAABUNDCUJUCUKTU
      L $.

    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 C x $.
    elrnmptd.f $e |- F = ( x e. A |-> B ) $.
    elrnmptd.x $e |- ( ph -> E. x e. A C = B ) $.
    elrnmptd.c $e |- ( ph -> C e. V ) $.
    $( The range of a function in maps-to notation.  (Contributed by Glauco
       Siliprandi, 17-Aug-2020.) $)
    elrnmptd $p |- ( ph -> C e. ran F ) $=
      ( crn wcel wceq wrex wb elrnmpt syl mpbird ) AEFKLZEDMBCNZIAEGLSTOJBCDEFG
      HPQR $.
  $}

  ${
    elrnmpt1d.1 $e |- F = ( x e. A |-> B ) $.
    elrnmpt1d.2 $e |- ( ph -> x e. A ) $.
    elrnmpt1d.3 $e |- ( ph -> B e. V ) $.
    $( Elementhood in an image set.  Deducion version of ~ elrnmpt1 .
       (Contributed by Glauco Siliprandi, 23-Oct-2021.) $)
    elrnmpt1d $p |- ( ph -> B e. ran F ) $=
      ( cv wcel crn elrnmpt1 syl2anc ) ABJCKDFKDELKHIBCDEFGMN $.
  $}

  ${
    $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 $.
  $}

  ${
    $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 $.
  $}

  ${
    $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 breq2 cbvexvw
      opelco 3imtr4i exlimi eldm2 eldm ssriv ) CABFZGZBGZCHZDHZIUEJZDKUHUIBLZDK
      ZUHUFJUHUGJUJULDUKDMUHEHZBLZUMUIALZNEKUNEKUJULUNUOEOEUHUIABCPZDPSUKUNDEUI
      UMUHBQRTUADUHUEUPUBDUHBUPUCTUD $.
  $}

  $( 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 A w x y z $.  $d B w x y z $.
    $( Domain of a composition.  (Contributed by NM, 28-May-1998.)  (Proof
       shortened by Andrew Salmon, 27-Aug-2011.)  Avoid ~ ax-11 .  (Revised by
       BTernaryTau, 23-Jun-2025.) $)
    dmcosseq $p |- ( ran B C_ dom A -> dom ( A o. B ) = dom B )
        $=
      ( vx vy vz vw crn cdm wss ccom dmcoss cv wbr wex wcel wa vex eldm eximdv
      wi a1i cop ssel elrn imbi12i 19.8a imim1i pm3.2 sylbi syl weq breq2 breq1
      sylcom anbi12d excomimw opelco exbii imbitrrdi eldm2 3imtr4g ssrdv eqssd
      syl6 ) BGZAHZIZABJZHZBHZVIVJIVGABKUAVGCVJVIVGCLZDLZBMZDNZVKELZUBVHOZENZVK
      VJOVKVIOVGVNVMVLVOAMZPZDNZENZVQVGVNVSENZDNWAVGVMWBDVGVLVEOZVLVFOZTZVMWBTZ
      VEVFVLUCWEVMCNZVRENZTZWFWCWGWDWHCVLBDQZUDEVLAWJRUEWIVMWHWBVMWGWHVMCUFUGVM
      VRVSEVMVRUHSUNUIUJSVSVKFLZBMZWKVOAMZPDEFDFUKVMWLVRWMVLWKVKBULVLWKVOAUMUOU
      PVDVPVTEDVKVOABCQZEQUQURUSDVKBWNREVKVHWNUTVAVBVC $.
    $( $j usage 'dmcosseq' avoids 'ax-11'; $)
  $}

  ${
    $d x y z A $.  $d x y z B $.
    $( Obsolete version of ~ dmcosseq as of 23-Jun-2025.  (Contributed by NM,
       28-May-1998.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    dmcosseqOLD $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.)  (Revised by NM, 23-Aug-2018.) $)
  csbres $p |- [_ A / x ]_ ( B |` C ) = ( [_ A / x ]_ B |` [_ A / x ]_ C ) $=
    ( cres csb cvv cxp cin df-res csbeq2i wcel csbxp csbconstg xpeq2d eqtrid wn
    wceq c0 csbprc 0xp a1i xpeq1d 3eqtr4rd pm2.61i ineq2i csbin 3eqtr4i eqtri )
    ABCDEZFABCDGHZIZFZABCFZABDFZEZABUJULCDJKUNABUKFZIUNUOGHZIUMUPUQURUNBGLZUQUR
    RUSUQUOABGFZHURABDGMUSUTGUOABGGNOPUSQZSGHZSURUQVBSRVAGUAUBVAUOSGABDTUCABUKT
    UDUEUFABCUKUGUNUOJUHUI $.

  $( 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 $.

  ${
    $d A x y z $.  $d B x y z $.
    $( Alternate definition of restriction.  (Contributed by Scott Fenton,
       17-Apr-2014.)  (Revised by Mario Carneiro, 19-Apr-2014.) $)
    dfres3 $p |- ( A |` B ) = ( A i^i ( B X. ran A ) ) $=
      ( vx vy vz cres cvv cxp cin crn df-res cv wcel wa cop wceq wex vex elxp
      wb eleq1 biantru opelrn biantrud bitr3id biimtrdi pm5.32d 2exbidv 3bitr4g
      com12 pm5.32i elin bitr4i ineqri eqtri ) ABFABGHZIABAJZHZIZABKCAUPUSCLZAM
      ZUTUPMZNVAUTURMZNUTUSMVAVBVCVAUTDLZELZOZPZVDBMZVEGMZNZNZEQDQVGVHVEUQMZNZN
      ZEQDQVBVCVAVKVNDEVAVGVJVMVGVAVJVMTZVGVAVFAMZVOUTVFAUAVJVHVPVMVIVHERZUBVPV
      LVHVDVEADRVQUCUDUEUFUJUGUHDEUTBGSDEUTBUQSUIUKUTAURULUMUNUO $.
  $}

  $( Ordered pair elementhood in a restriction.  Exercise 13 of [TakeutiZaring]
     p. 25.  (Contributed by NM, 13-Nov-1995.)  (Revised by BJ, 18-Feb-2022.)
     Commute the consequent.  (Revised by Peter Mazsa, 24-Sep-2022.) $)
  opelres $p |- ( C e. V ->
      ( <. B , C >. e. ( R |` A ) <-> ( B e. A /\ <. B , C >. e. R ) ) ) $=
    ( cop cres wcel cvv cxp wa df-res elin2 opelxp elex biantrud bitr4id bitrid
    anbi1cd ) BCFZDAGZHTDHZTAIJZHZKCEHZBAHZUBKTDUCUADALMUEUDUFUBUEUDUFCIHZKUFBC
    AINUEUGUFCEOPQSR $.

  $( Binary relation on a restriction.  (Contributed by Mario Carneiro,
     4-Nov-2015.)  Commute the consequent.  (Revised by Peter Mazsa,
     24-Sep-2022.) $)
  brres $p |- ( C e. V -> ( B ( R |` A ) C <-> ( B e. A /\ B R C ) ) ) $=
    ( wcel cop cres wa wbr opelres df-br anbi2i 3bitr4g ) CEFBCGZDAHZFBAFZODFZI
    BCPJQBCDJZIABCDEKBCPLSRQBCDLMN $.

  ${
    opelresi.1 $e |- C e. _V $.
    $( Ordered pair membership in a restriction.  Exercise 13 of
       [TakeutiZaring] p. 25.  (Contributed by NM, 13-Nov-1995.) $)
    opelresi $p |- ( <. B , C >. e. ( R |` A ) <->
                     ( B e. A /\ <. B , C >. e. R ) ) $=
      ( cvv wcel cop cres wa wb opelres ax-mp ) CFGBCHZDAIGBAGNDGJKEABCDFLM $.

    $( Binary relation on a restriction.  (Contributed by NM, 12-Dec-2006.) $)
    brresi $p |- ( B ( R |` A ) C <-> ( B e. A /\ B R C ) ) $=
      ( cvv wcel cres wbr wa wb brres ax-mp ) CFGBCDAHIBAGBCDIJKEABCDFLM $.
  $}

  ${
    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 opelresi baib ) ABFZCDGHADHKCHDABCEIJ $.
  $}

  ${
    $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.) $)
  opelidres $p |- ( A e. V -> ( <. A , A >. e. ( _I |` B ) <-> A e. B ) ) $=
    ( wcel cop cid cres wbr ididg df-br sylib opelres mpbiran2d ) ACDZAAEZFBGDA
    BDOFDZNAAFHPACIAAFJKBAAFCLM $.

  $( 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.)  (Proof shortened by JJ, 25-Aug-2021.) $)
    resiun1 $p |- ( U_ x e. A B |` C ) = U_ x e. A ( B |` C ) $=
      ( cvv cxp cin ciun cres iunin1 wceq cv wcel df-res a1i iuneq2i 3eqtr4ri )
      ABCDEFZGZHABCHZRGABCDIZHTDIABRCJABUASUASKALBMCDNOPTDNQ $.

    $( 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 $.
  $}

  $( 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 $.

  ${
    resabs1i.1 $e |- B C_ C $.
    $( Absorption law for restriction.  (Contributed by Glauco Siliprandi,
       2-Jan-2022.) $)
    resabs1i $p |- ( ( A |` C ) |` B ) = ( A |` B ) $=
      ( wss cres wceq resabs1 ax-mp ) BCEACFBFABFGDABCHI $.
  $}

  ${
    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 $.

  $( The domain of a restriction is a subset of the original domain.
     (Contributed by Glauco Siliprandi, 23-Oct-2021.)  Proof shortened and
     axiom usage reduced.  (Proof shortened by AV, 15-May-2025.) $)
  dmresss $p |- dom ( A |` B ) C_ dom A $=
    ( cres wss cdm resss dmss ax-mp ) ABCZADIEAEDABFIAGH $.

  ${
    $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 cop wex wa vex eldm2 19.42v opelresi exbii anbi2i
      wcel 3bitr4i bitr2i ineqri eqcomi ) BAEZFABGZEZCBUCUECHZUERUFDHZIZUDRZDJZ
      UFBRZUFUCRZKZDUFUDCLZMUKUHARZKZDJUKUODJZKUJUMUKUODNUIUPDBUFUGADLOPULUQUKD
      UFAUNMQSTUAUB $.
  $}

  $( 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 dfss2 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 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.)  (Proof
     shortened by JJ, 25-Aug-2021.) $)
  resima2 $p |- ( B C_ C -> ( ( A |` C ) " B ) = ( A " B ) ) $=
    ( wss cin cres crn cima wceq sseqin2 reseq2 sylbi rneqd df-ima resres rneqi
    eqtri 3eqtr4g ) BCDZACBEZFZGZABFZGACFZBHZABHSUAUCSTBIUAUCIBCJTBAKLMUEUDBFZG
    UBUDBNUFUAACBOPQABNR $.

  $( The range of a restriction is a subset of the whole range.  (Contributed
     by Glauco Siliprandi, 17-Aug-2020.) $)
  rnresss $p |- ran ( A |` B ) C_ ran A $=
    ( cres resss rnssi ) ABCAABDE $.

  $( Restriction of a constant function (or other Cartesian 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 cvv df-res inxp inv1 xpeq2i 3eqtri sseqin2 biimpi xpeq1d
    wceq eqtrid ) CADZABEZCFZACGZBEZCBEUATCHEGUBBHGZEUCTCIABCHJUDBUBBKLMSUBCBSU
    BCQCANOPR $.

  ${
    $d x y A $.  $d x y B $.  $d x y C $.  $d x y R $.
    $( Membership in an intersection with a Cartesian product.  (Contributed by
       Peter Mazsa, 9-Sep-2022.) $)
    elinxp $p |- ( C e. ( R i^i ( A X. B ) ) <->
      E. x e. A E. y e. B ( C = <. x , y >. /\ <. x , y >. e. R ) ) $=
      ( cxp cin wcel cv cop wceq wa wrex wex wrel relinxp elrel mpan eleq1 an12
      biimpd opelinxp biimpi syl6com ancld imbitrdi 2eximdv mpd sylibr simplbi2
      r2ex biimprd syl9 impd rexlimivv impbii ) EFCDGHZIZEAJZBJZKZLZVBFIZMZBDNA
      CNZUSUTCIVADIMZVEMZBOAOZVFUSVCBOAOZVIURPUSVJCDFQABEURRSUSVCVHABUSVCVCVGVD
      MZMVHUSVCVKVCUSVBURIZVKVCUSVLEVBURTZUBVLVKCDUTVAFUCZUDUEUFVCVGVDUAUGUHUIV
      EABCDULUJVEUSABCDVGVCVDUSVGVDVLVCUSVLVGVDVNUKVCUSVLVMUMUNUOUPUQ $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.
    $( Membership in a restriction.  (Contributed by Scott Fenton,
       17-Mar-2011.)  (Proof shortened by Peter Mazsa, 9-Sep-2022.) $)
    elres $p |- ( A e. ( B |` C )
          <-> E. x e. C E. y ( A = <. x , y >. /\ <. x , y >. e. B ) ) $=
      ( cres wcel cvv cxp cin cv cop wceq wa wrex wex df-res eleq2i elinxp rexv
      rexbii 3bitri ) CDEFZGCDEHIJZGCAKBKLZMUEDGNZBHOZAEOUFBPZAEOUCUDCDEQRABEHC
      DSUGUHAEUFBTUAUB $.

    $d x y A $.  $d x y B $.  $d x y C $.
    ${
      elsnres.1 $e |- C e. _V $.
      $( Membership 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 ancrd
      wi syl5 opelresi imbitrrdi adantl relssdv resss jctil eqss sylibr ) AEZAF
      ZBGZHZABIZAGZAUMGZHUMAJULUOUNULCDAUMUIUKKUKCLZDLZMZANZURUMNZSUIUKUSUPBNZU
      SHUTUKUSVAUSUPUJNUKVAUPUQACODOZPUJBUPQTRBUPUQAVBUAUBUCUDABUEUFUMAUGUH $.
  $}

  $( The domain of a restriction to a singleton is a singleton.  (Contributed
     by Alexander van der Vekens, 2-Jul-2017.) $)
  dmressnsn $p |- ( A e. dom F -> dom ( F |` { A } ) = { A } ) $=
    ( cdm wcel csn cres cin dmres wss wceq snssi dfss2 sylib eqtrid ) ABCZDZBAE
    ZFCQOGZQBQHPQOIRQJAOKQOLMN $.

  $( The element of the domain of a restriction to a singleton is the element
     of the singleton.  (Contributed by Alexander van der Vekens,
     2-Jul-2017.) $)
  eldmressnsn $p |- ( A e. dom F -> A e. dom ( F |` { A } ) ) $=
    ( cdm wcel csn cres snidg dmressnsn eleqtrrd ) ABCZDAAEZBKFCAJGABHI $.

  $( An element of the domain (of a relation) is an element of the domain of
     the restriction (of the relation) to the singleton containing this
     element.  (Contributed by Alexander van der Vekens, 22-Jul-2018.) $)
  eldmeldmressn $p |- ( X e. dom F <-> X e. dom ( F |` { X } ) ) $=
    ( cdm wcel csn cres eldmressnsn cin elinel2 dmres eleq2s impbii ) BACZDZBAB
    EZFCZDBAGNBOMHPBOMIAOJKL $.

  $( 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 $.

  ${
    resexd.1 $e |- ( ph -> A e. V ) $.
    $( The restriction of a set is a set.  (Contributed by Glauco Siliprandi,
       23-Oct-2021.) $)
    resexd $p |- ( ph -> ( A |` B ) e. _V ) $=
      ( wcel cres cvv resexg syl ) ABDFBCGHFEBCDIJ $.
  $}

  ${
    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 $.

  $( Split a relation into two disjoint parts based on its domain.
     (Contributed by Thierry Arnoux, 9-Oct-2023.) $)
  reldisjun $p |- ( ( Rel R /\ dom R = ( A u. B ) /\ ( A i^i B ) = (/) )
                 -> R = ( ( R |` A ) u. ( R |` B ) ) ) $=
    ( wrel cdm cun wceq cin w3a cres reseq2 3ad2ant2 resdm 3ad2ant1 resundi a1i
    c0 3eqtr3d ) CDZCEZABFZGZABHQGZIZCTJZCUAJZCCAJCBJFZUBSUEUFGUCTUACKLSUBUECGU
    CCMNUFUGGUDCABOPR $.

  $( Restriction of a disjoint union to the domain of the first term.
     (Contributed by Thierry Arnoux, 9-Dec-2021.) $)
  relresdm1 $p |- ( ( Rel A /\ ( dom A i^i dom B ) = (/) )
    -> ( ( A u. B ) |` dom A ) = A ) $=
    ( wrel cdm cin c0 wceq wa cun cres resundir resdm adantr dmres simpr eqtrid
    wb relres reldm0 ax-mp sylibr uneq12d un0 eqtrdi ) ACZADZBDEZFGZHZABIUFJAUF
    JZBUFJZIZAABUFKUIULAFIAUIUJAUKFUEUJAGUHALMUIUKDZFGZUKFGZUIUMUGFBUFNUEUHOPUK
    CUOUNQBUFRUKSTUAUBAUCUDP $.

  ${
    $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 $.

    $( 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 wb wal wa wi vex opeldm a1i biimtrid
      wcel wrel ssel jcad wbr df-br ideq bitr3i wex eldm2 opeq2 eleq1d biimprcd
      sylcom exlimdv imbi2d impcomd impbid opelresi bitr4di alrimivv reli relss
      syl5ibcom mpi relres eqrel sylancl mpbird resss sseq1 mpbiri impbii ) ADE
      ZADAFZGZHZVLVOBIZCIZJZASZVRVNSZKZCLBLZVLWABCVLVSVPVMSZVRDSZMZVTVLVSWEVLVS
      WCWDVSWCNVLVPVQABOZCOZPQADVRUAZUBVLWDWCVSWDVPVQHZVLWCVSNZWDVPVQDUCWIVPVQD
      UDVPVQWGUEUFZVLWCVPVPJZASZNWIWJWCVSCUGVLWMCVPAWFUHVLVSWMCVLVSWDWMWHWDWIVS
      WMWKWIWMVSWIWLVRAVPVQVPUIUJZUKRULUMRWIWMVSWCWNUNVBRUOUPVMVPVQDWGUQURUSVLA
      TZVNTVOWBKVLDTWOUTADVAVCDVMVDBCAVNVEVFVGVOVLVNDEDVMVHAVNDVIVJVK $.
  $}

  ${
    $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 wcel wbr copab relres relopabv cop vex weq eleq1w anbi12d
      wa breq1 breq2 anbi2d opelopab brresi df-br 3bitr2ri eqrelriiv ) EFDCGZAH
      ZCIZUIBHZDJZSZABKZDCLUMABMEHZFHZNZUNIUOCIZUOUPDJZSZUOUPUHJUQUHIUMURUOUKDJ
      ZSUTABUOUPEOFOZAEPUJURULVAAECQUIUOUKDTRBFPVAUSURUKUPUODUAUBUCCUOUPDVBUDUO
      UPUHUEUFUG $.
  $}

  ${
    $d A x $.  $d B x $.
    $( Sufficient condition for inclusion among two functions in maps-to
       notation.  (Contributed by Glauco Siliprandi, 17-Aug-2020.) $)
    mptss $p |- ( A C_ B -> ( x e. A |-> C ) C_ ( x e. B |-> C ) ) $=
      ( wss cmpt cres resmpt resss eqsstrrdi ) BCEABDFACDFZBGKACBDHKBIJ $.
  $}

  ${
    $d A x $.  $d C x $.  $d D x $.
    elimampt.f $e |- F = ( x e. A |-> B ) $.
    elimampt.c $e |- ( ph -> C e. W ) $.
    elimampt.d $e |- ( ph -> D C_ A ) $.
    $( Membership in the image of a mapping.  (Contributed by Thierry Arnoux,
       3-Jan-2022.) $)
    elimampt $p |- ( ph -> ( C e. ( F " D ) <-> E. x e. D C = B ) ) $=
      ( cima wcel cres crn wceq wrex cmpt wb syl df-ima eleq2i wss resmpt rneqd
      reseq1i eqtrid eleq2d eqid elrnmpt bitrd bitrid ) EGFLZMEGFNZOZMZAEDPBFQZ
      UMUOEGFUAUBAUPEBFDRZOZMZUQAFCUCZUPUTSKVAUOUSEVAUNURVAUNBCDRZFNURGVBFIUFBC
      FDUDUGUEUHTAEHMUTUQSJBFDEURHURUIUJTUKUL $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.
    $( Characterization of the elements of the intersection of the identity
       relation with a Cartesian product.  (Contributed by Peter Mazsa,
       9-Sep-2022.) $)
    elidinxp $p |- ( C e. ( _I i^i ( A X. B ) ) <->
                                       E. x e. ( A i^i B ) C = <. x , x >. ) $=
      ( vy cv wcel cop wceq wa cid cin cxp risset anbi2ci r19.42v opeq2 equcoms
      wrex rexbii eqeq2d pm5.32ri wbr ideq equcom 3bitr3i anbi2i bitr4i 3bitr2i
      vex df-br rexin elinxp 3bitr4ri ) AFZCGZDUOUOHZIZJZABSDUOEFZHZIZVAKGZJZEC
      SZABSURABCLSDKBCMLGUSVEABUSURUTUOIZECSZJURVFJZECSVEUPVGUREUOCNOURVFECPVHV
      DECVHVBVFJVDVFURVBVFUQVADUQVAIAEUOUTUOQRUAUBVCVFVBUOUTKUCUOUTIVCVFUOUTEUJ
      UDUOUTKUKAEUEUFUGUHTUITURABCULAEBCDKUMUN $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Characterization of the elements of the intersection of the identity
       relation with a Cartesian square.  (Contributed by Peter Mazsa,
       9-Sep-2022.) $)
    elidinxpid $p |- ( B e. ( _I i^i ( A X. A ) ) <->
                                                 E. x e. A B = <. x , x >. ) $=
      ( cid cxp cin wcel cv cop wceq wrex elidinxp inidm rexeqi bitri ) CDBBEFG
      CAHZPIJZABBFZKQABKABBCLQARBBMNO $.
  $}

  ${
    $d x A $.  $d x X $.
    $( Characterization of the elements of a restricted identity relation.
       (Contributed by BJ, 28-Aug-2022.)  (Proof shortened by Peter Mazsa,
       9-Sep-2022.) $)
    elrid $p |- ( A e. ( _I |` X ) <-> E. x e. X A = <. x , x >. ) $=
      ( cid cres wcel cvv cxp cin cop wceq df-res eleq2i elidinxp rexeqi 3bitri
      cv wrex inv1 ) BDCEZFBDCGHIZFBAQZUBJKZACGIZRUCACRTUABDCLMACGBNUCAUDCCSOP
      $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( The intersection of the identity relation with a cartesian product is
       the restriction of the identity relation to the intersection of the
       factors.  (Contributed by FL, 2-Aug-2009.)  (Proof shortened by Peter
       Mazsa, 9-Sep-2022.)  Generalize statement from cartesian square (now
       ~ idinxpresid ) to cartesian product.  (Revised by BJ, 23-Dec-2023.) $)
    idinxpres $p |- ( _I i^i ( A X. B ) ) = ( _I |` ( A i^i B ) ) $=
      ( vx vy cid cxp cin cres wcel cop wceq wrex elidinxp elrid bitr4i eqriv
      cv ) CEABFGZEABGZHZCQZRIUADQZUBJKDSLUATIDABUAMDUASNOP $.
  $}

  $( The intersection of the identity relation with the cartesian square of a
     class is the restriction of the identity relation to that class.
     (Contributed by FL, 2-Aug-2009.)  (Proof shortened by Peter Mazsa,
     9-Sep-2022.)  (Proof shortened by BJ, 23-Dec-2023.) $)
  idinxpresid $p |- ( _I i^i ( A X. A ) ) = ( _I |` A ) $=
    ( cid cxp cin cres idinxpres inidm reseq2i eqtri ) BAACDBAADZEBAEAAFJABAGHI
    $.

  $( A diagonal set as a subset of a Cartesian square.  (Contributed by Thierry
     Arnoux, 29-Dec-2019.)  (Proof shortened by BJ, 9-Sep-2022.) $)
  idssxp $p |- ( _I |` A ) C_ ( A X. A ) $=
    ( cid cres cxp cin idinxpresid inss2 eqsstrri ) BACBAADZEIAFBIGH $.

  ${
    $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 bitri df-br anbi2i 3bitr2ri opelresi opelxp
      pm5.32i 3bitr4i eqrelriiv ) BCDAEZFZULULGZDULHULULIBJZULKZUOCJZLZDKZMZUPU
      QULKZMZURUMKURUNKVBUOANZUQANZMZUPUOUQDOZMZUTUPVCVAVDBARZCARPVGVCUOUQNZMVE
      UPVCVFVIVHUOUQCQZSPVCVIVDVCVIAUQNVDUOAUQTAUQUAUBUIUCVFUSUPUOUQDUDUEUFULUO
      UQDVJUGUOUQULULUHUJUK $.
  $}

  ${
    $d A x y $.
    $( The identity function restricted to a class ` A ` is empty iff the class
       ` A ` is empty.  (Contributed by AV, 30-Jan-2024.) $)
    iresn0n0 $p |- ( A = (/) <-> ( _I |` A ) = (/) ) $=
      ( vx vy cv wcel weq wa copab c0 wceq wal cid opab0 opabresid eqeq1i nel02
      wn cres wo sylbi intnanrd alrimivv ianor albii id ax6v pm2.21i jaoi alimi
      19.32v eq0 sylibr impbii 3bitr4ri ) BDZAEZCBFZGZBCHZIJURQZCKZBKZLARZIJAIJ
      ZURBCMVCUSIBCANOVDVBVDUTBCVDUPUQAUOPUAUBVBUPQZBKVDVAVEBVAVEUQQZSZCKZVEUTV
      GCUPUQUCUDVHVEVFCKZSVEVEVFCUJVEVEVIVEUEVIVECBUFUGUHTTUIBAUKULUMUN $.
  $}

  $( 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.)  (Proof shortened by Peter Mazsa, 2-Oct-2022.) $)
    dfima2 $p |- ( A " B ) = { y | E. x e. B x A y } $=
      ( cima cres crn cv wbr wex cab wrex df-ima dfrn2 wcel wa wb cvv brres elv
      exbii df-rex bitr4i abbii 3eqtri ) CDECDFZGAHZBHZUFIZAJZBKUGUHCIZADLZBKCD
      MABUFNUJULBUJUGDOUKPZAJULUIUMAUIUMQBDUGUHCRSTUAUKADUBUCUDUE $.

    $( 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 $.
  $}

  ${
    $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 anbi1ci exbii eleq1w equsexvw bitri abbii abid2 3eqtri ) DAEBFZAGZUGCF
      ZHDGZRZBIZCJUIAGZCJABCDAKULUMCULBCLZUHRZBIUMUKUOBUJUNUHUJUGUIDMUNUGUIDNUG
      UICOPQSTUHUMBCBCAUAUBUCUDCAUEUF $.
  $}

  $( 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 x y $.  $d F y $.
    $( Move class substitution in and out of the image of a function.
       (Contributed by FL, 15-Dec-2006.)  (Revised by NM, 20-Aug-2018.) $)
    csbima12 $p |- [_ A / x ]_ ( F " B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) $=
      ( vy cvv wcel cima csb wceq csbeq1 imaeq12d eqeq12d nfcsb1v nfima csbeq1a
      cv vex c0 csbprc weq csbief vtoclg wn imaeq2d ima0 eqtr2di eqtrd pm2.61i
      ) BFGZABDCHZIZABDIZABCIZHZJZAEQZUKIZAUQDIZAUQCIZHZJUPEBFUQBJZURULVAUOAUQB
      UKKVBUSUMUTUNAUQBDKAUQBCKLMAUQUKVAERAUSUTAUQDNAUQCNOAEUADUSCUTAUQDPAUQCPL
      UBUCUJUDZULSUOABUKTVCUOUMSHSVCUNSUMABCTUEUMUFUGUHUI $.
  $}

  $( 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 $.

  ${
    imadisjlnd.1 $e |- ( ph -> ( dom A i^i B ) =/= (/) ) $.
    $( Deduction form of one negated side of ~ imadisj .  (Contributed by
       Stanislas Polu, 9-Mar-2020.) $)
    imadisjlnd $p |- ( ph -> ( A " B ) =/= (/) ) $=
      ( cdm cin c0 wne cima wceq imadisj biimpi necon3i syl ) ABECFZGHBCIZGHDPG
      OGPGJOGJBCKLMN $.
  $}

  $( 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 $.

    $( The image of a singleton.  (Contributed by NM, 20-May-1998.) $)
    relimasn $p |- ( Rel R -> ( R " { A } ) = { y | A R y } ) $=
      ( wrel cvv wcel csn cima cv wbr cab wceq wn wa c0 snprc imaeq2 sylbi ima0
      eqtrdi adantl wex brrelex1 stoic1a nexdv abn0 necon1bbii eqtr4d ex imasng
      sylib pm2.61d2 ) CDZBEFZCBGZHZBAIZCJZAKZLZUMUNMZUTUMVANZUPOUSVAUPOLUMVAUP
      COHZOVAUOOLUPVCLBPUOOCQRCSTUAVBURAUBZMUSOLVBURAUMURUNBUQCUCUDUEVDUSOURAUF
      UGUKUHUIABECUJUL $.

    $( 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 ) ) $=
      ( vx wrel csn cima cv wbr cab relimasn eleq2d cvv wi wb brrelex2 ex breq2
      wcel elab3g syl bitrd ) CEZBCAFGZSBADHZCIZDJZSZABCIZUCUDUGBDACKLUCUIBMSZN
      UHUIOUCUIUJABCPQUFUIDBMUEBACRTUAUB $.
  $}

  ${
    $d A x $.  $d B x $.  $d C x $.  $d V x $.  $d W x $.
    $( Membership in an image of a singleton.  (Contributed by Raph Levien,
       21-Oct-2006.)  Revise to use ~ df-br and to prove ~ elimasn1 from it.
       (Revised by BJ, 16-Oct-2024.) $)
    elimasng1 $p |- ( ( B e. V /\ C e. W ) ->
                                          ( C e. ( A " { B } ) <-> B A C ) ) $=
      ( vx wcel wa cv wbr csn cima simpr cab wceq imasng adantr breq2d elabd2 )
      BDGZCEGZHZBFIZAJZBCAJFCABKLZETUAMTUEUDFNOUAFBDAPQUBUCCOZHUCCBAUBUFMRS $.
  $}

  ${
    elimasn1.1 $e |- B e. _V $.
    elimasn1.2 $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.)  Use
       ~ df-br and shorten proof.  (Revised by BJ, 16-Oct-2024.) $)
    elimasn1 $p |- ( C e. ( A " { B } ) <-> B A C ) $=
      ( cvv wcel csn cima wbr wb elimasng1 mp2an ) BFGCFGCABHIGBCAJKDEABCFFLM
      $.
  $}

  $( Membership in an image of a singleton.  (Contributed by Raph Levien,
     21-Oct-2006.)  TODO: replace existing usages by usages of ~ elimasng1 ,
     remove, and relabel ~ elimasng1 to "elimasng". $)
  elimasng $p |- ( ( B e. V /\ C e. W ) ->
                   ( C e. ( A " { B } ) <-> <. B , C >. e. A ) ) $=
    ( wcel wa csn cima wbr cop elimasng1 df-br bitrdi ) BDFCEFGCABHIFBCAJBCKAFA
    BCDELBCAMN $.

  ${
    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.)  (Proof
       shortened by BJ, 16-Oct-2024.)  TODO: replace existing usages by usages
       of ~ elimasn1 , remove, and relabel ~ elimasn1 to "elimasn". $)
    elimasn $p |- ( C e. ( A " { B } ) <-> <. B , C >. e. A ) $=
      ( cvv wcel csn cima cop wb elimasng mp2an ) BFGCFGCABHIGBCJAGKDEABCFFLM
      $.
  $}

  $( Membership in an image of a singleton.  (Contributed by NM,
     5-Aug-2010.) $)
  elimasni $p |- ( C e. ( A " { B } ) -> B A C ) $=
    ( cvv wcel wa csn cima wbr wn c0 noel wceq snprc biimpi imaeq2d ima0 eqtrdi
    eleq2d mtbiri con4i elex jca elimasng1 biimpd mpcom ) BDEZCDEZFZCABGZHZEZBC
    AIZULUGUHUGULUGJZULCKECLUNUKKCUNUKAKHKUNUJKAUNUJKMBNOPAQRSTUACUKUBUCUIULUMA
    BCDDUDUEUF $.

  ${
    $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).  Observe the resemblance to the alternate
       definition ~ dffv4 of function value, which is based on the idea in
       Quine's definition.  (Contributed by NM, 8-May-2005.) $)
    args $p |- { x | E. y ( F " { x } ) = { y } } = { x | E! y x F y } $=
      ( csn cima wceq wex wbr weu cab cvv imasng elv eqeq1i exbii euabsn bitr4i
      cv abbii ) CARZDEZBRZDZFZBGZTUBCHZBIZAUEUFBJZUCFZBGUGUDUIBUAUHUCUAUHFABTK
      CLMNOUFBPQS $.
  $}

  $( Membership in the inverse image of a singleton.  (Contributed by NM,
     28-Apr-2004.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.)  Put in
     closed form and shorten proof.  (Revised by BJ, 16-Oct-2024.) $)
  elinisegg $p |-
             ( ( B e. V /\ C e. W ) -> ( C e. ( `' A " { B } ) <-> C A B ) ) $=
    ( wcel wa ccnv csn cima wbr elimasng1 brcnvg bitrd ) BDFCEFGCAHZBIJFBCOKCBA
    KOBCDELBCDEAMN $.

  ${
    eliniseg.1 $e |- C e. _V $.
    $( Membership in the inverse image of a singleton.  An application is to
       express initial segments for an order relation.  See 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 elinisegg mpan2 ) BDFCGFCAHBIJFCBAKLEABCD
      GMN $.
  $}

  ${
    $d A x $.  $d V x $.
    $( Any set is equal to its preimage under the converse membership relation.
       (Contributed by Mario Carneiro, 9-Mar-2013.)  Put in closed form.
       (Revised by BJ, 16-Oct-2024.) $)
    epin $p |- ( A e. V -> ( `' _E " { A } ) = A ) $=
      ( vx wcel cep ccnv csn cima cv wbr vex eliniseg epelg bitrd eqrdv ) ABDZC
      EFAGHZAPCIZQDRAEJRADEARBCKLRABMNO $.
  $}

  ${
    epini.1 $e |- A e. _V $.
    $( Any set is equal to its preimage under the converse membership relation.
       (Contributed by Mario Carneiro, 9-Mar-2013.) $)
    epini $p |- ( `' _E " { A } ) = A $=
      ( cvv wcel cep ccnv csn cima wceq epin ax-mp ) ACDEFAGHAIBACJK $.
  $}

  ${
    $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 eqabdv syl ) C
      DECFEZBGCHIZAJZCBKZALMCDNSUBATBCUAFAOPQR $.
  $}

  ${
    $d F a b $.  $d A a b $.
    $( Nonemptiness of an initial segment in terms of range.  (Contributed by
       Stefan O'Rear, 18-Jan-2015.) $)
    inisegn0 $p |- ( A e. ran F <-> ( `' F " { A } ) =/= (/) ) $=
      ( va vb crn wcel cvv ccnv csn cima c0 wne elex wn wceq snprc imaeq2d ima0
      biimpi cv eqtrdi necon1ai eleq1 sneq neeq1d wbr cab wex iniseg elv neeq1i
      abn0 vex elrn 3bitr4ri vtoclbg pm5.21nii ) ABEZFZAGFZBHZAIZJZKLZAURMUTVCK
      UTNZVCVAKJKVEVBKVAVEVBKOAPSQVARUAUBCTZURFZVAVFIZJZKLZUSVDCAGVFAURUCVFAOZV
      IVCKVKVHVBVAVFAUDQUEDTVFBUFZDUGZKLVLDUHVJVGVLDULVIVMKVIVMOCDBVFGUIUJUKDVF
      BCUMUNUOUPUQ $.
  $}

  ${
    $d x y z A $.  $d x y z R $.
    $( Alternate definition of well-founded relation.  Definition 6.21 of
       [TakeutiZaring] p. 30.  (Contributed by NM, 23-Apr-2004.)  (Revised by
       Mario Carneiro, 23-Jun-2015.) $)
    dffr3 $p |- ( R Fr A <-> A. x ( ( x C_ A /\ x =/= (/) ) ->
                E. y e. x ( x i^i ( `' R " { y } ) ) = (/) ) ) $=
      ( vz wfr cv wss c0 wne wa wbr crab wceq wrex wi wal ccnv csn cin cima cab
      dffr2 cvv iniseg ineq2i dfrab3 eqtr4i eqeq1i rexbii imbi2i albii bitr4i
      elv ) CDFAGZCHUOIJKZEGBGZDLZEUOMZINZBUOOZPZAQUPUODRUQSUAZTZINZBUOOZPZAQAB
      ECDUCVGVBAVFVAUPVEUTBUOVDUSIVDUOUREUBZTUSVCVHUOVCVHNBEDUQUDUEUNUFUREUOUGU
      HUIUJUKULUM $.
  $}

  ${
    $d x y A $.  $d x y R $.
    $( 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 wbr crab cvv wcel wral ccnv csn cima cin df-se cab dfrab3 iniseg
      cv wceq elv ineq2i eqtr4i eleq1i ralbii bitri ) BCEDSASZCFZDBGZHIZABJBCKU
      GLMZNZHIZABJADBCOUJUMABUIULHUIBUHDPZNULUHDBQUKUNBUKUNTADCUGHRUAUBUCUDUEUF
      $.
  $}

  $( 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.)  (Proof shortened by OpenAI, 3-Jul-2020.) $)
  ndmima $p |- ( -. A e. dom B -> ( B " { A } ) = (/) ) $=
    ( csn cima c0 wceq cdm cin wcel wn imadisj disjsn sylbbr ) BACZDEFBGZNHEFAO
    IJBNKOALM $.

  ${
    $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 relopabiv ) 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 brrelex12i a1i brrelex12 ancomd ex wb
    brcnvg pm5.21ndd ) CDZAEFZBEFZGZABCHZIZBACIZUEUCJTABUDCKLMTUFUCTUFGUBUABACN
    OPUCUEUFQJTABEECRMS $.

  $( 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 A x y z $.  $d B x y z $.
    $( 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 relopabiv ) CFDFZBGMEFAGHDICEABJCEDAB
      KL $.
  $}

  ${
    $d A w x y z $.  $d B w x y z $.  $d C w x y z $.
    $( Two ways of saying that the composition of two relations is included in
       a third relation.  See its special instance ~ cotr for the main
       application.  (Contributed by NM, 27-Dec-1996.)  (Proof shortened by
       Andrew Salmon, 27-Aug-2011.)  Generalized from its special instance
       ~ cotr .  (Revised by Richard Penner, 24-Dec-2019.)  (Proof shortened by
       SN, 19-Dec-2024.)  Avoid ~ ax-11 .  (Revised by BTernaryTau,
       29-Dec-2024.) $)
    cotrg $p |-
      ( ( A o. B ) C_ C <-> A. x A. y A. z ( ( x B y /\ y A z ) -> x C z ) ) $=
      ( vw ccom wss cv wbr wi wal wa wrel vex albii weq breq2 bitri relco ax-mp
      ssrel3 wex brco imbi1i 19.23v bitr4i anbi2d imbi12d anbi12d imbi1d alcomw
      wb breq1 ) DEHZFIZAJZCJZUPKZURUSFKZLZCMZAMZURBJZEKZVEUSDKZNZVALZCMBMZAMUP
      OUQVDUNDEUAACUPFUCUBVCVJAVCVIBMZCMVJVBVKCVBVHBUDZVALVKUTVLVABURUSDEAPCPUE
      UFVHVABUGUHQVIVFVEGJZDKZNZURVMFKZLURVMEKZVMUSDKZNZVALCBGGCGRZVHVOVAVPVTVG
      VNVFUSVMVEDSUIUSVMURFSUJBGRZVHVSVAWAVFVQVGVRVEVMURESVEVMUSDUOUKULUMTQT $.
    $( $j usage 'cotrg' avoids 'ax-11'; $)
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x y z C $.
    $( Obsolete version of ~ cotrg as of 29-Dec-2024.  (Contributed by NM,
       27-Dec-1996.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.)
       Generalized from its special instance ~ cotr .  (Revised by Richard
       Penner, 24-Dec-2019.)  (Proof shortened by SN, 19-Dec-2024.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    cotrgOLD $p |-
      ( ( A o. B ) C_ C <-> A. x A. y A. z ( ( x B y /\ y A z ) -> x C z ) ) $=
      ( ccom wss cv wbr wi wal wa wrel wb relco ssrel3 vex albii bitri wex brco
      ax-mp imbi1i 19.23v bitr4i alcom ) DEGZFHZAIZCIZUHJZUJUKFJZKZCLZALZUJBIZE
      JUQUKDJMZUMKZCLBLZALUHNUIUPODEPACUHFQUCUOUTAUOUSBLZCLUTUNVACUNURBUAZUMKVA
      ULVBUMBUJUKDEARCRUBUDURUMBUEUFSUSCBUGTST $.

    $( Obsolete version of ~ cotrg as of 19-Dec-2024.  (Contributed by NM,
       27-Dec-1996.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.)
       Generalized from its special instance ~ cotr .  (Revised by Richard
       Penner, 24-Dec-2019.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    cotrgOLDOLD $p |-
      ( ( A o. B ) C_ C <-> A. x A. y A. z ( ( x B y /\ y A z ) -> x C z ) ) $=
      ( ccom wss cv cop wcel wi wal wbr wa wrel wb vex albii bitri relco opelco
      ssrel ax-mp wex df-br bicomi imbi12i 19.23v bitr4i alcom ) DEGZFHZAIZCIZJ
      ZULKZUPFKZLZCMZAMZUNBIZENVBUODNOZUNUOFNZLZCMBMZAMULPUMVAQDEUAACULFUCUDUTV
      FAUTVEBMZCMVFUSVGCUSVCBUEZVDLVGUQVHURVDBUNUODEARCRUBVDURUNUOFUFUGUHVCVDBU
      IUJSVECBUKTST $.
  $}

  ${
    $d x y z R $.
    $( Two ways of saying a relation is transitive.  Definition of transitivity
       in [Schechter] p. 51.  Special instance of ~ cotrg .  (Contributed by
       NM, 27-Dec-1996.) $)
    cotr $p |-
      ( ( R o. R ) C_ R <-> A. x A. y A. z ( ( x R y /\ y R z ) -> x R z ) ) $=
      ( cotrg ) ABCDDDE $.
  $}

  ${
    $d x y R $.  $d x y A $.
    $( Alternate proof of ~ idref not relying on definitions related to
       functions.  Two ways to state that a relation is reflexive on a class.
       (Contributed by FL, 15-Jan-2012.)  (Proof shortened by Mario Carneiro,
       3-Nov-2015.)  (Revised by NM, 30-Mar-2016.)  (Proof shortened by BJ,
       28-Aug-2022.)  The "proof modification is discouraged" tag is here only
       because this is an *ALT result.  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    idrefALT $p |- ( ( _I |` A ) C_ R <-> A. x e. A x R x ) $=
      ( vy cid cres wss cv wcel wal wbr wral df-ss cop wceq wrex ralbii 3bitr2i
      wi elrid imbi1i r19.23v eleq1 df-br bitr4di pm5.74i albii ralcom4 ceqsalv
      opex biidd bitri ) EBFZCGDHZUMIZUNCIZSZDJZAHZUSCKZABLZDUMCMURUNUSUSNZOZUT
      SZABLZDJVDDJZABLVAUQVEDUQVCABPZUPSVCUPSZABLVEUOVGUPAUNBTUAVCUPABUBVHVDABV
      CUPUTVCUPVBCIUTUNVBCUCUSUSCUDUEUFQRUGVDADBUHVFUTABUTUTDVBUSUSUJVCUTUKUIQR
      UL $.
  $}

  ${
    $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 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.)  (Proof shortened by
       SN, 23-Dec-2024.)  Avoid ~ ax-11 .  (Revised by BTernaryTau,
       29-Dec-2024.) $)
    cnvsym $p |- ( `' R C_ R <-> A. x A. y ( x R y -> y R x ) ) $=
      ( vz ccnv wss cv wbr wi wal wrel wb relcnv ssrel3 ax-mp weq breq1 imbi12d
      breq2 vex alcomw brcnv imbi1i 2albii 3bitri ) CEZCFZBGZAGZUFHZUHUICHZIZAJ
      BJZULBJAJUIUHCHZUKIZBJAJUFKUGUMLCMBAUFCNOULDGZUIUFHZUPUICHZIUHUPUFHZUHUPC
      HZIBADDBDPUJUQUKURUHUPUIUFQUHUPUICQRADPUJUSUKUTUIUPUHUFSUIUPUHCSRUAULUOAB
      UJUNUKUHUICBTATUBUCUDUE $.
    $( $j usage 'cnvsym' avoids 'ax-11'; $)

    $( Obsolete version of ~ cnvsym as of 29-Dec-2024.  (Contributed by NM,
       28-Dec-1996.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.)  (Proof
       shortened by SN, 23-Dec-2024.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    cnvsymOLD $p |- ( `' R C_ R <-> A. x A. y ( x R y -> y R x ) ) $=
      ( ccnv wss cv wbr wi wal wrel relcnv ssrel3 ax-mp alcom vex imbi1i 2albii
      wb brcnv 3bitri ) CDZCEZBFZAFZUAGZUCUDCGZHZAIBIZUGBIAIUDUCCGZUFHZBIAIUAJU
      BUHRCKBAUACLMUGBANUGUJABUEUIUFUCUDCBOAOSPQT $.

    $( Obsolete version of ~ cnvsym as of 23-Dec-2024.  (Contributed by NM,
       28-Dec-1996.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    cnvsymOLDOLD $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 ideq anbi2i bibi12i pm5.32 19.21v relcnv ax-mp
      opelresi relin2 relres eqrel mp2an df-ral ) ADZBDZUAZCCUBZUCZEZVKFCGGZUDZ
      EZHZBIZAIZVIVOEZVIVJCJZVJVICJZKZVIVJLZHZBIZMZAIVMVPLZWGAVOUEVSWHAVSWAWFMZ
      BIWHVRWJBWDWAWEKZHWAWDKZWKHVRWJWDWLWKWDWAWBWAWCWBWAVJVOEZWBVKCEZWAWMKVIVJ
      CNZVIVJCASZBSZUFULUGUHUIUJVNWDVQWKVNWNVKVLEZKWDVKCVLUKWBWNWCWRWOWCVIVJVLJ
      WRVIVJCWPWQUMVIVJVLNOUNUOVQWAVKFEZKWKVOVIVJFWQVCWSWEWAWSVIVJFJWEVIVJFNVIV
      JWQUPOUQPURWAWDWEUSQRWAWFBUTPRVMTZVPTWIVTHVLTWTCVACVLVDVBFVOVEABVMVPVFVGW
      GAVOVHQ $.

    $( Two ways of saying a relation is antisymmetric and reflexive.
       (Contributed by NM, 6-May-2008.)  (Proof shortened by Mario Carneiro,
       4-Dec-2016.) $)
    asymref2 $p |- ( ( R i^i `' R ) = ( _I |` U. U. R ) <->
   ( A. x e. U. U. R x R x /\ A. x A. y ( ( x R y /\ y R x ) -> x = y ) ) ) $=
      ( ccnv cin cid cuni wceq cv wbr wa wral wi ralbii albii bitri wcel 3bitri
      wal vex cres asymref albiim r19.26 ancom equcom breq2 breq1 anbi12d anidm
      wb imbi1i bitrdi equsalvw df-ral wn cop df-br opeluu simpld sylbi pm2.24d
      adantr com12 alrimiv id ja ax-1 impbii anbi12i ) CCDEFCGGZUAHAIZBIZCJZVMV
      LCJZKZVLVMHZUKBSZAVKLVPVQMZBSZVQVPMZBSZKZAVKLZVLVLCJZAVKLZVTASZKZABCUBVRW
      CAVKVPVQBUCNWDVTAVKLZWBAVKLZKWJWIKWHVTWBAVKUDWIWJUEWJWFWIWGWBWEAVKWBVMVLH
      ZVPMZBSWEWAWLBVQWKVPABUFULOVPWEBAWKVPWEWEKWEWKVNWEVOWEVMVLVLCUGVMVLVLCUHU
      IWEUJUMUNPNWIVLVKQZVTMZASWGVTAVKUOWNVTAWNVTWMVTVTWMUPZVSBVPWOVQVPWMVQVNWM
      VOVNVLVMUQCQZWMVLVMCURWPWMVMVKQVLVMCATBTUSUTVAVCVBVDVEVTVFVGVTWMVHVIOPVJR
      R $.

    $( 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 incom 3bitr2i
      df-br eqeq1i disj2 wrel wb reli ssrel ax-mp 3bitri equcom vex wa biantrur
      ideq eldif bitr4i xchnxbir imbi12i 2albii breq2 notbid equsalvw albii
      opex ) BDEZFGZAHZCHZIZDJZVHKBLZJZMZCNANZVGVFGZVFVGBOZPZMZCNZANVFVFBOZPZAN
      VEDBEZFGDVJQZVMVDWAFBDRUADBUBDUCWBVMUDUEACDVJUFUGUHVQVLACVNVIVPVKVNVFVGGV
      FVGDOVICAUIVFVGCUJUMVFVGDTSVHBJZVKVOWCPZVHKJZWDUKVKWEWDVFVGVCULVHKBUNUOVF
      VGBTUPUQURVRVTAVPVTCAVNVOVSVGVFVFBUSUTVAVBS $.

    $( 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 wex wral wb cvv wss opelxp el2v
      df-br brcodir bitr3i imbi12i 2albii wrel relxp ssrel ax-mp r2al 3bitr4i )
      AGZBGZHZDEIZJZUQFKFLZJZMZBNANZUODJUPEJOZUOCGZFPUPVEFPOCQZMZBNANURUTUAZVFB
      ERADRVBVGABUSVDVAVFUOUPDEUBVAUOUPUTPZVFUOUPUTUDVIVFSABCUOUPFTTUEUCUFUGUHU
      RUIVHVCSDEUJABURUTUKULVFABDEUMUN $.

    $( 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 Cartesian square 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 wi wal wcel simplbi2com simplbiim com12
      brxp adantr sylbi imp ax-gen gen2 cotr mpbir ) AAEZUEFUEGBHZCHZUEIZUGDHZU
      EIZJUFUIUEIZKZDLZCLBLUMBCULDUHUJUKUHUFAMZUGAMZJUJUKKZUFUGAAQUNUPUOUJUNUKU
      JUOUIAMZUNUKKUGUIAAQUKUNUQUFUIAAQNOPRSTUAUBBCDUEUCUD $.

    $( 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 is irreflexive.  (Contributed by Mario Carneiro,
       2-Nov-2015.)  (Proof shortened by Peter Mazsa, 2-Oct-2022.) $)
    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 brresi poirr ideq breq2 notbid
      wb sylbi syl5ibcom expimpd con2d imnan sylib pm2.21d relssdv ss0 syl ) AB
      EZBFAGZHZIJVFIKVDCDVFIVELVFLVDFAMBVENUACOZDOZUBZVFPZVGVHBQZVGVHVEQZRZVDVI
      IPZVJVGVHVFQVMVGVHVFUCVGVHBVEUDUEVDVMVNVDVKVLSUFVMSVDVLVKVLVGAPZVGVHFQZRV
      DVKSZAVGVHFDUGZUHVDVOVPVQVDVORVGVGBQZSVPVQAVGBUIVPVSVKVPVGVHKVSVKUMVGVHVR
      UJVGVHVGBUKUNULUOUPTUQVKVLURUSUTTVAVFVBVC $.
  $}

  $( The relation induced by a transitive relation on a part of its field is
     transitive.  (Taking the intersection of a relation with a Cartesian
     square 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 $=
      ( wcel wa wbr wn wor sonr mpan adantl brel con3i pm2.61i ) ACFZQGZAABHZIZ
      QTQCBJQTDCABKLMSRAACCBENOP $.

    $( 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 ` A <_ B ` 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 ) $=
      ( wbr wcel wn wi brel simpld wa wceq wo wor wb sotric mpan con2bid biimpd
      breq1 sotri ex jaoi biimtrrdi com3r mpand 3imp231 ) BCDHZAEIZBADHZJZACDHZ
      UKBEIZULUNUOKUKUPCEIBCEEDGLMUPULNZUNUKUOUQUNBAOZABDHZPZUKUOKZUQUMUTEDQUQU
      MUTJRFEBADSTUAURVAUSURUKUOBACDUCUBUSUKUOABCDEFGUDUEUFUGUHUIUJ $.

    $( A transitivity relation.  (Read ` A < B ` and ` B <_ C ` implies
       ` A < C ` .)  (Contributed by Mario Carneiro, 10-May-2013.) $)
    sotri3 $p |- ( ( C e. S /\ A R B /\ -. C R B ) -> A R C ) $=
      ( wbr wcel wn wi brel simprd wa wceq wo wor wb sotric mpan con2bid expcom
      breq2 biimprd sotri jaoi biimtrrdi com3r mpan2d 3imp21 ) ABDHZCEIZCBDHZJZ
      ACDHZUKULBEIZUNUOKUKAEIUPABEEDGLMULUPNZUNUKUOUQUNCBOZBCDHZPZUKUOKZUQUMUTE
      DQUQUMUTJRFECBDSTUAURVAUSURUOUKCBADUCUDUKUSUOABCDEFGUEUBUFUGUHUIUJ $.
  $}

  $( 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 $.

  $( Property of a minimum in a strict order.  (Contributed by Stefan O'Rear,
     17-Jan-2015.) $)
  somin1 $p |- ( ( R Or X /\ ( A e. X /\ B e. X ) ) ->
      if ( A R B , A , B ) ( R u. _I ) A ) $=
    ( wor wcel wa wbr cif cid cun wceq wo iftrue olcd adantl wn sotric mpbird
    wb orcom eqcom orbi2i bitri notbii bitrdi con2bid biimpar iffalse breq1 syl
    eqeq1 orbi12d pm2.61dan poleloe ad2antrl ) DCEZADFZBDFZGGZABCHZABIZACJKHZVB
    ACHZVBALZMZUTVAVFVAVFUTVAVEVDVAABNOPUTVAQZGVFBACHZBALZMZUTVJVGUTVAVJUTVAABL
    ZVHMZQVJQDABCRVLVJVLVHVKMVJVKVHUAVKVIVHABUBUCUDUEUFUGUHVGVFVJTZUTVGVBBLZVMV
    AABUIVNVDVHVEVIVBBACUJVBBAULUMUKPSUNURVCVFTUQUSVBACDUOUPS $.

  $( Commutativity of minimum in a total order.  (Contributed by Stefan O'Rear,
     17-Jan-2015.) $)
  somincom $p |- ( ( R Or X /\ ( A e. X /\ B e. X ) ) ->
      if ( A R B , A , B ) = if ( B R A , B , A ) ) $=
    ( wor wcel wa wbr cif wn so2nr nan mpbi iffalsed eqcomd wceq sotric con2bid
    wi wo ifeq2 ifid eqtr2di iftrue jaoi biimtrrdi imp ifeqda ) DCEADFBDFGGZABC
    HZABBACHZBAIZUIUJGZULAUMUKBAUIUJUKGJSUMUKJSDABCKUIUJUKLMNOUIUJJZBULPZUIUNAB
    PZUKTZUOUIUJUQDABCQRUPUOUKUPULUKBBIBUKABBUAUKBUBUCUKULBUKBAUDOUEUFUGUH $.

  $( Property of a minimum in a strict order.  (Contributed by Stefan O'Rear,
     17-Jan-2015.) $)
  somin2 $p |- ( ( R Or X /\ ( A e. X /\ B e. X ) ) ->
      if ( A R B , A , B ) ( R u. _I ) B ) $=
    ( wor wcel wa wbr cif cid cun somincom somin1 ancom2s eqbrtrd ) DCEZADFZBDF
    ZGGABCHABIBACHBAIZBCJKZABCDLPRQSBTHBACDMNO $.

  $( Being less than a minimum, for a general total order.  (Contributed by
     Stefan O'Rear, 17-Jan-2015.) $)
  soltmin $p |- ( ( R Or X /\ ( A e. X /\ B e. X /\ C e. X ) ) ->
      ( A R if ( B R C , B , C ) <-> ( A R B /\ A R C ) ) ) $=
    ( wor wcel w3a wa wbr cif wpo cid cun 3jca syl12anc poltletr syl22anc breq2
    imp sopo ad2antrr simplr1 simplr2 simplr3 ifcld simpll somin1 somin2 jca ex
    simpr ifboth impbid1 ) EDFZAEGZBEGZCEGZHZIZABCDJZBCKZDJZABDJZACDJZIZUTVCVFU
    TVCIZVDVEVGEDLZUPVBEGZUQHZVCVBBDMNZJZVDUOVHUSVCEDUAUBZVGUPVIUQUPUQURUOVCUCZ
    VGVABCEUPUQURUOVCUDZUPUQURUOVCUEZUFZVOOUTVCULZVGUOUQURVLUOUSVCUGZVOVPBCDEUH
    PVHVJIVCVLIVDAVBBDEQTRVGVHUPVIURHZVCVBCVKJZVEVMVGUPVIURVNVQVPOVRVGUOUQURWAV
    SVOVPBCDEUIPVHVTIVCWAIVEAVBCDEQTRUJUKVAVDVEVCBCBVBADSCVBADSUMUN $.

  ${
    $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.)
       Avoid ~ ax-10 , ~ ax-12 .  (Revised by SN, 7-Jun-2025.) $)
    cnvopab $p |- `' { <. x , y >. | ph } = { <. y , x >. | ph } $=
      ( vz vw copab ccnv relcnv relopabv cv cop wcel wceq wa wex elopab weq vex
      opth 3bitr4i excom ancom anbi1i 2exbii 3bitri opelcnv eqrelriiv ) DEABCFZ
      GZACBFZUHHACBIEJZDJZKZUHLZULUKKZCJZBJZKMZANZBOCOZUOUILUOUJLUNUMUQUPKMZANZ
      COBOVBBOCOUTABCUMPVBBCUAVBUSCBVAURAEBQZDCQZNVDVCNVAURVCVDUBUKULUQUPERZDRZ
      SULUKUPUQVFVESTUCUDUEULUKUHVFVEUFACBUOPTUG $.
    $( $j usage 'cnvopab' avoids 'ax-10' 'ax-12'; $)

    $( Obsolete version of ~ cnvopab as of 7-Jun-2025.  (Contributed by NM,
       11-Dec-2003.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    cnvopabOLD $p |- `' { <. x , y >. | ph } = { <. y , x >. | ph } $=
      ( vz vw copab ccnv relcnv relopabv cv cop wcel wsb vopelopabsb sbcom2 vex
      bitri opelcnv 3bitr4i eqrelriiv ) DEABCFZGZACBFZUAHACBIEJZDJZKUALZABEMCDM
      ZUEUDKZUBLUHUCLUFACDMBEMUGABCEDNACDBEOQUEUDUADPEPRACBDENST $.
  $}

  ${
    $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 z $.
    $( The converse of the empty set.  (Contributed by NM, 6-Apr-1998.)  Remove
       dependency on ~ ax-sep , ~ ax-nul , ~ ax-pr .  (Revised by KP,
       25-Oct-2021.) $)
    cnv0 $p |- `' (/) = (/) $=
      ( vx vz vy c0 ccnv cv wcel cop wbr wa wex br0 intnan nex copab cab df-cnv
      wceq df-opab eqtri eqabri mtbir nel0 ) ADEZAFZUDGUEBFZCFZHRZUGUFDIZJZCKZB
      KZUKBUJCUIUHUGUFLMNNULAUDUDUIBCOULAPBCDQUIBCASTUAUBUC $.

    $( 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 class 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 ) $=
      ( cdif ccnv cin cnvdif difeq2i eqtri dfin4 cnveqi 3eqtr4i ) AABCZCZDZADZO
      BDZCZCZABEZDOPENOLDZCRALFTQOABFGHSMABIJOPIK $.
  $}

  $( 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 A y $.  $d F y $.
    $( Condition for the range of a relation to be the range of one its
       restrictions.  (Contributed by AV, 4-Oct-2025.) $)
    imadifssran $p |- ( ( Rel F /\ A C_ dom F )
                        -> ( ( F " ( dom F \ A ) ) C_ ran ( F |` A )
                             -> ran F = ran ( F |` A ) ) ) $=
      ( vy cdm cres crn wss wa wceq wi wb eqcomd adantr rneqd eleq2d cun eqtrdi
      wcel adantl ex cdif cima wrel df-ima sseq1i cv ssel wo resdm undif biimpi
      reseq2d rnun elun bitrdi bitrd pm2.27 jao1i com12 sylbid syl impcom ssrdv
      resundi rnresss a1i eqssd biimtrid ) BBDZAUAZUBZBAEZFZGBVJEZFZVMGZBUCZAVI
      GZHZBFZVMIZVKVOVMBVJUDUEVSVPWAVSVPHZVTVMWBCVTVMVPVSCUFZVTRZWCVMRZJZVPWCVO
      RZWEJZVSWFJVOVMWCUGWHVSWFWHVSHWDWEWGUHZWEVSWDWIKWHVSWDWCBVIEZFZRZWIVSVTWK
      WCVSBWJVQBWJIVRVQWJBBUILMNOVRWLWIKVQVRWLWCVMVOPZRWIVRWKWMWCVRWKVLVNPZFWMV
      RWJWNVRWJBAVJPZEWNVRVIWOBVRWOVIVRWOVIIAVIUJUKLULBAVJVDQNVLVNUMQOWCVMVOUNU
      OSUPSWHWIWEJVSWIWHWEWEWGWHWGWEUQURUSMUTTVAVBVCVMVTGWBBAVEVFVGTVH $.
  $}

  $( The preimage of a superset of the range of a class is the domain of the
     class.  Generalization of ~ cnvimarndm for subsets.  (Contributed by AV,
     18-Sep-2024.) $)
  cnvimassrndm $p |- ( ran F C_ A -> ( `' F " A ) = dom F ) $=
    ( crn wss cun wceq ccnv cdm ssequn1 imaeq2 imaundi eqtrdi cnvimarndm uneq1i
    cima cnvimass ssequn2 mpbi eqtri eqcoms sylbi ) BCZADUBAEZAFBGZAOZBHZFZUBAI
    UGAUCAUCFZUEUDUBOZUEEZUFUHUEUDUCOUJAUCUDJUDUBAKLUJUFUEEZUFUIUFUEBMNUEUFDUKU
    FFBAPUEUFQRSLTUA $.

  ${
    $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 elin cop a1i cvv elimasng elvd anbi12d 3bitr4rd
      wa wb bitr2id eqrdv ) CDFZEABGZCHZIZAUEIZBUEIZGZEJZUIFUJUGFZUJUHFZSZUCUJU
      FFZUJUGUHKUCCUJLZUDFZUOAFZUOBFZSZUNUMUPUSTUCUOABKMUCUNUPTEUDCUJDNOPUCUKUQ
      ULURUCUKUQTEACUJDNOPUCULURTEBCUJDNOPQRUAUB $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( The converse of a Cartesian 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 Cartesian 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 Cartesian product of nonempty classes is nonempty.  (Variation of a
       theorem contributed by Raph Levien, 30-Jun-2006.)  (Contributed by NM,
       30-Jun-2006.) $)
    xpnz $p |- ( ( A =/= (/) /\ B =/= (/) ) <-> ( A X. B ) =/= (/) ) $=
      ( vx vy vz c0 wne wa cxp wcel wex anbi12i exdistrv bitr4i cop wceq eqtrdi
      cv n0 necon3i eleq1 opelxp bitrdi spcev sylibr exlimivv sylbi xpeq1 xpeq2
      opex 0xp xp0 jca impbii ) AFGZBFGZHZABIZFGZUQCRZAJZDRZBJZHZDKCKZUSUQVACKZ
      VCDKZHVEUOVFUPVGCASDBSLVAVCCDMNVDUSCDVDERZURJZEKUSVIVDEUTVBOZUTVBUJVHVJPV
      IVJURJVDVHVJURUAUTVBABUBUCUDEURSUEUFUGUSUOUPAFURFAFPURFBIFAFBUHBUKQTBFURF
      BFPURAFIFBFAUIAULQTUMUN $.
  $}

  $( At least one member of an empty Cartesian product is empty.  (Contributed
     by NM, 27-Aug-2006.) $)
  xpeq0 $p |- ( ( A X. B ) = (/) <-> ( A = (/) \/ B = (/) ) ) $=
    ( cxp c0 wceq wne wa wn wo xpnz necon2bbii ianor nne orbi12i 3bitri ) ABCZD
    EADFZBDFZGZHQHZRHZIADEZBDEZISPDABJKQRLTUBUAUCADMBDMNO $.

  $( Cartesian 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 xpeq1 inxp 0xp eqcomi 3eqtr4g ) ABEZFGNCDEZHFOHZACHBDHEFN
    FOIACBDJPFOKLM $.

  $( Cartesian 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 xpeq2 inxp xp0 eqcomi 3eqtr4g ) ABEZFGCDEZNHOFHZCAHDBHEFN
    FOICADBJPFOKLM $.

  $( Cartesian 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 A x y $.  $d B x y $.  $d C x y $.  $d D x y $.
    $( Difference of Cartesian products, expressed in terms of a union of
       Cartesian products of differences.  (Contributed by Jeff Madsen,
       2-Sep-2009.)  (Revised by Mario Carneiro, 26-Jun-2014.)  (Proof
       shortened by Wolf Lammen, 16-May-2025.) $)
    difxp $p |- ( ( C X. D ) \ ( A X. B ) ) =
                ( ( ( C \ A ) X. D ) u. ( C X. ( D \ B ) ) ) $=
      ( vx vy cxp cdif cun wrel relxp cv wcel wn wa anbi2i opelxp eldif 3bitr4i
      wo wss difss relss mp2 relun mpbir2an cop ianor andi bitri notbii anbi12i
      anbi1i an32 3bitri anass orbi12i elun eqrelriiv ) EFCDGZABGZHZCAHZDGZCDBH
      ZGZIZVBUTUAUTJVBJUTVAUBCDKVBUTUCUDVGJVDJVFJVCDKCVEKVDVFUEUFELZFLZUGZUTMZV
      JVAMZNZOZVJVDMZVJVFMZTZVJVBMVJVGMVHCMZVIDMZOZVHAMZVIBMZOZNZOZVTWANZOZVTWB
      NZOZTZVNVQWEVTWFWHTZOWJWDWKVTWAWBUHPVTWFWHUIUJVKVTVMWDVHVICDQVLWCVHVIABQU
      KULVOWGVPWIVOVHVCMZVSOVRWFOZVSOWGVHVIVCDQWLWMVSVHCARUMVRWFVSUNUOVRVIVEMZO
      VRVSWHOZOVPWIWNWOVRVIDBRPVHVICVEQVRVSWHUPSUQSVJUTVARVJVDVFURSUS $.
  $}

  $( Difference law for Cartesian product.  (Contributed by Scott Fenton,
     18-Feb-2013.)  (Revised by Mario Carneiro, 26-Jun-2014.) $)
  difxp1 $p |- ( ( A \ B ) X. C ) = ( ( A X. C ) \ ( B X. C ) ) $=
    ( cxp cdif cun c0 difxp difid xpeq2i xp0 eqtri uneq2i un0 3eqtrri ) ACDBCDE
    ABECDZACCEZDZFPGFPBCACHRGPRAGDGQGACIJAKLMPNO $.

  $( Difference law for Cartesian product.  (Contributed by Scott Fenton,
     18-Feb-2013.)  (Revised by Mario Carneiro, 26-Jun-2014.) $)
  difxp2 $p |- ( A X. ( B \ C ) ) = ( ( A X. B ) \ ( A X. C ) ) $=
    ( cxp cdif cun c0 difxp difid xpeq1i 0xp eqtri uneq1i uncom un0 3eqtrri ) A
    BDACDEAAEZBDZABCEDZFGSFZSACABHRGSRGBDGQGBAIJBKLMTSGFSGSNSOLP $.

  ${
    $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 $.
  $}

  ${
    $d A i j p x y $.  $d B i j p x y $.
    $( The set of distinct couples in a Cartesian product.  (Contributed by
       Thierry Arnoux, 25-May-2019.) $)
    xpdifid $p |- U_ x e. A ( { x } X. ( B \ { x } ) ) = ( ( A X. B ) \ _I ) $=
      ( vi vj vy cv csn cdif cid wcel wrex wceq wa wex wne wn necon3bbii eleq2d
      velsn vp cxp ciun cop elxp rexbii rexcom4 exbii 3bitri eliun eldif opelxp
      wbr df-br vex ideq bitr3i anbi12i bitri anbi2i 2exbii eldifi elxpi 2eximi
      simpl 3syl ancli 19.42vv sylibr ancom eleq1 adantl pm5.32da 2exbidv mpbid
      bitrid biimpar impbii r19.42v simprl sylib eqeltrd simprr eldifad eldifbd
      wb exlimivv necomd eqnetrd jca31 adantll difeq2d anbi12d cbvrexvw r19.29a
      biimpi simpll vsnid a1i simplr simpr eldifd rspcev syl12anc 3bitr4i eqriv
      sneq ) UAABAGZHZCXIIZUBZUCZBCUBZJIZUAGZXKKZABLZXODGZEGZUDZMZXRXIKZXSXJKZN
      ZNZABLZEOZDOZXOXLKXOXNKZXQYEEOZDOZABLYJABLZDOYHXPYKABDEXOXIXJUEUFYJADBUGY
      LYGDYEAEBUGUHUIAXOBXKUJYAXTXNKZNZEODOZYAXRBKZXSCKZNZXRXSPZNZNZEODOYIYHYNU
      UADEYMYTYAYMXTXMKZXTJKZQZNYTXTXMJUKUUBYRUUDYSXRXSBCULUUCXRXSUUCXRXSJUMXRX
      SMXRXSJUNXRXSEUOUPUQRURUSUTVAYIYOYIYIYANZEODOZYOYIYIYAEODOZNUUFYIUUGYIXOX
      MKYAYRNZEODOUUGXOXMJVBDEXOBCVCUUHYADEYAYRVEVDVFVGYIYADEVHVIYIUUEYNDEUUEYA
      YINYIYNYIYAVJYIYAYIYMYAYIYMWFYIXOXTXNVKZVLVMVPVNVOYNYIDEYAYIYMUUIVQWGVRYF
      UUADEYFYAYDABLZNUUAYAYDABVSUUJYTYAUUJYTUUJXRFGZHZKZXSCUULIZKZNZYTFBUUKBKZ
      UUPYTUUJUUQUUPNZYPYQYSUURXRUUKBUURUUMXRUUKMUUQUUMUUOVTDUUKTWAZUUQUUPVEWBU
      URXSCUULUUQUUMUUOWCZWDUURXRUUKXSUUSUURXSUUKUURXSUULKZQXSUUKPUURXSCUULUUTW
      EUVAXSUUKEUUKTRWAWHWIWJWKUUJUUPFBLYDUUPAFBXHUUKMZYBUUMYCUUOUVBXIUULXRXHUU
      KXGZSUVBXJUUNXSUVBXIUULCUVCWLSWMWNWPWOYTYPXRXRHZKZXSCUVDIZKZUUJYPYQYSWQUV
      EYTDWRWSYTXSCUVDYPYQYSWTYTXSXRPXSUVDKZQYTXRXSYRYSXAWHUVHXSXREXRTRVIXBYDUV
      EUVGNAXRBXHXRMZYBUVEYCUVGUVIXIUVDXRXHXRXGZSUVIXJUVFXSUVIXIUVDCUVJWLSWMXCX
      DVRUTUSVAXEXEXF $.
  $}

  $( 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 reseq2 resres res0 eqcomi 3eqtr4g ) ABDZEFCMGCEGZCAGBGEM
    ECHCABINECJKL $.

  $( The range of a Cartesian product.  Part of Theorem 3.13(x) of [Monk1]
     p. 37.  (Contributed by NM, 12-Apr-2004.) $)
  rnxp $p |- ( A =/= (/) -> ran ( A X. B ) = B ) $=
    ( c0 wne cxp crn cdm ccnv df-rn cnvxp dmeqi eqtri dmxp eqtrid ) ACDABEZFZBA
    EZGZBPOHZGROISQABJKLBAMN $.

  $( The domain of a Cartesian product is included in its first factor.
     (Contributed by NM, 19-Mar-2007.) $)
  dmxpss $p |- dom ( A X. B ) C_ A $=
    ( cxp cdm wss c0 wceq xpeq2 xp0 eqtrdi dm0 0ss eqsstrdi wne dmxp eqimss syl
    dmeqd pm2.61ine ) ABCZDZAEZBFBFGZUAFAUCUAFDFUCTFUCTAFCFBFAHAIJRKJALMBFNUAAG
    UBABOUAAPQS $.

  $( The range of a Cartesian product is included in its 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 $.

  $( The range of a Cartesian square.  (Contributed by FL, 17-May-2010.) $)
  rnxpid $p |- ran ( A X. A ) = A $=
    ( cxp crn wceq c0 rn0 xpeq2 xp0 eqtrdi rneqd id 3eqtr4a rnxp pm2.61ine ) AA
    BZCZADAEAEDZECEPAFQOEQOAEBEAEAGAHIJQKLAAMN $.

  $( A Cartesian product subclass relationship is equivalent to the conjunction
     of the analogous relationships for the factors.  (Contributed by NM,
     17-Dec-2008.) $)
  ssxpb $p |- ( ( A X. B ) =/= (/) -> ( ( A X. B ) C_ ( C X. D ) <->
              ( A C_ C /\ B C_ D ) ) ) $=
    ( cxp c0 wne wss wa cdm wceq xpnz dmxp adantl sylbir adantr eqsstrrd sstrdi
    dmss crn dmxpss rnxp rnss rnxpss jca ex xpss12 impbid1 ) ABEZFGZUICDEZHZACH
    ZBDHZIZUJULUOUJULIZUMUNUPAUKJZCUPAUIJZUQUJURAKZULUJAFGZBFGZIZUSABLZVAUSUTAB
    MNOPULURUQHUJUIUKSNQCDUARUPBUKTZDUPBUITZVDUJVEBKZULUJVBVFVCUTVFVAABUBPOPULV
    EVDHUJUIUKUCNQCDUDRUEUFACBDUGUH $.

  $( The Cartesian product of nonempty classes is a one-to-one "function" of
     its two "arguments".  In other words, two Cartesian products, at least one
     with nonempty factors, are equal if and only if their respective factors
     are equal.  (Contributed by NM, 31-May-2008.) $)
  xp11 $p |- ( ( A =/= (/) /\ B =/= (/) )
      -> ( ( A X. B ) = ( C X. D ) <-> ( A = C /\ B = D ) ) ) $=
    ( c0 wne wa cxp wceq wi xpnz anidm neeq1 anbi2d bitr3id wss ssxpb syl5ibcom
    eqimss eqss eqimss2 anim12d an4 anbi12i bitr4i imbitrdi sylbid com12 xpeq12
    sylbi impbid1 ) AEFBEFGZABHZCDHZIZACIZBDIZGZULUMEFZUOURJABKUOUSURUOUSUSUNEF
    ZGZURUSUSUSGUOVAUSLUOUSUTUSUMUNEMNOUOVAACPZBDPZGZCAPZDBPZGZGZURUOUSVDUTVGUO
    UMUNPUSVDUMUNSABCDQRUOUNUMPUTVGUNUMUACDABQRUBVHVBVEGZVCVFGZGURVBVCVEVFUCUPV
    IUQVJACTBDTUDUEUFUGUHUJACBDUIUK $.

  $( Cancellation law for Cartesian product.  (Contributed by NM,
     30-Aug-2011.) $)
  xpcan $p |- ( C =/= (/) -> ( ( C X. A ) = ( C X. B ) <-> A = B ) ) $=
    ( c0 wne cxp wceq wb wa xp11 eqid biantrur bitr4di wn wi simpr xpeq2 eqtrdi
    nne xp0 eqeq1d eqcom bitrdi adantl df-ne xpeq0 orel1 biimtrid adantr sylbid
    wo sylbi eqtr3 syl6an sylan2b impbid1 pm2.61dan ) CDEZADEZCAFZCBFZGZABGZHUR
    USIVBCCGZVCIVCCACBJVDVCCKLMURUSNZIVBVCVEURADGZVBVCOADSURVFIZVFVBBDGZVCURVFP
    VGVBVADGZVHVFVBVIHURVFVBDVAGVIVFUTDVAVFUTCDFDADCQCTRUADVAUBUCUDURVIVHOZVFUR
    CDGZNZVJCDUEVIVKVHUKVLVHCBUFVKVHUGUHULUIUJABDUMUNUOABCQUPUQ $.

  $( Cancellation law for Cartesian product.  (Contributed by NM,
     30-Aug-2011.) $)
  xpcan2 $p |- ( C =/= (/) -> ( ( A X. C ) = ( B X. C ) <-> A = B ) ) $=
    ( c0 wne cxp wceq wb wa xp11 eqid biantru bitr4di wn nne simpl xpeq1 eqtrdi
    0xp eqeq1d eqcom bitrdi adantr wi df-ne wo xpeq0 orel2 biimtrid sylbi eqtr3
    adantl sylbid syl6an impbid1 sylanb pm2.61ian ) ADEZCDEZACFZBCFZGZABGZHZURU
    SIVBVCCCGZIVCACBCJVEVCCKLMURNADGZUSVDADOVFUSIZVBVCVGVFVBBDGZVCVFUSPVGVBVADG
    ZVHVFVBVIHUSVFVBDVAGVIVFUTDVAVFUTDCFDADCQCSRTDVAUAUBUCUSVIVHUDZVFUSCDGZNZVJ
    CDUEVIVHVKUFVLVHBCUGVKVHUHUIUJULUMABDUKUNABCQUOUPUQ $.

  ${
    $d x y A $.  $d x y B $.  $d x y C $.
    $( Two ways to express surjectivity of a restricted and corestricted binary
       relation (intersection of a binary relation with a Cartesian product):
       the LHS expresses inclusion in the range of the restricted relation,
       while the RHS expresses equality with the range of the restricted and
       corestricted relation.  (Contributed by NM, 16-Jan-2006.)  (Proof
       shortened by Peter Mazsa, 2-Oct-2022.) $)
    ssrnres $p |- ( B C_ ran ( C |` A ) <-> ran ( C i^i ( A X. B ) ) = B ) $=
      ( vy vx cxp cin crn wceq wss cres inss2 rnssi rnxpss sstri cv wcel wex wa
      elrn2 eqss mpbiran inxpssres sstr cop ssel vex imbitrdi opelinxp opelresi
      mpan2 ancld bianassc bitr4i exbii 19.42v 3bitri imbitrrdi impbii bitr2i
      ssrdv ) CABFZGZHZBIZBVDJZBCAKZHZJZVEVDBJVFVDVBHBVCVBCVBLMABNOVDBUAUBVFVIV
      FVDVHJVIVCVGABCUCMBVDVHUDUKVIDBVDVIDPZBQZVKEPZVJUEZVGQZERZSZVJVDQZVIVKVOV
      IVKVJVHQVOBVHVJUFEVJVGDUGZTUHULVQVMVCQZERVKVNSZERVPEVJVCVRTVSVTEVSVLAQZVK
      SVMCQZSVTABVLVJCUIVNWAWBVKAVLVJCVRUJUMUNUOVKVNEUPUQURVAUSUT $.
  $}

  ${
    $d x y A $.  $d y B $.  $d x y C $.
    $( Two ways to express surjectivity of a restricted and corestricted binary
       relation (intersection of a binary relation with a Cartesian 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 $.
    $( Two ways to express totality of a restricted and corestricted binary
       relation (intersection of a binary relation with a Cartesian 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 by a restricted and corestricted binary relation (intersection of a
     binary relation with a Cartesian product).  (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 $.

  $( Direct image by a Cartesian product.  (Contributed by Thierry Arnoux,
     4-Feb-2017.) $)
  xpima $p |- ( ( A X. B ) " C ) = if ( ( A i^i C ) = (/) , (/) , B ) $=
    ( cxp cima cin c0 wceq cif wa wn wo exmid crn cvv rneqi eqtrdi eqtrid ancli
    cres df-ima df-res eqtri inxp inv1 xpeq2i 3eqtri xpeq1 0xp rneqd df-ne rnxp
    rn0 wne sylbir orim12i ax-mp eqif mpbir ) ABDZCEZACFZGHZGBIHVCVAGHZJZVCKZVA
    BHZJZLZVCVFLVIVCMVCVEVFVHVCVDVCVAVBBDZNZGVAUTCODFZNZVBBOFZDZNVKVAUTCTZNVMUT
    CUAVPVLUTCUBPUCVLVOABCOUDPVOVJVNBVBBUEUFPUGZVCVKGNGVCVJGVCVJGBDGVBGBUHBUIQU
    JUMQRSVFVGVFVAVKBVQVFVBGUNVKBHVBGUKVBBULUORSUPUQVCVAGBURUS $.

  $( Direct image by a Cartesian product (case of empty intersection with the
     domain).  (Contributed by Thierry Arnoux, 16-Dec-2017.) $)
  xpima1 $p |- ( ( A i^i C ) = (/) -> ( ( A X. B ) " C ) = (/) ) $=
    ( cin c0 wceq cxp cima cif xpima iftrue eqtrid ) ACDEFZABGCHMEBIEABCJMEBKL
    $.

  $( Direct image by a Cartesian product (case of nonempty intersection with
     the domain).  (Contributed by Thierry Arnoux, 16-Dec-2017.) $)
  xpima2 $p |- ( ( A i^i C ) =/= (/) -> ( ( A X. B ) " C ) = B ) $=
    ( cin c0 wne cxp cima wceq cif xpima ifnefalse eqtrid ) ACDZEFABGCHNEIEBJBA
    BCKNEEBLM $.

  $( Direct image of a singleton by a Cartesian product.  (Contributed by
     Thierry Arnoux, 14-Jan-2018.)  (Proof shortened by BJ, 6-Apr-2019.) $)
  xpimasn $p |- ( X e. A -> ( ( A X. B ) " { X } ) = B ) $=
    ( wcel csn cin c0 wne cxp cima wceq disjsn necon3abii notnotb bitr4i xpima2
    wn sylbir ) CADZACEZFZGHZABITJBKUBSQZQSUCUAGACLMSNOABTPR $.

  ${
    $d x y A $.  $d x B $.  $d x y R $.
    $( The base set of a strict order is contained in the field of the
       relation, except possibly for one element (note that ` (/) Or { B } ` ).
       (Contributed by Mario Carneiro, 27-Apr-2015.) $)
    sossfld $p |- ( ( R Or A /\ B e. A ) ->
      ( A \ { B } ) C_ ( dom R u. ran R ) ) $=
      ( vx wor wcel wa csn cdif cdm crn cun cv wne eldifsn wbr wo wb wi 3expia
      sotrieq necon2abid anass1rs breldmg ancoms brelrng orim12d elun imbitrrdi
      adantll sylbird expimpd biimtrid ssrdv ) ACEZBAFZGZDABHIZCJZCKZLZDMZURFVB
      AFZVBBNZGUQVBVAFZVBABOUQVCVDVEUQVCGVDVBBCPZBVBCPZQZVEUOVCUPVHVDRUOVCUPGGV
      HVBBAVBBCUAUBUCUPVCVHVESUOUPVCGZVHVBUSFZVBUTFZQVEVIVFVJVGVKVCUPVFVJSVCUPV
      FVJVBBAACUDTUEUPVCVGVKBVBCAAUFTUGVBUSUTUHUIUJUKULUMUN $.

    $( The base set of a nonempty strict order is the same as the field of the
       relation.  (Contributed by Mario Carneiro, 15-May-2015.) $)
    sofld $p |- ( ( R Or A /\ R C_ ( A X. A ) /\ R =/= (/) ) ->
      A = ( dom R u. ran R ) ) $=
      ( vx vy wor cxp wss c0 cdm crn wa wrel ad2antlr wcel ssun1 sseqtrdi unssd
      cun cv ex wne w3a wn wceq relxp relss mpi cop wbr df-br csn cdif sseqtrri
      undif1 simpll dmxpid releldm sylancom sseldd sossfld syl2anc sselid snssd
      dmss sstrid biimtrrid con3dimp pm2.21d relssdv ss0 necon1ad 3impia rnxpid
      syl rnss 3ad2ant2 eqssd ) ABEZBAAFZGZBHUAZUBABIZBJZRZVRVTWAAWDGZVRVTKZWEB
      HWFWEUCZBHUDZWFWGKZBHGWHWICDBHVTBLZVRWGVTVSLWJAAUEBVSUFUGZMWICSZDSZUHZBNZ
      WNHNWFWOWEWOWLWMBUIZWFWEWLWMBUJWFWPWEWFWPKZAAWLUKZULZWRRZWDAAWRRWTAWROAWR
      UNUMWQWSWRWDWQVRWLANWSWDGVRVTWPUOWQWBAWLVTWBAGVRWPVTWBVSIABVSVDAUPPZMWFWP
      WJWLWBNVTWJVRWPWKMWLWMBUQURZUSAWLBUTVAWQWLWDWQWBWDWLWBWCOXBVBVCQVETVFVGVH
      VIBVJVNTVKVLVTVRWDAGWAVTWBWCAXAVTWCVSJABVSVOAVMPQVPVQ $.
  $}

  ${
    $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.  An
       analogue of ~ dffn5 for relations.  (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 $.
  $}

  ${
    $d a b x y $.  $d a b R $.
    dfrel4.1 $e |- F/_ x R $.
    dfrel4.2 $e |- F/_ y R $.
    $( A relation can be expressed as the set of ordered pairs in it.  An
       analogue of ~ dffn5 for relations.  (Contributed by Mario Carneiro,
       16-Aug-2015.)  (Revised by Thierry Arnoux, 11-May-2017.) $)
    dfrel4 $p |- ( Rel R <-> R = { <. x , y >. | x R y } ) $=
      ( va vb wrel cv wbr copab wceq dfrel4v nfcv nfbr nfv breq12 cbvopab bitri
      eqeq2i ) CHCFIZGIZCJZFGKZLCAIZBIZCJZABKZLFGCMUDUHCUCUGFGABAUAUBCAUANDAUBN
      OBUAUBCBUANEBUBNOUGFPUGGPUAUEUBUFCQRTS $.
  $}

  $( The double converse of a class strips out all elements that are not
     ordered pairs.  (Contributed by NM, 8-Dec-2003.)  (Proof shortened by BJ,
     26-Nov-2021.) $)
  cnvcnv $p |- `' `' A = ( A i^i ( _V X. _V ) ) $=
    ( ccnv cvv cxp cin cnvin cnveqi wceq wrel relcnv df-rel mpbi relxp sseqtrri
    wss dfrel2 dfss 3eqtr4ri relinxp eqtri ) ABZBZACCDZEZBZBZUDUAUCBZEZBUBUGBZE
    ZUFUBUAUGFUEUHAUCFGUBUIOUBUJHUBUCUIUBIUBUCOUAJUBKLUCIUIUCHCCMUCPLNUBUIQLRUD
    IUFUDHCCASUDPLT $.

  $( 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 $.

  $( Two ways to express the corestriction of a class.  (Contributed by BJ,
     28-Dec-2023.) $)
  cnvrescnv $p |- `' ( `' R |` B ) = ( R i^i ( _V X. B ) ) $=
    ( ccnv cres cvv cxp cin df-res cnveqi cnvin cnvcnv cnvxp ineq12i inass inxp
    inv1 eqcomi ssv ssid 3eqtri ssini inss2 eqssi xpeq12i eqtr4i ineq2i ) BCZAD
    ZCUGAEFZGZCUGCZUICZGZBEAFZGZUHUJUGAHIUGUIJUMBEEFZGZUNGBUPUNGZGUOUKUQULUNBKA
    ELMBUPUNNURUNBUREEGZEAGZFUNEEEAOEUSAUTUSEEPQAUTAEAARASUAEAUBUCUDUEUFTT $.

  $( 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 $.

  ${
    $d x A $.
    $( Characterization of the elements of the identity relation.  TODO:
       reorder theorems to move this theorem and ~ dfrel3 after ~ elrid .
       (Contributed by BJ, 28-Aug-2022.) $)
    elid $p |- ( A e. _I <-> E. x A = <. x , x >. ) $=
      ( cid wcel cvv cres cop wceq wrex wex wrel reli dfrel3 mpbi eqcomi eleq2i
      cv elrid rexv 3bitri ) BCDBCEFZDBAQZUBGHZAEIUCAJCUABUACCKUACHLCMNOPABERUC
      AST $.
  $}

  $( 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 $.

  $( Distribute proper substitution through the range of a class.  (Contributed
     by Alan Sare, 10-Nov-2012.) $)
  csbrn $p |- [_ A / x ]_ ran B = ran [_ A / x ]_ B $=
    ( cvv cima csb crn csbima12 wcel wceq csbconstg imaeq2d wn c0 eqcomi csbprc
    0ima imaeq1d eqtrdi dfrn4 3eqtr4a pm2.61i eqtri csbeq2i 3eqtr4i ) ABCDEZFZA
    BCFZDEZABCGZFUHGUGUHABDFZEZUIABDCHBDIZULUIJUMUKDUHABDDKLUMMZNNDEZULUIUONDQO
    UNULNUKENUNUHNUKABCPZRUKQSUNUHNDUPRUAUBUCABUJUFCTUDUHTUE $.

  $( 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 nonzero iff the singleton argument is an
       ordered pair.  (Contributed by NM, 14-Dec-2008.)  (Proof shortened by
       Andrew Salmon, 27-Aug-2011.) $)
    dmsnn0 $p |- ( A e. ( _V X. _V ) <-> dom { A } =/= (/) ) $=
      ( vx vy cv cop wceq wex csn cdm wcel cvv cxp wne wbr vex eldm df-br exbii
      c0 opex elsn eqcom 3bitri bitr2i elvv n0 3bitr4i ) ABDZCDZEZFZCGZBGUHAHZI
      ZJZBGAKKLJUNSMULUOBUOUHUIUMNZCGULCUHUMBOPUPUKCUPUJUMJUJAFUKUHUIUMQUJAUHUI
      TUAUJAUBUCRUDRBCAUEBUNUFUG $.
  $}

  $( The range of a singleton is nonzero iff the singleton argument is an
     ordered pair.  (Contributed by NM, 14-Dec-2008.) $)
  rnsnn0 $p |- ( A e. ( _V X. _V ) <-> ran { A } =/= (/) ) $=
    ( cvv cxp wcel csn cdm c0 wne crn dmsnn0 dm0rn0 necon3bii bitri ) ABBCDAEZF
    ZGHNIZGHAJOGPGNKLM $.

  $( The domain of the singleton of the empty set is empty.  (Contributed by
     NM, 30-Jan-2004.) $)
  dmsn0 $p |- dom { (/) } = (/) $=
    ( c0 csn cdm wceq cvv cxp wcel wn 0nelxp dmsnn0 necon2bbii mpbir ) ABCZADAE
    EFGZHEEINMAAJKL $.

  $( 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 $.

  $( 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 } = (/) ) $=
    ( c0 wcel csn cdm wne cvv cxp wn dmsnn0 0nelelxp sylbir necon4ai ) BACZADEZ
    BOBFAGGHCNIAJGGAKLM $.

  $( A singleton is a relation iff it has a nonempty domain.  (Contributed by
     NM, 25-Sep-2013.)  Make hypothesis an antecedent.  (Revised by BJ,
     12-Feb-2022.) $)
  relsn2 $p |- ( A e. V -> ( Rel { A } <-> dom { A } =/= (/) ) ) $=
    ( wcel csn wrel cvv cxp cdm c0 wne relsng dmsnn0 bitrdi ) ABCADZEAFFGCNHIJA
    BKALM $.

  ${
    $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
      ZDABGZHZIZAHZUJDJZEJZGZUKKZELZUOAKZUOUMFZUOUNFUJUSUTURUTEUOUPABDMZEMNOUTU
      OBGZUKKZUJUSUOABPURVDEBCUPBKUQVCUKUPBUOQRSTUAVAUQULFZELUSEUOULVBUBVEUREUQ
      UKUOUPUCUDUEUFDAUGUHUI $.

    $( The domain of a singleton of an ordered pair is a subset of the
       singleton of the first member (with no sethood assumptions on ` B ` ).
       (Contributed by Mario Carneiro, 30-Apr-2015.) $)
    dmsnopss $p |- dom { <. A , B >. } C_ { A } $=
      ( cvv wcel cop csn cdm wss dmsnopg eqimss syl wn opprc2 sneqd dmeqd dmsn0
      wceq c0 eqtrdi 0ss eqsstrdi pm2.61i ) BCDZABEZFZGZAFZHZUCUFUGQUHABCIUFUGJ
      KUCLZUFRUGUIUFRFZGRUIUEUJUIUDRABMNOPSUGTUAUB $.

    $( 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 >. } $=
      ( vx vy cop csn ccnv relcnv cv wcel vex opelcnv wceq wa opth 3bitr4i opex
      ancom elsn bitri eqrelriiv ) CDABEZFZGZGZBAEZFZGZUDHUGHCIZDIZEZUEJUJUIEZU
      DJZUKUHJZUIUJUDCKZDKZLUKUCJZULUGJZUMUNUKUBMZULUFMZUQURUIAMZUJBMZNVBVANUSU
      TVAVBRUIUJABUOUPOUJUIBAUPUOOPUKUBUIUJQSULUFUJUIQSPUJUIUCUPUOLUIUJUGUOUPLP
      TUA $.

    $( The domain of the singleton of the singleton of a singleton.
       (Contributed by NM, 15-Sep-2004.)  (Revised by Mario Carneiro,
       26-Apr-2015.) $)
    dmsnsnsn $p |- dom { { { A } } } = { A } $=
      ( vx cvv wcel csn cdm wceq cv cop vex opid sneq sneqd eqtrid dmeqd dmsnop
      eqeq12d vtoclg wn c0 0ex snid dmsn0el ax-mp snprc biimpi 3eqtr4a pm2.61i
      ) ACDZAEZEZEZFZUJGZBHZUOIZEZFZUOEZGUNBACUOAGZURUMUSUJUTUQULUTUPUKUTUPUSEU
      KUOBJZKUTUSUJUOALZMNMOVBQUOUOVAPRUISZTEZEZFZTUMUJTVDDVFTGTUAUBVDUCUDVCULV
      EVCUKVDVCUJTVCUJTGAUEUFZMMOVGUGUH $.
  $}

  $( 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 $.

  $( Converse of a singleton of an ordered pair.  (Contributed by NM,
     23-Jan-2015.)  (Proof shortened by BJ, 12-Feb-2022.) $)
  cnvsng $p |-
            ( ( A e. V /\ B e. W ) -> `' { <. A , B >. } = { <. B , A >. } ) $=
    ( wcel cop csn ccnv cnvcnvsn wrel wceq relsnopg ancoms dfrel2 sylib eqtr3id
    wa ) ACEZBDEZQZABFGHBAFGZHHZUABAITUAJZUBUAKSRUCBADCLMUANOP $.

  ${
    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, ~ op1stb for an alternate version, and ~ op1st for
       the preferred 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.)  (Proof
       shortened by BJ, 12-Feb-2022.) $)
    cnvsn $p |- `' { <. A , B >. } = { <. B , A >. } $=
      ( cvv wcel cop csn ccnv wceq cnvsng mp2an ) AEFBEFABGHIBAGHJCDABEEKL $.

    $( Extract the second member of an ordered pair.  Theorem 5.12(ii) of
       [Monk1] p. 52.  (See ~ op1stb to extract the first member, ~ op2nda for
       an alternate version, and ~ op2nd for the preferred version.)
       (Contributed by NM, 25-Nov-2003.) $)
    op2ndb $p |- |^| |^| |^| `' { <. A , B >. } = B $=
      ( cop csn ccnv cint cnvsn inteqi opex intsn eqtri op1stb ) ABEFGZHZHZHBAE
      ZHZHBQSPRPRFZHROTABCDIJRBAKLMJJBADCNM $.

    $( Extract the second member of an ordered pair.  (See ~ op1sta to extract
       the first member, ~ op2ndb for an alternate version, and ~ op2nd for the
       preferred 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 $.
  $}

  $( Swap the members of an ordered pair.  (Contributed by NM, 14-Dec-2008.)
     (Revised by Mario Carneiro, 30-Aug-2015.) $)
  opswap $p |- U. `' { <. A , B >. } = <. B , A >. $=
    ( cvv wcel wa cop ccnv cuni wceq cnvsng unieqd opex unisn eqtrdi wn c0 uni0
    csn opprc sneqd cnveqd cnvsn0 ancom sylnbi 3eqtr4a pm2.61i ) ACDZBCDZEZABFZ
    RZGZHZBAFZIUIUMUNRZHUNUIULUOABCCJKUNBALMNUIOZPHPUMUNQUPULPUPULPRZGPUPUKUQUP
    UJPABSTUAUBNKUIUHUGEUNPIUGUHUCBASUDUEUF $.

  ${
    $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 wcel cop wa wex 19.41v vex opelresi opelcnv
      3bitr4i elima3 anbi2ci bianass exbii anbi1i elin eqriv ) DCAFZGZBHZCGZBHZ
      AIZEJZBKZUMDJZLZUHKZMZENZUOUKKZUOAKZMZUOUIKUOULKUNUPUJKZMZVAMZENVDENZVAMU
      SVBVDVAEOURVEEUQVCVAUNUOUMLZUGKVAVGCKZMUQVCVAMAUOUMCEPZQUMUOUGVIDPZRVCVHV
      AUMUOCVIVJRUASUBUCUTVFVAEUOUJBVJTUDSEUOUHBVJTUOUKAUESUF $.
  $}

  $( 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 $.

  $( A restriction by an arbitrary set is a restriction by its domain.
     (Contributed by AV, 16-Nov-2020.) $)
  resresdm $p |- ( F = ( E |` A ) -> F = ( E |` dom F ) ) $=
    ( cres wceq cdm id dmeq reseq2d resdmres eqtr2di eqtrd ) CBADZEZCMBCFZDZNGN
    PBMFZDMNOQBCMHIBAJKL $.

  $( 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 $.

  $( Subset relationship for the domain of a restriction.  (Contributed by
     Scott Fenton, 9-Aug-2024.) $)
  resdmss $p |- dom ( A |` B ) C_ B $=
    ( cres cdm cin dmres inss1 eqsstri ) ABCDBADZEBABFBIGH $.

  $( Distributive law for restriction over difference.  (Contributed by
     BTernaryTau, 15-Aug-2024.) $)
  resdifdi $p |- ( A |` ( B \ C ) ) = ( ( A |` B ) \ ( A |` C ) ) $=
    ( cdif cres cvv cxp cin df-res difxp1 ineq2i indifdi 3eqtri difeq12i eqtr4i
    ) ABCDZEZABFGZHZACFGZHZDZABEZACEZDQAPFGZHARTDZHUBAPIUEUFABCFJKARTLMUCSUDUAA
    BIACINO $.

  $( Distributive law for restriction over difference.  (Contributed by
     BTernaryTau, 15-Aug-2024.) $)
  resdifdir $p |- ( ( A \ B ) |` C ) = ( ( A |` C ) \ ( B |` C ) ) $=
    ( cdif cvv cxp cin cres indifdir df-res difeq12i 3eqtr4i ) ABDZCEFZGANGZBNG
    ZDMCHACHZBCHZDABNIMCJQORPACJBCJKL $.

  ${
    $d x y C $.  $d y A $.  $d y B $.  $d y F $.  $d x V $.
    dmmpt.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 19.42v dfclel
      cres exbii bicomi anbi2i abbii rnopab df-rab 3eqtr4i ) EHZDIAJBKZGJZCLZMZ
      GANZDIZCDKZABOZVAVFDVAVEAGNZHVFEVJEABCUBVJFAGBCUCPUDVEAGUAPUEVGVFDUMZQZVI
      VFDUFVLVCDKZVEMZGANZQZVIVKVOVEGADUGUHVNGRZASVBVHMZASVPVIVQVRAVQVBVDVMMZMZ
      GRZVRVNVTGVNVEVMMVTVMVEUIVBVDVMUJTUNWAVBVSGRZMVRVBVSGUKWBVHVBVHWBGCDULUOU
      PTTUQVNGAURVHABUSUTPPP $.

    $( 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 $=
      ( cvv wcel cdm dmmpt ssrab3 ) CFGABDHABCDEIJ $.
  $}

  ${
    $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 A x $.
    rnmpt0f.1 $e |- F/ x ph $.
    rnmpt0f.2 $e |- ( ( ph /\ x e. A ) -> B e. V ) $.
    rnmpt0f.3 $e |- F = ( x e. A |-> B ) $.
    $( The range of a function in maps-to notation is empty if and only if its
       domain is empty.  (Contributed by Glauco Siliprandi, 8-Apr-2021.) $)
    rnmpt0f $p |- ( ph -> ( ran F = (/) <-> A = (/) ) ) $=
      ( c0 wceq cmpt cdm crn wcel wral cv eqcomd eqeq1d a1i ex dmmptg wb dm0rn0
      ralrimi syl rneqi 3bitrrd ) ACJKBCDLZMZJKZUINZJKZENZJKACUJJAUJCADFOZBCPUJ
      CKAUOBCGABQCOUOHUAUEBCDFUBUFRSUKUMUCAUIUDTAULUNJAUNULUNULKAEUIIUGTRSUH $.

    rnmptn0.a $e |- ( ph -> A =/= (/) ) $.
    $( The range of a function in maps-to notation is nonempty if the domain is
       nonempty.  (Contributed by Glauco Siliprandi, 8-Apr-2021.) $)
    rnmptn0 $p |- ( ph -> ran F =/= (/) ) $=
      ( crn c0 wceq neneqd rnmpt0f mtbird neqned ) AEKZLARLMCLMACLJNABCDEFGHIOP
      Q $.
  $}

  ${
    $d w x y z A $.  $d w x y z B $.  $d w x y z C $.
    $( 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 el2v wrex eliun opelxp
      rexv opelcnv anbi12i exbii 3bitrri eqrelriiv ) DEBCFZAGCHZAIZJZKZBVBKZLZU
      AZBCUBVFMVEMZAGAGVEUCVGVAGNVCVDUDUEUFDIZEIZOZUSNZVHVAOCNZVAVIOBNZTZAPZVJV
      FNZVKVOUGDEAVHVIBCGGUHUIVPVJVENZAGUJVQAPVOAVJGVEUKVQAUMVQVNAVQVHVCNZVIVDN
      ZTVNVHVIVCVDULVRVLVSVMVRVAVHOUTNVLUTVAVHAQZDQZRVAVHCVTWAUNSBVAVIVTEQRUOSU
      PUQSUR $.

    $( 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 wceq wbr wb eliniseg elv brelrn elimasn
      dfco2 opeldm anim12ci adantl exlimivv elxp elin ssel syl5 pm4.71rd exbidv
      3imtr4i rexv df-rex eqrdv eqtrid ) BUAZCUBZUCZDUDZBCUEAHCUFAIZUGZJZBVQJZU
      HZKZADVTKZABCUPVOEWAWBVOEIZVTLZAHMZWDADMZWCWALWCWBLVOWDANVPDLZWDOZANWEWFV
      OWDWHAVOWDWGWDVPVNLZVOWGWCFIZGIZPUIZWJVRLZWKVSLZOZOZGNFNVPVLLZVPVMLZOZWDW
      IWPWSFGWOWSWLWMWRWNWQWMWJVPCUJZWRWMWTUKACVPWJHFQZULUMWJVPCXAAQZUNRWNVPWKP
      BLWQBVPWKXBGQZUOVPWKBXBXCUQRURUSUTFGWCVRVSVAVPVLVMVBVGVNDVPVCVDVEVFWDAVHW
      DADVISAWCHVTTAWCDVTTSVJVK $.

    $( 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 brresi baib syl56 pm5.32d exbidv opabbidv 3eqtr4g ) BGZCHZDIZEIZBJZU
      LFIZACKZJZLZEMZDFNUMULUNAJZLZEMZDFNUOBOABOUJURVADFUJUQUTEUJUMUPUSUMULUIPU
      JULCPZUPUSQUKULBDRERSUICULUAUPVBUSCULUNAFRUBUCUDUEUFUGDFEUOBTDFEABTUH $.

    $( 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 wcel wbr wa wex anbi2i 19.42v brresi
      vex brco anbi1i anass bitr2i exbii 3bitr2i 3bitr4i eqbrriv ) DEABGZCHZABC
      HZGZUHCIAUJJDKZCLZULEKZUHMZNZULFKZUJMZUQUNAMZNZFOZULUNUIMULUNUKMUPUMULUQB
      MZUSNZFOZNUMVCNZFOVAUOVDUMFULUNABDSZESZTPUMVCFQVEUTFUTUMVBNZUSNVEURVHUSCU
      LUQBFSRUAUMVBUSUBUCUDUECULUNUHVGRFULUNAUJVFVGTUFUG $.

    $( Image of the composition of two classes.  (Contributed by Jason
       Orendorff, 12-Dec-2006.)  (Proof shortened by Wolf Lammen,
       16-May-2025.) $)
    imaco $p |- ( ( A o. B ) " C ) = ( A " ( B " C ) ) $=
      ( vx vy vz ccom cima cv wbr wrex wcel wa wex df-rex vex elima brco rexbii
      exbii rexcom4 r19.41v 3bitri anbi1i 3bitr4i 3bitr4ri eqriv ) DABGZCHZABCH
      ZHZEIZDIZAJZEUJKULUJLZUNMZENZUMUKLUMUILZUNEUJOEUMAUJDPZQFIZUMUHJZFCKZUTUL
      BJZFCKZUNMZENZURUQVBVCUNMZENZFCKVGFCKZENVFVAVHFCEUTUMABFPUSRSVGFECUAVIVEE
      VCUNFCUBTUCFUMUHCUSQUPVEEUOVDUNFULBCEPQUDTUEUFUG $.

    $( The range of the composition of two classes.  (Contributed by NM,
       12-Dec-2006.)  (Proof shortened by Peter Mazsa, 2-Oct-2022.) $)
    rnco $p |- ran ( A o. B ) = ran ( A |` ran B ) $=
      ( vy vx vz ccom crn cres cv wbr wex wcel vex brco exbii excom elrn anbi1i
      wa brresi 19.41v 3bitr4ri 3bitri 3bitr4i eqriv ) CABFZGZABGZHZGZDIZCIZUFJ
      ZDKZEIZULUIJZEKZULUGLULUJLUNUKUOBJZUOULAJZSZEKZDKUTDKZEKUQUMVADEUKULABDMC
      MZNOUTDEPVBUPEUOUHLZUSSURDKZUSSUPVBVDVEUSDUOBEMQRUHUOULAVCTURUSDUAUBOUCDU
      LUFVCQEULUIVCQUDUE $.
  $}

  $( 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 $.

  $( A composition of two relations is empty iff there is no overlap between
     the range of the second and the domain of the first.  Useful in
     combination with ~ coundi and ~ coundir to prune meaningless terms in the
     result.  (Contributed by Stefan O'Rear, 8-Oct-2014.) $)
  coeq0 $p |- ( ( A o. B ) = (/) <-> ( dom A i^i ran B ) = (/) ) $=
    ( ccom c0 wceq crn cres cdm wrel wb relco relrn0 ax-mp eqeq1i relres reldm0
    cin rnco dmres incom eqtri 3bitr3i 3bitri ) ABCZDEZUDFZDEZABFZGZFZDEZAHZUHQ
    ZDEZUDIUEUGJABKUDLMUFUJDABRNUIDEZUIHZDEZUKUNUIIZUOUQJAUHOZUIPMURUOUKJUSUILM
    UPUMDUPUHULQUMAUHSUHULTUANUBUC $.

  ${
    $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 cop wrex wbr wa wex eliun bitr4i
      reliun a1i mprgbir df-br rexbii 3bitr4i anbi1i r19.41v rexcom4 vex opelco
      exbii eqrelriiv ) EFBADCHZIZADBCIZHZBUNJUQKUPKZADADUPUAURALDMBCJUBUCELZFL
      ZNZUOMZVAUPMZADOZVAUQMUSGLZUNPZVEUTBPZQZGRZUSVECPZVGQZGRZADOZVBVDVIVKADOZ
      GRVMVHVNGVHVJADOZVGQVNVFVOVGUSVENZUNMVPCMZADOVFVOAVPDCSUSVEUNUDVJVQADUSVE
      CUDUEUFUGVJVGADUHTULVKAGDUITGUSUTBUNEUJZFUJZUKVCVLADGUSUTBCVRVSUKUEUFAVAD
      UPSTUM $.
  $}

  $( 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 cop wcel wbr wex br0 intnanr nex vex opelco
      cv wa mtbir noel 2false eqrelriiv ) BCAEFZEAEGHBRZCRZIZUDJZUGEJUHUEDRZEKZ
      UIUFAKZSZDLULDUJUKUEUIMNODUEUFAEBPCPQTUGUAUBUC $.

    $( 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 equsexvw df-br eqrelriv mpan ) AEFZGAGUH
      AHAEIBCUHABJZCJZKZUHLZUIUJAMZUKALULUIDJZEMZUNUJAMZRZDNZUMDUIUJAEBOCOPURUN
      UIHZUPRZDNUMUQUTDUOUSUPUOUIUNHUSUIUNDOSBDQTUAUBUPUMDBUNUIUJAUCUDTTUIUJAUE
      TUFUG $.

    $( 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 wceq ccom dfrel2 cnvco relcnv ax-mp cnveqi eqtr3i coeq2 coeq1
      cid coi1 cnvi sylan9eq mpan2 id 3eqtr3a sylbi ) ABACZCZADZNAEZADAFUDNCZUC
      EZUCUEAUBNEZCUGUCUBNGUHUBUBBUHUBDAHUBOIJKUDUFNDZUGUEDPUDUIUGUFAEUEUCAUFLU
      FNAMQRUDSTUA $.
  $}

  $( 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 $.
  $}

  $( General form of ~ relcnvtr .  (Contributed by Peter Mazsa,
     17-Oct-2023.) $)
  relcnvtrg $p |- ( ( Rel R /\ Rel S /\ Rel T ) ->
                    ( ( R o. S ) C_ T <-> ( `' S o. `' R ) C_ `' T ) ) $=
    ( wrel w3a ccom wss ccnv cnvco cnvss eqsstrrid sseq1 dfrel2 biimpi 3ad2ant1
    wceq wi 3ad2ant2 coeq12d 3ad2ant3 sseq12d biimpcd biimtrdi mpsyl impbid2
    com12 ) ADZBDZCDZEZABFZCGZBHZAHZFZCHZGZULUOUKHUPABIUKCJKUQUJULUOHZUNHZUMHZF
    ZPZUQURUPHZGZUJULQZUMUNIUOUPJVBVDVAVCGZVEURVAVCLUJVFULUJVAUKVCCUJUSAUTBUGUH
    USAPZUIUGVGAMNOUHUGUTBPZUIUHVHBMNRSUIUGVCCPZUHUIVICMNTUAUBUCUDUFUE $.

  $( A relation is transitive iff its converse is transitive.  (Contributed by
     FL, 19-Sep-2011.)  (Proof shortened by Peter Mazsa, 17-Oct-2023.) $)
  relcnvtr $p |- ( Rel R ->
     ( ( R o. R ) C_ R <-> ( `' R o. `' R ) C_ `' R ) ) $=
    ( wrel w3a ccom wss ccnv wb 3anidm relcnvtrg sylbir ) ABZKKKCAADAEAFZLDLEGK
    HAAAIJ $.

  ${
    $d x y A $.
    $( A relation is included in the Cartesian product of its domain and range.
       Exercise 4.12(t) of [Mendelson] p. 235.  (Contributed by NM,
       3-Aug-1994.)  (Proof shortened by SN, 23-Dec-2024.) $)
    relssdmrn $p |- ( Rel A -> A C_ ( dom A X. ran A ) ) $=
      ( vx vy wrel cdm crn cxp id cv cop wcel vex opeldm opelrn opelxpd relssdv
      wi a1i ) ADZBCAAEZAFZGZSHBIZCIZJZAKZUEUBKQSUFUCUDTUAUCUDABLZCLZMUCUDAUGUH
      NORP $.
    $( $j usage 'relssdmrn' avoids 'ax-12'; $)

    $( Obsolete version of ~ relssdmrn as of 23-Dec-2024.  (Contributed by NM,
       3-Aug-1994.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    relssdmrnOLD $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 $.
  $}

  $( If the ` R ` -image of a class ` A ` is a subclass of ` B ` , then the
     restriction of ` R ` to ` A ` is a subset of the Cartesian product of
     ` A ` and ` B ` .  (Contributed by RP, 24-Dec-2019.) $)
  resssxp $p |- ( ( R " A ) C_ B <-> ( R |` A ) C_ ( A X. B ) ) $=
    ( cima wss cres crn cdm cxp df-ima sseq1i dmres inss1 eqsstri biantrur wrel
    wa cin relres sstrdi relssdmrn xpss12 sstrid dmss dmxpss rnss rnxpss impbii
    ax-mp jca 3bitri ) CADZBECAFZGZBEZUMHZAEZUOQZUMABIZEZULUNBCAJKUQUOUPACHZRAC
    ALAVAMNOURUTURUMUPUNIZUSUMPUMVBECASUMUAUIUPAUNBUBUCUTUQUOUTUPUSHAUMUSUDABUE
    TUTUNUSGBUMUSUFABUGTUJUHUK $.

  $( 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 Cartesian 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 $.

  $( 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 ) ) $=
    ( wrel cdm wa cvv cxp wss df-rel anbi2i crn relssdmrn xpss12 mpan2 sylan9ss
    ssv xpss sstr sylibr dmss c0 wne wceq vn0 dmxp ax-mp sseqtrdi impbii bitri
    jca ) ABZACZBZDUJUKEEFZGZDZAUMEFZGZULUNUJUKHIUOUQUJUNAUKAJZFZUPAKUNUREGUSUP
    GUROUKUMURELMNUQUJUNUQAUMGZUJUQUPUMGUTUMEPAUPUMQMAHRUQUKUPCZUMAUPSETUAVAUMU
    BUCUMEUDUEUFUIUGUH $.

  ${
    $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 cdm crn cun wss wceq dmrnssfld simpr sylbir cores
    wa unss mp2b coi2 eqtrid ) ABCADDZEAFZCAFZAAGZAHZIUAJZUEUAJZUBUCKALUFUDUAJZ
    UGPUGUDUEUAQUHUGMNCAUAORAST $.

  $( Composition with the identity relation restricted to a relation's field.
     (Contributed by FL, 8-May-2011.)  (Proof shortened by OpenAI,
     3-Jul-2020.) $)
  relcoi1 $p |- ( Rel R -> ( R o. ( _I |` U. U. R ) ) = R ) $=
    ( wrel cid cuni cres ccom coires1 relresfld eqtrid ) ABACADDZEFAJEAAJGAHI
    $.

  $( 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 $.

  $( The double class union of a nonempty Cartesian product is the union of it
     members.  (Contributed by NM, 17-Sep-2006.) $)
  unixp $p |- ( ( A X. B ) =/= (/) -> U. U. ( A X. B ) = ( A u. B ) ) $=
    ( cxp c0 wne cuni cdm crn cun wrel wceq relxp relfld ax-mp xpeq2 xp0 eqtrdi
    necon3i xpeq1 0xp dmxp rnxp uneq12 syl2an syl2anc eqtrid ) ABCZDEZUGFFZUGGZ
    UGHZIZABIZUGJUIULKABLUGMNUHBDEZADEZULUMKZBDUGDBDKUGADCDBDAOAPQRADUGDADKUGDB
    CDADBSBTQRUNUJAKUKBKUPUOABUAABUBUJAUKBUCUDUEUF $.

  ${
    $d x y z A $.  $d x y z B $.
    $( A Cartesian product is empty iff its union is empty.  (Contributed by
       NM, 20-Sep-2006.) $)
    unixp0 $p |- ( ( A X. B ) = (/) <-> U. ( A X. B ) = (/) ) $=
      ( vz vx vy cxp c0 wceq cuni unieq uni0 eqtrdi wne cv wex n0 cop vex sylbi
      wcel elxp3 wss elssuni opnzi ssn0 sylancl adantl exlimivv exlimiv necon4i
      wa impbii ) ABFZGHZUMIZGHUNUOGIGUMGJKLUMGUOGUMGMCNZUMTZCOUOGMZCUMPUQURCUQ
      DNZENZQZUPHZVAUMTZUKZEODOURDEUPABUAVDURDEVCURVBVCVAUOUBVAGMURVAUMUCUSUTDR
      ERUDVAUOUEUFUGUHSUISUJUL $.
  $}

  $( Field of a Cartesian square.  (Contributed by FL, 10-Oct-2009.) $)
  unixpid $p |- U. U. ( A X. A ) = A $=
    ( c0 wceq cxp cuni xpeq1 eqtrdi wi unieq unieqd uni0 unieqi eqtri wa expcom
    0xp eqtr eqcoms syl5com wne sylancl mpcom wn df-ne cun unixp unidm sylancbr
    xpnz sylbi pm2.61i ) ABCZAADZEZEZACZUMBCZULUPULUMBADBABAFAPGUQUOBEZEZCZUSBC
    ZULUPHUQUNURUMBIJUSURBURBKLKMUTVANUOBCZULUPUOUSBQVBUPHBAVBBACUPUOBAQORSUAUB
    ULUCABTZVCUPABUDZVDVCVCNUMBTZUPAAUIVEUOAAUEAAAUFAUGGUJUHUK $.

  ${
    $d x y A $.  $d x y B $.
    $( Restriction of a class to a singleton.  (Contributed by Mario Carneiro,
       28-Dec-2014.)  (Proof shortened by Peter Mazsa, 2-Oct-2022.) $)
    ressn $p |- ( A |` { B } ) = ( { B } X. ( A " { B } ) ) $=
      ( vx vy csn cres cima cxp relres relxp cv wcel cop wa elimasn elsni sneqd
      vex imaeq2d eleq2d bitr3id pm5.32i opelresi opelxp 3bitr4i eqrelriiv ) CD
      ABEZFZUGAUGGZHZAUGIUGUIJCKZUGLZUKDKZMZALZNULUMUILZNUNUHLUNUJLULUOUPUOUMAU
      KEZGZLULUPAUKUMCRDRZOULURUIUMULUQUGAULUKBUKBPQSTUAUBUGUKUMAUSUCUKUMUGUIUD
      UEUF $.
  $}

  ${
    $d A a b x $.  $d B a b $.
    $( The converse of an intersection is the intersection of the converse.
       (Contributed by FL, 15-Oct-2012.) $)
    cnviin $p |- ( A =/= (/) -> `' |^|_ x e. A B = |^|_ x e. A `' B ) $=
      ( va vb ciin ccnv wrel relcnv cvv wss wral df-rel cv cop wcel eliin ax-mp
      wb opex c0 wne wceq cxp wrex mpbi rgenw r19.2z mpan2 iinss syl sylibr vex
      opelcnv ralbii bitri 3bitr4i eqrelriv sylancr ) BUAUBZABCFZGZHABCGZFZHZVB
      VDUCVAIUTVDJJUDZKZVEUTVCVFKZABUEZVGUTVHABLVIVHABVCHVHCIVCMUFUGVHABUHUIABV
      CVFUJUKVDMULDEVBVDENZDNZOZVAPZVLCPZABLZVKVJOZVBPVPVDPZVLJPVMVOSVJVKTAVLBC
      JQRVKVJVADUMZEUMZUNVQVPVCPZABLZVOVPJPVQWASVKVJTAVPBVCJQRVTVNABVKVJCVRVSUN
      UOUPUQURUS $.
  $}

  ${
    $d x y z A $.  $d x y z R $.
    $( The converse of a partial order is a partial order.  (Contributed by NM,
       15-Jun-2005.) $)
    cnvpo $p |- ( R Po A <-> `' R Po A ) $=
      ( vx vy vz cv wbr wn wa wi wral wpo r19.26 brcnv ralbii bitr2i c0 3bitr4i
      vex ralcom weq id breq12d bitrid notbid cbvralvw anbi12ci imbi12i anbi12i
      ccnv ralidm wb wceq rzal 2thd wne r19.3rzv ralbidv pm2.61ine anbi1i bitri
      df-po ) CFZVCBGZHZVCDFZBGZVFEFZBGZIZVCVHBGZJZIEAKZDAKCAKZVHVHBUJZGZHZVHVF
      VOGZVFVCVOGZIZVHVCVOGZJZIZCAKZDAKEAKZABLAVOLVMCAKZDAKWDEAKZDAKVNWEWFWGDAV
      ECAKZVLEAKZIZCAKZWCEAKZCAKWFWGWJWLCAWLVQEAKZWBEAKZIWJVQWBEAMWMWHWNWIVQVEE
      CAECUAZVPVDVPVHVHBGWOVDVHVHBESZWPNWOVHVCVHVCBWOUBZWQUCUDUEUFWBVLEAVTVJWAV
      KVRVIVSVGVHVFBWPDSZNVFVCBWRCSZNUGVHVCBWPWSNUHOUIPOVEEAKZWIIZCAKZWHCAKZWIC
      AKZIZWFWKXBWTCAKZXDIXEWTWICAMXFXCXDXCWHXFVECAUKWHXFULAQAQUMWHXFVECAUNWTCA
      UNUOAQUPVEWTCAVEEAUQURUSPUTVAVMXACAVEVLEAMOWHWICAMRWCECAATROVMCDAATWDEDAA
      TRCDEABVBEDCAVOVBR $.

    $( The converse of a strict order relation is a strict order relation.
       (Contributed by NM, 15-Jun-2005.) $)
    cnvso $p |- ( R Or A <-> `' R Or A ) $=
      ( vx vy wpo cv wbr weq w3o wral wa ccnv wor cnvpo ralcom vex brcnv equcom
      3orbi123i df-so 2ralbii bitr4i anbi12i 3bitr4i ) ABEZCFZDFZBGZCDHZUGUFBGZ
      IZDAJCAJZKABLZEZUGUFUMGZDCHZUFUGUMGZIZCAJDAJZKABMAUMMUEUNULUSABNULUKCAJDA
      JUSUKCDAAOURUKDCAAUOUHUPUIUQUJUGUFBDPZCPZQDCRUFUGBVAUTQSUAUBUCCDABTDCAUMT
      UD $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x y z C $.
    $( Composition of two Cartesian products.  (Contributed by Thierry Arnoux,
       17-Nov-2017.) $)
    xpco $p |- ( B =/= (/) -> ( ( B X. C ) o. ( A X. B ) ) = ( A X. C ) ) $=
      ( vx vy vz c0 wne cv cxp wbr wa wex copab wcel ccom biimpi biantrurd brxp
      n0 ancom anbi1i anbi12i anandi 3bitr4i exbii 19.41v bitr2i opabbidv df-co
      bitr2di df-xp 3eqtr4g ) BGHZDIZEIZABJZKZUPFIZBCJZKZLZEMZDFNUOAOZUSCOZLZDF
      NUTUQPACJUNVCVFDFUNVFUPBOZEMZVFLZVCUNVHVFUNVHEBTQRVCVGVFLZEMVIVBVJEVDVGLZ
      VGVELZLVGVDLZVLLVBVJVKVMVLVDVGUAUBURVKVAVLUOUPABSUPUSBCSUCVGVDVEUDUEUFVGV
      FEUGUHUKUIDFEUTUQUJDFACULUM $.
  $}

  $( Composition of two Cartesian squares.  (Contributed by Thierry Arnoux,
     14-Jan-2018.) $)
  xpcoid $p |- ( ( A X. A ) o. ( A X. A ) ) = ( A X. A ) $=
    ( cxp ccom wceq c0 co01 sqxpeqd 0xp eqtrdi coeq12d 3eqtr4a xpco pm2.61ine
    id ) AABZOCZODAEAEDZEECEPOEFQOEOEQOEEBEQAEQNGEHIZRJRKAAALM $.

  ${
    $d A x y $.  $d V y $.  $d X x y $.  $d Z x y $.
    $( Membership in a Cartesian product with a singleton.  (Contributed by
       Thierry Arnoux, 10-Apr-2020.)  (Proof shortened by JJ, 14-Jul-2021.) $)
    elsnxp $p |- ( X e. V -> ( Z e. ( { X } X. A )
      <-> E. y e. A Z = <. X , y >. ) ) $=
      ( vx wcel csn cxp cv cop wceq wrex wa wex elxp df-rex an13 exbii bitr4i
      elsni opeq1d eqeq2d biimpa reximi exlimiv sylbi snidg opelxpi sylan eleq1
      sylbir syl5ibrcom rexlimdva impbid2 ) DCGZEDHZBIZGZEDAJZKZLZABMZUSEFJZUTK
      ZLZVDUQGZUTBGZNNZAOZFOVCFAEUQBPVJVCFVJVGVFNZABMZVCVLVHVKNZAOVJVKABQVIVMAV
      FVGVHRSTVKVBABVGVFVBVGVEVAEVGVDDUTVDDUAUBUCUDUEULUFUGUPVBUSABUPVHNUSVBVAU
      RGZUPDUQGVHVNDCUHDUTUQBUIUJEVAURUKUMUNUO $.
  $}

  ${
    $d X a b p q $.  $d X p q x y $.  $d Y a b p q $.  $d Y p q x y $.
    $d ps a b q x y $.  $d ch p $.
    reu3op.a $e |- ( p = <. a , b >. -> ( ps <-> ch ) ) $.
    $( There is a unique ordered pair fulfilling a wff iff there are uniquely
       two sets fulfilling a corresponding wff.  (Contributed by AV,
       1-Jul-2023.) $)
    reu3op $p |- ( E! p e. ( X X. Y ) ps
                  <-> ( E. a e. X E. b e. Y ch
                     /\ E. x e. X E. y e. Y A. a e. X A. b e. Y
                                     ( ch -> <. x , y >. = <. a , b >. ) ) ) $=
      ( vq wrex wi wral wa cv cop wceq rexxp wcel cxp wreu eqeq2 imbi2d ralbidv
      weq reu3 eqeq1 imbi12d ralxp wb eqcom a1i 2ralbidva bitrid 2rexbiia bitri
      anbi12i ) AGEFUAZUBAGUSLZAGKUFZMZGUSNZKUSLZOBIFLHELZBCPZDPZQZHPZIPZQZRZMZ
      IFNHENZDFLCELZOAGKUSUGUTVEVDVOABGHIEFJSVDAGPZVHRZMZGUSNZDFLCELVOVCVSKCDEF
      KPZVHRZVBVRGUSWAVAVQAVTVHVPUCUDUESVSVNCDEFVSBVKVHRZMZIFNHENVFETVGFTOZVNVR
      WCGHIEFVPVKRABVQWBJVPVKVHUHUIUJWDWCVMHIEFWDVIETVJFTOOZWBVLBWBVLUKWEVKVHUL
      UMUDUNUOUPUQURUQ $.

    $d X a b c d p q x y $.  $d X p q w $.  $d Y c d $.  $d Y w $.
    $d c d ps $.  $d c ch d q $.  $d c d p q th $.  $d ps q w $.
    reuop.x $e |- ( p = <. x , y >. -> ( ps <-> th ) ) $.
    $( There is a unique ordered pair fulfilling a wff iff there are uniquely
       two sets fulfilling a corresponding wff.  (Contributed by AV,
       23-Jun-2023.) $)
    reuop $p |- ( E! p e. ( X X. Y ) ps
                  <-> E. a e. X E. b e. Y ( ch /\ A. x e. X A. y e. Y
                                     ( th -> <. x , y >. = <. a , b >. ) ) ) $=
      ( vq cv wsbc wceq wi wa wrex wcel vw vc cxp wreu wral cop nfsbc1v sbceq1a
      vd dfsbcq reu8nf elxp2 biimpcd adantr imp opelxpi wb eqeq2 imbi12d adantl
      rspcdv sbcie pm2.27 sylbir eqcom imbitrrdi com12 eqcoms imbi2d syl5ibrcom
      opex a1d syl6 expimpd imp4c impcom ralrimivva ex reximdvva sylbi rexlimiv
      jca simprl nfim opeq1 eqeq1d opeq2 rspc2 ad2antlr sbcop expcom rexlimdvva
      nfv impd biimpi impel ralrimiva nfcv nfralw nfan ralbidv anbi12d syl12anc
      eqeq1 rspce rexlimivv impbii bitri ) AHFGUCZUDAAHMNZOZHNZXJPZQZMXIUEZRZHX
      ISZBCDNZENZUFZINZJNZUFZPZQZEGUEDFUEZRZJGSIFSZAXKAHUANZOHMUAXIAHXJUGZAHYIU
      GAHYIUHAHYIXJUJUKXQYHXPYHHXIXLXITXLYCPZJGSIFSZXPYHQIJXLFGULXPYLYHXPYKYGIJ
      FGXPYAFTYBGTRZRZYKYGYNYKRZBYFYNYKBXPYKBQZYMAYPXOYKABKUMUNUNUOYOYEDEFGXRFT
      XSGTRZYOYEYQXPYMYKYEYQAXOYMYKYEQZQZYQARXOAHXTOZXLXTPZQZYSYQXOUUBQAYQXNUUB
      MXTXIXRXSFGUPXJXTPZXNUUBUQYQUUCXKYTXMUUAAHXJXTUJXJXTXLURUSUTVAUNUUBYRYMUU
      BYEYKCXTXLPZQCUUBUUDCUUBUUAUUDCYTUUBUUAQACHXTXRXSVKLVBYTUUAVCVDXTXLVEVFVG
      YKYDUUDCYDUUDUQYCXLYCXLXTURVHVIVJVLVMVNVOVPVQWBVRVSVGVTWAYGXQIJFGYMYGXQYM
      YGRZYCXITZBXKYCXJPZQZMXIUEZXQYMUUFYGYAYBFGUPUNYMBYFWCUUEUUHMXIUUEXJUBNZUI
      NZUFZPZUIGSUBFSZUUHXJXITZUUEUUMUUHUBUIFGUUEUUJFTUUKGTRZRUUHUUMAHUULOZYCUU
      LPZQZUUPUUEUUSUUPYMYGUUSYMUUPYGUUSQYMUUPRZBYFUUSUUTBRYFCDUUJOZEUUKOZUULYC
      PZQZUUSUUPYFUVDQYMBYEUVDUVAUUJXSUFZYCPZQDEUUJUUKFGUVAUVFDCDUUJUGUVFDWMWDU
      VBUVCEUVAEUUKUGUVCEWMWDXRUUJPZCUVAYDUVFCDUUJUHUVGXTUVEYCXRUUJXSWEWFUSXSUU
      KPZUVAUVBUVFUVCUVAEUUKUHUVHUVEUULYCXSUUKUUJWGWFUSWHWIUUQUVDUURUUQUVDUVCUU
      RUUQUVBUVDUVCQACDEHUBUILWJUVBUVCVCVDYCUULVEVFVGVMVNWKWNVPUUMXKUUQUUGUURAH
      XJUULUJXJUULYCURUSVJWLUUOUUNUBUIXJFGULWOWPWQXPBUUIRHYCXIBUUIHBHWMUUHHMXIH
      XIWRXKUUGHYJUUGHWMWDWSWTYKABXOUUIKYKXNUUHMXIYKXMUUGXKXLYCXJXDVIXAXBXEXCVR
      XFXGXH $.
  $}

  ${
    $d A a b p x y $.  $d B a b p x y $.  $d ph a b x y $.  $d ch p x y $.
    opreu2reurex.a $e |- ( p = <. a , b >. -> ( ph <-> ch ) ) $.
    $( There is a unique ordered pair fulfilling a wff iff there are uniquely
       two sets fulfilling a corresponding wff.  (Contributed by AV,
       24-Jun-2023.)  (Revised by AV, 1-Jul-2023.) $)
    opreu2reurex $p |- ( E! p e. ( A X. B ) ph
                  <-> ( E! a e. A E. b e. B ch /\ E! b e. B E. a e. A ch ) ) $=
      ( vx vy wrex cv cop wceq wi wral wa weq wreu wcel cxp wb eqcom opth bitri
      vex imbi2i a1i 2ralbidva 2rexbiia anbi2i reu3op 2reu4 3bitr4i ) BGDKZFCKZ
      BILZJLZMZFLZGLZMZNZOZGDPFCPZJDKICKZQUPBFIRGJRQZOZGDPFCPZJDKICKZQAECDUASUO
      FCSBFCKGDSQVFVJUPVEVIIJCDUQCTURDTQZVDVHFGCDVDVHUBVKUTCTVADTQQVCVGBVCVBUSN
      VGUSVBUCUTVAUQURFUFGUFUDUEUGUHUIUJUKABIJCDEFGHULBFGIJCDUMUN $.

    $( If there is a unique ordered pair fulfilling a wff, then there is a
       double restricted unique existential qualification fulfilling a
       corresponding wff.  (Contributed by AV, 25-Jun-2023.)  (Revised by AV,
       2-Jul-2023.) $)
    opreu2reu $p |- ( E! p e. ( A X. B ) ph -> E! a e. A E! b e. B ch ) $=
      ( cxp wreu wrex wa opreu2reurex 2rexreu sylbi ) AECDIJBGDKFCJBFCKGDJLBGDJ
      FCJABCDEFGHMBFGCDNO $.
  $}

  ${
    $d R x y z w $.  $d A x y z w $.
    $( Quantifier-free definition of a partial ordering.  (Contributed by Scott
       Fenton, 22-Feb-2013.)  (Proof shortened by Peter Mazsa, 2-Oct-2022.) $)
    dfpo2 $p |- ( R Po A <-> ( ( R i^i ( _I |` A ) ) = (/) /\
           ( ( R i^i ( A X. A ) ) o. ( R i^i ( A X. A ) ) ) C_ R ) ) $=
      ( vx vy vz vw cid cin c0 wceq wa wb wbr wn wi wral wcel wal eleq1 df-br
      wpo cres cxp ccom wss po0 res0 ineq2i in0 eqtri xp0 coeq2i eqsstri pm3.2i
      0ss poeq2 reseq2 ineq2d eqeq1d xpeq2 coeq2d sseq1d anbi12d bibi12d mpbiri
      co02 2th wne cv r19.28zv ralbidv bitrd r19.26 bitrdi disj df-ral cop opex
      df-po bitr4di cvv opelidres notbid imbi12d spcv con2d alrimiv wrel relres
      elv wex elrel mpan ancri weq breq12 anidms spvv breq2 biimpcd impcomd syl
      imbi2d brresi ideq anbi2i 3bitr3ri syl5ibrcom exlimdvv impd impbii bitr4i
      vex syl5 3bitri wrex ralcom r19.23v ralbii bitri anbi12i an4 ancom anbi1i
      brin brxp anandi 3bitr4i anass exbii bitr3i df-rex r19.42v imbi12i 2albii
      brco r2al impexp relco ssrel ax-mp bitr2i 3bitr4g pm2.61ine ) ABUAZBGAUBZ
      HZIJZBAAUCZHZUUJUDZBUEZKZLZAIAIJZUUNIBUAZBGIUBZHZIJZUUJBAIUCZHZUDZBUEZKZL
      UUPUVDBUFUUSUVCUURBIHZIUUQIBGUGUHBUIZUJUVBIBUVBUUJIUDIUVAIUUJUVAUVEIUUTIB
      AUKUHUVFUJULUUJVFUJBUOUMUNVGUUOUUEUUPUUMUVDAIBUPUUOUUHUUSUULUVCUUOUUGUURI
      UUOUUFUUQBAIGUQURUSUUOUUKUVBBUUOUUJUVAUUJUUOUUIUUTBAIAUTURVAVBVCVDVEAIVHZ
      CVIZUVHBMZNZUVHDVIZBMZUVKEVIZBMZKZUVHUVMBMZOZKEAPZDAPZCAPZUVJCAPZUVQEAPZD
      APZCAPZKZUUEUUMUVGUVTUVJUWCKZCAPUWEUVGUVSUWFCAUVGUVSUVJUWBKZDAPUWFUVGUVRU
      WGDAUVJUVQEAVJVKUVJUWBDAVJVLVKUVJUWCCAVMVNCDEABVSUUHUWAUULUWDUUHFVIZUUFQZ
      NZFBPUWHBQZUWJOZFRZUWAFBUUFVOUWJFBVPUWMUVHAQZUVJOZCRZUWAUWMUWPUWMUWOCUWMU
      VIUWNUWLUVIUWNNZOFUVHUVHVQZUVHUVHVRUWHUWRJZUWKUVIUWJUWQUWSUWKUWRBQUVIUWHU
      WRBSUVHUVHBTVTUWSUWIUWNUWSUWIUWRUUFQZUWNUWHUWRUUFSUWTUWNLCUVHAWAWBWJVNWCW
      DWEWFWGUWPUWLFUWPUWIUWKUWIUWHUVKUVMVQZJZEWKDWKZUWIKUWPUWKNZUWIUXCUUFWHUWI
      UXCGAWIDEUWHUUFWLWMWNUWPUXCUWIUXDUWPUXBUWIUXDOZDEUWPUXEUXBUVKAQZDEWOZKZUV
      NNZOZUWPUXFUVKUVKBMZNZOZUXJUWOUXMCDCDWOZUWNUXFUVJUXLUVHUVKASUXNUVIUXKUXNU
      VIUXKLUVHUVKUVHUVKBWPWQWCWDWRUXMUXGUXFUXIUXGUXMUXFUXIOUXGUXLUXIUXFUXGUXKU
      VNUVKUVMUVKBWSWCXCWTXAXBUXBUWIUXHUXDUXIUXBUWIUXAUUFQZUXHUWHUXAUUFSUVKUVMU
      UFMUXFUVKUVMGMZKUXOUXHAUVKUVMGEXMZXDUVKUVMUUFTUXPUXGUXFUVKUVMUXQXEXFXGVTU
      XBUWKUVNUXBUWKUXABQUVNUWHUXABSUVKUVMBTVTWCWDXHXIXJXNWFWGXKUVJCAVPXLXOUWDU
      VODAXPZUVPOZEAPZCAPZUULUWCUXTCAUWCUVQDAPZEAPUXTUVQDEAAXQUYBUXSEAUVOUVPDAX
      RXSXTXSUWNUVMAQZKZUXRKZUVPOZERCRZUVHUVMVQZUUKQZUYHBQZOZERCRZUYAUULUYFUYKC
      EUYEUYIUVPUYJUVHUVKUUJMZUVKUVMUUJMZKZDWKZUXFUYDUVOKZKZDWKZUYIUYEUYOUYRDUY
      OUVLUVHUVKUUIMZKZUVNUVKUVMUUIMZKZKZUXFUYDKZUVOKZUYRUYMVUAUYNVUCUVHUVKBUUI
      YEUVKUVMBUUIYEYAVUDUVOUYTVUBKZKVUGUVOKVUFUVLUYTUVNVUBYBUVOVUGYCVUGVUEUVOU
      WNUXFKZUXFUYCKZKUXFUWNKZVUIKVUGVUEVUHVUJVUIUWNUXFYCYDUYTVUHVUBVUIUVHUVKAA
      YFUVKUVMAAYFYAUXFUWNUYCYGYHYDXOUXFUYDUVOYIXOYJUYPUVHUVMUUKMUYIDUVHUVMUUJU
      UJCXMUXQYPUVHUVMUUKTYKUYSUYQDAXPUYEUYQDAYLUYDUVODAYMYKXGUVHUVMBTYNYOUYAUY
      DUXSOZERCRUYGUXSCEAAYQUYFVUKCEUYDUXRUVPYRYOXLUUKWHUULUYLLUUJUUJYSCEUUKBYT
      UUAYHUUBYAUUCUUD $.
  $}

  ${
    $d x y $.  $d A y $.  $d B y $.  $d C y $.
    $( Distribute proper substitution through a composition of relations.
       (Contributed by RP, 28-Jun-2020.) $)
    csbcog $p |- ( A e. V -> [_ A / x ]_ ( B o. C )
                             = ( [_ A / x ]_ B o. [_ A / x ]_ C ) ) $=
      ( vy cv ccom csb wceq csbeq1 coeq12d eqeq12d vex nfcsb1v nfco weq csbeq1a
      csbief vtoclg ) AFGZCDHZIZAUACIZAUADIZHZJABUBIZABCIZABDIZHZJFBEUABJZUCUGU
      FUJAUABUBKUKUDUHUEUIAUABCKAUABDKLMAUAUBUFFNAUDUEAUACOAUADOPAFQCUDDUEAUACR
      AUADRLST $.
  $}

  ${
    snres0.1 $e |- B e. _V $.
    $( Condition for restriction of a singleton to be empty.  (Contributed by
       Scott Fenton, 9-Aug-2024.) $)
    snres0 $p |- ( ( { <. A , B >. } |` C ) = (/) <-> -. A e. C ) $=
      ( cop csn cres c0 wceq cdm cin wcel wn wrel wb relres reldm0 ax-mp dmsnop
      dmres ineq2i eqtri eqeq1i disjsn 3bitri ) ABEFZCGZHIZUGJZHIZCAFZKZHIACLMU
      GNUHUJOUFCPUGQRUIULHUICUFJZKULUFCTUMUKCABDSUAUBUCCAUDUE $.
  $}

  ${
    $d R x y $.  $d A x y $.
    $( The image is unaffected by intersection with the domain.  (Contributed
       by Scott Fenton, 17-Dec-2021.) $)
    imaindm $p |- ( R " A ) = ( R " ( A i^i dom R ) ) $=
      ( vx vy cima cdm cin cv wbr wrex wcel wa vex breldm pm4.71ri rexbii rexin
      bitr4i elima 3bitr4i eqriv ) CBAEZBABFZGZEZDHZCHZBIZDAJZUHDUDJZUGUBKUGUEK
      UIUFUCKZUHLZDAJUJUHULDAUHUKUFUGBDMCMZNOPUHDAUCQRDUGBAUMSDUGBUDUMSTUA $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  The Predecessor Class
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c Pred $.
  $( The predecessors symbol. $)
  cpred $a class Pred ( R , A , X ) $.

  $( Define the predecessor class of a binary relation.  This is the class of
     all elements ` y ` of ` A ` such that ` y R X ` (see ~ elpred ).
     (Contributed by Scott Fenton, 29-Jan-2011.) $)
  df-pred $a |- Pred ( R , A , X ) = ( A i^i ( `' R " { X } ) ) $.

  $( Equality theorem for the predecessor class.  (Contributed by Scott Fenton,
     13-Jun-2018.) $)
  predeq123 $p |- ( ( R = S /\ A = B /\ X = Y ) ->
                    Pred ( R , A , X ) = Pred ( S , B , Y ) ) $=
    ( wceq w3a ccnv csn cima cin cpred simp2 3ad2ant1 3ad2ant3 imaeq12d df-pred
    cnveq sneq ineq12d 3eqtr4g ) CDGZABGZEFGZHZACIZEJZKZLBDIZFJZKZLACEMBDFMUFAB
    UIULUCUDUENUFUGUJUHUKUCUDUGUJGUECDSOUEUCUHUKGUDEFTPQUAACERBDFRUB $.

  $( Equality theorem for the predecessor class.  (Contributed by Scott Fenton,
     2-Feb-2011.) $)
  predeq1 $p |- ( R = S -> Pred ( R , A , X ) = Pred ( S , A , X ) ) $=
    ( wceq cpred eqid predeq123 mp3an23 ) BCEAAEDDEABDFACDFEAGDGAABCDDHI $.

  $( Equality theorem for the predecessor class.  (Contributed by Scott Fenton,
     2-Feb-2011.) $)
  predeq2 $p |- ( A = B -> Pred ( R , A , X ) = Pred ( R , B , X ) ) $=
    ( wceq cpred eqid predeq123 mp3an13 ) CCEABEDDEACDFBCDFECGDGABCCDDHI $.

  $( Equality theorem for the predecessor class.  (Contributed by Scott Fenton,
     2-Feb-2011.) $)
  predeq3 $p |- ( X = Y -> Pred ( R , A , X ) = Pred ( R , A , Y ) ) $=
    ( wceq cpred eqid predeq123 mp3an12 ) BBEAAECDEABCFABDFEBGAGAABBCDHI $.

  ${
    nfpred.1 $e |- F/_ x R $.
    nfpred.2 $e |- F/_ x A $.
    nfpred.3 $e |- F/_ x X $.
    $( Bound-variable hypothesis builder for the predecessor class.
       (Contributed by Scott Fenton, 9-Jun-2018.) $)
    nfpred $p |- F/_ x Pred ( R , A , X ) $=
      ( cpred ccnv csn cima cin df-pred nfcnv nfsn nfima nfin nfcxfr ) ABCDHBCI
      ZDJZKZLBCDMABUAFASTACENADGOPQR $.
  $}

  $( Move class substitution in and out of the predecessor class of a relation.
     (Contributed by ML, 25-Oct-2020.) $)
  csbpredg $p |- ( A e. V -> [_ A / x ]_ Pred ( R , D , X ) =
                   Pred ( [_ A / x ]_ R , [_ A / x ]_ D , [_ A / x ]_ X ) ) $=
    ( wcel ccnv csn cima cin cpred csbin csbima12 csbcnv imaeq1i csbsng df-pred
    csb eqtrid imaeq2d eqtr3id ineq2d csbeq2i 3eqtr4g ) BEGZABCDHZFIZJZKZSZABCS
    ZABDSZHZABFSZIZJZKZABCDFLZSULUMUOLUFUKULABUISZKURABCUIMUFUTUQULUFUTABUGSZAB
    UHSZJZUQABUHUGNUFVCUNVBJUQUNVAVBABDOPUFVBUPUNABFEQUAUBTUCTABUSUJCDFRUDULUMU
    ORUE $.

  $( If ` A ` is a subset of ` B ` , then their predecessor classes are also
     subsets.  (Contributed by Scott Fenton, 2-Feb-2011.) $)
  predpredss $p |- ( A C_ B -> Pred ( R , A , X ) C_ Pred ( R , B , X ) ) $=
    ( wss ccnv csn cima cin cpred ssrin df-pred 3sstr4g ) ABEACFDGHZIBNIACDJBCD
    JABNKACDLBCDLM $.

  $( The predecessor class of ` A ` is a subset of ` A ` .  (Contributed by
     Scott Fenton, 2-Feb-2011.) $)
  predss $p |- Pred ( R , A , X ) C_ A $=
    ( cpred ccnv csn cima cin df-pred inss1 eqsstri ) ABCDABECFGZHAABCIALJK $.

  $( Another subset/predecessor class relationship.  (Contributed by Scott
     Fenton, 6-Feb-2011.) $)
  sspred $p |- ( ( B C_ A /\ Pred ( R , A , X ) C_ B ) ->
                 Pred ( R , A , X ) = Pred ( R , B , X ) ) $=
    ( wss cin wceq ccnv csn cima cpred sseqin2 df-pred sseq1i dfss2 in32 eqeq1i
    3bitri wa ineq1 eqeq1d biimpa 3eqtr4g eqcomd syl2anb ) BAEABFZBGZUFCHDIJZFZ
    AUHFZGZACDKZBCDKZGULBEZBALUNUJBEUJBFZUJGUKULUJBACDMZNUJBOUOUIUJAUHBPQRUGUKS
    ZUMULUQBUHFZUJUMULUGUKURUJGUGUIURUJUFBUHTUAUBBCDMUPUCUDUE $.

  ${
    $d R y $.  $d X y $.
    dfpred2.1 $e |- X e. _V $.
    $( An alternate definition of predecessor class when ` X ` is a set.
       (Contributed by Scott Fenton, 8-Feb-2011.) $)
    dfpred2 $p |- Pred ( R , A , X ) = ( A i^i { y | y R X } ) $=
      ( cpred ccnv csn cima cin cv wbr cab df-pred cvv wcel iniseg ax-mp ineq2i
      wceq eqtri ) BCDFBCGDHIZJBAKDCLAMZJBCDNUBUCBDOPUBUCTEACDOQRSUA $.

    $d A y $.
    $( An alternate definition of predecessor class when ` X ` is a set.
       (Contributed by Scott Fenton, 13-Jun-2018.) $)
    dfpred3 $p |- Pred ( R , A , X ) = { y e. A | y R X } $=
      ( cv wbr cab cin cpred crab incom dfpred2 dfrab2 3eqtr4i ) BAFDCGZAHZIQBI
      BCDJPABKBQLABCDEMPABNO $.
  $}

  ${
    $d R x y $.  $d A x y $.  $d X x y $.
    $( An alternate definition of predecessor class when ` X ` is a set.
       (Contributed by Scott Fenton, 13-Jun-2018.) $)
    dfpred3g $p |- ( X e. V -> Pred ( R , A , X ) = { y e. A | y R X } ) $=
      ( vx cpred wbr crab wceq predeq3 breq2 rabbidv eqeq12d vex dfpred3 vtoclg
      cv ) BCFRZGZARZSCHZABIZJBCEGZUAECHZABIZJFEDSEJZTUDUCUFBCSEKUGUBUEABSEUACL
      MNABCSFOPQ $.
  $}

  $( Membership in a predecessor class.  (Contributed by Scott Fenton,
     17-Apr-2011.)  Generalize to closed form.  (Revised by BJ,
     16-Oct-2024.) $)
  elpredgg $p |- ( ( X e. V /\ Y e. W ) ->
                       ( Y e. Pred ( R , A , X ) <-> ( Y e. A /\ Y R X ) ) ) $=
    ( cpred wcel ccnv csn cima wa wbr df-pred elin2 elinisegg anbi2d bitrid ) F
    ABEGZHFAHZFBIEJKZHZLECHFDHLZTFEBMZLFAUASABENOUCUBUDTBEFCDPQR $.

  $( Membership in a predecessor class.  (Contributed by Scott Fenton,
     17-Apr-2011.)  (Proof shortened by BJ, 16-Oct-2024.) $)
  elpredg $p |-
           ( ( X e. B /\ Y e. A ) -> ( Y e. Pred ( R , A , X ) <-> Y R X ) ) $=
    ( wcel wa cpred wbr elpredgg wb ibar bicomd adantl bitrd ) DBFZEAFZGEACDHFQ
    EDCIZGZRACBADEJQSRKPQRSQRLMNO $.

  $( Membership in a predecessor class - implicative version.  (Contributed by
     Scott Fenton, 13-Apr-2011.)  (Revised by NM, 5-Apr-2016.)  (Proof
     shortened by BJ, 16-Oct-2024.) $)
  elpredimg $p |- ( ( X e. V /\ Y e. Pred ( R , A , X ) ) -> Y R X ) $=
    ( wcel cpred wbr wa elpredgg simpr biimtrdi syldbl2 ) DCFZEABDGZFZEDBHZNPIP
    EAFZQIQABCODEJRQKLM $.

  ${
    elpredim.1 $e |- X e. _V $.
    $( Membership in a predecessor class - implicative version.  (Contributed
       by Scott Fenton, 9-May-2012.)  (Proof shortened by BJ, 16-Oct-2024.) $)
    elpredim $p |- ( Y e. Pred ( R , A , X ) -> Y R X ) $=
      ( cvv wcel cpred wbr elpredimg mpan ) CFGDABCHGDCBIEABFCDJK $.
  $}

  ${
    elpred.1 $e |- Y e. _V $.
    $( Membership in a predecessor class.  (Contributed by Scott Fenton,
       4-Feb-2011.)  (Proof shortened by BJ, 16-Oct-2024.) $)
    elpred $p |- ( X e. D ->
                       ( Y e. Pred ( R , A , X ) <-> ( Y e. A /\ Y R X ) ) ) $=
      ( wcel cvv cpred wbr wa wb elpredgg mpan2 ) DBGEHGEACDIGEAGEDCJKLFACBHDEM
      N $.
  $}

  $( The predecessor class exists when ` A ` does.  (Contributed by Scott
     Fenton, 8-Feb-2011.)  Generalize to closed form.  (Revised by BJ,
     27-Oct-2024.) $)
  predexg $p |- ( A e. V -> Pred ( R , A , X ) e. _V ) $=
    ( wcel cpred ccnv csn cima cin cvv df-pred inex1g eqeltrid ) ACEABDFABGDHIZ
    JKABDLAOCMN $.

  ${
    $d x y A $.  $d x y R $.
    $( Alternate definition of well-founded relation.  (Contributed by Scott
       Fenton, 2-Feb-2011.) $)
    dffr4 $p |- ( R Fr A <-> A. x ( ( x C_ A /\ x =/= (/) ) ->
                                      E. y e. x Pred ( R , x , y ) = (/) ) ) $=
      ( wfr cv wss c0 wne wa ccnv csn cima cin wceq wrex wi wal cpred dffr3
      df-pred eqeq1i rexbii imbi2i albii bitr4i ) CDEAFZCGUGHIJZUGDKBFZLMNZHOZB
      UGPZQZARUHUGDUISZHOZBUGPZQZARABCDTUQUMAUPULUHUOUKBUGUNUJHUGDUIUAUBUCUDUEU
      F $.
  $}

  $( Membership in the predecessor class implies membership in the base class.
     (Contributed by Scott Fenton, 11-Feb-2011.) $)
  predel $p |- ( Y e. Pred ( R , A , X ) -> Y e. A ) $=
    ( wcel ccnv csn cima cin cpred elinel1 df-pred eleq2s ) DAEDABFCGHZIABCJDAN
    KABCLM $.

  ${
    $d A y z $.  $d R y z $.  $d X y z $.  $d Y y z $.
    $( If ` R ` is transitive over ` A ` and ` Y R X ` , then
       ` Pred ( R , A , Y ) ` is a subclass of ` Pred ( R , A , X ) ` .
       (Contributed by Scott Fenton, 28-Oct-2024.) $)
    predtrss $p |- (
     ( ( ( R i^i ( A X. A ) ) o. ( R i^i ( A X. A ) ) ) C_ R /\
       Y e. Pred ( R , A , X ) /\ X e. A ) ->
     Pred ( R , A , Y ) C_ Pred ( R , A , X ) ) $=
      ( vz vy cpred wcel cv wbr wa 3ad2ant2 adantr brxp sylanbrc brin wb mpan2d
      cvv elpred cxp cin ccom wss w3a simpr predel elpredimg ancoms 3adant1 wex
      simpl3 wi wceq breq2 breq1 anbi12d spcegv vex brcog sylancr sylibrd ssbrd
      simpl1 syld biimtrrid imdistanda 3ad2ant3 3imtr4d ssrdv ) BAAUAZUBZVLUCZB
      UDZDABCGZHZCAHZUEZEABDGZVOVREIZAHZVTDBJZKZWAVTCBJZKZVTVSHZVTVOHZVRWAWBWDV
      RWAKZWBVTDVKJZWDWHWADAHZWIVRWAUFVRWJWAVPVNWJVQABCDUGLMZVTDAANOWBWIKVTDVLJ
      ZWHWDVTDBVKPWHWLDCVLJZWDWHDCBJZDCVKJZWMVRWNWAVPVQWNVNVQVPWNABACDUHUIUJMWH
      WJVQWOWKVNVPVQWAULZDCAANODCBVKPOWHWLWMKZVTCVMJZWDWHWQVTFIZVLJZWSCVLJZKZFU
      KZWRVRWQXCUMZWAVPVNXDVQXBWQFDVOWSDUNWTWLXAWMWSDVTVLUOWSDCVLUPUQURLMWHVTSH
      VQWRXCQEUSZWPFVTCVLVLSAUTVAVBWHVMBVTCVNVPVQWAVDVCVERVFRVGVPVNWFWCQVQAVOBD
      VTXETLVQVNWGWEQVPAABCVTXETVHVIVJ $.
  $}

  $( Property of the predecessor class for partial orders.  (Contributed by
     Scott Fenton, 28-Apr-2012.)  (Proof shortened by Scott Fenton,
     28-Oct-2024.) $)
  predpo $p |- ( ( R Po A /\ X e. A ) ->
                 ( Y e. Pred ( R , A , X ) ->
                   Pred ( R , A , Y ) C_ Pred ( R , A , X ) ) ) $=
    ( wpo wcel wa cpred wss cxp cin ccom cres wceq dfpo2 simprbi ad2antrr simpr
    cid c0 simplr predtrss syl3anc ex ) ABEZCAFZGZDABCHZFZABDHUHIZUGUIGBAAJKZUK
    LBIZUIUFUJUEULUFUIUEBSAMKTNULABOPQUGUIRUEUFUIUAABCDUBUCUD $.

  $( Property of the predecessor class for strict total orders.  (Contributed
     by Scott Fenton, 11-Feb-2011.) $)
  predso $p |- ( ( R Or A /\ X e. A ) ->
                  ( Y e. Pred ( R , A , X ) ->
                    Pred ( R , A , Y ) C_ Pred ( R , A , X ) ) ) $=
    ( wor wpo wcel cpred wss wi sopo predpo sylan ) ABEABFCAGDABCHZGABDHNIJABKA
    BCDLM $.

  ${
    $d R x $.  $d A x $.  $d X x $.
    $( If ` R ` is set-like in ` A ` , then all predecessor classes of elements
       of ` A ` exist.  (Contributed by Scott Fenton, 20-Feb-2011.)  (Revised
       by Mario Carneiro, 26-Jun-2015.) $)
    setlikespec $p |- ( ( X e. A /\ R Se A ) -> Pred ( R , A , X ) e. _V ) $=
      ( vx wcel wse wa cv wbr crab cpred cvv wceq cab df-rab vex elpred eqtr4id
      eqabdv adantr seex ancoms eqeltrrd ) CAEZABFZGDHZCBIZDAJZABCKZLUDUHUIMUEU
      DUHUFAEUGGZDNUIUGDAOUDUJDUIAABCUFDPQSRTUEUDUHLEDACBUAUBUC $.
  $}

  $( Idempotent law for the predecessor class.  (Contributed by Scott Fenton,
     29-Mar-2011.) $)
  predidm $p |- Pred ( R , Pred ( R , A , X ) , X ) = Pred ( R , A , X ) $=
    ( cpred ccnv csn cima cin df-pred inidm ineq2i eqtr4i inass ineq1i ) ABCDZB
    CDOBECFGZHZOOBCIOAPHZPHZQOAPPHZHZSORUAABCIZTPAPJKLAPPMLORPUBNLL $.

  $( Intersection law for predecessor classes.  (Contributed by Scott Fenton,
     29-Mar-2011.) $)
  predin $p |- Pred ( R , ( A i^i B ) , X ) =
               ( Pred ( R , A , X ) i^i Pred ( R , B , X ) ) $=
    ( cin ccnv csn cima cpred inindir df-pred ineq12i 3eqtr4i ) ABEZCFDGHZEAOEZ
    BOEZENCDIACDIZBCDIZEABOJNCDKRPSQACDKBCDKLM $.

  $( Union law for predecessor classes.  (Contributed by Scott Fenton,
     29-Mar-2011.) $)
  predun $p |- Pred ( R , ( A u. B ) , X ) =
               ( Pred ( R , A , X ) u. Pred ( R , B , X ) ) $=
    ( cun ccnv csn cima cin cpred indir df-pred uneq12i 3eqtr4i ) ABEZCFDGHZIAP
    IZBPIZEOCDJACDJZBCDJZEABPKOCDLSQTRACDLBCDLMN $.

  $( Difference law for predecessor classes.  (Contributed by Scott Fenton,
     14-Apr-2011.) $)
  preddif $p |- Pred ( R , ( A \ B ) , X ) =
                  ( Pred ( R , A , X ) \ Pred ( R , B , X ) ) $=
    ( cdif ccnv csn cima cin cpred indifdir df-pred difeq12i 3eqtr4i ) ABEZCFDG
    HZIAPIZBPIZEOCDJACDJZBCDJZEABPKOCDLSQTRACDLBCDLMN $.

  ${
    $d X y $.  $d B y $.
    $( The predecessor under the membership relation is equivalent to an
       intersection.  (Contributed by Scott Fenton, 27-Mar-2011.)  (Proof
       shortened by Andrew Salmon, 27-Aug-2011.) $)
    predep $p |- ( X e. B -> Pred ( _E , A , X ) = ( A i^i X ) ) $=
      ( vy wcel cep cpred ccnv csn cima cin df-pred cv wbr cab wrel wceq relcnv
      relimasn eqtrid ax-mp wb cvv brcnvg elvd epelg bitrd eqabcdv ineq2d ) CBE
      ZAFCGAFHZCIJZKACKAFCLUJULCAUJULCDMZUKNZDOZCUKPULUOQFRDCUKSUAUJUNDCUJUNUMC
      FNZUMCEUJUNUPUBDCUMBUCFUDUEUMCBUFUGUHTUIT $.
  $}

  $( The class of predecessors of an element of a transitive class for the
     membership relation is that element.  (Contributed by BJ, 12-Oct-2024.) $)
  trpred $p |- ( ( Tr A /\ X e. A ) -> Pred ( _E , A , X ) = X ) $=
    ( wtr wcel wa cep cpred cin wceq predep adantl wss trss sseqin2 sylib eqtrd
    imp ) ACZBADZEZAFBGZABHZBSUAUBIRAABJKTBALZUBBIRSUCABMQBANOP $.

  ${
    $d A x y z $.  $d B x y z $.  $d R x y z $.  $d X y $.
    $( A property of classes that are downward closed under predecessor.
       (Contributed by Scott Fenton, 13-Apr-2011.) $)
    preddowncl $p |- ( ( B C_ A /\ A. x e. B Pred ( R , A , x ) C_ B ) ->
                     ( X e. B -> Pred ( R , B , X ) = Pred ( R , A , X ) ) ) $=
      ( vy vz wss cv cpred wral wa wcel wceq eleq1 predeq3 eqeq12d imbi2d vex
      wi imbi12d predpredss ad2antrr wbr weq sseq1d rspccva sseld elpredim jca2
      wb elpred adantl mpbird ssrdv adantll eqssd ex vtoclg pm2.43b ) CBHZBDAIZ
      JZCHZACKZLZECMZCDEJZBDEJZNZVFFIZCMZCDVKJZBDVKJZNZTZTVFVGVJTZTFECVKENZVPVQ
      VFVRVLVGVOVJVKECOVRVMVHVNVICDVKEPBDVKEPQUARVFVLVOVFVLLVMVNVAVMVNHVEVLCBDV
      KUBUCVEVLVNVMHVAVEVLLZGVNVMVSGIZVNMZVTVMMZTZWAVTCMZVTVKDUDZLZTZVSWAWDWEVS
      VNCVTVDVNCHAVKCAFUEVCVNCBDVBVKPUFUGUHBDVKVTFSUIUJVLWCWGUKVEVLWBWFWACCDVKV
      TGSULRUMUNUOUPUQURUSUT $.
  $}

  $( Given a partial ordering, a class is not a member of its predecessor
     class.  (Contributed by Scott Fenton, 17-Apr-2011.) $)
  predpoirr $p |- ( R Po A -> -. X e. Pred ( R , A , X ) ) $=
    ( wpo wcel cpred wn wa wbr poirr wb elpredg anidms notbid imbitrrid pm2.43b
    expd predel con3i pm2.61d1 ) ABDZCAEZCABCFEZGZUAUBUDUBUAUBUDUAUBHUDUBCCBIZG
    ACBJUBUCUEUBUCUEKAABCCLMNOQPUCUBABCCRST $.

  $( Given a well-founded relation, a class is not a member of its predecessor
     class.  (Contributed by Scott Fenton, 22-Apr-2011.) $)
  predfrirr $p |- ( R Fr A -> -. X e. Pred ( R , A , X ) ) $=
    ( wfr wcel cpred wn wa wbr frirr wb elpredg anidms notbid imbitrrid pm2.43b
    expd predel con3i pm2.61d1 ) ABDZCAEZCABCFEZGZUAUBUDUBUAUBUDUAUBHUDUBCCBIZG
    ACBJUBUCUEUBUCUEKAABCCLMNOQPUCUBABCCRST $.

  $( The predecessor class over ` (/) ` is always ` (/) ` .  (Contributed by
     Scott Fenton, 16-Apr-2011.)  (Proof shortened by AV, 11-Jun-2021.) $)
  pred0 $p |- Pred ( R , (/) , X ) = (/) $=
    ( c0 cpred ccnv csn cima cin df-pred 0in eqtri ) CABDCAEBFGZHCCABILJK $.

  ${
    $d A x $.  $d R x $.
    $( Alternate definition of set-like relationships.  (Contributed by Scott
       Fenton, 19-Aug-2024.) $)
    dfse3 $p |- ( R Se A <-> A. x e. A Pred ( R , A , x ) e. _V ) $=
      ( wse ccnv cv csn cima cin cvv wcel wral cpred dfse2 eleq1i ralbii bitr4i
      df-pred ) BCDBCEAFZGHIZJKZABLBCSMZJKZABLABCNUCUAABUBTJBCSROPQ $.
  $}

  $( Subset carries from relation to predecessor class.  (Contributed by Scott
     Fenton, 25-Nov-2024.) $)
  predrelss $p |- ( R C_ S -> Pred ( R , A , X ) C_ Pred ( S , A , X ) ) $=
    ( wss ccnv csn cima cin cpred cnvss imass1 sslin 3syl df-pred 3sstr4g ) BCE
    ZABFZDGZHZIZACFZSHZIZABDJACDJQRUBETUCEUAUDEBCKRUBSLTUCAMNABDOACDOP $.

  $( The predecessor of a proper class is empty.  (Contributed by Scott Fenton,
     25-Nov-2024.) $)
  predprc $p |- ( -. X e. _V -> Pred ( R , A , X ) = (/) ) $=
    ( cvv wcel wn cpred ccnv csn cima c0 df-pred wceq snprc biimpi imaeq2d ima0
    cin eqtrdi ineq2d in0 eqtrid ) CDEFZABCGABHZCIZJZRZKABCLUCUGAKRKUCUFKAUCUFU
    DKJKUCUEKUDUCUEKMCNOPUDQSTAUASUB $.

  ${
    $d R y $.  $d A y $.  $d X y $.
    $( Predecessor class is unaffected by restriction to the base class.
       (Contributed by Scott Fenton, 25-Nov-2024.) $)
    predres $p |- Pred ( R , A , X ) = Pred ( ( R |` A ) , A , X ) $=
      ( vy cvv wcel cpred cres wceq ccnv csn cima cin cv wbr cab iniseg df-pred
      ineq2d predprc crab wss ssrab2 sseqin2 mpbi dfrab2 eqtr2i incom eqtrdi wa
      brres abbidv df-rab eqtr4di eqtrd 3eqtr4a 3eqtr4g wn c0 eqtr4d pm2.61i )
      CEFZABCGZABAHZCGZIVBABJCKZLZMZAVDJVFLZMZVCVEVBDNZCBOZDPZAMZAVLDAUAZMZVHVJ
      VPVOVNVOAUBVPVOIVLDAUCVOAUDUEVLDAUFUGVBVHAVMMVNVBVGVMADBCEQSAVMUHUIVBVIVO
      AVBVIVKCVDOZDPZVODVDCEQVBVRVKAFVLUJZDPVOVBVQVSDAVKCBEUKULVLDAUMUNUOSUPABC
      RAVDCRUQVBURVCUSVEABCTAVDCTUTVA $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Well-founded induction (variant)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d A x y z w $.  $d R x y z w $.  $d B x y z w $.
    $( Every nonempty (possibly proper) subclass of a class ` A ` with a
       well-founded set-like partial order ` R ` has a minimal element.  The
       additional condition of partial order over ~ frmin enables avoiding the
       axiom of infinity.  (Contributed by Scott Fenton, 11-Feb-2022.) $)
    frpomin $p |- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\
    ( B C_ A /\ B =/= (/) ) ) -> E. x e. B A. y e. B -. y R x ) $=
      ( vz vw c0 cv wbr wn wral wrex wcel wa wi breq1 biimtrid expr ad2antrr n0
      wfr wpo wse w3a wss wne wex crab wceq rabeq0 simprr notbid cbvralvw breq2
      weq ralbidv bitrid rspcev ex syl cvv simprl simpl3 sess2 sylc seex simpl1
      syl2anc ssrab2 sstrid syl21anc rexrab ralrab simplrl simpll2 poss simpllr
      fri simplr simplrr potr syl13anc mp2and idd jad ralimdva expimpd reximdva
      con3d syld pm2.61dne exlimdv impr ) CEUBZCEUCZCEUDZUEZDCUFZDHUGZBIZAIZEJZ
      KZBDLZADMZWTFIZDNZFUHWRWSOZXFFDUAXIXHXFFWRWSXHXFWRWSXHOZOZXFGIZXGEJZGDUIZ
      HXNHUJXMKZGDLZXKXFXMGDUKXKXHXPXFPWRWSXHULZXHXPXFXEXPAXGDXEXLXBEJZKZGDLAFU
      PZXPXDXSBGDBGUPXCXRXAXLXBEQUMUNXTXSXOGDXTXRXMXBXGXLEUOUMUQURUSUTVARXKXNHU
      GZXDBXNLZAXNMZXFXKXNVBNZWOXNCUFZYAYCPXKDEUDZXHYDXKWSWQYFWRWSXHVCZWOWPWQXJ
      VDDCEVEVFXQGDXGEVGVIWOWPWQXJVHXKXNDCXMGDVJYGVKYDWOOYEYAYCABCXNVBEVSSVLYCX
      BXGEJZYBOZADMXKXFXMYHYBAGDXLXBXGEQVMXKYIXEADXKXBDNZOZYHYBXEYBXAXGEJZXDPZB
      DLYKYHOZXEXMYLXDBGDXLXAXGEQVNYNYMXDBDYNXADNZOZYLXDXDYPXCYLYNYOXCYLYNYOXCO
      ZOZXCYHYLYNYOXCULYKYHYQVTYRDEUCZYOYJXHXCYHOYLPYRWSWPYSYKWSYHYQWRWSXHYJVOT
      YKWPYHYQWOWPWQXJYJVPTDCEVQVFYNYOXCVCXKYJYHYQVRYKXHYHYQWRWSXHYJWATDXAXBXGE
      WBWCWDSWJYPXDWEWFWGRWHWIRWKWLSWMRWN $.

    $( Every nonempty (possibly proper) subclass of a class ` A ` with a
       well-founded set-like partial order ` R ` has a minimal element.  The
       additional condition of partial order over ~ frmin enables avoiding the
       axiom of infinity.  (Contributed by Scott Fenton, 11-Feb-2022.) $)
    frpomin2 $p |- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\
    ( B C_ A /\ B =/= (/) ) ) -> E. x e. B Pred ( R , B , x ) = (/) ) $=
      ( vy wfr wpo wse w3a wss c0 wne wa cv wbr wn wral wrex cpred wceq frpomin
      crab vex dfpred3 eqeq1i rabeq0 bitri rexbii sylibr ) BDFBDGBDHICBJCKLMMEN
      ANZDOZPECQZACRCDUJSZKTZACRAEBCDUAUNULACUNUKECUBZKTULUMUOKECDUJAUCUDUEUKEC
      UFUGUHUI $.
  $}

  ${
    $d A y $.  $d B y $.  $d R y $.
    $( The principle of well-founded induction over a partial order.  This
       theorem is a version of ~ frind that does not require the axiom of
       infinity and can be used to prove ~ wfi and ~ tfi .  (Contributed by
       Scott Fenton, 11-Feb-2022.) $)
    frpoind $p |- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\ ( B C_ A /\
    A. y e. A ( Pred ( R , A , y ) C_ B -> y e. B ) ) ) -> A = B ) $=
      ( wfr wpo wss cpred wcel wa wn cdif c0 ssdif0 wceq wrex cin df-pred incom
      eqtri wse w3a cv wi wral wne necon3bbii difss frpomin2 eldif anbi1i anass
      ccnv csn cima indif2 difeq1i 3eqtr4i eqeq1i bitr4i anbi1ci anbi2i rexbii2
      3bitri rexanali bitri sylib mpani biimtrid con4d imp adantrl simprl eqssd
      ex ) BDEBDFBDUAUBZCBGZBDAUCZHZCGZVRCIZUDABUEZJJBCVPWBBCGZVQVPWBWCVPWCWBWC
      KBCLZMUFZVPWBKZWCWDMBCNUGVPWDBGZWEWFBCUHVPWGWEJZWFVPWHJWDDVRHZMOZAWDPZWFA
      BWDDUIWKVTWAKZJZABPWFWJWMAWDBVRWDIZWJJVRBIZWLJZWJJWOWLWJJZJWOWMJWNWPWJVRB
      CUJUKWOWLWJULWQWMWOWJVTWLWJVSCLZMOVTWIWRMDUMVRUNUOZWDQZWSBQZCLWIWRWSBCUPW
      IWDWSQWTWDDVRRWDWSSTVSXACVSBWSQXABDVRRBWSSTUQURUSVSCNUTVAVBVDVCVTWAABVEVF
      VGVOVHVIVJVKVLVPVQWBVMVN $.
  $}

  ${
    $d A w y z $.  $d ph w z $.  $d R w y z $.
    frpoinsg.1 $e |- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\ y e. A ) ->
                       ( A. z e. Pred ( R , A , y ) [. z / y ]. ph -> ph ) ) $.
    $( Well-Founded Induction Schema (variant).  If a property passes from all
       elements less than ` y ` of a well-founded set-like partial order class
       ` A ` to ` y ` itself (induction hypothesis), then the property holds
       for all elements of ` A ` .  (Contributed by Scott Fenton,
       11-Feb-2022.) $)
    frpoinsg $p |- ( ( R Fr A /\ R Po A /\ R Se A ) -> A. y e. A ph ) $=
      ( vw wral wss cv cpred wcel wi wa wsbc nfcv elrabsf nfsbc1v nfim imbi12d
      wfr wpo wse w3a crab wceq dfss3 simprbi ralimi sylbi nfralw eleq1w anbi2d
      nfv predeq3 raleqdv sbceq1a chvarfv syl5 simpr jctild imbitrrdi ralrimiva
      weq ssrab2 jctil frpoind mpdan rabid2 sylib ) DEUADEUBDEUCUDZDABDUEZUFZAB
      DHVKVLDIZDEGJZKZVLIZVOVLLZMZGDHZNVMVKVTVNVKVSGDVKVODLZNZVQWAABVOOZNVRWBVQ
      WCWAVQABCJZOZCVPHZWBWCVQWDVLLZCVPHWFCVPVLUGWGWECVPWGWDDLWEABWDDBDPZQUHUIU
      JVKBJZDLZNZWECDEWIKZHZAMZMWBWFWCMZMBGWBWOBWBBUNWFWCBWEBCVPBVPPABWDRUKABVO
      RSSBGVDZWKWBWNWOWPWJWAVKBGDULUMWPWMWFAWCWPWECWLVPDEWIVOUOUPABVOUQTTFURUSV
      KWAUTVAABVODWHQVBVCABDVEVFGDVLEVGVHABDVIVJ $.
  $}

  ${
    $d A y z $.  $d ph z $.  $d R y z $.
    frpoins2fg.1 $e |- ( y e. A -> ( A. z e. Pred ( R , A , y ) ps -> ph ) ) $.
    frpoins2fg.2 $e |- F/ y ps $.
    frpoins2fg.3 $e |- ( y = z -> ( ph <-> ps ) ) $.
    $( Well-Founded Induction schema, using implicit substitution.
       (Contributed by Scott Fenton, 24-Aug-2022.) $)
    frpoins2fg $p |- ( ( R Fr A /\ R Po A /\ R Se A ) -> A. y e. A ph ) $=
      ( cv wsbc cpred wral wfr wpo wse w3a wcel wa wsb sbsbc bitr3i wi biimtrid
      sbiev ralbii adantl frpoinsg ) ACDEFACDJKZDEFCJZLZMBDUKMZEFNEFOEFPQZUJERZ
      SAUIBDUKUIACDTBACDUAABCDHIUEUBUFUNULAUCUMGUGUDUH $.
  $}

  ${
    $d A y z $.  $d ph z $.  $d R y z $.  $d ps y $.
    frpoins2g.1 $e |- ( y e. A -> ( A. z e. Pred ( R , A , y ) ps -> ph ) ) $.
    frpoins2g.3 $e |- ( y = z -> ( ph <-> ps ) ) $.
    $( Well-Founded Induction schema, using implicit substitution.
       (Contributed by Scott Fenton, 24-Aug-2022.) $)
    frpoins2g $p |- ( ( R Fr A /\ R Po A /\ R Se A ) -> A. y e. A ph ) $=
      ( nfv frpoins2fg ) ABCDEFGBCIHJ $.
  $}

  ${
    $d A x y $.  $d B x $.  $d ch x $.  $d ph y $.  $d ps x $.  $d R x y $.
    frpoins3g.1 $e |- ( x e. A -> ( A. y e. Pred ( R , A , x ) ps -> ph ) ) $.
    frpoins3g.2 $e |- ( x = y -> ( ph <-> ps ) ) $.
    frpoins3g.3 $e |- ( x = B -> ( ph <-> ch ) ) $.
    $( Well-Founded Induction schema, using implicit substitution.
       (Contributed by Scott Fenton, 19-Aug-2024.) $)
    frpoins3g $p |- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\ B e. A ) -> ch ) $=
      ( wfr wpo wse w3a wral wcel frpoins2g rspccva sylan ) FHLFHMFHNOADFPGFQCA
      BDEFHIJRACDGFKST $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Well-ordered induction
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d y A $.  $d y B $.  $d y R $.
    $( All nonempty subclasses of a class having a well-ordered set-like
       relation have minimal elements for that relation.  Proposition 6.26 of
       [TakeutiZaring] p. 31.  (Contributed by Scott Fenton, 29-Jan-2011.)
       (Revised by Mario Carneiro, 26-Jun-2015.)  (Proof shortened by Scott
       Fenton, 17-Nov-2024.) $)
    tz6.26 $p |- ( ( ( R We A /\ R Se A ) /\ ( B C_ A /\ B =/= (/) ) ) ->
                    E. y e. B Pred ( R , B , y ) = (/) ) $=
      ( wwe wse wa wfr wpo w3a wss c0 wne cv cpred wceq wrex wefr adantr wor
      weso sopo syl simpr 3jca frpomin2 sylan ) BDEZBDFZGZBDHZBDIZUIJCBKCLMGCDA
      NOLPACQUJUKULUIUHUKUIBDRSUHULUIUHBDTULBDUABDUBUCSUHUIUDUEABCDUFUG $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y R $.
    $( Obsolete version of ~ tz6.26 as of 17-Nov-2024.
       (New usage is discouraged.)  (Proof modification is discouraged.)
       (Contributed by Scott Fenton, 29-Jan-2011.)  (Revised by Mario Carneiro,
       26-Jun-2015.) $)
    tz6.26OLD $p |- ( ( ( R We A /\ R Se A ) /\ ( B C_ A /\ B =/= (/) ) ) ->
                    E. y e. B Pred ( R , B , y ) = (/) ) $=
      ( vx wwe wse wa wss c0 wne cv wbr wn wral wrex cpred wceq wreu wereu2 syl
      reurex crab rabeq0 cab cin dfrab3 vex dfpred2 eqtr4i eqeq1i bitr3i rexbii
      sylib ) BDFBDGHCBICJKHHZELALZDMZNECOZACPZCDUPQZJRZACPUOURACSUSAEBCDTURACU
      BUAURVAACURUQECUCZJRVAUQECUDVBUTJVBCUQEUEUFUTUQECUGECDUPAUHUIUJUKULUMUN
      $.
  $}

  ${
    $d A y $.  $d B y $.  $d R y $.
    tz6.26i.1 $e |- R We A $.
    tz6.26i.2 $e |- R Se A $.
    $( All nonempty subclasses of a class having a well-ordered set-like
       relation ` R ` have ` R `-minimal elements.  Proposition 6.26 of
       [TakeutiZaring] p. 31.  (Contributed by Scott Fenton, 14-Apr-2011.)
       (Revised by Mario Carneiro, 26-Jun-2015.) $)
    tz6.26i $p |- ( ( B C_ A /\ B =/= (/) ) ->
                    E. y e. B Pred ( R , B , y ) = (/) ) $=
      ( wwe wse wss c0 wne wa cv cpred wceq wrex tz6.26 mpanl12 ) BDGBDHCBICJKL
      CDAMNJOACPEFABCDQR $.
  $}

  ${
    $d A y $.  $d B y $.  $d R y $.
    $( The Principle of Well-Ordered Induction.  Theorem 6.27 of
       [TakeutiZaring] p. 32.  This principle states that if ` B ` is a
       subclass of a well-ordered class ` A ` with the property that every
       element of ` B ` whose inital segment is included in ` A ` is itself
       equal to ` A ` .  (Contributed by Scott Fenton, 29-Jan-2011.)  (Revised
       by Mario Carneiro, 26-Jun-2015.)  (Proof shortened by Scott Fenton,
       17-Nov-2024.) $)
    wfi $p |- ( ( ( R We A /\ R Se A ) /\
                    ( B C_ A /\
                      A. y e. A ( Pred ( R , A , y ) C_ B -> y e. B ) ) )
                  -> A = B ) $=
      ( wwe wse wa wfr wpo w3a wss cv cpred wcel wi wral wceq wefr adantr wor
      weso sopo syl simpr 3jca frpoind sylan ) BDEZBDFZGZBDHZBDIZUIJCBKBDALZMCK
      UMCNOABPGBCQUJUKULUIUHUKUIBDRSUHULUIUHBDTULBDUABDUBUCSUHUIUDUEABCDUFUG $.
  $}

  ${
    $d A y $.  $d B y $.  $d R y $.
    $( Obsolete version of ~ wfi as of 17-Nov-2024.
       (New usage is discouraged.)  (Proof modification is discouraged.)
       (Contributed by Scott Fenton, 29-Jan-2011.)  (Revised by Mario Carneiro,
       26-Jun-2015.) $)
    wfiOLD $p |- ( ( ( R We A /\ R Se A ) /\
                    ( B C_ A /\
                      A. y e. A ( Pred ( R , A , y ) C_ B -> y e. B ) ) )
                  -> A = B ) $=
      ( wwe wse wa wss cpred wcel wn cdif c0 ssdif0 wceq wrex cin df-pred incom
      eqtri cv wi wral wne necon3bbii difss tz6.26 eldif anbi1i anass ccnv cima
      indif2 difeq1i 3eqtr4i eqeq1i bitr4i anbi1ci anbi2i 3bitri rexanali bitri
      csn rexbii2 sylib ex mpani biimtrid con4d imp adantrl simprl eqssd ) BDEB
      DFGZCBHZBDAUAZIZCHZVPCJZUBABUCZGGBCVNVTBCHZVOVNVTWAVNWAVTWAKBCLZMUDZVNVTK
      ZWAWBMBCNUEVNWBBHZWCWDBCUFVNWEWCGZWDVNWFGWBDVPIZMOZAWBPZWDABWBDUGWIVRVSKZ
      GZABPWDWHWKAWBBVPWBJZWHGVPBJZWJGZWHGWMWJWHGZGWMWKGWLWNWHVPBCUHUIWMWJWHUJW
      OWKWMWHVRWJWHVQCLZMOVRWGWPMDUKVPVCULZWBQZWQBQZCLWGWPWQBCUMWGWBWQQWRWBDVPR
      WBWQSTVQWSCVQBWQQWSBDVPRBWQSTUNUOUPVQCNUQURUSUTVDVRVSABVAVBVEVFVGVHVIVJVK
      VNVOVTVLVM $.
  $}

  ${
    $d A y $.  $d B y $.  $d R y $.
    wfi.1 $e |- R We A $.
    wfi.2 $e |- R Se A $.
    $( The Principle of Well-Ordered Induction.  Theorem 6.27 of
       [TakeutiZaring] p. 32.  This principle states that if ` B ` is a
       subclass of a well-ordered class ` A ` with the property that every
       element of ` B ` whose inital segment is included in ` A ` is itself
       equal to ` A ` .  (Contributed by Scott Fenton, 29-Jan-2011.)  (Revised
       by Mario Carneiro, 26-Jun-2015.) $)
    wfii $p |- ( ( B C_ A /\
                    A. y e. A ( Pred ( R , A , y ) C_ B -> y e. B ) )
                  -> A = B ) $=
      ( wwe wse wss cv cpred wcel wi wral wa wceq wfi mpanl12 ) BDGBDHCBIBDAJZK
      CISCLMABNOBCPEFABCDQR $.
  $}

  ${
    $d A y z $.  $d ph z $.  $d R y z $.
    wfisg.1 $e |- ( y e. A ->
                       ( A. z e. Pred ( R , A , y ) [. z / y ]. ph -> ph ) ) $.
    $( Well-Ordered Induction Schema.  If a property passes from all elements
       less than ` y ` of a well-ordered class ` A ` to ` y ` itself (induction
       hypothesis), then the property holds for all elements of ` A ` .
       (Contributed by Scott Fenton, 11-Feb-2011.)  (Proof shortened by Scott
       Fenton, 17-Nov-2024.) $)
    wfisg $p |- ( ( R We A /\ R Se A ) -> A. y e. A ph ) $=
      ( wwe wse wa wfr wpo wral wefr adantr wor weso sopo syl simpr cv cpred wi
      wcel wsbc w3a adantl frpoinsg syl3anc ) DEGZDEHZIDEJZDEKZUJABDLUIUKUJDEMN
      UIULUJUIDEOULDEPDEQRNUIUJSABCDEBTZDUCABCTUDCDEUMUALAUBUKULUJUEFUFUGUH $.
  $}

  ${
    $d A w y $.  $d A z $.  $d ph w $.  $d ph z $.  $d R w y $.  $d R z $.
    $d w y z $.
    wfisgOLD.1 $e |- ( y e. A ->
                       ( A. z e. Pred ( R , A , y ) [. z / y ]. ph -> ph ) ) $.
    $( Obsolete version of ~ wfisg as of 17-Nov-2024.
       (New usage is discouraged.)  (Proof modification is discouraged.)
       (Contributed by Scott Fenton, 11-Feb-2011.) $)
    wfisgOLD $p |- ( ( R We A /\ R Se A ) -> A. y e. A ph ) $=
      ( vw wa wceq wral wss cv cpred wcel wi wsbc nfcv elrabsf nfsbc1v nfim wwe
      wse crab ssrab2 dfss3 simprbi ralimi sylbi nfralw predeq3 raleqdv sbceq1a
      nfv eleq1w imbi12d chvarfv syl5 anc2li imbitrrdi wfi mpanr12 rabid2 sylib
      rgen ) DEUADEUBHZDABDUCZIZABDJVEVFDKDEGLZMZVFKZVHVFNZOZGDJVGABDUDVLGDVHDN
      ZVJVMABVHPZHVKVMVJVNVJABCLZPZCVIJZVMVNVJVOVFNZCVIJVQCVIVFUEVRVPCVIVRVODNV
      PABVODBDQZRUFUGUHBLZDNZVPCDEVTMZJZAOZOVMVQVNOZOBGVMWEBVMBUMVQVNBVPBCVIBVI
      QABVOSUIABVHSTTVTVHIZWAVMWDWEBGDUNWFWCVQAVNWFVPCWBVIDEVTVHUJUKABVHULUOUOF
      UPUQURABVHDVSRUSVDGDVFEUTVAABDVBVC $.
  $}

  ${
    $d A y z $.  $d ph z $.  $d R y z $.
    wfis.1 $e |- R We A $.
    wfis.2 $e |- R Se A $.
    wfis.3 $e |- ( y e. A ->
                    ( A. z e. Pred ( R , A , y ) [. z / y ]. ph -> ph ) ) $.
    $( Well-Ordered Induction Schema.  If all elements less than a given set
       ` x ` of the well-ordered class ` A ` have a property (induction
       hypothesis), then all elements of ` A ` have that property.
       (Contributed by Scott Fenton, 29-Jan-2011.) $)
    wfis $p |- ( y e. A -> ph ) $=
      ( wwe wse wral wfisg mp2an rspec ) ABDDEIDEJABDKFGABCDEHLMN $.
  $}

  ${
    $d A y z $.  $d ph z $.  $d R y z $.
    wfis2fg.1 $e |- F/ y ps $.
    wfis2fg.2 $e |- ( y = z -> ( ph <-> ps ) ) $.
    wfis2fg.3 $e |- ( y e. A -> ( A. z e. Pred ( R , A , y ) ps -> ph ) ) $.
    $( Well-Ordered Induction Schema, using implicit substitution.
       (Contributed by Scott Fenton, 11-Feb-2011.)  (Proof shortened by Scott
       Fenton, 17-Nov-2024.) $)
    wfis2fg $p |- ( ( R We A /\ R Se A ) -> A. y e. A ph ) $=
      ( wwe wse wa wfr wpo wral wefr adantr wor weso sopo syl simpr frpoins2fg
      syl3anc ) EFJZEFKZLEFMZEFNZUFACEOUEUGUFEFPQUEUHUFUEEFRUHEFSEFTUAQUEUFUBAB
      CDEFIGHUCUD $.
  $}

  ${
    $d A y z $.  $d ph z $.  $d R y z $.
    wfis2fgOLD.1 $e |- F/ y ps $.
    wfis2fgOLD.2 $e |- ( y = z -> ( ph <-> ps ) ) $.
    wfis2fgOLD.3 $e |- ( y e. A -> ( A. z e. Pred ( R , A , y ) ps -> ph ) ) $.
    $( Obsolete version of ~ wfis2fg as of 17-Nov-2024.
       (New usage is discouraged.)  (Proof modification is discouraged.)
       (Contributed by Scott Fenton, 11-Feb-2011.) $)
    wfis2fgOLD $p |- ( ( R We A /\ R Se A ) -> A. y e. A ph ) $=
      ( cv wsbc cpred wral wcel wsb sbsbc sbiev bitr3i ralbii biimtrid wfisg )
      ACDEFACDJKZDEFCJZLZMBDUDMUCENAUBBDUDUBACDOBACDPABCDGHQRSITUA $.
  $}

  ${
    $d A y z $.  $d ph z $.  $d R y z $.
    wfis2f.1 $e |- R We A $.
    wfis2f.2 $e |- R Se A $.
    wfis2f.3 $e |- F/ y ps $.
    wfis2f.4 $e |- ( y = z -> ( ph <-> ps ) ) $.
    wfis2f.5 $e |- ( y e. A -> ( A. z e. Pred ( R , A , y ) ps -> ph ) ) $.
    $( Well-Ordered Induction schema, using implicit substitution.
       (Contributed by Scott Fenton, 29-Jan-2011.) $)
    wfis2f $p |- ( y e. A -> ph ) $=
      ( wwe wse wral wfis2fg mp2an rspec ) ACEEFLEFMACENGHABCDEFIJKOPQ $.
  $}

  ${
    $d A y z $.  $d ph z $.  $d ps y $.  $d R y z $.
    wfis2g.1 $e |- ( y = z -> ( ph <-> ps ) ) $.
    wfis2g.2 $e |- ( y e. A -> ( A. z e. Pred ( R , A , y ) ps -> ph ) ) $.
    $( Well-Ordered Induction Schema, using implicit substitution.
       (Contributed by Scott Fenton, 11-Feb-2011.) $)
    wfis2g $p |- ( ( R We A /\ R Se A ) -> A. y e. A ph ) $=
      ( nfv wfis2fg ) ABCDEFBCIGHJ $.
  $}

  ${
    $d A y z $.  $d ph z $.  $d ps y $.  $d R y z $.
    wfis2.1 $e |- R We A $.
    wfis2.2 $e |- R Se A $.
    wfis2.3 $e |- ( y = z -> ( ph <-> ps ) ) $.
    wfis2.4 $e |- ( y e. A -> ( A. z e. Pred ( R , A , y ) ps -> ph ) ) $.
    $( Well-Ordered Induction schema, using implicit substitution.
       (Contributed by Scott Fenton, 29-Jan-2011.) $)
    wfis2 $p |- ( y e. A -> ph ) $=
      ( wwe wse wral wfis2g mp2an rspec ) ACEEFKEFLACEMGHABCDEFIJNOP $.
  $}

  ${
    $d A y z $.  $d B y $.  $d ch y $.  $d ph z $.  $d ps y $.  $d R y z $.
    wfis3.1 $e |- R We A $.
    wfis3.2 $e |- R Se A $.
    wfis3.3 $e |- ( y = z -> ( ph <-> ps ) ) $.
    wfis3.4 $e |- ( y = B -> ( ph <-> ch ) ) $.
    wfis3.5 $e |- ( y e. A -> ( A. z e. Pred ( R , A , y ) ps -> ph ) ) $.
    $( Well-Ordered Induction schema, using implicit substitution.
       (Contributed by Scott Fenton, 29-Jan-2011.) $)
    wfis3 $p |- ( B e. A -> ch ) $=
      ( wfis2 vtoclga ) ACDGFLABDEFHIJKMNO $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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 $.

  $( Define the ordinal predicate, which is true for a class that is transitive
     and is well-ordered by the membership relation.  Variant of definition of
     [BellMachover] p. 468.

     Some sources will define a notation for ordinal order corresponding to
     ` < ` and ` <_ ` but we just use ` e. ` and ` C_ ` respectively.

     (Contributed by NM, 17-Sep-1993.) $)
  df-ord $a |- ( Ord A <-> ( Tr A /\ _E We A ) ) $.

  $( 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 a nonempty ordinal
     that 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.  See ~ dflim2 , ~ dflim3 , and dflim4
     for alternate definitions.  (Contributed by NM, 22-Apr-1994.) $)
  df-lim $a |- ( Lim A <-> ( Ord A /\ A =/= (/) /\ A = U. A ) ) $.

  $( Define the successor of a class.  When applied to an ordinal number, the
     successor means the same thing as "plus 1" (see ~ oa1suc ).  Definition
     7.22 of [TakeutiZaring] p. 41, who use "+ 1" to denote this function.
     Definition 1.4 of [Schloeder] p. 1, similarly.  Ordinal natural numbers
     defined using this successor function and 0 as the empty set are also
     called von Neumann ordinals; 0 is the empty set {}, 1 is {0, {0}}, 2 is
     {1, {1}}, and so on.  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 ), so that the successor of any
     ordinal class is still an ordinal class ( ~ ordsuc ), simplifying certain
     proofs.  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 } ) $.

  $( Equality theorem for the ordinal predicate.  (Contributed by NM,
     17-Sep-1993.) $)
  ordeq $p |- ( A = B -> ( Ord A <-> Ord B ) ) $=
    ( wceq wtr cep wwe wa word treq weeq2 anbi12d df-ord 3bitr4g ) ABCZADZAEFZG
    BDZBEFZGAHBHNOQPRABIABEJKALBLM $.

  ${
    $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.  Part of Definition 1.2 of
     [Schloeder] p. 1.  (Contributed by NM, 8-Feb-2004.) $)
  elon2 $p |- ( A e. On <-> ( Ord A /\ A e. _V ) ) $=
    ( con0 wcel word cvv elex elong biadanii biancomi ) ABCZADZAECZJLKABFAEGHI
    $.

  $( 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 wne cuni w3a wlim ordeq neeq1 unieq eqeq12d 3anbi123d df-lim
    id 3bitr4g ) ABCZADZAEFZAAGZCZHBDZBEFZBBGZCZHAIBIRSUCTUDUBUFABJABEKRABUAUER
    PABLMNAOBOQ $.

  $( Membership well-orders every ordinal.  Proposition 7.4 of [TakeutiZaring]
     p. 36.  (Contributed by NM, 3-Apr-1994.) $)
  ordwe $p |- ( Ord A -> _E We A ) $=
    ( word wtr cep wwe df-ord simprbi ) ABACADEAFG $.

  $( An ordinal class is transitive.  (Contributed by NM, 3-Apr-1994.) $)
  ordtr $p |- ( Ord A -> Tr A ) $=
    ( word wtr cep wwe df-ord simplbi ) ABACADEAFG $.

  $( Membership is well-founded on an ordinal class.  In other words, an
     ordinal class is well-founded.  (Contributed by NM, 22-Apr-1994.) $)
  ordfr $p |- ( Ord A -> _E Fr A ) $=
    ( word cep wwe wfr ordwe wefr syl ) ABACDACEAFACGH $.

  $( 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 $.

  $( 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 ) $=
    ( wtr wss word w3a cep wwe wa wess ordwe impel anim2i 3impb df-ord sylibr )
    ACZABDZBEZFQAGHZIZAEQRSUARSITQRBGHTSABGJBKLMNAOP $.

  $( No ordinal class is a member of itself.  In other words, the membership
     relation is irreflexive on ordinal classes.  Theorem 2.2(i) of
     [BellMachover] p. 469, generalized to classes.  Theorem 1.9(i) of
     [Schloeder] p. 1.  We prove this without invoking the Axiom of Regularity.
     (Contributed by NM, 2-Jan-1994.) $)
  ordirr $p |- ( Ord A -> -. A e. A ) $=
    ( word cep wfr wcel wn ordfr efrirr syl ) ABACDAAEFAGAHI $.
  $( $j usage 'ordirr' avoids 'ax-reg'; $)

  $( 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 $.

  ${
    $d x B $.
    $( A nonempty subclass of an ordinal class has a minimal element.
       Proposition 7.5 of [TakeutiZaring] p. 36.  (Contributed by NM,
       18-Feb-2004.)  (Revised by David Abernethy, 16-Mar-2011.) $)
    tz7.5 $p |- ( ( Ord A /\ B C_ A /\ B =/= (/) ) ->
               E. x e. B ( B i^i x ) = (/) ) $=
      ( word cep wwe wss c0 wne cv cin wceq wrex ordwe wefrc syl3an1 ) BDBEFCBG
      CHICAJKHLACMBNABCOP $.
  $}

  ${
    $d x y z A $.  $d x y z B $.
    $( An element of an ordinal class is ordinal.  Proposition 7.6 of
       [TakeutiZaring] p. 36.  Lemma 1.3 of [Schloeder] p. 1.  (Contributed by
       NM, 23-Apr-1994.) $)
    ordelord $p |- ( ( Ord A /\ B e. A ) -> Ord B ) $=
      ( vx vz vy word wcel cv wa wceq eleq1 anbi2d wtr cep wwe wel wal w3a syl
      wi ordeq imbi12d simpll 3anrot 3anass bitr3i ordtr biimtrrid impl expcomd
      trel3 trel imp31 adantrl simplr ordwe wetrep syl13anc ex pm2.43d alrimivv
      sylan dftr2 sylibr wss trss wess syl6ci df-ord sylanbrc vtoclg anabsi7
      imp ) AFZBAGZBFZVNCHZAGZIZVQFZTVNVOIZVPTCBAVQBJZVSWAVTVPWBVRVOVNVQBAKLVQB
      UAUBVSVQMZVQNOZVTVSDEPZECPZIZDCPZTZEQDQWCVSWIDEVSWGWHVSWGWIVSWGIVNDHZAGZE
      HZAGZVRWIVNVRWGUCVNVRWGWKVRWGIZWEWFVRRZVNWKWOVRWEWFRWNVRWEWFUDVRWEWFUEUFV
      NAMZWOWKTAUGZAWJWLVQUKSUHUIVSWFWMWEVNVRWFWMVNWFVRWMVNWPWFVRIWMTWQAWLVQULS
      UJUMUNVNVRWGUOVNANOZWKWMVRRWIAUPZDECAUQVBURUSUTVADEVQVCVDVNVRWDVNVRVQAVEZ
      WRWDVNWPVRWTTWQAVQVFSWSVQANVGVHVMVQVIVJVKVL $.
  $}

  ${
    $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.  Lemma 1.3 of
     [Schloeder] p. 1.  (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.  Lemma 1.3 of [Schloeder] p. 1.  (Contributed by
     NM, 26-Oct-2003.) $)
  onelon $p |- ( ( A e. On /\ B e. A ) -> B e. On ) $=
    ( con0 wcel word eloni ordelon sylan ) ACDAEBADBCDAFABGH $.

  ${
    $d x y A $.  $d x y B $.
    $( A transitive class belongs to an ordinal class iff it is strictly
       included in it.  Proposition 7.7 of [TakeutiZaring] p. 37.  (Contributed
       by NM, 5-May-1994.) $)
    tz7.7 $p |- ( ( Ord A /\ Tr B ) ->
                ( B e. A <-> ( B C_ A /\ B =/= A ) ) ) $=
      ( vx vy word wtr wa wcel wss wne wi cep c0 cv wceq imp32 wn nsyli adantll
      imp wfr ordtr ordfr tz7.2 3exp sylc adantr cdif pssdifn0 wrex difss tz7.5
      cin mp3an2 eldifi difin0ss com12 syl56 syl ad2antrr eleq1w biimpcd eldifn
      trss adantl expcomd ex adantld w3o wwe ordwe ssel2 anim12i wecmpep syl2an
      trel adantlr ecase23d exp44 com34 imp31 ssrdv adantrr ad2antrl rexlimdvaa
      eqssd eqeltrrd syl5 exp4b com23 adantrd pm2.43i syl7 exp4a pm2.43d impbid
      impd ) AEZBFZGZBAHZBAIZBAJZGZWRXAXDKZWSWRAFZALUAZXEAUBZAUCXFXGXAXDABUDUEU
      FUGWTXBXCXAWTXBXCXAKWTXBXBXCXAXDABUHZMJZWTXBXABAUIWTXBXJXAKZKZWTWRXLWSWTX
      BWRXKWTXBWRXJXAWRXJGXICNZUMMOZCXIUJZWTXBGZXAWRXIAIXJXOABUKCAXIULUNXPXNXAC
      XIXPXMXIHZXNGGZXMBAXRXMBXPXQXNXMBIZWRXQXNXSKZKZWSXBWRXFYAXHXQXMAHZXFXMAIZ
      XTXMABUOZAXMVDXNYCXSABXMUPUQURUSUTPXPXQBXMIXNXPXQGDBXMWTXBXQDNZBHZYEXMHZK
      WTXBYFXQYGWTXBYFXQYGWTXBYFGZXQGZGYGYEXMOZXMYEHZYIYJQZWTYFXQYLXBYFXQYLYFYJ
      XMBHZXQYJYFYMDCBVAVBXMABVCZRTSVEWSYIYKQZWRWSYHXQYOWSYFXQYOKZXBWSYFYPWSYFG
      YKYMXQWSYFYKYMKWSYKYFYMBXMYEVPVFTYNRVGVHPSWRYIYGYJYKVIZWSWRALVJYEAHZYBGYQ
      YIAVKYHYRXQYBBAYEVLYDVMDCAVNVOVQVRVSVTWAWBWCWFXQYBXPXNYDWDWGWEWHWIWJWKWLW
      MWNWOWQWP $.
  $}

  $( For ordinal classes, membership is equivalent to strict inclusion.
     Corollary 7.8 of [TakeutiZaring] p. 37.  (Contributed by NM,
     25-Nov-1995.) $)
  ordelssne $p |- ( ( Ord A /\ Ord B ) ->
              ( A e. B <-> ( A C_ B /\ A =/= B ) ) ) $=
    ( word wcel wss wne wa wb wtr ordtr tz7.7 sylan2 ancoms ) BCZACZABDABEABFGH
    ZONAIPAJBAKLM $.

  $( For ordinal classes, membership is equivalent to strict inclusion.
     Corollary 7.8 of [TakeutiZaring] p. 37.  (Contributed by NM,
     17-Jun-1998.) $)
  ordelpss $p |- ( ( Ord A /\ Ord B ) -> ( A e. B <-> A C. B ) ) $=
    ( word wa wcel wss wne wpss ordelssne df-pss bitr4di ) ACBCDABEABFABGDABHAB
    IABJK $.

  $( For ordinal classes, inclusion is equivalent to membership or equality.
     (Contributed by NM, 25-Nov-1995.)  (Proof shortened by Andrew Salmon,
     25-Jul-2011.) $)
  ordsseleq $p |- ( ( Ord A /\ Ord B ) ->
                  ( A C_ B <-> ( A e. B \/ A = B ) ) ) $=
    ( word wa wss wpss wceq wo wcel sspss ordelpss orbi1d bitr4id ) ACBCDZABEAB
    FZABGZHABIZPHABJNQOPABKLM $.

  $( 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 $.

  $( A trichotomy law for ordinals.  Proposition 7.10 of [TakeutiZaring] p. 38.
     Theorem 1.9(iii) of [Schloeder] p. 1.  (Contributed by NM, 10-May-1994.)
     (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
  ordtri3or $p |- ( ( Ord A /\ Ord B ) -> ( A e. B \/ A = B \/ B e. A ) ) $=
    ( word wa wcel wceq w3o wpss wss wo ordin sylib inss1 ordsseleq mpbii sylan
    cin wn ord dfss2 ordirr syl ianor elin incom eleq1i anbi2i xchnxbir anabss1
    bitri imbitrrdi anabss4 orim12d sspsstri ordelpss biidd wb ancoms 3orbi123d
    mpd mpbird ) ACZBCZDZABEZABFZBAEZGABHZVFBAHZGZVDABIZBAIZJZVJVDABQZAEZRZBAQZ
    BEZRZJZVMVDVNVNEZRZVTVDVNCZWBABKZVNUAUBVOVRDZVTWAVOVRUCWAVOVNBEZDWEVNABUDWF
    VRVOVNVQBABUEUFUGUJUHLVDVPVKVSVLVDVPVNAFZVKVDVOWGVBVCVOWGJZVDWCVBWHWDWCVBDV
    NAIWHABMVNANOPUISABTUKVDVSVQBFZVLVDVRWIVBVCVRWIJZVCVBDVQCZVCWJBAKWKVCDVQBIW
    JBAMVQBNOPULSBATUKUMUTABUNLVDVEVHVFVFVGVIABUOVDVFUPVCVBVGVIUQBAUOURUSVA $.

  $( A trichotomy law for ordinals.  (Contributed by NM, 25-Mar-1995.)  (Proof
     shortened by Andrew Salmon, 25-Jul-2011.) $)
  ordtri1 $p |- ( ( Ord A /\ Ord B ) -> ( A C_ B <-> -. B e. A ) ) $=
    ( word wa wss wcel wceq wo wn ordsseleq wi imnan sylibr ordirr eleq2 notbid
    ordn2lp syl5ibrcom jaao w3o ordtri3or df-3or sylib orcomd ord impbid bitrd
    ) ACZBCZDZABEABFZABGZHZBAFZIZABJUJUMUOUHUKUOUIULUHUKUNDIUKUOKABQUKUNLMUIUOU
    LBBFZIBNULUNUPABBOPRSUJUNUMUJUMUNUJUKULUNTUMUNHABUAUKULUNUBUCUDUEUFUG $.

  $( A trichotomy law for ordinal numbers.  (Contributed by NM, 6-Nov-2003.) $)
  ontri1 $p |- ( ( A e. On /\ B e. On ) -> ( A C_ B <-> -. B e. A ) ) $=
    ( con0 wcel word wss wn wb eloni ordtri1 syl2an ) ACDAEBEABFBADGHBCDAIBIABJ
    K $.

  $( A trichotomy law for ordinals.  (Contributed by NM, 25-Nov-1995.) $)
  ordtri2 $p |- ( ( Ord A /\ Ord B ) ->
               ( A e. B <-> -. ( A = B \/ B e. A ) ) ) $=
    ( word wa wceq wcel wo wn wb wss ordsseleq eqcom orbi2i orcom bitri ordtri1
    bitrdi bitr3d ancoms con2bid ) ACZBCZDABEZBAFZGZABFZUBUAUEUFHZIUBUADZBAJZUE
    UGUHUIUDBAEZGZUEBAKUKUDUCGUEUJUCUDBALMUDUCNOQBAPRST $.

  $( A trichotomy law for ordinals.  (Contributed by NM, 18-Oct-1995.)  (Proof
     shortened by Andrew Salmon, 25-Jul-2011.)  (Proof shortened by JJ,
     24-Sep-2021.) $)
  ordtri3 $p |- ( ( Ord A /\ Ord B ) ->
               ( A = B <-> -. ( A e. B \/ B e. A ) ) ) $=
    ( word wa wceq wcel wo ordirr adantl eleq2 notbid syl5ibrcom pm4.71d pm5.61
    wn pm4.52 bitr3i bitrdi ordtri2 orbi1d bitr4d ) ACZBCZDZABEZUEBAFZGZOZUFGZO
    ZABFZUFGZOUDUEUEUFOZDZUJUDUEUMUDUMUEBBFZOZUCUPUBBHIUEUFUOABBJKLMUNUGUMDUJUE
    UFNUGUFPQRUDULUIUDUKUHUFABSTKUA $.

  $( A trichotomy law for ordinals.  (Contributed by NM, 1-Nov-2003.)  (Proof
     shortened by Andrew Salmon, 25-Jul-2011.) $)
  ordtri4 $p |- ( ( Ord A /\ Ord B ) ->
               ( A = B <-> ( A C_ B /\ -. A e. B ) ) ) $=
    ( wceq wss wa word wcel wn eqss wb ordtri1 ancoms anbi2d bitrid ) ABCABDZBA
    DZEAFZBFZEZOABGHZEABISPTORQPTJBAKLMN $.

  $( 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 $.

  ${
    $d x y z $.
    $( The ordinal class is well-founded.  This proof does not require the
       axiom of regularity.  This lemma is used in ~ ordon (through ~ epweon )
       in order to eliminate the need for the axiom of regularity.
       (Contributed by NM, 17-May-1994.) $)
    onfr $p |- _E Fr On $=
      ( vx vz vy con0 cep wfr cv wss c0 wne cin wceq wrex dfepfr wel wex eqeq1d
      wa wi wcel n0 ineq2 rspcev adantll inss1 word ssel2 eloni ordfr inss2 vex
      3syl inex1 epfrc mp3an2 sylan wb inass syl elinel2 ordelss syl2an sseqin2
      sylib ineq2d eqtrid rexbidva adantr mpbid ssrexv mpsyl pm2.61dane exlimdv
      ex biimtrid imp mpgbir ) DEFAGZDHZVRIJZRVRBGZKZILZBVRMZSAABDNVSVTWDVTCAOZ
      CPVSWDCVRUAVSWEWDCVSWEWDVSWERZWDVRCGZKZIWEWHILZWDVSWCWIBWGVRWAWGLWBWHIWAW
      GVRUBQUCUDWHVRHWFWHIJZRZWCBWHMZWDVRWGUEWKWHWAKZILZBWHMZWLWFWGEFZWJWOWFWGD
      TZWGUFZWPVRDWGUGZWGUHZWGUIULWPWHWGHWJWOVRWGUJBWGWHVRWGAUKUMUNUOUPWFWOWLUQ
      WJWFWNWCBWHWFWAWHTZRZWMWBIXBWMVRWGWAKZKWBVRWGWAURXBXCWAVRXBWAWGHZXCWALWFW
      RBCOXDXAWFWQWRWSWTUSWAVRWGUTWGWAVAVBWAWGVCVDVEVFQVGVHVIWCBWHVRVJVKVLVNVMV
      OVPVQ $.
    $( $j usage 'onfr' avoids 'ax-reg'; $)
  $}

  $( Relationship between membership and proper subset of an ordinal number.
     (Contributed by NM, 15-Sep-1995.) $)
  onelpss $p |- ( ( A e. On /\ B e. On ) ->
               ( A e. B <-> ( A C_ B /\ A =/= B ) ) ) $=
    ( con0 wcel word wss wne wa wb eloni ordelssne syl2an ) ACDAEBEABDABFABGHIB
    CDAJBJABKL $.

  $( Relationship between subset and membership of an ordinal number.
     (Contributed by NM, 15-Sep-1995.) $)
  onsseleq $p |- ( ( A e. On /\ B e. On ) ->
                 ( A C_ B <-> ( A e. B \/ A = B ) ) ) $=
    ( con0 wcel word wss wceq wo wb eloni ordsseleq syl2an ) ACDAEBEABFABDABGHI
    BCDAJBJABKL $.

  $( 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 $.

  $( The elementhood relation on the ordinals is complete, so we have triality.
     Theorem 1.9(iii) of [Schloeder] p. 1.  See ~ ordtri3or .  (Contributed by
     RP, 15-Jan-2025.) $)
  oneltri $p |- ( ( A e. On /\ B e. On )
                 -> ( A e. B \/ B e. A \/ A = B ) ) $=
    ( con0 wcel wa wceq w3o word eloni ordtri3or syl2an 3orcomb sylib ) ACDZBCD
    ZEABDZABFZBADZGZPRQGNAHBHSOAIBIABJKPQRLM $.

  $( 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 classes.  (Contributed by NM, 12-Dec-2004.)
     (Proof shortened by Andrew Salmon, 25-Jul-2011.) $)
  ordtr2 $p |- ( ( Ord A /\ Ord C ) -> ( ( A C_ B /\ B e. C ) -> A e. C ) ) $=
    ( word wa wss wcel wpss wi ordelord ex ancld anc2li ordelpss sspsstr expcom
    biimtrdi com23 imp32 com12 syl9 impd adantl sylibrd ) ADZCDZEABFZBCGZEZACHZ
    ACGUFUIUJIUEUFUGUHUJUFUHUFUHBDZEZEZUGUJUFUHULUFUHUKUFUHUKCBJKLMUMUGUJUFUHUK
    UGUJIZUFUKUHUNUKUFUHUNIUKUFEUHBCHZUNBCNUGUOUJABCOPQPRSTUAUBUCACNUD $.

  $( Transitive law for ordinal classes.  (Contributed by Mario Carneiro,
     30-Dec-2014.)  (Proof shortened by JJ, 24-Sep-2021.) $)
  ordtr3 $p |- ( ( Ord B /\ Ord C ) -> ( A e. B -> ( A e. C \/ C e. B ) ) ) $=
    ( word wa wcel wo wn wss nelss adantl wb ordtri1 con2bid adantr mpbird expr
    orrd ex ) BDCDEZABFZACFZCBFZGTUAEUBUCTUAUBHZUCTUAUDEZEUCBCIZHZUEUGTABCJKTUC
    UGLUETUFUCBCMNOPQRS $.

  $( Transitive law for ordinal numbers.  Theorem 7M(b) of [Enderton] p. 192.
     Theorem 1.9(ii) of [Schloeder] p. 1.  (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 $.

  $( Transitive law for ordinal numbers.  Exercise 3 of [TakeutiZaring] p. 40.
     (Contributed by NM, 6-Nov-2003.) $)
  ontr2 $p |- ( ( A e. On /\ C e. On ) ->
              ( ( A C_ B /\ B e. C ) -> A e. C ) ) $=
    ( con0 wcel word wss wa wi eloni ordtr2 syl2an ) ADEAFCFABGBCEHACEICDEAJCJA
    BCKL $.

  ${
    $d A b $.  $d C b $.
    $( Ordinal less than is equivalent to having an ordinal between them.
       (Contributed by Scott Fenton, 8-Aug-2024.) $)
    onelssex $p |- ( ( A e. On /\ C e. On ) ->
     ( A e. C <-> E. b e. C A C_ b ) ) $=
      ( con0 wcel wa cv wss wrex ssid sseq2 rspcev mpan2 ontr2 expcomd rexlimdv
      impbid2 ) ADEBDEFZABEZACGZHZCBIZSAAHZUBAJUAUCCABTAAKLMRUASCBRUATBESATBNOP
      Q $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( The union of an ordinal stays the same if a subset equal to one of its
       elements is removed.  (Contributed by NM, 10-Dec-2004.) $)
    ordunidif $p |- ( ( Ord A /\ B e. A ) -> U. ( A \ B ) = U. A ) $=
      ( vx vy word wcel wa cv wss cdif wrex wi syl wn eldif adantl sseq2 rspcev
      cuni syl6 wral wceq con0 ordelon onelss eloni ordirr simplbi2 syl5 jctild
      mpd adantr biimpri ssid jctir ex pm2.61d ralrimiva unidif ) AEZBAFZGZCHZD
      HZIZDABJZKZCAUAVFSASUBVBVGCAVBVCAFZGZVCBFZVGVIVJBVFFZVCBIZGZVGVBVJVMLVHVB
      VJVLVKVBBUCFZVJVLLABUDZBVCUEMVBVNVKVOVAVNVKLUTVNBBFNZVAVKVNBEVPBUFBUGMVKV
      AVPBABOUHUIPUKUJULVEVLDBVFVDBVCQRTVHVJNZVGLVBVHVQVCVFFZVCVCIZGZVGVHVQVTVH
      VQGZVRVSVRWAVCABOUMVCUNUOUPVEVSDVCVFVDVCVCQRTPUQURCDABUSM $.

    $( If ` B ` is smaller than ` A ` , then it equals the intersection of the
       difference.  Exercise 11 in [TakeutiZaring] p. 44.  (Contributed by
       Andrew Salmon, 14-Nov-2011.) $)
    ordintdif $p |- ( ( Ord A /\ Ord B /\ ( A \ B ) =/= (/) )
      -> B = |^| ( A \ B ) ) $=
      ( vx cdif c0 wne word wss wn cint wceq ssdif0 necon3bbii w3a cv wcel crab
      dfdif2 wa ordtri1 inteqi con2bid wb ordelord an32s rabbidva inteqd intmin
      id syl2anr sylan9req ex sylbird 3impia eqtr2id syl3an3br ) ABDZEFAGZBGZAB
      HZIZBUQJZKUTUQEABLMURUSVANVBCOZBPIZCAQZJZBUQVECABRUAURUSVAVFBKZURUSSZVABA
      PZVGVHUTVIABTUBVHVIVGVHVIVFBVCHZCAQZJBVHVKVEVHVJVDCAURVCAPZUSVJVDUCZUSUSV
      CGVMURVLSUSUIAVCUDBVCTUJUEUFUGCBAUHUKULUMUNUOUP $.
  $}

  ${
    $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 $.
  $}

  ${
    $d x A $.  $d x B $.
    $( A way to show that an ordinal number equals the minimum of a collection
       of ordinal numbers: it must be in the collection, and it must not be
       larger than any member of the collection.  (Contributed by NM,
       14-Nov-2003.) $)
    oneqmini $p |- ( B C_ On -> ( ( A e. B /\ A. x e. A -. x e. B )
                    -> A = |^| B ) ) $=
      ( con0 wss wcel cv wn wral wa cint wceq ssint wi ssel anim12d ontri1 syl6
      wb expdimp pm5.74d con2b bitrdi bitrid biimprd expimpd intss1 a1i adantrd
      ralbidv2 jcad eqss imbitrrdi ) CDEZBCFZAGZCFZHZABIZJZBCKZEZVABEZJBVALUNUT
      VBVCUNUOUSVBUNUOJZVBUSVBBUPEZACIVDUSABCMVDVEURACBVDUQVENUQUPBFZHZNVFURNVD
      UQVEVGUNUOUQVEVGSZUNUOUQJBDFZUPDFZJVHUNUOVIUQVJCDBOCDUPOPBUPQRTUAUQVFUBUC
      UJUDUEUFUNUOVCUSUOVCNUNBCUGUHUIUKBVAULUM $.
  $}

  $( The empty set is an ordinal class.  Remark 1.5 of [Schloeder] p. 1.
     (Contributed by NM, 11-May-1994.) $)
  ord0 $p |- Ord (/) $=
    ( c0 word wtr cep wwe tr0 we0 df-ord mpbir2an ) ABACADEFDGAHI $.

  $( The empty set is an ordinal number.  Corollary 7N(b) of [Enderton] p. 193.
     Remark 1.5 of [Schloeder] p. 1.  (Contributed by NM, 17-Sep-1993.) $)
  0elon $p |- (/) e. On $=
    ( c0 con0 wcel word ord0 0ex elon mpbir ) ABCADEAFGH $.

  $( A nonempty ordinal contains the empty set.  Lemma 1.10 of [Schloeder]
     p. 2.  (Contributed by NM, 25-Nov-1995.) $)
  ord0eln0 $p |- ( Ord A -> ( (/) e. A <-> A =/= (/) ) ) $=
    ( word c0 wcel ne0i wceq wo wi ord0 wa wn noel ordtri2 con2bid mpbiri mpan2
    wne neor sylib impbid2 ) ABZCADZACQZACEUAACFUBGZUCUBHUACBZUDIUAUEJZUDACDZKA
    LUFUGUDACMNOPUBACRST $.

  $( An ordinal number contains zero iff it is nonzero.  (Contributed by NM,
     6-Dec-2004.) $)
  on0eln0 $p |- ( A e. On -> ( (/) e. A <-> A =/= (/) ) ) $=
    ( con0 wcel word c0 wne wb eloni ord0eln0 syl ) ABCADEACAEFGAHAIJ $.

  $( An alternate definition of a limit ordinal.  (Contributed by NM,
     4-Nov-2004.) $)
  dflim2 $p |- ( Lim A <-> ( Ord A /\ (/) e. A /\ A = U. A ) ) $=
    ( wlim word c0 wne cuni wceq w3a wcel df-lim ord0eln0 anbi1d pm5.32i 3anass
    wa 3bitr4i bitr4i ) ABACZADEZAAFGZHZRDAIZTHZAJRUBTOZORSTOZOUCUARUDUERUBSTAK
    LMRUBTNRSTNPQ $.

  $( 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 wne cuni wceq df-lim simp1bi ) ABACADEAAFGAHI $.

  $( A limit ordinal is its own supremum (union).  Lemma 2.13 of [Schloeder]
     p. 5.  (Contributed by NM, 4-May-1995.) $)
  limuni $p |- ( Lim A -> A = U. A ) $=
    ( wlim word c0 wne cuni wceq df-lim simp3bi ) ABACADEAAFGAHI $.

  $( 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 c0 wcel wne wceq nlim0 limeq mtbiri necon2ai word wb limord ord0eln0
    syl mpbird ) ABZCADZACEZQACACFQCBGACHIJQAKRSLAMANOP $.

  $( 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 0elon ne0ii ) ABCD $.

  $( 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.  Lemma 1.13 of [Schloeder] p. 2.
     (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 csn cun c0 csuc snprc biimpi uneq2d df-suc un0 eqcomi 3eqtr4g
    wceq ) ABCDZAAEZFAGFZAHAQRGAQRGPAIJKALSAAMNO $.

  ${
    $( The union of the successor of a set is equal to the binary union of that
       set with its union.  (Contributed by NM, 30-Aug-1993.)  Extract from
       ~ unisuc .  (Revised by BJ, 28-Dec-2024.) $)
    unisucs $p |- ( A e. V -> U. suc A = ( U. A u. A ) ) $=
      ( wcel csuc cuni csn cun wceq df-suc unieqi uniun unisng uneq2d 3eqtrd
      a1i ) ABCZADZEZAAFZGZEZAEZSEZGZUBAGRUAHPQTAIJOUAUDHPASKOPUCAUBABLMN $.
  $}

  $( A transitive class is equal to the union of its successor, closed form.
     Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton]
     p. 73.  (Contributed by NM, 30-Aug-1993.)  Generalize from ~ unisuc .
     (Revised by BJ, 28-Dec-2024.) $)
  unisucg $p |- ( A e. V -> ( Tr A <-> U. suc A = A ) ) $=
    ( wcel cuni wss cun wceq wtr csuc ssequn1 a1i df-tr unisucs eqeq1d 3bitr4d
    wb ) ABCZADZAEZRAFZAGZAHZAIDZAGSUAPQRAJKUBSPQALKQUCTAABMNO $.

  ${
    unisuc.1 $e |- A e. _V $.
    $( A transitive class is equal to the union of its successor, inference
       form.  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 ) $=
      ( cvv wcel wtr csuc cuni wceq wb unisucg ax-mp ) ACDAEAFGAHIBACJK $.
  $}

  $( 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).  Lemma
     1.7 of [Schloeder] p. 1.  (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 NM, 3-Apr-1995.) $)
  nsuceq0 $p |- suc A =/= (/) $=
    ( csuc c0 cvv wcel wceq wn noel sucidg eleq2 syl5ibcom mtoi 0ex eleq1 con3i
    mpbiri sucprc eqeq1d mtbird pm2.61i neir ) ABZCADEZUBCFZGUCUDACEZAHUCAUBEUD
    UEADIUBCAJKLUCGZUDACFZUGUCUGUCCDEMACDNPOUFUBACAQRSTUA $.

  ${
    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.)  (Proof shortened by JJ, 24-Sep-2021.) $)
    suctr $p |- ( Tr A -> Tr suc A ) $=
      ( vz vy wtr cv wcel csuc wa wi wal wceq elsuci trel expdimp eleq2 biimpcd
      wo adantl jaod syl5 expimpd elelsuc syl6 alrimivv dftr2 sylibr ) ADZBEZCE
      ZFZUIAGZFZHZUHUKFZIZCJBJUKDUGUOBCUGUMUHAFZUNUGUJULUPULUIAFZUIAKZQUGUJHZUP
      UIALUSUQUPURUGUJUQUPAUHUIMNUJURUPIUGURUJUPUIAUHOPRSTUAUHAUBUCUDBCUKUEUF
      $.
  $}

  $( 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 wa trel cvv wss sssucid ssexg mpan sucidg syl ancri impel )
    ACBBDZEZQAEZFBAESABQGSRSBHEZRBQISTBJBQAKLBHMNOP $.

  $( A member of the successor of a transitive class is a subclass of it.
     Lemma 1.13 of [Schloeder] p. 2.  (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 $.

  $( An ordinal is a subset of another ordinal if and only if it belongs to its
     successor.  (Contributed by NM, 28-Nov-2003.) $)
  ordsssuc $p |- ( ( A e. On /\ Ord B ) -> ( A C_ B <-> A e. suc B ) ) $=
    ( con0 wcel word wa wss wceq wo csuc wb eloni ordsseleq sylan elsucg adantr
    bitr4d ) ACDZBEZFABGZABDABHIZABJDZRAESTUAKALABMNRUBUAKSABCOPQ $.

  $( A subset of an ordinal number belongs to its successor.  (Contributed by
     NM, 15-Sep-1995.) $)
  onsssuc $p |- ( ( A e. On /\ B e. On ) -> ( A C_ B <-> A e. suc B ) ) $=
    ( con0 wcel word wss csuc wb eloni ordsssuc sylan2 ) BCDACDBEABFABGDHBIABJK
    $.

  $( An ordinal subset of an ordinal number belongs to its successor.
     (Contributed by NM, 1-Feb-2005.)  (Proof shortened by Andrew Salmon,
     12-Aug-2011.) $)
  ordsssuc2 $p |- ( ( Ord A /\ B e. On ) -> ( A C_ B <-> A e. suc B ) ) $=
    ( cvv wcel word con0 wa wss csuc wb wi elong biimprd anim1d onsssuc syl6 wn
    annim ssexg ex elex a1d pm5.21ni sylbi expcom adantld pm2.61i ) ACDZAEZBFDZ
    GZABHZABIZDZJZKUHUKAFDZUJGUOUHUIUPUJUHUPUIACLMNABOPUHQZUJUOUIUJUQUOUJUQGUJU
    HKZQUOUJUHRULURUNULUJUHABFSTUNUHUJAUMUAUBUCUDUEUFUG $.

  ${
    $d x A $.  $d x B $.
    $( When its successor is subtracted from a class of ordinal numbers, an
       ordinal number is less than the minimum of the resulting subclass.
       (Contributed by NM, 1-Dec-2003.) $)
    onmindif $p |- ( ( A C_ On /\ B e. On ) -> B e. |^| ( A \ suc B ) ) $=
      ( vx con0 wss wcel wa csuc cdif cint cv wral wn eldif wi wb ssel2 onsssuc
      ontri1 bitr3d con1bid sylan biimpd exp31 com23 imp4b elintg adantl mpbird
      biimtrid ralrimiv ) ADEZBDFZGZBABHZIZJFZBCKZFZCUPLZUNUSCUPURUPFURAFZURUOF
      ZMZGUNUSURAUONULUMVAVCUSULVAUMVCUSOZULVAUMVDULVAGZUMGVCUSVEURDFZUMVCUSPAD
      URQVFUMGZUSVBVGURBEUSMVBURBSURBRTUAUBUCUDUEUFUJUKUMUQUTPULCBUPDUGUHUI $.
  $}

  $( There is no set between an ordinal class and its successor.  Generalized
     Proposition 7.25 of [TakeutiZaring] p. 41.  Lemma 1.15 of [Schloeder]
     p. 2.  (Contributed by NM, 21-Jun-1998.)  (Proof shortened by JJ,
     24-Sep-2021.) $)
  ordnbtwn $p |- ( Ord A -> -. ( A e. B /\ B e. suc A ) ) $=
    ( word wcel wceq wo wa ordirr wn ordn2lp wi pm2.24 eleq2 biimpac a1d jaodan
    csuc syl5com mtod elsuci anim2i nsyl ) ACZABDZBADZBAEZFZGZUDBAQDZGUCUHAADZA
    HUCUDUEGZIZUHUJABJUDUEULUJKUFUKUJLUDUFGUJULUFUDUJBAAMNOPRSUIUGUDBATUAUB $.

  $( There is no set between an ordinal number and its successor.  Proposition
     7.25 of [TakeutiZaring] p. 41.  Lemma 1.15 of [Schloeder] p. 2.
     (Contributed by NM, 9-Jun-1994.) $)
  onnbtwn $p |- ( A e. On -> -. ( A e. B /\ B e. suc A ) ) $=
    ( con0 wcel word csuc wa wn eloni ordnbtwn syl ) ACDAEABDBAFDGHAIABJK $.

  $( 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 $.

  $( 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 $.

  $( 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 $.

  $( A trichotomy law for ordinal classes.  (Contributed by NM, 13-Sep-2003.)
     (Proof shortened by Andrew Salmon, 12-Aug-2011.) $)
  ordtri2or $p |- ( ( Ord A /\ Ord B ) -> ( A e. B \/ B C_ A ) ) $=
    ( word wa wcel wss wn wb ordtri1 ancoms biimprd orrd ) ACZBCZDZABEZBAFZOQPG
    ZNMQRHBAIJKL $.

  $( A trichotomy law for ordinal classes.  (Contributed by NM, 2-Nov-2003.) $)
  ordtri2or2 $p |- ( ( Ord A /\ Ord B ) -> ( A C_ B \/ B C_ A ) ) $=
    ( word wa wcel wss wo ordtri2or wi ordelss ex orim1d adantl mpd ) ACZBCZDAB
    EZBAFZGZABFZRGZABHPSUAIOPQTRPQTBAJKLMN $.

  $( A consequence of total ordering for ordinal classes.  Similar to
     ~ ordtri2or2 .  (Contributed by David Moews, 1-May-2017.) $)
  ordtri2or3 $p |- ( ( Ord A /\ Ord B ) ->
                     ( A = ( A i^i B ) \/ B = ( A i^i B ) ) ) $=
    ( word wa wss wo cin wceq ordtri2or2 dfss sseqin2 eqcom bitri orbi12i sylib
    ) ACBCDABEZBAEZFAABGZHZBRHZFABIPSQTABJQRBHTBAKRBLMNO $.

  $( The intersection of two ordinal classes is an element of a third if and
     only if either one of them is.  (Contributed by David Moews, 1-May-2017.)
     (Proof shortened by JJ, 24-Sep-2021.) $)
  ordelinel $p |- ( ( Ord A /\ Ord B /\ Ord C ) ->
                    ( ( A i^i B ) e. C <-> ( A e. C \/ B e. C ) ) ) $=
    ( word w3a cin wcel wo wceq ordtri2or3 3adant3 eleq1a orim12d syl5com ordin
    wi wa wss ordtr2 mpani inss1 inss2 jaod stoic3 impbid ) ADZBDZCDZEZABFZCGZA
    CGZBCGZHZUIAUJIZBUJIZHZUKUNUFUGUQUHABJKUKUOULUPUMUJCALUJCBLMNUFUGUJDZUHUNUK
    PABOURUHQZULUKUMUSUJARULUKABUAUJACSTUSUJBRUMUKABUBUJBCSTUCUDUE $.

  $( Property of a subclass of the maximum (i.e. union) of two ordinals.
     (Contributed by NM, 28-Nov-2003.) $)
  ordssun $p |- ( ( Ord B /\ Ord C ) ->
               ( A C_ ( B u. C ) <-> ( A C_ B \/ A C_ C ) ) ) $=
    ( word wa cun wss wo wi ordtri2or2 wceq wb ssequn1 sseq2 sylbi olc biimtrdi
    ssequn2 orc jaoi syl ssun impbid1 ) BDCDEZABCFZGZABGZACGZHZUDBCGZCBGZHUFUII
    ZBCJUJULUKUJUFUHUIUJUECKUFUHLBCMUECANOUHUGPQUKUFUGUIUKUEBKUFUGLCBRUEBANOUGU
    HSQTUAABCUBUC $.

  $( The maximum (i.e. union) of two ordinals is either one or the other.
     Similar to Exercise 14 of [TakeutiZaring] p. 40.  (Contributed by NM,
     28-Nov-2003.) $)
  ordequn $p |- ( ( Ord B /\ Ord C ) ->
               ( A = ( B u. C ) -> ( A = B \/ A = C ) ) ) $=
    ( word wa wss wo cun wceq ordtri2or2 orcomd ssequn2 bitr4id ssequn1 orbi12d
    eqeq1 syl5ibcom ) BDCDEZCBFZBCFZGABCHZIZABIZACIZGRTSBCJKUBSUCTUDUBSUABIUCCB
    LAUABPMUBTUACIUDBCNAUACPMOQ $.

  $( The maximum (i.e., union) of two ordinals is ordinal.  Exercise 12 of
     [TakeutiZaring] p. 40.  (Contributed by NM, 28-Nov-2003.) $)
  ordun $p |- ( ( Ord A /\ Ord B ) -> Ord ( A u. B ) ) $=
    ( word wa cun wceq wo eqid ordequn mpi ordeq biimprcd jaao mpd ) ACZBCZDZAB
    EZAFZRBFZGZRCZQRRFUARHRABIJOSUBPTSUBORAKLTUBPRBKLMN $.

  $( The union of two ordinals is in a third iff both of the first two are.
     (Contributed by Scott Fenton, 10-Sep-2024.) $)
  onunel $p |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( ( A u. B ) e. C <->
  ( A e. C /\ B e. C ) ) ) $=
    ( con0 wcel w3a wss cun wa wb wceq biimpi eleq1d adantl ontr2 expdimp bitrd
    wi word eloni ssequn1 3adant2 pm4.71rd ssequn2 3adant1 wo ordtri2or2 syl2an
    pm4.71d 3adant3 mpjaodan ) ADEZBDEZCDEZFZABGZABHZCEZACEZBCEZIZJBAGZUOUPIZUR
    UTVAUPURUTJUOUPUQBCUPUQBKABUALMNVCUTUSUOUPUTUSULUNUPUTIUSRUMABCOUBPUCQUOVBI
    ZURUSVAVBURUSJUOVBUQACVBUQAKBAUDLMNVDUSUTUOVBUSUTUMUNVBUSIUTRULBACOUEPUIQUL
    UMUPVBUFZUNULASBSVEUMATBTABUGUHUJUK $.

  ${
    $d x A $.  $d x B $.
    $( A subclass relationship for union and successor of ordinal classes.
       (Contributed by NM, 28-Nov-2003.) $)
    ordunisssuc $p |- ( ( A C_ On /\ Ord B ) ->
                      ( U. A C_ B <-> A C_ suc B ) ) $=
      ( vx con0 word wa cv wral csuc wcel cuni wb ssel2 ordsssuc sylan ralbidva
      wss an32s unissb dfss3 3bitr4g ) ADQZBEZFZCGZBQZCAHUEBIZJZCAHAKBQAUGQUDUF
      UHCAUBUEAJZUCUFUHLZUBUIFUEDJUCUJADUEMUEBNORPCABSCAUGTUA $.
  $}

  $( 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 wcel wa csuc wceq wn wo wi word eloni ordn2lp wss sucssel elsuci ord
    syl5 com12 syl9 pm3.13 3syl adantr eqimss eqimss2 eqcom imbitrdi jaao suceq
    mpd impbid1 ) ACDZBCDZEZAFZBFZGZABGZUNABDZHZBADZHZIZUQURJZULVCUMULAKUSVAEHV
    CALABMUSVAUAUBUCULUTVDUMVBULUQAUPDZUTURUQUOUPNULVEUOUPUDAUPCORVEUTURVEUSURA
    BPQSTUMUQBUODZVBURUQUPUONUMVFUPUOUEBUOCORVFVBURVFVBBAGZURVFVAVGBAPQBAUFUGST
    UHUJABUIUK $.

  $( The union of two ordinals is an ordinal.  (Contributed by Scott Fenton,
     9-Aug-2024.) $)
  onun2 $p |- ( ( A e. On /\ B e. On ) -> ( A u. B ) e. On ) $=
    ( con0 wcel wss cun wceq ssequn1 eleq1a adantl biimtrid ssequn2 adantr word
    wa wi wo eloni ordtri2or2 syl2an mpjaod ) ACDZBCDZOZABEZABFZCDZBAEZUEUFBGZU
    DUGABHUCUIUGPUBBCUFIJKUHUFAGZUDUGBALUBUJUGPUCACUFIMKUBANBNUEUHQUCARBRABSTUA
    $.

  $( An ordinal number is a transitive class.  (Contributed by NM,
     11-Jun-1994.)  Put in closed form.  (Resised by BJ, 28-Dec-2024.) $)
  ontr $p |- ( A e. On -> Tr A ) $=
    ( con0 wcel word wtr eloni ordtr syl ) ABCADAEAFAGH $.

  $( An ordinal number is equal to the union of its successor.  (Contributed by
     NM, 12-Jun-1994.)  Generalize from ~ onunisuci .  (Revised by BJ,
     28-Dec-2024.) $)
  onunisuc $p |- ( A e. On -> U. suc A = A ) $=
    ( con0 wcel wtr csuc cuni wceq ontr unisucg mpbid ) ABCADAEFAGAHABIJ $.

  ${
    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 $.

    $( Obsolete version of ~ ontr as of 28-Dec-2024.  (Contributed by NM,
       11-Jun-1994.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    ontrciOLD $p |- Tr A $=
      ( con0 wcel wtr ontr ax-mp ) ACDAEBAFG $.

    $( 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 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 ordering law for ordinal numbers.  (Contributed by NM,
       13-Jun-1994.) $)
    onssneli $p |- ( A C_ B -> -. B e. A ) $=
      ( wss wcel ssel con0 word wn oneli eloni ordirr 3syl nsyli pm2.01d ) ABDZ
      BAEZPQBBEZQABBFQBGEBHRIABCJBKBLMNO $.

    $( An ordering law for ordinal numbers.  (Contributed by NM,
       13-Jun-1994.) $)
    onssnel2i $p |- ( B C_ A -> -. A e. B ) $=
      ( wss wcel onirri ssel mtoi ) BADABEAAEACFBAAGH $.

    $( 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 $=
      ( con0 wcel csuc cuni wceq onunisuc ax-mp ) ACDAEFAGBAHI $.

    ${
      on.2 $e |- B e. On $.
      $( Subset is equivalent to membership or equality for ordinal numbers.
         (Contributed by NM, 15-Sep-1995.) $)
      onsseli $p |- ( A C_ B <-> ( A e. B \/ A = B ) ) $=
        ( con0 wcel wss wceq wo wb onsseleq mp2an ) AEFBEFABGABFABHIJCDABKL $.

      $( The union of two ordinal numbers is an ordinal number.  (Contributed
         by NM, 13-Jun-1994.) $)
      onun2i $p |- ( A u. B ) e. On $=
        ( con0 wcel cun onun2 mp2an ) AEFBEFABGEFCDABHI $.
    $}
  $}

  $( An ordinal equal to its own union is either zero or a limit ordinal.
     (Contributed by NM, 1-Oct-2003.) $)
  unizlim $p |- ( Ord A -> ( A = U. A <-> ( A = (/) \/ Lim A ) ) ) $=
    ( word cuni wceq c0 wlim wo wa wn wi wne df-ne w3a df-lim biimpri biimtrrid
    3exp com23 imp orrd ex uni0 eqcomi id unieq 3eqtr4a limuni jaoi impbid1 ) A
    BZAACZDZAEDZAFZGZUJULUOUJULHUMUNUJULUMIZUNJUJUPULUNUPAEKZUJULUNJAELUJUQULUN
    UNUJUQULMANOQPRSTUAUMULUNUMEECZAUKUREUBUCUMUDAEUEUFAUGUHUI $.

  $( An ordinal number either equals zero or contains zero.  (Contributed by
     NM, 1-Jun-2004.) $)
  on0eqel $p |- ( A e. On -> ( A = (/) \/ (/) e. A ) ) $=
    ( con0 wcel c0 wceq wo wss 0ss 0elon onsseleq mpan mpbii eqcom orbi2i orcom
    wb bitri sylib ) ABCZDACZDAEZFZADEZTFZSDAGZUBAHDBCSUEUBPIDAJKLUBTUCFUDUAUCT
    DAMNTUCOQR $.

  $( The singleton of the singleton of the empty set is not an ordinal (nor a
     natural number by ~ omsson ).  It can be used to represent an "undefined"
     value for a partial operation on natural or ordinal numbers.  See also
     ~ onxpdisj .  (Contributed by NM, 21-May-2004.)  (Proof shortened by
     Andrew Salmon, 12-Aug-2011.) $)
  snsn0non $p |- -. { { (/) } } e. On $=
    ( c0 csn con0 wcel wceq snex snid n0ii 0ex eqcom mtbir elsn pm3.2ni on0eqel
    wo mto ) ABZBZCDRAEZARDZOSTQRQAFGHTAQEZUAQAEAQAIGHAQJKAQILKMRNP $.

  $( Ordinal numbers and ordered pairs are disjoint collections.  This theorem
     can be used if we want to extend a set of ordinal numbers or ordered pairs
     with disjoint elements.  See also ~ snsn0non .  (Contributed by NM,
     1-Jun-2004.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
  onxpdisj $p |- ( On i^i ( _V X. _V ) ) = (/) $=
    ( vx con0 cvv cxp cin c0 wceq cv wcel wn disj on0eqel 0nelxp eleq1 0nelelxp
    wo mtbiri con2i jaoi syl mprgbir ) BCCDZEFGAHZUBIZJZABABUBKUCBIUCFGZFUCIZPU
    EUCLUFUEUGUFUDFUBICCMUCFUBNQUDUGCCUCORSTUA $.

  $( The class of ordinal numbers is not equal to the universe.  (Contributed
     by NM, 16-Jun-2007.)  (Proof shortened by Mario Carneiro, 10-Jan-2013.)
     (Proof shortened by Wolf Lammen, 27-May-2024.) $)
  onnev $p |- On =/= _V $=
    ( con0 cvv wceq c0 csn wcel snsn0non snex id eleqtrrid mto neir ) ABABCZDEZ
    EZAFGMOBANHMIJKL $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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
       ~ riotacl2 (or ~ iotacl for unbounded iota), as demonstrated in the
       proof of ~ supub .  This can be easier than applying ~ riotasbc or 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 x y $.  $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 wb wal df-iota absn abbii unieqi eqtri ) ABDAB
      ECFZGHZCEZIABFQHJBKZCEZIABCLSUARTCABQMNOP $.
  $}

  ${
    $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 ) $.
    $( Deduction version of ~ nfiotaw .  Version of ~ nfiotad with a disjoint
       variable condition, which does not require ~ ax-13 .  (Contributed by
       NM, 18-Feb-2013.)  Avoid ~ ax-13 .  (Revised by GG, 26-Jan-2024.) $)
    nfiotadw $p |- ( ph -> F/_ x ( iota y ps ) ) $=
      ( vz cio weq wb wal cab cuni dfiota2 nfv nfvd nfbid nfald nfabdw nfunid
      nfcxfrd ) ACBDHBDGIZJZDKZGLZMBDGNACUEAUDCGAGOAUCCDEABUBCFAUBCPQRSTUA $.
    $( $j usage 'nfiotadw' avoids 'ax-13'; $)
  $}

  ${
    $d x y $.
    nfiotaw.1 $e |- F/ x ph $.
    $( Bound-variable hypothesis builder for the ` iota ` class.  Version of
       ~ nfiota with a disjoint variable condition, which does not require
       ~ ax-13 .  (Contributed by NM, 23-Aug-2011.)  Avoid ~ ax-13 .  (Revised
       by GG, 26-Jan-2024.) $)
    nfiotaw $p |- F/_ x ( iota y ph ) $=
      ( cio wnfc wtru nftru wnf a1i nfiotadw mptru ) BACEFGABCCHABIGDJKL $.
    $( $j usage 'nfiotaw' avoids 'ax-13'; $)
  $}

  ${
    $d z ps $.  $d z ph $.  $d x z $.  $d y z $.
    nfiotad.1 $e |- F/ y ph $.
    nfiotad.2 $e |- ( ph -> F/ x ps ) $.
    $( Deduction version of ~ nfiota .  Usage of this theorem is discouraged
       because it depends on ~ ax-13 .  Use the weaker ~ nfiotadw when
       possible.  (Contributed by NM, 18-Feb-2013.)
       (New usage is discouraged.) $)
    nfiotad $p |- ( ph -> F/_ x ( iota y ps ) ) $=
      ( vz cio weq wb wal cab cuni dfiota2 nfv wn wa wnf adantr nfeqf1 nfcxfrd
      adantl nfbid nfald2 nfabd nfunid ) ACBDHBDGIZJZDKZGLZMBDGNACUJAUICGAGOAUH
      CDEACDICKPZQBUGCABCRUKFSUKUGCRACDGTUBUCUDUEUFUA $.
  $}

  ${
    nfiota.1 $e |- F/ x ph $.
    $( Bound-variable hypothesis builder for the ` iota ` class.  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  Use the weaker
       ~ nfiotaw when possible.  (Contributed by NM, 23-Aug-2011.)
       (New usage is discouraged.) $)
    nfiota $p |- F/_ x ( iota y ph ) $=
      ( cio wnfc wtru nftru wnf a1i nfiotad mptru ) BACEFGABCCHABIGDJKL $.
  $}

  ${
    $d z w x y $.  $d z w ph $.  $d z w ps $.
    cbviotaw.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    cbviotaw.2 $e |- F/ y ph $.
    cbviotaw.3 $e |- F/ x ps $.
    $( Change bound variables in a description binder.  Version of ~ cbviota
       with a disjoint variable condition, which does not require ~ ax-13 .
       (Contributed by Andrew Salmon, 1-Aug-2011.)  Avoid ~ ax-13 .  (Revised
       by GG, 26-Jan-2024.) $)
    cbviotaw $p |- ( iota x ph ) = ( iota y ps ) $=
      ( vw vz weq wb wal cab cuni cio nfv nfbi equequ1 bibi12d cbvalv1 nfs1v cv
      wsb sbequ12 nfsbv sbhypf bitri abbii unieqi dfiota2 3eqtr4i ) ACHJZKZCLZH
      MZNBDHJZKZDLZHMZNACOBDOUOUSUNURHUNACIUCZIHJZKZILURUMVBCIUMIPUTVACACIUAVAC
      PQCIJAUTULVAACIUDCIHRSTVBUQIDUTVADACIDFUEVADPQUQIPIDJUTBVAUPABCIDUBGEUFID
      HRSTUGUHUIACHUJBDHUJUK $.
    $( $j usage 'cbviotaw' avoids 'ax-13'; $)
  $}

  ${
    $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, which requires fewer axioms .
       (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 $.
    $( $j usage 'cbviotavw' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    $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.  Usage of this theorem
       is discouraged because it depends on ~ ax-13 .  Use the weaker
       ~ cbviotaw when possible.  (Contributed by Andrew Salmon, 1-Aug-2011.)
       (New usage is discouraged.) $)
    cbviota $p |- ( iota x ph ) = ( iota y ps ) $=
      ( vw vz weq wb wal cab cuni cio wsb nfv nfbi equequ1 bibi12d sbequ12 nfsb
      nfs1v cbvalv1 sbequ sbie bitrdi bitri abbii unieqi dfiota2 3eqtr4i ) ACHJ
      ZKZCLZHMZNBDHJZKZDLZHMZNACOBDOUPUTUOUSHUOACIPZIHJZKZILUSUNVCCIUNIQVAVBCAC
      IUCVBCQRCIJAVAUMVBACIUACIHSTUDVCURIDVAVBDACIDFUBVBDQRURIQIDJZVABVBUQVDVAA
      CDPBAIDCUEABCDGEUFUGIDHSTUDUHUIUJACHUKBDHUKUL $.
  $}

  ${
    $d ph y $.  $d ps x $.
    cbviotav.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Change bound variables in a description binder.  Usage of this theorem
       is discouraged because it depends on ~ ax-13 .  Use the weaker
       ~ cbviotavw when possible.  (Contributed by Andrew Salmon, 1-Aug-2011.)
       (New usage is discouraged.) $)
    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 .  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .
       (Contributed by NM, 18-Mar-2013.)  (New usage is discouraged.) $)
    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 cbvalv1 sblbis albii 3bitri abbii unieqi 3eqtr4i ) ABEGZTZB
      HZEIZJABCKZCEGZTZCHZEIZJABLUNCLUMURULUQEULUKBFKZFHUKBCKZCHUQUKBFUKFMNUSUT
      FCUSABFKZUJBFKZTCAUJBFOVAVBCABFCDPVBFEGZCBFEQVCCMRUARUTFMUKFCBUBUCUTUPCUJ
      UOABCBCEQUDUEUFUGUHABESUNCESUI $.
  $}

  ${
    $d y z $.  $d x z $.  $d ph z $.
    $( Equality theorem for descriptions.  Usage of this theorem is discouraged
       because it depends on ~ ax-13 .  (Contributed by Andrew Salmon,
       30-Jun-2011.)  (New usage is discouraged.) $)
    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 eqeq1d abbidv unieqd df-iota 3eqtr4g
      abbi ) ABECFZACGZDHIZJZDGZKBCGZUAJZDGZKACLBCLSUCUFSUBUEDSTUDUAABCRMNOACDP
      BCDPQ $.

    $( 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 ) $=
      ( weq wb wal cab cuni cv csn abbi df-sn eqtr4di unieqd unisnv eqtrdi ) AB
      CDZEBFZABGZHCIZJZHTRSUARSQBGUAAQBKBTLMNCOP $.
  $}

  ${
    $d w x y $.  $d ph w $.
    $( Version of ~ iotaval using ~ df-iota instead of ~ dfiota2 .
       (Contributed by SN, 6-Nov-2024.) $)
    iotaval2 $p |- ( { x | ph } = { y } -> ( iota x ph ) = y ) $=
      ( vw cab cv csn wceq cio cuni df-iota weq wal eqeq1 cvv sneqbg elv equcom
      wb bitri bitrdi alrimiv uniabio syl eqtrid ) ABEZCFZGZHZABIUFDFZGZHZDEJZU
      GABDKUIULDCLZSZDMUMUGHUIUODUIULUHUKHZUNUFUHUKNUPCDLZUNUPUQSCUGUJOPQCDRTUA
      UBULDCUCUDUE $.

    $d ph y $.
    $( Version of ~ iotauni using ~ df-iota instead of ~ dfiota2 .
       (Contributed by SN, 6-Nov-2024.) $)
    iotauni2 $p |-
      ( E. y { x | ph } = { y } -> ( iota x ph ) = U. { x | ph } ) $=
      ( cab cv csn wceq cio cuni iotaval2 unieq unisnv eqtr2di eqtrd exlimiv )
      ABDZCEZFZGZABHZPIZGCSTQUAABCJSUARIQPRKCLMNO $.

    $d v w x y $.  $d ph v $.
    $( Version of ~ iotanul using ~ df-iota instead of ~ dfiota2 .
       (Contributed by SN, 6-Nov-2024.) $)
    iotanul2 $p |- ( -. E. y { x | ph } = { y } -> ( iota x ph ) = (/) ) $=
      ( vw vv cab cv csn wceq wex wn cio cuni c0 df-iota wne wcel n0 wel sylbi
      wa eluni vex sneq eqeq2d elab biimpi adantl eximi exlimiv necon1bi eqtrid
      weq ) ABFZCGZHZIZCJZKABLUNDGZHZIZDFZMZNABDOURVCNVCNPEGZVCQZEJUREVCRVEUREV
      EECSZUOVBQZUAZCJURCVDVBUBVHUQCVGUQVFVGUQVAUQDUOCUCDCUMUTUPUNUSUOUDUEUFUGU
      HUITUJTUKUL $.
  $}

  ${
    $d x y $.
    $( Theorem 8.19 in [Quine] p. 57.  This theorem is the fundamental property
       of iota.  (Contributed by Andrew Salmon, 11-Jul-2011.)  Remove
       dependency on ~ ax-10 , ~ ax-11 , ~ ax-12 .  (Revised by SN,
       23-Nov-2024.) $)
    iotaval $p |- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y ) $=
      ( weq wb wal cab cv csn wceq cio abbi df-sn eqtr4di iotaval2 syl ) ABCDZE
      BFZABGZCHZIZJABKTJRSQBGUAAQBLBTMNABCOP $.
    $( $j usage 'iotaval' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $d x y $.  $d ph y $.
    $( The ` iota ` class is a subset of the union of all elements satisfying
       ` ph ` .  (Contributed by Mario Carneiro, 24-Dec-2016.)  Remove
       dependency on ~ ax-10 , ~ ax-11 , ~ ax-12 .  (Revised by SN,
       6-Nov-2024.) $)
    iotassuni $p |- ( iota x ph ) C_ U. { x | ph } $=
      ( vy cab cv csn wceq wex cio cuni wss iotauni2 eqimss syl wn iotanul2 0ss
      c0 eqsstrdi pm2.61i ) ABDZCEFGCHZABIZUAJZKZUBUCUDGUEABCLUCUDMNUBOUCRUDABC
      PUDQST $.
    $( $j usage 'iotassuni' avoids 'ax-10' 'ax-11' 'ax-12'; $)

    $( Theorem 8.23 in [Quine] p. 58.  This theorem proves the existence of the
       ` iota ` class under our definition.  (Contributed by Andrew Salmon,
       11-Jul-2011.)  Remove dependency on ~ ax-10 , ~ ax-11 , ~ ax-12 .
       (Revised by SN, 6-Nov-2024.) $)
    iotaex $p |- ( iota x ph ) e. _V $=
      ( vy cab cv csn wceq wex cio cvv wcel iotaval2 vex eqeltrdi exlimiv wn c0
      iotanul2 0ex pm2.61i ) ABDCEZFGZCHZABIZJKZUBUECUBUDUAJABCLCMNOUCPUDQJABCR
      SNT $.
    $( $j usage 'iotaex' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $d ph z $.  $d x y z $.
    $( Obsolete version of ~ iotaval as of 23-Dec-2024.  (Contributed by Andrew
       Salmon, 11-Jul-2011.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    iotavalOLD $p |- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y ) $=
      ( vz weq wb wal cio cab cuni cv dfiota2 wceq wa wi cvv sbeqalb ex equequ2
      elv bibi2d biimpd alimdv impbid equcom bitrdi alrimiv uniabio syl eqtrid
      com12 ) ABCEZFZBGZABHABDEZFZBGZDIJZCKZABDLUNUQDCEZFZDGURUSMUNVADUNUQCDEZU
      TUNUQVBUNUQVBUNUQNVBOCABUSDKPQTRVBUNUQVBUMUPBVBUMUPVBULUOACDBSUAUBUCUKUDC
      DUEUFUGUQDCUHUIUJ $.

    $( 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 cab cuni eu6 iotaval uniabio eqtr4d sylbi
      exlimiv ) ABDABECEZFGBHZCIABJZABKLZFZABCMTUCCTUASUBABCNABCOPRQ $.

    $( 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 wceq uniintab biimpi eqtrd ) ABCZABDABEZF
      ZNGZABHMOPIABJKL $.

    $( 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 eu6 sp iotaval eqeq2d bitr4d eqcom bitrdi
      exlimiv sylbi ) ABDABEZCEZFZGZBHZCIAABJZTFZGZABCKUDUGCUDATUEFZUFUDAUBUHUC
      BLUDUEUATABCMNOTUEPQRS $.

    $( 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 weq wb wal wex cio c0 wceq eu6 wn cab dfiota2 alnex dfnul2 equid
      cuni tbt biimpi con1bid abbi syl eqtr2id sylbir unieqd uni0 eqtrdi eqtrid
      alimi sylnbi ) ABDABCEFBGZCHZABIZJKABCLUNMZUOUMCNZSZJABCOUPURJSJUPUQJUPUM
      MZCGZUQJKUMCPUTJCCEZMZCNZUQCQUTVBUMFZCGVCUQKUSVDCUSUMVAUSUSVAFVAUSCRTUAUB
      UKVBUMCUCUDUEUFUGUHUIUJUL $.

    $( Obsolete version of ~ iotassuni as of 23-Dec-2024.  (Contributed by
       Mario Carneiro, 24-Dec-2016.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    iotassuniOLD $p |- ( iota x ph ) C_ U. { x | ph } $=
      ( weu cio cab cuni wss wceq iotauni eqimss syl wn c0 iotanul 0ss eqsstrdi
      pm2.61i ) ABCZABDZABEFZGZRSTHUAABISTJKRLSMTABNTOPQ $.

    $( Obsolete version of ~ iotaex as of 23-Dec-2024.  (Contributed by Andrew
       Salmon, 11-Jul-2011.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    iotaexOLD $p |- ( iota x ph ) e. _V $=
      ( vz weu cio cvv wcel cv wceq wb wal wex iotaval eqcomd eximi eu6 3imtr4i
      isset wn c0 iotanul 0ex eqeltrdi pm2.61i ) ABDZABEZFGZABHCHZIJBKZCLUHUFIZ
      CLUEUGUIUJCUIUFUHABCMNOABCPCUFRQUESUFTFABUAUBUCUD $.

    $( 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 eu6 wsb wi biimpr alimi sb6 sylibr cv iotaval
      wsbc wceq eqcomd dfsbcq2 syl mpbid exlimiv sylbi ) ABDABCEZFZBGZCHABABIZS
      ZABCJUIUKCUIABCKZUKUIUGALZBGULUHUMBAUGMNABCOPUICQZUJTULUKFUIUJUNABCRUAABC
      UJUBUCUDUEUF $.
  $}

  $( 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 wcel iotaex simpl sbcth ax-mp wb sbcimg mpbi
    syl ) ABDZCETCTCFZGZACUAGZTCHTAIZCUAGZUBUCIZUAJKZUETCLZUDCUAJABMNOUGUEUFPUH
    TACUAJQORS $.

  ${
    $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 be useful for expanding an unbounded iota-based definition (see
     ~ df-iota ).  If you have a bounded iota-based definition, ~ riotacl2 may
     be useful.

     (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 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 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 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 $.
  $}

  ${
    $d A x $.  $d ps x $.
    iotan0.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( Representation of "the unique element such that ` ph ` " with a class
       expression ` A ` which is not the empty set (that means that "the unique
       element such that ` ph ` " exists).  (Contributed by AV,
       30-Jan-2024.) $)
    iotan0 $p |- ( ( A e. V /\ A =/= (/) /\ A = ( iota x ph ) ) -> ps ) $=
      ( wcel c0 wne cio wceq w3a weu pm13.18 expcom iotanul necon1ai syl6 a1i
      wi 3imp wa eqcom iota2 biimprd biimtrid impancom 3adant2 mpd ) DEGZDHIZDA
      CJZKZLACMZBUJUKUMUNUKUMUNTTUJUKUMULHIZUNUMUKUODULHNOUNULHACPQRSUAUJUMUNBT
      UKUJUNUMBUMULDKZUJUNUBZBDULUCUQBUPABCDEFUDUEUFUGUHUI $.
  $}

  $( 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 cab cio nfeu1 nfab1 nfiota1 nfsn cv wceq wcel iota1 eqcom bitrdi abid
    csn velsn 3bitr4g eqrd ) ABCZBABDZABEZQZABFABGBUCABHIUAABJZUCKZUEUBLUEUDLUA
    AUCUEKUFABMUCUENOABPBUCRST $.

  $( The ` iota ` operation using the ` if ` operator.  (Contributed by Scott
     Fenton, 6-Oct-2017.)  (Proof shortened by JJ, 28-Oct-2021.) $)
  dfiota4 $p |- ( iota x ph ) = if ( E! x ph , U. { x | ph } , (/) ) $=
    ( cio weu cab cuni c0 cif wceq wi wn iotauni iotanul ifval mpbir2an ) ABCZA
    BDZABEFZGHIQPRIJQKPGIJABLABMQPRGNO $.

  ${
    $d A y z $.  $d x y z $.  $d ph z $.
    $( Class substitution within a description binder.  (Contributed by Scott
       Fenton, 6-Oct-2017.)  (Revised by NM, 23-Aug-2018.) $)
    csbiota $p |- [_ A / x ]_ ( iota y ph ) = ( iota y [. A / x ]. ph ) $=
      ( vz cvv wcel cio csb wsbc wceq cv wsb csbeq1 dfsbcq2 iotabidv eqeq12d wn
      c0 con3i vex nfs1v nfiotaw weq sbequ12 csbief vtoclg csbprc wex weu sbcex
      nexdv euex iotanul 3syl eqtr4d pm2.61i ) DFGZBDACHZIZABDJZCHZKZBELZUSIZAB
      EMZCHZKVCEDFVDDKZVEUTVGVBBVDDUSNVHVFVACABEDOPQBVDUSVGEUAVFBCABEUBUCBEUDAV
      FCABEUEPUFUGURRZUTSVBBDUSUHVIVACUIZRVACUJZRVBSKVIVACVAURABDUKTULVKVJVACUM
      TVACUNUOUPUQ $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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 or backtick, backquote (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 cos ` is true once we define cosine ( ~ df-cos ).  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 and
       ~ df-mpo ).  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 , ~ dffun3 ,
       ~ dffun4 , ~ dffun5 , ~ 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.  For alternate definitions, see ~ dffn2 , ~ dffn3 , ~ dffn4 , and
       ~ dffn5 .  (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. ` F : A --> B ` can be read as " ` F `
       is a function from ` A ` to ` B ` ".  For alternate definitions, see
       ~ dff2 , ~ dff3 , and ~ dff4 .  (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.  For equivalent definitions see ~ dff12
       and ~ dff13 .  Compare Definition 6.15(5) of [TakeutiZaring] p. 27.  We
       use their notation ("1-1" above the arrow).

       A one-to-one function is also called an "injection" or an "injective
       function", ` F : A -1-1-> B ` can be read as " ` F ` is an injection
       from ` A ` into ` B ` ".  Injections are precisely the monomorphisms in
       the category ` SetCat ` of sets and set functions, see ~ setcmon .
       (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).  For alternate
       definitions, see ~ dffo2 , ~ dffo3 , ~ dffo4 , and ~ dffo5 .

       An onto function is also called a "surjection" or a "surjective
       function", ` F : A -onto-> B ` can be read as " ` F ` is a surjection
       from ` A ` onto ` B ` ".  Surjections are precisely the epimorphisms in
       the category ` SetCat ` of sets and set functions, see ~ setcepi .
       (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.  For equivalent definitions see
       ~ dff1o2 , ~ dff1o3 , ~ dff1o4 , and ~ dff1o5 .  Compare Definition
       6.15(6) of [TakeutiZaring] p. 27.  We use their notation ("1-1" above
       the arrow and "onto" below the arrow).

       A one-to-one onto function is also called a "bijection" or a "bijective
       function", ` F : A -1-1-onto-> B ` can be read as " ` F ` is a bijection
       between ` A ` and ` B ` ".  Bijections are precisely the isomorphisms in
       the category ` SetCat ` of sets and set functions, see ~ setciso .
       Therefore, two sets are called "isomorphic" if there is a bijection
       between them.  According to ~ isof1oidb , two sets are isomorphic iff
       there is an isomorphism ` Isom ` regarding the identity relation.  In
       this case, the two sets are also "equinumerous", see ~ bren .
       (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, ` ( cos `` 0 ) = 1 ` (we prove this in ~ cos0
       after we define cosine in ~ df-cos ).  Typically, function ` F ` is
       defined using maps-to notation (see ~ df-mpt and ~ df-mpo ), but this is
       not required.  For example,
       ` F = { <. 2 , 6 >. , <. 3 , 9 >. } -> ( F `` 3 ) = 9 ` ( ~ ex-fv ).
       Note that ~ df-ov will 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 (as shown by ~ ndmfv and ~ fvprc ).
       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.  Alternate definitions are
       ~ dffv2 , ~ dffv3 , ~ fv2 , and ~ fv3 (the latter two previously
       required ` A ` to be a set.)  Restricted equivalents that require ` F `
       to be a function are shown in ~ funfv and ~ funfv2 .  For the familiar
       definition of function value in terms of ordered pair membership, see
       ~ funopfvb .  (Contributed by NM, 1-Aug-1994.)  Revised to use
       ` iota ` .  Original version is now Theorem ~ dffv4 .  (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 A w x y z $.
    $( Alternate definition of a function.  (Contributed by NM, 29-Dec-1996.)
       Avoid ~ ax-10 , ~ ax-12 .  (Revised by SN, 19-Dec-2024.)  Avoid
       ~ ax-11 .  (Revised by BTernaryTau, 29-Dec-2024.) $)
    dffun2 $p |- ( Fun A <-> ( Rel A /\
                 A. x A. y A. z ( ( x A y /\ x A z ) -> y = z ) ) ) $=
      ( vw wfun wrel ccnv ccom cid wa cv wbr weq wi wal breq1 albidv vex bitri
      df-fun cotrg anbi1d imbi12d breq2 anbi12d imbi1d alcomw brcnv anbi1i ideq
      wss imbi12i 3albii anbi2i ) DFDGZDDHZIJULZKUPALZBLZDMZUSCLZDMZKZBCNZOZCPB
      PAPZKDUAURVGUPURUTUSUQMZVCKZUTVBJMZOZCPZAPBPZVGBACDUQJUBVMVLBPAPVGVLELZUS
      UQMZVCKZVNVBJMZOZCPUTVNUQMZVNVBDMZKZVJOZCPBAEEBENZVKVRCWCVIVPVJVQWCVHVOVC
      UTVNUSUQQUCUTVNVBJQUDRAENZVKWBCWDVIWAVJWDVHVSVCVTUSVNUTUQUEUSVNVBDQUFUGRU
      HVKVFABCVIVDVJVEVHVAVCUTUSDBSASUIUJUTVBCSUKUMUNTTUOT $.
    $( $j usage 'dffun2' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}

  ${
    $d A x y z $.
    $( Obsolete version of ~ dffun2 as of 29-Dec-2024.  (Contributed by NM,
       29-Dec-1996.)  Avoid ~ ax-10 , ~ ax-12 .  (Revised by SN, 19-Dec-2024.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    dffun2OLD $p |- ( Fun A <-> ( Rel A /\
                 A. x A. y A. z ( ( x A y /\ x A z ) -> y = z ) ) ) $=
      ( wfun wrel ccnv ccom cid wss wa cv wbr weq wi wal df-fun cotrg vex bitri
      alcom brcnv anbi1i ideq imbi12i 3albii anbi2i ) DEDFZDDGZHIJZKUHALZBLZDMZ
      UKCLZDMZKZBCNZOZCPBPAPZKDQUJUSUHUJULUKUIMZUOKZULUNIMZOZCPZAPBPZUSBACDUIIR
      VEVDBPAPUSVDBAUAVCURABCVAUPVBUQUTUMUOULUKDBSASUBUCULUNCSUDUEUFTTUGT $.
    $( $j usage 'dffun2OLD' avoids 'ax-10' 'ax-12'; $)

    $( Obsolete version of ~ dffun2 as of 11-Dec-2024.  (Contributed by NM,
       29-Dec-1996.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    dffun2OLDOLD $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 weq df-fun df-id sseq2i df-co sseq1i ssopab2bw brcnv anbi1i exbii
      ccom imbi1i 19.23v bitr4i anbi2i ) DEDFZDDUAZULZGHZIUQAJZBJZDKZVACJDKZIZB
      CUBZTZCLZBLALZIDUCUTVIUQUTVBVAURKZVDIZAMZVFTZCLZBLZVHALZBLVIUTUSVFBCNZHVL
      BCNZVQHVOGVQUSBCUDUEUSVRVQBCADURUFUGVLVFBCUHOVNVPBVNVGALZCLVPVMVSCVMVEAMZ
      VFTVSVLVTVFVKVEAVJVCVDVBVADBPAPUIUJUKUMVEVFAUNUOQVGCARSQVHBAROUPS $.
  $}

  ${
    $d F x y z $.
    $( Alternate definition of a function using "at most one" notation.
       (Contributed by NM, 9-Mar-1995.)  Avoid ~ ax-10 , ~ ax-12 .  (Revised by
       SN, 19-Dec-2024.) $)
    dffun6 $p |- ( Fun F <-> ( Rel F /\ A. x E* y x F y ) ) $=
      ( vz wfun wrel cv wbr wa weq wal wmo dffun2 breq2 mo4 albii anbi2i bitr4i
      wi ) CECFZAGZBGZCHZUADGZCHZIBDJSDKBKZAKZITUCBLZAKZIABDCMUIUGTUHUFAUCUEBDU
      BUDUACNOPQR $.
    $( $j usage 'dffun6' avoids 'ax-10' 'ax-12'; $)
  $}

  ${
    $d A x y z $.
    $( Alternate definition of function.  (Contributed by NM, 29-Dec-1996.)
       (Proof shortened by SN, 19-Dec-2024.) $)
    dffun3 $p |- ( Fun A <-> ( Rel A /\
                 A. x E. z A. y ( x A y -> y = z ) ) ) $=
      ( wfun wrel cv wbr wmo wal wa weq wi wex dffun6 df-mo albii anbi2i bitri
      ) DEDFZAGBGDHZBIZAJZKTUABCLMBJCNZAJZKABDOUCUETUBUDAUABCPQRS $.

    $( Obsolete version of ~ dffun3 as of 19-Dec-2024.  Alternate definition of
       function.  (Contributed by NM, 29-Dec-1996.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    dffun3OLD $p |- ( Fun A <-> ( Rel A /\
                 A. x E. z A. y ( x A y -> y = z ) ) ) $=
      ( wfun wrel cv wbr wa weq wal wex dffun2 wmo breq2 mo4 df-mo bitr3i albii
      wi anbi2i bitri ) DEDFZAGZBGZDHZUDCGZDHZIBCJZTCKBKZAKZIUCUFUITBKCLZAKZIAB
      CDMUKUMUCUJULAUJUFBNULUFUHBCUEUGUDDOPUFBCQRSUAUB $.

    $( 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 $.

    $( Alternate definition of function.  (Contributed by NM, 29-Dec-1996.) $)
    dffun5 $p |- ( Fun A <-> ( Rel A /\
                 A. x E. z A. y ( <. x , y >. e. A -> y = z ) ) ) $=
      ( wfun wrel cv wbr weq wi wal wex wa wcel dffun3 df-br imbi1i albii exbii
      cop anbi2i bitri ) DEDFZAGZBGZDHZBCIZJZBKZCLZAKZMUCUDUETDNZUGJZBKZCLZAKZM
      ABCDOUKUPUCUJUOAUIUNCUHUMBUFULUGUDUEDPQRSRUAUB $.
  $}

  ${
    $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 wrel cv wbr weq wal wa wmo nfcv nfbr nfv albii wex dffun3
      wi breq2 cbvmow df-mo nfmov breq1 mobidv cbvalv1 3bitr3ri anbi2i bitr4i )
      CICJZFKZGKZCLZGHMUCGNHUAZFNZOUNAKZBKZCLZBPZANZOFGHCUBVDUSUNUQGPZFNUOVACLZ
      BPZFNUSVDVEVGFUQVFGBBUOUPCBUOQEBUPQRVFGSUPVAUOCUDUETVEURFUQGHUFTVGVCFAVFA
      BAUOVACAUOQDAVAQRUGVCFSFAMVFVBBUOUTVACUHUIUJUKULUM $.
  $}

  ${
    $d x y A $.  $d x y F $.
    $( Obsolete version of ~ dffun6 as of 19-Dec-2024.  (Contributed by NM,
       9-Mar-1995.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    dffun6OLD $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.)  (Proof shortened by SN, 19-Dec-2024.) $)
    funmo $p |- ( Fun F -> E* y A F y ) $=
      ( vx wfun cv wbr cvv wcel wa wi wal wrel dffun6 simplbi brrelex1 ex ancrd
      wmo syl alrimiv simprbi wceq breq1 mobidv spcgv syl5com moanimv moim sylc
      sylibr ) CEZBAFZCGZBHIZUNJZKZALUPASZUNASZULUQAULUNUOULCMZUNUOKULUTDFZUMCG
      ZASZDLZDACNZOUTUNUOBUMCPQTRUAULUOUSKURULVDUOUSULUTVDVEUBVCUSDBHVABUCVBUNA
      VABUMCUDUEUFUGUOUNAUHUKUNUPAUIUJ $.
    $( $j usage 'funmo' avoids 'ax-12'; $)

    $( Obsolete version of ~ funmo as of 19-Dec-2024.  (Contributed by NM,
       24-May-1998.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    funmoOLD $p |- ( Fun F -> E* y A F y ) $=
      ( vx wfun cv wbr cvv wcel wa wi wal wrel dffun6 simplbi brrelex1 ex ancrd
      wmo syl alrimiv breq1 mobidv imbi2d simprbi 19.21bi vtoclg moanimv sylibr
      wceq com12 moim sylc ) CEZBAFZCGZBHIZUPJZKZALURASZUPASZUNUSAUNUPUQUNCMZUP
      UQKUNVBDFZUOCGZASZDLZDACNZOVBUPUQBUOCPQTRUAUNUQVAKUTUQUNVAUNVEKUNVAKDBHVC
      BUJZVEVAUNVHVDUPAVCBUOCUBUCUDUNVEDUNVBVFVGUEUFUGUKUQUPAUHUIUPURAULUM $.
  $}

  $( 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 vz vy wrel cv wbr wi wal wex wa wsbc csb wfun sbcal csbconstg bitrid
      wcel weq sbcan sbcrel sbcex2 sbcimg sbcbr123 breq12d imbi12d bitrd albidv
      sbcg exbidv anbi12d dffun3 sbcbii 3bitr4g ) BDUAZCHZEIZFIZCJZFGUBZKZFLZGM
      ZELZNZABOZABCPZHZUTVAVJJZVCKZFLZGMZELZNZCQZABOVJQVIUSABOZVGABOZNURVQUSVGA
      BUCURVSVKVTVPABCDUDVTVFABOZELURVPVFEABRURWAVOEWAVEABOZGMURVOVEGABUEURWBVN
      GWBVDABOZFLURVNVDFABRURWCVMFURWCVBABOZVCABOZKVMVBVCABDUFURWDVLWEVCWDABUTP
      ZABVAPZVJJURVLABUTVACUGURWFUTWGVAVJABUTDSABVADSUHTVCABDULUIUJUKTUMTUKTUNT
      VRVHABEFGCUOUPEFGVJUOUQ $.
  $}

  ${
    $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 moeu sylib mpd ) DEZBCDFZGZBAHDFZAIZUJAJZUIBDKZLZUKUG
      DMUHUNDNBCDOPUNUKABDUMQRSUIUJATZUKULUAUGUOUHABDUBUCUJAUDUEUF $.

    $( 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 moeu 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 $.
  $}

  $( A class is a function if and only if it is a function on its domain.
     (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 on 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.  See also ~ idfn .  (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 universal class is not a function.  (Contributed by Raph Levien,
     27-Jan-2004.) $)
  nfunv $p |- -. Fun _V $=
    ( cvv wfun wrel nrelv funrel mto ) ABACDAEF $.

  ${
    $d t u v w x y z A $.  $d t u v w x y z B $.
    $( A Kuratowski ordered pair of sets is a function only if its components
       are equal.  (Contributed by NM, 5-Jun-2008.)  (Revised by Mario
       Carneiro, 26-Apr-2015.)  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.) $)
    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 zfpair2 3eqtr4ri eqeq2i bitr3i wal dffun4 simprbi opex
      vsnex prid1 eleq2 mpbiri jca w3a opeq12 3adant3 eleq1d 3adant2 anbi12d wb
      prid2 eqeq12 3adant1 spc3gv mp3an syl2im dfsn2 preq2 eqtr2id eqeq2d eqtr3
      biimtrid expcom biimtrdi com13 imp sylcom exlimdvv mpd vtocl2g 3impia ) A
      CLBDLABMZUCZABNZEOZFOZMZUCZEFPZQAXEMZUCZAXENZQXBXCQEFABCDXDANZXGXJXHXKXLX
      FXIXDAXEUDUAXDAXEUEUBXEBNZXJXBXKXCXMXIXAXEBAUFUAXEBAUGUBXGXDGOZUHZNZXEXNH
      OZRZNZUIZHUJGUJZXHXGXFUKZYAXFULGHXDXEESZFSZUMUNXGXTXHGHXGXTGHPZXHXTXFXNXN
      MZXNXQMZRZNZXGYEXTXFXOXRMZNYIXDXEXOXRYCYDUOYJYHXFYFXOXRRZRXOUHZYKRYHYJYFY
      LYKXNGSZUPUQYGYKYFXNXQYMHSZURUSXOXRGVHGHUTURVAVBVCXGIOZJOZMZXFLZYOKOZMZXF
      LZUIZJKPZQZKVDJVDIVDZYIYFXFLZYGXFLZUIZYEXGYBUUEIJKXFVEVFYIUUFUUGYIUUFYFYH
      LYFYGXNXNVGVIXFYHYFVJVKYIUUGYGYHLYFYGXNXQVGVTXFYHYGVJVKVLXNTLZUUIXQTLUUEU
      UHYEQZQYMYMYNUUDUUJIJKXNXNXQTTTIGPZJGPZKHPZVMZUUBUUHUUCYEUUNYRUUFUUAUUGUU
      NYQYFXFUUKUULYQYFNUUMYOYPXNXNVNVOVPUUNYTYGXFUUKUUMYTYGNUULYOYSXNXQVNVQVPV
      RUULUUMUUCYEVSUUKYPXNYSXQWAWBUBWCWDWEWKXPXSYEXHQYEXSXPXHYEXSXEXONZXPXHQYE
      XRXOXEYEXOXNXNRXRXNWFXNXQXNWGWHWIXPUUOXHXDXEXOWJWLWMWNWOWPWQWRWSWT $.
  $}

  ${
    $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 cv wbr wmo wal wrel relopabv nfopab1 nfopab2 dffun6f mpbiran
      cop wcel df-br opabidw bitri mobii albii ) ABCDZEZBFZCFZUCGZCHZBIZACHZBIU
      DUCJUIABCKBCUCABCLABCMNOUHUJBUGACUGUEUFPUCQAUEUFUCRABCSTUAUBT $.
  $}

  ${
    $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 vz vy wfun wa cv wbr wex copab ccom wmo wal alrimiv moexexvw syl2anr
      funmo funopab sylibr df-co funeqi ) AFZBFZGZCHZDHZBIZUGEHAIZGDJZCEKZFZABL
      ZFUEUJEMZCNULUEUNCUDUHDMUIEMZDNUNUCDUFBRUCUODEUGAROUHUIDEPQOUJCESTUMUKCED
      ABUAUBT $.
  $}

  $( Composition of two functions, generalization of ~ funco .  (Contributed by
     Alexander van der Vekens, 25-Jul-2017.) $)
  funresfunco $p |- ( ( Fun ( F |` ran G ) /\ Fun G ) -> Fun ( F o. G ) ) $=
    ( crn cres wfun ccom funco wss wceq ssid cores ax-mp eqcomi funeqi sylibr
    wa ) ABCZDZEBEPRBFZEABFZERBGTSSTQQHSTIQJABQKLMNO $.

  $( 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
      opelresi opeldm a1i ssel jcad adantl weu funeu2 eldm2 ancrd eximdv eupick
      biimtrid syl2an exp43 com23 com34 pm2.43d impcomd impbid bitr4id alrimivv
      relres funrel relss mpan9 eqrel sylancr mpbird ) AEZBAFZGZABHZIZBJZCKZDKZ
      LZVNMZVRBMZNZDOCOZVLWACDVLVSVPVMMZVRAMZGZVTVMVPVQADPZUAVLVTWEVKVTWEQVJVKV
      TWCWDVTWCQVKVPVQBCPZWFUBUCBAVRUDZUEUFVLWDWCVTVLWDWCVTQVLWDWCWDVTVJVKWDWCW
      DVTQZQZQVJWDVKWJVJWDVKWCWIVJWDGWDDUGWDVTGZDRZWIVKWCGDVPVQAUHVKWCWLWCVTDRV
      KWLDVPBWGUIVKVTWKDVKVTWDWHUJUKUMSWDVTDULUNUOUPSUQURUSUTVAVBVLVNTBTZVOWBNA
      VMVCVJATVKWMAVDBAVEVFCDVNBVGVHVI $.
  $}

  $( 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 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 $.
  $}

  $( A set of two pairs is a function if their first members are different.
     (Contributed by FL, 26-Jun-2011.)  (Proof shortened by JJ,
     14-Jul-2021.) $)
  funprg $p |- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) /\ A =/= B )
     -> Fun { <. A , C >. , <. B , D >. } ) $=
    ( wcel wa wne w3a cop csn wfun cdm cin c0 funsng dmsnopg cun wceq ineqan12d
    cpr anim12i an4s 3adant3 disjsn2 sylan9eq 3adant1 funun df-pr funeqi sylibr
    syl2anc ) AEIZBFIZJZCGIZDHIZJZABKZLZACMZNZBDMZNZUAZOZVDVFUDZOVCVEOZVGOZJZVE
    PZVGPZQZRUBZVIURVAVMVBUPUSUQUTVMUPUSJVKUQUTJVLACEGSBDFHSUEUFUGVAVBVQURVAVBV
    PANZBNZQRUSUTVNVRVOVSACGTBDHTUCABUHUIUJVEVGUKUOVJVHVDVFULUMUN $.

  $( A set of three pairs is a function if their first members are different.
     (Contributed by Alexander van der Vekens, 5-Dec-2017.)  (Proof shortened
     by JJ, 14-Jul-2021.) $)
  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 cpr wfun c0 wceq csn cun ctp cdm cin 3simpa simp1 funprg
    syl3an simp3 funsng syl2an 3adant3 dmpropg dmsnopg ineqan12d 3impa disjprsn
    wa 3adant1 sylan9eq funun syl21anc df-tp funeqi sylibr ) JDMZKHMZLIMZNZAEMZ
    BFMZCGMZNZJKOZJLOZKLOZNZNZJAPZKBPZQZLCPZUAZUBZRZVTWAWCUCZRVSWBRZWDRZWBUDZWD
    UDZUEZSTZWFVJVGVHUSVNVKVLUSZVRVOWHVGVHVIUFVKVLVMUFVOVPVQUGJKABDHEFUHUIVJVNW
    IVRVJVIVMWIVNVGVHVIUJVKVLVMUJLCIGUKULUMVNVRWMVJVNVRWLJKQZLUAZUEZSVKVLVMWLWQ
    TWNVMWJWOWKWPJAKBEFUNLCGUOUPUQVPVQWQSTVOJKLURUTVAUTWBWDVBVCWGWEVTWAWCVDVEVF
    $.

  ${
    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 df-fn sylanblrc ) ABMACM
      BCMNADOBEOCFOPZQUERABCPZSUEUFTABCDEFGHIJKLUAADBECFJKLUBUEUFUCUD $.
  $}

  $( The converse pair of ordered pairs is a function if the second members are
     different.  Note that the second members need not be sets.  (Contributed
     by AV, 23-Jan-2021.) $)
  funcnvpr $p |- ( ( A e. U /\ C e. V /\ B =/= D )
                   -> Fun `' { <. A , B >. , <. C , D >. } ) $=
    ( wcel cop csn ccnv cun wfun cdm cin c0 wceq funcnvsn crn df-rn rnsnopg wne
    w3a cpr pm3.2i eqtr3id ineqan12d 3adant3 disjsn2 3ad2ant3 eqtrd funun df-pr
    wa sylancr cnveqi cnvun eqtri funeqi sylibr ) AEGZCFGZBDUAZUBZABHZIZJZCDHZI
    ZJZKZLZVDVGUCZJZLVCVFLZVILZUMVFMZVIMZNZOPVKVNVOABQCDQUDVCVRBIZDIZNZOUTVAVRW
    APVBUTVAVPVSVQVTUTVPVERVSVESABETUEVAVQVHRVTVHSCDFTUEUFUGVBUTWAOPVABDUHUIUJV
    FVIUKUNVMVJVMVEVHKZJVJVLWBVDVGULUOVEVHUPUQURUS $.

  $( The converse triple of ordered pairs is a function if the second members
     are pairwise different.  Note that the second members need not be sets.
     (Contributed by AV, 23-Jan-2021.) $)
  funcnvtp $p |- ( ( ( A e. U /\ C e. V /\ E e. W )
                     /\ ( B =/= D /\ B =/= F /\ D =/= F ) )
                   -> Fun `' { <. A , B >. , <. C , D >. , <. E , F >. } ) $=
    ( wcel w3a wne wa cop cpr ccnv csn wfun c0 wceq cun ctp cdm cin simp1 simp2
    funcnvpr syl2an3an funcnvsn a1i crn rnpropg eqtr3id 3adant3 rnsnopg ineq12d
    df-rn 3ad2ant3 disjprsn 3adant1 sylan9eq funun syl21anc df-tp cnveqi funeqi
    cnvun eqtri sylibr ) AEJZCHJZFIJZKZBDLZBGLZDGLZKZMZABNZCDNZOZPZFGNZQZPZUAZR
    ZVSVTWCUBZPZRVRWBRZWERZWBUCZWEUCZUDZSTWGVMVJVKVQVNWJVJVKVLUEVJVKVLUFVNVOVPU
    EABCDEHUGUHWKVRFGUIUJVMVQWNBDOZGQZUDZSVMWLWOWMWPVJVKWLWOTVLVJVKMWLWAUKWOWAU
    QACBDEHULUMUNVLVJWMWPTVKVLWMWDUKWPWDUQFGIUOUMURUPVOVPWQSTVNBDGUSUTVAWBWEVBV
    CWIWFWIWAWDUAZPWFWHWRVSVTWCVDVEWAWDVGVHVFVI $.

  $( The converse quadruple of ordered pairs is a function if the second
     members are pairwise different.  Note that the second members need not be
     sets.  (Contributed by AV, 23-Jan-2021.)  (Proof shortened by JJ,
     14-Jul-2021.) $)
  funcnvqp $p |- ( ( ( ( A e. U /\ C e. V ) /\ ( E e. W /\ G e. T ) )
                     /\ ( ( B =/= D /\ B =/= F /\ B =/= H )
                          /\ ( D =/= F /\ D =/= H ) /\ F =/= H ) )
                   -> Fun `' ( { <. A , B >. , <. C , D >. }
                               u. { <. E , F >. , <. G , H >. } ) ) $=
    ( wcel wa wne cop cpr ccnv wfun c0 w3a cun cdm cin funcnvpr 3expa 3ad2antr1
    wceq ad2ant2r 3adantr2 ad2ant2l crn df-rn rnpropg eqtr3id ineqan12d disjpr2
    an4s 3adantl1 3adant3 sylan9eq funun syl21anc cnvun funeqi sylibr ) AFMZCKM
    ZNZGLMZIEMZNZNZBDOZBHOZBJOZUAZDHOZDJOZNZHJOZUAZNZABPCDPQZRZGHPIJPQZRZUBZSZW
    DWFUBRZSWCWESZWGSZWEUCZWGUCZUDZTUHWIVMVQWAWKVTVIVQWKVLWAVIVOVNWKVPVGVHVNWKA
    BCDFKUEUFUGUIUJVMVQWAWLVTVLWAWLVIVQVJVKWAWLGHIJLEUEUFUKUJVMWBWOBDQZHJQZUDZT
    VIVLWMWPWNWQVIWMWDULWPWDUMACBDFKUNUOVLWNWFULWQWFUMGIHJLEUNUOUPVQVTWRTUHZWAV
    OVPVTWSVNVOVRVPVSWSBDHJUQURUSUTVAWEWGVBVCWJWHWDWFVDVEVF $.

  $( 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 converse of the empty set is a function.  (Contributed by AV,
     7-Jan-2021.) $)
  funcnv0 $p |- Fun `' (/) $=
    ( c0 ccnv wfun fun0 cnv0 funeqi mpbir ) ABZCACDHAEFG $.

  $( 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 eqabri biimpi weu
      wa vex biantrurd ralbiia funcnv df-reu breldm pm4.71ri eubii df-eu ralbii
      3bitr2i 3bitr4i ) ADZBDZCEZAFZBCGZHUNAIZUOSZBUPHCJKUNACLZMZBUPHUOURBUPUMU
      PNZUQUOVAUQUQBUPABCOPQUAUBABCUCUTURBUPUTULUSNZUNSZARUNARURUNAUSUDUNVCAUNV
      BULUMCATBTUEUFUGUNAUHUJUIUK $.

    $( 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 $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y R $.
    $( Single-rootedness (see ~ funcnv ) of a class cut down by a Cartesian
       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
      brinxp2 an21 bitri mobii df-rmo imbi2i biimt bitr4id bitrdi bitrid r19.26
      ralbiia pm5.32i ancom reu5 ralbii 3bitr2i ) ECDFGZHZDIVLJZVLKZDLZMVMVKUBZ
      DLZMZANZBNZEOZACUCZBDPZVLDUAVQVOVMVPVNDVKUDUEUFVQVMMZWAACUGZWAACUHZMZBDPZ
      VRWCVQWFBDPZMWEBDPZWIMWDWHVQWJWIABCDEUIUJVQVMWIVMVSVTVKOZAQZBVPPZVQWIABVK
      UKVQWMWLBDPWIWLBVPDULWLWFBDVTDRZWLWNWFSZWFWNVSCRZWAMZMZAQWNWQAQZSWLWOWNWQ
      AUMWKWRAWKWPWNMWAMWRCDVSVTEUNWPWNWAUOUPUQWFWSWNWAACURUSTWNWFUTVAVEVBVCVFW
      EWFBDVDTVMVQVGWBWGBDWAACVHVITVJ $.
  $}

  ${
    $d x y z w A $.
    $( 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 3bitr2i breq1 anbi1d imbi1d equsalvw bitri
      breq2 bitr2i wmo fun2cnv mo4 funcnv2 alrot4 3bitr4i ) CFZBFZEGZUSDFZEGZHZ
      BDIZJZBKZDKZCKZAFZVBEGZVCHZACIZJZAKZDKCKZHZVJUTEGZVCHZVMVEUAZJZBKAKZDKCKZ
      ELZLMZWDMZHWADKCKBKAKWCVGVOHZDKCKVQWBWGCDWBVMVSVEJZJZVEVSVMJZJZHZBKAKWIBK
      AKZWKBKZAKZHWGWAWLABWAVSVMVEJZVEVMJZHZJVSWPJZVSWQJZHWLVTWRVSVMVEUBUCVSWPW
      QUDWSWIWTWKVSVMVENVSVEVMNOUEPWIWKABTWMVGWOVOWMWIAKZBKVGWIABQXAVFBWHVFACVM
      VSVDVEVMVRVAVCVJUSUTEUFUGUHUIRUJWNVNAWJVNBDVEVSVLVMVEVRVKVCUTVBVJEUKUGUHU
      IROSPVGVOCDTULWEVIWFVPWEVABUMZCKVFDKBKZCKVICBEUNXBXCCVAVCBDUTVBUSEUKUORXC
      VHCVFBDQRSWFVKAUMZDKVNCKAKZDKZVPADEUPXDXEDVKVCACVJUSVBEUFUORXFVOCKZDKVPXE
      XGDVNACQRVODCQUJSOWAABCDUQUR $.
  $}

  ${
    $d f g x y z v w A $.
    $( 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 r19.28v anim1d dffun4 simprbi 19.21bbi
      ssel 19.21bi syl9r adantl anim2d jaod 2ralimi funeq sseq1 orbi12d anbi12d
      sseq2 anbi2d cbvral2vw ralcom orcom bitrid bitri anbi12i anandir r19.26-2
      imp anidm 2ralbii bitr2i 3bitr3i eluni exdistrv an4 2exbii 3bitr2i imbi1i
      biancomi 19.23v impexp 2albii albii 3bitr2ri 3imtr4i alrimiv alrimivv syl
      r2al sylanbrc ) BIZJZWQCIZKZWSWQKZLZCAMZNZBAMZAUAZOZDIZEIUBZXFPZXHFIUBZXF
      PZNZEFQZRZFSZESDSZXFJXEWQOZBAMXGXDXRBAWRXRXCWQUCUDUEBAUFUGXEWRXBNZCAMZBAM
      ZXQXDXTBAWRXBCAUHUEYAXPDEYAXOFGIZJZHIZJZNZYBYDKZYDYBKZLZNZHAMGAMZXIYBPZXK
      YDPZNZXNRZHAMGAMZYAXOYJYOGHAAYFYIYOYFYGYOYHYEYGYORYCYGYNXIYDPZYMNZYEXNYGY
      LYQYMYBYDXIUMUIYEYRXNRZFYEYSFSZDEYEYDOYTESDSDEFYDUJUKULUNUOUPYCYHYORYEYHY
      NYLXKYBPZNZYCXNYHYMUUAYLYDYBXKUMUQYCUUBXNRZFYCUUCFSZDEYCYBOUUDESDSDEFYBUJ
      UKULUNUOUDURVNUSYAYANYCYINZHAMGAMZYEYINZHAMGAMZNZYAYKYAUUFYAUUHXSUUEYCYBW
      SKZWSYBKZLZNBCGHAABGQZWRYCXBUULWQYBUTUUMWTUUJXAUUKWQYBWSVAWQYBWSVDVBVCCHQ
      ZUULYIYCUUNUUJYGUUKYHWSYDYBVDWSYDYBVAVBVEVFYAXSBAMCAMUUHXSBCAAVGXSUUGWRYB
      WQKZWQYBKZLZNCBGHAACGQZXBUUQWRXBXAWTLUURUUQWTXAVHUURXAUUOWTUUPWSYBWQVAWSY
      BWQVDVBVIVEBHQZWRYEUUQYIWQYDUTUUSUUOYGUUPYHWQYDYBVDWQYDYBVAVBVCVFVJVKYAVO
      YKUUEUUGNZHAMGAMUUIYJUUTGHAAYCYEYIVLVPUUEUUGGHAAVMVQVRXOYBAPZYDAPZNZYNNZH
      TZGTZXNRUVEXNRZGSZYPXMUVFXNXMYLUVANZGTZYMUVBNZHTZNUVIUVKNZHTGTUVFXJUVJXLU
      VLGXIAVSHXKAVSVKUVIUVKGHVTUVMUVDGHUVMUVCYNYLUVAYMUVBWAWEWBWCWDUVEXNGWFYPU
      VCYORZHSGSUVDXNRZHSZGSUVHYOGHAAWOUVOUVNGHUVCYNXNWGWHUVPUVGGUVDXNHWFWIWJWC
      WKWLWMWNDEFXFUJWP $.
  $}

  $( 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 ) $=
    ( wfun cima ccnv wss crn funimacnv sseq2d inss1 sstr2 biimtrdi imass2 impel
    cin mpi ) CDZCAEZCCFBEZEZGZSBGZATGRUBSBCHZPZGZUCRUAUESBCIJUFUEBGUCBUDKSUEBL
    QMATCNO $.

  ${
    $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 ) ) ) $=
      ( vy vx cdif cima cv wcel wbr wa wex wn exbii wi wmo vex sylib wo elima2
      ccnv wfun anandir 19.40 sylbi wal nfv nfe1 funmo brcnv mobii mopick sylan
      nfan con2d imnan alrimi exancom alnex 3imtr3g anim2d syl5 19.29r sylan2br
      ex andi ianor anbi2i an32 pm3.24 intnan anass mtbir biorfri bitri 3bitr4i
      impbid1 eldif anbi1i notbii anbi12i 3bitr4g eqrdv ) CUAZUBZDCABFZGZCAGZCB
      GZFZWEEHZWFIZWKDHZCJZKZELZWMWHIZWMWIIZMZKZWMWGIWMWJIWEWKAIZWKBIZMZKZWNKZE
      LZXAWNKZELZXBWNKZELZMZKZWPWTWEXFXLXFXHXCWNKZELZKZWEXLXFXGXMKZELXOXEXPEXAX
      CWNUCNXGXMEUDUEWEXNXKXHWEWNXCKZELZXIMZEUFZXNXKWEXRXTWEXRKZXSEWEXREWEEUGXQ
      EUHUNYAXBWNMZOXSYAWNXBWEWNEPZXRWNXCOWEWMWKWDJZEPYCEWMWDUIYDWNEWMWKCDQZEQU
      JUKRWNXCEULUMUOXBWNUPRUQVEWNXCEURXIEUSZUTVAVBXLXGXSKZELZXFXKXHXTYHYFXGXSE
      VCVDYGXEEXGXCYBSZKXGXCKZXGYBKZSZYGXEXGXCYBVFXSYIXGXBWNVGVHXEYJYLXAXCWNVIY
      KYJYKXAWNYBKZKYMXAWNVJVKXAWNYBVLVMVNVOVPNRVQWOXEEWLXDWNWKABVRVSNWQXHWSXKE
      WMCAYETWRXJEWMCBYETVTWAWBEWMCWFYETWMWHWIVRWBWC $.
  $}

  $( 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 ) ) ) $=
    ( ccnv wfun cdif cima cin imadif difeq2d eqtrd dfin4 imaeq2i 3eqtr4g ) CDEZ
    CAABFZFZGZCAGZSCBGZFZFZCABHZGSTHORSCPGZFUBAPCIOUDUASABCIJKUCQCABLMSTLN $.

  $( Condition for the range of a one-to-one function to be the range of one
     its restrictions.  Variant of ~ imadifssran .  (Contributed by AV,
     4-Oct-2025.) $)
  f1imadifssran $p |- ( Fun `' F -> ( ( F " ( dom F \ A ) ) C_ ran ( F |` A )
                                      -> ran F = ran ( F |` A ) ) ) $=
    ( ccnv wfun cdm cdif cima wss crn wceq wa imadmrn imadif sseq1d cun ssundif
    cres unidm sseq2i id imassrn sseqtrri a1i eqssd sylbir biimtrdi imp eqtr3id
    sylbi ex df-ima eqcomi eqeq2i 3imtr4g ) BCDZBBEZAFGZBAGZHZBIZURJZUQBAQIZHUT
    VBJUOUSVAUOUSKUTBUPGZURBLZUOUSVCURJZUOUSVCURFZURHZVEUOUQVFURUPABMNVGVCURURO
    ZHZVEVCURURPVIVCURHZVEVHURVCURRSVJVCURVJTURVCHVJURUTVCBAUAVDUBUCUDUIUEUFUGU
    HUJVBURUQURVBBAUKULZSVBURUTVKUMUN $.

  ${
    $d A x y $.  $d B x y $.
    $( 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.)  Shorten proof and avoid ~ ax-10 , ~ ax-12 .  (Revised
       by SN, 19-Dec-2024.) $)
    funimaexg $p |- ( ( Fun A /\ B e. C ) -> ( A " B ) e. _V ) $=
      ( vx vy wcel wfun cima cvv cv wbr wmo wal wrel dffun6 simprbi wa wrex cab
      dfima2 axrep6g eqeltrid sylan2 ancoms ) BCFZAGZABHZIFZUFUEDJEJAKZELDMZUHU
      FANUJDEAOPUEUJQUGUIDBRESIDEABTUIDEBCUAUBUCUD $.
    $( $j usage 'funimaexg' avoids 'ax-10' 'ax-12'; $)
  $}

  ${
    $d w B $.  $d x y z w A $.
    $( Obsolete version of ~ funimaexg as of 19-Dec-2024.  (Contributed by NM,
       10-Sep-2006.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    funimaexgOLD $p |- ( ( Fun A /\ B e. C ) -> ( A " B ) e. _V ) $=
      ( vw vx vy vz wcel wfun cima cvv cv wceq imaeq2 eleq1d wal wex wel bitri
      wi imbi2d cop dffun5 wa wb nfv axrep4 isset cab dfima3 eqeq2i eqabb exbii
      wrel sylibr simplbiim vtoclg impcom ) BCHAIZABJZKHZUSADLZJZKHZTUSVATDBCVB
      BMZVDVAUSVEVCUTKVBBANOUAUSAUNELFLZUBAHZVFGLZMTFPGQEPZVDEFGAUCVIFGREDRVGUD
      EQZUEFPZGQZVDVGEFGDVGGUFUGVDVHVCMZGQVLGVCUHVMVKGVMVHVJFUIZMVKVCVNVHEFAVBU
      JUKVJFVHULSUMSUOUPUQUR $.
  $}

  ${
    zfrep5.1 $e |- B e. _V $.
    $( The image of a set under any function is also a set.  Equivalent of
       Axiom of Replacement ~ ax-rep .  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.)  (Proof shortened by SN, 19-Dec-2024.) $)
    isarep1 $p |- ( b e. ( { <. x , y >. | ph } " A ) <->
                 E. x e. A [ b / y ] ph ) $=
      ( vz cv copab cima wcel wbr wrex wsb elima df-br vopelopabsb bitri rexbii
      vex cop nfs1v nfv sbequ12r cbvrexw 3bitri ) EGZABCHZDIJFGZUFUGKZFDLACEMZB
      FMZFDLUJBDLFUFUGDESNUIUKFDUIUHUFTUGJUKUHUFUGOABCFEPQRUKUJFBDUJBFUAUJFUBUJ
      FBUCUDUE $.

    $( Obsolete version of ~ isarep1 as of 19-Dec-2024.  (Contributed by NM,
       26-Oct-2006.)  (Proof shortened by Mario Carneiro, 4-Dec-2016.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    isarep1OLD $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 cbvrexw ) ETZABCGZDH
      IFTZUIUJJZFDKACELZBFLZFDKUMBDKFUIUJDEMNULUNFDULUKUIOUJIACUIPZBUKPZUNUKUIU
      JQABCUKUIUAUNUOBFLUPUMUOBFACERUBUOBFRUCSUDUNUMFBDUMBFUEUMFUFUMFBUGUHS $.
  $}

  ${
    $d w x y A $.  $d x y $.  $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 $.

  ${
    fnfund.1 $e |- ( ph -> F Fn A ) $.
    $( A function with domain is a function, deduction form.  (Contributed by
       Jonathan Ben-Naim, 3-Jun-2011.) $)
    fnfund $p |- ( ph -> Fun F ) $=
      ( wfn wfun fnfun syl ) ACBECFDBCGH $.
  $}

  $( 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 wfun wcel cdm fnfun fndm eleq2d biimpar syl2an2r ) DBFZDGCBHZCDIZHZ
      ABDJORPOQBCBDKLMEN $.
  $}

  $( 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 exmoeub
      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 $.
  $}

  $( Restriction of a disjoint union to the domain of the first function.
     (Contributed by Thierry Arnoux, 9-Dec-2021.) $)
  fnunres1 $p |- ( ( F Fn A /\ G Fn B /\ ( A i^i B ) = (/) )
    -> ( ( F u. G ) |` A ) = F ) $=
    ( wfn cin c0 wceq w3a cun cdm cres fndm 3ad2ant1 reseq2d wrel fnrel ineq12d
    3ad2ant2 simp3 eqtrd relresdm1 syl2anc eqtr3d ) CAEZDBEZABFZGHZIZCDJZCKZLZU
    JALCUIUKAUJUEUFUKAHUHACMNZOUICPZUKDKZFZGHULCHUEUFUNUHACQNUIUPUGGUIUKAUOBUMU
    FUEUOBHUHBDMSRUEUFUHTUACDUBUCUD $.

  $( Restriction of a disjoint union to the domain of the second function.
     (Contributed by Thierry Arnoux, 12-Oct-2023.) $)
  fnunres2 $p |- ( ( F Fn A /\ G Fn B /\ ( A i^i B ) = (/) ) -> ( ( F u. G )
      |` B ) = G ) $=
    ( wfn cin c0 wceq w3a cun cres uncom reseq1i ineqcom fnunres1 3com12 eqtrid
    syl3an3b ) CAEZDBEZABFGHZICDJZBKDCJZBKZDUBUCBCDLMTSUAUDDHZUATSBAFGHUEABGNBA
    DCORPQ $.

  $( 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 $.

  ${
    fnund.1 $e |- ( ph -> F Fn A ) $.
    fnund.2 $e |- ( ph -> G Fn B ) $.
    fnund.3 $e |- ( ph -> ( A i^i B ) = (/) ) $.
    $( The union of two functions with disjoint domains, a deduction version.
       (Contributed by metakunt, 28-May-2024.) $)
    fnund $p |- ( ph -> ( F u. G ) Fn ( A u. B ) ) $=
      ( wfn cin c0 wceq cun fnun syl21anc ) ADBIECIBCJKLDEMBCMIFGHBCDENO $.
  $}

  ${
    fnunop.x $e |- ( ph -> X e. V ) $.
    fnunop.y $e |- ( ph -> Y e. W ) $.
    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.)  (Revised by
       AV, 16-Aug-2024.) $)
    fnunop $p |- ( ph -> G Fn E ) $=
      ( csn cun wfn wcel sylibr cop fnsng syl2anc wn cin c0 disjsn fnund fneq1i
      wceq fneq2i bitri ) ADHIUAPZQZBHPZQZRZECRZABUODUMLAHFSIGSUMUORJKHIFGUBUCA
      HBSUDBUOUEUFUJOBHUGTUHURUNCRUQCEUNMUICUPUNNUKULT $.
  $}

  $( 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 two functions with domains as a function with domain.
     (Contributed by NM, 22-May-2006.)  (Proof shortened by AV,
     20-Sep-2024.) $)
  fnco $p |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> ( F o. G ) Fn B ) $=
    ( wfn crn wss w3a ccom ccnv cima wfun fnfun fncofn 3adant3 cdm cnvimassrndm
    sylan2 wceq 3ad2ant3 fndm 3ad2ant2 eqtr2d fneq2d mpbird ) CAEZDBEZDFAGZHZCD
    IZBEUJDJAKZEZUFUGULUHUGUFDLULBDMACDNROUIBUKUJUIUKDPZBUHUFUKUMSUGADQTUGUFUMB
    SUHBDUAUBUCUDUE $.

  $( 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 funresd biantrurd ssdmres fndm sseq2d
    fnfun bitr3id bitr3d bitrid ) CBDZBEUAFZUAGBHZIZCAEZBAJZUABKUEUCUDUFUEUBUCU
    EBCACQLMUCBCGZJUEUFBCNUEUGABACOPRST $.

  $( 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.)  (Proof shortened by Peter
       Mazsa, 2-Oct-2022.) $)
    fnres $p |- ( ( F |` A ) Fn A <-> A. x e. A E! y x F y ) $=
      ( cres wfun cdm wceq wa cv wbr wmo wral wal vex bitri mpbiran 3bitr4i wss
      wcel wex wfn weu ancom wi brresi mobii moanimv albii relres dffun6 df-ral
      wrel cin dmres inss1 eqsstri eqss dfss3 elin2 baib bitrdi ralbiia anbi12i
      eldm r19.26 df-fn df-eu ralbii ) DCEZFZVJGZCHZIZAJZBJZDKZBUAZVQBLZIZACMZV
      JCUBVQBUCZACMVSACMZVRACMZIWDWCIVNWAWCWDUDVKWCVMWDVOVPVJKZBLZANZVOCTZVSUEZ
      ANVKWCWFWIAWFWHVQIZBLWIWEWJBCVOVPDBOUFUGWHVQBUHPUIVKVJUMWGDCUJABVJUKQVSAC
      ULRVMCVLSZWDVMVLCSWKVLCDGZUNCDCUOZCWLUPUQVLCURQWKVOVLTZACMWDACVLUSWNVRACW
      HWNVOWLTZVRWNWHWOVOCWLVLWMUTVABVODAOVEVBVCPPVDVRVSACVFRVJCVGWBVTACVQBVHVI
      R $.
  $}

  $( The identity relation is a function on the universal class.  See also
     ~ funi .  (Contributed by BJ, 23-Dec-2023.) $)
  idfn $p |- _I Fn _V $=
    ( cid cvv wfn wfun cdm wceq funi dmi df-fn mpbir2an ) ABCADAEBFGHABIJ $.

  $( The restricted identity relation is a function on the restricting class.
     (Contributed by NM, 27-Aug-2004.)  (Proof shortened by BJ,
     27-Dec-2023.) $)
  fnresi $p |- ( _I |` A ) Fn A $=
    ( cid cvv wfn wss cres idfn ssv fnssres mp2an ) BCDACEBAFADGAHCABIJ $.

  $( 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.
     _EDITORIAL_: usable in ~ fnwe2lem2 .  (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 dfss2 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 $.  $d y A $.  $d y B $.
    mptfnf.0 $e |- F/_ x A $.
    $( The maps-to notation defines a function with domain.  (Contributed by
       Scott Fenton, 21-Mar-2011.)  (Revised by Thierry Arnoux,
       10-May-2017.) $)
    mptfnf $p |- ( A. x e. A B e. _V <-> ( x e. A |-> B ) Fn A ) $=
      ( vy wcel wral cv wceq wfn ralbii wex wmo wa wal wi albii df-ral 3bitr4ri
      cab cvv weu cmpt eueq r19.26 df-eu copab wfun df-mpt fneq1i df-fn moanimv
      cdm bitri funopab eqcom dmopab 19.42v abbii eqtri eqeq1i wb eqabf 3bitr4i
      pm4.71 anbi12i ancom 3bitr2i bitr4i ) CUAFZABGEHCIZEUBZABGZABCUCZBJZVJVLA
      BECUDKVKELZVKEMZNZABGVPABGZVQABGZNZVMVOVPVQABUEVLVRABVKEUFKVOAHBFZVKNZAEU
      GZUHZWDUMZBIZNZVTVSNWAVOWDBJWHBVNWDAEBCUIUJWDBUKUNVTWEVSWGWCEMZAOWBVQPZAO
      WEVTWIWJAWBVKEULQWCAEUOVQABRSWBVPNZATZBIBWLIZWGVSWLBUPWFWLBWFWCELZATWLWCA
      EUQWNWKAWBVKEURUSUTVAWBVPPZAOWBWKVBZAOVSWMWOWPAWBVPVEQVPABRWKABDVCVDSVFVT
      VSVGVHSVI $.

    $( The maps-to notation defines a function with domain.  (Contributed by
       NM, 9-Apr-2013.)  (Revised by Thierry Arnoux, 10-May-2017.) $)
    fnmptf $p |- ( A. x e. A B e. V -> ( x e. A |-> B ) Fn A ) $=
      ( wcel wral cvv cmpt wfn elex ralimi mptfnf sylib ) CDFZABGCHFZABGABCIBJO
      PABCDKLABCEMN $.
  $}

  ${
    $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
      albii funopab df-ral 3bitr4ri dmopab3 anbi12i r19.26 df-fn df-eu biancomi
      wi ralbii fneq1i ) ACGZACHZIZBDJZBKDLZAIZBCMZDNZACOZBDJEDNUOBDJZUPBDJZIVA
      PZVAQDRZIURVBVDVFVEVGUTCGZBSUSUOULZBSVFVDVHVIBUSACUAUBUTBCUCUOBDUDUEABCDU
      FUGUOUPBDUHVADUITVCUQBDVCUOUPACUJUKUMDEVAFUNT $.
  $}

  ${
    $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 $.
  $}

  ${
    $d A x $.
    fnmptd.1 $e |- F/ x ph $.
    fnmptd.2 $e |- ( ( ph /\ x e. A ) -> B e. V ) $.
    fnmptd.3 $e |- F = ( x e. A |-> B ) $.
    $( The maps-to notation defines a function with domain.  (Contributed by
       Glauco Siliprandi, 23-Oct-2021.) $)
    fnmptd $p |- ( ph -> F Fn A ) $=
      ( wcel wral wfn cv ex ralrimi fnmpt syl ) ADFJZBCKECLARBCGABMCJRHNOBCDEFI
      PQ $.
  $}

  $( 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 the mapping operation.  (Contributed by NM, 6-Sep-2005.)
       (Revised by Mario Carneiro, 31-Aug-2015.) $)
    dmmpti $p |- dom F = A $=
      ( fnmpti fndmi ) BDABCDEFGH $.
  $}

  ${
    $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 $.
  $}

  $( Rewrite a function defined by parts, using a mapping and an if construct,
     into a union of functions on disjoint domains.  (Contributed by Thierry
     Arnoux, 30-Mar-2017.) $)
  partfun $p |- ( x e. A |-> if ( x e. B , C , D ) ) =
                ( ( x e. ( A i^i B ) |-> C ) u. ( x e. ( A \ B ) |-> D ) ) $=
    ( cin cdif cun cv wcel cif cmpt mptun inundif eqid mpteq12i elinel2 iftrued
    mpteq2ia eldifn iffalsed uneq12i 3eqtr3i ) ABCFZBCGZHZAIZCJZDEKZLAUDUILZAUE
    UILZHABUILAUDDLZAUEELZHAUDUEUIMAUFUIBUIBCNUIOPUJULUKUMAUDUIDUGUDJUHDEUGBCQR
    SAUEUIEUGUEJUHDEUGBCTUASUBUC $.

  $( 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 $.
  $}

  ${
    feq1dd.eq $e |- ( ph -> F = G ) $.
    feq1dd.f $e |- ( ph -> F : A --> B ) $.
    $( Equality deduction for functions.  (Contributed by Glauco Siliprandi,
       11-Dec-2019.) $)
    feq1dd $p |- ( ph -> G : A --> B ) $=
      ( wf feq1d mpbid ) ABCDHBCEHGABCDEFIJ $.
  $}

  ${
    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 feq3d bitrd ) ABDFKCDGKCEGKABCDFGHILADEGCJMN $.
  $}

  $( 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 $.
  $}

  ${
    feq12i.1 $e |- F = G $.
    feq12i.2 $e |- A = B $.
    $( Equality inference for functions.  (Contributed by AV, 7-Feb-2021.) $)
    feq12i $p |- ( F : A --> C <-> G : B --> C ) $=
      ( wceq wf wb eqid feq123 mp3an ) DEHABHCCHACDIBCEIJFGCKACBCDELM $.
  $}

  ${
    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 csbdm
      wceq eqeq1i bitrdi anbi12d sbcan 3bitr4g bitrd ) EDFZCBGZAEHCIZCJZBTZKZAE
      HZAECLZAEBLZGZUGUHULAEUHULMUGCBNOPUGUIAEHZUKAEHZKUNIZUNJZUOTZKUMUPUGUQUSU
      RVAAECDQUGURAEUJLZUOTVAAEUJBDRVBUTUOAECSUAUBUCUIUKAEUDUNUONUEUF $.

    $( 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 sbcssg csbrn
      sseq1i bitrdi anbi12d sbcan 3bitr4g bitrd ) FEGZBCDHZAFIDBJZDKZCLZMZAFIZA
      FBNZAFCNZAFDNZHZUHUIUMAFUIUMOUHBCDPQRUHUJAFIZULAFIZMUQUOJZUQKZUPLZMUNURUH
      USVAUTVCABDEFSUHUTAFUKNZUPLVCAFUKCETVDVBUPAFDUAUBUCUDUJULAFUEUOUPUQPUFUG
      $.
  $}

  ${
    elimf.1 $e |- G : A --> B $.
    $( Eliminate a mapping hypothesis for the weak deduction theorem ~ dedth ,
       when a special case ` G : A --> B ` is provable, in order to convert
       ` F : A --> B ` from a hypothesis to an antecedent.  (Contributed by NM,
       24-Aug-2006.) $)
    elimf $p |- if ( F : A --> B , F , G ) : A --> B $=
      ( wf cif feq1 elimhyp ) ABCFZABJCDGZFABDFCDABCKHABDKHEI $.
  $}

  $( A mapping is a function with domain.  (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 $.

  ${
    freld.1 $e |- ( ph -> F : A --> B ) $.
    $( A mapping is a relation.  (Contributed by Glauco Siliprandi,
       26-Jun-2021.) $)
    freld $p |- ( ph -> Rel F ) $=
      ( wf wrel frel syl ) ABCDFDGEBCDHI $.
  $}

  $( 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 $.
  $}

  $( The domain of a mapping.  (Contributed by NM, 2-Aug-1994.)  (Proof
     shortened by Wolf Lammen, 29-May-2024.) $)
  fdm $p |- ( F : A --> B -> dom F = A ) $=
    ( wf ffn fndmd ) ABCDACABCEF $.

  ${
    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 $.
    $( Inference associated with ~ fdm .  The domain of a mapping.
       (Contributed by NM, 28-Jul-2008.) $)
    fdmi $p |- dom F = A $=
      ( wf cdm wceq fdm ax-mp ) ABCECFAGDABCHI $.
  $}

  $( 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 $.

  $( Characterization of a function with domain and codomain (essentially using
     that the range is always included in the codomain).  Generalization of
     ~ ffrn .  (Contributed by BJ, 21-Sep-2024.) $)
  ffrnb $p |- ( F : A --> B <-> ( F : A --> ran F /\ ran F C_ B ) ) $=
    ( wf wfn crn wss wa df-f dffn3 anbi1i bitri ) ABCDCAEZCFZBGZHANCDZOHABCIMPO
    ACJKL $.

  ${
    ffrnbd.r $e |- ( ph -> ran F C_ B ) $.
    $( A function maps to its range iff the range is a subset of its codomain.
       Generalization of ~ ffrn .  (Contributed by AV, 20-Sep-2024.) $)
    ffrnbd $p |- ( ph -> ( F : A --> B <-> F : A --> ran F ) ) $=
      ( wf crn wss wa ffrnb biantrud bitr4id ) ABCDFBDGZDFZMCHZINBCDJAONEKL $.
  $}

  $( 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 $.
  $}

  $( The preimage of the codomain of a function is the function's domain.
     (Contributed by FL, 25-Jan-2007.)  (Proof shortened by AV,
     20-Sep-2024.) $)
  fimacnv $p |- ( F : A --> B -> ( `' F " B ) = A ) $=
    ( wf ccnv cima cdm crn wss wceq frn cnvimassrndm syl fdm eqtrd ) ABCDZCEBFZ
    CGZAPCHBIQRJABCKBCLMABCNO $.

  $( 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 $.

  $( Composition of two functions with domain and codomain as a function with
     domain and codomain.  (Contributed by NM, 29-Aug-1999.)  (Proof shortened
     by Andrew Salmon, 17-Sep-2011.)  (Proof shortened by AV, 20-Sep-2024.) $)
  fco $p |- ( ( F : B --> C /\ G : A --> B ) -> ( F o. G ) : A --> C ) $=
    ( wf ccom ccnv cima wfun ffun fcof sylan2 wceq fimacnv eqcomd adantl mpbird
    wa feq2d ) BCDFZABEFZSZACDEGZFEHBIZCUDFZUBUAEJUFABEKBCDELMUCAUECUDUBAUENUAU
    BUEAABEOPQTR $.

  ${
    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 $.

  $( 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 $.
  $}

  $( 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 $.

  ${
    funcofd.1 $e |- ( ph -> Fun F ) $.
    funcofd.2 $e |- ( ph -> Fun G ) $.
    $( Composition of two functions as a function with domain and codomain.
       (Contributed by Glauco Siliprandi, 26-Jun-2021.)  (Proof shortened by
       AV, 20-Sep-2024.) $)
    funcofd $p |- ( ph -> ( F o. G ) : ( `' G " dom F ) --> ran F ) $=
      ( cdm crn wf wfun ccnv cima ccom fdmrn sylib fcof syl2anc ) ABFZBGZBHZCIC
      JQKRBCLHABISDBMNEQRBCOP $. $( 6 s, 96 b $)
  $}

  $( 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 rnun unss12 df-f an4 impcom
    eqsstrid anim12d1 anbi12i bitri 3imtr4g ) ABGHIZACEJZBDFJZKZABLZCDLZEFLZJZU
    HEAMZFBMZKZENZCOZFNZDOZKZKZUNULMZUNNZUMOZKUKUOUHURVEVCVGURUHVEABEFPQVCVFUSV
    ALUMEFRUSCVADSUCUDUKUPUTKZUQVBKZKVDUIVHUJVIACETBDFTUEUPUTUQVBUAUFULUMUNTUGU
    B $.

  $( 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 wa cres wfn crn df-f fnssres resss rnssi sstr anim12i an32s sylanb
    mpan sylibr ) ABDEZCAFZGDCHZCIZUCJZBFZGZCBUCEUADAIZDJZBFZGUBUGABDKUHUBUJUGU
    HUBGUDUJUFACDLUEUIFUJUFUCDDCMNUEUIBOSPQRCBUCKT $.

  ${
    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, express their union as a
     union of three functions with pairwise disjoint domains.  (Contributed by
     Stefan O'Rear, 9-Oct-2014.) $)
  resasplit $p |- ( ( F Fn A /\ G Fn B /\
      ( F |` ( A i^i B ) ) = ( G |` ( A i^i B ) ) ) -> ( F u. G ) =
  ( ( F |` ( A i^i B ) ) u. ( ( F |` ( A \ B ) ) u. ( G |` ( B \ A ) ) ) ) ) $=
    ( wfn cin cres w3a cun fnresdm uneq12 syl2an inundif reseq2i resundi eqtr3i
    wceq cdif uneq1i eqtrdi 3adant3 incom eqtri uneq12i simp3 uneq1d uneq2d un4
    eqtr4id unidm eqtr3d ) CAEZDBEZCABFZGZDUNGZQZHZCAGZDBGZIZCDIZUOCABRZGZDBARZ
    GZIZIZULUMVAVBQZUQULUSCQUTDQVIUMACJBDJUSCUTDKLUAURVAUOUOIZVGIZVHURVAUOVDIZU
    OVFIZIZVKURVAVLUPVFIZIVNUSVLUTVOCUNVCIZGUSVLVPACABMNCUNVCOPDUNVEIZGUTVOVQBD
    VQBAFZVEIBUNVRVEABUBSBAMUCNDUNVEOPUDURVMVOVLURUOUPVFULUMUQUEUFUGUIUOVDUOVFU
    HTVJUOVGUOUJSTUK $.

  $( The union of two functions which agree on their common domain is a
     function.  (Contributed by Stefan O'Rear, 9-Oct-2014.) $)
  fresaun $p |- ( ( F : A --> C /\ G : B --> C /\
      ( F |` ( A i^i B ) ) = ( G |` ( A i^i B ) ) ) ->
    ( F u. G ) : ( A u. B ) --> C ) $=
    ( wf cin cres wceq cun wss fssres sylancl difss c0 disjdif 3eqtri a1i fun2d
    cdif simp1 inss1 simp2 indifdir difeq1i 0dif indi inass ineq2i incom ineq1i
    w3a in0 eqtri uneq12i un0 un12 uneq1i inundif uneq2i undif1 feq2i resasplit
    wfn ffn id syl3an feq1d bitr4id mpbid ) ACDFZBCEFZDABGZHZEVMHIZULZVMABTZBAT
    ZJZJZCVNDVQHZEVRHZJZJZFZABJZCDEJZFZVPVMVSCVNWCVPVKVMAKVMCVNFVKVLVOUAZABUBAC
    VMDLMVPVQVRCWAWBVPVKVQAKVQCWAFWIABNACVQDLMVPVLVRBKVRCWBFVKVLVOUCBANBCVRELMV
    QVRGZOIVPWJAVRGZBVRGZTOWLTOABVRUDWKOWLABPZUEWLUFQRSVMVSGZOIVPWNVMVQGZVMVRGZ
    JOOJOVMVQVRUGWOOWPOWOABVQGZGAOGOABVQUHWQOABAPUIAUMQWPBAGZVRGZOVMWRVRABUJZUK
    WSBWKGBOGOBAVRUHWKOBWMUIBUMQUNUOOUPQRSVPWEWFCWDFWHVTWFCWDVTVQVMVRJZJVQBJWFV
    MVQVRUQXABVQXAWRVRJBVMWRVRWTURBAUSUNUTABVAQVBVPWFCWGWDVKDAVDVLEBVDVOVOWGWDI
    ACDVEBCEVEVOVFABDEVCVGVHVIVJ $.

  $( From the union of two functions that agree on the domain overlap, either
     component can be recovered by restriction.  (Contributed by Stefan O'Rear,
     9-Oct-2014.) $)
  fresaunres2 $p |- ( ( F : A --> C /\ G : B --> C /\
      ( F |` ( A i^i B ) ) = ( G |` ( A i^i B ) ) ) ->
    ( ( F u. G ) |` B ) = G ) $=
    ( wf cin cres wceq cun cdif wfn ffn resundir wss ax-mp c0 cdm eqtri eqtrid
    w3a id resasplit syl3an reseq1d inss2 resabs2 uneq12i ineq2i disjdif ineq1i
    dmres inass 0in 3eqtr3i wb relres reldm0 mpbir difss uneq2i simp3 uneq1d wa
    wrel uncom un0 resundi incom uneq1i inundif reseq2i fnresdm eqtr3id 3adant3
    syl adantl eqtrd ) ACDFZBCEFZDABGZHZEWAHZIZUAZDEJZBHWBDABKZHZEBAKZHZJZJZBHZ
    EWEWFWLBVSDALVTEBLZWDWDWFWLIACDMBCEMZWDUBABDEUCUDUEWEWMWBBHZWKBHZJZEWBWKBNW
    EWRWBWHBHZWJBHZJZJZEWPWBWQXAWABOWPWBIABUFDWABUGPWHWJBNUHWEXBWBQWJJZJZEXAXCW
    BWSQWTWJWSQIZWSRZQIZXFBWHRZGZQWHBULXIBWGDRZGZGZQXHXKBDWGULUIBWGGZXJGQXJGXLQ
    XMQXJBAUJUKBWGXJUMXJUNUOSSWSVEXEXGUPWHBUQWSURPUSWIBOWTWJIBAUTEWIBUGPUHVAWEX
    DWCXCJZEWEWBWCXCVSVTWDVBVCVSVTXNEIWDVSVTVDZXNWCWJJZEXCWJWCXCWJQJWJQWJVFWJVG
    SVAXOXPEWAWIJZHZEEWAWIVHXOXREBHZEXQBEXQBAGZWIJBWAXTWIABVIVJBAVKSVLVTXSEIZVS
    VTWNYAWOBEVMVPVQTVNTVOVRTTTVR $.

  $( From the union of two functions that agree on the domain overlap, either
     component can be recovered by restriction.  (Contributed by Mario
     Carneiro, 16-Feb-2015.) $)
  fresaunres1 $p |- ( ( F : A --> C /\ G : B --> C /\
      ( F |` ( A i^i B ) ) = ( G |` ( A i^i B ) ) ) ->
    ( ( F u. G ) |` A ) = F ) $=
    ( wf cin cres w3a cun uncom reseq1i incom reseq2i eqeq12i eqcom fresaunres2
    wceq bitri 3com12 syl3an3b eqtrid ) ACDFZBCEFZDABGZHZEUEHZRZIDEJZAHEDJZAHZD
    UIUJADEKLUHUCUDEBAGZHZDULHZRZUKDRZUHUNUMRUOUFUNUGUMUEULDABMZNUEULEUQNOUNUMP
    SUDUCUOUPBACEDQTUAUB $.

  $( 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 pm4.71rd vex opelresi bitri
      opelcnv bitr4di relres opelf simpld simprd anbi2i bitr3d eqrelrdv ) ABCFZ
      DECAGZHZCHZBGZUIIUKBUAUHEJZDJZKZCLZUNUMKZUJLZUQULLZUHUPUMALZUPMZURUHUPUTU
      HUPUTUHUPMZUTUNBLZABUMUNCUBZUCNOURUOUILVAUNUMUIDPZEPZSAUMUNCVEQRTUHUPVCUP
      MZUSUHUPVCUHUPVCVBUTVCVDUDNOUSVCUQUKLZMVGBUNUMUKVFQVHUPVCUNUMCVEVFSUERTUF
      UG $.
  $}

  $( 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 $.
    fint.1 $e |- B =/= (/) $.
    $( Function into an intersection.  (Contributed by NM, 14-Oct-1999.)
       (Proof shortened by Andrew Salmon, 17-Sep-2011.) $)
    fint $p |- ( F : A --> |^| B <-> A. x e. B F : A --> x ) $=
      ( wfn crn cint wss wa cv wral wf ssint anbi2i c0 wne wb r19.28zv df-f
      ax-mp bitr4i ralbii 3bitr4i ) DBFZDGZCHZIZJZUEUFAKZIZJZACLZBUGDMBUJDMZACL
      UIUEUKACLZJZUMUHUOUEAUFCNOCPQUMUPREUEUKACSUAUBBUGDTUNULACBUJDTUCUD $.
  $}

  $( 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 $.

  $( 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 dfss2 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 Cartesian 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 Cartesian 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 Cartesian 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 fconstg ffnd ) BCDABEZAIFABCGH $.

  $( 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 fconstg snssi fssd ) BCDABEZCAJFABCGBCHI $.

  ${
    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 $.

  $( 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 f1f ffnd ) ABCDABCABCEF $.

  $( 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.)
     (Proof shortened by Wolf Lammen, 29-May-2024.) $)
  f1dm $p |- ( F : A -1-1-> B -> dom F = A ) $=
    ( wf1 f1fn fndmd ) ABCDACABCEF $.

  $( A function that is one-to-one is also one-to-one on some superset of its
     codomain.  (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 $.

  $( A function that is one-to-one is also one-to-one on some superset of its
     range.  (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 $.

  $( 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 fssres sylan df-f1 funres11 simplbiim adantr
    f1f sylanbrc ) ABDEZCAFZGCBDCHZIZUBJKZCBUBETABDIZUAUCABDRABCDLMTUDUATUEDJKU
    DABDNCDOPQCBUBNS $.

  $( 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 cres wf w3a crn f1ssres 3adant3 frn 3ad2ant3 f1ssr syl2anc ) ABEF
    ZCAGZCDECHZIZJCBTFZTKDGZCDTFRSUBUAABCELMUARUCSCDTNOCBDTPQ $.

  $( 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 two one-to-one functions.  Generalization of ~ f1co .
     (Contributed by AV, 18-Sep-2024.) $)
  f1cof1 $p |- ( ( F : C -1-1-> D /\ G : A -1-1-> B )
                -> ( F o. G ) : ( `' G " C ) -1-1-> D ) $=
    ( wf1 wa ccnv cima ccom wf wfun df-f1 ffun fcof sylan2 funco cnvco sylibr
    funeqi ancoms anim12i an4s syl2anb ) CDEGZABFGZHFIZCJZDEFKZLZUJIZMZHZUIDUJG
    UFCDELZEIZMZHABFLZUHMZHUNUGCDENABFNUOURUQUSUNUOURHUKUQUSHUMURUOFMUKABFOCDEF
    PQUSUQUMUSUQHUHUPKZMUMUHUPRULUTEFSUATUBUCUDUEUIDUJNT $.

  $( Composition of one-to-one functions when the codomain of the first matches
     the domain of the second.  Exercise 30 of [TakeutiZaring] p. 25.
     (Contributed by NM, 28-May-1998.)  (Proof shortened by AV,
     20-Sep-2024.) $)
  f1co $p |-
         ( ( F : B -1-1-> C /\ G : A -1-1-> B ) -> ( F o. G ) : A -1-1-> C ) $=
    ( wf1 wa ccom ccnv cima f1cof1 wceq wb f1f fimacnv syl adantl eqcomd f1eq2
    wf mpbird ) BCDFZABEFZGZACDEHZFZEIBJZCUEFZABBCDEKUDAUGLUFUHMUDUGAUCUGALZUBU
    CABETUIABENABEOPQRAUGCUESPUA $.

  $( 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 fof ffund ) ABCDABCABCEF $.

  $( 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 fof ffnd ) ABCDABCABCEF $.

  $( 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 fof fdmd imaeq2d forn 3eqtr3a ) ABCDZCCEZFCGCAFB
    CHNOACNABCABCIJKABCLM $.

  $( 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 an onto 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 $.

  $( Alternate proof of ~ fimadmfo , based on ~ fores .  A function is a
     function onto the image of its domain.  (Contributed by AV, 1-Dec-2022.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  fimadmfoALT $p |- ( F : A --> B -> F : A -onto-> ( F " A ) ) $=
    ( wf cres wceq cima wfo cdm fdm wrel resdm eqcomd syl reseq2 sylan9eq mpdan
    frel wa wfun wss ffun eqimss2 jca adantr fores wb foeq1 adantl mpbird ) ABC
    DZCCAEZFZACAGZCHZUKCIZAFZUMABCJZUKUQCCUPEZULUKCKZCUSFABCRUTUSCCLMNUPACOPQUK
    UMSZUOAUNULHZVACTZAUPUAZSZVBUKVEUMUKVCVDABCUBUKUQVDURAUPUCNUDUEACUFNUMUOVBU
    GUKAUNCULUHUIUJQ $.

  $( The preimage of the codomain of a surjection is its domain.  (Contributed
     by AV, 29-Sep-2024.) $)
  focnvimacdmdm $p |- ( G : A -onto-> B -> ( `' G " B ) = A ) $=
    ( wfo ccnv cima cdm crn forn eqcomd imaeq2d cnvimarndm eqtrdi fdmd eqtrd
    fof ) ABCDZCEZBFZCGZAQSRCHZFTQBUARQUABABCIJKCLMQABCABCPNO $.

  $( Composition of onto functions.  Generalisation of ~ foco .  (Contributed
     by AV, 29-Sep-2024.) $)
  focofo $p |- ( ( F : A -onto-> B /\ Fun G /\ A C_ ran G )
                 -> ( F o. G ) : ( `' G " A ) -onto-> B ) $=
    ( wfo wfun crn wss w3a ccnv cima ccom wceq fcof sylan 3adant3 cres 3ad2ant1
    wf fof rnco cdm freld fdm eqcomd syl sseq1d biimpa relssres rneqd 3imp3i2an
    wrel wa forn eqtrd eqtrid dffo2 sylanbrc ) ABCEZDFZADGZHZIZDJAKZBCDLZSZVEGZ
    BMVDBVEEUSUTVFVBUSABCSZUTVFABCTZABCDNOPVCVGCVAQZGZBCDUAVCVKCGZBUSUTVBCULZCU
    BZVAHZVKVLMUSUTVMVBUSABCVIUCRUSVBVOUSAVNVAUSVHAVNMVIVHVNAABCUDUEUFUGUHVMVOU
    MVJCCVAUIUJUKUSUTVLBMVBABCUNRUOUPVDBVEUQUR $.

  $( Composition of onto functions.  (Contributed by NM, 22-Mar-2006.)  (Proof
     shortened by AV, 29-Sep-2024.) $)
  foco $p |- ( ( F : B -onto-> C /\ G : A -onto-> B ) ->
             ( F o. G ) : A -onto-> C ) $=
    ( wfo ccom ccnv cima wfun crn wss simpl fofun adantl wceq forn eqimss2 syl
    wa focofo syl3anc wb focnvimacdmdm eqcomd foeq2 mpbird ) BCDFZABEFZTZACDEGZ
    FZEHBIZCUKFZUJUHEJZBEKZLZUNUHUIMUIUOUHABENOUIUQUHUIUPBPUQABEQBUPRSOBCDEUAUB
    UJAUMPZULUNUCUIURUHUIUMAABEUDUEOAUMCUKUFSUG $.

  $( A nonzero constant function is onto.  (Contributed by NM, 12-Jan-2007.) $)
  foconst $p |- ( ( F : A --> { B } /\ F =/= (/) ) -> F : A -onto-> { B } ) $=
    ( csn wf c0 wne wa crn wceq wfo wn wrel wb frel relrn0 necon3abid syl wss
    wo frn sssn sylib ord sylbid imdistani dffo2 sylibr ) ABDZCEZCFGZHUJCIZUIJZ
    HAUICKUJUKUMUJUKULFJZLZUMUJCMZUKUONAUICOUPUNCFCPQRUJUNUMUJULUISUNUMTAUICUAU
    LBUBUCUDUEUFAUICUGUH $.

  $( 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 f1of ffnd ) ABCDABCABCEF $.

  $( 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 $.

  $( 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 mpcom anim1ci
    wi dff1o4 3imtr4i ) CADZCEZBDZFUBUAEZADZFABCGBAUAGTUDUBCHZTUDACIUEUCCJZTUDQ
    CKUFUDTAUCCLMNOPABCRBAUARS $.

  $( 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 cima funcnvres crn df-ima wf1 dff1o5
    wceq simprbi eqtrid reseq2d sylan9eq f1oeq1d mpbid ) CDZEZBACBFZGZHZABUFDZG
    ZABUDAFZGUGUJUEBAUFIJUHABUIUKUEUGUIUDCBKZFUKBCLUGULAUDUGULUFMZACBNUGBAUFOUM
    AQBAUFPRSTUAUBUC $.

  $( 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 df-fn sylanblrc dfdm4 forn sseq2d biimpar ssdmres sylib df-fo
    sseqtrdi sylanbrc foima eqtr3d ) ABDEZCBFZGZDDHZCIZIDVJJZVJIZCDVJKVHVICJZHZ
    VJIZVLCVHVNVKVJVHDLZVNVKMVFVPVGABDNZOCDUAPUBVHVJCVNEZVOCMVHVNVJUCZVNQZCMVRV
    HVNLZVNRZVJMVSVFWAVGVNDFVFVPWAVNVIHZDVMVIFVNWCFVICUDVMVIUEUFDUGUHVQVNDUIUJO
    VJVMQWBVICUKVMSULVNVJUMUNVHVTVMRZCVMUOVHCVIRZFWDCMVHCDQZWEVFCWFFVGVFWFBCABD
    UPUQURDSVBCVIUSUTTVJCVNVAVCVJCVNVDPVET $.

  $( 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 $.

  $( The union of two one-to-one functions with disjoint domains and codomains.
     (Contributed by BTernaryTau, 3-Dec-2024.) $)
  f1un $p |- ( ( ( F : A -1-1-> B /\ G : C -1-1-> D )
           /\ ( ( A i^i C ) = (/) /\ ( B i^i D ) = (/) ) )
           -> ( F u. G ) : ( A u. C ) -1-1-> ( B u. D ) ) $=
    ( wf1 wa crn cun wss cin c0 wceq f1f frnd syl2an wf1o f1f1orn syl2an2r f1ss
    unss12 anim12i simprl ss2in sseq0 sylan adantrl jca f1oun f1of1 syl ancoms
    ) ABEGZCDFGZHZEIZFIZJZBDJZKZACLMNZBDLZMNZHZACJZUSEFJZGZVFUTVGGZUNUQBKZURDKZ
    VAUOUNABEABEOPZUOCDFCDFOPZUQBURDUBQUPVEHZVFUSVGRZVHUPAUQERZCURFRZHVEVBUQURL
    ZMNZHVOUNVPUOVQABESCDFSUCVNVBVSUPVBVDUDUPVDVSVBUPVRVCKZVDVSUNVJVKVTUOVLVMUQ
    BURDUEQVRVCUFUGUHUIAUQCUREFUJTVFUSVGUKULVHVAVIVFUSUTVGUAUMT $.

  $( 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 wb ax-mp crn df-ima wa forn eqtrid
    w3a cdm wss fofun difss fof fdmd sseqtrrid fores syl2anc cin resres reseq2i
    indif eqtri foeq1 rneqi 3eqtr4i foeq3 bitri funres11 dff1o3 biimpri syl2anr
    sylib 3adant3 anim12i imadif difeq12 sylan9eq sylan2 3impb f1oeq3d mpbid )
    EFGZACEAHZIZBDEBHZIZUAZABJZEWAKZEWAHZLZWACDJZWCLVOVQWDVSVQWAWBWCIZWCFGZWDVO
    VQWAVPWAKZVPWAHZIZWFVQVPGWAVPUBZUCWJACVPUDVQAWAWKABUEVQACVPACVPUFUGUHWAVPUI
    UJWJWAWHWCIZWFWIWCMWJWLNWIEAWAUKZHWCEAWAULWMWAEABUNUMUOZWAWHWIWCUPOWHWBMWLW
    FNWIPWCPWHWBWIWCWNUQVPWAQEWAQURWHWBWAWCUSOUTVEWAEVAWDWFWGRWAWBWCVBVCVDVFVTW
    BWEWAWCVOVQVSWBWEMZVQVSRVOEAKZCMZEBKZDMZRZWOVQWQVSWSVQWPVPPCEAQACVPSTVSWRVR
    PDEBQBDVRSTVGVOWTWBWPWRJWEABEVHWPCWRDVIVJVKVLVMVN $.

  $( The restriction of a one-to-one onto function to an intersection maps onto
     the intersection of the images.  (Contributed by Paul Chapman,
     11-Apr-2009.) $)
  resin $p |- ( ( Fun `' F /\ ( F |` A ) : A -onto-> C /\
                               ( F |` B ) : B -onto-> D ) ->
      ( F |` ( A i^i B ) ) : ( A i^i B ) -1-1-onto-> ( C i^i D ) ) $=
    ( ccnv wfun cres wfo w3a cdif wf1o cin resdif f1ofo syl wceq wb dfin4 ax-mp
    syld3an3 f1oeq3 f1oeq2 reseq2i f1oeq1 3bitrri sylib ) EFGZACEAHIZBDEBHIZJZA
    ABKZKZCCDKZKZEUMHZLZABMZCDMZEURHZLZUHUIUJULUNEULHZIZUQUKULUNVBLVCABCDENULUN
    VBOPAULCUNENUAVAURUOUTLZUMUOUTLZUQUSUOQVAVDRCDSUSUOURUTUBTURUMQVDVERABSZURU
    MUOUTUCTUTUPQVEUQRURUMEVFUDUMUOUTUPUETUFUG $.

  $( 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 reseq2i eqeq2i bitri sylib ) ABZAACZDZEFZUFEAGZHZIZU
    DAJUGAKLUGUFEUFMZHZIUJUFPULUIUFUKUHEUKUEMZUHAMUEGIUKUMIANAUEOQARSTUAUBUC $.

  $( 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 $.

  $( 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 funcnv0 df-f1 mpbir2an ) BABCBABDBEFAGHBABIJ $.

  ${
    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 fndmd dm0 eqtr3di biimpri mpbir fneq2 sylan9bb mpbiri
    eqid impbii bitri ) CABDBCEZBFZAEZGZBCHZACHZGZCABIUOURUOUPUQULUPUNULUPBJZKL
    UOCQACUOACULUNCAEZULAUMCULUPUMCHUSUPUMCFCBCMNOZPRSUAUBUCTURULUNUPULUQULUPUS
    UDLURUNCCEZVBCCHCUICJUEUPUNUTUQVBUPAUMCVARACCUFUGUHTUJUK $.

  $( 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 f1oeq2d opeq1 sneqd f1oeq1d bitrd opeq2
      f1oeq3d vex f1osn vtocl2g ) EGZHZFGZHZUCUEIZHZJZAHZUFAUEIZHZJZUJBHZABIZHZ
      JZEFABCDUCAKZUIUJUFUHJUMURUDUJUFUHUCALMURUJUFUHULURUGUKUCAUENOPQUEBKZUMUJ
      UNULJUQUSUFUNUJULUEBLSUSUJUNULUPUSUKUOUEBAROPQUCUEETFTUAUB $.

    $( 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 $.
  $}

  $( A two-element swap is a bijection on a pair.  (Contributed by Mario
     Carneiro, 23-Jan-2015.) $)
  f1oprswap $p |- ( ( A e. V /\ B e. W ) ->
      { <. A , B >. , <. B , A >. } : { A , B } -1-1-onto-> { A , B } ) $=
    ( wcel wa cpr cop wf1o wceq csn f1osng anidms ad2antrr dfsn2 eqtrid wfn cun
    ccnv cnvsng wb opeq2 opeq1 preq12d preq2 f1oeq123d adantl wne simpll simplr
    mpbid simpr fnprg syl221anc ancoms uneq12d uncom eqtrdi adantr df-pr cnveqi
    cnvun eqtri 3eqtr4g fneq1d mpbird dff1o4 sylanbrc pm2.61dane ) ACEZBDEZFZAB
    GZVMABHZBAHZGZIZABVLABJZFAKZVSAAHZKZIZVQVJWBVKVRVJWBAACCLMNVRWBVQUAVLVRVSVM
    VSVMWAVPVRWAVTVTGVPVTOVRVTVNVTVOABAUBABAUCUDPVRVSAAGVMAOABAUEPZWCUFUGUKVLAB
    UHZFZVPVMQZVPSZVMQZVQWEVJVKVKVJWDWFVJVKWDUIZVJVKWDUJZWJWIVLWDULABBACDDCUMUN
    ZWEWHWFWKWEVMWGVPWEVNKZSZVOKZSZRZWLWNRZWGVPVLWPWQJWDVLWPWNWLRWQVLWMWNWOWLAB
    CDTVKVJWOWLJBADCTUOUPWNWLUQURUSWGWQSWPVPWQVNVOUTZVAWLWNVBVCWRVDVEVFVMVMVPVG
    VHVI $.

  $( An unordered pair of ordered pairs with different elements is a one-to-one
     onto function, analogous to ~ f1oprswap .  (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 ` .
       See ~ brprcneuALT for a proof that uses ~ ax-pow instead of ~ ax-pr .
       (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 exnal sylibr opprc1 eleq1d df-br
      wal cvv wmo weu dtru equcom albii xchbinx mpbir jctr 19.42v anbi12d anidm
      bitrdi anbi1d exbidv imbi12d mpbiri anbi12i anbi1i 3imtr4g eximdv exanali
      exbii breq2 mo4 notbii 3bitr4ri imbitrrdi df-eu imnan bitr4i ) BUAEFZBAGZ
      CHZAIZVNAUBZFZJZVNAUCZFZVLVOVNBDGZCHZKZADLZFZKZDIZAIZVQVLVNWGAVLBVMMZCEZW
      JBWAMZCEZKZWEKZDIZVNWGVLWJWOJNCEZWPWEKZDIZJWPWPWEDIZKWRWPWSWSDALZDTZFDAUD
      WSWDDTXAWDDOWDWTDADUEUFUGUHUIWPWEDUJPVLWJWPWOWRVLWINCBVMQRZVLWNWQDVLWMWPW
      EVLWMWPWPKWPVLWJWPWLWPXBVLWKNCBWAQRUKWPULUMUNUOUPUQBVMCSZWFWNDWCWMWEVNWJW
      BWLXCBWACSURUSVCUTVAWCWDJDTZFZAIXDATZFWHVQXDAOWGXEAWCWDDVBVCVPXFVNWBADVMW
      ABCVDVEVFVGVHVTVOVPKZFVRVSXGVNAVIVFVOVPVJVKP $.
    $( $j usage 'brprcneu' avoids 'ax-pow'; $)
  $}

  ${
    $d x y A $.  $d x y F $.
    $( Alternate proof of ~ brprcneu using ~ ax-pow instead of ~ ax-pr .
       (Contributed by Scott Fenton, 7-Oct-2017.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    brprcneuALT $p |- ( -. A e. _V -> -. E! x A F x ) $=
      ( vy wcel wn cv wbr wex wi wa weq cop c0 exnal sylibr opprc1 eleq1d df-br
      wal cvv wmo dtruALT2 equcom albii xchbinx mpbir jctr 19.42v anbi12d anidm
      bitrdi anbi1d exbidv imbi12d mpbiri anbi12i anbi1i 3imtr4g eximdv exanali
      weu exbii breq2 mo4 notbii 3bitr4ri imbitrrdi df-eu imnan bitr4i ) BUAEFZ
      BAGZCHZAIZVNAUBZFZJZVNAVBZFZVLVOVNBDGZCHZKZADLZFZKZDIZAIZVQVLVNWGAVLBVMMZ
      CEZWJBWAMZCEZKZWEKZDIZVNWGVLWJWOJNCEZWPWEKZDIZJWPWPWEDIZKWRWPWSWSDALZDTZF
      DAUCWSWDDTXAWDDOWDWTDADUDUEUFUGUHWPWEDUIPVLWJWPWOWRVLWINCBVMQRZVLWNWQDVLW
      MWPWEVLWMWPWPKWPVLWJWPWLWPXBVLWKNCBWAQRUJWPUKULUMUNUOUPBVMCSZWFWNDWCWMWEV
      NWJWBWLXCBWACSUQURVCUSUTWCWDJDTZFZAIXDATZFWHVQXDAOWGXEAWCWDDVAVCVPXFVNWBA
      DVMWABCVDVEVFVGVHVTVOVPKZFVRVSXGVNAVIVFVOVPVJVKP $.
    $( $j usage 'brprcneuALT' avoids 'ax-pr'; $)
  $}

  ${
    $d A x $.  $d F x $.
    $( A function's value at a proper class is the empty set.  See ~ fvprcALT
       for a proof that uses ~ ax-pow instead of ~ ax-pr .  (Contributed by NM,
       20-May-1998.)  Avoid ~ ax-pow .  (Revised by BTernaryTau, 3-Aug-2024.)
       (Proof shortened by BTernaryTau, 3-Dec-2024.) $)
    fvprc $p |- ( -. A e. _V -> ( F ` A ) = (/) ) $=
      ( vx cvv wcel wn cv wbr weu cfv c0 wceq brprcneu tz6.12-2 syl ) ADEFACGBH
      CIFABJKLCABMCABNO $.
    $( $j usage 'fvprc' avoids 'ax-sep' 'ax-pow'; $)
  $}

  ${
    $d x A $.  $d x F $.
    $( Alternate proof of ~ fvprc using ~ ax-pow instead of ~ ax-pr .
       (Contributed by NM, 20-May-1998.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    fvprcALT $p |- ( -. A e. _V -> ( F ` A ) = (/) ) $=
      ( vx cvv wcel wn cv wbr weu cfv c0 wceq brprcneuALT tz6.12-2 syl ) ADEFAC
      GBHCIFABJKLCABMCABNO $.
    $( $j usage 'fvprcALT' avoids 'ax-sep' 'ax-pr'; $)
  $}

  ${
    rnfvprc.y $e |- Y = ( F ` X ) $.
    $( The range of a function value at a proper class is empty.  (Contributed
       by AV, 20-Aug-2022.) $)
    rnfvprc $p |- ( -. X e. _V -> ran Y = (/) ) $=
      ( cvv wcel wn crn c0 cfv fvprc eqtrid rneqd rn0 eqtrdi ) BEFGZCHIHIPCIPCB
      AJIDBAKLMNO $.
  $}

  ${
    $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 $.
    $( A definition of function value in terms of iota.  (Contributed by Scott
       Fenton, 19-Feb-2013.) $)
    dffv3 $p |- ( F ` A ) = ( iota x x e. ( F " { A } ) ) $=
      ( cvv wcel cfv cv csn cima cio wceq wbr df-fv wb wa iotabidv wn c0 eqtrdi
      cop elimasng df-br bitr4di elvd eqtr4id fvprc snprc biimpi imaeq2d eleq2d
      ima0 weu wex noel nex euex mto iotanul ax-mp eqtr4d pm2.61i ) BDEZBCFZAGZ
      CBHZIZEZAJZKVBVCBVDCLZAJVHABCMVBVGVIAVBVGVINAVBVDDEOVGBVDTCEVICBVDDDUABVD
      CUBUCUDPUEVBQZVCRVHBCUFVJVHVDREZAJZRVJVGVKAVJVFRVDVJVFCRIRVJVERCVJVERKBUG
      UHUICUKSUJPVKAULZQVLRKVMVKAUMVKAVDUNUOVKAUPUQVKAURUSSUTVA $.
  $}

  ${
    $d x y A $.  $d x y F $.
    $( 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.) $)
    dffv4 $p |- ( F ` A ) = U. { x | ( F " { A } ) = { x } } $=
      ( vy cfv csn cima wcel cio cab wceq cuni dffv3 df-iota abid2 eqeq1i abbii
      cv unieqi 3eqtri ) BCEDRCBFGZHZDIUBDJZARFZKZAJZLUAUDKZAJZLDBCMUBDANUFUHUE
      UGAUCUAUDDUAOPQST $.
  $}

  ${
    $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 A $.  $d x F $.
    $( The value of a class exists.  Corollary 6.13 of [TakeutiZaring] p. 27.
       (Contributed by NM, 30-Dec-1996.) $)
    fvex $p |- ( F ` A ) e. _V $=
      ( vx cfv cv wbr cio cvv df-fv iotaex eqeltri ) ABDACEBFZCGHCABILCJK $.
  $}

  ${
    fvexi.1 $e |- A = ( F ` B ) $.
    $( The value of a class exists.  Inference form of ~ fvex .  (Contributed
       by Glauco Siliprandi, 23-Oct-2021.) $)
    fvexi $p |- A e. _V $=
      ( cfv cvv fvex eqeltri ) ABCEFDBCGH $.
  $}

  $( The value of a class exists (as consequent of anything).  (Contributed by
     Glauco Siliprandi, 23-Oct-2021.) $)
  fvexd $p |- ( ph -> ( F ` A ) e. _V ) $=
    ( cfv cvv wcel fvex a1i ) BCDEFABCGH $.

  $( Move a conditional outside of a function.  (Contributed by Jeff Madsen,
     2-Sep-2009.) $)
  fvif $p |- ( F ` if ( ph , A , B ) ) = if ( ph , ( F ` A ) , ( F ` B ) ) $=
    ( cif cfv fveq2 ifsb ) ABCABCEZDFBDFCDFIBDGICDGH $.

  $( Move a conditional outside of a function.  (Contributed by Thierry Arnoux,
     28-Sep-2018.) $)
  iffv $p |- ( if ( ph , F , G ) ` A ) = if ( ph , ( F ` A ) , ( G ` A ) ) $=
    ( cif cfv fveq1 ifsb ) ACDBACDEZFBCFBDFBICGBIDGH $.

  ${
    $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 eu6
      alimi equsalvw anim2i eximi elequ2 anbi12d cbvexvw exsimpr sylibr jca nfv
      nfeu1 nfa1 nfan nfex nfim biimp syl6 impcomd anc2ri com12 eximdv biimtrid
      ax9 sps exlimi imp impbii bitri eqabi ) AFZBFZGZCVMDHZIZBJZVOBKZIZACDLZVL
      VTGVLEFZGZVOBEMZNZBOZIZEJZVSEBVLCDPWGVSWGVQVRWGWBCWADHZIZEJVQWFWIEWEWHWBW
      EWCVOQZBOWHWDWJBVOWCRUBVOWHBEVMWACDSUCTUDUEWIVPEBEBMWBVNWHVOEBAUFWAVMCDSU
      GUHTWGWEEJZVRWBWEEUIVOBEUAZUJUKVQVRWGVPVRWGQBVRWGBVOBUMWFBEWBWEBWBBULWDBU
      NUOUPUQVRWKVPWGWLVPWEWFEWEVPWFWEVPWBWDVPWBQBWDVOVNWBWDVOWCVNWBQVOWCURBEAV
      EUSUTVFVAVBVCVDVGVHVIVJVK $.
  $}

  ${
    $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 brresi baib iotabidv df-fv 3eqtr4g ) AB
      EZADFZCBGZHZDIARCHZDIASJACJQTUADTQUABARCDKLMNDASODACOP $.
  $}

  ${
    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 $.
    $( Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27.  (Contributed by
       NM, 30-Apr-2004.)  (Proof shortened by SN, 23-Dec-2024.) $)
    tz6.12c $p |- ( E! y A F y -> ( ( F ` A ) = y <-> A F y ) ) $=
      ( cv wbr weu cfv wceq cio df-fv eqeq1i iota1 bitr4id ) BADZCEZAFBCGZNHOAI
      ZNHOPQNABCJKOALM $.
  $}

  ${
    $d y F $.  $d y A $.
    $( Function value.  Theorem 6.12(1) of [TakeutiZaring] p. 27.  (Contributed
       by NM, 30-Apr-2004.)  (Proof shortened by SN, 23-Dec-2024.) $)
    tz6.12-1 $p |- ( ( A F y /\ E! y A F y ) -> ( F ` A ) = y ) $=
      ( cv wbr weu cfv wceq tz6.12c biimparc ) BADZCEZAFBCGKHLABCIJ $.

    $( Obsolete version of ~ tz6.12-1 as of 23-Dec-2024.  (Contributed by NM,
       30-Apr-2004.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    tz6.12-1OLD $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 A y z $.  $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 cbveuw a1i
      anbi12d eqeq2 imbi12d tz6.12 chvarvv ) BEFZGZCHZUHEIZJZBCKZUFLZMBAFZGZCHZ
      UOAIZJZUKUMLZMEAUFUMLZUJUQULURUSUHUOUIUPUSUGUNCUFUMBNOZUIUPPUSUHUOEAAUGCD
      QUOERUTSTUAUFUMUKUBUCEBCUDUE $.
  $}

  ${
    $d y F $.  $d y A $.
    $( Obsolete version of ~ tz6.12c as of 23-Dec-2024.  (Contributed by NM,
       30-Apr-2004.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    tz6.12cOLD $p |- ( E! y A F y -> ( ( F ` A ) = y <-> A F y ) ) $=
      ( wbr weu cfv wceq nfeu1 nfv euex tz6.12-1 expcom breq2 biimprd exlimimdd
      cv syli syl5ibcom impbid ) BAPZCDZAEZBCFZTGZUAUBBUCCDZUDUAUBUAUEAUAAHUEAI
      UAAJUAUBUDUEUAUBUDABCKLZUDUEUAUCTBCMZNQOUGRUFS $.
  $}

  ${
    $d y F $.  $d y A $.  $d y B $.
    $( Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27.  (Contributed by
       Mario Carneiro, 17-Nov-2014.) $)
    tz6.12i $p |- ( B =/= (/) -> ( ( F ` A ) = B -> A F B ) ) $=
      ( vy cfv wceq c0 wne wbr wi fvex cv neeq1 tz6.12-2 necon1ai tz6.12c breq2
      weu wb 3imtr3d syl biimpcd sylbird eqcoms vtocle a1i com12 ) ACEZBFZBGHZA
      BCIZUIUHGHZAUHCIZUJUKULUMJZUIUNDUHACKDLZUHFUOGHZAUOCIZULUMUPUQJUHUOUHUOFZ
      UPULUQUHUOGMULURUQULUQDRZURUQSUSUHGDACNODACPUAUBUCUDUOUHGMUOUHACQTUEUFUHB
      GMUHBACQTUG $.
  $}

  $( Two possibilities for the behavior of a function value.  (Contributed by
     Stefan O'Rear, 2-Nov-2014.)  (Proof shortened by Mario Carneiro,
     31-Aug-2015.) $)
  fvbr0 $p |- ( X F ( F ` X ) \/ ( F ` X ) = (/) ) $=
    ( cfv wbr c0 wceq wne eqid tz6.12i mpi necon1bi orri ) BBACZADZMEFNMEMEGMMF
    NMHBMAIJKL $.

  $( A function value is a member of the range plus null.  (Contributed by
     Scott Fenton, 8-Jun-2011.)  (Revised by Stefan O'Rear, 3-Jan-2015.) $)
  fvrn0 $p |- ( F ` X ) e. ( ran F u. { (/) } ) $=
    ( cfv c0 wceq crn csn cun wcel id ssun2 0ex snid sselii eqeltrdi wn cvv wbr
    ssun1 con1i fvprc fvexd fvbr0 ori brelrng syl3anc sselid pm2.61i ) BACZDEZU
    IAFZDGZHZIUJUIDUMUJJULUMDULUKKDLMNOUJPZUKUMUIUKULSUNBQIZUIQIBUIARZUIUKIUOUJ
    BAUATUNBAUBUPUJUPUJABUCUDTBUIAQQUEUFUGUH $.

  $( If the value of a function is not null, the value is an element of the
     range of the function.  (Contributed by Alexander van der Vekens,
     22-Jul-2018.)  (Proof shortened by SN, 13-Jan-2025.) $)
  fvn0fvelrn $p |- ( ( F ` X ) =/= (/) -> ( F ` X ) e. ran F ) $=
    ( cfv c0 wne crn csn cun wcel wn fvrn0 nelsn elunnel2 sylancr ) BACZDEOAFZD
    GZHIOQIJOPIABKODLOPQMN $.

  $( A function value is a subset of the union of the range.  (An artifact of
     our function value definition, compare ~ elfvdm ).  (Contributed by
     Thierry Arnoux, 13-Nov-2016.)  Remove functionhood antecedent.  (Revised
     by SN, 10-Jan-2025.) $)
  elfvunirn $p |- ( B e. ( F ` A ) -> B e. U. ran F ) $=
    ( cfv wcel crn cuni c0 wne wss ne0i fvn0fvelrn elssuni 3syl sseld pm2.43i )
    BACDZEZBCFZGZERQTBRQHIQSEQTJQBKCALQSMNOP $.

  ${
    $d F x $.  $d X x $.
    $( 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, 31-Aug-2015.)  (Proof
       shortened by SN, 13-Jan-2025.) $)
    fvssunirn $p |- ( F ` X ) C_ U. ran F $=
      ( vx cfv crn cuni cv elfvunirn ssriv ) CBADAEFBCGAHI $.
  $}

  $( Obsolete version of ~ fvssunirn as of 13-Jan-2025.  (Contributed by Stefan
     O'Rear, 2-Nov-2014.)  (Revised by Mario Carneiro, 31-Aug-2015.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  fvssunirnOLD $p |- ( F ` X ) C_ U. ran F $=
    ( cfv crn c0 csn cun cuni wcel wss fvrn0 elssuni ax-mp uniun 0ex uneq2i un0
    unisn 3eqtri sseqtri ) BACZADZEFZGZHZUBHZUAUDIUAUEJABKUAUDLMUEUFUCHZGUFEGUF
    UBUCNUGEUFEORPUFQST $.

  ${
    $d x A $.  $d x F $.
    $( The value of a class outside its domain is the empty set.  (An artifact
       of our function value definition.)  (Contributed by NM, 24-Aug-1995.) $)
    ndmfv $p |- ( -. A e. dom F -> ( F ` A ) = (/) ) $=
      ( vx cvv wcel cdm wn cfv c0 wceq wi cv wbr weu euex eldmg imbitrrid con3d
      wex tz6.12-2 syl6 fvprc a1d pm2.61i ) ADEZABFEZGZABHIJZKUEUGACLBMZCNZGUHU
      EUJUFUJUFUEUICSUICOCABDPQRCABTUAUEGUHUGABUBUCUD $.
  $}

  ${
    ndmfvrcl.1 $e |- dom F = S $.
    ndmfvrcl.2 $e |- -. (/) e. S $.
    $( Reverse closure law for function with the empty set not in its domain.
       (Contributed by NM, 26-Apr-1996.) $)
    ndmfvrcl $p |- ( ( F ` A ) e. S -> A e. S ) $=
      ( cfv wcel cdm wn c0 ndmfv eleq1d mtbiri con4i eleqtrdi ) ACFZBGZACHZBARG
      ZQSIZQJBGETPJBACKLMNDO $.
  $}

  $( If a function value has a member, then the argument belongs to the domain.
     (An artifact of our function value definition.)  (Contributed by NM,
     12-Feb-2007.)  (Proof shortened by BJ, 22-Oct-2022.) $)
  elfvdm $p |- ( A e. ( F ` B ) -> B e. dom F ) $=
    ( cfv wcel c0 wceq cdm n0i ndmfv nsyl2 ) ABCDZELFGBCHELAIBCJK $.

  $( If a function value has a member, then the argument is a set.  (An
     artifact of our function value definition.)  (Contributed by Mario
     Carneiro, 6-Nov-2015.) $)
  elfvex $p |- ( A e. ( F ` B ) -> B e. _V ) $=
    ( cfv wcel cdm elfvdm elexd ) ABCDEBCFABCGH $.

  ${
    elfvexd.1 $e |- ( ph -> A e. ( B ` C ) ) $.
    $( If a function value has a member, then its argument is a set.  Deduction
       form of ~ elfvex .  (An artifact of our function value definition.)
       (Contributed by David Moews, 1-May-2017.) $)
    elfvexd $p |- ( ph -> C e. _V ) $=
      ( cfv wcel cvv elfvex syl ) ABDCFGDHGEBDCIJ $.
  $}

  ${
    $d x A $.  $d x B $.  $d x F $.
    $( A nonempty function value is an element of the image of the function.
       (Contributed by Thierry Arnoux, 25-Jun-2019.) $)
    eliman0 $p |- ( ( A e. B /\ -. ( F ` A ) = (/) )
      -> ( F ` A ) e. ( F " B ) ) $=
      ( vx wcel cfv c0 wceq wn wa cv wbr wrex cima fvbr0 orcom mpbi ori breq1
      wo rspcev sylan2 fvex elima sylibr ) ABEZACFZGHZIZJDKZUGCLZDBMZUGCBNEUIUF
      AUGCLZULUHUMUMUHTUHUMTCAOUMUHPQRUKUMDABUJAUGCSUAUBDUGCBACUCUDUE $.
  $}

  $( The value of a non-member of a restriction is the empty set.  (An artifact
     of our function value definition.)  (Contributed by NM, 13-Nov-1995.) $)
  nfvres $p |- ( -. A e. B -> ( ( F |` B ) ` A ) = (/) ) $=
    ( cres cdm wcel cfv c0 wceq cin dmres inss1 eqsstri sseli ndmfv nsyl5 ) ACB
    DZEZFABFAQGHIRBARBCEZJBCBKBSLMNAQOP $.

  ${
    $d x y A $.  $d x y F $.
    $( If the restriction of a class to a singleton is not a function, then its
       value is the empty set.  (An artifact of our function value definition.)
       (Contributed by NM, 8-Aug-2010.)  (Proof shortened by Andrew Salmon,
       22-Oct-2011.) $)
    nfunsn $p |- ( -. Fun ( F |` { A } ) -> ( F ` A ) = (/) ) $=
      ( vx vy cfv c0 wceq csn cres wfun wn wrel cv wbr wmo wal weu eumo sylbi
      wa wcel vex brresi wb velsn breq1 biimpa moimi syl tz6.12-2 nsyl4 alrimiv
      relres jctil dffun6 sylibr con1i ) ABEFGZBAHZIZJZURKZUTLZCMZDMZUTNZDOZCPZ
      TVAVBVHVCVBVGCAVEBNZDQZVGURVJVIDOVGVIDRVFVIDVFVDUSUAZVDVEBNZTVIUSVDVEBDUB
      UCVKVLVIVKVDAGVLVIUDCAUEVDAVEBUFSUGSUHUIDABUJUKULBUSUMUNCDUTUOUPUQ $.
  $}

  $( If the "value of a class" at an argument is not the empty set, then the
     argument is in the domain of the class and the class restricted to the
     singleton formed on that argument is a function.  (Contributed by
     Alexander van der Vekens, 26-May-2017.)  (Proof shortened by BJ,
     13-Aug-2022.) $)
  fvfundmfvn0 $p |- ( ( F ` A ) =/= (/)
                      -> ( A e. dom F /\ Fun ( F |` { A } ) ) ) $=
    ( cfv c0 wne cdm wcel csn cres wfun ndmfv necon1ai nfunsn jca ) ABCZDEABFGZ
    BAHIJZPODABKLQODABMLN $.

  $( Function value of the empty set.  (Contributed by Stefan O'Rear,
     26-Nov-2014.) $)
  0fv $p |- ( (/) ` A ) = (/) $=
    ( c0 cdm wcel wn cfv wceq noel dm0 eleq2i mtbir ndmfv ax-mp ) ABCZDZEABFBGO
    ABDAHNBAIJKABLM $.

  $( 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
    $.

  $( If a function value of a function value has a member, then the first
     argument is a set.  (Contributed by AV, 8-Apr-2021.) $)
  elfv2ex $p |- ( A e. ( ( F ` B ) ` C ) -> B e. _V ) $=
    ( cvv wcel cfv wi ax-1 wn c0 fv2prc eleq2d noel pm2.21i biimtrdi pm2.61i )
    BEFZACBDGGZFZRHRTIRJZTAKFZRUASKABCDLMUBRANOPQ $.

  $( Equal values imply equal values in a restriction.  (Contributed by NM,
     13-Nov-1995.) $)
  fveqres $p |- ( ( F ` A ) = ( G ` A ) ->
                ( ( F |` B ) ` A ) = ( ( G |` B ) ` A ) ) $=
    ( wcel cfv wceq cres wi fvres eqeq12d biimprd wn nfvres eqtr4d a1d pm2.61i
    c0 ) ABEZACFZADFZGZACBHFZADBHFZGZISUEUBSUCTUDUAABCJABDJKLSMZUEUBUFUCRUDABCN
    ABDNOPQ $.

  ${
    $d y A $.  $d y B $.  $d y F $.  $d x y $.
    $( Move class substitution in and out of a function value.  (Contributed by
       NM, 11-Nov-2005.)  (Revised by NM, 20-Aug-2018.) $)
    csbfv12 $p |- [_ A / x ]_ ( F ` B ) = ( [_ A / x ]_ F ` [_ A / x ]_ B ) $=
      ( vy cvv wcel cfv csb wceq cv wbr cio csbiota sbcbr123 csbconstg df-fv c0
      wsbc csbprc breq2d bitrid iotabidv eqtrid csbeq2i 3eqtr4g wn fveq1d eqtrd
      0fv eqtr2di pm2.61i ) BFGZABCDHZIZABCIZABDIZHZJUMABCEKZDLZEMZIZUPUSUQLZEM
      ZUOURUMVBUTABSZEMVDUTAEBNUMVEVCEVEUPABUSIZUQLUMVCABCUSDOUMVFUSUPUQABUSFPU
      AUBUCUDABUNVAECDQUEEUPUQQUFUMUGZUORURABUNTVGURUPRHRVGUPUQRABDTUHUPUJUKUIU
      L $.
  $}

  ${
    $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 csbfv12 csbconstg fveq1d eqtrid ) BDFZABCEGHABCHZABEHZGNEG
      ABCEIMNOEABEDJKL $.

    $( Substitution for a function value.  (Contributed by NM, 1-Jan-2006.)
       (Revised by NM, 20-Aug-2018.) $)
    csbfv $p |- [_ A / x ]_ ( F ` x ) = ( F ` A ) $=
      ( cvv wcel cv cfv csb csbfv2g csbvarg fveq2d eqtrd wn csbprc fvprc eqtr4d
      wceq c0 pm2.61i ) BDEZABAFZCGZHZBCGZQTUCABUAHZCGUDABUADCITUEBCABDJKLTMUCR
      UDABUBNBCOPS $.
  $}

  ${
    $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 $.
    $( 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 cv wb wi fvex eqeq2 breq2 bibi12d syl
      imbi2d weu fneu tz6.12c vtocl mpbii syl5ibcom fnfun funbrfv adantr impbid
      wfun ) DAFZBAGZHZBDIZCJZBCDKZUOBUPDKZUQURUOUPUPJZUSUPLUOUPEMZJZBVADKZNZOU
      OUTUSNZOEUPBDPVAUPJZVDVEUOVFVBUTVCUSVAUPUPQVAUPBDRSUAUOVCEUBVDEABDUCEBDUD
      TUEUFUPCBDRUGUMURUQOZUNUMDULVGADUHBCDUITUJUK $.
  $}

  $( 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 $.

  ${
    $d A x $.  $d B x $.  $d C x $.  $d F x $.
    $( Function value in an image.  (Contributed by Glauco Siliprandi,
       2-Jan-2022.) $)
    fvelima2 $p |- ( ( F Fn A /\ B e. ( F " C ) )
                  -> E. x e. ( A i^i C ) ( F ` x ) = B ) $=
      ( cima wcel wfn cv wbr wa wex cfv wceq cin wrex elimag df-rex adantrl imp
      ibi sylib fnbr simprl elind wfun fnfun funbrfv sylan jca ex eximdv sylibr
      sylan2 ) CEDFZGZEBHZAIZDGZURCEJZKZALZUREMCNZABDOZPZUPUTADPZVBUPVFACEDUOQU
      AUTADRUBUQVBKURVDGZVCKZALZVEUQVBVIUQVAVHAUQVAVHUQVAKZVGVCVJBDURUQUTURBGUS
      BURCEUCSUQUSUTUDUEUQUTVCUSUQEUFZUTVCBEUGVKUTVCURCEUHTUISUJUKULTVCAVDRUMUN
      $.
  $}

  $( 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 $.  $d x Y $.
    $( 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.  (Contributed by
       FL, 14-Sep-2013.)  (Proof shortened by Mario Carneiro, 31-Aug-2015.) $)
    dffn5 $p |- ( F Fn A <-> F = ( x e. A |-> ( F ` x ) ) ) $=
      ( vy wfn cv cfv cmpt wceq wcel wa copab wbr wrel fnrel dfrel4v sylib fnbr
      ex pm4.71rd eqcom fnbrfvb bitrid pm5.32da bitr4d opabbidv eqtrd fvex eqid
      df-mpt eqtr4di fnmpti fneq1 mpbiri impbii ) CBEZCABAFZCGZHZIZUPCUQBJZDFZU
      RIZKZADLZUSUPCUQVBCMZADLZVEUPCNCVGIBCOADCPQUPVFVDADUPVFVAVFKVDUPVFVAUPVFV
      ABUQVBCRSTUPVAVCVFVCURVBIUPVAKVFVBURUABUQVBCUBUCUDUEUFUGADBURUJUKUTUPUSBE
      ABURUSUQCUHUSUIULBCUSUMUNUO $.

    $( 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 dffn5 rneq sylbi eqid rnmpt eqtrdi )
      DCEZDFZACAGDHZIZFZBGUAJACKBLSDUBJTUCJACDMDUBNOABCUAUBUBPQR $.

    $( 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 wfn crn wcel cv cfv wceq wrex cab fnrnfv eleq2d cvv fvex eleq1 mpbii
      bitrdi rexlimivw eqeq1 eqcom rexbidv elab3 ) DBFZCDGZHCEIZAIZDJZKZABLZEMZ
      HUJCKZABLZUFUGUMCAEBDNOULUOECPUNCPHZABUNUJPHUPUIDQUJCPRSUAUHCKZUKUNABUQUK
      CUJKUNUHCUJUBCUJUCTUDUET $.

    $( 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 $.

    $( 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 cdm wss wfun cima cv cfv wcel wral wb wi wal wa wceq wrex bitrid wbr
      df-ss vex elima eqcom ssel funbrfvb ex syl9 imp31 rexbidva bitr4id imbi1d
      r19.23v bitr4di albidv ralcom4 eleq1 ceqsalv ralbii bitr3i bitrdi ancoms
      fvex ) BDFZGZDHZDBIZCGZAJZDKZCLZABMZNVIEJZVHLZVNCLZOZEPZVFVGQZVMEVHCUBVSV
      RVNVKRZVPOZABMZEPZVMVSVQWBEVSVQVTABSZVPOWBVSVOWDVPVSVOVJVNDUAZABSWDAVNDBE
      UCUDVSVTWEABVTVKVNRZVSVJBLZQWEVNVKUEVFVGWGWFWENZVFWGVJVELZVGWHBVEVJUFVGWI
      WHVJVNDUGUHUIUJTUKULUMVTVPABUNUOUPWCWAEPZABMVMWAAEBUQWJVLABVPVLEVKVJDVDVN
      VKCURUSUTVAVBTVC $.

    $( 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 cv wbr wrex cfv wceq cima wcel funbrfv reximdv elimag ibi impel )
      DEZAFZBDGZACHZSDIBJZACHBDCKZLZRTUBACSBDMNUDUAABDCUCOPQ $.
  $}

  ${
    $d A x y $.  $d B x y $.  $d F x y $.  $d ph y $.
    funimassd.1 $e |- F/ x ph $.
    funimassd.2 $e |- ( ph -> Fun F ) $.
    funimassd.3 $e |- ( ( ph /\ x e. A ) -> ( F ` x ) e. B ) $.
    $( Sufficient condition for the image of a function being a subclass.
       (Contributed by Glauco Siliprandi, 23-Oct-2021.) $)
    funimassd $p |- ( ph -> ( F " A ) C_ B ) $=
      ( vy cima cv wcel wa cfv wceq wrex wfun fvelima nfv wi nfan w3a id eqcomd
      sylan 3ad2ant3 3adant3 eqeltrd 3exp adantr rexlimd mpd ex ssrdv ) AIECJZD
      AIKZUOLZUPDLZAUQMZBKZENZUPOZBCPZURAEQUQVCGBUPCERUEUSVBURBCAUQBFUQBSUAURBS
      AUTCLZVBURTTUQAVDVBURAVDVBUBUPVADVBAUPVAOVDVBVAUPVBUCUDUFAVDVADLVBHUGUHUI
      UJUKULUMUN $.
  $}

  ${
    $d A x y $.  $d B x y $.  $d C x y $.  $d F y $.  $d ph y $.
    fvelimad.x $e |- F/_ x F $.
    fvelimad.f $e |- ( ph -> F Fn A ) $.
    fvelimad.c $e |- ( ph -> C e. ( F " B ) ) $.
    $( Function value in an image.  (Contributed by Glauco Siliprandi,
       23-Oct-2021.) $)
    fvelimad $p |- ( ph -> E. x e. ( A i^i B ) ( F ` x ) = C ) $=
      ( vy cv cfv wceq cin wrex wcel syl nfv cvv adantr wbr cima elimag ibi w3a
      nfre1 wa cdm vex a1i simpr breldmd fndmd eleqtrd 3adant2 simp2 elind wfun
      wfn fnfun 3ad2ant1 simp3 funbrfv sylc rspe syl2anc 3exp rexlimd nfcv nffv
      mpd nfeq1 fveqeq2 cbvrexw sylibr ) AJKZFLZEMZJCDNZOZBKZFLEMZBVSOAVPEFUAZJ
      DOZVTAEFDUBZPZWDIWFWDJEFDWEUCUDQAWCVTJDAJRVRJVSUFAVPDPZWCVTAWGWCUEZVPVSPV
      RVTWHCDVPAWCVPCPWGAWCUGZVPFUHZCWIVPESWEFVPSPWIJUIUJAWFWCITAWCUKULAWJCMWCA
      CFHUMTUNUOAWGWCUPUQWHFURZWCVRAWGWKWCAFCUSWKHCFUTQVAAWGWCVBVPEFVCVDVRJVSVE
      VFVGVHVKWBVRBJVSWBJRBVQEBVPFGBVPVIVJVLWAVPEFVMVNVO $.
  $}

  ${
    $d x A $.  $d x C $.  $d x F $.
    feqmptd.1 $e |- ( ph -> F : A --> B ) $.
    $( Deduction form of ~ dffn5 .  (Contributed by Mario Carneiro,
       8-Jan-2015.) $)
    feqmptd $p |- ( ph -> F = ( x e. A |-> ( F ` x ) ) ) $=
      ( wfn cv cfv cmpt wceq ffnd dffn5 sylib ) AECGEBCBHEIJKACDEFLBCEMN $.

    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 fssresd feqmptd fvres mpteq2ia eqtrdi ) AFEIZBEBJZRKZL
      BESFKZLABEDRACDEFGHMNBETUASEFOPQ $.
  $}

  ${
    $d x y $.  $d y A $.  $d y F $.
    feqmptdf.1 $e |- F/_ x A $.
    feqmptdf.2 $e |- F/_ x F $.
    feqmptdf.3 $e |- ( ph -> F : A --> B ) $.
    $( Deduction form of ~ dffn5f .  (Contributed by Mario Carneiro,
       8-Jan-2015.)  (Revised by Thierry Arnoux, 10-May-2017.) $)
    feqmptdf $p |- ( ph -> F = ( x e. A |-> ( F ` x ) ) ) $=
      ( vy wf wfn cv cfv cmpt wceq ffn wcel wa copab wbr wrel fnrel nfcv dfrel4
      sylib nffn nfv fnbr pm4.71rd eqcom fnbrfvb bitrid pm5.32da bitr4d opabbid
      ex eqtrd df-mpt eqtr4di 3syl ) ACDEJECKZEBCBLZEMZNZOHCDEPVAEVBCQZILZVCOZR
      ZBISZVDVAEVBVFETZBISZVIVAEUAEVKOCEUBBIEGIEUCUDUEVAVJVHBIBCEGFUFVAIUGVAVJV
      EVJRVHVAVJVEVAVJVECVBVFEUHUPUIVAVEVGVJVGVCVFOVAVERVJVFVCUJCVBVFEUKULUMUNU
      OUQBICVCURUSUT $.
  $}

  ${
    $d x z $.  $d x z A $.  $d z F $.
    dffn5f.1 $e |- F/_ x F $.
    $( Representation of a function in terms of its values.  (Contributed by
       Mario Carneiro, 3-Jul-2015.) $)
    dffn5f $p |- ( F Fn A <-> F = ( x e. A |-> ( F ` x ) ) ) $=
      ( vz wfn cv cfv cmpt wceq dffn5 nfcv nffv fveq2 cbvmpt eqeq2i bitri ) CBF
      CEBEGZCHZIZJCABAGZCHZIZJEBCKTUCCEABSUBARCDARLMEUBLRUACNOPQ $.
  $}

  ${
    $d y A $.  $d x y B $.  $d x y C $.  $d x y 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 ) ) $=
      ( vy wfn wss wa cima wcel cv cfv wceq wrex cvv anim2i eleq1 wb wi rexbidv
      elex fvex mpbii rexlimivw eqeq2 bibi12d imbi2d wfun cdm fnfun fndm sseq2d
      cab biimpar dfimafn syl2an2r eqabrd vtoclg impcom pm5.21nd ) EBGZCBHZIZDE
      CJZKZALZEMZDNZACOZVDDPKZIVFVKVDDVEUBQVJVKVDVIVKACVIVHPKVKVGEUCVHDPRUDUEQV
      KVDVFVJSZVDFLZVEKZVHVMNZACOZSZTVDVLTFDPVMDNZVQVLVDVRVNVFVPVJVMDVERVRVOVIA
      CVMDVHUFUAUGUHVDVPFVEVBEUIVCCEUJZHZVEVPFUNNBEUKVBVTVCVBVSBCBEULUMUOAFCEUP
      UQURUSUTVA $.
  $}

  ${
    $d B x $.  $d C x $.  $d F x $.
    fvelimabd.1 $e |- ( ph -> F Fn A ) $.
    fvelimabd.2 $e |- ( ph -> B C_ A ) $.
    $( Deduction form of ~ fvelimab .  (Contributed by Stanislas Polu,
       9-Mar-2020.) $)
    fvelimabd $p |- ( ph -> ( C e. ( F " B ) <-> E. x e. B ( F ` x ) = C ) ) $=
      ( wfn wss cima wcel cv cfv wceq wrex wb fvelimab syl2anc ) AFCIDCJEFDKLBM
      FNEOBDPQGHBCDEFRS $.
  $}

  ${
    $d A y $.  $d B y $.  $d F x y $.  $d X x y $.
    $( Expressing the image of a set as a restricted abstract builder.
       (Contributed by Thierry Arnoux, 27-Jan-2020.) $)
    fimarab $p |- ( ( F : A --> B /\ X C_ A )
      -> ( F " X ) = { y e. B | E. x e. X ( F ` x ) = y } ) $=
      ( wf wss wa cima cv cfv wceq wrex crab nfv nfcv nfrab1 wcel wb wfn anbi2d
      ffn fvelimab sylan fimass adantr sseld pm4.71rd rabid a1i 3bitr4d eqrd )
      CDEGZFCHZIZBEFJZAKELBKZMAFNZBDOZUPBPBUQQUSBDRUPURDSZURUQSZIZVAUSIZVBURUTS
      ZUNECUAZUOVCVDTCDEUCVFUOIVBUSVAACFUREUDUBUEUPVBVAUPUQDURUNUQDHUOCDEFUFUGU
      HUIVEVDTUPUSBDUJUKULUM $.
  $}

  ${
    $d A y $.  $d B x y $.  $d C x y $.  $d F x y $.
    $( Image of a union.  (Contributed by Glauco Siliprandi, 11-Dec-2019.) $)
    unima $p |- ( ( F Fn A /\ B C_ A /\ C C_ A )
                -> ( F " ( B u. C ) ) = ( ( F " B ) u. ( F " C ) ) ) $=
      ( vy vx wfn wss w3a cun cima cv wcel wo cfv wceq wrex simp1 wb fvelimab
      simpl simpr unssd 3adant1 fvelimabd bitrdi 3adant3 3adant2 orbi12d bitr4d
      wa rexun elun bitr4di eqrdv ) DAGZBAHZCAHZIZEDBCJZKZDBKZDCKZJZUSELZVAMZVE
      VBMZVEVCMZNZVEVDMUSVFFLDOVEPZFBQZVJFCQZNZVIUSVFVJFUTQVMUSFAUTVEDUPUQURRUQ
      URUTAHUPUQURUKBCAUQURUAUQURUBUCUDUEVJFBCULUFUSVGVKVHVLUPUQVGVKSURFABVEDTU
      GUPURVHVLSUQFACVEDTUHUIUJVEVBVCUMUNUO $.
  $}

  $( 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 value of the identity function is a subset of the argument.  (An
       artifact of our function value definition.)  (Contributed by Mario
       Carneiro, 27-Feb-2016.) $)
    fviss $p |- ( _I ` A ) C_ A $=
      ( vx cid cfv cv wcel id cvv wceq elfvex fvi syl eleqtrd ssriv ) BACDZABEZ
      OFZPOAQGQAHFOAIPACJAHKLMN $.

    $( 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 fvex dfiin2 fnrnfv inteqd eqtr4id
      crn ) CBEZABAFZCGZHDFUAIABJDKZLCRZLADBUATCMNSUCUBADBCOPQ $.

    $( Singleton of function value.  (Contributed by NM, 22-May-1998.)  (Proof
       shortened by Scott Fenton, 8-Aug-2024.) $)
    fnsnfv $p |- ( ( F Fn A /\ B e. A ) -> { ( F ` B ) } = ( F " { B } ) ) $=
      ( vy wfn wcel wa csn cima cv wbr cab wceq imasng adantl velsn eqcom bitri
      cfv fnbrfvb bitr2id eqabcdv eqtr2d ) CAEZBAFZGZCBHIZBDJZCKZDLZBCSZHZUEUGU
      JMUDDBACNOUFUIDULUHULFZUKUHMZUFUIUMUHUKMUNDUKPUHUKQRABUHCTUAUBUC $.
  $}

  ${
    $d x y z B $.  $d x y z F $.  $d z ph $.  $d x z ps $.
    opabiota.1 $e |- F = { <. x , y >. | { y | ph } = { y } } $.
    $( Define a function whose value is "the unique ` y ` such that
       ` ph ( x , y ) ` ".  (Contributed by NM, 19-May-2015.) $)
    opabiotafun $p |- Fun F $=
      ( vz wfun cab cv csn wceq copab funopab cuni wi mo2icl unieq unisnv mpbir
      wmo eqtr2di mpg nfv nfab1 nfeq1 sneq eqeq2d cbvmow mpgbir funeqi ) DGACHZ
      CIZJZKZBCLZGZUPUNCTZBUNBCMUQUKFIZJZKZFTZUTURUKNZKOVAFUTFVBPUTVBUSNURUKUSQ
      FRUAUBUNUTCFUNFUCCUKUSACUDUEULURKUMUSUKULURUFUGUHSUIDUOEUJS $.

    $( Define a function whose value is "the unique ` y ` such that
       ` ph ( x , y ) ` ".  (Contributed by NM, 16-Nov-2013.) $)
    opabiotadm $p |- dom F = { x | E! y ph } $=
      ( cab cv csn wceq copab cdm wex weu dmopab dmeqi euabsn abbii 3eqtr4i ) A
      CFCGHIZBCJZKSCLZBFDKACMZBFSBCNDTEOUBUABACPQR $.

    opabiota.2 $e |- ( x = B -> ( ph <-> ps ) ) $.
    $( Define a function whose value is "the unique ` y ` such that
       ` ph ( x , y ) ` ".  (Contributed by NM, 16-Nov-2013.) $)
    opabiota $p |- ( B e. dom F -> ( F ` B ) = ( iota y ps ) ) $=
      ( cv cfv cio wceq cdm fveq2 iotabidv eqeq12d wcel vex sylib sylbi wbr wex
      eldm nfiota1 nfeq2 wfun opabiotafun funbrfv ax-mp cab csn cop copab df-br
      wi eleq2i opabidw 3bitri vsnid id eleqtrrid abid weu wb breldm opabiotadm
      eqabri iota1 syl mpbid eqtr4d exlimi vtoclga ) CIZFJZADKZLZEFJZBDKZLCEFMZ
      VNELZVOVRVPVSVNEFNWAABDHOPVNVTQZVNDIZFUAZDUBVQDVNFCRZUCWDVQDDVOVPADUDUEWD
      VOWCVPFUFWDVOWCLUOACDFGUGVNWCFUHUIWDAVPWCLZWDADUJZWCUKZLZAWDVNWCULZFQWJWI
      CDUMZQWIVNWCFUNFWKWJGUPWICDUQURWIWCWGQAWIWCWHWGDUSWIUTVAADVBSTWDADVCZAWFV
      DWDWBWLVNWCFWEDRVEWLCVTACDFGVFVGSADVHVIVJVKVLTVM $.
  $}

  $( 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 $.

  ${
    fnimatpd.1 $e |- ( ph -> F Fn D ) $.
    fnimatpd.2 $e |- ( ph -> A e. D ) $.
    fnimatpd.3 $e |- ( ph -> B e. D ) $.
    fnimatpd.4 $e |- ( ph -> C e. D ) $.
    $( The image of an unordered triple under a function.  (Contributed by
       Thierry Arnoux, 19-Sep-2023.) $)
    fnimatpd $p |- ( ph -> ( F " { A , B , C } )
                           = { ( F ` A ) , ( F ` B ) , ( F ` C ) } ) $=
      ( cpr cima csn cun cfv ctp wfn wcel wceq df-tp fnimapr syl3anc syl2anc
      fnsnfv eqcomd uneq12d imaeq2i imaundi eqtri 3eqtr4g ) AFBCKZLZFDMZLZNZBFO
      ZCFOZKZDFOZMZNFBCDPZLZUPUQUSPAULURUNUTAFEQZBERCERULURSGHIEBCFUAUBAUTUNAVC
      DERUTUNSGJEDFUDUCUEUFVBFUKUMNZLUOVAVDFBCDTUGFUKUMUHUIUPUQUSTUJ $.
  $}

  ${
    $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 jca2 reximdv2 adantl wb wfn
      simpr funfn inss2 sstri fvelimab mpan2 sylbi sylibrd syld adantlr biimpcd
      eleq1 rexlimiv impbid eqrdv inex1 rabex sseq1 imaeq2 eqeq2d anbi12d spcev
      mpd syl6 sylancr inss1 sstr anim1i eximi syl sylan2br ) CDBIZJDUAZCDBDKZU
      CZIZJZALZBJZCDXCIZMZNZAOZXAWQCDDBUDKZIXAWQXIWTDDBUEUBDBUFUGUHWRXBNZXCWTJZ
      XFNZAOZXHXJFLZDPZCQZFWTUNZWTJZCDXQIZMZXMXPFWTUIZXJGCXSXJGLZCQZYBXSQZXJYCY
      DXJYCNYBXAQZYDXBYCYEWRCXAYBUJUKWRYCYEYDRXBWRYCNZYEHLZDPZYBMZHWTUOZYDWRYEY
      JRYCWRYEYJHYBWTDULSTYFYJYIHXQUOZYDYCYJYKRWRYCYIYIHWTXQYCYGWTQZYINZYGXQQZY
      IYCYMYLYHCQZNZYNYCYIYOYLYBCYHUMUPXPYOFYGWTXNYGMXOYHCXNYGDUQURUSZUTYLYIVFV
      AVBVCWRYDYKVDZYCWRDWSVEZYRDVGYSXQWSJYRXQWTWSYABWSVHVIHWSXQYBDVJVKVLTVMVNV
      OWHSWRYDYCRXBWRYDYKYCWRYDYKHYBXQDULSYIYCHXQYNYPYIYCRZYQYOYTYLYIYOYCYHYBCV
      QVPVCVLVRWITVSVTXLXRXTNAXQXPFWTBWSEWAWBXCXQMZXKXRXFXTXCXQWTWCUUAXEXSCXCXQ
      DWDWEWFWGWJXLXGAXKXDXFXKWTBJXDBWSWKXCWTBWLVKWMWNWOWP $.
  $}

  ${
    $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 is a
     function.  (Contributed by NM, 22-May-1998.) $)
  funfv $p |- ( Fun F -> ( F ` A ) = U. ( F " { A } ) ) $=
    ( wfun cdm wcel cfv csn cima cuni wceq fvex unisn wfn df-fn mpbiran2 fnsnfv
    wa eqid unieqd c0 sylanbr eqtr3id ex wn ndmfv ndmima eqtrdi eqtr4d pm2.61d1
    uni0 ) BCZABDZEZABFZBAGHZIZJZUKUMUQUKUMQZUNUNGZIUPUNABKLURUSUOUKBULMZUMUSUO
    JUTUKULULJULRBULNOULABPUASUBUCUMUDZUNTUPABUEVAUPTITVAUOTABUFSUJUGUHUI $.

  ${
    $d y A $.  $d y F $.
    $( The value of a function.  Definition of function value in [Enderton]
       p. 43.  (Contributed by NM, 22-May-1998.) $)
    funfv2 $p |- ( Fun F -> ( F ` A ) = U. { y | A F y } ) $=
      ( wfun cfv csn cima cuni cv wbr cab funfv wrel funrel relimasn syl unieqd
      wceq eqtrd ) CDZBCECBFGZHBAICJAKZHBCLTUAUBTCMUAUBRCNABCOPQS $.
  $}

  ${
    $d w A $.  $d w F $.  $d w y $.
    funfv2f.1 $e |- F/_ y A $.
    funfv2f.2 $e |- F/_ y F $.
    $( The value of a function.  Version of ~ funfv2 using a bound-variable
       hypotheses instead of distinct variable conditions.  (Contributed by NM,
       19-Feb-2006.) $)
    funfv2f $p |- ( Fun F -> ( F ` A ) = U. { y | A F y } ) $=
      ( vw wfun cfv cv wbr cab cuni funfv2 nfcv nfbr breq2 cbvabw unieqi eqtrdi
      nfv ) CGBCHBFIZCJZFKZLBAIZCJZAKZLFBCMUCUFUBUEFAABUACDEAUANOUEFTUAUDBCPQRS
      $.
  $}

  $( Value of the union of two functions when the domains are separate.
     (Contributed by FL, 7-Nov-2011.) $)
  fvun $p |- ( ( ( Fun F /\ Fun G ) /\ ( dom F i^i dom G ) = (/) ) ->
                ( ( F u. G ) ` A ) = ( ( F ` A ) u. ( G ` A ) ) ) $=
    ( wfun wa cdm cin c0 wceq cun cfv csn cima cuni funun funfv imaundir eqcomd
    syl a1i unieqd uniun anim12i adantr uneq12 eqtrid 3eqtrd ) BDZCDZEZBFCFGHIZ
    EZABCJZKZUMALZMZNZBUOMZCUOMZJZNZABKZACKZJZULUMDUNUQIBCOAUMPSULUPUTUPUTIULBC
    UOQTUAULVAURNZUSNZJZVDURUSUBULVEVBIZVFVCIZEZVGVDIUJVJUKUHVHUIVIUHVBVEABPRUI
    VCVFACPRUCUDVEVBVFVCUESUFUG $.

  $( 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 cun cfv wfun cdm fnfun 3ad2ant1 fndm wn ndmfv
    3ad2ant2 ineqan12d eqeq1d biimprd 3impia fvun syl21anc disjel adantl eleq2d
    adantrd wb adantr mtbird 3adant1 syl uneq2d un0 eqtrdi eqtrd ) CAFZDBFZABGZ
    HIZEAJZKZLZECDMNZECNZEDNZMZVJVHCOZDOZCPZDPZGZHIZVIVLIVBVCVMVGACQRVCVBVNVGBD
    QUBVBVCVGVRVBVCKZVEVRVFVSVRVEVSVQVDHVBVCVOAVPBACSBDSZUCUDUEULUFECDUGUHVHVLV
    JHMVJVHVKHVJVHEVPJZTZVKHIVCVGWBVBVCVGKWAEBJZVGWCTVCABEUIUJVCWAWCUMVGVCVPBEV
    TUKUNUOUPEDUAUQURVJUSUTVA $.

  $( 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 z A $.  $d x y z F $.
    $( Alternate definition of function value ~ df-fv that doesn't require
       dummy variables.  (Contributed by NM, 4-Aug-2010.) $)
    dffv2 $p |- ( F ` A ) = U. ( ( F " { A } )
            \ U. U. ( ( ( F |` { A } ) o. `' ( F |` { A } ) ) \ _I ) ) $=
      ( vy vz vx cid cdif cuni wceq cvv wcel wn c0 wss unieqd eqtrdi wa wbr wex
      wal csn cres wfun cfv cima ccnv ccom snidb fvres sylbi fvprc eqtr4d funfv
      pm2.61i resima dif0 eqtr4i wrel df-fun simprbi sylib uni0 difeq2d eqtr4id
      ssdif0 eqtrd eqtr3id nfunsn cdm cv weq wi relres mpbiran iman albii alnex
      dffun3 bitri exbii exnal 3bitrri con1bii eximi snssi residm dmeqi ssdmres
      sp biimpi syl vex breldm eleq2 bitrdi biimpa syl2an breq1d biimpd pm2.43d
      velsn ex anim1d eximdv exlimdv mpan9 eleq2i elimasni sylbir cpr cop uniop
      opex unisn wb brrelex1i brcnvg sylancr biimpar adantl breq2 breq1 anbi12d
      wrex rspcev mpancom ancoms syldan anim1i an32s eldif rexv brco df-br ideq
      3bitr2ri equcom 3bitr3i notbii eqsstrrid anbi12i bitr2i uniss 3syl unissd
      prss sylibr simpld syl5 exlimiv ssrdv ndmima difeq1d 0dif pm2.61d1 ) BAUA
      ZUBZUCZABUDZBUUPUEZUUQUUQUFZUGZFGZHZHZGZHZIUURUUSAUUQUDZUVGAJKZUVHUUSIZUV
      IAUUPKUVJAUHAUUPBUIUJUVILUVHMUUSAUUQUKABUKULUNUURUVHUUQUUPUEZHUVGAUUQUMUU
      RUVKUVFUURUVKUUTMGZUVFUVKUUTUVLBUUPUOZUUTUPUQUURUVEMUUTUURUVEMHZMUURUVDMU
      URUVDUVNMUURUVCMUURUVBFNZUVCMIUURUUQURZUVOUUQUSUTUVBFVEVAOVBPOVBPVCVDOVFV
      GUURLZUUSMUVGABVHUVQUVGUVNMUVQUVFMUVQAUUQVIZKZUVFMIZUVQUVSUVTUVQUVSQZUUTU
      VENUVTUWACUUTUVEUWAADVJZUUQRZDCVKZLZQZDSZCVJZUUTKZUWHUVEKZVLZUVQEVJZUWBUU
      QRZUWEQZDSZESZUVSUWGUVQUWOCTZESZUWPUWRUURUURUWMUWDVLZDTZCSZETZUWQLZETUWRL
      UURUVPUXBBUUPVMZEDCUUQVRVNUXAUXCEUXAUWOLZCSUXCUWTUXECUWTUWNLZDTUXEUWSUXFD
      UWMUWDVOVPUWNDVQVSVTUWOCWAVSVPUWQEVQWBWCUWQUWOEUWOCWIWDUJUVSUWOUWGEUVSUWN
      UWFDUVSUWMUWCUWEUVSUWMUWCUVSUWMUWMUWCVLUVSUWMQZUWMUWCUXGUWLAUWBUUQUVSUVRU
      UPIZUWLUVRKZUWLAIZUWMUVSUUPUVRNZUXHAUVRWEUXKUVRUUQUUPUBZVIZUUPUXLUUQBUUPW
      FWGUXKUXMUUPIUUPUUQWHWJVGWKUWLUWBUUQEWLDWLZWMUXHUXIUXJUXHUXIUWLUUPKUXJUVR
      UUPUWLWNEAXAWOWPWQWRWSXBWTXCXDXEXFUWFUWKDUWIAUWHUUQRZUWFUWJUWIUWHUVKKUXOU
      VKUUTUWHUVMXGUUQAUWHXHXIUWFUXOUWJUWFUXOQZUWJUWBUVEKZUXPUWHUWBXJZUVENUWJUX
      QQUXPUXRUWHUWBXKZHUVEUWHUWBCWLZUXNXLUXPUXSUVDUXPUXSUXSUAZHZUVDUXSUWHUWBXM
      XNUXPUXSUVCKZUYAUVCNUYBUVDNUXPUWHUWLUVARZUWMQZEJYDZUWEQZUYCUWCUXOUWEUYGUW
      CUXOQUYFUWEUWCUXOUWHAUVARZUYFUWCUYHUXOUWCUWHJKUVIUYHUXOXOUXTAUWBUUQUXDXPZ
      UWHAJJUUQXQXRXSUYHUWCUYFUVIUYHUWCQZUYFUWCUVIUYHUYIXTUYEUYJEAJUXJUYDUYHUWM
      UWCUWLAUWHUVAYAUWLAUWBUUQYBYCYEYFYGYHYIYJUYCUXSUVBKZUXSFKZLZQUYGUXSUVBFYK
      UYKUYFUYMUWEUYFUYEESUWHUWBUVBRUYKUYEEYLEUWHUWBUUQUVAUXTUXNYMUWHUWBUVBYNYP
      UYLUWDUWHUWBFRCDVKUYLUWDUWHUWBUXNYOUWHUWBFYNCDYQYRYSUUAUUBVAUXSUVCWEUYAUV
      CUUCUUDYTUUEYTUWHUWBUVEUXTUXNUUFUUGUUHXBUUIUUJWKUUKUUTUVEVEVAXBUVSLZUVFMU
      VEGMUYNUUTMUVEUYNUUTUVKMUVMAUUQUULVGUUMUVEUUNPUUOOVBPULUN $.
  $}

  ${
    $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 ) ) $=
      ( vx vy wfun cdm wcel wa ccom cv cop wex cfv wb eldm2g exbidv wceq bitrid
      cvv opelco2g elvd bitrd adantl eldm2 opeq1 eleq1d ceqsexv funopfvb anbi1d
      fvex eqcom bitr3id bitr4d ) CFZACGZHZIZABCJZGHZADKZLCHZVAEKZLZBHZIZDMZEMZ
      ACNZBGHZUQUTVHOUOUQUTAVCLUSHZEMVHEAUSUPPUQVKVGEUQVKVGOEDAVCBCUPTUAUBQUCUD
      VJVIVCLZBHZEMURVHEVIBACUKZUEURVMVGEVMVAVIRZVEIZDMURVGVEVMDVIVNVOVDVLBVAVI
      VCUFUGUHURVPVFDURVOVBVEVOVIVARURVBVAVIULAVACUISUJQUMQSUN $.

    $( 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 imaco fnsnfv imaeq2d eqtr4id eleq2d
      cfv dffv3 iotabidv 3eqtr4g ) CAFDAGHZEIZBCJZDKZLZGZEMUDBDCSZKZLZGZEMDUESU
      IBSUCUHULEUCUGUKUDUCUGBCUFLZLUKBCUFNUCUJUMBADCOPQRUAEDUETEUIBTUB $.
  $}

  $( 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 $.

  ${
    fvco3d.1 $e |- ( ph -> G : A --> B ) $.
    fvco3d.2 $e |- ( ph -> C e. A ) $.
    $( Value of a function composition.  Deduction form of ~ fvco3 .
       (Contributed by Stanislas Polu, 9-Mar-2020.) $)
    fvco3d $p |- ( ph -> ( ( F o. G ) ` C ) = ( F ` ( G ` C ) ) ) $=
      ( wf wcel ccom cfv wceq fvco3 syl2anc ) ABCFIDBJDEFKLDFLELMGHBCDEFNO $.
  $}

  ${
    fvco4i.a $e |- (/) = ( F ` (/) ) $.
    fvco4i.b $e |- Fun G $.
    $( Conditions for a composition to be expandable without conditions on the
       argument.  (Contributed by Stefan O'Rear, 31-Mar-2015.) $)
    fvco4i $p |- ( ( F o. G ) ` X ) = ( F ` ( G ` X ) ) $=
      ( cdm wcel ccom cfv wceq wfn wfun funfn mpbi fvco2 mpan wn dmcoss ndmfv
      c0 sseli nsyl5 fveq2d 3eqtr4a pm2.61i ) CBFZGZCABHZIZCBIZAIZJZBUFKZUGULBL
      UMEBMNUFABCOPUGQZTTAIUIUKDCUHFZGUGUITJUOUFCABRUACUHSUBUNUJTACBSUCUDUE $.
  $}

  ${
    $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 B x y $.  $d X x y $.  $d Y x y $.  $d Z x y $.  $d ps x y $.
    brfvopabrbr.1 $e |- ( A ` Z ) = { <. x , y >. |
                                      ( x ( B ` Z ) y /\ ph ) } $.
    brfvopabrbr.2 $e |- ( ( x = X /\ y = Y ) -> ( ph <-> ps ) ) $.
    brfvopabrbr.3 $e |- Rel ( B ` Z ) $.
    $( The binary relation of a function value which is an ordered-pair class
       abstraction of a restricted binary relation is the restricted binary
       relation.  The first hypothesis can often be obtained by using
       ~ fvmptopab .  (Contributed by AV, 29-Oct-2021.) $)
    brfvopabrbr $p |- ( X ( A ` Z ) Y <-> ( X ( B ` Z ) Y /\ ps ) ) $=
      ( cfv wbr cvv wcel wa c0 wne brne0 w3a fvprc necon1ai relopabiv brrelex1i
      syl cv brrelex2i 3jca adantr copab wceq a1i rbropap pm5.21nii ) GHIEMZNZI
      OPZGOPZHOPZUAZGHIFMZNZBQUQURUSUTUQUPRSURGHUPTURUPRIEUBUCUFGHUPCUGDUGVBNAQ
      ZCDUPJUDZUEGHUPVEUHUIVCVABVCURUSUTVCVBRSURGHVBTURVBRIFUBUCUFGHVBLUEGHVBLU
      HUIUJURABHCGUPVBOODUPVDCDUKULURJUMKUNUO $.
  $}

  ${
    $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 $.

    $( Value of a function given in maps-to notation.  (Contributed by Mario
       Carneiro, 23-Apr-2014.) $)
    fvmpti $p |- ( A e. D -> ( F ` A ) = ( _I ` C ) ) $=
      ( wcel cvv cfv cid wceq wa fvmptg fvi adantl eqtr4d wn c0 cv eleq1d dmmpt
      cdm elrab2 baib notbid ndmfv biimtrrdi imp fvprc pm2.61dan ) BEIZDJIZBFKZ
      DLKZMUMUNNUODUPABCDEJFGHOUNUPDMUMDJPQRUMUNSZNUOTUPUMUQUOTMZUMUQBFUDZIZSUR
      UMUTUNUTUMUNCJIUNABEUSAUABMCDJGUBAECFHUCUEUFUGBFUHUIUJUQUPTMUMDLUKQRUL $.

    ${
      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 x y $.  $d y A $.  $d y B $.
    fvmpt2f.0 $e |- F/_ x A $.
    $( Value of a function given by the maps-to notation.  (Contributed by
       Thierry Arnoux, 9-Mar-2017.) $)
    fvmpt2f $p |- ( ( x e. A /\ B e. C ) -> ( ( x e. A |-> B ) ` x ) = B ) $=
      ( vy csb cmpt weq csbeq1 csbid eqtrdi nfcv nfcsb1v csbeq1a cbvmptf fvmptg
      cv ) FARZAFRZCGZCBDABCHFAIUAASCGCATSCJACKLAFBCUAEFBMFCMATCNATCOPQ $.
  $}

  ${
    $d B x $.  $d V x $.  $d X x $.
    fvtresfn.f $e |- F = ( x e. B |-> ( x |` V ) ) $.
    $( Functionality of a tuple-restriction function.  (Contributed by Stefan
       O'Rear, 24-Jan-2015.) $)
    fvtresfn $p |- ( X e. B -> ( F ` X ) = ( X |` V ) ) $=
      ( wcel cres cvv cfv wceq resexg cv reseq1 fvmptg mpdan ) EBGEDHZIGECJQKED
      BLAEAMZDHQBICREDNFOP $.
  $}

  ${
    $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 D x $.
    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 ) $.
    ${
      $d A y $.  $d B y $.  $d C y $.  $d ph y $.  $d x y $.
      fvmptdf.p $e |- F/ x ph $.
      fvmptdf.a $e |- F/_ x A $.
      fvmptdf.c $e |- F/_ x C $.
      $( Deduction version of ~ fvmptd using bound-variable hypotheses instead
         of distinct variable conditions.  (Contributed by AV, 29-Mar-2024.) $)
      fvmptdf $p |- ( ph -> ( F ` A ) = C ) $=
        ( vy cfv wcel wceq a1i cmpt csb fveq1d cv nfcsb1v weq csbeq1a adantl wa
        wnfc cvv nfeq2 nfan eqtr ancoms sylan2 anassrs csbiedf csbie2df eqeltrd
        vex eqid fvmpts syl2anc 3eqtrd ) ACGQCBFDUAZQZBCDUBZEACGVFIUCACFRVHHRVG
        VHSKAVHEHABPCDBPUDZDUBZEFMBVJUJABVIDUETBEUJZAOTKBPUFZDVJSABVIDUGUHAVICS
        ZUIZBVIDEUKAVMBMBVICNULUMVKVNOTVIUKRVNPVATAVMVLDESZVMVLUIABUDZCSZVOVLVM
        VQVPVICUNUOJUPUQURUSZLUTBCDFVFHVFVBVCVDVRVE $.
    $}

    $d x A $.  $d x C $.  $d x ph $.
    $( Deduction version of ~ fvmpt .  (Contributed by Scott Fenton,
       18-Feb-2013.)  (Revised by Mario Carneiro, 31-Aug-2015.)  (Proof
       shortened by AV, 29-Mar-2024.) $)
    fvmptd $p |- ( ph -> ( F ` A ) = C ) $=
      ( nfv nfcv fvmptdf ) ABCDEFGHIJKLABMBCNBENO $.
  $}

  ${
    $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 A x $.
    mptrcl.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.) $)
    mptrcl $p |- ( I e. ( F ` X ) -> X e. A ) $=
      ( cfv wcel c0 wceq wn n0i cdm dmmptss sseli ndmfv nsyl4 syl ) EFDHZITJKZL
      FBIZTEMFDNZIUBUAUCBFABCDGOPFDQRS $.

    $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
       Mario Carneiro, 23-Apr-2014.) $)
    fvmpt2i $p |- ( x e. A -> ( F ` x ) = ( _I ` B ) ) $=
      ( vy cv csb weq csbeq1 csbid eqtrdi cmpt nfcv nfcsb1v cbvmpt eqtri fvmpti
      csbeq1a ) FAGZAFGZCHZCBDFAIUBATCHCAUATCJACKLDABCMFBUBMEAFBCUBFCNAUACOAUAC
      SPQR $.

    $( 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 ) $=
      ( cv wcel cfv cid fvmpt2i fvi sylan9eq ) AGZBHCDHNEICJICABCEFKCDLM $.

    $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, even if ` D ` is not in the base set
       ` A ` .  (Contributed by Mario Carneiro, 13-Feb-2015.) $)
    fvmptss $p |- ( A. x e. A B C_ C -> ( F ` D ) C_ C ) $=
      ( vy wss wcel cfv cv wi wceq fveq2 sseq1d imbi2d nfcv wa c0 dmmptss sseli
      wral cdm nfra1 cmpt nfmpt1 nfcxfr nffv nfss nfim cvv reqabi fvmpt2 eqimss
      dmmpt syl sylbi wn ndmfv 0ss eqsstrdi pm2.61i rsp impcom vtoclgaf vtoclga
      sstrid ex sylan2 adantl pm2.61dan ) CDIZABUCZEFUDZJZEFKZDIZVPVNEBJZVRVOBE
      ABCFGUAUBVSVNVRVNHLZFKZDIZMZVNVRMHEBVTENZWBVRVNWDWAVQDVTEFOPQVNALZFKZDIZM
      WCAVTBAVTRZVNWBAVMABUEAWADAVTFAFABCUFGABCUGUHWHUIADRUJUKWEVTNZWGWBVNWIWFW
      ADWEVTFOPQWEBJZVNWGWJVNSWFCDWEVOJZWFCIZWKWJCULJZSZWLWMAVOBABCFGUPUMWNWFCN
      WLABCULFGUNWFCUOUQURWKUSWFTCWEFUTCVAVBVCVNWJVMVMABVDVEVHVIVFVGVEVJVNVPUSZ
      SVQTDWOVQTNVNEFUTVKDVAVBVL $.
  $}

  ${
    $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 id eqid fvmpt2 syl2an2 eqtrd ) A
      BIZCJZKUBELZUBBCDMZLZDAUDUFNUCAUBEUEGOPUCUCADFJUFDNUCQHBCDFUEUERSTUA $.
  $}

  ${
    $d x y A $.  $d y B $.  $d y C $.
    fvmptex.1 $e |- F = ( x e. A |-> B ) $.
    fvmptex.2 $e |- G = ( x e. A |-> ( _I ` B ) ) $.
    $( Express a function ` F ` whose value ` B ` may not always be a set in
       terms of another function ` G ` for which sethood is guaranteed.  (Note
       that ` ( _I `` B ) ` is just shorthand for
       ` if ( B e. _V , B , (/) ) ` , and it is always a set by ~ fvex .)  Note
       also that these functions are not the same; wherever ` B ( C ) ` is not
       a set, ` C ` is not in the domain of ` F ` (so it evaluates to the empty
       set), but ` C ` is in the domain of ` G ` , and ` G ( C ) ` is defined
       to be the empty set.  (Contributed by Mario Carneiro, 14-Jul-2013.)
       (Revised by Mario Carneiro, 23-Apr-2014.) $)
    fvmptex $p |- ( F ` C ) = ( G ` C ) $=
      ( vy wcel cfv wceq csb cid cv cmpt nfcv cbvmpt eqtri c0 nfcsb1v nffv fvex
      csbeq1 csbeq1a fvmpti fveq2d fvmpt eqtr4d wn cdm sseli ndmfv nsyl5 dmmpti
      dmmptss eleq2i sylnbir pm2.61i ) DBJZDEKZDFKZLUTVAADCMZNKZVBIDAIOZCMZVCBE
      AVEDCUDZEABCPIBVFPGAIBCVFICQAVECUAZAVECUEZRSUFIDVFNKZVDBFVEDLVFVCNVGUGFAB
      CNKZPIBVJPHAIBVKVJIVKQAVFNANQVHUBAOVELCVFNVIUGRSVCNUCUHUIUTUJVATVBDEUKZJU
      TVATLVLBDABCEGUPULDEUMUNUTDFUKZJVBTLVMBDABVKFCNUCHUOUQDFUMURUIUS $.
  $}

  ${
    $d x A $.  $d x D $.
    fvmptd2f.1 $e |- ( ph -> A e. D ) $.
    fvmptd2f.2 $e |- ( ( ph /\ x = A ) -> B e. V ) $.
    fvmptd2f.3 $e |- ( ( ph /\ x = A ) -> ( ( F ` A ) = B -> ps ) ) $.
    ${
      fvmptd3f.4 $e |- F/_ x F $.
      fvmptd3f.5 $e |- F/ x ps $.
      fvmptd3f.6 $e |- F/ x ph $.
      $( Alternate deduction version of ~ fvmpt with three nonfreeness
         hypotheses instead of distinct variable conditions.  (Contributed by
         AV, 19-Jan-2022.) $)
      fvmptd3f $p |- ( ph -> ( F = ( x e. D |-> B ) -> ps ) ) $=
        ( cv wceq cmpt wi wcel cfv nfmpt1 nfeq nfim cvv elexd isset sylib fveq1
        wa simpr fveq2d adantr eqeltrd eqid fvmpt2 syl2anc eqtr3d eqeq2d sylbid
        wex syl5 exlimdd ) ACOZDPZGCFEQZPZBRCNVFBCCGVELCFEUAUBMUCADUDSVDCUTADFI
        UECDUFUGVFDGTZDVETZPZAVDUIZBDGVEUHVJVIVGEPBVJVHEVGVJVCVETZVHEVJVCDVEAVD
        UJZUKVJVCFSEHSVKEPVJVCDFVLADFSVDIULUMJCFEHVEVEUNUOUPUQURKUSVAVB $.
    $}

    $d x ph $.
    ${
      fvmptd2f.4 $e |- F/_ x F $.
      fvmptd2f.5 $e |- F/ x ps $.
      $( Alternate deduction version of ~ fvmpt , suitable for iteration.
         (Contributed by Mario Carneiro, 7-Jan-2017.)  (Proof shortened by AV,
         19-Jan-2022.) $)
      fvmptd2f $p |- ( ph -> ( F = ( x e. D |-> B ) -> ps ) ) $=
        ( nfv fvmptd3f ) ABCDEFGHIJKLMACNO $.
    $}

    $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 fvmptd2f ) 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 wex elexd isset sylib eqeltrrd exlimddv
      wa fvmptd fveq1 eqeq1d syl5ibrcom ) ACGLZEMGBFDNZMZCUKLZEMABCDEFUKOAUKPKI
      ABQCMZEORBACORUNBSACFITBCUAUBAUNUEZDEOKUODHJTUCUDUFULUJUMECGUKUGUHUI $.
  $}

  ${
    $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 cvv cfv wceq cv wi nfel1 cmpt nfmpt1 nfcxfr nffv nfeq eleq1d fveq2
      nfim eqeq12d imbi12d fvmpt2 ex vtoclgaf elex impel ) BELDMLZBFNZDOZDGLCML
      ZAPZFNZCOZQUNUPQABEHUNUPAADMIRAUODABFAFAECSKAECTUAHUBIUCUFURBOZUQUNUTUPVA
      CDMJUDVAUSUOCDURBFUEJUGUHURELUQUTAECMFKUIUJUKDGULUM $.

    $( The value of a function given by an ordered-pair class abstraction is
       the empty set when the class it would otherwise map to is a proper
       class.  This version of ~ fvmptn uses bound-variable hypotheses instead
       of distinct variable conditions.  (Contributed by NM, 21-Oct-2003.)
       (Revised by Mario Carneiro, 11-Sep-2015.) $)
    fvmptnf $p |- ( -. C e. _V -> ( F ` A ) = (/) ) $=
      ( cvv wcel wn cdm cfv c0 wceq dmmptss sseli cid cmpt eqid fvmptex fvex cv
      nfcv nffv fveq2d fvmptf mpan2 eqtrid fvprc sylan9eq expcom ndmfv pm2.61d1
      syl5 ) DKLMZBFNZLZBFOZPQZUTBELZURVBUSEBAECFJRSVCURVBVCURVADTOZPVCVABAECTO
      ZUAZOZVDAECBFVFJVFUBZUCVCVDKLVGVDQDTUDABVEVDEVFKGADTATUFHUGAUEBQCDTIUHVHU
      IUJUKDTULUMUNUQBFUOUP $.
  $}

  ${
    $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 fvmptg syl2anc ) ACFMEHMCGNEOKLBCDEFHGJIPQ $.
  $}

  ${
    $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 x A $.  $d x C $.  $d x D $.
    fvmptn.1 $e |- ( x = D -> B = C ) $.
    fvmptn.2 $e |- F = ( x e. A |-> B ) $.
    $( This somewhat non-intuitive theorem tells us the value of its function
       is the empty set when the class ` C ` it would otherwise map to is a
       proper class.  This is a technical lemma that can help eliminate
       redundant sethood antecedents otherwise required by ~ fvmptg .
       (Contributed by NM, 21-Oct-2003.)  (Revised by Mario Carneiro,
       9-Sep-2013.) $)
    fvmptn $p |- ( -. C e. _V -> ( F ` D ) = (/) ) $=
      ( nfcv fvmptnf ) AECDBFAEIADIGHJ $.

    $( 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.) $)
    fvmptss2 $p |- ( F ` D ) C_ C $=
      ( cdm wcel cfv wss cvv wa cv wceq eleq1d dmmpt elrab2 fvmptg eqimss sylbi
      syl wn c0 ndmfv 0ss eqsstrdi pm2.61i ) EFIZJZEFKZDLZUKEBJDMJZNZUMCMJUNAEB
      UJAOEPCDMGQABCFHRSUOULDPUMAECDBMFGHTULDUAUCUBUKUDULUEDEFUFDUGUHUI $.
  $}

  ${
    $d M x y $.  $d V x $.  $d X x y $.  $d Y y $.  $d m x y $.
    elfvmptrab1w.f $e |- F = ( x e. V |-> { y e. [_ x / m ]_ M | ph } ) $.
    elfvmptrab1w.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.
       Version of ~ elfvmptrab1 with a disjoint variable condition, which does
       not require ~ ax-13 .  (Contributed by Alexander van der Vekens,
       15-Jul-2018.)  Avoid ~ ax-13 .  (Revised by GG, 26-Jan-2024.) $)
    elfvmptrab1w $p |- ( Y e. ( F ` X )
                        -> ( X e. V /\ Y e. [_ X / m ]_ M ) ) $=
      ( cdm wcel cfv csb wa crab cvv wceq nfcv elfvdm wsbc dmmptss sseli rabexg
      3syl nfsbc1v nfcsbw nfrabw csbeq1 sbceq1a rabeqbidv fvmptf syl2anc eleq2d
      cv elrabi anim12i ex sylbid mpcom ) HELZMZIHENZMZHGMZIDHFOZMZPZIHEUAVCVEI
      ABHUBZCVGQZMZVIVCVDVKIVCVFVKRMZVDVKSVBGHBGACDBUPZFOZQZEJUCUDZVCVFVGRMVMVQ
      KVJCVGRUEUFBHVPVKGERBHTZVJBCVGABHUGBDHFVRBFTUHUIVNHSAVJCVOVGDVNHFUJABHUKU
      LJUMUNUOVCVLVIVCVFVLVHVQVJCIVGUQURUSUTVA $.
    $( $j usage 'elfvmptrab1w' avoids 'ax-13'; $)
  $}

  ${
    $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.
       Usage of this theorem is discouraged because it depends on ~ ax-13 .
       Use the weaker ~ elfvmptrab1w when possible.  (Contributed by Alexander
       van der Vekens, 15-Jul-2018.)  (New usage is discouraged.) $)
    elfvmptrab1 $p |- ( Y e. ( F ` X )
                        -> ( X e. V /\ Y e. [_ X / m ]_ M ) ) $=
      ( cfv wcel csb c0 crab cvv wceq 3syl nfcv wa wne ne0i ndmfv necon1ai wsbc
      cdm wi dmmptss sseli rabexg nfsbc1v nfcsb csbeq1 sbceq1a rabeqbidv fvmptf
      cv nfrab syl2anc eleq2d elrabi anim12i ex sylbid pm2.43i ) IHELZMZHGMZIDH
      FNZMZUAZVHVGOUBHEUGZMZVHVLUHVGIUCVNVGOHEUDUEVNVHIABHUFZCVJPZMZVLVNVGVPIVN
      VIVPQMZVGVPRVMGHBGACDBURZFNZPZEJUIUJZVNVIVJQMVRWBKVOCVJQUKSBHWAVPGEQBHTZV
      OBCVJABHULBDHFWCBFTUMUSVSHRAVOCVTVJDVSHFUNABHUOUPJUQUTVAVNVQVLVNVIVQVKWBV
      OCIVJVBVCVDVESVF $.
  $}

  ${
    $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 elfvmptrab1w eleq2d biimpd imdistani ) HGDLMGFMZHKGENZ
      MZOULHEMZOABCKDEFGHDBFACEPZQBFACKBUAZENZPZQIBFUPUSUQFMZEURRUPUSRUTUREKUQE
      FSUEACEURUBTUCUDULUMEUFKGEFSZJUGUHULUNUOULUNUOULUMEHVAUIUJUKT $.
  $}

  ${
    $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 ndmfv
      nsyl5 ) EFHZIEDIEFJKLUBDEUBBMDIANBCOZHDFUCGPABCDQRSEFTUA $.
  $}

  ${
    $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 G x y $.  $d X x y $.  $d ps x $.
    fvmptrabfv.f $e |- F = ( x e. _V |-> { y e. ( G ` x ) | ph } ) $.
    fvmptrabfv.r $e |- ( x = X -> ( ph <-> ps ) ) $.
    $( Value of a function mapping a set to a class abstraction restricting the
       value of another function.  (Contributed by AV, 18-Feb-2022.) $)
    fvmptrabfv $p |- ( F ` X ) = { y e. ( G ` X ) | ps } $=
      ( cvv wcel cfv crab wceq cv fveq2 rabeqbidv fvex c0 fvprc wn rabeqdv rab0
      rabex fvmpt eqtr2di eqtrd pm2.61i ) GJKZGELZBDGFLZMZNCGADCOZFLZMULJEUMGNA
      BDUNUKUMGFPIQHBDUKGFRUDUEUIUAZUJSULGETUOULBDSMSUOBDUKSGFTUBBDUCUFUGUH $.
  $}

  ${
    $d x y z A $.  $d z F $.  $d x ps $.
    fvopab5.1 $e |- F = { <. x , y >. | ph } $.
    fvopab5.2 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( The value of a function that is expressed as an ordered pair
       abstraction.  (Contributed by NM, 19-Feb-2006.)  (Revised by Mario
       Carneiro, 11-Sep-2015.) $)
    fvopab5 $p |- ( A e. V -> ( F ` A ) = ( iota y ps ) ) $=
      ( vz wcel cvv cio wceq cv wbr nfcv nfcxfr nfbr nfv elex df-fv breq2 copab
      cfv nfopab2 cbviotaw eqtri wb nfopab1 nfbi breq1 bibi12d cop df-br eleq2i
      opabidw 3bitri vtoclg1f iotabidv eqtrid syl ) EGKELKZEFUEZBDMZNEGUAVCVDED
      OZFPZDMZVEVDEJOZFPZJMVHJEFUBVJVGJDVIVFEFUCDEVIFDEQDFACDUDZHACDUFRDVIQSVGJ
      TUGUHVCVGBDCOZVFFPZAUIVGBUICELVGBCCEVFFCEQCFVKHACDUJRCVFQSBCTUKVLENVMVGAB
      VLEVFFULIUMVMVLVFUNZFKVNVKKAVLVFFUOFVKVNHUPACDUQURUSUTVAVB $.
  $}

  ${
    $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 dffn5 eqeq12 syl2anb cvv wcel rgenw mpteqb
      wb fvex ax-mp bitrdi ) CBEZDBEZFCDGZABAHZCIZJZABUFDIZJZGZUGUIGABKZUCCUHGD
      UJGUEUKSUDABCLABDLCUHDUJMNUGOPZABKUKULSUMABUFCTQABUGUIORUAUB $.

    $( 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 cfv wral wss wcel eqfnfv2 eqss biancomi anbi1i anass r19.26
      cv dfss3 bitr4i anbi2i 3bitri bitrdi ) DBFECFGDEHBCHZASZDIUFEIHZABJZGZCBK
      ZUFCLZUGGABJZGZABCDEMUIUJBCKZGZUHGUJUNUHGZGUMUEUOUHUEUJUNBCNOPUJUNUHQUPUL
      UJUPUKABJZUHGULUNUQUHABCTPUKUGABRUAUBUCUD $.

    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
      cbvralw bitrdi ) CBHDBHICDJGKZCLZUCDLZJZGBMAKZCLZUGDLZJZABMGBCDNUFUJGABAU
      DUEAUCCEAUCOZPAUCDFUKPQUJGRUCUGJUDUHUEUIUCUGCSUCUGDSTUAUB $.
  $}

  ${
    $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 F x $.  $d G x $.  $d A x $.  $d B x $.
    $( Two functions on ` A u. B ` are equal if and only if they have equal
       restrictions to both ` A ` and ` B ` .  (Contributed by Jeff Madsen,
       19-Jun-2011.) $)
    eqfnun $p |- ( ( F Fn ( A u. B ) /\ G Fn ( A u. B ) ) -> ( F = G <->
                ( ( F |` A ) = ( G |` A ) /\ ( F |` B ) = ( G |` B ) ) ) ) $=
      ( vx cun wfn wa wceq cres reseq1 jca cv wcel fveq1 fvres sylan9req adantl
      cfv eqtr3d wral wo elun adantlr adantll jaodan ralrimiva eqfnfv imbitrrid
      sylan2b impbid2 ) CABFZGDULGHZCDIZCAJZDAJZIZCBJZDBJZIZHZUNUQUTCDAKCDBKLVA
      UNUMEMZCSZVBDSZIZEULUAVAVEEULVBULNVAVBANZVBBNZUBVEVBABUCVAVFVEVGUQVFVEUTU
      QVFHVBUPSZVCVDUQVFVHVBUOSVCVBUOUPOVBACPQVFVHVDIUQVBADPRTUDUTVGVEUQUTVGHVB
      USSZVCVDUTVGVIVBURSVCVBURUSOVBBCPQVGVIVDIUTVBBDPRTUEUFUJUGEULCDUHUIUK $.
  $}

  ${
    $d x B $.  $d x F $.  $d x G $.
    $( Equality of restricted functions is determined by their values (for
       functions with different domains).  (Contributed by AV, 6-Jan-2019.) $)
    fvreseq0 $p |- ( ( ( F Fn A /\ G Fn C ) /\ ( B C_ A /\ B C_ C ) ) ->
           ( ( F |` B ) = ( G |` B ) <-> A. x e. B ( F ` x ) = ( G ` x ) ) ) $=
      ( wfn wss cres wceq cv cfv wral wb wa fnssres eqfnfv wcel fvres eqeq12d
      ralbiia bitrdi syl2an an4s ) EBGZCBHZFDGZCDHZECIZFCIZJZAKZELZULFLZJZACMZN
      ZUEUFOUICGZUJCGZUQUGUHOBCEPDCFPURUSOUKULUILZULUJLZJZACMUPACUIUJQVBUOACULC
      RUTUMVAUNULCESULCFSTUAUBUCUD $.

    $( Equality of a function restricted to the domain of another function.
       (Contributed by AV, 6-Jan-2019.) $)
    fvreseq1 $p |- ( ( ( F Fn A /\ G Fn B ) /\ B C_ A ) ->
                    ( ( F |` B ) = G <-> A. x e. B ( F ` x ) = ( G ` x ) ) ) $=
      ( wfn wa wss cres wceq cv cfv wral fnresdm ad2antlr eqcomd eqeq2d wb ssid
      fvreseq0 mpanr2 bitrd ) DBFZECFZGZCBHZGZDCIZEJUHECIZJZAKZDLUKELJACMZUGEUI
      UHUGUIEUDUIEJUCUFCENOPQUEUFCCHUJULRCSABCCDETUAUB $.
  $}

  ${
    $d x B $.  $d x F $.  $d x G $.
    $( Equality of restricted functions is determined by their values.
       (Contributed by NM, 3-Aug-1994.)  (Proof shortened by AV,
       4-Mar-2019.) $)
    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 fvreseq0 anabsan2 ) DBFEBFGCBHDCIEC
      IJAKZDLQELJACMNABCBDEOP $.
  $}

  ${
    $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 sseqin2 sylib vex
      eldm adantll necon3abid breq2 notbid ceqsexv bitr4di adantlr anbi1d brdif
      fvex exbidv bitr2d rabbi2dva eqtr3d ) CBFZDBFZGZBCDUAZHZUBZVMAIZCJZVODJZU
      CZABUDVKVMBKVNVMLVKCHZVMBVLCKVMVSKCDUEVLCUFUGVIVSBLVJBCUHUIUJVMBUKULVKVRA
      BVMVOVMMVOEIZVLNZEOZVKVOBMZGZVREVOVLAUMUNWDVRVTVPLZVOVTDNZPZGZEOZWBWDVRVO
      VPDNZPZWIWDWJVPVQVJWCVPVQLZWJQVIWLVQVPLVJWCGWJVPVQRBVOVPDSTUOUPWGWKEVPVOC
      VDWEWFWJVTVPVODUQURUSUTWDWHWAEWDWHVOVTCNZWGGWAWDWEWMWGVIWCWEWMQVJWEVPVTLV
      IWCGWMVTVPRBVOVTCSTVAVBVOVTCDVCUTVEVFTVGVH $.

    $( 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 necom rabbii fndmdif wceq ancoms 3eqtr4a
      cdm ) BAEZCAEZFDGZBHZUACHZIZDAJUCUBIZDAJZBCKRCBKRZUDUEDAUBUCLMDABCNTSUGUF
      ODACBNPQ $.

    $( The difference set of two functions is empty if and only if the
       functions are equal.  (Contributed by Stefan O'Rear, 17-Jan-2015.) $)
    fndmdifeq0 $p |- ( ( F Fn A /\ G Fn A ) ->
        ( dom ( F \ G ) = (/) <-> F = G ) ) $=
      ( vx wfn wa cdif cdm c0 wceq cv cfv wne crab fndmdif eqeq1d wn rabeq0 nne
      wral ralbii bitri eqfnfv bitr4id bitrd ) BAECAEFZBCGHZIJDKZBLZUHCLZMZDANZ
      IJZBCJZUFUGULIDABCOPUFUMUIUJJZDATZUNUMUKQZDATUPUKDARUQUODAUIUJSUAUBDABCUC
      UDUE $.

    $( 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 cfv wceq copab cmpt dffn5 biimpi df-mpt eqtrdi wex
      cab wcel ineqan12d inopab dmeqd 19.42v anandi exbii eqeq1 ceqsexv 3bitr3i
      crab fvex anbi2i abbii dmopab df-rab 3eqtr4i ) CBFZDBFZGZCDHZIAJZBUAZEJZV
      BCKZLZGZVCVDVBDKZLZGZGZAEMZIZVEVHLZABUKZUTVAVLUTVAVGAEMZVJAEMZHVLURUSCVPD
      VQURCABVENZVPURCVRLABCOPAEBVEQRUSDABVHNZVQUSDVSLABDOPAEBVHQRUBVGVJAEUCRUD
      VKESZATVCVNGZATVMVOVTWAAVCVFVIGZGZESVCWBESZGVTWAVCWBEUEWCVKEVCVFVIUFUGWDV
      NVCVIVNEVEVBCULVDVEVHUHUIUMUJUNVKAEUOVNABUPUQR $.
  $}

  ${
    $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 dfss2 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 cfv csn wss snssi funimass2 sylan2 fvex snss cdm wceq
    wa cnvimass sseli wfn funfn fnsnfv sylanb sseq1d bitrid mpbird ) CDZACEBFZG
    ZRZACHZBGZCAIZFZBJZUJUHUNUIJUPAUIKUNBCLMUMULIZBJUKUPULBACNOUKUQUOBUJUHACPZG
    ZUQUOQZUIURACBSTUHCURUAUSUTCUBURACUCUDMUEUFUG $.

  $( 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 wb fvex mpan adantl mpbird
    cvv wss opelcnvg elimasng imass2 sylbi sseld syl5com wi fvimacnvi ex adantr
    snss impbid ) CDZACEZFZGZACHZBFZACIZBJZFZUPAUSUQKZJZFZURVAUPVDUQALUSFZUPVEA
    UQLCFZACMUOVEVFNZUMUQSFZUOVGACOZUQASUNCUAPQRUOVDVENZUMVHUOVJVIUSUQASUNUBPQR
    URVCUTAURVBBTVCUTTUQBVIUKVBBUSUCUDUEUFUMVAURUGUOUMVAURABCUHUIUJUL $.

  ${
    $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 cv cfv wceq wral cima csn ccnv wcel funimass4 fvex elsn
      ralbii bitr2di funimass3 bitrd ) DEBDFGHZAIZDJZCKZABLZDBMCNZGZBDOUIMGUDUJ
      UFUIPZABLUHABUIDQUKUGABUFCUEDRSTUABUIDUBUC $.
  $}

  $( Alternate proof of ~ fvimacnv , based on ~ funimass3 .  If ~ funimass3 is
     ever proved directly, as opposed to using ~ funimacnv pointwise, then the
     proof of ~ funimacnv should be replaced with this one.  (Contributed by
     Raph Levien, 20-Nov-2006.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  fvimacnvALT $p |- ( ( Fun F /\ A e. dom F ) ->
                    ( ( F ` A ) e. B <-> A e. ( `' F " B ) ) ) $=
    ( wfun cdm wcel wa csn cima wss ccnv cfv wb snssi funimass3 sylan2 fvex wfn
    snss wceq eqid df-fn biimpri mpan2 fnsnfv sylan sseq1d bitrid snssg 3bitr4d
    adantl ) CDZACEZFZGZCAHZIZBJZUPCKBIZJZACLZBFZAUSFZUNULUPUMJURUTMAUMNUPBCOPV
    BVAHZBJUOURVABACQSUOVDUQBULCUMRZUNVDUQTULUMUMTZVEUMUAVEULVFGCUMUBUCUDUMACUE
    UFUGUHUNVCUTMULAUSUMUIUKUJ $.

  $( 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 $.

  ${
    elpreimad.f $e |- ( ph -> F Fn A ) $.
    elpreimad.b $e |- ( ph -> B e. A ) $.
    elpreimad.c $e |- ( ph -> ( F ` B ) e. C ) $.
    $( Membership in the preimage of a set under a function.  (Contributed by
       Glauco Siliprandi, 23-Oct-2021.) $)
    elpreimad $p |- ( ph -> B e. ( `' F " C ) ) $=
      ( ccnv cima wcel cfv wfn wa wb elpreima syl mpbir2and ) ACEIDJKZCBKZCELDK
      ZGHAEBMSTUANOFBCDEPQR $.
  $}

  $( 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 fvex elsn anbi2i bitrdi ) DAE
    CDFBGZHICAIZCDJZRIZKSTBLZKACRDMUAUBSTBCDNOPQ $.

  ${
    $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 eqabdv 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 cfv wcel crab wceq fncnvima2 fvex elsn rabbii eqtrdi
      cv ) DBEDFCGZHARZDIZSJZABKUACLZABKABSDMUBUCABUACTDNOPQ $.
  $}

  ${
    $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 wb funfn cfv wa biancomi elpreima
      wcel elin anbi1i fvres eleq1d adantl pm5.32i bitri an32 bitrdi wceq fnfun
      a1i funresd dmres df-fn sylanblrc syl anbi1d bitrid 3bitr4d sylbi eqrdv )
      CEZDCBFZGAHZCGAHZBIZVBCCJZKZDLZVDSZVIVFSZMCNVHVIBVGIZSZVIVCOZASZPZVIVGSZV
      ICOZASZPZVIBSZPZVJVKVHVPVQWAPZVSPZWBVPWDMVHVPWCVOPWDVMWCVOVMVQWAVIBVGTQUA
      WCVOVSWAVOVSMVQWAVNVRAVIBCUBUCUDUEUFUKVQWAVSUGUHVHVCVLKZVJVPMVHVCEVCJVLUI
      WEVHBCVGCUJULCBUMVCVLUNUOVLVIAVCRUPVKVIVESZWAPVHWBVIVEBTVHWFVTWAVGVIACRUQ
      URUSUTVA $.
  $}

  $( The preimage of the intersection of the range of a class and a class ` A `
     is the preimage of the class ` A ` .  (Contributed by AV, 17-Sep-2024.) $)
  cnvimainrn $p |- ( Fun F -> ( `' F " ( ran F i^i A ) ) = ( `' F " A ) ) $=
    ( wfun ccnv crn cin cima inpreima wss wceq cdm cnvimass cnvimarndm sseqtrri
    dfss2 mpbi ineqcomi eqtrdi ) BCBDZBEZAFGSTGZSAGZFUBTABHUBUAUBUBUAIUBUAFUBJU
    BBKUABALBMNUBUAOPQR $.

  $( The preimage of a subset is a subset of the preimage.  (Contributed by
     Brendan Leahy, 23-Sep-2017.) $)
  sspreima $p |- ( ( Fun F /\ A C_ B ) -> ( `' F " A ) C_ ( `' F " B ) ) $=
    ( wfun wss wa ccnv cima wceq inpreima dfss2 biimpi imaeq2d sylan9req sylibr
    cin ) CDZABEZFCGZAHZSBHZPZTITUAEQRUBSABPZHTABCJRUCASRUCAIABKLMNTUAKO $.

  ${
    $d A x y $.  $d B y $.  $d F x y $.
    $( Preimage of an intersection.  (Contributed by FL, 16-Apr-2012.) $)
    iinpreima $p |- ( ( Fun F /\ A =/= (/) )
        -> ( `' F " |^|_ x e. A B ) = |^|_ x e. A ( `' F " B ) ) $=
      ( vy ciin cima wcel wral simpll cnvimass sseli adantl cvv fvimacnvi eliin
      wa biimpa wb wi wfun c0 wne ccnv cv cdm cfv fvex adantlr sylancr fvimacnv
      ralbidv syl21anc elv sylibr biimpd ralimdv sylc ax-mp wrex r19.2zb biimpi
      ex rexlimivw syl6 biimtrid imp syl2anc mpbid impbida eqrdv ) DUAZBUBUCZQZ
      EDUDZABCFZGZABVOCGZFZVNEUEZVQHZVTVSHZVNWAQZVTVRHZABIZWBWCVLVTDUFZHZVTDUGZ
      CHZABIZWEVLVMWAJWAWGVNVQWFVTDVPKLMWCWHNHZWHVPHZWJVTDUHZVLWAWLVMVTVPDOUIWK
      WLWJAWHBCNPZRUJVLWGQZWJWEWOWIWDABVTCDUKULRUMWBWESEAVTBVRNPZUNZUOVNWBQZWLW
      AWRWJWLWRVLWEWJVLVMWBJZWBWEVNWBWETEVTNHWBWEWPUPUNMVLWDWIABVLWDWIVTCDOVCUQ
      URWKWLWJSWMWNUSUOWRVLWGWLWASWSVNWBWGVMWBWGTVLWBWEVMWGWQVMWEWDABUTZWGVMWEW
      TTWDABVAVBWDWGABVRWFVTDCKLVDVEVFMVGVTVPDUKVHVIVJVK $.

    $( Preimage of an intersection.  (Contributed by FL, 28-Apr-2012.) $)
    intpreima $p |- ( ( Fun F /\ A =/= (/) )
        -> ( `' F " |^| A ) = |^|_ x e. A ( `' F " x ) ) $=
      ( wfun c0 wne wa ccnv cint cima cv ciin intiin imaeq2i iinpreima eqtrid )
      CDBEFGCHZBIZJQABAKZLZJABQSJLRTQABMNABSCOP $.
  $}

  $( Taking the converse image of a set can be limited to the range of the
     function used.  (Contributed by Thierry Arnoux, 21-Jan-2017.) $)
  fimacnvinrn $p |- ( Fun F -> ( `' F " A ) = ( `' F " ( A i^i ran F ) ) ) $=
    ( wfun ccnv crn cin cima cdm inpreima wf wceq wfo funforn fof sylbi fimacnv
    syl ineq2d cres cnvresima resdm2 funrel dfrel2 sylib eqtrid imaeq1d eqtr3id
    wrel cnveqd 3eqtrrd ) BCZBDZABEZFGULAGZULUMGZFUNBHZFZUNAUMBIUKUOUPUNUKUPUMB
    JZUOUPKUKUPUMBLURBMUPUMBNOUPUMBPQRUKUQBUPSZDZAGUNUPABTUKUTULAUKUSBUKUSULDZB
    BUAUKBUHVABKBUBBUCUDUEUIUFUGUJ $.

  $( Taking the converse image of a set can be limited to the range of the
     function used.  (Contributed by Thierry Arnoux, 17-Feb-2017.) $)
  fimacnvinrn2 $p |- ( ( Fun F /\ ran F C_ B ) ->
                                    ( `' F " A ) = ( `' F " ( A i^i B ) ) ) $=
    ( wfun crn wss ccnv cin cima inass wceq sseqin2 biimpi adantl ineq2d eqtrid
    wa imaeq2d fimacnvinrn adantr 3eqtr4rd ) CDZCEZBFZQZCGZABHZUCHZIZUFAUCHZIZU
    FUGIZUFAIZUEUHUJUFUEUHABUCHZHUJABUCJUEUNUCAUDUNUCKZUBUDUOUCBLMNOPRUBULUIKUD
    UGCSTUBUMUKKUDACSTUA $.

  ${
    rescnvimafod.f $e |- ( ph -> F Fn A ) $.
    rescnvimafod.e $e |- ( ph -> E = ( ran F i^i B ) ) $.
    rescnvimafod.d $e |- ( ph -> D = ( `' F " B ) ) $.
    $( The restriction of a function to a preimage of a class is a function
       onto the intersection of this class and the range of the function.
       (Contributed by AV, 13-Sep-2024.)  (Revised by AV, 29-Sep-2024.) $)
    rescnvimafod $p |- ( ph -> ( F |` D ) : D -onto-> E ) $=
      ( cres wfn crn wceq wfo ccnv cima cdm wss a1i cin cnvimass eqcomd 3sstr4d
      fndmd fnssresd df-ima imaeq2d fnfun funimacnv incom 3eqtrd eqtr3id eqtr4d
      wfun 3syl df-fo sylanbrc ) AFDJZDKURLZEMDEURNABDFGAFOCPZFQZDBUTVARAFCUASI
      AVABABFGUDUBUCUEAUSFLZCTZEAUSFDPZVCFDUFAVDFUTPZCVBTZVCADUTFIUGAFBKFUNVEVF
      MGBFUHCFUIUOVFVCMACVBUJSUKULHUMDEURUPUQ $.
  $}

  ${
    $d D a p $.  $d D x y z $.  $d F p $.  $d F a w x y z $.
    $( If a class' function values for certain arguments is not the empty set,
       the arguments are contained in the domain of the class, and the class
       restricted to the arguments is a function, analogous to ~ fvfundmfvn0 .
       (Contributed by AV, 27-Jan-2020.)  (Proof shortened by Peter Mazsa,
       2-Oct-2022.) $)
    fvn0ssdmfun $p |- ( A. a e. D ( F ` a ) =/= (/)
                        -> ( D C_ dom F /\ Fun ( F |` D ) ) ) $=
      ( vx vz vy vw cv wral wcel cres wfun wa weq wi wal wex sylibr albii exbii
      vp cfv wne cdm csn wss fvfundmfvn0 ralimi r19.26 eleq1w rspccv ssrdv ciun
      wrel cop funrel reliun sneq reseq2d funeqd rspcva dffun5 wceq vex elsnres
      imbi1i equcom opeq12 biimtrid adantr impcom equcoms eleq1d biimpcd adantl
      c0 ex opeq2 jca imim1d alimdv eximdv spimvw sylbi simplbiim expcom impexp
      spimevw 19.21v 19.37v 3bitri alrimiv resiun2 eqcomi eleq2i iunid opelresi
      syl reseq2i sylanbrc funeqi bitri anim12i ) CHZBUBVPUCZCAIXDBUDZJZBXDUEZK
      ZLZMZCAIZAXFUFZBAKZLZMZXEXKCAXDBUGUHXLXGCAIZXJCAIZMXPXGXJCAUIXQXMXRXOXQUA
      AXFXGUAHZXFJCXSACUAXFUJUKULXRCAXIUMZLZXOXRXTUNZDHZEHZUOZXTJZEFNZOZEPZFQZD
      PZYAXRXIUNZCAIYBXJYLCAXIUPUHCAXIUQRXRYCAJZYEBJZMZYGOZEPZFQZDPYKXRYRDXRYMY
      NYGOZEPZFQZOZYRYMXRUUAYMXRMBYCUEZKZLZUUAXJUUECYCACDNZXIUUDUUFXHUUCBXDYCUR
      USUTVAUUEUUDUNGHZYDUOZUUDJZYGOZEPZFQZGPZUUAGEFUUDVBUUMUUHYCXDUOZVCZUUNBJZ
      MZCQZYGOZEPZFQZGPUUAUULUVAGUUKUUTFUUJUUSEUUIUURYGCUUHBYCDVDVEVFSTSUVAUUAG
      DGDNZUUTYTFUVBUUSYSEUVBYNUURYGUVBYNUURUVBYNMZUUQCECENZUVCUUQUVDUVCMUUOUUP
      UVCUVDUUOUVBUVDUUOOYNUVDECNZUVBUUOCEVGUVBUVEUUOUUGYDYCXDVHVQVIVJVKUVCUVDU
      UPYNUVDUUPOUVBUVDYNUUPUVDYEUUNBYEUUNVCECYDXDYCVRVLVMVNVOVKVSVQWHVQVTWAWBW
      CWDWEWRWFYRYMYSOZEPZFQYMYTOZFQUUBYQUVGFYPUVFEYMYNYGWGSTUVGUVHFYMYSEWITYMY
      TFWJWKRWLYJYRDYIYQFYHYPEYFYOYGYFYEBCAXHUMZKZJYEXNJYOXTUVJYEUVJXTCAXHBWMZW
      NWOUVJXNYEUVIABCAWPZWSWOAYCYDBEVDWQWKVFSTSRDEFXTVBWTXOUVJLYAXNUVJAUVIBUVI
      AUVLWNWSXAUVJXTUVKXAXBRXCWDWR $.
  $}

  $( 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 cv wa wi wceq eleq1 anbi2d fveq2 eleq1d imbi12d
      crn cop wex funfvop vex opeq1 spcev syl fvex elrn2 sylibr vtoclg anabsi7
      ) BEZABFZGZABHZBRZGZUKCIZULGZJZUQBHZUOGZKUKUMJZUPKCAULUQALZUSVBVAUPVCURUM
      UKUQAULMNVCUTUNUOUQABOPQUSDIZUTSZBGZDTZVAUSUQUTSZBGZVGUQBUAVFVIDUQCUBVDUQ
      LVEVHBVDUQUTUCPUDUEDUTBUQBUFUGUHUIUJ $.
  $}

  $( A function value cannot be any element not contained in the range of the
     function.  (Contributed by AV, 28-Jan-2020.) $)
  nelrnfvne $p |- ( ( Fun F /\ X e. dom F /\ Y e/ ran F )
                    -> ( F ` X ) =/= Y ) $=
    ( wfun cdm wcel cfv crn wnel wne fvelrn elnelne2 stoic3 ) ADBAEFBAGZAHZFCOI
    NCJBAKNCOLM $.

  ${
    $d A a b x $.  $d B a b x $.  $d D a x $.
    fveqdmss.1 $e |- D = dom B $.
    $( If the empty set is not contained in the range of a function, and the
       function values of another class (not necessarily a function) are equal
       to the function values of the function for all elements of the domain of
       the function, then the domain of the function is contained in the domain
       of the class.  (Contributed by AV, 28-Jan-2020.) $)
    fveqdmss $p |- ( ( Fun B /\ (/) e/ ran B
                       /\ A. x e. D ( A ` x ) = ( B ` x ) ) -> D C_ dom A ) $=
      ( va vb wfun c0 crn wnel cv cfv wceq w3a cdm wcel wi fveq2 com12 wral weq
      wa eqeq12d rspcva wne nelrnfvne wex n0 eleq2 eqcoms elfvdm biimtrdi sylbi
      wb exlimiv syl 3exp eleq2s com24 adantr mpd ex com23 com14 3imp ssrdv ) C
      HZICJKZALZBMZVJCMZNZADUAZOFDBPZVHVIVNFLZDQZVPVOQZRVQVIVNVHVRVQVNVIVHVRRZV
      QVNVIVSRZVQVNUCVPBMZVPCMZNZVTVMWCAVPDAFUBVKWAVLWBVJVPBSVJVPCSUDUEVQWCVTRV
      NVQVHVIWCVRVHVIWCVRRZRZRVPCPZDVHVPWFQZWEVHWGVIWDVHWGVIOWBIUFZWDCVPIUGWHGL
      ZWBQZGUHWDGWBUIWJWDGWCWJVRWCWJWIWAQZVRWJWKUOWBWAWBWAWIUJUKWIVPBULUMTUPUNU
      QURTEUSUTVAVBVCVDVEVFVG $.

    $( If the empty set is not contained in the range of a function, and the
       function values of another class (not necessarily a function) are equal
       to the function values of the function for all elements of the domain of
       the function, then the class restricted to the domain of the function is
       the function itself.  (Contributed by AV, 28-Jan-2020.) $)
    fveqressseq $p |- ( ( Fun B /\ (/) e/ ran B
                    /\ A. x e. D ( A ` x ) = ( B ` x ) ) -> ( A |` D ) = B ) $=
      ( wfun c0 cfv wceq wral cdm cin biimpi eqtrid syl wa wcel adantl ralimdva
      3impia crn wnel cv w3a cres wss fveqdmss dmres incom sseqin2 eqtrdi fvres
      id sylan9eq ex raleqtrrdv wb simpll eleq2i simplr nelrnfvne syl3anc neeq1
      wne syl5ibrcom fvn0ssdmfun simprd simp1 eqfunfv syl2anc mpbir2and ) CFZGC
      UAUBZAUCZBHZVNCHZIZADJZUDZBDUEZCIZVTKZCKZIZVNVTHZVPIZAWBJZVSDBKZUFZWDABCD
      EUGZWIWBDWCWIWBDWHLZDBDUHZWIWKWHDLZDDWHUIWIWMDIDWHUJMNZNEUKOVSWFADWBVLVMV
      RWFADJVLVMPZVQWFADWOVNDQZPZVQWFWQVQWEVOVPWPWEVOIWOVNDBULRVQUMUNUOSTVSWBWK
      DWLVSWIWKDIWJWNONUPVSVTFZVLWAWDWGPUQVSVOGVDZADJZWRVLVMVRWTWOVQWSADWQWSVQV
      PGVDZWQVLVNWCQZVMXAVLVMWPURWPXBWOWPXBDWCVNEUSMRVLVMWPUTCVNGVAVBVOVPGVCVES
      TWTWIWRDBAVFVGOVLVMVRVHAVTCVIVJVK $.
  $}

  $( 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 $.

  ${
    fnfvelrnd.1 $e |- ( ph -> F Fn A ) $.
    fnfvelrnd.2 $e |- ( ph -> B e. A ) $.
    $( A function's value belongs to its range.  (Contributed by Glauco
       Siliprandi, 2-Jan-2022.) $)
    fnfvelrnd $p |- ( ph -> ( F ` B ) e. ran F ) $=
      ( wfn wcel cfv crn fnfvelrn syl2anc ) ADBGCBHCDIDJHEFBCDKL $.
  $}

  ${
    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 $.
  $}

  $( A class is an element of the domain iff it's function value is an element
     of the codomain of a function.  (Contributed by AV, 22-Apr-2025.) $)
  feldmfvelcdm $p |- ( ( F : A --> B /\ (/) e/ B )
                     -> ( X e. A <-> ( F ` X ) e. B ) ) $=
    ( wf c0 wnel wa wcel cfv simpl ffvelcdmda ex wne wn wi df-nel nelelne sylbi
    cdm wceq csn cres wfun fdm fvfundmfvn0 simprl eleqtrd syl2im sylan9r impbid
    ) ABCEZFBGZHZDAIZDCJZBIZUNUOUQUNABDCULUMKLMUMUQUPFNZULUOUMFBIOUQURPFBQFBUPR
    SULCTZAUAZURDUSIZCDUBUCUDZHZUOABCUEDCUFUTVCUOUTVCHDUSAUTVAVBUGUTVCKUHMUIUJU
    K $.

  ${
    $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 wa fvexd wceq wrex fvelrnb eqcom rexbii bitrdi
      wb adantl rexxfr2d ) FEHZABCDDIZFJZFKZELUEUFEMNUFFOUECIZUHMUGUIPZDEQUIUGP
      ZDEQDEUIFRUJUKDEUGUISTUAUKABUBUEGUCUD $.

    $( 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 wa fvexd wceq wrex fvelrnb eqcom rexbii bitrdi
      wb adantl ralxfr2d ) FEHZABCDDIZFJZFKZELUEUFEMNUFFOUECIZUHMUGUIPZDEQUIUGP
      ZDEQDEUIFRUJUKDEUGUISTUAUKABUBUEGUCUD $.
  $}

  ${
    $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 $.

    $( For any element in the domain of a function, there is an element in the
       range of the function which is the value of the function at that
       element.  Because of the definition ~ df-fv of the value of a function,
       the theorem is only valid in general if the empty set is not contained
       in the range of the function (the implication "to the right" is always
       valid).  Indeed, with the definition ~ df-fv of the value of a function,
       ` ( F `` Y ) = (/) ` may mean that the value of ` F ` at ` Y ` is the
       empty set or that ` F ` is not defined at ` Y ` .  (Contributed by
       Alexander van der Vekens, 17-Dec-2017.) $)
    eldmrexrnb $p |- ( ( Fun F /\ (/) e/ ran F )
                         -> ( Y e. dom F <-> E. x e. ran F x = ( F ` Y ) ) ) $=
      ( vy wfun c0 crn wnel wa cdm wcel cv cfv wceq wrex eldmrexrn adantr eleq1
      wi wne elnelne2 wex elfvdm exlimiv sylbi syl expcom adantl com12 biimtrdi
      n0 com13 rexlimdv impbid ) BEZFBGZHZIZCBJKZALZCBMZNZAUPOZUOUSVCSUQABCPQUR
      VBUSAUPVBUTUPKZURUSVBVDVAUPKZURUSSUTVAUPRURVEUSUQVEUSSUOVEUQUSVEUQIVAFTZU
      SVAFUPUAVFDLZVAKZDUBUSDVAUKVHUSDVGCBUCUDUEUFUGUHUIUJULUMUN $.
  $}

  ${
    $d x F $.  $d x G $.  $d x H $.  $d x K $.  $d x X $.
    $( The values of two function compositions are equal if the values of the
       composed functions are pairwise equal.  (Contributed by AV,
       26-Jan-2019.) $)
    fvcofneq $p |- ( ( G Fn A /\ K Fn B )
                  -> ( ( X e. ( A i^i B ) /\ ( G ` X ) = ( K ` X )
                         /\ A. x e. ( ran G i^i ran K ) ( F ` x ) = ( H ` x ) )
                       -> ( ( F o. G ) ` X ) = ( ( H o. K ) ` X ) ) ) $=
      ( wfn wa cin wcel cfv wceq crn ccom 3ad2ant1 fvco2 fveq2 ex cv wral simpl
      elinel1 syl2an simpr elinel2 eqcoms 3ad2ant2 adantl wi id syl2anr anim12d
      w3a fnfvelrn wb anbi2d elin biimpri biimtrdi sylan9 eqeq12d rspcva eqcomd
      eleq1 syl6 com23 3impia impcom 3eqtrrd eqtrd ) EBIZGCIZJZHBCKLZHEMZHGMZNZ
      AUAZDMZVTFMZNZAEOZGOZKZUBZUOZHDEPMZHFGPMZNVOWHJZWIVQDMZWJVOVMHBLZWIWLNWHV
      MVNUCVPVSWMWGHBCUDZQBDEHRUEWKWJVRFMZVQFMZWLVOVNHCLZWJWONWHVMVNUFVPVSWQWGH
      BCUGZQCFGHRUEWHWOWPNZVOVSVPWSWGWSVRVQVRVQFSUHUIUJWHVOWPWLNZVPVSWGVOWTUKVP
      VSJZVOWGWTXAVOVQWFLZWGWTUKVPVOVQWDLZVRWELZJZVSXBVPVMXCVNXDVPVMXCVMVMWMXCV
      PVMULWNBHEUPUMTVPVNXDVNVNWQXDVPVNULWRCHGUPUMTUNVSXEXCVQWELZJZXBVSXDXFXCXD
      XFUQVRVQVRVQWEVFUHURXBXGVQWDWEUSUTVAVBXBWGWTXBWGJWLWPWCWLWPNAVQWFVTVQNWAW
      LWBWPVTVQDSVTVQFSVCVDVETVGVHVIVJVKVLT $.
  $}

  ${
    $d w z A $.  $d x y z $.  $d x z A $.  $d w y z $.  $d y B $.  $d y ch $.
    $d w y z F $.  $d w z ps $.  $d x z ps $.
    ralrnmptw.1 $e |- F = ( x e. A |-> B ) $.
    ralrnmptw.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
    $( A restricted quantifier over an image set.  Version of ~ ralrnmpt with a
       disjoint variable condition, which does not require ~ ax-13 .
       (Contributed by Mario Carneiro, 20-Aug-2015.)  Avoid ~ ax-13 .  (Revised
       by GG, 26-Jan-2024.) $)
    ralrnmptw $p |- ( A. x e. A B e. V ->
      ( A. y e. ran F ps <-> A. x e. A ch ) ) $=
      ( vw vz wcel wral cv cfv wsbc wb syl nfv crn fnmpt dfsbcq nfsbc1v sbceq2a
      wfn ralrn cbvralw cmpt nfmpt1 nfcxfr nfcv nffv nfsbcw weq sbceq1d 3bitr3g
      fveq2 wa fvmpt2 sbcieg adantl bitrd ralimiaa ralbi ) FHMZCENZADGUAZNZADCO
      ZGPZQZCENZBCENZVGADKOZQZKVHNZADLOZGPZQZLENZVIVMVGGEUFVQWARCEFGHIUBVPVTKLE
      GADVOVSUCUGSVPAKDVHADVOUDAKTADVOUEUHVTVLLCEACDVSCVRGCGCEFUIICEFUJUKCVRULU
      MACTUNVLLTLCUOADVSVKVRVJGURUPUHUQVGVLBRZCENVMVNRVFWBCEVJEMZVFUSZVLADFQZBW
      DADVKFCEFHGIUTUPVFWEBRWCABDFHJVAVBVCVDVLBCEVESVC $.
    $( $j usage 'ralrnmptw' avoids 'ax-13'; $)
  $}

  ${
    $d x y $.  $d A x $.  $d B y $.  $d ch y $.  $d F y $.  $d ps x $.
    rexrnmptw.1 $e |- F = ( x e. A |-> B ) $.
    rexrnmptw.2 $e |- ( y = B -> ( ps <-> ch ) ) $.
    $( A restricted quantifier over an image set.  Version of ~ rexrnmpt with a
       disjoint variable condition, which does not require ~ ax-13 .
       (Contributed by Mario Carneiro, 20-Aug-2015.)  Avoid ~ ax-13 .  (Revised
       by GG, 26-Jan-2024.) $)
    rexrnmptw $p |- ( A. x e. A B e. V ->
      ( E. y e. ran F ps <-> E. x e. A ch ) ) $=
      ( wcel wral wn crn wrex cv wceq notbid ralrnmptw dfrex2 3bitr4g ) FHKCELZ
      AMZDGNZLZMBMZCELZMADUDOBCEOUBUEUGUCUFCDEFGHIDPFQABJRSRADUDTBCETUA $.
    $( $j usage 'rexrnmptw' avoids 'ax-13'; $)
  $}

  ${
    $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.  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  Use the weaker ~ ralrnmptw
       when possible.  (Contributed by Mario Carneiro, 20-Aug-2015.)
       (New usage is discouraged.) $)
    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 cfv wsbc wb syl nfv crn fnmpt dfsbcq nfsbc1v sbceq1a
      wfn ralrn cbvral bicomi cmpt nfmpt1 nfcxfr nfcv nfsbc weq sbceq1d 3bitr3g
      nffv fveq2 wa fvmpt2 sbcieg adantl bitrd ralimiaa ralbi ) FHMZCENZADGUAZN
      ZADCOZGPZQZCENZBCENZVHADKOZQZKVINZADLOZGPZQZLENZVJVNVHGEUFVRWBRCEFGHIUBVQ
      WAKLEGADVPVTUCUGSVJVRAVQDKVIAKTADVPUDADVPUEUHUIWAVMLCEACDVTCVSGCGCEFUJICE
      FUKULCVSUMURACTUNVMLTLCUOADVTVLVSVKGUSUPUHUQVHVMBRZCENVNVORVGWCCEVKEMZVGU
      TZVMADFQZBWEADVLFCEFHGIVAUPVGWFBRWDABDFHJVBVCVDVEVMBCEVFSVD $.

    $( A restricted quantifier over an image set.  Usage of this theorem is
       discouraged because it depends on ~ ax-13 .  Use the weaker ~ rexrnmptw
       when possible.  (Contributed by Mario Carneiro, 20-Aug-2015.)
       (New usage is discouraged.) $)
    rexrnmpt $p |- ( A. x e. A B e. V ->
      ( E. y e. ran F ps <-> E. x e. A ch ) ) $=
      ( wcel wral wn crn wrex cv wceq notbid ralrnmpt dfrex2 3bitr4g ) FHKCELZA
      MZDGNZLZMBMZCELZMADUDOBCEOUBUEUGUCUFCDEFGHIDPFQABJRSRADUDTBCETUA $.
  $}

  ${
    f0cl.1 $e |- F : A --> B $.
    f0cl.2 $e |- (/) e. B $.
    $( Unconditional closure of a function when the codomain includes the empty
       set.  (Contributed by Mario Carneiro, 12-Sep-2013.) $)
    f0cli $p |- ( F ` C ) e. B $=
      ( wcel cfv ffvelcdmi cdm fdmi eleq2i wn c0 ndmfv eqeltrdi sylnbir pm2.61i
      ) CAGZCDHZBGZABCDEISCDJZGZUAUBACABDEKLUCMTNBCDOFPQR $.
  $}

  $( 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 f g x y z A $.  $d f g x y z B $.  $d x y z F $.
    $( Alternate definition of a mapping.  (Contributed by NM, 20-Mar-2007.) $)
    dff3 $p |- ( F : A --> B <->
              ( F C_ ( A X. B ) /\ A. x e. A E! y x F y ) ) $=
      ( vz cxp wss cv wbr weu wa wcel wex cop sylibr adantr sylanbrc wi syl6 wf
      wral fssxp wmo cfv wfun cdm ffun fdm eleq2d biimpar funfvop syl2an2r fvex
      df-br breq2 spcev syl funmo df-eu ralrimiva jca wfn crn wceq wrel wal cvv
      xpss sstr mpan2 df-rel df-ral eumo imim2i adantl wn ssel biimtrid opelxp1
      exlimdv con3d nexmo pm2.61d ex alimdv imp dffun6 dmss dmxpss sstrdi breq1
      weq eubidv rspccv euex vex eldm ssrdv anim12i eqss df-fn rnss rnxpss df-f
      impbii ) CDEUAZECDGZHZAIZBIZEJZBKZACUBZLZXGXIXNCDEUCXGXMACXGXJCMZLZXLBNZX
      LBUDZXMXQXJXJEUEZEJZXRXQXJXTOEMZYAXGEUFZXPXJEUGZMZYBCDEUHZXGYEXPXGYDCXJCD
      EUIUJUKXJEULUMXJXTEUOPXLYABXTXJEUNXKXTXJEUPUQURXGXSXPXGYCXSYFBXJEUSURQXLB
      UTRVAVBXOECVCZEVDZDHZXGXOYCYDCVEZYGXOEVFZXSAVGZYCXIYKXNXIEVHVHGZHZYKXIXHY
      MHYNCDVIEXHYMVJVKEVLPQXIXNYLXNXPXMSZAVGXIYLXMACVMXIYOXSAXIYOXSXIYOLXPXSYO
      XPXSSXIXMXSXPXLBVNVOVPXIXPVQZXSSYOXIYPXRVQXSXIXRXPXIXLXPBXIXLXJXKOZXHMZXP
      XLYQEMXIYRXJXKEUOEXHYQVRVSXJXKCDVTTWAWBXLBWCTQWDWEWFVSWGABEWHRXOYDCHZCYDH
      ZLYJXIYSXNYTXIYDXHUGCEXHWICDWJWKXNFCYDXNFIZCMUUAXKEJZBKZUUAYDMZXMUUCAUUAC
      AFWMXLUUBBXJUUAXKEWLWNWOUUCUUBBNUUDUUBBWPBUUAEFWQWRPTWSWTYDCXAPECXBRXIYIX
      NXIYHXHVDDEXHXCCDXDWKQCDEXERXF $.

    $( Alternate definition of a mapping.  (Contributed by NM, 20-Mar-2007.) $)
    dff4 $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 dff3 wcel cop df-br ssel opelxp2
      syl6 biimtrid pm4.71rd eubidv df-reu bitr4di ralbidv pm5.32i bitri ) 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 ffvelcdm adantr eqeltrd rexlimdva2 biantrurd bitr4id
      simpr albidv eqabcb df-ral 3bitr4g bitrd pm5.32i bitri ) CDEFCDEGZEHZDIZJ
      VABKZAKZELZIZACMZBDNZJCDEOVAVCVIVAVCVHBPZDIZVIVAECUAZVCVKQCDEUCVLVBVJDABC
      EUBUDUEVAVHVDDRZQZBSVMVHTZBSVKVIVAVNVOBVAVNVHVMTZVOJVOVHVMUFVAVPVOVAVGVMA
      CVAVECRJZVGJVDVFDVQVGUMVQVFDRVGCDVEEUGUHUIUJUKULUNVHBDUOVHBDUPUQURUSUT $.

    $( 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 $.

    $( A relation equivalent to the existence of an onto mapping.  The
       right-hand ` f ` is not necessarily a function.  (Contributed by NM,
       20-Mar-2007.) $)
    exfo $p |- ( E. f f : A -onto-> B <->
           E. f ( A. x e. A E! y e. B x f y /\ A. x e. B E. y e. A y f x ) ) $=
      ( vg cv wfo wex wbr wreu wral wrex wa wf dffo4 wss dff4 sylibr foeq1 wceq
      cxp simprbi anim1i sylbi eximi cin crn wcel brinxp reubidva ralimia inss2
      biimpd jctil rninxp biimpri anim12i dffo2 vex inex1 spcev exlimiv cbvexvw
      syl sylib impbii ) CDEGZHZEIZAGZBGZVHJZBDKZACLZVLVKVHJBCMADLZNZEIZVIVQEVI
      CDVHOZVPNVQBACDVHPVSVOVPVSVHCDUBZQVOABCDVHRUCUDUEUFVRCDFGZHZFIZVJVQWCEVQC
      DVHVTUGZHZWCVQCDWDOZWDUHDUAZNWEVOWFVPWGVOWDVTQZVKVLWDJZBDKZACLZNWFVOWKWHV
      NWJACVKCUIZVNWJWLVMWIBDVKVLCDVHUJUKUNULVHVTUMUOABCDWDRSWGVPBACDVHUPUQURCD
      WDUSSWBWEFWDVHVTEUTVACDWAWDTVBVEVCWBVIFECDWAVHTVDVFVG $.
  $}

  ${
    $d A w x y $.  $d B x y $.  $d F w y $.
    dffo3f.1 $e |- F/_ x F $.
    $( An onto mapping expressed in terms of function values.  As ~ dffo3 but
       with less disjoint vars constraints.  (Contributed by Glauco Siliprandi,
       17-Aug-2020.) $)
    dffo3f $p |- ( F : A -onto-> B
                  <-> ( F : A --> B /\ A. y e. B E. x e. A y = ( F ` x ) ) ) $=
      ( vw wfo wceq wa cv cfv wrex cab wb nfcv nfv wcel wal wi wf crn dffo2 wfn
      wral ffn fnrnfv nffv nfeq2 fveq2 eqeq2d cbvrexw abbii eqtrdi eqeq1d dfbi2
      syl nff simpr ffvelcdm adantr eqeltrd exp31 rexlimd bitr4id albidv eqabcb
      biantrurd df-ral 3bitr4g bitrd pm5.32i bitri ) CDEHCDEUAZEUBZDIZJVNBKZAKZ
      ELZIZACMZBDUEZJCDEUCVNVPWBVNVPWABNZDIZWBVNECUDZVPWDOCDEUFWEVOWCDWEVOVQGKZ
      ELZIZGCMZBNWCGBCEUGWIWABWHVTGACAVQWGAWFEFAWFPUHUIVTGQWFVRIWGVSVQWFVREUJUK
      ULUMUNUOUQVNWAVQDRZOZBSWJWATZBSWDWBVNWKWLBVNWKWAWJTZWLJWLWAWJUPVNWMWLVNVT
      WJACACDEFACPADPURWJAQVNVRCRZVTWJVNWNJZVTJVQVSDWOVTUSWOVSDRVTCDVREUTVAVBVC
      VDVHVEVFWABDVGWABDVIVJVKVLVM $.
  $}

  ${
    $d F x y $.  $d A x y $.  $d B x y $.  $d C x y $.
    $( Property of a surjective function.  (Contributed by Jeff Madsen,
       4-Jan-2011.) $)
    foelrn $p |- ( ( F : A -onto-> B /\ C e. B )
                                  -> E. x e. A C = ( F ` x ) ) $=
      ( vy wfo cv wceq wrex wral wcel dffo3 simprbi eqeq1 rexbidv rspccva sylan
      cfv wf ) BCEGZFHZAHESZIZABJZFCKZDCLDUCIZABJZUABCETUFAFBCEMNUEUHFDCUBDIUDU
      GABUBDUCOPQR $.
  $}

  ${
    $d A x y $.  $d B x y $.  $d C x y $.  $d F y $.
    foelrnf.1 $e |- F/_ x F $.
    $( Property of a surjective function.  As ~ foelrn but with less disjoint
       vars constraints.  (Contributed by Glauco Siliprandi, 17-Aug-2020.) $)
    foelrnf $p |- ( ( F : A -onto-> B /\ C e. B ) -> E. x e. A C = ( F ` x ) )
               $=
      ( vy wfo cv cfv wceq wrex wral wcel dffo3f simprbi eqeq1 rexbidv rspccva
      wf sylan ) BCEHZGIZAIEJZKZABLZGCMZDCNDUDKZABLZUBBCETUGAGBCEFOPUFUIGDCUCDK
      UEUHABUCDUDQRSUA $.
  $}

  ${
    $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.)
       (Proof shortened by JJ, 14-Jul-2021.) $)
    foco2 $p |- ( ( F : B --> C /\ G : A --> B /\
                        ( F o. G ) : A -onto-> C ) -> F : B -onto-> C ) $=
      ( vy vx vz wf ccom wfo wa cv cfv wceq wrex wral w3a wcel foelrn rexlimdva
      ffvelcdm fvco3 fveq2 syl2anc eqeq1 rexbidv syl5ibrcom syl5 impl ralrimiva
      rspceeqv anim2i 3anass dffo3 3imtr4i ) BCDIZABEIZACDEJZKZLZLUQFMZGMZDNZOZ
      GBPZFCQZLUQURUTRBCDKVAVGUQVAVFFCURUTVBCSZVFUTVHLVBHMZUSNZOZHAPURVFHACVBUS
      TURVKVFHAURVIASLZVFVKVJVDOZGBPZVLVIENZBSVJVODNZOVNABVIEUBABVIDEUCGVOBVDVP
      VJVCVODUDULUEVKVEVMGBVBVJVDUFUGUHUAUIUJUKUMUQURUTUNGFBCDUOUP $.
  $}

  ${
    $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 bitrdi cbveuw brcnv eubii df-reu ralbii
      opabidw wrel cdm wss relcnv crn df-rn frn relssres sylancr fneq1d bitr3id
      eqsstrrid bitr4d pm5.32i f1of pm4.71ri fmpt anbi1i ) CDFUAZCDFUBZIWIBJZEK
      ZACUCZBDLZIWJEDMACLZWNIWIWJWNWIWJFUDZDNZWNWIFCNZWJWQUECDFUFWJWRWQCDFUGUHU
      IWNWPDUJZDNZWIWQWTWKHJZWPOZHPZBDLWNBHDWPUKXCWMBDXAWKFOZHPAJZCMWLIZAPXCWMX
      DXFHAAXAWKFAXAQAFACEULZGACEUMUNAWKQURXFHUOXAXEKXDXEWKFOZXFXAXEWKFUPXHXEWK
      XFABUQZOZXFXEWKFXIFXGXIGABCEUSUTVAXJXEWKVBXIMXFXEWKXIVCXFABVJRRVDVEXBXDHW
      KXAFBSHSVFVGWLACVHTVIRWIDWSWPWIWPVKWPVLZDVMWSWPKFVNWIXKFVODFVPCDFVQWBWPDV
      RVSVTWAWCWDWJWICDFWEWFWOWIWNACDEFGWGWHT $.

    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 A x y $.  $d B x y $.  $d C y $.  $d F y $.
    fompt.1 $e |- F = ( x e. A |-> C ) $.
    $( Express being onto for a mapping operation.  (Contributed by Glauco
       Siliprandi, 17-Aug-2020.) $)
    fompt $p |- ( F : A -onto-> B
                     <-> ( A. x e. A C e. B /\ A. y e. B E. x e. A y = C ) ) $=
      ( wcel wral cv wceq wrex nfcv simpr syl2anc adantr eqtrd reximdai nfra1
      wa wfo wf fof fmpt sylibr cfv cmpt nfmpt1 nfcxfr foelrnf wi nffo r19.21bi
      fvmpt2 exp31 mpd ralrimiva jca biimpi nfv nfan simpll rspa adantll eqcomd
      w3a simp3 3adant3 3exp sylc ralrimia dffo3f sylanbrc impbii ) CDFUAZEDHZA
      CIZBJZEKZACLZBDIZTZVOVQWAVOCDFUBZVQCDFUCACDEFGUDZUEZVOVTBDVOVRDHZTVRAJZFU
      FZKZACLZVTACDVRFAFACEUGGACEUHUIZUJVOWJVTUKWFVOWIVSACACDFWKACMADMULVOWGCHZ
      WIVSVOWLTZWITVRWHEWMWINWMWHEKZWIWMWLVPWNVOWLNVOVPACWEUMACEDFGUNZOPQUORPUP
      UQURWBWCWJBDIVOVQWCWAVQWCWDUSPWBWJBDVQWABVQBUTVTBDSVAWBWFTVQVTWJVQWAWFVBW
      AWFVTVQVTBDVCVDVQVSWIACVPACSVQWLVSWIVQWLVSVFVREWHVQWLVSVGVQWLEWHKVSVQWLTZ
      WHEWPWLVPWNVQWLNVPACVCWOOVEVHQVIRVJVKABCDFWKVLVMVN $.
  $}

  ${
    $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
      cbvralfw anbi2i bitri ) BCDIDBJZHKZDLZCMZHBNZOUGAKZDLZCMZABNZOHBCDPUKUOUG
      UJUNHABHBQEAUICAUHDGAUHQRFSUNHTHAUAUIUMCUHULDUBUCUDUEUF $.
  $}

  ${
    $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 A k $.  $d F k $.  $d V k $.
    $( A function is a function into a subset of its codomain if all of its
       values are elements of this subset.  (Contributed by AV, 7-Feb-2021.) $)
    fcdmssb $p |- ( ( V C_ W /\ A. k e. A ( F ` k ) e. V )
                   -> ( F : A --> W <-> F : A --> V ) ) $=
      ( wss cv cfv wcel wa wf wfn simpr ffn anim12ci ffnfv sylibr simpl anim1ci
      wral fss syl impbida ) DEFZBGCHDIBATZJZAECKZADCKZUFUGJCALZUEJUHUFUEUGUIUD
      UEMAECNOBADCPQUFUHJUHUDJUGUFUDUHUDUERSADECUAUBUC $.
  $}

  ${
    $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 C x $.  $d D x $.  $d F x $.
    fssrescdmd.f $e |- ( ph -> F : A --> B ) $.
    fssrescdmd.c $e |- ( ph -> C C_ A ) $.
    fssrescdmd.d $e |- ( ph -> ( F " C ) C_ D ) $.
    $( Restriction of a function to a subclass of its domain as a function with
       domain and codomain.  (Contributed by AV, 13-May-2025.) $)
    fssrescdmd $p |- ( ph -> ( F |` C ) : C --> D ) $=
      ( vx cres wfn cv cfv cima wss cdm wb wceq mpbid wcel wral fnssresd resima
      ffnd eqsstrid wfun ffund funresd fdmd sseqtrrd ssdmres a1i bitrdi eqimssd
      wf eqcom funimass4 syl2anc ffnfv sylanbrc ) AFDKZDLJMVBNEUAJDUBZDEVBUPABD
      FABCFGUEHUCAVBDOZEPZVCAVDFDOEFDUDIUFAVBUGDVBQZPVEVCRADFABCFGUHUIADVFADFQZ
      PZDVFSZADBVGHABCFGUJUKAVHVFDSZVIVHVJRADFULUMVFDUQUNTUOJDEVBURUSTJDEVBUTVA
      $.
  $}

  ${
    $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 wfun 3syl wcel wb f1ofun funcnvcnv wf1 f1ocnv f1of1
      cima wss ssrab2 f1ores sylancl mptpreima cv wa wceq wi wal 3expia alrimiv
      wf wral f1of fmpt sylibr r19.21bi elrab3t syl2anc rabbidva eqtrid f1oeq3d
      syl mpbid f1orescnv rescnvcnv f1oeq1 ax-mp sylib ) ABDFMZCEGMZINZNZVQOZPZ
      VQVRIVQOZPZAVTQZVRVQVSVROZPZWBAFGIPZIQWEKFGIUAIUBRAVRVSVRUFZWFPZWGAGFVSUC
      ZVRGUGWJAWHGFVSPWKKFGIUDGFVSUERCEGUHGFVRVSUIUJAWIVQVRWFAWIHVRSZDFMVQDFHVR
      IJUKAWLBDFADULFSZUMZEULHUNZCBTZUOZEUPHGSZWLBTWNWQEAWMWOWPLUQURAWRDFAFGIUS
      ZWRDFUTAWHWSKFGIVAVJDFGHIJVBVCVDCBEHGVEVFVGVHVIVKVQVRVSVLVFWAWCUNWBWDTIVQ
      VMVQVRWAWCVNVOVP $.
  $}

  ${
    $d A w x $.  $d B y $.  $d C x y $.  $d X x $.  $d Y x y $.  $d ph x y $.
    $d ps x $.  $d ch x y $.  $d ta x $.  $d w x y $.
    f1ossf1o.x $e |- X = { w e. A | ( ps /\ ch ) } $.
    f1ossf1o.y $e |- Y = { w e. A | ps } $.
    f1ossf1o.f $e |- F = ( x e. X |-> B ) $.
    f1ossf1o.g $e |- G = ( x e. Y |-> B ) $.
    f1ossf1o.b $e |- ( ph -> G : Y -1-1-onto-> C ) $.
    f1ossf1o.s $e |- ( ( ph /\ x e. Y /\ y = B )
                       -> ( ta <-> [ x / w ] ch ) ) $.
    $( Restricting a bijection, which is a mapping from a restricted class
       abstraction, to a subset is a bijection.  (Contributed by AV,
       7-Aug-2022.) $)
    f1ossf1o $p |- ( ph -> F : X -1-1-onto-> { y e. C | ta } ) $=
      ( crab wf1o wsb cres f1oresrab cmpt wss wa wi cv wcel a1i ss2rabi 3sstr4i
      simpl resmptd wceq rabeqi nfs1v sbequ12 elrabf anbi1i anass bitri rabbia2
      nfcv nfan anbi12d cbvrabw eqtr2i 3eqtri reseq12d 3eqtr4rd eqidd f1oeq123d
      nfv mpbird ) AMDFJUAZKUBCGEUCZENUAZVRLVTUDZUBAVSDEFNJILRSTUEAMVTVRVRKWAAE
      NIUFZMUDEMIUFZWAKAENMIMNUGABCUHZGHUAZBGHUAZMNWDBGHWDBUIGUJZHUKBCUOULUMOPU
      NULUPALWBVTMLWBUQARULVTMUQAVTVSEWFUAZBGEUCZVSUHZEHUAZMVSENWFPURZVSWJEWFHE
      UJZWFUKZVSUHWMHUKZWIUHZVSUHWOWJUHWNWPVSBWIGWMHGWMVFGHVFZBGEUSZBGEUTZVAVBW
      OWIVSVCVDVEZMWEWKOWDWJGEHWQEHVFWDEVPWIVSGWRCGEUSVGWGWMUQBWICVSWSCGEUTVHVI
      ZVJVKULVLKWCUQAQULVMMVTUQAMWEWKVTOXAVTWHWKWLWTVJVKULAVRVNVOVQ $.
  $}

  ${
    $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 mptrel wex csb cop wfun
      cfv fmpt3d ffund funbrfv imp sylan eqcomd a1d expimpd pm4.71rd fvex breq2
      exbidv breq1 anbi12d ceqsexv cdm wb funfvbrb syl fdmd eleq2d bitr3d eqidd
      fveq1d breq123d wi nfcv nfv nffvmpt1 nfbr nfcsb1v nfeq2 nfbi fveq2 breq1d
      nfim csbeq1a eqeq2d bibi12d imbi2d cvv vex simpl eleq1d eqeqan12rd df-mpt
      id brabga sylancl fvmpt2 syl2an2 biantrurd 3bitr4d expcom vtoclgaf impcom
      eqid pm5.32da bitrd bitrid opelco copab eleq2i nfan eqeq1 opelopabf bitri
      eleq1w anbi2d 3bitr4g eqrelrdv ) AUAUBJIUDZBDHUEZJIUFBDHUGAUAPZOPZIQZYGUB
      PZJQZRZOUHZYFDSZYIBYFHUIZTZRZYFYIUJZYDSYQYESZAYLYGYFIULZTZYKRZOUHZYPAYKUU
      AOAYKYTAYHYJYTAYHRZYTYJUUCYSYGAIUKZYHYSYGTZADEIABDFEILKUMZUNZUUDYHUUEYFYG
      IUOUPUQURUSUTVAVDUUBYFYSIQZYSYIJQZRZAYPYKUUJOYSYFIVBYTYHUUHYJUUIYGYSYFIVC
      YGYSYIJVEVFVGAUUJYMYFBDFUEZULZYICEGUEZQZRYPAUUHYMUUIUUNAYFIVHZSZUUHYMAUUD
      UUPUUHVIUUGYFIVJVKAUUODYFADEIUUFVLVMVNAYSUULYIYIJUUMAYFIUUKLVPMAYIVOVQVFA
      YMUUNYOYMAUUNYOVIZABPZUUKULZYIUUMQZYIHTZVIZVRAUUQVRBYFDBYFVSAUUQBABVTUUNY
      OBBUULYIUUMBDFYFWABUUMVSBYIVSWBBYIYNBYFHWCZWDWEWHUURYFTZUVBUUQAUVDUUTUUNU
      VAYOUVDUUSUULYIUUMUURYFUUKWFWGUVDHYNYIBYFHWIZWJWKWLAUURDSZUVBAUVFRZFYIUUM
      QZFESZUVARZUUTUVAUVGUVIYIWMSUVHUVJVIKUBWNZCPZESZYGGTZRUVJCOFYIUUMEWMUVLFT
      ZYGYITZRZUVMUVIUVNUVAUVQUVLFEUVOUVPWOWPUVPUVOYGYIGHUVPWSNWQVFCOEGWRWTXAUV
      GUUSFYIUUMUVFUVFAUVIUUSFTUVFWSKBDFEUUKUUKXIXBXCWGUVGUVIUVAKXDXEXFXGXHXJXK
      XLXKOYFYIJIUAWNZUVKXMYRYQUVFUCPZHTZRZBUCXNZSYPYEUWBYQBUCDHWRXOUWAYMUVSYNT
      ZRYPBUCYFYIYMUWCBYMBVTBUVSYNUVCWDXPYPUCVTUVRUVKUVDUVFYMUVTUWCBUADXTUVDHYN
      UVSUVEWJVFUVSYITUWCYOYMUVSYIYNXQYAXRXSYBYC $.
  $}

  ${
    $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 nfcsbw csbeq1d eqtr4di csbiegf
        eqid nfcvd ralimi mpteq12 sylancr syl eqtrd ) AJIUAZBDCFGQZRZBDHRZAVFOD
        CBOUBZFQZGQZRVHAOPDEVKCPUBZGQZVLIJAFEUCZBDUDZVJDUCVKEUCZKVOVQBVJDBVKEBV
        JFUEZUFBUBVJSZFVKEBVJFUGZUIUHUJAIBDFRODVKRLBODFVKOFTVRVTUKULAJCEGRPEVNR
        MCPEGVNPGTCVMGUECVMGUGUKULCVMVKGUMUNBODVGVLOVGTBCVKGVRBGTUOVSCFVKGVTUPU
        KUQAVPVHVISZKVPDDSVGHSZBDUDWADUSVOWBBDCFGHEVOCHUTNURVABDVGDHVBVCVDVE $.
    $}

    $( 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 dffn5 sylib
      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 bitr2i wrel opelf
      wf anbi12i sylib ex wreu snid feu mpan2 anbi1i opeq2 biancomi eubii eueqi
      pm5.32i biantru euanv bitr4i df-reu 3bitr4i sylibr opeq12 syl5ibrcom opex
      impbid elsn opth2 bitrdi alrimivv frel relsnop eqrel sylancl mpbird f1osn
      wf1o f1oeq1 mpbiri f1of syl impbii ) AHZBHZCUBZCABIZHZJZWDWGFKZGKZIZCLZWJ
      WFLZMZGNFNZWDWMFGWDWKWHAJZWIBJZOZWLWDWKWQWDWKWQWDWKOWHWBLZWIWCLZOWQWBWCWH
      WICUAWRWOWSWPFAPGBPZUCUDUEWDWKWQWECLZWDAWIIZCLZGWCUFZXAWDAWBLXDADUGGWBWCA
      CUHUIXAWPOZGQZWSXCOZGQXAXDXEXGGXGWPXCOZXEWSWPXCWTUJXHXAWPWPXCXAWPXBWECWIB
      AUKRUOULSUMXAXAWPGQZOXFXIXAGBEUNUPXAWPGUQURXCGWCUSUTVAWQWJWECWHWIABVBRVCV
      EWLWJWEJWQWJWEWHWIVDVFWHWIABDEVGSVHVIWDCTWFTWGWNMWBWCCVJABDEVKFGCWFVLVMVN
      WGWBWCCVPZWDWGXJWBWCWFVPABDEVOWBWCCWFVQVRWBWCCVSVTWA $.
  $}

  ${
    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 wcel wa cop wceq snid ffvelcdm mpan2 wfn ffn crn biimpi cima
      dffn3 cdm imadmrn imaeq2d eqtr3id fnsnfv eqtr4d feq3d mpbid syl jca snssi
      fndm wss fss ancoms sylan impbii fvex fsn anbi2i bitri ) AEZBCFZACGZBHZVB
      VDEZCFZIZVECAVDJEKZIVCVHVCVEVGVCAVBHZVEADLZVBBACMNVCCVBOZVGVBBCPVLVBCQZCF
      ZVGVLVNVBCTRVLVMVFCVBVLVMCVBSZVFVLVMCCUAZSVOCUBVLVPVBCVBCULUCUDVLVJVFVOKV
      KVBACUENUFUGUHUIUJVEVFBUMZVGVCVDBUKVGVQVCVBVFBCUNUOUPUQVGVIVEAVDCDACURUSU
      TVA $.
  $}

  ${
    $d A a b $.  $d B a 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
      feq3d opeq2 fsn vtocl2g ) FHZIZGHZIZEJZEUEUGKZIZLZMAIZUHEJZEAUGKZIZLZMUMB
      IZEJZEABKZIZLZMFGABCDUEALZUIUNULUQVCUFUMUHEUEANOVCUKUPEVCUJUOUEAUGPQRSUGB
      LZUNUSUQVBVDUHUREUMUGBNUAVDUPVAEVDUOUTUGBAUBQRSUEUGEFTGTUCUD $.

    $( 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 Cartesian product of two singletons is the singleton consisting in the
     associated ordered pair.  (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 $.

  $( The Cartesian product of an unordered pair and a singleton.  (Contributed
     by AV, 20-May-2019.) $)
  xpprsng $p |- ( ( A e. V /\ B e. W /\ C e. U ) ->
    ( { A , B } X. { C } ) = { <. A , C >. , <. B , C >. } ) $=
    ( wcel w3a cpr csn cxp cun df-pr xpeq1i wceq xpsng 3adant2 3adant1 uneq12d
    cop xpundir 3eqtr4g eqtrid ) AEGZBFGZCDGZHZABIZCJZKAJZBJZLZUIKZACTZBCTZIZUH
    ULUIABMNUGUJUIKZUKUIKZLUNJZUOJZLUMUPUGUQUSURUTUDUFUQUSOUEACEDPQUEUFURUTOUDB
    CFDPRSUJUKUIUAUNUOMUBUC $.

  ${
    xpsn.1 $e |- A e. _V $.
    xpsn.2 $e |- B e. _V $.
    $( The Cartesian product of two singletons is the singleton consisting in
       the associated ordered pair.  (Contributed by NM, 4-Nov-2006.) $)
    xpsn $p |- ( { A } X. { B } ) = { <. A , B >. } $=
      ( cvv wcel csn cxp cop wceq xpsng mp2an ) AEFBEFAGBGHABIGJCDABEEKL $.
  $}

  $( A singleton consisting in a nested ordered pair is a one-to-one function
     from the cartesian product of two singletons onto a singleton (case where
     the two singletons are equal).  (Contributed by AV, 15-Aug-2019.) $)
  f1o2sn $p |- ( ( E e. V /\ X e. W )
       -> { <. <. E , E >. , X >. } : ( { E } X. { E } ) -1-1-onto-> { X } ) $=
    ( wcel wa cop csn wf1o cxp cvv opex simpr f1osng sylancr wceq anidms eqcomd
    xpsng adantr f1oeq2d mpbid ) ABEZDCEZFZAAGZHZDHZUFDGHZIZAHZUKJZUHUIIUEUFKEU
    DUJAALUCUDMUFDKCNOUEUGULUHUIUCUGULPUDUCULUGUCULUGPAABBSQRTUAUB $.

  $( Restriction of the identity to a pair.  (Contributed by AV,
     11-Dec-2018.) $)
  residpr $p |- ( ( A e. V /\ B e. W ) -> ( _I |` { A , B } )
                                          = { <. A , A >. , <. B , B >. } ) $=
    ( wcel cid cpr cres csn cun cop df-pr reseq2i resundi cxp wceq xpsng anidms
    wa restidsing eqtri adantr adantl uneq12d uneq12i 3eqtr4g eqtrid ) ACEZBDEZ
    SZFABGZHZFAIZHZFBIZHZJZAAKZBBKZGZULFUMUOJZHUQUKVAFABLMFUMUONUAUJUMUMOZUOUOO
    ZJURIZUSIZJUQUTUJVBVDVCVEUHVBVDPZUIUHVFAACCQRUBUIVCVEPZUHUIVGBBDDQRUCUDUNVB
    UPVCATBTUEURUSLUFUG $.

  ${
    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 $.
  $}

  ${
    $d A x $.  $d R x $.
    $( Two ways to state that a relation is reflexive on a class.  (Contributed
       by FL, 15-Jan-2012.)  (Proof shortened by Mario Carneiro, 3-Nov-2015.)
       (Revised by NM, 30-Mar-2016.) $)
    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
      df-f mpbiran bitri df-br ralbii mptresid vex fnasrn eqtri sseq1i 3bitr4ri
      ) ADZUKEZCFZABGZABULHZIZCJZUKUKCKZABGLBMZCJUNBCUONZUQABCULUOUOOZPUTUOBQUQ
      ABULUOUKUKRVASBCUOTUAUBURUMABUKUKCUCUDUSUPCUSABUKHUPABUEABUKAUFUGUHUIUJ
      $.
  $}

  ${
    $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 dffn5 sylbb fvex dfmpt
      eqtrdi ) BCZBABDZAEZBFZGZATUAUBHIJSBTKBUCLBMATBNOATUBUABPQR $.
  $}

  ${
    $d F a x $.  $d X a x $.  $d Y a x $.
    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 >. } ) ) $=
      ( vx cop wceq cv cfv csn ciun wa wex wb adantl c0 wne wi cdm funiun eqeq1
      wfun eqcom bitrdi opnzi neeq1 eqcoms wrel funrel reldm0 syl biimprd com12
      necon3d biimtrdi com3l impd ax-mp fvex iunopeqop sylbid imp iuneq1 vex id
      weq fveq2 opeq12d sneqd iunxsn eqtrdi eqeq2d snopeqop sylbb simpr3 eqcomd
      w3a simp1 opeq2d biimpac jca ex a1dd mpd impancom eximdv mpidan ) AUDZABC
      HZIZAGAUAZGJZWNAKZHZLZMZIZBDJZLZIZAWTWTHZLZIZNZDOZGAUBWJWLNZWSNZWMXAIZDOZ
      XGXHWSXKXHWSWRWKIZXKWLWSXLPWJWLWSWKWRIXLAWKWRUCWKWRUEUFQXHWMRSZXLXKTWKRSZ
      XHXMTBCEFUGXNWJWLXMWLXNWJXMWLXNARSZWJXMTXNXOPWKAWKARUHUIWJXOXMWJWMRARWJAR
      IZWMRIZWJAUJXPXQPAUKAULUMUNUPUOUQURUSUTGDWMWOBCWNAVAEFVBUMVCVDXIXJXFDXHXJ
      WSXFXHXJNZWSAWTWTAKZHZLZIZXFXRWRYAAXJWRYAIXHXJWRGXAWQMYAGWMXAWQVEGWTWQYAD
      VFZGDVHZWPXTYDWNWTWOXSYDVGWNWTAVIVJVKVLVMQVNXHYBXJXFXHYBNZWTXSIZBCIZXBVSZ
      XJXFTXHYBYHXHYBWKYAIZYHWLYBYIPWJAWKYAUCQYIYAWKIYHWKYAUEWTXSBCYCWTAVAVOVPU
      QVDYEYHXFXJYBYHXFTXHYBYHXFYBYHNXBXEYBYFYGXBVQYHYBXEYHYAXDAYHXTXCYHXSWTWTY
      HWTXSYFYGXBVTVRWAVKVNWBWCWDQWEWFWGVCWGWHWFWI $.

    $( 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 $.
  $}

  ${
    $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 $.
  $}

  ${
    $d A a x y $.  $d B a x y $.  $d G a x y $.
    funsndifnop.a $e |- A e. _V $.
    funsndifnop.b $e |- B e. _V $.
    funsndifnop.g $e |- G = { <. A , B >. } $.
    $( A singleton of an ordered pair is not an ordered pair if the components
       are different.  (Contributed by AV, 23-Sep-2020.)  (Avoid depending on
       this detail.) $)
    funsndifnop $p |- ( A =/= B -> -. G e. ( _V X. _V ) ) $=
      ( vx vy va cvv cv cop wceq wex wfun csn funeq ax-mp wa wi wcel elvv funsn
      cxp mpbiri vex funop eqeq2 eqeq1 opex sneqr opth eqtr3 a1d sylbi biimtrdi
      bitrdi syl com23 impcom exlimiv com12 sylbid mpi exlimivv necon3ai ) CJJU
      DUAZABVGCGKZHKZLZMZHNGNABMZGHCUBVKVLGHVKCOZVLCABLZPZMZVMFVPVMVOOABDEUCCVO
      QUERVKVMVHIKZPMZVJVQVQLZPZMZSZINZVLVKVMVJOWCCVJQVHVIIGUFHUFUGUQWCVKVLWBVK
      VLTZIWAVRWDWAVKVRVLWAVKCVTMZVRVLTZVJVTCUHVPWEWFTFVPWEVOVTMZWFCVOVTUIWGVNV
      SMZWFVNVSABUJUKWHAVQMBVQMSZWFABVQVQDEULWIVLVRABVQUMUNUOURUPRUPUSUTVAVBVCV
      DVEUOVF $.

    $( A singleton of an ordered pair is an ordered pair iff the components are
       equal.  (Contributed by AV, 24-Sep-2020.)  (Avoid depending on this
       detail.) $)
    funsneqopb $p |- ( A = B <-> G e. ( _V X. _V ) ) $=
      ( wceq cvv cxp wcel csn cop opeq1 sneqd snopeqopsnid eqtrdi eqtrid opelvv
      snex eqeltrdi funsndifnop necon4ai impbii ) ABGZCHHIZJZUDCBKZUGLZUEUDCABL
      ZKZUHFUDUJBBLZKUHUDUIUKABBMNBEOPQUGUGBSZULRTUFABABCDEFUAUBUC $.
  $}

  $( 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 ) = (/) ) $=
    ( cop cvv cxp wcel csn cres c0 wceq opelxp1 cin df-res incom disjsn biimpri
    wn eqtri eqtrid nsyl5 ) ABDZCEFZGZACGUBHZCIZJKABCELUDRZUFUCUEMZJUFUEUCMUHUE
    CNUEUCOSUHJKUGUCUBPQTUA $.

  ${
    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 wss wf sylanblrc csn cun df-pr rnsnop wfun cdm wceq
      funpr dmprop df-fn rneqi rnun uneq12i eqtr4i 3eqtri eqimssi df-f ) ABIZAC
      JZBDJZKZABKZLZUQMZCDKZNURVAUQOUNUQUAUQUBURUCUSABCDEFGHUDACBDGHUEUQURUFPUT
      VAUTUOQZUPQZRZMVBMZVCMZRZVAUQVDUOUPSUGVBVCUHVGCQZDQZRVAVEVHVFVIACETBDFTUI
      CDSUJUKULURVAUQUMP $.
  $}

  $( 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 cpr cop wf cvv wi elex c0 cif wceq 0ex anim12i neeq1 opeq1 preq1d
    wne preq1 feq12d imbi12d neeq2 preq2d preq2 opeq2 feq123d imbi2d elimel fpr
    eqidd dedth4h syl2an 3impia ) AEIZBFIZJZCGIZDHIZJZABUEZABKZCDKZACLZBDLZKZMZ
    VCANIZBNIZJCNIZDNIZJVGVMOZVFVAVNVBVOAEPBFPUAVDVPVEVQCGPDHPUAVNVOVPVQVRVNAQR
    ZBUEZVSBKZVIVSCLZVKKZMZOVSVOBQRZUEZVSWEKZVIWBWEDLZKZMZOWFWGVPCQRZDKZVSWKLZW
    HKZMZOWFWGWKVQDQRZKZWMWEWPLZKZMZOABCDQQQQAVSSZVGVTVMWDAVSBUBXAVHWAVIVLWCXAV
    JWBVKAVSCUCUDAVSBUFUGUHBWESZVTWFWDWJBWEVSUIXBWAWGVIWCWIXBVKWHWBBWEDUCUJBWEV
    SUKUGUHCWKSZWJWOWFXCWGWGVIWLWIWNXCWBWMWHCWKVSULUDXCWGUQCWKDUFUMUNDWPSZWOWTW
    FXDWGWGWLWQWNWSXDWHWRWMDWPWEULUJXDWGUQDWPWKUKUMUNVSWEWKWPAQNTUOBQNTUOCQNTUO
    DQNTUOUPURUSUT $.

  $( 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 fsng syl mpbird elpri eqcom nne orbi12i
    bitr4i ianor sylbb2 con2i 3adant1 3ad2ant3 disjsn sylibr fun syl21anc df-tp
    wo feq1i feq23i bitri ) JDMZKHMZLIMZNZAEMZBFMZCGMZNZJKOZJLOZKLOZNZNZJKPZLUA
    ZUBZABPZCUAZUBZJAUCZKBUCZPZLCUCZUAZUBZQZJKLUDZABCUDZWQWRWTUDZQZWJWKWNWSQZWL
    WOXAQZWKWLUEUFRZXCWAVRVSSWEWBWCSWIWFXHVRVSVTUGWBWCWDUGWFWGWHUHJKABDHEFUJUIW
    JXIXAXARZWJXAUKWJVTWDSZXIXKULWAWEXLWIWAVTWEWDVRVSVTUMWBWCWDUMUNUOLCIGXAUPUQ
    URWJLWKMZTZXJWIWAXNWEWGWHXNWFXMWGWHSZXMLJRZLKRZVNZXOTZLJKUSXRWGTZWHTZVNXSXP
    XTXQYAXPJLRXTLJUTJLVAVCXQKLRYALKUTKLVAVCVBWGWHVDVEUQVFVGVHWKLVIVJWKWLWNWOWS
    XAVKVLXGXDXEXBQXCXDXEXFXBWQWRWTVMVOXDXEWMWPXBJKLVMABCVMVPVQVJ $.

  ${
    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 wf fsn2 biantrur
      dffn2 vsnid fvres ax-mp opeq2i eqeq2i bitr3i 3bitri expcom vtoclga impcom
      fvex sneqi sylib ) BAECAFZCBGZHZBBCIZJZGZKZVFCDLZGZHZVMVMCIZJZGZKZMVFVLMD
      BAVMBKZVSVLVFVTVOVHVRVKVTVNVGCVMBNOVTVQVJVTVPVIKVQVJKVMBCPVMVPBVIQRUAUBUC
      VFVMAEZVSVFWASVOVNFZVSWAVFVNAUDWBVMADUEZUFAVNCUGUHWBVNTVOUIVMVOIZTEZVOVMW
      DJZGZKZSZVSVNVOULVMTVOWCUJWIWHVSWEWHVMVOVCUKWGVRVOWFVQWDVPVMVMVNEWDVPKDUM
      VMVNCUNUOUPVDUQURUSVEUTVAVB $.

    $( A function restricted to a singleton.  (Contributed by Mario Carneiro,
       16-Nov-2014.) $)
    funressn $p |- ( Fun F -> ( F |` { B } ) C_ { <. B , ( F ` B ) >. } ) $=
      ( wfun cdm wcel csn cres cfv cop wss wceq wfn funfn fnressn sylanb eqimss
      wa syl wn c0 cin disjsn fnresdisj sylbi bitr3id biimpa eqsstrdi pm2.61dan
      wb 0ss ) BCZABDZEZBAFZGZAABHIFZJZUKUMQUOUPKZUQUKBULLZUMURBMZULABNOUOUPPRU
      KUMSZQUOTUPUKVAUOTKZVAULUNUATKZUKVBULAUBUKUSVCVBUIUTULUNBUCUDUEUFUPUJUGUH
      $.

    $( 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 $.
  $}

  ${
    $d F x $.  $d X x $.
    $( If the value of a function is in the range of the function restricted to
       the singleton containing the argument, then the value of the function is
       in the range of the function.  (Contributed by Alexander van der Vekens,
       22-Jul-2018.) $)
    fvrnressn $p |- ( X e. V -> ( ( F ` X ) e. ran ( F |` { X } )
                                  -> ( F ` X ) e. ran F ) ) $=
      ( vx cfv csn cres crn wcel cima df-ima eleq2i cop cv wex opeq1 eleq1d cvv
      wceq wb spcegv fvex elimasng mpan2 elrn2g mp1i 3imtr4d biimtrrid ) CAEZAC
      FZGHZIUIAUJJZIZCBIZUIAHIZULUKUIAUJKLUNCUIMZAIZDNZUIMZAIZDOZUMUOUTUQDCBURC
      SUSUPAURCUIPQUAUNUIRIZUMUQTCAUBZACUIBRUCUDVBUOVATUNVCDUIARUEUFUGUH $.
  $}

  $( The value of a function restricted to the singleton containing the
     argument equals the value of the function for this argument.  (Contributed
     by Alexander van der Vekens, 22-Jul-2018.) $)
  fvressn $p |- ( X e. V -> ( ( F |` { X } ) ` X ) = ( F ` X ) ) $=
    ( wcel csn snidg fvresd ) CBDCCEACBFG $.

  $( Obsolete version of ~ fvn0fvelrn as of 13-Jan-2025.  (Contributed by
     Alexander van der Vekens, 22-Jul-2018.)  (New usage is discouraged.)
     (Proof modification is discouraged.) $)
  fvn0fvelrnOLD $p |- ( ( F ` X ) =/= (/) -> ( F ` X ) e. ran F ) $=
    ( cfv c0 wne cdm wcel csn cres wfun wa crn fvfundmfvn0 wi eldmressnsn pm3.2
    fvelrn syl ex com13 mpd imp fvressn eleq1d fvrnressn sylbid impcom 3syl ) B
    ACZDEBAFZGZABHIZJZKBULCZULLZGZUKKZUIALGZBAMUKUMUQUKBULFGZUMUQNBAOUMUSUKUQUM
    USUKUQNZUMUSKUPUTBULQUPUKPRSTUAUBUKUPURUKUPUIUOGURUKUNUIUOAUJBUCUDAUJBUEUFU
    GUH $.

  $( 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
    $.

  $( If a class belongs to a function on a singleton, then that class is the
     obvious ordered pair.  Note that this theorem also holds when ` A ` is a
     proper class, but its meaning is then different.  (Contributed by Jonathan
     Ben-Naim, 3-Jun-2011.)  (Proof shortened by Mario Carneiro,
     22-Dec-2016.) $)
  fnsnr $p |- ( F Fn { A } -> ( B e. F -> B = <. A , ( F ` A ) >. ) ) $=
    ( csn wfn wcel cfv cop wceq cres fnresdm wfun fnfun funressn eqsstrrd sseld
    wss syl elsni syl6 ) CADZEZBCFBAACGHZDZFBUCIUBCUDBUBCCUAJZUDUACKUBCLUEUDQUA
    CMACNROPBUCST $.

  ${
    $d A x $.  $d F x $.  $d V x $.
    $( A function's domain is a singleton iff the function is a singleton.
       (Contributed by Steven Nguyen, 18-Aug-2023.)  Relax condition for being
       in the universal class.  (Revised by Zhi Wang, 21-Oct-2025.) $)
    fnsnbg $p |- ( A e. V ->
                        ( F Fn { A } <-> F = { <. A , ( F ` A ) >. } ) ) $=
      ( vx wcel csn wfn cfv cop wceq wa cv wi adantl wfun cdm syl5ibrcom impbid
      fnsnr cvv fnfun snidg adantr eleqtrrd funfvop syl2an2 eleq1 velsn bitr4di
      fndm eqrdv ex fvex fnsng mpan2 fneq1 ) ACEZBAFZGZBAABHZIZFZJZUQUSVCUQUSKZ
      DBVBVDDLZBEZVEVAJZVEVBEVDVFVGUSVFVGMUQAVEBSNVDVFVGVABEZUSBOUQABPZEVHURBUA
      VDAURVIUQAUREUSACUBUCUSVIURJUQURBUJNUDABUEUFVEVABUGQRDVAUHUIUKULUQUSVCVBU
      RGZUQUTTEVJABUMAUTCTUNUOURBVBUPQR $.
  $}

  ${
    $d x A $.  $d x F $.
    fnsnb.1 $e |- A e. _V $.
    $( A function whose domain is a singleton can be represented as a singleton
       of an ordered pair.  (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
       Revised to add reverse implication.  (Revised by NM, 29-Dec-2018.)
       (Proof shortened by Zhi Wang, 21-Oct-2025.) $)
    fnsnb $p |- ( F Fn { A } <-> F = { <. A , ( F ` A ) >. } ) $=
      ( cvv wcel csn wfn cfv cop wceq wb fnsnbg ax-mp ) ADEBAFGBAABHIFJKCABDLM
      $.

    $( Obsolete version of ~ fnsnb as of 21-Oct-2025.  A function whose domain
       is a singleton can be represented as a singleton of an ordered pair.
       (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)  Revised to add reverse
       implication.  (Revised by NM, 29-Dec-2018.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    fnsnbOLD $p |- ( F Fn { A } <-> F = { <. A , ( F ` A ) >. } ) $=
      ( vx csn wfn cfv cop wceq cv wcel fnsnr wfun cdm df-fn snid mpbiri anim2i
      wa eleq2 sylbi funfvop syl eleq1 syl5ibrcom velsn bitr4di eqrdv fvex fnsn
      impbid fneq1 impbii ) BAEZFZBAABGZHZEZIZUODBURUODJZBKZUTUQIZUTURKUOVAVBAU
      TBLUOVAVBUQBKZUOBMZABNZKZSZVCUOVDVEUNIZSVGBUNOVHVFVDVHVFAUNKACPVEUNATQRUA
      ABUBUCUTUQBUDUEUKDUQUFUGUHUSUOURUNFAUPCABUIUJUNBURULQUM $.
  $}

  ${
    $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 A p x y $.  $d B p y $.  $d C p x y $.  $d V p $.  $d W p $.
    fmptsng.1 $e |- ( x = A -> B = C ) $.
    $( Express a singleton function in maps-to notation.  Version of ~ fmptsn
       allowing the value ` B ` to depend on the variable ` x ` .  (Contributed
       by AV, 27-Feb-2019.) $)
    fmptsng $p |- ( ( A e. V /\ C e. W ) ->
        { <. A , C >. } = ( x e. { A } |-> B ) ) $=
      ( vy vp wcel wa cv wceq copab csn cop velsn eqidd eqeq1 adantr opabbii wb
      cmpt bicomi anbi1i eqeq2d sylan9bbr opelopabga mpbir2and eleq1 syl5ibrcom
      anbi12d biimtrid elopab wi opeq12 opeq2d opex snid eqeltrdi sylbid impcom
      wex exlimivv a1i impbid eqrdv df-mpt 3eqtr4a ) BEJDFJKZALZBMZHLZCMZKZAHNZ
      VKBOZJZVNKZAHNZBDPZOZAVQCUCZVOVSAHVLVRVNVRVLABQUDUEUAVJIWBVPVJILZWBJZWDVP
      JZWEWDWAMZVJWFIWAQVJWFWGWAVPJZVJWHBBMZDDMZVJBRVJDRVOWIWJKAHBDEFVLVMDMZKVL
      WIVNWJVLVLWIUBWKVKBBSTWKVNDCMVLWJVMDCSVLCDDGUFUGULUHUIWDWAVPUJUKUMWFWDVKV
      MPZMZVOKZHVCAVCZVJWEVOAHWDUNWOWEUOVJWNWEAHVOWMWEVOWMWDBCPZMZWEVOWLWPWDVKV
      MBCUPUFVOWEWQWPWBJVOWPWAWBVOCDBVLCDMVNGTUQWABDURUSUTWDWPWBUJUKVAVBVDVEUMV
      FVGWCVTMVJAHVQCVHVEVI $.
  $}

  ${
    $d A p x y $.  $d B p y $.  $d C p x y $.  $d V p $.  $d W p $.
    $d ph p x y $.
    fmptsnd.1 $e |- ( ( ph /\ x = A ) -> B = C ) $.
    fmptsnd.2 $e |- ( ph -> A e. V ) $.
    fmptsnd.3 $e |- ( ph -> C e. W ) $.
    $( Express a singleton function in maps-to notation.  Deduction form of
       ~ fmptsng .  (Contributed by AV, 4-Aug-2019.) $)
    fmptsnd $p |- ( ph -> { <. A , C >. } = ( x e. { A } |-> B ) ) $=
      ( vy vp cv wceq wa copab wcel cop wsbc wb csn velsn bicomi anbi1i opabbii
      cmpt eqidd sbcan eqsbc1 anbi12d bitrid sbcbidv eqeq1 adantl eqeq2d sbcied
      sbcg syl bitrd mpbir2and opelopabsb sylibr syl5ibrcom biimtrid wex elopab
      eleq1 wi opeq12 adantrr opeq2d opex snid eqeltrdi sylbid impcomd exlimdvv
      ex impbid eqrdv df-mpt a1i 3eqtr4a ) ABMZCNZKMZDNZOZBKPZWDCUAZQZWGOZBKPZC
      ERZUAZBWJDUFZWHWLBKWEWKWGWKWEBCUBUCUDUEALWOWIALMZWOQZWQWIQZWRWQWNNZAWSLWN
      UBAWSWTWNWIQZAWHKESZBCSZXAAXCCCNZEENZACUGAEUGAXCWEEDNZOZBCSXDXEOZAXBXGBCX
      BWEKESZWGKESZOAXGWEWGKEUHAXIWEXJXFAEGQZXIWETJWEKEGUQURAXKXJXFTJKEDGUIURUJ
      UKULAXGXHBCFIAWEOZWEXDXFXEWEWEXDTAWDCCUMUNXLDEEHUOUJUPUSUTWHBKCEVAVBWQWNW
      IVGVCVDWSWQWDWFRZNZWHOZKVEBVEAWRWHBKWQVFAXOWRBKAWHXNWRAWHXNWRVHAWHOZXNWQC
      DRZNZWRXPXMXQWQWHXMXQNAWDWFCDVIUNUOXPWRXRXQWOQXPXQWNWOXPDECAWEDENWGHVJVKW
      NCEVLVMVNWQXQWOVGVCVOVRVPVQVDVSVTWPWMNABKWJDWAWBWC $.
  $}

  ${
    $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 mpteq1i 3eqtr2i ) AEDKZBCLMZNUIABMZDKZNAEUKNZDKAFDKUJULUIUJ
      AUKCKZULBOPCOPUJUNQGHABCOORSAUKDCATZUKPUOBQDCQUOBUAJUBUCUEUFAEUKDUDAUMFDI
      UGUH $.
  $}

  ${
    $d x A $.  $d x B $.  $d x R $.  $d x S $.  $d x ph $.
    fmptapd.a $e |- ( ph -> A e. V ) $.
    fmptapd.b $e |- ( ph -> B e. W ) $.
    fmptapd.s $e |- ( ph -> ( R u. { A } ) = S ) $.
    fmptapd.c $e |- ( ( ph /\ x = A ) -> C = B ) $.
    $( Append an additional value to a function.  (Contributed by Thierry
       Arnoux, 3-Jan-2017.)  (Revised by AV, 10-Aug-2024.) $)
    fmptapd $p |- ( ph ->
      ( ( x e. R |-> C ) u. { <. A , B >. } ) = ( x e. S |-> C ) ) $=
      ( cmpt cop csn cun fmptsnd uneq2d wceq mptun a1i mpteq1d 3eqtr2d ) ABFENZ
      CDOPZQUEBCPZENZQZBFUGQZENZBGENAUFUHUEABCEDHIMJKRSUKUITABFUGEUAUBABUJGELUC
      UD $.
  $}

  ${
    $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 ) ) $=
      ( cop csn cun cpr cmpt wceq df-pr fmptsnd uneq1d eqcomi fmptapd 3eqtrd
      a1i ) ACERZDFRZUAZUKSZULSZTZBCSZGUBZUOTBCDUAZGUBUMUPUCAUKULUDUJAUNURUOABC
      GEHJPLNUEUFABDFGUQUSIKMOUQDSTZUSUCAUSUTCDUDUGUJQUHUI $.
  $}

  $( 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 $.

  ${
    $d F x $.  $d A x $.  $d X x $.
    $( Express the class of fixed points of a function.  (Contributed by Stefan
       O'Rear, 1-Feb-2015.) $)
    fninfp $p |- ( F Fn A -> dom ( F i^i _I ) = { x e. A | ( F ` x ) = x } ) $=
      ( wfn cid cin cdm cres cv cfv wceq crab fnresdm inres incom reseq1i eqtri
      ineq1d 3eqtr4i eqtr3di dmeqd fnresi fndmin wcel fvresi eqeq2d rabbiia a1i
      mpan2 3eqtrd ) CBDZCEFZGCEBHZFZGZAIZCJZUPUMJZKZABLZUQUPKZABLZUKULUNUKCBHZ
      EFZULUNUKVCCEBCMREVCFZULBHZVDUNVEECFZBHVFECBNVGULBECOPQVCEOCEBNSTUAUKUMBD
      UOUTKBUBABCUMUCUIUTVBKUKUSVAABUPBUDURUPUQBUPUEUFUGUHUJ $.

    $( Property of a fixed point of a function.  (Contributed by Stefan O'Rear,
       1-Feb-2015.) $)
    fnelfp $p |- ( ( F Fn A /\ X e. A ) -> ( X e. dom ( F i^i _I ) <->
          ( F ` X ) = X ) ) $=
      ( vx wfn cid cin cdm wcel cv cfv wceq crab fninfp eleq2d fveq2 id eqeq12d
      elrab3 sylan9bb ) BAEZCBFGHZICDJZBKZUCLZDAMZICAICBKZCLZUAUBUFCDABNOUEUHDC
      AUCCLZUDUGUCCUCCBPUIQRST $.

    $( Express the class of non-fixed points of a function.  (Contributed by
       Stefan O'Rear, 14-Aug-2015.) $)
    fndifnfp $p |- ( F Fn A -> dom ( F \ _I ) =
          { x e. A | ( F ` x ) =/= x } ) $=
      ( wfn cid cdif cdm cres cv cfv wne crab cvv cxp cun c0 wss wceq wf dffn2
      fssxp sylbi ssdif0 sylib uneq2d un0 eqtr2di difeq2i difindi eqtri eqtr4di
      cin df-res dmeqd fnresi fndmdif mpan2 wcel fvresi neeq2d rabbiia 3eqtrd
      a1i ) CBDZCEFZGCEBHZFZGZAIZCJZVIVFJZKZABLZVJVIKZABLZVDVEVGVDVEVECBMNZFZOZ
      VGVDVRVEPOVEVDVQPVEVDCVPQZVQPRVDBMCSVSBCTBMCUAUBCVPUCUDUEVEUFUGVGCEVPULZF
      VRVFVTCEBUMUHCEVPUIUJUKUNVDVFBDVHVMRBUOABCVFUPUQVMVORVDVLVNABVIBURVKVIVJB
      VIUSUTVAVCVB $.

    $( Property of a non-fixed point of a function.  (Contributed by Stefan
       O'Rear, 15-Aug-2015.) $)
    fnelnfp $p |- ( ( F Fn A /\ X e. A ) -> ( X e. dom ( F \ _I ) <->
          ( F ` X ) =/= X ) ) $=
      ( vx wfn cid cdif cdm wcel cv cfv crab fndifnfp eleq2d wceq fveq2 neeq12d
      wne id elrab3 sylan9bb ) BAEZCBFGHZICDJZBKZUDRZDALZICAICBKZCRZUBUCUGCDABM
      NUFUIDCAUDCOZUEUHUDCUDCBPUJSQTUA $.

    $( A function is the identity iff it moves no points.  (Contributed by
       Stefan O'Rear, 25-Aug-2015.) $)
    fnnfpeq0 $p |- ( F Fn A -> ( dom ( F \ _I ) = (/) <->
          F = ( _I |` A ) ) ) $=
      ( vx wfn cv cfv wne crab c0 wceq cid cres wral cdif cdm wn rabeq0 wcel wa
      wb nne fvresi eqeq2d adantl ralbidva bitrid fndifnfp eqeq1d fnresi eqfnfv
      bitr4id mpan2 3bitr4d ) BADZCEZBFZUOGZCAHZIJZUPUOKALZFZJZCAMZBKNOZIJBUTJZ
      USUQPZCAMUNVCUQCAQUNVFVBCAUNUOARZSVFUPUOJZVBUPUOUAVGVBVHTUNVGVAUOUPAUOUBU
      CUDUKUEUFUNVDURICABUGUHUNUTADVEVCTAUICABUTUJULUM $.
  $}

  $( Remove an ordered pair not participating in a function value.
     (Contributed by NM, 1-Oct-2013.)  (Revised by Mario Carneiro,
     28-May-2014.) $)
  fvunsn $p |- ( B =/= D
       -> ( ( A u. { <. B , C >. } ) ` D ) = ( A ` D ) ) $=
    ( wne cop csn cun cres cfv resundir c0 wcel wceq cvv fvressn eqtr4d pm2.61i
    wn fvprc nelsn ressnop0 syl uneq2d un0 eqtrdi eqtrid fveq1d 3eqtr3g ) BDEZD
    ABCFGZHZDGZIZJZDAUMIZJZDULJZDAJZUJDUNUPUJUNUPUKUMIZHZUPAUKUMKUJVAUPLHUPUJUT
    LUPUJBUMMSUTLNBDUABCUMUBUCUDUPUEUFUGUHDOMZUOURNULODPVBSZUOLURDUNTDULTQRVBUQ
    USNAODPVCUQLUSDUPTDATQRUI $.

  $( The value of a singleton of an ordered pair is the second member.
     (Contributed by NM, 26-Oct-2012.)  (Proof shortened by BJ,
     25-Feb-2023.) $)
  fvsng $p |- ( ( A e. V /\ B e. W ) -> ( { <. A , B >. } ` A ) = B ) $=
    ( wcel wa cop csn wfun cfv wceq funsng opex snid funopfv mpisyl ) ACEBDEFAB
    GZHZIQREARJBKABCDLQABMNABROP $.

  ${
    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.)  (Proof shortened by BJ,
       25-Feb-2023.) $)
    fvsn $p |- ( { <. A , B >. } ` A ) = B $=
      ( cvv wcel cop csn cfv wceq fvsng mp2an ) AEFBEFAABGHIBJCDABEEKL $.
  $}

  ${
    fvsnun.1 $e |- ( ph -> A e. V ) $.
    fvsnun.2 $e |- ( ph -> B e. W ) $.
    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.)  Put in deduction form.  (Revised by BJ, 25-Feb-2023.) $)
    fvsnun1 $p |- ( ph -> ( G ` A ) = B ) $=
      ( csn cres cfv cun c0 wceq eqtri wcel fvresd cop reseq1i resundir resdisj
      cdif cin disjdifr ax-mp uneq2i un0 fveq1i snidg syl fvsng syl2anc 3eqtr3a
      eqtrd ) ABFBLZMZNBBCUALZURMZNZBFNCBUSVAUSUTEDURUEZMZOZURMZVAFVEURKUBVFVAV
      DURMZOZVAUTVDURUCVHVAPOVAVGPVAVCURUFPQVGPQURDUGVCUREUDUHUIVAUJRRRUKABURFA
      BGSZBURSIBGULUMZTAVBBUTNZCABURUTVJTAVICHSVKCQIJBCGHUNUOUQUP $.

    fvsnun2.4 $e |- ( ph -> D e. ( C \ { A } ) ) $.
    $( 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.)  Put in deduction form.  (Revised by
       BJ, 25-Feb-2023.) $)
    fvsnun2 $p |- ( ph -> ( G ` D ) = ( F ` D ) ) $=
      ( csn cres cfv cun wceq a1i c0 cdif cop reseq1i resundir cin disjdif wcel
      wfn wb fnsng syl2anc fnresdisj syl mpbii residm uneq12d un0 3eqtrd fveq1d
      uncom fvresd 3eqtr3d ) AEGDBNZUAZOZPEFVDOZPEGPEFPAEVEVFAVEBCUBNZVFQZVDOZV
      GVDOZVFVDOZQZVFVEVIRAGVHVDLUCSVIVLRAVGVFVDUDSAVLTVFQZVFTQZVFAVJTVKVFAVCVD
      UETRZVJTRZVCDUFAVGVCUHZVOVPUIABHUGCIUGVQJKBCHIUJUKVCVDVGULUMUNVKVFRAFVDUO
      SUPVMVNRATVFUTSVNVFRAVFUQSURURUSAEVDGMVAAEVDFMVAVB $.
  $}

  $( Split a function into a single point and all the rest.  (Contributed by
     Stefan O'Rear, 27-Feb-2015.) $)
  fnsnsplit $p |- ( ( F Fn A /\ X e. A ) ->
      F = ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) ) $=
    ( wfn wcel wa cres csn cdif cfv cop cun wceq fnresdm adantr resundi difsnid
    adantl reseq2d fnressn uneq2d 3eqtr3a eqtr3d ) BADZCAEZFZBAGZBBACHZIZGZCCBJ
    KHZLZUDUGBMUEABNOUFBUIUHLZGUJBUHGZLUGULBUIUHPUFUMABUEUMAMUDACQRSUFUNUKUJACB
    TUAUBUC $.

  $( 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 simp2l simp3 syl2anc
    f1osng f1of syl simp2r disjsn fun syl21anc wss snssi 3ad2ant3 ssequn2 sylib
    sylibr feq3d mpbid ) ABCGZEDHZEAHIZJZFBHZKZAELZMZBFLZMZCEFNLZMZGZVEBVIGVCUR
    VDVFVHGZAVDOPQZVJURVAVBRVCVDVFVHSZVKVCUSVBVMURUSUTVBTURVAVBUAEFDBUCUBVDVFVH
    UDUEVCUTVLURUSUTVBUFAEUGUOAVDBVFCVHUHUIVCVGBVIVEVCVFBUJZVGBQVBURVNVAFBUKULV
    FBUMUNUPUQ $.

  $( Adjoining a point to a punctured function gives a function.  (Contributed
     by Stefan O'Rear, 28-Feb-2015.) $)
  fsnunf2 $p |- ( ( F : ( S \ { X } ) --> T /\ X e. S /\ Y e. T ) ->
      ( F u. { <. X , Y >. } ) : S --> T ) $=
    ( csn cdif wf wcel w3a cun cop simp1 simp2 neldifsnd simp3 fsnunf syl121anc
    wn wceq difsnid 3ad2ant2 feq2d mpbid ) ADFZGZBCHZDAIZEBIZJZUFUEKZBCDELFKZHZ
    ABULHUJUGUHDUFISUIUMUGUHUIMUGUHUINUJDAOUGUHUIPUFBCADEQRUJUKABULUHUGUKATUIAD
    UAUBUCUD $.

  $( 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 incom 3adant3 wrel
    eqtri disjsn biimpri eqtrid 3ad2ant3 relres reldm0 ax-mp sylibr wfn fnresdm
    wb fnsng syl uneq12d resundir uncom un0 eqtr2i fveq1d snidg 3ad2ant1 fvresd
    3eqtr4g fvsng 3eqtr3d ) DBFZECFZDAGZFHZIZDADEJKZLZDKZMZNDVMNZDVNNEVLDVPVMVL
    AVOMZVMVOMZLOVMLZVPVMVLVROVSVMVLVRGZOPZVROPZVKVHWBVIVKWAVJVOQZOWAVOVJQWDAVO
    RVOVJSUBWDOPVKVJDUCUDUEUFVRUAWCWBUMAVOUGVRUHUIUJVLVMVOUKZVSVMPVHVIWEVKDEBCU
    NTVOVMULUOUPAVMVOUQVTVMOLVMOVMURVMUSUTVEVAVLDVOVNVHVIDVOFVKDBVBVCVDVHVIVQEP
    VKDEBCVFTVG $.

  $( 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 the function itself.  (Contributed by AV,
     2-Dec-2018.) $)
  funresdfunsn $p |- ( ( Fun F /\ X e. dom F ) ->
                ( ( F |` ( _V \ { X } ) ) u. { <. X , ( F ` X ) >. } ) = F ) $=
    ( wfun cdm wcel wa cvv csn cdif cres cfv cop cun wceq wrel funrel resdmdfsn
    syl adantr uneq1d wfn funfn fnsnsplit sylanb eqtr4d ) ACZBADZEZFZAGBHZIJZBB
    AKLHZMAUGUJIJZULMZAUIUKUMULUFUKUMNZUHUFAOUOAPABQRSTUFAUGUAUHAUNNAUBUGABUCUD
    UE $.

  $( 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 cun df-pr fveq1i necom fvunsn sylbi
    eqtrid 3ad2ant3 fvsng 3adant3 eqtrd ) AEGZCFGZABHZIAACJZBDJZKZLZAUIMZLZCUHU
    FULUNNUGUHULAUMUJMOZLZUNAUKUOUIUJPQUHBAHUPUNNABRUMBDASTUAUBUFUGUNCNUHACEFUC
    UDUE $.

  $( The value of a function with a domain of (at most) two elements.
     (Contributed by Alexander van der Vekens, 3-Dec-2017.)  (Proof shortened
     by BJ, 26-Sep-2024.) $)
  fvpr2g $p |- ( ( B e. V /\ D e. W /\ A =/= B )
                 -> ( { <. A , C >. , <. B , D >. } ` B ) = D ) $=
    ( wcel wne w3a cop cpr cfv prcom fveq1i wceq necom fvpr1g syl3an3b eqtrid )
    BEGZDFGZABHZIBACJZBDJZKZLBUDUCKZLZDBUEUFUCUDMNUBTUABAHUGDOABPBADCEFQRS $.

  ${
    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.)  (Proof shortened by BJ, 26-Sep-2024.) $)
    fvpr1 $p |- ( A =/= B -> ( { <. A , C >. , <. B , D >. } ` A ) = C ) $=
      ( cvv wcel wne cop cpr cfv wceq fvpr1g mp3an12 ) AGHCGHABIAACJBDJKLCMEFAB
      CDGGNO $.
  $}

  ${
    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.)  (Proof shortened by BJ, 26-Sep-2024.) $)
    fvpr2 $p |- ( A =/= B -> ( { <. A , C >. , <. B , D >. } ` B ) = D ) $=
      ( cvv wcel wne cop cpr cfv wceq fvpr2g mp3an12 ) BGHDGHABIBACJBDJKLDMEFAB
      CDGGNO $.
  $}

  ${
    $d A x y $.  $d B x y $.  $d F x y $.  $d R x y $.
    fprb.1 $e |- A e. _V $.
    fprb.2 $e |- B e. _V $.
    $( A condition for functionhood over a pair.  (Contributed by Scott Fenton,
       16-Sep-2013.) $)
    fprb $p |- ( A =/= B -> ( F : { A , B } --> R <->
       E. x e. R E. y e. R F = { <. A , x >. , <. B , y >. } ) ) $=
      ( cpr wf cv cop wceq wrex wa cfv wcel ffvelcdm mpan2 fveq2 wne prid1 wral
      adantr prid2 fvex fvpr1 fvpr2 eqeq12d eqcom bitrdi sylanbrc adantl wfn wb
      ralpr ffn eqfnfv syl2an mpbird opeq2 preq1d eqeq2d preq2d rspc2ev syl3anc
      fpr ffnd expcom wi wss vex prssi fss feq1 biimprcd syl6 rexlimdvv impbid
      ex ) CDUAZCDIZEFJZFCAKZLZDBKZLZIZMZBENAENZWCWAWJWCWAOZCFPZEQZDFPZEQZFCWLL
      ZDWNLZIZMZWJWCWMWAWCCWBQWMCDGUBWBECFRSUDWCWOWAWCDWBQWOCDHUEWBEDFRSUDWKWSW
      DFPZWDWRPZMZAWBUCZWAXCWCWACWRPZWLMZDWRPZWNMZXCCDWLWNGCFUFZUGCDWLWNHDFUFZU
      HXBXEXGACDGHWDCMZXBWLXDMXEXJWTWLXAXDWDCFTWDCWRTUIWLXDUJUKWDDMZXBWNXFMXGXK
      WTWNXAXFWDDFTWDDWRTUIWNXFUJUKUPULUMWCFWBUNWRWBUNWSXCUOWAWBEFUQWAWBWLWNIWR
      CDWLWNGHXHXIVGVHAWBFWRURUSUTWIWSFWPWGIZMABWLWNEEWDWLMZWHXLFXMWEWPWGWDWLCV
      AVBVCWFWNMZXLWRFXNWGWQWPWFWNDVAVDVCVEVFVIWAWIWCABEEWAWDEQWFEQOZWBEWHJZWIW
      CVJWAXOXPWAWBWDWFIZWHJXQEVKXPXOCDWDWFGHAVLBVLVGWDWFEVMWBXQEWHVNUSVTWIWCXP
      WBEFWHVOVPVQVRVS $.
  $}

  ${
    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 ) $=
      ( wne wa cop ctp cfv cpr csn cun df-tp fveq1i wceq necom fvunsn sylan9eqr
      sylbi fvpr1 eqtrid ) ABIZACIZJAADKZBEKZCFKZLZMAUHUINZUJOPZMZDAUKUMUHUIUJQ
      RUGUFUNAULMZDUGCAIUNUOSACTULCFAUAUCABDEGHUDUBUE $.
  $}

  ${
    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 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 fvunsn fvpr1g
    sylbi ad2antll 3expa adantrr eqtrd eqtrid ) AGIZDHIZJZABKZACKZJJZAADLZBELZC
    FLZMZNAUPUQOZURPQZNZDAUSVAUPUQURRSUOVBAUTNZDUNVBVCTZULUMUNCAKVDACUAUTCFAUBU
    DUEULUMVCDTZUNUJUKUMVEABDEGHUCUFUGUHUI $.

  $( 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 $.

  ${
    $d A a b x $.  $d B a b x $.  $d C a b x $.  $d D a b x $.  $d E a b x $.
    $d F a b x $.  $d ph a b x $.  $d T x $.
    tpres.t $e |- ( ph -> T = { <. A , D >. , <. B , E >. , <. C , F >. } ) $.
    tpres.b $e |- ( ph -> B e. V ) $.
    tpres.c $e |- ( ph -> C e. V ) $.
    tpres.e $e |- ( ph -> E e. V ) $.
    tpres.f $e |- ( ph -> F e. V ) $.
    tpres.1 $e |- ( ph -> B =/= A ) $.
    tpres.2 $e |- ( ph -> C =/= A ) $.
    $( An unordered triple of ordered pairs restricted to all but one first
       components of the pairs is an unordered pair of ordered pairs.
       (Contributed by AV, 14-Mar-2020.) $)
    tpres $p |- ( ph -> ( T |` ( _V \ { A } ) )
                        = { <. B , E >. , <. C , F >. } ) $=
      ( cvv wcel wa wceq vx va vb csn cdif cres cxp cin cop cpr df-res elin wex
      cv elxp anbi2i ctp wi eleq2d wo w3o vex eltp eldifsn wb eqeq1 adantl opth
      wne eqneqall com12 impd biimtrid adantr sylbid ex impcom exlimdvv orc a1d
      sylbi olc 3jaoi elpr imbitrrdi expd 3mix2 3mix3 bitrdi mpbird elexd jca31
      jaoi anim2i opeq12 eqeq2d eleq1 neeq1 anbi12d bi2anan9 spc2egv mpd jaoian
      syl2anc anbi1i 2exbii sylibr jca impbid bitrid eqrdv eqtrid ) AFQBUDUEZUF
      FXMQUGZUHZCGUIZDHUIZUJZFXMUKAUAXOXRUAUNZXORXSFRZXSXNRZSZAXSXRRZXSFXNULYBX
      TXSUBUNZUCUNZUIZTZYDXMRZYEQRZSZSZUCUMUBUMZSZAYCYAYLXTUBUCXSXMQUOUPAYMYCAX
      TYLYCAXTXSBEUIZXPXQUQZRZYLYCURZAFYOXSJUSZYPAYQYPAYLYCYPAYLSZXSXPTZXSXQTZU
      TZYCYPXSYNTZYTUUAVAZYSUUBURZXSYNXPXQUAVBZVCZUUCUUEYTUUAUUCAYLUUBUUCAYLUUB
      URUUCASZYKUUBUBUCYKUUHUUBYJYGUUHUUBURZYHYGUUIURZYIYHYDQRZYDBVIZSZUUJYDQBV
      DZUULUUJUUKUULYGUUIUULYGSZUUCAUUBUUOUUCYFYNTZAUUBURZYGUUCUUPVEUULXSYFYNVF
      VGUULUUPUUQURYGUUPYDBTZYEETZSUULUUQYDYEBEUBVBUCVBVHUULUURUUSUUQUURUULUUSU
      UQURZUUTYDBVJVKVLVMVNVOVLVPVGWAVNVQVKVRVPVLYTUUBYSYTUUAVSVTUUAUUBYSUUAYTW
      BVTWCWAXSXPXQUUFWDZWEWFVKVOVLYCAYMYCUUBAYMURUVAUUBAYMUUBASZXTYLUVBXTUUDUU
      BUUDAYTUUDUUAYTUUCUUAWGUUAUUCYTWHWMVNAXTUUDVEUUBAXTYPUUDYRUUGWIVGWJUVBYGU
      UMYISZSZUCUMUBUMZYLYTAUVEUUAYTASYTCQRZCBVIZSZGQRZSZSZUVEAUVJYTAUVFUVGUVIA
      CIKWKOAGIMWKWLWNAUVKUVEURZYTACIRGIRUVLKMUVDUVKUBUCCGIIYDCTZYEGTZSZYGYTUVC
      UVJUVOYFXPXSYDYECGWOWPUVMUUMUVHUVNYIUVIUVMUUKUVFUULUVGYDCQWQYDCBWRWSYEGQW
      QWTWSXAXDVGXBUUAASUUADQRZDBVIZSZHQRZSZSZUVEAUVTUUAAUVPUVQUVSADILWKPAHINWK
      WLWNAUWAUVEURZUUAADIRHIRUWBLNUVDUWAUBUCDHIIYDDTZYEHTZSZYGUUAUVCUVTUWEYFXQ
      XSYDYEDHWOWPUWCUUMUVRUWDYIUVSUWCUUKUVPUULUVQYDDQWQYDDBWRWSYEHQWQWTWSXAXDV
      GXBXCYKUVDUBUCYJUVCYGYHUUMYIUUNXEUPXFXGXHVPWAVKXIXJXJXKXL $.
  $}

  $( 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 Cartesian 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 Cartesian 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 $.
  $}

  $( Two ways to express that a function is constant.  (Contributed by NM,
     27-Nov-2007.) $)
  fconst5 $p |- ( ( F Fn A /\ A =/= (/) ) -> ( F = ( A X. { B } ) <->
                 ran F = { B } ) ) $=
    ( wfn c0 wne wa csn cxp wceq crn wi rneq eqeq2d imbitrid adantl cvv adantrd
    wb adantr rnxp wcel wf wfo df-fo fof sylbir fconst2g expd wrel fnrel relrn0
    wn snprc biimprd eqeq2 xpeq2 xp0 eqtrdi 3imtr4d sylbi syl5 pm2.61i impbid
    ex ) CADZAEFZGZCABHZIZJZCKZVIJZVGVKVMLVFVKVLVJKZJVGVMCVJMVGVNVIVLAVIUANOPBQ
    UBZVHVMVKLZLVOVFVPVGVOVFVMVKVFVMGZAVICUCZVOVKVQAVICUDVRAVICUEAVICUFUGABQCUH
    OUIRVOUMZVFVPVGVFCUJZVSVPACUKVSVIEJZVTVPLBUNWAVTVPWAVTGVLEJZCEJZVMVKVTWBWCL
    WAVTWCWBCULUOPWAVMWBSVTVIEVLUPTWAVKWCSVTWAVJECWAVJAEIEVIEAUQAURUSNTUTVEVAVB
    RVCVD $.

  ${
    $d A x $.  $d B x $.
    rnmptc.f $e |- F = ( x e. A |-> B ) $.
    rnmptc.a $e |- ( ph -> A =/= (/) ) $.
    $( Range of a constant function in maps-to notation.  (Contributed by
       Glauco Siliprandi, 11-Dec-2019.)  Remove extra hypothesis.  (Revised by
       SN, 17-Apr-2024.) $)
    rnmptc $p |- ( ph -> ran F = { B } ) $=
      ( c0 wne crn csn wceq cxp cmpt fconstmpt eqtr4i rneqi rnxp eqtrid syl ) A
      CHIZEJZDKZLGUAUBCUCMZJUCEUDEBCDNUDFBCDOPQCUCRST $.
  $}

  ${
    $d F x $.  $d A x $.  $d B x $.
    fnprb.a $e |- A e. _V $.
    fnprb.b $e |- B e. _V $.
    $( A function whose domain has at most two elements can be represented as a
       set of at most two ordered pairs.  (Contributed by FL, 26-Jun-2011.)
       (Proof shortened by Scott Fenton, 12-Oct-2017.)  Eliminate unnecessary
       antecedent ` A =/= B ` .  (Revised by NM, 29-Dec-2018.) $)
    fnprb $p |- ( F Fn { A , B }
        <-> F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) $=
      ( vx cpr wfn cfv cop wceq wb csn dfsn2 fveq2 wa cdm fvex adantl wcel wral
      fnsnb fneq2i eqeq2i 3bitr3i preq2 fneq2d id opeq12d preq2d eqeq2d 3bitr3d
      a1i wne cv fndm dmprop eqtr4di eleq2d wo elpr fvpr1 adantr eqcomd eqeq12d
      syl5ibrcom fvpr2 jaod biimtrid sylbid ralrimiv wfun fnfun eqfunfv syl2anr
      vex funpr mpbir2and df-fn sylanblrc fneq1 biimprd mpan9 impbida pm2.61ine
      ) CABGZHZCAACIZJZBBCIZJZGZKZLABABKZCAAGZHZCWIWIGZKZWGWMWPWRLWNCAMZHCWIMZK
      WPWRACDUBWSWOCANUCWTWQCWINUDUEUMWNWOWFCABAUFUGWNWQWLCWNWIWKWIWNABWHWJWNUH
      ABCOUIUJUKULABUNZWGWMXAWGPZWMCQZWLQZKZFUOZCIZXFWLIZKZFXCUAZWGXEXAWGXCWFXD
      WFCUPZAWHBWJACRZBCRZUQZURSXBXIFXCXBXFXCTXFWFTZXIXBXCWFXFWGXCWFKXAXKSUSXOX
      FAKZXFBKZUTXBXIXFABFVPVAXBXPXIXQXBXIXPWHAWLIZKXBXRWHXAXRWHKWGABWHWJDXLVBV
      CVDXPXGWHXHXRXFACOXFAWLOVEVFXBXIXQWJBWLIZKXBXSWJXAXSWJKWGABWHWJEXMVGVCVDX
      QXGWJXHXSXFBCOXFBWLOVEVFVHVIVJVKWGCVLWLVLZWMXEXJPLXAWFCVMABWHWJDEXLXMVQZF
      CWLVNVOVRXAWLWFHZWMWGXAXTXDWFKYBYAXNWLWFVSVTWMWGYBWFCWLWAWBWCWDWE $.

    $d C x $.
    fntpb.c $e |- C e. _V $.
    $( A function whose domain has at most three elements can be represented as
       a set of at most three ordered pairs.  (Contributed by AV,
       26-Jan-2021.) $)
    fntpb $p |- ( F Fn { A , B , C } <-> F = { <. A , ( F ` A ) >. ,
                                               <. B , ( F ` B ) >. ,
                                               <. C , ( F ` C ) >. } ) $=
      ( vx ctp wfn cfv cop wceq wb wne wa cpr eqcomi fneq2i fveq2 fnprb tpidm23
      wi eqeq2i 3bitr3i a1i tpeq3 fneq2d opeq12d tpeq3d eqeq2d 3bitr3d a1d wral
      id cdm fndm fvex dmtpop eqtr4di adantl wcel eleq2d w3o vex fvtp1 ad2antrr
      cv eqcomd eqeq12d syl5ibrcom fvtp2 ad4ant13 fvtp3 ad4ant23 3jaod biimtrid
      eltp ralrimiv wfun fnfun funtp 3expa eqfunfv syl2anr mpbir2and fntp fneq1
      sylbid biimprd impbida expcom pm2.61ine tpidm12 tpeq2d tpidm13 pm2.61iine
      mpan9 tpeq2 ) DABCIZJZDAADKZLZBBDKZLZCCDKZLZIZMZNZAABCABOZACOZPZXJUCBCBCM
      ZXJXMXNDABBIZJZDXCXEXEIZMZXAXIXPXRNXNDABQZJZDXCXEQZMZXPXRABDEFUAZXSXODXOX
      SABUBRSYAXQDXQYAXCXEUBRUDUEUFXNXOWTDBCABUGUHXNXQXHDXNXEXGXCXEXNBCXDXFXNUO
      BCDTUIUJUKULUMXMBCOZXJXMYDPZXAXIYEXAPZXIDUPZXHUPZMZHVHZDKZYJXHKZMZHYGUNZX
      AYIYEXAYGWTYHWTDUQZAXBBXDCXFADURZBDURZCDURZUSUTVAYFYMHYGYFYJYGVBYJWTVBZYM
      YFYGWTYJXAYGWTMYEYOVAVCYSYJAMZYJBMZYJCMZVDYFYMYJABCHVEVRYFYTYMUUAUUBYFYMY
      TXBAXHKZMYFUUCXBXMUUCXBMYDXAABCXBXDXFEYPVFVGVIYTYKXBYLUUCYJADTYJAXHTVJVKY
      FYMUUAXDBXHKZMYFUUDXDXKYDUUDXDMXLXAABCXBXDXFFYQVLVMVIUUAYKXDYLUUDYJBDTYJB
      XHTVJVKYFYMUUBXFCXHKZMYFUUEXFXLYDUUEXFMXKXAABCXBXDXFGYRVNVOVIUUBYKXFYLUUE
      YJCDTYJCXHTVJVKVPVQWIVSXADVTXHVTZXIYIYNPNYEWTDWAXKXLYDUUFABCXBXDXFEFGYPYQ
      YRWBWCHDXHWDWEWFYEXHWTJZXIXAXKXLYDUUGABCXBXDXFEFGYPYQYRWGWCXIXAUUGWTDXHWH
      WJWRWKWLWMABMZDAACIZJZDXCXCXGIZMZXAXIUUJUULNUUHDACQZJDXCXGQZMUUJUULACDEGU
      AUUMUUIDUUIUUMACWNRSUUNUUKDUUKUUNXCXGWNRUDUEUFUUHUUIWTDABACWSUHUUHUUKXHDU
      UHXCXEXCXGUUHABXBXDUUHUOABDTUIWOUKULACMZDABAIZJZDXCXEXCIZMZXAXIUUQUUSNUUO
      XTYBUUQUUSYCXSUUPDUUPXSABWPRSYAUURDUURYAXCXEWPRUDUEUFUUOUUPWTDACABUGUHUUO
      UURXHDUUOXCXGXCXEUUOACXBXFUUOUOACDTUIUJUKULWQ $.
  $}

  ${
    $d A a b $.  $d B b $.  $d F a b $.
    $( A function whose domain has at most two elements can be represented as a
       set of at most two ordered pairs.  (Contributed by Thierry Arnoux,
       12-Jul-2020.) $)
    fnpr2g $p |- ( ( A e. V /\ B e. W ) -> ( F Fn { A , B }
        <-> F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) ) $=
      ( va vb cv cpr wfn cfv cop wceq wb fneq2d id fveq2 opeq12d eqeq2d bibi12d
      preq1 preq1d preq2 preq2d vex fnprb vtocl2g ) CFHZGHZIZJZCUHUHCKZLZUIUICK
      ZLZIZMZNCAUIIZJZCAACKZLZUOIZMZNCABIZJZCVABBCKZLZIZMZNFGABDEUHAMZUKUSUQVCV
      JUJURCUHAUIUAOVJUPVBCVJUMVAUOVJUHAULUTVJPUHACQRUBSTUIBMZUSVEVCVIVKURVDCUI
      BAUCOVKVBVHCVKUOVGVAVKUIBUNVFVKPUIBCQRUDSTUHUICFUEGUEUFUG $.
  $}

  $( A function that maps a pair to a class is a pair of ordered pairs.
     (Contributed by Thierry Arnoux, 12-Jul-2020.) $)
  fpr2g $p |- ( ( A e. V /\ B e. W ) -> ( F : { A , B } --> C
    <-> ( ( F ` A ) e. C /\ ( F ` B ) e. C
    /\ F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) ) ) $=
    ( wcel wa cpr wf cfv cop wceq w3a simpr ad2antrr ffvelcdmd ad2antlr opelxpd
    prid1g prid2g wfn ffn adantl fnpr2g adantr mpbid 3jca cxp biimpar 3ad2antr3
    wb wss simpr3 simpr1 simpr2 prssd eqsstrd dff2 sylanbrc impbida ) AEGZBFGZH
    ZABIZCDJZADKZCGZBDKZCGZDAVGLZBVILZIZMZNZVDVFHZVHVJVNVPVECADVDVFOZVBAVEGZVCV
    FABETZPQVPVECBDVQVCBVEGZVBVFABFUAZRQVPDVEUBZVNVFWBVDVECDUCUDVDWBVNULVFABDEF
    UEZUFUGUHVDVOHZWBDVECUIZUMVFVDVHVNWBVJVDWBVNWCUJUKWDDVMWEVDVHVJVNUNWDVKVLWE
    WDAVGVECVBVRVCVOVSPVDVHVJVNUOSWDBVIVECVCVTVBVOWARVDVHVJVNUPSUQURVECDUSUTVA
    $.

  ${
    $d x A $.  $d x B $.  $d x F $.
    $( A constant function expressed in terms of its functionality, domain, and
       value.  See also ~ fconst2 .  (Contributed by NM, 27-Aug-2004.)  (Proof
       shortened by OpenAI, 25-Mar-2020.) $)
    fconstfv $p |- ( F : A --> { B } <->
                   ( F Fn A /\ A. x e. A ( F ` x ) = B ) ) $=
      ( csn wf wfn cv cfv wcel wral wa wceq ffnfv fvex elsn ralbii anbi2i bitri
      ) BCEZDFDBGZAHZDIZTJZABKZLUAUCCMZABKZLABTDNUEUGUAUDUFABUCCUBDOPQRS $.

    $( Two ways to express a constant function.  (Contributed by NM,
       15-Mar-2007.) $)
    fconst3 $p |- ( F : A --> { B } <->
                  ( F Fn A /\ A C_ ( `' F " { B } ) ) ) $=
      ( vx csn wf wfn cv cfv wceq wral wa ccnv cima wss fconstfv wfun cdm fnfun
      wb fndm eqimss2 syl funconstss syl2anc pm5.32i bitri ) ABEZCFCAGZDHCIBJDA
      KZLUIACMUHNOZLDABCPUIUJUKUICQACRZOZUJUKTACSUIULAJUMACUAAULUBUCDABCUDUEUFU
      G $.
  $}

  $( Two ways to express a constant function.  (Contributed by NM,
     8-Mar-2007.) $)
  fconst4 $p |- ( F : A --> { B } <->
                ( F Fn A /\ ( `' F " { B } ) = A ) ) $=
    ( csn wf wfn ccnv cima wss wa wceq fconst3 cnvimass fndm sseqtrid biantrurd
    cdm eqss bitr4di pm5.32i bitri ) ABDZCECAFZACGUBHZIZJUCUDAKZJABCLUCUEUFUCUE
    UDAIZUEJUFUCUGUEUCCQUDACUBMACNOPUDARSTUA $.

  ${
    $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 wa cres cdm cv cfv cop cmpt cima cvv crn wfn funres adantr
      wceq funfnd dffn5 sylib fvex fnasrn eqtrdi dmmpti imaeq2i imadmrn eqtr4di
      opex eqid eqtr3i funmpt dmresexg adantl funimaexg sylancr eqeltrd ) AEZBC
      FZGZABHZDVCIZDJZVEVCKZLZMZVDNZOVBVCVHPZVIVBVCDVDVFMZVJVBVCVDQVCVKTVBVCUTV
      CEVABARSUADVDVCUBUCDVDVFVEVCUDUEUFVHVHIZNVIVJVLVDVHDVDVGVHVEVFUKVHULUGUHV
      HUIUMUJVBVHEVDOFZVIOFDVDVGUNVAVMUTABCUOUPVHVDOUQURUS $.
  $}

  ${
    resiexd.b $e |- ( ph -> B e. V ) $.
    $( The restriction of the identity relation to a set is a set.
       (Contributed by AV, 15-Feb-2020.) $)
    resiexd $p |- ( ph -> ( _I |` B ) e. _V ) $=
      ( cid wfun wcel cres cvv funi resfunexg sylancr ) AEFBCGEBHIGJDEBCKL $.
  $}

  $( 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 .  See ~ fnexALT for alternate
     proof.  (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 wrel wcel cdm cres cvv fnrel wfun wceq eleq1a impcom resfunexg sylan2
    wa df-fn anassrs sylanb resdm eleq1d biimpa syl2an2r ) CADZCEZABFZCCGZHZIFZ
    CIFZACJUECKZUHALZQUGUJCARULUMUGUJUMUGQULUHBFZUJUGUMUNABUHMNCUHBOPSTUFUJUKUF
    UICICUAUBUCUD $.

  ${
    fnexd.1 $e |- ( ph -> F Fn A ) $.
    fnexd.2 $e |- ( ph -> A e. V ) $.
    $( If the domain of a function is a set, the function is a set.
       (Contributed by Glauco Siliprandi, 23-Oct-2021.) $)
    fnexd $p |- ( ph -> F e. _V ) $=
      ( wfn wcel cvv fnex syl2anc ) ACBGBDHCIHEFBDCJK $.
  $}

  $( 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 $.
  $}

  ${
    mptexgf.a $e |- F/_ 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.)  (Revised by Thierry Arnoux, 17-May-2020.) $)
    mptexgf $p |- ( A e. V -> ( x e. A |-> B ) e. _V ) $=
      ( wcel cmpt wfun cdm cvv funmpt wss crab eqid dmmpt wtru tru 2a1i ss2rabi
      cv rabtru sseqtri eqsstri ssexg mpan funex sylancr ) BDFZABCGZHUIIZJFZUIJ
      FABCKUJBLUHUKUJCJFZABMZBABCUIUINOUMPABMBULPABPATBFULQRSABEUAUBUCUJBDUDUEJ
      UIUFUG $.
  $}

  ${
    $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.  Inference version of ~ mptexg .  (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 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 A y $.  $d V x y $.  $d W y $.  $d Y y $.
    mptfvmpt.y $e |- ( y = Y -> M = ( x e. V |-> A ) ) $.
    mptfvmpt.g $e |- G = ( y e. W |-> M ) $.
    mptfvmpt.v $e |- V = ( F ` X ) $.
    $( A function in maps-to notation as the value of another function in
       maps-to notation.  (Contributed by AV, 20-Aug-2022.) $)
    mptfvmpt $p |- ( Y e. W -> ( G ` Y ) = ( x e. V |-> A ) ) $=
      ( cmpt fvexi mptex fvmpt ) BJFAGCNHEKLAGCGIDMOPQ $.
  $}

  ${
    $d f x z A $.  $d f z 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 ) $=
      ( vz cv wfn cfv wceq wral wa weu wb wal wex cmpt mptex cvv eqeq2 pm4.71ri
      bibi2d albidv spcev eqid fnmpti fneq1 mpbiri dffn5 eqeq1 sylbi wcel rgenw
      fvex mpteqb ax-mp bitrdi pm5.32i bitr2i mpg eu6 mpbir ) DHZBIZAHZVDJZCKAB
      LZMZDNVIVDGHZKZOZDPZGQZVIVDABCRZKZOZVNDVMVQDPGVOABCESVJVOKZVLVQDVRVKVPVIV
      JVOVDUAUCUDUEVPVEVPMVIVPVEVPVEVOBIABCVOFVOUFUGBVDVOUHUIUBVEVPVHVEVPABVGRZ
      VOKZVHVEVDVSKVPVTOABVDUJVDVSVOUKULVGTUMZABLVTVHOWAABVFVDUOUNABVGCTUPUQURU
      SUTVAVIDGVBVC $.
  $}

  $( 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 wcel cdm wi cfv cima wss funfvima ex com23 a2d ssel impel ) CDZBAEZB
    CFZEZGRBCHCAIEZGZASJQRTUAQTRUAQTUBABCKLMNASBOP $.

  ${
    funfvima2d.1 $e |- ( ph -> F : A --> B ) $.
    $( A function's value in a preimage belongs to the image.  (Contributed by
       Stanislas Polu, 9-Mar-2020.)  (Revised by AV, 23-Mar-2024.) $)
    funfvima2d $p |- ( ( ph /\ X e. A ) -> ( F ` X ) e. ( F " A ) ) $=
      ( wcel cfv cima wfun cdm wss wi ffund fdmd sseqtrrd funfvima2 syl2anc imp
      ssidd ) AEBGZEDHDBIGZADJBDKZLUAUBMABCDFNABBUCABTABCDFOPBEDQRS $.
  $}

  $( 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 $.
  $}

  ${
    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 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 ssel funfvop impel wb wceq
      cv sneq imaeq2d eleq2d opeq1 eleq1d vex fvex elimasn vtoclbg mpbird exp32
      ad2antll impcom ) BCEZBFZABGZHZABIZCAJZKZHZLUNUOUQVAUNUOUQMZMVAAURNZCHZUN
      VCBHVDVBBCVCOABPQUQVAVDRUNUOURCDTZJZKZHVEURNZCHVAVDDAUPVEASZVGUTURVIVFUSC
      VEAUAUBUCVIVHVCCVEAURUDUECVEURDUFABUGUHUIULUJUKUM $.
  $}

  ${
    $d ph y $.  $d ps x $.  $d F x y $.  $d B x y $.
    ralima.x $e |- ( x = ( F ` y ) -> ( ph <-> ps ) ) $.
    $( Universal quantification under an image in terms of the base set.
       (Contributed by Stefan O'Rear, 21-Jan-2015.)  Reduce DV conditions.
       (Revised by Matthew House, 14-Aug-2025.) $)
    ralima $p |- ( ( F Fn A /\ B C_ A ) ->
        ( A. x e. ( F " B ) ph <-> A. y e. B ps ) ) $=
      ( wfn cdm wss cima wral wb fnfun wa cv wcel wceq wrex funfnd fndm biimpar
      sseq2d cfv fvexd fvelimab eqcom rexbii bitrdi adantl ralxfr2d syl2an2r
      cvv ) GEIZGGJZIZFEKZFUPKZACGFLZMBDFMNUOGEGOUAUOUSURUOUPEFEGUBUDUCUQUSPZAB
      CDDQZGUEZUTFUNVAVBFRPVBGUFVACQZUTRVCVDSZDFTVDVCSZDFTDUPFVDGUGVEVFDFVCVDUH
      UIUJVFABNVAHUKULUM $.

    $( Existential quantification under an image in terms of the base set.
       (Contributed by Stefan O'Rear, 21-Jan-2015.)  Reduce DV conditions.
       (Revised by Matthew House, 14-Aug-2025.) $)
    rexima $p |- ( ( F Fn A /\ B C_ A ) ->
        ( E. x e. ( F " B ) ph <-> E. y e. B ps ) ) $=
      ( wfn wss wa wn cima wral wrex cv cfv wceq notbid dfrex2 ralima 3bitr4g )
      GEIFEJKZALZCGFMZNZLBLZDFNZLACUEOBDFOUCUFUHUDUGCDEFGCPDPGQRABHSUASACUETBDF
      TUB $.
  $}

  ${
    $d ph y $.  $d ps x $.  $d F x y $.  $d B x y $.  $d A x y $.
    reximaOLD.x $e |- ( x = ( F ` y ) -> ( ph <-> ps ) ) $.
    $( Obsolete version of ~ rexima as of 14-Aug-2025.  (Contributed by Stefan
       O'Rear, 21-Jan-2015.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    reximaOLD $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 fvexd wceq wrex fvelimab eqcom rexxfr2d
      rexbii bitrdi wb adantl ) GEIFEJKZABCDDLZGMZGFNZFOUGUHFPKUHGQUGCLZUJPUIUK
      RZDFSUKUIRZDFSDEFUKGTULUMDFUIUKUAUCUDUMABUEUGHUFUB $.

    $( Obsolete version of ~ ralima as of 14-Aug-2025.  (Contributed by Stefan
       O'Rear, 21-Jan-2015.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    ralimaOLD $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 fvexd wceq wrex fvelimab eqcom ralxfr2d
      rexbii bitrdi wb adantl ) GEIFEJKZABCDDLZGMZGFNZFOUGUHFPKUHGQUGCLZUJPUIUK
      RZDFSUKUIRZDFSDEFUKGTULUMDFUIUKUAUCUDUMABUEUGHUFUB $.
  $}

  ${
    $d x y F $.
    $( Upper bound for the class of values of a class.  (Contributed by NM,
       9-Nov-1995.) $)
    fvclss $p |- { y | E. x y = ( F ` x ) } C_ ( ran F u. { (/) } ) $=
      ( cv cfv wceq wex cab crn wcel c0 csn wo cun wn wne wbr tz6.12i imbitrrdi
      eqcom biimtrid eximdv vex elrn com12 necon1bd velsn ss2abi df-un sseqtrri
      orrd ) BDZADZCEZFZAGZBHULCIZJZULKLZJZMZBHUQUSNUPVABUPURUTUPUROULKFUTUPURU
      LKULKPZUPURVBUPUMULCQZAGURVBUOVCAUOUNULFVBVCULUNTUMULCRUAUBAULCBUCUDSUEUF
      BKUGSUKUHBUQUSUIUJ $.
  $}

  ${
    $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 elab sylibr nfv nfcsb1v nfeq2 eqeq2d cbvrexw eleqtrrdi
      abbii ) AGZCHZDBGZAFGZDIZJZFCKZBLZUNDJZACKZBLUMDUPJZFCKZDUSHUMMVCNVBMFULC
      UOULJZVBMVBAFAUODOZPVDQTRSURVCBDEUTUQVBFCUNDUPUAUBUCUDVAURBUTUQAFCUTFUEAU
      NUPAUODUFUGULUOJDUPUNVEUHUIUKUJ $.
  $}

  ${
    $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 fvex dfiun2 fnrnfv unieqd eqtr4id
      crn ) CBEZABAFZCGZHDFUAIABJDKZLCRZLADBUATCMNSUCUBADBCOPQ $.

    $( The indexed union of a function's values is the union of its image under
       the index class.

       Note:  This theorem depends on the fact that our function value is the
       empty set outside of its domain.  If the antecedent is changed to
       ` F Fn A ` , the theorem can be proved without this dependency.
       (Contributed by NM, 26-Mar-2006.)  (Proof shortened by Mario Carneiro,
       31-Aug-2015.) $)
    funiunfv $p |- ( Fun F -> U_ x e. A ( F ` x ) = U. ( F " A ) ) $=
      ( wfun cres cdm cv cfv ciun crn cuni cima wfn wceq syl eqtri wcel iuneq2i
      cun c0 funres funfnd fniunfv cdif undif2 wss cin dmres inss1 eqsstri mpbi
      ssequn1 iuneq1 ax-mp iunxun eldifn ndmfv uneq2i un0 fvres 3eqtr3ri df-ima
      wn iun0 unieqi 3eqtr4g ) CDZACBEZFZAGZVHHZIZVHJZKZABVJCHZIZCBLZKVGVHVIMVL
      VNNVGVHBCUAUBAVIVHUCOAVIBVIUDZSZVKIZABVKIZVLVPVSBNVTWANVSVIBSZBVIBUEVIBUF
      WBBNVIBCFZUGBCBUHBWCUIUJVIBULUKPAVSBVKUMUNVTVLAVRVKIZSZVLAVIVRVKUOWEVLTSV
      LWDTVLWDAVRTITAVRVKTVJVRQVJVIQVCVKTNVJBVIUPVJVHUQORAVRVDPURVLUSPPABVKVOVJ
      BCUTRVAVQVMCBVBVEVF $.
  $}

  ${
    $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 ~ funiunfv uses a bound-variable
       hypothesis in place of a distinct variable condition.  (Contributed by
       NM, 26-Mar-2006.)  (Revised by David Abernethy, 15-Apr-2013.) $)
    funiunfvf $p |- ( Fun F -> U_ x e. A ( F ` x ) = U. ( F " A ) ) $=
      ( vz wfun cv cfv ciun cima cuni nfcv nffv fveq2 cbviun funiunfv eqtr3id )
      CFABAGZCHZIEBEGZCHZICBJKEABUASATCDATLMESLTRCNOEBCPQ $.
  $}

  ${
    $d x A $.  $d x B $.  $d x F $.
    $( Membership in the union of an image of a function.  (Contributed by NM,
       28-Sep-2006.) $)
    eluniima $p |- ( Fun F ->
                   ( B e. U. ( F " A ) <-> E. x e. A B e. ( F ` x ) ) ) $=
      ( wfun cv cfv ciun wcel cima cuni wrex funiunfv eleq2d eliun bitr3di ) DE
      ZCABAFDGZHZICDBJKZICRIABLQSTCABDMNACBROP $.
  $}

  ${
    $d x y A $.  $d x y F $.
    $( Membership in the union of the range of a function.  See ~ elunirnALT
       for a shorter proof which uses ~ ax-pow .  See ~ elfvunirn for a more
       general version of the reverse direction.  (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 funfn eleq2
      wb fvelrnb anbi2d r19.42v bitr4di biimparc reximi biimtrdi exlimdv fvelrn
      sylbi a1d ancld fvex eleq1 anbi12d spcev syl6 rexlimdva impbid bitrid ) B
      CEZFGBDHZGZVBVAGZIZDJZCKZBAHZCLZGZACMZNZDBVAOVGVFVLVGVEVLDVGVEVCVIVBPZIZA
      VKNZVLVGVEVCVMAVKNZIVOVGVDVPVCVGCVKQVDVPTCRAVKVBCUAUJUBVCVMAVKUCUDVNVJAVK
      VMVJVCVIVBBSUEUFUGUHVGVJVFAVKVGVHVKGIZVJVJVIVAGZIZVFVQVJVRVQVRVJVHCUIUKUL
      VEVSDVIVHCUMVBVIPVCVJVDVRVBVIBSVBVIVAUNUOUPUQURUSUT $.
  $}

  ${
    $d x A $.  $d x F $.
    $( Alternate proof of ~ elunirn .  It is shorter but requires ~ ax-pow
       (through ~ eluniima , ~ funiunfv , ~ ndmfv ).  (Contributed by NM,
       24-Sep-2006.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    elunirnALT $p |- ( Fun F ->
                        ( A e. U. ran F <-> E. x e. dom F A e. ( F ` x ) ) ) $=
      ( crn cuni wcel cdm cima wfun cv cfv wrex imadmrn unieqi eluniima bitr3id
      eleq2i ) BCDZEZFBCCGZHZEZFCIBAJCKFATLUBSBUARCMNQATBCOP $.
  $}

  ${
    $d x A $.  $d x B $.  $d x F $.
    $( Obsolete version of ~ elfvunirn as of 12-Jan-2025.  (Contributed by
       Thierry Arnoux, 13-Nov-2016.)  (New usage is discouraged.)
       (Proof modification is discouraged.) $)
    elunirn2OLD $p |- ( ( Fun F /\ B e. ( F ` A ) ) -> B e. U. ran F ) $=
      ( vx wfun cfv wcel wa crn cuni cdm wrex elfvdm wceq eleq2d rspcev mpancom
      cv fveq2 adantl wb elunirn adantr mpbird ) CEZBACFZGZHBCIJGZBDRZCFZGZDCKZ
      LZUGUMUEAULGUGUMBACMUKUGDAULUIANUJUFBUIACSOPQTUEUHUMUAUGDBCUBUCUD $.
  $}

  ${
    $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 bitri dff12
      wf1 wmo wfn ffn cdm fndm imbitrid anim12d pm4.71rd eqcom fnbrfvb bi2anan9
      wf bitrid anandis pm5.32da bitr4d imbi1d impexp bitrdi albidv 19.21v fvex
      wex 19.23v eqvinc imbi1i bitr4i imbi2i 2albidv breq1 albii alrot3 3bitr4g
      mo4 r2al syl pm5.32i ) CDEUBCDEUNZAGZFGZEHZAUCZFIZJVTWAEKZBGZEKZLZWAWGLZM
      ZBCNACNZJAFCDEUAVTWEWLVTECUDZWEWLOCDEUEWMWCWGWBEHZJZWJMZFIZBIAIZWACPZWGCP
      ZJZWKMZBIAIWEWLWMWQXBABWMWQXAWBWFLZWBWHLZJZWJMZMZFIZXBWMWPXGFWMWPXAXEJZWJ
      MXGWMWOXIWJWMWOXAWOJXIWMWOXAWMWCWSWNWTWCWAEUFZPWMWSWAWBEAQFQZRWMXJCWACEUG
      ZSUHWNWGXJPWMWTWGWBEBQXKRWMXJCWGXLSUHUIUJWMXAXEWOWMWSWTXEWOOWMWSJZXCWCWMW
      TJZXDWNXCWFWBLXMWCWBWFUKCWAWBEULUOXDWHWBLXNWNWBWHUKCWGWBEULUOUMUPUQURUSXA
      XEWJUTVAVBXHXAXFFIZMXBXAXFFVCXOWKXAXOXEFVEZWJMWKXEWJFVFWIXPWJFWFWHWAEVDVG
      VHVIVJTVAVKWEWPBIAIZFIWRWDXQFWCWNABWAWGWBEVLVPVMWPFABVNTWKABCCVQVOVRVST
      $.
  $}

  ${
    $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 cbvralw ralbii nfralw fveqeq2 equequ1 bitri anbi2i
      wf1 ralbidv ) CDEUOCDEUAZHJZEKZIJZEKZLZHIMZNZICOZHCOZPUQAJZEKBJZEKZLZABMZ
      NZBCOZACOZPHICDEUBVFVNUQVFUSVILZHBMZNZBCOZHCOVNVEVRHCVDVQIBCVBVCBBUSVABUR
      EGBURQRBUTEGBUTQRSVCBTUCVQITIBMZVBVOVCVPVSVAVIUSUTVHEUDUEIBHUFUGUHUIVRVMH
      ACVQABCACQVOVPAAUSVIAUREFAURQRAVHEFAVHQRSVPATUCUJVMHTHAMZVQVLBCVTVOVJVPVK
      URVGVIEUKHABULUGUPUHUMUNUM $.
  $}

  ${
    $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 fveqeq2 imbi12d imp
      eqeq1 fveq2 eqeq2d eqeq2 rspc2v com12 simplbiim ) ABEHZCAIDAIJZCEKZDEKZLZ
      CDLZMZUIABENFOZEKGOZEKZLZFGPZMZGAQFAQZUJUOMFGABERUJVBUOVAUOUKURLZCUQLZMFG
      CDAAUPCLUSVCUTVDUPCURESUPCUQUBTUQDLZVCUMVDUNVEURULUKUQDEUCUDUQDCUETUFUGUH
      UA $.
  $}

  $( If the values of a composition of one-to-one functions for two arguments
     are equal, the arguments themselves must be equal.  (Contributed by AV,
     3-Feb-2021.) $)
  f1cofveqaeq $p |- ( ( ( F : B -1-1-> C /\ G : A -1-1-> B )
                        /\ ( X e. A /\ Y e. A ) )
                      -> ( ( F ` ( G ` X ) ) = ( F ` ( G ` Y ) )
                           -> X = Y ) ) $=
    ( wf1 wa wcel cfv wceq wi simpl wf f1f ffvelcdm ex anim12d f1veqaeq syl imp
    adantl syl2an2r adantll syld ) BCDHZABEHZIZFAJZGAJZIZIFEKZDKGEKZDKLZUMUNLZF
    GLZUIUGULUMBJZUNBJZIZUOUPMUGUHNUIULUTUHULUTMZUGUHABEOZVAABEPVBUJURUKUSVBUJU
    RABFEQRVBUKUSABGEQRSUAUCUBBCUMUNDTUDUHULUPUQMUGABFGETUEUF $.

  $( Alternate proof of ~ f1cofveqaeq , 1 essential step shorter, but having
     more bytes (305 versus 282).  (Contributed by AV, 3-Feb-2021.)
     (New usage is discouraged.)  (Proof modification is discouraged.) $)
  f1cofveqaeqALT $p |- ( ( ( F : B -1-1-> C /\ G : A -1-1-> B )
                           /\ ( X e. A /\ Y e. A ) )
                         -> ( ( F ` ( G ` X ) ) = ( F ` ( G ` Y ) )
                              -> X = Y ) ) $=
    ( wf1 wa wcel cfv wceq ccom wb wi wf f1f fvco3 adantrr adantrl eqeq12d f1co
    ex syl adantl imp f1veqaeq sylan sylbird ) BCDHZABEHZIZFAJZGAJZIZIFEKDKZGEK
    DKZLZFDEMZKZGUSKZLZFGLZULUOVBURNZUKUOVDOZUJUKABEPZVEABEQVFUOVDVFUOIUTUPVAUQ
    VFUMUTUPLUNABFDERSVFUNVAUQLUMABGDERTUAUCUDUEUFULACUSHUOVBVCOABCDEUBACFGUSUG
    UHUI $.

  $( If two one-to-one functions are applied on different arguments, also the
     values are different.  (Contributed by Alexander van der Vekens,
     25-Jan-2018.) $)
  2f1fvneq $p |- ( ( ( E : D -1-1-> R /\ F : C -1-1-> D )
                       /\ ( A e. C /\ B e. C ) /\ A =/= B )
                   -> ( ( ( E ` ( F ` A ) ) = X /\ ( E ` ( F ` B ) ) = Y )
                        -> X =/= Y ) ) $=
    ( wf1 wa wcel wne cfv wceq wn wi f1veqaeq ffvelcdm ex adantll 3impia simpll
    w3a necon3ad wf f1f anim12dan syl adantl imp syl2anc con3dimp eqeq12 notbid
    neqne biimtrdi syl5com 3adant3 mpd ) DEFJZCDGJZKZACLZBCLZKZABMZUDAGNZBGNZOZ
    PZVHFNZHOVIFNZIOKZHIMZQZVCVFVGVKVCVFKZVJABVBVFVJABOQVACDABGRUAUEUBVCVFVKVPQ
    VGVQVKVPVQVKKVLVMOZPZVNVOVQVRVJVQVAVHDLZVIDLZKZVRVJQVAVBVFUCVCVFWBVBVFWBQZV
    AVBCDGUFZWCCDGUGWDVFWBWDVDVTVEWACDAGSCDBGSUHTUIUJUKDEVHVIFRULUMVNVSHIOZPVOV
    NVRWEVLHVMIUNUOHIUPUQURTUSUT $.

  ${
    $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 cbvralvw raaanv fvmpt2 fvmptg eqeqan12d an4s imbi1d
      ex ralimdva syl6 ralimia syl sylbir sylan2b anidms pm5.32i 3bitr2i ) CDGJ
      CDGKZALZGMZBLZGMZNZVGVINZOZBCPZACPZQEDRZACPZVOQVQEFNZVLOZBCPZACPZQABCDGAG
      ACEUEHACEUAUBBGUFUCVQVFVOACDEGHUGUDVQVOWAVQVOWASZVQVQFDRZBCPZWBVPWCABCVLE
      FDIUHUIVQWDQVPWCQZBCPZACPZWBVPWCABCUJWGVNVTSZACPWBWFWHACVGCRZWFVMVSSZBCPW
      HWIWEWJBCWIVICRZQZWEWJWLWEQVKVRVLWIVPWKWCVKVRSWIVPQWKWCQVHEVJFACEDGHUKAVI
      EFCDGIHULUMUNUOUPUQVMVSBCTURUSVNVTACTUTVAVBVCVDVE $.
  $}

  $( 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 fveqeq2 rspcev mpan2 impbid1 bitrd ) AB
      CGZDAHZEAIZJZDCKZCELHZFMZCKVGNZFEOZDEHZVCVEVHVKPZVDVCCAQVEVMABCUBFAEVGCUA
      UCUDVFVKVLVCVDVEVKVLRVCVDSZVESVJVLFEVNVEVIEHZVJVLRZVEVOSVNVIAHZVOSVPVEVOV
      QEAVIUEUFVNVQVOVPVNVQSVJVIDNZVOVLVCVDVQVJVRRVCVDVQSSVJVRVCVQVDVJVRPABVIDC
      UGUHUITVRVOVLVIDEUMUJUKULUNTUOUPVLVGVGNZVKVGUQVJVSFDEVIDVGCURUSUTVAVB $.
  $}

  ${
    $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 $.

    $( Taking images under a one-to-one function preserves proper subsets.
       (Contributed by Stefan O'Rear, 30-Oct-2014.) $)
    f1imapss $p |- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) ->
      ( ( F " C ) C. ( F " D ) <-> C C. D ) ) $=
      ( wf1 wss wa cima wceq wpss f1imass f1imaeq notbid anbi12d dfpss2 3bitr4g
      wn ) ABEFCAGDAGHHZECIZEDIZGZTUAJZRZHCDGZCDJZRZHTUAKCDKSUBUEUDUGABCDELSUCU
      FABCDEMNOTUAPCDPQ $.
  $}

  ${
    $d F x y $.  $d X x y $.  $d Y x y $.
    fpropnf1.f $e |- F = { <. X , Z >. , <. Y , Z >. } $.
    $( A function, given by an unordered pair of ordered pairs, which is not
       injective/one-to-one.  (Contributed by Alexander van der Vekens,
       22-Oct-2017.)  (Revised by AV, 8-Jan-2021.) $)
    fpropnf1 $p |- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ X =/= Y )
                     -> ( Fun F /\ -. Fun `' F ) ) $=
      ( vx vy w3a wa adantr syl wceq cfv wi wral imbi12d eqeq2d wcel wne wn cop
      wfun ccnv cpr 3adant3 jca 3ad2ant3 simpr 3jca funprg funeqi sylibr adantl
      id neneq wf fprg eqcomi feq1i sylib wf1 df-f1 cv dff13 wb fveqeq2 ralbidv
      eqeq1 ralprg fveq2 eqeq2 fveq1i 3simpb anim1i df-3an fvpr1g eqtrid 3simpc
      weq anbi12d fvpr2g eqtr2id eqtrd embantd adantld adantrd sylbid biimtrrid
      idd biimtrid mpand mtod ) EAUAZFCUAZGDUAZKZEFUBZLZBUEZBUFUEZUCXAEGUDFGUDU
      GZUEZXBXAWPWQLZWRWRLZWTKZXEXAXFXGWTWSXFWTWPWQXFWRXFUQUHMWSXGWTWRWPXGWQWRW
      RWRWRUQZXIUIUJMWSWTUKULZEFGGACDDUMNBXDHUNUOXAXCEFOZWTXKUCWSEFURUPXAEFUGZG
      GUGZBUSZXCXKXAXLXMXDUSZXNXAXHXOXJEFGGACDDUTNXLXMXDBBXDHVAVBVCXNXCLXLXMBVD
      ZXAXKXLXMBVEXPXNIVFZBPJVFZBPZOZIJWBZQZJXLRZIXLRZLXAXKIJXLXMBVGXAYDXKXNXAY
      DEBPZXSOZEXROZQZJXLRZFBPZXSOZFXROZQZJXLRZLZXKWSYDYOVHZWTWPWQYPWRYCYIYNIEF
      ACXQEOZYBYHJXLYQXTYFYAYGXQEXSBVIXQEXRVKSVJXQFOZYBYMJXLYRXTYKYAYLXQFXSBVIX
      QFXRVKSVJVLUHMXAYOYEYEOZEEOZQZYEYJOZXKQZLZYJYEOZFEOZQZYJYJOZFFOZQZLZLZXKW
      SYOUULVHZWTWPWQUUMWRXFYIUUDYNUUKYHUUAUUCJEFACXREOZYFYSYGYTUUNXSYEYEXREBVM
      ZTXREEVNSXRFOZYFUUBYGXKUUPXSYJYEXRFBVMZTXRFEVNSVLYMUUGUUJJEFACUUNYKUUEYLU
      UFUUNXSYEYJUUOTXREFVNSUUPYKUUHYLUUIUUPXSYJYJUUQTXRFFVNSVLWCUHMXAUUDXKUUKX
      AUUCXKUUAXAUUBXKXKXAYEGYJXAYEEXDPZGEBXDHVOXAWPWRWTKZUURGOXAWPWRLZWTLUUSWS
      UUTWTWPWQWRVPVQWPWRWTVRUOEFGGADVSNVTXAYJFXDPZGFBXDHVOXAWQWRWTKZUVAGOXAWQW
      RLZWTLUVBWSUVCWTWPWQWRWAVQWQWRWTVRUOEFGGCDWDNWEWFXAXKWLWGWHWIWJWJWHWMWKWN
      WOUI $.
  $}

  ${
    f1dom3fv3dif.v $e |- ( ph -> ( A e. X /\ B e. Y /\ C e. Z ) ) $.
    f1dom3fv3dif.n $e |- ( ph -> ( A =/= B /\ A =/= C /\ B =/= C ) ) $.
    f1dom3fv3dif.f $e |- ( ph -> F : { A , B , C } -1-1-> R ) $.
    $( The function values for a 1-1 function from a set with three different
       elements are different.  (Contributed by AV, 20-Mar-2019.) $)
    f1dom3fv3dif $p |- ( ph -> ( ( F ` A ) =/= ( F ` B )
                   /\ ( F ` A ) =/= ( F ` C ) /\ ( F ` B ) =/= ( F ` C ) ) ) $=
      ( cfv wne wcel wceq wb syl mpbird f1fveq simp1d ctp wf1 w3o 3mix1d 3mix2d
      eqidd eltpg simp2d syl12anc necon3bid simp3d tpid3g 3jca ) ABFMZCFMZNZUOD
      FMZNZUPURNZAUQBCNZAVABDNZCDNZKUAAUOUPBCABCDUBZEFUCZBVDOZCVDOZUOUPPBCPZQLA
      VFBBPZVHBDPZUDZAVIVHVJABUGUEABGOZVFVKQAVLCHOZDIOZJUABBCDGUHRSZAVGCBPZCCPZ
      CDPZUDZAVQVPVRACUGUFAVMVGVSQAVLVMVNJUICBCDHUHRSZVDEBCFTUJUKSAUSVBAVAVBVCK
      UIAUOURBDAVEVFDVDOZUOURPVJQLVOAVNWAAVLVMVNJULDIBCUMRZVDEBDFTUJUKSAUTVCAVA
      VBVCKULAUPURCDAVEVGWAUPURPVRQLVTWBVDECDFTUJUKSUN $.

    $d A x y z $.  $d B x y z $.  $d C x y z $.  $d F x y z $.  $d R x y z $.
    $( The codomain of a 1-1 function from a set with three different elements
       has (at least) three different elements.  (Contributed by AV,
       20-Mar-2019.) $)
    f1dom3el3dif $p |- ( ph -> E. x e. R E. y e. R E. z e. R
                               ( x =/= y /\ x =/= z /\ y =/= z ) ) $=
      ( wcel w3a wne wceq syl cfv cv wrex ctp wf1 wf wi f1f wa simpr w3o 3mix1d
      eqidd wb simp1d eltpg mpbird adantr ffvelcdmd 3mix2d simp2d simp3d tpid3g
      3jca expcom mpcom f1dom3fv3dif 3anbi12d 3anbi13d 3anbi23d rspc3ev syl2anc
      neeq1 neeq2 ) AEIUAZHPZFIUAZHPZGIUAZHPZQZVOVQRZVOVSRZVQVSRZQZBUBZCUBZRZWF
      DUBZRZWGWIRZQZDHUCCHUCBHUCEFGUDZHIUEZAWAOWNWMHIUFZAWAUGWMHIUHAWOWAAWOUIZV
      PVRVTWPWMHEIAWOUJZAEWMPZWOAWREESZEFSZEGSZUKZAWSWTXAAEUMULAEJPZWRXBUNAXCFK
      PZGLPZMUOEEFGJUPTUQURUSWPWMHFIWQAFWMPZWOAXFFESZFFSZFGSZUKZAXHXGXIAFUMUTAX
      DXFXJUNAXCXDXEMVAFEFGKUPTUQURUSWPWMHGIWQAGWMPZWOAXEXKAXCXDXEMVBGLEFVCTURU
      SVDVETVFAEFGHIJKLMNOVGWLWEVOWGRZVOWIRZWKQWBXMVQWIRZQBCDVOVQVSHHHWFVOSWHXL
      WJXMWKWFVOWGVMWFVOWIVMVHWGVQSXLWBWKXNXMWGVQVOVNWGVQWIVMVIWIVSSXMWCXNWDWBW
      IVSVOVNWIVSVQVNVJVKVL $.
  $}

  ${
    $d x y A $.  $d x y F $.
    $( A one-to-one function in terms of different function values for
       different arguments.  (Contributed by Alexander van der Vekens,
       26-Jan-2018.) $)
    dff14a $p |- ( F : A -1-1-> B <-> ( F : A --> B /\
             A. x e. A A. y e. A ( x =/= y -> ( F ` x ) =/= ( F ` y ) ) ) ) $=
      ( wf1 wf cv cfv wceq weq wi wral wa wne dff13 wn df-ne bicomi bitri
      con34b imbi12i 2ralbii anbi2i ) CDEFCDEGZAHZEIZBHZEIZJZABKZLZBCMACMZNUEUF
      UHOZUGUIOZLZBCMACMZNABCDEPUMUQUEULUPABCCULUKQZUJQZLUPUJUKUAURUNUSUOUNURUF
      UHRSUOUSUGUIRSUBTUCUDT $.

    $( A one-to-one function in terms of different function values for
       different arguments.  (Contributed by Alexander van der Vekens,
       26-Jan-2018.) $)
    dff14b $p |- ( F : A -1-1-> B <-> ( F : A --> B /\
             A. x e. A A. y e. ( A \ { x } ) ( F ` x ) =/= ( F ` y ) ) ) $=
      ( wf1 wf cv wne cfv wi wral wa csn cdif dff14a necom imbi1i ralbii bitri
      raldifsnb anbi2i ) CDEFCDEGZAHZBHZIZUDEJUEEJIZKZBCLZACLZMUCUGBCUDNOLZACLZ
      MABCDEPUJULUCUIUKACUIUEUDIZUGKZBCLUKUHUNBCUFUMUGUDUEQRSUGBCUDUATSUBT $.
  $}

  ${
    $d A a b x y $.  $d B x y $.  $d G a b x y $.  $d V x y $.  $d W x y $.
    $d X x y $.  $d Y x y $.
    f1ounsn.f $e |- F = ( G u. { <. X , Y >. } ) $.
    $( Extension of a bijection by an ordered pair.  (Contributed by AV,
       15-Sep-2025.) $)
    f1ounsn $p |- ( ( G : A -1-1-onto-> B /\ ( X e. V /\ Y e. W )
                      /\ ( X e/ A /\ Y e/ B ) )
                    -> F : ( A u. { X } ) -1-1-onto-> ( B u. { Y } ) ) $=
      ( vx vy wcel wa wne cfv wi wral wceq adantr syl wf1o wnel w3a csn cun cop
      va vb wf cv crn wn f1of wss ssun1 a1i fssd 3ad2ant1 df-nel biimpi anim12i
      simpl 3adant1 wo eqid olci elunsn mpbiri adantl 3ad2ant2 fsnunf wf1 f1of1
      3jca dff14a weq neeq1 fveq2 neeq1d imbi12d neeq2 neeq2d expcom sylbi impl
      rspc2va imp nelne2 necomd 3ad2ant3 fvunsn 3netr4d ex ralrimiva ffvelcdmda
      syl2anc simpr f1odm eqcomd eleq2d notbid bitrid biimpd adantrd 3adant2 wb
      cdm fsnunfv ralsng mpbird ralun jca eqneqall ax-mp ralbidv rnun wfo f1ofo
      forn rnsnopg uneq12d eqtrid dff1o5 bianbi sylibr f1oeq1 ) ABDUAZGELZHFLZM
      ZGAUBZHBUBZMZUCZAGUDZUEZBHUDZUEZDGHUFUDZUEZUAZYPYRCUAZYNYPYRYTUIZJUJZKUJZ
      NZUUDYTOZUUEYTOZNZPZKYPQZJYPQZMZYTUKZYRRZMUUAYNUUMUUOYNUUCUULYNAYRDUIZYHG
      ALZULZMZHYRLZUCUUCYNUUPUUSUUTYGYJUUPYMYGABYRDABDUMZBYRUNYGBYQUOUPUQURYJYM
      UUSYGYJYHYMUURYHYIVBZYKUURYLYKUURGAUSZUTSVAVCYJYGUUTYMYIUUTYHYIUUTHBLZHHR
      ZVDUVEUVDHVEVFHBHFVGVHVIVJVNAYRDEGHVKTYNUUKJAQUUKJYOQZUULYNUUKJAYNUUDALZM
      ZUUJKAQUUJKYOQZUUKUVHUUJKAUVHUUEALZMZUUFUUIUVKUUFMZUUDDOZUUEDOZUUGUUHUVKU
      UFUVMUVNNZYNUVGUVJUUFUVOPZYGYJUVGUVJMZUVPPZYMYGABDVLZUVRABDVMUVSABDUIZUGU
      JZUHUJZNZUWADOZUWBDOZNZPZUHAQUGAQZMUVRUGUHABDVOUWHUVRUVTUVQUWHUVPUWGUVPUU
      DUWBNZUVMUWENZPUGUHUUDUUEAAUGJVPZUWCUWIUWFUWJUWAUUDUWBVQUWKUWDUVMUWEUWAUU
      DDVRVSVTUHKVPZUWIUUFUWJUVOUWBUUEUUDWAUWLUWEUVNUVMUWBUUEDVRWBVTWFWCVIWDTUR
      WEWGUVLGUUDNZUUGUVMRZUVKUWMUUFUVHUWMUVJYNUVGUWMYMYGUVGUWMPZYJYKUWOYLYKUUR
      UWOUVCUVGUURUWMUVGUURMUUDGUUDGAWHWIWCWDSWJWGSSDGHUUDWKZTUVLGUUENZUUHUVNRZ
      UVKUWQUUFUVHUVJUWQYNUVJUWQPZUVGYMYGUWSYJYKUWSYLYKUURUWSUVCUVJUURUWQUVJUUR
      MUUEGUUEGAWHWIWCWDSWJSWGSDGHUUEWKZTWLWMWNUVHUVIUUDGNZUUGGYTOZNZPZUVHUXAUX
      CUVHUXAMZUVMHUUGUXBUVHUVMHNZUXAUVHUVMBLUVDULZUXFYNABUUDDYGYJUVTYMUVAURZWO
      YNUXGUVGYMYGUXGYJYLUXGYKYLUXGHBUSUTVIWJZSUVMHBWHWPSUXEUWMUWNUXEUUDGUVHUXA
      WQWIUWPTUXEYHYIGDXGZLZULZUCZUXBHRZUVHUXMUXAYNUXMUVGYNYHYIUXLYJYGYHYMUVBVJ
      ZYJYGYIYMYHYIWQVJYGYMUXLYJYGYMUXLYGYKUXLYLYGYKUXLYKUURYGUXLUVCYGUUQUXKYGA
      UXJGYGUXJAABDWRWSWTXAXBXCXDWGXEVNZSSDEFGHXHZTWLWMUVHYHUVIUXDXFYNYHUVGUXOS
      UUJUXDKGEUUEGRZUUFUXAUUIUXCUUEGUUDWAUXRUUHUXBUUGUUEGYTVRZWBVTXITXJUUJKAYO
      XKWPWNYNUVFUWQUXBUUHNZPZKYPQZYNUYAKAQUYAKYOQZUYBYNUYAKAYNUVJMZUWQUXTUYDUW
      QMZHUVNUXBUUHUYEUVNBLZUXGMZHUVNNUYDUYGUWQUYDUYFUXGYNABUUEDUXHWOYNUXGUVJUX
      ISXLSUYGUVNHUVNHBWHWITUYEUXMUXNUYDUXMUWQYNUXMUVJUXPSSUXQTUWQUWRUYDUWTVIWL
      WMWNYNUYCGGNZUXBUXBNZPZGGRUYJGVEUYIGGXMXNYNYHUYCUYJXFUXOUYAUYJKGEUXRUWQUY
      HUXTUYIUUEGGWAUXRUUHUXBUXBUXSWBVTXITVHUYAKAYOXKWPYNYHUVFUYBXFUXOUUKUYBJGE
      UUDGRZUUJUYAKYPUYKUUFUWQUUIUXTUUDGUUEVQUYKUUGUXBUUHUUDGYTVRVSVTXOXITXJUUK
      JAYOXKWPXLYNUUNDUKZYSUKZUEYRDYSXPYNUYLBUYMYQYGYJUYLBRZYMYGABDXQUYNABDXRAB
      DXSTURYJYGUYMYQRZYMYHUYOYIGHEXTSVJYAYBXLUUAYPYRYTVLUUOUUMYPYRYTYCJKYPYRYT
      VOYDYECYTRUUBUUAXFIYPYRCYTYFXNYE $.
  $}

  ${
    $d x y A $.  $d x y F $.  $d x y X $.  $d x y Y $.
    f12dfv.a $e |- A = { X , Y } $.
    $( A one-to-one function with a domain with at least two different elements
       in terms of function values.  (Contributed by Alexander van der Vekens,
       2-Mar-2018.) $)
    f12dfv $p |- ( ( ( X e. U /\ Y e. V ) /\ X =/= Y )
      -> ( F : A -1-1-> B <-> ( F : A --> B /\ ( F ` X ) =/= ( F ` Y ) ) ) ) $=
      ( vx vy cfv wne csn cdif wral wa wb wceq fveq2 adantr wf1 wcel dff14b cpr
      wf cv raleqi sneq difeq2d neeq1d raleqbidv ralprg difeq1i difprsn1 eqtrid
      adantl raleqdv neeq2d ralsng bitrd difprsn2 anbi12d necom pm4.71i bitr4di
      biimpi bitrid anbi2d ) ABDUAABDUEZIUFZDKZJUFZDKZLZJAVJMZNZOZIAOZPFCUBZGEU
      BZPZFGLZPZVIFDKZGDKZLZPIJABDUCWCVRWFVIVRVQIFGUDZOZWCWFVQIAWGHUGWCWHWDVMLZ
      JAFMZNZOZWEVMLZJAGMZNZOZPZWFWAWHWQQWBVQWLWPIFGCEVJFRZVNWIJVPWKWRVOWJAVJFU
      HUIWRVKWDVMVJFDSUJUKVJGRZVNWMJVPWOWSVOWNAVJGUHUIWSVKWEVMVJGDSUJUKULTWCWQW
      FWEWDLZPWFWCWLWFWPWTWCWLWIJWNOZWFWCWIJWKWNWBWKWNRWAWBWKWGWJNWNAWGWJHUMFGU
      NUOUPUQWAXAWFQZWBVTXBVSWIWFJGEVLGRVMWEWDVLGDSURUSUPTUTWCWPWMJWJOZWTWCWMJW
      OWJWBWOWJRWAWBWOWGWNNWJAWGWNHUMFGVAUOUPUQWAXCWTQZWBVSXDVTWMWTJFCVLFRVMWDW
      EVLFDSURUSTTUTVBWFWTWFWTWDWEVCVFVDVEUTVGVHVG $.
  $}

  ${
    $d x y A $.  $d x y F $.  $d x y X $.  $d x y Y $.  $d x y Z $.
    f13dfv.a $e |- A = { X , Y , Z } $.
    $( A one-to-one function with a domain with at least three different
       elements in terms of function values.  (Contributed by Alexander van der
       Vekens, 26-Jan-2018.) $)
    f13dfv $p |- ( ( ( X e. U /\ Y e. V /\ Z e. W )
                     /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) )
        -> ( F : A -1-1-> B <-> ( F : A --> B /\ ( ( F ` X ) =/= ( F ` Y )
              /\ ( F ` X ) =/= ( F ` Z ) /\ ( F ` Y ) =/= ( F ` Z ) ) ) ) ) $=
      ( vx vy wne cdif wral wa w3a wceq fveq2 necom wf1 cfv csn wcel dff14b ctp
      wf cv raleqi wb sneq difeq2d neeq1d raleqbidv raltpg adantr difeq1i tprot
      cpr anbi12i biimpi 3adant3 diftpsn3 syl eqtrid adantl neeq2d ralprg bitrd
      raleqdv 3adant1 tpcomb anim2i 3adant2 3anbi123d ancom 3anbi2i 3an6 3anrot
      bicomi 3anbi123i anidm 3ancoma bitri 3bitri bitrdi bitrid anbi2d ) ABDUAA
      BDUGZKUHZDUBZLUHZDUBZMZLAWJUCZNZOZKAOZPGCUDZHEUDZIFUDZQZGHMZGIMZHIMZQZPZW
      IGDUBZHDUBZMZXHIDUBZMZXIXKMZQZPKLABDUEXGWRXNWIWRWQKGHIUFZOZXGXNWQKAXOJUIX
      GXPXHWMMZLAGUCZNZOZXIWMMZLAHUCZNZOZXKWMMZLAIUCZNZOZQZXNXBXPYIUJXFWQXTYDYH
      KGHICEFWJGRZWNXQLWPXSYJWOXRAWJGUKULYJWKXHWMWJGDSUMUNWJHRZWNYALWPYCYKWOYBA
      WJHUKULYKWKXIWMWJHDSUMUNWJIRZWNYELWPYGYLWOYFAWJIUKULYLWKXKWMWJIDSUMUNUOUP
      XGYIXJXLPZXIXHMZXMPZXKXHMZXKXIMZPZQZXNXGXTYMYDYOYHYRXGXTXQLHIUSZOZYMXGXQL
      XSYTXFXSYTRXBXFXSXOXRNZYTAXOXRJUQXFUUBHIGUFZXRNZYTXOUUCXRGHIURUQXFHGMZIGM
      ZPZUUDYTRXCXDUUGXEXCXDPUUGXCUUEXDUUFGHTGITUTVAVBHIGVCVDVEVEVFVJXBUUAYMUJZ
      XFWTXAUUHWSXQXJXLLHIEFWLHRZWMXIXHWLHDSZVGWLIRZWMXKXHWLIDSZVGVHVKUPVIXGYDY
      ALGIUSZOZYOXGYALYCUUMXFYCUUMRXBXFYCXOYBNZUUMAXOYBJUQXFUUOGIHUFZYBNZUUMXOU
      UPYBGHIVLUQXFXCIHMZPZUUQUUMRXCXEUUSXDXEUURXCXEUURHITVAVMVNGIHVCVDVEVEVFVJ
      XBUUNYOUJZXFWSXAUUTWTYAYNXMLGICFWLGRZWMXHXIWLGDSZVGUUKWMXKXIUULVGVHVNUPVI
      XGYHYELGHUSZOZYRXGYELYGUVCXFYGUVCRXBXFYGXOYFNZUVCAXOYFJUQXDXEUVEUVCRXCGHI
      VCVKVEVFVJXBUVDYRUJZXFWSWTUVFXAYEYPYQLGHCEUVAWMXHXKUVBVGUUIWMXIXKUUJVGVHV
      BUPVIVOYSYMXMYNPZYRQXJXMYPQZXLYNYQQZPZXNYOUVGYMYRYNXMVPVQXJXLXMYNYPYQVRUV
      JYPXJXMQZUVKPUVKXNUVHUVKUVIUVKUVKUVHYPXJXMVSVTXLYPYNXJYQXMXHXKTXIXHTXKXIT
      WAUTUVKWBUVKXJYPXMQXNYPXJXMWCYPXLXJXMXKXHTVQWDWEWEWFVIWGWHWG $.
  $}

  ${
    $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 $.

  $( An involution is a bijection.  (Contributed by Thierry Arnoux,
     7-Dec-2016.) $)
  nvof1o $p |- ( ( F Fn A /\ `' F = F ) -> F : A -1-1-onto-> A ) $=
    ( wfn ccnv wceq wa wf1 wfo wf1o wfun cdm crn fnfun fdmrn sylib adantr df-rn
    wf fndm sylanbrc eqtrid sylan9eqr feq23d mpbid wb funeq adantl mpbird df-f1
    dmeq simpl df-fo df-f1o ) BACZBDZBEZFZAABGZAABHZAABIUQAABRZUOJZURUQBKZBLZBR
    ZUTUNVDUPUNBJZVDABMZBNOPUQVBVCAABUNVBAEUPABSZPUPUNVCVBAUPVCUOKVBBQUOBUJUAVG
    UBZUCUDUQVAVEUNVEUPVFPUPVAVEUEUNUOBUFUGUHAABUITUQUNVCAEUSUNUPUKVHAABULTAABU
    MT $.

  ${
    $d x y z A $.  $d x y z F $.
    $( The converse of an involution is the function itself.  (Contributed by
       Thierry Arnoux, 7-May-2019.) $)
    nvocnv $p |- ( ( F : A --> A /\ A. x e. A ( F ` ( F ` x ) ) = x )
      -> `' F = F ) $=
      ( vz vy cv cfv wceq cmpt ccnv wcel simprr simpll simprl ffvelcdmd eqeltrd
      wa fveq2d 2fveq3 id wral eqeq12d simplr rspcdva eqtr2d jca impbida mptcnv
      wf wfn ffn dffn5 biimpi adantr sylan cnveqd 3eqtr4d ) BBCUIZAFZCGCGZUSHZA
      BUAZQZDBDFZCGZIZJEBEFZCGZIZCJCVCDEBVEBVHVCVDBKZVGVEHZQZVGBKZVDVHHZQZVCVLQ
      ZVMVNVPVGVEBVCVJVKLZVPBBVDCURVBVLMVCVJVKNZOPVPVHVECGZVDVPVGVECVQRVPVAVSVD
      HABVDUSVDHZUTVSUSVDUSVDCCSVTTUBURVBVLUCVRUDUEUFVCVOQZVJVKWAVDVHBVCVMVNLZW
      ABBVGCURVBVOMVCVMVNNZOPWAVEVHCGZVGWAVDVHCWBRWAVAWDVGHABVGUSVGHZUTWDUSVGUS
      VGCCSWETUBURVBVOUCWCUDUEUFUGUHVCCVFURCBUJZVBCVFHZBBCUKZWFWGVBWFWGDBCULUMU
      NUOUPURWFVBCVIHZWHWFWIVBWFWIEBCULUMUNUOUQ $.
  $}

  ${
    $d A x y $.  $d A y z $.  $d B y z $.  $d F y z $.
    $( If a one-to-one function with a nonempty domain has a singleton as its
       codomain, its domain must also be a singleton.  (Contributed by
       BTernaryTau, 1-Dec-2024.) $)
    f1cdmsn $p |- ( ( F : A -1-1-> { B } /\ A =/= (/) ) -> E. x A = { x } ) $=
      ( vy vz csn wf1 c0 wne wa cv wceq wral wrex wex wcel w3a cfv fvconst issn
      wf f1f 3adant3 3adant2 eqtr4d syl3an1 f1veqaeq 3impb mpd 3expia reximdva0
      wi ralrimiv syl ) BCGZDHZBIJKELZFLZMZFBNZEBOBALGMAPUQVAEBUQURBQZKUTFBUQVB
      USBQZUTUQVBVCRURDSZUSDSZMZUTUQBUPDUBZVBVCVFBUPDUCVGVBVCRVDCVEVGVBVDCMVCBC
      URDTUDVGVCVECMVBBCUSDTUEUFUGUQVBVCVFUTUMBUPURUSDUHUIUJUKUNULEFABUAUO $.
  $}

  ${
    $d A f x $.  $d D f x $.  $d V f x $.  $d f ps $.  $d ph x $.
    $( TODO: there must be a more elegant way to prove this theorem using
       ~ exxfr $)
    fsnex.1 $e |- ( x = ( f ` A ) -> ( ps <-> ph ) ) $.
    $( Relate a function with a singleton as domain and one variable.
       (Contributed by Thierry Arnoux, 12-Jul-2020.) $)
    fsnex $p |- ( A e. V
      -> ( E. f ( f : { A } --> D /\ ph ) <-> E. x e. D ps ) ) $=
      ( wcel csn cv wf wa wex cfv cop wceq ex cvv elvd wrex simprbda adantrr wb
      fsn2g adantl simprr rspcedvd exlimdv imp nfre1 nfan wf1o f1osng ad3antrrr
      nfv f1of syl simplr snssd fssd fvsng eqcomd snex feq1 fveq1 anbi12d spcev
      eqeq2d syl2anc simprl simpllr simplrr mpbid mpdan jca eximdv mpd r19.29af
      simpr impbida ) DGIZDJZEFKZLZAMZFNZBCEUAZWBWGWHWBWFWHFWBWFWHWBWFMZBACDWDO
      ZEWBWEWJEIZAWBWEWKWDDWJPJQDEWDGUEUBUCCKZWJQZBAUDZWIHUFWBWEAUGUHRUIUJWBWHM
      ZBWGCEWBWHCWBCUPBCEUKULWOWLEIZMZBMZWEWMMZFNZWGWRWCEDWLPZJZLZWLDXBOZQZWTWR
      WCWLJZEXBWRWCXFXBUMZWCXFXBLWBXGWHWPBWBXGCDWLGSUNTUOWCXFXBUQURWRWLEWOWPBUS
      UTVAWBXEWHWPBWBXDWLWBXDWLQCDWLGSVBTVCUOWSXCXEMFXBXAVDWDXBQZWEXCWMXEWCEWDX
      BVEXHWJXDWLDWDXBVFVIVGVHVJWRWSWFFWRWSWFWRWSMZWEAWRWEWMVKZXIWEAXJXIWEMZBAW
      QBWSWEVLXKWMWNWRWEWMWEVMHURVNVOVPRVQVRWBWHVTVSWA $.
  $}

  ${
    $d A f x y $.  $d B f x y $.  $d D f x y $.  $d V f x y $.  $d W f x y $.
    $d ch x $.  $d ps f $.  $d ph x y $.
    f1prex.1 $e |- ( x = ( f ` A ) -> ( ps <-> ch ) ) $.
    f1prex.2 $e |- ( y = ( f ` B ) -> ( ch <-> ph ) ) $.
    $( Relate a one-to-one function with a pair as domain and two different
       variables.  (Contributed by Thierry Arnoux, 12-Jul-2020.) $)
    f1prex $p |- ( ( A e. V /\ B e. W /\ A =/= B )
      -> ( E. f ( f : { A , B } -1-1-> D /\ ph )
        <-> E. x e. D E. y e. D ( x =/= y /\ ps ) ) ) $=
      ( wcel wne w3a wa syl wceq jca cpr cv wf1 wex cfv wf simpl1 simpl2 simprl
      f1f cop fpr2g biimpa simp1d syl21anc simp2d prid1g prid2g simpl3 f1veqaeq
      wrex necon3d imp simprr neeq1 anbi12d rspc2ev syl3anc ex exlimdv wss wf1o
      neeq2 simpll1 simplrl simpll2 simplrr simpll3 f1oprg syl22anc f1of1 prssd
      f1ss syl2anc fvpr1g eqcomd fvpr2g prex f1eq1 fveq1 spcev syl12anc simprrl
      eqeq2d wb mpbid simprrr eximdv mpd rexlimdvva impbid ) FJNZGKNZFGOZPZFGUA
      ZHIUBZUCZAQZIUDZDUBZEUBZOZBQZEHVADHVAZXEXIXOIXEXIXOXEXIQZFXGUEZHNZGXGUEZH
      NZXQXSOZAQZXOXPXBXCXFHXGUFZXRXBXCXDXIUGZXBXCXDXIUHZXPXHYCXEXHAUIZXFHXGUJR
      ZXBXCQZYCQZXRXTXGFXQUKGXSUKUASZYHYCXRXTYJPFGHXGJKULUMZUNUOXPXBXCYCXTYDYEY
      GYIXRXTYJYKUPUOXPYAAXPXHFXFNZGXFNZQZXDYAYFXPYLYMXPXBYLYDFGJUQRXPXCYMYEFGK
      URRTXBXCXDXIUSXHYNQZXDYAYOXQXSFGXFHFGXGUTVBVCUOXEXHAVDTXNYBXQXLOZCQDEXQXS
      HHXKXQSZXMYPBCXKXQXLVELVFXLXSSZYPYACAXLXSXQVMMVFVGVHVIVJXEXNXJDEHHXEXKHNZ
      XLHNZQZQZXNXJUUBXNQZXHYQYRQZQZIUDZXJUUCXFHFXKUKZGXLUKZUAZUCZXKFUUIUEZSZXL
      GUUIUEZSZUUFUUCXFXKXLUAZUUIUCZUUOHVKUUJUUCXFUUOUUIVLZUUPUUCXBYSQZXCYTQZXD
      XMUUQUUCXBYSXBXCXDUUAXNVNZXEYSYTXNVOZTUUCXCYTXBXCXDUUAXNVPZXEYSYTXNVQZTXB
      XCXDUUAXNVRZUUBXMBUIUURUUSQXDXMQUUQFXKGXLJHKHVSVCVTXFUUOUUIWARUUCXKXLHUVA
      UVCWBXFUUOHUUIWCWDUUCXBYSXDUULUUTUVAUVDXBYSXDPUUKXKFGXKXLJHWEWFVHUUCXCYTX
      DUUNUVBUVCUVDXCYTXDPUUMXLFGXKXLKHWGWFVHUUEUUJUULUUNQZQIUUIUUGUUHWHXGUUISZ
      XHUUJUUDUVEXFHXGUUIWIUVFYQUULYRUUNUVFXQUUKXKFXGUUIWJWNUVFXSUUMXLGXGUUIWJW
      NVFVFWKWLUUCUUEXIIUUCUUEXIUUCUUEQZXHAUUCXHUUDUIUVGCAUVGBCUUBXMBUUEVQUVGYQ
      BCWOUUCXHYQYRWMLRWPUVGYRCAWOUUCXHYQYRWQMRWPTVIWRWSVIWTXA $.
  $}

  $( 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 fvco3 fvresi 3eqtr3rd fveq2 rspceeqv syl2anc
      adantl ralrimiva dffo3 sylanbrc ) ABDGZBACGZDCHZIBJZKZLZUOETZFTZDMZKFANZE
      BOABDPUOUPUSQUTVDEBUTVABRZUAZVACMZARZVAVGDMZKVDUPUOVEVHUSBAVACUBSVFVAUQMZ
      VAURMZVIVAVFVAUQURUOUPUSVEUCUDUPUOVEVJVIKUSBAVADCUESVEVKVAKUTBVAUFUKUGFVG
      AVCVIVAVBVGDUHUIUJULFEABDUMUN $.
  $}

  ${
    $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 wn wral wrex cv cfv wceq notbid cbvfo dfrex2 3bitr4g ) EFGIZAJZCEKZ
      JBJZDFKZJACELBDFLTUBUDUAUCCDEFGCMGNDMOABHPQPACERBDFRS $.
  $}

  ${
    $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 3ad2antl3 wb ffnd syl2anc
      wfn wf1 w3a cv eqeq12d simpl1 ffvelcdm f1fveq syl12anc bitrd ralbidva f1f
      wa 3ad2ant1 simp2 fnfco simp3 eqfnfv 3bitr4d ) BCDUAZABEHZABFHZUBZGUCZDEI
      ZJZVCDFIZJZKZGALZVCEJZVCFJZKZGALZVDVFKZEFKZVBVHVLGAVBVCAMZULZVHVJDJZVKDJZ
      KZVLVQVEVRVGVSUTUSVPVEVRKVAABVCDENOVAUSVPVGVSKUTABVCDFNPUDVQUSVJBMZVKBMZV
      TVLQUSUTVAVPUEUTUSVPWAVAABVCEUFOVAUSVPWBUTABVCFUFPBCVJVKDUGUHUIUJVBVDATZV
      FATZVNVIQVBDBTZUTWCVBBCDUSUTBCDHVABCDUKUMRZUSUTVAUNZBADEUOSVBWEVAWDWFUSUT
      VAUPZBADFUOSGAVDVFUQSVBEATFATVOVMQVBABEWGRVBABFWHRGAEFUQSUR $.
  $}

  ${
    $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 $.
  $}

  ${
    fcof1oinvd.f $e |- ( ph -> F : A -1-1-onto-> B ) $.
    fcof1oinvd.g $e |- ( ph -> G : B --> A ) $.
    fcof1oinvd.b $e |- ( ph -> ( F o. G ) = ( _I |` B ) ) $.
    $( Show that a function is the inverse of a bijective function if their
       composition is the identity function.  Formerly part of proof of
       ~ fcof1o .  (Contributed by Mario Carneiro, 21-Mar-2015.)  (Revised by
       AV, 15-Dec-2019.) $)
    fcof1oinvd $p |- ( ph -> `' F = G ) $=
      ( ccnv ccom cid cres coeq2d coass wf1o wceq f1ococnv1 syl coeq1d wf fcoi2
      eqtrd eqtr3id f1ocnv f1of fcoi1 4syl 3eqtr3rd ) ADIZDEJZJZUIKCLZJZEUIAUJU
      LUIHMAUKUIDJZEJZEUIDENAUOKBLZEJZEAUNUPEABCDOZUNUPPFBCDQRSACBETUQEPGCBEUAR
      UBUCAURCBUIOCBUITUMUIPFBCDUDCBUIUECBUIUFUGUH $.
  $}

  ${
    fcof1od.f $e |- ( ph -> F : A --> B ) $.
    fcof1od.g $e |- ( ph -> G : B --> A ) $.
    fcof1od.a $e |- ( ph -> ( G o. F ) = ( _I |` A ) ) $.
    fcof1od.b $e |- ( ph -> ( F o. G ) = ( _I |` B ) ) $.
    $( A function is bijective if a "retraction" and a "section" exist, see
       comments for ~ fcof1 and ~ fcofo .  Formerly part of proof of ~ fcof1o .
       (Contributed by Mario Carneiro, 21-Mar-2015.)  (Revised by AV,
       15-Dec-2019.) $)
    fcof1od $p |- ( ph -> F : A -1-1-onto-> B ) $=
      ( wf1 wfo wf1o wf ccom cid cres wceq fcof1 syl2anc fcofo syl3anc sylanbrc
      df-f1o ) ABCDJZBCDKZBCDLABCDMZEDNOBPQUDFHBCEDRSAUFCBEMDENOCPQUEFGIBCEDTUA
      BCDUCUB $.

    $( Show that a function is the inverse of a function if their compositions
       are the identity functions.  (Contributed by Mario Carneiro,
       21-Mar-2015.)  (Revised by AV, 15-Dec-2019.) $)
    2fcoidinvd $p |- ( ph -> `' F = G ) $=
      ( fcof1od fcof1oinvd ) ABCDEABCDEFGHIJGIK $.
  $}

  $( Show that two functions are inverse to each other by computing their
     compositions.  (Contributed by Mario Carneiro, 21-Mar-2015.)  (Proof
     shortened by AV, 15-Dec-2019.) $)
  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 ccnv simpll simplr simprr simprl 2fcoidinvd
    fcof1od jca ) ABCEZBADEZFZCDGHBIJZDCGHAIJZFZFZABCKCLDJUFABCDTUAUEMZTUAUENZU
    BUCUDOZUBUCUDPZRUFABCDUGUHUIUJQS $.

  ${
    $d A a x $.  $d B x $.  $d F a x $.  $d G a x $.  $d ph x $.
    2fvcoidd.f $e |- ( ph -> F : A --> B ) $.
    2fvcoidd.g $e |- ( ph -> G : B --> A ) $.
    2fvcoidd.i $e |- ( ph -> A. a e. A ( G ` ( F ` a ) ) = a ) $.
    $( Show that the composition of two functions is the identity function by
       applying both functions to each value of the domain of the first
       function.  (Contributed by AV, 15-Dec-2019.) $)
    2fvcoidd $p |- ( ph -> ( G o. F ) = ( _I |` A ) ) $=
      ( vx ccom cv cfv cmpt cid cres wf wceq fcompt syl2anc wcel wral wi 2fveq3
      weq id eqeq12d rspccv syl imp mpteq2dva mptresid eqtr4di eqtrd ) AEDKZJBJ
      LZDMEMZNZOBPZACBEQBCDQUOURRHGJEDBCBSTAURJBUPNUSAJBUQUPAUPBUAZUQUPRZAFLZDM
      EMZVBRZFBUBUTVAUCIVDVAFUPBFJUEZVCUQVBUPVBUPEDUDVEUFUGUHUIUJUKJBULUMUN $.

    $d B b $.  $d F b $.  $d G b $.
    2fvidf1od.i $e |- ( ph -> A. b e. B ( F ` ( G ` b ) ) = b ) $.
    $( A function is bijective if it has an inverse function.  (Contributed by
       AV, 15-Dec-2019.) $)
    2fvidf1od $p |- ( ph -> F : A -1-1-onto-> B ) $=
      ( 2fvcoidd fcof1od ) ABCDEHIABCDEFHIJLACBEDGIHKLM $.

    $( Show that two functions are inverse to each other by applying them twice
       to each value of their domains.  (Contributed by AV, 13-Dec-2019.) $)
    2fvidinvd $p |- ( ph -> `' F = G ) $=
      ( 2fvcoidd 2fcoidinvd ) ABCDEHIABCDEFHIJLACBEDGIHKLM $.
  $}

  ${
    $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 fofn ad2antlr cv wcel cfv sylibr
      wbr cid wi fococnv2 cnveq coeq2d eqeq1d syl5ibcom ad2antrr wex cop adantl
      cres fnopfv sylan vex brcnv df-br bitri breq2 breq1 anbi12d spcev syl2anc
      fvex weq brco adantlr wb mpbid wf fof ffvelcdmda resieq eqcomd eqfnfvd ex
      breq impbid ) ABCGZABDGZHZCDIZCDJZKZUABULZIZVSWBWFUBVTVSCCJZKZWEIWBWFABCU
      CWBWHWDWEWBWGWCCCDUDUEUFUGLWAWFWBWAWFHZEACDVSCAMZVTWFABCNZUHVTDAMZVSWFABD
      NZOWIEPZAQZHZWNDRZWNCRZWPWQWRWETZWQWRIZWPWQWRWDTZWSWAWOXAWFWAWOHZWQFPZWCT
      ZXCWRCTZHZFUIZXAXBWQWNWCTZWNWRCTZXGXBWNWQUJDQZXHWAWLWOXJVTWLVSWMUKAWNDUMU
      NXHWNWQDTXJWQWNDWNDVDZEUOZUPWNWQDUQURSXBWNWRUJCQZXIWAWJWOXMVSWJVTWKLAWNCU
      MUNWNWRCUQSXFXHXIHFWNXLFEVEXDXHXEXIXCWNWQWCUSXCWNWRCUTVAVBVCFWQWRCWCXKWNC
      VDVFSVGWFXAWSVHWAWOWQWRWDWEVQOVIWAWOWSWTVHZWFXBWQBQWRBQXNWAABWNDVTABDVJVS
      ABDVKUKVLWAABWNCVSABCVJVTABCVKLVLBWQWRVMVCVGVIVNVOVPVR $.

    $( Condition for function equality in terms of vanishing of the composition
       with the inverse.  (Contributed by Stefan O'Rear, 12-Feb-2015.)  (Proof
       shortened by Wolf Lammen, 29-May-2024.) $)
    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 cfv wb cid
      ccnv cres f1cocnv1 coeq2 eqeq1d syl5ibcom equid resieq mpbiri anidms breq
      ad2antlr mpbird wex cop wfun cdm fnfun syl fndmd eleq2d funopfvb syl2an2r
      biimpar bicomd df-br eqcom 3bitr4g biimpd bitr4id vex anim12d eximdv brco
      brcnv fvex eqvinc 3imtr4g adantlr mpd eqfnfvd eqcomd ex impbid ) ABCGZABD
      GZHZCDIZCUBZDJZUAAUCZIZWFWIWMKWGWFWJCJZWLIWIWMABCUDWIWNWKWLCDWJUEUFUGLWHW
      MWIWHWMHZDCWOEADCWHDAMZWMWGWPWFABDNOZLWHCAMZWMWFWRWGABCNLZLWOEPZAQZHZWTWT
      WKRZWTDSZWTCSZIZXBXCWTWTWLRZXAXGWOXAXGXAXAHXGWTWTIEUHAWTWTUIUJUKOWMXCXGTW
      HXAWTWTWKWLULUMUNWHXAXCXFKWMWHXAHZWTFPZDRZXIWTWJRZHZFUOXIXDIZXIXEIZHZFUOX
      CXFXHXLXOFXHXJXMXKXNXHXJXMXHWTXIUPZDQZXDXIIZXJXMXHXRXQWHDUQZXAWTDURZQZXRX
      QTWHWPXSWQADUSUTWHYAXAWHXTAWTWHADWQVAVBVEWTXIDVCVDVFWTXIDVGXIXDVHVIVJXHXK
      XNXHWTXICRZXEXIIZXKXNXHYBXPCQZYCWTXICVGWHCUQZXAWTCURZQZYCYDTWHWRYEWSACUSU
      TWHYGXAWHYFAWTWHACWSVAVBVEWTXICVCVDVKXIWTCFVLEVLZVPXIXEVHVIVJVMVNFWTWTWJD
      YHYHVOFXDXEWTDVQVRVSVTWAWBWCWDWE $.
  $}

  ${
    fveqf1o.1 $e |- G = ( F o. ( ( _I |` ( A \ { C , ( `' F ` D ) } ) ) u.
      { <. C , ( `' F ` D ) >. , <. ( `' F ` D ) , C >. } ) ) $.
    $( Given a bijection ` F ` , produce another bijection ` G ` which
       additionally maps two specified points.  (Contributed by Mario Carneiro,
       30-May-2015.) $)
    fveqf1o $p |- ( ( F : A -1-1-onto-> B /\ C e. A /\ D e. B ) ->
      ( G : A -1-1-onto-> B /\ ( G ` C ) = D ) ) $=
      ( wf1o wcel cfv wceq cpr cop cun a1i wf f1of syl2anc syl eqtrd ccnv simp1
      w3a cid cdif cres ccom cin c0 f1oi simp2 f1ocnv simp3 ffvelcdmd f1oprswap
      3syl disjdifr f1oun syl22anc uncom prssd undif sylib eqtrid f1oeq2d mpbid
      wss f1oeq3d f1oco wb f1oeq1 ax-mp sylibr fveq1i fvco3 fnresi f1ofn prid1g
      wfn fvun2 syl112anc wfun f1ofun prid1 funopfv mpisyl fveq2d f1ocnvfv2 jca
      opex ) ABEHZCAIZDBIZUCZABFHZCFJZDKWNABEUDACDEUAZJZLZUEZUFZCWRMZWRCMZLZNZU
      GZHZWOWNWKAAXEHZXGWKWLWMUBZWNAWTWSNZXEHZXHWNXJXJXEHZXKWNWTWTXAHZWSWSXDHZW
      TWSUHUIKZXOXLXMWNWTUJOWNWLWRAIXNWKWLWMUKZWNBADWQWNWKBAWQHBAWQPXIABEULBAWQ
      QUPWKWLWMUMZUNZCWRAAUORZXOWNWSAUQOZXTWTWTWSWSXAXDURUSWNXJAXJXEWNXJWSWTNZA
      WTWSUTWNWSAVGYAAKWNCWRAXPXRVAWSAVBVCVDZVEVFWNXJAAXEYBVHVFZAABEXEVIRFXFKWO
      XGVJGABFXFVKVLVMWNWPCXEJZEJZDWNWPCXFJZYECFXFGVNWNAAXEPZWLYFYEKWNXHYGYCAAX
      EQSXPAACEXEVORVDWNYEWREJZDWNYDWREWNYDCXDJZWRWNXAWTVSZXDWSVSZXOCWSIZYDYIKY
      JWNWTVPOWNXNYKXSWSWSXDVQSXTWNWLYLXPCWRAVRSWTWSXAXDCVTWAWNXDWBZXBXDIYIWRKW
      NXNYMXSWSWSXDWCSXBXCCWRWJWDCWRXDWEWFTWGWNWKWMYHDKXIXQABDEWHRTTWI $.
  $}

  $( The composition of two bijections as bijection onto the image of the range
     of the first bijection.  (Contributed by AV, 15-Aug-2025.) $)
  f1ocoima $p |- ( ( F : A -1-1-onto-> B /\ G : C -1-1-onto-> D /\ B C_ C )
                   -> ( G o. F ) : A -1-1-onto-> ( G " B ) ) $=
    ( wf1o wss w3a cima ccom cres wf1 wa f1of1 anim1i 3adant1 f1ores syl wceq
    simp1 f1oco syl2anc crn wfo f1ofo forn eqimssd 3ad2ant1 cores eqcomd mpbird
    f1oeq1d ) ABEGZCDFGZBCHZIZAFBJZFEKZGAURFBLZEKZGZUQBURUTGZUNVBUQCDFMZUPNZVCU
    OUPVEUNUOVDUPCDFOPQCDBFRSUNUOUPUAABURUTEUBUCUQAURUSVAUQEUDZBHZUSVATUNUOVGUP
    UNVFBUNABEUEVFBTABEUFABEUGSUHUIVGVAUSFEBUJUKSUMUL $.

  ${
    $d A x y $.  $d F x y $.  $d X x y $.  $d Y x y $.
    $( A constant function from at least two elements is not one-to-one.
       (Contributed by AV, 30-Mar-2024.) $)
    nf1const $p |- ( ( F : A --> { B } /\ ( X e. A /\ Y e. A /\ X =/= Y ) )
                   -> -. F : A -1-1-> C ) $=
      ( vx vy wf wcel wa wn cv cfv wceq wi wral wrex fvconst sylan2 csn wne w3a
      weq wo wf1 simp1 simp2 eqtr4d neneq 3ad2ant3 adantl fveqeq2 eqeq1 imbi12d
      jcnd notbid fveq2 eqeq2 rspc2ev syl2an23an rexnal2 sylib olcd ianor dff13
      eqeq2d xchnxbir sylibr ) ABUADIZEAJZFAJZEFUBZUCZKZACDIZLZGMZDNHMZDNZOZGHU
      DZPZHAQGAQZLZUEZACDUFZLVOWEVQVOWCLZHARGARZWEVNVKVLVJEDNZFDNZOZEFOZPZLZWIV
      KVLVMUGZVKVLVMUHZVOWLWMVOWJBWKVNVJVKWJBOWPABEDSTVNVJVLWKBOWQABFDSTUIVNWML
      ZVJVMVKWRVLEFUJUKULUPWHWOWJVTOZEVSOZPZLGHEFAAVREOZWCXAXBWAWSWBWTVREVTDUMV
      REVSUNUOUQVSFOZXAWNXCWSWLWTWMXCVTWKWJVSFDURVGVSFEUSUOUQUTVAWCGHAAVBVCVDVP
      WDKWFWGVPWDVEGHACDVFVHVI $.
  $}

  $( A constant function from at least two elements is not bijective.
     (Contributed by AV, 30-Mar-2024.) $)
  nf1oconst $p |- ( ( F : A --> { B } /\ ( X e. A /\ Y e. A /\ X =/= Y ) )
                    -> -. F : A -1-1-onto-> C ) $=
    ( csn wf wcel wne w3a wa wf1 wn wfo wo wf1o nf1const orcd ianor xchnxbir
    df-f1o sylibr ) ABGDHEAIFAIEFJKLZACDMZNZACDOZNZPZACDQZNUDUFUHABCDEFRSUEUGLU
    IUJUEUGTACDUBUAUC $.

  $( Swapping two values in a bijection between two classes yields another
     bijection between those classes.  (Contributed by BTernaryTau,
     31-Aug-2024.) $)
  f1ofvswap $p |- ( ( F : A -1-1-onto-> B /\ X e. A /\ Y e. A ) ->
      ( ( F |` ( A \ { X , Y } ) ) u.
      { <. X , ( F ` Y ) >. , <. Y , ( F ` X ) >. } ) : A -1-1-onto-> B ) $=
    ( wf1o wcel w3a cpr cres cop cun ccom cfv wa wceq csn cxp xpsng cvv cid cin
    cdif f1oi f1oprswap c0 disjdifr f1oun mpanr12 sylancr wb prssi undifr sylib
    wss f1oeq23 syl2anc mpbid f1oco sylan2 3impb coundi fcoconst 3adant2 coeq2d
    wfn f1ofn 3adant1 fvex mpan2 3ad2ant2 3eqtr3d 3adant3 ancoms 3ad2ant3 df-pr
    uneq12d coeq2i 3eqtr4g uneq2d eqtrid coires1 uneq1i eqtrdi syl3an1 f1oeq1d
    eqtri ) ABCFZDAGZEAGZHZABCUAADEIZUCZJZDEKZEDKZIZLZMZFZABCWMJZDECNZKZEDCNZKZ
    IZLZFWHWIWJWTWIWJOZWHAAWRFZWTXHWMWLLZXJWRFZXIXHWMWMWNFZWLWLWQFZXKWMUDDEAAUE
    XLXMOWMWLUBUFPZXNXKWLAUGZXOWMWMWLWLWNWQUHUIUJXHXJAPZXPXKXIUKXHWLAUOXPDEAULW
    LAUMUNZXQXJAXJAWRUPUQURAABCWRUSUTVAWKABWSXGWHCAVFZWIWJWSXGPABCVGXRWIWJHZWSC
    WNMZXFLZXGXSWSXTCWQMZLYACWNWQVBXSYBXFXTXSCWOQZMZCWPQZMZLZXCQZXEQZLYBXFXSYDY
    HYFYIXSCDQZEQZRZMZYJXBQRZYDYHXRWJYMYNPWICYJAEVCVDWIWJYMYDPXRXHYLYCCDEAASVEV
    HWIXRYNYHPZWJWIXBTGYOECVIDXBATSVJVKVLXSCYKYJRZMZYKXDQRZYFYIXRWIYQYRPWJCYKAD
    VCVMWIWJYQYFPZXRWJWIYSWJWIOYPYECEDAASVEVNVHWJXRYRYIPZWIWJXDTGYTDCVIEXDATSVJ
    VOVLVQYBCYCYELZMYGWQUUACWOWPVPVRCYCYEVBWGXCXEVPVSVTWAXTXAXFCWMWBWCWDWEWFUR
    $.

  $( Values of a one-to-one function between two sets with two elements.
     Actually, such a function is a bijection.  (Contributed by AV,
     22-Jul-2025.) $)
  fvf1pr $p |- ( ( ( A e. V /\ B e. W /\ A =/= B )
                   /\ F : { A , B } -1-1-> { X , Y } )
                 -> ( ( ( F ` A ) = X /\ ( F ` B ) = Y )
                   \/ ( ( F ` A ) = Y /\ ( F ` B ) = X ) ) ) $=
    ( wcel cpr wa cfv wceq wo ffvelcdm syl2anr elpri wi eqtr3 syl5 a1i 3ad2ant1
    wne w3a wf1 wf f1f prid1g prid2g 3ad2ant2 f1veqaeq sylan2 ex eqneqall com12
    jca 3ad2ant3 syldd impcom olc orc ccased syl2ani mp2and ) ADHZBEHZABUBZUCZA
    BIZFGIZCUDZJZACKZVIHZBCKZVIHZVLFLZVNGLZJZVLGLZVNFLZJZMZVJVHVICUEZAVHHZVMVGV
    HVICUFZVDVEWDVFABDUGUAZVHVIACNOVJWCBVHHZVOVGWEVEVDWGVFABEUHUIZVHVIBCNOVMVKV
    PVSMVTVQMWBVOVLFGPVNFGPVKVPVTVSVQWBVJVGVPVTJZWBQVJVGWIABLZWBVJVGWIWJQWIVLVN
    LZVJVGJZWJVLVNFRVGVJWDWGJWKWJQVGWDWGWFWHUOVHVIABCUJUKZSULVGWJWBQZQVJVFVDWNV
    EWJVFWBWBABUMUNUPTZUQURWAWBQVKWAVRUSTVRWBQVKVRWAUTTVJVGVSVQJZWBQVJVGWPWJWBV
    JVGWPWJQWPWKWLWJVLVNGRWMSULWOUQURVAVBVC $.

  ${
    $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 cv wcel wa opelxpd fmpttd frnd eqsstrid ) AGBHCDLZMZNE
      FOZIAHUEUDABHUCUEABPHQRCDEFJKSTUAUB $.

    $( 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 ) ) ) $=
      ( wbr cop wceq wrex wa wcel cmpt crn df-br eleq2i eqid elrnmpti 3bitri cv
      opex wb opthg2 syl2anc rexbidva bitrid ) EFINZEFOZCDOZPZBJQZAECPFDPRZBJQU
      NUOISUOBJUPTZUAZSUREFIUBIVAUOKUCBJUPUOUTUTUDCDUHUEUFAUQUSBJABUGJSRCGSDHSU
      QUSUILMEFCDGHUJUKULUM $.

    $( 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 opex eqid mpan2 adantl eleqtrrdi sylibr
      elrnmpt1 df-br ) ABLHMZNZCDOZGMCDGPUIUJBHUJQZRZGUHUJULMZAUHUJSMUMCDTBHUJU
      KSUKUAUFUBUCIUDCDGUGUE $.

    $( 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
        eqeq12d anassrs ralrimiva exp31 ralrimd wal bitrdi biimpd reeanv r19.29
        wb cbvrexvw eqtr2 imim1i simprlr simprrr 3eqtr4d rexlimivw ex biimtrrid
        imp syl syl9 alrimdv cxp wss fliftrel relxp relss mpisyl dffun2 sylibrd
        wrel baib impbid ) AJUDZDFQZEGQZUEZCKUFZBKUFZAXIXMBKABUGBJBJBKDEUHZUIZU
        MLBXPBKXOUJUKULUNAXIBRZKUOZXMAXISZXRSXLCKXSXRCRZKUOZXLXJDJUPZFJUPZQXSXR
        YASZSZXKDFJUQYEYBEYCGYEXIDEJURZYBEQAXIYDUSZAXRYFXIYAABDEHIJKLMNUTVADEJV
        EVBYEXIFGJURZYCGQYGYEYHFDQZGEQZSZBKTZYEYAFFQZGGQZYLXSXRYAVCYEFVDYEGVDYK
        YMYNSBXTKXQXTQZYIYMYJYNYODFFOVFYOEGGPVFVGVHVIAYHYLWDXIYDABDEFGHIJKLMNVJ
        VKVLFGJVEVBVNVMVOVPVQVRAXNUARZUBRZJURZYPUCRZJURZSZYQYSQZUEZUCVSZUBVSZUA
        VSZXIAXNUUEUAAXNUUDUBAXNUUCUCAUUAYPDQZYQEQZSZBKTZYPFQZYSGQZSZCKTZSZXNUU
        BAUUAUUOAYRUUJYTUUNABDEYPYQHIJKLMNVJAYTUUGYSEQZSZBKTUUNABDEYPYSHIJKLMNV
        JUUQUUMBCKYOUUGUUKUUPUULYODFYPOVFYOEGYSPVFVGWEVTVGWAUUOUUIUUMSZCKTZBKTZ
        XNUUBUUIUUMBCKKWBXNUUTUUBXNUUTSXMUUSSZBKTUUBXMUUSBKWCUVAUUBBKUVAXLUURSZ
        CKTUUBXLUURCKWCUVBUUBCKUVBEGYQYSXLUURXKUURXJXKUUGUUKXJUUHUULYPDFWFVAWGW
        NXLUUGUUHUUMWHXLUUIUUKUULWIWJWKWOWKWOWLWMWPWQWQWQAJXFZXIUUFWDAJHIWRZWSU
        VDXFUVCABDEHIJKLMNWTHIXAJUVDXBXCXIUVCUUFUAUBUCJXDXGWOXEXH $.

      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
      nfralw nfan nfxfr ) BCDEFUABCFUBZLNZMNZDOZULFPZUMFPZEOZUCZMBQZLBQZUDALMBC
      DEFUEUKUTAABCFGJKUFUSALBJURAMBJUNUQAAULUMDAULRZHAUMRZSAUOUPEAULFGVATIAUMF
      GVBTSUGUHUHUIUJ $.
  $}

  ${
    $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 A x y $.  $d B x y $.  $d H x y $.
    $( A function is a bijection iff it is an isomorphism regarding the
       identity relation.  (Contributed by AV, 9-May-2021.) $)
    isof1oidb $p |- ( H : A -1-1-onto-> B <-> H Isom _I , _I ( A , B ) ) $=
      ( vx vy wf1o cv cid wbr cfv wb wral wiso wcel wceq weq wf1 f1of1 f1fveq
      wa sylan fvex ideq a1i ideqg ad2antll 3bitr4rd ralrimivva pm4.71i df-isom
      bitr4i ) ABCFZULDGZEGZHIZUMCJZUNCJZHIZKZEALDALZTABHHCMULUTULUSDEAAULUMANZ
      UNANZTZTZUPUQOZDEPZURUOULABCQVCVEVFKABCRABUMUNCSUAURVEKVDUPUQUNCUBUCUDVBU
      OVFKULVAUMUNAUEUFUGUHUIDEABHHCUJUK $.

    $( A function is a bijection iff it is an isomorphism regarding the
       universal class of ordered pairs as relations.  (Contributed by AV,
       9-May-2021.) $)
    isof1oopb $p |- ( H : A -1-1-onto-> B
                    <-> H Isom ( _V X. _V ) , ( _V X. _V ) ( A , B ) ) $=
      ( vx vy wf1o cv cvv cxp wbr cfv wb wral wiso wcel cop fvex opelvv df-br
      wa mpbir a1i opelvvg sylibr a1d impbid2 adantl ralrimivva pm4.71i df-isom
      bitr4i ) ABCFZULDGZEGZHHIZJZUMCKZUNCKZUOJZLZEAMDAMZTABUOUOCNULVAULUTDEAAU
      MAOUNAOTZUTULVBUPUSUSUPUSUQURPUOOUQURUMCQUNCQRUQURUOSUAUBVBUPUSVBUMUNPUOO
      UPUMUNAAUCUMUNUOSUDUEUFUGUHUIDEABUOUOCUJUK $.
  $}

  ${
    $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 w x y z A $.  $d w z B $.  $d w z C $.  $d w x y z F $.  $d w x y z R $.
    $d w x y z S $.
    $( Express the condition of isomorphism on two strict orders for a
       function's restriction.  (Contributed by Mario Carneiro,
       22-Jan-2015.) $)
    soisores $p |- ( ( ( R Or B /\ S Or C ) /\ ( F : B --> C /\ A C_ B ) ) ->
      ( ( F |` A ) Isom R , S ( A , ( F " A ) ) <->
        A. x e. A A. y e. A ( x R y -> ( F ` x ) S ( F ` y ) ) ) ) $=
      ( vz vw wa cv wbr cfv wi wral wcel wb weq fveq2 wor wf wss cima cres wiso
      isorel fvres breqan12d adantl biimpd ralrimivva w3a wf1o wfn crn wceq ffn
      bitrd ad2antrl simprr fnssres syl2anc 3adant3 df-ima eqcomi a1i eqeqan12d
      wo simprl simpl3 breq1 breq1d imbi12d breq2 breq2d rspc2va syl21anc con3d
      orim12d simpl1r simpl2l simpl2r sseldd ffvelcdmd sotrieq syl12anc simpl1l
      wn 3imtr4d sylbid dff1o6 syl3anbrc sotric impbid df-isom sylanbrc impbid2
      bitr4d 3expia ) DFUAZEGUAZKZDEHUBZCDUCZKZKZCHCUDZFGHCUEZUFZALZBLZFMZXKHNZ
      XLHNZGMZOZBCPACPZXJXQABCCXJXKCQZXLCQZKZKZXMXPYBXMXKXINZXLXINZGMZXPCXHXKXL
      FGXIUGYAYEXPRXJXSXTYCXNYDXOGXKCHUHXLCHUHUIUJUSUKULXCXFXRXJXCXFXRUMZCXHXIU
      NZILZJLZFMZYHXINZYIXINZGMZRZJCPICPXJYFXICUOZXIUPZXHUQZYKYLUQZIJSZOZJCPICP
      YGXCXFYOXRXGHDUOZXEYOXDUUAXCXEDEHURUTXCXDXEVADCHVBVCVDYQYFXHYPHCVEVFVGYFY
      TIJCCYFYHCQZYICQZKZKZYRYHHNZYIHNZUQZYSUUDYRUUHRYFUUBUUCYKUUFYLUUGYHCHUHZY
      ICHUHZVHUJUUEUUFUUGGMZUUGUUFGMZVIZWIZYJYIYHFMZVIZWIZUUHYSUUEUUPUUMUUEYJUU
      KUUOUULUUEUUBUUCXRYJUUKOZYFUUBUUCVJZYFUUBUUCVAZXCXFXRUUDVKZXQUURYHXLFMZUU
      FXOGMZOABYHYICCAISZXMUVBXPUVCXKYHXLFVLUVDXNUUFXOGXKYHHTVMVNBJSZUVBYJUVCUU
      KXLYIYHFVOUVEXOUUGUUFGXLYIHTVPVNVQVRZUUEUUCUUBXRUUOUULOZUUTUUSUVAXQUVGYIX
      LFMZUUGXOGMZOABYIYHCCAJSZXMUVHXPUVIXKYIXLFVLUVJXNUUGXOGXKYIHTVMVNBISZUVHU
      UOUVIUULXLYHYIFVOUVKXOUUFUUGGXLYHHTVPVNVQVRZVTVSUUEXBUUFEQZUUGEQZUUHUUNRX
      AXBXFXRUUDWAZUUEDEYHHXDXEXCXRUUDWBZUUECDYHXDXEXCXRUUDWCZUUSWDZWEZUUEDEYIH
      UVPUUECDYIUVQUUTWDZWEZEUUFUUGGWFWGUUEXAYHDQZYIDQZYSUUQRXAXBXFXRUUDWHZUVRU
      VTDYHYIFWFWGWJWKULIJCXHXIWLWMYFYNIJCCUUEYJUUKYMUUEYJUUKUVFUUEUUHUULVIZWIZ
      YSUUOVIZWIZUUKYJUUEUWGUWEUUEYSUUHUUOUULYSUUHOUUEYHYIHTVGUVLVTVSUUEXBUVMUV
      NUUKUWFRUVOUVSUWAEUUFUUGGWNWGUUEXAUWBUWCYJUWHRUWDUVRUVTDYHYIFWNWGWJWOUUDY
      MUUKRYFUUBUUCYKUUFYLUUGGUUIUUJUIUJWSULIJCXHFGXIWPWQWTWR $.
  $}

  ${
    $d R x y a b $.  $d S x y a b $.  $d H x y a b $.  $d A x y a b $.
    $d B x y a b $.
    $( Infer isomorphism from one direction of an order proof for isomorphisms
       between strict orders.  (Contributed by Stefan O'Rear, 2-Nov-2014.) $)
    soisoi $p |- ( ( ( R Or A /\ S Po B ) /\ ( H : A -onto-> B /\
          A. x e. A A. y e. A ( x R y -> ( H ` x ) S ( H ` y ) ) ) ) ->
        H Isom R , S ( A , B ) ) $=
      ( va vb wa cv wbr cfv wi wral weq wcel wn fveq2 imbi12d wor wpo wf1o wiso
      wfo wb wf1 wf wceq simprl fof syl wo sotrieq ad4ant14 simprr breq1 breq1d
      con2bid breq2 breq2d rspc2va ancoms sylan simpllr simplrl ffvelcdmd poirr
      syl5ibrcom syl2anc con2d syld ancom2s jaod sylbird con4d ralrimivva dff13
      notbid sylanbrc df-f1o sotric po2nr sylibr syl12anc impcon4bid df-isom
      imnan ) CEUAZDFUBZJZCDGUEZAKZBKZELZWMGMZWNGMZFLZNZBCOACOZJZJZCDGUCZHKZIKZ
      ELZXDGMZXEGMZFLZUFZICOHCOCDEFGUDXBCDGUGZWLXCXBCDGUHZXGXHUIZHIPZNZICOHCOXK
      XBWLXLWKWLWTUJZCDGUKZULXBXOHICCXBXDCQZXECQZJZJZXNXMYAXNRZXFXEXDELZUMZXMRZ
      WIXTYDYBUFWJXAWIXTJZXNYDCXDXEEUNUSUOYAXFYEYCYAXFXIYEXBWTXTXFXINZWKWLWTUPZ
      XTWTYGWSYGXDWNELZXGWQFLZNABXDXECCAHPZWOYIWRYJWMXDWNEUQYKWPXGWQFWMXDGSURTB
      IPZYIXFYJXIWNXEXDEUTYLWQXHXGFWNXEGSVATVBVCVDZYAXMXIYAWJXHDQZXMXIRZNWIWJXA
      XTVEZYACDXEGYAWLXLWKWLWTXTVFXQULZXBXRXSUPVGZWJYNJZYOXMXHXHFLZRZDXHFVHZXMX
      IYTXGXHXHFUQVSVIVJVKVLYAYCXHXGFLZYEXBWTXTYCUUCNZYHWTXSXRUUDXSXRJWTUUDWSUU
      DXEWNELZXHWQFLZNABXEXDCCAIPZWOUUEWRUUFWMXEWNEUQUUGWPXHWQFWMXEGSURTBHPZUUE
      YCUUFUUCWNXDXEEUTUUHWQXGXHFWNXDGSVATVBVCVMVDZYAXMUUCYAWJYNXMUUCRZNYPYRYSU
      UJXMUUAUUBXMUUCYTXGXHXHFUTVSVIVJVKVLVNVOVPVQHICDGVRVTXPCDGWAVTXBXJHICCYAX
      FXIYMYAXFRZXNYCUMZYOWIXTUULUUKUFWJXAYFXFUULCXDXEEWBUSUOYAXNYOYCYAWJYNXNYO
      NYPYRYSYOXNUUAUUBXNXIYTXNXGXHXHFXDXEGSURVSVIVJYAYCUUCYOUUIYAWJYNXGDQZUUCY
      ONZYPYRYACDXDGYQXBXRXSUJVGWJYNUUMJJUUCXIJRUUNDXHXGFWCUUCXIWHWDWEVLVNVOWFV
      QHICDEFGWGVT $.
  $}

  ${
    $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 rgen2 df-isom mpbir2an ) AABBEAFZGAAUCHCQZDQZBIZUDUCJZUEUCJZBIZR
      ZDAKCAKALUJCDAAUDAMZUEAMZSUIUFUKULUGUDUHUEBAUDNAUENOPTCDAABBUCUAUB $.
  $}

  ${
    $d w x y z A $.  $d w x y z B $.  $d x y C $.  $d x y D $.  $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 wf1o cv wbr cfv wb wral wa ccnv wiso vex brcnv fvex df-isom
      ralcom bibi12i 2ralbii bitr4i anbi2i 3bitr4i ) ABEHZFIZGIZCJZUHEKZUIEKZDJ
      ZLZGAMFAMZNUGUIUHCOZJZULUKDOZJZLZFAMGAMZNABCDEPABUPUREPUOVAUGUOUNFAMGAMVA
      UNFGAAUAUTUNGFAAUQUJUSUMUIUHCGQFQRULUKDUIESUHESRUBUCUDUEFGABCDETGFABUPURE
      TUF $.

    isocnv3.1 $e |- C = ( ( A X. A ) \ R ) $.
    isocnv3.2 $e |- D = ( ( B X. B ) \ S ) $.
    $( Complementation law for isomorphism.  (Contributed by Mario Carneiro,
       9-Sep-2015.) $)
    isocnv3 $p |- ( H Isom R , S ( A , B ) <-> H Isom C , D ( A , B ) ) $=
      ( vx vy cv wbr cfv wb wral wa wiso wcel wn wf1o notbi cxp brxp cdif breqi
      brdif bitri baib sylbir adantl wf ffvelcdm anim12dan sylibr sylan bibi12d
      f1of syl bitr4id 2ralbidva pm5.32i df-isom 3bitr4i ) ABGUAZJLZKLZEMZVFGNZ
      VGGNZFMZOZKAPJAPZQVEVFVGCMZVIVJDMZOZKAPJAPZQABEFGRABCDGRVEVMVQVEVLVPJKAAV
      EVFASZVGASZQZQZVLVHTZVKTZOVPVHVKUBWAVNWBVOWCVTVNWBOZVEVTVFVGAAUCZMZWDVFVG
      AAUDVNWFWBVNVFVGWEEUEZMWFWBQVFVGCWGHUFVFVGWEEUGUHUIUJUKWAVIVJBBUCZMZVOWCO
      VEABGULZVTWIABGURWJVTQVIBSZVJBSZQWIWJVRWKVSWLABVFGUMABVGGUMUNVIVJBBUDUOUP
      VOWIWCVOVIVJWHFUEZMWIWCQVIVJDWMIUFVIVJWHFUGUHUIUSUQUTVAVBJKABEFGVCJKABCDG
      VCVD $.
  $}

  ${
    $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 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 $.
    $( Isomorphisms preserve minimal elements.  Note that ` ( ``' R " { D } ) `
       is Takeuti and Zaring's idiom for the initial segment
       ` { x | x R D } ` .  Proposition 6.31(1) of [TakeutiZaring] p. 33.
       (Contributed by NM, 19-Apr-2004.) $)
    isomin $p |- ( ( H Isom R , S ( A , B ) /\ ( C C_ A /\ D e. A ) ) ->
               ( ( C i^i ( `' R " { D } ) ) = (/) <->
               ( ( H " C ) i^i ( `' S " { ( H ` D ) } ) ) = (/) ) ) $=
      ( vy vx wcel wa cima wceq wex wrex wbr wb wi syl9r cvv wss csn cin c0 cfv
      wiso ccnv wn cv neq0 vex elima ssel wf1o wfn isof1o f1ofn fnbrfvb ex 3syl
      imp31 rexbidva bitr4id fvex eliniseg mp1i anbi12d r19.41v 3bitr4g adantrr
      breq1 biimpar ad2antll isorel bitrd imbitrrid exp32 com34 imp32 reximdvai
      elin sylbid exbii df-rex 3bitr4i imbitrrdi exlimdv biimtrid con4d fnfvima
      syl 3expia sylan adantrd biimpd ax-mp impd jcad 3imtr4g syl6 impcon4bid
      n0i ) ABEFGUFZCAUAZDAJZKZKZCEUGDUBLZUCZUDMZGCLZFUGDGUEZUBLZUCZUDMZXGXOXJX
      OUHZHUIZXNJZHNXGXJUHZHXNUJXGXRXSHXGXRIUIZXHJZICOZXSXGXRXTGUEZXQMZXQXLFPZK
      ZICOZYBXCXDXRYGQXEXCXDKZXQXKJZXQXMJZKYDICOZYEKXRYGYHYIYKYJYEYHYIXTXQGPZIC
      OYKIXQGCHUKZULYHYDYLICXCXDXTCJZYDYLQZXDYNXTAJZXCYOCAXTUMZXCABGUNZGAUOZYPY
      ORABEFGUPZABGUQZYSYPYOAXTXQGURUSUTSVAVBVCXLTJZYJYEQYHDGVDZFXLXQTYMVEVFVGX
      QXKXMWAYDYEICVHVIVJXGYFYAICXCXDXEYNYFYARZRXCXDYNXEUUDXDYNYPXCXEUUDRYQXCYP
      XEUUDYFYAXCYPXEKKZYCXLFPZYDUUFYEYCXQXLFVKVLUUEYAXTDEPZUUFXEYAUUGQXCYPEDXT
      AIUKVEVMZABXTDEFGVNZVOVPVQSVRVSVTWBXTXIJZINZYNYAKZINXSYBUUJUULIXTCXHWAZWC
      IXIUJZYAICWDWEWFWGWHWIXSUUKXGXPUUNXGUUJXPIXGUUJYCXNJZXPXGUULYCXKJZYCXMJZK
      UUJUUOXGUULUUPUUQXGYNUUPYAXCYSXFYNUUPRZXCYRYSYTUUAWKYSXDUURXEYSXDYNUUPACG
      XTWJWLVJWMWNXGYNYAUUQXCXDXEYNYAUUQRZRXCXDYNXEUUSXDYNYPXCXEUUSRYQXCYPXEUUS
      UUEYAUUGUUQUUHUUEUUGUUFUUQUUEUUGUUFUUIWOUUBUUQUUFQUUCFXLYCTXTGVDVEWPWFWBV
      QSVRVSWQWRUUMYCXKXMWAWSXNYCXBWTWGWHXA $.
  $}

  ${
    $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 cfv wb bitrdi wiso cab elin
      dfima2 wceq crn wf1o wfo isof1o f1ofo forn eleq2d 3syl wfn fvelrnb bitr3d
      f1ofn cvv fvex vex eliniseg mp1i adantr anbi2d bitrid anbi1d anass adantl
      anbi12d isorel syl fnbrfvb bicomd sylan adantrr ancom breq1 pm5.32i bitri
      wi exp32 com23 imp pm5.32d bitrd rexbidv2 r19.41v bitr4d eqabdv eqtr4id )
      ABDEFUAZCAIZJZFADKCLMZNZMGOZHOZFPZGWOQZHUBBEKCFRZLMZNZGHFWOUDWMWSHXBWQXBI
      WQBIZWQXAIZJZWMWSWQBXAUCWMXEWPFRZWQUEZGAQZWQWTEPZJZWSWKXEXJSWLWKXCXHXDXIW
      KWQFUFZIZXCXHWKABFUGZABFUHZXLXCSABDEFUIZABFUJXNXKBWQABFUKULUMWKXMFAUNZXLX
      HSXOABFUQZGAWQFUOUMUPWTURIXDXISWKCFUSEWTWQURHUTVAVBVIVCWMWSXGXIJZGAQXJWMW
      RXRGWOAWMWPWOIZWRJZWPAIZWPCDPZWRJZJZYAXRJWLXTYDSWKWLXTYAYBJZWRJYDWLXSYEWR
      XSYAWPWNIZJWLYEWPAWNUCWLYFYBYADCWPAGUTVAVDVEVFYAYBWRVGTVHWMYAYCXRWKWLYAYC
      XRSZVTWKYAWLYGWKYAWLYGWKYAWLJJZYCXFWTEPZXGJZXRYHYBYIWRXGABWPCDEFVJWKYAWRX
      GSZWLWKXPYAYKWKXMXPXOXQVKXPYAJXGWRAWPWQFVLVMVNVOVIYJXGYIJXRYIXGVPXGYIXIXF
      WQWTEVQVRVSTWAWBWCWDWEWFXGXIGAWGTWHVEWIWJ $.
  $}

  ${
    $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 wa wf1o wbr cfv wral cima cres cv wb wf1 wss isof1o syl
      f1of1 adantr ccnv csn inss1 eqsstri f1ores sylancl isoini imaeq2i 3eqtr4g
      f1oeq3d mpbid df-isom simprbi ssralv ralimdv mpsyl fvres breqan12d bibi2d
      cin ralbidva ralbiia sylibr sylanbrc ) ABEFGMZHANZOZCDGCUAZPZKUBZLUBZEQZV
      SVQRZVTVQRZFQZUCZLCSZKCSZCDEFVQMVPCGCTZVQPZVRVPABGUDZCAUEZWIVNWJVOVNABGPZ
      WJABEFGUFABGUHUGUICAEUJHUKTZVIZAIAWMULUMZABCGUNUOVPWHDCVQVPGWNTBFUJHGRUKT
      VIWHDABHEFGUPCWNGIUQJURUSUTVPWAVSGRZVTGRZFQZUCZLCSZKCSZWGWKVPWTKASZXAWOWK
      VPWSLASZKASZXBWOVNXDVOVNWLXDKLABEFGVAVBUIWKXCWTKAWSLCAVCVDVEWTKCAVCVEWFWT
      KCVSCNZWEWSLCXEVTCNZOWDWRWAXEXFWBWPWCWQFVSCGVFVTCGVFVGVHVJVKVLKLCDEFVQVAV
      M $.
  $}

  ${
    $d w x y z A $.  $d w x y z B $.  $d w x y z H $.  $d w x y z ph $.
    $d w x y z R $.  $d w x y z S $.
    isofrlem.1 $e |- ( ph -> H Isom R , S ( A , B ) ) $.
    isofrlem.2 $e |- ( ph -> ( H " x ) e. _V ) $.
    $( Lemma for ~ isofr .  (Contributed by NM, 29-Apr-2004.)  (Revised by
       Mario Carneiro, 18-Nov-2014.) $)
    isofrlem $p |- ( ph -> ( S Fr B -> R Fr A ) ) $=
      ( vy vw vz cv c0 wa cima cin wceq wi syl wfr wss wne ccnv csn wrex isof1o
      wal wf1o wiso wfn f1ofn wcel wex n0 w3a cfv fnfvima ne0d exlimdv biimtrid
      3expia expimpd wfo f1ofo crn imassrn forn sseqtrid jctild dffr3 cvv sseq1
      neeq1 anbi12d ineq1 eqeq1d rexeqbi1dv imbi12d spcgv adantr f1ofun fvelima
      syl5d wfun simpl syl2an simpr ssel imdistani isomin sneq imaeq2d sylan9bb
      wb ineq2d imbitrrid exp42 imp com3l com4t reximdvai rexlimdvaa ex adantrd
      mpd a2d syld alrimdv imbitrrdi ) ADFUAZBMZCUBZXLNUCZOZXLEUDJMZUEPQNRZJXLU
      FZSZBUHCEUAAXKXSBAXKXOGXLPZFUDZKMZUEZPZQZNRZKXTUFZSXSAXOXTDUBZXTNUCZOZXKY
      GACDGUIZXOYJSACDEFGUJZYKHCDEFGUGTZYKXOYIYHYKGCUKZXOYISCDGULYNXMXNYIXNXPXL
      UMZJUNYNXMOZYIJXLUOYPYOYIJYNXMYOYIYNXMYOUPXTXPGUQZCXLGXPURUSVBUTVAVCTYKCD
      GVDZYHCDGVEYRGVFXTDGXLVGCDGVHVITVJTXKLMZDUBZYSNUCZOZYSYDQZNRZKYSUFZSZLUHZ
      AYJYGSZLKDFVKAXTVLUMUUGUUHSIUUFUUHLXTVLYSXTRZUUBYJUUEYGUUIYTYHUUAYIYSXTDV
      MYSXTNVNVOUUDYFKYSXTUUIUUCYENYSXTYDVPVQVRVSVTTVAWDAXOYGXRAXMYGXRSZXNAXMUU
      JAXMOZYFXRKXTUUKYBXTUMZYFOZOZYQYBRZJXLUFZXRUUKGWEZUULUUPUUMUUKYKUUQAYKXMY
      MWACDGWBTUULYFWFJYBXLGWCWGUUNUUOXQJXLUUKUUMYOUUOXQSSYOUUOUUKUUMXQUUKYOUUO
      UUMXQSZAXMYOUUOUURSSAXMYOUUOUURUUMXQAXMYOOZOZUUOOYFUULYFWHUUTXQXTYAYQUEZP
      ZQZNRZUUOYFAYLXMXPCUMZOXQUVDWOUUSHXMYOUVEXLCXPWIWJCDXLXPEFGWKWGUUOUVCYENU
      UOUVBYDXTUUOUVAYCYAYQYBWLWMWPVQWNWQWRWSWTXAWSXBXFXCXDXEXGXHXIBJCEVKXJ $.

    $( 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 $.  $d x V $.
    $( An isomorphism preserves well-foundedness.  Proposition 6.32(1) of
       [TakeutiZaring] p. 33.  (Contributed by NM, 30-Apr-2004.)  (Revised by
       Mario Carneiro, 18-Nov-2014.) $)
    isofr $p |- ( H Isom R , S ( A , B ) -> ( R Fr A <-> S Fr B ) ) $=
      ( vx wiso wfr ccnv id wf1o wfun cima cvv wcel isof1o f1ofun funimaex 3syl
      isofrlem wi isocnv cv vex syl impbid ) ABCDEGZACHZBDHZUGBADCEIZGZUHUIUAAB
      CDEUBUKFBADCUJUKJUKBAUJKUJLUJFUCZMNOBADCUJPBAUJQUJULFUDZRSTUEUGFABCDEUGJU
      GABEKELEULMNOABCDEPABEQEULUMRSTUF $.

    $( 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 $.

    $( A weak form of ~ isofr that does not need Replacement.  (Contributed by
       Mario Carneiro, 18-Nov-2014.) $)
    isofr2 $p |- ( ( H Isom R , S ( A , B ) /\ B e. V ) ->
                   ( S Fr B -> R Fr A ) ) $=
      ( vx wiso wcel wa simpl cv cima wss cvv crn imassrn wf1o wf isof1o sstrid
      f1of frn 3syl ssexg sylan isofrlem ) ABCDEHZBFIZJGABCDEUHUIKUHEGLZMZBNUIU
      KOIUHUKEPZBEUJQUHABERABESULBNABCDETABEUBABEUCUDUAUKBFUEUFUG $.
  $}

  ${
    $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 the property of being a partial order.
       (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 $.

    $( 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 ) ) $=
      ( va vb vc vd wpo cv wbr weq w3o wral wa wor wcel wceq 3orbi123d isopolem
      wiso cfv wi wf1o wf isof1o f1of ffvelcdm ex anim12d imp breq1 eqeq1 breq2
      3syl eqeq2 rspc2v syl isorel wb f1of1 f1fveq sylan bicomd ancom2s sylibrd
      wf1 ralrimdvva df-so 3imtr4g ) ABCDEUBZBDJZFKZGKZDLZFGMZVOVNDLZNZGBOFBOZP
      ACJZHKZIKZCLZHIMZWCWBCLZNZIAOHAOZPBDQACQVLVMWAVTWHABCDEUAVLVTWGHIAAVLWBAR
      ZWCARZPZPZVTWBEUCZWCEUCZDLZWMWNSZWNWMDLZNZWGWLWMBRZWNBRZPZVTWRUDVLWKXAVLA
      BEUEZABEUFZWKXAUDABCDEUGZABEUHXCWIWSWJWTXCWIWSABWBEUIUJXCWJWTABWCEUIUJUKU
      PULVSWRWMVODLZWMVOSZVOWMDLZNFGWMWNBBVNWMSVPXEVQXFVRXGVNWMVODUMVNWMVOUNVNW
      MVODUOTVOWNSXEWOXFWPXGWQVOWNWMDUOVOWNWMUQVOWNWMDUMTURUSWLWDWOWEWPWFWQABWB
      WCCDEUTWLWPWEVLABEVHZWKWPWEVAVLXBXHXDABEVBUSABWBWCEVCVDVEVLWJWIWFWQVAABWC
      WBCDEUTVFTVGVIUKFGBDVJHIACVJVK $.

    $( An isomorphism preserves the property of being a strict total order.
       (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 ) ABCDEFZACGZBDGZNBADCEHZFO
      PIABCDEJBADCQKLABCDEKM $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y R $.  $d x y S $.  $d x y H $.
    $( An isomorphism preserves the property of being a well-ordering.
       Proposition 6.32(3) of [TakeutiZaring] p. 33.  (Contributed by NM,
       30-Apr-2004.)  (Revised by Mario Carneiro, 18-Nov-2014.) $)
    isowe $p |- ( H Isom R , S ( A , B ) -> ( R We A <-> S We B ) ) $=
      ( wiso wfr wor wa wwe isofr isoso anbi12d df-we 3bitr4g ) ABCDEFZACGZACHZ
      IBDGZBDHZIACJBDJPQSRTABCDEKABCDELMACNBDNO $.

    $( A weak form of ~ isowe that does not need Replacement.  (Contributed by
       Mario Carneiro, 18-Nov-2014.) $)
    isowe2 $p |- ( ( H Isom R , S ( A , B ) /\ A. x ( H " x ) e. _V ) ->
                   ( S We B -> R We A ) ) $=
      ( vy wiso cv cima cvv wcel wal wa wfr wor wwe simpl weq df-we imaeq2 spvv
      eleq1d adantl isofrlem wi isosolem adantr anim12d 3imtr4g ) BCDEFHZFAIZJZ
      KLZAMZNZCEOZCEPZNBDOZBDPZNCEQBDQUPUQUSURUTUPGBCDEFUKUORUOFGIZJZKLZUKUNVCA
      GAGSUMVBKULVAFUAUCUBUDUEUKURUTUFUOBCDEFUGUHUICETBDTUJ $.
  $}

  ${
    $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 eqeq1 2rexbidv anbi2d opelopab weq anass f1fveq
      fvex equcom bitrdi anassrs bitrid rexbidv r19.42v rexbidva breq1 ceqsrexv
      adantl bitrd breq2 sylan9bb anandis sylan9bbr an32s bitr2id sylan df-isom
      ralrimivva sylanbrc ) EFIUAZHCLZALZIMZNZDLZBLZIMZNZOZWBWFGPZOZBEQAEQZCDUB
      ZNZOVTJLZKLZGPZWOIMZWPIMZHPZRZKEUCJEUCZEFGHIUDVTWNUEVTEFIUFZWNXBEFIUGXCWN
      OZXAJKEEWTWRWSUHZHSZXDWOESZWPESZOZOWQWRWSHUIXCXIWNXFWQRWNXFXEWMSZXCXIOZWQ
      HWMXEUJXJWRWCNZWSWGNZOZWJOZBEQZAEQZXKWQWLXLWHOZWJOZBEQAEQXQCDWRWSWOIURWPI
      URWAWRNZWKXSABEEXTWIXRWJXTWDXLWHWAWRWCUKTTULWEWSNZXSXOABEEYAXRXNWJYAWHXMX
      LWEWSWGUKUMTULUNXCXGXHXQWQRXCXGOZXQXMWOWFGPZOZBEQZXCXHOZWQYBXQAJUOZXMWJOZ
      BEQZOZAEQZYEYBXPYJAEYBWBESZOZXPYGYHOZBEQYJYMXOYNBEXOXLYHOYMYNXLXMWJUPYMXL
      YGYHXCXGYLXLYGRXCXGYLOOXLJAUOYGEFWOWBIUQJAUSUTVATVBVCYGYHBEVDUTVEXGYKYERX
      CYIYEAWOEYGYHYDBEYGWJYCXMWBWOWFGVFUMVCVGVHVIYFYEBKUOZYCOZBEQZWQYFYDYPBEYF
      WFESZOXMYOYCXCXHYRXMYORXCXHYROOXMKBUOYOEFWPWFIUQKBUSUTVATVEXHYQWQRXCYCWQB
      WPEWFWPWOGVJVGVHVIVKVLVBVMVNVOVRVPJKEFGHIVQVS $.
  $}

  ${
    $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 $.
  $}

  ${
    $d z w R $.  $d x y z w S $.  $d z w A $.  $d z w B $.  $d x y z w F $.
    f1owe.1 $e |- R = { <. x , y >. | ( F ` x ) S ( F ` y ) } $.
    $( Well-ordering of isomorphic relations.  (Contributed by NM,
       4-Mar-1997.) $)
    f1owe $p |- ( F : A -1-1-onto-> B -> ( S We B -> R We A ) ) $=
      ( vz vw wf1o wwe cv wbr cfv wb wral weq fveq2 breq1d breq2d brabg wa wiso
      rgen2 df-isom isowe sylbir mpan2 biimprd ) CDGKZCELZDFLZUKIMZJMZENUNGOZUO
      GOZFNZPZJCQICQZULUMPZUSIJCCAMZGOZBMZGOZFNUPVEFNURABUNUOCCEAIRVCUPVEFVBUNG
      STBJRVEUQUPFVDUOGSUAHUBUEUKUTUCCDEFGUDVAIJCDEFGUFCDEFGUGUHUIUJ $.
  $}

  ${
    $d A a b c $.  $d R b c $.  $d F a b c $.
    $( A set-like well-ordering has no nontrivial automorphisms.  (Contributed
       by Stefan O'Rear, 16-Nov-2014.)  (Revised by Mario Carneiro,
       25-Jun-2015.) $)
    weniso $p |- ( ( R We A /\ R Se A /\ F Isom R , R ( A , A ) ) ->
        F = ( _I |` A ) ) $=
      ( va vc vb wceq cfv wral wn wbr wa syl wcel wi fveq2 id eqeq12d wb adantr
      wwe wse wiso w3a cid cres cv crab c0 wne wrex rabn0 rexnal bitri wreu wss
      simpl1 simpl2 ssrab2 a1i simpr wereu2 syl22anc reurex notbid elrab ralrab
      ex con34b bicomi ralbii wf1o wf simpl3 isof1o f1of simprl ffvelcdmd breq1
      imbi12d rspcv com23 imp f1of1 f1fveq syl12anc pm2.21 ad2antll sylbid syld
      wf1 ccnv f1ocnv 3syl isorel f1ocnvfv2 syl2anc breq1d bitr2d biimpa adantl
      sylc simplrr fveq2d 3eqtr3d wo simprr wor sotrieq con2bid mpbird mpjaodan
      weso biimtrid rexlimdv pm2.18d fvresi eqeq2d biimprd ralimia wfn 3ad2ant3
      biimtrrid f1ofn fnresi eqfnfv ) ABUAZABUBZAABBCUCZUDZCUEAUFZGZDUGZCHZYMYK
      HZGZDAIZYJYNYMGZDAIZYQYJYSYSJZYRJZDAUHZUIUJZYJYSUUCUUADAUKYTUUADAULYRDAUM
      UNYJUUCEUGZFUGZBKZJZEUUBIZFUUBUKZYSYJUUCUUIYJUUCLZUUHFUUBUOZUUIUUJYGYHUUB
      AUPZUUCUUKYGYHYIUUCUQYGYHYIUUCURUULUUJUUADAUSUTYJUUCVAFEAUUBBVBVCUUHFUUBV
      DMVHYJUUHYSFUUBUUEUUBNUUEANZUUECHZUUEGZJZLZYJUUHYSOZUUAUUPDUUEAYMUUEGZYRU
      UOUUSYNUUNYMUUEYMUUECPUUSQRVEVFYJUUQUURUUHUUFUUDCHZUUDGZOZEAIZYJUUQLZYSUU
      HUVAJZUUGOZEAIUVCUUAUVEUUGEDAYMUUDGZYRUVAUVGYNUUTYMUUDYMUUDCPUVGQRVEVGUVF
      UVBEAUVBUVFUUFUVAVIVJVKUNUVDUUNUUEBKZUVCYSOUUEUUNBKZUVDUVHLUVCUUNCHZUUNGZ
      YSUVDUVHUVCUVKOUVDUVCUVHUVKUVDUUNANZUVCUVHUVKOZOUVDAAUUECUVDAACVLZAACVMUV
      DYIUVNYGYHYIUUQVNZAABBCVOZMZAACVPMYJUUMUUPVQZVRZUVBUVMEUUNAUUDUUNGZUUFUVH
      UVAUVKUUDUUNUUEBVSUVTUUTUVJUUDUUNUUDUUNCPUVTQRVTWAMWBWCUVDUVKYSOUVHUVDUVK
      UUOYSUVDAACWKZUVLUUMUVKUUOSUVDUVNUWAUVQAACWDMUVSUVRAAUUNUUECWEWFUUPUUOYSO
      YJUUMUUOYSWGZWHWITWJUVDUVILZUVCUUECWLZHZCHZUWEGZYSUWCUWEANZUWEUUEBKZUVCUW
      GOUVDUWHUVIUVDAAUUEUWDUVDUVNAAUWDVLAAUWDVMUVQAACWMAAUWDVPWNUVRVRZTUVDUVIU
      WIUVDUWIUWFUUNBKZUVIUVDYIUWHUUMUWIUWKSUVOUWJUVRAAUWEUUEBBCWOWFUVDUWFUUEUU
      NBUVDUVNUUMUWFUUEGZUVQUVRAAUUECWPWQZWRWSWTUWHUVCUWIUWGUVBUWIUWGOEUWEAUUDU
      WEGZUUFUWIUVAUWGUUDUWEUUEBVSUWNUUTUWFUUDUWEUUDUWECPUWNQRVTWAWBXBUVDUWGYSO
      UVIUVDUWGYSUVDUWGLZUUPUUOYSYJUUMUUPUWGXCUWOUWFCHZUWFUUNUUEUWGUWPUWFGUVDUW
      FUWECPXAUVDUWPUUNGUWGUVDUWFUUECUWMXDTUVDUWLUWGUWMTXEUWBXBVHTWJUVDUVHUVIXF
      ZUUPYJUUMUUPXGUVDUUOUWQUVDABXHZUVLUUMUUOUWQJSUVDYGUWRYGYHYIUUQUQABXMMUVSU
      VRAUUNUUEBXIWFXJXKXLXNVHXNXOWJYCXPYRYPDAYMANZYPYRUWSYOYMYNAYMXQXRXSXTMYJC
      AYAZYKAYAZYLYQSYJUVNUWTYIYGUVNYHUVPYBAACYDMUXAYJAYEUTDACYKYFWQXK $.

    $( Thus, there is at most one isomorphism between any two set-like
       well-ordered classes.  Class version of ~ wemoiso .  (Contributed by
       Mario Carneiro, 25-Jun-2015.) $)
    weisoeq $p |- ( ( ( R We A /\ R Se A ) /\
      ( F Isom R , S ( A , B ) /\ G Isom R , S ( A , B ) ) ) -> F = G ) $=
      ( wwe wse wa wiso wceq ccnv ccom cid cres wf1 wf1o isof1o f1of1 3syl id
      isocnv isotr syl2anr weniso sylan2 simprl simprr f1eqcocnv syl2anc mpbird
      3expa wb ) ACGZACHZIZABCDEJZABCDFJZIZIZEFKZELZFMZNAOKZUSUPAACCVCJZVDURURB
      ADCVBJVEUQURUAABCDEUBABACDCVBFUCUDUNUOVEVDACVCUEULUFUTABEPZABFPZVAVDUMUTU
      QABEQVFUPUQURUGABCDERABESTUTURABFQVGUPUQURUHABCDFRABFSTABEFUIUJUK $.

    $( Thus, there is at most one isomorphism between any two set-like
       well-ordered classes.  Class version of ~ wemoiso2 .  (Contributed by
       Mario Carneiro, 25-Jun-2015.) $)
    weisoeq2 $p |- ( ( ( S We B /\ S Se B ) /\
      ( F Isom R , S ( A , B ) /\ G Isom R , S ( A , B ) ) ) -> F = G ) $=
      ( wwe wse wa wiso wceq ccnv isocnv anim12i weisoeq wrel wf1o isof1o 3syl
      f1orel sylan2 wb simprl simprr cnveqb syl2anc mpbird ) BDGBDHIZABCDEJZABC
      DFJZIZIZEFKZELZFLZKZUKUHBADCUNJZBADCUOJZIUPUIUQUJURABCDEMABCDFMNBADCUNUOO
      UAULEPZFPZUMUPUBULUIABEQUSUHUIUJUCABCDERABETSULUJABFQUTUHUIUJUDABCDFRABFT
      SEFUEUFUG $.
  $}

  ${
    $d w x y z A $.  $d w x y z F $.  $d w V $.  $d w x y X $.
    knatar.1 $e |- X = |^| { z e. ~P A | ( F ` z ) C_ z } $.
    $( The Knaster-Tarski theorem says that every monotone function over a
       complete lattice has a (least) fixpoint.  Here we specialize this
       theorem to the case when the lattice is the powerset lattice ` ~P A ` .
       (Contributed by Mario Carneiro, 11-Jun-2015.) $)
    knatar $p |- ( ( A e. V /\ ( F ` A ) C_ A /\
        A. x e. ~P A A. y e. ~P x ( F ` y ) C_ ( F ` x ) ) ->
      ( X C_ A /\ ( F ` X ) = X ) ) $=
      ( vw wcel cfv wss cv cpw wral wceq fveq2 id rspcdva sylibr w3a crab pwidg
      cint 3ad2ant1 simp2 sseq12d intminss syl2anc eqsstrid wi wa sseq1d sseq2d
      pweq raleqbidv simpl3 simprl adantl vex elpw2 simprr sstrd expr ralrimiva
      ssintrab cbvrabv inteqi eqtri sseqtrrdi simp3 sselpwd fvex elpw eqssd jca
      ) DFJZDEKZDLZBMZEKZAMZEKZLZBWBNZOZADNZOZUAZGDLGEKZGPWIGCMZEKZWKLZCWGUBZUD
      ZDHWIDWGJZVSWODLVQVSWPWHDFUCUEZVQVSWHUFZWMVSCDWGWKDPZWLVRWKDWKDEQWSRUGUHU
      IUJZWIWJGWIWJIMZEKZXALZIWGUBZUDZGWIXCWJXALZUKZIWGOWJXELWIXGIWGWIXAWGJZXCX
      FWIXHXCULZULZWJXBXAXJWAXBLZWJXBLBXANZGVTGPZWAWJXBVTGEQZUMXJWFXKBXLOAWGXAW
      BXAPZWDXKBWEXLWBXAUOXOWCXBWAWBXAEQUNUPVQVSWHXIUQWIXHXCURSXJGXALGXLJXJGWOX
      AHXIWOXALWIWMXCCXAWGWKXAPZWLXBWKXAWKXAEQXPRUGZUHUSUJGXAIUTVATSWIXHXCVBVCV
      DVEXCIWJWGVFTGWOXEHWNXDWMXCCIWGXQVGVHVIZVJZWIGXEWJXRWIWJWGJZWJEKZWJLZXEWJ
      LWIWJDLXTWIWJVRDWIWAVRLZWJVRLBWGGXMWAWJVRXNUMWIWFYCBWGOAWGDWBDPZWDYCBWEWG
      WBDUOYDWCVRWAWBDEQUNUPVQVSWHVKZWQSWIGDWGWQWTVLZSWRVCWJDGEVMZVNTWIWAWJLZYB
      BGNZWJVTWJPWAYAWJVTWJEQUMWIWFYHBYIOAWGGWBGPZWDYHBWEYIWBGUOYJWCWJWAWBGEQUN
      UPYEYFSWIWJGLWJYIJXSWJGYGVNTSXCYBIWJWGXAWJPZXBYAXAWJXAWJEQYKRUGUHUIUJVOVP
      $.
  $}

  $( The value of a restricted function at a class is either the empty set or
     the value of the unrestricted function at that class.  (Contributed by
     Scott Fenton, 4-Sep-2011.) $)
  fvresval $p |- ( ( ( F |` B ) ` A ) = ( F ` A ) \/
                   ( ( F |` B ) ` A ) = (/) ) $=
    ( wcel wn wo cres cfv wceq c0 exmid fvres nfvres orim12i ax-mp ) ABDZPEZFAC
    BGHZACHIZRJIZFPKPSQTABCLABCMNO $.

  $( If ` (/) ` is not part of the range of a function ` F ` , then ` A ` is in
     the domain of ` F ` iff ` ( F `` A ) =/= (/) ` .  (Contributed by Scott
     Fenton, 7-Dec-2021.) $)
  funeldmb $p |- ( ( Fun F /\ -. (/) e. ran F ) ->
  ( A e. dom F <-> ( F ` A ) =/= (/) ) ) $=
    ( wfun c0 crn wcel wn wa cdm cfv wi fvelrn ex adantr wb eleq1 adantl sylibd
    wceq con3d impancom ndmfv impbid1 necon2abid ) BCZDBEZFZGZHZABIFZABJZDUIUKD
    SZUJGZUEULUHUMUEULHZUJUGUNUJUKUFFZUGUEUJUOKULUEUJUOABLMNULUOUGOUEUKDUFPQRTU
    AABUBUCUD $.

  ${
    $d F x y $.  $d G x y $.  $d X x y $.  $d Y x y $.
    $( Law for adjoining an element to restrictions of functions.  (Contributed
       by Scott Fenton, 6-Dec-2021.) $)
    eqfunresadj $p |- ( ( ( Fun F /\ Fun G ) /\ ( F |` X ) = ( G |` X ) /\
     ( Y e. dom F /\ Y e. dom G /\ ( F ` Y ) = ( G ` Y ) ) ) ->
     ( F |` ( X u. { Y } ) ) = ( G |` ( X u. { Y } ) ) ) $=
      ( vx vy wfun wa cres wceq cdm wcel cfv w3a cun relres cv wbr wo wb simp33
      breq 3ad2ant2 eqeq1d simp1l simp31 funbrfvb syl2anc simp1r simp32 3bitr3d
      csn velsn bibi12d syl5ibrcom biimtrid pm5.32d vex 3bitr4g orbi12d resundi
      breq1 brresi breqi brun bitri eqbrrdiv ) AGZBGZHZACIZBCIZJZDAKLZDBKLZDAMZ
      DBMZJZNZNZEFACDULZOZIZBWBIZAWBPBWBPVTEQZFQZVKRZWEWFAWAIZRZSZWEWFVLRZWEWFB
      WAIZRZSZWEWFWCRZWEWFWDRZVTWGWKWIWMVMVJWGWKTVSWEWFVKVLUBUCVTWEWALZWEWFARZH
      WQWEWFBRZHWIWMVTWQWRWSWQWEDJZVTWRWSTZEDUMVTXAWTDWFARZDWFBRZTVTVPWFJZVQWFJ
      ZXBXCVTVPVQWFVJVMVNVOVRUAUDVTVHVNXDXBTVHVIVMVSUEVJVMVNVOVRUFDWFAUGUHVTVIV
      OXEXCTVHVIVMVSUIVJVMVNVOVRUJDWFBUGUHUKWTWRXBWSXCWEDWFAVBWEDWFBVBUNUOUPUQW
      AWEWFAFURZVCWAWEWFBXFVCUSUTWOWEWFVKWHOZRWJWEWFWCXGACWAVAVDWEWFVKWHVEVFWPW
      EWFVLWLOZRWNWEWFWDXHBCWAVAVDWEWFVLWLVEVFUSVG $.
  $}

  $( Law for equality of restriction to successors.  This is primarily useful
     when ` X ` is an ordinal, but it does not require that.  (Contributed by
     Scott Fenton, 6-Dec-2021.) $)
  eqfunressuc $p |- ( ( ( Fun F /\ Fun G ) /\ ( F |` X ) = ( G |` X ) /\
   ( X e. dom F /\ X e. dom G /\ ( F ` X ) = ( G ` X ) ) ) ->
   ( F |` suc X ) = ( G |` suc X ) ) $=
    ( wfun wa cres wceq cdm wcel cfv w3a csn eqfunresadj df-suc reseq2i 3eqtr4g
    cun csuc ) ADBDEACFBCFGCAHICBHICAJCBJGKKACCLQZFBSFACRZFBTFABCCMTSACNZOTSBUA
    OP $.

  ${
    $d A y $.  $d B x y $.  $d C x y $.  $d F x y $.
    $( Condition for subset of an intersection of an image.  (Contributed by
       Scott Fenton, 16-Aug-2024.) $)
    fnssintima $p |- ( ( F Fn A /\ B C_ A ) -> ( C C_ |^| ( F " B ) <->
     A. x e. B C C_ ( F ` x ) ) ) $=
      ( vy cima cint wss cv wcel wi wal wfn wa cfv wral bitri wceq albii df-ral
      ssint wrex fvelimab imbi1d albidv ralcom4 eqcom imbi1i fvex sseq2 ceqsalv
      ralbii r19.23v 3bitr3ri bitrdi bitrid ) DECGZHIZFJZURKZDUTIZLZFMZEBNCBIOZ
      DAJZEPZIZACQZUSVBFURQVDFDURUBVBFURUARVEVDVGUTSZACUCZVBLZFMZVIVEVCVLFVEVAV
      KVBABCUTEUDUEUFVJVBLZFMZACQVNACQZFMVIVMVNAFCUGVOVHACVOUTVGSZVBLZFMVHVNVRF
      VJVQVBVGUTUHUITVBVHFVGVFEUJUTVGDUKULRUMVPVLFVJVBACUNTUOUPUQ $.
  $}

  ${
    $d A x $.  $d B x y $.  $d F x y $.  $d ph y $.  $d ps x $.
    imaeqsexvOLD.1 $e |- ( x = ( F ` y ) -> ( ph <-> ps ) ) $.
    $( Obsolete version of ~ rexima as of 14-Aug-2025.  Duplicate version of
       ~ rexima .  (Contributed by Scott Fenton, 27-Sep-2024.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    imaeqsexvOLD $p |- ( ( F Fn A /\ B C_ A ) ->
    ( E. x e. ( F " B ) ph <-> E. y e. B ps ) ) $=
      ( wfn wss wa cima wrex cv cfv wceq wex wcel df-rex exbii fvelimab rexcom4
      anbi1d exbidv bitrid eqcom anbi1i ceqsexv rexbii r19.41v 3bitr3ri bitrdi
      fvex bitri ) GEIFEJKZACGFLZMZDNZGOZCNZPZDFMZAKZCQZBDFMZUQUTUPRZAKZCQUOVDA
      CUPSUOVGVCCUOVFVBADEFUTGUAUCUDUEVAAKZCQZDFMVHDFMZCQVEVDVHDCFUBVIBDFVIUTUS
      PZAKZCQBVHVLCVAVKAUSUTUFUGTABCUSURGUMHUHUNUIVJVCCVAADFUJTUKUL $.

    $( Obsolete version of ~ ralima as of 14-Aug-2025.  Duplicate version of
       ~ ralima .  (Contributed by Scott Fenton, 27-Sep-2024.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    imaeqsalvOLD $p |- ( ( F Fn A /\ B C_ A ) ->
    ( A. x e. ( F " B ) ph <-> A. y e. B ps ) ) $=
      ( wfn wss wa wn cima wrex wral cv cfv wceq notbid dfral2 imaeqsexvOLD
      3bitr4g ) GEIFEJKZALZCGFMZNZLBLZDFNZLACUEOBDFOUCUFUHUDUGCDEFGCPDPGQRABHSU
      ASACUETBDFTUB $.
  $}

$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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.  For the equinumerosity version, see ~ canth2 .  Note
       that ` A ` must be a set: this theorem does not hold when ` A ` is too
       large to be a set; see ~ ncanth for a counterexample.  (Use ~ nex if you
       want the form ` -. E. f f : A -onto-> ~P A ` .)  (Contributed by NM,
       7-Aug-1994.)  (Proof shortened by Mario Carneiro, 7-Jun-2016.) $)
    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 id fveq2 eleq12d notbid elrab baibr nbbn sylib eleq2 nsyl wfn
      nrex fofn fvelrnb syl mtbiri pm2.65i ) AAFZBGZDHZVABIZJZKZDALZBMZJZUTVEUS
      VFVEASCVDDANOAUSBPQUTVGEHZBIZVERZEAUAZVJEAVHAJZVHVIJZVHVEJZTZVJVLVMKZVNTV
      OKVNVLVPVDVPDVHAVAVHRZVCVMVQVAVHVBVIVQUBVAVHBUCUDUEUFUGVMVNUHUIVIVEVHUJUK
      UMUTBAULVGVKTAUSBUNEAVEBUOUPUQUR $.
  $}

  $( Cantor's theorem fails for the universal class (which is not a set but a
     proper class by ~ vprc ).  Specifically, the identity function maps the
     universe onto its power class.  Compare ~ canth that works for sets.

     This failure comes from a limitation of the collection principle (which is
     necessary to avoid Russell's paradox ~ ru ): ` ~P _V ` , being a class,
     cannot contain proper classes, so it is no larger than ` _V ` , which is
     why the identity function "succeeds" in being surjective onto ` ~P _V `
     (see ~ pwv ).  See also the remark in ~ ru about NF, in which Cantor's
     theorem fails for sets that are "too large".  This theorem gives some
     intuition behind that failure: in NF the universal class is a set, and it
     equals its own power set.  (Contributed by NM, 29-Jun-2004.)  (Proof
     shortened by BJ, 29-Dec-2023.) $)
  ncanth $p |- _I : _V -onto-> ~P _V $=
    ( cvv cpw cid wfo wf1o f1ovi f1ofo ax-mp wceq wb pwv foeq3 mpbir ) AABZCDZA
    ACDZAACEPFAACGHNAIOPJKNAACLHM $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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 anbi2d iotabidv df-riota 3eqtr4g ) ADGEHZBIZDJPCIZD
      JBDEKCDEKAQRDABCPFLMBDENCDENO $.
  $}

  ${
    $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 $.
  $}

  $( Restricted iota is a set.  (Contributed by NM, 15-Sep-2011.) $)
  riotaex $p |- ( iota_ x e. A ps ) e. _V $=
    ( crio cv wcel wa cio cvv df-riota iotaex eqeltri ) ABCDBECFAGZBHIABCJMBKL
    $.

  $( 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 w $.  $d A w $.  $d ps w $.
    nfriotadw.1 $e |- F/ y ph $.
    nfriotadw.2 $e |- ( ph -> F/ x ps ) $.
    nfriotadw.3 $e |- ( ph -> F/_ x A ) $.
    $( Deduction version of ~ nfriota with a disjoint variable condition, which
       contrary to ~ nfriotad does not require ~ ax-13 .  (Contributed by NM,
       18-Feb-2013.)  Avoid ~ ax-13 .  (Revised by GG, 26-Jan-2024.) $)
    nfriotadw $p |- ( ph -> F/_ x ( iota_ y e. A ps ) ) $=
      ( vw crio cv wcel wa cio df-riota wal wnfc adantr wnf df-nfc weq wn nfcvd
      nfnaew adantl nfeld nfand nfiotadw ex nfiota1 biidd drnf1v albidv 3bitr4g
      nfan mpbiri pm2.61d2 nfcxfrd ) ACBDEJDKZELZBMZDNZBDEOACDUACPZCVBQZAVCUBZV
      DAVEMZVACDAVEDFCDDUDUOVFUTBCVFCUSEVECUSQAVECUSUCUEACEQVEHRUFABCSVEGRUGUHU
      IVCVDDVBQZVADUJVCIKVBLZCSZIPVHDSZIPVDVGVCVIVJIVHVHCDVCVHUKULUMCIVBTDIVBTU
      NUPUQUR $.
    $( $j usage 'nfriotadw' avoids 'ax-13'; $)
  $}

  ${
    $d x z A $.  $d y z A $.  $d z ph $.  $d z ps $.  $d x y $.
    cbvriotaw.1 $e |- F/ y ph $.
    cbvriotaw.2 $e |- F/ x ps $.
    cbvriotaw.3 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Change bound variable in a restricted description binder.  Version of
       ~ cbvriota with a disjoint variable condition, which does not require
       ~ ax-13 .  (Contributed by NM, 18-Mar-2013.)  Avoid ~ ax-13 .  (Revised
       by GG, 26-Jan-2024.) $)
    cbvriotaw $p |- ( iota_ x e. A ph ) = ( iota_ y e. A ps ) $=
      ( vz cv wcel wa cio crio weq eleq1w anbi12d nfv nfan cbviotaw wsb sbequ12
      nfs1v sbhypf nfsbv eqtri df-riota 3eqtr4i ) CJEKZALZCMZDJZEKZBLZDMZACENBD
      ENUKIJEKZACIUAZLZIMUOUJURCICIOUIUPAUQCIEPACIUBQUJIRUPUQCUPCRACIUCSTURUNID
      IDOUPUMUQBIDEPABCIULGHUDQUPUQDUPDRACIDFUESUNIRTUFACEUGBDEUGUH $.
    $( $j usage 'cbvriotaw' avoids 'ax-13'; $)
  $}

  ${
    $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, which requires fewer
       axioms .  (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 $.
    $( $j usage 'cbvriotavw' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    nfriotad.1 $e |- F/ y ph $.
    nfriotad.2 $e |- ( ph -> F/ x ps ) $.
    nfriotad.3 $e |- ( ph -> F/_ x A ) $.
    $( Deduction version of ~ nfriota .  Usage of this theorem is discouraged
       because it depends on ~ ax-13 .  Use the weaker ~ nfriotadw when
       possible.  (Contributed by NM, 18-Feb-2013.)  (Revised by Mario
       Carneiro, 15-Oct-2016.)  (New usage is discouraged.) $)
    nfriotad $p |- ( ph -> F/_ x ( iota_ y e. A ps ) ) $=
      ( crio cv wcel wa cio df-riota weq wal wnfc wn nfnae adantr nfcvf nfiotad
      nfan adantl nfeld wnf nfand nfiota1 eqidd drnfc1 mpbiri pm2.61d2 nfcxfrd
      ex ) ACBDEIDJZEKZBLZDMZBDENACDOCPZCURQZAUSRZUTAVALZUQCDAVADFCDDSUCVBUPBCV
      BCUOEVACUOQACDUAUDACEQVAHTUEABCUFVAGTUGUBUNUSUTDURQUQDUHCDURURUSURUIUJUKU
      LUM $.
  $}

  ${
    $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 nfriotadw mptru ) BACDGHIABCDCJABKIELBDHIF
      LMN $.
  $}

  ${
    $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.  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  Use the weaker
       ~ cbvriotaw when possible.  (Contributed by NM, 18-Mar-2013.)  (Revised
       by Mario Carneiro, 15-Oct-2016.)  (New usage is discouraged.) $)
    cbvriota $p |- ( iota_ x e. A ph ) = ( iota_ y e. A ps ) $=
      ( vz cv wcel wa cio crio wsb weq eleq1w anbi12d nfv nfan nfs1v sbequ sbie
      sbequ12 cbviota bitrdi nfsb eqtri df-riota 3eqtr4i ) CJEKZALZCMZDJEKZBLZD
      MZACENBDENUMIJEKZACIOZLZIMUPULUSCICIPUKUQAURCIEQACIUDRULISUQURCUQCSACIUAT
      UEUSUOIDIDPZUQUNURBIDEQUTURACDOBAIDCUBABCDGHUCUFRUQURDUQDSACIDFUGTUOISUEU
      HACEUIBDEUIUJ $.
  $}

  ${
    $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.  Usage of this
       theorem is discouraged because it depends on ~ ax-13 .  Use the weaker
       ~ cbvriotavw when possible.  (Contributed by NM, 18-Mar-2013.)  (Revised
       by Mario Carneiro, 15-Oct-2016.)  (New usage is discouraged.) $)
    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.)  (Revised by NM, 2-Sep-2018.) $)
    csbriota $p |- [_ A / x ]_ ( iota_ y e. B ph )
          = ( iota_ y e. B [. A / x ]. ph ) $=
      ( vz cvv wcel crio csb wsbc wceq wsb csbeq1 dfsbcq2 riotabidv eqeq12d vex
      cv c0 nfs1v nfcv nfriota weq sbequ12 csbief vtoclg wn csbprc cio df-riota
      weu wex euex sbcex adantl exlimiv syl iotanul nsyl5 eqtr2id eqtrd pm2.61i
      wa ) DGHZBDACEIZJZABDKZCEIZLZBFSZVFJZABFMZCEIZLVJFDGVKDLZVLVGVNVIBVKDVFNV
      OVMVHCEABFDOPQBVKVFVNFRVMBCEABFUABEUBUCBFUDAVMCEABFUEPUFUGVEUHZVGTVIBDVFU
      IVPVICSEHZVHVDZCUJZTVHCEUKVRCULZVEVSTLVTVRCUMVEVRCUNVRVECVHVEVQABDUOUPUQU
      RVRCUSUTVAVBVC $.
  $}

  $( 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.  Compare ~ iotasbc .  (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 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 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.  (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 B $.  $d x C $.  $d x y A $.  $d x y ph $.  $d ps y $.  $d ch x $.
    riotaxfrd.1 $e |- F/_ y C $.
    riotaxfrd.2 $e |- ( ( ph /\ y e. A ) -> B e. A ) $.
    riotaxfrd.3 $e |- ( ( ph /\ ( iota_ y e. A ch ) e. A ) -> C e. A ) $.
    riotaxfrd.4 $e |- ( x = B -> ( ps <-> ch ) ) $.
    riotaxfrd.5 $e |- ( y = ( iota_ y e. A ch ) -> B = C ) $.
    riotaxfrd.6 $e |- ( ( ph /\ x e. A ) -> E! y e. A x = B ) $.
    $( Change the variable ` x ` in the expression for "the unique ` x ` such
       that ` ps ` " to another variable ` y ` contained in expression ` B ` .
       Use ~ reuhypd to eliminate the last hypothesis.  (Contributed by NM,
       16-Jan-2012.)  (Revised by Mario Carneiro, 15-Oct-2016.) $)
    riotaxfrd $p |- ( ( ph /\ E! x e. A ps )
         -> ( iota_ x e. A ps ) = C ) $=
      ( wreu wa crio crab wcel adantl cv baib riotabiia wceq reuxfr1ds riotacl2
      rabid wb riotacl nfriota1 rabxfrd sylan2 mpbird sylbid syl5 baibr reubiia
      ex imp biimpi nfcv nfrab1 nfel2 eleq1 riota2f syl2anc mpbid eqtr3id ) ABD
      FOZPZBDFQDUAZBDFRZSZDFQZHVMBDFVMVKFSZBBDFUGZUBUCVJHVLSZVNHUDZAVIVQAVICEFO
      ZVQABCDEGFFJNLUEZAVSVQAVSPVQCEFQZCEFRSZVSWBACEFUFTVSAWAFSZVQWBUHCEFUIZABC
      DEGWAHFCEFUJIJLMUKULUMURUNUSVJHFSZVMDFOZVQVRUHAVIWEAVIVSWEVTVSWCAWEWDAWCW
      EKURUOUNUSVIWFAVIWFBVMDFVMVOBVPUPUQUTTVMVQDFHDHVADHVLBDFVBVCVKHVLVDVEVFVG
      VH $.
  $}

  ${
    $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 c0 wne euex rexn0 r19.2z 3syl
      ex impbid ) AGZDHZBCIZAJZVHVGBCKZVIVHALZFGZMZHZFNVHVJOZVHAFPVNVOFVNVHVFVL
      HZVJVNVFVKQVFVMQVHVPVKVMVFRVHASAVLUAUBVNVJVPVLDHZBCKVNVQBCBVKVMVHBAVGBCUC
      UDUEBGCQDVKQZVNVQBACDEUFVNVRDVMQZVQVKVMDRVSDVLHVQDVLEUGDVLUHULUIUJUKVPVGV
      QBCVFVLDUMUNUOUPTUQVIVHANCURUSZVJVHOVHAUTVHVTAVGBCVATVTVJVHVGBCVBVDVCVE
      $.

    $( 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 $.

    $( The restricted iota class is limited in size by the base set.
       (Contributed by Mario Carneiro, 24-Dec-2016.) $)
    riotassuni $p |- ( iota_ x e. A ph ) C_ ( ~P U. A u. U. A ) $=
      ( wreu crio cuni cpw cun crab riotauni ssrab2 unissi ssun2 sstri eqsstrdi
      wss wn c0 riotaund 0ss pm2.61i ) ABCDZABCEZCFZGZUDHZPUBUCABCIZFZUFABCJUHU
      DUFUGCABCKLUDUEMNOUBQUCRUFABCSUFTOUA $.

    $( Bidirectional closure of restricted iota when domain is not empty.
       (Contributed by NM, 28-Feb-2013.)  (Revised by Mario Carneiro,
       24-Dec-2016.)  (Revised by NM, 13-Sep-2018.) $)
    riotaclb $p |- ( -. (/) e. A
          -> ( E! x e. A ph <-> ( iota_ x e. A ph ) e. A ) ) $=
      ( c0 wcel wreu crio riotacl riotaund eleq1d notbid biimprcd con4d impbid2
      wn ) DCEZOZABCFZABCGZCEZABCHQRTROZTOQUATPUASDCABCIJKLMN $.
  $}

  ${
    $d A y $.  $d ph y $.  $d x y $.
    riotarab.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( Restricted iota of a restricted abstraction.  (Contributed by Scott
       Fenton, 8-Aug-2024.) $)
    riotarab $p |- ( iota_ x e. { y e. A | ps } ch ) =
     ( iota_ x e. A ( ph /\ ch ) ) $=
      ( cv crab wcel wa cio crio wb weq bicomd equcoms elrab anbi1i df-riota
      anass bitri iotabii 3eqtr4i ) DHZBEFIZJZCKZDLUEFJZACKZKZDLCDUFMUJDFMUHUKD
      UHUIAKZCKUKUGULCBAEUEFBANDEDEOABGPQRSUIACUAUBUCCDUFTUJDFTUD $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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.  (For an example of how syntax could become ambiguous if we
     are not careful, see the comment in ~ cneg .) $)
  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 ` (see
     ~ 3p2e5 ).  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, such as ` +o ` in ~ oav .
     ` 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 an operation given by a class
       abstraction 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
       ` C ( 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 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 $.
    $d x y z $.
    $( The law of concretion.  Special case of Theorem 9.5 of [Quine] p. 61.
       Version of ~ oprabid with a disjoint variable condition, which does not
       require ~ ax-13 .  (Contributed by Mario Carneiro, 20-Mar-2013.)  Avoid
       ~ ax-13 .  (Revised by GG, 26-Jan-2024.) $)
    oprabidw $p |- ( <. <. x , y >. , z >. e.
        { <. <. x , y >. , z >. | ph } <-> ph ) $=
      ( vw va vt vr vs cv cop wceq wa wex wi vex weq 19.8a anim2i eximi eqvinop
      coprab opex biimpi eqeq1 opth1 biimtrdi opeq1 eqeq2d w3a otth2 weu eupick
      euequ mpan syl6 3impd biimtrid df-3an bitri anbi1i 3bitri 3exbii nfe1 wal
      anass biidd drex1v imbitrrid 19.40 nfvd 19.9d anim1d syl56 pm2.61i exlimi
      excom 3imtr4i 3syl sylbi syl11 eqcom bitrdi anbi1d 3exbidv imbi1d imbi12d
      wn mpbiri adantr exlimivv com3l mpdd mpcom ex impbid df-oprab elab2 ) EJZ
      BJZCJZKZDJZKZLZAMZDNZCNZBNZAEXDABCDUBXBXCUCXEXIAWSFJZGJZKZLZXLXDLZMZGNFNZ
      XEXIAOZXEXPFGWSXBXCWTXAUCDPZUAUDXOXEXQOZFGXMXSXNXMXEXJXBLZXQXMXEXNXTWSXLX
      DUEXJXKXBXCFPGPUFUGXTXMXEXQXTXJHJZIJZKZLZYCXBLZMZINHNXMXSOZHIXJWTXABPZCPZ
      UAYFYGHIYDYGYEYDXMWSYCXKKZLZXSYDXLYJWSXJYCXKUHUIYKXSXDYJLZYLAMZDNCNBNZAOZ
      OBHQZCIQZDGQZAMZDNZMZCNZMZBNZYLAYNYLYPYQYRUJZUUDAWTXAYAYBXCXKYHYIXRUKZUUD
      YPYQYRAUUDYPUUBYQYRAOZOYPBULUUDYPUUBOBHUNYPUUBBUMUOUUBYQYTUUGYQCULUUBYQYT
      OCIUNYQYTCUMUOYRDULYTUUGDGUNYRADUMUOUPUPUQURYNYPYQYSMZMZDNZCNBNZUUDYMUUIB
      CDYMYPYQMZYRMZAMUULYSMUUIYLUUMAYLUUEUUMUUFYPYQYRUSUTVAUULYRAVFYPYQYSVFVBV
      CUUKYPUUHDNZMZCNZBNZYPUUNCNZMZBNZUUDUUJBNZCNUUOBNZCNUUKUUQUVAUVBCUUJUVBBU
      UOBVDBDQBVEZUUJUVBOUUJUVBUVCUUODNUUIUUODUUHUUNYPUUHDRSTUUOUUOBDUVCUUOVGVH
      VIUUJYPDNZUUNMUVCWHZUUOUVBYPUUHDVJUVEUVDYPUUNYPUVEDUVEYPDVKVLVMUUOBRVNVOV
      PTUUJBCVQUUOBCVQVRUUPUUTBUUSBVDBCQBVEZUUPUUTOUUPUUTUVFUUSCNUUOUUSCUUNUURY
      PUUNCRSTUUSUUSBCUVFUUSVGVHVIUUPYPCNZUURMUVFWHZUUSUUTYPUUNCVJUVHUVGYPUURYP
      UVHCUVHYPCVKVLVMUUSBRVNVOVPUUSUUCBUURUUBYPUUNUUBCUUACVDCDQCVEZUUNUUBOUUNU
      UBUVIUUADNUUHUUADYSYTYQYSDRSTUUAUUACDUVIUUAVGVHVIUUNYQDNZYTMUVIWHZUUAUUBY
      QYSDVJUVKUVJYQYTYQUVKDUVKYQDVKVLVMUUACRVNVOVPSTVSVTWAYKXEYLXQYOYKXEYJXDLY
      LWSYJXDUEYJXDWBWCZYKXIYNAYKXFYMBCDYKXEYLAUVLWDWEWFWGWIUGWJWKVTWLWMWJWKWNX
      EAXIXFXGXHXIXFDRXGCRXHBRVSWOWPABCDEWQWR $.
    $( $j usage 'oprabidw' avoids 'ax-13'; $)
  $}

  ${
    $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.
       Usage of this theorem is discouraged because it depends on ~ ax-13 .
       Use the weaker ~ oprabidw when possible.  (Contributed by Mario
       Carneiro, 20-Mar-2013.)  (New usage is discouraged.) $)
    oprabid $p |- ( <. <. x , y >. , z >. e.
        { <. <. x , y >. , z >. | ph } <-> ph ) $=
      ( vw va vt vr vs cv cop wceq wa wex wi vex weq weu euequ eupick eqeq1 w3a
      coprab opex eqvinop biimpi opth1 biimtrdi opeq1 eqeq2d mpan syl6 biimtrid
      otth2 3impd df-3an bitri anbi1i anass 3bitri 3exbii wn nfcvf2 nfcvd nfeqd
      exdistrf eximi excom 3imtr4i anim2i 3syl sylbi syl11 eqcom bitrdi 3exbidv
      wal anbi1d imbi1d imbi12d mpbiri adantr exlimivv com3l mpcom 19.8a impbid
      mpdd ex df-oprab elab2 ) EJZBJZCJZKZDJZKZLZAMZDNZCNZBNZAEWQABCDUCWOWPUDWR
      XBAWLFJZGJZKZLZXEWQLZMZGNFNZWRXBAOZWRXIFGWLWOWPWMWNUDDPZUEUFXHWRXJOZFGXFX
      LXGXFWRXCWOLZXJXFWRXGXMWLXEWQUAXCXDWOWPFPGPUGUHXMXFWRXJXMXCHJZIJZKZLZXPWO
      LZMZINHNXFXLOZHIXCWMWNBPZCPZUEXSXTHIXQXTXRXQXFWLXPXDKZLZXLXQXEYCWLXCXPXDU
      IUJYDXLWQYCLZYEAMZDNCNBNZAOZOBHQZCIQZDGQZAMZDNZMCNZMZBNZYEAYGYEYIYJYKUBZY
      PAWMWNXNXOWPXDYAYBXKUNZYPYIYJYKAYPYIYNYJYKAOZOYIBRYPYIYNOBHSYIYNBTUKYNYJY
      MYSYJCRYNYJYMOCISYJYMCTUKYKDRYMYSDGSYKADTUKULULUOUMYGYIYJYLMZMZDNZCNBNZYP
      YFUUABCDYFYIYJMZYKMZAMUUDYLMUUAYEUUEAYEYQUUEYRYIYJYKUPUQURUUDYKAUSYIYJYLU
      SUTVAUUCYIYTDNZMZCNBNZYIUUFCNZMZBNYPUUBBNZCNUUGBNZCNUUCUUHUUKUULCYIYTBDBD
      QBVQVBZDWMXNBDVCUUMDXNVDVEVFVGUUBBCVHUUGBCVHVIYIUUFBCBCQBVQVBZCWMXNBCVCUU
      NCXNVDVEVFUUJYOBUUIYNYIYJYLCDCDQCVQVBZDWNXOCDVCUUODXOVDVEVFVJVGVKVLVMYDWR
      YEXJYHYDWRYCWQLYEWLYCWQUAYCWQVNVOZYDXBYGAYDWSYFBCDYDWRYEAUUPVRVPVSVTWAUHW
      BWCVLWDWHWBWCWEWRAXBWSWTXAXBWSDWFWTCWFXABWFVKWIWGABCDEWJWK $.
  $}

  $( The result of an operation is a set.  (Contributed by NM, 13-Mar-1995.) $)
  ovex $p |- ( A F B ) e. _V $=
    ( co cop df-ov fvexi ) ABCDABECABCFG $.

  ${
    ovexi.1 $e |- A = ( B F C ) $.
    $( The result of an operation is a set.  (Contributed by Glauco Siliprandi,
       26-Jun-2021.) $)
    ovexi $p |- A e. _V $=
      ( co cvv ovex eqeltri ) ABCDFGEBCDHI $.
  $}

  $( The result of an operation is a set.  (Contributed by Glauco Siliprandi,
     23-Oct-2021.) $)
  ovexd $p |- ( ph -> ( A F B ) e. _V ) $=
    ( co cvv wcel ovex a1i ) BCDEFGABCDHI $.

  $( The result of an operation value is always a subset of the union of the
     range.  (Contributed by Mario Carneiro, 12-Jan-2017.) $)
  ovssunirn $p |- ( X F Y ) C_ U. ran F $=
    ( co cop cfv crn cuni df-ov fvssunirn eqsstri ) BCADBCEZAFAGHBCAIALJK $.

  $( Operation value of the empty set.  (Contributed by AV, 15-May-2021.) $)
  0ov $p |- ( A (/) B ) = (/) $=
    ( c0 co cop cfv df-ov 0fv eqtri ) ABCDABEZCFCABCGJHI $.

  ${
    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 c0 df-ov cdm wceq wbr df-br brrelex12i sylbir
      ndmfv nsyl5 eqtrid ) AEFBEFGZHABCIABJZCKZLABCMUDCNZFZUCUELOUGABUFPUCABUFQ
      ABUFDRSUDCTUAUB $.

    $( 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 wa co c0 wceq simpl ovprc nsyl5 ) AEFZBEFZGNABCHIJNOKABCDLM $.

    $( 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 wa co c0 wceq simpr ovprc nsyl5 ) AEFZBEFZGOABCHIJNOKABCDLM $.

    $( Reverse closure for an operation value.  (Contributed by Mario Carneiro,
       5-May-2015.) $)
    ovrcl $p |- ( C e. ( A F B ) -> ( A e. _V /\ B e. _V ) ) $=
      ( co wcel c0 wceq cvv wa n0i ovprc nsyl2 ) CABDFZGOHIAJGBJGKOCLABDEMN $.
  $}

  ${
    elfvov1.o $e |- Rel dom O $.
    elfvov1.s $e |- S = ( I O R ) $.
    elfvov1.x $e |- ( ph -> X e. ( S ` Y ) ) $.
    $( Utility theorem: reverse closure for any operation that results in a
       function.  (Contributed by SN, 18-May-2025.) $)
    elfvov1 $p |- ( ph -> I e. _V ) $=
      ( cfv c0 wceq cvv wcel wn n0i syl co ovprc1 eqtrid fveq1d eqtrdi nsyl2
      0fv ) AGCKZLMZDNOZAFUFOUGPJUFFQRUHPZUFGLKLUIGCLUICDBESLIDBEHTUAUBGUEUCUD
      $.

    $( Utility theorem: reverse closure for any operation that results in a
       function.  (Contributed by SN, 4-Aug-2025.) $)
    elfvov2 $p |- ( ph -> R e. _V ) $=
      ( cfv c0 wceq cvv wcel wn n0i syl co ovprc2 eqtrid fveq1d eqtrdi nsyl2
      0fv ) AGCKZLMZBNOZAFUFOUGPJUFFQRUHPZUFGLKLUIGCLUICDBESLIDBEHTUAUBGUEUCUD
      $.
  $}

  ${
    $d y A $.  $d y B $.  $d y C $.  $d y F $.  $d x y $.
    $( Move class substitution in and out of an operation.  (Contributed by NM,
       12-Nov-2005.)  (Revised by NM, 23-Aug-2018.) $)
    csbov123 $p |- [_ A / x ]_ ( B F C ) =
           ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ C ) $=
      ( vy cvv wcel co csb cv csbeq1 oveq123d eqeq12d nfcsb1v csbeq1a c0 csbprc
      wceq cfv vex nfov weq csbief vtoclg wn cop df-ov fveq1d 0fv eqtr2id eqtrd
      eqtrdi pm2.61i ) BGHZABCDEIZJZABCJZABDJZABEJZIZSZAFKZUPJZAVCCJZAVCDJZAVCE
      JZIZSVBFBGVCBSZVDUQVHVAAVCBUPLVIVEURVFUSVGUTAVCBELAVCBCLAVCBDLMNAVCUPVHFU
      AAVEVFVGAVCCOAVCEOAVCDOUBAFUCCVEDVFEVGAVCEPAVCCPAVCDPMUDUEUOUFZUQQVAABUPR
      VJVAURUSUGZUTTZQURUSUTUHVJVLVKQTQVJVKUTQABERUIVKUJUMUKULUN $.
  $}

  ${
    $d x B $.  $d x C $.
    $( Move class substitution in and out of an operation.  (Contributed by NM,
       23-Aug-2018.) $)
    csbov $p |- [_ A / x ]_ ( B F C ) = ( B [_ A / x ]_ F C ) $=
      ( co csb csbov123 cvv wcel wceq csbconstg oveq12d wn c0 cop cfv 0fv df-ov
      oveqd 0ov 3eqtr4ri csbprc 3eqtr4a pm2.61i eqtri ) ABCDEFGABCGZABDGZABEGZF
      ZCDUIFZABCDEHBIJZUJUKKULUGCUHDUIABCILABDILMULNZUGUHOFZCDOFZUJUKCDPZOQOUOU
      NUPRCDOSUGUHUAUBUMUIOUGUHABEUCZTUMUIOCDUQTUDUEUF $.
  $}

  ${
    $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 csbov123 csbconstg oveqd eqtrid ) BFGZABCDEHIABCIZABDIZABEI
      ZHOPEHABCDEJNQEOPABEFKLM $.
  $}

  ${
    $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.)  (Proof shortened by BJ, 15-Feb-2022.) $)
  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 co wceq cotp wb wa cop wbr fnbrovb df-br a1i df-ot eqcomi
    eleq1i 3bitrd 3impb ) FABGHZCAIZDBIZCDFJEKZCDELZFIZMUEUFUGNNZUHCDOZEFPZULEO
    ZFIZUJCDEFABQUMUOMUKULEFRSUOUJMUKUNUIFUIUNCDETUAUBSUCUD $.

  $( A collection of ordered pairs with an extension of a binary relation is a
     set.  (Contributed by Alexander van der Vekens, 1-Nov-2017.)  (Revised by
     BJ/AV, 20-Jun-2019.)  (Proof shortened by OpenAI, 25-Mar-2020.) $)
  opabbrex $p |- ( ( A. x A. y ( x R y -> ph ) /\ { <. x , y >. | ph } e. V )
                   -> { <. x , y >. | ( x R y /\ ps ) } e. _V ) $=
    ( cv wbr wi wal copab wcel wa simpr wss pm3.41 2alimi adantr ssopab2 syl
    ssexd ) CGDGEHZAIZDJCJZACDKZFLZMZUBBMZCDKZUEFUDUFNUGUHAIZDJCJZUIUEOUDUKUFUC
    UJCDUBBAPQRUHACDSTUA $.

  ${
    $d W x z $.  $d W y z $.  $d G x z $.  $d G y $.  $d th z $.
    $( Restrictions of a collection of ordered pairs of related elements are
       sets.  (Contributed by Alexander van der Vekens, 1-Nov-2017.)  (Revised
       by AV, 15-Jan-2021.)  Add disjoint variable conditions betweem ` W , G `
       and ` x , y ` to remove hypotheses.  (Revised by SN, 13-Dec-2024.) $)
    opabresex2 $p |- { <. x , y >. | ( x ( W ` G ) y /\ th ) } e. _V $=
      ( vz cv cfv wbr wa copab fvex elopabran ssriv ssexi ) BGCGDEHZIAJBCKZPDEL
      FQPABCFGPMNO $.
  $}

  ${
    $d ph x $.  $d ph y $.
    opabresex2d.1 $e |- ( ( ph /\ x ( W ` G ) y ) -> ps ) $.
    opabresex2d.2 $e |- ( ph -> { <. x , y >. | ps } e. V ) $.
    $( Obsolete version of ~ opabresex2 as of 13-Dec-2024.  (Contributed by
       Alexander van der Vekens, 1-Nov-2017.)  (Revised by AV, 15-Jan-2021.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    opabresex2d $p |- ( ph -> { <. x , y >. |
                                ( x ( W ` G ) y /\ th ) } e. _V ) $=
      ( cv cfv wbr wi wal copab wcel wa cvv ex alrimivv opabbrex syl2anc ) ADKE
      KFHLZMZBNZEODOBDEPGQUECRDEPSQAUFDEAUEBITUAJBCDEUDGUBUC $.
  $}

  ${
    $d F x y z $.  $d Z x y z $.  $d ps z $.
    fvmptopab.1 $e |- ( z = Z -> ( ph <-> ps ) ) $.
    fvmptopab.m $e |- M = ( z e. _V |-> { <. x , y >. |
                                             ( x ( F ` z ) y /\ ph ) } ) $.
    $( The function value of a mapping ` M ` to a restricted binary relation
       expressed as an ordered-pair class abstraction:  The restricted binary
       relation is a binary relation given as value of a function ` F `
       restricted by the condition ` ps ` .  (Contributed by AV, 31-Jan-2021.)
       (Revised by AV, 29-Oct-2021.)  Add disjoint variable condition on
       ` F , x , y ` to remove a sethood hypothesis.  (Revised by SN,
       13-Dec-2024.) $)
    fvmptopab $p |- ( M ` Z ) = { <. x , y >. | ( x ( F ` Z ) y /\ ps ) } $=
      ( cvv wcel cfv cv wbr wa copab wceq c0 fvprc fveq2 breqd anbi12d opabbidv
      opabresex2 fvmpt wn wss elopabran ssriv sseqtrid ss0 syl eqtr4d pm2.61i )
      HKLZHGMZCNZDNZHFMZOZBPZCDQZREHURUSENZFMZOZAPZCDQVCKGVDHRZVGVBCDVHVFVAABVH
      VEUTURUSVDHFUAUBIUCUDJBCDHFUEUFUPUGZUQSVCHGTVIVCSUHVCSRVIUTVCSEVCUTBCDVDU
      TUIUJHFTUKVCULUMUNUO $.
  $}

  ${
    $d F z $.  $d Z x y z $.  $d ph x y z $.  $d ps z $.
    fvmptopabOLD.1 $e |- ( ( ph /\ z = Z ) -> ( ch <-> ps ) ) $.
    fvmptopabOLD.2 $e |- ( ph -> { <. x , y >. | x ( F ` Z ) y } e. _V ) $.
    fvmptopabOLD.3 $e |- M = ( z e. _V |-> { <. x , y >. |
                                             ( x ( F ` z ) y /\ ch ) } ) $.
    $( Obsolete version of ~ fvmptopab as of 13-Dec-2024.  (Contributed by AV,
       31-Jan-2021.)  (Revised by AV, 29-Oct-2021.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    fvmptopabOLD $p |- ( ph -> ( M ` Z )
                             = { <. x , y >. | ( x ( F ` Z ) y /\ ps ) } ) $=
      ( cvv wcel cfv cv wbr wa wal c0 copab wceq wi fveq2 breqd adantll anbi12d
      wb adantl opabbidv simpl id gen2 opabbrex sylancr fvmptd2 ex wn fvprc br0
      mtbiri intnanrd alrimivv opab0 sylibr eqtr4d a1d pm2.61i ) IMNZAIHOZDPZEP
      ZIGOZQZBRZDEUAZUBZUCVIAVQVIARZFIVKVLFPZGOZQZCRZDEUAVPMHMLVRVSIUBZRZWBVODE
      WDWAVNCBWCWAVNUHVRWCVTVMVKVLVSIGUDUEUIAWCCBUHVIJUFUGUJVIAUKVRVNVNUCZESDSV
      NDEUAMNZVPMNWEDEVNULUMAWFVIKUIVNBDEVMMUNUOUPUQVIURZVQAWGVJTVPIHUSWGVOURZE
      SDSVPTUBWGWHDEWGVNBWGVNVKVLTQVKVLUTWGVMTVKVLIGUSUEVAVBVCVODEVDVEVFVGVH $.
  $}

  ${
    $d A r s t u v w $.  $d B r s t u v w $.  $d F r s t u v w $.
    $( Condition for an operation to be one-to-one.  (Contributed by Jeff
       Madsen, 17-Jun-2010.) $)
    f1opr $p |- ( F : ( A X. B ) -1-1-> C <-> ( F : ( A X. B ) --> C /\
                          A. r e. A A. s e. B A. t e. A A. u e. B
                    ( ( r F s ) = ( t F u ) -> ( r = t /\ s = u ) ) ) ) $=
      ( vv vw cv cfv wceq weq wi wral wa co cop fveq2 cxp wf dff13 df-ov eqeq1d
      wf1 eqtr4di eqeq1 imbi12d ralbidv ralxp eqeq2d eqeq2 bitrdi 2ralbii bitri
      vex opth anbi2i ) CDUAZEFUFUTEFUBZIKZFLZJKZFLZMZIJNZOZJUTPZIUTPZQVAHKZGKZ
      FRZBKZAKZFRZMZHBNGANQZOZADPBCPZGDPHCPZQIJUTEFUCVJWAVAVJVMVEMZVKVLSZVDMZOZ
      JUTPZGDPHCPWAVIWFIHGCDVBWCMZVHWEJUTWGVFWBVGWDWGVCVMVEWGVCWCFLVMVBWCFTVKVL
      FUDUGUEVBWCVDUHUIUJUKWFVTHGCDWEVSJBACDVDVNVOSZMZWBVQWDVRWIVEVPVMWIVEWHFLV
      PVDWHFTVNVOFUDUGULWIWDWCWHMVRVDWHWCUMVKVLVNVOHUQGUQURUNUIUKUOUPUSUP $.
  $}

  ${
    brfvopab.1 $e |- ( X e. _V -> ( F ` X ) = { <. y , z >. | ph } ) $.
    $( The classes involved in a binary relation of a function value which is
       an ordered-pair class abstraction are sets.  (Contributed by AV,
       7-Jan-2021.) $)
    brfvopab $p |- ( A ( F ` X ) B -> ( X e. _V /\ A e. _V /\ B e. _V ) ) $=
      ( cvv wcel cfv wbr w3a wi wa copab breqd brabv biimtrdi c0 imdistani wceq
      3anass sylibr ex wn fvprc breq br0 pm2.21i syl pm2.61i ) GIJZDEGFKZLZUMDI
      JZEIJZMZNZUMUOURUMUOOUMUPUQOZOURUMUOUTUMUODEABCPZLUTUMUNVADEHQABCDERSUAUM
      UPUQUCUDUEUMUFUNTUBZUSGFUGVBUODETLZURDEUNTUHVCURDEUIUJSUKUL $.
  $}

  ${
    $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
      eqeq2d pm5.32ri anbi1i anass an32 exbii opex isseti 19.42v 3exbii 19.42vv
      mpbiran2 2exbii abbii df-oprab df-opab 3eqtr4i ) FGZBGZCGZHZDGZHZIZAJZDKC
      KBKZFLUQEGZVAHZIZVFUTIZAJZCKBKZJZDKEKZFLABCDMVKEDNVEVMFVHVJJZCKBKZEKDKZVO
      DKEKVEVMVODEOVPVNEKZDKCKBKVEVNDEBCPVQVDBCDVQVDVIJZEKZVDVNVREVHVIJZAJVCVIJ
      ZAJVNVRVTWAAVIVHVCVIVGVBUQVFUTVAQTUAUBVHVIAUCVCVIAUDRUEVSVDVIEKEUTURUSUFU
      GVDVIEUHUKSUISVOVLEDVHVJBCUJULRUMABCDFUNVKEDFUOUP $.

    $( 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 relopabiv ) EFBFCFGHAICJBJEDABCDK
      ABCDELM $.
  $}

  ${
    $d X w x y z $.  $d Y w x y z $.  $d Z w x y z $.  $d ph w $.
    $( If a pair and a class are in a relationship given by a class abstraction
       of a collection of nested ordered pairs, the involved classes are sets.
       (Contributed by Alexander van der Vekens, 8-Jul-2018.) $)
    oprabv $p |- ( <. X , Y >. { <. <. x , y >. , z >. | ph } Z
                   -> ( X e. _V /\ Y e. _V /\ Z e. _V ) ) $=
      ( vw cop cvv wcel wa wi cv wceq wex nfcv nfeq1 nfan nfex wbr w3a reloprab
      coprab brrelex12i df-br copab wsbc opex nfv nfsbc1v eqeq1 2exbidv sbceq1a
      anbi1d anbi2d opelopabgf mpan eqcom vex opth bitri eqvisset anim12i sylbi
      wb adantr exlimivv anim1i df-3an sylibr expcom sylbid com12 eleq2s adantl
      dfoprab2 mpcom ) EFIZJKZGJKZLVSGABCDUDZUAZEJKZFJKZWAUBZVSGWBABCDUCUEWAWCW
      FMVTWCWAWFWCVSGIZWBKWAWFMZVSGWBUFWHWGHNZBNZCNZIZOZALZCPBPZHDUGZWBWAWGWPKZ
      WFWAWQVSWLOZADGUHZLZCPZBPZWFVTWAWQXBVFEFUIWOWRALZCPZBPXBHDVSGJJXDHBXCHCWR
      AHHVSWLHVSQRAHUJSTTXADBWTDCWRWSDDVSWLDVSQRADGUKSTTWIVSOZWNXCBCXEWMWRAWIVS
      WLULUOUMDNGOZXCWTBCXFAWSWRADGUNUPUMUQURXBWAWFXBWALWDWELZWALWFXBXGWAWTXGBC
      WRXGWSWRWJEOZWKFOZLZXGWRWLVSOXJVSWLUSWJWKEFBUTCUTVAVBXHWDXIWEBEVCCFVCVDVE
      VGVHVIWDWEWAVJVKVLVMVNABCDHVQVOVEVNVPVR $.
  $}

$(
  @{
    @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 w ph $.  $d y z w ph $.  $d w ps $.  $d w ch $.
    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 } ) $=
      ( vw cv cop wceq wa wex cab coprab anbi2d exbidv abbidv df-oprab 3eqtr4g
      ) AHIDIEIJFIJKZBLZFMZEMZDMZHNUACLZFMZEMZDMZHNBDEFOCDEFOAUEUIHAUDUHDAUCUGE
      AUBUFFABCUAGPQQQRBDEFHSCDEFHST $.
    $( $j usage 'oprabbidv' avoids 'ax-12'; $)
  $}

  ${
    $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 anim2d aleximi ss2abdv df-oprab
      id 3sstr4g ) ABGZEHZDHZCHZFICIDIJEIJKZALZEMZDMZCMZFNUFBLZEMZDMZCMZFNACDEO
      BCDEOUEUJUNFUDUIUMCUCUHULDUBUGUKEUBABUFUBTPQQQRACDEFSBCDEFSUA $.
  $}

  $( Equivalence of ordered pair abstraction subclass and implication.  Compare
     ~ ssopab2b .  Usage of this theorem is discouraged because it depends on
     ~ ax-13 .  (Contributed by FL, 6-Nov-2013.)  (Proof shortened by Mario
     Carneiro, 11-Dec-2016.)  (New usage is discouraged.) $)
  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 $.

  ${
    $d x y z $.
    $( Equivalence of ordered pair abstraction subclass and biconditional.
       Version of ~ eqoprab2b with a disjoint variable condition, which does
       not require ~ ax-13 .  (Contributed by Mario Carneiro, 4-Jan-2017.)
       Avoid ~ ax-13 .  (Revised by GG, 26-Jan-2024.) $)
    eqoprab2bw $p |- ( { <. <. x , y >. , z >. | ph } =
      { <. <. x , y >. , z >. | ps } <-> A. x A. y A. z ( ph <-> ps ) ) $=
      ( coprab wss wa wi wal nfoprab1 nfss nfoprab2 nfoprab3 wcel ssel oprabidw
      cv cop alrimi wceq wb 3imtr3g ssoprab2 impbii anbi12i 2albiim albii 19.26
      eqss bitri 3bitr4i ) ACDEFZBCDEFZGZUNUMGZHABIZEJZDJZCJZBAIZEJZDJZCJZHZUMU
      NUAABUBEJDJZCJZUOUTUPVDUOUTUOUSCCUMUNACDEKZBCDEKZLUOURDDUMUNACDEMZBCDEMZL
      UOUQEEUMUNACDENZBCDENZLUOCRDRSERSZUMOZVNUNOZABUMUNVNPACDEQZBCDEQZUCTTTABC
      DEUDUEUPVDUPVCCCUNUMVIVHLUPVBDDUNUMVKVJLUPVAEEUNUMVMVLLUPVPVOBAUNUMVNPVRV
      QUCTTTBACDEUDUEUFUMUNUJVGUSVCHZCJVEVFVSCABDEUGUHUSVCCUIUKUL $.
    $( $j usage 'eqoprab2bw' avoids 'ax-13'; $)
  $}

  $( Equivalence of ordered pair abstraction subclass and biconditional.
     Compare ~ eqopab2b .  Usage of this theorem is discouraged because it
     depends on ~ ax-13 .  Use the weaker ~ eqoprab2bw when possible.
     (Contributed by Mario Carneiro, 4-Jan-2017.)
     (New usage is discouraged.) $)
  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 nfcv nfralw eqeq2
      syl6 pm5.32d eleq2 anbi1d sylan9bbr anass 3bitr4g oprabbid df-mpo 3eqtr4g
      rsp ) CFJZDGJZEHJZBDKZLZACKZLZAMZCNZBMZDNZLIMZEJZLZABIOVBFNZVDGNZLVFHJZLZ
      ABIOABCDEPABFGHPVAVHVLABIUOUTAUOAQUSACRSUOUTBUOBQUSBACBCUAUPURBUPBQUQBDRS
      UBSVAIQVAVCVEVGLZLZVIVJVKLZLZVHVLUTVNVCVOLUOVPUTVCVMVOUTVCUSVMVOTUSACUNUR
      VMVEVKLUPVOURVEVGVKURVEUQVGVKTUQBDUNEHVFUCUDUEUPVEVJVKDGVDUFUGUHUDUEUOVCV
      IVOCFVBUFUGUHVCVEVGUIVIVJVKUIUJUKABICDEULABIFGHULUM $.
  $}

  ${
    $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 A w y z v $.  $d A w x z $.  $d B w x z v $.  $d B w y z $.
    $d C w z v $.
    $( A mapping operation with empty domain is empty.  Generalization of
       ~ mpo0 .  (Contributed by AV, 27-Jan-2024.) $)
    0mpo0 $p |- ( ( A = (/) \/ B = (/) )
                  -> ( x e. A , y e. B |-> C ) = (/) ) $=
      ( vz vw vv c0 wceq wo cv cop wcel wa wex wn nel02 sylibr nexdv cab coprab
      cmpo df-mpo df-oprab eqtri orim12i ianor simprl nsyl alrimiv eqeq1 anbi1d
      wal weq 3exbidv ab0w eqtrid ) CIJZDIJZKZABCDEUCZFLZALZBLZMGLZMZJZVDCNZVED
      NZOZVFEJZOZOZGPBPAPZFUAZIVBVMABGUBVPABGCDEUDVMABGFUEUFVAHLZVGJZVMOZGPZBPZ
      APZQZHUNVPIJVAWCHVAWAAVAVTBVAVSGVAVKVSVAVIQZVJQZKVKQUSWDUTWECVDRDVERUGVIV
      JUHSVRVKVLUIUJTTTUKVOWBFHFHUOZVNVSABGWFVHVRVMVCVQVGULUMUPUQSUR $.
  $}

  ${
    $d B x $.  $d B y $.
    $( A mapping operation with empty domain.  (Contributed by Stefan O'Rear,
       29-Jan-2015.)  (Revised by Mario Carneiro, 15-May-2015.)  (Proof
       shortened by AV, 27-Jan-2024.) $)
    mpo0v $p |- ( x e. (/) , y e. B |-> C ) = (/) $=
      ( c0 wceq wo cmpo eqid orci 0mpo0 ax-mp ) EEFZCEFZGABECDHEFMNEIJABECDKL
      $.
  $}

  ${
    $d w x z $.  $d w y z $.  $d w z B $.  $d w z C $.
    $( A mapping operation with empty domain.  In this version of ~ mpo0v , the
       class of the second operator may depend on the first operator.
       (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 cbvexv1
      opeq1 exbidv opabbii dfoprab2 3eqtr4i ) JKZCKZDKZLZMZANZDOZCOZJEPUIFKZUKL
      ZMZBNZDOZFOZJEPACDEQBFDEQUPVBJEUOVACFUNFDUMAFUMFRGSTUTCDUSBCUSCRHSTUJUQMZ
      UNUTDVCUMUSABVCULURUIUJUQUKUDUAIUBUEUCUFACDEJUGBFDEJUGUH $.
  $}

  ${
    $d v w x y $.  $d v w 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 exbii abbii
      eqeq2d anbi12d exbidv cbvexv1 df-oprab 3eqtr4i ) JKZCKZDKZLZEKZLZMZANZEOZ
      DOZCOZJPUKULFKZLZUOLZMZBNZEOZFOZCOZJPACDEQBCFEQVAVIJUTVHCUSVGDFURFEUQAFUQ
      FRGSTVFDEVEBDVEDRHSTUMVBMZURVFEVJUQVEABVJUPVDUKVJUNVCUOUMVBULUAUBUEIUFUGU
      HUCUDACDEJUIBCFEJUIUJ $.
  $}

  ${
    $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 anbi12d cbvex2v opabbii dfoprab2 3eqtr4i
      copab coprab opeq12 eqeq2d ) MNZCNZDNZOZPZAQZDRCRZMEUFUJFNZGNZOZPZBQZGRFR
      ZMEUFACDEUGBFGEUGUPVBMEUOVACDFGUNAFUNFSHTUNAGUNGSITUTBCUTCSJTUTBDUTDSKTUK
      UQPULURPQZUNUTABVCUMUSUJUKULUQURUHUILUAUBUCACDEMUDBFGEMUDUE $.
  $}

  ${
    $d x y z w v u $.  $d w v u ph $.  $d x y u 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 } $=
      ( vu cv cop wceq wa wex cab coprab weq opeq12 opeq1d df-oprab cbvex2vw
      eqeq2d anbi12d exbidv abbii 3eqtr4i ) IJZCJZDJZKZEJZKZLZAMZENZDNCNZIOUGFJ
      ZGJZKZUKKZLZBMZENZGNFNZIOACDEPBFGEPUPVDIUOVCCDFGCFQDGQMZUNVBEVEUMVAABVEUL
      UTUGVEUJUSUKUHUIUQURRSUBHUCUDUAUEACDEITBFGEITUF $.
    $( $j usage 'cbvoprab12v' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13' 'ax-sep'
       'ax-nul' 'ax-pr'; $)
  $}

  ${
    $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 v $.  $d y z w v $.  $d w v ph $.  $d z v 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 } $=
      ( vv cv cop wceq wa wex cab coprab weq opeq2 eqeq2d anbi12d df-oprab
      cbvexvw 2exbii abbii 3eqtr4i ) HIZCIDIJZEIZJZKZALZEMZDMCMZHNUEUFFIZJZKZBL
      ZFMZDMCMZHNACDEOBCDFOULURHUKUQCDUJUPEFEFPZUIUOABUSUHUNUEUGUMUFQRGSUAUBUCA
      CDEHTBCDFHTUD $.
    $( $j usage 'cbvoprab3v' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13' 'ax-sep'
       'ax-nul' 'ax-pr'; $)
  $}

  ${
    $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 wceq coprab cmpo nfv nfcri nfeq2 nfcv weq eleq1w adantr
      wa wb eleq2d sylan9bb anbi12d eqeq2d cbvoprab12 df-mpo 3eqtr4i ) ASEUAZBS
      ZFUAZULZRSZGUBZULZABRUCCSEUAZDSHUAZULZVEIUBZULZCDRUCABEFGUDCDEHIUDVGVLABR
      CDVDVFCVAVCCVACUECBFJUFTCVEGLUGTVDVFDVAVCDVADUEDBFDFUHUFTDVEGMUGTVJVKAVHV
      IAVHAUEADHKUFTAVEINUGTVJVKBVJBUEBVEIOUGTACUIZBDUIZULZVDVJVFVKVOVAVHVCVIVM
      VAVHUMVNACEUJUKVMVCVBHUAVNVIVMFHVBPUNBDHUJUOUPVOGIVEQUQUPURABREFGUSCDREHI
      USUT $.
  $}

  ${
    $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 v A $.  $d w x y z v B $.  $d w z v C $.  $d x y v 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 ) $=
      ( vv cv wcel wa wceq coprab cmpo weq eleq1w bi2anan9 sylan9eq cbvoprab12v
      eqeq2d anbi12d df-mpo 3eqtr4i ) AMENZBMFNZOZLMZGPZOZABLQCMENZDMFNZOZUKHPZ
      OZCDLQABEFGRCDEFHRUMURABLCDACSZBDSZOZUJUPULUQUSUHUNUTUIUOACETBDFTUAVAGHUK
      USUTGIHJKUBUDUEUCABLEFGUFCDLEFHUFUG $.
    $( $j usage 'cbvmpov' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13' 'ax-sep'
       'ax-nul' 'ax-pr'; $)
  $}

  ${
    elimdelov.1 $e |- ( ph -> C e. ( A F B ) ) $.
    elimdelov.2 $e |- Z e. ( X F Y ) $.
    $( Eliminate a hypothesis which is a predicate expressing membership in the
       result of an operator (deduction version).  (Contributed by Paul
       Chapman, 25-Mar-2008.) $)
    elimdelov $p |- if ( ph , C , Z ) e.
                     ( if ( ph , A , X ) F if ( ph , B , Y ) ) $=
      ( cif co wcel iftrue oveq12d 3eltr4d wn iffalse eqeltrdi eleqtrrd pm2.61i
      ) AADHKZABFKZACGKZELZMADBCELUBUEIADHNAUCBUDCEABFNACGNOPAQZUBFGELZUEUFUBHU
      GADHRJSUFUCFUDGEABFRACGROTUA $.
  $}

  $( Move a relation inside and outside the conditional operator.  (Contributed
     by SN, 14-Aug-2024.) $)
  brif1 $p |- ( if ( ph , A , B ) R C <-> if- ( ph , A R C , B R C ) ) $=
    ( cif wbr iftrue breq1d wn iffalse casesifp ) AABCFZDEGBDEGCDEGAMBDEABCHIAJ
    MCDEABCKIL $.

  $( Move a conditional outside of an operation.  (Contributed by Thierry
     Arnoux, 25-Jan-2017.) $)
  ovif $p |- ( if ( ph , A , B ) F C ) = if ( ph , ( A F C ) , ( B F C ) ) $=
    ( cif co oveq1 ifsb ) ABCABCFZDEGBDEGCDEGJBDEHJCDEHI $.

  $( Move a conditional outside of an operation.  (Contributed by Thierry
     Arnoux, 1-Oct-2018.) $)
  ovif2 $p |- ( A F if ( ph , B , C ) ) = if ( ph , ( A F B ) , ( A F C ) ) $=
    ( cif co oveq2 ifsb ) ACDBACDFZEGBCEGBDEGJCBEHJDBEHI $.

  $( Move a conditional outside of an operation.  (Contributed by Thierry
     Arnoux, 25-Jan-2017.) $)
  ovif12 $p |- ( if ( ph , A , B ) F if ( ph , C , D ) )
    = if ( ph , ( A F C ) , ( B F D ) ) $=
    ( cif co wceq iftrue oveq12d eqtr4d wn iffalse pm2.61i ) AABCGZADEGZFHZABDF
    HZCEFHZGZIARSUAAPBQDFABCJADEJKASTJLAMZRTUAUBPCQEFABCNADENKASTNLO $.

  $( Move a conditional outside of an operation.  (Contributed by AV,
     11-Nov-2019.) $)
  ifov $p |- ( A if ( ph , F , G ) B ) = if ( ph , ( A F B ) , ( A G B ) ) $=
    ( cif co oveq ifsb ) ADEBCADEFZGBCDGBCEGBCJDHBCJEHI $.

  ${
    $d ph x $.
    $( Move a conditional inside and outside a function in maps-to notation.
       (Contributed by SN, 16-Oct-2025.) $)
    ifmpt2v $p |- ( x e. A |-> if ( ph , B , C ) )
                    = if ( ph , ( x e. A |-> B ) , ( x e. A |-> C ) ) $=
      ( cif cmpt wceq iftrue mpteq2dv eqtr4d wn iffalse pm2.61i ) ABCADEFZGZABC
      DGZBCEGZFZHAPQSABCODADEIJAQRIKALZPRSTBCOEADEMJAQRMKN $.
  $}

  ${
    $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
      opex isseti 19.41v mpbiran 2exbii bitri abbii 3eqtri ) ABCDFZGEHBHZCHZIZJ
      ZAKZCLBLZEDMZGUMELZDNACLBLZDNUGUNABCDEOPUMEDQUOUPDUOULELZCLBLUPULEBCRUQAB
      CUQUKELAEUJUHUISTUKAEUAUBUCUDUEUF $.
  $}

  ${
    $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 relopabiv ) 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.)  Remove unnecessary
       distinct variable conditions.  (Revised by David Abernethy,
       19-Jun-2012.)  (Revised by Mario Carneiro, 19-Dec-2013.)  Avoid
       ~ ax-10 , ~ ax-11 .  (Revised by Wolf Lammen, 15-Oct-2024.) $)
    eloprabga $p |- ( ( A e. V /\ B e. W /\ C e. X ) ->
       ( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } <-> ps ) ) $=
      ( vw wcel cvv cop cv wceq wa wex coprab wb elex w3a wi simpr eqeq1d eqcom
      opex vex otth2 bitri bitrdi anbi1d pm5.32i 3exbidv df-oprab eleq2i bitr2i
      cab eleq1 bitrid adantl 19.41vvv elisset 3anim123i 3exdistr 19.41v anbi2i
      3bitri 3anass bitr4i sylibr biantrurd bitr4id adantr expcom vtocle syl3an
      abid 3bitr3d ) FINFONZGJNGONZHKNHONZFGPZHPZACDEUAZNZBUBZFIUCGJUCHKUCWBWCW
      DUDZWIUEMWFWEHUIWJMQZWFRZWIWJWLSZWKCQZDQZPEQZPZRZASZETDTCTZWNFRZWOGRZWPHR
      ZUDZBSZETDTCTZWHBWMWSXECDEWMWSXDASXEWMWRXDAWMWRWFWQRZXDWMWKWFWQWJWLUFUGXG
      WQWFRXDWFWQUHWNWOFGWPHCUJDUJEUJUKULUMUNXDABLUOUMUPWLWTWHUBWJWTWKWGNZWLWHX
      HWKWTMUTZNWTWGXIWKACDEMUQURWTMVTUSWKWFWGVAVBVCWJXFBUBWLWJXFXDETDTCTZBSBXD
      BCDEVDWJXJBWJXACTZXBDTZXCETZUDZXJWBXKWCXLWDXMCFOVEDGOVEEHOVEVFXJXKXLXMSZS
      ZXNXJXAXBXMSDTZSCTXKXQSXPXAXBXCCDEVGXAXQCVHXQXOXKXBXMDVHVIVJXKXLXMVKVLVMV
      NVOVPWAVQVRVS $.
    $( $j usage 'eloprabga' avoids 'ax-10' 'ax-11'; $)
  $}

  ${
    $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 iunxpconst mpteq1i mpomptx eqtr3i ) CADAIJEKL
      ZFMCDEKZFMABDEGNCSTFADEOPABCDEFGHQR $.
  $}

  $( 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 eldifsnneq adantr iffalsed mpoeq3ia ) EFA
    GHIZBEJZGKZCDLDUATMZFJBMZNUBCDUCUBOUDUAAGPQRS $.

  $( 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 z $.  $d B z $.  $d C z $.  $d x y 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.)  (Proof shortened by SN,
       23-Jul-2024.) $)
    mpofun $p |- Fun F $=
      ( vz wfun cv wcel wa wceq coprab moeq moani funoprab cmpo df-mpo eqtri
      funeqi mpbir ) FIAJCKBJDKLZHJEMZLZABHNZIUEABHUDUCHHEOPQFUFFABCDERUFGABHCD
      ESTUAUB $.
  $}

  ${
    $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
       NM, 7-Feb-2004.)  (Revised by Mario Carneiro, 31-Aug-2015.) $)
    fnov $p |- ( F Fn ( A X. B ) <->
                    F = ( x e. A , y e. B |-> ( x F y ) ) ) $=
      ( vz cxp wfn cv cfv cmpt wceq cmpo dffn5 cop fveq2 df-ov eqtr4di mpompt
      co eqeq2i bitri ) 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 eqoprab2bw albii 19.21v bitr3i 2albii bitr4i 3bitri ralbi bitr4id
      pm13.183 syl ) EGIZBDJZACJZABCDEKZABCDFKZLZHMZELZVAFLZNZHOZBDJZACJZEFLZBD
      JZACJZUTAMCIBMDIPZVBPZABHQZVKVCPZABHQZLVLVNNZHOZBOAOZVGURVMUSVOABHCDERABH
      CDFRUCVLVNABHUDVRVKVESZBOAOVGVQVSABVQVKVDSZHOVSVTVPHVKVBVCUAUEVKVDHUFUGUH
      VEABCDUBUIUJUQVIVFNZACJVJVGNUPWAACUPVHVENZBDJWAUOWBBDHEFGUMTVHVEBDUKUNTVI
      VFACUKUNUL $.
  $}

  ${
    $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
        JCDEGHQRUJUNJFSUMFSLZACULUOBDULUOESLIFESUAUBTTUHFNUIULABCDUHFEUCUDUEUF
        $.
    $}

    ${
      $d A x $.  $d B x $.  $d B y $.  $d X x $.  $d X y $.  $d Y x $.
      $d Y y $.
      elimampo.d $e |- ( ph -> D e. V ) $.
      elimampo.x $e |- ( ph -> X C_ A ) $.
      elimampo.y $e |- ( ph -> Y C_ B ) $.
      $( Membership in the image of an operation.  (Contributed by SN,
         27-Apr-2025.) $)
      elimampo $p |- ( ph -> ( D e. ( F " ( X X. Y ) ) <->
        E. x e. X E. y e. Y D = C ) ) $=
        ( wcel cmpo crn wceq wrex cxp cima df-ima eleq2i reseq1i resmpo syl2anc
        cres wss eqtrid rneqd eleq2d bitrid wb eqid elrnmpog syl bitrd ) AGHJKU
        AZUBZPZGBCJKFQZRZPZGFSCKTBJTZVAGHUSUHZRZPAVDUTVGGHUSUCUDAVGVCGAVFVBAVFB
        CDEFQZUSUHZVBHVHUSLUEAJDUIKEUIVIVBSNOBCDEJKFUFUGUJUKULUMAGIPVDVEUNMBCJK
        FGVBIVBUOUPUQUR $.
    $}

    ${
      $d x y z $.  $d A p y z $.  $d B p z $.  $d C p z $.  $d F z $.
      $d D x y z $.  $d R p x y z $.
      $( Membership in the range of a restricted operation class abstraction.
         (Contributed by Thierry Arnoux, 25-May-2019.) $)
      elrnmpores $p |- ( D e. V -> ( D e. ran ( F |` R )
        <-> E. x e. A E. y e. B ( D = C /\ x R y ) ) ) $=
        ( vz vp wcel cres cv wa wceq wex anbi2d copab crn wbr wrex eqeq1 anbi1d
        2exbidv coprab cab cop ancom eleq1 df-br bitr4di bitr3id bitrid pm5.32i
        an12 bitri 2exbii 19.42vv bitr3i opabbii dfoprab2 df-mpo 3eqtri reseq1i
        cmpo resopab eqtri 3eqtr4ri rneqi rnoprab elab2g r2ex ) FIMFHGNZUAZMAOZ
        CMBOZDMPZFEQZVQVRGUBZPZPZBRARZWBBDUCACUCVSKOZEQZWAPZPZBRARZWDKFVPIWEFQZ
        WHWCABWJWGWBVSWJWFVTWAWEFEUDUESUFVPWHABKUGZUAWIKUHVOWKLOZVQVRUIZQZWHPZB
        RARZLKTWLGMZWNVSWFPZPZBRARZPZLKTZWKVOWPXALKWPWQWSPZBRARXAXCWOABXCWNWQWR
        PZPWOWQWNWRUQWNXDWHXDVSWQWFPZPWNWHWQVSWFUQWNXEWGVSXEWFWQPWNWGWFWQUJWNWQ
        WAWFWNWQWMGMWAWLWMGUKVQVRGULUMSUNSUOUPURUSWQWSABUTVAVBWHABKLVCVOWTLKTZG
        NXBHXFGHABCDEVGWRABKUGXFJABKCDEVDWRABKLVCVEVFWTLKGVHVIVJVKWHABKVLVIVMWB
        ABCDVNUM $.
    $}

    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 ) ) $=
      ( wcel wral wn crn wrex cv notbid dfrex2 wceq rexbii rexnal bitri 3bitr4g
      ralrnmpo ) HJMDGNCFNZAOZEIPZNZOBOZDGNZCFNZOZAEUIQBDGQZCFQZUGUJUMUHUKCDEFG
      HIJKERHUAABLSUFSAEUITUPULOZCFQUNUOUQCFBDGTUBULCFUCUDUE $.
  $}

  ${
    $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 opabidw biimpri fnopfvb sylancr eleq2i oprabidw bitri baib bitrd
      bitrid ) BJZCJZGKZDJZLUPUQMZGNZUSLZUPEOUQFOPZAURVAUSUPUQGQRVCVBUTUSMZGOZA
      VCGVCBCSZTZUTVFOZVBVEUDVGVCAPZBCDUAZVFTVCABCDHUBVFGVJIUCUEVHVCVCBCUFUGVFU
      TUSGUHUIVEVCAVEVDVJOVIGVJVDIUJVIBCDUKULUMUNUO $.
  $}

  ${
    $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 cop cfv df-ov wfun wcel wceq coprab funoprab funeqi mpbir oprabidw
      co biimpri eleqtrrdi funopfv mpsyl eqtrid ) ABHZCHZETUFUGIZEJZDHZUFUGEKEL
      ZAUHUJIZEMUIUJNUKABCDOZLABCDFPEUMGQRAULUMEULUMMAABCDSUAGUBUHUJEUCUDUE $.
  $}

  ${
    $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.  Compared with ~ ovig , the
       condition ` ( x e. R /\ y e. S ) ` is 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).
       (Contributed by NM, 14-Sep-1999.)  Remove unnecessary distinct variable
       conditions.  (Revised by David Abernethy, 19-Jun-2012.)  (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 analogue 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 nfel1 elex nfmpo1 nfcxfr nfmpo2
      wa cmpo nfov nfeq nfim csbeq1a eleq1d oveq1 eqeq12d imbi12d oveq2 ovmpt4g
      3expia vtocl2gaf csbcom eleq1i eqeq2i 3imtr4g syl5 3impia ) CEKZDFKZACBDG
      LLZIKZCDHMZVGNZVHVGOKZVEVFUEZVJVGIUAVLBDACGLZLZOKZVIVNNZVKVJGOKZAPZBPZHMZ
      GNZQVMOKZCVSHMZVMNZQVOVPQABCDEFACRZBCRZBDRZWBWDAAVMOACGSZTAWCVMACVSHWEAHA
      BEFGUFZJABEFGUBUCAVSRUGWHUHUIVOVPBBVNOBDVMSZTBVIVNBCDHWFBHWIJABEFGUDUCWGU
      GWJUHUIVRCNZVQWBWAWDWKGVMOACGUJZUKWKVTWCGVMVRCVSHULWLUMUNVSDNZWBVOWDVPWMV
      MVNOBDVMUJZUKWMWCVIVMVNVSDCHUOWNUMUNVREKVSFKVQWAABEFGHOJUPUQURVGVNOABCDGU
      SZUTVGVNVIWOVAVBVCVD $.
  $}

  ${
    $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 simplr ad2antrr eqeltrd simpr 3eltr4d anassrs elexd biimt
        wb syl3anc oveq12d eqeq12d bitr3d nfeq2 nfan wnf nfcv nfov nfeq sbciedf
        nfmpo2 nfmpo1 mpbid eqtrd ) ADEJUEDEBCFGHUFZUEZIAJWFDEMUGABUHZFUIZCUHZG
        UIZHUJUIZUKZWHWJWFUEZHULZUMZCEUNZBDUNWGIULZAWQBDFPAWQBSAWPCEKQAWPCTWPAB
        CFGHWFUJWFUOUPUSUQURUQURAWQWRBDFPAWHDULZUTZWPWRCEKAEKUIZWSQVAWTWJEULZUT
        ZWOWPWRXCWIWKWLWOWPVJXCWHDFAWSXBVBZADFUIWSXBPVCVDXCEKWJGAXAWSXBQVCWTXBV
        EZWTGKULXBOVAVFXCHIUJAWSXBHIULNVGZAIUJUIWSXBAILRVHVCVDWMWOVIVKXCWNWGHIX
        CWHDWJEWFXDXEVLXFVMVNAWSCTCWHDUAVOVPWRCVQWTCWGICDEWFUABCFGHWBCEVRVSUDVT
        USWASWRBVQABWGIBDEWFBDVRBCFGHWCUBVSUCVTUSWAWDWE $.
    $}

    $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 cvv wex elexd isset sylib nfmpo2
        nfim wa co simprl simprr oveq12d adantr eqeltrd adantrr ovmpt4g syl3anc
        oveq eqid eqtr3d eqeq2d sylbid syl5 expr exlimimdd exlimdd ) ACUAZEUBZJ
        CDGHIUCZUBZBUDZCACUEVQBCCJVPPCDGHIUFUGQUNAEUHTVOCUIAEGLUJCEUKULAVOUOZDU
        AZFUBZVRDVSDUEVQBDDJVPRCDGHIUMUGSUNVSFUHTWADUIVSFHMUJDFUKULAVOWAVRVQEFJ
        UPZEFVPUPZUBZAVOWAUOZUOZBEFJVPVEWFWDWBIUBBWFWCIWBWFVNVTVPUPZWCIWFVNEVTF
        VPAVOWAUQZAVOWAURZUSWFVNGTVTHTIKTWGIUBWFVNEGWHAEGTWELUTVAWFVTFHWIAVOFHT
        WAMVBVANCDGHIVPKVPVFVCVDVGVHOVIVJVKVLVM $.
    $}

    $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 x y A $.  $d x y B $.  $d x y C $.  $d x y D $.  $d x y F $.
    $( The value of an operation is equal to the value of the same operation
       expressed in maps-to notation.  (Contributed by GG, 16-Mar-2025.)
       (Revised by GG, 13-Apr-2025.) $)
    ovmpot $p |- ( ( A e. C /\ B e. D ) ->
                     ( A ( x e. C , y e. D |-> ( x F y ) ) B ) = ( A F B ) ) $=
      ( cv co cmpo oveq12 eqid ovex ovmpoa ) ABCDEFAHZBHZGIZCDGIABEFQJZOCPDGKRL
      CDGMN $.
  $}

  ${
    $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 $.
    ov3.1 $e |- S e. _V $.
    ov3.2 $e |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) ->
         R = S ) $.
    ov3.3 $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.) $)
    ov3 $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 isseti nfv nfcv cxp coprab nfoprab3 nfcxfr nfov
      co nfeq1 eqeq2d copsex4g wi opelxpi nfoprab1 nfim nfoprab2 anbi1d 4exbidv
      eqeq1 oveq1 eqeq1d imbi12d anbi2d oveq2 moeq mosubop anass 2exbii 19.42vv
      wmo bitri mobii mpbir a1i ovidi vtocl2gaf syl2an sylbird eqeq2 mpbidi mpi
      exlimd ) GOUAHOUASZIOUAJOUASZSZCUBZLUCZCTGHUDZIJUDZNUMZLUCZCLPUEWJWLWPCWJ
      CUFCWOLCWMWNNCWMUGCNAUBZOOUHZUABUBZWRUASZWQDUBZEUBZUDZUCZWSFUBZMUBZUDZUCZ
      SZWKKUCZSZMTFTZETDTZSZABCUIZRXNABCUJUKCWNUGULUNWLWOWKUCZWPWJWJWLWMXCUCZWN
      XGUCZSZXJSZMTFTETDTZXPXJWLDEFMGHIJOOXAGUCXBHUCSXEIUCXFJUCSSKLWKQUOUPWHWMW
      RUAWNWRUAYAXPUQZWIGHOOURIJOOURXMWQWSNUMZWKUCZUQXQXHSZXJSZMTFTETDTZWMWSNUM
      ZWKUCZUQYBABWMWNWRWRAWMUGZBWMUGZBWNUGZYGYIAYGAUFAYHWKAWMWSNYJANXORXNABCUS
      UKAWSUGULUNUTYAXPBYABUFBWOWKBWMWNNYKBNXORXNABCVAUKYLULUNUTWQWMUCZXMYGYDYI
      YMXKYFDEFMYMXIYEXJYMXDXQXHWQWMXCVDVBVBVCYMYCYHWKWQWMWSNVEVFVGWSWNUCZYGYAY
      IXPYNYFXTDEFMYNYEXSXJYNXHXRXQWSWNXGVDVHVBVCYNYHWOWKWSWNWMNVIVFVGXMABCWRWR
      NXMCVOZWTYOXDXHXJSZMTFTZSZETDTZCVOYQCDEWQXJCFMWSCKVJVKVKXMYSCXLYRDEXLXDYP
      SZMTFTYRXKYTFMXDXHXJVLVMXDYPFMVNVPVMVQVRVSRVTWAWBWCWKLWOWDWEWGWF $.
  $}

  ${
    $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 opelxpi fvresd df-ov 3eqtr4g ) ACFBDFGZABHZEC
    DIZJZKREKABTLABELQRSEABCDMNABTOABEOP $.

  ${
    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 $.
  $}

  ${
    $d F x y $.  $d G x y $.  $d Y x y $.  $d ph x y $.
    oprres.v $e |- ( ( ph /\ x e. Y /\ y e. Y ) -> ( x F y ) = ( x G y ) ) $.
    oprres.s $e |- ( ph -> Y C_ X ) $.
    oprres.f $e |- ( ph -> F : ( Y X. Y ) --> R ) $.
    oprres.g $e |- ( ph -> G : ( X X. X ) --> S ) $.
    $( The restriction of an operation is an operation.  (Contributed by NM,
       1-Feb-2008.)  (Revised by AV, 19-Oct-2021.) $)
    oprres $p |- ( ph -> F = ( G |` ( Y X. Y ) ) ) $=
      ( cxp wceq cv co wa wfn syl2anc cres 3expb ovres adantl eqtr4d ralrimivva
      wral wcel eqid jctil wb ffnd wss xpss12 fnssres eqfnov mpbird ) AFGIINZUA
      ZOZURUROZBPZCPZFQZVBVCUSQZOZCIUGBIUGZRZAVGVAAVFBCIIAVBIUHZVCIUHZRZRVDVBVC
      GQZVEAVIVJVDVLOJUBVKVEVLOAVBVCIIGUCUDUEUFURUIUJAFURSUSURSZUTVHUKAURDFLULA
      GHHNZSURVNUMZVMAVNEGMULAIHUMZVPVOKKIHIHUNTVNURGUOTBCIIIIFUSUPTUQ $.
  $}

  $( 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 eleq2d cvv rexlimivw
      ovex eleq1 mpbiri eqeq1 2rexbidv elab3 bitrdi ) FCDHIZEFJZKEGLZALZBLZFMZN
      ZBDOACOZGPZKEUMNZBDOZACOZUHUIUPEABGCDFQRUOUSGESURESKZACUQUTBDUQUTUMSKUKUL
      FUAEUMSUBUCTTUJENUNUQABCDUJEUMUDUEUFUG $.

    $( 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 $.
  $}

  $( An operation value is a member of the image plus null.  (Contributed by
     Thierry Arnoux, 25-Jun-2019.) $)
  ovima0 $p |- ( ( X e. A /\ Y e. B )
    -> ( X R Y ) e. ( ( R " ( A X. B ) ) u. { (/) } ) ) $=
    ( wcel wa co c0 wceq cxp cima csn cun simpr ssun2 0ex snid sselii wn eqeq1i
    eqeltrdi ssun1 cop cfv opelxpi notbii biimpi eliman0 syl2an eqeltrid sselid
    df-ov pm2.61dan ) DAFEBFGZDECHZIJZUPCABKZLZIMZNZFUOUQGUPIVAUOUQOUTVAIUTUSPI
    QRSUBUOUQTZGZUSVAUPUSUTUCVCUPDEUDZCUEZUSDECUMZUOVDURFVEIJZTZVEUSFVBDEABUFVB
    VHUQVGUPVEIVFUAUGUHVDURCUIUJUKULUN $.

  ${
    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 S $.  $d x y F $.
    oprssdm.1 $e |- -. (/) e. S $.
    oprssdm.2 $e |- ( ( x e. S /\ y e. S ) -> ( x F y ) e. S ) $.
    $( Domain of closure of an operation.  (Contributed by NM, 24-Aug-1995.) $)
    oprssdm $p |- ( S X. S ) C_ dom F $=
      ( cxp cdm relxp cv cop wcel wa opelxp cfv co df-ov eqeltrrid wn c0 eleq1d
      ndmfv mtbiri con4i syl sylbi relssi ) ABCCGZDHZCCIAJZBJZKZUHLUJCLUKCLMZUL
      UILZUJUKCCNUMULDOZCLZUNUMUOUJUKDPCUJUKDQFRUNUPUNSZUPTCLEUQUOTCULDUBUAUCUD
      UEUFUG $.
  $}

  $( The value of an operation outside its domain.  (Contributed by Alexander
     van der Vekens, 7-Sep-2017.) $)
  nssdmovg $p |- ( ( dom F C_ ( R X. S ) /\ -. ( A e. R /\ B e. S ) )
              -> ( A F B ) = (/) ) $=
    ( cdm cxp wss wcel wa wn co cop cfv c0 df-ov wceq ssel2 opelxp sylib eqtrid
    stoic1a ndmfv syl ) EFZCDGZHZACIBDIJZKJZABELABMZENZOABEPUIUJUEIZKUKOQUGULUH
    UGULJUJUFIUHUEUFUJRABCDSTUBUJEUCUDUA $.

  $( The value of an operation outside its domain.  (Contributed by NM,
     28-Mar-2008.) $)
  ndmovg $p |- ( ( dom F = ( R X. S ) /\ -. ( A e. R /\ B e. S ) )
              -> ( A F B ) = (/) ) $=
    ( cdm cxp wceq wcel wa wn co cop cfv c0 df-ov eleq2 opelxp bitrdi notbid
    ndmfv biimtrrdi imp eqtrid ) EFZCDGZHZACIBDIJZKZJABELABMZENZOABEPUGUIUKOHZU
    GUIUJUEIZKULUGUMUHUGUMUJUFIUHUEUFUJQABCDRSTUJEUAUBUCUD $.

  ${
    ndmov.1 $e |- dom F = ( S X. S ) $.
    $( The value of an operation outside its domain.  (Contributed by NM,
       24-Aug-1995.) $)
    ndmov $p |- ( -. ( A e. S /\ B e. S ) -> ( A F B ) = (/) ) $=
      ( cdm cxp wceq wcel wa wn co c0 ndmovg mpan ) DFCCGHACIBCIJKABDLMHEABCCDN
      O $.

    ${
      ndmovcl.2 $e |- ( ( A e. S /\ B e. S ) -> ( A F B ) e. S ) $.
      ndmovcl.3 $e |- (/) e. S $.
      $( The closure of an operation outside its domain, when the domain
         includes the empty set.  This technical lemma can make the operation
         more convenient to work in some cases.  It is dependent on our
         particular definitions of operation value, function value, and ordered
         pair.  (Contributed by NM, 24-Sep-2004.) $)
      ndmovcl $p |- ( A F B ) e. S $=
        ( wcel wa co wn c0 ndmov eqeltrdi pm2.61i ) ACHBCHIZABDJZCHFPKQLCABCDEM
        GNO $.
    $}

    ${
      ndmovrcl.3 $e |- -. (/) e. S $.
      $( Reverse closure law, when an operation's domain doesn't contain the
         empty set.  (Contributed by NM, 3-Feb-1996.) $)
      ndmovrcl $p |- ( ( A F B ) e. S -> ( A e. S /\ B e. S ) ) $=
        ( wcel wa co wn c0 ndmov eleq1d mtbiri con4i ) ACGBCGHZABDIZCGZPJZRKCGF
        SQKCABCDELMNO $.
    $}

    $( Any operation is commutative outside its domain.  (Contributed by NM,
       24-Aug-1995.) $)
    ndmovcom $p |- ( -. ( A e. S /\ B e. S ) -> ( A F B ) = ( B F A ) ) $=
      ( wcel wa wn co c0 ndmov wceq ancom sylnbi eqtr4d ) ACFZBCFZGZHABDIJBADIZ
      ABCDEKRQPGSJLPQMBACDEKNO $.

    ${
      ndmov.5 $e |- -. (/) e. S $.
      $( Any operation is associative outside its domain, if the domain doesn't
         contain the empty set.  (Contributed by NM, 24-Aug-1995.) $)
      ndmovass $p |- ( -. ( A e. S /\ B e. S /\ C e. S ) ->
              ( ( A F B ) F C ) = ( A F ( B F C ) ) ) $=
        ( wcel w3a wn co c0 wa wceq ndmovrcl anim1i df-3an sylibr ndmov nsyl5
        anim2i 3anass eqtr4d ) ADHZBDHZCDHZIZJABEKZCEKZLABCEKZEKZUHDHZUFMZUGUIL
        NUMUDUEMZUFMUGULUNUFABDEFGOPUDUEUFQRUHCDEFSTUDUJDHZMZUGUKLNUPUDUEUFMZMU
        GUOUQUDBCDEFGOUAUDUEUFUBRAUJDEFSTUC $.

      ndmov.6 $e |- dom G = ( S X. S ) $.
      $( Any operation is distributive outside its domain, if the domain
         doesn't contain the empty set.  (Contributed by NM, 24-Aug-1995.) $)
      ndmovdistr $p |- ( -. ( A e. S /\ B e. S /\ C e. S ) ->
          ( A G ( B F C ) ) = ( ( A G B ) F ( A G C ) ) ) $=
        ( wcel w3a wn co c0 wa wceq ndmovrcl sylibr ndmov nsyl5 anim12i anandi3
        anim2i 3anass eqtr4d ) ADJZBDJZCDJZKZLABCEMZFMZNABFMZACFMZEMZUFUJDJZOZU
        IUKNPUPUFUGUHOZOUIUOUQUFBCDEGHQUCUFUGUHUDRAUJDFISTULDJZUMDJZOZUIUNNPUTU
        FUGOZUFUHOZOUIURVAUSVBABDFIHQACDFIHQUAUFUGUHUBRULUMDEGSTUE $.
    $}

    ndmovord.4 $e |- R C_ ( S X. S ) $.
    ndmovord.5 $e |- -. (/) e. S $.
    ndmovord.6 $e |- ( ( A e. S /\ B e. S /\ C e. S ) ->
                   ( A R B <-> ( C F A ) R ( C F B ) ) ) $.
    $( Elimination of redundant antecedents in an ordering law.  (Contributed
       by NM, 7-Mar-1996.) $)
    ndmovord $p |- ( C e. S -> ( A R B <-> ( C F A ) R ( C F B ) ) ) $=
      ( wcel wa wbr co wb wi 3expia brel ndmovrcl simprd wn anim12i syl pm2.61i
      pm5.21ni a1d ) AEKZBEKZLZCEKZABDMZCAFNZCBFNZDMZOZPUGUHUJUOJQUIUAUOUJUKUIU
      NABEEDHRUNULEKZUMEKZLUIULUMEEDHRUPUGUQUHUPUJUGCAEFGISTUQUJUHCBEFGISTUBUCU
      EUFUD $.
  $}

  ${
    ndmovordi.2 $e |- dom F = ( S X. S ) $.
    ndmovordi.4 $e |- R C_ ( S X. S ) $.
    ndmovordi.5 $e |- -. (/) e. S $.
    ndmovordi.6 $e |- ( C e. S -> ( A R B <-> ( C F A ) R ( C F B ) ) ) $.
    $( Elimination of redundant antecedent in an ordering law.  (Contributed by
       NM, 25-Jun-1998.) $)
    ndmovordi $p |- ( ( C F A ) R ( C F B ) -> A R B ) $=
      ( wcel co wbr brel simpld ndmovrcl syl biimprd mpcom ) CEKZCAFLZCBFLZDMZA
      BDMZUCUAEKZTUCUEUBEKUAUBEEDHNOUETAEKCAEFGIPOQTUDUCJRS $.
  $}


$(
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
  Variable-to-class conversion for operations
-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
$)

  ${
    $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 $.

      ${
        caov.4 $e |- D e. _V $.
        $( Rearrange arguments in a commutative, associative operation.
           (Contributed by NM, 26-Aug-1995.) $)
        caov4 $p |- ( ( A F B ) F ( C F D ) ) = ( ( A F C ) F ( B F D ) ) $=
          ( co caov12 oveq2i ovex caovass 3eqtr4i ) DEFGHOZHOZHODFEGHOZHOZHODEH
          OUAHODFHOUCHOUBUDDHABCEFGHJKNLMPQABCDEUAHIJFGHRMSABCDFUCHIKEGHRMST $.

        $( Rearrange arguments in a commutative, associative operation.
           (Contributed by NM, 26-Aug-1995.) $)
        caov411 $p |- ( ( A F B ) F ( C F D ) ) = ( ( C F B ) F ( A F D ) ) $=
          ( co caov31 oveq1i ovex caovass 3eqtr3i ) DEHOZFHOZGHOFEHOZDHOZGHOUAF
          GHOHOUCDGHOHOUBUDGHABCDEFHIJKLMPQABCUAFGHDEHRKNMSABCUCDGHFEHRINMST $.

        $( Rearrange arguments in a commutative, associative operation.
           (Contributed by NM, 26-Aug-1995.) $)
        caov42 $p |- ( ( A F B ) F ( C F D ) ) =
                        ( ( A F C ) F ( D F B ) ) $=
          ( co caov4 caovcom oveq2i eqtri ) DEHOFGHOHODFHOZEGHOZHOTGEHOZHOABCDE
          FGHIJKLMNPUAUBTHABEGHJNLQRS $.
      $}
    $}

    ${
      caovdir.1 $e |- A e. _V $.
      caovdir.2 $e |- B e. _V $.
      caovdir.3 $e |- C e. _V $.
      caovdir.com $e |- ( x G y ) = ( y G x ) $.
      caovdir.distr $e |- ( x G ( y F z ) ) = ( ( x G y ) F ( x G z ) ) $.
      $( Reverse distributive law.  (Contributed by NM, 26-Aug-1995.) $)
      caovdir $p |- ( ( A F B ) G C ) = ( ( A G C ) F ( B G C ) ) $=
        ( co caovdi ovex caovcom oveq12i 3eqtr3i ) FDEGNZHNFDHNZFEHNZGNTFHNDFHN
        ZEFHNZGNABCFDEGHKIJMOABFTHKDEGPLQUAUCUBUDGABFDHKILQABFEHKJLQRS $.

      $d x y z H $.  $d x y z R $.
      caovdl.4 $e |- D e. _V $.
      caovdl.5 $e |- H e. _V $.
      caovdl.ass $e |- ( ( x G y ) G z ) = ( x G ( y G z ) ) $.
      $( Lemma used by real number construction.  (Contributed by NM,
         26-Aug-1995.) $)
      caovdilem $p |- ( ( ( A G C ) F ( B G D ) ) G H ) =
                       ( ( A G ( C G H ) ) F ( B G ( D G H ) ) ) $=
        ( co ovex caovdir caovass oveq12i eqtri ) DFISZEGISZHSJISUEJISZUFJISZHS
        DFJISISZEGJISISZHSABCUEUFJHIDFITEGITQNOUAUGUIUHUJHABCDFJIKMQRUBABCEGJIL
        PQRUBUCUD $.

      caovdl2.6 $e |- R e. _V $.
      caovdl2.com $e |- ( x F y ) = ( y F x ) $.
      caovdl2.ass $e |- ( ( x F y ) F z ) = ( x F ( y F z ) ) $.
      $( Lemma used in real number construction.  (Contributed by NM,
         26-Aug-1995.) $)
      caovlem2 $p |- ( ( ( ( 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 ) ) ) ) $=
        ( co ovex caov42 caovdilem oveq12i caovdi 3eqtr4i ) DFKJUCZJUCZEGKJUCZJ
        UCZIUCZDGHJUCZJUCZEFHJUCZJUCZIUCZIUCUKUPIUCZURUMIUCZIUCDFJUCEGJUCIUCKJU
        CZDGJUCEFJUCIUCHJUCZIUCDUJUOIUCJUCZEUQULIUCJUCZIUCABCUKUMUPURIDUJJUDEUL
        JUDDUOJUDUAUBEUQJUDUEVBUNVCUSIABCDEFGIJKLMNOPQRSUFABCDEGFIJHLMQOPNTSUFU
        GVDUTVEVAIABCDUJUOIJLFKJUDGHJUDPUHABCEUQULIJMFHJUDGKJUDPUHUGUI $.
    $}

    ${
      $d u w A $.  $d u v w x y B $.  $d u v w x y z F $.  $d w x S $.
      $( Identity element. $)
      caovmo.2 $e |- B e. S $.
      caovmo.dom $e |- dom F = ( S X. S ) $.
      caovmo.3 $e |- -. (/) e. S $.
      caovmo.com $e |- ( x F y ) = ( y F x ) $.
      caovmo.ass $e |- ( ( x F y ) F z ) = ( x F ( y F z ) ) $.
      caovmo.id $e |- ( x e. S -> ( x F B ) = x ) $.
      $( Uniqueness of inverse element in commutative, associative operation
         with identity.  Remark in proof of Proposition 9-2.4 of [Gleason]
         p. 119.  (Contributed by NM, 4-Mar-1996.) $)
      caovmo $p |- E* w ( A F w ) = B $=
        ( vv wcel cv co wceq wa vu wmo oveq1 eqeq1d mobidv wal oveq2 mo4 oveq2d
        wi simpr simpl oveq1d caovass caov12 eqtri elexi caovcom 3eqtr3g eqtr3d
        vex eqeltrdi ndmovrcl id eqeq12d vtoclga simpl2im 3eqtr3d ax-gen mpgbir
        syl vtoclg moanimv mpbir eleq1 mpbiri simpld ancri moimi ax-mp ) EGPZED
        QZHRZFSZTZDUBZWDDUBZWFWAWGUJUAQZWBHRZFSZDUBZWGUAEGWHESZWJWDDWLWIWCFWHEW
        BHUCUDUEWKWJWHOQZHRZFSZTZWBWMSZUJZOUFDWJWODOWQWIWNFWBWMWHHUGUDUHWROWPWB
        FHRZWMFHRZWBWMWPWBWNHRZWSWTWPWNFWBHWJWOUKZUIWPWIWMHRZFWMHRXAWTWPWIFWMHW
        JWOULZUMXCWHWBWMHRHRXAABCWHWBWMHUAVAZDVAZOVAZMUNABCWHWBWMHXEXFXGLMUOUPA
        BFWMHFGIUQXGLURUSUTWPWHGPZWBGPZWSWBSZWPWIGPXHXITWPWIFGXDIVBWHWBGHJKVCVK
        AQZFHRZXKSZXJAWBGXKWBSZXLWSXKWBXKWBFHUCXNVDVENVFVGWPXHWMGPZWTWMSZWPWNGP
        XHXOTWPWNFGXBIVBWHWMGHJKVCVKXMXPAWMGXKWMSZXLWTXKWMXKWMFHUCXQVDVENVFVGVH
        VIVJVLWAWDDVMVNWDWEDWDWAWDWAXIWDWCGPZWAXITWDXRFGPIWCFGVOVPEWBGHJKVCVKVQ
        VRVSVT $.
    $}
  $}


  ${
    $d A x y z $.  $d B x y z $.  $d C x y z $.  $d F x y z $.  $d ph y z $.
    $d ps x $.
    imaeqexov.1 $e |- ( x = ( y F z ) -> ( ph <-> ps ) ) $.
    $( Substitute an operation value into an existential quantifier over an
       image.  (Contributed by Scott Fenton, 20-Jan-2025.) $)
    imaeqexov $p |- ( ( F Fn A /\ ( B X. C ) C_ A ) ->
    ( E. x e. ( F " ( B X. C ) ) ph <-> E. y e. B E. z e. C ps ) ) $=
      ( cxp wrex cv wa wex r19.41v rexbii bitrdi rexcom4 bitr3i cima wfn df-rex
      wcel wss co wceq ovelimab anbi1d bitr2i exbidv ovex ceqsexv bitrid ) ACIG
      HKZUAZLCMZUPUDZANZCOZIFUBUOFUENZBEHLZDGLZACUPUCVAUTUQDMZEMZIUFZUGZANZEHLZ
      DGLZCOZVCVAUSVJCVAUSVGEHLZDGLZANZVJVAURVMADEFGHUQIUHUIVJVLANZDGLVNVIVODGV
      GAEHPQVLADGPUJRUKVKVICOZDGLVCVIDCGSVPVBDGVPVHCOZEHLVBVHECHSVQBEHABCVFVDVE
      IULJUMQTQTRUN $.

    $( Substitute an operation value into a universal quantifier over an image.
       (Contributed by Scott Fenton, 20-Jan-2025.) $)
    imaeqalov $p |- ( ( F Fn A /\ ( B X. C ) C_ A ) ->
   ( A. x e. ( F " ( B X. C ) ) ph <-> A. y e. B A. z e. C ps ) ) $=
      ( wral cv wrex wi wal ralcom4 r19.23v ralbii bitri bitr3i wfn cxp wa cima
      wss co wceq wcel df-ral ovelimab imbi1d albidv bitrid ovex ceqsalv bitrdi
      albii ) IFUAGHUBZFUEUCZACIURUDZKZCLZDLZELZIUFZUGZEHMZDGMZANZCOZBEHKZDGKZV
      AVBUTUHZANZCOUSVJACUTUIUSVNVICUSVMVHADEFGHVBIUJUKULUMVJVFANZEHKZCOZDGKZVL
      VRVPDGKZCOVJVPDCGPVSVICVSVGANZDGKVIVPVTDGVFAEHQRVGADGQSUQSVQVKDGVQVOCOZEH
      KVKVOECHPWABEHABCVEVCVDIUNJUORTRTUP $.
  $}

$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Maps-to notation
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x y z $.  $d X x y z $.  $d Y x y z $.  $d C z $.
    mpondm0.f $e |- F = ( x e. X , y e. Y |-> C ) $.
    $( The value of an operation given by a maps-to rule is the empty set if
       the arguments are not contained in the base sets of the rule.
       (Contributed by Alexander van der Vekens, 12-Oct-2017.) $)
    mpondm0 $p |- ( -. ( V e. X /\ W e. Y ) -> ( V F W ) = (/) ) $=
      ( vz cdm cxp wss wcel wa wn co c0 wceq cv coprab cmpo eqtri dmeqi eqsstri
      df-mpo dmoprabss nssdmovg mpan ) DKZGHLZMEGNFHNOPEFDQRSUJATGNBTHNOJTCSZOA
      BJUAZKUKDUMDABGHCUBUMIABJGHCUFUCUDULABJGHUGUEEFGHDUHUI $.
  $}

  ${
    $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 not empty, 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 elfvdm df-ov eleq2s sselid opelxp sylib ) IFGHLZMZFGNZCDOZ
      MFCMGDMPUOHQZUQUPURARCMBRDMPKRESZPABKTZQUQHUTHABCDEUEUTJABKCDEUAUBUFUSABK
      CDUCUDUPURMIUPHUGUNIUPHUHFGHUIUJUKFGCDULUM $.

    $( If a two-parameter class is not empty, 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 not empty, 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.

       _EDITORIAL_: can simplify ~ islmhm .  (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 biadanii df-3an bitr4i ) FGHDNZOZGAOZHBOZPZFEOZPUOUPURUAUNUQ
      URIJABCGHDFKUBUQUMEFUOUPEQOZUMERUQCQOZJSISUSUTIJLUHUTUSIJGHABITGRJTHRPCEQ
      MUCUDUEIJGHABCEDQMKUFUIUGUJUOUPURUKUL $.
  $}

  ${
    $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.
       Version of ~ elovmporab1 with a disjoint variable condition, which does
       not require ~ ax-13 .  (Contributed by Alexander van der Vekens,
       15-Jul-2018.)  Avoid ~ ax-13 .  (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 $.
    $( $j usage 'elovmporab1w' avoids 'ax-13'; $)
  $}

  ${
    $d M x y z $.  $d X x y z $.  $d Y x y z $.  $d Z z $.  $d m z $.
    elovmporab1.o $e |- O = ( x e. _V , y e. _V
                              |-> { z e. [_ x / m ]_ M | ph } ) $.
    elovmporab1.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.
       Usage of this theorem is discouraged because it depends on ~ ax-13 .
       Use the weaker ~ elovmporab1w when possible.  (Contributed by Alexander
       van der Vekens, 15-Jul-2018.)  (New usage is discouraged.) $)
    elovmporab1 $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 nfsbc1v nfcsb nfrab nfsbc ovmpodxf eleq2d df-3an simplbi2com elrabi
      nfan syl11 sylbid mpcom ) HMNZIMNZOZJHIGUAZNZVMVNJEHFPZNZUBZBCMMADEBQZFPZ
      UCZHIGJKUDVOVQJACIUEZBHUEZDVRUCZNZVTVOVPWFJVOBCHIMMWCWFGMMGBCMMWCUFRVOKUH
      VOWAHRZCQIRZOZOAWEDWBVRWHWBVRRVOWIEWAHFUGUIWJAWEUJVOWIAWDWHWEACIUKWDBHUKU
      LUMUNVOWHOMUOVMVNUPVMVNUSVOVRMNWFMNLWEDVRMUQURVMVNBBHMBHSZTBIMBISZTVIVMVN
      CCHMCHSZTCIMCISTVIWMWLWEBDVRWDBHUTBEHFWKBFSVAVBWECDVRWDCBHWMACIUTVCCEHFWM
      CFSVAVBVDVEVSVOVTWGVTVOVSVMVNVSVFVGWEDJVRVHVJVKVL $.
  $}

  ${
    $d A x y $.  $d B x y $.  $d C s t $.  $d D s t $.
    2mpo0.o $e |- O = ( x e. A , y e. B |-> E ) $.
    2mpo0.u $e |- ( ( X e. A /\ Y e. B )
                     -> ( X O Y ) = ( s e. C , t e. D |-> F ) ) $.
    $( If the operation value of the operation value of two nested maps-to
       notation is not empty, all involved arguments belong to the
       corresponding base classes of the maps-to notations.  (Contributed by
       AV, 21-May-2021.) $)
    2mpo0 $p |- ( -. ( ( X e. A /\ Y e. B ) /\ ( S e. C /\ T e. D ) )
                   -> ( S ( X O Y ) T ) = (/) ) $=
      ( wcel wa c0 wn wo co wceq ianor mpondm0 oveqd eqtrdi notnotb cmpo adantr
      0ov eqid adantl eqtrd sylanbr jaoi3 sylbi ) MDRNERSZHFRIGRSZSUAUSUAZUTUAZ
      UBHIMNLUCZUCZTUDZUSUTUEVAVEVBVAVDHITUCTVAVCTHIABJLMNDEPUFUGHIULUHVAUAUSVB
      VEUSUIUSVBSZVDHIOCFGKUJZUCZTVFVCVGHIUSVCVGUDVBQUKUGVBVHTUDUSOCKVGHIFGVGUM
      UFUNUOUPUQUR $.
  $}

  ${
    $d x A $.
    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 ) $=
      ( cfv wrel cvv cxp wss copab fvmptss cv wcel relopab df-rel mpbi a1i mprg
      mpbir ) FGIZJUDKKLZMZACDNZUEMZUFBEBEUGUEFGHOUHBPEQUGJUHACDRUGSTUAUBUDSUC
      $.
  $}

  ${
    $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 cvv crn imassrn wceq syl sseqtrid adantr ccnv cmpt wfo
      eqid wf1o f1ofo forn elpwd cdm dfdm4 f1odm eqtr3id wa wss adantl foimacnv
      elpwi syl2an eqcomd imaeq2 eqeq2d syl5ibrcom f1of1 f1imacnv impbid f1o2d
      wf1 ) AFEBJZCJZDFKZLZDUAZEKZLZFVHVKUBZVOUDAVKVIMVJVHMZAVKCNIADOZVKCDVJPAB
      CDUCZVQCQABCDUEZVRGBCDUFRZBCDUGRSUHTAVNVHMVMVIMZAVNBNHAVLOZVNBVLVMPAWBDUI
      ZBDUJAVSWCBQGBCDUKRULSUHTAVPWAUMZUMZVJVNQZVMVKQZWEWGWFVMDVNLZQWEWHVMAVRVM
      CUNZWHVMQWDVTWAWIVPVMCUQUOBCVMDUPURUSWFVKWHVMVJVNDUTVAVBWEWFWGVJVLVKLZQWE
      WJVJABCDVGZVJBUNZWJVJQWDAVSWKGBCDVCRVPWLWAVJBUQTBCVJDVDURUSWGVNWJVJVMVKVL
      UTVAVBVEVF $.
  $}

  ${
    $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 wfo ccnv wfun cv cima cvv wcel dff1o3 vex funimaex simplbiim
      f1ofun syl f1opw2 ) ABCFZABCEDUBGUBABCHCIZJUCEKZLMNABCOUCUDEPQRUBCJCDKZLM
      NABCSCUEDPQTUA $.
  $}

  ${
    $d x y $.
    elovmpt3imp.o $e |- O = ( x e. _V , y e. _V |-> ( z e. M |-> B ) ) $.
    $( If the value of a function which is the result of an operation defined
       by the maps-to notation is not empty, the operands must be sets.
       Remark: a function which is the result of an operation can be regared as
       operation with 3 operands - therefore the abbreviation "mpt3" is used in
       the label.  (Contributed by AV, 16-May-2019.) $)
    elovmpt3imp $p |- ( A e. ( ( X O Y ) ` Z ) -> ( X e. _V /\ Y e. _V ) ) $=
      ( co cfv wcel c0 wne cvv wa ne0i wceq wi ax-1 wn mpondm0 fveq1 0fv eqtrdi
      cmpt eqneqall 3syl pm2.61i syl ) DJHIGLZMZNUNOPZHQNIQNRZUNDSUPUOUPUAZUPUO
      UBUPUCUMOTZUNOTUQABCFEUHGHIQQKUDURUNJOMOJUMOUEJUFUGUPUNOUIUJUKUL $.
  $}

  ${
    $d K x y z $.  $d L a x y $.  $d N a $.  $d V x y $.  $d W x y $.
    $d U x y $.  $d X a x y z $.  $d Y a x y z $.
    ovmpt3rab1.o $e |- O = ( x e. _V , y e. _V
                             |-> ( z e. M |-> { a e. N | ph } ) ) $.
    ovmpt3rab1.m $e |- ( ( x = X /\ y = Y ) -> M = K ) $.
    ovmpt3rab1.n $e |- ( ( x = X /\ y = Y ) -> N = L ) $.
    ${
      ovmpt3rab1.p $e |- ( ( x = X /\ y = Y ) -> ( ph <-> ps ) ) $.
      ovmpt3rab1.x $e |- F/ x ps $.
      ovmpt3rab1.y $e |- F/ y ps $.
      $( The value of an operation defined by the maps-to notation with a
         function into a class abstraction as a result.  The domain of the
         function and the base set of the class abstraction may depend on the
         operands, using implicit substitution.  (Contributed by AV,
         16-Jul-2018.)  (Revised by AV, 16-May-2019.) $)
      ovmpt3rab1 $p |- ( ( X e. V /\ Y e. W /\ K e. U )
                         -> ( X O Y ) = ( z e. K |-> { a e. L | ps } ) ) $=
        ( wcel w3a cvv crab cmpt cmpo wceq a1i cv wa rabeqbidv mpteq12dv adantl
        eqidd elex 3ad2ant1 3ad2ant2 mptexg 3ad2ant3 nfcv nfrabw nfmpt ovmpodxf
        nfv ) NLUCZOMUCZGFUCZUDZCDNOUEUEEIAPJUFZUGZEGBPHUFZUGZKUEUEKCDUEUEVLUHU
        IVJQUJCUKNUIZDUKOUIULZVLVNUIVJVPEIVKGVMRVPABPJHSTUMUNUOVJVOULUEUPVGVHNU
        EUCVINLUQURVHVGOUEUCVIOMUQUSVIVGVNUEUCVHEGVMFUTVAVJCVFVJDVFDNVBCOVBCEGV
        MCGVBBCPHUACHVBVCVDDEGVMDGVBBDPHUBDHVBVCVDVE $.
    $}

    $d L z $.  $d T z $.  $d U z $.  $d V z $.  $d W z $.
    $( If the value of a function which is the result of an operation defined
       by the maps-to notation is not empty, the operands and the argument of
       the function must be sets.  (Contributed by AV, 16-May-2019.) $)
    ovmpt3rabdm $p |- ( ( ( X e. V /\ Y e. W /\ K e. U ) /\ L e. T )
                        -> dom ( X O Y ) = K ) $=
      ( wcel w3a wa co cdm wsbc crab cmpt wceq sbceq1a sylan9bbr nfsbc1v nfsbcw
      cv nfcv ovmpt3rab1 adantr dmeqd wral rabexg adantl ralrimivw dmmptg eqtrd
      cvv syl ) NLTOMTGFTUAZHETZUBZNOKUCZUDDGACOUEZBNUEZPHUFZUGZUDZGVHVIVMVFVIV
      MUHVGAVKBCDFGHIJKLMNOPQRSCUMOUHAVJBUMNUHVKACOUIVJBNUIUJVJBNUKVJCBNCNUNACO
      UKULUOUPUQVHVLVDTZDGURVNGUHVHVODGVGVOVFVKPHEUSUTVADGVLVDVBVEVC $.

    $d A a $.  $d Z a z $.
    $( Implications for the value of an operation defined by the maps-to
       notation with a function into a class abstraction as a result having an
       element.  The domain of the function and the base set of the class
       abstraction may depend on the operands, using implicit substitution.
       (Contributed by AV, 16-Jul-2018.)  (Revised by AV, 16-May-2019.) $)
    elovmpt3rab1 $p |- ( ( K e. U /\ L e. T ) -> ( A e. ( ( X O Y ) ` Z )
                   -> ( ( X e. _V /\ Y e. _V ) /\ ( Z e. K /\ A e. L ) ) ) ) $=
      ( wcel co cfv wa cvv crab elovmpt3imp simprl cdm elfvdm wceq simpl adantr
      wi simplr simprr ovmpt3rabdm syl31anc eleq2d biimpcd imp wsbc adantl cmpt
      w3a anim2i df-3an sylibr ad2antll cv sbceq1a sylan9bbr nfsbc1v ovmpt3rab1
      nfcv nfsbcw fveq1d rabexg nfrabw rabbidv eqid fvmptf sylan2 eqtr2d elrabi
      syl eleqtrrd jca mpancom exp31 mpcom exp32 mpd com12 ) EOMNLUAZUBZTZHGTZI
      FTZUCZMUDTZNUDTZUCZOHTZEITZUCZUCZWPXBWSXFUMBCDEAPKUEJLMNOQUFWPXBWSXFWPXBW
      SUCZUCXBXEWPXBWSUGWPXGXEOWNUHZTZWPXGXEUMEOWNUIXIWPXGXEXCXIWPUCZXGUCZXEXJX
      GXCXIXGXCUMWPXGXIXCXGXHHOXGWTXAWQWRXHHUJXBWTWSWTXAUKULWTXAWSUNXBWQWRUGXBW
      QWRUOABCDFGHIJKLUDUDMNPQRSUPUQURUSULUTXCXKUCZXCXDXCXKUKXLEACNVAZBMVAZDOVA
      ZPIUEZTXDXLEWOXPXKWPXCXIWPXGUNVBXLWOODHXNPIUEZVCZUBZXPXLWTXAWQVDZWOXSUJXG
      XTXCXJXGXBWQUCXTWSWQXBWQWRUKVEWTXAWQVFVGVHXTOWNXRAXNBCDGHIJKLUDUDMNPQRSCV
      INUJAXMBVIMUJXNACNVJXMBMVJVKXMBMVLXMCBMCMVNACNVLVOVMVPWEXKXCXPUDTZXSXPUJW
      SYAXJXBWRYAWQXOPIFVQVBVHDOXQXPHXRUDDOVNXODPIXNDOVLDIVNVRDVIOUJXNXOPIXNDOV
      JVSXRVTWAWBWCWFXOPEIWDWEWGWHWIWJUTWGWKWLWM $.
  $}

  ${
    $d A a $.  $d M a x y z $.  $d N a x y z $.  $d T z $.  $d U x y z $.
    $d X a x y z $.  $d Y a x y z $.  $d Z a z $.
    elovmpt3rab.o $e |- O = ( x e. _V , y e. _V
                              |-> ( z e. M |-> { a e. N | ph } ) ) $.
    $( 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 AV, 17-Jul-2018.)  (Revised by AV, 16-May-2019.) $)
    elovmpt3rab $p |- ( ( M e. U /\ N e. T ) -> ( A e. ( ( X O Y ) ` Z )
                   -> ( ( X e. _V /\ Y e. _V ) /\ ( Z e. M /\ A e. N ) ) ) ) $=
      ( cv wceq wa eqidd elovmpt3rab1 ) ABCDEFGHIHIJKLMNOBPKQCPLQRZHSUAIST $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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 ` oR 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 ) $=
      ( wceq id ofeqd ) ABCZABFDE $.

    $( 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 cfv wbr cdm cin wral copab df-ofr nfcv nfbr nfralw nfopab
      cv nfcxfr ) ABGDSZESZHZUAFSZHZBIZDUBJUDJKZLZEFMDBEFNUHEFAUFADUGAUGOAUCUEB
      AUCOCAUEOPQRT $.
  $}

  ${
    $d A x $.  $d F f g x $.  $d G f g x $.  $d ph x $.  $d S x $.
    $d R f g x $.
    ofrfvalg.1 $e |- ( ph -> F Fn A ) $.
    ofrfvalg.2 $e |- ( ph -> G Fn B ) $.
    ofrfvalg.3 $e |- ( ph -> F e. V ) $.
    ofrfvalg.4 $e |- ( ph -> G e. W ) $.
    ofrfvalg.5 $e |- ( A i^i B ) = S $.
    ofrfvalg.6 $e |- ( ( ph /\ x e. A ) -> ( F ` x ) = C ) $.
    ofrfvalg.7 $e |- ( ( ph /\ x e. B ) -> ( G ` x ) = D ) $.
    $( Value of a relation applied to two functions.  Originally part of
       ~ ofrfval , this version assumes the functions are sets rather than
       their domains, avoiding ~ ax-rep .  (Contributed by SN, 5-Aug-2024.) $)
    ofrfvalg $p |- ( ph -> ( F oR R G <-> A. x e. S C R D ) ) $=
      ( wcel vf vg cofr wbr cv cfv cdm cin wral wb wceq wa dmeq ineqan12d fveq1
      breqan12d raleqbidv df-ofr brabga syl2anc ineq12d eqtrdi raleqdv eqsstrri
      fndmd inss1 sseli sylan2 inss2 breq12d ralbidva 3bitrd ) AIJGUCZUDZBUEZIU
      FZVOJUFZGUDZBIUGZJUGZUHZUIZVRBHUIEFGUDZBHUIAIKTJLTVNWBUJOPVOUAUEZUFZVOUBU
      EZUFZGUDZBWDUGZWFUGZUHZUIWBUAUBIJVMKLWDIUKZWFJUKZULWHVRBWKWAWLWMWIVSWJVTW
      DIUMWFJUMUNWLWMWEVPWGVQGVOWDIUOVOWFJUOUPUQBGUAUBURUSUTAVRBWAHAWACDUHZHAVS
      CVTDACIMVEADJNVEVAQVBVCAVRWCBHAVOHTZULVPEVQFGWOAVOCTVPEUKHCVOHWNCQCDVFVDV
      GRVHWOAVODTVQFUKHDVOHWNDQCDVIVDVGSVHVJVKVL $.
    $( $j usage 'ofrfvalg' avoids 'ax-rep'; $)
  $}

  ${
    $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 cv cfv cmpt wcel wceq wfn fnex syl2anc fndmd
        ineq12d eqtrdi mpteq1d inex1g eqeltrrid 3syl eqeltrd wa ineqan12d fveq1
        mptexg oveqan12d mpteq12dv df-of ovmpoga syl3anc eleq2i adantrr adantrl
        dmeq elin bitr3i oveq12d sylan2b mpteq2dva 3eqtrd ) AIJGUCZUDZBIUEZJUEZ
        UFZBUGZIUHZWGJUHZGUDZUIZBHWJUIZBHEFGUDZUIAITUJZJTUJZWKTUJWCWKUKAICULCKU
        JZWNMOCKIUMUNAJDULDLUJWONPDLJUMUNAWKWLTABWFHWJAWFCDUFZHAWDCWEDACIMUOADJ
        NUOUPQUQURZAWPHTUJWLTUJOWPHWQTQCDKUSUTBHWJTVFVAVBUAUBIJTTBUAUGZUEZUBUGZ
        UEZUFZWGWSUHZWGXAUHZGUDZUIWKWBTWSIUKZXAJUKZVCBXCXFWFWJXGXHWTWDXBWEWSIVO
        XAJVOVDXGXHXDWHXEWIGWGWSIVEWGXAJVEVGVHBGUAUBVIVJVKWRABHWJWMWGHUJZAWGCUJ
        ZWGDUJZVCZWJWMUKXIWGWQUJXLWQHWGQVLWGCDVPVQAXLVCWHEWIFGAXJWHEUKXKRVMAXKW
        IFUKXJSVNVRVSVTWA $.

      $( 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 ) ) $=
        ( cvv fnexd ofrfvalg ) ABCDEFGHIJTTMNACIKMOUAADJLNPUAQRSUB $.
    $}

    ${
      ofval.6 $e |- ( ( ph /\ X e. A ) -> ( F ` X ) = C ) $.
      ofval.7 $e |- ( ( ph /\ X e. B ) -> ( G ` X ) = D ) $.
      $( Evaluate a function operation at a point.  (Contributed by Mario
         Carneiro, 20-Jul-2014.) $)
      ofval $p |- ( ( ph /\ X e. S ) -> ( ( F oF R G ) ` X ) = ( C R D ) ) $=
        ( cfv vx wcel wa cof co cv cmpt wceq eqidd offval fveq1d adantr oveq12d
        fveq2 eqid ovex fvmpt adantl inss1 eqsstrri sseli sylan2 inss2 3eqtrd
        cin ) ALGUBZUCZLHIFUDUEZTZLUAGUAUFZHTZVJITZFUEZUGZTZLHTZLITZFUEZDEFUEAV
        IVOUHVFALVHVNAUABCVKVLFGHIJKMNOPQAVJBUBUCVKUIAVJCUBUCVLUIUJUKULVFVOVRUH
        AUALVMVRGVNVJLUHVKVPVLVQFVJLHUNVJLIUNUMVNUOVPVQFUPUQURVGVPDVQEFVFALBUBV
        PDUHGBLGBCVEZBQBCUSUTVARVBVFALCUBVQEUHGCLGVSCQBCVCUTVASVBUMVD $.

      $( 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 $.
    $}

    $( The function operation produces a function.  (Contributed by Mario
       Carneiro, 22-Jul-2014.) $)
    offn $p |- ( ph -> ( F oF R G ) Fn S ) $=
      ( vx co wfn cfv wcel wa cof cv cmpt ovex eqid fnmpti offval fneq1d mpbiri
      eqidd ) AFGDUAPZEQOEOUBZFRZULGRZDPZUCZEQOEUOUPUMUNDUDUPUEUFAEUKUPAOBCUMUN
      DEFGHIJKLMNAULBSTUMUJAULCSTUNUJUGUHUI $.
  $}

  ${
    offun.1 $e |- ( ph -> F Fn A ) $.
    offun.2 $e |- ( ph -> G Fn B ) $.
    offun.3 $e |- ( ph -> A e. V ) $.
    offun.4 $e |- ( ph -> B e. W ) $.
    $( The function operation produces a function.  (Contributed by SN,
       23-Jul-2024.) $)
    offun $p |- ( ph -> Fun ( F oF R G ) ) $=
      ( cof co cin wfn wfun eqid offn fnfun syl ) AEFDMNZBCOZPUBQABCDUCEFGHIJKL
      UCRSUCUBTUA $.
  $}

  ${
    $d y A $.  $d y B $.  $d y C $.  $d y F $.  $d y G $.  $d y ph $.
    $d x y R $.
    offval2f.0 $e |- F/ x ph $.
    offval2f.a $e |- F/_ x A $.
    offval2f.1 $e |- ( ph -> A e. V ) $.
    offval2f.2 $e |- ( ( ph /\ x e. A ) -> B e. W ) $.
    offval2f.3 $e |- ( ( ph /\ x e. A ) -> C e. X ) $.
    offval2f.4 $e |- ( ph -> F = ( x e. A |-> B ) ) $.
    offval2f.5 $e |- ( ph -> G = ( x e. A |-> C ) ) $.
    $( The function operation expressed as a mapping.  (Contributed by Thierry
       Arnoux, 23-Jun-2017.) $)
    offval2f $p |- ( ph -> ( F oF R G ) = ( x e. A |-> ( B R C ) ) ) $=
      ( vy cmpt cof co cv cfv wfn wcel wral ex ralrimi fnmptf syl fneq1d mpbird
      inidm wa wceq adantr fveq1d offval nfcv nffvmpt1 nfov fveq2 oveq12d simpr
      cbvmptf fvmpt2f syl2anc mpteq2da eqtrid eqtrd ) AGHFUAUBSCSUCZBCDTZUDZVLB
      CETZUDZFUBZTZBCDEFUBZTZASCCVNVPFCGHIIAGCUEVMCUEZADJUFZBCUGWAAWBBCLABUCZCU
      FZWBOUHUIBCDJMUJUKACGVMQULUMAHCUEVOCUEZAEKUFZBCUGWEAWFBCLAWDWFPUHUIBCEKMU
      JUKACHVORULUMNNCUNAVLCUFZUOZVLGVMAGVMUPWGQUQURWHVLHVOAHVOUPWGRUQURUSAVRBC
      WCVMUDZWCVOUDZFUBZTVTSBCVQWKSCUTMBVNVPFBCDVLVABFUTBCEVLVAVBSWKUTVLWCUPVNW
      IVPWJFVLWCVMVCVLWCVOVCVDVFABCWKVSLAWDUOZWIDWJEFWLWDWBWIDUPAWDVEZOBCDJMVGV
      HWLWDWFWJEUPWMPBCEKMVGVHVDVIVJVK $.
  $}

  ${
    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 $.
  $}

  $( Function value of a pointwise composition.  (Contributed by Stefan O'Rear,
     5-Oct-2014.)  (Proof shortened by Mario Carneiro, 5-Jun-2015.) $)
  fnfvof $p |- ( ( ( F Fn A /\ G Fn A ) /\ ( A e. V /\ X e. A ) ) ->
      ( ( F oF R G ) ` X ) = ( ( F ` X ) R ( G ` X ) ) ) $=
    ( wfn wa wcel cof co cfv wceq simpll simplr simpr inidm eqidd ofval anasss
    ) CAGZDAGZHZAEIZFAIZFCDBJKLFCLZFDLZBKMUCUDHZAAUFUGBACDEEFUAUBUDNUAUBUDOUCUD
    PZUIAQUHUEHZUFRUJUGRST $.

  ${
    $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 cv cfv co cof ffnd wcel wa eqidd offval wral wf inss1 eqsstrri sseli
      cin ffvelcdm syl2an inss2 ralrimivva adantr ovrspc2v syl21anc fmpt3d ) AU
      AFUAUBZKUCZVELUCZGUDZJKLGUEUDAUADEVFVGGFKLMNADHKPUFAEILQUFRSTAVEDUGZUHVFU
      IAVEEUGZUHVGUIUJAVEFUGZUHVFHUGZVGIUGZBUBCUBGUDJUGZCIUKBHUKZVHJUGADHKULVIV
      LVKPFDVEFDEUPZDTDEUMUNUODHVEKUQURAEILULVJVMVKQFEVEFVPETDEUSUNUOEIVELUQURA
      VOVKAVNBCHIOUTVABCHIJGVFVGVBVCVD $.
  $}

  ${
    $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 cbvralw simpr fvmpt2 syl2anc ralbidva bitrid bitrd ) AGHFUARQUKZBCDUB
      ZUCZVKBCEUBZUCZFRZQCSZDEFRZBCSZAQCCVMVOFCGHIIAGCUDVLCUDZADJUEZBCSVTAWABCM
      UFBCDVLJVLUGZUHUIACGVLOUJULAHCUDVNCUDZAEKUEZBCSWCAWDBCNUFBCEVNKVNUGZUHUIA
      CHVNPUJULLLCUMAVKCUEZUNZVKGVLAGVLTWFOUOUPWGVKHVNAHVNTWFPUOUPUQVQBUKZVLUCZ
      WHVNUCZFRZBCSAVSVPWKQBCBVMVOFBCDVKURBFUSBCEVKURUTWKQVCVKWHTVMWIVOWJFVKWHV
      LVAVKWHVNVAVBVDAWKVRBCAWHCUEZUNZWIDWJEFWMWLWAWIDTAWLVEZMBCDJVLWBVFVGWMWLW
      DWJETWNNBCEKVNWEVFVGVBVHVIVJ $.
  $}

  ${
    $d A x $.  $d F x $.  $d G x $.  $d ph x $.  $d R x $.
    offvalfv.a $e |- ( ph -> A e. V ) $.
    offvalfv.f $e |- ( ph -> F Fn A ) $.
    offvalfv.g $e |- ( ph -> G Fn A ) $.
    $( The function operation expressed as a mapping with function values.
       (Contributed by AV, 6-Apr-2019.) $)
    offvalfv $p |- ( ph -> ( F oF R G )
                           = ( x e. A |-> ( ( F ` x ) R ( G ` x ) ) ) ) $=
      ( cfv crn wfn wcel fnfvelrn sylan cmpt wceq dffn5 sylib cv offval2 ) ABCB
      UAZEKZUCFKZDEFGELZFLZHAECMZUCCNZUDUFNICUCEOPAFCMZUIUEUGNJCUCFOPAUHEBCUDQR
      IBCESTAUJFBCUEQRJBCFSTUB $.
  $}

  ${
    $d A x a $.  $d V a $.  $d B a $.  $d C a $.  $d R a x $.
    $( Value of a pointwise operation on two functions defined using maps-to
       notation.  (Contributed by Stefan O'Rear, 5-Oct-2014.) $)
    ofmpteq $p |- ( ( A e. V /\ ( x e. A |-> B ) Fn A
        /\ ( x e. A |-> C ) Fn A )
  -> ( ( x e. A |-> B ) oF R ( x e. A |-> C ) ) = ( x e. A |-> ( B R C ) ) ) $=
      ( va wcel cmpt wfn co cv csb cvv wral eqid mptfng wceq nfcv cbvmpt w3a wa
      cof simp1 simpr simpl2 sylibr nfcsb1v csbeq1a eleq1d rspc sylc simpl3 a1i
      nfel1 offval2 nfov oveq12d eqtr4di ) BFHZABCIZBJZABDIZBJZUAZVAVCEUCKGBAGL
      ZCMZAVFDMZEKZIABCDEKZIVEGBVGVHEVAVCFNNUTVBVDUDVEVFBHZUBZVKCNHZABOZVGNHZVE
      VKUEZVLVBVNUTVBVDVKUFABCVAVAPQUGVMVOAVFBAVGNAVFCUHZUOALVFRZCVGNAVFCUIZUJU
      KULVLVKDNHZABOZVHNHZVPVLVDWAUTVBVDVKUMABDVCVCPQUGVTWBAVFBAVHNAVFDUHZUOVRD
      VHNAVFDUIZUJUKULVAGBVGIRVEAGBCVGGCSVQVSTUNVCGBVHIRVEAGBDVHGDSWCWDTUNUPAGB
      VJVIGVJSAVGVHEVQAESWCUQVRCVGDVHEVSWDURTUS $.
  $}

  ${
    $d A x $.  $d B b c x $.  $d F b c x $.  $d G b c x $.  $d H b c x $.
    $d R b c x $.  $d S b c x $.  $d ph b c x $.
    coof.f $e |- ( ph -> F : A --> B ) $.
    coof.g $e |- ( ph -> G : A --> B ) $.
    coof.h $e |- ( ph -> H Fn B ) $.
    coof.a $e |- ( ph -> A e. V ) $.
    coof.1 $e |- ( ( ph /\ ( b e. B /\ c e. B ) ) -> ( b R c ) e. B ) $.
    coof.2 $e |- ( ( ph /\ ( b e. B /\ c e. B ) ) ->
           ( H ` ( b R c ) ) = ( ( H ` b ) S ( H ` c ) ) ) $.
    $( The composition of a _homomorphism_ with a function operation.
       (Contributed by SN, 20-May-2025.) $)
    coof $p |-
      ( ph -> ( H o. ( F oF R G ) ) = ( ( H o. F ) oF S ( H o. G ) ) ) $=
      ( vx cfv co cv cmpt cof ccom wcel wceq wral ffvelcdmda ralrimivva fvoveq1
      wa adantr fveq2 oveq1d eqeq12d oveq2 fveq2d oveq2d rspc2va mpteq2dva ffnd
      syl21anc inidm eqidd offval coeq2d crn wfn dffn3 sylib jca caovclg syldan
      wf cofmpt eqtrd fnfco syl2anc fvco2 sylan 3eqtr4d ) ARBRUAZFSZWBGSZDTZHSZ
      UBZRBWCHSZWDHSZETZUBHFGDUCTZUDZHFUDZHGUDZEUCTARBWFWJAWBBUEZUKZWCCUEZWDCUE
      ZJUAZKUAZDTHSZWSHSZWTHSZETZUFZKCUGJCUGZWFWJUFZABCWBFLUHZABCWBGMUHZAXFWOAX
      EJKCCQUIULXEXGWCWTDTZHSZWHXCETZUFJKWCWDCCWSWCUFZXAXKXDXLWSWCWTHDUJXMXBWHX
      CEWSWCHUMUNUOWTWDUFZXKWFXLWJXNXJWEHWTWDWCDUPUQXNXCWIWHEWTWDHUMURUOUSVBUTA
      WLHRBWEUBZUDWGAWKXOHARBBWCWDDBFGIIABCFLVAZABCGMVAZOOBVCZWPWCVDWPWDVDVEVFA
      RBWECHVGZHAHCVHZCXSHVNNCHVIVJAWOWQWRUKWECUEWPWQWRXHXIVKAJKWCWDCCCDPVLVMVO
      VPARBBWHWIEBWMWNIIAXTBCFVNWMBVHNLCBHFVQVRAXTBCGVNWNBVHNMCBHGVQVROOXRAFBVH
      WOWBWMSWHUFXPBHFWBVSVTAGBVHWOWBWNSWIUFXQBHGWBVSVTVEWA $.
  $}

  ${
    $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 ffnd fvco2 sylan eqtr4d ) AGHFUCZUDZIUEUAEUAUFZITZGTZVQ
      HTZFUDZUGGIUEZHIUEZVNUDAUAUBEDVQUBUFZGTZWCHTZFUDVTIVOAEDVPIOUHAUAEDIOUIAU
      BBCWDWEFDGHJKMNPQSAWCBUJUNWDUKAWCCUJUNWEUKULWCVQUMWDVRWEVSFWCVQGUOWCVQHUO
      UPUQAUAEEVRVSFEWAWBLLAGBUREBIUSZWAEURMAEDIUSZDBUTWFODBCVAZBSBCVBVCEDBIVDV
      EBEGIVGVFAHCURECIUSZWBEURNAWGDCUTWIODWHCSBCVHVCEDCIVDVECEHIVGVFRREVIAIEUR
      ZVPEUJZVPWATVRUMAEDIOVJZEGIVPVKVLAWJWKVPWBTVSUMWLEHIVPVKVLULVM $.
  $}

  ${
    $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 ) $.
    ${
      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 ) $=
        ( cof co cfv inidm offn cv wcel wa ofval eqtrd eqfnfvd ) ABCGHFRSZIACCF
        CGHJJLMKKCUAZUBNABUCZCUDUEUKUITDEFSUKITACCDEFCGHJJUKLMKKUJOPUFQUGUH $.
    $}

    $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 dffn5 sylib inidm offval eqeq12d cvv wcel
      cv wb fvexd ralrimivw mpteqb syl bitrd ) AIGHFUAQZRBCBUKZIUBZSZBCDEFQZSZR
      ZUTVBRBCTZAIVAURVCAICUCIVARNBCIUDUEABCCDEFCGHJJLMKKCUFOPUGUHAUTUIUJZBCTVD
      VEULAVFBCAUSIUMUNBCUTVBUIUOUPUQ $.
  $}

  ${
    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 ) $.
    $( Left operation by a constant.  (Contributed by Mario Carneiro,
       24-Jul-2014.) $)
    ofc1 $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 ofval ) AB
      BCDEBBCNOZFGGIACHPZUFBQKBCHRSLJJBTAUGIBPIUFUACUBKBCIHUCUDMUE $.
  $}

  ${
    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 ) $.
    $( Right operation by a constant.  (Contributed by NM, 7-Oct-2014.) $)
    ofc2 $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 ofval ) AB
      BDCEBFBCNOZGGILACHPZUFBQKBCHRSJJBTMAUGIBPIUFUACUBKBCIHUCUDUE $.
  $}

  ${
    $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 C $.  $d w x y z F $.  $d w x y z G $.  $d w x y z H $.
    $d w x y z O $.  $d w x y z P $.  $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 ffnd inidm eqidd ofrfval mpbird ) AFFDLMKNZFOZULDMZKCPAUM
        KCAUKCQZRZBNZUPDMZUMBEULUPULSZUPULUPULDURUDZUSUAAUQBEPUNAUQBEJTUBACEUKF
        IUEUCTAKCCULULDCFFGGACEFIUFZUTHHCUGUOULUHZVAUIUJ $.
    $}

    ${
      $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 wa adantr ffvelcdmda ffvelcdmd fmpt3d
          wcel wf wfn wceq fvex eqid fnmpti fneq1d mpbiri dffn5 feqmptd offval2
          sylib fveq1d 2fveq3 fvmpt sylan9eq oveq1d id oveq12d eqeq1d ralrimiva
          fveq2 wral rspcdva eqtrd mpteq2dva fconstmpt eqtr4di ) AIHFUAUBZSDEUC
          ZDEUDUEAWASDSUFZITZWCHTZFUBZUCWBASDWDWEFIHKGGMADGWCIACDCUFZHTZJTZGIQA
          WGDULZUGGGWHJAGGJUMWJPUHADGWGHNUIUJUKUIADGWCHNUIZAIDUNZISDWDUCUOAWLCD
          WIUCZDUNCDWIWMWHJUPWMUQZURADIWMQUSUTSDIVAVDASDGHNVBVCASDWFEAWCDULZUGZ
          WFWEJTZWEFUBZEWPWDWQWEFAWOWDWCWMTWQAWCIWMQVECWCWIWQDWMWGWCJHVFWNWEJUP
          VGVHVIWPBUFZJTZWSFUBZEUOZWREUOBGWEWSWEUOZXAWREXCWTWQWSWEFWSWEJVNXCVJV
          KVLAXBBGVOWOAXBBGRVMUHWKVPVQVRVQSDEVSVT $.
      $}
    $}

    ${
      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 ) $.
      ${
        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 $.
      $}

      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
        $.
    $}

    ${
      $d .0. w x y $.
      caofidlcan.4 $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
                           ( ( x R y ) = y <-> x = .0. ) ) $.
      $( Transfer a cancellation/identity law to the function operation.
         (Contributed by SN, 16-Oct-2025.) $)
      caofidlcan $p |-
        ( ph -> ( ( F oF R G ) = G <-> F = ( A X. { .0. } ) ) ) $=
        ( vw co cmpt wceq wral wb cv cfv cof csn cxp wcel ffvelcdmda ralrimivva
        wa jca oveq1 eqeq1d eqeq1 bibi12d oveq2 id eqeq12d bibi1d rspc2v syldan
        mpan9 ralbidva cvv ovexd ralrimiva mpteqb syl 3bitr4d feqmptd fconstmpt
        offval2 a1i ) AODOUAZGUBZVMHUBZEPZQZODVOQZRZODVNQZODJQZRZGHEUCPZHRGDJUD
        UEZRAVPVORZODSZVNJRZODSZVSWBAWEWGODAVMDUFZVNFUFZVOFUFZUIZWEWGTZAWIUIZWJ
        WKADFVMGLUGZADFVMHMUGZUJABUAZCUAZEPZWRRZWQJRZTZCFSBFSWLWMAXBBCFFNUHXBWM
        VNWREPZWRRZWGTBCVNVOFFWQVNRZWTXDXAWGXEWSXCWRWQVNWREUKULWQVNJUMUNWRVORZX
        DWEWGXFXCVPWRVOWRVOVNEUOXFUPUQURUSVAUTVBAVPVCUFZODSVSWFTAXGODWNVNVOEVDV
        EODVPVOVCVFVGAWJODSWBWHTAWJODWOVEODVNJFVFVGVHAWCVQHVRAODVNVOEGHIFFKWOWP
        AODFGLVIZAODFHMVIZVKXIUQAGVTWDWAXHWDWARAODJVJVLUQVH $.
    $}

    ${
      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 ffnd inidm ofrfval 3imtr4d
        eqidd ) AOPZHUAZUSIUAZEQZODRUTVAGQZODRHIEUBQHIGUBQAVBVCODAUSDSZUCZUTFSV
        AFSBPZCPZEQZVFVGGQZTZCFRBFRZVBVCTZADFUSHLUDADFUSIMUDAVKVDAVJBCFFNUEUFVJ
        VLUTVGEQZUTVGGQZTBCUTVAFFVFUTUHVHVMVIVNVFUTVGEUGVFUTVGGUGUIVGVAUHVMVBVN
        VCVGVAUTEUJVGVAUTGUJUIUKULUMAODDUTVAEDHIJJADFHLUNZADFIMUNZKKDUOZVEUTURZ
        VEVAURZUPAODDUTVAGDHIJJVOVPKKVQVRVSUPUQ $.
    $}

    caofass.4 $e |- ( ph -> H : A --> S ) $.
    ${
      caofass.5 $e |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) ->
                        ( ( x R y ) T z ) = ( x O ( y P z ) ) ) $.
      $( Transfer an associative law to the function operation.  (Contributed
         by Mario Carneiro, 26-Jul-2014.) $)
      caofass $p |- ( ph ->
        ( ( F oF R G ) oF T H ) = ( F oF O ( G oF P H ) ) ) $=
        ( co vw cv cfv cmpt cof wcel wa wceq wral ralrimivvva adantr ffvelcdmda
        wi oveq1 oveq1d eqeq12d oveq2 oveq2d rspc3v syl3anc mpd mpteq2dva ovexd
        cvv feqmptd offval2 3eqtr4d ) AUAEUAUBZJUCZVHKUCZGTZVHLUCZITZUDUAEVIVJV
        LFTZMTZUDJKGUETZLIUETJKLFUETZMUETAUAEVMVOAVHEUFZUGZBUBZCUBZGTZDUBZITZVT
        WAWCFTZMTZUHZDHUICHUIBHUIZVMVOUHZAWHVRAWGBCDHHHSUJUKVSVIHUFVJHUFVLHUFWH
        WIUMAEHVHJPULZAEHVHKQULZAEHVHLRULZWGWIVIWAGTZWCITZVIWEMTZUHVKWCITZVIVJW
        CFTZMTZUHBCDVIVJVLHHHVTVIUHZWDWNWFWOWSWBWMWCIVTVIWAGUNUOVTVIWEMUNUPWAVJ
        UHZWNWPWOWRWTWMVKWCIWAVJVIGUQUOWTWEWQVIMWAVJWCFUNURUPWCVLUHZWPVMWRVOWCV
        LVKIUQXAWQVNVIMWCVLVJFUQURUPUSUTVAVBAUAEVKVLIVPLNVDHOVSVIVJGVCWLAUAEVIV
        JGJKNHHOWJWKAUAEHJPVEZAUAEHKQVEZVFAUAEHLRVEZVFAUAEVIVNMJVQNHVDOWJVSVJVL
        FVCXBAUAEVJVLFKLNHHOWKWLXCXDVFVFVG $.
    $}

    ${
      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 anbi12d imbi1d anbi2d rspc3v syl3anc ralimdva ffnd
        breq2 mpd inidm eqidd ofrfval r19.26 bitr4di 3imtr4d ) ASUAZJUBZVJKUBZF
        TZVLVJLUBZHTZUCZSEUDZVKVNITZSEUDJKFUETZKLHUETZUCZJLIUETAVPVRSEAVJEUFZUC
        ZBUAZCUAZFTZWEDUAZHTZUCZWDWGITZUGZDGUDCGUDBGUDZVPVRUGZAWLWBAWKBCDGGGRUH
        UIWCVKGUFVLGUFVNGUFWLWMUGAEGVJJOUJAEGVJKPUJAEGVJLQUJWKWMVKWEFTZWHUCZVKW
        GITZUGVMVLWGHTZUCZWPUGBCDVKVLVNGGGWDVKUKZWIWOWJWPWSWFWNWHWDVKWEFULUMWDV
        KWGIULUNWEVLUKZWOWRWPWTWNVMWHWQWEVLVKFVBWEVLWGHULUOUPWGVNUKZWRVPWPVRXAW
        QVOVMWGVNVLHVBUQWGVNVKIVBUNURUSVCUTAWAVMSEUDZVOSEUDZUCVQAVSXBVTXCASEEVK
        VLFEJKMMAEGJOVAZAEGKPVAZNNEVDZWCVKVEZWCVLVEZVFASEEVLVNHEKLMMXEAEGLQVAZN
        NXFXHWCVNVEZVFUOVMVOSEVGVHASEEVKVNIEJLMMXDXINNXFXGXJVFVI $.
    $}
  $}

  ${
    $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 $.
    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 ) $.
    ${
      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.) $)
      caofdi $p |- ( ph ->
        ( F oF T ( G oF R H ) ) = ( ( F oF T G ) oF O ( F oF T H ) ) ) $=
        ( co vw cfv cmpt cof wcel w3a wceq adantlr ffvelcdmda caovdid mpteq2dva
        cv wa cvv ovexd feqmptd offval2 3eqtr4d ) AUAEUAULZIUBZUSJUBZUSKUBZFTZH
        TZUCUAEUTVAHTZUTVBHTZMTZUCIJKFUDTZHUDZTIJVITZIKVITZMUDTAUAEVDVGAUSEUEZU
        MZBCDUTVAVBGFHMLABULZLUECULZGUEDULZGUEUFVNVOVPFTHTVNVOHTVNVPHTMTUGVLSUH
        AELUSIPUIZAEGUSJQUIZAEGUSKRUIZUJUKAUAEUTVCHIVHNLUNOVQVMVAVBFUOAUAELIPUP
        ZAUAEVAVBFJKNGGOVRVSAUAEGJQUPZAUAEGKRUPZUQUQAUAEVEVFMVJVKNUNUNOVMUTVAHU
        OVMUTVBHUOAUAEUTVAHIJNLGOVQVRVTWAUQAUAEUTVBHIKNLGOVQVSVTWBUQUQUR $.
    $}

    ${
      caofdir.5 $e |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. K ) ) ->
                          ( ( x R y ) T z ) = ( ( x T z ) O ( y T z ) ) ) $.
      $( Transfer a reverse distributive law to the function operation.
         (Contributed by NM, 19-Oct-2014.) $)
      caofdir $p |- ( ph ->
        ( ( G oF R H ) oF T F ) = ( ( G oF T F ) oF O ( H oF T F ) ) ) $=
        ( co vw cv cfv cmpt cof wcel wceq adantlr ffvelcdmda caovdird mpteq2dva
        wa w3a cvv ovexd feqmptd offval2 3eqtr4d ) AUAEUAUBZJUCZUSKUCZFTZUSIUCZ
        HTZUDUAEUTVCHTZVAVCHTZMTZUDJKFUETZIHUEZTJIVITZKIVITZMUETAUAEVDVGAUSEUFZ
        ULZBCDUTVAVCGFHMLABUBZGUFCUBZGUFDUBZLUFUMVNVOFTVPHTVNVPHTVOVPHTMTUGVLSU
        HAEGUSJQUIZAEGUSKRUIZAELUSIPUIZUJUKAUAEVBVCHVHINUNLOVMUTVAFUOVSAUAEUTVA
        FJKNGGOVQVRAUAEGJQUPZAUAEGKRUPZUQAUAELIPUPZUQAUAEVEVFMVJVKNUNUNOVMUTVCH
        UOVMVAVCHUOAUAEUTVCHJINGLOVQVSVTWBUQAUAEVAVCHKINGLOVRVSWAWBUQUQUR $.
    $}
  $}

  ${
    $d ph x y z $.  $d A x y z $.  $d B y z $.  $d I z $.  $d M x y z $.
    $d S x y $.
    caonncan.i $e |- ( ph -> I e. V ) $.
    caonncan.a $e |- ( ph -> A : I --> S ) $.
    caonncan.b $e |- ( ph -> B : I --> S ) $.
    caonncan.z $e |- ( ( ph /\ ( x e. S /\ y e. S ) ) ->
        ( x M ( x M y ) ) = y ) $.
    $( Transfer ~ nncan -shaped laws to vectors of numbers.  (Contributed by
       Stefan O'Rear, 27-Mar-2015.) $)
    caonncan $p |- ( ph -> ( A oF M ( A oF M B ) ) = B ) $=
      ( vz cv cfv co wcel wceq cvv cmpt wa wral ffvelcdmda ralrimivva adantr id
      oveq1 oveq12d eqeq1d oveq2 eqeq12d rspc2va syl21anc mpteq2dva fvexd ovexd
      cof oveq2d feqmptd offval2 3eqtr4d ) ANGNOZDPZVDVCEPZHQZHQZUANGVEUADDEHUR
      ZQZVHQEANGVGVEAVCGRZUBZVDFRVEFRBOZVLCOZHQZHQZVMSZCFUCBFUCZVGVESZAGFVCDKUD
      AGFVCELUDAVQVJAVPBCFFMUEUFVPVRVDVDVMHQZHQZVMSBCVDVEFFVLVDSZVOVTVMWAVLVDVN
      VSHWAUGVLVDVMHUHUIUJVMVESZVTVGVMVEWBVSVFVDHVMVEVDHUKUSWBUGULUMUNUOANGVDVF
      HDVIITTJVKVCDUPZVKVDVEHUQANGFDKUTZANGVDVEHDEITTJWCVKVCEUPWDANGFELUTZVAVAW
      EVB $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Proper subset relation
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c [C.] $.

  $( Extend class notation to include the reified proper subset relation. $)
  crpss $a class [C.] $.

  ${
    $d x y A $.  $d x y B $.
    $( Define a relation which corresponds to proper subsethood ~ df-pss on
       sets.  This allows to use proper subsethood with general concepts that
       require relations, such as strict ordering, see ~ sorpss .  (Contributed
       by Stefan O'Rear, 2-Nov-2014.) $)
    df-rpss $a |- [C.] = { <. x , y >. | x C. y } $.

    $( The proper subset relation is a relation.  (Contributed by Stefan
       O'Rear, 2-Nov-2014.) $)
    relrpss $p |- Rel [C.] $=
      ( vx vy cv wpss crpss df-rpss relopabiv ) ACBCDABEABFG $.

    $( The proper subset relation on sets is the same as class proper
       subsethood.  (Contributed by Stefan O'Rear, 2-Nov-2014.) $)
    brrpssg $p |- ( B e. V -> ( A [C.] B <-> A C. B ) ) $=
      ( vx vy wcel crpss wbr wpss cvv elex relrpss brrelex1i anim12i adantr wss
      wa pssss ssexg cv syl2anr jca psseq1 psseq2 df-rpss brabg ancoms pm5.21nd
      wb ) BCFZABGHZABIZBJFZAJFZQUJUMUKUNBCKZABGLMNUJULQUMUNUJUMULUOOULABPUMUNU
      JABRUOABJSUAUBUNUMUKULUIDTZETZIAUQIULDEABJJGUPAUQUCUQBAUDDEUEUFUGUH $.
  $}

  ${
    brrpss.a $e |- B e. _V $.
    $( The proper subset relation on sets is the same as class proper
       subsethood.  (Contributed by Stefan O'Rear, 2-Nov-2014.) $)
    brrpss $p |- ( A [C.] B <-> A C. B ) $=
      ( cvv wcel crpss wbr wpss wb brrpssg ax-mp ) BDEABFGABHICABDJK $.
  $}

  ${
    $d x y z A $.
    $( Every class is partially ordered by proper subsets.  (Contributed by
       Stefan O'Rear, 2-Nov-2014.) $)
    porpss $p |- [C.] Po A $=
      ( vx vy vz crpss wpo cv wbr wn wa wi wral wpss pssirr psstr brrpss notbii
      vex anbi12i imbi12i mpbir2an rgenw rgen2w df-po mpbir ) AEFBGZUFEHZIZUFCG
      ZEHZUIDGZEHZJZUFUKEHZKZJZDALZCALBALUQBCAAUPDAUPUFUFMZIZUFUIMZUIUKMZJZUFUK
      MZKZUFNUFUIUKOUHUSUOVDUGURUFUFBRPQUMVBUNVCUJUTULVAUFUICRPUIUKDRZPSUFUKVEP
      TSUAUBUCBCDAEUDUE $.

    $( Express strict ordering under proper subsets, i.e. the notion of a chain
       of sets.  (Contributed by Stefan O'Rear, 2-Nov-2014.) $)
    sorpss $p |- ( [C.] Or A <-> A. x e. A A. y e. A ( x C_ y \/ y C_ x ) ) $=
      ( cv crpss wbr weq w3o wral wpo wss wor porpss biantrur wpss sspsstri vex
      wa wo brrpss biid 3orbi123i bitr4i 2ralbii df-so 3bitr4ri ) ADZBDZEFZABGZ
      UHUGEFZHZBCIACIZCEJZUMRUGUHKUHUGKSZBCIACICELUNUMCMNUOULABCCUOUGUHOZUJUHUG
      OZHULUGUHPUIUPUJUJUKUQUGUHBQTUJUAUHUGAQTUBUCUDABCEUEUF $.
  $}

  $( Property of a chain of sets.  (Contributed by Stefan O'Rear,
     2-Nov-2014.) $)
  sorpssi $p |- ( ( [C.] Or A /\ ( B e. A /\ C e. A ) ) ->
      ( B C_ C \/ C C_ B ) ) $=
    ( crpss wor wcel wa wpss wceq w3o wss wo wbr solin wb elex ad2antll brrpssg
    cvv syl biidd ad2antrl 3orbi123d mpbid sspsstri sylibr ) ADEZBAFZCAFZGGZBCH
    ZBCIZCBHZJZBCKCBKLUJBCDMZULCBDMZJUNABCDNUJUOUKULULUPUMUJCSFZUOUKOUIUQUGUHCA
    PQBCSRTUJULUAUJBSFZUPUMOUHURUGUIBAPUBCBSRTUCUDBCUEUF $.

  $( A chain of sets is closed under binary union.  (Contributed by Mario
     Carneiro, 16-May-2015.) $)
  sorpssun $p |- ( ( [C.] Or A /\ ( B e. A /\ C e. A ) ) ->
      ( B u. C ) e. A ) $=
    ( crpss wor wcel wa wss cun simprr wb ssequn1 eleq1 sylbi syl5ibrcom simprl
    wceq ssequn2 sorpssi mpjaod ) ADEZBAFZCAFZGGZBCHZBCIZAFZCBHZUDUGUEUCUAUBUCJ
    UEUFCQUGUCKBCLUFCAMNOUDUGUHUBUAUBUCPUHUFBQUGUBKCBRUFBAMNOABCST $.

  $( A chain of sets is closed under binary intersection.  (Contributed by
     Mario Carneiro, 16-May-2015.) $)
  sorpssin $p |- ( ( [C.] Or A /\ ( B e. A /\ C e. A ) ) ->
      ( B i^i C ) e. A ) $=
    ( crpss wor wcel wa wss cin simprl wceq dfss2 eleq1 sylbi syl5ibrcom simprr
    wb sseqin2 sorpssi mpjaod ) ADEZBAFZCAFZGGZBCHZBCIZAFZCBHZUDUGUEUBUAUBUCJUE
    UFBKUGUBQBCLUFBAMNOUDUGUHUCUAUBUCPUHUFCKUGUCQCBRUFCAMNOABCST $.

  ${
    $d Y u v $.
    $( In a chain of sets, a maximal element is the union of the chain.
       (Contributed by Stefan O'Rear, 2-Nov-2014.) $)
    sorpssuni $p |- ( [C.] Or Y -> ( E. u e. Y A. v e. Y -. u C. v <->
        U. Y e. Y ) ) $=
      ( crpss wor cv wpss wn wral wrex cuni wcel w3a wss wa sorpssi syl elssuni
      wo wi anassrs sspss orel1 eqimss2 syl6com sylbi ax-1 jaoi ralimdva 3impia
      weq unissb sylibr 3ad2ant2 eqssd simp2 rexlimdv3a ssnpss rgen wceq psseq1
      eqeltrd notbid ralbidv rspcev mpan2 impbid1 ) CDEZBFZAFZGZHZACIZBCJZCKZCL
      ZVHVMVPBCVHVICLZVMMZVOVICVRVOVIVRVJVINZACIZVOVINVHVQVMVTVHVQOZVLVSACWAVJC
      LZOVIVJNZVSSZVLVSTZVHVQWBWDCVIVJPUAWCWEVSWCVKBAUKZSZWEVIVJUBVLWGWFVSVKWFU
      CVJVIUDUEUFVSVLUGUHQUIUJACVIULUMVQVHVIVONVMVICRUNUOVHVQVMUPVBUQVPVOVJGZHZ
      ACIZVNWIACWBVJVONWIVJCRVJVOURQUSVMWJBVOCVIVOUTZVLWIACWKVKWHVIVOVJVAVCVDVE
      VFVG $.

    $( In a chain of sets, a minimal element is the intersection of the chain.
       (Contributed by Stefan O'Rear, 2-Nov-2014.) $)
    sorpssint $p |- ( [C.] Or Y -> ( E. u e. Y A. v e. Y -. v C. u <->
        |^| Y e. Y ) ) $=
      ( crpss wor cv wpss wn wral wrex cint wcel w3a wss intss1 3ad2ant2 wa syl
      wo wi sorpssi anassrs weq sspss orel1 eqimss2 sylbi jao1i ralimdva 3impia
      syl6com ssint sylibr eqssd simp2 eqeltrd rexlimdv3a ssnpss psseq2 ralbidv
      rgen wceq notbid rspcev mpan2 impbid1 ) CDEZAFZBFZGZHZACIZBCJZCKZCLZVGVLV
      OBCVGVICLZVLMZVNVICVQVNVIVPVGVNVINVLVICOPVQVIVHNZACIZVIVNNVGVPVLVSVGVPQZV
      KVRACVTVHCLZQVRVHVINZSZVKVRTZVGVPWAWCCVIVHUAUBVRWBVKWBVJABUCZSZWDVHVIUDVK
      WFWEVRVJWEUEVIVHUFUKUGUHRUIUJAVICULUMUNVGVPVLUOUPUQVOVHVNGZHZACIZVMWHACWA
      VNVHNWHVHCOVNVHURRVAVLWIBVNCVIVNVBZVKWHACWJVJWGVIVNVHUSVCUTVDVEVF $.
  $}

  ${
    $d Y x y $.  $d A x y $.  $d Y u $.  $d A u $.  $d u x y $.
    $( The componentwise complement of a chain of sets is also a chain of sets.
       (Contributed by Stefan O'Rear, 2-Nov-2014.) $)
    sorpsscmpl $p |- ( [C.] Or Y -> [C.] Or { u e. ~P A | ( A \ u ) e. Y } ) $=
      ( vx vy crpss wor cv wss wo cdif wcel wral wa difeq2 eleq1d elrab syl2anb
      weq wi cpw crab biimpi sorpssi expcom wceq velpw dfss4 bitri sscon sseq12
      an4 imbitrid ancoms orim12d com12 orcoms syl6 com3l impd ralrimivv sorpss
      wb syl5 sylibr ) CFGZDHZEHZIZVHVGIZJZEBAHZKZCLZABUAZUBZMDVPMVPFGVFVKDEVPV
      PVGVPLZVHVPLZNVGVOLZVHVOLZNZBVGKZCLZBVHKZCLZNZNZVFVKVQVSWCNZVTWENZWGVRVNW
      CAVGVOADSVMWBCVLVGBOPQVNWEAVHVOAESVMWDCVLVHBOPQWHWINWGVSWCVTWEULUCRVFWAWF
      VKWFVFWAVKWFVFWBWDIZWDWBIZJZWAVKTZVFWFWLCWBWDUDUEWKWJWMWAWKWJJZVKVSBWBKZV
      GUFZBWDKZVHUFZWNVKTVTVSVGBIWPDBUGVGBUHUIVTVHBIWREBUGVHBUHUIWPWRNZWKVIWJVJ
      WKWOWQIWSVIWDWBBUJWOVGWQVHUKUMWJWQWOIZWSVJWBWDBUJWRWPWTVJVCWQVHWOVGUKUNUM
      UORUPUQURUSUTVDVADEVPVBVE $.
  $}


$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
  ZF Set Theory - add the Axiom of Union
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Introduce the Axiom of Union
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d w x y z $.
    $( Axiom of Union.  An axiom of 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 .

       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.)  Use ~ ax-un instead.
       (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 imbi1i albii exbii
      cbvexvw mpbi ) BDEZDCEZFZDGZBAEZHZBIZAGUDACEZFZAGZUDHZBIZAGCABDJUFUKAUEUJ
      BUCUIUDUBUHDADAKTUDUAUGDABLDACMNROPQS $.

    $( 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 sepexi ) CDEDAEFDGCBBABCDHI $.

    $( The Axiom of Union using the standard abbreviation for union.  Given any
       set ` x ` , its union ` y ` exists.  (Contributed by NM, 4-Jun-2006.) $)
    uniex2 $p |- E. y y = U. x $=
      ( vz vw cv cuni wceq wex wel wcel wb wi wa ax-un eluni imbi1i albii exbii
      wal mpbir sepexi dfcleq ) BEZAEZFZGZBHCBIZCEZUEJZKCSZBHUICBBUIUGLZCSZBHCD
      IDAIMDHZUGLZCSZBHABCDNULUOBUKUNCUIUMUGDUHUDOPQRTUAUFUJBCUCUEUBRT $.
  $}

  ${
    $d x y $.
    $( The union of a setvar is a set.  (Contributed by BJ, 3-May-2021.)
       (Revised by BJ, 6-Apr-2024.) $)
    vuniex $p |- U. x e. _V $=
      ( vy cv cuni uniex2 issetri ) BACDABEF $.
  $}

  ${
    $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 vuniex vtoclg ) CDZEZFGAEZFGCABMA
      HNOFMAIJCKL $.
  $}

  ${
    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 $=
      ( cvv wcel cuni uniexg ax-mp ) ACDAECDBACFG $.
  $}

  ${
    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 $.
  $}

  ${
    $( The union of two sets is a set.  Corollary 5.8 of [TakeutiZaring] p. 16.
       (Contributed by NM, 18-Sep-2006.)  Prove ~ unexg first and then ~ unex
       and ~ unexb from it.  (Revised by BJ, 21-Jul-2025.) $)
    unexg $p |- ( ( A e. V /\ B e. W ) -> ( A u. B ) e. _V ) $=
      ( wcel wa cpr cuni cun cvv uniprg prex a1i uniexd eqeltrrd ) ACEBDEFZABGZ
      HABIJABCDKPQJQJEPABLMNO $.
  $}

  ${
    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 $=
      ( cvv wcel cun unexg mp2an ) AEFBEFABGEFCDABEEHI $.

    $( Obsolete version of ~ unex as of 21-Jul-2025.  (Contributed by NM,
       1-Jul-1994.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    unexOLD $p |- ( A u. B ) e. _V $=
      ( cpr cuni cun cvv unipr prex uniex eqeltrri ) ABEZFABGHABCDIMABJKL $.
  $}

  $( An unordered triple of classes exists.  (Contributed by NM,
     10-Apr-1994.) $)
  tpex $p |- { A , B , C } e. _V $=
    ( ctp cpr csn cun cvv df-tp prex snex unex eqeltri ) ABCDABEZCFZGHABCINOABJ
    CKLM $.

  $( 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 ) $=
    ( cvv wcel wa cun unexg wss ssun1 ssexg mpan ssun2 jca impbii ) ACDZBCDZEAB
    FZCDZABCCGROPAQHROABIAQCJKBQHRPBALBQCJKMN $.

  ${
    $d x y A $.  $d x y B $.
    $( Obsolete version of ~ unexb as of 21-Jul-2025.  (Contributed by NM,
       11-Jun-1998.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    unexbOLD $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 $.
  $}

  $( Obsolete version of ~ unexg as of 21-Jul-2025.  (Contributed by NM,
     18-Sep-2006.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  unexgOLD $p |- ( ( A e. V /\ B e. W ) -> ( A u. B ) e. _V ) $=
    ( wcel cvv cun elex wa unexb biimpi syl2an ) ACEAFEZBFEZABGFEZBDEACHBDHMNIO
    ABJKL $.

  $( The Cartesian product of two sets is a set.  Proposition 6.2 of
     [TakeutiZaring] p. 23.  See also ~ xpexgALT .  (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 $.
  $}

  $( The Cartesian product of three sets is a set.  (Contributed by Alexander
     van der Vekens, 21-Feb-2018.) $)
  3xpexg $p |- ( V e. W -> ( ( V X. V ) X. V ) e. _V ) $=
    ( cxp cvv wcel xpexg anidms mpancom ) AACZDEZABEZIACDEKJAABBFGIADBFH $.

  ${
    xpex.1 $e |- A e. _V $.
    xpex.2 $e |- B e. _V $.
    $( The Cartesian 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 $.

  ${
    $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 $.
    $( The class of all singletons is a proper class.  See also ~ pwnex .
       (Contributed by NM, 10-Oct-2008.)  (Proof shortened by Eric Schmidt,
       7-Dec-2008.)  (Proof shortened by BJ, 5-Dec-2021.) $)
    snnex $p |- { x | E. y x = { y } } e/ _V $=
      ( cv csn cvv wcel wa wceq wex cab wnel wn abnex df-nel sylibr vsnex vsnid
      wal pm3.2i mpg ) BCZDZEFZUAUBFZGZACUBHBIAJZEKZBUEBRUFEFLUGBAUBEMUFENOUCUD
      BPBQST $.
  $}

  ${
    $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 $.
  $}

  $( If the subtrahend of a class difference exists, then the minuend exists
     iff the difference exists.  (Contributed by NM, 12-Nov-2003.)  (Proof
     shortened by Andrew Salmon, 12-Aug-2011.) $)
  difex2 $p |- ( B e. C -> ( A e. _V <-> ( A \ B ) e. _V ) ) $=
    ( wcel cvv cdif difexg wa cun ssun2 uncom undif2 eqtr2i sseqtri unexg ssexg
    wss sylancr expcom impbid2 ) BCDZAEDZABFZEDZABEGUDUAUBUDUAHAUCBIZQUEEDUBABA
    IZUEABJUEBUCIUFUCBKBALMNUCBECOAUEEPRST $.

  $( If the difference of a class and a singleton is a set, the class itself is
     a set.  (Contributed by AV, 15-Jan-2019.) $)
  difsnexi $p |- ( ( N \ { K } ) e. _V -> N e. _V ) $=
    ( wcel csn cdif cvv wi wa simpr snex unexg sylancl wb difsnid eqcomd eleq1d
    cun adantr mpbird ex wn difsn biimpd pm2.61i ) ABCZBADZEZFCZBFCZGUEUHUIUEUH
    HZUIUGUFQZFCZUJUHUFFCULUEUHIAJUGUFFFKLUEUIULMUHUEBUKFUEUKBBANOPRSTUEUAZUHUI
    UMUGBFABUBPUCUD $.

  ${
    $d A x y v z $.  $d A x y u z $.
    $( Expression for double union that moves union into a class abstraction.
       (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 ceqsexv exancom 3bitr2i
      vuniex eleq2 bitri vex eqeq1 anbi1d exbidv elab abbii df-uni 3eqtr4i ) DE
      GZEHZCIZJZSZEKZDLDFGZFHZAHZBHZIZMZVJCJZSZBKZALZJZSZFKZDLVCIVPIVFVSDVFVAEB
      GZVMSZBKZSZEKZVMDHZVKJZSZBKZVSVEWCEVDWBVABVBCNOPWDVAWASZBKZEKZVMVAVTSZEKZ
      SZBKZWHWCWJEWJWCVAWABQRPWKWIEKBKVMWLSZEKBKWOWIEBUAWIWPBEWIWLVMSWPVAVTVMUB
      WLVMUCUDUEVMWLBEUFTWNWGBWMWFVMWFWMEWEVJNROPTWHVGVHVKMZVMSZSZFKZBKWSBKFKZV
      SWGWTBWGVMVGWQSZFKZSVMXBSZFKWTWFXCVMWFWQVGSFKXCVGWFFVKBUJVHVKWEUKUGWQVGFU
      HUDOVMXBFQXDWSFXDXBVMSWSVMXBUCVGWQVMUBULPUIPWSBFUAXAVGWRBKZSZFKVSVGWRFBUF
      XFVRFXEVQVGVQXEVOXEAVHFUMVIVHMZVNWRBXGVLWQVMVIVHVKUNUOUPUQROPULTTURDEVCUS
      DFVPUSUT $.
  $}

  $( Converse of the Axiom of Union.  Note that it does not require ~ ax-un .
     (Contributed by NM, 11-Nov-2003.) $)
  uniexr $p |- ( U. A e. V -> A e. _V ) $=
    ( cuni wcel cpw wss cvv pwuni pwexg ssexg sylancr ) ACZBDALEZFMGDAGDAHLBIAM
    GJK $.

  $( 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 uniexr impbii ) ABCADBCABEABFG $.

  $( Converse of the Axiom of Power Sets.  Note that it does not require
     ~ ax-pow .  (Contributed by NM, 11-Nov-2003.) $)
  pwexr $p |- ( ~P A e. V -> A e. _V ) $=
    ( cpw wcel cuni cvv unipw uniexg eqeltrrid ) ACZBDAJEFAGJBHI $.

  $( 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 ) $=
    ( cvv wcel cpw pwexg pwexr impbii ) ABCADBCABEABFG $.

  $( 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 $.

  ${
    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 $.

    $( Membership in the power class of a union.  (Contributed by NM,
       26-Mar-2007.) $)
    elpwun $p |- ( A e. ~P ( B u. C ) <-> ( A \ C ) e. ~P B ) $=
      ( cun cpw wcel cvv cdif elex difex2 ax-mp sylibr wss elpwg sseq2i ssundif
      wb uncom bitri difexg syl bitr4id bitrd pm5.21nii ) ABCEZFZGZAHGZACIZBFZG
      ZAUGJULUJHGZUIUJUKJCHGUIUMRDACHKLMUIUHAUFNZULAUFHOUIUNUJBNZULUNACBEZNUOUF
      UPABCSPACBQTUIUMULUORACHUAUJBHOUBUCUDUE $.
  $}

  $( Power classes are closed under union.  (Contributed by AV,
     27-Feb-2024.) $)
  pwuncl $p |- ( ( A e. ~P X /\ B e. ~P X ) -> ( A u. B ) e. ~P X ) $=
    ( cpw wcel wa cun cvv unexg wss elpwi unss biimpi syl2an elpwd ) ACDZEZBPEZ
    FABGZCHABPPIQACJZBCJZSCJZRACKBCKTUAFUBABCLMNO $.

  ${
    $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 cv cuni wceq wrex cpw ciun wss wcel sseq2 biimprcd com12 ssiun velpw
      reximdv eliun wa uniiun sseqtrrdi impbid1 rexbii bitri 3bitr4g eqrdv ssid
      uniex elpw eleq2 bitr3id mpbii sylib elssuni elpwi anim12i eqss sylibr ex
      reximia syl impbii ) AEZBFZGZABHZVEIZABVDIZJZGZVGDVHVJVGDEZVEKZVLVDKZABHZ
      VLVHLVLVJLZVGVMVOVMVGVOVMVFVNABVFVNVMVDVEVLMNROVOVLABVDJVEABVDVLPABUAUBUC
      DVEQVPVLVILZABHVOAVLBVISVQVNABDVDQUDUEUFUGVKVEVILZABHZVGVKVEVJLZVSVKVEVEK
      ZVTVEUHWAVEVHLVKVTVEVEBCUIUJVHVJVEUKULUMAVEBVISUNVRVFABVDBLZVRVFWBVRTVDVE
      KZVEVDKZTVFWBWCVRWDVDBUOVEVDUPUQVDVEURUSUTVAVBVC $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.  $d x y D $.  $d x y R $.
    $( A well-founded relation has no 3-cycle loops.  Special case of
       Proposition 6.23 of [TakeutiZaring] p. 30.  (Contributed by NM,
       10-Apr-1994.)  (Revised by Mario Carneiro, 22-Jun-2015.) $)
    fr3nr $p |- ( ( R Fr A /\ ( B e. A /\ C e. A /\ D e. A ) ) ->
                -. ( B R C /\ C R D /\ D R B ) ) $=
      ( vy vx wcel w3a wbr wn wral wss csn wb wceq breq2 notbid ralbidv syl wfr
      wa w3o cv ctp wrex cvv c0 wne a1i simpl cpr cun df-tp simpr1 simpr2 prssd
      tpex simpr3 snssd unssd eqsstrid tpnzd fri syl22anc rextpg adantl snsstp3
      mpbid wi snssg mpbiri breq1 rspcv snsstp1 snsstp2 3orim123d 3ianor sylibr
      mpd 3anrot sylnib ) AEUAZBAHZCAHZDAHZIZUBZDBEJZBCEJZCDEJZIZWJWKWIIWHWIKZW
      JKZWKKZUCZWLKWHFUDZBEJZKZFBCDUEZLZWQCEJZKZFWTLZWQDEJZKZFWTLZUCZWPWHWQGUDZ
      EJZKZFWTLZGWTUFZXHWHWTUGHZWCWTAMWTUHUIXMXNWHBCDURUJWCWGUKWHWTBCULZDNZUMAB
      CDUNWHXOXPAWHBCAWCWDWEWFUOZWCWDWEWFUPZUQWHDAWCWDWEWFUSZUTVAVBWHBCDAXQVCGF
      AWTUGEVDVEWGXMXHOWCXLXAXDXGGBCDAAAXIBPZXKWSFWTXTXJWRXIBWQEQRSXICPZXKXCFWT
      YAXJXBXICWQEQRSXIDPZXKXFFWTYBXJXEXIDWQEQRSVFVGVIWHXAWMXDWNXGWOWHDWTHZXAWM
      VJWHYCXPWTMZBCDVHWHWFYCYDOXSDWTAVKTVLWSWMFDWTWQDPWRWIWQDBEVMRVNTWHBWTHZXD
      WNVJWHYEBNWTMZBCDVOWHWDYEYFOXQBWTAVKTVLXCWNFBWTWQBPXBWJWQBCEVMRVNTWHCWTHZ
      XGWOVJWHYGCNWTMZBCDVPWHWEYGYHOXRCWTAVKTVLXFWOFCWTWQCPXEWKWQCDEVMRVNTVQVTW
      IWJWKVRVSWIWJWKWAWB $.
  $}

  $( A well-founded class contains no 3-cycle loops.  (Contributed by NM,
     19-Apr-1994.)  (Revised by Mario Carneiro, 22-Jun-2015.) $)
  epne3 $p |- ( ( _E Fr A /\ ( B e. A /\ C e. A /\ D e. A ) ) ->
                -. ( B e. C /\ C e. D /\ D e. B ) ) $=
    ( cep wfr wcel w3a wa wbr fr3nr wb epelg 3ad2ant2 3ad2ant3 3anbi123d adantl
    3ad2ant1 mtbid ) AEFZBAGZCAGZDAGZHZIBCEJZCDEJZDBEJZHZBCGZCDGZDBGZHZABCDEKUD
    UHULLTUDUEUIUFUJUGUKUBUAUEUILUCBCAMNUCUAUFUJLUBCDAMOUAUBUGUKLUCDBAMRPQS $.

  ${
    $d x y z R $.  $d x y z A $.
    $( Alternate definition of well-ordering.  Definition 6.24(2) of
       [TakeutiZaring] p. 30.  (Contributed by NM, 16-Mar-1997.)  (Proof
       shortened by Andrew Salmon, 12-Aug-2011.) $)
    dfwe2 $p |- ( R We A <-> ( R Fr A /\ A. x e. A A. y e. A
                ( x R y \/ x = y \/ y R x ) ) ) $=
      ( vz wa cv wbr weq w3o wral wn wi wcel w3a wal breq2 r3al 3imtr4g ralbii
      wwe wfr wor df-we df-so simpr ax-1 fr2nr 3adantr3 anbi2d notbid syl5ibcom
      wpo pm2.21 syl6 fr3nr df-3an biimpri ancoms nsyl pm2.21d expd 3jaod frirr
      a1i 3ad2antr1 jctild ex a2d alimdv 2alimdv equequ2 breq1 ralidmw cbvralvw
      3orbi123d bitr3i df-po ancrd impbid2 bitrid pm5.32i bitri ) CDUACDUBZCDUC
      ZFWDAGZBGZDHZABIZWGWFDHZJZBCKZACKZFCDUDWDWEWMWECDUMZWMFZWDWMABCDUEWDWOWMW
      NWMUFWDWMWNWDWFEGZDHZAEIZWPWFDHZJZECKZBCKZACKZWFWFDHLZWHWGWPDHZFZWQMZFZEC
      KBCKACKZWMWNWDWFCNZWGCNZWPCNZOZWTMZEPZBPAPXMXHMZEPZBPAPXCXIWDXOXQABWDXNXP
      EWDXMWTXHWDXMWTXHMWDXMFZWTXGXDXRWQXGWRWSWQXGMXRWQXFUGVEXRWRXFLZXGXRWHWJFZ
      LZWRXSWDXJXKYAXLCWFWGDUHUIWRXTXFWRWJXEWHWFWPWGDQUJUKULXFWQUNUOXRWSXFWQXRW
      SXFFZWQXRWHXEWSOZYBCWFWGWPDUPXFWSYCYCXFWSFWHXEWSUQURUSUTVAVBVCWDXKXJXDXLC
      WFDVDVFVGVHVIVJVKWTABECCCRXHABECCCRSWLXBACWLWLBCKXBWKWTBECBEIWHWQWIWRWJWS
      WGWPWFDQBEAVLWGWPWFDVMVPZVNWLXABCWKWTBECYDVOTVQTABECDVRSVSVTWAWBWC $.
    $( $j usage 'dfwe2' avoids 'ax-10' 'ax-11' 'ax-12'; $)
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Ordinals (continued)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x y z $.
    $( The membership relation well-orders the class of ordinal numbers.  This
       proof does not require the axiom of regularity.  Proposition 4.8(g) of
       [Mendelson] p. 244.  For a shorter proof requiring ~ ax-un , see
       ~ epweonALT .  (Contributed by NM, 1-Nov-2003.)  Avoid ~ ax-un .
       (Revised by BTernaryTau, 30-Nov-2024.) $)
    epweon $p |- _E We On $=
      ( vx vy vz con0 cep wwe wfr wor cv wbr w3o wral wn wa wcel wel word eloni
      epel mpbir2an onfr wpo weq df-po ordirr syl sylnibr ontr1 anbi12i 3imtr4g
      anim12i ralrimiva ralrimivw mprgbir ordtri3or biid 3orbi123i sylibr rgen2
      wi syl2an df-so df-we ) DEFDEGDEHZUAVDDEUBZAIZBIZEJZABUCZVGVFEJZKZBDLADLV
      EVFVFEJZMZVHVGCIZEJZNZVFVNEJZUTZNZCDLZBDLADABCDEUDVFDOZVTBDWAVSCDWAVMVNDO
      ZVRWAAAPZVLWAVFQZWCMVFRZVFUEUFAVFSUGWBABPZBCPZNACPVPVQVFVGVNUHVHWFVOWGBVF
      SZCVGSUICVFSUJUKULUMUNVKABDDWAWDVGQZVKVGDOWEVGRWDWINWFVIBAPZKVKVFVGUOVHWF
      VIVIVJWJWHVIUPAVGSUQURVAUSABDEVBTDEVCT $.
    $( $j usage 'epweon' avoids 'ax-un' 'ax-reg'; $)
  $}

  ${
    $d x y $.
    $( Alternate proof of ~ epweon , shorter but requiring ~ ax-un .
       (Contributed by NM, 1-Nov-2003.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    epweonALT $p |- _E We On $=
      ( vx vy con0 cep wwe wfr cv wbr weq w3o wral onfr wcel eloni wa ordtri3or
      word wel epel biid 3orbi123i sylibr syl2an rgen2 dfwe2 mpbir2an ) CDECDFA
      GZBGZDHZABIZUHUGDHZJZBCKACKLULABCCUGCMUGQZUHQZULUHCMUGNUHNUMUNOABRZUJBARZ
      JULUGUHPUIUOUJUJUKUPBUGSUJTAUHSUAUBUCUDABCDUEUF $.
    $( $j usage 'epweonALT' avoids 'ax-reg'; $)
  $}

  $( 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 $=
    ( con0 word wtr cep wwe tron epweon df-ord mpbir2an ) ABACADEFGAHI $.
  $( $j usage 'ordon' avoids 'ax-reg'; $)

  $( No set contains all ordinal numbers.  Proposition 7.13 of [TakeutiZaring]
     p. 38, but without using the Axiom of Regularity.  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 $.
  $( $j usage 'onprc' avoids 'ax-reg'; $)

  ${
    $d x y A $.
    $( The union of a class of ordinal numbers is ordinal.  Proposition 7.19 of
       [TakeutiZaring] p. 40.  Lemma 2.7 of [Schloeder] p. 4.  (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.  Lemma 2.7 of [Schloeder] p. 4.  (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 $.
  $}

  $( A way to express the ordinal property of a class in terms of the class of
     ordinal numbers.  Corollary 7.14 of [TakeutiZaring] p. 38 and its
     converse.  (Contributed by NM, 1-Jun-2003.) $)
  ordeleqon $p |- ( Ord A <-> ( A e. On \/ A = On ) ) $=
    ( word con0 wcel wceq wo cvv onprc elex mto w3o ordon ordtri3or mpan2 sylib
    df-3or ord mt3i eloni ordeq mpbiri jaoi impbii ) ABZACDZACEZFZUDUGCADZUHCGD
    HCAIJUDUGUHUDUEUFUHKZUGUHFUDCBZUILACMNUEUFUHPOQRUEUDUFASUFUDUJLACTUAUBUC $.

  $( 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.)  (Proof shortened by Andrew Salmon, 12-Aug-2011.) $)
  ordsson $p |- ( Ord A -> A C_ On ) $=
    ( word con0 wss ordon wa wcel wceq ordeleqon biimpi adantr ordsseleq mpbird
    wo mpan2 ) ABZCBZACDZEPQFRACGACHNZPSQPSAIJKACLMO $.

  $( A class is ordinal iff it is a subclass of ` On ` and transitive.
     (Contributed by Scott Fenton, 21-Nov-2021.) $)
  dford5 $p |- ( Ord A <-> ( A C_ On /\ Tr A ) ) $=
    ( word con0 wss wtr wa ordsson ordtr jca cep wwe epweon wess df-ord biimpri
    mpi ancoms sylan impbii ) ABZACDZAEZFTUAUBAGAHIUAAJKZUBTUACJKUCLACJMPUBUCTT
    UBUCFANOQRS $.

  $( 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 predecessor of an ordinal under ` _E ` and ` On ` is itself.
     (Contributed by Scott Fenton, 27-Mar-2011.)  (Proof shortened by BJ,
     16-Oct-2024.) $)
  predon $p |- ( A e. On -> Pred ( _E , On , A ) = A ) $=
    ( con0 wtr wcel cep cpred wceq tron trpred mpan ) BCABDBEAFAGHBAIJ $.

  $( Two ways of saying a class of ordinals is unbounded.  (Contributed by
     Mario Carneiro, 8-Jun-2013.) $)
  ssonprc $p |- ( A C_ On -> ( A e/ _V <-> U. A = On ) ) $=
    ( cvv wnel wcel wn con0 wss cuni wceq df-nel word ssorduni ordeleqon orcomd
    wo sylib ord uniexr syl6 con1d onprc uniexg eleq1 imbitrid impbid1 bitrid
    mtoi ) ABCABDZEZAFGZAHZFIZABJUJUIULUJULUHUJULEUKFDZUHUJULUMUJUMULUJUKKUMULO
    ALUKMPNQAFRSTULUHFBDZUAUHUKBDULUNABUBUKFBUCUDUGUEUF $.

  $( 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 z A $.
    $( The intersection (infimum) of a nonempty class of ordinal numbers
       belongs to the class.  Compare Exercise 4 of [TakeutiZaring] p. 45.
       (Contributed by NM, 31-Jan-1997.) $)
    onint $p |- ( ( A C_ On /\ A =/= (/) ) -> |^| A e. A ) $=
      ( vx vy vz con0 wss c0 wne cint wcel wa cv cin wceq wrex wi wel ssel wral
      exp32 ordon tz7.5 mp3an1 imdistani wn ontri1 biimtrrdi sylan9 com4r imp31
      word ex ralimdva disj vex elint2 sylan2 com4l imp32 ssrdv intss1 ad2antrl
      3imtr4g eqssd eleq1d biimpd com34 pm2.43d rexlimdv syl5 anabsi5 ) AEFZAGH
      ZAIZAJZVLVMKABLZMGNZBAOZVLVOEUKVLVMVRUABEAUBUCVLVQVOBAVLVPAJZVQVOPVLVSVQV
      SVOVLVSVQVSVOPVLVSVQKKZVSVOVTVPVNAVTVPVNVTCVPVNVLVSVQCBQZCLZVNJZPWAVLVSVQ
      WCWAVLVSVQWCPZVLVSKWAVLVPEJZKZWDVLVSWEAEVPRUDWAWFKZDBQUEZDASCDQZDASVQWCWG
      WHWIDAWAWFDLZAJZWHWIPWFWKWHWAWIVLWKWJEJZWEWHWAWIPZPZAEWJRWEWLWNWEWLKWHVPW
      JFWMVPWJUFVPWJWBRUGULUHUIUJUMDAVPUNDWBACUOUPVCUQTURUSUTVSVNVPFVLVQVPAVAVB
      VDVEVFTVGVHVIVJVK $.
  $}

  $( The intersection of a class of ordinal numbers is zero iff the class
     contains zero.  (Contributed by NM, 24-Apr-2004.) $)
  onint0 $p |- ( A C_ On -> ( |^| A = (/) <-> (/) e. A ) ) $=
    ( con0 wss cint c0 wceq wcel wa wne cvv 0ex eleq1 mpbiri intex sylibr onint
    sylan2 wb adantl mpbid ex int0el impbid1 ) ABCZADZEFZEAGZUDUFUGUDUFHUEAGZUG
    UFUDAEIZUHUFUEJGZUIUFUJEJGKUEEJLMANOAPQUFUHUGRUDUEEALSTUAAUBUC $.

  ${
    $d x y A $.
    $( A nonempty class of ordinal numbers has the smallest member.  Exercise 9
       of [TakeutiZaring] p. 40.  (Contributed by NM, 3-Oct-2003.) $)
    onssmin $p |- ( ( A C_ On /\ A =/= (/) ) -> E. x e. A A. y e. A x C_ y ) $=
      ( con0 wss c0 wne wa cint wcel cv wral wrex onint rgen wceq sseq1 ralbidv
      intss1 rspcev sylancl ) CDECFGHCIZCJUBBKZEZBCLZAKZUCEZBCLZACMCNUDBCUCCSOU
      HUEAUBCUFUBPUGUDBCUFUBUCQRTUA $.
  $}

  $( If a property is true for some ordinal number, it is true for a minimal
     ordinal number.  This version uses explicit substitution.  Theorem Schema
     62 of [Suppes] p. 228.  (Contributed by NM, 29-Sep-2003.) $)
  onminesb $p |- ( E. x e. On ph -> [. |^| { x e. On | ph } / x ]. ph ) $=
    ( con0 wrex crab cint wcel wsbc wne rabn0 wss ssrab2 onint mpan sylbir nfcv
    c0 elrabsf simprbi syl ) ABCDZABCEZFZUBGZABUCHZUAUBQIZUDABCJUBCKUFUDABCLUBM
    NOUDUCCGUEABUCCBCPRST $.

  ${
    onminsb.1 $e |- F/ x ps $.
    onminsb.2 $e |- ( x = |^| { x e. On | ph } -> ( ph <-> ps ) ) $.
    $( If a property is true for some ordinal number, it is true for a minimal
       ordinal number.  This version uses implicit substitution.  Theorem
       Schema 62 of [Suppes] p. 228.  (Contributed by NM, 3-Oct-2003.) $)
    onminsb $p |- ( E. x e. On ph -> ps ) $=
      ( con0 wrex crab cint wcel wne rabn0 wss ssrab2 onint sylbir nfrab1 nfint
      c0 mpan nfcv elrabf simprbi syl ) ACFGZACFHZIZUFJZBUEUFSKZUHACFLUFFMUIUHA
      CFNUFOTPUHUGFJBABCUGFCUFACFQRCFUADEUBUCUD $.
  $}

  $( The intersection of a nonempty collection of ordinal numbers is an ordinal
     number.  Compare Exercise 6 of [TakeutiZaring] p. 44.  (Contributed by NM,
     29-Jan-1997.) $)
  oninton $p |- ( ( A C_ On /\ A =/= (/) ) -> |^| A e. On ) $=
    ( con0 wss c0 wne cint wcel onint ex ssel syld imp ) ABCZADEZAFZBGZMNOAGZPM
    NQAHIABOJKL $.

  $( The intersection of a class of ordinal numbers exists iff it is an ordinal
     number.  (Contributed by NM, 6-Nov-2003.) $)
  onintrab $p |- ( |^| { x e. On | ph } e. _V <->
                 |^| { x e. On | ph } e. On ) $=
    ( con0 crab cint cvv wcel c0 wne wss ssrab2 oninton mpan sylbir elex impbii
    intex ) ABCDZEZFGZSCGZTRHIZUARQRCJUBUAABCKRLMNSCOP $.

  $( An existence condition equivalent to an intersection's being an ordinal
     number.  (Contributed by NM, 6-Nov-2003.) $)
  onintrab2 $p |- ( E. x e. On ph <-> |^| { x e. On | ph } e. On ) $=
    ( con0 wrex crab cint cvv wcel intexrab onintrab bitri ) ABCDABCEFZGHLCHABC
    IABJK $.

  $( No member of a set of ordinal numbers belongs to its minimum.
     (Contributed by NM, 2-Feb-1997.) $)
  onnmin $p |- ( ( A C_ On /\ B e. A ) -> -. B e. |^| A ) $=
    ( con0 wss wcel wa cint wn intss1 adantl wb c0 wne ne0i sylan2 ssel2 ontri1
    oninton syl2anc mpbid ) ACDZBAEZFZAGZBDZBUDEHZUBUEUABAIJUCUDCEZBCEUEUFKUBUA
    ALMUGABNAROACBPUDBQST $.

  ${
    $d x A $.  $d x ps $.
    onnminsb.1 $e |- ( x = A -> ( ph <-> ps ) ) $.
    $( An ordinal number smaller than the minimum of a set of ordinal numbers
       does not have the property determining that set. ` ps ` is the wff
       resulting from the substitution of ` A ` for ` x ` in wff ` ph ` .
       (Contributed by NM, 9-Nov-2003.) $)
    onnminsb $p |- ( A e. On -> ( A e. |^| { x e. On | ph } -> -. ps ) ) $=
      ( con0 wcel crab cint wn wa elrab wss ssrab2 onnmin mpan sylbir ex con2d
      ) DFGZBDACFHZIGZTBUBJZTBKDUAGZUCABCDFELUAFMUDUCACFNUADOPQRS $.
  $}

  ${
    $d x A $.  $d x B $.
    $( A way to show that an ordinal number equals the minimum of a nonempty
       collection of ordinal numbers: it must be in the collection, and it must
       not be larger than any member of the collection.  (Contributed by NM,
       14-Nov-2003.) $)
    oneqmin $p |- ( ( B C_ On /\ B =/= (/) ) -> ( A = |^| B <->
                  ( A e. B /\ A. x e. A -. x e. B ) ) ) $=
      ( con0 wss c0 wne wa cint wceq wcel cv wn onint eleq1 syl5ibrcom wi eleq2
      wral adantr biimpd onnmin ex con2d syl9r ralrimdv jcad oneqmini impbid )
      CDEZCFGZHZBCIZJZBCKZALZCKZMZABSZHZULUNUOUSULUOUNUMCKCNBUMCOPUJUNUSQUKUJUN
      URABUNUPBKZUPUMKZUJURUNVAVBBUMUPRUAUJUQVBUJUQVBMCUPUBUCUDUEUFTUGUJUTUNQUK
      ABCUHTUI $.
  $}

  ${
    $d x y A $.
    uniordint.1 $e |- A e. _V $.
    $( The union of a set of ordinals is equal to the intersection of its upper
       bounds.  Problem 2.5(ii) of [BellMachover] p. 471.  (Contributed by NM,
       20-Sep-2003.) $)
    uniordint $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 unissb rabbii
      inteqi eqtr3di syl ) CEFCGZEHZUABIAIZFBCJZAEKZLZMCDNUBUAUCFZAEKZLUAUFAUAE
      OUHUEUGUDAEBCUCPQRST $.
  $}

  ${
    $d x y z $.  $d y z ph $.  $d x z ps $.
    onminex.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
    $( If a wff is true for an ordinal number, then there is the smallest
       ordinal number for which it is true.  (Contributed by NM, 2-Feb-1997.)
       (Proof shortened by Mario Carneiro, 20-Nov-2016.) $)
    onminex $p |- ( E. x e. On ph -> E. x e. On ( ph /\ A. y e. x -. ps ) ) $=
      ( vz con0 wrex wsb wn cv wral wa crab cint wcel wss raleq anbi12d nfv wne
      wsbc c0 ssrab2 rabn0 biimpri oninton sylancr onminesb onss sseld onnminsb
      syl syli ralrimiv wceq dfsbcq2 rspcev syl12anc nfs1v nfan sbequ12 cbvrexw
      weq sylibr ) ACGHZACFIZBJZDFKZLZMZFGHZAVHDCKZLZMZCGHVFACGNZOZGPZACVQUBZVH
      DVQLZVLVFVPGQVPUCUAZVRACGUDWAVFACGUEUFVPUGUHZACUIVFVHDVQDKZVQPVFWCGPVHVFV
      QGWCVFVRVQGQWBVQUJUMUKABCWCEULUNUOVKVSVTMFVQGVIVQUPVGVSVJVTACFVQUQVHDVIVQ
      RSURUSVOVKCFGVOFTVGVJCACFUTVJCTVACFVDAVGVNVJACFVBVHDVMVIRSVCVE $.
  $}

  $( 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 $.

  $( 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 snex biantru df-suc eleq1i 3bitr4i ) ABCZA
    DZBCZEAOFZBCNAGZBCAOHPNAIJRQBAKLM $.

  $( 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 minimum of a class of ordinal numbers is less than the minimum of
       that class with its minimum removed.  (Contributed by NM,
       20-Nov-2003.) $)
    onmindif2 $p |- ( ( A C_ On /\ A =/= (/) ) ->
                   |^| A e. |^| ( A \ { |^| A } ) ) $=
      ( vx con0 wss c0 wne wa cint cdif wcel cv wral eldifsn wn wceq wo adantlr
      csn wb cvv onnmin oninton ssel2 ontri1 onsseleq bitr3d syl2an2r mpbid ord
      eqcom imbitrdi necon1ad expimpd biimtrid intex elintg sylbi adantl mpbird
      ralrimiv ) ACDZAEFZGZAHZAVDRIZHJZVDBKZJZBVELZVCVHBVEVGVEJVGAJZVGVDFZGVCVH
      VGAVDMVCVJVKVHVCVJGZVHVGVDVLVHNVDVGOZVGVDOVLVHVMVLVGVDJNZVHVMPZVAVJVNVBAV
      GUAQVCVDCJZVJVGCJZVNVOSAUBVAVJVQVBACVGUCQVPVQGVDVGDVNVOVDVGUDVDVGUEUFUGUH
      UIVDVGUJUKULUMUNUTVBVFVISZVAVBVDTJVRAUOBVDVETUPUQURUS $.
  $}

  $( The successor of an ordinal class is an ordinal class.  Remark 1.5 of
     [Schloeder] p. 1.  (Contributed by NM, 6-Jun-1994.)  Extract and adapt
     from a subproof of ~ onsuc .  (Revised by BTernaryTau, 6-Jan-2025.)
     (Proof shortened by BJ, 11-Jan-2025.) $)
  ordsuci $p |- ( Ord A -> Ord suc A ) $=
    ( word csuc wtr con0 wss ordtr suctr syl csn cun df-suc ordsson cvv wcel wa
    elon2 snssi sylbir c0 wceq snprc biimpi 0ss eqsstrdi adantl pm2.61dan unssd
    wn eqsstrid ordon a1i trssord syl3anc ) ABZACZDZUPEFEBZUPBUOADUQAGAHIUOUPAA
    JZKEALUOAUSEAMUOANOZUSEFZUOUTPAEOVAAQAERSUTUIZVAUOVBUSTEVBUSTUAAUBUCEUDUEUF
    UGUHUJURUOUKULUPEUMUN $.
  $( $j usage 'ordsuci' avoids 'ax-un'; $)

  $( If the successor of an ordinal number exists, it is an ordinal number.
     This variation of ~ onsuc does not require ~ ax-un .  (Contributed by
     BTernaryTau, 30-Nov-2024.)  (Proof shortened by BJ, 11-Jan-2025.) $)
  sucexeloni $p |- ( ( A e. On /\ suc A e. V ) -> suc A e. On ) $=
    ( con0 wcel csuc word cvv eloni ordsuci syl elex elong biimparc syl2an ) AC
    DZAEZFZPGDZPCDZPBDOAFQAHAIJPBKRSQPGLMN $.
  $( $j usage 'sucexeloni' avoids 'ax-un'; $)

  ${
    $d A x $.
    $( Obsolete version of ~ sucexeloni as of 6-Jan-2025.  (Contributed by
       BTernaryTau, 30-Nov-2024.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    sucexeloniOLD $p |- ( ( A e. On /\ suc A e. V ) -> suc A e. On ) $=
      ( vx con0 wcel csuc wa word wtr wss cv wral csn wo onelss wi velsn eqimss
      wceq sylbi a1i orim12d cun df-suc eleq2i elun oridm 3imtr3g sssucid sstr2
      bitr2i syl6mpi ralrimiv dftr3 sylibr onss snssi unssd eqsstrid ordon 3exp
      trssord mpii sylc adantr wb elong adantl mpbird ) ADEZAFZBEZGVKDEZVKHZVJV
      NVLVJVKIZVKDJZVNVJCKZVKJZCVKLVOVJVRCVKVJVQVKEZVQAJZAVKJVRVJVQAEZVQAMZEZNZ
      VTVTNVSVTVJWAVTWCVTAVQOWCVTPVJWCVQASVTCAQVQARTUAUBVSVQAWBUCZEWDVKWEVQAUDZ
      UEVQAWBUFUKVTUGUHAUIVQAVKUJULUMCVKUNUOVJVKWEDWFVJAWBDAUPADUQURUSVOVPDHZVN
      UTVOVPWGVNVKDVBVAVCVDVEVLVMVNVFVJVKBVGVHVI $.
  $}

  $( 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.  Remark 1.5 of [Schloeder] p. 1.  (Contributed by
     NM, 6-Jun-1994.)  (Proof shortened by BTernaryTau, 30-Nov-2024.) $)
  onsuc $p |- ( A e. On -> suc A e. On ) $=
    ( con0 wcel csuc cvv sucexg sucexeloni mpdan ) ABCADZECIBCABFAEGH $.

  ${
    $d x A $.
    $( Obsolete version of ~ onsuc as of 30-Nov-2024.  (Contributed by NM,
       6-Jun-1994.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    suceloniOLD $p |- ( A e. On -> suc A e. On ) $=
      ( vx con0 wcel csuc word wtr wss cv wral csn wo onelss velsn eqimss sylbi
      wi wceq a1i cvv orim12d cun df-suc eleq2i elun bitr2i oridm 3imtr3g sstr2
      sssucid syl6mpi ralrimiv dftr3 sylibr onss snssi unssd ordon trssord 3exp
      eqsstrid mpii sylc wb sucexg elong syl mpbird ) ACDZAEZCDZVJFZVIVJGZVJCHZ
      VLVIBIZVJHZBVJJVMVIVPBVJVIVOVJDZVOAHZAVJHVPVIVOADZVOAKZDZLZVRVRLVQVRVIVSV
      RWAVRAVOMWAVRQVIWAVOARVRBANVOAOPSUAVQVOAVTUBZDWBVJWCVOAUCZUDVOAVTUEUFVRUG
      UHAUJVOAVJUIUKULBVJUMUNVIVJWCCWDVIAVTCAUOACUPUQVAVMVNCFZVLURVMVNWEVLVJCUS
      UTVBVCVIVJTDVKVLVDACVEVJTVFVGVH $.
  $}

  $( A class is ordinal if and only if its successor is ordinal.  (Contributed
     by NM, 3-Apr-1995.)  Avoid ~ ax-un .  (Revised by BTernaryTau,
     6-Jan-2025.) $)
  ordsuc $p |- ( Ord A <-> Ord suc A ) $=
    ( word csuc ordsuci cvv wcel wi sucidg ordelord ex syl5com wn sucprc eqcomd
    wceq wb ordeq syl biimprd pm2.61i impbii ) ABZACZBZADAEFZUDUBGUEAUCFZUDUBAE
    HUDUFUBUCAIJKUELZUBUDUGAUCOUBUDPUGUCAAMNAUCQRSTUA $.
  $( $j usage 'ordsuc' avoids 'ax-un'; $)

  $( Obsolete version of ~ ordsuc as of 6-Jan-2025.  (Contributed by NM,
     3-Apr-1995.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  ordsucOLD $p |- ( Ord A <-> Ord suc A ) $=
    ( cvv wcel word csuc wb con0 elong onsuc eloni biimtrrdi sucidg ordelord ex
    syl syl5com impbid wn wceq sucprc eqcomd ordeq pm2.61i ) ABCZADZAEZDZFZUDUE
    UGUDUEAGCZUGABHUIUFGCUGAIUFJOKUDAUFCZUGUEABLUGUJUEUFAMNPQUDRZAUFSUHUKUFAATU
    AAUFUBOUC $.

  ${
    $d x A $.
    $( The collection of ordinals in the power class of an ordinal is its
       successor.  (Contributed by NM, 30-Jan-2005.) $)
    ordpwsuc $p |- ( Ord A -> ( ~P A i^i On ) = suc A ) $=
      ( vx word cpw con0 cin csuc cv wcel wss elin velpw anbi2ci bitri ordsssuc
      wa wb expcom pm5.32d simpr wi ordsuc ordelon ex sylbi ancrd impbid2 bitrd
      bitrid eqrdv ) ACZBADZEFZAGZBHZUMIZUOEIZUOAJZPZUKUOUNIZUPUOULIZUQPUSUOULE
      KVAURUQBALMNUKUSUQUTPZUTUKUQURUTUQUKURUTQUOAORSUKVBUTUQUTTUKUTUQUKUNCZUTU
      QUAAUBVCUTUQUNUOUCUDUEUFUGUHUIUJ $.

    $( The collection of ordinal numbers in the power set of an ordinal number
       is its successor.  (Contributed by NM, 19-Oct-2004.) $)
    onpwsuc $p |- ( A e. On -> ( ~P A i^i On ) = suc A ) $=
      ( con0 wcel word cpw cin csuc wceq eloni ordpwsuc syl ) ABCADAEBFAGHAIAJK
      $.
  $}

  $( 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 ) $=
    ( word cvv wcel wa csuc con0 ordsuc sucexb anbi12i elon2 3bitr4i ) ABZACDZE
    AFZBZOCDZEAGDOGDMPNQAHAIJAKOKL $.

  $( The successor of an element of an ordinal class is a subset of it.  Lemma
     1.14 of [Schloeder] p. 2.  (Contributed by NM, 21-Jun-1998.) $)
  ordsucss $p |- ( Ord B -> ( A e. B -> suc A C_ B ) ) $=
    ( word wcel csuc wss wi wa ordelord wn ordnbtwn imnan sylibr adantr ordtri1
    wb ordsuc sylanb sylibrd pm2.43b sylan exp31 ) BCZABDZAEZBFZUDUCUDUFGZUCUDU
    CUGUCUDHACZUCUGBAIUHUCHUDBUEDZJZUFUHUDUJGZUCUHUDUIHJUKABKUDUILMNUHUECUCUFUJ
    PAQUEBORSUAUBTT $.

  $( An ordinal number is a proper subset of its successor.  (Contributed by
     Stefan O'Rear, 18-Nov-2014.) $)
  onpsssuc $p |- ( A e. On -> A C. suc A ) $=
    ( con0 wcel csuc wpss sucidg word eloni ordsuc sylib ordelpss syl2anc mpbid
    wb ) ABCZAADZCZAPEZABFOAGZPGZQRNAHZOSTUAAIJAPKLM $.

  $( 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 $.

  ${
    $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 successors.  Generalization of Exercise 9 of
     [TakeutiZaring] p. 42.  (Contributed by NM, 22-Jun-1998.)  (Proof
     shortened by Andrew Salmon, 12-Aug-2011.) $)
  ordsucelsuc $p |- ( Ord B -> ( A e. B <-> suc A e. suc B ) ) $=
    ( word wcel csuc wa simpl ordelord jca ordsuc sylibr sylanb cvv wb wss wceq
    wi wo adantr elex ancoms adantl ordsucss sucssel impbid sucexb elsucg sylbi
    ordsseleq ad2antrl 3bitr4d ex wn pm5.21ni a1d pm2.61i pm5.21nd ) BCZABDZAEZ
    BEZDZURACZFZURUSFURVCURUSGBAHIURVBFURVCURVBGURVACZVBVCBJVEVBFUTCZVCVAUTHAJZ
    KLIAMDZVDUSVBNZQVHVDVIVHVDFZUTBOZUTBDUTBPRZUSVBVDVKVLNZVHVCURVMVCVFURVMVGUT
    BUILUAUBVJUSVKURUSVKQVHVCABUCUJVHVKUSQVDABMUDSUEVHVBVLNZVDVHUTMDZVNAUFZUTBM
    UGUHSUKULVHUMVIVDUSVHVBABTVBVOVHUTVATVPKUNUOUPUQ $.

  $( The subclass relationship between two ordinal classes is inherited by
     their successors.  (Contributed by NM, 4-Oct-2003.) $)
  ordsucsssuc $p |- ( ( Ord A /\ Ord B ) -> ( A C_ B <-> suc A C_ suc B ) ) $=
    ( word wa wcel wn csuc wss ordsucelsuc notbid adantr ordtri1 ordsuc syl2anb
    wb 3bitr4d ) ACZBCZDBAEZFZBGZAGZEZFZABHUBUAHZQTUDORQSUCBAIJKABLQUBCUACUEUDO
    RAMBMUBUALNP $.

  $( Given an element ` A ` of the union of an ordinal ` B ` , ` suc A ` is an
     element of ` B ` itself.  (Contributed by Scott Fenton, 28-Mar-2012.)
     (Proof shortened by Mario Carneiro, 29-May-2015.) $)
  ordsucuniel $p |- ( Ord B -> ( A e. U. B <-> suc A e. B ) ) $=
    ( word cuni wcel csuc wi orduni ordelord ex syl wa ordsuc sylibr wb wn con0
    wss sylan ordtri1 ordsson ordunisssuc sylan2b 3bitr3d con4bid pm5.21ndd ) B
    CZACZABDZEZAFZBEZUGUICZUJUHGBHZUMUJUHUIAIJKUGULUHUGULLUKCZUHBUKIAMZNJUGUHUJ
    ULOUGUHLZUJULUQUIARZBUKRZUJPZULPZUGBQRUHURUSOBUABAUBSUGUMUHURUTOUNUIATSUHUG
    UOUSVAOUPBUKTUCUDUEJUF $.

  ${
    $d x A $.  $d x B $.
    $( The successor of the maximum (i.e. union) of two ordinals is the maximum
       of their successors.  (Contributed by NM, 28-Nov-2003.) $)
    ordsucun $p |- ( ( Ord A /\ Ord B ) ->
               suc ( A u. B ) = ( suc A u. suc B ) ) $=
      ( vx word wa cun csuc cv con0 wcel wi ordun ordsuc ordelon ex wb ordsssuc
      syl wo wss syl2anb ordssun adantl adantrr adantrl orbi12d 3bitr3d bitr4di
      sylbi sylan2 elun expcom pm5.21ndd eqrdv ) ADZBDZEZCABFZGZAGZBGZFZUQCHZIJ
      ZVCUSJZVCVBJZUQURDZVEVDKZABLZVGUSDZVHURMVJVEVDUSVCNOUIRUOUTDZVADZVFVDKZUP
      AMBMVKVLEVBDZVMUTVALVNVFVDVBVCNORUAVDUQVEVFPVDUQEZVEVCUTJZVCVAJZSZVFVOVCU
      RTZVCATZVCBTZSZVEVRUQVSWBPVDVCABUBUCUQVDVGVSVEPVIVCURQUJVOVTVPWAVQVDUOVTV
      PPUPVCAQUDVDUPWAVQPUOVCBQUEUFUGVCUTVAUKUHULUMUN $.
  $}

  $( The maximum of two ordinals is equal to one of them.  (Contributed by
     Mario Carneiro, 25-Jun-2015.) $)
  ordunpr $p |- ( ( B e. On /\ C e. On ) -> ( B u. C ) e. { B , C } ) $=
    ( con0 wcel wa cun cpr wceq wss word eloni ordtri2or2 syl2an orcomd ssequn2
    wo ssequn1 orbi12i sylib cvv wb unexg elprg syl mpbird ) ACDZBCDZEZABFZABGD
    ZUIAHZUIBHZPZUHBAIZABIZPUMUHUOUNUFAJBJUOUNPUGAKBKABLMNUNUKUOULBAOABQRSUHUIT
    DUJUMUAABCCUBUIABTUCUDUE $.

  $( The maximum of two ordinals belongs to a third if each of them do.
     (Contributed by NM, 18-Sep-2006.)  (Revised by Mario Carneiro,
     25-Jun-2015.) $)
  ordunel $p |- ( ( Ord A /\ B e. A /\ C e. A ) -> ( B u. C ) e. A ) $=
    ( word wcel w3a cpr cun wss prssi 3adant1 ordelon 3adant3 ordunpr 3imp3i2an
    con0 sseldd ) ADZBAEZCAEZFBCGZABCHZSTUAAIRBCAJKRSTBPEZCPEUBUAERSUCTABLMACLB
    CNOQ $.

  $( A class of ordinal numbers is a subclass of the successor of its union.
     Similar to Proposition 7.26 of [TakeutiZaring] p. 41.  (Contributed by NM,
     19-Sep-2003.) $)
  onsucuni $p |- ( A C_ On -> A C_ suc U. A ) $=
    ( con0 wss cuni word csuc ssorduni wa ssid ordunisssuc mpbii mpdan ) ABCZAD
    ZEZANFCZAGMOHNNCPNIANJKL $.

  $( An ordinal class is a subclass of the successor of its union.
     (Contributed by NM, 12-Sep-2003.) $)
  ordsucuni $p |- ( Ord A -> A C_ suc U. A ) $=
    ( word con0 wss cuni csuc ordsson onsucuni syl ) ABACDAAEFDAGAHI $.

  $( An ordinal class is either its union or the successor of its union.  If we
     adopt the view that zero is a limit ordinal, this means every ordinal
     class is either a limit or a successor.  (Contributed by NM,
     13-Sep-2003.) $)
  orduniorsuc $p |- ( Ord A -> ( A = U. A \/ A = suc U. A ) ) $=
    ( word cuni wceq csuc wne wss wa wn wcel orduniss wb orduni mpancom biimprd
    ordelssne mpand ordsucss syld ordsucuni jctild df-ne necom bitr3i eqss orrd
    3imtr4g ) ABZAACZDZAUIEZDZUHUIAFZAUKGZUKAGZHUJIZULUHUMUOUNUHUMUIAJZUOUHUIAG
    ZUMUQAKUHUQURUMHZUIBUHUQUSLAMUIAPNOQUIARSATUAUPAUIFUMAUIUBAUIUCUDAUKUEUGUF
    $.

  ${
    $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 $.

    $( An ordinal class is equal to the union of its successor.  (Contributed
       by NM, 10-Dec-2004.)  (Proof shortened by Andrew Salmon,
       27-Aug-2011.) $)
    ordunisuc $p |- ( Ord A -> U. suc A = A ) $=
      ( vx word con0 wcel wceq wo csuc ordeleqon cv suceq unieqd id eqeq12d wtr
      cuni eloni ordtr syl vex unisuc sylib vtoclga sucon unon eqtri jaoi sylbi
      unieqi 3eqtr4a ) ACADEZADFZGAHZPZAFZAIUKUOULBJZHZPZUPFZUOBADUPAFZURUNUPAU
      TUQUMUPAKLUTMNUPDEZUPOZUSVAUPCVBUPQUPRSUPBTUAUBUCULDHZPZDUNAVDDPDVCDUDUIU
      EUFULUMVCADKLULMUJUGUH $.

    $( The union of the ordinal subsets of an ordinal number is that number.
       (Contributed by NM, 30-Jan-2005.) $)
    orduniss2 $p |- ( Ord A -> U. { x e. On | x C_ A } = A ) $=
      ( word cv wss con0 crab cuni csuc cpw cin wcel wa df-rab incom inab df-pw
      cab eqcomi abid2 ineq12i 3eqtr3i eqtri ordpwsuc eqtrid unieqd ordunisuc
      eqtrd ) BCZADZBEZAFGZHBIZHBUIULUMUIULBJZFKZUMULUJFLZUKMARZUOUKAFNUPARZUKA
      RZKUSURKUQUOURUSOUPUKAPUSUNURFUNUSABQSAFTUAUBUCBUDUEUFBUGUH $.
  $}

  $( 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 cuni word eleq1 biimpac eloni ordsuc ordunisuc sylbir
    wa suceq syl eqtr4d 3syl unieq unieqd eqeq12d imbitrrid anabsi7 adantr
    eqtrd ) ACDZABEZFZOZAGZEZAEZGZAUGUIULUNFZUJUOUIUHGZEZUHEZGZFZUJUHCDZUHHZUTU
    IUGVAAUHCIJUHKVBUQUHUSVBUPBFZUQUHFVBBHVCBLBMNUPBPQUHMRSUIULUQUNUSUIUKUPFULU
    QFAUHTUKUPPQUIUMURAUHPUAUBUCUDUGUNAFZUIUGAHVDAKAMQUEUF $.

  $( The successor of an ordinal class contains the empty set.  (Contributed by
     NM, 4-Apr-1995.) $)
  0elsuc $p |- ( Ord A -> (/) e. suc A ) $=
    ( word csuc c0 wcel ordsuc wne nsuceq0 ord0eln0 mpbiri sylbi ) ABACZBZDLEZA
    FMNLDGAHLIJK $.

  $( The class of ordinal numbers is a limit ordinal.  (Contributed by NM,
     24-Mar-1995.) $)
  limon $p |- Lim On $=
    ( con0 wlim word c0 wne cuni wceq ordon onn0 unon eqcomi df-lim mpbir3an )
    ABACADEAAFZGHINAJKALM $.

  $( An ordinal number is either its own union (if zero or a limit ordinal) or
     the successor of its union.  (Contributed by NM, 13-Jun-1994.)  Put in
     closed form.  (Revised by BJ, 11-Jan-2025.) $)
  onuniorsuc $p |- ( A e. On -> ( A = U. A \/ A = suc U. A ) ) $=
    ( con0 wcel word cuni wceq csuc wo eloni orduniorsuc syl ) ABCADAAEZFALGFHA
    IAJK $.

  ${
    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 $.

    $( Obsolete version of ~ onuniorsuc as of 11-Jan-2025.  (Contributed by NM,
       13-Jun-1994.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    onuniorsuciOLD $p |- ( A = U. A \/ A = suc U. A ) $=
      ( con0 wcel cuni wceq csuc wo onuniorsuc ax-mp ) ACDAAEZFAKGFHBAIJ $.

    ${
      $d x A $.
      $( An ordinal is equal to its union if and only if it is not the
         successor of an ordinal.  A closed-form generalization of this result
         is ~ orduninsuc .  (Contributed by NM, 18-Feb-2004.) $)
      onuninsuci $p |- ( A = U. A <-> -. E. x e. On A = suc x ) $=
        ( cv csuc wceq con0 wrex cuni wn wa onirri id cun csn wtr sylancr sylib
        wcel ax-mp df-suc eqeq2i unieq sylbi uniun unisnv uneq2i eqtri wss tron
        eqtrdi eleq1 mpbii trsuc ontr df-tr ssequn1 eqtrd sylan9eqr sucid eleq2
        syl vex mpbiri adantr eqeltrd mto imnani rexlimivw onuni onuniorsuc ori
        wo suceq rspceeqv impbii con2bii ) BADZEZFZAGHZBBIZFZWAWCJZVTWDAGVTWCVT
        WCKZBBSBCLWEBVRBWCVTBWBVRWCMVTWBVRIZVRNZVRVTWBVRVROZNZIZWGVTBWIFWBWJFVS
        WIBVRUAUBBWIUCUDWJWFWHIZNWGVRWHUEWKVRWFAUFUGUHUKVTWFVRUIZWGVRFVTVRGSZWL
        VTGPVSGSZWMUJVTBGSZWNCBVSGULUMGVRUNQWMVRPWLVRUOVRUPRVBWFVRUQRURUSVTVRBS
        ZWCVTWPVRVSSVRAVCUTBVSVRVAVDVEVFVGVHVIWDWBGSZBWBEZFZWAWOWQCBVJTWCWSWOWC
        WSVMCBVKTVLAWBGVSWRBVRWBVNVOQVPVQ $.
    $}

    ${
      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 ) AEFBGABFAHBIJCBDKA
        BELM $.
    $}
  $}

  $( 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 ) $=
    ( csuc wlim wcel word cuni wceq limord ordsuc sylibr limuni ordunisuc eleq2
    wn eqeq2d ordirr notbid syl5ibrcom sucidg con3i syl6 sylbid sylc con2i ) AC
    ZDZABEZUGAFZUFUFGZHZUHOZUGUFFUIUFIAJKUFLUIUKUFAHZULUIUJAUFAMPUIUMAUFEZOZULU
    IUOUMAAEZOAQUMUNUPUFAANRSUHUNABTUAUBUCUDUE $.

  ${
    $d x A $.
    $( An ordinal class is equal to its union if and only if it is not the
       successor of an ordinal.  Closed-form generalization of ~ onuninsuci .
       (Contributed by NM, 18-Feb-2004.) $)
    orduninsuc $p |- ( Ord A -> ( A = U. A <-> -. E. x e. On A = suc x ) ) $=
      ( word con0 wcel wceq cuni wn wb c0 id unieq eqeq12d eqeq1 rexbidv notbid
      wrex bibi12d cvv mpbiri wo cv ordeleqon cif 0elon elimel onuninsuci dedth
      csuc unon eqcomi onprc vex sucex eleq1 mto a1i nrex 2th jaoi sylbi ) BCBD
      EZBDFZUABBGZFZBAUBZUIZFZADQZHZIZBUCVBVKVCVBVKVBBJUDZVLGZFZVLVGFZADQZHZIBJ
      BVLFZVEVNVJVQVRBVLVDVMVRKBVLLMVRVIVPVRVHVOADBVLVGNOPRAVLBJDUEUFUGUHVCVKDD
      GZFZDVGFZADQZHZIVTWCVSDUJUKWAADWAHVFDEWADSEZULWAWDVGSEVFAUMUNDVGSUOTUPUQU
      RUSVCVEVTVJWCVCBDVDVSVCKBDLMVCVIWBVCVHWAADBDVGNOPRTUTVA $.

    $( An ordinal equal to its union contains the successor of each of its
       members.  (Contributed by NM, 1-Feb-2005.) $)
    ordunisuc2 $p |- ( Ord A -> ( A = U. A <-> A. x e. A suc x e. A ) ) $=
      ( word cuni wceq cv csuc con0 wn wcel wral wi wa wo eloni adantr simpr ex
      bitr3id bitrd wrex orduninsuc ralnex wb onsuc syl ordtri3 con2bid onnbtwn
      sylan2 imnan sylibr con2d pm2.21 syl6 adantl ax-1 a1i wss ordtri2or sylan
      jaod ancoms orcomd ordsssuc2 biimpd orim12d impbid bitr3d pm5.74da impexp
      mpd ordelon ancrd impbid2 imbi1d ralbidv2 ) BCZBBDEBAFZGZEZAHUAIZVTBJZABK
      ZABUBWBWAIZAHKVRWDWAAHUCVRWEWCAHBVRVSHJZWELWFVSBJZWCLZLZWHVRWFWEWHVRWFMZB
      VTJZWCNZWEWHWJWAWLWFVRVTCZWAWLIUDWFVTHJWMVSUEVTOUFBVTUGUJUHWJWLWHWJWKWHWC
      WFWKWHLVRWFWKWGIWHWFWGWKWFWGWKMIWGWKILVSBUIWGWKUKULUMWGWCUNUOUPWCWHLWJWCW
      GUQURVBWJWHWLWJWHMZBVSUSZWGNZWLWJWPWHWJWGWOWFVRWGWONZWFVSCVRWQVSOVSBUTVAV
      CVDPWNWOWKWGWCWJWOWKLWHWJWOWKBVSVEVFPWJWHQVGVLRVHVIVJWIWFWGMZWCLVRWHWFWGW
      CVKVRWRWGWCVRWRWGWFWGQVRWGWFVRWGWFBVSVMRVNVOVPSTVQST $.

    $( An ordinal is zero, a successor ordinal, or a limit ordinal.  Remark
       1.12 of [Schloeder] p. 2.  (Contributed by NM, 1-Oct-2003.) $)
    ordzsl $p |- ( Ord A <->
                  ( A = (/) \/ E. x e. On A = suc x \/ Lim A ) ) $=
      ( word c0 wceq cv csuc con0 wrex wlim w3o cuni orduninsuc biimprd unizlim
      wo wn sylibd orrd wcel 3orass or12 bitri ord0 ordeq onsuc eleq1 imbitrrid
      sylibr mpbiri eloni syl6com rexlimiv limord 3jaoi impbii ) BCZBDEZBAFZGZE
      ZAHIZBJZKZUQVBURVCPZPZVDUQVBVEUQVBQZBBLEZVEUQVHVGABMNBORSVDURVBVCPPVFURVB
      VCUAURVBVCUBUCUIURUQVBVCURUQDCUDBDUEUJVAUQAHVAUSHTZBHTZUQVIVJVAUTHTUSUFBU
      THUGUHBUKULUMBUNUOUP $.

    $( An ordinal number is zero, a successor ordinal, or a limit ordinal
       number.  (Contributed by NM, 1-Oct-2003.)  (Proof shortened by Andrew
       Salmon, 27-Aug-2011.) $)
    onzsl $p |- ( A e. On <->
               ( A = (/) \/ E. x e. On A = suc x \/ ( A e. _V /\ Lim A ) ) ) $=
      ( con0 wcel c0 wceq cv csuc wrex cvv wlim wa word elex eloni ordzsl 3mix1
      w3o adantl eleq1 3mix2 3mix3 3jaodan sylan2b syl2anc 0elon onsuc rexlimiv
      mpbiri syl5ibrcom limelon 3jaoi impbii ) BCDZBEFZBAGZHZFZACIZBJDZBKZLZRZU
      NUTBMZVCBCNBOVDUTUOUSVARVCABPUTUOVCUSVAUOVCUTUOUSVBQSUSVCUTUSUOVBUASVBUOU
      SUBUCUDUEUOUNUSVBUOUNECDUFBECTUIURUNACUPCDUNURUQCDUPUGBUQCTUJUHBJUKULUM
      $.

    $( An alternate definition of a limit ordinal, which is any ordinal that is
       neither zero nor a successor.  (Contributed by NM, 1-Nov-2004.)  (Proof
       shortened by Andrew Salmon, 27-Aug-2011.) $)
    dflim3 $p |- ( Lim A <->
                 ( Ord A /\ -. ( A = (/) \/ E. x e. On A = suc x ) ) ) $=
      ( wlim word c0 wne cuni wceq w3a wa cv csuc con0 wrex wo wn df-lim 3anass
      wb df-ne a1i orduninsuc anbi12d ioran bitr4di pm5.32i 3bitri ) BCBDZBEFZB
      BGHZIUHUIUJJZJUHBEHZBAKLHAMNZOPZJBQUHUIUJRUHUKUNUHUKULPZUMPZJUNUHUIUOUJUP
      UIUOSUHBETUAABUBUCULUMUDUEUFUG $.

    $( An alternate definition of a limit ordinal.  (Contributed by NM,
       1-Feb-2005.) $)
    dflim4 $p |- ( Lim A <->
                 ( Ord A /\ (/) e. A /\ A. x e. A suc x e. A ) ) $=
      ( wlim word wcel cuni wceq w3a csuc wral dflim2 ordunisuc2 anbi2d pm5.32i
      c0 cv wa 3anass 3bitr4i bitri ) BCBDZOBEZBBFGZHZUAUBAPIBEABJZHZBKUAUBUCQZ
      QUAUBUEQZQUDUFUAUGUHUAUCUEUBABLMNUAUBUCRUAUBUERST $.
  $}

  ${
    $d x A $.  $d x B $.
    $( The successor of a member of a limit ordinal is also a member.
       (Contributed by NM, 3-Sep-2003.) $)
    limsuc $p |- ( Lim A -> ( B e. A <-> suc B e. A ) ) $=
      ( vx wlim wcel csuc word c0 cv wral w3a dflim4 wceq suceq eleq1d 3ad2ant3
      wi rspccv sylbi wtr limord ordtr trsuc ex 3syl impbid ) ADZBAEZBFZAEZUGAG
      ZHAEZCIZFZAEZCAJZKUHUJQZCALUPUKUQULUOUJCBAUMBMUNUIAUMBNORPSUGUKATZUJUHQAU
      AAUBURUJUHABUCUDUEUF $.

    $( A class includes a limit ordinal iff the successor of the class includes
       it.  (Contributed by NM, 30-Oct-2003.) $)
    limsssuc $p |- ( Lim A -> ( A C_ B <-> A C_ suc B ) ) $=
      ( vx wlim wss csuc sssucid sstr2 wa cv wcel wceq wn wi eleq1 biimpcd word
      mpi sylib ex limsuc biimpa limord ordelord sylan ordtri1 syl2an2r con2bid
      wb ordsuc mpbid sylan9r con2d com23 imp31 ssel2 vex elsuc ord adantll mpd
      wo con1d ssrdv impbid2 ) ADZABEZABFZEZVGBVHEVIBGABVHHRVFVIVGVFVIIZCABVJCJ
      ZAKZVKBKZVJVLIVKBLZMZVMVFVIVLVOVFVLVIVOVFVLVIVONVFVLIVNVIVLVNBAKZVFVIMZVN
      VLVPVKBAOPVFVPVQVFVPIZVHAKZVQVFVPVSABUAUBVRVIVSVFAQZVPVHQZVIVSMUIAUCZVRBQ
      ZWAVFVTVPWCWBABUDUEBUJSAVHUFUGUHUKTULUMTUNUOVIVLVOVMNVFVIVLIZVMVNWDVMVNWD
      VKVHKVMVNVBAVHVKUPVKBCUQURSUSVCUTVATVDTVE $.
  $}

  ${
    $d x y $.
    $( Two ways to express the class of non-limit ordinal numbers.  Part of
       Definition 7.27 of [TakeutiZaring] p. 42, who use the symbol K_I for
       this class.  (Contributed by NM, 1-Nov-2004.) $)
    nlimon $p |- { x e. On | ( x = (/) \/ E. y e. On x = suc y ) } =
                   { x e. On | -. Lim x } $=
      ( cv c0 wceq csuc con0 wrex wo wlim wn wcel word wb eloni dflim3 baib syl
      con2bid rabbiia ) ACZDEUABCFEBGHIZUAJZKZAGUAGLUAMZUBUDNUAOUEUCUBUCUEUBKBU
      APQSRT $.
  $}

  ${
    $d x y z A $.
    $( The union of a nonempty class of limit ordinals is a limit ordinal.
       (Contributed by NM, 1-Feb-2005.) $)
    limuni3 $p |- ( ( A =/= (/) /\ A. x e. A Lim x ) -> Lim U. A ) $=
      ( vy vz c0 wne cv wlim wral wa cuni word wcel csuc con0 wss cvv adantl wi
      elunii limeq rspcv vex limelon mpan syl6com ssrdv ssorduni syl wex 0ellim
      n0 expcom syl5 syld exlimiv imp wrex eluni2 rspccv limsuc anbi1d biimtrdi
      sylbi expd com3r sylcom rexlimdv biimtrid ralrimiv dflim4 syl3anbrc ) BEF
      ZAGZHZABIZJBKZLZEVQMZCGZNZVQMZCVQIZVQHVPVRVMVPBOPVRVPDBODGZBMZVPWDHZWDOMZ
      VOWFAWDBVNWDUAZUBZWDQMWFWGDUCWDQUDUEUFUGBUHUIRVMVPVSVMWEDUJVPVSSZDBULWEWJ
      DWEVPWFVSWIWFEWDMZWEVSWDUKWKWEVSEWDBTUMUNUOUPVDUQVPWCVMVPWBCVQVTVQMVTWDMZ
      DBURVPWBDVTBUSVPWLWBDBVPWEWFWLWBSVOWFAWDBWHUTWFWLWEWBWFWLWEWBWFWLWEJWAWDM
      ZWEJWBWFWLWMWEWDVTVAVBWAWDBTVCVEVFVGVHVIVJRCVQVKVL $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Transfinite induction
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x 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 ` .

       See Theorem ~ tfindes or ~ tfinds for the version involving basis and
       induction hypotheses.  (Contributed by NM, 18-Feb-2004.) $)
    tfi $p |- ( ( A C_ On /\ A. x e. On ( x C_ A -> x e. A ) ) -> A = On ) $=
      ( con0 wss cv wcel wi wral wa wceq cdif cin c0 wrex wn eldifn adantl onss
      difin0ss syl5com imim1d a2i eldifi impel mtod ex ralimi2 ralnex sylib wne
      ssdif0 necon3bbii word ordon difss tz7.5 mp3an12 sylbi anim2i eqss sylibr
      nsyl2 ) BCDZAEZBDZVDBFZGZACHZIVCCBDZIBCJVHVIVCVHCBKZVDLMJZAVJNZVIVHVKOZAV
      JHVLOVGVMACVJVDCFZVGGZVDVJFZVMVOVPIVKVFVPVFOVOVDCBPQVOVNVKVFGZVPVNVGVQVNV
      KVEVFVNVDCDVKVEVDRCBVDSTUAUBVDCBUCUDUEUFUGVKAVJUHUIVIOVJMUJZVLVIVJMCBUKUL
      CUMVJCDVRVLUNCBUOACVJUPUQURVBUSBCUTVA $.
  $}

  ${
    $d ph y z $.  $d x y z $.
    $( A closed form of ~ tfis .  (Contributed by Scott Fenton, 8-Jun-2011.) $)
    tfisg $p |- ( A. x e. On ( A. y e. x [. y / x ]. ph -> ph ) ->
                 A. x e. On ph ) $=
      ( vz cv wsbc wral wi con0 crab wceq wss wcel ssrab2 wa dfss3 nfcv elrabsf
      simprbi nfsbc1v ralimi sylbi nfralw nfim sbceq1a imbi12d rspc impcom syl5
      raleq simpr jctild imbitrrdi ralrimiva tfi sylancr eqcomd rabid2 sylib )
      ABCEZFZCBEZGZAHZBIGZIABIJZKABIGVEVFIVEVFILDEZVFLZVGVFMZHZDIGVFIKABINVEVJD
      IVEVGIMZOZVHVKABVGFZOVIVLVHVMVKVHVACVGGZVLVMVHUTVFMZCVGGVNCVGVFPVOVACVGVO
      UTIMVAABUTIBIQZRSUAUBVKVEVNVMHZVDVQBVGIVNVMBVABCVGBVGQABUTTUCABVGTUDVBVGK
      VCVNAVMVACVBVGUJABVGUEUFUGUHUIVEVKUKULABVGIVPRUMUNDVFUOUPUQABIURUS $.
  $}

  ${
    $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 nfim dfss3 sseq1 rabid eleq1w imbi12d sbequ sbequ12
      nfs1v cbvrabw elrab2 ralimi syl5 anc2li vtoclgaf rgen mp2an eqcomi reqabi
      nfv tfi ) BGZHIZVFAABHHABHJZHVGHKEGZVGKZVHVGIZLZEHMVGHNABHOVKEHCGZVGIZCVE
      MZVFAPZLVKBVHHBVHQZVIVJBBVHVGVPABHUAZUBBEVGVQUCUDVEVHNZVNVIVOVJVNVEVGKVRV
      ICVEVGUEVEVHVGUFRVOVEVGIVRVJABHUGBEVGUHRUIVFVNAVNABCTZCVEMVFAVMVSCVEVMVLH
      IVSABFTZVSFVLHVGAFCBUJAVTBFHBHQFHQAFVCABFULABFUKUMUNSUODUPUQURUSEVGVDUTVA
      VBS $.
  $}

  ${
    $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 sbiev ralbii biimtrid tfis ) ACDACDHZDCIZJBDRJRKL
      AQBDRABCDEFMNGOP $.
  $}

  ${
    $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
      sseq1 imbi12d albidv eqeq1d imbi2d cbvalvw bitrdi w3a simp3l simp2 simp1l
      wral eqeltrd simp3r eqsstrd csb wsb simpl3l simpl1l simpl2 eleqtrd onelss
      cv simpr sylc simpl3r sstrd simpl1r rspcdva eqidd nfcv csbhypf equcoms wb
      eqcomd nfv sbhypf bicomd spvv mp2and alrimiv eleq1d sylib syl121anc tfis3
      ex 3exp syl spcgv mpi expd pm2.43i ) AJJUCZDJUDAXGDTAAXGDAJJUEZAXGUFZDTZJ
      UGAGKUHHJUEZXIBTZTZEUIZXHXJTZLAJUJUHXNMHUAVQZUEZAXPJUCZUFZBTZTZEUIZIUBVQZ
      UEZAYCJUCZUFZCTZTZFUIZXNUAUBJUAUBUKZYBHYCUEZYFBTZTZEUIYIYJYAYMEYJXQYKXTYL
      XPYCHULYJXSYFBYJXRYEAXPYCJUOUMUNUPUQYMYHEFEFUKZYKYDYLYGYNHIYCQURYNBCYFOUS
      UPUTVAXPJUEZYAXMEYOXQXKXTXLXPJHULYOXSXIBYOXRXGAXPJJUOUMUNUPUQXPUJUHZYIUBX
      PVFZYBYPYQUFZYAEYRXQXSBYRXQXSVBZAHUJUHHJUCIHUHZCTZFUIZBYRXQAXRVCYSHXPUJYR
      XQXSVDZYPYQXQXSVEVGYSHXPJUUCYRXQAXRVHVIYSESVQHVJZHUHZBESVKZTZSUIUUBYSUUGS
      YSUUEUUFYSUUEUFZAUUDJUCZUUFAXRYRXQUUEVLUUHUUDXPJUUHYPUUDXPUHUUDXPUCYPYQXQ
      XSUUEVMUUHUUDHXPYSUUEVRYRXQXSUUEVNVOZXPUUDVPVSAXRYRXQUUEVTWAUUHIUUDUEZAUU
      IUFZCTZTZFUIZUUDUUDUEZUULUUFTZUUHYIUUOUBXPUUDYCUUDUEZYHUUNFUURYDUUKYGUUMY
      CUUDIULUURYFUULCUURYEUUIAYCUUDJUOUMUNUPUQYPYQXQXSUUEWBUUJWCUUHUUDWDUUNUUP
      UUQTFSFSUKZUUKUUPUUMUUQUUSIUUDUUDUUKSFSFUKZUUDIESFVQZHIEUVAWEEIWEQWFZWIWG
      URUUSCUUFUULCUUFWHSFUUTUUFCBCESUVACEWJOWKZWLWGUSUPWMVSWNWTWOUUGUUASFUUTUU
      EYTUUFCUUTUUDIHUVBWPUVCUPUTWQNWRXAWOWTWSXBXMXOEGKEVQGUEZXKXHXLXJUVDHJJRUR
      UVDBDXIPUSUPXCVSXDXEXFXD $.
  $}

  ${
    $d x y z $.  $d x A $.  $d x z ch $.  $d x ta $.  $d y z ph $.
    $( Substitutions. $)
    tfinds.1 $e |- ( x = (/) -> ( ph <-> ps ) ) $.
    tfinds.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
    tfinds.3 $e |- ( x = suc y -> ( ph <-> th ) ) $.
    tfinds.4 $e |- ( x = A -> ( ph <-> ta ) ) $.
    $( Basis. $)
    tfinds.5 $e |- ps $.
    $( Induction step for successors. $)
    tfinds.6 $e |- ( y e. On -> ( ch -> th ) ) $.
    $( Induction step for limit ordinals. $)
    tfinds.7 $e |- ( Lim x -> ( A. y e. x ch -> ph ) ) $.
    $( Principle of Transfinite Induction (inference schema), using implicit
       substitutions.  The first four hypotheses establish the substitutions we
       need.  The last three are the basis, the induction step for successors,
       and the induction step for limit ordinals.  Theorem Schema 4 of [Suppes]
       p. 197.  Theorem 1.19 of [Schloeder] p. 3.  (Contributed by NM,
       16-Apr-1995.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
    tfinds $p |- ( A e. On -> ta ) $=
      ( vz cv con0 wral wi wcel wlim wn word c0 wceq csuc wrex wo dflim3 notbii
      wa iman eloni pm2.27 syl mpbiri a1d nfra1 nfv nfim sucid rspcv ax-mp syl5
      vex wsb raleq sbiev bitr3id cbvralvw cbvralsvw 3bitr4g syl5ibrcom biimprd
      sbequ imbi1d syldd rexlimi jaoi syl6 biimtrrid biimtrid pm2.61d2 tfis3
      a1i ) ACEFGHJLFQZRUAZWGUBZCGWGSZATZWIUCWGUDZWGUEUFZWGGQZUGZUFZGRUHZUIZUCU
      LZUCZWHWKWIWSGWGUJUKWTWLWRTZWHWKWLWRUMWHXAWRWKWHWLXAWRTWGUNWLWRUOUPWMWKWQ
      WMAWJWMABMIUQURWPWKGRWJAGCGWGUSAGUTVAWNRUAZWPWJDAXBWJDTWPAFWOSZDTXCCXBDWN
      WOUAXCCTWNGVFVBACFWNWOJVCVDNVEWPWJXCDWPAFPVGZPWGSXDPWOSWJXCXDPWGWOVHCXDGP
      WGCAFGVGWNPQUFXDACFGCFUTJVIAGPFVPVJVKAFPWOVLVMVQVNWPDATTXBWPADKVOWFVRVSVT
      WAWBWCOWDWE $.
  $}

  ${
    $d x A $.  $d x y B $.  $d x ch $.  $d x th $.  $d x ta $.  $d y ph $.
    $( Substitutions. $)
    tfindsg.1 $e |- ( x = B -> ( ph <-> ps ) ) $.
    tfindsg.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
    tfindsg.3 $e |- ( x = suc y -> ( ph <-> th ) ) $.
    tfindsg.4 $e |- ( x = A -> ( ph <-> ta ) ) $.
    $( Basis. $)
    tfindsg.5 $e |- ( B e. On -> ps ) $.
    $( Induction step for successor ordinals. $)
    tfindsg.6 $e |- ( ( ( y e. On /\ B e. On ) /\ B C_ y ) -> ( ch -> th ) ) $.
    $( Induction step for limit ordinals. $)
    tfindsg.7 $e |- ( ( ( Lim x /\ B e. On ) /\ B C_ x ) ->
                   ( A. y e. x ( B C_ y -> ch ) -> ph ) ) $.
    $( Transfinite Induction (inference schema), using implicit substitutions.
       The first four hypotheses establish the substitutions we need.  The last
       three are the basis, the induction step for successors, and the
       induction step for limit ordinals.  The basis of this version is an
       arbitrary ordinal ` B ` instead of zero.  Remark in [TakeutiZaring]
       p. 57.  (Contributed by NM, 5-Mar-2004.) $)
    tfindsg $p |- ( ( ( A e. On /\ B e. On ) /\ B C_ A ) -> ta ) $=
      ( wcel wi wceq wa con0 wss cv c0 csuc wb sseq2 adantl eqeq2 biimtrrdi imp
      imbi12d imbi1d ss0 pm2.21d pm5.74d sylan9bbr pm2.61ian imbi2d a1d wex vex
      wn con3i sucex eqvinc imbitrrid biimpd sylan9r exlimiv sylbi eqcoms com4r
      imim2i df-ne anbi2i annim bitri onsssuc onsuc onelpss sylan2 bitrd ancoms
      wne ex a1ddd a2d com23 sylbird biimtrrid pm2.61d wlim wral pm2.27 ralimdv
      ad2antlr syld exp31 com3l com4t tfinds imp31 ) HUAQIUAQZIHUBZEXDIFUCZUBZA
      RZRXDIUDUBZBRZRXDIGUCZUBZCRZRZXDIXKUEZUBZDRZRXDXEERZRFGHXFUDSZXHXJXDIUDSZ
      XSXHXJUFXTXSTXGXIABXSXGXIUFXTXFUDIUGZUHXTXSABUFZXTXSXFISZYBIUDXFUIJUJUKUL
      XSXHXIARXTVCZXJXSXGXIAYAUMYDXIABYDXIYBXIXTIUNVDUOUPUQURUSXFXKSZXHXMXDYEXG
      XLACXFXKIUGKULUSXFXOSZXHXQXDYFXGXPADXFXOIUGLULUSXFHSZXHXRXDYGXGXEAEXFHIUG
      MULUSXDBXINUTXKUAQZXDXMXQYHXDXMXQRZYHXDTZXPIXOSZRZYIXDYLYIRYHYLXMXPXDDYLX
      PXDDRZRXMYKYMXPYMXOIXOISYFYCTZFVAYMFXOIXKGVBVEVFYNYMFYCXDAYFDXDAYCBNJVGYF
      ADLVHVIVJVKVLVNUTVMUHYLVCZXPIXOWEZTZYJYIYQXPYKVCZTYOYPYRXPIXOVOVPXPYKVQVR
      YJYQXLYIXDYHXLYQUFXDYHTXLIXOQZYQIXKVSYHXDXOUAQYSYQUFXKVTIXOWAWBWCWDYJXMXL
      XQYJXLCXQYJXLCXPDYJXLCDROWFWGWHWIWJWKWLWFWHXDXGXFWMZXNGXFWNZAYTXDXGUUAARZ
      YTXDXGUUBYTXDTXGTUUAXMGXFWNZAXDUUAUUCRYTXGXDXNXMGXFXDXMWOWPWQPWRWSWTXAXBX
      C $.
  $}

  ${
    $d x A $.  $d x y B $.  $d x ch $.  $d x th $.  $d x ta $.  $d y ph $.
    $( Substitutions. $)
    tfindsg2.1 $e |- ( x = suc B -> ( ph <-> ps ) ) $.
    tfindsg2.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
    tfindsg2.3 $e |- ( x = suc y -> ( ph <-> th ) ) $.
    tfindsg2.4 $e |- ( x = A -> ( ph <-> ta ) ) $.
    $( Basis. $)
    tfindsg2.5 $e |- ( B e. On -> ps ) $.
    $( Induction step for successor ordinals. $)
    tfindsg2.6 $e |- ( ( y e. On /\ B e. y ) -> ( ch -> th ) ) $.
    $( Induction step for limit ordinals. $)
    tfindsg2.7 $e |- ( ( Lim x /\ B e. x ) ->
                   ( A. y e. x ( B e. y -> ch ) -> ph ) ) $.
    $( Transfinite Induction (inference schema), using implicit substitutions.
       The first four hypotheses establish the substitutions we need.  The last
       three are the basis, the induction step for successors, and the
       induction step for limit ordinals.  The basis of this version is an
       arbitrary ordinal ` suc B ` instead of zero.  (Contributed by NM,
       5-Jan-2005.)  Remove unnecessary distinct variable conditions.  (Revised
       by David Abernethy, 19-Jun-2012.) $)
    tfindsg2 $p |- ( ( A e. On /\ B e. A ) -> ta ) $=
      ( con0 wcel wa wi csuc wss onelon onsucb sylib word eloni ordsucss sylbir
      syl imp cv wb ordelsuc sylan2 ancoms ex adantr sylbird sylan2br wlim wral
      cvv vex limelon mpan anassrs imbi1d ralbidva imbi12d mpbid tfindsg mp2and
      expl ) HQRZIHRZSZIUAZQRZVRHUBZEVQIQRZVSHIUCIUDZUEVOVPVTVOHUFVPVTTHUGIHUHU
      JUKVOVSVTSETVPVOVSVTEABCDEFGHVRJKLMVSWABWBNUIGULZQRZVSSVRWCUBZCDTZVSWDWAW
      EWFTWBWDWASWEIWCRZWFWAWDWGWEUMZWDWAWCUFZWHWCUGZIWCQUNZUOUPWDWGWFTWAWDWGWF
      OUQURUSUTUKFULZVAZVSSVRWLUBZWECTZGWLVBZATZVSWMWAWNWQTZWBWMWASIWLRZWGCTZGW
      LVBZATZTZWRWMXCWAWMWSXBPUQURWAWMXCWRUMZWMWAWLQRZXDWLVCRWMXEFVDWLVCVEVFWAX
      ESZWSWNXBWQXEWAWLUFWSWNUMWLUGIWLQUNUOXFXAWPAXFWTWOGWLXFWCWLRZSWGWECWAXEXG
      WHXEXGSZWAWIWHXHWDWIWLWCUCWJUJWKUOVGVHVIVHVJUOUPVKUTUKVLVNURVM $.
  $}

  ${
    $d x y z $.  $d y z ph $.
    tfindes.1 $e |- [. (/) / x ]. ph $.
    tfindes.2 $e |- ( x e. On -> ( ph -> [. suc x / x ]. ph ) ) $.
    tfindes.3 $e |- ( Lim y -> ( A. x e. y ph -> [. y / x ]. ph ) ) $.
    $( Transfinite Induction with explicit substitution.  The first hypothesis
       is the basis, the second is the induction step for successors, and the
       third is the induction step for limit ordinals.  Theorem Schema 4 of
       [Suppes] p. 197.  (Contributed by NM, 5-Mar-2004.) $)
    tfindes $p |- ( x e. On -> ph ) $=
      ( vz cv wsbc c0 csuc dfsbcq sbceq2a con0 wcel nfsbc1v nfim imbi12d wral
      wi nfv weq eleq1w sbceq1a suceq sbceq1d chvarfv wlim wsb cbvralsvw ralbii
      sbsbc bitri biimtrrid tfinds ) ABCHZIZABJIABGHZIZABURKZIZACGBHZABUPJLABUP
      URLABUPUTLABUPMDVBNOZAABVBKZIZTZTURNOZUSVATZTBGVGVHBVGBUAUSVABABURPABUTPQ
      QBGUBZVCVGVFVHBGNUCVIAUSVEVAABURUDVIABVDUTVBURUEUFRREUGUSGUPSZABUPSZUPUHU
      QVKABGUIZGUPSVJABGUPUJVLUSGUPABGULUKUMFUNUO $.
  $}

  ${
    $d x y ta $.  $d x ps $.  $d x ch $.  $d x th $.  $d y ph $.
    $( Substitutions. $)
    tfinds2.1 $e |- ( x = (/) -> ( ph <-> ps ) ) $.
    tfinds2.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
    tfinds2.3 $e |- ( x = suc y -> ( ph <-> th ) ) $.
    $( Basis. $)
    tfinds2.4 $e |- ( ta -> ps ) $.
    $( Induction step for successors. $)
    tfinds2.5 $e |- ( y e. On -> ( ta -> ( ch -> th ) ) ) $.
    $( Induction step for limit ordinals. $)
    tfinds2.6 $e |- ( Lim x -> ( ta -> ( A. y e. x ch -> ph ) ) ) $.
    $( Transfinite Induction (inference schema), using implicit substitutions.
       The first three hypotheses establish the substitutions we need.  The
       last three are the basis and the induction hypotheses (for successor and
       limit ordinals respectively).  Theorem Schema 4 of [Suppes] p. 197.  The
       wff ` ta ` is an auxiliary antecedent to help shorten proofs using this
       theorem.  (Contributed by NM, 4-Sep-2004.) $)
    tfinds2 $p |- ( x e. On -> ( ta -> ph ) ) $=
      ( wi wsbc imbi2d sbcie cvv elv wb c0 0ex cv wceq mpbir con0 wcel csuc a2d
      sbcth sbcimg sbcel1v 3imtr3i vex bicomd equcoms sucex sbcbii suceq sbcco2
      mpbi weq bitr3i 3imtr3g wral wlim wsb sbralie sbsbc bitr2i biimtrid limeq
      r19.21v biimtrrid tfindes ) EANZFGVPFUAOEBNZKVPVQFUAUBFUCZUAUDABEHPQUEVRU
      FUGZECNZGVROZEDNZGVROZVPVPFVRUHZOZGUCZUFUGZGVROZVTWBNZGVROZVSWAWCNZWGWINZ
      GVROZWHWJNZWMFWLGVRRWGECDLUIUJSWMWNTFWGWIGVRRUKSVAGVRUFULWJWKTFVTWBGVRRUK
      SUMVTVPGVRFUNGFVBCAECATFGFGVBZACIUOUPPQWCVPFWFUHZOZGVROWEWQWBGVRVPWBFWPWF
      GUNZUQVRWPUDADEJPQURVPFGWDWPVRWFUSUTVCVDVPFWFVEZVTGVRVEZFWFOZWFVFZVPFWFOZ
      WSWTFGVGXAVPVTFGWOACEIPVHWTFGVIVJVRVFZFWFOZWTVPNZFWFOZXBXAXCNZXDXFNZFWFOZ
      XEXGNZXJGXIFWFRWTECGVRVEZNXDVPECGVRVMXDEXLAMUIVKUJSXJXKTGXDXFFWFRUKSVAXDX
      BFWFWRVRWFVLQXGXHTGWTVPFWFRUKSUMVNVO $.
  $}

  ${
    $d x A $.  $d y ph $.  $d x ch $.  $d x ta $.  $d x y et $.
    $( Substitutions. $)
    tfinds3.1 $e |- ( x = (/) -> ( ph <-> ps ) ) $.
    tfinds3.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
    tfinds3.3 $e |- ( x = suc y -> ( ph <-> th ) ) $.
    tfinds3.4 $e |- ( x = A -> ( ph <-> ta ) ) $.
    $( Basis. $)
    tfinds3.5 $e |- ( et -> ps ) $.
    $( Induction step for successors. $)
    tfinds3.6 $e |- ( y e. On -> ( et -> ( ch -> th ) ) ) $.
    $( Induction step for limit ordinals. $)
    tfinds3.7 $e |- ( Lim x -> ( et -> ( A. y e. x ch -> ph ) ) ) $.
    $( Principle of Transfinite Induction (inference schema), using implicit
       substitutions.  The first four hypotheses establish the substitutions we
       need.  The last three are the basis, the induction step for successors,
       and the induction step for limit ordinals.  (Contributed by NM,
       6-Jan-2005.)  (Revised by David Abernethy, 21-Jun-2011.) $)
    tfinds3 $p |- ( A e. On -> ( et -> ta ) ) $=
      ( wi cv wceq imbi2d csuc con0 wcel a2d wral wlim r19.21v biimtrid tfinds
      c0 ) FAQZFBQFCQZFDQFEQGHIGRZUJSABFJTUMHRZSACFKTUMUNUASADFLTUMISAEFMTNUNUB
      UCFCDOUDULHUMUEFCHUMUEZQUMUFZUKFCHUMUGUPFUOAPUDUHUI $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  The natural numbers (i.e., finite ordinals)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare new symbol. $)
  $c _om $.  $( Upright lowercase 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, which are all ordinal numbers that
       are less than every limit ordinal, i.e., all finite ordinals.  Our
       definition is a variant of the Definition of N of [BellMachover] p. 471.
       See ~ dfom2 for an alternate definition.  Later, when we assume the
       Axiom of Infinity, we show ` _om ` is a set in ~ omex , and ` _om ` can
       then be defined per ~ dfom3 (the smallest inductive set) and ~ dfom4 .

       _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-nn ) with analogous
       properties and operations, but they will be different sets.
       (Contributed by NM, 15-May-1994.) $)
    df-om $a |- _om = { x e. On | A. y ( Lim y -> x e. y ) } $.
  $}

  ${
    $d x z $.  $d y z $.
    $( An alternate definition of the set of natural numbers ` _om ` .
       Definition 7.28 of [TakeutiZaring] p. 42, who use the symbol K_I for the
       restricted class abstraction of non-limit ordinal numbers (see
       ~ nlimon ).  (Contributed by NM, 1-Nov-2004.) $)
    dfom2 $p |- _om = { x e. On | suc x C_ { y e. On | -. Lim y } } $=
      ( vz com cv wlim wcel wi wal con0 crab csuc wn wss df-om wa cvv impexp wb
      vex limelon mpan pm4.71ri imbi1i con34b ibar imbi2d bitrid pm5.74i 3bitri
      onsssuc ontri1 bitr3d ancoms weq limeq elrab a1i imbi12d pm5.74da bitr4id
      notbid simpr onsuc onelon ex syl ancrd impbid2 imbi1d bitr3id bitrd df-ss
      albidv bitr4di rabbiia eqtri ) DCEZFZAEZVRGZHZCIZAJKVTLZBEZFZMZBJKZNZAJKA
      COWCWIAJVTJGZWCVRWDGZVRWHGZHZCIWIWJWBWMCWJWBVRJGZWMHZWMWJWBWNWAMZWNVSMZPZ
      HZHZWOWBWNVSPZWAHWNWBHWTVSXAWAVSWNVRQGVSWNCTVRQUAUBUCUDWNVSWARWNWBWSWBWPW
      QHWNWSVSWAUEWNWQWRWPWNWQUFUGUHUIUJWJWNWMWSWJWNPZWKWPWLWRWNWJWKWPSWNWJPVRV
      TNWKWPVRVTUKVRVTULUMUNWLWRSXBWGWQBVRJBCUOWFVSWEVRUPVBUQURUSUTVAWOWNWKPZWL
      HWJWMWNWKWLRWJXCWKWLWJXCWKWNWKVCWJWKWNWJWDJGZWKWNHVTVDXDWKWNWDVRVEVFVGVHV
      IVJVKVLVNCWDWHVMVOVPVQ $.
  $}

  ${
    $d A x y $.
    $( Membership in omega.  The left conjunct can be eliminated if we assume
       the Axiom of Infinity; see ~ elom3 .  (Contributed by NM,
       15-May-1994.) $)
    elom $p |- ( A e. _om <-> ( A e. On /\ A. x ( Lim x -> A e. x ) ) ) $=
      ( vy cv wlim wcel wi wal con0 com wceq eleq1 imbi2d albidv df-om elrab2 )
      ADZEZCDZQFZGZAHRBQFZGZAHCBIJSBKZUAUCAUDTUBRSBQLMNCAOP $.
  $}

  ${
    $d x y $.
    $( Omega is a subset of ` On ` .  (Contributed by NM, 13-Jun-1994.)  (Proof
       shortened by Andrew Salmon, 27-Aug-2011.) $)
    omsson $p |- _om C_ On $=
      ( vy vx cv wlim wel wi wal con0 com df-om ssrab3 ) ACDBAEFAGBHIBAJK $.
  $}

  ${
    $d x y A $.
    $( The class of natural numbers is a subclass of any (infinite) limit
       ordinal.  Exercise 1 of [TakeutiZaring] p. 44.  Remarkably, our proof
       does not require the Axiom of Infinity.  (Contributed by NM,
       30-Oct-2003.) $)
    limomss $p |- ( Lim A -> _om C_ A ) $=
      ( vx vy word wlim com wss limord con0 wcel wceq wo wi ordeleqon wa cv wal
      elom simprbi limeq eleq2 imbi12d spcgv syl5 com23 imp ssrdv omsson mpbiri
      ex sseq2 a1d jaoi sylbi mpcom ) ADZAEZFAGZAHUPAIJZAIKZLUQURMZANUSVAUTUSUQ
      URUSUQOBFAUSUQBPZFJZVBAJZMUSVCUQVDVCCPZEZVBVEJZMZCQZUSUQVDMZVCVBIJVICVBRS
      VHVJCAIVEAKVFUQVGVDVEATVEAVBUAUBUCUDUEUFUGUJUTURUQUTURFIGUHAIFUKUIULUMUNU
      O $.
  $}

  $( A natural number is an ordinal number.  (Contributed by NM,
     27-Jun-1994.) $)
  nnon $p |- ( A e. _om -> A e. On ) $=
    ( com con0 omsson sseli ) BCADE $.

  ${
    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 $.

  ${
    $d x y z $.
    $( The class of finite ordinals ` _om ` is a transitive class.
       (Contributed by NM, 18-Oct-1995.)  (Proof shortened by Andrew Salmon,
       27-Aug-2011.) $)
    trom $p |- Tr _om $=
      ( vy vx vz com wtr wel cv wcel wa wi dftr2 con0 wlim onelon expcom limord
      wal word ordtr elom trel 3syl com12 a2d alimdv anim12d 3imtr4g imp ax-gen
      expd mpgbir ) DEABFZBGZDHZIAGZDHZJZBQAABDKUQBULUNUPULUMLHZCGZMZBCFZJZCQZI
      UOLHZUTACFZJZCQZIUNUPULURVDVCVGURULVDUMUONOULVBVFCULUTVAVEUTULVAVEJUTULVA
      VEUTUSRUSEULVAIVEJUSPUSSUSUOUMUAUBUJUCUDUEUFCUMTCUOTUGUHUIUK $.
  $}

  $( The class of finite ordinals ` _om ` is ordinal.  Theorem 7.32 of
     [TakeutiZaring] p. 43.  Theorem 1.22 of [Schloeder] p. 3.  (Contributed by
     NM, 18-Oct-1995.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
  ordom $p |- Ord _om $=
    ( com wtr con0 wss word trom omsson ordon trssord mp3an ) ABACDCEAEFGHACIJ
    $.

  $( 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 wtr wcel wa wi trom trel ax-mp ) CDABEBCEFACEGHCABIJ $.

  $( The class of natural numbers ` _om ` is either an ordinal number (if we
     accept the Axiom of Infinity) or the proper class of all ordinal numbers
     (if we deny the Axiom of Infinity).  Remark in [TakeutiZaring] p. 43.
     (Contributed by NM, 10-May-1998.) $)
  omon $p |- ( _om e. On \/ _om = On ) $=
    ( com word con0 wcel wceq wo ordom ordeleqon mpbi ) ABACDACEFGAHI $.

  $( Omega is an ordinal number.  (Contributed by Mario Carneiro,
     30-Jan-2013.) $)
  omelon2 $p |- ( _om e. _V -> _om e. On ) $=
    ( com con0 wcel cvv wn wceq omon ori onprc eleq1 mtbiri syl con4i ) ABCZADC
    ZNEABFZOENPGHPOBDCIABDJKLM $.

  ${
    $d x A $.
    $( A natural number is not a limit ordinal.  (Contributed by NM,
       18-Oct-1995.) $)
    nnlim $p |- ( A e. _om -> -. Lim A ) $=
      ( vx com wcel wlim word wn nnord ordirr syl cv wal con0 elom simprbi wceq
      wi limeq eleq2 imbi12d spcgv mpd mtod ) ACDZAEZAADZUDAFUFGAHAIJUDBKZEZAUG
      DZQZBLZUEUFQZUDAMDUKBANOUJULBACUGAPUHUEUIUFUGARUGAASTUAUBUC $.

    $( The class of natural numbers is a subclass of the class of non-limit
       ordinal numbers.  Exercise 4 of [TakeutiZaring] p. 42.  (Contributed by
       NM, 2-Nov-2004.)  (Proof shortened by Andrew Salmon, 27-Aug-2011.) $)
    omssnlim $p |- _om C_ { x e. On | -. Lim x } $=
      ( com cv wlim wn con0 crab wss wral omsson nnlim rgen ssrab mpbir2an ) BA
      CZDEZAFGHBFHPABIJPABOKLPAFBMN $.
  $}

  $( Omega is a limit ordinal.  Theorem 2.8 of [BellMachover] p. 473.  Theorem
     1.23 of [Schloeder] p. 4.  Our proof, however, does not require the Axiom
     of Infinity.  (Contributed by NM, 26-Mar-1995.)  (Proof shortened by Mario
     Carneiro, 2-Sep-2015.) $)
  limom $p |- Lim _om $=
    ( vx com word wlim ordom con0 wcel wceq wo ordeleqon cv wi wal ordirr ax-mp
    wn elom baib mtbii limeq limomss wb limord ordsseleq sylancr mpbid biimprcd
    wss ord syld con1d com12 alrimiv nsyl2 limon mpbiri jaoi sylbi ) BCZBDZEUSB
    FGZBFHZIUTBJVAUTVBVAAKZDZBVCGZLZAMZUTVABBGZVGUSVHPEBNOVHVAVGABQRSUTPZVFAVDV
    IVEVDVEUTVDVEPBVCHZUTVDVEVJVDBVCUHZVEVJIZVCUAVDUSVCCVKVLUBEVCUCBVCUDUEUFUIV
    JUTVDBVCTUGUJUKULUMUNVBUTFDUOBFTUPUQURO $.

  $( 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 wlim wcel csuc wb limom limsuc ax-mp ) BCABDAEBDFGBAHI $.

  ${
    $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 ) $=
      ( com wcel c0 wne wa cv csuc wceq con0 wrex wlim wn nnlim adantr wi nnord
      word cuni orduninsuc w3a df-lim biimpri 3expia sylbird sylan mt3d biimpcd
      wb eleq1 peano2b imbitrrdi ancrd adantld reximdv2 mpd ) BCDZBEFZGZBAHZIZJ
      ZAKLZVCACLZUTVDBMZURVFNUSBOPURBSZUSVDNZVFQBRVGUSGVHBBTJZVFVGVIVHUJUSABUAP
      VGUSVIVFVFVGUSVIUBBUCUDUEUFUGUHURVDVEQUSURVCVCAKCURVCVACDZVCGVAKDURVCVJUR
      VCVBCDZVJVCURVKBVBCUKUIVAULUMUNUOUPPUQ $.
  $}

  $( A natural number is not the successor of itself.  (Contributed by AV,
     17-Oct-2023.) $)
  omsucne $p |- ( A e. _om -> A =/= suc A ) $=
    ( com wcel csn cun wne csuc cin c0 wceq wpss word nnord orddisj syl disjpss
    snnzg syl2anc pssned df-suc neeq2i sylibr ) ABCZAAADZEZFAAGZFUCAUEUCAUDHIJZ
    UDIFAUEKUCALUGAMANOABQAUDPRSUFUEAATUAUB $.

  ${
    $d x A $.
    $( An ordinal subclass of non-limit ordinals is a class of natural numbers.
       Exercise 7 of [TakeutiZaring] p. 42.  (Contributed by NM,
       2-Nov-2004.) $)
    ssnlim $p |- ( ( Ord A /\ A C_ { x e. On | -. Lim x } ) -> A C_ _om ) $=
      ( word cv wlim wn con0 crab wss wa com wcel limom ssel limeq notbid elrab
      wceq simprbi syl6 mt2i adantl wb ordom ordtri1 mpan2 adantr mpbird ) BCZB
      ADZEZFZAGHZIZJBKIZKBLZFZUNUQUIUNUPKEZMUNUPKUMLZURFZBUMKNUSKGLUTULUTAKGUJK
      RUKURUJKOPQSTUAUBUIUOUQUCZUNUIKCVAUDBKUEUFUGUH $.
  $}

  ${
    $d A x $.  $d ch x $.  $d ph y $.  $d ps x $.  $d x y $.
    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 the finite ordinals.
       (Contributed by Scott Fenton, 17-Jul-2015.)  (Proof shortened by BJ,
       16-Oct-2024.) $)
    omsinds $p |- ( A e. _om -> ch ) $=
      ( com cep con0 wss wwe omsson epweon wess mp2 epse wral cv wcel cpred wtr
      wceq trom trpred mpan raleqdv sylbid wfis3 ) ABCDEJFKJLMLKNJKNOPJLKQRJSGH
      DUAZJUBZBEJKULUCZTBEULTAUMBEUNULJUDUMUNULUEUFJULUGUHUIIUJUK $.
  $}

  $( The union of two finite ordinals is a finite ordinal.  (Contributed by
     Scott Fenton, 15-Mar-2025.) $)
  omun $p |- ( ( A e. _om /\ B e. _om ) -> ( A u. B ) e. _om ) $=
    ( com wcel wa wss cun wceq ssequn1 wi eleq1a adantl biimtrid ssequn2 adantr
    word wo nnord ordtri2or2 syl2an mpjaod ) ACDZBCDZEZABFZABGZCDZBAFZUEUFBHZUD
    UGABIUCUIUGJUBBCUFKLMUHUFAHZUDUGBANUBUJUGJUCACUFKOMUBAPBPUEUHQUCARBRABSTUA
    $.

$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Peano's postulates
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Zero is a natural number.  One of Peano's five postulates for arithmetic.
     Proposition 7.30(1) of [TakeutiZaring] p. 42.  Note:  Unlike most
     textbooks, our proofs of ~ peano1 through ~ peano5 do not use the Axiom of
     Infinity.  Unlike Takeuti and Zaring, they also do not use the Axiom of
     Regularity.  (Contributed by NM, 15-May-1994.)  Avoid ~ ax-un .  (Revised
     by BTernaryTau, 29-Nov-2024.) $)
  peano1 $p |- (/) e. _om $=
    ( vx c0 com wcel con0 cv wlim wi wal 0elon 0ellim ax-gen elom mpbir2an ) BC
    DBEDAFZGBODHZAIJPAOKLABMN $.
  $( $j usage 'peano1' avoids 'ax-un' 'ax-inf' 'ax-inf2' 'ax-reg'; $)

  $( Obsolete version of ~ peano1 as of 29-Nov-2024.  (Contributed by NM,
     15-May-1994.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  peano1OLD $p |- (/) e. _om $=
    ( com wlim c0 wcel limom 0ellim ax-mp ) ABCADEAFG $.

  $( 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 ) $=
    ( com wcel csuc peano2b biimpi ) ABCADBCAEF $.
  $( $j usage 'peano2' avoids 'ax-inf' 'ax-inf2' 'ax-reg'; $)

  $( 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 =/= (/) ) $=
    ( csuc c0 wne com wcel nsuceq0 a1i ) ABCDAEFAGH $.
  $( $j usage 'peano3' avoids 'ax-inf' 'ax-inf2' 'ax-reg'; $)

  $( 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 wcel con0 csuc wceq wb nnon suc11 syl2an ) ACDAEDBEDAFBFGABGHBCDAIBIA
    BJK $.
  $( $j usage 'peano4' avoids 'ax-inf' 'ax-inf2' 'ax-reg'; $)

  ${
    $d x y z 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, except our proof does not require the Axiom of Infinity.  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.)  Avoid
       ~ ax-10 , ~ ax-12 .  (Revised by GG, 3-Oct-2024.) $)
    peano5 $p |- ( ( (/) e. A /\
                 A. x e. _om ( x e. A -> suc x e. A ) ) -> _om C_ A ) $=
      ( vz vy c0 wcel cv csuc wi com wral wa cdif wceq wss wrex wne eleq1 exp32
      wn cin eldifn adantl eldifi biimpcd necon3bd mpan9 nnsuc syl2an2 ad4ant13
      elndif weq eleq1w suceq eleq1d imbi12d rspccv wel vex sucid eleq2 peano2b
      mpbiri bitr4di minel neldif sylan2 biimtrdi mpid syl5 impd eleq1a imim12d
      com12 com13 sylan9 rexlimdv a1i imp41 mtand nrexdv word ordom difss tz7.5
      mpd mp3an12 necon1bi syl ssdif0 sylibr ) EBFZAGZBFZWMHZBFZIZAJKZLZJBMZENZ
      JBOWSWTCGZUAENZCWTPZTXAWSXCCWTWSXBWTFZLZXCXBBFZXEXGTWSXBJBUBUCXFXCLXBDGZH
      ZNZDJPZXGWLXEXKWRXCXEXBJFZWLXBEQZXKXBJBUDZWLEWTFZTXEXMEBJUKXEXOXBEXBENXEX
      OXBEWTRUEUFUGDXBUHUIUJWLWRXEXCXKXGIZWRXEXCXPIIIWLWRXEXCXPWRXEXCLZLXJXGDJW
      RXHJFZXHBFZXIBFZIZXQXJXGIWQYAAXHJADULZWNXSWPXTADBUMYBWOXIBWMXHUNUOUPUQXJY
      AXQXGXJXQXSXTXGXJXEXCXSXEXLXJXCXSIZXNXJXLDCURZYCXJYDXHXIFXHDUSUTXBXIXHVAV
      CXJXLXRYDYCIXJXLXIJFXRXBXIJRXHVBVDXRYDXCXSYDXCLXRXHWTFTXSXHXBWTVEXHJBVFVG
      SVHVIVJVKXTXJXGXIBXBVLVNVMVOVPVQSVRVSWFVTWAXDWTEJWBWTJOWTEQXDWCJBWDCJWTWE
      WGWHWIJBWJWK $.
    $( $j usage 'peano5' avoids 'ax-10' 'ax-12' 'ax-inf' 'ax-inf2' 'ax-reg'; $)
  $}

  ${
    $d x A $.
    $( A natural number is either 0 or a successor.  (Contributed by NM,
       27-May-1998.) $)
    nn0suc $p |- ( A e. _om -> ( A = (/) \/ E. x e. _om A = suc x ) ) $=
      ( com wcel c0 wceq cv csuc wrex wn wne df-ne nnsuc sylan2br ex orrd ) BCD
      ZBEFZBAGHFACIZQRJZSTQBEKSBELABMNOP $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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.)
       (Proof shortened by Wolf Lammen, 28-May-2024.) $)
    find $p |- A = _om $=
      ( com wss c0 wcel cv csuc wral simp1i wi wa w3a 3simpc df-ral alral sylbi
      wal anim2i mp2b peano5 ax-mp eqssi ) BDBDEZFBGZAHZIBGZABJZCKUFUGBGUHLZADJ
      ZMZDBEUEUFUINUFUIMULCUEUFUIOUIUKUFUIUJASUKUHABPUJADQRTUAABUBUCUD $.
  $}

  ${
    $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 A $.  $d x y B $.  $d x ps $.  $d x ch $.  $d x th $.  $d x ta $.
    $d y ph $.
    $( Substitutions. $)
    findsg.1 $e |- ( x = B -> ( ph <-> ps ) ) $.
    findsg.2 $e |- ( x = y -> ( ph <-> ch ) ) $.
    findsg.3 $e |- ( x = suc y -> ( ph <-> th ) ) $.
    findsg.4 $e |- ( x = A -> ( ph <-> ta ) ) $.
    $( Basis. $)
    findsg.5 $e |- ( B e. _om -> ps ) $.
    $( Induction step. $)
    findsg.6 $e |- ( ( ( y e. _om /\ B e. _om ) /\ B C_ y )
        -> ( 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.  The basis of
       this version is an arbitrary natural number ` B ` instead of zero.
       (Contributed by NM, 16-Sep-1995.) $)
    findsg $p |- ( ( ( A e. _om /\ B e. _om ) /\ B C_ A ) -> ta ) $=
      ( wcel wss wi wceq wa com cv c0 csuc sseq2 adantl eqeq2 biimtrrdi imbi12d
      wb imp wn imbi1d ss0 con3i pm2.21d pm5.74d sylan9bbr pm2.61ian imbi2d a1d
      wex vex sucex eqvinc imbitrrid biimpd sylan9r exlimiv sylbi eqcoms imim2i
      com4r wne df-ne anbi2i annim bitri con0 nnon onsssuc onsuc onelpss sylan2
      bitrd syl2anr ex a1ddd a2d com23 sylbird biimtrrid pm2.61d finds imp31 )
      HUAPIUAPZIHQZEWPIFUBZQZARZRWPIUCQZBRZRWPIGUBZQZCRZRWPIXCUDZQZDRZRWPWQERZR
      FGHWRUCSZWTXBWPIUCSZXJWTXBUJXKXJTWSXAABXJWSXAUJXKWRUCIUEZUFXKXJABUJZXKXJW
      RISZXMIUCWRUGJUHUKUIXJWTXAARXKULZXBXJWSXAAXLUMXOXAABXOXAXMXAXKIUNUOUPUQUR
      USUTWRXCSZWTXEWPXPWSXDACWRXCIUEKUIUTWRXFSZWTXHWPXQWSXGADWRXFIUELUIUTWRHSZ
      WTXIWPXRWSWQAEWRHIUEMUIUTWPBXANVAXCUAPZWPXEXHXSWPXEXHRZXSWPTZXGIXFSZRZXTW
      PYCXTRXSYCXEXGWPDYCXGWPDRZRXEYBYDXGYDXFIXFISXQXNTZFVBYDFXFIXCGVCVDVEYEYDF
      XNWPAXQDWPAXNBNJVFXQADLVGVHVIVJVKVLVAVMUFYCULZXGIXFVNZTZYAXTYHXGYBULZTYFY
      GYIXGIXFVOVPXGYBVQVRYAYHXDXTWPIVSPZXCVSPZXDYHUJXSIVTXCVTYJYKTXDIXFPZYHIXC
      WAYKYJXFVSPYLYHUJXCWBIXFWCWDWEWFYAXEXDXHYAXDCXHYAXDCXGDYAXDCDROWGWHWIWJWK
      WLWMWGWIWNWO $.
  $}

  ${
    $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.  See ~ tfindes for the transfinite version.  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 eleq1w sbequ12 suceq sbceq1d chvarfv finds ) ABEG
      ABHIABFGZABFSZJZIZAEFBSZABEHKAEFBLABEULKAEBMCUNNOZAABUNJZIZTZTUKNOZUJUMTZ
      TBFUSUTBUSBPUJUMBABFUAABULUBQQBFUCZUOUSURUTBFNUDVAAUJUQUMABFUEVAABUPULUNU
      KUFUGRRDUHUI $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Relations and functions (cont.)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( 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 $.
  $}

  ${
    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 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 $.

  $( Alternate proof of ~ dmfex : shorter but using ~ ax-rep .  (Contributed by
     NM, 27-Aug-2006.)  (Proof shortened by Andrew Salmon, 17-Sep-2011.)
     (Proof shortened by AV, 23-Aug-2024.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  dmfexALT $p |- ( ( F e. C /\ F : A --> B ) -> A e. _V ) $=
    ( wcel cvv wf elex fdmexb biimprd mpan9 ) DCEDFEZABDGZAFEZDCHMNLABDIJK $.

  ${
    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, as in ~ idcn .  (Contributed by NM,
     1-Jan-2007.) $)
  iprc $p |- -. _I e. _V $=
    ( cid cvv wcel cdm dmi vprc eqneltri dmexg mto ) ABCADZBCJBBEFGABHI $.

  $( The existence of a restricted identity function, proved without using the
     Axiom of Replacement (unlike ~ resfunexg ).  (Contributed by NM,
     13-Jan-2007.)  (Proof shortened by Peter Mazsa, 2-Oct-2022.) $)
  resiexg $p |- ( A e. V -> ( _I |` A ) e. _V ) $=
    ( wcel cid cres cxp wss cvv idssxp sqxpexg ssexg sylancr ) ABCDAEZAAFZGNHCM
    HCAIABJMNHKL $.

  $( 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 $.
  $}

  ${
    rnexd.1 $e |- ( ph -> A e. V ) $.
    $( The range of a set is a set.  Deduction version of ~ rnexd .
       (Contributed by Thierry Arnoux, 14-Feb-2025.) $)
    rnexd $p |- ( ph -> ran A e. _V ) $=
      ( wcel crn cvv rnexg syl ) ABCEBFGEDBCHI $.

    $( The image of a set is a set.  Deduction version of ~ imaexg .
       (Contributed by Thierry Arnoux, 14-Feb-2025.) $)
    imaexd $p |- ( ph -> ( A " B ) e. _V ) $=
      ( wcel cima cvv imaexg syl ) ABDFBCGHFEBCDIJ $.
  $}

  ${
    $d x y A $.  $d x y R $.  $d x V $.
    $( 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 $.
  $}

  $( If a Cartesian product is a set, one of its components must be a set.
     (Contributed by NM, 27-Aug-2006.) $)
  xpexr $p |- ( ( A X. B ) e. C -> ( A e. _V \/ B e. _V ) ) $=
    ( cxp wcel cvv wn wi wceq 0ex eleq1 mpbiri pm2.24d a1d wne crn rnexg eleq1d
    c0 rnxp imbitrid a1dd pm2.61ine orrd ) ABDZCEZAFEZBFEZUFUGGZUHHZHASASIZUJUF
    UKUGUHUKUGSFEJASFKLMNASOZUFUHUIUFUEPZFEULUHUECQULUMBFABTRUAUBUCUD $.

  $( If a nonempty Cartesian product is a set, so are both of its components.
     (Contributed by NM, 27-Aug-2006.) $)
  xpexr2 $p |- ( ( ( A X. B ) e. C /\ ( A X. B ) =/= (/) ) ->
               ( A e. _V /\ B e. _V ) ) $=
    ( cxp c0 wne wcel wa cvv xpnz cdm wceq dmxp adantl adantr eqeltrrd crn rnxp
    dmexg rnexg anim12dan ancom2s sylan2br ) ABDZEFUDCGZAEFZBEFZHAIGZBIGZHZABJU
    EUGUFUJUEUGUHUFUIUEUGHUDKZAIUGUKALUEABMNUEUKIGUGUDCSOPUEUFHUDQZBIUFULBLUEAB
    RNUEULIGUFUDCTOPUAUBUC $.

  $( 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.) $)
  xpexcnv $p |- ( ( B =/= (/) /\ ( A X. B ) e. _V ) -> A e. _V ) $=
    ( c0 wne cxp cvv wcel cdm dmexg dmxp eleq1d imbitrid imp ) BCDZABEZFGZAFGZP
    OHZFGNQOFINRAFABJKLM $.

  ${
    $d x A $.  $d x R $.  $d x V $.
    $( If the relation in a strict order is a set, then the base field is also
       a set.  (Contributed by Mario Carneiro, 27-Apr-2015.) $)
    soex $p |- ( ( R Or A /\ R e. V ) -> A e. _V ) $=
      ( vx wor wcel wa cvv c0 wceq simpr 0ex eqeltrdi wne cv wex cun unexg wss
      n0 csn cdm crn vsnex dmexg rnexg syl2anc sylancr ad2antlr sossfld adantlr
      cdif ssundif sylibr ssexd ex exlimdv imp sylan2b pm2.61dane ) ABEZBCFZGZA
      HFZAIVCAIJZGAIHVCVEKLMAINVCDOZAFZDPZVDDATVCVHVDVCVGVDDVCVGVDVCVGGZAVFUAZB
      UBZBUCZQZQZHVBVNHFZVAVGVBVJHFVMHFZVODUDVBVKHFVLHFVPBCUEBCUFVKVLHHRUGVJVMH
      HRUHUIVIAVJULVMSZAVNSVAVGVQVBAVFBUJUKAVJVMUMUNUOUPUQURUSUT $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.
    $( Membership in a Cartesian product.  This version requires no quantifiers
       or dummy variables.  See also ~ elxp5 , ~ elxp6 , and ~ elxp7 .
       (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 cv cop wceq wa wex csn cdm cuni crn sneq unieqd vex pm4.71ri
      eqtr2di elxp rneqd op2nda anbi1i anass bitri exbii snex rnex uniex eqeq2d
      opeq2 eleq1 anbi2d anbi12d ceqsexv dmeqd op1sta 3bitri dmex opeq1 anbi1d
      cxp ) ABCVCFADGZEGZHZIZVDBFZVECFZJZJZEKZDKVDALZMZNZIZAVDVMOZNZHZIZVHVRCFZ
      JZJZJZDKAVOVRHZIZVOBFZWAJZJZDEABCUAVLWDDVLWCVPVTJZWBJWDVLVEVRIZVKJZEKWCVK
      WLEVKWKVGJZVJJWLVGWMVJVGWKVGVRVFLZOZNVEVGVQWOVGVMWNAVFPUBQVDVEDRZERUCTSUD
      WKVGVJUEUFUGVKWCEVRVQVMAUHZUIUJZWKVGVTVJWBWKVFVSAVEVRVDULUKWKVIWAVHVEVRCU
      MUNUOUPUFVTWJWBVTVPVTVOVSLZMZNVDVTVNWTVTVMWSAVSPUQQVDVRWPWRURTSUDVPVTWBUE
      USUGWCWIDVOVNVMWQUTUJVPVTWFWBWHVPVSWEAVDVOVRVAUKVPVHWGWAVDVOBUMVBUOUPUS
      $.

    $( Membership in a Cartesian 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 cv cop wceq wa wex cint csn crn cuni eqtr2di pm4.71ri anbi1i
      vex anass elxp sneq rneqd unieqd op2nda bitri exbii snex rnex uniex opeq2
      cxp eqeq2d eleq1 anbi2d anbi12d ceqsexv inteqd op1stb 3bitri cvv eqvisset
      inteq adantr exlimiv elex ad2antrl opeq1 anbi1d ceqsexgv pm5.21nii ) ABCU
      LFADGZEGZHZIZVLBFZVMCFZJZJZEKZDKVLALZLZIZAVLAMZNZOZHZIZVPWFCFZJZJZJZDKZAW
      BWFHZIZWBBFZWIJZJZDEABCUAVTWLDVTWKWCWHJZWJJWLVTVMWFIZVSJZEKWKVSXAEVSWTVOJ
      ZVRJXAVOXBVRVOWTVOWFVNMZNZOVMVOWEXDVOWDXCAVNUBUCUDVLVMDSZESUEPQRWTVOVRTUF
      UGVSWKEWFWEWDAUHUIUJZWTVOWHVRWJWTVNWGAVMWFVLUKUMWTVQWIVPVMWFCUNUOUPUQUFWH
      WSWJWHWCWHWBWGLZLVLWHWAXGAWGVCURVLWFXEXFUSPQRWCWHWJTUTUGWMWBVAFZWRWLXHDWC
      XHWKDWBVBVDVEWPXHWOWIWBBVFVGWKWRDWBVAWCWHWOWJWQWCWGWNAVLWBWFVHUMWCVPWPWIV
      LWBBUNVIUPVJVKUT $.
  $}

  $( 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 xpexd ssexg sylancr ) ABCZADZUCEZUCFZGZHZUFICUCICUCJUGAKU
    CLMUBUDUEIIUBUDAFIANABPOUBUEAEIAQABROSUCUFITUA $.

  ${
    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 $.

  $( If the range of a 1-1 onto function is a set, the function itself is a
     set.  (Contributed by AV, 2-Jun-2019.) $)
  f1oexrnex $p |- ( ( F : A -1-1-onto-> B /\ B e. V ) -> F e. _V ) $=
    ( wf1o wcel wa cvv ccnv wf f1ocnv f1of 3syl fex sylancom wrel f1orel adantr
    simpl wb relcnvexb syl mpbird ) ABCEZBDFZGZCHFZCIZHFZUDUEBAUHJZUIUFUDBAUHEU
    JUDUESABCKBAUHLMBADUHNOUFCPZUGUITUDUKUEABCQRCUAUBUC $.

  ${
    $d A f g $.  $d B f g $.
    $( There is a one-to-one onto function from a set to a second set iff there
       is a one-to-one onto function from the second set to the first set.
       (Contributed by Alexander van der Vekens, 30-Sep-2018.) $)
    f1oexbi $p |- ( E. f f : A -1-1-onto-> B <-> E. g g : B -1-1-onto-> A ) $=
      ( cv wf1o wex ccnv cvv wcel vex cnvex f1ocnv f1oeq1 spcegv exlimiv impbii
      mpsyl ) ABCEZFZCGZBADEZFZDGZTUDCSHZIJTBAUEFZUDSCKLABSMUCUFDUEIBAUBUENORPU
      CUADUBHZIJUCABUGFZUAUBDKLBAUBMTUHCUGIABSUGNORPQ $.
  $}

  $( 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 $.
  $}

  ${
    coexd.1 $e |- ( ph -> A e. V ) $.
    coexd.2 $e |- ( ph -> B e. W ) $.
    $( The composition of two sets is a set.  (Contributed by SN,
       7-Feb-2025.) $)
    coexd $p |- ( ph -> ( A o. B ) e. _V ) $=
      ( wcel ccom cvv coexg syl2anc ) ABDHCEHBCIJHFGBCDEKL $.
  $}

  ${
    $d f g x y z w v A $.
    $( 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 orbi12d
      cuni cnveq eqeq2d cbvrexvw funeqd sseq1 sseq2 ralbidv anbi12d rspcv funeq
      cab wcel biimprcd cnvss orim12i wb sseq12 ancoms syl5ibrcom expd rexlimdv
      syl6com com23 alrimdv anim12ii biimtrid alrimiv df-ral eqeq1 rexbidv elab
      vex ralab anbi2i imbi12i albii bitr2i sylib fununi syl ciun cnvuni dfiun2
      cnvex eqtri funeqi sylibr ) BIZJZKZWJCIZLZWMWJLZMZCANZOZBANZDIZEIZJZPZEAQ
      ZDUMZUBZKZAUBJZKWSFIZKZXIGIZLZXKXILZMZGXENZOZFXENZXGWSXIXBPZEAQZXJXKXBPZE
      AQZXNRZGSZOZRZFSZXQWSYEFXSXIHIZJZPZHAQWSYDXRYIEHAEHTXBYHXIXAYGUCUDUEWSYIY
      DHAYGAUNWSYHKZYGWMLZWMYGLZMZCANZOZYIYDRWRYOBYGABHTZWLYJWQYNYPWKYHWJYGUCUF
      YPWPYMCAYPWNYKWOYLWJYGWMUGWJYGWMUHUAUIUJUKYJYIXJYNYCYIXJYJXIYHULUOYNYIYBG
      YNYAYIXNYNXTYIXNRZEAXAAUNYNYGXALZXAYGLZMZXTYQRYMYTCXAACETYKYRYLYSWMXAYGUH
      WMXAYGUGUAUKYTXTYIXNYTXNXTYIOZYHXBLZXBYHLZMYRUUBYSUUCYGXAUPXAYGUPUQUUAXLU
      UBXMUUCYIXTXLUUBURXIYHXKXBUSUTXKXBXIYHUSUAVAVBVDVCVEVFVGVDVCVHVIXQXIXEUNZ
      XPRZFSYFXPFXEVJUUEYEFUUDXSXPYDXDXSDXIFVNDFTXCXREAWTXIXBVKVLVMXOYCXJXDYAXN
      GDDGTXCXTEAWTXKXBVKVLVOVPVQVRVSVTXEFGWAWBXHXFXHEAXBWCXFEAWDEDAXBXAEVNWFWE
      WGWHWI $.

    $( 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 $.
  $}

  ${
    $d A x y z $.  $d B x y z $.  $d C x y z $.  $d C w x y $.  $d F x y z $.
    $d F w $.  $d X x y z $.
    $( Extension of an injection which is a restriction of a function.
       (Contributed by AV, 3-Oct-2025.) $)
    resf1extb $p |- ( ( F : A --> B /\ X e. ( A \ C ) /\ C C_ A )
                -> ( ( ( F |` C ) : C -1-1-> B /\ ( F ` X ) e/ ( F " C ) )
                   <-> ( F |` ( C u. { X } ) ) : ( C u. { X } ) -1-1-> B ) ) $=
      ( vy vz vx wcel cfv wa wne wi wral adantr wceq fveq2 fvresd neeq12d syl
      vw wf cdif wss w3a cres wf1 cima wnel csn cun cv simp1 simp3 eldifi snssd
      3ad2ant2 unssd fssresd wo elun anbi12i dff14a neeq1 neeq1d imbi12d neeq2d
      neeq2 rspc2v simpl simpr imbi2d bi23imp13 wb elun1 adantl 3ad2ant1 mpbird
      3exp syldc a1i biimtrid a1dd imp32 wn wrex fvelimabd notbid df-nel ralnex
      wfn ffn 3bitr4g df-ne bitr3id rspcv ad2antll eqcomd elsni ad2antrl fveq2d
      elun2 eqtr4d biimpa necomd a1d syld com24 sylbid impcomd imp biimpd velsn
      eqtr3 eqneqall syl2anb ccased ralrimivv sylanbrc ccnv wfun fssres 3adant2
      df-f1 funres11 simplbiim ssun1 resabs1i eqcomi cnveqi funeqi sylibr snidg
      ex syl2anr eldifn nelelne pm2.27 sylibd expimpd impancom neneqd ralrimiva
      jca impbida ) ABDUBZEACUCZIZCAUDZUEZCBDCUFZUGZEDJZDCUHZUIZKZCEUJZUKZBDUUR
      UFZUGZUUJUUPKZUURBUUSUBZFULZGULZLZUVCUUSJZUVDUUSJZLZMZGUURNFUURNZUUTUUJUV
      BUUPUUJABUURDUUFUUHUUIUMUUJCUUQAUUFUUHUUIUNZUUHUUFUUQAUDUUIUUHEAEACUOUPUQ
      URUSOUVAUVIFGUURUURUVCUURIZUVDUURIZKUVCCIZUVCUUQIZUTZUVDCIZUVDUUQIZUTZKUV
      AUVIUVLUVPUVMUVSUVCCUUQVAUVDCUUQVAVBUVAUVNUVQUVOUVRUVIUUJUULUUOUVNUVQKZUV
      IMZUUJUULUWAUUOUULCBUUKUBZUAULZHULZLZUWCUUKJZUWDUUKJZLZMZHCNUACNZKZUUJUWA
      UAHCBUUKVCUWKUWAMUUJUWJUWAUWBUVTUWJUVEUVCUUKJZUVDUUKJZLZMZUVIUWIUWOUVCUWD
      LZUWLUWGLZMUAHUVCUVDCCUWCUVCPZUWEUWPUWHUWQUWCUVCUWDVDUWRUWFUWLUWGUWCUVCUU
      KQVEVFUWDUVDPZUWPUVEUWQUWNUWDUVDUVCVHUWSUWGUWMUWLUWDUVDUUKQVGVFVIUVTUWOUV
      EUVHUVTUWOUVEUEUVHUVCDJZUVDDJZLZUVTUWOUVEUXBUVTUWNUXBUVEUVTUWLUWTUWMUXAUV
      TUVCCDUVNUVQVJRUVTUVDCDUVNUVQVKRSVLVMUVTUWOUVHUXBVNUVEUVTUVFUWTUVGUXAUVTU
      VCUURDUVNUVLUVQUVCCUUQVOZORUVTUVDUURDUVQUVMUVNUVDCUUQVOZVPRSVQVRVSVTVPWAW
      BWCWDUUJUUPUVOUVQKZUVIMZUUJUUOUULUXFUUJUUOUWDDJZUUMPZWEZHCNZUULUXFMUUJUUM
      UUNIZWEUXHHCWFZWEUUOUXJUUJUXKUXLUUJHACUUMDUUFUUHDAWKUUIABDWLVQUVKWGWHUUMU
      UNWIUXHHCWJWMZUUJUXEUULUXJUVIUUJUXEUULUXJUVIMZMZUUJUXEKZUXNUULUXPUXJUXAUU
      MLZUVIUVQUXJUXQMUUJUVOUXIUXQHUVDCUXIUXGUUMLZUWSUXQUXGUUMWNZUWSUXGUXAUUMUW
      DUVDDQVEWOWPWQUXPUXQUVIUXPUXQKZUVHUVEUXTUVGUVFUXPUXQUVGUVFLUXPUXAUVGUUMUV
      FUXPUVGUXAUXPUVDUURDUVQUVMUUJUVOUXDWQRWRUXPUUMUWTUVFUXPEUVCDUVOEUVCPUUJUV
      QUVOUVCEUVCEWSWRWTXAUXPUVCUURDUVOUVLUUJUVQUVCUUQCXBWTRXCSXDXEXFYNXGXFYNXH
      XIXJXKUUJUUPUVNUVRKZUVIMZUUJUUOUULUYBUUJUUOUXJUULUYBMUXMUUJUYAUULUXJUVIUU
      JUYAUXOUUJUYAKZUXNUULUYCUXJUWTUUMLZUVIUVNUXJUYDMUUJUVRUXIUYDHUVCCUXIUXRUW
      DUVCPZUYDUXSUYEUXGUWTUUMUWDUVCDQVEWOWPWTUYCUYDUVHUVEUYCUYDUVHUYCUWTUVFUUM
      UVGUYCUVFUWTUYCUVCUURDUVNUVLUUJUVRUXCWTRWRUYCUUMUXAUVGUYCEUVDDUVREUVDPUUJ
      UVNUVRUVDEUVDEWSWRWQXAUYCUVDUURDUVRUVMUUJUVNUVDUUQCXBWQRXCSXLWCXGXFYNXHXI
      XJXKUVOUVRKUVIMUVAUVOUVCEPZUVDEPZUVIUVRFEXMGEXMUYFUYGKUVCUVDPUVIUVCUVDEXN
      UVHUVCUVDXOTXPWAXQWBXRFGUURBUUSVCZXSUUJUUTKZUULUUOUYIUWBUUKXTZYAZUULUUJUW
      BUUTUUFUUIUWBUUHABCDYBYCOUYIUUSCUFZXTZYAZUYKUUTUYNUUJUUTUVBUUSXTYAUYNUURB
      UUSYDCUUSYEYFVPUYJUYMUUKUYLUYLUUKDCUURCUUQYGYHYIYJYKYLCBUUKYDXSUYIUUOUXJU
      YIUXIHCUYIUWDCIZKUXGUUMUYIUYOUXRUUJUYOUUTUXRUUTUVBUVJKUUJUYOKZUXRUYHUYPUV
      BUVJUXRUYPUVBKZUVJUWDELZUWDUUSJZEUUSJZLZMZUXRUYPUVJVUBMZUVBUYOUWDUURIZEUU
      RIZVUCUUJUWDCUUQVOZUUJEUUQIZVUEUUHUUFVUGUUIEUUGYMZUQEUUQCXBZTUVIVUBUWDUVD
      LZUYSUVGLZMFGUWDEUURUURUVCUWDPZUVEVUJUVHVUKUVCUWDUVDVDVULUVFUYSUVGUVCUWDU
      USQVEVFUYGVUJUYRVUKVUAUVDEUWDVHUYGUVGUYTUYSUVDEUUSQVGVFVIYOOUYQVUBVUAUXRU
      YQUYRVUBVUAMUYPUYRUVBUUJUYOUYRUUHUUFUYOUYRMZUUIUUHECIWEVUMEACYPECUWDYQTUQ
      XKOUYRVUAYRTUYQUYSUXGUYTUUMUYQUWDUURDUYPVUDUVBUYOVUDUUJVUFVPORUYPUYTUUMPU
      VBUYPEUURDUUJVUEUYOUUHUUFVUEUUIUUHVUGVUEVUHVUITUQOROSYSXGYTWBUUAXKUUBUUCU
      UJUUOUXJVNUUTUXMOVRUUDUUE $.
  $}

  $( Extension of an injection which is a restriction of a function.
     (Contributed by AV, 3-Oct-2025.) $)
  resf1ext2b $p |- ( ( F : A --> B /\ X e. ( A \ C ) /\ C C_ A )
                     -> ( ( Fun `' ( F |` C ) /\ ( F ` X ) e/ ( F " C ) )
                        <-> Fun `' ( F |` ( C u. { X } ) ) ) ) $=
    ( wf cdif wcel wss w3a cres ccnv wfun cfv wf1 df-f1 simprbi biimtrrid mpand
    wa cima wnel csn wi fssres 3adant2 resf1extb biimtrdi expd impd simp1 simp3
    cun eldifi snssd 3ad2ant2 unssd fssresd anim1i biimtrrdi impbid ) ABDFZEACG
    HZCAIZJZDCKZLMZEDNDCUAUBZTZDCEUCZUMZKZLMZVEVGVHVMVECBVFFZVGVHVMUDZVBVDVNVCA
    BCDUEUFVNVGTCBVFOZVEVOCBVFPZVEVPVHVMVEVPVHTZVKBVLOZVMABCDEUGZVSVKBVLFZVMVKB
    VLPZQUHUIRSUJVEWAVMVIVEABVKDVBVCVDUKVECVJAVBVCVDULVCVBVJAIVDVCEAEACUNUOUPUQ
    URWAVMTVSVEVIWBVEVSVRVIVTVPVGVHVPVNVGVQQUSUTRSVA $.

  $( A function with bounded domain and codomain is a set.  This version of
     ~ fex is proven without the Axiom of Replacement ~ ax-rep , but depends on
     ~ ax-un , which is not required for the proof of ~ fex .  (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 $.

  ${
    $d X f $.  $d Y f $.  $d ph f $.
    fabexd.f $e |- ( ( ph /\ ps ) -> f : X --> Y ) $.
    fabexd.x $e |- ( ph -> X e. V ) $.
    fabexd.y $e |- ( ph -> Y e. W ) $.
    $( Existence of a set of functions.  In contrast to ~ fabex or ~ fabexg ,
       the condition in the class abstraction does not contain the function
       explicitly, but the function can be derived from it.  Therefore, this
       theorem is also applicable for more special functions like one-to-one,
       onto or one-to-one onto functions.  (Contributed by AV, 20-May-2025.) $)
    fabexd $p |- ( ph -> { f | ps } e. _V ) $=
      ( cab cxp cpw cvv xpexd pwexd cv wcel wa wf wss fssxp velpw sylibr syl ex
      abssdv ssexd ) ABCKFGLZMZNAUINAFGDEIJOPABCUJABCQZUJRZABSFGUKTZULHUMUKUIUA
      ULFGUKUBCUIUCUDUEUFUGUH $.
  $}

  ${
    $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.)  (Proof shortened by AV, 9-Jun-2025.) $)
    fabexg $p |- ( ( A e. C /\ B e. D ) -> F e. _V ) $=
      ( wcel cvv elex wa cv wf cab simprl simpl simpr fabexd eqeltrid syl2an )
      CEICJIZDJIZGJIDFICEKDFKUBUCLZGCDBMNZALZBOJHUDUFBJJCDUDUEAPUBUCQUBUCRSTUA
      $.

    $( Obsolete version of ~ fabexg as of 9-Jun-2025.  (Contributed by Paul
       Chapman, 25-Feb-2008.)  (New usage is discouraged.)
       (Proof modification is discouraged.) $)
    fabexgOLD $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 $.
  $}

  ${
    $d f A $.  $d f B $.
    $( 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.)  (Proof shortened by AV, 16-Jun-2025.) $)
    mapex $p |- ( ( A e. C /\ B e. D ) -> { f | f : A --> B } e. _V ) $=
      ( wcel wa cv wf cab cvv eqid fabexg wss id ancli ss2abi a1i ssexd ) ACFBD
      FGZABEHIZEJZUAUAGZEJZKUAEABCDUDUDLMUBUDNTUAUCEUAUAUAOPQRS $.
  $}

  ${
    $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.)  (Proof shortened by AV,
       9-Jun-2025.) $)
    f1oabexg $p |- ( ( A e. C /\ B e. D ) -> F e. _V ) $=
      ( wcel cvv elex wa cv wf1o cab wf f1of ad2antrl simpl simpr fabexd syl2an
      eqeltrid ) BDIBJIZCJIZGJICEIBDKCEKUDUELZGBCFMZNZALZFOJHUFUIFJJBCUHBCUGPUF
      ABCUGQRUDUESUDUETUAUCUB $.

    $( Obsolete version of ~ f1oabexg as of 9-Jun-2025.  (Contributed by Paul
       Chapman, 25-Feb-2008.)  (New usage is discouraged.)
       (Proof modification is discouraged.) $)
    f1oabexgOLD $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 $.
  $}

  ${
    $d A v x z $.  $d A v y $.  $d B v y $.  $d B z $.  $d C v x $.  $d D v $.
    $d S v $.  $d u v y $.  $d x y $.
    fiun.1 $e |- ( x = y -> B = C ) $.
    $( Lemma for ~ fiun and ~ f1iun .  Formerly part of ~ f1iun .  (Contributed
       by AV, 6-Oct-2023.) $)
    fiunlem $p |- ( ( ( B : D --> S /\ A. y e. A ( B C_ C \/ C C_ B ) )
                    /\ u = B ) -> A. v e. { z | E. x e. A z = B }
                                     ( u C_ v \/ v C_ u ) ) $=
      ( wss wo wa cv wceq wrex weq sseq12 sylan2b wf wral cab vex eqeq1 rexbidv
      wcel elab eqeq2d cbvrexvw r19.29 wi wb orbi12d biimprcd expdimp rexlimivw
      ancoms imp sylan an32s adantlll ralrimiva ) IJGUAZGHLZHGLZMZBFUBZNEOZGPZN
      ZVIDOZLZVLVILZMZDCOZGPZAFQZCUCZVLVSUGVKVLGPZAFQZVOVRWACVLDUDCDRVQVTAFVPVL
      GUEUFUHWAVKVLHPZBFQZVOVTWBABFABRGHVLKUIUJVHVJWCVOVDVHWCVJVOVHWCNVGWBNZBFQ
      ZVJVOVGWBBFUKWEVJVOWDVJVOULBFVGWBVJVOWBVJNZVOVGWFVMVEVNVFVJWBVMVEUMVIGVLH
      SURVLHVIGSUNUOUPUQUSUTVAVBTTVC $.

    $d A u x z $.  $d B u $.  $d C u $.  $d D u $.  $d S u x $.
    fiun.2 $e |- B e. _V $.
    $( The union of a chain (with respect to inclusion) of functions is a
       function.  Analogous to ~ f1iun .  (Contributed by AV, 6-Oct-2023.) $)
    fiun $p |- ( A. x e. A ( B : D --> S /\ A. y e. A ( B C_ C \/ C C_ B ) )
                 -> U_ x e. A B : U_ x e. A D --> S ) $=
      ( vz vu vv wss wral wa ciun wfun cv wrex wcel wf wo wfn crn cdm wceq cuni
      cab vex weq eqeq1 rexbidv elab r19.29 nfv nfre1 nfab nfralw nfan wi funeq
      wb bianir syl2an adantlr fiunlem jca a1i rexlimi sylan2b ralrimiva fununi
      ffun syl dfiun2 funeqi sylibr cop wex fdm eleq2d bitr3id adantr ralrexbid
      eldm2 eliun exbii rexcom4 3bitr4i 3bitr4g eqrdv df-fn sylanbrc frn ralimi
      rniun iunss eqsstrid df-f ) FGDUAZDEMEDMUBBCNZOZACNZACDPZACFPZUCZXDUDZGMX
      EGXDUAXCXDQZXDUEZXEUFXFXCJRZDUFZACSZJUHZUGZQZXHXCKRZQZXPLRZMXRXPMUBZLXMNZ
      OZKXMNXOXCYAKXMXPXMTXCXPDUFZACSZYAXLYCJXPKUIZJKUJXKYBACXJXPDUKULUMXCYCOXB
      YBOZACSYAXBYBACUNYEYAACXQXTAXQAUOXSALXMXLAJXKACUPUQXSAUOURUSYEYAUTARCTYEX
      QXTWTYBXQXAWTDQZXQYFVBXQYBFGDVMXPDVAYFXQVCVDVEABJLKCDEFGHVFVGVHVIVNVJVKXM
      KLVLVNXDXNAJCDIVOVPVQXCKXIXEXCXPXRVRZDTZLVSZACSZXPFTZACSXPXITZXPXETXBYIYK
      ACWTYIYKVBXAYIXPDUEZTWTYKLXPDYDWEWTYMFXPFGDVTWAWBWCWDYGXDTZLVSYHACSZLVSYL
      YJYNYOLAYGCDWFWGLXPXDYDWEYHALCWHWIAXPCFWFWJWKXDXEWLWMXCXGACDUDZPZGACDWPXC
      YPGMZACNYQGMXBYRACWTYRXAFGDWNWCWOACYPGWQVQWRXEGXDWSWM $.

    $( 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.)  (Proof shortened by AV,
       5-Nov-2023.) $)
    f1iun $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 nfralw wi f1eq1
      nfan biimparc df-f1 ffun anim1i sylbi syl adantlr f1f fiunlem sylanl1 jca
      a1i rexlimi sylan2b ralrimiva fun11uni simpld dfiun2 funeqi sylibr cop wb
      eldm2 eleq2d bitr3id adantr ralrexbid eliun exbii rexcom4 3bitr4i 3bitr4g
      f1dm eqrdv df-fn sylanbrc rniun frnd ralimi iunss eqsstrid simprd cnveqi
      wex df-f ) FGDUAZDEMEDMUBBCNZOZACNZACFUCZGACDUCZUDZXMUEZPZXLGXMUAXKXMXLUF
      ZXMUGZGMXNXKXMPZXMUHZXLQXQXKJRZDQZACSZJUIZUJZPZXSXKYFYEUEZPZXKKRZPZYIUEPZ
      OZYILRZMYMYIMUBZLYDNZOZKYDNYFYHOXKYPKYDYIYDTXKYIDQZACSZYPYCYRJYIKUKZYAYIQ
      YBYQACYAYIDULUMUNXKYROXJYQOZACSYPXJYQACUOYTYPACYLYOAYLAUPYNALYDYCAJYBACUQ
      URYNAUPUSVBYTYPUTARCTYTYLYOXHYQYLXIXHYQOFGYIUAZYLYQUUAXHFGYIDVAVCUUAFGYIU
      DZYKOYLFGYIVDUUBYJYKFGYIVEVFVGVHVIXHFGDUDXIYQYOFGDVJZABJLKCDEFGHVKVLVMVNV
      OVHVPVQYDKLVRVHZVSXMYEAJCDIVTZWAWBXKKXTXLXKYIYMWCZDTZLXFZACSZYIFTZACSYIXT
      TZYIXLTXJUUHUUJACXHUUHUUJWDXIUUHYIDUHZTXHUUJLYIDYSWEXHUULFYIFGDWOWFWGWHWI
      UUFXMTZLXFUUGACSZLXFUUKUUIUUMUUNLAUUFCDWJWKLYIXMYSWEUUGALCWLWMAYICFWJWNWP
      XMXLWQWRXKXRACDUGZUCZGACDWSXKUUOGMZACNUUPGMXJUUQACXHUUQXIXHFGDUUCWTWHXAAC
      UUOGXBWBXCXLGXMXGWRXKYHXPXKYFYHUUDXDXOYGXMYEUUEXEWAWBXLGXMVDWR $.
  $}

  ${
    $d F i $.  $d I i $.  $d J i $.
    fviunfun.u $e |- U = U_ i e. I ( F ` i ) $.
    $( The function value of an indexed union is the value of one of the
       indexed functions.  (Contributed by AV, 4-Nov-2023.) $)
    fviunfun $p |- ( ( Fun U /\ J e. I /\ X e. dom ( F ` J ) )
                     -> ( U ` X ) = ( ( F ` J ) ` X ) ) $=
      ( wcel wfun cfv wss cdm wceq ciun fveq2 ssiun2s sseqtrrdi funssfv syl3an2
      cv ) EDHZAIECJZAKFUBLHFAJFUBJMUAUBBDBTZCJZNABDUDEUBUCECOPGQFAUBRS $.
  $}

  ${
    $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 $.
  $}

  $( Alternate proof of ~ resfunexg , shorter but requiring ~ 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 eqeltrid xpexd
    syl ) ADZBCEZFZABGZUMHZUMIZJZKZUPREUMREUMLUQABMUMNOULUNUORRUKUNREZUJUKUNBHZ
    KUSREURABPBCUAUNUSRQSUBULUOABIZUCZIZRABUDULVAREZVBREUKUJUTREVCBCTAUTRUEUFVA
    RTUIUGUHUMUPRQS $.

  $( 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 $.

  $( Alternate proof of ~ fnex , derived using the Axiom of Replacement in the
     form of ~ funimaexg .  This version 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-rep .  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-rep' ; $)

  ${
    $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-rep .  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-rep' ; $)
  $}

  $( 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 $.

  ${
    $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 in place of our ~ ax-rep .  (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 ) $=
      ( cv wral wrex wcel wa cvv wex cab wceq sylibr vex wi wal wmo 3imtr4i weu
      copab crn wfun crab 19.42v abbii dmopab df-rab 3eqtr4i euex ralimi rabid2
      cdm eqtr4id eqeltrdi eumo imim2i moanimv alimi funopab funrnex sylc nfra1
      df-ral eleq2d cop opabidw opelrn sylbir impac eximi eqabri df-rex ralrimi
      ex biimtrrdi nfopab1 nfrn nfeq2 nfcv nfopab2 rexeqf ralbid spcedv ) ACUAZ
      BDFZGZACEFZHZBWGGACBFZWGIZAJZBCUBZUCZHZBWGGEKWOWHWNUNZKIWNUDZWOKIWHWQWGKW
      HWQACLZBWGUEZWGWMCLZBMWLWSJZBMWQWTXAXBBWLACUFUGWMBCUHZWSBWGUIUJWHWSBWGGWG
      WTNWFWSBWGACUKULWSBWGUMOUOZDPUPWLWFQZBRWMCSZBRWHWRXEXFBXEWLACSZQXFWFXGWLA
      CUQURWLACUSOUTWFBWGVEWMBCVATKWNVBVCWHWPBWGWFBWGVDWHWLWKWQIZWPWHWQWGWKXDVF
      XACFZWOIZAJZCLXHWPWMXKCWLAXJWLAXJWMWKXIVGWNIXJWMBCVHWKXIWNBPCPVIVJVPVKVLX
      ABWQXCVMACWOVNTVQVOWIWONWJWPBWGBWIWOBWNWMBCVRVSVTACWIWOCWIWACWNWMBCWBVSWC
      WDWE $.
  $}

  $( 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 fofun funrnex syl5com fdmd eleq1d 3imtr3d com12
    fof forn ) ABDEZACFZBGFZTDHZCFZDIZGFZUAUBTDJUDUFABDKCDLMTUCACTABDABDRNOTUEB
    GABDSOPQ $.

  $( If the codomain of a one-to-one function exists, so does its domain.  This
     theorem is equivalent to the Axiom of Replacement ~ ax-rep .  (Contributed
     by NM, 4-Sep-2004.) $)
  f1dmex $p |- ( ( F : A -1-1-> B /\ B e. C ) -> A e. _V ) $=
    ( wf1 wcel cvv crn ccnv wfo wss f1f frnd ssexg sylan wf1o f1cnv f1ofo syl
    ex focdmex syl6ci imp ) ABDEZBCFZAGFZUDUEDHZGFZUGADIZJZUFUDUEUHUDUGBKUEUHUD
    ABDABDLMUGBCNOTUDUGAUIPUJABDQUGAUIRSUGAGUIUAUBUC $.

  $( The codomain/range of a 1-1 onto function is a set iff its domain is a
     set.  (Contributed by AV, 21-Mar-2019.) $)
  f1ovv $p |- ( F : A -1-1-onto-> B -> ( A e. _V <-> B e. _V ) ) $=
    ( wf1o cvv wcel wfo f1ofo focdmex syl5com wf1 wi f1of1 f1dmex ex syl impbid
    ) ABCDZAEFZBEFZRABCGSTABCHABECIJRABCKZTSLABCMUATSABECNOPQ $.

  ${
    $d x y F $.
    fvclex.1 $e |- F e. _V $.
    $( Existence of the class of values of a set.  (Contributed by NM,
       9-Nov-1995.) $)
    fvclex $p |- { y | E. x y = ( F ` x ) } e. _V $=
      ( cv cfv wceq wex cab crn c0 csn cun rnex snex unex fvclss ssexi ) BEAECF
      GAHBICJZKLZMSTCDNKOPABCQR $.
  $}

  ${
    $d x y z A $.  $d x y z F $.
    fvresex.1 $e |- A e. _V $.
    $( Existence of the class of values of a restricted class.  (Contributed by
       NM, 14-Nov-1995.)  (Revised by Mario Carneiro, 11-Sep-2015.) $)
    fvresex $p |- { y | E. x y = ( ( F |` A ) ` x ) } e. _V $=
      ( vz cv cfv cmpt wceq wex cab cres cvv wss ssv resmpt ax-mp fveq1i fveq2
      eqid fvex fvmpt fveqres eqtr3i eqeq2i exbii abbii mptex fvclex eqeltrri
      elv ) BGZAGZFCFGZDHZIZHZJZAKZBLUMUNDCMHZJZAKZBLNUTVCBUSVBAURVAUMUNFNUPIZC
      MZHZURVAUNVEUQCNOVEUQJCPFNCUPQRSUNVDHUNDHZJZVFVAJVHAFUNUPVGNVDUOUNDTVDUAU
      NDUBUCULUNCVDDUDRUEUFUGUHABUQFCUPEUIUJUK $.
  $}

  ${
    $d x y A $.  $d y B $.
    $( Existence of a class abstraction of existentially restricted sets.  The
       class ` B ` can be thought of as an expression in ` x ` (which is
       typically a free variable in the class expression substituted for
       ` B ` ) and the class abstraction appearing in the statement as the
       class of values ` B ` as ` x ` varies through ` A ` .  If the "domain"
       ` A ` is a set, then the abstraction is also a set.  Therefore, this
       statement is a kind of Replacement.  This can be seen by tracing back
       through the path ~ axrep6g , ~ axrep6 , ~ ax-rep .  See also
       ~ abrexex2g .  There are partial converses under additional conditions,
       see for instance ~ abnexg .  (Contributed by NM, 3-Nov-2003.)  (Proof
       shortened by Mario Carneiro, 31-Aug-2015.)  Avoid ~ ax-10 , ~ ax-11 ,
       ~ ax-12 , ~ ax-pr , ~ ax-un and shorten proof.  (Revised by SN,
       11-Dec-2024.) $)
    abrexexg $p |- ( A e. V -> { y | E. x e. A y = B } e. _V ) $=
      ( wcel cv wceq wmo wal wrex cab cvv moeq ax-gen axrep6g mpan2 ) CEFBGDHZB
      IZAJRACKBLMFSABDNORABCEPQ $.
    $( $j usage 'abrexexg' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-pr' 'ax-un'; $)

    $( Obsolete version of ~ abrexexg as of 11-Dec-2024.  EDITORIAL:  Comment
       kept since the line of equivalences to ~ ax-rep is different.

       Existence of a class abstraction of existentially restricted sets.  The
       class ` B ` can be thought of as an expression in ` x ` (which is
       typically a free variable in the class expression substituted for
       ` B ` ) and the class abstraction appearing in the statement as the
       class of values ` B ` as ` x ` varies through ` A ` .  If the "domain"
       ` A ` is a set, then the abstraction is also a set.  Therefore, this
       statement is a kind of Replacement.  This can be seen by tracing back
       through the path ~ mptexg , ~ funex , ~ fnex , ~ resfunexg , and
       ~ funimaexg .  See also ~ abrexex2g .  There are partial converses under
       additional conditions, see for instance ~ abnexg .  (Contributed by NM,
       3-Nov-2003.)  (Proof shortened by Mario Carneiro, 31-Aug-2015.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    abrexexgOLD $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 $.
    abrexex.1 $e |- A e. _V $.
    $( Existence of a class abstraction of existentially restricted sets.  See
       the comment of ~ abrexexg .  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 $=
      ( cvv wcel cv wceq wrex cab abrexexg ax-mp ) CFGBHDIACJBKFGEABCDFLM $.
  $}

  ${
    $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 uniexd
      wa cv adantr eqeltrd ) BDGZCEGABHZSABCIZFTCJABKFLZMZNUDUEUGJUCAFBCEOPUCUG
      NGUDUCUFNAFBCDQRUAUB $.
  $}

  ${
    $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 weq eqeltrrid
      sbequ12 rexbidv cbvabw df-clab rexbii abbii eqtr4i df-iun iunexg eqeltrid
      ciun ) DEHACIZFHBDJKZABDLZCIZGMUMHZBDLZGIZNUPACGOZBDLZGIUSUOVACGUOGPUTCBD
      CDQACGRSCGTAUTBDACGUBUCUDURVAGUQUTBDAGCUEUFUGUHUNUSBDUMULNBGDUMUIBDUMEFUJ
      UAUK $.
  $}

  ${
    $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.)  (Revised by AV,
       9-Aug-2024.) $)
    opabex3d $p |- ( ph -> { <. x , y >. | ( x e. A /\ ps ) } e. _V ) $=
      ( vz vv vw cv wcel wa cvv wex cop wceq exbii bitri copab csn cab cxp ciun
      19.42v an12 elxp excom velsn anbi1i opeq1 eqeq2d anbi1d equsexvw nfv nfan
      nfsab1 opeq2 wsb df-clab wb sbequ12 equcoms bitr4id anbi12d 3bitri anbi2i
      cbvexv1 3bitr4ri wrex eliun df-rex 3bitr4i eqriv wral vsnex xpexg sylancr
      elopab ralrimiva iunexg syl2anc eqeltrrid ) ACLZEMZBNZCDUAZCEWEUBZBDUCZUD
      ZUEZOIWLWHWFILZWKMZNZCPZWMWEDLZQZRZWGNZDPZCPWMWLMZWMWHMWOXACWFWSBNZNZDPWF
      XCDPZNXAWOWFXCDUFWTXDDWSWFBUGSWNXEWFWNWMJLZKLZQZRZXFWIMZXGWJMZNNZKPJPZWMW
      EXGQZRZXKNZKPZXEJKWMWIWJUHXMXLJPZKPXQXLJKUIXRXPKXRXFWERZXIXKNZNZJPXPXLYAJ
      XLXJXTNYAXIXJXKUGXJXSXTJWEUJUKTSXTXPJCXSXIXOXKXSXHXNWMXFWEXGULUMUNUOTSTXP
      XCKDXOXKDXODUPBDKURUQXCKUPXGWQRZXOWSXKBYBXNWRWMXGWQWEUSUMYBXKBDKUTZBBKDVA
      BYCVBDKBDKVCVDVEVFVIVGVHVJSXBWNCEVKWPCWMEWKVLWNCEVMTWGCDWMVTVNVOAEFMWKOMZ
      CEVPWLOMGAYDCEAWFNWIOMWJOMYDCVQHWIWJOOVRVSWACEWKFOWBWCWD $.
  $}

  ${
    $d A x y v w z $.  $d ps v w z $.  $d ph y $.
    opabex3rd.1 $e |- ( ph -> A e. V ) $.
    opabex3rd.2 $e |- ( ( ph /\ y e. A ) -> { x | ps } e. _V ) $.
    $( Existence of an ordered pair abstraction if the second components are
       elements of a set.  (Contributed by AV, 17-Sep-2023.)  (Revised by AV,
       9-Aug-2024.) $)
    opabex3rd $p |- ( ph -> { <. x , y >. | ( y e. A /\ ps ) } e. _V ) $=
      ( vz vw vv cv wcel wa cvv wex cop wceq exbii bitri copab cab csn cxp ciun
      19.42v an12 ancom anbi2i 2exbii velsn anbi1i opeq2 eqeq2d anbi1d equsexvw
      elxp nfv nfsab1 nfan opeq1 wsb df-clab wb sbequ12 equcoms bitr4id anbi12d
      cbvexv1 3bitri 3bitr4ri excom wrex eliun df-rex elopab 3bitr4i eqriv wral
      vsnex xpexg sylancl ralrimiva iunexg syl2anc eqeltrrid ) ADLZEMZBNZCDUAZD
      EBCUBZWGUCZUDZUEZOIWNWJWHILZWMMZNZDPZWOCLZWGQZRZWINZDPCPZWOWNMZWOWJMWRXBC
      PZDPXCWQXEDWHXABNZNZCPWHXFCPZNXEWQWHXFCUFXBXGCXAWHBUGSWPXHWHWPWOJLZKLZQZR
      ZXJWLMZXIWKMZNZNZKPZJPZWOXIWGQZRZXNNZJPXHWPXLXNXMNZNZKPJPXRJKWOWKWLUQYCXP
      JKYBXOXLXNXMUHUIUJTXQYAJXQXJWGRZXLXNNZNZKPYAXPYFKXPXMYENYFXLXMXNUGXMYDYEK
      WGUKULTSYEYAKDYDXLXTXNYDXKXSWOXJWGXIUMUNUOUPTSYAXFJCXTXNCXTCURBCJUSUTXFJU
      RXIWSRZXTXAXNBYGXSWTWOXIWSWGVAUNYGXNBCJVBZBBJCVCBYHVDCJBCJVEVFVGVHVIVJUIV
      KSXBDCVLTXDWPDEVMWRDWOEWMVNWPDEVOTWICDWOVPVQVRAEFMWMOMZDEVSWNOMGAYIDEAWHN
      WKOMWLOMYIHDVTWKWLOOWAWBWCDEWMFOWDWEWF $.
  $}

  ${
    $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 elxp excom velsn anbi1i opeq1 anbi1d equsexvw nfv nfan
      nfsab1 opeq2 wsb df-clab wb sbequ12 equcoms bitr4id anbi12d 3bitri anbi2i
      cbvexv1 3bitr4ri wrex eliun df-rex 3bitr4i eqriv wral vsnex xpexg sylancr
      elopab rgen iunexg mp2an eqeltrri ) BDBJZUCZACUDZUEZUAZWCDKZALZBCUBZMGWGW
      JWHGJZWFKZLZBNZWKWCCJZOZPZWILZCNZBNWKWGKZWKWJKWMWSBWHWQALZLZCNWHXACNZLWSW
      MWHXACUFWRXBCWQWHAQRWLXCWHWLWKHJZIJZOZPZXDWDKZXEWEKZLLZINHNZWKWCXEOZPZXIL
      ZINZXCHIWKWDWEUGXKXJHNZINXOXJHIUHXPXNIXPXDWCPZXGXILZLZHNXNXJXSHXJXHXRLXSX
      GXHXIQXHXQXRHWCUIUJSRXRXNHBXQXGXMXIXQXFXLWKXDWCXEUKTULUMSRSXNXAICXMXICXMC
      UNACIUPUOXAIUNXEWOPZXMWQXIAXTXLWPWKXEWOWCUQTXTXIACIURZAAICUSAYAUTCIACIVAV
      BVCVDVGVEVFVHRWTWLBDVIWNBWKDWFVJWLBDVKSWIBCWKVRVLVMDMKWFMKZBDVNWGMKEYBBDW
      HWDMKWEMKYBBVOFWDWEMMVPVQVSBDWFMMVTWAWB $.
  $}

  ${
    $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 A $.
    abrexex2.1 $e |- A e. _V $.
    abrexex2.2 $e |- { y | ph } e. _V $.
    $( Existence of an existentially restricted class abstraction. ` ph `
       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 $=
      ( cvv wcel cab wral wrex rgenw abrexex2g mp2an ) DGHACIGHZBDJABDKCIGHEOBD
      FLABCDGGMN $.

    $( 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 abstraction 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 z w v u f R $.  $d x y z w v u f S $.  $d z w v u f A $.
    $d z w v u f B $.  $d x y z w v u f F $.
    f1oweALT.1 $e |- R = { <. x , y >. | ( F ` x ) S ( F ` y ) } $.
    $( Alternate proof of ~ f1owe , more direct since not using the isomorphism
       predicate, but requiring ~ ax-un .  (Contributed by NM, 4-Mar-1997.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    f1oweALT $p |- ( F : A -1-1-onto-> B -> ( S We B -> R We A ) ) $=
      ( vu vf vw vv cv wbr wceq wral wa wi wcel cfv vz wf1o wfr w3o wwe wfo wfn
      f1ofo crn df-fo freq2 biimprd wfun cdm df-fn wss c0 wne wn wrex wal df-fr
      cima cvv vex funimaex wex n0 funfvima2 ne0i syl6 exlimdv biimtrid imassrn
      imp jctil sseq1 neeq1 anbi12d raleq rexeqbi1dv imbi12d spcgv syl7 anabsi5
      com23 expd impd cres wb fores fveq2 breq1d breq2d brab breqan12rd bitr4id
      syl fvres notbid ralbidva rexbiia breq1 cbvfo rexbidv breq2 ralbidv bitrd
      cbvexfo bitrid sylibrd exp4b com34 imp4a alrimdv imbitrrdi biimpd sylan9r
      sylan9 sylbi wf1 df-f1o a1i f1fveq bicomd 3orbi123d 2ralbidva eqeq1 eqeq2
      sylan9bb anim12d dfwe2 3imtr4g ) CDGUBZDFUCZIMZJMZFNZYPYQOZYQYPFNZUDZJDPZ
      IDPZQCEUCZKMZLMZENZUUEUUFOZUUFUUEENZUDZLCPKCPZQDFUECEUEYNYOUUDUUCUUKYNCDG
      UFZYOUUDRZCDGUHUULGCUGZGUIZDOZQUUMCDGUJUUPYOUUOFUCZUUNUUDUUPUUQYOUUODFUKU
      LUUNGUMZGUNZCOZQUUQUUDRGCUOUURUUQUUSEUCZUUTUUDUURUUQUAMZUUSUPZUVBUQURZQUU
      IUSZLUVBPZKUVBUTZRZUAVAUVAUURUUQUVHUAUURUUQUVCUVDUVGUURUVCUUQUVDUVGRUURUV
      CUVDUUQUVGUURUVCUVDUUQUVGUURUVCQZUVDUUQQYTUSZJGUVBVCZPZIUVKUTZUVGUVIUVDUU
      QUVMUURUVCUVDUUQUVMRZRUURUVIUVDUVNUURUUQUVIUVDQZUVMUUQUUEUUOUPZUUEUQURZQZ
      UVJJUUEPZIUUEUTZRZKVAZUURUVOUVMRZKIJUUOFVBUURUVKVDSZUWBUWCRGUVBUAVEVFUVOU
      VKUUOUPZUVKUQURZQZUWDUWBUVMUVOUWFUWEUVIUVDUWFUVDUUEUVBSZKVGUVIUWFKUVBVHUV
      IUWHUWFKUVIUWHUUEGTZUVKSUWFUVBUUEGVIUVKUWIVJVKVLVMVOGUVBVNVPUWAUWGUVMRKUV
      KVDUUEUVKOZUVRUWGUVTUVMUWJUVPUWEUVQUWFUUEUVKUUOVQUUEUVKUQVRVSUVSUVLIUUEUV
      KUVJJUUEUVKVTWAWBWCWDWRVMWFWGWEWHUVIUVBUVKGUVBWIZUFZUVGUVMWJUVBGWKUVGUUFU
      WKTZUUEUWKTZFNZUSZLUVBPZKUVBUTZUWLUVMUVFUWQKUVBUWHUVEUWPLUVBUWHUUFUVBSZQZ
      UUIUWOUWTUUIUUFGTZUWIFNZUWOAMZGTZBMZGTZFNZUXAUXFFNUXBABUUFUUEELVEZKVEZUXC
      UUFOUXDUXAUXFFUXCUUFGWLWMUXEUUEOUXFUWIUXAFUXEUUEGWLWNHWOZUWSUWHUWMUXAUWNU
      WIFUUFUVBGWSUUEUVBGWSWPWQWTXAXBUWLUWRYQUWNFNZUSZJUVKPZKUVBUTUVMUWLUWQUXMK
      UVBUWPUXLLJUVBUVKUWKUWMYQOUWOUXKUWMYQUWNFXCWTXDXEUXMUVLKIUVBUVKUWKUWNYPOZ
      UXLUVJJUVKUXNUXKYTUWNYPYQFXFWTXGXIXHXJWRXKXLXMWFXNXOUAKLUUSEVBXPUUTUVAUUD
      UUSCEUKXQXSXTXRXTWRYNUUKUUCYNCDGYAZUULQUUKUUCWJCDGYBUXOUUKUWIUXAFNZUWIUXA
      OZUXBUDZLCPZKCPZUULUUCUXOUUJUXRKLCCUXOUUECSUUFCSQQZUUGUXPUUHUXQUUIUXBUUGU
      XPWJUYAUXGUWIUXFFNUXPABUUEUUFEUXIUXHUXCUUEOUXDUWIUXFFUXCUUEGWLWMUXEUUFOUX
      FUXAUWIFUXEUUFGWLWNHWOYCUYAUXQUUHCDUUEUUFGYDYEUUIUXBWJUYAUXJYCYFYGUULUXTY
      PUXAFNZYPUXAOZUXAYPFNZUDZLCPZIDPUUCUXSUYFKICDGUWIYPOZUXRUYELCUYGUXPUYBUXQ
      UYCUXBUYDUWIYPUXAFXCUWIYPUXAYHUWIYPUXAFXFYFXGXDUULUYFUUBIDUYEUUALJCDGUXAY
      QOUYBYRUYCYSUYDYTUXAYQYPFXFUXAYQYPYIUXAYQYPFXCYFXDXGXHYJXTULYKIJDFYLKLCEY
      LYM $.
  $}

  ${
    $d R f g $.  $d A f g $.  $d S f g $.  $d B f g $.
    $( Thus, there is at most one isomorphism between any two well-ordered
       sets.  TODO:  Shorten ~ finnisoeu .  (Contributed by Stefan O'Rear,
       12-Feb-2015.)  (Revised by Mario Carneiro, 25-Jun-2015.) $)
    wemoiso $p |- ( R We A -> E* f f Isom R , S ( A , B ) ) $=
      ( vg wwe cv wiso wa weq wi wal wmo wse simpl cvv wcel wf syl isof1o dmfex
      vex wf1o f1of sylancr ad2antrl exse jca weisoeq sylancom ex isoeq1 sylibr
      alrimivv mo4 ) ACGZABCDEHZIZABCDFHZIZJZEFKZLZFMEMUSENUQVDEFUQVBVCUQVBUQAC
      OZJVCUQVBJZUQVEUQVBPVFAQRZVEUSVGUQVAUSURQRABURSZVGEUCUSABURUDVHABCDURUAAB
      URUETABQURUBUFUGACQUHTUIABCDURUTUJUKULUOUSVAEFABCDUTURUMUPUN $.

    $( Thus, there is at most one isomorphism between any two well-ordered
       sets.  (Contributed by Stefan O'Rear, 12-Feb-2015.)  (Revised by Mario
       Carneiro, 25-Jun-2015.) $)
    wemoiso2 $p |- ( S We B -> E* f f Isom R , S ( A , B ) ) $=
      ( vg wwe cv wiso wa wceq wi wal wmo wse simpl cvv wcel crn wf1o wfo f1ofo
      isof1o forn 3syl vex rnex eqeltrrdi ad2antrl syl jca weisoeq2 sylancom ex
      exse alrimivv isoeq1 mo4 sylibr ) BDGZABCDEHZIZABCDFHZIZJZVAVCKZLZFMEMVBE
      NUTVGEFUTVEVFUTVEUTBDOZJVFUTVEJZUTVHUTVEPVIBQRZVHVBVJUTVDVBBVASZQVBABVATA
      BVAUAVKBKABCDVAUCABVAUBABVAUDUEVAEUFUGUHUIBDQUOUJUKABCDVAVCULUMUNUPVBVDEF
      ABCDVCVAUQURUS $.
  $}

  ${
    $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. W ) $.
    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.)  (Revised by AV, 9-Aug-2024.) $)
    oprabexd $p |- ( ph -> F e. _V ) $=
      ( cv wcel wa cvv wmo wal coprab wfun cdm wi ex moanimv alrimivv funoprabg
      sylibr syl cxp wss dmoprabss xpexd ssexg sylancr funex syl2anc eqeltrd )
      AHCOFPDOGPQZBQZCDEUAZRNAVBUBZVBUCZRPZVBRPAVAESZDTCTVCAVFCDAUTBESZUDVFAUTV
      GMUEUTBEUFUIUGVACDEUHUJAVDFGUKZULVHRPVEBCDEFGUMAFGIJKLUNVDVHRUOUPRVBUQURU
      S $.
  $}

  ${
    $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 $.
    $( Alternate proof of ~ xpexg requiring Replacement ( ~ ax-rep ) but not
       Power Set ( ~ ax-pow ).  (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 elinel2 oveq12d mpteq2ia inindi ineq12i mpteq1i resmpt3 reseq1d
      syl incom resexg syl2an 3eqtr4a ) CEHZDFHZUAZGCIZDIZJZGUBZCKZVCDKZBLZMZAN
      ZGCANZIZDANZIZJZVCVIKZVCVKKZBLZMZCDBUCZLZANVIVKVRLZGVBAJZVPMGWAVFMVQVHGWA
      VPVFVCWAHVCAHZVPVFOVCVBAUDWBVNVDVOVEBVCACPVCADPUEULUFGVMWAVPAVBJAUTJZAVAJ
      ZJWAVMAUTVAUGVBAUMVJWCVLWDCAQDAQUHRUIGVBAVFUJRUSVSVGAGBCDEFSUKUQVITHVKTHV
      TVQOURCAEUNDAFUNGBVIVKTTSUOUP $.
  $}

  ${
    $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 xpexd ofexg syl ) ABCIZJKDLQMJKABCEFGHNQDJOP $.
  $}

  ${
    $d A x y z $.  $d F x y z $.  $d M x $.  $d ph x y z $.
    mptcnfimad.m $e |- M = ( x e. A |-> ( F " x ) ) $.
    mptcnfimad.f $e |- ( ph -> F : V -1-1-onto-> W ) $.
    mptcnfimad.a $e |- ( ph -> A C_ ~P V ) $.
    mptcnfimad.r $e |- ( ph -> ran M C_ ~P W ) $.
    mptcnfimad.v $e |- ( ph -> V e. U ) $.
    $( The converse of a mapping of subsets to their image of a bijection.
       (Contributed by AV, 23-Apr-2025.) $)
    mptcnfimad $p |- ( ph -> `' M = ( y e. ran M |-> ( `' F " y ) ) ) $=
      ( cima wcel wceq wa cvv syl vz ccnv cv cmpt crn cnveqi simpr wf1o wf f1of
      fexd imaexd adantr elrnmpt1d wf1 wss f1of1 cpw wi ssel elpwi imp f1imacnv
      syl6 eqcomd syl2an2r jca eleq1 imaeq2 anbi12d syl5ibrcom expimpd cfv wrex
      eqeq2d wfn wb ralrimiva fnmpt fvelrnb cbvmptv eqtri imaeq2d fvmptd eqeq1d
      wral a1i eqcoms adantl eqeltrd sylbid rexlimdva wfo f1ofo foimacnv impbid
      ex mptcnv eqtrid ) AGUBBDFBUCZOZUDZUBCGUEZFUBZCUCZOZUDGXBJUFABCDXAXCXFAWT
      DPZXEXAQZRZXEXCPZWTXFQZRZAXGXHXLAXGRZXLXHXAXCPZWTXDXAOZQZRXMXNXPXMBDXAGSJ
      AXGUGZAXASPZXGAFWTSAHIEFAHIFUHZHIFUIKHIFUJTNUKULUMZUNAHIFUOZXGWTHUPZXPAXS
      YAKHIFUQTZAXGYBADHURZUPZXGYBUSLYEXGWTYDPYBDYDWTUTWTHVAVDTVBZYAYBRXOWTHIWT
      FVCZVEVFVGXHXJXNXKXPXEXAXCVHXHXFXOWTXEXAXDVIZVOVJVKVLAXJXKXIAXJRZXIXKXFDP
      ZXEFXFOZQZRYIYJYLAXJYJAXJWTGVMZXEQZBDVNZYJAGDVPZXJYOVQAXRBDWFYPAXRBDXTVRB
      DXAGSJVSTBDXEGVTTAYNYJBDXMYNXAXEQZYJXMYMXAXEXMUAWTFUAUCZOZXADGSGUADYSUDZQ
      XMGXBYTJBUADXAYSWTYRFVIWAWBWGXMYRWTQZRYRWTFXMUUAUGWCXQXTWDWEXMYQYJXMYQRXF
      XODYQXFXOQZXMUUBXEXAYHWHWIXMXODPYQXMXOWTDAYAXGYBXOWTQYCYFYGVFXQWJUMWJWQWK
      WLWKVBYIYKXEAHIFWMZXJXEIUPZYKXEQAXSUUCKHIFWNTAXJUUDAXCIURZUPZXJUUDUSMUUFX
      JXEUUEPUUDXCUUEXEUTXEIVAVDTVBHIXEFWOVFVEVGXKXGYJXHYLWTXFDVHXKXAYKXEWTXFFV
      IVOVJVKVLWPWRWS $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  First and second members of an ordered pair
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c 1st $.  $( First component of an ordered pair $)
  $c 2nd $.  $( Second component 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.) $)
    1stval $p |- ( 1st ` A ) = U. dom { A } $=
      ( vx cvv wcel c1st cfv csn cdm cuni wceq cv sneq dmeqd unieqd df-1st snex
      dmex uniex c0 eqtrdi fvmpt wn fvprc snprc biimpi dm0 uni0 eqtr4d pm2.61i
      ) ACDZAEFZAGZHZIZJBABKZGZHZIUNCEUOAJZUQUMURUPULUOALMNBOUMULAPQRUAUJUBZUKS
      UNAEUCUSUNSISUSUMSUSUMSHSUSULSUSULSJAUDUEMUFTNUGTUHUI $.

    $( 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.) $)
    2ndval $p |- ( 2nd ` A ) = U. ran { A } $=
      ( vx cvv wcel c2nd cfv csn crn cuni wceq cv sneq rneqd unieqd df-2nd snex
      rnex uniex c0 eqtrdi fvmpt wn fvprc snprc biimpi rn0 uni0 eqtr4d pm2.61i
      ) ACDZAEFZAGZHZIZJBABKZGZHZIUNCEUOAJZUQUMURUPULUOALMNBOUMULAPQRUAUJUBZUKS
      UNAEUCUSUNSISUSUMSUSUMSHSUSULSUSULSJAUDUEMUFTNUGTUHUI $.
  $}

  $( Value of the first-member function at non-pairs.  (Contributed by Thierry
     Arnoux, 22-Sep-2017.) $)
  1stnpr $p |- ( -. A e. ( _V X. _V ) -> ( 1st ` A ) = (/) ) $=
    ( cvv cxp wcel wn c1st cfv csn cdm cuni c0 1stval wne dmsnn0 biimpri unieqd
    necon1bi uni0 eqtrdi eqtrid ) ABBCDZEZAFGAHIZJZKALUBUDKJKUBUCKUAUCKUAUCKMAN
    OQPRST $.

  $( Value of the second-member function at non-pairs.  (Contributed by Thierry
     Arnoux, 22-Sep-2017.) $)
  2ndnpr $p |- ( -. A e. ( _V X. _V ) -> ( 2nd ` A ) = (/) ) $=
    ( cvv cxp wcel wn c2nd cfv csn crn cuni c0 2ndval wne rnsnn0 biimpri unieqd
    necon1bi uni0 eqtrdi eqtrid ) ABBCDZEZAFGAHIZJZKALUBUDKJKUBUCKUAUCKUAUCKMAN
    OQPRST $.

  $( The value of the first-member function at the empty set.  (Contributed by
     NM, 23-Apr-2007.) $)
  1st0 $p |- ( 1st ` (/) ) = (/) $=
    ( c0 c1st cfv csn cdm cuni 1stval dmsn0 unieqi uni0 3eqtri ) ABCADEZFAFAAGL
    AHIJK $.

  $( 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 2ndval wceq dmsn0 dm0rn0 mpbi unieqi uni0 3eqtri
    cdm ) ABCADZEZFAFAAGQAPOAHQAHIPJKLMN $.

  ${
    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 1stval op1sta eqtri ) ABEZFGNHIJANKABCDLM $.

    $( 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 2ndval op2nda eqtri ) ABEZFGNHIJBNKABCDLM $.

    $( 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 fveqeq2d op1st
      vex vtocl2g ) EGZFGZHZIJZUAKAUBHZIJZAKABHZIJAKEFABCDUAAKZUDUFUAAUHUCUEIUA
      AUBLMUHNOUBBKUEUGAIUBBAPQUAUBESFSRT $.

    $( 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 fveqeq2d opeq2 fveq2d id eqeq12d op2nd
      vex vtocl2g ) EGZFGZHZIJUBKAUBHZIJZUBKABHZIJZBKEFABCDUAAKUCUDUBIUAAUBLMUB
      BKZUEUGUBBUHUDUFIUBBANOUHPQUAUBESFSRT $.

    $( 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 , ~ ot3rdg .)  (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 cotp c1st cfv wceq wa cop df-ot fveq2i cvv opex op1stg mpan eqtrid
      fveq2d sylan9eqr 3impa ) ADGZBEGZCFGZABCHZIJZIJZAKUFUDUELUIABMZIJAUFUHUJI
      UFUHUJCMZIJZUJUGUKIABCNOUJPGUFULUJKABQUJCPFRSTUAABDERUBUC $.

    $( 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 cotp c1st cfv c2nd wceq wa cop df-ot fveq2i cvv opex op1stg mpan
      eqtrid fveq2d op2ndg sylan9eqr 3impa ) ADGZBEGZCFGZABCHZIJZKJZBLUHUFUGMUK
      ABNZKJBUHUJULKUHUJULCNZIJZULUIUMIABCOPULQGUHUNULLABRULCQFSTUAUBABDEUCUDUE
      $.

    $( Extract the third member of an ordered triple.  (See ~ ot1stg comment.)
       (Contributed by NM, 3-Apr-2015.) $)
    ot3rdg $p |- ( C e. V -> ( 2nd ` <. A , B , C >. ) = C ) $=
      ( wcel cotp c2nd cfv cop df-ot fveq2i cvv wceq opex op2ndg mpan eqtrid )
      CDEZABCFZGHABIZCIZGHZCSUAGABCJKTLERUBCMABNTCLDOPQ $.

    $( 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 $.
  $}

  $( The components of an ordered triple.  (Contributed by Alexander van der
     Vekens, 2-Mar-2018.) $)
  oteqimp $p |- ( T = <. A , B , C >. -> ( ( A e. X /\ B e. Y /\ C e. Z )
                         -> ( ( 1st ` ( 1st ` T ) ) = A
                           /\ ( 2nd ` ( 1st ` T ) ) = B
                           /\ ( 2nd ` T ) = C ) ) ) $=
    ( wcel w3a c1st wceq c2nd cotp ot1stg ot2ndg ot3rdg 3ad2ant3 2fveq3 eqeq1d
    cfv 3jca fveqeq2 3anbi123d imbitrrid ) AEHZBFHZCGHZIZDJTZJTZAKZUILTZBKZDLTC
    KZIDABCMZKZUOJTZJTZAKZUQLTZBKZUOLTCKZIUHUSVAVBABCEFGNABCEFGOUGUEVBUFABCGPQU
    AUPUKUSUMVAUNVBUPUJURADUOJJRSUPULUTBDUOLJRSDUOCLUBUCUD $.

  ${
    $d x y $.
    $( 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 cv csn cdm cuni vsnex dmex uniex df-1st
      fnmpti wrex cab wcel rnmpt vex cop opex op1sta eqcomi sneq dmeqd rspceeqv
      unieqd mp2an 2th eqabi eqtr4i df-fo mpbir2an ) CCDEDCFDGZCHACAIZJZKZLZDUT
      USAMNOAPZQUQBIZVAHACRZBSCABCVADVBUAVDBCVCCTVDBUBZVCVCUCZCTVCVFJZKZLZHVDVC
      VCUDVIVCVCVCVEVEUEUFAVFCVAVIVCURVFHZUTVHVJUSVGURVFUGUHUJUIUKULUMUNCCDUOUP
      $.

    $( 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 csn cuni vsnex rnex uniex df-2nd fnmpti
      cv wrex cab rnmpt wcel vex cop opex op2nda eqcomi sneq rneqd unieqd mp2an
      rspceeqv 2th eqabi eqtr4i df-fo mpbir2an ) CCDEDCFDGZCHACAPZIZGZJZDUSURAK
      LMANZOUPBPZUTHACQZBRCABCUTDVASVCBCVBCTVCBUAZVBVBUBZCTVBVEIZGZJZHVCVBVBUCV
      HVBVBVBVDVDUDUEAVECUTVHVBUQVEHZUSVGVIURVFUQVEUFUGUHUJUIUKULUMCCDUNUO $.
  $}

  $( Uniqueness condition for the binary relation ` 1st ` .  (Contributed by
     Scott Fenton, 2-Jul-2020.)  Revised to remove sethood hypothesis on
     ` C ` .  (Revised by Peter Mazsa, 17-Jan-2022.) $)
  br1steqg $p |-
                 ( ( A e. V /\ B e. W ) -> ( <. A , B >. 1st C <-> C = A ) ) $=
    ( wcel wa cop c1st cfv wceq wbr op1stg eqeq1d cvv wfn wb wfo fo1st fofn
    ax-mp opex fnbrfvb mp2an eqcom 3bitr3g ) ADFBEFGZABHZIJZCKZACKUHCILZCAKUGUI
    ACABDEMNIOPZUHOFUJUKQOOIRULSOOITUAABUBOUHCIUCUDACUEUF $.

  $( Uniqueness condition for the binary relation ` 2nd ` .  (Contributed by
     Scott Fenton, 2-Jul-2020.)  Revised to remove sethood hypothesis on
     ` C ` .  (Revised by Peter Mazsa, 17-Jan-2022.) $)
  br2ndeqg $p |-
                 ( ( A e. V /\ B e. W ) -> ( <. A , B >. 2nd C <-> C = B ) ) $=
    ( wcel wa cop c2nd cfv wceq wbr op2ndg eqeq1d cvv wfn wb wfo fo2nd fofn
    ax-mp opex fnbrfvb mp2an eqcom 3bitr3g ) ADFBEFGZABHZIJZCKZBCKUHCILZCBKUGUI
    BCABDEMNIOPZUHOFUJUKQOOIRULSOOITUAABUBOUHCIUCUDBCUEUF $.

  ${
    $d x y z A $.  $d x y z B $.
    $( 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 $.

    $( Onto mapping of a restriction of the ` 1st ` (first member of an ordered
       pair) function.  (Contributed by NM, 14-Dec-2008.) $)
    fo1stres $p |- ( B =/= (/) ->
                 ( 1st |` ( A X. B ) ) : ( A X. B ) -onto-> A ) $=
      ( vx vy c0 wne cxp c1st cres wf crn wa wss cv wcel cfv ax-mp jctil sylibr
      vex wfo wex wi n0 cop opelxp fvres op1st eqtr2di wfn f1stres ffn fnfvelrn
      wceq mpan eqeltrd sylbir expcom exlimiv sylbi ssrdv frn eqss dffo2 ) BEFZ
      ABGZAHVFIZJZVGKZAUNZLVFAVGUAVEVJVHVEVIAMZAVIMZLVJVEVLVKVECAVIVEDNZBOZDUBC
      NZAOZVOVIOZUCZDBUDVNVRDVPVNVQVPVNLVOVMUEZVFOZVQVOVMABUFVTVOVSVGPZVIVTWAVS
      HPVOVSVFHUGVOVMCTDTUHUIVGVFUJZVTWAVIOVHWBABUKZVFAVGULQVFVSVGUMUOUPUQURUSU
      TVAVHVKWCVFAVGVBQRVIAVCSWCRVFAVGVDS $.

    $( Onto mapping of a restriction of the ` 2nd ` (second member of an
       ordered pair) function.  (Contributed by NM, 14-Dec-2008.) $)
    fo2ndres $p |- ( A =/= (/) ->
                 ( 2nd |` ( A X. B ) ) : ( A X. B ) -onto-> B ) $=
      ( vy vx c0 wne cxp c2nd cres wf crn wa wss cv wcel cfv ax-mp jctil sylibr
      vex wfo wex wi n0 cop opelxp fvres op2nd eqtr2di wfn f2ndres ffn fnfvelrn
      wceq mpan eqeltrd sylbir ex exlimiv sylbi ssrdv frn eqss dffo2 ) AEFZABGZ
      BHVFIZJZVGKZBUNZLVFBVGUAVEVJVHVEVIBMZBVIMZLVJVEVLVKVECBVIVEDNZAOZDUBCNZBO
      ZVOVIOZUCZDAUDVNVRDVNVPVQVNVPLVMVOUEZVFOZVQVMVOABUFVTVOVSVGPZVIVTWAVSHPVO
      VSVFHUGVMVODTCTUHUIVGVFUJZVTWAVIOVHWBABUKZVFBVGULQVFVSVGUMUOUPUQURUSUTVAV
      HVKWCVFBVGVBQRVIBVCSWCRVFBVGVDS $.
  $}

  ${
    $d x y z w v $.  $d v w A $.
    $( Value of an alternate definition of the ` 1st ` function.  (Contributed
       by NM, 14-Oct-2004.)  (Revised by Mario Carneiro, 30-Dec-2014.) $)
    1st2val $p |- ( { <. <. x , y >. , z >. | z = x } ` A ) = ( 1st ` A ) $=
      ( vw vv cvv wcel weq cfv wceq cv wex vex eqtrdi eqtr4d cdm cuni c0 copab
      cxp coprab c1st cop elvv fveq2 df-ov simpl cmpo mpov eqcomi ovmpoa eqtr3i
      co op1std exlimivv sylbi wn csn wa pm3.2i ax6ev 2th opabbii df-xp dmoprab
      el2v 3eqtr4ri eleq2i ndmfv sylnbir wne dmsnn0 necon1bi unieqd uni0 1stval
      biimpri eqtr4di pm2.61i ) DGGUAZHZDCAIZABCUBZJZDUCJZKZWBDELZFLZUDZKZFMEMW
      GEFDUEWKWGEFWKWEWHWFWKWEWJWDJZWHDWJWDUFWHWIWDUNZWLWHWHWIWDUGWMWHKEFABWHWI
      GGALZWHWDAEIBFIUHABGGWNUIWDABCWNUJUKENZULVGUMOWHWIDWOFNUOPUPUQWBURZWEDUSQ
      ZRZWFWPWESWRWBDWDQZHWESKWSWADWNGHZBLGHZUTZABTWCCMZABTWAWSXBXCABXBXCWTXAAN
      BNVACAVBVCVDABGGVEWCABCVFVHVIDWDVJVKWPWRSRSWPWQSWBWQSWBWQSVLDVMVRVNVOVPOP
      DVQVSVT $.

    $( Value of an alternate definition of the ` 2nd ` function.  (Contributed
       by NM, 10-Aug-2006.)  (Revised by Mario Carneiro, 30-Dec-2014.) $)
    2nd2val $p |- ( { <. <. x , y >. , z >. | z = y } ` A ) = ( 2nd ` A ) $=
      ( vw vv cvv cxp wcel weq cfv wceq cv wex vex eqtrdi eqtr4d cuni c0 copab
      coprab c2nd elvv fveq2 co df-ov simpr cmpo mpov eqcomi ovmpoa el2v eqtr3i
      cop op2ndd exlimivv sylbi wn csn crn cdm pm3.2i ax6ev 2th opabbii dmoprab
      wa df-xp 3eqtr4ri eleq2i ndmfv sylnbir wne rnsnn0 biimpri necon1bi unieqd
      uni0 2ndval eqtr4di pm2.61i ) DGGHZIZDCBJZABCUAZKZDUBKZLZWCDEMZFMZUNZLZFN
      ENWHEFDUCWLWHEFWLWFWJWGWLWFWKWEKZWJDWKWEUDWIWJWEUEZWMWJWIWJWEUFWNWJLEFABW
      IWJGGBMZWJWEAEJBFJUGABGGWOUHWEABCWOUIUJFOZUKULUMPWIWJDEOWPUOQUPUQWCURZWFD
      USUTZRZWGWQWFSWSWCDWEVAZIWFSLWTWBDAMGIZWOGIZVGZABTWDCNZABTWBWTXCXDABXCXDX
      AXBAOBOVBCBVCVDVEABGGVHWDABCVFVIVJDWEVKVLWQWSSRSWQWRSWCWRSWCWRSVMDVNVOVPV
      QVRPQDVSVTWA $.
  $}

  $( 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 $.
  $}

  $( Membership in a Cartesian 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 csn cdm cuni crn cop wceq wa c1st cfv elxp4 1stval 2ndval eleq1i
    c2nd anbi12i opeq12i eqeq2i bitr4i ) ABCDEAAFZGHZUDIHZJZKZUEBEZUFCEZLZLAAMN
    ZASNZJZKZULBEZUMCEZLZLABCOUOUHURUKUNUGAULUEUMUFAPZAQZUAUBUPUIUQUJULUEBUSRUM
    UFCUTRTTUC $.

  $( Membership in a Cartesian 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 c1st cfv c2nd cop wceq wa cvv fvex pm3.2i mpbiran2 anbi1i bitr4i
    elxp6 ) ABCDEAAFGZAHGZIJZSBETCEKZKALLDEZUBKABCRUCUAUBUCUASLEZTLEZKUDUEAFMAH
    MNALLROPQ $.

  $( 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 Cartesian 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 eqabi df-rab eqtr4i ) B
      CDZAEZFFDZGSHIBGSJICGKZKZALUAATMUBARSBCNOUAATPQ $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( The membership relation for a Cartesian 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 Cartesian 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 $.

  $( Reconstruction of an ordered pair in terms of its components.
     (Contributed by NM, 25-Feb-2014.) $)
  1st2ndb $p |- ( A e. ( _V X. _V )
      <-> A = <. ( 1st ` A ) , ( 2nd ` A ) >. ) $=
    ( cvv cxp wcel c1st cfv c2nd cop wceq 1st2nd2 fvex opelvv eqeltrdi impbii
    id ) ABBCZDAAEFZAGFZHZIZABBJTASPTOQRAEKAGKLMN $.

  $( An ordered pair theorem for members of Cartesian 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 fvex opth bitr2di )
    ACDGHZBEFGHZIABJAKLZAMLZNZBKLZBMLZNZJUBUEJUCUFJITUAAUDBUGACDOBEFOPUBUCUEUFA
    KQAMQRS $.

  $( 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 fvex opth bitrdi ) ADEF
    GZABCHZIAJKZALKZHZTIUABIUBCIMSAUCTADENOUAUBBCAJPALPQR $.

  ${
    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 opelvv eleq1 mpbiri eqop biadanii
      wa ) ABCFZGZAHHIZJZAKLBGAMLCGSUAUCTUBJBCDENATUBOPABCHHQR $.
  $}

  ${
    $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 ex
      eqopi mpanr2 fvex opeq2 eqeq2d spcev eqop simpl biimtrdi exlimdv impbid )
      BDEFZGBHHFZGZBIJCKZBCALZMZKZANZOUNUOBDEPQUPUQVAUPUQVAUPUQRBCBSJZMZKZVAUPU
      QVBVBKVDVBUABCVBHHUCUDUTVDAVBBSUEURVBKUSVCBURVBCUFUGUHTUBUPUTUQAUPUTUQVBU
      RKZRUQBCURHHUIUQVEUJUKULUMT $.
  $}

  ${
    $d A a b p $.  $d B a b p $.  $d ph a b $.
    opreuopreu.a $e |- ( ( a = ( 1st ` p ) /\ b = ( 2nd ` p ) )
                         -> ( ps <-> ph ) ) $.
    $( There is a unique ordered pair fulfilling a wff iff its components
       fulfil a corresponding wff.  (Contributed by AV, 2-Jul-2023.) $)
    opreuopreu $p |- ( E! p e. ( A X. B ) ph
                <-> E! p e. ( A X. B ) E. a E. b ( p = <. a , b >. /\ ps ) ) $=
      ( cv wceq wa wex wcel wi c1st cfv c2nd vex eqcomi fveq2 cop cxp simprl wb
      elxpi op1st pm3.2i eqeq2d anbi12d mpbiri syl biimprd adantr impcom jca ex
      op2nd 2eximdv syl5com biimpa a1i exlimdvv impbid reubiia ) AEIZFIZGIZUAZJ
      ZBKZGLFLZECDUBZVEVLMZAVKVMVIVFCMVGDMKZKZGLFLAVKFGVECDUEAVOVJFGAVOVJAVOKVI
      BAVIVNUCVOABVIABNVNVIBAVIVFVEOPZJZVGVEQPZJZKZBAUDVIVTVFVHOPZJZVGVHQPZJZKW
      BWDWAVFVFVGFRZGRZUFSWCVGVFVGWEWFUQSUGVIVQWBVSWDVIVPWAVFVEVHOTUHVIVRWCVGVE
      VHQTUHUIUJHUKZULUMUNUOUPURUSVMVJAFGVJANVMVIBAWGUTVAVBVCVD $.
  $}

  ${
    $d A p x y z $.  $d B p x y z $.  $d C p x y z $.  $d D p x y z $.
    $( A member of a nested Cartesian product is an ordered triple.
       (Contributed by Alexander van der Vekens, 15-Feb-2018.) $)
    el2xptp $p |- ( A e. ( ( B X. C ) X. D )
                    <-> E. x e. B E. y e. C E. z e. D A = <. x , y , z >. ) $=
      ( vp cxp wcel cv cop wceq wrex cotp elxp2 opeq1 eqeq2d rexbidv rexbii
      rexxp df-ot eqcomi eqeq2i 3bitri ) DEFIZGIJDHKZCKZLZMZCGNZHUFNDAKZBKZLZUH
      LZMZCGNZBFNZAENDULUMUHOZMZCGNZBFNZAENHCDUFGPUKUQHABEFUGUNMZUJUPCGVCUIUODU
      GUNUHQRSUAURVBAEUQVABFUPUTCGUOUSDUSUOULUMUHUBUCUDTTTUE $.
  $}

  $( A member of a nested Cartesian product is an ordered triple.  (Contributed
     by Alexander van der Vekens, 15-Feb-2018.) $)
  el2xptp0 $p |- ( ( X e. U /\ Y e. V /\ Z e. W )
                -> ( ( A e. ( ( U X. V ) X. W ) /\ ( ( 1st ` ( 1st ` A ) ) = X
                           /\ ( 2nd ` ( 1st ` A ) ) = Y /\ ( 2nd ` A ) = Z ) )
                     <-> A = <. X , Y , Z >. ) ) $=
    ( wcel w3a cxp c1st cfv wceq c2nd wa cop ad2antrl adantl jca sylan9eqr cotp
    xp1st 3simpa eqopi syl2anc simprr3 df-ot eqeq2i eqop bitrid opelxpi 3adant3
    wb mpbird simp3 opelxpd eqeltrid adantr eleq1 2fveq3 ot1stg ot2ndg 3ad2ant3
    fveq2 ot3rdg 3jca impbida ) EBHZFCHZGDHZIZABCJZDJZHZAKLZKLZEMZVONLZFMZANLZG
    MZIZOZAEFGUAZMZVKWCOZWEVOEFPZMZWAOZWFWHWAWFVOVLHZVQVSOZWHVNWJVKWBAVLDUBQWCW
    KVKWBWKVNVQVSWAUCRRVOEFBCUDUEVQVSWAVNVKUFSVNWEWIUMVKWBWEAWGGPZMVNWIWDWLAEFG
    UGZUHAWGGVLDUIUJQUNVKWEOZVNWBWNVNWDVMHZVKWOWEVKWDWLVMWMVKWGGVLDVHVIWGVLHVJE
    FBCUKULVHVIVJUOUPUQURWEVNWOUMVKAWDVMUSRUNWNVQVSWAWEVKVPWDKLZKLEAWDKKUTEFGBC
    DVATWEVKVRWPNLFAWDNKUTEFGBCDVBTWEVKVTWDNLZGAWDNVDVJVHWQGMVIEFGDVEVCTVFSVG
    $.

  ${
    $d A x y z $.  $d B x y z $.  $d C x y z $.  $d D x y z $.
    $( Version of ~ elrel for triple Cartesian products.  (Contributed by Scott
       Fenton, 1-Feb-2025.) $)
    el2xpss $p |- ( ( A e. R /\ R C_ ( ( B X. C ) X. D ) ) ->
     E. x E. y E. z A = <. x , y , z >. ) $=
      ( wcel cxp wss wa cv cotp wceq wex wrex rexex reximi syl ancoms el2xptp
      ssel2 sylbi ) DHIZHEFJGJZKZLDUFIZDAMBMCMNOZCPZBPZAPZUGUEUHHUFDUCUAUHUICGQ
      ZBFQZAEQZULABCDEFGUBUOUKAEQULUNUKAEUNUJBFQUKUMUJBFUICGRSUJBFRTSUKAERTUDT
      $.
  $}

  $( 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 1st2nd2 sneqd cnveqd unieqd opswap eqtrdi
    c2nd cop ) ABCDEZAFZGZHAIJZAQJZRZFZGZHUCUBRSUAUFSTUESAUDABCKLMNUBUCOP $.

  $( 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 c2nd cop crn 1st2nd simpr eqeltrrd fvex opelrn syl
    ) BCZABDZEZAFGZAHGZIZBDUABJDSAUBBABKQRLMTUABAFNAHNOP $.

  $( 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 fvex wex
      wb eqeltrrdi rexlimivw cop eldm2g adantl cxp wss wi df-rel ssel sylbi imp
      op1steq syl rexbidva adantr rexcom4 risset bitr4i bitrdi bitr4d pm5.21nd
      exbii ) BEZCBFZGZAHZIJZCKZABLZVDCMGZNZVFVKVDCVEOPVJVKVDVIVKABVICVHMVIQVGI
      RUAUBPVLVFCDHUCZBGZDSZVJVKVFVOTVDDCBMUDUEVLVJVGVMKZDSZABLZVOVDVJVRTVKVDVI
      VQABVDVGBGZNVGMMUFZGZVIVQTVDVSWAVDBVTUGVSWAUHBUIBVTVGUJUKULDVGCMMUMUNUOUP
      VRVPABLZDSVOVPADBUQVNWBDAVMBURVCUSUTVAVB $.
  $}

  ${
    $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 cv c1st cfv cmpt crn wcel wceq wrex releldm2 wfn wb fvex
      eqid fnmpti fvelrnb ax-mp fveq2 fvmpt eqeq1d rexbiia bitr2id bitrd eqrdv
      a1i ) BEZCBFZABAGZHIZJZKZUKCGZULLDGZHIZUQMZDBNZUQUPLZDBUQOVBURUOIZUQMZDBN
      ZUKVAUOBPVBVEQABUNUOUMHRUOSZTDBUQUOUAUBVEVAQUKVDUTDBURBLVCUSUQAURUNUSBUOU
      MURHUCVFURHRUDUEUFUJUGUHUI $.
  $}

  ${
    $d A x $.  $d B x $.  $d C x $.
    $( One way of expressing membership in the difference of domains of two
       nested relations.  (Contributed by AV, 26-Oct-2023.) $)
    releldmdifi $p |- ( ( Rel A /\ B C_ A ) -> ( C e. ( dom A \ dom B )
                                    -> E. x e. ( A \ B ) ( 1st ` x ) = C ) ) $=
      ( wrel wss wa cdm cdif wcel cv c1st wceq wrex wn eldif wb releldm2 adantr
      cfv anbi1d bitrid wral simprl relss 1stdm sylan eleq1 syl5ibcom rexlimdva
      impcom con3d ralnex imbitrrdi adantld imp rexdifi syl2anc ex sylbid ) BEZ
      CBFZGZDBHZCHZIJZAKZLTZDMZABNZDVEJZOZGZVIABCINZVFDVDJZVLGVCVMDVDVEPVCVOVJV
      LVAVOVJQVBABDRSUAUBVCVMVNVCVMGVJVIOACUCZVNVCVJVLUDVCVMVPVCVLVPVJVCVLVIACN
      ZOVPVCVQVKVCVIVKACVCVGCJZGVHVEJZVIVKVCCEZVRVSVBVAVTCBUEUKVGCUFUGVHDVEUHUI
      UJULVIACUMUNUOUPVIABCUQURUSUT $.
  $}

  $( The function value for the first component of an ordered pair is the
     second component of the ordered pair.  (Contributed by AV,
     17-Oct-2023.) $)
  funfv1st2nd $p |- ( ( Fun F /\ X e. F )
                      -> ( F ` ( 1st ` X ) ) = ( 2nd ` X ) ) $=
    ( wfun wcel wa c1st cfv c2nd cop wceq wrel funrel 1st2nd sylan eleq1 adantl
    wb wi funopfv adantr sylbid impancom mpd ) ACZBADZEBBFGZBHGZIZJZUFAGUGJZUDA
    KUEUIALBAMNUDUIUEUJUDUIEUEUHADZUJUIUEUKQUDBUHAOPUDUKUJRUIUFUGASTUAUBUC $.

  $( If the first component of an element of a function is in the domain of a
     subset of the function, the element is a member of this subset.
     (Contributed by AV, 27-Oct-2023.) $)
  funelss $p |- ( ( Fun A /\ B C_ A /\ X e. A )
                  -> ( ( 1st ` X ) e. dom B -> X e. B ) ) $=
    ( wfun wss wcel c1st cfv cdm wi wa c2nd cop wceq wrel sylan wb eleq1 adantr
    imp funrel 1st2nd w3a simpl1l simpl3 simpr funssfv syl3anc funopfv impancom
    adantl sylbid 3adant3 eqtr3d wfn funss com12 3adant2 fnopfvb mpbid 3ad2ant2
    funfnd mpbird 3exp1 mpd ex com23 3imp ) ADZBAEZCAFZCGHZBIZFZCBFZJZVIVKVJVPV
    IVKVJVPJZVIVKKZCVLCLHZMZNZVQVIAOVKWAAUACAUBPVRWAVJVNVOVRWAVJUCZVNKZVOVTBFZW
    CVLBHZVSNZWDWCVLAHZWEVSWCVIVJVNWGWENVIVKWAVJVNUDVRWAVJVNUEWBVNUFVLABUGUHWBW
    GVSNZVNVRWAWHVJVRWAWHVIWAVKWHVIWAKVKVTAFZWHWAVKWIQVICVTARUKVIWIWHJWAVLVSAUI
    SULUJTUMSUNWBBVMUOZVNWFWDQVRVJWJWAVRVJKBVRVJBDZVIVJWKJVKVJVIWKBAUPUQSTVBURV
    MVLVSBUSPUTWBVOWDQZVNWAVRWLVJCVTBRVASVCVDVEVFVGVH $.

  ${
    $d A x $.  $d B x $.  $d C x $.
    $( Two ways of expressing membership in the difference of domains of two
       nested functions.  (Contributed by AV, 27-Oct-2023.) $)
    funeldmdif $p |- ( ( Fun A /\ B C_ A ) -> ( C e. ( dom A \ dom B )
                                   <-> E. x e. ( A \ B ) ( 1st ` x ) = C ) ) $=
      ( wfun wss wa cdm cdif wcel cv c1st cfv wceq wrex wrel wi funrel adantr
      wn releldmdifi sylan eldif w3a 1stdm syl com12 impcom funelss 3expa con3d
      ex impr eldifd 3adant3 eleq1 3ad2ant3 mpbid 3exp biimtrid rexlimdv impbid
      wb ) BEZCBFZGZDBHZCHZIZJZAKZLMZDNZABCIZOZVDBPZVEVJVOQBRZABCDUAUBVFVMVJAVN
      VKVNJVKBJZVKCJZTZGZVFVMVJQVKBCUCVFWAVMVJVFWAVMUDVLVIJZVJVFWAWBVMVFWAGVLVG
      VHWAVFVLVGJZVRVFWCQVTVFVRWCVDVRWCQZVEVDVPWDVQVPVRWCVKBUEULUFSUGSUHVFVRVTV
      LVHJZTVFVRGWEVSVDVEVRWEVSQBCVKUIUJUKUMUNUOVMVFWBVJVCWAVLDVIUPUQURUSUTVAVB
      $.
  $}

  $( Equality theorem for substitution of a class for an ordered pair (analogue
     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 ` (analogue 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 $.

  $( Equality theorem for substitution of a class for an ordered triple.
     (Contributed by Scott Fenton, 22-Aug-2024.) $)
  sbcoteq1a $p |- ( A = <. x , y , z >. ->
  ( [. ( 1st ` ( 1st ` A ) ) / x ]. [. ( 2nd ` ( 1st ` A ) ) / y ].
    [. ( 2nd ` A ) / z ]. ph <-> ph ) ) $=
    ( cv wceq c2nd cfv wsbc c1st cvv eqtr2di sbceq1a syl 2fveq3 wcel vex mp3an
    wb cotp fveq2 ot3rdg elv ot2ndg ot1stg 3bitrrd ) EBFZCFZDFZUAZGZAADEHIZJZUN
    CEKIZHIZJZUQBUOKIZJZULUJUMGAUNTULUMUKHIZUJEUKHUBUTUJGDUHUIUJLUCUDMADUMNOULU
    IUPGUNUQTULUPUKKIZHIZUIEUKHKPUHLQZUILQZUJLQZVBUIGBRZCRZDRZUHUIUJLLLUESMUNCU
    PNOULUHURGUQUSTULURVAKIZUHEUKKKPVCVDVEVIUHGVFVGVHUHUIUJLLLUFSMUQBURNOUG $.

  ${
    $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 nfcv nfsbcw bitr3i anbi1i 3bitr4i df-opab
      elvv abbii df-rab 3eqtr4i ) DEZBECEFGZATZCHZBHZDIUOJJKZLZACUOMNZOZBUOPNZO
      ZTZDIABCUAVEDUTUBUSVFDUPCHZVETZBHVGBHZVETUSVFVGVEBVCBVDQRURVHBURUPVETZCHV
      HVJUQCUPVEAABCUOUCUDSUPVECVCCBVDCVDUEACVBQUFRUGSVAVIVEBCUOUKUHUIULABCDUJV
      EDUTUMUN $.
  $}

  ${
    $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 nfsbcw bitr3i anbi1i 3bitr4i
      elvv opabbii eqtri ) ABCDFEGZBGCGHIZATZCJZBJZEDKUNLLMNZACUNUAOZPZBUNUBOZP
      ZTZEDKABCDEUCURVDEDUOCJZVCTZBJVEBJZVCTURVDVEVCBVABVBQRUQVFBUQUOVCTZCJVFVH
      UPCUOVCAABCUNUDUESUOVCCVACBVBCVBUFACUTQUGRUHSUSVGVCBCUNUKUIUJULUM $.
  $}

  ${
    $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 c2nd cfv wsbc c1st wa copab fvex wceq wb eqcom wcel eqopi
      cxp dfoprab3s cop anbi12i sylan2b syl bicomd ex sbc2iedv pm5.32i opabbii
      eqtr2i ) BCDEHFIZJJUCUAZBDUOKLZMCUONLZMZOZFEPUPAOZFEPBCDEFUDUTVAFEUPUSAUP
      BACDURUQUONQUOKQUPCIZURRZDIZUQRZOZBASUPVFOZABVGUOVBVDUERZABSVFUPURVBRZUQV
      DRZOVHVCVIVEVJVBURTVDUQTUFUOVBVDJJUBUGGUHUIUJUKULUMUN $.
  $}

  ${
    $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.
       (Contributed by NM, 20-Dec-2008.)  Remove unnecessary distinct variable
       conditions.  (Revised by David Abernethy, 19-Jun-2012.)  (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
      wi eqeq2d sbequ12 bibi2d imbi12d opeq2 chvarfv dfoprab4 nfan nfsbv eleq1w
      wsb bi2anan9 sylan9bbr anbi12d cbvoprab12 eqtr4i ) FNZGHUAOAPFEUBLNZGOZMN
      ZHOZPZBDMUSZCLUSZPZLMEUCCNZGOZDNZHOZPZBPZCDEUCAVLLMEFGHVEVNVHQZRZAVKSZUHZ
      VEVFVHQZRZAVLSZUHCLWEWFCWECTAVLCIVKCLUDZUEUFVNVFRZWAWEWBWFWHVTWDVEVNVFVHU
      GUIWHVKVLAVKCLUJZUKULVEVNVPQZRZABSZUHWCDMWAWBDWADTAVKDJBDMUDZUEUFVPVHRZWK
      WAWLWBWNWJVTVEVPVHVNUMUIWNBVKABDMUJZUKULKUNUNUOVSVMCDELMVSLTVSMTVJVLCVJCT
      WGUPVJVLDVJDTVKCLDWMUQUPWHWNPVRVJBVLWHVOVGWNVQVICLGURDMHURUTWNBVKWHVLWOWI
      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 A $.  $d x y z B $.  $d x y z C $.  $d y ch $.  $d z ph $.
    $d x y z D $.  $d x ps $.
    opiota.1 $e |- I =
      ( iota z E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) ) $.
    opiota.2 $e |- X = ( 1st ` I ) $.
    opiota.3 $e |- Y = ( 2nd ` I ) $.
    opiota.4 $e |- ( x = C -> ( ph <-> ps ) ) $.
    opiota.5 $e |- ( y = D -> ( ps <-> ch ) ) $.
    $( The property of a uniquely specified ordered pair.  The proof uses
       properties of the ` iota ` description binder.  (Contributed by Mario
       Carneiro, 21-May-2015.) $)
    opiota $p |- ( E! z E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) ->
      ( ( C e. A /\ D e. B /\ ch ) <-> ( C = X /\ D = Y ) ) ) $=
      ( wceq wa cv cop wrex weu wcel c1st cfv c2nd w3a ceqsrex2v bicomd cio cvv
      opex a1i id wb eqeq1 eqcom opth bitri bitrdi anbi1d 2rexbidv adantl nfeu1
      vex nfcvd iota2df eqeq1i bitr4di sylan9bbr pm5.32da cxp cab opelxpi simpl
      nfvd eleq1d syl5ibrcom rexlimivv abssi iotacl sselid opelxp eleq1 bitr3id
      eqeltrid pm4.71rd 1st2nd2 eqeq2d 3bitr2d df-3an eqeq2i anbi12i fvex opth2
      syl bitr4i 3bitr4g ) FUAZDUAZEUAZUBZSZATZEHUCDGUCZFUDZIGUEZJHUEZTZCTZIJUB
      ZKUFUGZKUHUGZUBZSZXIXJCUIILSZJMSZTZXHXLXKXMKSZTYAXQXHXKCYAXKCXBISXCJSTZAT
      ZEHUCDGUCZXHYAXKYDCABCDEIJGHQRUJUKXHYDXGFULZXMSZYAXHXGYDFXMUMXMUMUEXHIJUN
      UOXHUPXAXMSZXGYDUQXHYGXFYCDEGHYGXEYBAYGXEXMXDSZYBXAXMXDURYHXDXMSYBXMXDUSX
      BXCIJDVGEVGUTVAVBVCVDVEXGFVFXHYDFVRXHFXMVHVIYAKXMSYFXMKUSKYEXMNVJVAVKVLVM
      XHYAXKXHXKYAKGHVNZUEZXHKYEYINXHXGFVOYIYEXGFYIXFXAYIUEZDEGHXBGUEXCHUETYKXF
      XDYIUEXBXCGHVPXFXAXDYIXEAVQVSVTWAWBXGFWCWDWHZXKXMYIUEYAYJIJGHWEXMKYIWFWGV
      TWIXHKXPXMXHYJKXPSYLKGHWJWRWKWLXIXJCWMXTIXNSZJXOSZTXQXRYMXSYNLXNIOWNMXOJP
      WNWOIJXNXOKUFWPKUHWPWQWSWT $.
  $}

  ${
    $d a x y z $.  $d a ph $.  $d x y ps $.
    cnvoprab.1 $e |- ( a = <. x , y >. -> ( ps <-> ph ) ) $.
    cnvoprab.2 $e |- ( ps -> a e. ( _V X. _V ) ) $.
    $( The converse of a class abstraction of nested ordered pairs.
       (Contributed by Thierry Arnoux, 17-Aug-2017.)  (Proof shortened by
       Thierry Arnoux, 20-Feb-2022.) $)
    cnvoprab $p |- `' { <. <. x , y >. , z >. | ph } = { <. z , a >. | ps } $=
      ( cv cvv cxp wcel wa copab ccnv coprab dfoprab3 cnveqi cin cnvopab inopab
      wss wceq ssopab2i sseqin2 mpbi 3eqtr2i eqtr3i ) FIJJKLZBMZFENZOZACDEPZOBE
      FNZUKUMBACDEFGQRULUJEFNUIEFNZUNSZUNUJFETUIBEFUAUNUOUBUPUNUCBUIEFHUDUNUOUE
      UFUGUH $.
  $}

  ${
    $d x y z u A $.  $d x y z u B $.  $d x y z u C $.
    $( Define the Cartesian 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 relopabv 1st2nd mpan fvex sylib
      id eqeltrrd opelopab ) FADEIZJZFKLZFMLZNZUEJCUFFUIUEUEOUFFUIPADEQFUERSUFU
      BUCABCDEUGUHFKTFMTGHUDUA $.
  $}

  ${
    $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 opex vex op1std fveq2d op1st eqtr2di op2nd op2ndd
      eqcomd 3bitrd biimpa exlimiv ) HAEFGUAZUBZHEMZFMZUGZGMZUGZNZAUCZGOZFOZEOZ
      DVDVNLMZVINZAUCZGOFOEOVNLHVCVCVOHNZVQVKEFGVRVPVJAVOHVIUDUEUFAEFGLUHUIUJVM
      DEVLDFVKDGVJADVJABCDVJVEHPQZPQZNABRVJVTVGPQVEVJVSVGPVGVHHVEVFUKZGULZUMZUN
      VEVFEULZFULZUOUPISVJVFVSTQZNBCRVJWFVGTQVFVJVSVGTWCUNVEVFWDWEUQUPJSVJVHHTQ
      ZNCDRVJWGVHVGVHHWAWBURUSKSUTVAVBVBVBS $.
  $}

  ${
    $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 mpteq1i nfcsbw sylan9eqr cbvmpox 3eqtr4ri ) CGDGIZJZAUTEKZLZMZACI
      ZUBNZBVEUCNZFKZKZOGHDVBAUTBHIZFKZKZPCADAIZJZELZMZVIOABDEFPGHCDVBVIVLVEUTV
      JUDQZVIAUTVHKVLVQAVFUTVHUTVJVEGUEZHUEZUFUGVQAUTVHVKVQBVGVJFUTVJVEVRVSUHUG
      UIUAUJCVPVDVIAGDVOVCGVORAVAVBAVARAUTESZUKVMUTQZVNVAEVBVMUTULAUTETZUNUMUOA
      BGHDEFVBVLGERVTGFRHFRAUTVKSBAUTVKBUTRBVJFSUPWBBIVJQWAFVKVLBVJFTAUTVKTUQUR
      US $.

    $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 csb cmpt mpomptsx iunxpconst mpteq1i
      eqtri ) ABDEFGCADAHIEJKZACHZLMBUBNMFOOZPCDEJZUCPABCDEFQCUAUDUCADERST $.
  $}

  ${
    $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 nfcsbw sylan9eqr cop vex op1std csbeq1d csbeq2dv eqtrd
      op2ndd mpomptx 3eqtr4i dmmptss nfxp sneq xpeq12d cbviun sseqtrri ) FUCHCH
      KZLZAVDDMZNZOZACAKZLZDNZOIVHAIKZUAPZBVLUDPZEMZMZFABCDEUEHJCVFAVDBJKZEMZMZ
      UEFIVHVPUFABHJCDEVFVSHDQAVDDRZHEQJEQAVDVRRBAVDVRBVDQBVQERUGAVDDSZBKVQTVIV
      DTZEVRVSBVQESAVDVRSUHUBGHJICVFVPVSVLVDVQUITZVPAVDVOMVSWCAVMVDVOVDVQVLHUJZ
      JUJZUKULWCAVDVOVRWCBVNVQEVDVQVLWDWEUOULUMUNUPUQURAHCVKVGHVKQAVEVFAVEQVTUS
      WBVJVEDVFVIVDUTWAVAVBVC $.

    $( 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 nfcsbw wb eleq1w adantr csbeq1a eleq2d sylan9bbr
      nfcv anbi12d sylan9eqr eqeq2d cbvoprab12 df-mpo 3eqtr4i fmpt bitr3i nfel1
      mpomptx nfralw cbvralw raleqbidv bitrid nfxp xpeq12d cbviun feq2i 3bitr4i
      sneq ) AILZBJLZEMZMZFNZJAWJDMZOZICOZICWJUAZWOUEZUBZFGUQZEFNZBDOZACOACALZU
      AZDUEZUBZFGUQWQAKLZUCUFZBXHUDUFZEMZMZFNZKWTOXAXMWNKIJCWOXHWJWKUGPZXLWMFXN
      XLAWJXKMWMXNAXIWJXKWJWKXHIUHZJUHZUIUJXNAWJXKWLXNBXJWKEWJWKXHXOXPUKUJULUMZ
      QUNKWTFXLGABCDEUOZIJCWOWMUOZGKWTXLURXDCNZBLZDNZRZXHEPZRZABKUPWJCNZWKWONZR
      ZXHWMPZRZIJKUPXRXSYEYJABKIJYEISYEJSYHYIAYFYGAYFASAJWOAWJDTZUSUTAXHWMAWJWL
      TZVAUTYHYIBYHBSBXHWMBAWJWLBWJVIBWKETZVBVAUTXDWJPZYAWKPZRZYCYHYDYIYPXTYFYB
      YGYNXTYFVCYOAICVDVEYOYBWKDNYNYGBJDVDYNDWOWKAWJDVFZVGVHVJYPEWMXHYOYNEWLWMB
      WKEVFZAWJWLVFZVKVLVJVMABKCDEVNIJKCWOWMVNVOHIJKCWOXLWMXQVSVOVPVQXCWPAICXCI
      SWNAJWOYKAWMFYLVRVTXCWLFNZJDOYNWPXBYTBJDXBJSBWLFYMVRYOEWLFYRQWAYNYTWNJDWO
      YQYNWLWMFYSQWBWCWAXGWTFGAICXFWSIXFVIAWRWOAWRVIYKWDYNXEWRDWOXDWJWIYQWEWFWG
      WH $.
  $}

  ${
    $d A x y $.  $d B x y $.  $d D x y $.
    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 2ralimi wf fmpo dffn2 bitr4i sylib ) EGIZBDJ
      ACJEKIZBDJACJZFCDLZMZUAUBABCDEGNOUCUDKFPUEABCDEKFHQUDFRST $.

    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 fnmpoi fndmi ) CDIFABCDEFGHJK $.
  $}

  ${
    $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.)
       (Proof shortened by Lammen, 29-May-2024.) $)
    dmmpoga $p |- ( A. x e. A A. y e. B C e. V -> dom F = ( A X. B ) ) $=
      ( wcel wral cxp fnmpo fndmd ) EGIBDJACJCDKFABCDEFGHLM $.

    $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 vsnex
      wss xpexg mpan ralimi iunexg sylan2 ssexg funex ) CFJZDGJZACKZLZHMHNZOJZH
      OJABCDEHIPUOUPACAQRZDSZUAZUDUTOJZUQABCDEHIUBUNULUSOJZACKVAUMVBACUROJUMVBA
      UCURDOGUEUFUGACUSFOUHUIUPUTOUJTOHUKT $.

    $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-rep .  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-rep' ; $)
  $}

  ${
    $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 g $.  $d B a b g $.  $d D a b f g h $.  $d G a b f g h $.
    $d W g $.  $d X a b f g h $.  $d Y a b f g h $.  $d ph f h $.
    mptmpoopabbrd.g $e |- ( ph -> G e. W ) $.
    mptmpoopabbrd.x $e |- ( ph -> X e. ( A ` G ) ) $.
    mptmpoopabbrd.y $e |- ( ph -> Y e. ( B ` G ) ) $.
    ${
      $d ta g $.  $d th a b $.
      mptmpoopabbrd.1 $e |- ( ( a = X /\ b = Y ) -> ( ta <-> th ) ) $.
      mptmpoopabbrd.2 $e |- ( g = G -> ( ch <-> ta ) ) $.
      mptmpoopabbrd.m $e |- M = ( g e. _V
                                    |-> ( a e. ( A ` g ) , b e. ( B ` g )
                           |-> { <. f , h >. | ( ch /\ f ( D ` g ) h ) } ) ) $.
      $( The operation value of a function value of a collection of ordered
         pairs of elements related in two ways.  (Contributed by Alexander van
         Vekens, 8-Nov-2017.)  (Revised by AV, 15-Jan-2021.)  Add disjoint
         variable condition on ` D , f , h ` to remove hypotheses; avoid
         ~ ax-rep .  (Revised by SN, 7-Apr-2025.) $)
      mptmpoopabbrd $p |- ( ph -> ( X ( M ` G ) Y )
                               = { <. f , h >. | ( th /\ f ( D ` G ) h ) } ) $=
        ( cfv co cv wbr copab cmpo wceq fveq2 breqd anbi12d opabbidv mpoeq123dv
        cvv elexd wcel cpw fvex pwex simpr ssopab2i opabss elpwi2 rgen2w mpoexw
        wa sstri a1i fvmptd3 oveqd anbi1d eqid ancom opabbii opabresex2 eqeltri
        ovmpoa syl2anc eqtrd ) ANOKLUDZUENOPQKEUDZKFUDZDHUFZJUFZKGUDZUGZVHZHJUH
        ZUIZUEZCWHVHZHJUHZAWBWKNOAIKPQIUFZEUDZWOFUDZBWEWFWOGUDZUGZVHZHJUHZUIWKU
        PLUPUCWOKUJZPQWPWQXAWCWDWJWOKEUKWOKFUKXBWTWIHJXBBDWSWHUBXBWRWGWEWFWOKGU
        KULUMUNUOAKMRUQWKUPURAPQWCWDWJWGUSZKEUTKFUTWGKGUTZVAWJXCURPQWCWDWJWGUPX
        DWJWHHJUHWGWIWHHJDWHVBVCHJWGVDVIVEVFVGVJVKVLANWCUROWDURWLWNUJSTPQNOWCWD
        WJWNWKPUFNUJQUFOUJVHZWIWMHJXEDCWHUAVMUNWKVNWNWHCVHZHJUHUPWMXFHJCWHVOVPC
        HJKGVQVRVSVTWA $.
      $( $j usage 'mptmpoopabbrd' avoids 'ax-rep'; $)

      $( Obsolete version of ~ mptmpoopabbrd as of 7-Apr-2025.  (Contributed by
         Alexander van Vekens, 8-Nov-2017.)  (New usage is discouraged.)
         (Proof modification is discouraged.) $)
      mptmpoopabbrdOLD $p |- ( ph -> ( X ( M ` G ) Y )
                               = { <. f , h >. | ( th /\ f ( D ` G ) h ) } ) $=
        ( cfv co cv wbr copab cmpo wceq fveq2 breqd anbi12d opabbidv mpoeq123dv
        wa cvv elexd wcel fvex mpoex fvmptd3 oveqd anbi1d eqid ancom opabresex2
        a1i opabbii eqeltri ovmpoa syl2anc eqtrd ) ANOKLUDZUENOPQKEUDZKFUDZDHUF
        ZJUFZKGUDZUGZUPZHJUHZUIZUEZCVTUPZHJUHZAVNWCNOAIKPQIUFZEUDZWGFUDZBVQVRWG
        GUDZUGZUPZHJUHZUIWCUQLUQUCWGKUJZPQWHWIWMVOVPWBWGKEUKWGKFUKWNWLWAHJWNBDW
        KVTUBWNWJVSVQVRWGKGUKULUMUNUOAKMRURWCUQUSAPQVOVPWBKEUTKFUTVAVHVBVCANVOU
        SOVPUSWDWFUJSTPQNOVOVPWBWFWCPUFNUJQUFOUJUPZWAWEHJWODCVTUAVDUNWCVEWFVTCU
        PZHJUHUQWEWPHJCVTVFVICHJKGVGVJVKVLVM $.
    $}

    $d C a b g $.
    mptmpoopabovd.m $e |- M = ( g e. _V |-> ( a e. ( A ` g ) , b e. ( B ` g )
                              |-> { <. f , h >. | ( f ( a ( C ` g ) b ) h
                                                    /\ f ( D ` g ) h ) } ) ) $.
    $( The operation value of a function value of a collection of ordered pairs
       of related elements.  (Contributed by Alexander van der Vekens,
       8-Nov-2017.)  (Revised by AV, 15-Jan-2021.)  Add disjoint variable
       condition on ` D , f , h ` to remove hypotheses.  (Revised by SN,
       13-Dec-2024.) $)
    mptmpoopabovd $p |- ( ph -> ( X ( M ` G ) Y ) = { <. f , h >. |
                              ( f ( X ( C ` G ) Y ) h /\ f ( D ` G ) h ) } ) $=
      ( cv cfv co wbr wceq wa oveq12 breqd fveq2 oveqd mptmpoopabbrd ) AFTZHTZN
      TZOTZGTZDUAZUBZUCUKULLMIDUAZUBZUCUKULUMUNURUBZUCBCEFGHIJKLMNOPQRUMLUDUNMU
      DUEUTUSUKULUMLUNMURUFUGUOIUDZUQUTUKULVAUPURUMUNUOIDUHUIUGSUJ $.
  $}

  ${
    $d A a b g $.  $d B a b g $.  $d D a b g $.  $d G a b f g h $.  $d W g $.
    $d X a b f g h $.  $d Y a b f g h $.  $d ph f h $.
    mptmpoopabbrdOLD.g $e |- ( ph -> G e. W ) $.
    mptmpoopabbrdOLD.x $e |- ( ph -> X e. ( A ` G ) ) $.
    mptmpoopabbrdOLD.y $e |- ( ph -> Y e. ( B ` G ) ) $.
    mptmpoopabbrdOLD.v $e |- ( ph -> { <. f , h >. | ps } e. V ) $.
    mptmpoopabbrdOLD.r $e |- ( ( ph /\ f ( D ` G ) h ) -> ps ) $.
    ${
      $d ta g $.  $d th a b $.
      mptmpoopabbrdOLD.1 $e |- ( ( a = X /\ b = Y ) -> ( ta <-> th ) ) $.
      mptmpoopabbrdOLD.2 $e |- ( g = G -> ( ch <-> ta ) ) $.
      mptmpoopabbrdOLD.m $e |- M = ( g e. _V
                                    |-> ( a e. ( A ` g ) , b e. ( B ` g )
                           |-> { <. f , h >. | ( ch /\ f ( D ` g ) h ) } ) ) $.
      $( Obsolete version of ~ mptmpoopabbrd as of 13-Dec-2024.  (Contributed
         by Alexander van Vekens, 8-Nov-2017.)  (Revised by AV, 15-Jan-2021.)
         (Proof modification is discouraged.)  (New usage is discouraged.) $)
      mptmpoopabbrdOLDOLD $p |- ( ph -> ( X ( M ` G ) Y )
                               = { <. f , h >. | ( th /\ f ( D ` G ) h ) } ) $=
        ( cfv co cv wbr copab cmpo wcel wceq cvv fveq2 breqd anbi12d mpoeq123dv
        wa opabbidv elex adantr fvex pm3.2i mpoexga fvmptd3 syl2anc oveqd ancom
        mp1i opabbii opabresex2d eqeltrid anbi1d eqid ovmpoga syl3anc eqtrd ) A
        PQLMUHZUIPQRSLFUHZLGUHZEIUJZKUJZLHUHZUKZVAZIKULZUMZUIZDWGVAZIKULZAWAWJP
        QALOUNZWNWAWJUOTTWNWNVAZJLRSJUJZFUHZWPGUHZCWDWEWPHUHZUKZVAZIKULZUMWJUPM
        UPUGWPLUOZRSWQWRXBWBWCWIWPLFUQWPLGUQXCXAWHIKXCCEWTWGUFXCWSWFWDWEWPLHUQU
        RUSVBUTWNLUPUNWNLOVCVDWBUPUNZWCUPUNZVAWJUPUNWOXDXELFVELGVEVFRSWBWCWIUPU
        PVGVLVHVIVJAPWBUNQWCUNWMUPUNWKWMUOUAUBAWMWGDVAZIKULUPWLXFIKDWGVKVMABDIK
        LNHUDUCVNVORSPQWBWCWIWMWJUPRUJPUOSUJQUOVAZWHWLIKXGEDWGUEVPVBWJVQVRVSVT
        $.
    $}

    $d C a b g $.
    mptmpoopabovdOLD.m $e |- M = ( g e. _V |->
      ( a e. ( A ` g ) , b e. ( B ` g )
        |-> { <. f , h >. | ( f ( a ( C ` g ) b ) h /\ f ( D ` g ) h ) } ) ) $.
    $( Obsolete version of ~ mptmpoopabovd as of 13-Dec-2024.  (Contributed by
       Alexander van der Vekens, 8-Nov-2017.)  (Revised by AV, 15-Jan-2021.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    mptmpoopabovdOLD $p |- ( ph -> ( X ( M ` G ) Y ) = { <. f , h >. |
                              ( f ( X ( C ` G ) Y ) h /\ f ( D ` G ) h ) } ) $=
      ( cv cfv co wbr wceq wa oveq12 breqd fveq2 oveqd mptmpoopabbrdOLDOLD ) AB
      GUDZIUDZPUDZQUDZHUDZEUEZUFZUGUOUPNOJEUEZUFZUGUOUPUQURVBUFZUGCDFGHIJKLMNOP
      QRSTUAUBUQNUHUROUHUIVDVCUOUPUQNUROVBUJUKUSJUHZVAVDUOUPVEUTVBUQURUSJEULUMU
      KUCUN $.
  $}

  ${
    $d A a b s t x y $.  $d B a b s t x y $.  $d C a b s t $.  $d D a b s t $.
    $d E a b $.  $d U a b x y $.  $d V a b x y $.  $d X a b s t x y $.
    $d Y a b s t x y $.
    el2mpocsbcl.o $e |- O = ( x e. A , y e. B
                               |-> ( s e. C , t e. D |-> E ) ) $.
    $( If the operation value of the operation value of two nested maps-to
       notation is not empty, all involved arguments belong to the
       corresponding base classes of the maps-to notations.  (Contributed by
       AV, 21-May-2021.) $)
    el2mpocsbcl $p |- ( A. x e. A A. y e. B ( C e. U /\ D e. V )
                      -> ( W e. ( S ( X O Y ) T )
                           -> ( ( X e. A /\ Y e. B )
                                /\ ( S e. [_ X / x ]_ [_ Y / y ]_ C
                                /\ T e. [_ X / x ]_ [_ Y / y ]_ D ) ) ) ) $=
      ( wcel csb va vb wa wral co wi simpl cmpo cv cvv wceq nfcv nfcsb1v nfcsbw
      nfmpo csbeq1a csbeq2dv sylan9eq mpoeq123dv cbvmpo eqtri a1i csbeq1 adantr
      adantl eqtrd simpr ralimi rspcsbela syl2an ralimdv impcom syl2anc mpoexga
      ex ovmpod oveqd eleq2d eqid elmpocl biimtrdi impancom jca wn mpondm0 noel
      c0 pm2.21i 0ov eleq2s adantld pm2.61i ) FJSZGMSZUCZBEUDZADUDZNHIOPLUEZUEZ
      SZODSZPESZUCZHAOBPFTZTZSIAOBPGTZTZSUCZUCZXCWQWTUCZXIUFXCXJXIXCXJUCXCXHXCX
      JUGXJXCXHWQXCWTXHWQXCUCZWTNHIQCXEXGAOBPKTZTZUHZUEZSXHXKWSXONXKWRXNHIXKUAU
      BOPDEQCAUAUIZBUBUIZFTZTZAXPBXQGTZTZAXPBXQKTZTZUHZXNLUJLUAUBDEYDUHZUKXKLAB
      DEQCFGKUHZUHYERABUAUBDEYFYDUAYFULUBYFULQCAXSYAYCAXPXRUMAXPXTUMAXPYBUMUOQC
      BXSYAYCBAXPXRBXPULZBXQFUMUNBAXPXTYGBXQGUMUNBAXPYBYGBXQKUMUNUOAUIXPUKZBUIX
      QUKZUCQCFGKXSYAYCYHYIFAXPFTXSAXPFUPYIAXPFXRBXQFUPUQURYHYIGAXPGTYAAXPGUPYI
      AXPGXTBXQGUPUQURYHYIKAXPKTYCAXPKUPYIAXPKYBBXQKUPUQURUSUTVAVBXPOUKZXQPUKZU
      CZYDXNUKXKYLQCXSYAYCXEXGXMYLXSAOXRTZXEYJXSYMUKYKAXPOXRVCVDYLAOXRXDYKXRXDU
      KYJBXQPFVCVEUQVFYLYAAOXTTZXGYJYAYNUKYKAXPOXTVCVDYLAOXTXFYKXTXFUKYJBXQPGVC
      VEUQVFYLYCAOYBTZXMYJYCYOUKYKAXPOYBVCVDYLAOYBXLYKYBXLUKYJBXQPKVCVEUQVFUSVE
      XCXAWQXAXBUGVEZXCXBWQXAXBVGZVEXKXEJSZXGMSZXNUJSXKXAXDJSZADUDZYRYPXCWQUUAX
      CWPYTADXCWPYTXCXBWMBEUDYTWPYQWOWMBEWMWNUGVHBPEFJVIVJVOVKVLAODXDJVIVMXKXAX
      FMSZADUDZYSYPXCWQUUCXCWPUUBADXCWPUUBXCXBWNBEUDUUBWPYQWOWNBEWMWNVGVHBPEGMV
      IVJVOVKVLAODXFMVIVMQCXEXGXMJMVNVMVPVQVRQCXEXGXMHIXNNXNVSVTWAWBVLWCVOXCWDZ
      WTXIWQUUDWTNHIWGUEZSXIUUDWSUUENUUDWRWGHIABYFLOPDERWEVQVRXINWGUUENWGSXINWF
      WHHIWIWJWAWKWLVO $.
  $}

  ${
    $d A s t x y $.  $d B s t x y $.  $d C s t $.  $d D s t $.  $d F x y $.
    $d G x y $.  $d U x y $.  $d V x y $.  $d X s t x y $.  $d Y s t x y $.
    el2mpocl.o $e |- O = ( x e. A , y e. B |-> ( s e. C , t e. D |-> E ) ) $.
    el2mpocl.e $e |- ( ( x = X /\ y = Y ) -> ( C = F /\ D = G ) ) $.
    $( If the operation value of the operation value of two nested maps-to
       notation is not empty, all involved arguments belong to the
       corresponding base classes of the maps-to notations.  Using implicit
       substitution.  (Contributed by AV, 21-May-2021.) $)
    el2mpocl $p |- ( A. x e. A A. y e. B ( C e. U /\ D e. V )
                -> ( W e. ( S ( X O Y ) T )
                     -> ( ( X e. A /\ Y e. B ) /\ ( S e. F /\ T e. G ) ) ) ) $=
      ( wcel wa wral co csb el2mpocsbcl simpl wceq simplr simpld adantll csbied
      cv eleq2d simprd anbi12d biimpd imdistani syl6 ) FJUBGOUBUCBEUDADUDPHIQRN
      UEUEUBQDUBZREUBZUCZHAQBRFUFZUFZUBZIAQBRGUFZUFZUBZUCZUCVCHLUBZIMUBZUCZUCAB
      CDEFGHIJKNOPQRSTUGVCVJVMVCVJVMVCVFVKVIVLVCVELHVCAQVDLDVAVBUHZVCAUNQUIZUCZ
      BRFLEVAVBVOUJZVOBUNRUIZFLUIZVCVOVRUCZVSGMUIZUAUKULUMUMUOVCVHMIVCAQVGMDVNV
      PBRGMEVQVOVRWAVCVTVSWAUAUPULUMUMUOUQURUSUT $.
  $}

  ${
    $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 ph x y z $.  $d A x y z $.  $d B x y z $.  $d R x y z $.  $d C z $.
    $d D z $.
    offval22.a $e |- ( ph -> A e. V ) $.
    offval22.b $e |- ( ph -> B e. W ) $.
    offval22.c $e |- ( ( ph /\ x e. A /\ y e. B ) -> C e. X ) $.
    offval22.d $e |- ( ( ph /\ x e. A /\ y e. B ) -> D e. Y ) $.
    offval22.f $e |- ( ph -> F = ( x e. A , y e. B |-> C ) ) $.
    offval22.g $e |- ( ph -> G = ( x e. A , y e. B |-> D ) ) $.
    $( The function operation expressed as a mapping, variation of ~ offval2 .
       (Contributed by SO, 15-Jul-2018.) $)
    offval22 $p |- ( ph -> ( F oF R G ) =
        ( x e. A , y e. B |-> ( C R D ) ) ) $=
      ( vz cof co cxp cv c1st cfv c2nd csb cmpt cmpo cvv xpexd wcel xp1st xp2nd
      wa jca w3a wi fvex nfcv nfv nfcsb1v nfel1 nfim wceq eleq1 3anbi3d csbeq1a
      eleq1d imbi12d 3anbi2d elexd vtocl2gf mp2an sylan2 mpompts eqtrdi offval2
      3expb csbov12g csbeq2dv ax-mp eqtr2i mpteq2i eqtr4i ) AIJHUBUCUADEUDZBUAU
      EZUFUGZCWIUHUGZFUIZUIZBWJCWKGUIZUIZHUCZUJZBCDEFGHUCZUKZAUAWHWMWOHIJULULUL
      ADEKLOPUMWIWHUNZAWJDUNZWKEUNZUQZWMULUNZWTXAXBWIDEUOWIDEUPURZAXAXBXDWKULUN
      ZWJULUNZAXAXBUSZXDUTZWIUHVAZWIUFVAZABUEZDUNZCUEZEUNZUSZFULUNZUTAXMXBUSZWL
      ULUNZUTXICBWKWJULULCWKVBZBWKVBZBWJVBZXRXSCXRCVCZCWLULCWKFVDVEVFXHXDBXHBVC
      ZBWMULBWJWLVDVEVFXNWKVGZXPXRXQXSYEXOXBAXMXNWKEVHVIZYEFWLULCWKFVJVKVLXLWJV
      GZXRXHXSXDYGXMXAAXBXLWJDVHVMZYGWLWMULBWJWLVJVKVLXPFMQVNVOVPWAVQWTAXCWOULU
      NZXEAXAXBYIXFXGXHYIUTZXJXKXPGULUNZUTXRWNULUNZUTYJCBWKWJULULXTYAYBXRYLCYCC
      WNULCWKGVDVEVFXHYIBYDBWOULBWJWNVDVEVFYEXPXRYKYLYFYEGWNULCWKGVJVKVLYGXRXHY
      LYIYHYGWNWOULBWJWNVJVKVLXPGNRVNVOVPWAVQAIBCDEFUKUAWHWMUJSBCUADEFVRVSAJBCD
      EGUKUAWHWOUJTBCUADEGVRVSVTWQUAWHBWJCWKWRUIZUIZUJWSUAWHWPYNYNBWJWLWNHUCZUI
      ZWPXFYNYPVGXJXFBWJYMYOCWKFGHULWBWCWDXGYPWPVGXKBWJWLWNHULWBWDWEWFBCUADEWRV
      RWGVS $.
  $}

  $( If a binary relation holds for the result of an operation, the operands
     are in the domain of the operation.  (Contributed by AV, 31-Dec-2020.) $)
  brovpreldm $p |- ( D ( B A C ) E -> <. B , C >. e. dom A ) $=
    ( co wbr cop wcel cdm df-br c0 wne ne0i wn cfv df-ov ndmfv eqtrid necon1ai
    syl sylbi ) DEBCAFZGDEHZUCIZBCHZAJIZDEUCKUEUCLMUGUCUDNUGUCLUGOUCUFAPLBCAQUF
    ARSTUAUB $.

  ${
    $d E a b e f p v $.  $d V a b c e f p v $.  $d ps c e v $.
    bropopvvv.o $e |- O = ( v e. _V , e e. _V |-> ( a e. v , b e. v |->
                            { <. f , p >. | ph } ) ) $.
    bropopvvv.p $e |- ( ( v = V /\ e = E ) -> ( ph <-> ps ) ) $.
    bropopvvv.oo $e |- ( ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ B e. V ) )
                         -> ( A ( V O E ) B ) = { <. f , p >. | th } ) $.
    $( If a binary relation holds for the result of an operation which is a
       result of an operation, the involved classes are sets.  (Contributed by
       Alexander van der Vekens, 12-Dec-2017.)  (Proof shortened by AV,
       3-Jan-2021.) $)
    bropopvvv $p |- ( F ( A ( V O E ) B ) P
                      -> ( ( V e. _V /\ E e. _V )
                        /\ ( F e. _V /\ P e. _V )
                        /\ ( A e. V /\ B e. V ) ) ) $=
      ( wcel vc co wbr cvv wa w3a cop wi brovpreldm copab cmpo cv wceq opabbidv
      simpl mpoeq123dv ovmpoga dmeqd eleq2d coprab dmoprabss sseli opelxp df-br
      cdm cxp c0 wne ne0i breqd brabv anim2i ex adantr sylbid com23 a1d mpondm0
      cfv df-ov fveq1 eqtrid 0fv eqtrdi eqneqall 3syl pm2.61i syl sylbi pm2.43i
      wn com12 anc2ri df-3an imbitrrdi df-mpo eleq2s biimtrdi w3o 3ianor df-3or
      dmeqi wo ianor dm0 eleq2i bitrdi noel pm2.21i sylbir anor mpoexga pm2.24d
      id ancri imp jaoi3 ) KGEFMJLUBZUBZUCZMUDTZJUDTZUEZKUDTGUDTUEZEMTFMTUEZUFZ
      XTEFUGZXRVEZTZXTYFUHZXREFKGUIYAYBOPMMBINUJZUKZUDTZUFZYIYJUHZYNYIYGYLVEZTY
      JYNYHYPYGYNXRYLDHMJUDUDOPDULZYQAINUJZUKZYLLUDYQMUMZHULJUMZUEZOPYQYQYRMMYK
      YTUUAUOZUUCUUBABINRUNUPQUQURUSYJYGOULMTPULMTUEUAULYKUMZUEOPUAUTZVEZYPYGUU
      FTYGMMVFZTZYJUUFUUGYGUUDOPUAMMVAVBUUHYEYJEFMMVCYEXTYCYDUEZYEUEYFYEXTUUIXT
      YEUUIXTYEUUIUHZXTKGUGZXSTZXTUUJUHZKGXSVDUULXSVGVHZUUMXSUUKVIYCUUNUUMUHZYC
      UUMUUNYCYEXTUUIYCYEXTUUIUHYCYEUEZXTKGCINUJZUCZUUIUUPXSUUQKGSVJYCUURUUIUHY
      EYCUURUUIUURYDYCCINKGVKVLVMVNVOVMVPVQYCWKZXRVGUMZXSVGUMUUODHYSLMJUDUDQVRZ
      UUTXSYGVGVSZVGUUTXSYGXRVSUVBEFXRVTYGXRVGWAWBYGWCWDUUMXSVGWEWFWGWHWIWJWLWM
      YCYDYEWNWOWIWHYLUUEOPUAMMYKWPXBWQWRYNWKYAWKZYBWKZYMWKZWSZYOYAYBYMWTUVFUVC
      UVDXCZUVEXCYOUVCUVDUVEXAUVGYOUVEUVGUUSYOYAYBXDUUSYIYGVGTZYJUUSYIYGVGVEZTU
      VHUUSYHUVIYGUUSXRVGUVAURUSUVIVGYGXEXFXGUVHYJYGXHXIWRXJUVGWKZUVEYOUVJYCUVE
      YOUHYAYBXKYCYMYOYCYAYAUEZYMYAUVKYBYAYAYAXNXOVNOPMMYKUDUDXLWHXMXJXPXQWIWIW
      GWHWJ $.
  $}

  ${
    $d U a $.
    bropfvvvv.o $e |- O = ( a e. U |-> ( b e. V , c e. W
                                         |-> { <. d , e >. | ph } ) ) $.
    bropfvvvv.oo $e |- ( ( A e. U /\ B e. S /\ C e. T )
                         -> ( B ( O ` A ) C ) = { <. d , e >. | th } ) $.
    $( Lemma for ~ bropfvvvv .  (Contributed by AV, 31-Dec-2020.)  (Revised by
       AV, 16-Jan-2021.) $)
    bropfvvvvlem $p |- ( ( <. B , C >. e. ( S X. T ) /\ D ( B ( O ` A ) C ) E )
           -> ( A e. U /\ ( B e. S /\ C e. T ) /\ ( D e. _V /\ E e. _V ) ) ) $=
      ( cop cxp wcel cfv co wbr wa cvv w3a wi opelxp wne brne0 copab wceq 3expb
      c0 breqd brabv anim2i ex adantr sylbid com23 wn cmpo fvmptndm df-ov fveq1
      a1d eqtrid 0fv eqtrdi eqneqall 3syl pm2.61i mpcom com12 3anan32 imbitrrdi
      anc2ri sylbi imp ) DEUAZGHUBUCZFKDECLUDZUEZUFZCIUCZDGUCZEHUCZUGZFUHUCKUHU
      CUGZUIZWEWLWHWNUJDEGHUKWLWHWIWMUGZWLUGWNWLWHWOWHWLWOWGUQULZWHWLWOUJZFKWGU
      MWIWPWHWQUJZUJZWIWRWPWIWLWHWOWIWLWHWOUJWIWLUGZWHFKBRJUNZUFZWOWTWGXAFKWIWJ
      WKWGXAUOTUPURWIXBWOUJWLWIXBWOXBWMWIBRJFKUSUTVAVBVCVAVDVJWIVEWFUQUOZWGUQUO
      WSOIPQMNARJUNVFLCSVGXCWGWDUQUDZUQXCWGWDWFUDXDDEWFVHWDWFUQVIVKWDVLVMWRWGUQ
      VNVOVPVQVRWAWIWLWMVSVTWBWC $.

    $d A a b c d e $.  $d S a $.  $d S b c z $.  $d T a $.  $d T b c z $.
    $d ps a $.  $d ps z $.  $d d e z $.
    bropfvvvv.s $e |- ( a = A -> V = S ) $.
    bropfvvvv.t $e |- ( a = A -> W = T ) $.
    bropfvvvv.p $e |- ( a = A -> ( ph <-> ps ) ) $.
    $( If a binary relation holds for the result of an operation which is a
       function value, the involved classes are sets.  (Contributed by AV,
       31-Dec-2020.)  (Revised by AV, 16-Jan-2021.) $)
    bropfvvvv $p |- ( ( S e. X /\ T e. Y ) -> ( D ( B ( O ` A ) C ) E
         -> ( A e. U /\ ( B e. S /\ C e. T ) /\ ( D e. _V /\ E e. _V ) ) ) ) $=
      ( vz wcel wa cfv co wbr cop cdm cvv w3a brovpreldm copab cmpo wi opabbidv
      cv mpoeq123dv fvmptg dmeqd eleq2d coprab cxp dmoprabss sseli bropfvvvvlem
      ex syl df-mpo dmeqi eleq2s biimtrdi com23 a1d wn wo ianor c0 fvmptndm dm0
      wceq eleq2i bitrdi noel pm2.21i notnotb anim12i adantl mpoexga sylbir imp
      elex pm2.24d jaoi3 sylbi com34 pm2.61i mpdi ) HPUHZIQUHZUIZGLEFDMUJZUKULZ
      EFUMZXGUNZUHZDJUHZEHUHFIUHUIGUOUHLUOUHUIUPZXGEFGLUQXLSTHIBUAKURZUSZUOUHZU
      IZXFXHXKXMUTUTZUTXQXRXFXQXKXHXMXQXKXIXOUNZUHXHXMUTZXQXJXSXIXQXGXORDSTNOAU
      AKURZUSZXOJUOMRVBDWFZSTNOYAHIXNUDUEYCABUAKUFVAVCUBVDVEVFXTXISVBHUHTVBIUHU
      IUGVBXNWFZUISTUGVGZUNZXSXIYFUHXIHIVHZUHZXTYFYGXIYDSTUGHIVIVJYHXHXMACDEFGH
      IJKLMNORSTUAUBUCVKVLVMXOYESTUGHIXNVNVOVPVQVRVSXQVTZXFXKXHXMYIXLVTZXPVTZWA
      XFXKXTUTZUTZXLXPWBYJYMYKYJYLXFYJXKXIWCUHZXTYJXKXIWCUNZUHYNYJXJYOXIYJXGWCR
      JYBMDUBWDVEVFYOWCXIWEWGWHYNXTXIWIWJVQVSYJVTZYKYMYPXLYKYMUTXLWKXLXFYKYLXLX
      FYKYLUTXLXFUIZXPYLYQHUOUHZIUOUHZUIZXPXFYTXLXDYRXEYSHPWQIQWQWLWMSTHIXNUOUO
      WNVMWRVLVRWOWPWSWTXAXBXC $.
  $}

  ${
    $d u v x y z A $.  $d u v y z B $.  $d u v z C $.  $d u v x y z X $.
    ovmptss.1 $e |- F = ( x e. A , y e. B |-> C ) $.
    $( If all the values of the mapping are subsets of a class ` X ` , then so
       is any evaluation of the mapping.  (Contributed by Mario Carneiro,
       24-Dec-2016.) $)
    ovmptss $p |- ( A. x e. A A. y e. B C C_ X -> ( E F G ) C_ X ) $=
      ( vz vu vv cv cfv csb wss wral sseq1d nfcv c1st c2nd csn cxp ciun co cmpo
      cop cmpt mpomptsx eqtri fvmptss wceq op1std csbeq1d op2ndd csbeq2dv eqtrd
      vex raliunxp nfcsb1v nfxp weq sneq csbeq1a xpeq12d cbviun raleqi nfv nfss
      nfralw cbvralw raleqbidv bitrid 3bitr4ri df-ov sseq1i 3imtr4i ) AKNZUAOZB
      VSUBOZEPZPZIQZKACANZUCZDUDZUEZRZFHUHZGOZIQEIQZBDRZACRZFHGUFZIQKWHWCIWJGGA
      BCDEUGKWHWCUIJABKCDEUJUKULWDKLCLNZUCZAWPDPZUDZUEZRAWPBMNZEPZPZIQZMWRRZLCR
      WIWNWDXDKLMCWRVSWPXAUHUMZWCXCIXFWCAWPWBPXCXFAVTWPWBWPXAVSLUSZMUSZUNUOXFAW
      PWBXBXFBWAXAEWPXAVSXGXHUPUOUQURSUTWDKWHWTALCWGWSLWGTAWQWRAWQTAWPDVAZVBALV
      CZWFWQDWRWEWPVDAWPDVEZVFVGVHWMXEALCWMLVIXDAMWRXIAXCIAWPXBVAAITVJVKWMXBIQZ
      MDRXJXEWLXLBMDWLMVIBXBIBXAEVABITVJBMVCEXBIBXAEVESVLXJXLXDMDWRXKXJXBXCIAWP
      XBVESVMVNVLVOWOWKIFHGVPVQVR $.
  $}

  ${
    $d w x y z $.  $d y B $.  $d x y A $.
    relmpoopab.1 $e |- F = ( x e. A , y e. B |-> { <. z , w >. | ph } ) $.
    $( Any function to sets of ordered pairs produces a relation on function
       value unconditionally.  (Contributed by Mario Carneiro, 9-Feb-2015.) $)
    relmpoopab $p |- Rel ( C F D ) $=
      ( co wrel cvv cxp wss copab wral relopabv df-rel mpbi ovmptss ax-mp mpbir
      rgen2w ) HIJLZMUFNNOZPZADEQZUGPZCGRBFRUHUJBCFGUIMUJADESUITUAUEBCFGUIHJIUG
      KUBUCUFTUD $.
  $}

  ${
    $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 wf fmpo sylib nfcv nfcsb1v nfcsbw weq csbeq1a sylan9eq cbvmpo cop wceq
      vex op2ndd csbeq1d op1std csbeq2dv eqtrd mpompt eqtr4i fmpt sylibr eqtrdi
      fmptcos wa w3a 3impb nfcvd csbiegf syl mpoeq3dva eqtrid ) ALKUAQEFUBZDCQU
      LZUCUDZBWFUEUDZHTZTZITZUFZBCEFJUGZAQDWEGWJIKLAWEGBCEFHUGZUMZWJGUHQWEUIAHG
      UHZCFUIBEUIWOAWPBCEFMUJBCEFHGWNWNUKUNUOQWEGWJWNWNRSEFCSULZBRULZHTZTZUGQWE
      WJUFZBCRSEFHWTRHUPSHUPBCWQWSBWQUPBWRHUQURZCWQWSUQZBRUSZCSUSZHWSWTBWRHUTCW
      QWSUTVAZVBRSQEFWJWTWFWRWQVCVDZWJCWQWITWTXGCWGWQWIWRWQWFRVEZSVEZVFVGXGCWQW
      IWSXGBWHWRHWRWQWFXHXIVHVGVIVJZVKVLZVMVNAKWNXANXKVOOVPAWLBCEFDHITZUGZWMWLR
      SEFDWTITZUGXMRSQEFWKXNXGDWJWTIXJVGVKBCRSEFXLXNRXLUPSXLUPBDWTIXBBIUPURCDWT
      IXCCIUPURXDXEVQDHWTIXFVGVBVLABCEFXLJABULEUHZCULFUHZVRWPXLJVDAXOXPWPMVSDHI
      JGWPDJVTPWAWBWCWDVJ $.
  $}

  ${
    $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 wceq wa adantl a1i cmpt dffn5 biimpi fveq2
      fmpoco eqtr4id ) IFNZHABCDEIOZPIGQLUJABMCDFEMRZIOZUKGIARCSBRDSUAEFSUJJUBG
      ABCDEPTUJKUCUJIMFUMUDTMFIUEUFULEIUGUHUI $.
  $}

  ${
    $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 opelxpd co cfv df-ov a1i mpoeq3ia eqtri oprabco
      wceq ) ABCDEFPZGHQIJKARCSBRDSUAZEFGHLMUBNJABCDEFKUCZTABCDUKKUDZTOABCDUMUN
      UMUNUJULEFKUEUFUGUHUI $.
  $}

  ${
    $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 wfn wfo fo1st vex
      cmpt fofn ax-mp dffn5 mpbi mptv eqtri reseq1i resopab cop op1std dfoprab3
      eqeq2d 3eqtrri ) EFFGZHCIZDIZEJZKZDCLZUNHUPUNMURNDCLUOAIZKZABCOEUSUNEDFUQ
      TZUSEFPZEVBKFFEQVCRFFEUAUBDFEUCUDDCUQUEUFUGURDCUNUHURVAABCDUPUTBIZUIKUQUT
      UOUTVDUPASBSUJULUKUM $.

    $( 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 wfn wfo fo2nd vex
      cmpt fofn ax-mp dffn5 mpbi mptv eqtri reseq1i resopab cop op2ndd dfoprab3
      eqeq2d 3eqtrri ) EFFGZHCIZDIZEJZKZDCLZUNHUPUNMURNDCLUOBIZKZABCOEUSUNEDFUQ
      TZUSEFPZEVBKFFEQVCRFFEUAUBDFEUCUDDCUQUEUFUGURDCUNUHURVAABCDUPAIZUTUIKUQUT
      UOVDUTUPASBSUJULUKUM $.
  $}

  ${
    $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.)  (Proof shortened by Peter
       Mazsa, 2-Oct-2022.) $)
    1stconst $p |- ( B e. V ->
             ( 1st |` ( A X. { B } ) ) : ( A X. { B } ) -1-1-onto-> A ) $=
      ( vx vy wcel csn cxp c1st cres wfo cv wbr wmo wceq wa vex cfv cvv jca wne
      ccnv wfun wf1o c0 snnzg fo1stres syl wal cop moeq moani brresi fo1st fofn
      wfn wb ax-mp fnbrfvb mp2an c2nd elxp7 eleq1 biimpac adantlr adantll elsni
      anbi2i eqopi anass1rs sylanl2 adantlrl sylanb adantl simprr simprl adantr
      snidg opelxpd eqeltrd fveq2d op1stg syl2anc impbida bitr3id bitrid mobidv
      simpl eqtrd mpbiri alrimiv funcnv2 sylibr dff1o3 sylanbrc ) BCFZABGZHZAIW
      RJZKZWSUBUCZWRAWSUDWPWQUEUAWTBCUFAWQUGUHWPDLZELZWSMZDNZEUIXAWPXEEWPXEXCAF
      ZXBXCBUJZOZPZDNXHXFDDXGUKULWPXDXIDXDXBWRFZXBXCIMZPZWPXIWRXBXCIEQUMXLXJXBI
      RZXCOZPZWPXIXNXKXJISUPZXBSFXNXKUQSSIKXPUNSSIUOURDQSXBXCIUSUTVHWPXOXIXOXIW
      PXJXBSSHFZXMAFZXBVARZWQFZPZPZXNXIXBAWQVBYBXNPXFXHYAXNXFXQXRXNXFXTXNXRXFXM
      XCAVCVDVEVFXQXTXNXHXRXTXQXSBOZXNXHXSBVGXQXNYCXHXBXCBSSVIVJVKVLTVMVNWPXIPZ
      XJXNYDXBXGWRWPXFXHVOZYDXCBAWQWPXFXHVPZWPBWQFXIBCVRVQVSVTYDXMXGIRZXCYDXBXG
      IYEWAYDXFWPYGXCOYFWPXIWHXCBACWBWCWITWDWEWFWGWJWKDEWSWLWMWRAWSWNWO $.

    $( The mapping of a restriction of the ` 2nd ` function to a converse
       constant function.  (Contributed by NM, 27-Mar-2008.)  (Proof shortened
       by Peter Mazsa, 2-Oct-2022.) $)
    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 adantll jca
      cres ccnv wfun wf1o c0 wne snnzg fo2ndres syl wal cop moani brresi wfn wb
      moeq fo2nd fofn ax-mp fnbrfvb mp2an anbi2i c1st elxp7 eleq1 biimpac elsni
      eqopi anassrs sylanl2 adantlrr sylanb adantl simprr adantr simprl opelxpd
      snidg eqeltrd op2ndg elvd sylan9eqr adantrl impbida bitr3id bitrid mobidv
      fveq2 mpbiri alrimiv funcnv2 sylibr dff1o3 sylanbrc ) ACFZAGZBHZBIWQUAZJZ
      WRUBUCZWQBWRUDWOWPUEUFWSACUGWPBUHUIWODKZEKZWRLZDMZEUJWTWOXDEWOXDXBBFZXAAX
      BUKZNZOZDMXGXEDDXFUPULWOXCXHDXCXAWQFZXAXBILZOZWOXHWQXAXBIEPUMXKXIXAIQZXBN
      ZOZWOXHXMXJXIIRUNZXARFXMXJUORRIJXOUQRRIURUSDPRXAXBIUTVAVBWOXNXHXNXHWOXIXA
      RRHFZXAVCQZWPFZXLBFZOZOZXMXHXAWPBVDYAXMOXEXGXTXMXEXPXSXMXEXRXMXSXEXLXBBVE
      VFSSXPXRXMXGXSXRXPXQANZXMXGXQAVGXPYBXMXGXAAXBRRVHVIVJVKTVLVMWOXHOZXIXMYCX
      AXFWQWOXEXGVNYCAXBWPBWOAWPFXHACVRVOWOXEXGVPVQVSWOXGXMXEXGWOXLXFIQZXBXAXFI
      WHWOYDXBNEAXBCRVTWAWBWCTWDWEWFWGWIWJDEWRWKWLWQBWRWMWN $.
  $}

  ${
    $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 cmpo cxp cv cfv csb cop csn ciun csbex nfcv nfcsb1v nfop nfsn nfcsbw
      c1st c2nd cmpt mpompts dfmpt wceq csbopeq1a opeq12d sneqd iunxpf 3eqtri
      id ) ABCDEHGCDIZAGJZUBKZBUOUCKZELZLZUDGUNUOUSMZNZOACBDAJBJMZEMZNZOOABGCDE
      UEGUNUSAUPURBUQEFPPUFGABCDVAVDAUTAUOUSAUOQAUPURRSTBUTBUOUSBUOQBAUPURBUPQB
      UQERUASTGVDQUOVBUGZUTVCVEUOVBUSEVEUMABUOEUHUIUJUKUL $.
  $}

  ${
    $d A p x y $.  $d B p x y $.  $d C p $.  $d E p x y $.  $d U p x y $.
    $d V p x y $.  $d W p x y $.
    mposn.f $e |- F = ( x e. { A } , y e. { B } |-> C ) $.
    mposn.a $e |- ( x = A -> C = D ) $.
    mposn.b $e |- ( y = B -> D = E ) $.
    $( An operation (in maps-to notation) on two singletons.  (Contributed by
       AV, 4-Aug-2019.) $)
    mposn $p |- ( ( A e. V /\ B e. W /\ E e. U )
                   -> F = { <. <. A , B >. , E >. } ) $=
      ( vp wcel csn cfv csb wceq w3a cxp cv c1st c2nd cop xpsng 3adant3 mpteq1d
      cmpt mpompts eqtri a1i cvv wa wi op2ndg fveq2 eqcomd eqeq1d syl5ibcom imp
      cmpo op1stg simp1 simpl2 adantl sylan9eq csbied adantr wb csbeq1 csbeq2dv
      bitrd syl5ibrcom mp2and opex simp3 fmptsnd 3eqtr4d ) CJPZDKPZHGPZUAZOCQZD
      QZUBZAOUCZUDRZBWHUERZESZSZUJZOCDUFZQZWLUJIWNHUFQWDOWGWOWLWAWBWGWOTWCCDJKU
      GUHUIIWMTWDIABWEWFEVCWMLABOWEWFEUKULUMWDOWNWLHUNGWDWHWNTZUOZWJDTZWICTZWLH
      TZWDWPWRWAWBWPWRUPWCWAWBUOZWNUERZDTWPWRCDJKUQWPXBWJDWPWJXBWHWNUEURUSUTVAU
      HVBWDWPWSWAWBWPWSUPWCXAWNUDRZCTWPWSCDJKVDWPXCWICWPWIXCWHWNUDURUSUTVAUHVBW
      QWTWRWSUOZACBDESZSZHTZWDXGWPWDACXEHJWAWBWCVEWDAUCCTZUOZBDEHKWAWBWCXHVFXIB
      UCDTEFHXHEFTWDMVGNVHVIVIVJXDWTACWKSZHTZXGWSWTXKVKWRWSWLXJHAWICWKVLUTVGXDX
      JXFHXDACWKXEWRWKXETWSBWJDEVLVJVMUTVNVOVPWNUNPWDCDVQUMWAWBWCVRVSVT $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.  $d x D $.  $d x F $.  $d x G $.
    curry1.1 $e |- G = ( F o. `' ( 2nd |` ( { C } X. _V ) ) ) $.
    $( Composition with ` ``' ( 2nd |`` ( { C } X. _V ) ) ` turns any binary
       operation ` F ` with a constant first operand into a function ` G ` of
       the second operand only.  This transformation is called "currying".
       (Contributed by NM, 28-Mar-2008.)  (Revised by Mario Carneiro,
       26-Dec-2014.) $)
    curry1 $p |- ( ( F Fn ( A X. B ) /\ C e. A ) ->
                 G = ( x e. B |-> ( C F x ) ) ) $=
      ( cxp wfn wcel wa cfv wceq c2nd cvv cres adantr cin eqtrid eqtrd cmpt csn
      cv co ccnv ccom wfun cdm fnfun wf1o 2ndconst wfo dff1o3 simprbi syl funco
      syl2an cima dmco fndm crn imacnvcnv df-ima resres rneqi 3eqtri inxp incom
      imaeq2d inv1 eqtri xpeq2i wss snssi dfss2 sylib xpeq1d reseq2d rneqd forn
      f1ofo 3syl adantl fneq1i df-fn bitri sylanbrc dffn5 fveq1i fvco2 simpl2im
      dff1o4 ad2antlr cop snidg vex opelxp sylanblrc jca fvresd op2ndg f1ocnvfv
      elvd sylc fveq2d adantll df-ov eqtr4di mpteq2dva ) EBCHZIZDBJZKZFACAUCZFL
      ZUAZACDXNEUDZUAXMFCIZFXPMXMENDUBZOHZPZUEZUFZUGZYCUHZCMZXRXKEUGYBUGZYDXLXJ
      EUIXLXTOYAUJZYGDOBUKZYHXTOYAULYGXTOYAUMUNUOEYBUPUQXMYEYBUEZEUHZURZCEYBUSX
      MYLYJXJURZCXMYKXJYJXKYKXJMXLXJEUTQVIXLYMCMXKXLYMNXTXJRZPZVAZCYMYAXJURYAXJ
      PZVAYPYAXJVBYAXJVCYQYONXTXJVDVEVFXLYPNXSCHZPZVAZCXLYOYSXLYNYRNXLYNXSBRZCH
      ZYRYNUUAOCRZHUUBXSOBCVGUUCCUUAUUCCORCOCVHCVJVKVLVKXLUUAXSCXLXSBVMUUAXSMDB
      VNXSBVOVPVQSVRVSXLYRCYSUJYRCYSULYTCMDCBUKYRCYSWAYRCYSVTWBTSWCTSXRYCCIYDYF
      KCFYCGWDYCCWEWFWGACFWHVPXMACXOXQXMXNCJZKZXOXNYBLZELZXQUUEXOXNYCLZUUGXNFYC
      GWIXLUUHUUGMZXKUUDXLYAXTIZYBOIZUUIXLYHUUJUUKKYIXTOYAWLVPUUKUUIAOEYBXNWJXC
      WKWMSUUEUUGDXNWNZELZXQXLUUDUUGUUMMXKXLUUDKZUUFUULEUUNYHUULXTJZKUULYALZXNM
      ZUUFUULMUUNYHUUOXLYHUUDYIQXLUUOUUDXLDXSJXNOJUUODBWOAWPDXNXSOWQWRZQWSXLUUQ
      UUDXLUUPUULNLZXNXLUULXTNUURWTXLUUSXNMADXNBOXAXCTQXTOUULXNYAXBXDXEXFDXNEXG
      XHTXIT $.

    $( The value of a curried function with a constant first argument.
       (Contributed by NM, 28-Mar-2008.)  (Revised by Mario Carneiro,
       26-Apr-2015.) $)
    curry1val $p |- ( ( F Fn ( A X. B ) /\ C e. A ) ->
        ( G ` D ) = ( C F D ) ) $=
      ( vx cxp wfn wcel wa cfv cv co cmpt curry1 wceq wn c0 fveq1d fvmptndm cdm
      eqid adantl fndm adantr simpr con3i ndmovg syl2an eqtr4d oveq2 ovex fvmpt
      ex pm2.61d2 eqtrd ) EABIZJZCAKZLZDFMDHBCHNZEOZPZMZCDEOZVBDFVEHABCEFGQUAVB
      DBKZVFVGRZVBVHSZVIVBVJLVFTVGVJVFTRVBHBVDVEDVEUDZUBUEVBEUCUSRZVAVHLZSVGTRV
      JUTVLVAUSEUFUGVMVHVAVHUHUICDABEUJUKULUPHDVDVGBVEVCDCEUMVKCDEUNUOUQUR $.

    $( Functionality of a curried function with a constant first argument.
       (Contributed by NM, 29-Mar-2008.) $)
    curry1f $p |- ( ( F : ( A X. B ) --> D /\ C e. A ) ->
                 G : B --> D ) $=
      ( vx cxp wf wcel wa cv co wfn cmpt wceq ffn curry1 sylan fovcdm fmpt3d
      3expa ) ABIZDEJZCAKZLHBCHMZENZDFUEEUDOUFFHBUHPQUDDERHABCEFGSTUEUFUGBKUHDK
      CUGDABEUAUCUB $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.  $d x D $.  $d x F $.  $d x G $.
    curry2.1 $e |- G = ( F o. `' ( 1st |` ( _V X. { C } ) ) ) $.
    $( Composition with ` ``' ( 1st |`` ( _V X. { C } ) ) ` turns any binary
       operation ` F ` with a constant second operand into a function ` G ` of
       the first operand only.  This transformation is called "currying".  (If
       this becomes frequently used, we can introduce a new notation for the
       hypothesis.)  (Contributed by NM, 16-Dec-2008.) $)
    curry2 $p |- ( ( F Fn ( A X. B ) /\ C e. B ) ->
                 G = ( x e. A |-> ( x F C ) ) ) $=
      ( cxp wfn wcel wa cfv wceq c1st cvv cres adantr cin eqtrid eqtrd cmpt csn
      cv co ccnv ccom wfun cdm fnfun wf1o 1stconst wfo dff1o3 simprbi syl funco
      syl2an cima dmco fndm crn imacnvcnv df-ima resres rneqi 3eqtri inxp incom
      imaeq2d inv1 eqtri xpeq1i wss snssi dfss2 sylib xpeq2d reseq2d rneqd forn
      f1ofo adantl fneq1i df-fn bitri sylanbrc dffn5 fveq1i dff1o4 simprd fvco2
      3syl vex sylancl ad2antlr cop snidg opelxpd fvresd op1stg ancoms f1ocnvfv
      a1i jca sylc fveq2d adantll df-ov eqtr4di mpteq2dva ) EBCHZIZDCJZKZFABAUC
      ZFLZUAZABXODEUDZUAXNFBIZFXQMXNENODUBZHZPZUEZUFZUGZYDUHZBMZXSXLEUGYCUGZYEX
      MXKEUIXMYAOYBUJZYHODCUKZYIYAOYBULYHYAOYBUMUNUOEYCUPUQXNYFYCUEZEUHZURZBEYC
      USXNYMYKXKURZBXNYLXKYKXLYLXKMXMXKEUTQVIXMYNBMXLXMYNNYAXKRZPZVAZBYNYBXKURY
      BXKPZVAYQYBXKVBYBXKVCYRYPNYAXKVDVEVFXMYQNBXTHZPZVAZBXMYPYTXMYOYSNXMYOBXTC
      RZHZYSYOOBRZUUBHUUCOXTBCVGUUDBUUBUUDBORBOBVHBVJVKVLVKXMUUBXTBXMXTCVMUUBXT
      MDCVNXTCVOVPVQSVRVSXMYSBYTUJYSBYTULUUABMBDCUKYSBYTWAYSBYTVTWLTSWBTSXSYDBI
      YEYGKBFYDGWCYDBWDWEWFABFWGVPXNABXPXRXNXOBJZKZXPXOYCLZELZXRUUFXPXOYDLZUUHX
      OFYDGWHXMUUIUUHMZXLUUEXMYCOIZXOOJZUUJXMYBYAIZUUKXMYIUUMUUKKYJYAOYBWIVPWJA
      WMZOEYCXOWKWNWOSUUFUUHXODWPZELZXRXMUUEUUHUUPMXLXMUUEKZUUGUUOEUUQYIUUOYAJZ
      KUUOYBLZXOMUUGUUOMUUQYIUURXMYIUUEYJQUUQXODOXTUULUUQUUNXCXMDXTJUUEDCWQZQWR
      XDUUQUUSUUONLZXOXMUUSUVAMUUEXMUUOYANXMXODOXTUULXMUUNXCUUTWRWSQUUEXMUVAXOM
      XODBCWTXATYAOUUOXOYBXBXEXFXGXODEXHXITXJT $.

    $( Functionality of a curried function with a constant second argument.
       (Contributed by NM, 16-Dec-2008.) $)
    curry2f $p |- ( ( F : ( A X. B ) --> D /\ C e. B ) ->
                 G : A --> D ) $=
      ( vx cxp wf wcel wa cv co wfn cmpt wceq ffn curry2 sylan fovcdm 3com23
      3expa fmpt3d ) ABIZDEJZCBKZLHAHMZCENZDFUFEUEOUGFHAUIPQUEDERHABCEFGSTUFUGU
      HAKZUIDKZUFUJUGUKUHCDABEUAUBUCUD $.

    $( The value of a curried function with a constant second argument.
       (Contributed by NM, 16-Dec-2008.) $)
    curry2val $p |- ( ( F Fn ( A X. B ) /\ C e. B ) ->
        ( G ` D ) = ( D F C ) ) $=
      ( vx cxp wfn wcel wa cfv cv co cmpt curry2 wceq wn c0 fveq1d fvmptndm cdm
      wi eqid adantl fndm simpl con3i ndmovg syl2an eqtr4d ex adantr oveq1 ovex
      fvmpt pm2.61d2 eqtrd ) EABIZJZCBKZLZDFMDHAHNZCEOZPZMZDCEOZVCDFVFHABCEFGQU
      AVCDAKZVGVHRZVAVISZVJUDVBVAVKVJVAVKLVGTVHVKVGTRVAHAVEVFDVFUEZUBUFVAEUCUTR
      VIVBLZSVHTRVKUTEUGVMVIVIVBUHUIDCABEUJUKULUMUNHDVEVHAVFVDDCEUOVLDCEUPUQURU
      S $.
  $}

  $( 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 c1st cop simprr sneqd cnveqd unieqd
    wa opswap fvex 1st2nd adantrr eqtrdi simprl eqeltrrd opelcnv sylibr eqeltrd
    eqtrd eqcomi 3eqtr4a jca ) ADZBAEZCBFZGZHZIZRRZCAGZEBCFZGZHZIUSCBJKZBLKZMZU
    TUSCVEVDMZFZGZHZVFUSCUQVJUMUNURNUSUPVIUSUOVHUSBVGUMUNBVGIURBAUAUBZOPQUIVEVD
    SUCZUSVGAEVFUTEUSBVGAVKUMUNURUDUEVDVEABJTBLTUFUGUHUSVGVFFZGZHZBVCVOVGVDVESU
    JVKUSVBVNUSVAVMUSCVFVLOPQUKUL $.

  ${
    $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 cvv eqid wcel vsnex cnvex uniex cnvf1olem
      wa a1i wceq wb relcnv simpr sylancr dfrel2 eleq2 sylbi anbi1d adantr f1od
      mpbid impbida ) BDZACBBEZAFZGZEZHZCFZGZEZHZABUQIZJJVBKUQJLULUNBLZQUPUOAMN
      ORVAJLULURUMLZQUTUSCMNORULVCURUQSZQZVDUNVASQZBUNURPULVGQZUNUMEZLZVEQZVFVH
      UMDVGVKBUAULVGUBUMURUNPUCULVKVFTVGULVJVCVEULVIBSVJVCTBUDVIBUNUEUFUGUHUJUK
      UI $.
  $}

  ${
    $d x A $.  $d y B $.  $d x y F $.  $d x y G $.
    $( Lemma for ~ fpar .  (Contributed by NM, 22-Dec-2008.)  (Revised by Mario
       Carneiro, 28-Apr-2015.) $)
    fparlem1 $p |- ( `' ( 1st |` ( _V X. _V ) ) " { x } ) = ( { x } X. _V ) $=
      ( vy c1st cvv cxp cres ccnv cv csn cima wcel cfv wceq wa fvres eqeq1d vex
      c2nd elsn2 fvex biantru bitr3i bitrdi pm5.32i wf wfn f1stres ffn fniniseg
      wb mp2b elxp7 3bitr4i eqriv ) BCDDEZFZGAHZIZJZURDEZBHZUOKZVAUPLZUQMZNZVBV
      ACLZURKZVARLDKZNZNVAUSKZVAUTKVBVDVIVBVDVFUQMZVIVBVCVFUQVAUOCOPVKVGVIVFUQA
      QSVHVGVARTUAUBUCUDUODUPUEUPUOUFVJVEUJDDUGUODUPUHUOUQVAUPUIUKVAURDULUMUN
      $.

    $( Lemma for ~ fpar .  (Contributed by NM, 22-Dec-2008.)  (Revised by Mario
       Carneiro, 28-Apr-2015.) $)
    fparlem2 $p |- ( `' ( 2nd |` ( _V X. _V ) ) " { y } ) = ( _V X. { y } ) $=
      ( vx c2nd cvv cxp cres ccnv cv csn cima wcel cfv wceq wa fvres eqeq1d vex
      c1st elsn2 fvex biantrur bitr3i bitrdi pm5.32i wf wfn wb f2ndres fniniseg
      ffn mp2b elxp7 3bitr4i eqriv ) BCDDEZFZGAHZIZJZDUREZBHZUOKZVAUPLZUQMZNZVB
      VARLDKZVACLZURKZNZNVAUSKZVAUTKVBVDVIVBVDVGUQMZVIVBVCVGUQVAUOCOPVKVHVIVGUQ
      AQSVFVHVARTUAUBUCUDUODUPUEUPUOUFVJVEUGDDUHUODUPUJUOUQVAUPUIUKVADURULUMUN
      $.

    $( Lemma for ~ fpar .  (Contributed by NM, 22-Dec-2008.)  (Revised by Mario
       Carneiro, 28-Apr-2015.) $)
    fparlem3 $p |- ( F Fn A
          -> ( `' ( 1st |` ( _V X. _V ) ) o. ( F o. ( 1st |` ( _V X. _V ) ) ) )
            = U_ x e. A ( ( { x } X. _V ) X. ( { ( F ` x ) } X. _V ) ) ) $=
      ( vy cvv cxp cres ccnv cv csn cima ciun ccom cdm crn cin wceq ax-mp eqtri
      wss wfn c1st cfv coiun inss1 fndm sseqtrid dfco2a syl coeq2d dmxpss sstri
      wcel wa fvex fparlem1 sneq xpeq1d eqtrid imaeq2d df-ima xpssres rneqi wne
      ssid c0 snnz rnxp eqtrdi xpeq12d iunxsn cnveqi cnvco cnvxp 3eqtr3i xpeq2i
      fnsnfv eqtr3id cnveqd iuneq2dv 3eqtr4a ) CBUAZUBEEFGZHZABWDAIZJZKZCWFKZFZ
      LZMABWDWIMZLWDCWCMZMABWFEFZWECUCZJZEFZFZLAWDWIBUDWBWLWJWDWBCNZWCOZPZBTWLW
      JQWBWRWTBWRWSUEBCUFUGACWCBUHUIUJWBABWQWKWBWEBUMUNZWQWDWOWMFZHZMZWKXBWCMZH
      WPWMFZHXDWQXEXFXEDWOWDDIZJZKZXBXHKZFZLZXFXBNZWSPZWOTXEXLQXNXMWOXMWSUEWOWM
      UKULDXBWCWOUHRDWNXKXFWECUOZXGWNQZXIWPXJWMXPXIXHEFWPDUPXPXHWOEXGWNUQZURUSX
      PXJXBWOKZWMXPXHWOXBXQUTXRXBWOGZOZWMXBWOVAXTXBOZWMXSXBWOWOTXSXBQWOVEWOWMWO
      VBRVCWOVFVDYAWMQWNXOVGWOWMVHRSSVIVJVKSVLXBWCVMWPWMVNVOXAXCWIWDXAXCWHWGFZH
      WIXAXBYBXAXBWOWGFYBWGWMWOAUPVPXAWOWHWGBWECVQURVRVSWHWGVNVIUJVRVTWA $.

    $( Lemma for ~ fpar .  (Contributed by NM, 22-Dec-2008.)  (Revised by Mario
       Carneiro, 28-Apr-2015.) $)
    fparlem4 $p |- ( G Fn B
          -> ( `' ( 2nd |` ( _V X. _V ) ) o. ( G o. ( 2nd |` ( _V X. _V ) ) ) )
            = U_ y e. B ( ( _V X. { y } ) X. ( _V X. { ( G ` y ) } ) ) ) $=
      ( vx cvv cxp cres ccnv cv csn cima ciun ccom cdm crn cin wceq ax-mp eqtri
      wss wfn c2nd cfv coiun inss1 fndm sseqtrid dfco2a syl coeq2d dmxpss sstri
      wcel wa fvex fparlem2 sneq xpeq2d eqtrid imaeq2d df-ima xpssres rneqi wne
      ssid c0 snnz rnxp eqtrdi xpeq12d iunxsn cnveqi cnvco cnvxp 3eqtr3i xpeq2i
      fnsnfv xpeq1d eqtr3id cnveqd iuneq2dv 3eqtr4a ) CBUAZUBEEFGZHZABWEAIZJZKZ
      CWGKZFZLZMABWEWJMZLWECWDMZMABEWGFZEWFCUCZJZFZFZLAWEWJBUDWCWMWKWEWCCNZWDOZ
      PZBTWMWKQWCWSXABWSWTUEBCUFUGACWDBUHUIUJWCABWRWLWCWFBUMUNZWRWEWPWNFZHZMZWL
      XCWDMZHWQWNFZHXEWRXFXGXFDWPWEDIZJZKZXCXIKZFZLZXGXCNZWTPZWPTXFXMQXOXNWPXNW
      TUEWPWNUKULDXCWDWPUHRDWOXLXGWFCUOZXHWOQZXJWQXKWNXQXJEXIFWQDUPXQXIWPEXHWOU
      QZURUSXQXKXCWPKZWNXQXIWPXCXRUTXSXCWPGZOZWNXCWPVAYAXCOZWNXTXCWPWPTXTXCQWPV
      EWPWNWPVBRVCWPVFVDYBWNQWOXPVGWPWNVHRSSVIVJVKSVLXCWDVMWQWNVNVOXBXDWJWEXBXD
      WIWHFZHWJXBXCYCXBXCWPWHFYCWHWNWPAUPVPXBWPWIWHBWFCVQVRVSVTWIWHVNVIUJVSWAWB
      $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y F $.  $d x y G $.
    fpar.1 $e |- H = ( ( `' ( 1st |` ( _V X. _V ) )
                        o. ( F o. ( 1st |` ( _V X. _V ) ) ) )
                i^i ( `' ( 2nd |` ( _V X. _V ) )
                       o. ( G o. ( 2nd |` ( _V X. _V ) ) ) ) ) $.
    $( Merge two functions in parallel.  Use as the second argument of a
       composition with a binary operation to build compound functions such as
       ` ( x e. ( 0 [,) +oo ) , y e. RR |-> `
       ` ( ( sqrt `` x ) + ( sin `` y ) ) ) ` , see also ~ ex-fpar .
       (Contributed by NM, 17-Sep-2007.)  (Proof shortened by Mario Carneiro,
       28-Apr-2015.) $)
    fpar $p |- ( ( F Fn A /\ G Fn B )
       -> H = ( x e. A , y e. B |-> <. ( F ` x ) , ( G ` y ) >. ) ) $=
      ( cvv cxp ccom cin csn ciun cop inxp inv1 xpeq12i xpsn 3eqtri wfn wa c1st
      cres ccnv c2nd cfv cmpo fparlem3 fparlem4 ineqan12d opex dfmpo wceq incom
      cv wcel eqtri vex fvex a1i iuneq2i 2iunin 3eqtr2i 3eqtr4g ) ECUAZFDUAZUBU
      CIIJZUDZUEEVIKKZUFVHUDZUEFVKKKZLACAUPZMZIJZVMEUGZMZIJZJZNZBDIBUPZMZJZIWAF
      UGZMZJZJZNZLZGABCDVPWDOZUHZVFVGVJVTVLWHACEUIBDFUJUKHWKACBDVMWAOZWJOMZNZNA
      CBDVSWGLZNZNWIABCDWJVPWDULZUMACWPWNWPWNUNVMCUQBDWOWMWOWMUNWADUQWOVOWCLZVR
      WFLZJWLMZWJMZJWMVOVRWCWFPWRWTWSXAWRVNILZIWBLZJVNWBJWTVNIIWBPXBVNXCWBVNQXC
      WBILWBIWBUOWBQURRVMWAAUSBUSSTWSVQILZIWELZJVQWEJXAVQIIWEPXDVQXEWEVQQXEWEIL
      WEIWEUOWEQURRVPWDVMEUTWAFUTSTRWLWJVMWAULWQSTVAVBVAVBABCDVSWGVCVDVE $.
  $}

  ${
    $d x y z t $.
    $( A function that can be used to feed a common value to both operands of
       an operation.  Use as the second argument of a composition with the
       function of ~ fpar in order to build compound functions such as
       ` ( x e. ( 0 [,) +oo ) |-> ( ( sqrt `` x ) + ( sin `` x ) ) ) ` .
       (Contributed by NM, 17-Sep-2007.)  Replace use of ~ dfid2 with ~ df-id .
       (Revised by BJ, 31-Dec-2023.) $)
    fsplit $p |- `' ( 1st |` _I ) = ( x e. _V |-> <. x , x >. ) $=
      ( vy vz vt cv c1st cid wbr copab cop wceq cvv wcel wa exbii eqeq2d 3bitri
      vex wex equsexvw cres ccnv brcnv brresi cfv 19.42v op1std eqeq1d pm5.32ri
      cmpt wfn wb wfo fo1st fofn ax-mp fnbrfvb mp2an df-id eleq2i elopab equcom
      ancom opeq2 pm5.32i biidd bitri 3bitrri anbi12ci 3bitr3ri opeq12d opabbii
      anbi1i id wrel relcnv dfrel4v mpbi mptv 3eqtr4i ) AEZBEZFGUAZUBZHZABIZWBW
      AWAJZKZABIWDALWGUJWEWHABWEWBWAWCHWBGMZWBWAFHZNZWHWAWBWCARZBRZUCGWBWAFWLUD
      WKCEZWAKZWBWNWNJZKZNZCSZWHWBFUEZWAKZWQNZCSXAWQCSZNWSWKXAWQCUFXBWRCWQXAWOW
      QWTWNWAWNWNWBCRZXDUGUHUIOXAWJXCWIFLUKZWBLMXAWJULLLFUMXEUNLLFUOUPWMLWBWAFU
      QURWIWBWNDEZKZCDIZMWBWNXFJZKZXGNZDSZCSXCGXHWBCDUSUTXGCDWBVAXLWQCXLXFWNKZW
      QNZDSWQXKXNDXKXGXJNXMXJNXNXJXGVCXGXMXJCDVBVMXMXJWQXMXIWPWBXFWNWNVDPVEQOWQ
      WQDCXMWQVFTVGOVHVIVJWQWHCAWOWPWGWBWOWNWAWNWAWOVNZXOVKPTVGQVLWDVOWDWFKWCVP
      ABWDVQVRABWGVSVT $.
  $}

  ${
    $d A a p $.  $d A a x y $.  $d F a x y $.  $d G a x y $.  $d H a $.
    $d S a $.
    fsplitfpar.h $e |- H = ( ( `' ( 1st |` ( _V X. _V ) )
                        o. ( F o. ( 1st |` ( _V X. _V ) ) ) )
                i^i ( `' ( 2nd |` ( _V X. _V ) )
                       o. ( G o. ( 2nd |` ( _V X. _V ) ) ) ) ) $.
    fsplitfpar.s $e |- S = ( `' ( 1st |` _I ) |` A ) $.
    $( Merge two functions with a common argument in parallel.  Combination of
       ~ fsplit and ~ fpar .  (Contributed by AV, 3-Jan-2024.) $)
    fsplitfpar $p |- ( ( F Fn A /\ G Fn A )
                -> ( H o. S ) = ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) $=
      ( va vy wfn wa cfv cop wceq wral wcel cvv a1i adantl vp ccom cv cmpt cres
      c1st cid ccnv fsplit reseq1i eqtri fveq1i fvres eqidd weq id opeq12d elex
      opex fvmptd eqtrd fveq2d co df-ov cmpo fpar adantr fveq2 simpr ovmpod wss
      eqtr3id eqid fnmpt mprg ssv fnssres mp2an mpbir fvco2 sylan fvmpt 3eqtr4d
      fneq1i ralrimiva wb cxp crn ralrimivva fnmpo fneq1d mpbird sylancl sylibr
      syl cin wrex rneqi cima mptima df-ima rnmpt 3eqtr3i elinel2 simpl opelxpd
      cab wi eleq1 ex rexlimiv abssi eqsstrid fnco syl3anc eqfnfv syl2anc ) DBK
      EBKLZFCUBZABAUCZDMZXTEMZNZUDZOZIUCZXSMZYFYDMZOZIBPZXRYIIBXRYFBQZLZYFCMZFM
      ZYFDMZYFEMZNZYGYHYLYNYFYFNZFMZYQYLYMYRFYLYMYFARXTXTNZUDZBUEZMZYRYMUUCOYLY
      FCUUBCUFUGUEUHZBUEZUUBHUUDUUABAUIUJUKULSYKUUCYROXRYKUUCYFUUAMYRYFBUUAUMYK
      AYFYTYRRUUARYKUUAUNAIUOZYTYROYKUUFXTYFXTYFUUFUPZUUGUQTYFBURYRRQZYKYFYFUSZ
      SUTVATVAVBYLYSYFYFFVCYQYFYFFVDYLAJYFYFBBYAJUCZEMZNZYQFRXRFAJBBUULVEZOYKAJ
      BBDEFGVFZVGUUFJIUOZLZUULYQOYLUUPYAYOUUKYPUUFYAYOOUUOXTYFDVHZVGUUOUUKYPOUU
      FUUJYFEVHTUQTXRYKVIZUURYQRQYLYOYPUSZSVJVLVAXRCBKZYKYGYNOUUTXRUUTIRYRUDZBU
      EZBKZUVARKZBRVKZUVCUUHUVDIRIRYRUVARUVAVMVNZUUHYFRQZUUISVOBVPZRBUVAVQZVRBC
      UVBCUUEUVBHUUDUVABIUIUJUKZWDZVSSBFCYFVTWAYKYHYQOXRAYFYCYQBYDUUFYAYOYBYPUU
      QXTYFEVHUQYDVMZUUSWBTWCWEXRXSBKZYDBKZYEYJWFXRFBBWGZKZUUTCWHZUVOVKUVMXRUVP
      UUMUVOKZXRUULRQZJBPABPUVRXRUVSAJBBUVSXRXTBQZUUJBQLLYAUUKUSSWIAJBBUULUUMRU
      UMVMWJWOXRUVOFUUMUUNWKWLXRUVCUUTXRUVDUVEUVCXRUUHIRPUVDXRUUHIRUUHXRUVGLUUI
      SWEUVFWOUVHUVIWMUVKWNXRUVQUAUCZYROZIRBWPZWQZUAXGZUVOUVQUVBWHZUWECUVBUVJWR
      UVABWSIUWCYRUDZWHUWFUWEIRYRBWTUVABXAIUAUWCYRUWGUWGVMXBXCUKUWEUVOVKXRUWDUA
      UVOUWBUWAUVOQZIUWCYFUWCQYKUWBUWHXHYFRBXDYKUWBUWHYKUWBLZUWHYRUVOQZUWIYFYFB
      BYKUWBXEZUWKXFUWBUWHUWJWFYKUWAYRUVOXITWLXJWOXKXLSXMUVOBFCXNXOXRYCRQZABPUV
      NXRUWLABUWLXRUVTLYAYBUSSWEABYCYDRUVLVNWOIBXSYDXPXQWL $.

    $d C a $.  $d .+ a $.  $d V a $.  $d W a $.
    $( Express the function operation map ` oF ` by the functions defined in
       ~ fsplit and ~ fpar .  (Contributed by AV, 4-Jan-2024.) $)
    offsplitfpar $p |- ( ( ( F Fn A /\ G Fn A ) /\ ( F e. V /\ G e. W )
                           /\ ( .+ Fn C /\ ( ran F X. ran G ) C_ C ) )
                         -> ( .+ o. ( H o. S ) ) = ( F oF .+ G ) ) $=
      ( va wfn wa wcel crn ccom cfv cmpt co cxp wss w3a cop cof wceq fsplitfpar
      cv coeq2d 3ad2ant1 wf dffn3 biimpi adantr 3ad2ant3 simpl3r fnfvelrn sylan
      simp1l simp1r opelxpd sseldd cofmpt eqcomi mpteq2i eqtrdi cdm cin offval3
      df-ov fndm ineqan12d inidm mpteq1d sylan9eqr eqcomd 3adant3 3eqtrd ) EAMZ
      FAMZNZEHOFIONZCBMZEPZFPZUAZBUBZNZUCZCGDQZQZCLALUHZERZWLFRZUDZSZQZLAWMWNCT
      ZSZEFCUETZWAWBWKWQUFWHWAWJWPCLADEFGJKUGUIUJWIWQLAWOCRZSWSWILAWOBCPZCWHWAB
      XBCUKZWBWCXCWGWCXCBCULUMUNUOWIWLAOZNZWFBWOWCWGWAWBXDUPXEWMWNWDWEWIVSXDWMW
      DOVSVTWBWHUSAWLEUQURWIVTXDWNWEOVSVTWBWHUTAWLFUQURVAVBVCLAXAWRWRXAWMWNCVJV
      DVEVFWAWBWSWTUFWHWAWBNWTWSWBWAWTLEVGZFVGZVHZWRSWSLCEFHIVIWALXHAWRWAXHAAVH
      AVSVTXFAXGAAEVKAFVKVLAVMVFVNVOVPVQVR $.
  $}

  $( 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 cxp wss f2ndres fssxp fssres sylancr resabs1d eqcomd feq1d mpbird
    cres ) ABCDZCBECPZDCBEABFZPZCPZDZQSBTDCSGUBABHABCIZSBCTJKQCBRUAQUARQECSUCLM
    NO $.

  ${
    $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 ffrn f2ndf syl wfn ffn cv wcel cfv vex
      dffn3 sylbi frnd cop wex elrn2g ibi wa fvres adantl eqtr2di ffnd fnfvelrn
      op2nd sylan eqeltrd ex exlimdv syl5 ssrdv eqssd dffo2 sylanbrc ) ABCFZCCG
      ZHCIZFZVFGZVEJCVEVFKVDAVECFZVGABCLAVECMZNVDVHVEVDCVEVFVDCAOZVGABCPVKVIVGA
      CUAVJUBNUCVDDVEVHDQZVERZEQZVLUDZCRZEUEZVDVLVHRZVMVQEVLCVEUFUGVDVPVREVDVPV
      RVDVPUHZVLVOVFSZVHVSVTVOHSZVLVPVTWAJVDVOCHUIUJVNVLETDTUNUKVDVFCOVPVTVHRVD
      CBVFABCMULCVOVFUMUOUPUQURUSUTVACVEVFVBVC $.
  $}

  ${
    $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 ad2antrr ad2antlr eqeq12d op2nd eqeq12i f1fun
      wf vex funopfv anim12d eqcom biimpi eqeqan12d simpl anim12i sylan2 opeq12
      f1veqaeq com14 biimtrdi pm2.43i com13 impcom biimtrid sylbid adantl com12
      syl6 ad4ant13 wb eleq1 bi2anan9 anbi2d fveq2 simpllr simpr imbi12d imbi2d
      syld 3imtr4d rexlimdvva rexlimivv imp mpcom ralrimivv dff13 df-f1 simprbi
      sylanbrc dff1o3 ) ABCUAZCCUBZJCUCZUDZXNUEUFZCXMXNUGXLABCVHZXOABCUHZABCUIK
      XLCBXNUAZXPXLCBXNVHZDLZXNMZELZXNMZNZDEUJZOZECUKDCUKXSXLXQXTXRABCULKXLYGDE
      CCCABUMZUNZXLYACPZYCCPZQZYGOXLXQYIXRABCUOKYIYLXLYGYIYLXLYGOZYAFLZGLZUPZNZ
      GBRFARZYCHLZILZUPZNZIBRHARZQYIYLQZYMYIYJYRYKUUCYIYJQYAYHPYRCYHYAUQFGYAABU
      RUSYIYKQYCYHPUUCCYHYCUQHIYCABURUSUTYRUUCUUDYMOZYQUUCUUEOZFGABYNAPZYOBPZQZ
      YQUUFUUIYQQZUUBUUEHIABUUJYSAPZYTBPZQZQZUUBUUEUUNUUBQZYIYPCPZUUACPZQZQZXLY
      PXNMZUUAXNMZNZYPUUANZOZOZUUDYMUUIUUMUUSUVEOYQUUBUUSUUIUUMQZUVEUURUVFUVEOY
      IUURUVFUVEUURUVFQZUVBXLUVCUVGUVBYPJMZUUAJMZNZXLUVCOZUVGUUTUVHUVAUVIUUPUUT
      UVHNUUQUVFYPCJVAVBUUQUVAUVINUUPUVFUUACJVAVCVDUVJGIUJZUVGUVKUVHYOUVIYTYNYO
      FVIGVIVEYSYTHVIIVIVEVFUVGXLUVLUVCUVFUURXLUVLUVCOZOXLUURUVFUVMXLUURYNCMZYO
      NZYSCMZYTNZQZUVFUVMOXLCUFZUURUVROABCVGUVSUUPUVOUUQUVQYNYOCVJYSYTCVJVKKXLU
      VFUVRUVMUVLUVFUVRXLUVCUVLUVFUVRUVKOOUVRUVLUVFUVLUVKUVRUVLUVNUVPNZUVFUVLUV
      KOOUVOUVQYOUVNYTUVPUVOYOUVNNUVNYOVLVMUVQYTUVPNUVPYTVLVMVNXLUVFUVLUVTUVCXL
      UVFUVLUVTUVCOOXLUVFQZUVTUVLUVCUWAUVTFHUJZUVMUVFXLUUGUUKQUVTUWBOUUIUUGUUMU
      UKUUGUUHVOUUKUULVOVPABYNYSCVSVQUWBUVLUVCYNYOYSYTVRSWITSVTWAVTWBVTTWTWCWDT
      WEWFTSWGWHWJUUOYLUURYIUUNYJUUPUUBYKUUQYQYJUUPWKUUIUUMYAYPCWLVCYCUUACWLWMW
      NUUOYGUVDXLUUOYEUVBYFUVCUUNUUBYBUUTYDUVAYQYBUUTNUUIUUMYAYPXNWOVCYCUUAXNWO
      VNUUOYAYPYCUUAUUIYQUUMUUBWPUUNUUBWQVDWRWSXASXBSXCXDXESTXEXFDECBXNXGXJXSXT
      XPCBXNXHXIKCXMXNXKXJ $.
  $}

  ${
    opco1.exa $e |- ( ph -> A e. V ) $.
    opco1.exb $e |- ( ph -> B e. W ) $.
    $( Value of an operation precomposed with the projection on the first
       component.  (Contributed by Mario Carneiro, 28-May-2014.)  Generalize to
       closed form.  (Revised by BJ, 27-Oct-2024.) $)
    opco1 $p |- ( ph -> ( A ( F o. 1st ) B ) = ( F ` A ) ) $=
      ( c1st ccom co cop cfv wceq df-ov a1i cvv wfo wf wcel fo1st fof mp1i opex
      fvco3d op1stg syl2anc fveq2d 3eqtrd ) ABCDIJZKZBCLZUJMZULIMZDMBDMUKUMNABC
      UJOPAQQULDIQQIRQQISAUAQQIUBUCULQTABCUDPUEAUNBDABETCFTUNBNGHBCEFUFUGUHUI
      $.

    $( Value of an operation precomposed with the projection on the second
       component.  (Contributed by BJ, 27-Oct-2024.) $)
    opco2 $p |- ( ph -> ( A ( F o. 2nd ) B ) = ( F ` B ) ) $=
      ( c2nd ccom co cop cfv wceq df-ov a1i cvv wfo wf wcel fo2nd fof mp1i opex
      fvco3d op2ndg syl2anc fveq2d 3eqtrd ) ABCDIJZKZBCLZUJMZULIMZDMCDMUKUMNABC
      UJOPAQQULDIQQIRQQISAUAQQIUBUCULQTABCUDPUEAUNCDABETCFTUNCNGHBCEFUFUGUHUI
      $.
  $}

  ${
    opco1i.1 $e |- B e. _V $.
    opco1i.2 $e |- C e. _V $.
    $( Inference form of ~ opco1 .  (Contributed by Mario Carneiro,
       28-May-2014.)  (Revised by Mario Carneiro, 30-Apr-2015.) $)
    opco1i $p |- ( B ( F o. 1st ) C ) = ( F ` B ) $=
      ( c1st ccom co cfv wceq wtru cvv wcel a1i opco1 mptru ) ABCFGHACIJKABCLLA
      LMKDNBLMKENOP $.
  $}

  ${
    $d A a b c s v w x y z $.  $d B a b d s v w x y z $.
    $d R a b c s v w x y $.  $d S a b d s t u v w x y $.  $d T a b s w z $.
    frxp.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 well-founded classes.  (Contributed by
       Scott Fenton, 17-Mar-2011.)  (Revised by Mario Carneiro, 7-Mar-2013.)
       (Proof shortened by Wolf Lammen, 4-Oct-2014.) $)
    frxp $p |- ( ( R Fr A /\ S Fr B ) -> T Fr ( A X. B ) ) $=
      ( vw vz vb wa cv c0 wbr wn wi wceq wcel wex vs vc va vv vd vt wfr cxp wss
      vu wne wral wrex wal cdm ssn0 xpnz biimpri simprd syl dmxp sseq2 imbitrid
      dmss impcom syldan wrel relxp relss mpi reldm0 necon3bid biimpa jca df-fr
      wb vex dmex sseq1 neeq1 anbi12d raleq rexeqbi1dv imbi12d spcv syl5 adantr
      sylbi w3a cop csn cima crn imassrn wo xpeq0 orcs ss0 biimtrdi rn0 eqsstri
      rneq 0ss eqsstrdi syl6 rnxp rnss pm2.61ine sstrid eldm elimasn df-br c1st
      bitr4i cfv c2nd breq1 notbid rspccv expd weq adantl imbitrrid com3l eleq1
      op1std eqeq1d op2ndd breq1d mpdd ianor fveq2 orbi12d anbi2d breq2d df-rex
      ex bitri exbii sylibr ne0i sylbir exlimiv imaex syl2ani 1stdm elrel opeq1
      biimtrrid eleq1d imbi1d exlimivv adantlr ralrimiv sylan olc ralimi anbi1d
      jcad opex eqeq2d brab ioran pm4.62 anbi2i orbi2i ralbii imbitrrdi reximdv
      xchnxbir com23 sylcom impl expimpd 3adant3 cres resss eqeq1 breq2 ralbidv
      spcev sylanb eximi excom sylib elsnres anbi1i 19.41v anass 3bitr2i ssrexv
      eqid mpan mpsyl rexlimdv 3expib alrimiv ) CEUGZDFUGZLZUAMZCDUHZUIZUXANUKZ
      LZIMZJMZGOZPZIUXAULZJUXAUMZQZUAUNUXBGUGUWTUXLUAUWTUXEUBMZUCMZEOZPZUBUXAUO
      ZULZUCUXQUMZUXKUWRUXEUXSQUWSUXEUXQCUIZUXQNUKZLZUWRUXSUXEUXTUYAUXCUXDDNUKZ
      UXTUXEUXBNUKZUYCUXAUXBUPUYDCNUKZUYCUYEUYCLUYDCDUQURUSUTUYCUXCUXTUYCUXBUOZ
      CRZUXCUXTQCDVAUXCUXQUYFUIUYGUXTUXAUXBVDUYFCUXQVBVCUTVEVFUXCUXDUYAUXCUXANU
      XQNUXCUXAVGZUXANRZUXQNRVPUXCUXBVGUYHCDVHUXAUXBVIVJZUXAVKUTVLVMVNUWRUDMZCU
      IZUYKNUKZLZUXPUBUYKULZUCUYKUMZQZUDUNUYBUXSQZUDUCUBCEVOUYQUYRUDUXQUXAUAVQZ
      VRUYKUXQRZUYNUYBUYPUXSUYTUYLUXTUYMUYAUYKUXQCVSUYKUXQNVTWAUYOUXRUCUYKUXQUX
      PUBUYKUXQWBWCWDWEWHWFWGUWSUXEUXSUXKQZQUWRUWSUXCUXDVUAUWSUXCUXDWIZUXRUXKUC
      UXQVUBUXNUXQSZUXRUXKVUBVUCUXRLZUXFUXNKMZWJZGOZPZIUXAULZKUXAUXNWKZWLZUMZUX
      KUWSUXCVUDVULQUXDUWSUXCLVUCUXRVULUWSUXCVUCUXRVULQZUWSUXCVUCLUEMZVUEFOZPZU
      EVUKULZKVUKUMZVUMUXCUWSVUKDUIZVUKNUKZVURVUCUXCVUKUXAWMZDUXAVUJWNUXCVVADUI
      ZQZCNCNRZUXCUYIVVBVVDUXBNRZUXCUYIQVVDDNRZVVEVVEVVDVVFWOCDWPURWQVVEUXCUXAN
      UIUYIUXBNUXAVBUXAWRWSUTUYIVVANWMZDUXANXBVVGNDWTDXCXAXDXEUYEUXBWMZDRZVVCCD
      XFUXCVVAVVHUIVVIVVBUXAUXBXGVVHDVVAVBVCUTXHXIVUCUXNVUEUXAOZKTVUTKUXNUXAUCV
      QZXJVVJVUTKVVJVUEVUKSZVUTVVLVUFUXASZVVJUXAUXNVUEVVKKVQZXKZUXNVUEUXAXLXNVU
      KVUEUUAUUBUUCWHUWSUYKDUIZUYMLZVUPUEUYKULZKUYKUMZQZUDUNVUSVUTLZVURQZUDKUED
      FVOVVTVWBUDVUKUXAVUJUYSUUDUYKVUKRZVVQVWAVVSVURVWCVVPVUSUYMVUTUYKVUKDVSUYK
      VUKNVTWAVVRVUQKUYKVUKVUPUEUYKVUKWBWCWDWEWHUUEUXCVURVUMQVUCUXCUXRVURVULUXC
      UXRVURVULQUXCUXRLZVUQVUIKVUKVWDVUQUXFUXBSZVUFUXBSZLZPZUXFXMXOZUXNEOZPZVWI
      UXNRZUXFXPXOZVUEFOZPZQZLZWOZIUXAULZVUIVWDVUQVWQIUXAULZVWSUXCUYHUXRVUQVWTQ
      UYJUYHUXRLZVUQVWTVXAVUQLZVWQIUXAVXBUXFUXASZVWKVWPVXAVXCVWKQZVUQUXRUYHVXDU
      XRUYHVXCVWKUYHVXCLVWIUXQSUXRVWKUXFUXAUUFUXPVWKUBVWIUXQUXMVWIRUXOVWJUXMVWI
      UXNEXQXRXSWFXTVEWGUYHVUQVXCVWPQZUXRUYHVUQLZVXCUXFUFMZUJMZWJZRZUJTUFTZVWPU
      YHVXCVXKQVUQUYHVXCVXKUFUJUXFUXAUUGYQWGVXKVXFVXCVWPVXJVXFVXEQUFUJVXFVXEVXJ
      VXIUXASZUFUCYAZVXHVUEFOZPZQZQVXMVXFVXLVXOVXFVXLVXOQVXMUXNVXHWJZUXASZVXOQZ
      VUQVXSUYHVXRVXHVUKSVUQVXOUXAUXNVXHVVKUJVQZXKVUPVXOUEVXHVUKUEUJYAVUOVXNVUN
      VXHVUEFXQXRXSUUIYBVXMVXLVXRVXOVXMVXIVXQUXAVXGUXNVXHUUHUUJUUKYCYDVXJVXCVXL
      VWPVXPUXFVXIUXAYEVXJVWLVXMVWOVXOVXJVWIVXGUXNVXGVXHUXFUFVQZVXTYFYGVXJVWNVX
      NVXJVWMVXHVUEFVXGVXHUXFVYAVXTYHYIXRWDWDYCUULYDYJUUMUUSUUNYQUUOVWQVWRIUXAV
      WQVWHUUPUUQXEVUHVWRIUXAVUHVWHVWJVWLVWNLZWOZPZWOZVWRVWGVYCLZVYEVUGVWGVYCYK
      AMZUXBSZBMZUXBSZLZVYGXMXOZVYIXMXOZEOZVYLVYMRZVYGXPXOZVYIXPXOZFOZLZWOZLVWE
      VYJLZVWIVYMEOZVWIVYMRZVWMVYQFOZLZWOZLVYFABUXFVUFGIVQUXNVUEUUTZAIYAZVYKWUA
      VYTWUFWUHVYHVWEVYJVYGUXFUXBYEUURWUHVYNWUBVYSWUEWUHVYLVWIVYMEVYGUXFXMYLZYI
      WUHVYOWUCVYRWUDWUHVYLVWIVYMWUIYGWUHVYPVWMVYQFVYGUXFXPYLYIWAYMWAVYIVUFRZWU
      AVWGWUFVYCWUJVYJVWFVWEVYIVUFUXBYEYNWUJWUBVWJWUEVYBWUJVYMUXNVWIEUXNVUEVYIV
      VKVVNYFZYOWUJWUCVWLWUDVWNWUJVYMUXNVWIWUKUVAWUJVYQVUEVWMFUXNVUEVYIVVKVVNYH
      YOWAYMWAHUVBUVJVYDVWQVWHVYDVWKVYBPZLVWQVWJVYBUVCWULVWPVWKWULVWLPVWOWOVWPV
      WLVWNYKVWLVWNUVDXNUVEYRUVFYRUVGUVHUVIYQUVKWGUVLUVMUVNUVOUXAVUJUVPZUXAUIVU
      LUXJJWUMUMZUXKUXAVUJUVQVULUXGVUFRZVVMUXJLZLZKTZJTZWUNVULWUQJTZKTZWUSVULVV
      LVUILZKTWVAVUIKVUKYPWVBWUTKVVLVVMVUIWUTVVOVUFVUFRZVVMVUILZWUTVUFUWLWUQWVC
      WVDLJVUFWUGWUOWUOWVCWUPWVDUXGVUFVUFUVRWUOUXJVUIVVMWUOUXIVUHIUXAWUOUXHVUGU
      XGVUFUXFGUVSXRUVTYNWAUWAUWMUWBUWCWHWUQKJUWDUWEWUNUXGWUMSZUXJLZJTWUSUXJJWU
      MYPWVFWURJWVFWUOVVMLZKTZUXJLWVGUXJLZKTWURWVEWVHUXJKUXGUXAUXNVVKUWFUWGWVGU
      XJKUWHWVIWUQKWUOVVMUXJUWIYSUWJYSYRYTUXJJWUMUXAUWKUWNXEXTUWOUWPYBYJUWQUAJI
      UXBGVOYT $.
  $}

  ${
    $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
      VMVNGOWLVIVLGUFGWKVNLUGUHWJWOVSPZVGWBENZVGWBRZVHWFFNZPZUDZPXEABVIVLVGVHUI
      VJVKUIVOVIRZVTXGWIXLXMVQWOVSXMVQVIVPOWOVOVIVPUJVGVHCDUKULUMXMWCXHWHXKXMWA
      VGWBEVGVHVOHSZISZUNZUOXMWDXIWGXJXMWAVGWBXPUPXMWEVHWFFVGVHVOXNXOUQUOTURTVR
      VLRZXGWSXLXDXQVSWRWOXQVSVLVPOWRVRVLVPUJVJVKCDUKULUTXQXHWTXKXCXQWBVJVGEVJV
      KVRJSZKSZUNZVAXQXIXAXJXBXQWBVJVGXTVBXQWFVKVHFVJVKVRXRXSUQVATURTVCWSXFXDWM
      WNWPWQVDVEVF $.
  $}

  ${
    $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 A a b c d t u x y $.  $d B a b c d t u x y $.  $d R a b c d t u x y $.
    $d S a b c d t u x y $.  $d T a b c d t u $.
    soxp.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 strictly ordered classes.
       (Contributed by Scott Fenton, 17-Mar-2011.)  (Revised by Mario Carneiro,
       7-Mar-2013.) $)
    soxp $p |- ( ( R Or A /\ S Or B ) -> T Or ( A X. B ) ) $=
      ( va vb vc vd wa cv wbr wceq w3o wi wo wn vt wor cxp wpo wral sopo syl2an
      vu poxp wcel cop elxp ioran ianor anbi2i bitri anbi12i solin 3orass df-or
      wex sylib orim2d im2anan9 pm2.53 orc syl6 adantr orel1 imor biimpri com12
      anim2d equcomi anim1i olcd syld a1d jaoi imp syl6com jaod biimtrid df-3or
      ex impd sylibr pm3.2 ad2ant2l idd ancomd 3orim123d mpd an4s expcom breq12
      simpr eqeq12 wb ancoms 3orbi123d xporderlem vex 3orbi123i bitrdi biimprcd
      opth com3r exlimivv syl2anb ralrimivv df-so sylanbrc ) CEUBZDFUBZMZCDUCZG
      UDZUANZUHNZGOZXSXTPZXTXSGOZQZUHXQUEUAXQUEXQGUBXNCEUDDFUDXRXOCEUFDFUFABCDE
      FGHUIUGXPYDUAUHXQXQXSXQUJZXTXQUJZMXPYDYEXSINZJNZUKZPZYGCUJZYHDUJZMZMZJVAI
      VAZXTKNZLNZUKZPZYPCUJZYQDUJZMZMZLVAKVAZXPYDRZYFIJXSCDULKLXTCDULYOUUDUUEYN
      UUDUUERIJUUDYNUUEUUCYNUUERKLYNUUCUUEYJYSYMUUBUUEYJYSMZYMUUBMZUUEUUGXPUUFY
      DUUGXPYKYTMZYLUUAMZMZYGYPEOZYGYPPZYHYQFOZMZSZMZUULYHYQPZMZYTYKMZUUAYLMZMZ
      YPYGEOZYPYGPZYQYHFOZMZSZMZQZUUFYDRYKYTYLUUAXPUVHRXPUUJUVHXNUUHXOUUIUVHXNU
      UHMZXOUUIMZMZUUOUURUVFQZUVHUVKUUOUURSZTZUVFRZUVLUVNUUKTZUULTZUUMTZSZMZUVQ
      UUQTZSZMZUVKUVFUVNUUOTZUURTZMUWCUUOUURUMUWDUVTUWEUWBUWDUVPUUNTZMUVTUUKUUN
      UMUWFUVSUVPUULUUMUNUOUPUULUUQUNUQUPUVKUVTUWBUVFUVKUVTUULUVBSZUVQUUQUVDSZS
      ZMZUWBUVFRUVIUVPUWGUVJUVSUWIUVIUUKUULUVBQZUVPUWGRZCYGYPEURUWKUUKUWGSUWLUU
      KUULUVBUSUUKUWGUTUPVBUVJUVRUWHUVQUVJUUMUUQUVDQZUVRUWHRZDYHYQFURUWMUUMUWHS
      UWNUUMUUQUVDUSUUMUWHUTUPVBVCVDUWJUVQUVFUWAUWGUVQUVFRUWIUWGUVQUVBUVFUULUVB
      VEUVBUVEVFZVGVHUWAUWJUWGUVQUVDSZMUVFUWAUWIUWPUWGUWAUWHUVDUVQUUQUVDVIVCVMU
      WGUWPUVFUULUWPUVFRUVBUULUWPUVDUVFUWPUULUVDUULUVDRUWPUULUVDVJVKVLUULUVDUVF
      UULUVDMUVEUVBUULUVCUVDIKVNVOVPWEVQUVBUVFUWPUWOVRVSVTWAWBVGWFWCUVLUVMUVFSU
      VOUUOUURUVFWDUVMUVFUTUPWGUVKUUOUUPUURUURUVFUVGUUHUUIUUOUUPRXNXOUUJUUOWHWI
      UVKUURWJUVIUUSUUTUVFUVGRUVJUVIYKYTXNUUHWQWKUVJYLUUAXOUUIWQWKUVAUVFWHUGWLW
      MWNWOWNUUFYDUVHUUFYDYIYRGOZYIYRPZYRYIGOZQUVHUUFYAUWQYBUWRYCUWSXSYIXTYRGWP
      XSYIXTYRWRYSYJYCUWSWSXTYRXSYIGWPWTXAUWQUUPUWRUURUWSUVGABCDEFGIJKLHXBYGYHY
      PYQIXCJXCXGABCDEFGKLIJHXBXDXEXFVGXHVTWNWOXIVLXIVTXJVLXKUAUHXQGXLXM $.
  $}

  ${
    $d A x y $.  $d B x y $.  $d R x y $.  $d S x y $.
    wexp.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 well-ordered classes.  (Contributed by
       Scott Fenton, 17-Mar-2011.)  (Revised by Mario Carneiro, 7-Mar-2013.) $)
    wexp $p |- ( ( R We A /\ S We B ) -> T We ( A X. B ) ) $=
      ( wwe wa cxp wfr wor wefr frxp syl2an weso soxp df-we sylanbrc ) CEIZDFIZ
      JCDKZGLZUCGMZUCGIUACELDFLUDUBCENDFNABCDEFGHOPUACEMDFMUEUBCEQDFQABCDEFGHRP
      UCGST $.
  $}

  ${
    $d u v w x y z A $.  $d u v w x y z B $.  $d w x y G $.  $d w x z ph $.
    $d u v w x y z F $.  $d w x y Q $.  $d u v w x y R $.  $d u v w x y S $.
    $d w T $.
    fnwe.1 $e |- T = { <. x , y >. | ( ( x e. A /\ y e. A ) /\
      ( ( F ` x ) R ( F ` y ) \/ ( ( F ` x ) = ( F ` y ) /\ x S y ) ) ) } $.
    fnwe.2 $e |- ( ph -> F : A --> B ) $.
    fnwe.3 $e |- ( ph -> R We B ) $.
    fnwe.4 $e |- ( ph -> S We A ) $.
    fnwe.5 $e |- ( ph -> ( F " w ) e. _V ) $.
    ${
      fnwelem.6 $e |- Q = { <. u , v >. |
        ( ( u e. ( B X. A ) /\ v e. ( B X. A ) ) /\
          ( ( 1st ` u ) R ( 1st ` v ) \/
            ( ( 1st ` u ) = ( 1st ` v ) /\ ( 2nd ` u ) S ( 2nd ` v ) ) ) ) } $.
      fnwelem.7 $e |- G = ( z e. A |-> <. ( F ` z ) , z >. ) $.
      $( Lemma for ~ fnwe .  (Contributed by Mario Carneiro, 10-Mar-2013.)
         (Revised by Mario Carneiro, 18-Nov-2014.) $)
      fnwelem $p |- ( ph -> T We A ) $=
        ( crn wwe cxp wss wf cv cfv cop wcel wa ffvelcdm simpr opelxpd frn 3syl
        wexp syl2anc wess sylc ccnv wiso cima cvv wal wi wf1o wf1 wceq weq wral
        fmptd fveq2 opeq12d opex fvmpt eqeqan12d fvex vex opth simprbi biimtrdi
        id rgen2 dff13 sylanblrc f1f1orn f1ocnv 4syl copab eqid f1oiso2 wb wrel
        wbr wo frel dfrel2 fveq1d breq12d syl adantr breqan12d adantl c1st c2nd
        opelxp bitrdi anbi1d op1std breq1d eqeq1d op2ndd anbi12d orbi12d anbi2d
        sylib eleq1 breq2d eqeq2d brab jca anim12dan biantrurd bitr4id pm5.32da
        3bitrrd opabbidv eqtrid isoeq3 imbitrrid isocnv imacnvcnv cres imadmres
        sylancl cdm dmres elin2 xpexg simprr f1dm eleqtrd wfn ffnd cin sseqtrid
        inss2 fndmd eqsstrid eleqtrrd sylanbrc fnfvima syl3anc eleqtrdi eqeltrd
        simprl ralrimiva wfun f1fun resss dmss ax-mp funimass4 mpbird eqsstrrid
        sylan2b ssexd eqeltrid alrimiv isowe2 mpd ) AOUCZJUDZHMUDZAUVNIHUEZUFZU
        VQJUDZUVOAHINUGZHUVQOUGZUVRQUVTDHDUHZNUIZUWBUJZUVQOUVTUWBHUKZULUWCUWBIH
        HIUWBNUMUVTUWEUNUOUBVMZHUVQOUPUQAIKUDHLUDUVSRSGFIHKLJUAURUSUVNUVQJUTVAA
        HUVNMJOVBZVBZVCZUWHEUHZVDZVEUKZEVFUVOUVPVGAUVNHJMUWGVCZUWIAUVTUVNHUWGVH
        ZUWMQAUVTHUVQOVIZHUVNOVHUWNQUVTUWABUHZOUIZCUHZOUIZVJZBCVKZVGZCHVLBHVLUW
        OUWFUXBBCHHUWPHUKZUWRHUKZULZUWTUWPNUIZUWPUJZUWRNUIZUWRUJZVJZUXAUXCUXDUW
        QUXGUWSUXIDUWPUWDUXGHODBVKZUWCUXFUWBUWPUWBUWPNVNUXKWDVOUBUXFUWPVPZVQZDU
        WRUWDUXIHODCVKZUWCUXHUWBUWRUWBUWRNVNUXNWDVOUBUXHUWRVPZVQZVRUXJUXFUXHVJZ
        UXAUXFUWPUXHUWRUWPNVSZBVTZWAWBWCWEBCHUVQOWFWGZHUVQOWHHUVNOWIWJUWNUWMUVT
        UVNHJUXEUWPUWHUIZUWRUWHUIZJWPZULZBCWKZUWGVCZBCUVNHJUYEUWGUYEWLWMUVTMUYE
        VJUWMUYFWNUVTMUXEUXFUXHKWPZUXQUWPUWRLWPZULZWQZULZBCWKUYEPUVTUYKUYDBCUVT
        UXEUYJUYCUVTUXEULZUYCUWQUWSJWPZUXGUXIJWPZUYJUVTUYCUYMWNZUXEUVTUWAUYOUWF
        UWAUYAUWQUYBUWSJUWAUWPUWHOUWAOWOUWHOVJHUVQOWROWSXRZWTUWAUWRUWHOUYPWTXAX
        BXCUXEUYMUYNWNUVTUXCUXDUWQUXGUWSUXIJUXMUXPXDXEUYLUYNUXFIUKZUXCULZUXHIUK
        ZUXDULZULZUYJULZUYJGUHZUVQUKZFUHZUVQUKZULZVUCXFUIZVUEXFUIZKWPZVUHVUIVJZ
        VUCXGUIZVUEXGUIZLWPZULZWQZULUYRVUFULZUXFVUIKWPZUXFVUIVJZUWPVUMLWPZULZWQ
        ZULVUBGFUXGUXIJUXLUXOVUCUXGVJZVUGVUQVUPVVBVVCVUDUYRVUFVVCVUDUXGUVQUKUYR
        VUCUXGUVQXSUXFUWPIHXHXIXJVVCVUJVURVUOVVAVVCVUHUXFVUIKUXFUWPVUCUXRUXSXKZ
        XLVVCVUKVUSVUNVUTVVCVUHUXFVUIVVDXMVVCVULUWPVUMLUXFUWPVUCUXRUXSXNXLXOXPX
        OVUEUXIVJZVUQVUAVVBUYJVVEVUFUYTUYRVVEVUFUXIUVQUKUYTVUEUXIUVQXSUXHUWRIHX
        HXIXQVVEVURUYGVVAUYIVVEVUIUXHUXFKUXHUWRVUEUWRNVSZCVTZXKZXTVVEVUSUXQVUTU
        YHVVEVUIUXHUXFVVHYAVVEVUMUWRUWPLUXHUWRVUEVVFVVGXNXTXOXPXOUAYBUYLVUAUYJU
        VTUXCUYRUXDUYTUVTUXCULUYQUXCHIUWPNUMUVTUXCUNYCUVTUXDULUYSUXDHIUWRNUMUVT
        UXDUNYCYDYEYFYHYGYIYJUVNHJMUYEUWGYKXBYLVAUVNHJMUWGYMXBAUWLEAUWKOUWJVDZV
        EOUWJYNAVVINUWJVDZUWJUEZVEAVVJVEUKUWJVEUKVVKVEUKTEVTVVJUWJVEVEUUAYQAVVI
        OOUWJYOZYRZVDZVVKOUWJYPAVVNVVKUFZUWQVVKUKZBVVMVLZAVVPBVVMUWPVVMUKAUWPUW
        JUKZUWPOYRZUKZULZVVPUWPUWJVVSVVMOUWJYSYTAVWAULZUWQUXGVVKVWBUXCUWQUXGVJV
        WBUWPVVSHAVVRVVTUUBAVVSHVJZVWAAUVTUWOVWCQUXTHUVQOUUCUQXCUUDZUXMXBVWBUXF
        UWPVVJUWJVWBUXFNNUWJYOYRZVDZVVJVWBNHUUEZVWEHUFUWPVWEUKZUXFVWFUKAVWGVWAA
        HINQUUFXCZVWBVWEUWJNYRZUUGZHNUWJYSZVWBVWJVWKHUWJVWJUUIVWBHNVWIUUJZUUHUU
        KVWBVVRUWPVWJUKVWHAVVRVVTUURZVWBUWPHVWJVWDVWMUULUWPUWJVWJVWEVWLYTUUMHVW
        ENUWPUUNUUONUWJYPUUPVWNUOUUQUVHUUSAOUUTZVVMVVSUFZVVOVVQWNAUVTUWOVWOQUXT
        HUVQOUVAUQVVLOUFVWPOUWJUVBVVLOUVCUVDBVVMVVKOUVEYQUVFUVGUVIUVJUVKEHUVNMJ
        UWHUVLUSUVM $.
    $}

    $( A variant on lexicographic order, which sorts first by some function of
       the base set, and then by a "backup" well-ordering when the function
       value is equal on both elements.  (Contributed by Mario Carneiro,
       10-Mar-2013.)  (Revised by Mario Carneiro, 18-Nov-2014.) $)
    fnwe $p |- ( ph -> T We A ) $=
      ( vz vv vu cv cfv cxp wcel wa c1st wbr wceq c2nd wo cop cmpt eqid fnwelem
      copab ) ABCPDQREFRSZFEUAZUBQSZUOUBUCUNUDTZUPUDTZGUEUQURUFUNUGTUPUGTHUEUCU
      HUCRQUMZGHIJPEPSZJTUTUIUJZKLMNOUSUKVAUKUL $.
  $}

  ${
    $d x y z A $.  $d u w B $.  $d u w x y F $.  $d w z ph $.  $d u w x y R $.
    $d u z $.  $d x y S $.  $d w z T $.
    fnse.1 $e |- T = { <. x , y >. | ( ( x e. A /\ y e. A ) /\
      ( ( F ` x ) R ( F ` y ) \/ ( ( F ` x ) = ( F ` y ) /\ x S y ) ) ) } $.
    fnse.2 $e |- ( ph -> F : A --> B ) $.
    fnse.3 $e |- ( ph -> R Se B ) $.
    fnse.4 $e |- ( ph -> ( `' F " w ) e. _V ) $.
    $( Condition for the well-order in ~ fnwe to be set-like.  (Contributed by
       Mario Carneiro, 25-Jun-2015.) $)
    fnse $p |- ( ph -> T Se A ) $=
      ( vz cv cvv wcel wa wbr vu ccnv csn cima cin wral wse cfv crab ffvelcdmda
      cun seex syl2an2r snex sylancl wi wceq imaeq2 eleq1d imbi2d vtoclg impcom
      unexg syldan wss inss2 wb vex eliniseg elv wo breqan12d eqeqan12d anbi12d
      fveq2 breq12 orbi12d brab2a adantrr breq1 elrab3 syl biimprd simpl sylibr
      fvex elsn a1i orim12d imbitrrdi simprl jctild wfn adantr elpreima sylibrd
      elun ffnd expimpd biimtrid ssrdv sstrid ssexd ralrimiva dfse2 ) AEIUBOPZU
      CUDZUEZQRZOEUFEIUGAXIOEAXFERZSZXHJUBZUAPZXFJUHZGTZUAFUIZXNUCZUKZUDZQAXJXR
      QRZXSQRZXKXPQRZXQQRXTAFGUGXJXNFRYBMAEFXFJLUJUAFXNGULUMXNUNXPXQQQVCUOXTAYA
      AXLDPZUDZQRZUPAYAUPDXRQYCXRUQZYEYAAYFYDXSQYCXRXLURUSUTNVAVBVDAXHXSVEXJAXH
      XGXSEXGVFADXGXSYCXGRZYCXFITZAYCXSRZYGYHVGOIXFYCQDVHVIVJYHYCERZXJSZYCJUHZX
      NGTZYLXNUQZYCXFHTZSZVKZSAYIBPZJUHZCPZJUHZGTZYSUUAUQZYRYTHTZSZVKYQBCYCXFEE
      IYRYCUQZYTXFUQZSZUUBYMUUEYPUUFUUGYSYLUUAXNGYRYCJVOZYTXFJVOZVLUUHUUCYNUUDY
      OUUFUUGYSYLUUAXNUUIUUJVMYRYCYTXFHVPVNVQKVRAYKYQYIAYKSZYQYJYLXRRZSZYIUUKYQ
      UULYJUUKYQYLXPRZYLXQRZVKUULUUKYMUUNYPUUOUUKUUNYMUUKYLFRZUUNYMVGAYJUUPXJAE
      FYCJLUJVSXOYMUAYLFXMYLXNGVTWAWBWCYPUUOUPUUKYPYNUUOYNYOWDYLXNYCJWFWGWEWHWI
      YLXPXQWQWJAYJXJWKWLUUKJEWMZYIUUMVGAUUQYKAEFJLWRWNEYCXRJWOWBWPWSWTWTXAXBWN
      XCXDOEIXEWE $.
  $}

  ${
    $d A a b x y z $.  $d B a b x y z $.  $d F a b c x y z $.
    $d G a b c x y z $.  $d X a b c z $.  $d Y a b c z $.
    fvproj.h $e |- H = ( x e. A , y e. B |-> <. ( F ` x ) , ( G ` y ) >. ) $.
    ${
      fvproj.x $e |- ( ph -> X e. A ) $.
      fvproj.y $e |- ( ph -> Y e. B ) $.
      $( Value of a function on ordered pairs with values expressed as ordered
         pairs.  Note that ` F ` and ` G ` are the projections of ` H ` to the
         first and second coordinate respectively.  (Contributed by Thierry
         Arnoux, 30-Dec-2019.) $)
      fvproj $p |- ( ph -> ( H ` <. X , Y >. ) = <. ( F ` X ) , ( G ` Y ) >. )
        $=
        ( va vb cop cfv wceq cv fveq2 co df-ov wcel opeq1d opeq2d cbvmpov eqtri
        cmpo opex ovmpo syl2anc eqtr3id ) AIJPHQIJHUAZIFQZJGQZPZIJHUBAIDUCJEUCU
        MUPRLMNOIJDENSZFQZOSZGQZPZUPHUNUTPUQIRURUNUTUQIFTUDUSJRUTUOUNUSJGTUEHBC
        DEBSZFQZCSZGQZPZUHNODEVAUHKBCNODEVFVAURVEPVBUQRVCURVEVBUQFTUDVDUSRVEUTU
        RVDUSGTUEUFUGUNUOUIUJUKUL $.
    $}

    ${
      $d H a b c x y z $.  $d ph a b c z $.
      fimaproj.f $e |- ( ph -> F Fn A ) $.
      fimaproj.g $e |- ( ph -> G Fn B ) $.
      fimaproj.x $e |- ( ph -> X C_ A ) $.
      fimaproj.y $e |- ( ph -> Y C_ B ) $.
      $( Image of a cartesian product for a function on ordered pairs with
         values expressed as ordered pairs.  Note that ` F ` and ` G ` are the
         projections of ` H ` to the first and second coordinate respectively.
         (Contributed by Thierry Arnoux, 30-Dec-2019.) $)
      fimaproj $p |- ( ph -> ( H " ( X X. Y ) ) = ( ( F " X ) X. ( G " Y ) ) )
        $=
        ( vz wcel cfv wceq wa vc va vb cxp cima wrex wfn wss c1st c2nd cop opex
        cv wb cmpo vex op1std fveq2d op2ndd opeq12d mpompt eqtr4i fnmpti xpss12
        cmpt syl2anc sylancr simp-4r simplr opelxpi simpllr simpr sseldd fvproj
        fvelimab ad5antr 1st2nd2 ad5antlr 3eqtr4d fveqeq2 rspcev wfun ad3antrrr
        fnfun syl xp2nd ad3antlr fvelima r19.29a adantr xp1st adantl cvv fvmpt2
        ad2antrr sylancl fnfvima syl3anc eqeltrd eqeltrrd r19.29an bitr4d eqrdv
        impbida ) AUAHIJUDZUEZFIUEZGJUEZUDZAUAUMZXFQZPUMZHRZXJSZPXEUFZXJXIQZAHD
        EUDZUGXEXQUHZXKXOUNPXQXLUIRZFRZXLUJRZGRZUKZHXTYBULZHBCDEBUMZFRZCUMZGRZU
        KZUOPXQYCVEKBCPDEYCYIXLYEYGUKSZXTYFYBYHYJXSYEFYEYGXLBUPZCUPZUQURYJYAYGG
        YEYGXLYKYLUSURUTVAVBZVCAIDUHZJEUHZXRNOIDJEVDVFZPXQXEXJHVOVGAXPXOAXPTZUB
        UMZFRZXJUIRZSZXOUBIYQYRIQZTZUUATZUCUMZGRZXJUJRZSZXOUCJUUDUUEJQZTZUUHTZY
        RUUEUKZXEQZUULHRZXJSZXOUUKUUBUUIUUMYQUUBUUAUUIUUHVHZUUDUUIUUHVIZYRUUEIJ
        VJVFUUKYSUUFUKYTUUGUKZUUNXJUUKYSYTUUFUUGUUCUUAUUIUUHVKUUJUUHVLUTUUKBCDE
        FGHYRUUEKUUKIDYRAYNXPUUBUUAUUIUUHNVPUUPVMUUKJEUUEAYOXPUUBUUAUUIUUHOVPUU
        QVMVNXPXJUURSAUUBUUAUUIUUHXJXGXHVQVRVSXNUUOPUULXEXLUULXJHVTWAVFUUDGWBZU
        UGXHQZUUHUCJUFUUDGEUGZUUSAUVAXPUUBUUAMWCEGWDWEXPUUTAUUBUUAXJXGXHWFWGUCU
        UGJGWHVFWIYQFWBZYTXGQZUUAUBIUFYQFDUGZUVBAUVDXPLWJDFWDWEXPUVCAXJXGXHWKWL
        UBYTIFWHVFWIAXNXPPXEAXLXEQZTZXNTZXMXJXIUVFXNVLUVGXMYCXIUVGXLXQQYCWMQXMY
        CSUVGXEXQXLAXRUVEXNYPWOAUVEXNVIZVMYDPXQYCWMHYMWNWPUVGXTXGQZYBXHQZYCXIQU
        VGUVDYNXSIQZUVIAUVDUVEXNLWOAYNUVEXNNWOUVGUVEUVKUVHXLIJWKWEDIFXSWQWRUVGU
        VAYOYAJQZUVJAUVAUVEXNMWOAYOUVEXNOWOUVGUVEUVLUVHXLIJWFWEEJGYAWQWRXTYBXGX
        HVJVFWSWTXAXDXBXC $.
    $}
  $}

  ${
    $d A x y $.  $d B x y z $.  $d ph x $.
    $( A version of ~ ralxp with explicit substitution.  (Contributed by Scott
       Fenton, 21-Aug-2024.) $)
    ralxpes $p |- ( A. x e. ( A X. B ) [. ( 1st ` x ) / y ].
     [. ( 2nd ` x ) / z ]. ph <-> A. y e. A A. z e. B ph ) $=
      ( cv c2nd cfv wsbc c1st nfsbc1v nfcv nfsbcw nfv sbcopeq1a ralxpf ) ADBGZH
      IZJZCRKIZJABCDEFTCUALTDCUADUAMADSLNABOACDRPQ $.
  $}

  ${
    $d A w x y z $.  $d B w x y z $.  $d C w x y z $.
    ralxp3f.1 $e |- F/ y ph $.
    ralxp3f.2 $e |- F/ z ph $.
    ralxp3f.3 $e |- F/ w ph $.
    ralxp3f.4 $e |- F/ x ps $.
    ralxp3f.5 $e |- ( x = <. y , z , w >. -> ( ph <-> ps ) ) $.
    $( Restricted for all over a triple Cartesian product.  (Contributed by
       Scott Fenton, 22-Aug-2024.) $)
    ralxp3f $p |- ( A. x e. ( ( A X. B ) X. C ) ph <->
     A. y e. A A. z e. B A. w e. C ps ) $=
      ( wral cv wi wal wrex ralbii wcel cotp df-ral el2xptp imbi1i r19.23 bitri
      cxp wceq bitr2i albii ralcom4 otex ceqsal bitr3i 3bitri ) ACGHUHIUHZOCPZU
      QUAZAQZCRURDPZEPZFPZUBZUIZAQZFIOZEHOZDGOZCRZBFIOZEHOZDGOZACUQUCUTVICUTVEF
      ISZEHSZDGSZAQZVIUSVPADEFURGHIUDUEVIVOAQZDGOVQVHVRDGVHVNAQZEHOVRVGVSEHVEAF
      ILUFTVNAEHKUFUGTVOADGJUFUJUGUKVJVHCRZDGOVMVHDCGULVTVLDGVTVGCRZEHOVLVGECHU
      LWAVKEHWAVFCRZFIOVKVFFCIULWBBFIABCVDMVAVBVCUMNUNTUOTUOTUOUP $.
  $}

  ${
    $d A w x y z $.  $d B w x y z $.  $d C w x y z $.  $d ph w y z $.
    $d ps x $.
    ralxp3.1 $e |- ( x = <. y , z , w >. -> ( ph <-> ps ) ) $.
    $( Restricted for all over a triple Cartesian product.  (Contributed by
       Scott Fenton, 2-Feb-2025.) $)
    ralxp3 $p |- ( A. x e. ( ( A X. B ) X. C ) ph <->
     A. y e. A A. z e. B A. w e. C ps ) $=
      ( nfv ralxp3f ) ABCDEFGHIADKAEKAFKBCKJL $.
  $}

  ${
    $d A w x y z $.  $d B w x y z $.  $d C w x y z $.  $d ph x $.
    $( Restricted for-all over a triple Cartesian product with explicit
       substitution.  (Contributed by Scott Fenton, 22-Aug-2024.) $)
    ralxp3es $p |- ( A. x e. ( ( A X. B ) X. C )
     [. ( 1st ` ( 1st ` x ) ) / y ]. [. ( 2nd ` ( 1st ` x ) ) / z ].
     [. ( 2nd ` x ) / w ]. ph <->
     A. y e. A A. z e. B A. w e. C ph ) $=
      ( cv c2nd cfv wsbc c1st nfsbc1v nfcv nfsbcw nfv sbcoteq1a ralxp3f ) AEBIZ
      JKZLZDTMKZJKZLZCUCMKZLABCDEFGHUECUFNUEDCUFDUFOUBDUDNPUEECUFEUFOUBEDUDEUDO
      AEUANPPABQACDETRS $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Induction on Cartesian products
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d A p q x y z w $.  $d B p q w x y z $.  $d ch q y $.  $d ph p q z $.
    $d ps x w $.  $d R p q w x y z $.  $d ta y $.  $d th x $.  $d X x y $.
    $d Y y $.
    frpoins3xpg.1 $e |- ( ( x e. A /\ y e. B ) ->
    ( A. z A. w ( <. z , w >. e. Pred ( R , ( A X. B ) , <. x , y >. ) -> ch )
      -> ph ) ) $.
    frpoins3xpg.2 $e |- ( x = z -> ( ph <-> ps ) ) $.
    frpoins3xpg.3 $e |- ( y = w -> ( ps <-> ch ) ) $.
    frpoins3xpg.4 $e |- ( x = X -> ( ph <-> th ) ) $.
    frpoins3xpg.5 $e |- ( y = Y -> ( th <-> ta ) ) $.
    $( Special case of founded partial induction over a Cartesian product.
       (Contributed by Scott Fenton, 22-Aug-2024.) $)
    frpoins3xpg $p |- ( ( ( R Fr ( A X. B ) /\ R Po ( A X. B ) /\
    R Se ( A X. B ) ) /\ ( X e. A /\ Y e. B ) ) -> ta ) $=
      ( vq vp wcel wa cxp wfr wpo wse w3a wral c2nd cfv wsbc c1st cop wceq wrex
      cv cpred wi elxp2 nfcv nfsbc1v nfralw nfv nfsbcw wal weq sbcbidv cbvsbcvw
      nfim sbcbii bitri ralbii impexp cin elin wss predss sseqin2 eleq2i bitr3i
      imbi1i albii df-ral 3bitr4ri eleq1 sbcopeq1a imbi12d ralxpf opelxp 2albii
      mpbi r2al biimtrid predeq3 raleqdv syl5ibrcom rexlimd rexlimi sylbi fveq2
      3bitri ex sbceq1d sbceqbid frpoins2g ralxpes sylib rspc2va sylan2 ancoms
      ) MJUBNKUBUCZJKUDZLUEXMLUFXMLUGUHZEXNXLAGKUIFJUIZEXNAGUAUQZUJUKZULZFXPUMU
      KZULZUAXMUIXOXTAGTUQZUJUKZULZFYAUMUKZULZUATXMLXPXMUBXPFUQZGUQZUNZUOZGKUPZ
      FJUPYETXMLXPURZUIZXTUSZFGXPJKUTYJYMFJYLXTFYEFTYKFYKVAYCFYDVBVCXRFXSVBVJYF
      JUBZYIYMGKYNGVDYLXTGYEGTYKGYKVAYCGFYDGYDVAAGYBVBVEVCXRGFXSGXSVAAGXQVBVEVJ
      YNYGKUBZYIYMUSYNYOUCZYMYIYETXMLYHURZUIZAUSYRHUQZIUQZUNZYQUBZCUSZIVFHVFZYP
      AYRCIYBULZHYDULZTYQUIZYAYQUBZUUFUSZTXMUIZUUDYEUUFTYQYEBGYBULZHYDULUUFYCUU
      KFHYDFHVGABGYBPVHVIUUKUUEHYDBCGIYBQVIVKVLVMYAXMUBZUUIUSZTVFUUITVFUUJUUGUU
      MUUITUUMUULUUHUCZUUFUSUUIUULUUHUUFVNUUNUUHUUFUUNYAXMYQVOZUBUUHYAXMYQVPUUO
      YQYAYQXMVQUUOYQUOXMLYHVRYQXMVSWLZVTWAWBWAWCUUITXMWDUUFTYQWDWEUUJUUCIKUIHJ
      UIYSJUBYTKUBUCZUUCUSZIVFHVFUUDUUIUUCTHIJKUUHUUFHUUHHVDUUEHYDVBVJUUHUUFIUU
      HIVDUUEIHYDIYDVACIYBVBVEVJUUCTVDYAUUAUOUUHUUBUUFCYAUUAYQWFCHIYAWGWHWIUUCH
      IJKWMUURUUCHIUURUUAXMUBZUUBUCZCUSZUUCUVAUUSUUCUSUURUUSUUBCVNUUSUUQUUCYSYT
      JKWJWBVLUUTUUBCUUTUUAUUOUBUUBUUAXMYQVPUUOYQUUAUUPVTWAWBWAWKXBXBOWNYIYLYRX
      TAYIYETYKYQXMLXPYHWOWPAFGXPWGWHWQXCWRWSWTUATVGZXRYCFXSYDXPYAUMXAUVBAGXQYB
      XPYAUJXAXDXEXFAUAFGJKXGXHAEDFGMNJKRSXIXJXK $.
  $}

  ${
    $d A p q t u w x y z $.  $d B p q t u w x y z $.  $d C p q t u w x y z $.
    $d et y $.  $d ph p q w $.  $d R p q t u w x y z $.  $d ta x $.
    $d X x y z $.  $d Y y z $.  $d Z z $.  $d z ze $.  $d ch u y $.
    $d ps p q x t $.  $d th q x y z $.
    frpoins3xp3g.1 $e |- ( ( x e. A /\ y e. B /\ z e. C ) ->
     ( A. w A. t A. u
     ( <. w , t , u >. e. Pred ( R , ( ( A X. B ) X. C ) ,
     <. x , y , z >. ) -> th ) -> ph ) ) $.
    frpoins3xp3g.2 $e |- ( x = w -> ( ph <-> ps ) ) $.
    frpoins3xp3g.3 $e |- ( y = t -> ( ps <-> ch ) ) $.
    frpoins3xp3g.4 $e |- ( z = u -> ( ch <-> th ) ) $.
    frpoins3xp3g.5 $e |- ( x = X -> ( ph <-> ta ) ) $.
    frpoins3xp3g.6 $e |- ( y = Y -> ( ta <-> et ) ) $.
    frpoins3xp3g.7 $e |- ( z = Z -> ( et <-> ze ) ) $.
    $( Special case of founded partial recursion over a triple Cartesian
       product.  (Contributed by Scott Fenton, 22-Aug-2024.) $)
    frpoins3xp3g $p |- ( ( ( R Fr ( ( A X. B ) X. C )
     /\ R Po ( ( A X. B ) X. C )
     /\ R Se ( ( A X. B ) X. C ) ) /\ ( X e. A /\ Y e. B /\ Z e. C ) )
     -> ze ) $=
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      3expia sylbi 2fveq3 fveq2 sbceq1d sbceqbid frpoins2g ralxp3es sylib mpan9
      ex rspc3v ) NOUJPUJZQUKYAQULYAQUMUNZAJPUOIOUOHNUOZRNUPSOUPTPUPUNGYBAJUHUQ
      ZURUSZUTZIYDVAUSZURUSZUTZHYGVAUSZUTZUHYAUOYCYKAJUIUQZURUSZUTZIYLVAUSZURUS
      ZUTZHYOVAUSZUTZUHUIYAQYSUIYAQYDVBZUODLYMUTZMYPUTZKYRUTZUIYTUOZYDYAUPZYKYS
      UUCUIYTYSBJYMUTZIYPUTZKYRUTUUCYQUUGHKYRHKVDZYNUUFIYPUUHABJYMUBVCVCVEUUGUU
      BKYRUUGCJYMUTZMYPUTUUBUUFUUIIMYPIMVDBCJYMUCVCVEUUIUUAMYPCDJLYMUDVEVFVGVFV
      GVHUUEYDHUQZIUQZJUQZVIZVJZJPVKZIOVKZHNVKUUDYKVPZHIJYDNOPVLUUPUUQHNUUDYKHU
      UDHVQYIHYJVMVNUUJNUPZUUOUUQIOUURIVQUUDYKIUUDIVQYIIHYJIYJVOYFIYHVMVRVNUURU
      UKOUPZUUOUUQVPUURUUSWIZUUNUUQJPUUTJVQUUDYKJUUDJVQYIJHYJJYJVOYFJIYHJYHVOAJ
      YEVMVRVRVNUURUUSUULPUPZUUNUUQVPUURUUSUVAUNZUUQUUNUUCUIYAQUUMVBZUOZAVPUVDK
      UQZMUQZLUQZVIZUVCUPZDVPZLVSMVSKVSZUVBAYLYAUPZYLUVCUPZUUCVPZVPZUIVSZUVNUIV
      SZUVKUVDUVQUVPUVNUVOUIUVNUVLUVMWIZUUCVPUVOUVMUVRUUCUVMYLYAUVCVTZUPZUVRUVT
      UVMUVSUVCYLUVCYAWAUVSUVCVJYAQUUMWBUVCYAWCWDZWEWFYLYAUVCWJVGWGUVLUVMUUCWHV
      GWKWFUVKUVNUIYAUOZUVPUVJLPUOMOUOKNUOUVENUPUVFOUPUVGPUPUNZUVJVPZLVSMVSKVSU
      WBUVKUVJKMLNOPWLUVNUVJUIKMLNOPUVMUUCKUVMKVQUUBKYRVMVNUVMUUCMUVMMVQUUBMKYR
      MYRVOUUAMYPVMVRVNUVMUUCLUVMLVQUUBLKYRLYRVOUUALMYPLYPVODLYMVMVRVRVNUVJUIVQ
      YLUVHVJUVMUVIUUCDYLUVHUVCWTDKMLYLWMWNWOUVJUWDKMLUVJUWCUVIWIZDVPUWDUVIUWED
      UVHUVSUPUVHYAUPZUVIWIUVIUWEUVHYAUVCWJUVSUVCUVHUWAWEUWFUWCUVIUVEUVFUVGNOPW
      PWQWRWGUWCUVIDWHVGWSXAUVNUIYAXBVGUUCUIUVCXBXAUAXCUUNUUDUVDYKAUUNUUCUIYTUV
      CYAQYDUUMXDXEAHIJYDWMWNXFXIXGXSXGXHXJXCUHUIVDZYIYQHYJYRYDYLVAVAXKUWGYFYNI
      YHYPYDYLURVAXKUWGAJYEYMYDYLURXLXMXNXNXOAUHHIJNOPXPXQAGEFHIJRSTNOPUEUFUGXT
      XR $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Ordering on Cartesian products
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    xpord2.1 $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 ) \/ ( 2nd ` x ) = ( 2nd ` y ) ) /\
          x =/= y ) ) } $.

    ${
      $d A x y $.  $d B x y $.  $d a x y $.  $d b x y $.  $d c x y $.
      $d d x y $.  $d R x y $.  $d S x y $.
      $( Lemma for Cartesian product ordering.  Calculate the value of the
         Cartesian product relation.  (Contributed by Scott Fenton,
         19-Aug-2024.) $)
      xpord2lem $p |- ( <. a , b >. T <. c , d >. <->
       ( ( a e. A /\ b e. B ) /\ ( c e. A /\ d e. B ) /\
         ( ( a R c \/ a = c ) /\ ( b S d \/ b = d ) /\
           ( a =/= c \/ b =/= d ) ) ) ) $=
        ( cv wcel cfv wbr wceq wo wne w3a cxp c1st c2nd wa cop weq eleq1 opelxp
        bitrdi vex op1std breq1d eqeq1d orbi12d op2ndd neeq1 3anbi123d 3anbi13d
        opex breq2d eqeq2d neeq2 opthne 3anbi23d brab ) AMZCDUAZNZBMZVGNZVFUBOZ
        VIUBOZEPZVKVLQZRZVFUCOZVIUCOZFPZVPVQQZRZVFVISZTZTHMZCNIMZDNUDZVJWCVLEPZ
        WCVLQZRZWDVQFPZWDVQQZRZWCWDUEZVISZTZTWEJMZCNKMZDNUDZWCWOEPZHJUFZRZWDWPF
        PZIKUFZRZWCWOSWDWPSRZTZTABWLWOWPUEZGWCWDUSWOWPUSVFWLQZVHWEWBWNVJXGVHWLV
        GNWEVFWLVGUGWCWDCDUHUIXGVOWHVTWKWAWMXGVMWFVNWGXGVKWCVLEWCWDVFHUJZIUJZUK
        ZULXGVKWCVLXJUMUNXGVRWIVSWJXGVPWDVQFWCWDVFXHXIUOZULXGVPWDVQXKUMUNVFWLVI
        UPUQURVIXFQZVJWQWNXEWEXLVJXFVGNWQVIXFVGUGWOWPCDUHUIXLWHWTWKXCWMXDXLWFWR
        WGWSXLVLWOWCEWOWPVIJUJZKUJZUKZUTXLVLWOWCXOVAUNXLWIXAWJXBXLVQWPWDFWOWPVI
        XMXNUOZUTXLVQWPWDXPVAUNXLWMWLXFSXDVIXFWLVBWCWDWOWPXHXIVCUIUQVDLVE $.
    $}

    ${
      $d A a b c p q r s t u x y $.  $d B a b c p q r s t u x y $.
      $d ph a b c p q r s t u $.  $d R a b c p q r s t u x y $.
      $d S a b c p q r s t u x y $.  $d T a b c p q r s t u $.
      poxp2.1 $e |- ( ph -> R Po A ) $.
      poxp2.2 $e |- ( ph -> S Po B ) $.
      $( Another way of partially ordering a Cartesian product of two classes.
         (Contributed by Scott Fenton, 19-Aug-2024.) $)
      poxp2 $p |- ( ph -> T Po ( A X. B ) ) $=
        ( vp vq vr vs vt wbr wrex wa wo va vb vc vu cxp cv wcel wn cop wceq weq
        elxp2 wne w3a equid pm3.2i neorian con2bii simp3 mto xpord2lem mtbir wb
        mpbi breq12 anidms mtbiri rexlimivw sylbi adantl 3reeanv rexbii 2rexbii
        wi 3anbi123i 3bitr4ri df-3an simpr1l adantr simpr2r jca simpr2l simpr3r
        wpo ad2antrr simpr1r potr syl13anc orc syl6 breq1 biimtrrdi a1i simprl1
        breq2 equequ2 orbi12d syl5ibcom simprr1 simpr3l simprl2 simprr2 simprr3
        expd mpjaod sylib neeq1 3anbi123d anbi1d anbi2d orcom ordi equcom bitri
        equequ1 orbi2i anbi12i sylanbrc po2nr syl12anc orcnd biimtrdi biimtrrid
        3bitri mt3d 3jca syl2ani 3adant3 bitrdi 3adant1 anbi12d 3adant2 imbi12d
        com12 ex syl5ibrcom sylan2br anassrs rexlimdvva imp sylan2b ispod ) AUA
        UBUCDEUEZHUAUFZUUCUGZUUDUUDHQZUHZAUUEUUDLUFZMUFZUIZUJZMERZLDRZUUGLMUUDD
        EULZUULUUGLDUUKUUGMEUUKUUFUUJUUJHQZUUOUUHDUGZUUIEUGZSZUURUUHUUHFQLLUKZT
        ZUUIUUIGQMMUKZTZUUHUUHUMUUIUUIUMTZUNZUNZUVEUVDUVDUVCUUSUVASZUVCUHUUSUVA
        LUOMUOUPUVCUVFUUHUUHUUIUUIUQURVDUUTUVBUVCUSUTUURUURUVDUSUTBCDEFGHLMLMIV
        AVBUUKUUFUUOVCUUDUUJUUDUUJHVEVFVGVHVHVIVJUUEUBUFZUUCUGZUCUFZUUCUGZUNZAU
        UKUVGNUFZOUFZUIZUJZUVIPUFZUDUFZUIZUJZUNZUDEROERZMERZPDRZNDRLDRZUUDUVGHQ
        ZUVGUVIHQZSZUUDUVIHQZVNZUULUVOOERZUVSUDERZUNZPDRZNDRLDRUUMUWJNDRZUWKPDR
        ZUNUWDUVKUULUWJUWKLNPDDDVKUWCUWMLNDDUWBUWLPDUUKUVOUVSMOUDEEEVKVLVMUUEUU
        MUVHUWNUVJUWOUUNNOUVGDEULPUDUVIDEULVOVPAUWDUWIAUWCUWILNDDAUUPUVLDUGZSZS
        UWAUWIPMDEAUWQUVPDUGZUUQSZUWAUWIVNAUWQUWSSZSUVTUWIOUDEEAUWTUVMEUGZUVQEU
        GZSZUVTUWIVNZUWTUXCSAUWQUWSUXCUNZUXDUWQUWSUXCVQAUXESZUWIUVTUURUWPUXASZU
        UHUVLFQZLNUKZTZUUIUVMGQZMOUKZTZUUHUVLUMZUUIUVMUMZTZUNZUNZUXGUWRUXBSZUVL
        UVPFQZNPUKZTZUVMUVQGQZOUDUKZTZUVLUVPUMUVMUVQUMTZUNZUNZSZUURUXSUUHUVPFQZ
        LPUKZTZUUIUVQGQZMUDUKZTZUUHUVPUMUUIUVQUMTZUNZUNZVNUXRUXFUXQUYGUYRUYHUUR
        UXGUXQUSUXGUXSUYGUSUXFUXQUYGSZUYRUXFUYSSZUURUXSUYQUYTUUPUUQUXFUUPUYSUUP
        UWPUWSUXCAVRVSZUXFUUQUYSUWRUUQUWQUXCAVTVSZWAUYTUWRUXBUXFUWRUYSUWRUUQUWQ
        UXCAWBZVSZUXFUXBUYSUXAUXBUWQUWSAWCZVSZWAUYTUYLUYOUYPUYTUXTUYLUYAUYTUXHU
        XTUYLVNZUXIUYTUXHUXTUYLUYTUXHUXTSZUYJUYLUYTDFWDZUUPUWPUWRVUHUYJVNAVUIUX
        EUYSJWEVUAUXFUWPUYSUUPUWPUWSUXCAWFZVSVUDDUUHUVLUVPFWGWHUYJUYKWIZWJXDUXI
        VUGVNUYTUXIUXTUYJUYLUUHUVLUVPFWKVUKWLWMUXJUXMUXPUYGUXFWNZXEUYTUXJUYAUYL
        VULUYAUXHUYJUXIUYKUVLUVPUUHFWONPLWPWQWRUYBUYEUYFUXQUXFWSXEUYTUYCUYOUYDU
        YTUXKUYCUYOVNZUXLUYTUXKUYCUYOUYTUXKUYCSZUYMUYOUYTEGWDZUUQUXAUXBVUNUYMVN
        AVUOUXEUYSKWEVUBUXFUXAUYSUXAUXBUWQUWSAWTZVSVUFEUUIUVMUVQGWGWHUYMUYNWIZW
        JXDUXLVUMVNUYTUXLUYCUYMUYOUUIUVMUVQGWKVUQWLWMUXJUXMUXPUYGUXFXAZXEUYTUXM
        UYDUYOVURUYDUXKUYMUXLUYNUVMUVQUUIGWOOUDMWPWQWRUYBUYEUYFUXQUXFXBXEUYTUYP
        UYAUYDSZUYTUYFVUSUHUYBUYEUYFUXQUXFXCUVLUVPUVMUVQUQXFUYPUHUYKUYNSZUYTVUS
        UYPVUTUUHUVPUUIUVQUQURVUTUYTVUSVUTUYTUXFUVPUVLFQZPNUKZTZUVQUVMGQZUDOUKZ
        TZUVPUVLUMZUVQUVMUMZTZUNZUYGSZSZVUSVUTUYSVVKUXFVUTUXQVVJUYGVUTUXJVVCUXM
        VVFUXPVVIUYKUXJVVCVCUYNUYKUXHVVAUXIVVBUUHUVPUVLFWKLPNXOWQVSUYNUXMVVFVCU
        YKUYNUXKVVDUXLVVEUUIUVQUVMGWKMUDOXOWQVJVUTUXNVVGUXOVVHUYKUXNVVGVCUYNUUH
        UVPUVLXGVSUYNUXOVVHVCUYKUUIUVQUVMXGVJWQXHXIXJVVLUYAUYDVVLVVAUXTSZUYAVVL
        VVCUYBVVMUYATZVVCVVFVVIUYGUXFWNUYBUYEUYFVVJUXFWSVVNUYAVVMTUYAVVATZUYAUX
        TTZSVVCUYBSVVMUYAXKUYAVVAUXTXLVVOVVCVVPUYBVVOVVAUYATVVCUYAVVAXKUYAVVBVV
        ANPXMXPXNUYAUXTXKXQYDXRVVLVUIUWRUWPVVMUHAVUIUXEVVKJWEUXFUWRVVKVUCVSUXFU
        WPVVKVUJVSDUVPUVLFXSXTYAVVLVVDUYCSZUYDVVLVVFUYEVVQUYDTZVVCVVFVVIUYGUXFX
        AUYBUYEUYFVVJUXFXBVVRUYDVVQTUYDVVDTZUYDUYCTZSVVFUYESVVQUYDXKUYDVVDUYCXL
        VVSVVFVVTUYEVVSVVDUYDTVVFUYDVVDXKUYDVVEVVDOUDXMXPXNUYDUYCXKXQYDXRVVLVUO
        UXBUXAVVQUHAVUOUXEVVKKWEUXFUXBVVKVUEVSUXFUXAVVKVUPVSEUVQUVMGXSXTYAWAYBY
        NYCYEYFYFYOYGUVTUWGUYIUWHUYRUVTUWEUXRUWFUYHUVTUWEUUJUVNHQZUXRUUKUVOUWEV
        WAVCUVSUUDUUJUVGUVNHVEYHBCDEFGHLMNOIVAYIUVTUWFUVNUVRHQZUYHUVOUVSUWFVWBV
        CUUKUVGUVNUVIUVRHVEYJBCDEFGHNOPUDIVAYIYKUVTUWHUUJUVRHQZUYRUUKUVSUWHVWCV
        CUVOUUDUUJUVIUVRHVEYLBCDEFGHLMPUDIVAYIYMYPYQYRYSYRYSYSYTUUAUUB $.
    $}

    ${
      $d A a b c d e f p q s x y $.  $d B a b c d e f p q s x y $.
      $d R a b c d e f p q s x y $.  $d S a b c d e f p q s x y $.
      $d T a b c d e f p q s $.  $d ph a b c d e f p q s $.
      frxp2.1 $e |- ( ph -> R Fr A ) $.
      frxp2.2 $e |- ( ph -> S Fr B ) $.
      $( Another way of giving a well-founded order to a Cartesian product of
         two classes.  (Contributed by Scott Fenton, 19-Aug-2024.) $)
      frxp2 $p |- ( ph -> T Fr ( A X. B ) ) $=
        ( va vc ve cv c0 wa wn wi wcel vs vq vp vb vd cxp wss wne wbr wral wrex
        vf wal wfr cdm dmss ad2antrl dmxpss sstrdi simprr wrel wceq relxp relss
        wb mpi reldm0 syl necon3bid mpbid adantr df-fr sylib dmex sseq1 anbi12d
        vex neeq1 raleq rexeqbi1dv imbi12d spcv mp2and csn cima crn rnss rnxpss
        imassrn sstrid simprl cin imadisj bitri necon2abii ad2antrr imaex breq2
        disjsn cop notbid ralbidv elimasn wel wex elrel sylan weq breq1 simplrr
        wo biimpri adantl rspcdva intnanrd opeq1 eleq1d anbi2d 3anass biantrurd
        w3a olc orbi1d neirr biorfi bitr4di andir nonconne bitr4i bitrdi bitr3d
        biorfri bitrid mpbiri impcom opeldm simpr neneqd ioran rexlimddv mpd ex
        sylanbrc intn3an1d pm2.61dane intn3an3d eleq1 xpord2lem com12 ralrimiva
        exlimdvv rspcedvdw alrimiv sylibr ) AUAOZDEUFZUGZUUOPUHZQZUBOZUCOZHUIZR
        ZUBUUOUJZUCUUOUKZSZUAUMUUPHUNAUVFUAAUUSUVEAUUSQZUDOZLOZFUIZRZUDUUOUOZUJ
        ZUVELUVLUVGUVLDUGZUVLPUHZUVMLUVLUKZUVGUVLUUPUOZDUUQUVLUVQUGAUURUUOUUPUP
        UQDEURUSUVGUURUVOAUUQUURUTUVGUUOPUVLPUVGUUOVAZUUOPVBUVLPVBVEUUQUVRAUURU
        UQUUPVAUVRDEVCUUOUUPVDVFUQZUUOVGVHVIVJUVGMOZDUGZUVTPUHZQZUVKUDUVTUJZLUV
        TUKZSZMUMZUVNUVOQZUVPSZUVGDFUNZUWGAUWJUUSJVKMLUDDFVLVMUWFUWIMUVLUUOUAVQ
        ZVNUVTUVLVBZUWCUWHUWEUVPUWLUWAUVNUWBUVOUVTUVLDVOUVTUVLPVRVPUWDUVMLUVTUV
        LUVKUDUVTUVLVSVTWAWBVHWCUVGUVIUVLTZUVMQZQZUEOZUVTGUIZRZUEUUOUVIWDZWEZUJ
        ZUVEMUWTUWOUWTEUGZUWTPUHZUXAMUWTUKZUWOUWTUUOWFZEUUOUWSWIUWOUXEUUPWFZEUV
        GUXEUXFUGZUWNUUQUXGAUURUUOUUPWGUQVKDEWHUSWJUWOUWMUXCUVGUWMUVMWKUWMUWTPU
        WTPVBUVLUWSWLPVBUWMRUUOUWSWMUVLUVIWSWNWOVMUWONOZEUGZUXHPUHZQZUWRUEUXHUJ
        ZMUXHUKZSZNUMZUXBUXCQZUXDSZAUXOUUSUWNAEGUNUXOKNMUEEGVLVMWPUXNUXQNUWTUUO
        UWSUWKWQUXHUWTVBZUXKUXPUXMUXDUXRUXIUXBUXJUXCUXHUWTEVOUXHUWTPVRVPUXLUXAM
        UXHUWTUWRUEUXHUWTVSVTWAWBVHWCUWOUVTUWTTZUXAQZQZUVDUUTUVIUVTWTZHUIZRZUBU
        UOUJUCUYBUUOUVAUYBVBZUVCUYDUBUUOUYEUVBUYCUVAUYBUUTHWRXAXBUYAUXSUYBUUOTU
        WOUXSUXAWKUUOUVIUVTLVQZMVQXCVMUYAUYDUBUUOUYAUBUAXDZQZUUTUXHULOZWTZVBZUL
        XENXEZUYDUYAUVRUYGUYLUVGUVRUWNUXTUVSWPNULUUTUUOXFXGUYHUYKUYDNULUYKUYHUY
        DUYKUYHUYDSUYAUYJUUOTZQZUXHDTUYIETQZUVIDTUVTETQZUXHUVIFUIZNLXHZXKZUYIUV
        TGUIZULMXHZXKZUXHUVIUHZUYIUVTUHZXKZYAZYAZRZSUYNVUFUYOUYPUYNVUFRZUXHUVIU
        YRUYNVUIUYRUYNVUISUYAUVIUYIWTZUUOTZQZUYTVUDQZRZSVULUYTVUDVULUWRUYTRUEUW
        TUYIUEULXHUWQUYTUWPUYIUVTGXIXAUWOUXSUXAVUKXJVUKUYIUWTTZUYAVUOVUKUUOUVIU
        YIUYFULVQZXCXLXMXNXOUYRUYNVULVUIVUNUYRUYMVUKUYAUYRUYJVUJUUOUXHUVIUYIXPX
        QXRUYRVUFVUMVUFUYSVUBVUEQZQZUYRVUMUYSVUBVUEXSUYRVUQVURVUMUYRUYSVUQUYRUY
        QYBXTUYRVUQVUBVUDQZVUMUYRVUEVUDVUBUYRVUEUVIUVIUHZVUDXKVUDUYRVUCVUTVUDUX
        HUVIUVIVRYCVUTVUDUVIYDYEYFXRVUSVUMVUAVUDQZXKVUMUYTVUAVUDYGVVAVUMUYIUVTY
        HYLYIYJYKYMXAWAYNYOUYNVUCQZUYSVUBVUEVVBUYQRZUYRRUYSRVVBUVKVVCUDUVLUXHUD
        NXHUVJUYQUVHUXHUVIFXIXAUYAUVMUYMVUCUVGUWMUVMUXTXJWPUYNUXHUVLTZVUCUYMVVD
        UYAUXHUYIUUONVQVUPYPXMVKXNVVBUXHUVIUYNVUCYQYRUYQUYRYSUUCUUDUUEUUFUYKUYH
        UYNUYDVUHUYKUYGUYMUYAUUTUYJUUOUUGXRUYKUYCVUGUYKUYCUYJUYBHUIVUGUUTUYJUYB
        HXIBCDEFGHNULLMIUUHYJXAWAYNUUIUUKUUAUUJUULYTYTUUBUUMUAUCUBUUPHVLUUN $.
    $}

    ${
      $d A a b c d e x y $.  $d B a b c d e x y $.  $d R a b c d e x y $.
      $d S a b c d e x y $.  $d T a b c d e $.  $d X a b $.  $d Y b $.
      $( Calculate the predecessor class in ~ frxp2 .  (Contributed by Scott
         Fenton, 22-Aug-2024.) $)
      xpord2pred $p |- ( ( X e. A /\ Y e. B ) ->
      Pred ( T , ( A X. B ) , <. X , Y >. ) =
      ( ( ( Pred ( R , A , X ) u. { X } ) X. ( Pred ( S , B , Y ) u. { Y } ) )
      \ { <. X , Y >. } ) ) $=
        ( va vb vc vd cpred csn wceq wcel wa wo ve cxp cv cop cun opeq1 predeq3
        cdif syl sneq uneq12d xpeq1d sneqd difeq12d eqeq12d opeq2 xpeq2d predel
        wi a1i eldifi predss snssi unssd xpss12 syl2an sseld syl5 wrex wb elxp2
        wss wbr cvv elpred ax-mp opelxpi adantl biantrurd weq wne w3a xpord2lem
        opex eldif opelxp elun vex elv velsn orbi12i bitri anbi12i notbii df-ne
        elsn opthne 3bitr2i simprl orbi1d simprr 3anbi12d df-3an bitr2di bitrid
        wn pm3.22 bitr4di bitr2d bitr3d bibi12d syl5ibrcom rexlimdvva pm5.21ndd
        eleq1 biimtrid eqrdv vtocl2ga ) CDUBZGKUCZLUCZUDZOZCEXTOZXTPZUEZDFYAOZY
        APZUEZUBZYBPZUHZQXSGHYAUDZOZCEHOZHPZUEZYIUBZYMPZUHZQXSGHIUDZOZYQDFIOZIP
        ZUEZUBZUUAPZUHZQKLHICDXTHQZYCYNYLYTUUIYBYMQYCYNQXTHYAUFZXSGYBYMUGUIUUIY
        JYRYKYSUUIYFYQYIUUIYDYOYEYPCEXTHUGXTHUJUKULUUIYBYMUUJUMUNUOYAIQZYNUUBYT
        UUHUUKYMUUAQYNUUBQYAIHUPZXSGYMUUAUGUIUUKYRUUFYSUUGUUKYIUUEYQUUKYGUUCYHU
        UDDFYAIUGYAIUJUKUQUUKYMUUAUULUMUNUOXTCRZYADRZSZUAYCYLUUOUAUCZXSRZUUPYCR
        ZUUPYLRZUURUUQUSUUOXSGYBUUPURUTUUSUUPYJRUUOUUQUUPYJYKVAUUOYJXSUUPUUMYFC
        VLYIDVLYJXSVLUUNUUMYDYECYDCVLUUMCEXTVBUTXTCVCVDUUNYGYHDYGDVLUUNDFYAVBUT
        YADVCVDYFCYIDVEVFVGVHUUQUUPMUCZNUCZUDZQZNDVIMCVIUUOUURUUSVJZMNUUPCDVKUU
        OUVCUVDMNCDUUOUUTCRZUVADRZSZSZUVDUVCUVBYCRZUVBYLRZVJUVIUVBXSRZUVBYBGVMZ
        SZUVHUVJYBVNRUVIUVMVJXTYAWDXSVNGYBUVBUUTUVAWDZVOVPUVHUVLUVMUVJUVHUVKUVL
        UVGUVKUUOUUTUVACDVQVRVSUVLUVGUUOUUTXTEVMZMKVTZTZUVAYAFVMZNLVTZTZUUTXTWA
        UVAYAWATZWBZWBZUVHUVJABCDEFGMNKLJWCUVHUVJUWBUWCUVJUVEUVOSZUVPTZUVFUVRSZ
        UVSTZSZUWASZUVHUWBUVJUVBYJRZUVBYKRZXFZSUWIUVBYJYKWEUWJUWHUWLUWAUWJUUTYF
        RZUVAYIRZSUWHUUTUVAYFYIWFUWMUWEUWNUWGUWMUUTYDRZUUTYERZTUWEUUTYDYEWGUWOU
        WDUWPUVPUWOUWDVJKCVNEXTUUTMWHZVOWIMXTWJWKWLUWNUVAYGRZUVAYHRZTUWGUVAYGYH
        WGUWRUWFUWSUVSUWRUWFVJLDVNFYAUVANWHZVOWINYAWJWKWLWMWLUWLUVBYBQZXFUVBYBW
        AUWAUWKUXAUVBYBUVNWPWNUVBYBWOUUTUVAXTYAUWQUWTWQWRWMWLUVHUWBUWEUWGUWAWBU
        WIUVHUVQUWEUVTUWGUWAUVHUVOUWDUVPUVHUVEUVOUUOUVEUVFWSVSWTUVHUVRUWFUVSUVH
        UVFUVRUUOUVEUVFXAVSWTXBUWEUWGUWAXCXDXEUVHUWBUVGUUOSZUWBSUWCUVHUXBUWBUUO
        UVGXGVSUVGUUOUWBXCXHXIXEXJXEUVCUURUVIUUSUVJUUPUVBYCXOUUPUVBYLXOXKXLXMXP
        XNXQXR $.
    $}

    ${
      $d A a b p x y $.  $d B a b p x y $.  $d R a b p x y $.  $d ph a b p $.
      $d S a b p x y $.  $d T a b p $.
      sexp2.1 $e |- ( ph -> R Se A ) $.
      sexp2.2 $e |- ( ph -> S Se B ) $.
      $( Condition for the relation in ~ frxp2 to be set-like.  (Contributed by
         Scott Fenton, 19-Aug-2024.) $)
      sexp2 $p |- ( ph -> T Se ( A X. B ) ) $=
        ( vp va vb cv cpred cvv wcel wse csn cxp wral cop wceq wrex wa cun cdif
        elxp2 xpord2pred adantl setlikespec sylan adantrr vsnex a1i unexd xpexd
        ancoms adantrl difexd eqeltrd eleq1d syl5ibrcom biimtrid ralrimiv dfse3
        predeq3 rexlimdvva sylibr ) ADEUAZHLOZPZQRZLVKUBVKHSAVNLVKVLVKRVLMOZNOZ
        UCZUDZNEUEMDUEAVNMNVLDEUIAVRVNMNDEAVODRZVPERZUFZUFZVNVRVKHVQPZQRWBWCDFV
        OPZVOTZUGZEGVPPZVPTZUGZUAZVQTZUHZQWAWCWLUDABCDEFGHVOVPIUJUKWBWJWKQWBWFW
        IQQWBWDWEQQAVSWDQRZVTADFSZVSWMJVSWNWMDFVOULUSUMUNWEQRWBMUOUPUQWBWGWHQQA
        VTWGQRZVSAEGSZVTWOKVTWPWOEGVPULUSUMUTWHQRWBNUOUPUQURVAVBVRVMWCQVKHVLVQV
        HVCVDVIVEVFLVKHVGVJ $.
    $}

    ${
      $d A a b c d $.  $d ps a $.  $d ta a $.  $d A x y $.  $d B a b c d $.
      $d ch b $.  $d et b $.  $d B x y $.  $d ph c $.  $d th c $.  $d ps d $.
      $d R c d $.  $d R x y $.  $d S c d $.  $d S x y $.  $d T a b c d $.
      $d X a b $.  $d Y b $.
      xpord2indlem.1 $e |- R Fr A $.
      xpord2indlem.2 $e |- R Po A $.
      xpord2indlem.3 $e |- R Se A $.
      xpord2indlem.4 $e |- S Fr B $.
      xpord2indlem.5 $e |- S Po B $.
      xpord2indlem.6 $e |- S Se B $.
      xpord2indlem.7 $e |- ( a = c -> ( ph <-> ps ) ) $.
      xpord2indlem.8 $e |- ( b = d -> ( ps <-> ch ) ) $.
      xpord2indlem.9 $e |- ( a = c -> ( th <-> ch ) ) $.
      xpord2indlem.11 $e |- ( a = X -> ( ph <-> ta ) ) $.
      xpord2indlem.12 $e |- ( b = Y -> ( ta <-> et ) ) $.
      xpord2indlem.i $e |- ( ( a e. A /\ b e. B ) ->
       ( ( A. c e. Pred ( R , A , a ) A. d e. Pred ( S , B , b ) ch /\
           A. c e. Pred ( R , A , a ) ps /\
           A. d e. Pred ( S , B , b ) th ) -> ph ) ) $.
      $( Induction over the Cartesian product ordering.  Note that the
         substitutions cover all possible cases of membership in the
         predecessor class.  (Contributed by Scott Fenton, 22-Aug-2024.) $)
      xpord2indlem $p |- ( ( X e. A /\ Y e. B ) -> et ) $=
        ( cxp wfr wpo wse w3a wcel wa wtru a1i frxp2 poxp2 sexp2 3jca mptru cop
        cv cpred wi wal wne wo csn cun wral cdif xpord2pred eleq2d imbi1d eldif
        wn opelxp wceq opex elsn notbii df-ne vex opthne 3bitr2i anbi12i imbi1i
        bitri impexp bitrdi 2albidv r2al bitr4di wss ssun1 ssralv ax-mp syl weq
        ralimi predpoirr eleq1w mtbiri necon2ai olcd pm2.27 ralimia ssun2 neeq1
        orbi2d wb equcoms bicomd imbi12d ralsn ralbii sylib orcd orbi1d ralbidv
        syl5 sylbid frpoins3xpg mpan ) IJUMZMUNZYKMUOZYKMUPZUQZNIUROJURUSFYOUTY
        LYMYNUTGHIJKLMTIKUNUTUAVAJLUNUTUDVAVBUTGHIJKLMTIKUOZUTUBVAJLUOZUTUEVAVC
        UTGHIJKLMTIKUPUTUCVAJLUPUTUFVAVDVEVFABCEFPQRSIJMNOPVHZIURQVHZJURUSZRVHZ
        SVHZVGZYKMYRYSVGZVIZURZCVJZSVKRVKZUUAYRVLZUUBYSVLZVMZCVJZSJLYSVIZYSVNZV
        OZVPZRIKYRVIZYRVNZVOZVPZAYTUUHUUAUUSURUUBUUOURUSZUULVJZSVKRVKUUTYTUUGUV
        BRSYTUUGUUCUUSUUOUMZUUDVNZVQZURZCVJZUVBYTUUFUVFCYTUUEUVEUUCGHIJKLMYRYST
        VRVSVTUVGUVAUUKUSZCVJUVBUVFUVHCUVFUUCUVCURZUUCUVDURZWBZUSUVHUUCUVCUVDWA
        UVIUVAUVKUUKUUAUUBUUSUUOWCUVKUUCUUDWDZWBUUCUUDVLUUKUVJUVLUUCUUDUUAUUBWE
        WFWGUUCUUDWHUUAUUBYRYSRWISWIWJWKWLWNWMUVAUUKCWOWNWPWQUULRSUUSUUOWRWSUUT
        CSUUMVPZRUUQVPZBRUUQVPZDSUUMVPZUQYTAUUTUVNUVOUVPUUTUULSUUMVPZRUUQVPZUVN
        UUTUUPRUUQVPZUVRUUQUUSWTUUTUVSVJUUQUURXAUUPRUUQUUSXBXCZUUPUVQRUUQUUMUUO
        WTUUPUVQVJUUMUUNXAUULSUUMUUOXBXCZXFXDUVQUVMRUUQUULCSUUMUUBUUMURZUUKUULC
        VJUWBUUJUUIUWBUUBYSSQXEZUWBYSUUMURZYQUWDWBUEJLYSXGXCSQUUMXHXIXJZXKUUKCX
        LXDXMXFXDUUTUUIYSYSVLZVMZBVJZRUUQVPZUVOUUTUULSUUNVPZRUUQVPZUWIUUTUVSUWK
        UVTUUPUWJRUUQUUNUUOWTUUPUWJVJUUNUUMXNUULSUUNUUOXBXCXFXDUWJUWHRUUQUULUWH
        SYSQWIUWCUUKUWGCBUWCUUJUWFUUIUUBYSYSXOXPUWCBCBCXQQSUHXRXSXTYAYBYCUWHBRU
        UQUUAUUQURZUWGUWHBVJUWLUUIUWFUWLUUAYRRPXEZUWLYRUUQURZYPUWNWBUBIKYRXGXCR
        PUUQXHXIXJYDUWGBXLXDXMXDUUTYRYRVLZUUJVMZDVJZSUUMVPZUVPUUTUVQRUURVPZUWRU
        UTUUPRUURVPZUWSUURUUSWTUUTUWTVJUURUUQXNUUPRUURUUSXBXCUUPUVQRUURUWAXFXDU
        VQUWRRYRPWIUWMUULUWQSUUMUWMUUKUWPCDUWMUUIUWOUUJUUAYRYRXOYEUWMDCDCXQPRUI
        XRXSXTYFYAYCUWQDSUUMUWBUWPUWQDVJUWBUUJUWOUWEXKUWPDXLXDXMXDVEULYGYHUGUHU
        JUKYIYJ $.
    $}

  $}

  ${
    $d A a b c d x y $.  $d ps a $.  $d ta a $.  $d B a b c d $.  $d ch b $.
    $d et b $.  $d B x y $.  $d ph c $.  $d th c $.  $d ps d $.
    $d R a b c d x y $.  $d S a b c d x y $.  $d X a b $.  $d Y b $.
    xpord2ind.1 $e |- R Fr A $.
    xpord2ind.2 $e |- R Po A $.
    xpord2ind.3 $e |- R Se A $.
    xpord2ind.4 $e |- S Fr B $.
    xpord2ind.5 $e |- S Po B $.
    xpord2ind.6 $e |- S Se B $.
    xpord2ind.7 $e |- ( a = c -> ( ph <-> ps ) ) $.
    xpord2ind.8 $e |- ( b = d -> ( ps <-> ch ) ) $.
    xpord2ind.9 $e |- ( a = c -> ( th <-> ch ) ) $.
    xpord2ind.11 $e |- ( a = X -> ( ph <-> ta ) ) $.
    xpord2ind.12 $e |- ( b = Y -> ( ta <-> et ) ) $.
    xpord2ind.i $e |- ( ( a e. A /\ b e. B ) ->
       ( ( A. c e. Pred ( R , A , a ) A. d e. Pred ( S , B , b ) ch /\
           A. c e. Pred ( R , A , a ) ps /\
           A. d e. Pred ( S , B , b ) th ) -> ph ) ) $.
    $( Induction over the Cartesian product ordering.  Note that the
       substitutions cover all possible cases of membership in the predecessor
       class.  (Contributed by Scott Fenton, 22-Aug-2024.) $)
    xpord2ind $p |- ( ( X e. A /\ Y e. B ) -> et ) $=
      ( vx vy cv cxp wcel c1st cfv wbr wceq wo c2nd wne copab eqid xpord2indlem
      w3a ) ABCDEFUIUJGHIJUIUKZGHULZUMUJUKZVFUMVEUNUOZVGUNUOZIUPVHVIUQURVEUSUOZ
      VGUSUOZJUPVJVKUQURVEVGUTVDVDUIUJVAZKLMNOPVLVBQRSTUAUBUCUDUEUFUGUHVC $.

  $}

  ${
    xpord3.1 $e |- U = { <. x , y >. | ( x e. ( ( A X. B ) X. C )
     /\ y e. ( ( A X. B ) X. C )
     /\ ( ( ( ( 1st ` ( 1st ` x ) ) R ( 1st ` ( 1st ` y ) ) \/
              ( 1st ` ( 1st ` x ) ) = ( 1st ` ( 1st ` y ) ) ) /\
            ( ( 2nd ` ( 1st ` x ) ) S ( 2nd ` ( 1st ` y ) ) \/
              ( 2nd ` ( 1st ` x ) ) = ( 2nd ` ( 1st ` y ) ) ) /\
            ( ( 2nd ` x ) T ( 2nd ` y ) \/ ( 2nd ` x ) = ( 2nd ` y ) ) ) /\
          x =/= y ) ) } $.

    ${
      $d a x y $.  $d A x y $.  $d b x y $.  $d B x y $.  $d c x y $.
      $d C x y $.  $d d x y $.  $d e x y $.  $d f x y $.  $d R x y $.
      $d S x y $.  $d T x y $.

      $( Lemma for triple ordering.  Calculate the value of the relation.
         (Contributed by Scott Fenton, 21-Aug-2024.) $)
      xpord3lem $p |- ( <. a , b , c >. U <. d , e , f >. <->
     ( ( a e. A /\ b e. B /\ c e. C ) /\ ( d e. A /\ e e. B /\ f e. C ) /\
     ( ( ( a R d \/ a = d ) /\ ( b S e \/ b = e ) /\ ( c T f \/ c = f ) ) /\
       ( a =/= d \/ b =/= e \/ c =/= f ) ) ) ) $=
        ( wcel c1st cfv cvv cv cotp wbr cxp weq wo w3a wne wceq c2nd otex eleq1
        wa w3o 2fveq3 vex ot1stg mp3an eqtrdi breq1d eqeq1d ot2ndg fveq2 ot3rdg
        orbi12d elv 3anbi123d neeq1 anbi12d 3anbi13d breq2d neeq2 3anbi23d brab
        eqeq2d otelxp otthne anbi2i 3anbi123i bitri ) LUAZMUAZNUAZUBZOUAZJUAZKU
        AZUBZIUCWDCDUDEUDZQZWHWIQZWAWEFUCZLOUEZUFZWBWFGUCZMJUEZUFZWCWGHUCZNKUEZ
        UFZUGZWDWHUHZUMZUGZWACQWBDQWCEQUGZWECQWFDQWGEQUGZXAWAWEUHWBWFUHWCWGUHUN
        ZUMZUGAUAZWIQZBUAZWIQZXIRSZRSZXKRSZRSZFUCZXNXPUIZUFZXMUJSZXOUJSZGUCZXTY
        AUIZUFZXIUJSZXKUJSZHUCZYEYFUIZUFZUGZXIXKUHZUMZUGWJXLWAXPFUCZWAXPUIZUFZW
        BYAGUCZWBYAUIZUFZWCYFHUCZWCYFUIZUFZUGZWDXKUHZUMZUGXDABWDWHIWAWBWCUKWEWF
        WGUKXIWDUIZXJWJYLUUDXLXIWDWIULUUEYJUUBYKUUCUUEXSYOYDYRYIUUAUUEXQYMXRYNU
        UEXNWAXPFUUEXNWDRSZRSZWAXIWDRRUOWATQZWBTQZWCTQZUUGWAUILUPZMUPZNUPZWAWBW
        CTTTUQURUSZUTUUEXNWAXPUUNVAVEUUEYBYPYCYQUUEXTWBYAGUUEXTUUFUJSZWBXIWDUJR
        UOUUHUUIUUJUUOWBUIUUKUULUUMWAWBWCTTTVBURUSZUTUUEXTWBYAUUPVAVEUUEYGYSYHY
        TUUEYEWCYFHUUEYEWDUJSZWCXIWDUJVCUUQWCUINWAWBWCTVDVFUSZUTUUEYEWCYFUURVAV
        EVGXIWDXKVHVIVJXKWHUIZXLWKUUDXCWJXKWHWIULUUSUUBXAUUCXBUUSYOWNYRWQUUAWTU
        USYMWLYNWMUUSXPWEWAFUUSXPWHRSZRSZWEXKWHRRUOWETQZWFTQZWGTQZUVAWEUIOUPZJU
        PZKUPZWEWFWGTTTUQURUSZVKUUSXPWEWAUVHVOVEUUSYPWOYQWPUUSYAWFWBGUUSYAUUTUJ
        SZWFXKWHUJRUOUVBUVCUVDUVIWFUIUVEUVFUVGWEWFWGTTTVBURUSZVKUUSYAWFWBUVJVOV
        EUUSYSWRYTWSUUSYFWGWCHUUSYFWHUJSZWGXKWHUJVCUVKWGUIKWEWFWGTVDVFUSZVKUUSY
        FWGWCUVLVOVEVGXKWHWDVLVIVMPVNWJXEWKXFXCXHWAWBWCCDEVPWEWFWGCDEVPXBXGXAWA
        WBWCWEWFWGUUKUULUUMVQVRVSVT $.
    $}

    ${
      $d A a b c d e f g h i p q r x y $.  $d B a b c d e f g h i p q r x y $.
      $d C a b c d e f g h i p q r x y $.  $d R a b c d e f g h i p q r x y $.
      $d S a b c d e f g h i p q r x y $.  $d T a b c d e f g h i p q r x y $.
      $d U a b c d e f g h i p q r $.  $d ph a b c d e f g h i p q r $.
      poxp3.1 $e |- ( ph -> R Po A ) $.
      poxp3.2 $e |- ( ph -> S Po B ) $.
      poxp3.3 $e |- ( ph -> T Po C ) $.
      $( Triple Cartesian product partial ordering.  (Contributed by Scott
         Fenton, 21-Aug-2024.) $)
      poxp3 $p |- ( ph -> U Po ( ( A X. B ) X. C ) ) $=
        ( va vb wbr wrex w3a wa vp vq vr vc vd ve vf vg vh vi cv wcel cotp wceq
        cxp wn el2xptp weq wo wne w3o neirr 3pm3.2ni intnan simp3 mto wb breq12
        anidms xpord3lem bitrdi mtbiri rexlimivw sylbi adantl wi 3reeanv rexbii
        2rexbii 3anbi123i 3bitr4ri simpr1l simpr2r simp1l1 simp2l1 simp2r1 3jca
        bitri wpo potr syl2an expd biimprd a1i simpll1 3ad2ant3 mpjaod orc syl6
        breq1 equequ2 orbi12d syl5ibcom simprl1 simp1l2 simp2l2 simp2r2 simpll2
        breq2 simprl2 simp1l3 simp2l3 simp2r3 simpll3 simprl3 adantr equcom imp
        simp3rr equequ1 ordir sylanbrc po2nr syl12anc necon3d 3orim123d mpd jca
        orcnd ex 3adant3 3adant1 anbi12d an6 3adant2 imbi12d rexlimdvw biimtrid
        syl5ibrcom ispod ) AUAUBUCDEUOFUOZJUAUKZUUAULZUUBUUBJQZUPZAUUCUUBOUKZPU
        KZUDUKZUMZUNZUDFRZPERZODRZUUEOPUDUUBDEFUQZUULUUEODUUKUUEPEUUJUUEUDFUUJU
        UDUUFDULZUUGEULZUUHFULZSZUURUUFUUFGQOOURUSUUGUUGHQPPURUSUUHUUHIQUDUDURU
        SSZUUFUUFUTZUUGUUGUTZUUHUUHUTZVAZTZSZUVEUVDUVCUUSUUTUVAUVBUUFVBUUGVBUUH
        VBVCVDUURUURUVDVEVFUUJUUDUUIUUIJQZUVEUUJUUDUVFVGUUBUUIUUBUUIJVHVIBCDEFG
        HIJPUDOPUDOKVJVKVLVMVMVMVNVOAUUCUBUKZUUAULZUCUKZUUAULZSZUUBUVGJQZUVGUVI
        JQZTZUUBUVIJQZVPZUVKUUJUVGUEUKZUFUKZUGUKZUMZUNZUVIUHUKZUIUKZUJUKZUMZUNZ
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        ZUUGUWCUTZUUHUWDUTZVAZTZSZVPAVUDVUTAVUDTZUURUXOVUSUURUXJUXPVUCAWBUXJUXO
        UXKVUCAWCVVAVUNVURVVAVUGVUJVUMVVAUYHVUGUYIVVAUYHVUEVUGVVAUXQUYHVUEVPZUX
        RVVAUXQUYHVUEADGWIZUUOUXGUXLSUXQUYHTVUEVPVUDLVUDUUOUXGUXLUUOUUPUUQUXJUX
        PVUCWDUXGUXHUXIUXOUXKVUCWEZUXLUXMUXNUXJUXKVUCWFZWGDUUFUVQUWBGWJWKWLUXRV
        VBVPVVAUXRVUEUYHUUFUVQUWBGWTWMWNVUDUXSAVUCUXKUXSUXPUXSUYBUYEUYFVUBWOWPV
        OZWQVUEVUFWRWSVVAUXSUYIVUGVVFUYIUXQVUEUXRVUFUVQUWBUUFGXIUEUHOXAXBXCVUDU
        YJAVUCUXKUYJUXPUYJUYMUYPVUAUYGXDWPVOZWQVVAUYKVUJUYLVVAUYKVUHVUJVVAUXTUY
        KVUHVPZUYAVVAUXTUYKVUHAEHWIZUUPUXHUXMSUXTUYKTVUHVPVUDMVUDUUPUXHUXMUUOUU
        PUUQUXJUXPVUCXEUXGUXHUXIUXOUXKVUCXFZUXLUXMUXNUXJUXKVUCXGZWGEUUGUVRUWCHW
        JWKWLUYAVVHVPVVAUYAVUHUYKUUGUVRUWCHWTWMWNVUDUYBAVUCUXKUYBUXPUXSUYBUYEUY
        FVUBXHWPVOZWQVUHVUIWRWSVVAUYBUYLVUJVVLUYLUXTVUHUYAVUIUVRUWCUUGHXIUFUIPX
        AXBXCVUDUYMAVUCUXKUYMUXPUYJUYMUYPVUAUYGXJWPVOZWQVVAUYNVUMUYOVVAUYNVUKVU
        MVVAUYCUYNVUKVPZUYDVVAUYCUYNVUKAFIWIZUUQUXIUXNSUYCUYNTVUKVPVUDNVUDUUQUX
        IUXNUUOUUPUUQUXJUXPVUCXKUXGUXHUXIUXOUXKVUCXLZUXLUXMUXNUXJUXKVUCXMZWGFUU
        HUVSUWDIWJWKWLUYDVVNVPVVAUYDVUKUYNUUHUVSUWDIWTWMWNVUDUYEAVUCUXKUYEUXPUX
        SUYBUYEUYFVUBXNWPVOZWQVUKVULWRWSVVAUYEUYOVUMVVRUYOUYCVUKUYDVULUVSUWDUUH
        IXIUGUJUDXAXBXCVUDUYPAVUCUXKUYPUXPUYJUYMUYPVUAUYGXOWPVOZWQWGVVAVUAVURVU
        DVUAAUYQVUAUYGUXKUXPXSVOVVAUYRVUOUYSVUPUYTVUQVVAUUFUWBUVQUWBVVAVUFUYIVV
        AVUFTZUYHUWBUVQGQZTZUYIVVTUYJVWAUYIUSZVWBUYIUSVVAUYJVUFVVGXPVVAVUFVWCVV
        AUXSVUFVWCVVFVUFUXQVWAUXRUYIUUFUWBUVQGWTVUFUXRUHUEURUYIOUHUEXTUHUEXQVKX
        BXCXRUYHVWAUYIYAYBVVAVWBUPZVUFVVAVVCUXGUXLVWDAVVCVUDLXPVUDUXGAVVDVOVUDU
        XLAVVEVODUVQUWBGYCYDXPYIYJYEVVAUUGUWCUVRUWCVVAVUIUYLVVAVUITZUYKUWCUVRHQ
        ZTZUYLVWEUYMVWFUYLUSZVWGUYLUSVVAUYMVUIVVMXPVVAVUIVWHVVAUYBVUIVWHVVLVUIU
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        CYDXPYIYJYEVVAUUHUWDUVSUWDVVAVULUYOVVAVULTZUYNUWDUVSIQZTZUYOVWJUYPVWKUY
        OUSZVWLUYOUSVVAUYPVULVVSXPVVAVULVWMVVAUYEVULVWMVVRVULUYCVWKUYDUYOUUHUWD
        UVSIWTVULUYDUJUGURUYOUDUJUGXTUJUGXQVKXBXCXRUYNVWKUYOYAYBVVAVWLUPZVULVVA
        VVOUXIUXNVWNAVVOVUDNXPVUDUXIAVVPVOVUDUXNAVVQVOFUVSUWDIYCYDXPYIYJYEYFYGY
        HWGYJUWGUVNVUDUVOVUTUWGUVNUURUXJUYGSZUXJUXOVUBSZTVUDUWGUVLVWOUVMVWPUWGU
        VLUUIUVTJQZVWOUUJUWAUVLVWQVGUWFUUBUUIUVGUVTJVHYKBCDEFGHIJUFUGOPUDUEKVJV
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        GUHKVJVKYMUURUXJUYGUXJUXOVUBYNVKUWGUVOUUIUWEJQZVUTUUJUWFUVOVWSVGUWAUUBU
        UIUVIUWEJVHYOBCDEFGHIJUIUJOPUDUHKVJVKYPYSYQYQYQYQYQYQYQYQYQYRXRYT $.
    $}

    ${
      $d A a b c d e f g h i p q s x y $.  $d B a b c d e f g h i p q s x y $.
      $d C a b c d e f g h i p q s x y $.  $d R a b c d e f g h i p q s x y $.
      $d S a b c d e f g h i p q s x y $.  $d T a b c d e f g h i p q s x y $.
      $d ph a b c d e f g h i p q s $.  $d U a b c g h i p q s $.
      frxp3.1 $e |- ( ph -> R Fr A ) $.
      frxp3.2 $e |- ( ph -> S Fr B ) $.
      frxp3.3 $e |- ( ph -> T Fr C ) $.
      $( Give well-foundedness over a triple Cartesian product.  (Contributed
         by Scott Fenton, 21-Aug-2024.) $)
      frxp3 $p |- ( ph -> U Fr ( ( A X. B ) X. C ) ) $=
        ( cv wss c0 wa wn wcel vs vq vp vd va ve vb vf vc vg vh vi cxp wne wral
        wbr wrex wi wal wfr cdm cvv adantr dmss ad2antrl dmxpss sstrdi syl dmex
        vex a1i wceq wrel wb relxp relss mpi adantl reldm0 mp2 mpisyl bitrd frd
        csn cima ad2antrr crn imassrn rnss rnxpss sstrid imaex cin disjsn bitri
        imadisj necon2abii biimpi cop ad3antrrr simprl elimasn sylib cotp df-ot
        wex w3a weq wo w3o breq1 notbid simplrr simplr eqeltrrid sylibr rspcdva
        simpr neneqd ioran sylanbrc intn3an3d intnanrd pm2.61dane eleq1d anbi2d
        opex neeq1 neirr orel1 ax-mp impbii bitrdi imbi12d mpbiri impcom opeldm
        olc biidd rexlimddv necon3bid biimpa anasss eqeltrid wel simplrl ancoms
        el2xpss sylan df-ne con2bii intnand oteq2 orbi1d intn3an2d oteq1 3orass
        3orbi123d 3syl intn3an1d eleq1 xpord3lem exlimdv exlimdvv mpd ralrimiva
        com12 breq2 ralbidv rspcev syl2anc ex alrimiv df-fr ) AUAOZDEUMZFUMZPZU
        VOQUNZRZUBOZUCOZJUPZSZUBUVOUOZUCUVOUQZURZUAUSUVQJUTAUWGUAAUVTUWFAUVTRZU
        DOZUEOZGUPZSZUDUVOVAZVAZUOZUWFUEUWNUWHUEUDDUWNGVBADGUTUVTLVCUWHUWNUVPVA
        ZDUWHUWMUVPPZUWNUWPPUWHUWMUVQVAZUVPUVRUWMUWRPZAUVSUVOUVQVDZVEUVPFVFZVGZ
        UWMUVPVDVHDEVFVGUWNVBTUWHUWMUVOUAVJZVIZVIVKAUVRUVSUWNQUNZAUVRRZUVSUXEUX
        FUVOQUWNQUXFUVOQVLZUWMQVLZUWNQVLZUXFUVOVMZUXGUXHVNUVRUXJAUVRUVQVMUXJUVP
        FVOUVOUVQVPVQVRUVOVSVHUXFUWMVMZUXHUXIVNUVRUXKAUVRUWSUWRVMZUXKUWTUWRUVPP
        UVPVMUXLUXADEVOUWRUVPVPVTUWMUWRVPWAVRUWMVSVHWBUUAUUBUUCWCUWHUWJUWNTZUWO
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        DEWJVGWKVCUYAVBTUXOUWMUXTUXDWLVKUXMUYAQUNZUWHUWOUXMUYEUXMUYAQUYAQVLUWNU
        XTWMQVLUXMSUWMUXTWPUWNUWJWNWOWQWRVEWCUXOUXQUYATZUYBRZRZUHOZUIOZIUPZSZUH
        UVOUWJUXQWSZWDZWEZUOZUWFUIUYOUYHUIUHFUYOIVBAFIUTUVTUXNUYGNWTUWHUYOFPUXN
        UYGUWHUYOUVOWGZFUVOUYNWHUWHUYQUVQWGZFUVRUYQUYRPAUVSUVOUVQWIVEUVPFWJVGWK
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        YOTZUYPRZRZUWJUXQUYJXDZUVOTUWAVUDJUPZSZUBUVOUOZUWFVUCVUDUYMUYJWSZUVOUWJ
        UXQUYJXEVUCVUAVUHUVOTUYHVUAUYPXAUVOUYMUYJUWJUXQYGZUIVJXBXCUUDVUCVUFUBUV
        OVUCUBUAUUEZRZUWAUJOZUKOZULOZXDZVLZULXFZUKXFUJXFZVUFVUCUVRVUJVURUXOUVRU
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        UKVUPVUFULVUPVUKVUFVUPVUKVUFURVUCVUOUVOTZRZVULDTVUMETVUNFTXGZUWJDTUXQET
        UYJFTXGZVULUWJGUPZUJUEXHZXIZVUMUXQHUPZUKUGXHZXIZVUNUYJIUPZULUIXHZXIZXGZ
        VULUWJUNZVUMUXQUNZVUNUYJUNZXJZRZXGZSZURVUTVVQVVAVVBVUTVVQSZVULUWJVVDVUT
        VVTVVDVUTVVTURVUCUWJVUMVUNXDZUVOTZRZVVLVVNVVOXIZRZSZURZVWCVWFVUMUXQVVGV
        WCVWFVVGVWGVUCUWJUXQVUNXDZUVOTZRZVVLVVORZSZURVWJVWLVUNUYJVVJVWLVWJVVJVV
        OVVLVVJVVOSVVOVVJVUNUYJUUJUUKWRUULVRVWJVVORZVVLVVOVWMVVKVVEVVHVWMVVISZV
        VJSVVKSVWMUYLVWNUHUYOVUNUHULXHUYKVVIUYIVUNUYJIXKXLVWJUYPVVOUYHVUAUYPVWI
        XMVCVWMUYMVUNWSZUVOTVUNUYOTVWMVWOVWHUVOUWJUXQVUNXEVUCVWIVVOXNXOUVOUYMVU
        NVUIULVJZXBXPXQVWMVUNUYJVWJVVOXRXSVVIVVJXTYAYBYCYDVVGVWCVWJVWFVWLVVGVWB
        VWIVUCVVGVWAVWHUVOVUMUXQUWJVUNUUMYEYFVVGVWEVWKVVGVWDVVOVVLVVGVWDUXQUXQU
        NZVVOXIZVVOVVGVVNVWQVVOVUMUXQUXQYHUUNVWRVVOVWQSVWRVVOURUXQYIVWQVVOYJYKV
        VOVWQYRYLYMYFXLYNYOYPVWCVVNRZVVLVWDVWSVVHVVEVVKVWSVVFSZVVGSVVHSVWSUXSVW
        TUFUYAVUMUFUKXHUXRVVFUXPVUMUXQHXKXLVUCUYBVWBVVNUXOUYFUYBVUBXMWFVWSUWJVU
        MWSZUWMTZVUMUYATVWSVXAVUNWSZUVOTVXBVWSVXCVWAUVOUWJVUMVUNXEVUCVWBVVNXNXO
        VXAVUNUVOUWJVUMYGVWPYQVHUWMUWJVUMUYTUKVJZXBXPXQVWSVUMUXQVWCVVNXRXSVVFVV
        GXTYAUUOYCYDVVDVUTVWCVVTVWFVVDVUSVWBVUCVVDVUOVWAUVOVULUWJVUMVUNUUPYEYFV
        VDVVQVWEVVDVVPVWDVVLVVDVVPUWJUWJUNZVVNVVOXJZVWDVVDVVMVXEVVNVVNVVOVVOVUL
        UWJUWJYHVVDVVNYSVVDVVOYSUURVXFVXEVWDXIZVWDVXEVVNVVOUUQVXGVWDVXESVXGVWDU
        RUWJYIVXEVWDYJYKVWDVXEYRYLWOYMYFXLYNYOYPVUTVVMRZVVLVVPVXHVVEVVHVVKVXHVV
        CSZVVDSVVESVXHUWLVXIUDUWNVULUDUJXHUWKVVCUWIVULUWJGXKXLUYHUWOVUBVUSVVMUW
        HUXMUWOUYGXMWTVXHVULVUMWSZVUNWSZUVOTVXJUWMTVULUWNTVXHVXKVUOUVOVULVUMVUN
        XEVUCVUSVVMXNXOVXJVUNUVOVULVUMYGVWPYQVULVUMUWMUJVJVXDYQUUSXQVXHVULUWJVU
        TVVMXRXSVVCVVDXTYAUUTYCYDYBVUPVUKVUTVUFVVSVUPVUJVUSVUCUWAVUOUVOUVAYFVUP
        VUEVVRVUPVUEVUOVUDJUPVVRUWAVUOVUDJXKBCDEFGHIJUGUIUJUKULUEKUVBYMXLYNYOUV
        GUVCUVDUVEUVFUWEVUGUCVUDUVOUWBVUDVLZUWDVUFUBUVOVXLUWCVUEUWBVUDUWAJUVHXL
        UVIUVJUVKYTYTYTUVLUVMUAUCUBUVQJUVNXP $.
    $}

    ${
      $d A a b c d e f q x y $.  $d B a b c d e f q x y $.
      $d C a b c d e f q x y $.  $d R a b c d e f q x y $.
      $d S a b c d e f q x y $.  $d T a b c d e f q x y $.
      $d U a b c d e f q $.  $d X a b c $.  $d Y b c $.  $d Z c $.
      $( Calculate the predecsessor class for the triple order.  (Contributed
         by Scott Fenton, 31-Jan-2025.) $)
      xpord3pred $p |- ( ( X e. A /\ Y e. B /\ Z e. C ) ->
       Pred ( U , ( ( A X. B ) X. C ) , <. X , Y , Z >. ) =
       ( ( ( ( Pred ( R , A , X ) u. { X } ) X.
             ( Pred ( S , B , Y ) u. { Y } ) ) X.
             ( Pred ( T , C , Z ) u. { Z } ) ) \
             { <. X , Y , Z >. } ) ) $=
        ( cxp cpred csn wceq wcel wa wo va vb vc vq vd ve vf cv cotp cdif oteq1
        cun predeq3 syl sneq uneq12d xpeq1d sneqd difeq12d eqeq12d oteq2 xpeq2d
        oteq3 w3a cin wbr wrex el2xptp weq wne w3o df-3an simplrl simplrr simpr
        wb simpll jca biantrurd orbi1d 3anbi123d anbi1d bitr3d bitrid xpord3lem
        3jca breq1 bitrdi eleq1 eldifsn otelxp cvv vex elpred elv velsn orbi12i
        bitri 3anbi123i otthne anbi12i syl5ibrcom rexlimdva rexlimdvva biimtrid
        elun bibi12d pm5.32d otex ax-mp elin 3bitr4g eqrdv wss predss a1i snssi
        unssd 3ad2ant1 3ad2ant2 xpss12 syl2anc 3ad2ant3 ssdifssd sylib vtocl3ga
        sseqin2 eqtrd ) CDNZENZIUAUHZUBUHZUCUHZUIZOZCFYKOZYKPZULZDGYLOZYLPZULZN
        ZEHYMOZYMPZULZNZYNPZUJZQYJIJYLYMUIZOZCFJOZJPZULZUUANZUUENZUUIPZUJZQYJIJ
        KYMUIZOZUUMDGKOZKPZULZNZUUENZUURPZUJZQYJIJKLUIZOZUVCEHLOZLPZULZNZUVGPZU
        JZQUAUBUCJKLCDEYKJQZYOUUJUUHUUQUVOYNUUIQYOUUJQYKJYLYMUKZYJIYNUUIUMUNUVO
        UUFUUOUUGUUPUVOUUBUUNUUEUVOYRUUMUUAUVOYPUUKYQUULCFYKJUMYKJUOUPUQUQUVOYN
        UUIUVPURUSUTYLKQZUUJUUSUUQUVFUVQUUIUURQUUJUUSQYLKJYMVAZYJIUUIUURUMUNUVQ
        UUOUVDUUPUVEUVQUUNUVCUUEUVQUUAUVBUUMUVQYSUUTYTUVADGYLKUMYLKUOUPVBUQUVQU
        UIUURUVRURUSUTYMLQZUUSUVHUVFUVNUVSUURUVGQUUSUVHQYMLJKVCZYJIUURUVGUMUNUV
        SUVDUVLUVEUVMUVSUUEUVKUVCUVSUUCUVIUUDUVJEHYMLUMYMLUOUPVBUVSUURUVGUVTURU
        SUTYKCRZYLDRZYMERZVDZYOYJUUHVEZUUHUWDUDYOUWEUWDUDUHZYJRZUWFYNIVFZSZUWGU
        WFUUHRZSUWFYORZUWFUWERUWDUWGUWHUWJUWGUWFUEUHZUFUHZUGUHZUIZQZUGEVGZUFDVG
        UECVGUWDUWHUWJVPZUEUFUGUWFCDEVHUWDUWQUWRUEUFCDUWDUWLCRZUWMDRZSZSZUWPUWR
        UGEUXBUWNERZSZUWRUWPUWSUWTUXCVDZUWDUWLYKFVFZUEUAVIZTZUWMYLGVFZUFUBVIZTZ
        UWNYMHVFZUGUCVIZTZVDZUWLYKVJUWMYLVJUWNYMVJVKZSZVDZUWSUXFSZUXGTZUWTUXISZ
        UXJTZUXCUXLSZUXMTZVDZUXPSZVPUXRUXEUWDSZUXQSZUXDUYFUXEUWDUXQVLUXDUXQUYHU
        YFUXDUYGUXQUXDUXEUWDUXDUWSUWTUXCUWDUWSUWTUXCVMZUWDUWSUWTUXCVNZUXBUXCVOZ
        WFUWDUXAUXCVQVRVSUXDUXOUYEUXPUXDUXHUXTUXKUYBUXNUYDUXDUXFUXSUXGUXDUWSUXF
        UYIVSVTUXDUXIUYAUXJUXDUWTUXIUYJVSVTUXDUXLUYCUXMUXDUXCUXLUYKVSVTWAWBWCWD
        UWPUWHUXRUWJUYFUWPUWHUWOYNIVFUXRUWFUWOYNIWGABCDEFGHIUBUCUEUFUGUAMWEWHUW
        PUWJUWOUUHRZUYFUWFUWOUUHWIUYLUWOUUFRZUWOYNVJZSUYFUWOUUFYNWJUYMUYEUYNUXP
        UYMUWLYRRZUWMUUARZUWNUUERZVDUYEUWLUWMUWNYRUUAUUEWKUYOUXTUYPUYBUYQUYDUYO
        UWLYPRZUWLYQRZTUXTUWLYPYQXFUYRUXSUYSUXGUYRUXSVPUACWLFYKUWLUEWMZWNWOUEYK
        WPWQWRUYPUWMYSRZUWMYTRZTUYBUWMYSYTXFVUAUYAVUBUXJVUAUYAVPUBDWLGYLUWMUFWM
        ZWNWOUFYLWPWQWRUYQUWNUUCRZUWNUUDRZTUYDUWNUUCUUDXFVUDUYCVUEUXMVUDUYCVPUC
        EWLHYMUWNUGWMZWNWOUGYMWPWQWRWSWRUWLUWMUWNYKYLYMUYTVUCVUFWTXAWRWHXGXBXCX
        DXEXHYNWLRUWKUWIVPYKYLYMXIYJWLIYNUWFUDWMWNXJUWFYJUUHXKXLXMUWDUUHYJXNUWE
        UUHQUWDUUFYJUUGUWDUUBYIXNZUUEEXNZUUFYJXNUWDYRCXNZUUADXNZVUGUWAUWBVUIUWC
        UWAYPYQCYPCXNUWACFYKXOXPYKCXQXRXSUWBUWAVUJUWCUWBYSYTDYSDXNUWBDGYLXOXPYL
        DXQXRXTYRCUUADYAYBUWCUWAVUHUWBUWCUUCUUDEUUCEXNUWCEHYMXOXPYMEXQXRYCUUBYI
        UUEEYAYBYDUUHYJYGYEYHYF $.
    $}

    ${
      $d A a b c p x y $.  $d B a b c p x y $.  $d C a b c p x y $.
      $d R a b c p x y $.  $d S a b c p x y $.  $d T a b c p x y $.
      $d ph a b c p $.  $d U a b c p $.  $d x y $.
      sexp3.1 $e |- ( ph -> R Se A ) $.
      sexp3.2 $e |- ( ph -> S Se B ) $.
      sexp3.3 $e |- ( ph -> T Se C ) $.
      $( Show that the triple order is set-like.  (Contributed by Scott Fenton,
         21-Aug-2024.) $)
      sexp3 $p |- ( ph -> U Se ( ( A X. B ) X. C ) ) $=
        ( va vb vc cpred cvv wcel vp cxp cv wral wse cotp wceq wrex el2xptp w3a
        df-3an csn cun cdif xpord3pred adantl simpr1 adantr setlikespec syl2anc
        wa wi vsnex a1i unexd simpr2 xpexd simpr3 difexd eqeltrd predeq3 eleq1d
        syl5ibrcom sylan2br rexlimdva rexlimdvva biimtrid ralrimiv dfse3 sylibr
        anassrs ) ADEUBFUBZJUAUCZRZSTZUAWBUDWBJUEAWEUAWBWCWBTWCOUCZPUCZQUCZUFZU
        GZQFUHZPEUHODUHAWEOPQWCDEFUIAWKWEOPDEAWFDTZWGETZVAZVAWJWEQFAWNWHFTZWJWE
        VBZWNWOVAAWLWMWOUJZWPWLWMWOUKAWQVAZWEWJWBJWIRZSTWRWSDGWFRZWFULZUMZEHWGR
        ZWGULZUMZUBZFIWHRZWHULZUMZUBZWIULZUNZSWQWSXLUGABCDEFGHIJWFWGWHKUOUPWRXJ
        XKSWRXFXISSWRXBXESSWRWTXASSWRWLDGUEZWTSTAWLWMWOUQAXMWQLURDGWFUSUTXASTWR
        OVCVDVEWRXCXDSSWRWMEHUEZXCSTAWLWMWOVFAXNWQMUREHWGUSUTXDSTWRPVCVDVEVGWRX
        GXHSSWRWOFIUEZXGSTAWLWMWOVHAXOWQNURFIWHUSUTXHSTWRQVCVDVEVGVIVJWJWDWSSWB
        JWCWIVKVLVMVNWAVOVPVQVRUAWBJVSVT $.
    $}

    ${
      $d A a b c d e f $.  $d f si $.  $d a e ps $.  $d d ph $.  $d a rh $.
      $d U a b c d e f $.  $d a b c d e f ka $.  $d b mu $.  $d a b c th $.
      $d d ta $.  $d X a b c $.  $d Y b c $.  $d b ch f $.  $d Z c $.
      $d T x y $.  $d T d e f $.  $d S d e f $.  $d B a b c d e f $.
      $d R d e f $.  $d C a b c d e f $.  $d e ze $.  $d S x y $.  $d C x y $.
      $d c la $.  $d e et $.  $d A x y $.  $d B x y $.  $d R x y $.
      xpord3inddlem.x $e |- ( ka -> X e. A ) $.
      xpord3inddlem.y $e |- ( ka -> Y e. B ) $.
      xpord3inddlem.z $e |- ( ka -> Z e. C ) $.
      xpord3inddlem.1 $e |- ( ka -> R Fr A ) $.
      xpord3inddlem.2 $e |- ( ka -> R Po A ) $.
      xpord3inddlem.3 $e |- ( ka -> R Se A ) $.
      xpord3inddlem.4 $e |- ( ka -> S Fr B ) $.
      xpord3inddlem.5 $e |- ( ka -> S Po B ) $.
      xpord3inddlem.6 $e |- ( ka -> S Se B ) $.
      xpord3inddlem.7 $e |- ( ka -> T Fr C ) $.
      xpord3inddlem.8 $e |- ( ka -> T Po C ) $.
      xpord3inddlem.9 $e |- ( ka -> T Se C ) $.
      xpord3inddlem.10 $e |- ( a = d -> ( ph <-> ps ) ) $.
      xpord3inddlem.11 $e |- ( b = e -> ( ps <-> ch ) ) $.
      xpord3inddlem.12 $e |- ( c = f -> ( ch <-> th ) ) $.
      xpord3inddlem.13 $e |- ( a = d -> ( ta <-> th ) ) $.
      xpord3inddlem.14 $e |- ( b = e -> ( et <-> ta ) ) $.
      xpord3inddlem.15 $e |- ( b = e -> ( ze <-> th ) ) $.
      xpord3inddlem.16 $e |- ( c = f -> ( si <-> ta ) ) $.
      xpord3inddlem.17 $e |- ( a = X -> ( ph <-> rh ) ) $.
      xpord3inddlem.18 $e |- ( b = Y -> ( rh <-> mu ) ) $.
      xpord3inddlem.19 $e |- ( c = Z -> ( mu <-> la ) ) $.
      xpord3inddlem.i $e |- ( ( ka /\ ( a e. A /\ b e. B /\ c e. C ) ) ->
       ( ( ( A. d e. Pred ( R , A , a ) A. e e. Pred ( S , B , b )
             A. f e. Pred ( T , C , c ) th /\
             A. d e. Pred ( R , A , a ) A. e e. Pred ( S , B , b ) ch /\
             A. d e. Pred ( R , A , a ) A. f e. Pred ( T , C , c ) ze )
             /\
             ( A. d e. Pred ( R , A , a ) ps /\
               A. e e. Pred ( S , B , b ) A. f e. Pred ( T , C , c ) ta /\
               A. e e. Pred ( S , B , b ) si ) /\
             A. f e. Pred ( T , C , c ) et )
           -> ph ) ) $.
      $( Induction over the triple Cartesian product ordering.  Note that the
         substitutions cover all possible cases of membership in the
         predecessor class.  (Contributed by Scott Fenton, 2-Feb-2025.) $)
      xpord3inddlem $p |- ( ka -> la ) $=
        ( cxp wfr wpo wse wcel wi frxp3 poxp3 sexp3 cv cotp cpred bi2.04 3albii
        wal w3a 19.21v 2albii albii bitri wa wne w3o csn cun wral adantl bitrdi
        vex wss ssun1 ssralv ax-mp ralimi syl wn weq predpoirr eleq1 syl5ibrcom
        wceq notbid necon2ad imp 3mix3d pm2.27 ralimdva ralimdv mpd ssun2 biidd
        adantr neeq1 3orbi123d equcoms bicomd imbi12d ralsn sylib 3mix2d ralbii
        wb ralbidv 3jca 2ralbidv imbi2d xpord3pred eleq2d imbi1d eldifsn otelxp
        cdif otthne anbi12i imbi1i impexp albidv 2albidv bicomi 2ralbii equcomi
        r3al 3mix1d bicom1 3syl ex syld sylbid expcom a2d biimtrid frpoins3xp3g
        syl33anc pm2.43i ) LKLOPVOQVOZUAVPUVIUAVQUVIUAVRUDOVSUEPVSUFQVSLKVTZLMN
        OPQRSTUAUKUOURVAWALMNOPQRSTUAUKUPUSVBWBLMNOPQRSTUAUKUQUTVCWCULUMUNLAVTZ
        LBVTLCVTLDVTZLIVTLJVTUVJUGUHUIUJUCUBOPQUAUDUEUFUJWDZUBWDZUCWDZWEZUVIUAU
        GWDZUHWDZUIWDZWEZWFZVSZUVLVTZUCWIUBWIUJWIZLUWBDVTZUCWIZUBWIZUJWIZVTZUVQ
        OVSUVRPVSUVSQVSWJZUVKUWDLUWEVTZUCWIZUBWIUJWIZUWIUWCUWKUJUBUCUWBLDWGWHUW
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        WTVWRVVEVVBUBUXIVVOYOVVBVWRUBUVRVWMVVRVVAVWQCBVVRUWSUWSUWTVWBVUTVUTVWNV
        WOVVRVUTYEYHVVRBCBCYPUHUBVEYIYJYKYLWNYOYMXALVWSUYOVTUXOLVWRBUJUXLLUVMUX
        LVSZWOZVWQVWRBVTVXDUWSVWBVUTLVXCUWSLVXCUVMUVQLVXCXJUJUGXKZUVQUXLVSZXJZL
        ORVQVXGUPORUVQXLXIVXEVXCVXFUVMUVQUXLXMXPXNXQXRUUQVWQBXTXIYAYFYCVUBUVQUV
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        KUXMUXNXDZVUFVXMVTUXMUXLYDZVUDUJUXMUXNXFXGXIVUDVXLUJUVQUGXCZVXEUXCVXJUB
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        EEDEDYPUGUJVGYIYJYKZYSYLYMXALVXLUYQVTUXOLVXKUYPUBUXHLVXJEUCUXDVUOVXIVXJ
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        TLUXOVVFUJUXMWTZVXTUXOVVHVYAVVKVXNVVHVYAVTVXOVVFUJUXMUXNXFXGXIVYAVXJUCU
        XEWTZUBUXHWTZVXTVVFVYCUJUVQVXPVXEUXCVXJUBUCUXHUXEVXQYSYLVYBVXSUBUXHVXJV
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        KZHEVJYJYIYKYLYOWNYMXALVXTUYRVTUXOLVXSHUBUXHVVQVXRVXSHVTVVQUWTVXHVUTVWA
        YNVXRHXTXIYAYFYCYRVUBVXHVWBUXAWQZFVTZUCUXDWTZUYTUXOVYGLUXOVWGUJUXMWTZVY
        GUXOVWIVYHVWLVXNVWIVYHVTVXOVWGUJUXMUXNXFXGXIVYHVXKUBUXIWTZVYGVWGVYIUJUV
        QVXPVXEUXCVXJUBUCUXIUXDVXQYSYLVXKVYGUBUVRVWMVVRVXJVYFUCUXDVVRVXIVYEEFVV
        RVXHVXHUWTVWBUXAUXAVVRVXHYEVWOVWPYHVVRUHUBXKZFEYPEFYPUBUHUUOVHFEUURUUSY
        KYQYLWNYMXALVYGUYTVTUXOLVYFFUCUXDVUOVYEVYFFVTVUOUXAVXHVWBVUSXSVYEFXTXIY
        AYFYCYRUUTYFVNUVAUVBUVCUVDUVEUGUJXKABLVDYTVYJBCLVEYTVYDCDLVFYTUVQUDXOAI
        LVKYTUVRUEXOIJLVLYTUVSUFXOJKLVMYTUVFUVGUVH $.
    $}
  $}

  ${
    $d A a b c d e f x y $.  $d B a b c d e f x y $.  $d C a b c d e f x y $.
    $d R a b c d e f x y $.  $d S a b c d e f x y $.  $d T a b c d e f x y $.
    $d X a b c $.  $d Y b c $.  $d Z c $.  $d ka a b c d e f $.  $d ps a $.
    $d rh a $.  $d th a $.  $d ch b f $.  $d mu b $.  $d th b $.  $d la c $.
    $d th c $.  $d ph d $.  $d ta d $.  $d et e $.  $d ps e $.  $d ze e $.
    $d ka f $.  $d si f $.
    xpord3indd.x $e |- ( ka -> X e. A ) $.
    xpord3indd.y $e |- ( ka -> Y e. B ) $.
    xpord3indd.z $e |- ( ka -> Z e. C ) $.
    xpord3indd.1 $e |- ( ka -> R Fr A ) $.
    xpord3indd.2 $e |- ( ka -> R Po A ) $.
    xpord3indd.3 $e |- ( ka -> R Se A ) $.
    xpord3indd.4 $e |- ( ka -> S Fr B ) $.
    xpord3indd.5 $e |- ( ka -> S Po B ) $.
    xpord3indd.6 $e |- ( ka -> S Se B ) $.
    xpord3indd.7 $e |- ( ka -> T Fr C ) $.
    xpord3indd.8 $e |- ( ka -> T Po C ) $.
    xpord3indd.9 $e |- ( ka -> T Se C ) $.
    xpord3indd.10 $e |- ( a = d -> ( ph <-> ps ) ) $.
    xpord3indd.11 $e |- ( b = e -> ( ps <-> ch ) ) $.
    xpord3indd.12 $e |- ( c = f -> ( ch <-> th ) ) $.
    xpord3indd.13 $e |- ( a = d -> ( ta <-> th ) ) $.
    xpord3indd.14 $e |- ( b = e -> ( et <-> ta ) ) $.
    xpord3indd.15 $e |- ( b = e -> ( ze <-> th ) ) $.
    xpord3indd.16 $e |- ( c = f -> ( si <-> ta ) ) $.
    xpord3indd.17 $e |- ( a = X -> ( ph <-> rh ) ) $.
    xpord3indd.18 $e |- ( b = Y -> ( rh <-> mu ) ) $.
    xpord3indd.19 $e |- ( c = Z -> ( mu <-> la ) ) $.
    xpord3indd.i $e |- ( ( ka /\ ( a e. A /\ b e. B /\ c e. C ) ) ->
       ( ( ( A. d e. Pred ( R , A , a ) A. e e. Pred ( S , B , b )
             A. f e. Pred ( T , C , c ) th /\
             A. d e. Pred ( R , A , a ) A. e e. Pred ( S , B , b ) ch /\
             A. d e. Pred ( R , A , a ) A. f e. Pred ( T , C , c ) ze )
             /\
             ( A. d e. Pred ( R , A , a ) ps /\
               A. e e. Pred ( S , B , b ) A. f e. Pred ( T , C , c ) ta /\
               A. e e. Pred ( S , B , b ) si ) /\
             A. f e. Pred ( T , C , c ) et )
           -> ph ) ) $.
    $( Induction over the triple Cartesian product ordering.  Note that the
       substitutions cover all possible cases of membership in the predecessor
       class.  (Contributed by Scott Fenton, 2-Feb-2025.) $)
    xpord3indd $p |- ( ka -> la ) $=
      ( vx vy cv cxp wcel c1st cfv wbr wceq wo c2nd w3a wne copab xpord3inddlem
      wa eqid ) ABCDEFGHIJKLVKVLMNOPQRVKVMZMNVNOVNZVOVLVMZWIVOWHVPVQZVPVQZWJVPV
      QZVPVQZPVRWLWNVSVTWKWAVQZWMWAVQZQVRWOWPVSVTWHWAVQZWJWAVQZRVRWQWRVSVTWBWHW
      JWCWFWBVKVLWDZSTUAUBUCUDUEUFUGWSWGUHUIUJUKULUMUNUOUPUQURUSUTVAVBVCVDVEVFV
      GVHVIVJWE $.
  $}

  ${
    $d A a b c d e f $.  $d B a b c d e f $.  $d C a b c d e f $.
    $d R a b c d e f $.  $d S a b c d e f $.  $d T a b c d e f $.
    $d X a b c d e f $.  $d Y a b c d e f $.  $d Z a b c d e f $.  $d a ps $.
    $d a rh $.  $d a th $.  $d b ch $.  $d b mu $.  $d b th $.  $d c la $.
    $d c th $.  $d ch f $.  $d d ph $.  $d d ta $.  $d e et $.  $d e ps $.
    $d e ze $.  $d f si $.
    xpord3ind.1 $e |- R Fr A $.
    xpord3ind.2 $e |- R Po A $.
    xpord3ind.3 $e |- R Se A $.
    xpord3ind.4 $e |- S Fr B $.
    xpord3ind.5 $e |- S Po B $.
    xpord3ind.6 $e |- S Se B $.
    xpord3ind.7 $e |- T Fr C $.
    xpord3ind.8 $e |- T Po C $.
    xpord3ind.9 $e |- T Se C $.
    xpord3ind.10 $e |- ( a = d -> ( ph <-> ps ) ) $.
    xpord3ind.11 $e |- ( b = e -> ( ps <-> ch ) ) $.
    xpord3ind.12 $e |- ( c = f -> ( ch <-> th ) ) $.
    xpord3ind.13 $e |- ( a = d -> ( ta <-> th ) ) $.
    xpord3ind.14 $e |- ( b = e -> ( et <-> ta ) ) $.
    xpord3ind.15 $e |- ( b = e -> ( ze <-> th ) ) $.
    xpord3ind.16 $e |- ( c = f -> ( si <-> ta ) ) $.
    xpord3ind.17 $e |- ( a = X -> ( ph <-> rh ) ) $.
    xpord3ind.18 $e |- ( b = Y -> ( rh <-> mu ) ) $.
    xpord3ind.19 $e |- ( c = Z -> ( mu <-> la ) ) $.
    xpord3ind.i $e |- ( ( a e. A /\ b e. B /\ c e. C ) ->
       ( ( ( A. d e. Pred ( R , A , a ) A. e e. Pred ( S , B , b )
             A. f e. Pred ( T , C , c ) th /\
             A. d e. Pred ( R , A , a ) A. e e. Pred ( S , B , b ) ch /\
             A. d e. Pred ( R , A , a ) A. f e. Pred ( T , C , c ) ze )
             /\
             ( A. d e. Pred ( R , A , a ) ps /\
               A. e e. Pred ( S , B , b ) A. f e. Pred ( T , C , c ) ta /\
               A. e e. Pred ( S , B , b ) si ) /\
             A. f e. Pred ( T , C , c ) et )
           -> ph ) ) $.
    $( Induction over the triple Cartesian product ordering.  Note that the
       substitutions cover all possible cases of membership in the predecessor
       class.  (Contributed by Scott Fenton, 4-Sep-2024.) $)
    xpord3ind $p |- ( ( X e. A /\ Y e. B /\ Z e. C ) -> la ) $=
      ( wcel w3a simp1 simp2 simp3 wfr wi ax-1 ax-mp wpo a1i wse cv wral adantl
      cpred xpord3indd ) ABCDEFGHIJKTLVGZUAMVGZUBNVGZVHZLMNOPQRSTUAUBUCUDUEUFWD
      WEWFVIWDWEWFVJWDWEWFVKLOVLZWGWHVMUGWHWGVNVOLOVPWGUHVQLOVRWGUIVQMPVLWGUJVQ
      MPVPWGUKVQMPVRWGULVQNQVLWGUMVQNQVPWGUNVQNQVRWGUOVQUPUQURUSUTVAVBVCVDVEUCV
      SZLVGUDVSZMVGUEVSZNVGVHDSNQWKWBZVTRMPWJWBZVTUFLOWIWBZVTCRWMVTUFWNVTGSWLVT
      UFWNVTVHBUFWNVTESWLVTRWMVTHRWMVTVHFSWLVTVHAVMWGVFWAWC $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Ordering Ordinal Sequences
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d A f x $.  $d G f x $.  $d X x $.
    orderseqlem.1 $e |- F = { f | E. x e. On f : x --> A } $.
    $( Lemma for ~ poseq and ~ soseq .  The function value of a sequence is
       either in ` A ` or null.  (Contributed by Scott Fenton, 8-Jun-2011.) $)
    orderseqlem $p |- ( G e. F -> ( G ` X ) e. ( A u. { (/) } ) ) $=
      ( wcel cv wf con0 wrex cfv c0 csn cun wceq feq1 wss syl rexbidv ibi unss1
      elab2g crn frn fvrn0 ssel mpisyl rexlimivw ) EDHZAIZBEJZAKLZFEMZBNOZPZHZU
      KUNULBCIZJZAKLUNCEDDUSEQUTUMAKULBUSERUAGUDUBUMURAKUMEUEZUPPZUQSZUOVBHURUM
      VABSVCULBEUFVABUPUCTEFUGVBUQUOUHUIUJT $.
  $}

  ${
    $d A b f x $.  $d a b c f g t w x y z $.  $d F a b c f g t w x z $.
    $d R f g t w x z $.  $d S a b c $.
    poseq.1 $e |- R Po ( A u. { (/) } ) $.
    poseq.2 $e |- F = { f | E. x e. On f : x --> A } $.
    poseq.3 $e |- S = { <. f , g >. | ( ( f e. F /\ g e. F ) /\
                        E. x e. On ( A. y e. x ( f ` y ) = ( g ` y ) /\
                           ( f ` x ) R ( g ` x ) ) ) } $.
    $( A partial ordering of ordinal sequences.  (Contributed by Scott Fenton,
       8-Jun-2011.) $)
    poseq $p |- S Po F $=
      ( vz vw wbr wa wral cfv con0 wrex anbi12d va vb vc vt wpo cv wn wcel wceq
      wi w3a c0 csn cun cab feq2 cbvrexvw abbii eqtri orderseqlem poirr sylancr
      wf intnand adantr nrexdv imnan vex weq eleq1w anbi1d fveq1 eqeq1d ralbidv
      mpbi breq1d rexbidv anbi2d eqeq2d breq2d mtbir raleq fveq2 breq12d bitrid
      brab simplll simplrr an4 2rexbii reeanv bitri word eloni ordtri3or syl2an
      wel w3o simp1l wss onelss imp adantll ssralv anim2d r19.26 imbitrrdi eqtr
      ralimi syl6 syl adantrd 3impia eqeq12d rspcv breq2 biimpd com3l ad2ant2lr
      impcom 3adant1 syl12anc 3exp ad2antrr ad2antlr ad2antll 3jca potr anim12i
      rspcev anassrs sylan2 exp32 imbi1d syl5ibcom simp1r adantlr anim1d sylbir
      a1d breq1 biimprd ad2ant2rl 3jaod mpd rexlimivv jca31 an4s syl2anb sylibr
      pm3.2i a1i rgen3 df-po mpbir ) HEUEUAUFZUUPENZUGZUUPUBUFZENZUUSUCUFZENZOZ
      UUPUVAENZUJZOZUCHPUBHPUAHPUVFUAUBUCHHHUVFUUPHUHZUUSHUHZUVAHUHZUKUURUVEUUQ
      UVGUVGOZBUFZUUPQZUVLUIZBAUFZPZUVNUUPQZUVPDNZOZARSZOZUVJUVSUGZUJUVTUGUVGUW
      AUVGUVGUVRARUVGUVRUGUVNRUHUVGUVQUVOUVGCULUMUNZDUEZUVPUWBUHUVQUGIUBCFHUUPU
      VNHUVNCFUFZVCZARSZFUOUUSCUWDVCZUBRSZFUOJUWFUWHFUWEUWGAUBRUVNUUSCUWDUPUQUR
      USUTUWBUVPDVAVBVDVEVFVEUVJUVSVGVOUWDHUHZGUFZHUHZOZUVKUWDQZUVKUWJQZUIZBUVN
      PZUVNUWDQZUVNUWJQZDNZOZARSZOZUVGUWKOZUVLUWNUIZBUVNPZUVPUWRDNZOZARSZOUVTFG
      UUPUUPEUAVHZUXIFUAVIZUWLUXCUXAUXHUXJUWIUVGUWKFUAHVJVKZUXJUWTUXGARUXJUWPUX
      EUWSUXFUXJUWOUXDBUVNUXJUWMUVLUWNUVKUWDUUPVLVMZVNUXJUWQUVPUWRDUVNUWDUUPVLV
      PTVQTGUAVIZUXCUVJUXHUVSUXMUWKUVGUVGGUAHVJVRUXMUXGUVRARUXMUXEUVOUXFUVQUXMU
      XDUVMBUVNUXMUWNUVLUVLUVKUWJUUPVLVSVNUXMUWRUVPUVPDUVNUWJUUPVLVTTVQTKWFWAUV
      CUVGUVIOZUVLUVKUVAQZUIZBUDUFZPZUXQUUPQZUXQUVAQZDNZOZUDRSZOZUVDUUTUVGUVHOZ
      UVLUVKUUSQZUIZBLUFZPZUYHUUPQZUYHUUSQZDNZOZLRSZOZUVHUVIOZUYFUXOUIZBMUFZPZU
      YRUUSQZUYRUVAQZDNZOZMRSZOZUYDUVBUXBUXCUXDBUYHPZUYJUYHUWJQZDNZOZLRSZOUYOFG
      UUPUUSEUXIUBVHZUXJUWLUXCUXAVUJUXKUXAUWOBUYHPZUYHUWDQZVUGDNZOZLRSUXJVUJUWT
      VUOALRALVIZUWPVULUWSVUNUWOBUVNUYHWBVUPUWQVUMUWRVUGDUVNUYHUWDWCUVNUYHUWJWC
      WDTUQUXJVUOVUILRUXJVULVUFVUNVUHUXJUWOUXDBUYHUXLVNUXJVUMUYJVUGDUYHUWDUUPVL
      VPTVQWETGUBVIZUXCUYEVUJUYNVUQUWKUVHUVGGUBHVJVRVUQVUIUYMLRVUQVUFUYIVUHUYLV
      UQUXDUYGBUYHVUQUWNUYFUVLUVKUWJUUSVLVSVNVUQVUGUYKUYJDUYHUWJUUSVLVTTVQTKWFU
      XBUVHUWKOZUYFUWNUIZBUYRPZUYTUYRUWJQZDNZOZMRSZOVUEFGUUSUVAEVUKUCVHZFUBVIZU
      WLVURUXAVVDVVFUWIUVHUWKFUBHVJVKUXAUWOBUYRPZUYRUWDQZVVADNZOZMRSVVFVVDUWTVV
      JAMRAMVIZUWPVVGUWSVVIUWOBUVNUYRWBVVKUWQVVHUWRVVADUVNUYRUWDWCUVNUYRUWJWCWD
      TUQVVFVVJVVCMRVVFVVGVUTVVIVVBVVFUWOVUSBUYRVVFUWMUYFUWNUVKUWDUUSVLVMVNVVFV
      VHUYTVVADUYRUWDUUSVLVPTVQWETGUCVIZVURUYPVVDVUDVVLUWKUVIUVHGUCHVJZVRVVLVVC
      VUCMRVVLVUTUYSVVBVUBVVLVUSUYQBUYRVVLUWNUXOUYFUVKUWJUVAVLZVSVNVVLVVAVUAUYT
      DUYRUWJUVAVLVTTVQTKWFUYEUYPUYNVUDUYDUYEUYPOZUYNVUDOZOUVGUVIUYCUVGUVHUYPVV
      PWGUYEUVHUVIVVPWHVVPVVOUYCVVPUYIUYSOZUYLVUBOZOZMRSLRSZVVOUYCUJZVVTUYMVUCO
      ZMRSLRSVVPVVSVWBLMRRUYIUYSUYLVUBWIWJUYMVUCLMRRWKWLVVSVWALMRRUYHRUHZUYRRUH
      ZOZLMWQZLMVIZMLWQZWRZVVSVWAUJZVWCUYHWMUYRWMVWIVWDUYHWNUYRWNUYHUYRWOWPVWEV
      WFVWJVWGVWHVWEVWFVVSVWAVWEVWFVVSUKZUYCVVOVWKVWCUXPBUYHPZUYJUYHUVAQZDNZUYC
      VWCVWDVWFVVSWSVWEVWFVVSVWLVWEVWFOZVVQVWLVVRVWOUYHUYRWTZVVQVWLUJVWDVWFVWPV
      WCVWDVWFVWPUYRUYHXAXBXCVWPVVQUYGUYQOZBUYHPZVWLVWPVVQUYIUYQBUYHPZOZVWRVWPU
      YSVWSUYIUYQBUYHUYRXDXEUYGUYQBUYHXFZXGVWQUXPBUYHUVLUYFUXOXHZXIZXJXKXLXMVWF
      VVSVWNVWEVVSVWFVWNUYSUYLVWFVWNUJZUYIVUBUYSUYLVXDVWFUYSUYLVWNVWFUYSUYKVWMU
      IZUYLVWNUJUYQVXEBUYHUYRBLVIUYFUYKUXOVWMUVKUYHUUSWCUVKUYHUVAWCXNXOVXEUYLVW
      NUYKVWMUYJDXPXQXJXRXBXSXTYAUYBVWLVWNOZUDUYHRUDLVIZUXRVWLUYAVWNUXPBUXQUYHW
      BVXGUXSUYJUXTVWMDUXQUYHUUPWCUXQUYHUVAWCWDTYJZYBYTYCVWCVWGVWJUJVWDVWCVWRUY
      LUYKVWMDNZOZOZVWAUJVWGVWJVWCVXKVVOUYCVXKVVOOVWCVXFUYCVWRVXJVVOVXFVWRVWLVX
      JVVOOVWNVXCVVOVXJVWNVVOUWCUYJUWBUHZUYKUWBUHZVWMUWBUHZUKVXJVWNUJIVVOVXLVXM
      VXNUVGVXLUVHUYPACFHUUPUYHJUTYDUVHVXMUVGUYPACFHUUSUYHJUTYEUVIVXNUYEUVHACFH
      UVAUYHJUTYFYGUWBUYJUYKVWMDYHVBXTYIYKVXHYLYMVWGVXKVVSVWAVWGVWRVVQVXJVVRVWR
      VWTVWGVVQVXAVWGVWSUYSUYIUYQBUYHUYRWBVRWEVWGVXIVUBUYLVWGUYKUYTVWMVUADUYHUY
      RUUSWCUYHUYRUVAWCWDVRTYNYOVEVWEVWHVVSVWAVWEVWHVVSUKZUYCVVOVXOVWDUXPBUYRPZ
      UYRUUPQZVUADNZUYCVWCVWDVWHVVSYPVWEVWHVVSVXPVWEVWHOUYRUYHWTZVVSVXPUJVWCVWH
      VXSVWDVWCVWHVXSUYHUYRXAXBYQVXSVVQVXPVVRVXSVVQUYGBUYRPZUYSOZVXPVXSUYIVXTUY
      SUYGBUYRUYHXDYRVYAVWQBUYRPVXPUYGUYQBUYRXFVWQUXPBUYRVXBXIYSXJXLXKXMVWHVVSV
      XRVWEVVSVWHVXRUYIVUBVWHVXRUJZUYSUYLUYIVUBVYBVWHUYIVUBVXRVWHUYIVXQUYTUIZVU
      BVXRUJUYGVYCBUYRUYHBMVIUVLVXQUYFUYTUVKUYRUUPWCUVKUYRUUSWCXNXOVYCVXRVUBVXQ
      UYTVUADUUAUUBXJXRXBUUCXTYAUYBVXPVXROUDUYRRUDMVIZUXRVXPUYAVXRUXPBUXQUYRWBV
      YDUXSVXQUXTVUADUXQUYRUUPWCUXQUYRUVAWCWDTYJYBYTYCUUDUUEUUFYSXTUUGUUHUUIUXB
      UXCUXDBUXQPZUXSUXQUWJQZDNZOZUDRSZOUYDFGUUPUVAEUXIVVEUXJUWLUXCUXAVYIUXKUXA
      UWOBUXQPZUXQUWDQZVYFDNZOZUDRSUXJVYIUWTVYMAUDRAUDVIZUWPVYJUWSVYLUWOBUVNUXQ
      WBVYNUWQVYKUWRVYFDUVNUXQUWDWCUVNUXQUWJWCWDTUQUXJVYMVYHUDRUXJVYJVYEVYLVYGU
      XJUWOUXDBUXQUXLVNUXJVYKUXSVYFDUXQUWDUUPVLVPTVQWETVVLUXCUXNVYIUYCVVLUWKUVI
      UVGVVMVRVVLVYHUYBUDRVVLVYEUXRVYGUYAVVLUXDUXPBUXQVVLUWNUXOUVLVVNVSVNVVLVYF
      UXTUXSDUXQUWJUVAVLVTTVQTKWFUUJUUKUULUUMUAUBUCHEUUNUUO $.
  $}

  ${
    $d a b p q y $.  $d A f p q x $.  $d A y $.  $d F a b f g x $.
    $d f g x y $.  $d R f g x $.  $d S a b $.
    soseq.1 $e |- R Or ( A u. { (/) } ) $.
    soseq.2 $e |- F = { f | E. x e. On f : x --> A } $.
    soseq.3 $e |- S = { <. f , g >. | ( ( f e. F /\ g e. F ) /\
                        E. x e. On ( A. y e. x ( f ` y ) = ( g ` y ) /\
                           ( f ` x ) R ( g ` x ) ) ) } $.
    soseq.4 $e |- -. (/) e. A $.
    $( A linear ordering of ordinal sequences.  (Contributed by Scott Fenton,
       8-Jun-2011.) $)
    soseq $p |- S Or F $=
      ( wral wcel wa cfv wceq con0 wrex wi va vb vp wor wpo wbr weq w3o csn cun
      vq cv c0 sopo ax-mp poseq wo wn eleq1w anbi1d fveq1 eqeq1d ralbidv breq1d
      anbi12d rexbidv anbi2d eqeq2d breq2d brabg bianabs ancoms notbid ralinexa
      wb orbi12d andi eqcom ralbii anbi1i orbi2i bitri rexbii r19.43 xchbinx wf
      cab feq2 cbvrexvw abbii eqtri orderseqlem sotrieq mpan syl2an imbi2d wsbc
      vex fveq2 eqeq12d sbcie imbi1i tfisg sylbir feq1 elab2 anbi12i reeanv wss
      bitr4i onss ssralv syl ad2antlr rspcv a1i ffvelcdm cdm eloni ordirr eleq2
      fdm word biimparc sylan ndmfv eleq1 ex com23 sylan2 exp4b imp32 syldd imp
      syl5 mtoi syld jcad wfn ffn eqtr2 biimprd 3syl adantlr eqtr biimpd expcom
      ad2antrr adantll ordtri3or adantr 3orel13 syl6ci eqfnfv2 adantl rexlimivv
      sylibrd sylbi sylbird sylbid orrd 3orcomb df-3or bitr2i sylib rgen2 df-so
      biimtrrid mpbir2an ) HEUDHEUEUAULZUBULZEUFZUAUBUGZUVKUVJEUFZUHZUBHMUAHMAB
      CDEFGHCUMUIUJZDUDZUVPDUEIUVPDUNUOJKUPUVOUAUBHHUVJHNZUVKHNZOZUVLUVNUQZUVMU
      QZUVOUVTUWAUVMUVTUWAURBULZUVJPZUWCUVKPZQZBAULZMZUWGUVJPZUWGUVKPZDUFZOZARS
      ZUWEUWDQZBUWGMZUWJUWIDUFZOZARSZUQZURZUVMUVTUWAUWSUVTUVLUWMUVNUWRUVTUVLUWM
      FULZHNZGULZHNZOZUWCUXAPZUWCUXCPZQZBUWGMZUWGUXAPZUWGUXCPZDUFZOZARSZOZUVRUX
      DOZUWDUXGQZBUWGMZUWIUXKDUFZOZARSZOUVTUWMOFGUVJUVKHHEFUAUGZUXEUXPUXNUYAUYB
      UXBUVRUXDFUAHUSUTUYBUXMUXTARUYBUXIUXRUXLUXSUYBUXHUXQBUWGUYBUXFUWDUXGUWCUX
      AUVJVAVBVCUYBUXJUWIUXKDUWGUXAUVJVAVDVEVFVEGUBUGZUXPUVTUYAUWMUYCUXDUVSUVRG
      UBHUSVGUYCUXTUWLARUYCUXRUWHUXSUWKUYCUXQUWFBUWGUYCUXGUWEUWDUWCUXCUVKVAVHVC
      UYCUXKUWJUWIDUWGUXCUVKVAVIVEVFVEKVJVKUVSUVRUVNUWRVOUVSUVROZUVNUWRUXOUVSUX
      DOZUWEUXGQZBUWGMZUWJUXKDUFZOZARSZOUYDUWROFGUVKUVJHHEFUBUGZUXEUYEUXNUYJUYK
      UXBUVSUXDFUBHUSUTUYKUXMUYIARUYKUXIUYGUXLUYHUYKUXHUYFBUWGUYKUXFUWEUXGUWCUX
      AUVKVAVBVCUYKUXJUWJUXKDUWGUXAUVKVAVDVEVFVEGUAUGZUYEUYDUYJUWRUYLUXDUVRUVSG
      UAHUSVGUYLUYIUWQARUYLUYGUWOUYHUWPUYLUYFUWNBUWGUYLUXGUWDUWEUWCUXCUVJVAVHVC
      UYLUXKUWIUWJDUWGUXCUVJVAVIVEVFVEKVJVKVLVPVMUWTUWHUWKUWPUQZURZTZARMZUVTUVM
      UYPUWHUYMOZARSZUWSUWHUYMARVNUYRUWLUWQUQZARSUWSUYQUYSARUYQUWLUWHUWPOZUQUYS
      UWHUWKUWPVQUYTUWQUWLUWHUWOUWPUWFUWNBUWGUWDUWEVRVSVTWAWBWCUWLUWQARWDWBWEUV
      TUYPUWHUWIUWJQZTZARMZUVMUVTVUBUYOARUVTVUAUYNUWHUVRUWIUVPNZUWJUVPNZVUAUYNV
      OZUVSBCFHUVJUWGHUWGCUXAWFZARSZFWGUWCCUXAWFZBRSZFWGJVUHVUJFVUGVUIABRUWGUWC
      CUXAWHWIWJWKZWLBCFHUVKUWGVUKWLUVQVUDVUEOVUFIUVPUWIUWJDWMWNWOWPVCVUCVUAARM
      ZUVTUVMVUCVUAAUWCWQZBUWGMZVUATZARMVULVUOVUBARVUNUWHVUAVUMUWFBUWGVUAUWFAUW
      CBWRABUGUWIUWDUWJUWEUWGUWCUVJWSUWGUWCUVKWSWTXAVSXBVSVUAABXCXDUVTUCULZCUVJ
      WFZUKULZCUVKWFZOZUKRSUCRSZVULUVMTZUVTVUQUCRSZVUSUKRSZOVVAUVRVVCUVSVVDUVRU
      WGCUVJWFZARSZVVCVUHVVFFUVJHUAWRUYBVUGVVEARUWGCUXAUVJXEVFJXFVVEVUQAUCRUWGV
      UPCUVJWHWIWBUVSUWGCUVKWFZARSZVVDVUHVVHFUVKHUBWRUYKVUGVVGARUWGCUXAUVKXEVFJ
      XFVVGVUSAUKRUWGVURCUVKWHWIWBXGVUQVUSUCUKRRXHXJVUTVVBUCUKRRVUPRNZVURRNZOZV
      UTVVBVVKVUTOZVULUCUKUGZVUAAVUPMZOZUVMVVLVULVVMVVNVVLVULVUPVURNZURZVURVUPN
      ZURZOVVPVVMVVRUHZVVMVVLVULVVQVVSVVLVULVUAAVURMZVVQVVJVULVWATZVVIVUTVVJVUR
      RXIVWBVURXKVUAAVURRXLXMXNVVLVWAVVQVVLVWAOVVPUMCNZLVVLVWAVVPVWCTVVLVVPVWAV
      WCVVLVVPVWAVUPUVJPZVUPUVKPZQZVWCVVPVWAVWFTTVVLVUAVWFAVUPVURAUCUGUWIVWDUWJ
      VWEUWGVUPUVJWSUWGVUPUVKWSWTXOXPVVKVUQVUSVVPVWFVWCTZTVVKVUQVUSVVPVWGVUSVVP
      OVWECNZVVKVUQOVWGVURCVUPUVKXQVVIVUQVWHVWGTZVVJVUQVVIUVJXRZVUPQZVWIVUPCUVJ
      YBVVIVWKOZVWFVWHVWCVWLVUPVWJNZURZVWDUMQZVWFVWHVWCTZTVVIVUPVUPNZURZVWKVWNV
      VIVUPYCZVWRVUPXSZVUPXTXMVWKVWNVWRVWKVWMVWQVWJVUPVUPYAVMYDYEVUPUVJYFVWOVWF
      VWPVWOVWFOUMVWEQZVWPVWDUMVWEUUAVXAVWCVWHUMVWECYGUUBXMYHUUCYIYJUUDYOYKYLYM
      YIYNYPYHYQVVLVULVVNVVSVVIVULVVNTZVVJVUTVVIVUPRXIVXBVUPXKVUAAVUPRXLXMUUHZV
      VLVVNVVSVVLVVNOVVRVWCLVVLVVNVVRVWCTVVLVVRVVNVWCVVLVVRVVNVURUVJPZVURUVKPZQ
      ZVWCVVRVVNVXFTTVVLVUAVXFAVURVUPAUKUGUWIVXDUWJVXEUWGVURUVJWSUWGVURUVKWSWTX
      OXPVVKVUQVUSVVRVXFVWCTZTZVVKVUSVUQVXHVVKVUSVUQVVRVXGVUQVVROVXDCNZVVKVUSOV
      XGVUPCVURUVJXQVUSVVKUVKXRZVURQZVXIVXGTZVURCUVKYBVVJVXKVXLVVIVVJVXKOVXEUMQ
      ZVXLVVJVURVURNZURZVXKVXMVVJVURYCZVXOVURXSZVURXTXMVXOVXKOVURVXJNZURZVXMVXK
      VXSVXOVXKVXRVXNVXJVURVURYAVMYDVURUVKYFXMYEVXMVXFVXIVWCVXFVXMVXIVWCTZVXFVX
      MOVXDUMQZVXTVXDVXEUMUUEVYAVXIVWCVXDUMCYGUUFXMUUGYIXMUUIYJYOYKYIYLYMYIYNYP
      YHYQYRVVKVVTVUTVVIVWSVXPVVTVVJVWTVXQVUPVURUUJWOUUKVVPVVMVVRUULUUMVXCYRVUT
      UVMVVOVOZVVKVUQUVJVUPYSUVKVURYSVYBVUSVUPCUVJYTVURCUVKYTAVUPVURUVJUVKUUNWO
      UUOUUQYHUUPUURYOUUSUVHUUTUVAUVOUVLUVNUVMUHUWBUVLUVMUVNUVBUVLUVNUVMUVCUVDU
      VEUVFUAUBHEUVGUVI $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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
  ~ suppimacnv ).  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.  According to
       Wikipedia ("Support (mathematics)", 31-Mar-2019,
       ~ https://en.wikipedia.org/wiki/Support_(mathematics) ) "In mathematics,
       the support of a real-valued function f is the subset of the domain
       containing those elements which are not mapped to zero." and "The notion
       of support also extends in a natural way to functions taking values in
       more general sets than R [the real numbers] and to other objects."  The
       following definition allows for such extensions, being applicable for
       any sets (which usually are functions) and any element (even not
       necessarily from the range of the function) regarded as "zero".
       (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 $.

    $( The support of a class is empty if either the class or the "zero" is a
       proper class.  (Contributed by AV, 28-May-2019.) $)
    supp0prc $p |- ( -. ( X e. _V /\ Z e. _V ) -> ( X supp Z ) = (/) ) $=
      ( vx vz vi cv csn cima wne cdm crab csupp cvv df-supp mpondm0 ) CDCFZEFGH
      DFGIEPJKLABMMCDENO $.
  $}

  ${
    $d R x y $.  $d Z x y $.
    $( The value of the operation constructing the support of a function
       expressed by binary relations.  (Contributed by AV, 7-Apr-2019.) $)
    suppvalbr $p |- ( ( R e. V /\ Z e. W ) -> ( R supp Z )
                    = { x | ( E. y x R y /\ E. y ( x R y <-> y =/= Z ) ) } ) $=
      ( wcel wa cv wbr wex wceq wn wb cab csn cima wne abbii a1i cdm crab csupp
      df-rab vex eldm cvv imasng elv neeq1i neeq2i nabbib 3bitri anbi12i eqtr2i
      co df-sn df-ne bibi2i exbii anbi2i suppval 3eqtr4rd ) CDGFEGHZAIZBIZCJZBK
      ZVGVFFLZMZNZBKZHZAOZCVEPQZFPZRZACUAZUBZVHVGVFFRZNZBKZHZAOZCFUCUPVNVSLVDVS
      VEVRGZVQHZAOVNVQAVRUDWFVMAWEVHVQVLBVECAUEUFVQVGBOZVPRWGVIBOZRVLVOWGVPVOWG
      LABVEUGCUHUIUJVPWHWGBFUQUKVGVIBULUMUNSUOTWDVNLVDWCVMAWBVLVHWAVKBVTVJVGVFF
      URUSUTVASTADECFVBVC $.
  $}

  ${
    $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 fvex mp1i
      sneqbg necon3bid bitrd rabbidva eqtrd ) DFZDBGZECGZHZDEIJZDAKZLMZELZNZADO
      ZPZVDDQZENZAVHPUTVAVCVIRUSABCDESUBVBVGVKAVHVBVDVHGZUAZVGVJLZVFNVKVMVEVNVF
      VMVNVEVBDVHUCZVLVNVERUSUTVOVAUSVODUDUEUFVHVDDUGUHUIUJVMVNVFVJEVJTGVNVFRVJ
      ERUKVMVDDULVJETUNUMUOUPUQUR $.

    $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-rep .  (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-rep'; $)

    $( 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-rep .  (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-rep'; $)

    $( 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 $.
  $}

  ${
    fvdifsupp.1 $e |- ( ph -> F Fn A ) $.
    fvdifsupp.2 $e |- ( ph -> A e. V ) $.
    fvdifsupp.3 $e |- ( ph -> Z e. W ) $.
    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.) $)
    fvdifsupp $p |- ( ph -> ( F ` X ) = Z ) $=
      ( cfv wceq csupp co wcel wn eldifbd wne eldifad elsuppfn syl3anc mpbirand
      wfn wa wb necon2bbid mpbird ) AFCLZGMFCGNOZPZQAFBUJKRAUKUIGAUKFBPZUIGSZAF
      BUJKTACBUDBDPGEPUKULUMUEUFHIJFCDEBGUAUBUCUGUH $.
  $}

  ${
    $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 R x y $.  $d V x $.  $d W x $.  $d Z x y $.
    $( The support of functions "defined" by inverse images is a subset of the
       support defined by ~ df-supp .  (Contributed by AV, 7-Apr-2019.) $)
    suppimacnvss $p |- ( ( R e. V /\ Z e. W )
                          -> ( `' R " ( _V \ { Z } ) ) C_ ( R supp Z ) ) $=
      ( vx vy wcel wa cv wbr wne wex cab wb ccnv cvv csn cdif cima a1i csupp co
      wi exsimpl pm5.1 eximi jca ss2abdv wceq cnvimadfsn suppvalbr 3sstr4d ) AB
      GDCGHZEIFIZAJZUNDKZHZFLZEMZUOFLZUOUPNZFLZHZEMAOPDQRSZADUAUBUMURVCEURVCUCU
      MURUTVBUOUPFUDUQVAFUOUPUEUFUGTUHVDUSUIUMEFADUJTEFABCDUKUL $.

    $d R s t x y $.  $d Z s t $.
    $( Support sets of functions expressed by inverse images.  (Contributed by
       AV, 31-Mar-2019.)  (Revised by AV, 7-Apr-2019.) $)
    suppimacnv $p |- ( ( R e. V /\ Z e. W )
                       -> ( R supp Z ) = ( `' R " ( _V \ { Z } ) ) ) $=
      ( vx vt vy vs wcel wa cv wbr wex wne wb wi breq2 wceq ex com13 csupp ccnv
      co cvv csn cdif cab cbvexvw anbi1d bianir vex weq neeq1 anbi12d spcev syl
      cima pm2.43a adantld wn nne notbi eqcoms pm2.24 biimtrdi biimtrid pm2.43i
      imp sylbi pm2.61i adantr com12 pm2.61ine expcom exlimiv ss2abdv suppvalbr
      a1i cnvimadfsn 3sstr4d suppimacnvss eqssd ) ABIDCIJZADUAUCZAUBUDDUEUFUQZW
      CEKZFKZALZFMZWHWGDNZOZFMZJZEUGWFGKZALZWNDNZJZGMZEUGZWDWEWCWMWREWMWRPWCWIW
      LWRWIWFHKZALZHMWLWRPZWHXAFHWGWTWFAQUHXAXBHWLXAWRWKXAWRPFXAWKWRXAWKJZWRPWT
      DWTDRZXCWFDALZWKJZWRXDXAXEWKWTDWFAQUIWJXFWRPZWJWKWRXEWKWJWRWJWKWJWRPZWJWK
      JWHXHWJWHUJWHWJWRWQWHWJJGWGFUKGFULWOWHWPWJWNWGWFAQWNWGDUMUNUOSUPSURUSWJUT
      ZXGXIWGDRZXIXGPWGDVAXFXIXJWRXEWKXIXJWRPZPXIWKXEXKWKWHUTZXIOZXIXEXKPZWHWJV
      BXIXMXNXIXMJXLXNXIXLUJXJXEXLWRXJXEWHXLWRPXEWHODWGDWGWFAQVCWHWRVDVETUPSVFT
      VHTVIVGVJVEXCWTDNZWRXAXOWRPWKXAXOWRWQXAXOJGWTHUKGHULWOXAWPXOWNWTWFAQWNWTD
      UMUNUOSVKVLVMVNVOVLVOVIVHVRVPEFABCDVQWEWSRWCEGADVSVRVTABCDWAWB $.
  $}

  $( 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 wi cin eqtrd fex imp wfun
    expcom adantr suppimacnv syl2anc ffun inpreima syl wss cdm cnvimass fimacnv
    simplr fdm eqtr4d sseqtrid sseqin2 sylib invdif imaeq2i eqtr3di adantl ex )
    CDGZFEGZHZCABIZBFJKZBLZAFMZNZOZPVHVIHZVJVKQVLNZOZVNVOBQGZVGVJVQPVHVIVRVFVIV
    RRVGVIVFVRCADBUAUDUEUBVFVGVIUOBQEFUFUGVIVQVNPVHVIVKAVPSZOZVQVNVIVTVKAOZVQSZ
    VQVIBUCVTWBPCABUHAVPBUIUJVIVQWAUKWBVQPVIBULZVQWABVPUMVIWCCWACABUPCABUNUQURV
    QWAUSUTTVSVMVKAVLVAVBVCVDTVE $.

  $( Version of ~ fsuppeq avoiding ~ ax-rep 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 suppimacnv cin wfun sylib
    ffun inpreima syl wss cdm cnvimass fdm fimacnv eqtr4d sseqtrid eqtrd invdif
    sseqin2 imaeq2i eqtr3di sylan9eq ex ) BDGFEGHZCABIZBFJKZBLZAFMZNZOZPUSUTVAV
    BQVCNZOZVEBDEFRUTVBAVFSZOZVGVEUTVIVBAOZVGSZVGUTBTVIVKPCABUBAVFBUCUDUTVGVJUE
    VKVGPUTBUFZVGVJBVFUGUTVLCVJCABUHCABUIUJUKVGVJUNUAULVHVDVBAVCUMUOUPUQUR $.
  $( $j usage 'fsuppeqg' avoids 'ax-rep'; $)

  ${
    $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.) $)
    suppssdm $p |- ( F supp Z ) C_ dom F $=
      ( vi cvv wcel wa csupp co cdm wss cv csn cima wne suppval ssrab2 eqsstrdi
      crab wn c0 supp0prc 0ss pm2.61i ) ADEBDEFZABGHZAIZJUDUEACKLMBLNZCUFRUFCDD
      ABOUGCUFPQUDSUETUFABUAUFUBQUC $.
  $}

  ${
    $d F x $.  $d X x $.  $d Z x $.
    suppsnop.f $e |- F = { <. X , Y >. } $.
    $( The support of a singleton of an ordered pair.  (Contributed by AV,
       12-Apr-2019.) $)
    suppsnop $p |- ( ( X e. V /\ Y e. W /\ Z e. U )
                   -> ( F supp Z ) = if ( Y = Z , (/) , { X } ) ) $=
      ( vx wcel csn cima wne crab c0 cif wceq cvv wf neeq1d w3a csupp co cv cdm
      cop wa wf1o f1osng f1of syl 3adant3 feq1i sylibr snex fex sylancl suppval
      simp3 syl2anc fdmd rabeqdv sneq imaeq2d rabsnif eqtrdi cfv wfn ffnd snidg
      3ad2ant1 fnsnfv eqcomd fveq1i fvsng eqtrid sneqd sneqbg necon3abid 3bitrd
      wn wb 3ad2ant2 ifbid ifnot 3eqtrd ) ECJZFDJZGAJZUAZBGUBUCZBIUDZKZLZGKZMZI
      BUEZNZBEKZLZWOMZWSOPZFGQZOWSPZWJBRJZWIWKWRQWJWSFKZBSZWSRJXEWJWSXFEFUFKZSZ
      XGWGWHXIWIWGWHUGWSXFXHUHXIEFCDUIWSXFXHUJUKULWSXFBXHHUMUNZEUOWSXFRBUPUQWGW
      HWIUSIRABGURUTWJWRWPIWSNXBWJWPIWQWSWJWSXFBXJVAVBWPXAIEWLEQZWNWTWOXKWMWSBW
      LEVCVDTVEVFWJXBXCWAZWSOPXDWJXAXLWSOWJXAEBVGZKZWOMXFWOMXLWJWTXNWOWJBWSVHZE
      WSJZWTXNQWJWSXFBXJVIWGWHXPWIECVJVKXOXPUGXNWTWSEBVLVMUTTWJXNXFWOWJXMFWJXME
      XHVGZFEBXHHVNWGWHXQFQWIEFCDVOULVPVQTWJXCXFWOWHWGXFWOQXCWBWIFGDVRWCVSVTWDX
      CWSOWEVFWF $.
  $}

  $( The support of a singleton containing an ordered pair is a subset of the
     singleton containing the first element of the ordered pair, i.e. it is
     empty or the singleton itself.  (Contributed by AV, 19-Jul-2019.) $)
  snopsuppss $p |- ( { <. X , Y >. } supp Z ) C_ { X } $=
    ( cop csn csupp co cdm suppssdm dmsnopss sstri ) ABDEZCFGLHAELCIABJK $.

  $( 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 co cab eleq2s ad2antrl wi wceq
      cres cv cin elinel2 dmres wss snssi resima2 neeq1d biimpd adantld wn elin
      syl pm2.24 adantr sylbi pm2.61i jca ex ss2abdv df-rab 3sstr4g cvv suppval
      com12 resexg sylan 3sstr4d ) BCGZEDGZHZBAUAZFUBZIZJZEIZKZFVMLZMZBVOJZVQKZ
      FBLZMZVMENOZBENOVLVNVSGZVRHZFPVNWCGZWBHZFPVTWDVLWGWIFVLWGWIVLWGHZWHWBWFWH
      VLVRWHVNAWCUCZVSVNAWCUDBAUEZQRVNAGZWJWBSWMWGWBVLWMVRWBWFWMVRWBWMVPWAVQWMV
      OAUFVPWATVNAUGBVOAUHUNUIUJUKUKWJWMULZWBWFWNWBSZVLVRWOVNWKVSVNWKGWMWHHWOVN
      AWCUMWMWOWHWMWBUOUPUQWLQRVFURUSUTVAVRFVSVBWBFWCVBVCVJVMVDGVKWEVTTBACVGFVD
      DVMEVEVHFCDBEVEVI $.
  $}

  ${
    suppun.g $e |- ( ph -> G e. V ) $.
    $( The support of a class/function is a subset of the support of the union
       of this class/function with another class/function.  (Contributed by AV,
       4-Jun-2019.) $)
    suppun $p |- ( ph -> ( F supp Z ) C_ ( ( F u. G ) supp Z ) ) $=
      ( cvv wcel wa csupp co cun wss wi ccnv csn cdif cima wceq suppimacnv a1i
      ssun1 cnvun imaeq1i imaundir eqtri sseqtrri adantr adantlr sylan2 syl2anc
      unexg simplr 3sstr4d ex wn c0 supp0prc 0ss eqsstrdi a1d pm2.61i ) BGHZEGH
      ZIZABEJKZBCLZEJKZMZNVEAVIVEAIZBOZGEPQZRZVGOZVLRZVFVHVMVOMVJVMVMCOZVLRZLZV
      OVMVQUBVOVKVPLZVLRVRVNVSVLBCUCUDVKVPVLUEUFUGUAVEVFVMSABGGETUHVJVGGHZVDVHV
      OSAVECDHZVTFVCWAVTVDBCGDULUIUJVCVDAUMVGGGETUKUNUOVEUPZVIAWBVFUQVHBEURVHUS
      UTVAVB $.
  $}

  ${
    $d B x z $.  $d F x z $.  $d V x $.  $d W x $.  $d Z x z $.
    $( The support of the restriction of a function is a subset of the support
       of the function itself.  (Contributed by AV, 22-Apr-2019.) $)
    ressuppssdif $p |- ( ( F e. V /\ Z e. W ) -> ( F supp Z )
                         C_ ( ( ( F |` B ) supp Z ) u. ( dom F \ B ) ) ) $=
      ( vz vx wcel wa cv csn cima wne cdm crab wss wi wn ex wceq cres cun csupp
      cdif co eldif wo weq sneq imaeq2d neeq1d elrab ianor xchnxbir dmres elin2
      simpl anim2i ancomd sylibr pm2.24 adantr com12 sylbi adantl snssi resima2
      syl eqcomd simpr eqtrd necon3d impancom con3d impcom eldifd syl2anb ssrdv
      jaoi a1i ssundif suppval cvv resexg sylan uneq1d 3sstr4d ) BCHZEDHZIZBFJZ
      KZLZEKZMZFBNZOZBAUAZWLLZWNMZFWRNZOZWPAUDZUBZBEUCUEWREUCUEZXCUBWJWQXBUDZXC
      PWQXDPWJGXFXCGJZXFHZXGXCHZQWJXHXGWQHZXGXBHZRZIXIXGWQXBUFXJXGWPHZBXGKZLZWN
      MZIZXGXAHZRZWRXNLZWNMZRZUGZXIXLWOXPFXGWPFGUHZWMXOWNYDWLXNBWKXGUIZUJUKULXR
      YAIYCXKXRYAUMWTYAFXGXAYDWSXTWNYDWLXNWRYEUJUKULUNYCXQXIXSXQXIQZYBXSXGAHZRZ
      XMRZUGZYFYGXMIYJXRYGXMUMXGAWPXABAUOUPUNYHYFYIYHXQXIYHXQIZXMYHIXIYKYHXMXQX
      MYHXMXPUQZURUSXGWPAUFUTSXQYIXIXMYIXIQXPXMXIVAVBVCVSVDYBXQXIYBXQIXGWPAXQXM
      YBYLVEXQYBYHXQYGYAXMYGXPYAXMYGIZXTWNXOWNYMXTWNTZXOWNTYMYNIXOXTWNYMXOXTTYN
      YMXTXOYMXNAPZXTXOTYGYOXMXGAVFVEBXNAVGVHVIVBYMYNVJVKSVLVMVNVOVPSVSVOVQVDVT
      VRWQXBXCWAUTFCDBEWBWJXEXBXCWHWRWCHWIXEXBTBACWDFWCDWREWBWEWFWG $.
  $}

  ${
    $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 wceq cmpt eqeltrid suppimacnv
      mptexd syl2anc mptpreima eqtrdi ) AEHLMZENOHPQZRZDUJSBCTAEOSHGSUIUKUAAEBC
      DUBOIABCDFJUEUCKEOGHUDUFBCDUJEIUGUH $.

    $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 n $.  $d B n $.  $d Z n $.  $d ph n $.
    extmptsuppeq.b $e |- ( ph -> B e. W ) $.
    extmptsuppeq.a $e |- ( ph -> A C_ B ) $.
    extmptsuppeq.z $e |- ( ( ph /\ n e. ( B \ A ) ) -> X = Z ) $.
    $( The support of an extended function is the same as the original.
       (Contributed by Mario Carneiro, 25-May-2015.)  (Revised by AV,
       30-Jun-2019.) $)
    extmptsuppeq $p |- ( ph -> ( ( n e. A |-> X ) supp Z ) =
                               ( ( n e. B |-> X ) supp Z ) ) $=
      ( cvv wcel cmpt csupp co wceq wa adantl wn c0 wi csn cdif crab wss anim1d
      cv sseld eldif adantll sylan2br expr elsn2g biimtrrdi ad2antrr syld con4d
      elndif impr simprr jca ex impbid rabbidva2 eqid ssexd mptsuppdifd 3eqtr4d
      simpl simpr supp0prc nsyl5 eqtr4d a1d pm2.61i ) GKLZADBFMZGNOZDCFMZGNOZPZ
      UAVPAWAVPAQZFKGUBZUCLZDBUDWDDCUDVRVTWBWDWDDBCWBDUGZBLZWDQZWECLZWDQZWBWFWH
      WDWBBCWEABCUEVPIRUHUFWBWIWGWBWIQWFWDWBWHWDWFWBWHQZWFWDWJWFSZFGPZWDSZWBWHW
      KWLWHWKQWBWECBUCLZWLWECBUIAWNWLVPJUJUKULVPWLWMUAAWHVPWLFWCLWMFGKUMFWCKURU
      NUOUPUQUSWBWHWDUTVAVBVCVDWBDBFVQKKGVQVEABKLVPABCEHIVFRVPAVIZVGWBDCFVSEKGV
      SVEACELVPHRWOVGVHVBVPSZWAAWPVRTVTVQKLZVPQVPVRTPWQVPVJVQGVKVLVSKLZVPQVPVTT
      PWRVPVJVSGVKVLVMVNVO $.
  $}

  ${
    $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 $.
    $( 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 cvv wcel wfun wss w3a csupp co wi wa cdm wfn wceq adantr simpr c0 cv
      cfv wral funss impcom funfnd funfn biimpi jca 3adant3 dmss 3ad2ant2 dmexg
      3ad2ant3 3jca funssfv 3expa eqeq1 biimpd syl ralrimiva suppfnss expcom wn
      sylc ssid supp0prc nsyl5 sseq12d mpbiri a1d pm2.61i ) DFGZBHZABIZBCGZJZAD
      KLZBDKLZIZMVQVMVTVQVMNZAAOZPZBBOZPZNZWBWDIZWDFGZVMJZNEUAZBUBZDQZWJAUBZDQZ
      MZEWBUCZVTWAWFWIVQWFVMVNVOWFVPVNVONZWCWEWQAVOVNAHABUDUEUFVNWEVOVNWEBUGUHR
      UIUJRWAWGWHVMVQWGVMVOVNWGVPABUKULRVQWHVMVPVNWHVOBCUMUNRVQVMSUOUIVQWPVMVNV
      OWPVPWQWOEWBWQWJWBGZNWKWMQZWOVNVOWRWSWJBAUPUQWSWLWNWKWMDURUSUTVAUJREWBWDA
      BFFDVBVEVCVMVDZVTVQWTVTTTITVFWTVRTVSTAFGZVMNVMVRTQXAVMSADVGVHBFGZVMNVMVST
      QXBVMSBDVGVHVIVJVKVL $.
  $}

  ${
    $d F a $.  $d V a $.  $d A a $.  $d B a $.  $d Z a $.  $d W a $.
    $( Two ways to express restriction of a support set.  (Contributed by
       Stefan O'Rear, 5-Feb-2015.)  (Revised by AV, 28-May-2019.) $)
    fnsuppres $p |- ( ( F Fn ( A u. B ) /\ ( F e. W /\ Z e. V )
                        /\ ( A i^i B ) = (/) ) -> ( ( F supp Z ) C_ A
                                         <-> ( F |` B ) = ( B X. { Z } ) ) ) $=
      ( va cun wfn wcel wa wceq w3a cfv crab wss wral sseq1d wn wb cin wne cres
      c0 cv cdm csn cxp csupp fndm rabeqdv 3ad2ant1 unss ssrab2 biantrur rabun2
      co sseq1i 3bitr4ri wi rabss fvres adantl simp2r fvconst2g eqeq12d nne a1i
      sylan id simp3 minel syl2anr mtt syl 3bitr2rd ralbidva bitrid bitrd fnfun
      wfun 3anim1i 3expb suppval1 3adant3 simp1 ssun2 fnssres fnconstg 3ad2ant2
      syl2anc eqfnfv 3bitr4d ) CABHZIZCEJZFDJZKZABUAUDLZMZGUEZCNZFUBZGCUFZOZAPZ
      XACBUCZNZXABFUGUHZNZLZGBQZCFUIUQZAPXGXILZWTXFXCGWNOZAPZXLWTXEXOAWOWRXEXOL
      WSWOXCGXDWNWNCUJUKULRXPXCGBOZAPZWTXLXCGAOZAPZXRKXSXQHZAPXRXPXSXQAUMXTXRXC
      GAUNUOXOYAAXCGABUPURUSXRXCXAAJZUTZGBQWTXLXCGBAVAWTYCXKGBWTXABJZKZXKXBFLZX
      CSZYCYEXHXBXJFYDXHXBLWTXABCVBVCWTWQYDXJFLWOWPWQWSVDBFXADVEVIVFYGYFTYEXBFV
      GVHYEYBSZYGYCTYDYDWSYHWTYDVJWOWRWSVKXABAVLVMYBXCVNVOVPVQVRVRVSWTXMXEAWOWR
      XMXELZWSWOWRKCWAZWPWQMZYIWOWPWQYKWOYJWPWQWNCVTWBWCGEDCFWDVOWERWTXGBIZXIBI
      ZXNXLTWTWOBWNPZYLWOWRWSWFYNWTBAWGVHWNBCWHWKWRWOYMWSWQYMWPBFDWIVCWJGBXGXIW
      LWKWM $.

    $( The support of a function is empty iff it is identically zero.
       (Contributed by Stefan O'Rear, 22-Mar-2015.)  (Revised by AV,
       28-May-2019.) $)
    fnsuppeq0 $p |- ( ( F Fn A /\ A e. W /\ Z e. V )
                      -> ( ( F supp Z ) = (/) <-> F = ( A X. { Z } ) ) ) $=
      ( wfn wcel w3a csupp co c0 wceq cres csn cxp wss ss0b cun cvv 3ad2ant1 wb
      cin un0 uncom eqtr3i fneq2i biimpi fnex 3adant3 simp3 fnsuppres syl121anc
      0in a1i bitr3id fnresdm eqeq1d bitrd ) BAFZADGZECGZHZBEIJZKLZBAMZAENOZLZB
      VFLVDVCKPZVBVGVCQVBBKARZFZBSGZVAKAUBKLZVHVGUAUSUTVJVAUSVJAVIBAKRAVIAUCAKU
      DUEUFUGTUSUTVKVAADBUHUIUSUTVAUJVLVBAUMUNKABCSEUKULUOVBVEBVFUSUTVEBLVAABUP
      TUQUR $.
  $}

  $( The support of a constant function with value zero is empty.  (Contributed
     by AV, 30-Jun-2019.) $)
  fczsupp0 $p |- ( ( B X. { Z } ) supp Z ) = (/) $=
    ( csn cxp cvv wcel wa csupp co c0 wceq eqidd wb fnconstg adantl snnzg simpl
    wfn wne xpexcnv syl2an2 simpr fnsuppeq0 syl3anc mpbird supp0prc pm2.61i ) A
    BCZDZEFZBEFZGZUIBHIJKZULUMUIUIKZULUILULUIARZAEFZUKUMUNMUKUOUJABENOUKUHJSUJU
    JUPBEPUJUKQAUHTUAUJUKUBAUIEEBUCUDUEUIBUFUG $.

  ${
    $d k F $.  $d k ph $.  $d k W $.  $d k Z $.
    suppss.f $e |- ( ph -> F : A --> B ) $.
    suppss.n $e |- ( ( ph /\ k e. ( A \ W ) ) -> ( F ` k ) = Z ) $.
    $( 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.) $)
    suppss $p |- ( ph -> ( F supp Z ) C_ W ) $=
      ( cvv wcel wa csupp co wss wi cv cfv wne wn wfn ffnd adantl simpll simplr
      wb elsuppfng syl3anc wceq cdif eldif adantll sylan2br expr expimpd sylbid
      necon1ad ssrdv ex c0 supp0prc 0ss eqsstrdi a1d pm2.61i ) EJKZGJKZLZAEGMNZ
      FOZPVHAVJVHALZDVIFVKDQZVIKZVLBKZVLERZGSZLZVLFKZVKEBUAZVFVGVMVQUFAVSVHABCE
      HUBUCVFVGAUDVFVGAUEVLEJJBGUGUHVKVNVPVRVKVNLVRVOGVKVNVRTZVOGUIZVNVTLVKVLBF
      UJKZWAVLBFUKAWBWAVHIULUMUNUQUOUPURUSVHTZVJAWCVIUTFEGVAFVBVCVDVE $.
    $( $j usage 'suppss' avoids 'ax-rep'; $)
  $}

  ${
    suppssr.f $e |- ( ph -> F : A --> B ) $.
    suppssr.n $e |- ( ph -> ( F supp Z ) C_ W ) $.
    suppssr.a $e |- ( ph -> A e. V ) $.
    suppssr.z $e |- ( ph -> Z e. U ) $.
    $( A function is zero outside its support.  (Contributed by Mario Carneiro,
       19-Dec-2014.)  (Revised by AV, 28-May-2019.) $)
    suppssr $p |- ( ( ph /\ X e. ( A \ W ) ) -> ( F ` X ) = Z ) $=
      ( cdif wcel wn wa cfv cvv wb wceq eldif wne fvex eldifsn mpbiran csupp co
      csn wfn ffnd elsuppfn syl3anc ibar bitr4di pm5.32da bitrd sylbird expdimp
      mp1i sseld biimtrrid necon1bd impr sylan2b ) HBGNOAHBOZHGOZPZQHERZIUAZHBG
      UBAVFVHVJAVFQZVGVIIVIIUCZVISIUINOZVKVGVMVISOZVLHEUDZVISIUEZUFAVFVMVGAVFVM
      QZHEIUGUHZOZVGAVSVFVLQZVQAEBUJBFOIDOVSVTTABCEJUKLMHEFDBIULUMAVFVLVMVKVLVN
      VLQZVMVNVLWATVKVOVNVLUNUTVPUOUPUQAVRGHKVAURUSVBVCVDVE $.
  $}

  ${
    suppssrg.f $e |- ( ph -> F : A --> B ) $.
    suppssrg.n $e |- ( ph -> ( F supp Z ) C_ W ) $.
    suppssrg.a $e |- ( ph -> F e. V ) $.
    suppssrg.z $e |- ( ph -> Z e. U ) $.
    $( A function is zero outside its support.  Version of ~ suppssr avoiding
       ~ ax-rep by assuming ` F ` is a set rather than its domain ` A ` .
       (Contributed by SN, 5-May-2024.) $)
    suppssrg $p |- ( ( ph /\ X e. ( A \ W ) ) -> ( F ` X ) = Z ) $=
      ( cdif wcel wn wa cfv wceq eldif wne csupp co wfn wb ffnd elsuppfng sseld
      syl3anc sylbird expdimp necon1bd impr sylan2b ) HBGNOAHBOZHGOZPZQHERZISZH
      BGTAUOUQUSAUOQUPURIAUOURIUAZUPAUOUTQZHEIUBUCZOZUPAEBUDEFOIDOVCVAUEABCEJUF
      LMHEFDBIUGUIAVBGHKUHUJUKULUMUN $.
  $}

  ${
    $d ph v $.  $d ph x $.  $d B v $.  $d D x $.  $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 ) supp 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 ) $.
    suppssov1.y $e |- ( ph -> Y e. W ) $.
    $( Formula building theorem for support restrictions: operator with left
       annihilator.  (Contributed by Stefan O'Rear, 9-Mar-2015.)  (Revised by
       AV, 28-May-2019.)  (Proof shortened by SN, 11-Apr-2025.) $)
    suppssov1 $p |- ( ph -> ( ( x e. D |-> ( A O B ) ) supp Z ) C_ L ) $=
      ( cvv wcel wa co cmpt csupp wss csn cdif crab cv wne elexd adantlr adantr
      wceq oveq2 eqeq1d wral ad2antrr rspcdva oveq1 syl5ibrcom necon3d eldifsni
      ralrimiva impel eldifsn sylanbrc ss2rabdv eqid simprl mptsuppdifd 3sstr4d
      ex simprr sstrd wn c0 mptexg cdm ovex rgenw dmmptg ax-mp eqeltrrid impbii
      dmexg anbi1i supp0prc sylnbi 0ss eqsstrdi adantl pm2.61dan ) AFSTZMSTZUAZ
      BFDEIUBZUCZMUDUBZHUEZAWPUAZWSBFDUCZLUDUBZHXAWQSMUFUGTZBFUHDSLUFUGTZBFUHWS
      XCXAXDXEBFXABUIFTZUAZXDXEXGXDUADSTZDLUJZXEXGXHXDAXFXHWPAXFUADJPUKULUMXGWQ
      MUJXIXDXGDLWQMXGWQMUNDLUNZLEIUBZMUNZXGLCUIZIUBZMUNZXLCGEXMEUNXNXKMXMELIUO
      UPAXOCGUQWPXFAXOCGOVDURAXFEGTWPQULUSXJWQXKMDLEIUTUPVAVBWQSMVCVEDSLVFVGVMV
      HXABFWQWRSSMWRVIAWNWOVJZAWNWOVNVKXABFDXBSKLXBVIXPALKTWPRUMVKVLAXCHUEWPNUM
      VOWPVPZWTAXQWSVQHWPWRSTZWOUAWSVQUNWNXRWOWNXRBFWQSVRXRFWRVSZSWQSTZBFUQXSFU
      NXTBFDEIVTWABFWQSWBWCWRSWFWDWEWGWRMWHWIHWJWKWLWM $.
  $}

  ${
    $d ph v $.  $d ph x $.  $d A v $.  $d D x $.  $d O v $.  $d R v $.
    $d Y v $.  $d Y x $.  $d Z v $.  $d Z x $.
    suppssov2.s $e |- ( ph -> ( ( x e. D |-> B ) supp Y ) C_ L ) $.
    suppssov2.o $e |- ( ( ph /\ v e. R ) -> ( v O Y ) = Z ) $.
    suppssov2.a $e |- ( ( ph /\ x e. D ) -> A e. R ) $.
    suppssov2.b $e |- ( ( ph /\ x e. D ) -> B e. V ) $.
    suppssov2.y $e |- ( ph -> Y e. W ) $.
    $( Formula building theorem for support restrictions: operator with right
       annihilator.  (Contributed by SN, 11-Apr-2025.) $)
    suppssov2 $p |- ( ph -> ( ( x e. D |-> ( A O B ) ) supp Z ) C_ L ) $=
      ( cvv wcel wa co cmpt csupp wss csn cdif crab cv wne elexd adantlr adantr
      wceq oveq1 eqeq1d wral ad2antrr rspcdva oveq2 syl5ibrcom necon3d eldifsni
      ralrimiva impel eldifsn sylanbrc ss2rabdv eqid simprl mptsuppdifd 3sstr4d
      ex simprr sstrd wn c0 mptexg cdm ovex rgenw dmmptg ax-mp eqeltrrid impbii
      dmexg anbi1i supp0prc sylnbi 0ss eqsstrdi adantl pm2.61dan ) AFSTZMSTZUAZ
      BFDEIUBZUCZMUDUBZHUEZAWPUAZWSBFEUCZLUDUBZHXAWQSMUFUGTZBFUHESLUFUGTZBFUHWS
      XCXAXDXEBFXABUIFTZUAZXDXEXGXDUAESTZELUJZXEXGXHXDAXFXHWPAXFUAEJQUKULUMXGWQ
      MUJXIXDXGELWQMXGWQMUNELUNZDLIUBZMUNZXGCUIZLIUBZMUNZXLCGDXMDUNXNXKMXMDLIUO
      UPAXOCGUQWPXFAXOCGOVDURAXFDGTWPPULUSXJWQXKMELDIUTUPVAVBWQSMVCVEESLVFVGVMV
      HXABFWQWRSSMWRVIAWNWOVJZAWNWOVNVKXABFEXBSKLXBVIXPALKTWPRUMVKVLAXCHUEWPNUM
      VOWPVPZWTAXQWSVQHWPWRSTZWOUAWSVQUNWNXRWOWNXRBFWQSVRXRFWRVSZSWQSTZBFUQXSFU
      NXTBFDEIVTWABFWQSWBWCWRSWFWDWEWGWRMWHWIHWJWKWLWM $.
  $}

  ${
    $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 supp 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 ) $.
    suppssof1.y $e |- ( ph -> Y e. U ) $.
    $( Formula building theorem for support restrictions: vector operation with
       left annihilator.  (Contributed by Stefan O'Rear, 9-Mar-2015.)  (Revised
       by AV, 28-May-2019.) $)
    suppssof1 $p |- ( ph -> ( ( A oF O B ) supp Z ) C_ L ) $=
      ( vx cof co csupp cv cfv cmpt ffnd inidm wcel eqidd offval oveq1d feqmptd
      wa cvv eqsstrrd fvexd ffvelcdmda suppssov1 eqsstrd ) ACDIUAUBZMUCUBTETUDZ
      CUEZVBDUEZIUBUFZMUCUBHAVAVEMUCATEEVCVDIECDKKAEJCPUGAEFDQUGRREUHAVBEUIUNZV
      CUJVFVDUJUKULATBVCVDEFHIUOGLMATEVCUFZLUCUBCLUCUBHACVGLUCATEJCPUMULNUPOVFV
      BCUQAEFVBDQURSUSUT $.
  $}

  ${
    $d k A $.  $d k ph $.  $d k W $.  $d k Z $.
    suppss2.n $e |- ( ( ph /\ k e. ( A \ W ) ) -> B = Z ) $.
    suppss2.a $e |- ( ph -> A e. V ) $.
    $( Show that the support of a function is contained in a set.  (Contributed
       by Mario Carneiro, 19-Dec-2014.)  (Revised by Mario Carneiro,
       22-Mar-2015.)  (Revised by AV, 28-May-2019.) $)
    suppss2 $p |- ( ph -> ( ( k e. A |-> B ) supp Z ) C_ W ) $=
      ( cvv wcel cmpt csupp co wss wa cdif wn wceq c0 wi crab eqid adantl simpl
      csn mptsuppdifd cv wne eldifsni eldif adantll sylan2br expr necon1ad syl5
      3impia rabssdv eqsstrd ex intnand supp0prc syl 0ss eqsstrdi a1d pm2.61i
      id ) GJKZADBCLZGMNZFOZUAVIAVLVIAPZVKCJGUFQKZDBUBFVMDBCVJEJGVJUCABEKVIIUDV
      IAUEUGVMVNDBFVMDUHZBKZVNVOFKZVNCGUIVMVPPZVQCJGUJVRVQCGVMVPVQRZCGSZVPVSPVM
      VOBFQKZVTVOBFUKAWAVTVIHULUMUNUOUPUQURUSUTVIRZVLAWBVKTFWBVJJKZVIPRVKTSWBVI
      WCWBVHVAVJGVBVCFVDVEVFVG $.
  $}

  ${
    $d k A $.  $d k ph $.  $d k W $.  $d k Z $.
    suppsssn.n $e |- ( ( ph /\ k e. A /\ k =/= W ) -> B = Z ) $.
    suppsssn.a $e |- ( ph -> A e. V ) $.
    $( Show that the support of a function is a subset of a singleton.
       (Contributed by AV, 21-Jul-2019.) $)
    suppsssn $p |- ( ph -> ( ( k e. A |-> B ) supp Z ) C_ { W } ) $=
      ( csn cv cdif wcel wne wa wceq eldifsn 3expb sylan2b suppss2 ) ABCDEFJZGD
      KZBUALMAUBBMZUBFNZOCGPZUBBFQAUCUDUEHRSIT $.
  $}

  ${
    $d ph x $.  $d D x $.  $d Y x $.  $d Z x $.
    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 ) $.
    $( 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.) $)
    suppssfv $p |- ( ph -> ( ( x e. D |-> ( F ` A ) ) supp Z ) C_ L ) $=
      ( cvv wcel wa cfv cmpt wceq csupp co wss wi csn cdif crab wne cv eldifsni
      ad4ant23 fveqeq2 syl5ibrcom necon3d ad2antlr imp eldifsn sylanbrc ex syl5
      elexd ss2rabdv eqid simpll simplr mptsuppdifd adantl 3sstr4d sstrd mptexg
      wn c0 wral fvex rgenw dmmptg ax-mp dmexg eqeltrrid impbii anbi1i supp0prc
      cdm sylnbi 0ss eqsstrdi a1d pm2.61i ) DOPZJOPZQZABDCFRZSZJUAUBZGUCZUDWKAW
      OWKAQZWNBDCSZIUAUBZGWPWLOJUEUFPZBDUGCOIUEUFPZBDUGWNWRWPWSWTBDWSWLJUHZWPBU
      IDPZQZWTWLOJUJXCXAWTXCXAQCOPZCIUHZWTAXBXDWKXAAXBQCHMVAUKXCXAXEAXAXEUDWKXB
      ACIWLJAWLJTCITIFRJTLCIJFULUMUNUOUPCOIUQURUSUTVBWPBDWLWMOOJWMVCWIWJAVDZWIW
      JAVEVFWPBDCWQOEIWQVCXFAIEPWKNVGVFVHAWRGUCWKKVGVIUSWKVKZWOAXGWNVLGWKWMOPZW
      JQWNVLTWIXHWJWIXHBDWLOVJXHDWMWCZOWLOPZBDVMXIDTXJBDCFVNVOBDWLOVPVQWMOVRVSV
      TWAWMJWBWDGWEWFWGWH $.
  $}

  ${
    $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 $.
    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 ) $.
    ${
      $d G k $.  $d B y $.  $d F k $.  $d X k $.  $d Z k $.  $d k ph $.
      suppofssd.5 $e |- ( ph -> ( Z X Z ) = Z ) $.
      $( Condition for the support of a function operation to be a subset of
         the union of the supports of the left and right function terms.
         (Contributed by Steven Nguyen, 28-Aug-2023.) $)
      suppofssd $p |- ( ph -> ( ( F oF X G ) supp Z ) C_
                                ( ( F supp Z ) u. ( G supp Z ) ) ) $=
        ( co cv wcel wa wn wceq adantr vk vx cvv cof csupp cun ovexd inidm cdif
        vy off cfv eldif ioran elun xchnxbir anbi2i bitri wne wfn ffnd elsuppfn
        wo wb syl3anc notbid biimpd anim12d anim2d imp pm3.2 necon1bd imdistani
        simprl fnfvof syl22anc oveq12 ad2antll 3eqtrd sylan2 syldan sylan2b
        suppss ) ABUCUADEGUDNZDHUENZEHUENZUFZHAUBUJBBBGCCUCDEFFAUBOZCPUJOZCPQQW
        HWIGUGKLIIBUHUKUAOZBWGUIPZAWJBPZWJWEPZRZWJWFPZRZQZQZWJWDULZHSZWKWLWJWGP
        ZRZQWRWJBWGUMXBWQWLWMWOVCWQXAWMWOUNWJWEWFUOUPUQURAWRWLWLWJDULZHUSZQZRZW
        LWJEULZHUSZQZRZQZQZWTAWRXLAWQXKWLAWNXFWPXJAWNXFAWMXEADBUTZBFPZHCPZWMXEV
        DABCDKVAZIJWJDFCBHVBVEVFVGAWPXJAWOXIAEBUTZXNXOWOXIVDABCELVAZIJWJEFCBHVB
        VEVFVGVHVIVJXLAWLXCHSZXGHSZQZQZWTWLXKYAWLXFXSXJXTWLXEXCHWLXDVKVLWLXIXGH
        WLXHVKVLVHVMAYBQZWSXCXGGNZHHGNZHYCXMXQXNWLWSYDSAXMYBXPTAXQYBXRTAXNYBITA
        WLYAVNBGDEFWJVOVPYAYDYESAWLXCHXGHGVQVRAYEHSYBMTVSVTWAWBWC $.
    $}

    ${
      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.) $)
      suppofss1d $p |- ( ph -> ( ( F oF X G ) supp Z ) C_ ( F supp Z ) ) $=
        ( vy cfv wceq co wcel wa cv cof wral csupp wss inidm eqidd ofval adantr
        wi simpr oveq1d ralrimiva ffvelcdmda oveq2d eqeq1d rspcdv mpd 3eqtrd ex
        ffnd wfn offn ssidd suppfnss syl23anc ) AOUAZEPZIQZVGEFHUBRZPZIQZUJZOCU
        CZVJIUDREIUDRUEZAVMOCAVGCSZTZVIVLVQVITZVKVHVGFPZHRZIVSHRZIVQVKVTQVIACCV
        HVSHCEFGGVGACDELVAZACDFMVAZJJCUFZVQVHUGVQVSUGUHUIVRVHIVSHVQVIUKULVQWAIQ
        ZVIVQIBUAZHRZIQZBDUCZWEAWIVPAWHBDNUMUIVQWHWEBVSDACDVGFMUNVQWFVSQZTZWGWA
        IWKWFVSIHVQWJUKUOUPUQURUIUSUTUMAVJCVBECVBCCUECGSIDSVNVOUJACCHCEFGGWBWCJ
        JWDVCWBACVDJKOCCVJEGDIVEVFUR $.
    $}

    ${
      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.) $)
      suppofss2d $p |- ( ph -> ( ( F oF X G ) supp Z ) C_ ( G supp Z ) ) $=
        ( vy cfv wceq co wcel wa cv cof wral csupp wss inidm eqidd ofval adantr
        wi simpr oveq2d ralrimiva ffvelcdmda oveq1d eqeq1d rspcdv mpd 3eqtrd ex
        ffnd wfn offn ssidd suppfnss syl23anc ) AOUAZFPZIQZVGEFHUBRZPZIQZUJZOCU
        CZVJIUDRFIUDRUEZAVMOCAVGCSZTZVIVLVQVITZVKVGEPZVHHRZVSIHRZIVQVKVTQVIACCV
        SVHHCEFGGVGACDELVAZACDFMVAZJJCUFZVQVSUGVQVHUGUHUIVRVHIVSHVQVIUKULVQWAIQ
        ZVIVQBUAZIHRZIQZBDUCZWEAWIVPAWHBDNUMUIVQWHWEBVSDACDVGELUNVQWFVSQZTZWGWA
        IWKWFVSIHVQWJUKUOUPUQURUIUSUTUMAVJCVBFCVBCCUECGSIDSVNVOUJACCHCEFGGWBWCJ
        JWDVCWCACVDJKOCCVJFGDIVEVFUR $.
    $}
  $}

  $( 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.)  Extract this statement from the proof of
     ~ supp0cosupp0 .  (Revised by SN, 15-Sep-2023.) $)
  suppco $p |- ( ( F e. V /\ G e. W ) ->
                       ( ( F o. G ) supp Z ) = ( `' G " ( F supp Z ) ) ) $=
    ( cvv wcel wa ccom csupp co ccnv cima wceq suppimacnv imaeq2d wn c0 intnand
    supp0prc wi csn cdif coexg simpl syl2an2 cnvco imaeq1i imaco simprl syl2anc
    a1i eqtr4id 3eqtrd ex prcnel syl ima0 eqtrdi eqtr4d a1d pm2.61i ) EFGZACGZB
    DGZHZABIZEJKZBLZAEJKZMZNZUAVCVFVLVCVFHZVHVGLZFEUBUCZMZVIALZIZVOMZVKVFVGFGZV
    CVCVHVPNABCDUDVCVFUEZVGFFEOUFVPVSNVMVNVRVOABUGUHULVMVSVIVQVOMZMVKVIVQVOUIVM
    VJWBVIVMVDVCVJWBNVCVDVEUJWAACFEOUKPUMUNUOVCQZVLVFWCVHRVKWCVTVCHQVHRNWCVCVTE
    FUPZSVGETUQWCVKVIRMRWCVJRVIWCAFGZVCHQVJRNWCVCWEWDSAETUQPVIURUSUTVAVB $.

  ${
    $d ph k $.  $d F k $.  $d G k $.  $d Y k $.  $d Z k $.
    suppcoss.f $e |- ( ph -> F Fn A ) $.
    suppcoss.g $e |- ( ph -> G : B --> A ) $.
    suppcoss.b $e |- ( ph -> B e. W ) $.
    suppcoss.y $e |- ( ph -> Y e. V ) $.
    suppcoss.1 $e |- ( ph -> ( F ` Y ) = Z ) $.
    $( The support of the composition of two functions is a subset of the
       support of the inner function if the outer function preserves zero.
       Compare ~ suppssfv , which has a sethood condition on ` A ` instead of
       ` B ` .  (Contributed by SN, 25-May-2024.) $)
    suppcoss $p |- ( ph -> ( ( F o. G ) supp Z ) C_ ( G supp Y ) ) $=
      ( vk wcel cfv wceq wn wa crn ccom csupp co wfn wf dffn3 sylib fcod cv wne
      cdif eldif ffnd elsuppfn syl3anc notbid anbi2d annotanannot bitrdi bitrid
      wb nne anbi2i adantr simprl fvco3d simprr fveq2d 3eqtrd sylbid imp suppss
      ex biimtrid ) ACDUAZODEUBZEHUCUDZIACBVPDEADBUEBVPDUFJBDUGUHKUIAOUJZCVRULP
      ZVSVQQZIRZAVTVSCPZVSEQZHUKZSZTZWBVTWCVSVRPZSZTZAWGVSCVRUMAWJWCWCWETZSZTWG
      AWIWLWCAWHWKAECUECGPHFPWHWKVBACBEKUNLMVSEGFCHUOUPUQURWCWEUSUTVAWGWCWDHRZT
      ZAWBWFWMWCWDHVCVDAWNWBAWNTZWAWDDQHDQZIWOCBVSDEACBEUFWNKVEAWCWMVFVGWOWDHDA
      WCWMVHVIAWPIRWNNVEVJVNVOVKVLVM $.
  $}

  $( 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.) $)
  supp0cosupp0 $p |- ( ( F e. V /\ G e. W )
                  -> ( ( F supp Z ) = (/) -> ( ( F o. G ) supp Z ) = (/) ) ) $=
    ( wcel wa csupp co c0 wceq ccom ccnv cima suppco imaeq2 eqtrdi sylan9eq ex
    ima0 ) ACFBDFGZAEHIZJKZABLEHIZJKUAUCUDBMZUBNZJABCDEOUCUFUEJNJUBJUEPUETQRS
    $.

  $( The image of the support of the composition of two functions is the
     support of the outer function.  (Contributed by AV, 30-May-2019.) $)
  imacosupp $p |- ( ( F e. V /\ G e. W )
                    -> ( ( Fun G /\ ( F supp Z ) C_ ran G )
                         -> ( G " ( ( F o. G ) supp Z ) ) = ( F supp Z ) ) ) $=
    ( wcel wa wfun csupp co crn wss ccom cima wceq ccnv suppco imaeq2d cdm wfo
    funforn foimacnv sylanb sylan9eq ex ) ACFBDFGZBHZAEIJZBKZLZGZBABMEIJZNZUHOU
    FUKUMBBPUHNZNZUHUFULUNBABCDEQRUGBSZUIBTUJUOUHOBUAUPUIUHBUBUCUDUE $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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 C x y $.  $d D y $.  $d X x $.  $d Y x $.
    mpoxeldm.f $e |- F = ( x e. C , y e. D |-> R ) $.
    $( If there is an element of the value of an operation given by a maps-to
       rule, then the first argument is an element of the first component of
       the domain and the second argument is an element of the second component
       of the domain depending on the first argument.  (Contributed by AV,
       25-Oct-2020.) $)
    mpoxeldm $p |- ( N e. ( X F Y ) -> ( X e. C /\ Y e. [_ X / x ]_ D ) ) $=
      ( co wcel cop cv csn cxp ciun csb wa cdm dmmpossx cfv elfvdm df-ov eleq2s
      sselid nfcsb1v csbeq1a opeliunxp2f sylib ) GHIFKZLZHIMZACANODPQZLHCLIAHDR
      ZLSULFTZUNUMABCDEFJUAUMUPLGUMFUBUKGUMFUCHIFUDUEUFACDHIUOAHDUGAHDUHUIUJ $.

    $d C n $.  $d D n $.  $d F n $.  $d X n x $.  $d Y n $.
    $( If the first argument of an operation given by a maps-to rule is not an
       element of the first component of the domain or the second argument is
       not an element of the second component of the domain depending on the
       first argument, then the value of the operation is the empty set.
       (Contributed by AV, 25-Oct-2020.) $)
    mpoxneldm $p |- ( ( X e/ C \/ Y e/ [_ X / x ]_ D ) -> ( X F Y ) = (/) ) $=
      ( vn wnel csb wo wcel wa wn co c0 df-nel sylbi wceq orbi12i bitr4i cv wex
      ianor neq0 mpoxeldm exlimiv con1i ) GCKZHAGDLZKZMZGCNZHULNZOZPZGHFQZRUAZU
      NUOPZUPPZMURUKVAUMVBGCSHULSUBUOUPUFUCUTUQUTPJUDZUSNZJUEUQJUSUGVDUQJABCDEF
      VCGHIUHUITUJT $.
  $}

  ${
    $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 cdm dmmpossx elfvdm df-ov eleq2s
      ciun sselid fveq2 opeliunxp2 simprbi syl op1stg eleq2d imbitrid ) FGHLZED
      MZNZEUPOPZNZGINHJNQZEGNURUPELZARASZTVCOPZUAUGZNZUTURDUBZVEVBABRVDCDKUCVBV
      GNFVBDPUQFVBDUDUPEDUEUFUHVFUPRNUTARVDUPEUSVCUPOUIUJUKULVAUSGEGHIJUMUNUO
      $.

    $d K n $.  $d F n $.  $d V n $.  $d W n $.  $d X n $.  $d Y n $.
    $( If the second argument 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 is not element of the first
       component of the first argument, then the value of the operation is the
       empty set.  (Contributed by Alexander van der Vekens, 10-Oct-2017.) $)
    mpoxopynvov0g $p |- ( ( ( V e. X /\ W e. Y ) /\ K e/ V )
                          -> ( <. V , W >. F K ) = (/) ) $=
      ( vn wcel wa wnel cop co c0 wceq wn cv wex neq0 mpoxopn0yelv nnel exlimdv
      imbitrrdi biimtrid con4d imp ) FHLGILMZEFNZFGOEDPZQRZUJUMUKUMSKTZULLZKUAU
      JUKSZKULUBUJUOUPKUJUOEFLUPABCDEUNFGHIJUCEFUDUFUEUGUHUI $.

    $d F x $.
    $( If the first argument 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, is not an ordered pair, then the
       value of the operation is the empty set.  (Contributed by Alexander van
       der Vekens, 10-Oct-2017.) $)
    mpoxopxnop0 $p |- ( -. V e. ( _V X. _V ) -> ( V F K ) = (/) ) $=
      ( vn c0 cvv cxp wcel cv wex csn c1st cfv cdm wa sylbi co wceq wn neq0 cop
      ciun dmmpossx elfvdm df-ov eleq2s sselid fveq2 opeliunxp2 cuni eluni ne0i
      wi wne ad2antlr dmsnn0 sylibr ex exlimiv 1stval impcom syl con1i ) FEDUAZ
      IUBZFJJKLZVIUCAMZVHLZANVJAVHUDVLVJAVLFEUEZAJVKOVKPQZKUFZLZVJVLDRZVOVMABJV
      NCDGUGVMVQLVKVMDQVHVKVMDUHFEDUIUJUKVPFJLZEFPQZLZSVJAJVNFEVSVKFPULUMVTVRVJ
      VRVJUQZEFORZUNZVSEWCLEHMZLZWDWBLZSZHNWAHEWBUOWGWAHWGVRVJWGVRSWBIURZVJWFWH
      WEVRWBWDUPUSFUTVAVBVCTFVDUJVETVFVCTVG $.

    $( If the first argument 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, is the empty set, then the value
       of the operation is the empty set.  (Contributed by Alexander van der
       Vekens, 10-Oct-2017.) $)
    mpoxopx0ov0 $p |- ( (/) F K ) = (/) $=
      ( c0 cvv cxp wcel wn co wceq 0nelxp mpoxopxnop0 ax-mp ) GHHIJKGEDLGMHHNAB
      CDEGFOP $.

    $( If the components of the first argument 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, are not
       sets, then the value of the operation is the empty set.  (Contributed by
       Alexander van der Vekens, 10-Oct-2017.) $)
    mpoxopxprcov0 $p |- ( -. ( V e. _V /\ W e. _V )
                           -> ( <. V , W >. F K ) = (/) ) $=
      ( cvv wcel wa cop cxp co c0 wceq opelxp mpoxopxnop0 sylnbir ) FIJGIJKFGLZ
      IIMJTEDNOPFGIIQABCDETHRS $.

    $( If the second argument 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 is not element of the first
       component of the first argument, then the value of the operation is the
       empty set.  (Contributed by Alexander van der Vekens, 10-Oct-2017.) $)
    mpoxopynvov0 $p |- ( K e/ V -> ( <. V , W >. F K ) = (/) ) $=
      ( cvv wcel wa wnel cop co c0 wceq wi mpoxopynvov0g ex wn mpoxopxprcov0
      a1d pm2.61i ) FIJGIJKZEFLZFGMEDNOPZQUDUEUFABCDEFGIIHRSUDTUFUEABCDEFGHUAUB
      UC $.
  $}

  ${
    $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 } ) $=
      ( vz wcel wa cvv cv c1st wceq adantl nfcv cop crab wsbc cmpo fveq2 op1stg
      cfv adantr sylan9eqr adantrr wb sbceq1a bitrd rabeqbidv opex simpr rabexg
      a1i ad2antrr weq wnf equid nfvd ax-mp nfsbc1v nfrabw nfsbcw ovmpodxf ) GI
      MZHJMZNZFGMZNZBCGHUAZFOBPZQUGZADVPUBZACFUCZBVNUCZDGUBZEGOEBCOVPVQUDRVMKUR
      VMVOVNRZCPFRZNZNZAVSDVPGVMWAVPGRWBWAVMVPVNQUGZGVOVNQUEVKWEGRVLGHIJUFUHUIZ
      UJWDAVRVSWCAVRUKZVMWBWGWAACFULSSWCVRVSUKZVMWAWHWBVRBVNULUHSUMUNWFVNOMVMGH
      UOURVKVLUPVIVTOMVJVLVSDGIUQUSLLUTZVMBVALVBZWIVMBVCVDWIVMCVAWJWIVMCVCVDCVN
      TZBFTVSBDGVRBVNVEBGTVFVSCDGVRCBVNWKACFVEVGCGTVFVH $.

    $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 x $.
    mpoxopoveqd.1 $e |- ( ps -> ( V e. X /\ W e. Y ) ) $.
    mpoxopoveqd.2 $e |- ( ( ps /\ -. K e. V )
                -> { n e. V | [. <. V , W >. / x ]. [. K / y ]. 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, deduction version.  (Contributed by Alexander van
       der Vekens, 11-Oct-2017.) $)
    mpoxopoveqd $p |- ( ps -> ( <. V , W >. F K )
                       = { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph } ) $=
      ( wcel wsbc crab wceq wa c0 co wi mpoxopoveq ex syl11 wn wnel df-nel c1st
      cop cv cfv mpoxopynvov0 sylbir adantr eqcomd ancoms eqtrd pm2.61i ) GHOZB
      HIUJZGFUAZADGPCVAPEHQZRZUBHJOIKOSZUTVDBVEUTVDACDEFGHIJKLUCUDMUEUTUFZBVDVF
      BSVBTVCVFVBTRZBVFGHUGVGGHUHCDAECUKUIULQFGHILUMUNUOBVFTVCRBVFSVCTNUPUQURUD
      US $.
  $}

  ${
    $d x y $.
    brovex.1 $e |- O = ( x e. _V , y e. _V |-> C ) $.
    brovex.2 $e |- ( ( V e. _V /\ E e. _V ) -> Rel ( V O E ) ) $.
    $( A binary relation of the value of an operation given by the maps-to
       notation.  (Contributed by Alexander van der Vekens, 21-Oct-2017.) $)
    brovex $p |- ( F ( V O E ) P
                   -> ( ( V e. _V /\ E e. _V ) /\ ( F e. _V /\ P e. _V ) ) ) $=
      ( co wbr cvv wcel wa cop wi df-br c0 wne ne0i mpondm0 necon1ai wrel sylan
      brrelex12 id syldan ex 3syl sylbi pm2.43i ) FDHEGKZLZHMNEMNOZFMNDMNOZOZUN
      FDPZUMNZUNUQQZFDUMRUSUMSTUOUTUMURUAUOUMSABCGHEMMIUBUCUOUNUQUOUNUPUQUOUMUD
      UNUPJFDUMUFUEUQUGUHUIUJUKUL $.
  $}

  ${
    $d w x y z $.
    brovmpoex.1 $e |- O = ( x e. _V , y e. _V |-> { <. z , w >. | ph } ) $.
    $( A binary relation of the value of an operation given by the maps-to
       notation.  (Contributed by Alexander van der Vekens, 21-Oct-2017.) $)
    brovmpoex $p |- ( F ( V O E ) P
                   -> ( ( V e. _V /\ E e. _V ) /\ ( F e. _V /\ P e. _V ) ) ) $=
      ( copab co wrel cvv wcel wa relmpoopab a1i brovex ) BCADELFGHIJKJGIMNJOPG
      OPQABCDEOOJGIKRST $.
  $}

  ${
    $d E e v x y $.  $d R e v $.  $d V e v x y $.  $d ph e v x y $.
    $d ps e v $.
    sprmpod.1 $e |- M = ( v e. _V , e e. _V
                          |-> { <. x , y >. | ( x ( v R e ) y /\ ch ) } ) $.
    sprmpod.2 $e |- ( ( ph /\ v = V /\ e = E ) -> ( ch <-> ps ) ) $.
    sprmpod.3 $e |- ( ph -> ( V e. _V /\ E e. _V ) ) $.
    sprmpod.4 $e |- ( ph -> A. x A. y ( x ( V R E ) y -> th ) ) $.
    sprmpod.5 $e |- ( ph -> { <. x , y >. | th } e. _V ) $.
    $( The extension of a binary relation which is the value of an operation
       given in maps-to notation.  (Contributed by Alexander van der Vekens,
       30-Oct-2017.)  (Revised by AV, 20-Jun-2019.) $)
    sprmpod $p |- ( ph -> ( V M E )
                           = { <. x , y >. | ( x ( V R E ) y /\ ps ) } ) $=
      ( cvv cv wa co wbr copab cmpo wceq a1i oveq12 breqd adantl 3expb opabbidv
      wb anbi12d wcel simpld simprd wi wal opabbrex syl2anc ovmpod ) AGILJRRESZ
      FSZGSZISZHUAZUBZCTZEFUCZVBVCLJHUAZUBZBTZEFUCZKRKGIRRVIUDUEAMUFAVDLUEZVEJU
      EZTZTZVHVLEFVQVGVKCBVPVGVKULAVPVFVJVBVCVDLVEJHUGUHUIAVNVOCBULNUJUMUKALRUN
      ZJRUNZOUOAVRVSOUPAVKDUQFUREURDEFUCRUNVMRUNPQDBEFVJRUSUTVA $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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 Cartesian 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 an ordered 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 a1i wb cv wceq bitri reltpos
      ctpos c0 cun brrelex1i elex adantr cmpt wex ccom df-tpos breqi bitrid cfv
      brcog wfun funmpt funbrfv2b ax-mp cnvex uniex dmmpti eleq2i eqcom anbi12i
      snex cnveqd unieqd fvmpt eqeq2d pm5.32i biancomi anbi1i anass exbii breq1
      eqid sneq anbi2d ceqsexv bitrdi expcom pm5.21ndd ) BDGZAHGZABCUBZIZACJKUC
      LUDZGZALZKZMZBCIZNZWGWEOWDABWFCUAUEPWNWEOWDWIWEWMAWHUFUGPWEWDWGWNQWEWDNZW
      GAERZFWHFRZLZKZMZUHZIZWPBCIZNZEUIZWNWGABCXAUJZIWOXEABWFXFFCUKULEABCXAHDUO
      UMXEWPWLSZWIXCNZNZEUIWNXDXIEXDXGWINZXCNXIXBXJXCXBXGWIXBAXAJZGZAXAUNZWPSZN
      ZWIXGNZXAUPXBXOQFWHWTUQAWPXAURUSXOWIWPXMSZNXPXLWIXNXQXKWHAFWHWTXAWSWRWQVF
      UTVAXAVQZVBVCXMWPVDVEWIXQXGWIXMWLWPFAWTWLWHXAWQASZWSWKXSWRWJWQAVRVGVHXRWK
      WJAVFUTVAZVIVJVKTTVLVMXGWIXCVNTVOXHWNEWLXTXGXCWMWIWPWLBCVPVSVTTWAWBWC $.

    $( 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).  This allows to eliminate
       sethood hypotheses on ` A , B ` in ~ brtpos .  (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 wb cvv vex breldm sylbi wss wi
      eldm ctpos brtpos0 elv 0nelrel0 nsyl3 sylbir exlimiv con2i wa ccnv relcnv
      cxp csn df-rel mpbi sseli a1i elsni breq1d pm2.24d biimtrdi impcom wo cun
      com3l cuni brtpos2 ax-mp simplbi elun sylib adantl exlimdv biimtrid ssrdv
      mpjaod ex sylibr impbii ) AUAZDZEZFADZGZHZWDWBWDFBIZAJZBKWBHZBFALTWGWHBWG
      FWFVTJZWHWIWGMBWFANUBUCZWBFWAGWIWAUDFWFVTLBOZPUEUFUGQUHWEWANNULZRWBWECWAW
      LCIZWAGWMWFVTJZBKWEWMWLGZBWMVTCOTWEWNWOBWEWNWOWEWNUIZWMWCUJZGZWOWMFUMZGZW
      RWOSWPWQWLWMWQEWQWLRWCUKWQUNUOUPUQWNWEWTWOSWTWNWEWOWTWNWIWEWOSZWTWMFWFVTW
      MFURUSWIWGXAWJWGWDWOFWFALWKPUTQVAVEVBWNWRWTVCZWEWNWMWQWSVDGZXBWNXCWMUMUJV
      FWFAJZWFNGWNXCXDUIMWKWMWFANVGVHVIWMWQWSVJVKVLVPVQVMVNVOWAUNVRVS $.

    $( The transposition swaps arguments of a three-parameter relation.
       (Contributed by Mario Carneiro, 10-Sep-2015.) $)
    brtpos $p |- ( C e. V -> ( <. A , B >. tpos F C <-> <. B , A >. F C ) ) $=
      ( wcel cvv wa cop ctpos wbr wb cdm ccnv c0 csn adantr wn opprc breq1d cun
      cuni brtpos2 wi opex breldmg 3expia mpan opelcnvg adantl sylibrd pm4.71rd
      elun1 opswap breq1i anbi2i bitr4di bitr4d ex brtpos0 ancom sylnbi bibi12d
      syl6 syl5ibrcom pm2.61d ) CEFZAGFZBGFZHZABIZCDJZKZBAIZCDKZLZVGVJVPVGVJHZV
      MVKDMZNZOPZUAFZVKPNUBZCDKZHZVOVGVMWDLVJVKCDEUCQVQVOWAVOHWDVQVOWAVQVOVKVSF
      ZWAVQVOVNVRFZWEVGVOWFUDZVJVNGFZVGWGBAUEWHVGVOWFVNCGEDUFUGUHQVJWEWFLVGABGG
      VRUIUJUKVKVSVTUMVDULWCVOWAWBVNCDABUNUOUPUQURUSVGVPVJRZOCVLKZOCDKZLCDEUTWI
      VMWJVOWKWIVKOCVLABSTVJVIVHHZVOWKLVHVIVAWLRVNOCDBASTVBVCVEVF $.

    $( The transposition swaps the first two elements in a collection of
       ordered triples.  (Contributed by Mario Carneiro, 1-Dec-2014.) $)
    ottpos $p |- ( C e. V ->
      ( <. A , B , C >. e. tpos F <-> <. B , A , C >. e. F ) ) $=
      ( wcel cop ctpos cotp wbr brtpos df-br 3bitr3g df-ot eleq1i 3bitr4g ) CEF
      ZABGZCGZDHZFZBAGZCGZDFZABCIZTFBACIZDFQRCTJUBCDJUAUDABCDEKRCTLUBCDLMUESTAB
      CNOUFUCDBACNOP $.

    $( The transposition swaps arguments of a three-parameter relation.
       (Contributed by Mario Carneiro, 3-Nov-2015.) $)
    relbrtpos $p |- ( Rel F ->
      ( <. A , B >. tpos F C <-> <. B , A >. F C ) ) $=
      ( wrel cop ctpos wbr cvv wcel reltpos a1i brrelex2 sylan brtpos pm5.21nd
      ) DEZABFZCDGZHZBAFZCDHCIJZQSEZTUBUCQDKLRCSMNUACDMABCDIOP $.

    $( 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 df-rel reldmtpos 3imtr4i relcnv jctir brtpos
      mtoi wb mp1i exbidv opelcnv bitri 3bitr4g eqrelrdv2 mpancom ) AEZFZGZAFZH
      ZGZIVDGZVBVEJVGVCVFVDKKUAZUBZSVDLZUCVGVCVIVJSVHLKKUDVDVHSUEULVDUFAUGUHVDU
      IUJVGBCVBVEVGBTZCTZMZDTZVANZDOVLVKMZVNANZDOZVMVBLVMVELZVGVOVQDVNKLVOVQUMV
      GDPVKVLVNAKUKUNUODVMVAVKVLQRVSVPVDLVRVKVLVDBPCPUPDVPAVLVKQRUQURUSUT $.

    $( 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 ) $=
      ( vz vw vx vy cdm wrel crn cv wcel wbr wex vex elrn cop wceq breldm elrel
      wi breq1 ctpos ccnv dmtpos eleq2d imbitrid relcnv mpan syl6 wb cvv brtpos
      elv bitrdi opex brelrn biimtrdi exlimivv syli exlimdv biimtrid ex bitr4di
      syl5 impbid eqrdv ) AFZGZBAUAZHZAHZVGBIZVIJZVKVJJZVLCIZVKVHKZCLVGVMCVKVHB
      MZNVGVOVMCVOVGVNDIZEIZOZPZELDLZVMVGVOVNVFUBZJZWAVOVNVHFZJVGWCVNVKVHCMZVPQ
      VGWDWBVNAUCUDUEWBGWCWAVFUFDEVNWBRUGUHVTVOVMSDEVTVOVRVQOZVKAKZVMVTVOVSVKVH
      KZWGVNVSVKVHTWHWGUIBVQVRVKAUJUKULZUMWFVKAVRVQUNVPUOUPUQURUSUTVMVNVKAKZCLV
      GVLCVKAVPNVGWJVLCWJVGVNWFPZDLELZVLWJVNVFJZVGWLVNVKAWEVPQVGWMWLEDVNVFRVAVC
      WKWJVLSEDWKWJWHVLWKWJWGWHVNWFVKATWIVBVSVKVHVQVRUNVPUOUPUQURUSUTVDVE $.

    $( 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 xpexd ssexg sylancr ) ABCZADZAEZFZGHZIZAJZKZLUL
      MCUFMCANUEUJUKMMUEUHMCZUIMCUJMCUEUGMCUMABRUGMOPQUHUIMMSTABUAUBUFULMUCUD
      $.

    $( 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 ` RR ` or some ring, ` tpos F ` is the
       transposition of ` F ` , which is where the name comes from.
       (Contributed by Mario Carneiro, 10-Sep-2015.) $)
    ovtpos $p |- ( A tpos F B ) = ( B F A ) $=
      ( vy cop ctpos cfv co cv wbr cio wb cvv brtpos elv iotabii df-fv 3eqtr4i
      df-ov ) ABEZCFZGZBAEZCGZABUAHBACHTDIZUAJZDKUCUECJZDKUBUDUFUGDUFUGLDABUECM
      NOPDTUAQDUCCQRABUASBACSR $.

    $( 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 cab wcel cvv cxp bitr3i bitrdi
      coprab wss relcnv dmtpos releqd mpbiri reltpos jctil relrelss sylib sseld
      elvvv imbitrdi pm4.71rd 19.41vvv eleq1 df-br wb brtpos elv pm5.32i 3exbii
      ccnv eqabdv df-oprab eqtr4di ) DFZGZDHZEIZAIZBIZJZCIZJZKZVLVKJVNDLZMZCNBN
      ANZEOVQABCUAVHVSEVIVHVJVIPZVPCNBNANZVTMZVSVHVTWAVHVTVJQQRQRZPWAVHVIWCVJVH
      VIGZVIFZGZMVIWCUBVHWFWDVHWFVGVCZGVGUCVHWEWGDUDUEUFDUGUHVIUIUJUKABCVJULUMU
      NWBVPVTMZCNBNANVSVPVTABCUOWHVRABCVPVTVQVPVTVOVIPZVQVJVOVIUPWIVMVNVILZVQVM
      VNVIUQWJVQURCVKVLVNDQUSUTSTVAVBSTVDVQABCEVEVF $.

    $( 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 opswap eleq1i opelcnv bitr4i eleq1d bibi12d mpbiri exlimivv biimpcd
      elvv sylbi elun1 syl6 syl elun2 a1i wo simpll sylib mpjaod simpr eqbrtrrd
      elun jca sylanb brtpos2 sylibr df-br exlimiv relssi eqssi ) BUAZBAFFUBZUN
      GZHZAIZGZJZKZUCZUDZXIBABUEZJZXKHZXPUCZUDZXRABUFYBXQLYCXRLYBXQYAUGZXQXTXJL
      ZYAXLLYDYBMXTUHYEXSUIXTUJUKXTXJXKULAXLYAXPUMUOXQYAUPUQYBXQBURUSUTCDXRXIBX
      QVACIZDIZNZXROYFEIZXQPZYIYGBPZQZERYHXIOZEYFYGBXQCSZDSZVBYLYMEYLYFYGXIPZYM
      YLYFYAOZYFGZJZKZYGBPZQZYPYJYFXLOZYIYTMZQZYKUUBXMXLOZYGXPMZQUUCYGYTMZQUUEA
      DYFYIXQYNESZXMYFMZUUFUUCUUGUUHXMYFXLVCUUJXPYTYGUUJXOYSUUJXNYRXMYFVDVEVFVG
      VHYGYIMUUHUUDUUCYGYIYTVQVIADXLXPVJVKUUEYKQZYQUUAUUKYFXJOZYQYFXKOZUUKYTXSO
      ZUULYQVLUUKYIYTXSUUCUUDYKVMZYKYIXSOUUEYIYGBUUIYOVNVOVPUUNUULYFXTOZYQUULUU
      NUUPUULYFYGYINZMZERDRUUNUUPTZDEYFWGUURUUSDEUURUUSUUQGZJZKZXSOZUUQXTOZTUVC
      YIYGNZXSOUVDUVBUVEXSYGYIVRVSYGYIXSYOUUIVTWAUURUUNUVCUUPUVDUURYTUVBXSUURYS
      UVAUURYRUUTYFUUQVDVEVFWBYFUUQXTVCWCWDWEWHWFYFXTXKWIWJWKUUMYQVLUUKYFXKXTWL
      WMUUKUUCUULUUMWNUUCUUDYKWOYFXJXKWTWPWQUUKYIYTYGBUUOUUEYKWRWSXAXBYGFOYPUUB
      TYOYFYGBFXCUSXDYFYGXIXEWPXFWHXGXH $.

    $( 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 cv ccnv wcel cuni wbr wa wo wb bitrdi eqtrdi breq1d
      vex elv ctpos cxp cun cin reltpos relinxp cdm wrel wss relcnv df-rel mpbi
      vw simpl sselid simpr cop wceq wex elvv opelcnv sneq cnveqd unieqd opswap
      eleq1 anbi12d opex breldm pm4.71ri brtpos bitr3i breq1 exlimivv sylbi iba
      bitr4d bitrd pm5.21nii elsni cnvsn0 uni0 brtpos0 pm5.32i biancomi orbi12i
      sneqd andir andi 3bitr4i elun anbi1i brxp mpbiran2 anbi2i brtpos2 eqbrriv
      bitri brin ) UMBAUAZUAZAEEUBZFGZUCZEUBZUDZWTUEXDEAUFUMHZWTUGZIZXCUCJZXGGZ
      IZKZBHZWTLZMZXGXNALZXGXNXELZMZXGXNXALZXGXNXFLXGXIJZXGXCJZNZXOMZXQXGXBJZYB
      NZMZXPXSYAXOMZYBXOMZNXQYEMZXQYBMZNYDYGYHYJYIYKYHYEYJYHXIXBXGXIUHXIXBUIXHU
      JXIUKULYAXOUNUOXQYEUPYEYHXQYJYEXGCHZDHZUQZURZDUSCUSYHXQOZCDXGUTYOYPCDYOYH
      YNXNALZXQYOYHYMYLUQZXHJZYRXNWTLZMZYQYOYAYSXOYTYOYAYNXIJYSXGYNXIVFYLYMXHCS
      DSVAPYOXMYRXNWTYOXMYNGZIZKYRYOXLUUCYOXKUUBXGYNVBVCVDYLYMVEQRVGUUAYTYQYTYS
      YRXNWTYMYLVHBSZVIVJYTYQOBYMYLXNAEVKTVLPXGYNXNAVMVQVNVOYEXQVPVRVSYIXQYBYBX
      OXQYBXOFXNALZXQYBXOFXNWTLZUUEYBXMFXNWTYBXMFKFYBXLFYBXLXCIFYBXKXCYBXGFXGFV
      TZWGVCWAQVDWBQRUUFUUEOBXNAEWCTPYBXGFXNAUUGRVQWDWEWFYAYBXOWHXQYEYBWIWJXJYC
      XOXGXIXCWKWLXRYFXQXRXGXDJZYFXRUUHXNEJUUDXGXNXDEWMWNXGXBXCWKWRWOWJXTXPOBXG
      XNWTEWPTXGXNAXEWSWJWQ $.

    $( 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 dfss2 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 wa crn wfo tposfo2 wfn ffn dffn4 sylib impel fof syl
      wss frn adantl fssd ex ) ADZABCEZAFZBCGZEUDUEHZUFCIZBUGUHUFUIUGJZUFUIUGEU
      DAUICJZUJUEAUICKUECALUKABCMACNOPUFUIUGQRUEUIBSUDABCTUAUBUC $.

    $( 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 ctpos wceq cv co wral wb tposfn eqfnov2 mpancom eqcom ovtpos eqeq2i
      wfn bitri 2ralbii bitrdi ) DCCEZRZDFZDGZAHZBHZUDIZUFUGDIZGZBCJACJZUIUGUFD
      IZGZBCJACJUDUBRUCUEUKKCCDLABCCUDDMNUJUMABCCUJUIUHGUMUHUIOUHULUIUFUGDPQSTU
      A $.
  $}

  ${
    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 oprabidw 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 $.
  $}

  ${
    $d A x y $.  $d B x y $.  $d C x y $.
    $( The transposition of a constant operation using the relation
       representation.  (Contributed by SO, 11-Jul-2018.) $)
    tposconst $p |- tpos ( ( A X. B ) X. { C } ) = ( ( B X. A ) X. { C } ) $=
      ( vy vx cxp csn ctpos cmpo fconstmpo tposmpo eqtr4i ) ABFCGZFZHDEBACIBAFM
      FEDABCNEDABCJKDEBACJL $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Curry and uncurry
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c curry $.
  $c uncurry $.

  $( Extend class notation to include the currying function. $)
  ccur $a class curry A $.

  $( Extend class notation to include the uncurrying function. $)
  cunc $a class uncurry A $.

  ${
    $d x y z F $.
    $( Define the currying of ` F ` , which splits a function of two arguments
       into a function of the first argument, producing a function over the
       second argument.  (Contributed by Mario Carneiro, 7-Jan-2017.) $)
    df-cur $a |- curry F = ( x e. dom dom F |->
      { <. y , z >. | <. x , y >. F z } ) $.

    $( Define the uncurrying of ` F ` , which takes a function producing
       functions, and transforms it into a two-argument function.  (Contributed
       by Mario Carneiro, 7-Jan-2017.) $)
    df-unc $a |- uncurry F = { <. <. x , y >. , z >. | y ( F ` x ) z } $.
  $}

  ${
    $d C a b $.  $d C z $.  $d F x y z $.  $d V x y z $.  $d X a b $.
    $d X x y z $.  $d Y a b $.  $d Y x y z $.  $d ph a b x y $.  $d ph z $.
    mpocurryd.f $e |- F = ( x e. X , y e. Y |-> C ) $.
    mpocurryd.c $e |- ( ph -> A. x e. X A. y e. Y C e. V ) $.
    mpocurryd.n $e |- ( ph -> Y =/= (/) ) $.
    $( The currying of an operation given in maps-to notation, splitting the
       operation (function of two arguments) into a function of the first
       argument, producing a function over the second argument.  (Contributed
       by AV, 27-Oct-2019.) $)
    mpocurryd $p |- ( ph -> curry F = ( x e. X |-> ( y e. Y |-> C ) ) ) $=
      ( vz va vb cv cmpt wcel wceq wa weq ccur cdm cop wbr copab df-cur dmmpoga
      cxp wral syl dmeqd c0 wne dmxp eqtrd mpteq1d df-mpt cfv wfun wb funbrfv2b
      mpofun mp1i adantr eleq2d opelxp bitrdi anbi1d an21 ibar bicomd adantl co
      df-ov csb cmpo nfcv nfcsb1v nfcsbw csbeq1a sylan9eq cbvmpo eqcomd equcoms
      eqtri a1i csbeq2dv simpr wi rsp2 impl ovmpod eqtr3id eqcom bitrd pm5.32da
      eqeq1d bitrid 3bitrrd opabbidv eqtr2id mpteq2dva eqtrid ) AEUABEUBZUBZBOZ
      COZUCZLOZEUDZCLUEZPZBGCHDPZPZBCLEUFAXLBGXKPXNABXEGXKAXEGHUHZUBZGAXDXOADFQ
      ZCHUIBGUIZXDXORZJBCGHDEFIUGUJZUKAHULUMXPGRKGHUNUJUOUPABGXKXMAXFGQZSZXMXGH
      QZXIDRZSZCLUEXKCLHDUQYBYEXJCLYBXJXHXDQZXHEURZXIRZSZYAYCSZYHSZYEEUSXJYIUTY
      BBCGHDEIVBXHXIEVAVCYBYFYJYHYBYFXHXOQYJYBXDXOXHAXSYAXTVDVEXFXGGHVFVGVHYKYC
      YAYHSZSYBYEYAYCYHVIYBYCYLYDYBYCSZYLYHYDYBYLYHUTZYCYAYNAYAYHYLYAYHVJVKVLVD
      YMYHDXIRYDYMYGDXIYMYGXFXGEVMDXFXGEVNYMMNXFXGGHCNOZBMOZDVOZVOZDEFEMNGHYRVP
      ZRYMEBCGHDVPYSIBCMNGHDYRMDVQNDVQBCYOYQBYOVQBYPDVRVSCYOYQVRBMTZCNTZDYQYRBY
      PDVTZCYOYQVTWAWBWEWFMBTZNCTZSYRDRYMUUCUUDYRCYODVOZDUUCCYOYQDYQDRBMYTDYQUU
      BWCWDWGUUEDRCNUUADUUECYODVTWCWDWAVLYBYAYCAYAWHVDYBYCWHAYAYCXQAXRYJXQWIJXQ
      BCGHWJUJWKWLWMWQDXIWNVGWOWPWRWSWTXAXBUOXC $.

    $d A a x y $.
    mpocurryvald.y $e |- ( ph -> Y e. W ) $.
    mpocurryvald.a $e |- ( ph -> A e. X ) $.
    $( The value of a curried operation given in maps-to notation is a function
       over the second argument of the original operation.  (Contributed by AV,
       27-Oct-2019.) $)
    mpocurryvald $p |- ( ph -> ( curry F ` A )
                                 = ( y e. Y |-> [_ A / x ]_ C ) ) $=
      ( va csb cmpt nfcv mpteq2dv cv cvv mpocurryd nfcsb1v nfmpt csbeq1a cbvmpt
      ccur weq eqtrdi wceq wa csbeq1 adantl mptexd fvmptd ) APDCJBPUAZEQZRZCJBD
      EQZRIFUHZUBAVABICJERZRPIUSRABCEFGIJKLMUCBPIVBUSPVBSBCJURBJSBUQEUDUEBPUICJ
      EURBUQEUFTUGUJAUQDUKZULCJURUTVCURUTUKABUQDEUMUNTOACJUTHNUOUP $.
  $}

  ${
    $d A a b x y $.  $d B a b x y $.  $d C a b $.  $d F a b $.  $d V a b x y $.
    $d X a b x y $.  $d Y a b x y $.  $d ph a b x y $.
    fvmpocurryd.f $e |- F = ( x e. X , y e. Y |-> C ) $.
    fvmpocurryd.c $e |- ( ph -> A. x e. X A. y e. Y C e. V ) $.
    fvmpocurryd.y $e |- ( ph -> Y e. W ) $.
    fvmpocurryd.a $e |- ( ph -> A e. X ) $.
    fvmpocurryd.b $e |- ( ph -> B e. Y ) $.
    $( The value of the value of a curried operation given in maps-to notation
       is the operation value of the original operation.  (Contributed by AV,
       27-Oct-2019.) $)
    fvmpocurryd $p |- ( ph -> ( ( curry F ` A ) ` B ) = ( A F B ) ) $=
      ( vb va csb wcel cv cmpt cfv ccur wceq csbcom csbcow csbeq2i eqtri 3eqtri
      co wral wa nfcsb1v nfel1 csbeq1a eleq1d imp syl21anc eqeltrid eqid fvmpts
      rspc2 syl2anc cmpo nfcv nfcsbw weq sylan9eq mpan9 ralrimivva mpocurryvald
      cbvmpo ne0d fveq1d a1i csbid eqtr2i csbeq2dv 3eqtr3g csbeq1 adantr adantl
      eqtrd eqidd nfv nfcxfr ovmpodxf 3eqtr4d ) AEQKRDCQUAZBRUAZFSZSZSZUBZUCZQE
      WNSZEDGUDUCZUCDEGUKAEKTZWQHTWPWQUEPAWQCEBDFSZSZHWQRDQEWMSZSRDCEWLSZSZXAQR
      EDWMUFRDXBXCCQEWLUGUHXDCERDWLSZSXARCDEWLUFCEXEWTBRDFUGUHUIUJZADJTZWSFHTZC
      KULBJULZXAHTZOPMXGWSUMXIXJXHXJWTHTBCDEJKBWTHBDFUNUOCXAHCEWTUNZUOBUAZDUEZF
      WTHBDFUPUQCUAZEUEZWTXAHCEWTUPUQVCURUSUTZQEWNKWOHWOVAVBVDAEWRWOARQDWMGHIJK
      GBCJKFVEZRQJKWMVELBCRQJKFWMRFVFQFVFBCWJWLBWJVFBWKFUNZVGZCWJWLUNZBRVHZCQVH
      ZFWLWMBWKFUPZCWJWLUPZVIVMUIAWMHTZRQJKAXIWKJTWJKTUMYEMXHYEWLHTBCWKWJJKBWLH
      XRUOCWMHXTUOYAFWLHYCUQYBWLWMHYDUQVCVJVKAKEPVNNOVLVOABCDEJKFWQGKHGXQUEALVP
      AXMXOUMZFQXNRXLWMSZSZWQARXLWLSZRXLQXNWMSZSFYHARXLWLYJWLYJUEAYJCXNWLSWLCQX
      NWLUGCWLVQVRVPVSYIBXLFSFBRXLFUGBFVQUIRQXLXNWMUFVTYFYHQXNWNSZWQYFQXNYGWNXM
      YGWNUEXORXLDWMWAWBVSXOYKWQUEXMQXNEWNWAWCWDVIAXMUMKWEOPXPABWFACWFCDVFBEVFZ
      BQEWNYLBRDWMBDVFXSVGVGCWQXAXFXKWGWHWI $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Undefined values
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c Undef $.

  $( Extend class notation with undefined value function. $)
  cund $a class Undef $.

  $( Define the undefined value function, whose value at set ` s ` is
     guaranteed not to be a member of ` s ` (see ~ pwuninel ).  (Contributed by
     NM, 15-Sep-2011.) $)
  df-undef $a |- Undef = ( s e. _V |-> ~P U. s ) $.

  $( Proof of ~ pwuninel under the assumption that the union of the given class
     is a set, avoiding ~ ax-pr and ~ ax-un .  (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 powerclass of the union of a class does not belong to that class.
     This theorem provides a way of constructing a new set that does not belong
     to a given set.  See also ~ pwuninel2 .  (Contributed by NM, 27-Jun-2008.)
     (Proof shortened by Mario Carneiro, 23-Dec-2016.) $)
  pwuninel $p |- -. ~P U. A e. A $=
    ( cuni cpw wcel wn cvv pwexr pwuninel2 syl id pm2.61i ) ABZCADZMEZMLFDNLAGA
    FHINJK $.

  ${
    $d s S $.
    $( Value of the undefined value function.  Normally we will not reference
       the explicit value but will use ~ undefnel instead.  (Contributed by NM,
       15-Sep-2011.)  (Revised by Mario Carneiro, 24-Dec-2016.) $)
    undefval $p |- ( S e. V -> ( Undef ` S ) = ~P U. S ) $=
      ( vs wcel cuni cpw cvv cund df-undef wceq unieq pweqd elex uniexg fvmptd3
      cv pwexd ) ABDZCACPZEZFAEZFGHGCISAJTUASAKLABMRUAGABNQO $.
  $}

  $( The undefined value generated from a set is not a member of the set.
     (Contributed by NM, 15-Sep-2011.) $)
  undefnel2 $p |- ( S e. V -> -. ( Undef ` S ) e. S ) $=
    ( wcel cund cfv cuni cpw pwuninel undefval eleq1d mtbiri ) ABCZADEZACAFGZAC
    AHLMNAABIJK $.

  $( The undefined value generated from a set is not a member of the set.
     (Contributed by NM, 15-Sep-2011.) $)
  undefnel $p |- ( S e. V -> ( Undef ` S ) e/ S ) $=
    ( wcel cund cfv wn wnel undefnel2 df-nel sylibr ) ABCADEZACFKAGABHKAIJ $.
  $( The undefined value generated from a set is not empty.  (Contributed by
     NM, 3-Sep-2018.) $)
  undefne0 $p |- ( S e. V -> ( Undef ` S ) =/= (/) ) $=
    ( wcel cund cfv cuni cpw c0 undefval wne pwne0 a1i eqnetrd ) ABCZADEAFZGZHA
    BIPHJNOKLM $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Well-founded recursion
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare new constant symbols. $)
  $c frecs $.

  $( Declare the syntax for the well-founded recursion generator.  See
     ~ df-frecs . $)
  cfrecs $a class frecs ( R , A , F ) $.

  ${
    $d R f x y $.  $d A f x y $.  $d F f x y $.
    $( This is the definition for the well-founded recursion generator.
       Similar to ~ df-wrecs and ~ df-recs , it is a direct definition form of
       normally recursive relationships.  Unlike the former two definitions, it
       only requires a well-founded set-like relationship for its properties,
       not a well-ordered relationship.  This proof requires either a partial
       order or the axiom of infinity.  We develop the theorems twice, once
       with a partial order and once without.  The second development occurs
       later in the database, after ~ ax-inf has been introduced.  (Contributed
       by Scott Fenton, 23-Dec-2021.) $)
    df-frecs $a |- frecs ( R , A , F ) = U. { f | E. x ( f Fn x /\ ( x C_ A /\
               A. y e. x Pred ( R , A , y ) C_ x ) /\
               A. y e. x ( f ` y ) = ( y F ( f |` Pred ( R , A , y ) ) ) ) } $.
  $}

  ${
    $d R f x y $.  $d S f x y $.  $d A f x y $.  $d B f x y $.  $d F f x y $.
    $d G f x y $.
    $( Equality theorem for the well-founded recursion generator.  (Contributed
       by Scott Fenton, 23-Dec-2021.) $)
    frecseq123 $p |- ( ( R = S /\ A = B /\ F = G ) ->
      frecs ( R , A , F ) = frecs ( S , B , G ) ) $=
      ( vf vx vy wceq w3a cv wss cpred wral wa cres co wex cab wfn cfrecs simp2
      cfv cuni sseq2d weq equid predeq123 mp3an3 3adant3 sseq1d ralbidv anbi12d
      simp3 oveqd reseq2d oveq2d eqeq2d 3anbi23d exbidv abbidv df-frecs 3eqtr4g
      eqtrd unieqd ) CDJZABJZEFJZKZGLZHLZUAZVLAMZACILZNZVLMZIVLOZPZVOVKUDZVOVKV
      PQZERZJZIVLOZKZHSZGTZUEVMVLBMZBDVONZVLMZIVLOZPZVTVOVKWIQZFRZJZIVLOZKZHSZG
      TZUEACEUBBDFUBVJWGWSVJWFWRGVJWEWQHVJVSWLWDWPVMVJVNWHVRWKVJABVLVGVHVIUCUFV
      JVQWJIVLVJVPWIVLVGVHVPWIJZVIVGVHIIUGWTIUHABCDVOVOUIUJUKZULUMUNVJWCWOIVLVJ
      WBWNVTVJWBVOWAFRWNVJEFVOWAVGVHVIUOUPVJWAWMVOFVJVPWIVKXAUQURVEUSUMUTVAVBVF
      HIACGEVCHIBDGFVCVD $.
  $}

  ${
    $d R f y z $.  $d A f y z $.  $d F f y z $.  $d x f y z $.
    nffrecs.1 $e |- F/_ x R $.
    nffrecs.2 $e |- F/_ x A $.
    nffrecs.3 $e |- F/_ x F $.
    $( Bound-variable hypothesis builder for the well-founded recursion
       generator.  (Contributed by Scott Fenton, 23-Dec-2021.) $)
    nffrecs $p |- F/_ x frecs ( R , A , F ) $=
      ( vf vy vz cfrecs cv wfn wss cpred wral wa nfcv nfss nfralw cfv cres wceq
      co w3a wex cab cuni df-frecs nfpred nfan nfres nfov nfeq2 nf3an nfex nfab
      nfv nfuni nfcxfr ) ABCDKHLZILZMZVBBNZBCJLZOZVBNZJVBPZQZVEVAUAZVEVAVFUBZDU
      DZUCZJVBPZUEZIUFZHUGZUHIJBCHDUIAVQVPAHVOAIVCVIVNAVCAURVDVHAAVBBAVBRZFSVGA
      JVBVRAVFVBABCVEEFAVERZUJZVRSTUKVMAJVBVRAVJVLAVEVKDVSGAVAVFAVARVTULUMUNTUO
      UPUQUSUT $.
  $}

  ${
    $d A f y z $.  $d D f y z $.  $d F f y z $.  $d R f y z $.  $d V f y z $.
    $d f x y z $.
    $( Move class substitution in and out of the well-founded recursive
       function generator.  (Contributed by Scott Fenton, 18-Nov-2024.) $)
    csbfrecsg $p |- ( A e. V -> [_ A / x ]_ frecs ( R , D , F ) =
                   frecs ( [_ A / x ]_ R , [_ A / x ]_ D , [_ A / x ]_ F ) ) $=
      ( vf vz vy cv wss cpred wral wa wceq csb wsbc csbconstg bitrd eqtrid wcel
      wfn cfv cres co w3a wex cuni cfrecs csbuni csbab sbcex2 sbc3an sbcg sbcan
      cab sbcssg sseq1d sbcralg csbpredg predeq3 sseq12d ralbidv anbi12d bitrid
      syl eqtrd sbceqg csbov123 csbres reseq12d oveq12d 3anbi123d exbidv abbidv
      eqeq12d unieqd df-frecs csbeq2i 3eqtr4g ) BFUAZABGJZHJZUBZWCCKZCDIJZLZWCK
      ZIWCMZNZWFWBUCZWFWBWGUDZEUEZOZIWCMZUFZHUGZGUPZUHZPZWDWCABCPZKZXAABDPZWFLZ
      WCKZIWCMZNZWKWFWBXDUDZABEPZUEZOZIWCMZUFZHUGZGUPZUHZABCDEUIZPXAXCXIUIWAWTA
      BWRPZUHXPABWRUJWAXRXOWAXRWQABQZGUPXOWQAGBUKWAXSXNGXSWPABQZHUGWAXNWPHABULW
      AXTXMHXTWDABQZWJABQZWOABQZUFWAXMWDWJWOABUMWAYAWDYBXGYCXLWDABFUNYBWEABQZWI
      ABQZNWAXGWEWIABUOWAYDXBYEXFWAYDABWCPZXAKXBABWCCFUQWAYFWCXAABWCFRZURSWAYEW
      HABQZIWCMXFWHAIBWCFUSWAYHXEIWCWAYHABWGPZYFKXEABWGWCFUQWAYIXDYFWCWAYIXAXCA
      BWFPZLZXDABCDFWFUTWAYJWFOYKXDOABWFFRZXAXCYJWFVAVFVGZYGVBSVCSVDVEWAYCWNABQ
      ZIWCMXLWNAIBWCFUSWAYNXKIWCWAYNABWKPZABWMPZOXKABWKWMFVHWAYOWKYPXJABWKFRWAY
      PYJABWLPZXIUEXJABWFWLEVIWAYJWFYQXHXIYLWAYQABWBPZYIUDXHABWBWGVJWAYRWBYIXDA
      BWBFRYMVKTVLTVPSVCSVMVEVNVEVOTVQTABXQWSHICDGEVRVSHIXAXCGXIVRVT $.
  $}

  ${
    $d A w y z $.  $d F w y z $.  $d G w y z $.  $d H w y z $.  $d R w y z $.
    $( Functions defined by well-founded recursion over a partial order are
       identical up to relation, domain, and characteristic function.  This
       version of ~ frr3g does not require infinity.  (Contributed by Scott
       Fenton, 24-Aug-2022.) $)
    fpr3g $p |- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\
                     ( F Fn A /\
                       A. y e. A ( F ` y ) =
                                 ( y H ( F |` Pred ( R , A , y ) ) ) ) /\
                     ( G Fn A /\
                       A. y e. A ( G ` y ) =
                                 ( y H ( G |` Pred ( R , A , y ) ) ) ) )
                   -> F = G ) $=
      ( vz vw w3a wfn cv cfv cres co wceq wral wa wi fveq2 eqeq12d wfr wpo wcel
      wse cpred eqidd r19.21v simprll simprrl wss predss fvreseq mpan2 biimp3ar
      wb syl2anc oveq2d predeq3 reseq2d oveq12d simp2lr rspcdva simp2rr 3eqtr4d
      weq simp1 3exp a2d biimtrid imbi2d frpoins2g 3impib simp2l simp3l eqfnfv2
      id sylib mpbir2and ) BCUABCUBBCUDIZDBJZAKZDLZWADBCWAUEZMZFNZOZABPZQZEBJZW
      AELZWAEWCMZFNZOZABPZQZIZDEOZBBOZGKZDLZWSELZOZGBPZWPBUFVSWHWOXCVSWHWOQZXBR
      ZGBPXDXCRXEXDHKZDLZXFELZOZRZGHBCXJHBCWSUEZPXDXIHXKPZRWSBUCZXEXDXIHXKUGXMX
      DXLXBXMXDXLXBXMXDXLIZWSDXKMZFNZWSEXKMZFNZWTXAXNXOXQWSFXMXDXOXQOZXLXMXDQVT
      WIXSXLUOZXMVTWGWOUHXMWHWIWNUIVTWIQXKBUJXTBCWSUKHBXKDEULUMUPUNUQXNWFWTXPOA
      BWSAGVEZWBWTWEXPWAWSDSYAWAWSWDXOFYAVPZYAWCXKDBCWAWSURZUSUTTVTWGWOXMXLVAXM
      XDXLVFZVBXNWMXAXROABWSYAWJXAWLXRWAWSESYAWAWSWKXQFYBYAWCXKEYCUSUTTWIWNWHXM
      XLVCYDVBVDVGVHVIGHVEZXBXIXDYEWTXGXAXHWSXFDSWSXFESTVJVKXDXBGBUGVQVLWPVTWIW
      QWRXCQUOVSVTWGWOVMVSWHWIWNVNGBBDEVOUPVR $.
  $}

  ${
    $d A f g w x y z $.  $d f G g w x y z $.  $d R f g w x y z $.
    frrlem1.1 $e |- B = { f | E. x ( f Fn x /\
                           ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x ) /\
                             A. y e. x ( f ` y ) =
                                     ( y G ( f |` Pred ( R , A , y ) ) ) ) } $.
    $( Lemma for well-founded recursion.  The final item we are interested in
       is the union of acceptable functions ` B ` .  This lemma just changes
       bound variables for later use.  (Contributed by Paul Chapman,
       21-Apr-2012.) $)
    frrlem1 $p |- B = { g | E. z ( g Fn z /\
                         ( z C_ A /\ A. w e. z Pred ( R , A , w ) C_ z ) /\
                           A. w e. z ( g ` w ) =
                                   ( w G ( g |` Pred ( R , A , w ) ) ) ) } $=
      ( cv wfn wss wral cfv cres co wceq w3a cpred wa wex cab weq reseq1 oveq2d
      fneq1 fveq1 eqeq12d ralbidv 3anbi13d fneq2 sseq1 sseq2 raleqbi1dv predeq3
      exbidv sseq1d cbvralvw bitrdi anbi12d raleq fveq2 reseq2d oveq12d cbvexvw
      id 3anbi123d cbvabv eqtri ) FHLZALZMZVMENZEGBLZUAZVMNZBVMOZUBZVPVLPZVPVLV
      QQZJRZSZBVMOZTZAUCZHUDILZCLZMZWIENZEGDLZUAZWINZDWIOZUBZWLWHPZWLWHWMQZJRZS
      ZDWIOZTZCUCZIUDKWGXCHIHIUEZWGWHVMMZVTVPWHPZVPWHVQQZJRZSZBVMOZTZAUCXCXDWFX
      KAXDVNXEWEXJVTVMVLWHUHXDWDXIBVMXDWAXFWCXHVPVLWHUIXDWBXGVPJVLWHVQUFUGUJUKU
      LURXKXBACACUEZXEWJVTWPXJXAVMWIWHUMXLVOWKVSWOVMWIEUNXLVSVQWINZBWIOWOVRXMBV
      MWIVMWIVQUOUPXMWNBDWIBDUEZVQWMWIEGVPWLUQZUSUTVAVBXLXJXIBWIOXAXIBVMWIVCXIW
      TBDWIXNXFWQXHWSVPWLWHVDXNVPWLXGWRJXNVHXNVQWMWHXOVEVFUJUTVAVIVGVAVJVK $.

    $( Lemma for well-founded recursion.  An acceptable function is a function.
       (Contributed by Paul Chapman, 21-Apr-2012.) $)
    frrlem2 $p |- ( g e. B -> Fun g ) $=
      ( vz vw cv wcel wfn wss cpred wral wa cfv cres co wceq w3a frrlem1 eqabri
      wex wfun fnfun 3ad2ant1 exlimiv sylbi ) GLZDMULJLZNZUMCOCEKLZPZUMOKUMQRZU
      OULSUOULUPTHUAUBKUMQZUCZJUFZULUGZUTGDABJKCDEFGHIUDUEUSVAJUNUQVAURUMULUHUI
      UJUK $.

    $( Lemma for well-founded recursion.  An acceptable function's domain is a
       subset of ` A ` .  (Contributed by Paul Chapman, 21-Apr-2012.) $)
    frrlem3 $p |- ( g e. B -> dom g C_ A ) $=
      ( vz vw cv wcel wfn wss cpred wral wa cfv cres co wceq w3a wex cdm eqabri
      frrlem1 fndm sseq1d biimpar adantrr 3adant3 exlimiv sylbi ) GLZDMUOJLZNZU
      PCOZCEKLZPZUPOKUPQZRZUSUOSUSUOUTTHUAUBKUPQZUCZJUDZUOUEZCOZVEGDABJKCDEFGHI
      UGUFVDVGJUQVBVGVCUQURVGVAUQVGURUQVFUPCUPUOUHUIUJUKULUMUN $.
  $}

  ${
    $d A a f g $.  $d A a h x y $.  $d B a $.  $d f h x y $.  $d G a f g $.
    $d G h $.  $d G g x y $.  $d R a f g $.  $d R h x y $.  $d A a f g b c $.
    $d h x y b c $.  $d G b c $.  $d R b c $.
    frrlem4.1 $e |- B = { f | E. x ( f Fn x /\
                            ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x ) /\
                              A. y e. x ( f ` y ) =
                                     ( y G ( f |` Pred ( R , A , y ) ) ) ) } $.
    $( Lemma for well-founded recursion.  Properties of the restriction of an
       acceptable function to the domain of another acceptable function.
       (Contributed by Paul Chapman, 21-Apr-2012.) $)
    frrlem4 $p |- ( ( g e. B /\ h e. B ) ->
       ( ( g |` ( dom g i^i dom h ) ) Fn ( dom g i^i dom h ) /\
         A. a e. ( dom g i^i dom h ) ( ( g |` ( dom g i^i dom h ) ) ` a ) =
            ( a G ( ( g |` ( dom g i^i dom h ) ) |`
                    Pred ( R , ( dom g i^i dom h ) , a ) ) ) ) ) $=
      ( vb vc cv wcel wa cres wceq wral wss cdm cin wfn cpred co frrlem2 funfnd
      cfv fnresin1 syl adantr w3a wex frrlem1 fndm raleqdv biimp3ar rsp exlimiv
      wi eqabri sylbi elinel1 impel adantlr simpr fvresd resres predss exdistrv
      sseqin2 mpbi inss1 simpl2l sstrid simp2r nfan wel syl5com elinel2 anim12d
      nfra1 biimpi syl6com ralrimi syl2an wb simpl1 simpr1 ineq12 sseq1d sseq2d
      ssin fndmd raleqbidv anbi12d syl2anc mpbir2and exlimivv sylbir preddowncl
      syl2anb sylc eqtrid reseq2d oveq2d 3eqtr4d ralrimiva jca ) GNZDOZHNZDOZPZ
      XJXJUAZXLUAZUBZQZXQUCZJNZXRUHZXTXRXQEXTUDZQZIUEZRZJXQSXKXSXMXKXJXOUCXSXKX
      JABCDEFGIKUFUGXOXPXJUIUJUKXNYEJXQXNXTXQOZPZXTXJUHZXTXJCEXTUDZQZIUEZYAYDXK
      YFYHYKRZXMXKXTXOOZYLYFXKXJLNZUCZYNCTZYIYNTZJYNSZPZYLJYNSZULZLUMZYMYLUTZUU
      BGDABLJCDEFGIKUNVAZUUAUUCLUUAYLJXOSZUUCYOYSUUEYTYOYSPYLJXOYNYOXOYNRZYSYNX
      JUOUKUPUQYLJXOURUJUSVBXTXOXPVCVDVEYGXTXQXJXNYFVFZVGYGYCYJXTIYGYCXJXQYBUBZ
      QYJXJXQYBVHYGUUHYIXJYGUUHYBYIYBXQTUUHYBRXQEXTVIYBXQVKVLYGXQCTZYIXQTZJXQSZ
      PZYFYBYIRXNUULYFXKUUBXLMNZUCZUUMCTZYIUUMTZJUUMSZPZXTXLUHXTXLYIQIUERJUUMSZ
      ULZMUMZUULXMUUDUVAHDABMJCDEFHIKUNVAUUBUVAPUUAUUTPZMUMLUMUULUUAUUTLMVJUVBU
      ULLMUVBUULYNUUMUBZCTZYIUVCTZJUVCSZUVBUVCYNCYNUUMVMYPYRYOYTUUTVNVOUUAYRUUQ
      UVFUUTYOYPYRYTVPUUNUUOUUQUUSVPYRUUQPZUVEJUVCYRUUQJYQJYNWBUUPJUUMWBVQXTUVC
      OZUVGYQUUPPZUVEUVHYRYQUUQUUPUVHJLVRYRYQXTYNUUMVCYQJYNURVSUVHJMVRUUQUUPXTY
      NUUMVTUUPJUUMURVSWAUVIUVEYIYNUUMWMWCWDWEWFUVBUUFXPUUMRZUULUVDUVFPWGUVBYNX
      JYOYSYTUUTWHWNUVBUUMXLUUAUUNUURUUSWIWNUUFUVJPZUUIUVDUUKUVFUVKXQUVCCXOYNXP
      UUMWJZWKUVKUUJUVEJXQUVCUVLUVKXQUVCYIUVLWLWOWPWQWRWSWTXBUKUUGJCXQEXTXAXCXD
      XEXDXFXGXHXI $.
  $}

  ${
    $d A f g x y $.  $d G f g x y $.  $d R f g x y $.  $d B g $.
    frrlem5.1 $e |- B = { f | E. x ( f Fn x /\
                            ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x ) /\
                            A. y e. x ( f ` y ) =
                                    ( y G ( f |` Pred ( R , A , y ) ) ) ) } $.
    frrlem5.2 $e |- F = frecs ( R , A , G ) $.
    $( Lemma for well-founded recursion.  State the well-founded recursion
       generator in terms of the acceptable functions.  (Contributed by Scott
       Fenton, 27-Aug-2022.) $)
    frrlem5 $p |- F = U. B $=
      ( cfrecs cv wfn wss cpred wral wa cfv cres cuni wceq w3a wex cab df-frecs
      co unieqi 3eqtr4i ) CEHKFLZALZMUJCNCEBLZOZUJNBUJPQUKUIRUKUIULSHUFUABUJPUB
      AUCFUDZTGDTABCEFHUEJDUMIUGUH $.

    $( Lemma for well-founded recursion.  The well-founded recursion generator
       is a relation.  (Contributed by Scott Fenton, 27-Aug-2022.) $)
    frrlem6 $p |- Rel F $=
      ( vg wrel cv cuni wral frrlem5 releqi reluni bitri wcel wfun frrlem2 syl
      funrel mprgbir ) GLZKMZLZKDUFDNZLUHKDOGUIABCDEFGHIJPQKDRSUGDTUGUAUHABCDEF
      KHIUBUGUDUCUE $.

    $( Lemma for well-founded recursion.  The well-founded recursion
       generator's domain is a subclass of ` A ` .  (Contributed by Scott
       Fenton, 27-Aug-2022.) $)
    frrlem7 $p |- dom F C_ A $=
      ( vg cdm wss cv ciun wral cuni frrlem5 dmeqi dmuni sseq1i frrlem3 mprgbir
      eqtri iunss bitri ) GLZCMZKNLZCMZKDUHKDUIOZCMUJKDPUGUKCUGDQZLUKGULABCDEFG
      HIJRSKDTUDUAKDUICUEUFABCDEFKHIUBUC $.

    $d F a g w y $.  $d A a f x w y z $.  $d R a w y z $.  $d R a w y z $.
    $d G a g z $.
    $( Lemma for well-founded recursion. ` dom F ` is closed under predecessor
       classes.  (Contributed by Scott Fenton, 6-Dec-2022.) $)
    frrlem8 $p |- ( z e. dom F -> Pred ( R , A , z ) C_ dom F ) $=
      ( vw vg va cv wcel wex wss wa exlimiv cdm cop cpred vex wfn wral cfv cres
      eldm2 wceq w3a cab cuni frrlem5 frrlem1 unieqi eqtri eleq2i eluniab bitri
      co wel simpr2r opeldm adantr simpr1 fndmd eleqtrd rsp sylc sseqtrrd 19.8a
      eqabri sylibr adantl elssuni syl sseqtrrdi dmss sstrd expcom impcom sylbi
      wi ) COZHUAZPWELOZUBZHPZLQDFWEUCZWFRZLWEHCUDZUIWIWKLWIWHMOZPZWMNOZUEZWODR
      ZWJWORZCWOUFZSZWEWMUGWEWMWJUHIVAUJCWOUFZUKZNQZSZMQZWKWIWHXCMULZUMZPXEHXGW
      HHEUMZXGABDEFGHIJKUNZEXFABNCDEFGMIJUOZUPUQURXCMWHUSUTXDWKMXCWNWKXBWNWKWDN
      WNXBWKWNXBSZWJWMUAZWFXKWJWOXLXKWSCNVBWRWQWSWPXAWNVCXKWEXLWOWNWEXLPXBWEWGW
      MWLLUDVDVEXKWOWMWNWPWTXAVFVGZVHWRCWOVIVJXMVKXKWMHRXLWFRXKWMXHHXKWMEPZWMXH
      RXBXNWNXBXCXNXBNVLXCMEXJVMVNVOWMEVPVQXIVRWMHVSVQVTWATWBTWCTWC $.
  $}

  ${
    $d A f x y z $.  $d G f x y z $.  $d R f x y z $.  $d B g h z $.
    $d F x u v z $.  $d ph f z $.  $d F f $.  $d ph g h x u v $.
    frrlem9.1 $e |- B = { f | E. x ( f Fn x /\
                            ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x ) /\
                            A. y e. x ( f ` y ) =
                                    ( y G ( f |` Pred ( R , A , y ) ) ) ) } $.
    frrlem9.2 $e |- F = frecs ( R , A , G ) $.
    frrlem9.3 $e |- ( ( ph /\ ( g e. B /\ h e. B ) ) ->
    ( ( x g u /\ x h v ) -> u = v ) ) $.
    $( Lemma for well-founded recursion.  Show that the well-founded recursive
       generator produces a function.  Hypothesis three will be eliminated
       using different induction rules depending on if we use partial orders or
       the axiom of infinity.  (Contributed by Scott Fenton, 27-Aug-2022.) $)
    frrlem9 $p |- ( ph -> Fun F ) $=
      ( cv wbr wrex wcel wa weq wi wal wfun cop cuni eluni2 df-br frrlem5 bitri
      eleq2i rexbii 3bitr4i anbi12i reeanv rexlimdvva biimtrid alrimiv alrimivv
      bitr4i wrel frrlem6 dffun2 mpbiran sylibr ) ABQZEQZLRZVGDQZLRZUAZEDUBZUCZ
      DUDZEUDBUDZLUEZAVOBEAVNDVLVGVHJQZRZVGVJKQZRZUAZKGSJGSZAVMVLVSJGSZWAKGSZUA
      WCVIWDVKWEVGVHUFZGUGZTZWFVRTZJGSVIWDJWFGUHVIWFLTWHVGVHLUILWGWFBCFGHILMNOU
      JZULUKVSWIJGVGVHVRUIUMUNVGVJUFZWGTZWKVTTZKGSVKWEKWKGUHVKWKLTWLVGVJLUILWGW
      KWJULUKWAWMKGVGVJVTUIUMUNUOVSWAJKGGUPVAAWBVMJKGGPUQURUSUTVQLVBVPBCFGHILMN
      OVCBEDLVDVEVF $.

    $( Lemma for well-founded recursion.  Under the compatibility hypothesis,
       compute the value of ` F ` within its domain.  (Contributed by Scott
       Fenton, 6-Dec-2022.) $)
    frrlem10 $p |- ( ( ph /\ y e. dom F ) ->
     ( F ` y ) = ( y G ( F |` Pred ( R , A , y ) ) ) ) $=
      ( vz cv wcel wss cdm cfv cpred cres co wceq cop wex vex eldm2 wfn wral wa
      w3a cuni frrlem5 unieqi eqtri eleq2i eluniab bitri wi 19.8a 3ad2ant2 abid
      cab sylibr elssuni syl sseqtrrdi wel simpl23 simpl3 simpl21 fndmd eleqtrd
      opeldm rsp sylc wfun simpl1 frrlem9 simpr funssfv syl3anc adantr sseqtrrd
      simp22r fun2ssres oveq2d 3eqtr4d mpdan 3exp exlimdv impcomd biimtrid imp
      ) ACRZLUASZWRLUBZWRLFHWRUCZUDZMUEZUFZWSWRQRZUGZLSZQUHAXDQWRLCUIZUJAXGXDQX
      GXFIRZSZXIBRZUKZXKFTZXAXKTZCXKULZUMZWRXIUBZWRXIXAUDZMUEZUFZCXKULZUNZBUHZU
      MZIUHZAXDXGXFYCIVFZUOZSYELYGXFLGUOYGBCFGHILMNOUPGYFNUQURZUSYCIXFUTVAAYDXD
      IAYCXJXDAYBXJXDVBBAYBXJXDAYBXJUNZXILTZXDYIXIYGLYIXIYFSZXIYGTYIYCYKYBAYCXJ
      YBBVCVDYCIVEVGXIYFVHVIYHVJYIYJUMZXQXSWTXCYLYACBVKZXTXLXPYAAXJYJVLYLWRXIUA
      ZXKYLXJWRYNSZAYBXJYJVMWRXEXIXHQUIVQVIZYLXKXIXLXPYAAXJYJVNVOZVPZXTCXKVRVSY
      LLVTZYJYOWTXQUFYLAYSAYBXJYJWAABCDEFGHIJKLMNOPWBVIZYIYJWCZYPWRLXIWDWEYLXBX
      RWRMYLYSYJXAYNTXBXRUFYTUUAYLXAXKYNYLXOYMXNYIXOYJXMXOXLYAAXJWHWFYRXNCXKVRV
      SYQWGXALXIWIWEWJWKWLWMWNWOWNWPWNWPWQ $.
  $}

  ${
    $d A f x y z $.  $d G f x y z $.  $d R f x y z $.  $d B g h z $.
    $d F x u v z $.  $d ph f z $.  $d F f $.  $d ph g h x u v $.
    frrlem11.1 $e |- B = { f | E. x ( f Fn x /\
                             ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x ) /\
                             A. y e. x ( f ` y ) =
                                     ( y G ( f |` Pred ( R , A , y ) ) ) ) } $.
    frrlem11.2 $e |- F = frecs ( R , A , G ) $.
    frrlem11.3 $e |- ( ( ph /\ ( g e. B /\ h e. B ) ) ->
    ( ( x g u /\ x h v ) -> u = v ) ) $.
    frrlem11.4 $e |- C = ( ( F |` S ) u.
    { <. z , ( z G ( F |` Pred ( R , A , z ) ) ) >. } ) $.
    $( Lemma for well-founded recursion.  For the next several theorems we will
       be aiming to prove that ` dom F = A ` .  To do this, we set up a
       function ` C ` that supposedly contains an element of ` A ` that is not
       in ` dom F ` and we show that the element must be in ` dom F ` .  Our
       choice of what to restrict ` F ` to depends on if we assume partial
       orders or the axiom of infinity.  To begin with, we establish the
       functionality of ` C ` .  (Contributed by Scott Fenton, 7-Dec-2022.) $)
    frrlem11 $p |- ( ( ph /\ z e. ( A \ dom F ) ) ->
    C Fn ( ( S i^i dom F ) u. { z } ) ) $=
      ( cv cdm cdif wcel wa cres cpred co cop csn cun cin wfn wceq wfun frrlem9
      c0 funresd dmres df-fn mpbiran2 sylibr vex ovex fnsn jctir eldifn elinel2
      wn nsyl disjsn fnun syl2an fneq1i ) ADUAZGOUBZUCUDZUEOKUFZVOVOOGJVOUGUFZP
      UHZUIUJZUKZKVPULZVOUJZUKZUMZIWEUMAVRWCUMZWAWDUMZUEWCWDULUQUNZWFVQAWGWHAVR
      UOZWGAKOABCEFGHJLMNOPQRSUPURWGWJVRUBWCUNOKUSVRWCUTVAVBVOVTDVCVOVSPVDVEVFV
      QVOWCUDZVIWIVQVOVPUDWKVOGVPVGVOKVPVHVJWCVOVKVBWCWDVRWAVLVMWEIWBTVNVB $.

    $d A f p q h x y w $.  $d G p q w $.  $d R p q w $.  $d B q $.
    $d ph p q $.  $d F y $.  $d B x $.  $d F p q $.  $d q u v $.  $d g q $.
    frrlem12.5 $e |- ( ph -> R Fr A ) $.
    frrlem12.6 $e |- ( ( ph /\ z e. A ) -> Pred ( R , A , z ) C_ S ) $.
    frrlem12.7 $e |- ( ( ph /\ z e. A ) ->
    A. w e. S Pred ( R , A , w ) C_ S ) $.
    $( Lemma for well-founded recursion.  Next, we calculate the value of
       ` C ` .  (Contributed by Scott Fenton, 7-Dec-2022.) $)
    frrlem12 $p |- ( ( ph /\ z e. ( A \ dom F ) /\
    w e. ( ( S i^i dom F ) u. { z } ) ) -> ( C ` w ) =
    ( w G ( C |` Pred ( R , A , w ) ) ) ) $=
      ( vq vp cv cdm cdif wcel cin csn cun cfv cpred cres co wceq wo elun velsn
      weq wa orbi2i bitri elinel2 frrlem1 wbr wi anbi12d imbi1d imbi2d frrlem10
      breq1 chvarvv sylan2 adantlr cop fveq1i wfn c0 wfun frrlem9 funresd dmres
      df-fn sylanblrc adantr vex ovex fnsn a1i eldifn nsyl disjsn sylibr adantl
      wn simpr fvun1 syl112anc eqtrid elinel1 fvresd eqtrd frrlem11 fnfun ssun1
      wss sseqtrri wral eldifi syl2an frrlem8 ssind sseqtrrdi fun2ssres syl3anc
      syl rspa resabs1d oveq2d 3eqtr4d ex fvsn vsnid fvun2 reseq1i resundir wfr
      eqtri predfrirr ressnop0 uneq12d un0 eqtrdi 3eqtr4a fveq2 id predeq3 jaod
      reseq2d oveq12d eqeq12d syl5ibrcom biimtrid 3impia ) ADUGZHPUHZUIUJZEUGZL
      UUIUKZUUHULZUMZUJZUUKJUNZUUKJHKUUKUOZUPZQUQZURZUUOUUKUULUJZEDVBZUSZAUUJVC
      ZUUTUUOUVAUUKUUMUJZUSUVCUUKUULUUMUTUVEUVBUVAEUUHVAVDVEUVDUVAUUTUVBUVDUVAU
      UTUVDUVAVCZUUKPUNZUUKPUUQUPZQUQZUUPUUSAUVAUVGUVIURZUUJUVAAUUKUUIUJZUVJUUK
      LUUIVFZAUEEFGHIKUFNOPQBCUEEHIKMUFQRVGSANUGZIUJOUGZIUJVCVCZBUGZGUGZUVMVHZU
      VPFUGZUVNVHZVCZGFVBZVIZVIUVOUEUGZUVQUVMVHZUWDUVSUVNVHZVCZUWBVIZVIBUEBUEVB
      ZUWCUWHUVOUWIUWAUWGUWBUWIUVRUWEUVTUWFUVPUWDUVQUVMVNUVPUWDUVSUVNVNVJVKVLTV
      OVMVPVQUVFUUPUUKPLUPZUNZUVGUVFUUPUUKUWJUUHUUHPHKUUHUOZUPZQUQZVRULZUMZUNZU
      WKUUKJUWPUAVSUVFUWJUULVTZUWOUUMVTZUULUUMUKWAURZUVAUWQUWKURUVDUWRUVAAUWRUU
      JAUWJWBUWJUHZUULURUWRALPABCFGHIKMNOPQRSTWCWDPLWEZUWJUULWFWGWHZWHUWSUVFUUH
      UWNDWIZUUHUWMQWJZWKZWLUVDUWTUVAUUJUWTAUUJUUHUULUJZWRUWTUUJUUHUUIUJUXGUUHH
      UUIWMUUHLUUIVFWNUULUUHWOWPWQZWHUVDUVAWSUULUUMUWJUWOUUKWTXAXBUVFUUKLPUVAUU
      KLUJZUVDUUKLUUIXCZWQXDXEUVFUURUVHUUKQUVFUURUWJUUQUPZUVHUVFJWBZUWJJXIZUUQU
      XAXIUURUXKURUVDUXLUVAUVDJUUNVTUXLABCDFGHIJKLMNOPQRSTUAXFUUNJXGXSWHUXMUVFU
      WJUWPJUWJUWOXHUAXJWLUVFUUQUULUXAUVFUUQLUUIUVDUUQLXIZELXKZUXIUXNUVAUUJAUUH
      HUJZUXOUUHHUUIXLZUDVPUXJUXNELXTXMZUVAUUQUUIXIZUVDUVAUVKUXSUVLBCEHIKMPQRSX
      NXSWQXOUXBXPUUQJUWJXQXRUVFPUUQLUXRYAXEYBYCYDUVDUUTUVBUUHJUNZUUHJUWLUPZQUQ
      ZURUVDUUHUWOUNZUWNUXTUYBUUHUWNUXDUXEYEUVDUXTUUHUWPUNZUYCUUHJUWPUAVSUVDUWR
      UWSUWTUUHUUMUJZUYDUYCURUXCUWSUVDUXFWLUXHUYEUVDDYFWLUULUUMUWJUWOUUHYGXAXBU
      VDUYAUWMUUHQUVDUYAUWJUWLUPZUWOUWLUPZUMZUWMUYAUWPUWLUPUYHJUWPUWLUAYHUWJUWO
      UWLYIYKUVDUYHUWMWAUMUWMUVDUYFUWMUYGWAUVDPUWLLUUJAUXPUWLLXIUXQUCVPYAUVDUUH
      UWLUJWRZUYGWAURAUYIUUJAHKYJUYIUBHKUUHYLXSWHUUHUWNUWLYMXSYNUWMYOYPXBYBYQUV
      BUUPUXTUUSUYBUUKUUHJYRUVBUUKUUHUURUYAQUVBYSUVBUUQUWLJHKUUKUUHYTUUBUUCUUDU
      UEUUAUUFUUG $.

    frrlem13.8 $e |- ( ( ph /\ z e. A ) -> S e. _V ) $.
    frrlem13.9 $e |- ( ( ph /\ z e. A ) -> S C_ A ) $.

    $d A c $.  $d A t $.  $d C c $.  $d c f $.  $d c t $.  $d C t $.  $d c w $.
    $d C w $.  $d c x $.  $d c y $.  $d f t $.  $d F t $.  $d F w $.  $d G c $.
    $d G t $.  $d ph w $.  $d R c $.  $d R t $.  $d S t $.  $d S w $.
    $d t w $.  $d t x $.  $d t y $.  $d t z $.  $d w z $.
    $( Lemma for well-founded recursion.  Assuming that ` S ` is a subset of
       ` A ` and that ` z ` is ` R ` -minimal, then ` C ` is an acceptable
       function.  (Contributed by Scott Fenton, 7-Dec-2022.) $)
    frrlem13 $p |- ( ( ph /\
    ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) ) ->
    C e. B ) $=
      ( vt vc cv cdm cdif wcel cpred c0 wceq wa wfn wss wral cfv co w3a wex cin
      cres csn cun cvv eldifi sylan2 adantrr inex1g snex unexg sylancl frrlem11
      syl inss1 sstrid adantl snssd unssd weq elun elin velsn orbi12i bitri rsp
      frrlem8 anim12d1 ssin imbitrdi preddif eqeq1i ssdif0 sylbb2 predss sstrdi
      wo ssind predeq3 sseq1d syl5ibrcom jaod biimtrid ssun1 ralrimiva frrlem12
      imp jca 3expa fneq2 sseq1 sseq2 raleqbi1dv anbi12d raleq 3anbi123d spcegv
      wi syl13anc wb cop frrlem9 resfunexg syl2an2r eqeltrid fneq1 fveq1 reseq1
      wfun oveq2d eqeq12d ralbidv 3anbi13d exbidv frrlem1 elab2g mpbird ) ADUIZ
      HPUJZUKZULZUUCKUUAUMZUNUOZUPZUPZJIULZJUGUIZUQZUUJHURZHKEUIZUMZUUJURZEUUJU
      SZUPZUUMJUTZUUMJUUNVEZQVAZUOZEUUJUSZVBZUGVCZUUHLUUBVDZUUAVFZVGZVHULZJUVGU
      QZUVGHURZUUNUVGURZEUVGUSZUPZUVAEUVGUSZUVDUUHLVHULZUVHAUUDUVOUUFUUDAUUAHUL
      ZUVOUUAHUUBVIZUEVJVKZUVOUVEVHULUVFVHULUVHLUUBVHVLUUAVMUVEUVFVHVHVNVOVQAUU
      DUVIUUFABCDFGHIJKLMNOPQRSTUAVPVKUUHUVJUVLUUHUVEUVFHUUHUVELHLUUBVRAUUDLHUR
      ZUUFUUDAUVPUVSUVQUFVJVKVSUUHUUAHAUUDUVPUUFUUDUVPAUVQVTVKWAWBUUHUVKEUVGUUH
      UUMUVGULZUPUUNUVEUVGUUHUVTUUNUVEURZUVTUUMLULZUUMUUBULZUPZEDWCZWTZUUHUWAUV
      TUUMUVEULZUUMUVFULZWTUWFUUMUVEUVFWDUWGUWDUWHUWEUUMLUUBWEEUUAWFWGWHUUHUWDU
      WAUWEUUHUWDUUNLURZUUNUUBURZUPUWAUUHUWBUWIUWCUWJUUHUWIELUSZUWBUWIYAAUUDUWK
      UUFUUDAUVPUWKUVQUDVJVKUWIELWIVQBCEHIKMPQRSWJWKUUNLUUBWLWMUUHUWAUWEHKUUAUM
      ZUVEURUUHUWLLUUBAUUDUWLLURZUUFUUDAUVPUWMUVQUCVJVKUUGUWLUUBURZAUUFUWNUUDUU
      FUWLUUBKUUAUMZUUBUUFUWLUWOUKZUNUOUWLUWOURUUEUWPUNHUUBKUUAWNWOUWLUWOWPWQUU
      BKUUAWRWSVTVTXAUWEUUNUWLUVEHKUUMUUAXBXCXDXEXFXJUVEUVFXGWSXHXKAUUDUVNUUFAU
      UDUPUVAEUVGAUUDUVTUVAABCDEFGHIJKLMNOPQRSTUAUBUCUDXIXLXHVKUVHUVIUVMUVNVBZU
      VDUVCUWQUGUVGVHUUJUVGUOZUUKUVIUUQUVMUVBUVNUUJUVGJXMUWRUULUVJUUPUVLUUJUVGH
      XNUUOUVKEUUJUVGUUJUVGUUNXOXPXQUVAEUUJUVGXRXSXTXJYBUUHJVHULUUIUVDYCUUHJPLV
      EZUUAUUAPUWLVEQVAYDZVFZVGZVHUAUUHUWSVHULZUXAVHULUXBVHULAPYLUUGUVOUXCABCFG
      HIKMNOPQRSTYEUVRPLVHYFYGUWTVMUWSUXAVHVHVNVOYHUHUIZUUJUQZUUQUUMUXDUTZUUMUX
      DUUNVEZQVAZUOZEUUJUSZVBZUGVCUVDUHJIVHUXDJUOZUXKUVCUGUXLUXEUUKUXJUVBUUQUUJ
      UXDJYIUXLUXIUVAEUUJUXLUXFUURUXHUUTUUMUXDJYJUXLUXGUUSUUMQUXDJUUNYKYMYNYOYP
      YQBCUGEHIKMUHQRYRYSVQYT $.

    frrlem14.10 $e |- ( ( ph /\ ( A \ dom F ) =/= (/) ) ->
    E. z e. ( A \ dom F ) Pred ( R , ( A \ dom F ) , z ) = (/) ) $.
    $( Lemma for well-founded recursion.  Finally, we tie all these threads
       together and show that ` dom F = A ` when given the right ` S ` .
       Specifically, we prove that there can be no ` R ` -minimal element of
       ` ( A \ dom F ) ` .  (Contributed by Scott Fenton, 7-Dec-2022.) $)
    frrlem14 $p |- ( ph -> dom F = A ) $=
      ( cdm wss frrlem7 a1i cdif c0 wceq cv cpred wrex wcel wa wn eldifn adantl
      cuni frrlem13 elssuni syl frrlem5 sseqtrrdi dmss cres csn cun ssun2 vsnid
      sselii cop dmeqi dmun ovex dmsnop uneq2i 3eqtri eleqtrri sseldd expr mtod
      co nrexdv wne df-ne sylan2br ex mt3d ssdif0 sylibr eqssd ) APUHZHWQHUIABC
      HIKMPQRSUJUKAHWQULZUMUNZHWQUIAWSWRKDUOZUPUMUNZDWRUQZAXADWRAWTWRURZUSXAWTW
      QURZXCXDUTAWTHWQVAVBAXCXAXDAXCXAUSUSZJUHZWQWTXEJPUIXFWQUIXEJIVCZPXEJIURJX
      GUIABCDEFGHIJKLMNOPQRSTUAUBUCUDUEUFVDJIVEVFBCHIKMPQRSVGVHJPVIVFWTXFURXEWT
      PLVJZUHZWTVKZVLZXFXJXKWTXJXIVMDVNVOXFXHWTWTPHKWTUPVJZQWGZVPVKZVLZUHXIXNUH
      ZVLXKJXOUAVQXHXNVRXPXJXIWTXMWTXLQVSVTWAWBWCUKWDWEWFWHAWSUTZXBXQAWRUMWIXBW
      RUMWJUGWKWLWMHWQWNWOWP $.
  $}

  ${
    fprlem.1 $e |- B = { f | E. x ( f Fn x /\
                          ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x ) /\
                          A. y e. x ( f ` y ) =
                                  ( y G ( f |` Pred ( R , A , y ) ) ) ) } $.
    fprlem.2 $e |- F = frecs ( R , A , G ) $.

    $d A a f x y g h u v $.  $d R a f x y g h u v $.  $d G a f x y g h u v $.
    $d B a $.
    $( Lemma for well-founded recursion with a partial order.  Two acceptable
       functions are compatible.  (Contributed by Scott Fenton,
       11-Sep-2023.) $)
    fprlem1 $p |- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\
      ( g e. B /\ h e. B ) ) -> ( ( x g u /\ x h v ) -> u = v ) ) $=
      ( va cv wbr wa cres wceq cdm cin wfr wpo wse w3a wcel weq vex breldm elin
      biimpri syl2an brresi anbi12i an4 bitri syl21anbrc wfn cfv cpred wral wss
      id inss2 frrlem3 sstrid adantl simpl1 frss sylc simpl2 poss sess2 frrlem4
      co simpl3 incom reseq2i fneq12 mp2an fveq1i predeq2 ax-mp reseq12i oveq2i
      eqeq12i raleqbii sylibr ancoms fpr3g syl311anc breqd biimprd wfun frrlem2
      wb wi ad2antrl funres wrel wal dffun2 2sp sps simplbiim 3syl sylan2d syl5
      ) APZDPZIPZQZXJCPZJPZQZRZXJXKXLXLUAZXOUAZUBZSZQZXJXNXOXTSZQZRZEGUCZEGUDZE
      GUEZUFZXLFUGZXOFUGZRZRZDCUHZXQXJXTUGZYOXQYEXMXJXRUGZXJXSUGZYOXPXJXKXLAUIZ
      DUIZUJXJXNXOYRCUIZUJYOYPYQRXJXRXSUKULUMZUUAXQVDYEYOXMRZYOXPRZRYOYORXQRYBU
      UBYDUUCXTXJXKXLYSUNXTXJXNXOYTUNUOYOXMYOXPUPUQURYMYDXJXNYAQZYBYNYMUUDYDYMY
      AYCXJXNYMXTGUCZXTGUDZXTGUEZYAXTUSOPZYAUTUUHYAXTGUUHVAZSLVPTOXTVBRZYCXTUSZ
      UUHYCUTZUUHYCUUISZLVPZTZOXTVBZRZYAYCTYMXTEVCZYFUUEYLUURYIYKUURYJYKXTXSEXR
      XSVEABEFGHJLMVFVGVHVHZYFYGYHYLVIXTEGVJVKYMUURYGUUFUUSYFYGYHYLVLXTEGVMVKYM
      UURYHUUGUUSYFYGYHYLVQXTEGVNVKYLUUJYIABEFGHIJLOMVOVHYLUUQYIYKYJUUQYKYJRXOX
      SXRUBZSZUUTUSZUUHUVAUTZUUHUVAUUTGUUHVAZSZLVPZTZOUUTVBZRUUQABEFGHJILOMVOUU
      KUVBUUPUVHYCUVATXTUUTTZUUKUVBWQXTUUTXOXRXSVRZVSZUVJXTUUTYCUVAVTWAUUOUVGOX
      TUUTUVJUULUVCUUNUVFUUHYCUVAUVKWBUUMUVEUUHLYCUVAUUIUVDUVKUVIUUIUVDTUVJXTUU
      TGUUHWCWDWEWFWGWHUOWIWJVHOXTGYAYCLWKWLWMWNYMXLWOZYAWOZYBUUDRYNWRZYJUVLYIY
      KABEFGHILMWPWSXTXLWTUVMYAXAUVNCXBDXBZAXBUVNADCYAXCUVOUVNAUVNDCXDXEXFXGXHX
      I $.
  $}

  ${
    $d R w x z $.  $d A w x z $.
    $( Lemma for well-founded recursion with a partial order.  Establish a
       subset relation.  (Contributed by Scott Fenton, 11-Sep-2023.) $)
    fprlem2 $p |- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\ z e. A ) ->
       A. w e. Pred ( R , A , z ) Pred ( R , A , w ) C_ Pred ( R , A , z ) ) $=
      ( vx wfr w3a cv wcel wa cpred wbr wb cvv vex elpred elv simprl adantr jca
      wpo wse wss simpll2 simpllr 3jca simprr simplrr potr sylc 3imtr4g sylan2b
      ex ssrdv ralrimiva ) CDFZCDUAZCDUBZGZAHZCIZJZCDBHZKZCDUTKZUCZBVEVCVEIZVBV
      CCIZVCUTDLZJZVFVGVJMACNDUTVCBOPQVBVJJZEVDVEVKEHZCIZVLVCDLZJZVMVLUTDLZJZVL
      VDIZVLVEIZVKVOVQVKVOJZVMVPVKVMVNRZVTUQVMVHVAGZJVNVIJVPVTUQWBVKUQVOUPUQURV
      AVJUDSVTVMVHVAWAVKVHVOVBVHVIRSUSVAVJVOUEUFTVTVNVIVKVMVNUGVBVHVIVOUHTCVLVC
      UTDUIUJTUMVRVOMBCNDVCVLEOZPQVSVQMACNDUTVLWCPQUKUNULUO $.
  $}

  ${
    $d A f g h u v x y $.  $d F f g h u v x y $.  $d G f g h u v x y $.
    $d R f g h u v x y $.  $d X y $.
    fpr2a.1 $e |- F = frecs ( R , A , G ) $.
    $( Weak version of ~ fpr2 which is useful for proofs that avoid the axiom
       of replacement.  (Contributed by Scott Fenton, 18-Nov-2024.) $)
    fpr2a $p |- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\ X e. dom F ) ->
    ( F ` X ) = ( X G ( F |` Pred ( R , A , X ) ) ) ) $=
      ( vy vx vv vu vf vg vh wcel w3a cfv cres co wceq cv cdm wfr wpo wse cpred
      wi fveq2 predeq3 reseq2d oveq12d eqeq12d imbi2d wfn wss wral wex cab eqid
      id wa fprlem1 frrlem10 expcom vtoclga impcom ) ECUAZNABUBABUCABUDOZECPZEC
      ABEUEZQZDRZSZVGGTZCPZVMCABVMUEZQZDRZSZUFVGVLUFGEVFVMESZVRVLVGVSVNVHVQVKVM
      ECUGVSVMEVPVJDVSUSVSVOVICABVMEUHUIUJUKULVGVMVFNVRVGHGIJAKTZHTZUMWAAUNVOWA
      UNGWAUOUTVMVTPVMVTVOQDRSGWAUOOHUPKUQZBKLMCDWBURZFHGIJAWBBKLMCDWCFVAVBVCVD
      VE $.
    $( $j usage 'fpr2a' avoids 'ax-rep'; $)
  $}

  ${
    $d F x y z u v a b c f g h $.  $d R x y z u v a b c f g h $.
    $d A x y z u v a b c f g h $.  $d G x y z u v a b c f g h $.
    fprr.1 $e |- F = frecs ( R , A , G ) $.
    $( Law of well-founded recursion over a partial order, part one.  Establish
       the functionality and domain of the recursive function generator.  Note
       that by requiring a partial order we can avoid using the axiom of
       infinity.  (Contributed by Scott Fenton, 11-Sep-2023.) $)
    fpr1 $p |- ( ( R Fr A /\ R Po A /\ R Se A ) -> F Fn A ) $=
      ( vb vc vv vu vf vx vy va vg vh vz wceq cv wss wa wfr wpo wse w3a cdm wfn
      wfun cpred wral cfv cres wex cab eqid frrlem1 fprlem1 frrlem9 cop csn cun
      simp1 wcel ssidd fprlem2 cvv setlikespec ancoms 3ad2antl3 predss a1i cdif
      co c0 wne wrex difssd simpr jca frpomin2 syldan frrlem14 df-fn sylanbrc )
      ABUAZABUBZABUCZUDZCUGCUEZAQCAUFWGFGHIAJRZKRZUFWJASABLRZUHZWJSLWJUITWKWIUJ
      WKWIWLUKDVLQLWJUIUDKULJUMZBMNOCDKLFGAWMBJMDWMUNUOZEFGHIAWMBMNOCDWNEUPZUQW
      GFGPLHIAWMCABPRZUHZUKZWPWPWRDVLURUSUTZBWQMNOCDWNEWOWSUNWDWEWFVAWGWPAVBZTZ
      WQVCPLABVDWFWDWTWQVEVBZWEWTWFXBABWPVFVGVHWQASXAABWPVIVJWGAWHVKZVMVNZXCASZ
      XDTXCBWPUHVMQPXCVOWGXDTZXEXDXFAWHVPWGXDVQVRPAXCBVSVTWACAWBWC $.

    $d X z $.
    $( Law of well-founded recursion over a partial order, part two.  Now we
       establish the value of ` F ` within ` A ` .  (Contributed by Scott
       Fenton, 11-Sep-2023.)  (Proof shortened by Scott Fenton,
       18-Nov-2024.) $)
    fpr2 $p |- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\ X e. A ) ->
      ( F ` X ) = ( X G ( F |` Pred ( R , A , X ) ) ) ) $=
      ( wfr wpo wse w3a wcel cdm cfv cpred cres co wceq fpr1 fndmd eleq2d fpr2a
      biimpar syldan ) ABGABHABIJZEAKZECLZKZECMECABENODPQUDUGUEUDUFAEUDACABCDFR
      STUBABCDEFUAUC $.

    $d H z $.
    $( Law of well-founded recursion over a partial order, part three.
       Finally, we show that ` F ` is unique.  We do this by showing that any
       function ` H ` with the same properties we proved of ` F ` in ~ fpr1 and
       ~ fpr2 is identical to ` F ` .  (Contributed by Scott Fenton,
       11-Sep-2023.) $)
    fpr3 $p |- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\
    ( H Fn A /\
    A. z e. A ( H ` z ) = ( z G ( H |` Pred ( R , A , z ) ) ) ) ) -> F = H ) $=
      ( wfr wpo wse w3a wfn cv cfv cpred cres co wceq wral wa simpl fpr1 adantr
      fpr2 ralrimiva jca simpr fpr3g syl3anc ) BCHBCIBCJKZFBLAMZFNUKFBCUKOZPEQR
      ABSTZTUJDBLZUKDNUKDULPEQRZABSZTZUMDFRUJUMUAUJUQUMUJUNUPBCDEGUBUJUOABBCDEU
      KGUDUEUFUCUJUMUGABCDFEUHUI $.
  $}

  ${
    $d A f x y z $.  $d R f x y z $.  $d G f x y z $.  $d X f x y z $.
    $d F y z $.
    frrrel.1 $e |- F = frecs ( R , A , G ) $.
    $( Show without using the axiom of replacement that the well-founded
       recursion generator gives a relation.  (Contributed by Scott Fenton,
       18-Nov-2024.) $)
    frrrel $p |- Rel F $=
      ( vx vy vf cv wfn wss cpred wral wa cfv cres co wceq w3a wex eqid frrlem6
      cab ) FGAHIZFIZJUEAKABGIZLZUEKGUEMNUFUDOUFUDUGPDQRGUEMSFTHUCZBHCDUHUAEUB
      $.
    $( $j usage 'frrrel' avoids 'ax-rep'; $)

    $( Show without using the axiom of replacement that the domain of the
       well-founded recursion generator is a subclass of ` A ` .  (Contributed
       by Scott Fenton, 18-Nov-2024.) $)
    frrdmss $p |- dom F C_ A $=
      ( vx vy vf cv wfn wss cpred wral wa cfv cres co wceq w3a wex eqid frrlem7
      cab ) FGAHIZFIZJUEAKABGIZLZUEKGUEMNUFUDOUFUDUGPDQRGUEMSFTHUCZBHCDUHUAEUB
      $.
    $( $j usage 'frrdmss' avoids 'ax-rep'; $)

    $( Show without using the axiom of replacement that for a "function"
       defined by well-founded recursion, the predecessor class of an element
       of its domain is a subclass of its domain.  (Contributed by Scott
       Fenton, 21-Apr-2011.)  (Proof shortened by Scott Fenton,
       17-Nov-2024.) $)
    frrdmcl $p |- ( X e. dom F -> Pred ( R , A , X ) C_ dom F ) $=
      ( vz vx vy vf cv cpred cdm wss wceq predeq3 sseq1d wfn wral wa cfv co w3a
      cres wex cab eqid frrlem8 vtoclga ) ABGKZLZCMZNABELZULNGEULUJEOUKUMULABUJ
      EPQHIGAJKZHKZRUOANABIKZLZUONIUOSTUPUNUAUPUNUQUDDUBOIUOSUCHUEJUFZBJCDURUGF
      UHUI $.
    $( $j usage 'frrdmcl' avoids 'ax-rep'; $)
  $}

  ${
    $d A f g h u v x y z $.  $d F f u v x z $.  $d G f g h u v x y z $.
    $d R f g h u v x y z $.
    fprfung.1 $e |- F = frecs ( R , A , G ) $.
    $( A "function" defined by well-founded recursion is indeed a function when
       the relation is a partial order.  Avoids the axiom of replacement.
       (Contributed by Scott Fenton, 18-Nov-2024.) $)
    fprfung $p |- ( ( R Fr A /\ R Po A /\ R Se A ) -> Fun F ) $=
      ( vx vy vv vu vf vg vh wfr wpo wse w3a cv wfn wss wral cpred wa cres wceq
      cfv co wex cab eqid fprlem1 frrlem9 ) ABMABNABOPFGHIAJQZFQZRUMASABGQZUAZU
      MSGUMTUBUNULUEUNULUOUCDUFUDGUMTPFUGJUHZBJKLCDUPUIZEFGHIAUPBJKLCDUQEUJUK
      $.
    $( $j usage 'fprfung' avoids 'ax-rep'; $)

    $d A w $.  $d F g w $.  $d f w $.  $d G w $.  $d R w $.  $d w x y z $.
    $d X g w z $.
    $( The restriction of a function defined by well-founded recursion to the
       predecessor of an element of its domain is a set.  Avoids the axiom of
       replacement.  (Contributed by Scott Fenton, 18-Nov-2024.) $)
    fprresex $p |- ( ( ( R Fr A /\ R Po A /\ R Se A ) /\ X e. dom F ) ->
    ( F |` Pred ( R , A , X ) ) e. _V ) $=
      ( vg vf vx vy vz vw wcel wa cv wss wral cres wceq cvv wfr wpo wse w3a cdm
      cfv cop wfn cpred wex cab wfun fprfung funfvop sylan cuni cfrecs df-frecs
      eqtri eleq2i eluni bitri sylib eqid frrlem1 eqabri biimpi adantl ad2antrr
      co 3simpa simprlr elssuni syl sseqtrrdi predeq3 sseq1d simprrr wbr simplr
      simprll df-br sylibr breldmg mp3an2 syl2anc simprrl fndmd eleqtrd rspcdva
      fvex sseqtrrd fun2ssres syl3anc resex eqeltrdi expr syl5 exlimdv exlimddv
      vex mpd ) ABUAABUBABUCUDZECUEZMZNZEECUFZUGZGOZMZXIHOZIOZUHXLAPABJOZUIZXLP
      JXLQNXMXKUFXMXKXNRDVJSJXLQUDIUJHUKZMZNZCABEUIZRZTMZGXFXHCMZXQGUJZXCCULZXE
      YAABCDFUMZECUNUOYAXHXOUPZMYBCYEXHCABDUQYEFIJABHDURUSZUTGXHXOVAVBVCXFXQNZX
      IKOZUHZYHAPZABLOZUIZYHPZLYHQZNZYKXIUFYKXIYLRDVJSLYHQZUDZKUJZXTXQYRXFXPYRX
      JXPYRYRGXOIJKLAXOBHGDXOVDVEVFVGVHVHYGYQXTKYQYIYONZYGXTYIYOYPVKXFXQYSXTXFX
      QYSNZNZXSXIXRRZTUUAYCXICPXRXIUEZPXSUUBSXCYCXEYTYDVIUUAXIYECUUAXPXIYEPXFXJ
      XPYSVLXIXOVMVNYFVOUUAXRYHUUCUUAYMXRYHPLYHEYKESYLXRYHABYKEVPVQYTYNXFXQYIYJ
      YNVRVHUUAEUUCYHUUAXEEXGXIVSZEUUCMZXCXEYTVTUUAXJUUDXFXJXPYSWAEXGXIWBWCXEXG
      TMUUDUUEECWKEXGXDTXIWDWEWFUUAYHXIXFXQYIYOWGWHZWIWJUUFWLXRCXIWMWNXIXRGXAWO
      WPWQWRWSXBWT $.
    $( $j usage 'fprresex' avoids 'ax-rep'; $)
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Well-ordered recursion
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c wrecs $.

  $( Declare syntax for the well-ordered recursive function generator. $)
  cwrecs $a class wrecs ( R , A , F ) $.

  $( Define the well-ordered recursive function generator.  This function takes
     the usual expressions from recursion theorems and forms a unified
     definition.  Specifically, given a function ` F ` , a relation ` R ` , and
     a base set ` A ` , this definition generates a function
     ` G = wrecs ( R , A , F ) ` that has property that, at any point
     ` x e. A ` , ` ( G `` x ) = ( F `` ( G |`` Pred ( R , A , x ) ) ) ` .  See
     ~ wfr1 , ~ wfr2 , and ~ wfr3 .  (Contributed by Scott Fenton, 7-Jun-2018.)
     (Revised by BJ, 27-Oct-2024.) $)
  df-wrecs $a |- wrecs ( R , A , F ) = frecs ( R , A , ( F o. 2nd ) ) $.

  ${
    $d R f x y $.  $d A f x y $.  $d F f x y $.
    $( Obsolete definition of the well-ordered recursive function generator as
       of 18-Nov-2024.  (New usage is discouraged.)
       (Proof modification is discouraged.)  (Contributed by Scott Fenton,
       7-Jun-2018.) $)
    dfwrecsOLD $p |- wrecs ( R , A , F ) = U. { f | E. x ( f Fn x /\
    ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x ) /\
      A. y e. x ( f ` y ) = ( F ` ( f |` Pred ( R , A , y ) ) ) ) } $=
      ( c2nd ccom cv wfn wss cpred wral wa cfv cres co wceq w3a wex cwrecs cuni
      cfrecs cab df-wrecs df-frecs wtru cvv vex resex opco2 mptru eqeq2i ralbii
      wcel a1i 3anbi3i exbii abbii unieqi 3eqtri ) CDFUACDFGHUCEIAIJAICKCDBILAI
      KBAIMNBIEIOBIEICDBILPFGHQRBAIMSATEUDUBEIAIJAICKCDBILAIKBAIMNBIEIOEICDBILP
      FORBAIMSATEUDUBCDFUEABCDEFGHUFEIAIJAICKCDBILAIKBAIMNBIEIOBIEICDBILPFGHQRB
      AIMSATEUDEIAIJAICKCDBILAIKBAIMNBIEIOEICDBILPFORBAIMSATEUDEIAIJAICKCDBILAI
      KBAIMNBIEIOBIEICDBILPFGHQRBAIMSATEIAIJAICKCDBILAIKBAIMNBIEIOEICDBILPFORBA
      IMSATEEIAIJAICKCDBILAIKBAIMNBIEIOBIEICDBILPFGHQRBAIMSEIAIJAICKCDBILAIKBAI
      MNBIEIOEICDBILPFORBAIMSABIEIOBIEICDBILPFGHQRBAIMBIEIOEICDBILPFORBAIMEIAIJ
      AICKCDBILAIKBAIMNBIEIOBIEICDBILPFGHQRBIEIOEICDBILPFORBAIBIEICDBILPFGHQEIC
      DBILPFOBIEIOBIEICDBILPFGHQEICDBILPFORUGBIEICDBILPFUHUHBIUHUOUGBUIUPEICDBI
      LPUHUOUGEICDBILEUIUJUPUKULUMUNUQURUSUTVA $.
  $}

  $( General equality theorem for the well-ordered recursive function
     generator.  (Contributed by Scott Fenton, 7-Jun-2018.)  (Proof shortened
     by Scott Fenton, 17-Nov-2024.) $)
  wrecseq123 $p |- ( ( R = S /\ A = B /\ F = G ) ->
                     wrecs ( R , A , F ) = wrecs ( S , B , G ) ) $=
    ( wceq c2nd ccom cfrecs cwrecs coeq1 frecseq123 syl3an3 df-wrecs 3eqtr4g
    w3a ) CDGZABGZEFGZQACEHIZJZBDFHIZJZACEKBDFKTRSUAUCGUBUDGEFHLABCDUAUCMNACEOB
    DFOP $.

  ${
    $d R f x y $.  $d S f x y $.  $d A f x y $.  $d B f x y $.  $d F f x y $.
    $d G f x y $.
    $( Obsolete version of ~ wrecseq123 as of 17-Nov-2024.
       (New usage is discouraged.)  (Proof modification is discouraged.)
       (Contributed by Scott Fenton, 7-Jun-2018.) $)
    wrecseq123OLD $p |- ( ( R = S /\ A = B /\ F = G ) ->
                     wrecs ( R , A , F ) = wrecs ( S , B , G ) ) $=
      ( vf vx vy wceq w3a cv wss cpred wral wa cfv cres wex cab wfn cuni cwrecs
      wb sseq2 3ad2ant2 predeq1 predeq2 sylan9eq 3adant3 sseq1d ralbidv anbi12d
      reseq2d fveq12d eqeq2d 3anbi23d exbidv abbidv unieqd dfwrecsOLD 3eqtr4g
      simp3 ) CDJZABJZEFJZKZGLZHLZUAZVIAMZACILZNZVIMZIVIOZPZVLVHQZVHVMRZEQZJZIV
      IOZKZHSZGTZUBVJVIBMZBDVLNZVIMZIVIOZPZVQVHWFRZFQZJZIVIOZKZHSZGTZUBACEUCBDF
      UCVGWDWPVGWCWOGVGWBWNHVGVPWIWAWMVJVGVKWEVOWHVEVDVKWEUDVFABVIUEUFVGVNWGIVI
      VGVMWFVIVDVEVMWFJVFVDVEVMADVLNWFACDVLUGABDVLUHUIZUJUKULUMVGVTWLIVIVGVSWKV
      QVGVRWJEFVDVEVFVCVDVEVRWJJVFVDVEPVMWFVHWQUNUJUOUPULUQURUSUTHIACGEVAHIBDGF
      VAVB $.
  $}

  ${
    nfwrecs.1 $e |- F/_ x R $.
    nfwrecs.2 $e |- F/_ x A $.
    nfwrecs.3 $e |- F/_ x F $.
    $( Bound-variable hypothesis builder for the well-ordered recursive
       function generator.  (Contributed by Scott Fenton, 9-Jun-2018.)  (Proof
       shortened by Scott Fenton, 17-Nov-2024.) $)
    nfwrecs $p |- F/_ x wrecs ( R , A , F ) $=
      ( cwrecs c2nd ccom cfrecs df-wrecs nfcv nfco nffrecs nfcxfr ) ABCDHBCDIJZ
      KBCDLABCQEFADIGAIMNOP $.
  $}

  ${
    $d R f y z $.  $d A f y z $.  $d F f y z $.  $d x f y z $.
    nfwrecsOLD.1 $e |- F/_ x R $.
    nfwrecsOLD.2 $e |- F/_ x A $.
    nfwrecsOLD.3 $e |- F/_ x F $.
    $( Obsolete version of ~ nfwrecs as of 17-Nov-2024.
       (New usage is discouraged.)  (Proof modification is discouraged.)
       (Contributed by Scott Fenton, 9-Jun-2018.) $)
    nfwrecsOLD $p |- F/_ x wrecs ( R , A , F ) $=
      ( vf vy vz cwrecs cv wfn wss cpred wral cfv nfcv nfss nfralw wa cres wceq
      w3a wex cab cuni dfwrecsOLD nfpred nfan nfres nffv nfeq2 nf3an nfex nfuni
      nfv nfab nfcxfr ) ABCDKHLZILZMZVABNZBCJLZOZVANZJVAPZUAZVDUTQZUTVEUBZDQZUC
      ZJVAPZUDZIUEZHUFZUGIJBCHDUHAVPVOAHVNAIVBVHVMAVBAUQVCVGAAVABAVARZFSVFAJVAV
      QAVEVAABCVDEFAVDRUIZVQSTUJVLAJVAVQAVIVKAVJDGAUTVEAUTRVRUKULUMTUNUOURUPUS
      $.
  $}

  $( Equality theorem for the well-ordered recursive function generator.
     (Contributed by Scott Fenton, 7-Jun-2018.) $)
  wrecseq1 $p |- ( R = S -> wrecs ( R , A , F ) = wrecs ( S , A , F ) ) $=
    ( wceq cwrecs eqid wrecseq123 mp3an23 ) BCEAAEDDEABDFACDFEAGDGAABCDDHI $.

  $( Equality theorem for the well-ordered recursive function generator.
     (Contributed by Scott Fenton, 7-Jun-2018.) $)
  wrecseq2 $p |- ( A = B -> wrecs ( R , A , F ) = wrecs ( R , B , F ) ) $=
    ( wceq cwrecs eqid wrecseq123 mp3an13 ) CCEABEDDEACDFBCDFECGDGABCCDDHI $.

  $( Equality theorem for the well-ordered recursive function generator.
     (Contributed by Scott Fenton, 7-Jun-2018.) $)
  wrecseq3 $p |- ( F = G -> wrecs ( R , A , F ) = wrecs ( R , A , G ) ) $=
    ( wceq cwrecs eqid wrecseq123 mp3an12 ) BBEAAECDEABCFABDFEBGAGAABBCDHI $.

  ${
    $( Move class substitution in and out of the well-founded recursive
       function generator.  (Contributed by ML, 25-Oct-2020.)  (Revised by
       Scott Fenton, 18-Nov-2024.) $)
    csbwrecsg $p |- ( A e. V -> [_ A / x ]_ wrecs ( R , D , F ) =
                   wrecs ( [_ A / x ]_ R , [_ A / x ]_ D , [_ A / x ]_ F ) ) $=
      ( wcel c2nd ccom cfrecs csb cwrecs csbfrecsg wceq csbcog csbconstg coeq2d
      eqid eqtrd df-wrecs frecseq123 mp3an12i csbeq2i 3eqtr4g ) BFGZABCDEHIZJZK
      ZABCKZABDKZABEKZHIZJZABCDELZKUIUJUKLUEUHUIUJABUFKZJZUMABCDUFFMUJUJNUIUINU
      EUOULNUPUMNUJRUIRUEUOUKABHKZIULABEHFOUEUQHUKABHFPQSUIUIUJUJUOULUAUBSABUNU
      GCDETUCUIUJUKTUD $.
  $}

  ${
    $d A w y z $.  $d F w y z $.  $d G w y z $.  $d H w y z $.  $d R w y z $.
    $( Functions defined by well-ordered recursion are identical up to
       relation, domain, and characteristic function.  (Contributed by Scott
       Fenton, 11-Feb-2011.) $)
    wfr3g $p |- ( ( ( R We A /\ R Se A ) /\ ( F Fn A /\
                 A. y e. A ( F ` y ) = ( H ` ( F |` Pred ( R , A , y ) ) ) ) /\
                     ( G Fn A /\
                 A. y e. A ( G ` y ) = ( H ` ( G |` Pred ( R , A , y ) ) ) ) )
                     -> F = G ) $=
      ( vz vw wa wfn cv cfv cpred cres wceq wral wi fveq2 eqeq12d fveq2d r19.26
      wwe wse w3a imbi2d wcel ra4v predeq3 reseq2d anbi12d rspcva ancoms ex syl
      eqtr3 expimpd wss wb predss fvreseq mpan2 biimpar eqcomd expd impcomd a2d
      syl11 syl5 wfis2g r19.21v sylib com12 sylan2br an4s 3impib jctil ad2ant2r
      eqid eqfnfv2 3adant1 mpbird ) BCUBBCUCIZDBJZAKZDLZDBCWDMZNZFLZOZABPZIZEBJ
      ZWDELZEWFNZFLZOZABPZIZUDZDEOZBBOZGKZDLZXBELZOZGBPZIZWSXFXAWBWKWRXFWKWRIWB
      XFWCWLWJWQWBXFQZWJWQIWCWLIZWIWPIZABPZXHWIWPABUAWBXIXKIZXFWBXLXEQZGBPXLXFQ
      XMXLHKZDLZXNELZOZQZGHBCXBXNOZXEXQXLXSXCXOXDXPXBXNDRXBXNERSUEXRHBCXBMZPXLX
      QHXTPZQXBBUFZXMXLXQHXTUGYBXLYAXEYBXKXIYAXEQZYBXKXIYCQZYBXKIXCDXTNZFLZOZXD
      EXTNZFLZOZIZYDXJYKAXBBWDXBOZWIYGWPYJYLWEXCWHYFWDXBDRYLWGYEFYLWFXTDBCWDXBU
      HZUITSYLWMXDWOYIWDXBERYLWNYHFYLWFXTEYMUITSUJUKYKXIYAXEYIYFOZYKXEXIYAIZYNY
      GYJXEYNYGIXCYIOZYJXEQYGYNYPXCYIYFUOULYPYJXEXCXDYIUOUMUNUPYOYHYEFYOYEYHXIY
      EYHOZYAXIXTBUQYQYAURBCXBUSHBXTDEUTVAVBVCTVGVDUNUMVEVFVHVIXLXEGBVJVKVLVMVN
      VLVOBVRVPWKWRWTXGURZWBWCWLYRWJWQGBBDEVSVQVTWA $.
  $}

  ${
    $d A f g w x y z $.  $d F f g w x y z $.  $d R f g w x y z $.
    wfrlem1OLD.1 $e |- B = { f | E. x ( f Fn x /\ ( x C_ A /\
               A. y e. x Pred ( R , A , y ) C_ x ) /\
               A. y e. x ( f ` y ) = ( F ` ( f |` Pred ( R , A , y ) ) ) ) } $.
    $( Obsolete version as of 18-Nov-2024.  Lemma for well-ordered recursion.
       The final item we are interested in is the union of acceptable functions
       ` B ` .  This lemma just changes bound variables for later use.
       (New usage is discouraged.)  (Proof modification is discouraged.)
       (Contributed by Scott Fenton, 21-Apr-2011.) $)
    wfrlem1OLD $p |- B = { g | E. z ( g Fn z /\ ( z C_ A /\
               A. w e. z Pred ( R , A , w ) C_ z ) /\
               A. w e. z ( g ` w ) = ( F ` ( g |` Pred ( R , A , w ) ) ) ) } $=
      ( cv wfn wss wral cfv cres wceq w3a wex cpred cab weq fneq1 reseq1 fveq2d
      fveq1 eqeq12d ralbidv 3anbi13d exbidv fneq2 sseq1 sseq2 raleqbi1dv sseq1d
      wa predeq3 cbvralvw bitrdi anbi12d raleq reseq2d 3anbi123d cbvexvw cbvabv
      fveq2 eqtri ) FHLZALZMZVJENZEGBLZUAZVJNZBVJOZUQZVMVIPZVIVNQZJPZRZBVJOZSZA
      TZHUBILZCLZMZWFENZEGDLZUAZWFNZDWFOZUQZWIWEPZWEWJQZJPZRZDWFOZSZCTZIUBKWDWT
      HIHIUCZWDWEVJMZVQVMWEPZWEVNQZJPZRZBVJOZSZATWTXAWCXHAXAVKXBWBXGVQVJVIWEUDX
      AWAXFBVJXAVRXCVTXEVMVIWEUGXAVSXDJVIWEVNUEUFUHUIUJUKXHWSACACUCZXBWGVQWMXGW
      RVJWFWEULXIVLWHVPWLVJWFEUMXIVPVNWFNZBWFOWLVOXJBVJWFVJWFVNUNUOXJWKBDWFBDUC
      ZVNWJWFEGVMWIURZUPUSUTVAXIXGXFBWFOWRXFBVJWFVBXFWQBDWFXKXCWNXEWPVMWIWEVGXK
      XDWOJXKVNWJWEXLVCUFUHUSUTVDVEUTVFVH $.

    $( Obsolete version as of 18-Nov-2024.  Lemma for well-ordered recursion.
       An acceptable function is a function.  (New usage is discouraged.)
       (Proof modification is discouraged.)  (Contributed by Scott Fenton,
       21-Apr-2011.) $)
    wfrlem2OLD $p |- ( g e. B -> Fun g ) $=
      ( vz vw cv wcel wfn wss cpred wral wa cfv cres wceq w3a wfrlem1OLD eqabri
      wex wfun fnfun 3ad2ant1 exlimiv sylbi ) GLZDMUKJLZNZULCOCEKLZPZULOKULQRZU
      NUKSUKUOTHSUAKULQZUBZJUEZUKUFZUSGDABJKCDEFGHIUCUDURUTJUMUPUTUQULUKUGUHUIU
      J $.

    $( Obsolete version as of 18-Nov-2024.  Lemma for well-ordered recursion.
       An acceptable function's domain is a subset of ` A ` .
       (New usage is discouraged.)  (Proof modification is discouraged.)
       (Contributed by Scott Fenton, 21-Apr-2011.) $)
    wfrlem3OLD $p |- ( g e. B -> dom g C_ A ) $=
      ( vz vw cv wcel wfn wss cpred wral wa cfv cres wceq w3a wfrlem1OLD eqabri
      wex cdm fndm sseq1d biimpar adantrr 3adant3 exlimiv sylbi ) GLZDMUNJLZNZU
      OCOZCEKLZPZUOOKUOQZRZURUNSUNUSTHSUAKUOQZUBZJUEZUNUFZCOZVDGDABJKCDEFGHIUCU
      DVCVFJUPVAVFVBUPUQVFUTUPVFUQUPVEUOCUOUNUGUHUIUJUKULUM $.

    $d G f x y z w $.  $d G g $.
    wfrlem3OLDa.2 $e |- G e. _V $.
    $( Obsolete version as of 18-Nov-2024.  Lemma for well-ordered recursion.
       Show membership in the class of acceptable functions.
       (New usage is discouraged.)  (Proof modification is discouraged.)
       (Contributed by Scott Fenton, 31-Jul-2020.) $)
    wfrlem3OLDa $p |- ( G e. B <-> E. z ( G Fn z /\ ( z C_ A /\
               A. w e. z Pred ( R , A , w ) C_ z ) /\
               A. w e. z ( G ` w ) = ( F ` ( G |` Pred ( R , A , w ) ) ) ) ) $=
      ( vg cv wfn wss wral cfv cres wceq cpred wa w3a fneq1 fveq1 reseq1 fveq2d
      wex eqeq12d ralbidv 3anbi13d exbidv wfrlem1OLD elab2 ) MNZCNZOZUPEPEGDNZU
      AZUPPDUPQUBZURUORZUOUSSZIRZTZDUPQZUCZCUHJUPOZUTURJRZJUSSZIRZTZDUPQZUCZCUH
      MJFLUOJTZVFVMCVNUQVGVEVLUTUPUOJUDVNVDVKDUPVNVAVHVCVJURUOJUEVNVBVIIUOJUSUF
      UGUIUJUKULABCDEFGHMIKUMUN $.
  $}

  ${
    $d A a b c f g h x y $.  $d B a $.  $d F a b c f g h x y $.
    $d R a b c f g h x y $.
    wfrlem4OLD.2 $e |- B = { f | E. x ( f Fn x /\ ( x C_ A /\
               A. y e. x Pred ( R , A , y ) C_ x ) /\
               A. y e. x ( f ` y ) = ( F ` ( f |` Pred ( R , A , y ) ) ) ) } $.
    $( Obsolete version as of 18-Nov-2024.  Lemma for well-ordered recursion.
       Properties of the restriction of an acceptable function to the domain of
       another one.  (New usage is discouraged.)
       (Proof modification is discouraged.)  (Contributed by Scott Fenton,
       21-Apr-2011.)  (Revised by AV, 18-Jul-2022.) $)
    wfrlem4OLD $p |- ( ( g e. B /\ h e. B ) ->
                     ( ( g |` ( dom g i^i dom h ) ) Fn ( dom g i^i dom h ) /\
                       A. a e. ( dom g i^i dom h )
                        ( ( g |` ( dom g i^i dom h ) ) ` a ) =
                        ( F ` ( ( g |` ( dom g i^i dom h ) ) |`
                              Pred ( R , ( dom g i^i dom h ) , a ) ) ) ) ) $=
      ( vb vc wcel wa cfv wceq wral wss wex cv cdm cres cpred wfrlem2OLD funfnd
      cin wfn fnresin1 syl adantr w3a wi wfrlem1OLD eqabri fndm raleqdv biimpar
      rsp 3adant2 exlimiv sylbi elinel1 impel adantlr adantl resres predss mpbi
      fvres sseqin2 3an6 2exbii exdistrv bitri ssinss1 ad2antrr syl5com elinel2
      nfra1 nfan anim12d biimpi syl6com ralrimi ad2ant2l jca wb ineqan12d sseq1
      ssin raleqbi1dv anbi12d imbi2d mpbiri imp 3adant3 exlimivv sylbir syl2anb
      sseq2 simpr preddowncl sylc eqtrid reseq2d fveq2d 3eqtr4d ralrimiva ) GUA
      ZDNZHUAZDNZOZXJXJUBZXLUBZUGZUCZXQUHZJUAZXRPZXRXQEXTUDZUCZIPZQZJXQRXKXSXMX
      KXJXOUHXSXKXJABCDEFGIKUEUFXOXPXJUIUJUKXNYEJXQXNXTXQNZOZXTXJPZXJCEXTUDZUCZ
      IPZYAYDXKYFYHYKQZXMXKXTXONZYLYFXKXJLUAZUHZYNCSZYIYNSZJYNRZOZYLJYNRZULZLTZ
      YMYLUMZUUBGDABLJCDEFGIKUNUOZUUAUUCLYOYTUUCYSYOYTOYLJXORZUUCYOUUEYTYOYLJXO
      YNYNXJUPZUQURYLJXOUSUJUTVAVBXTXOXPVCVDVEYFYAYHQXNXTXQXJVJVFYGYCYJIYGYCXJX
      QYBUGZUCYJXJXQYBVGYGUUGYIXJYGUUGYBYIYBXQSUUGYBQXQEXTVHYBXQVKVIYGXQCSZYIXQ
      SZJXQRZOZYFYBYIQXNUUKYFXKUUBXLMUAZUHZUULCSZYIUULSZJUULRZOZXTXLPXLYIUCIPQJ
      UULRZULZMTZUUKXMUUDUUTHDABMJCDEFHIKUNUOUUBUUTOZYOUUMOZYSUUQOZYTUUROZULZMT
      LTZUUKUVFUUAUUSOZMTLTUVAUVEUVGLMYOUUMYSUUQYTUURVLVMUUAUUSLMVNVOUVEUUKLMUV
      BUVCUUKUVDUVBUVCUUKUVBUVCUUKUMZUVCYNUULUGZCSZYIUVISZJUVIRZOZUMZUVCUVJUVLY
      PUVJYRUUQYNUULCVPVQYRUUPUVLYPUUNYRUUPOZUVKJUVIYRUUPJYQJYNVTUUOJUULVTWAXTU
      VINZUVOYQUUOOZUVKUVPYRYQUUPUUOUVPXTYNNYRYQXTYNUULVCYQJYNUSVRUVPXTUULNUUPU
      UOXTYNUULVSUUOJUULUSVRWBUVQUVKYIYNUULWKWCWDWEWFWGUVBXQUVIQZUVHUVNWHYOUUMX
      OYNXPUULUUFUULXLUPWIUVRUUKUVMUVCUVRUUHUVJUUJUVLXQUVICWJUUIUVKJXQUVIXQUVIY
      IXAWLWMWNUJWOWPWQWRWSWTUKXNYFXBJCXQEXTXCXDXEXFXEXGXHXIWG $.
  $}

  ${
    $d A a f g h x y $.  $d B a $.  $d F a f g h x y $.  $d R a f g h x y $.
    $d g h u v x $.
    wfrlem5OLD.1 $e |- R We A $.
    wfrlem5OLD.2 $e |- R Se A $.
    wfrlem5OLD.3 $e |- B = { f | E. x ( f Fn x /\ ( x C_ A /\
               A. y e. x Pred ( R , A , y ) C_ x ) /\
               A. y e. x ( f ` y ) = ( F ` ( f |` Pred ( R , A , y ) ) ) ) } $.
    $( Obsolete version as of 18-Nov-2024.  Lemma for well-ordered recursion.
       The values of two acceptable functions agree within their domains.
       (New usage is discouraged.)  (Proof modification is discouraged.)
       (Contributed by Scott Fenton, 21-Apr-2011.)  (Revised by Mario Carneiro,
       26-Jun-2015.) $)
    wfrlem5OLD $p |- ( ( g e. B /\ h e. B ) ->
                     ( ( x g u /\ x h v ) -> u = v ) ) $=
      ( va cv wa cres wceq cfv wbr cdm cin wcel vex breldm sylibr anandi brresi
      anim12i elin anbi12i sylbb2 mpancom wwe wse wfn cpred wral wss wfrlem3OLD
      ssinss1 wess mpi sess2 3syl adantr wfrlem4OLD ancoms incom reseq2i fneq1i
      fneq2i bitri fveq1i predeq2 ax-mp reseq12i eqeq12i raleqbii wfr3g syl3anc
      jca fveq2i breqd biimprd wi wfun wal wfrlem2OLD funres dffun2 simprbi 2sp
      wrel sps 4syl sylan2d syl5 ) APZDPZIPZUAZWTCPZJPZUAZQZWTXAXBXBUBZXEUBZUCZ
      RZUAZWTXDXEXJRZUAZQZXBFUDZXEFUDZQZXAXDSZWTXJUDZXGXOXGWTXHUDZWTXIUDZQXTXCY
      AXFYBWTXAXBAUEZDUEZUFWTXDXEYCCUEZUFUJWTXHXIUKUGXTXGQXTXCQZXTXFQZQXOXTXCXF
      UHXLYFXNYGXJWTXAXBYDUIXJWTXDXEYEUIULUMUNXRXNWTXDXKUAZXLXSXRYHXNXRXKXMWTXD
      XRXJGUOZXJGUPZQZXKXJUQOPZXKTXKXJGYLURZRKTSOXJUSQXMXJUQZYLXMTZXMYMRZKTZSZO
      XJUSZQZXKXMSXPYKXQXPXHEUTXJEUTZYKABEFGHIKNVAXHXIEVBUUAYIYJUUAEGUOYILXJEGV
      CVDUUAEGUPYJMXJEGVEVDWCVFVGABEFGHIJKONVHXRXEXIXHUCZRZUUBUQZYLUUCTZUUCUUBG
      YLURZRZKTZSZOUUBUSZQZYTXQXPUUKABEFGHJIKONVHVIYNUUDYSUUJYNUUCXJUQUUDXJXMUU
      CXJUUBXEXHXIVJZVKZVLXJUUBUUCUULVMVNYRUUIOXJUUBUULYOUUEYQUUHYLXMUUCUUMVOYP
      UUGKXMUUCYMUUFUUMXJUUBSYMUUFSUULXJUUBGYLVPVQVRWDVSVTULUGOXJGXKXMKWAWBWEWF
      XPXLYHQXSWGZXQXPXBWHXKWHZUUNCWIDWIZAWIZUUNABEFGHIKNWJXJXBWKUUOXKWOUUQADCX
      KWLWMUUPUUNAUUNDCWNWPWQVGWRWS $.
  $}

  ${
    $d A f g w x y z $.  $d R f g w x y z $.  $d G f g w x y z $.
    $d X f g w x y z $.
    wfrlem6OLD.1 $e |- F = wrecs ( R , A , G ) $.
    $( Obsolete version of ~ wfrrel as of 17-Nov-2024.
       (New usage is discouraged.)  (Proof modification is discouraged.)
       (Contributed by Scott Fenton, 8-Jun-2018.) $)
    wfrrelOLD $p |- Rel F $=
      ( vf vx vy vg wrel cv wfn wss cpred wral wa cfv cres wceq w3a cuni reluni
      wex cab wcel wfun eqid wfrlem2OLD funrel mprgbir cwrecs dfwrecsOLD releqi
      syl eqtri mpbir ) CJFKZGKZLURAMABHKZNZURMHUROPUSUQQUQUTRDQSHUROTGUCFUDZUA
      ZJZVCIKZJZIVAIVAUBVDVAUEVDUFVEGHAVABFIDVAUGUHVDUIUNUJCVBCABDUKVBEGHABFDUL
      UOUMUP $.

    $( Obsolete version of ~ wfrdmss as of 17-Nov-2024.
       (New usage is discouraged.)  (Proof modification is discouraged.)
       (Contributed by Scott Fenton, 21-Apr-2011.) $)
    wfrdmssOLD $p |- dom F C_ A $=
      ( vg vf vx vy cdm cv wfn wss cpred wral wa cfv cres wceq eqtri w3a cwrecs
      wex cab ciun cuni dfwrecsOLD dmeqi dmuni iunss wfrlem3OLD mprgbir eqsstri
      eqid ) CJZFGKZHKZLUQAMABIKZNZUQMIUQOPURUPQUPUSRDQSIUQOUAHUCGUDZFKJZUEZAUO
      UTUFZJVBCVCCABDUBVCEHIABGDUGTUHFUTUITVBAMVAAMFUTFUTVAAUJHIAUTBGFDUTUNUKUL
      UM $.

    $( Obsolete version as of 18-Nov-2024.  Lemma for well-ordered recursion.
       Compute the predecessor class for an ` R ` minimal element of
       ` ( A \ dom F ) ` .  (New usage is discouraged.)
       (Proof modification is discouraged.)  (Contributed by Scott Fenton,
       21-Apr-2011.) $)
    wfrlem8OLD $p |- ( Pred ( R , ( A \ dom F ) , X ) = (/) <->
                   Pred ( R , A , X ) = Pred ( R , dom F , X ) ) $=
      ( cpred cdm wss wa cdif wceq wfrdmssOLD predpredss biantru preddif eqeq1i
      c0 ax-mp ssdif0 bitr4i eqss 3bitr4i ) ABEGZCHZBEGZIZUGUFUDIZJAUEKBEGZRLZU
      DUFLUHUGUEAIUHABCDFMUEABENSOUJUDUFKZRLUGUIUKRAUEBEPQUDUFTUAUDUFUBUC $.

    $( Obsolete version of ~ wfrdmcl as of 17-Nov-2024.
       (New usage is discouraged.)  (Proof modification is discouraged.)
       (Contributed by Scott Fenton, 21-Apr-2011.) $)
    wfrdmclOLD $p |- ( X e. dom F -> Pred ( R , A , X ) C_ dom F ) $=
      ( vg vf vx vy vz vw cdm wcel cpred cv wss wral wa cfv cres wceq ciun wrex
      wfn w3a wex cab cwrecs dfwrecsOLD eqtri dmeqi dmuni eleq2i eliun bitri wi
      cuni eqid wfrlem1OLD eqabri predeq3 sseq1d rspccv adantl wb eleq2d sseq2d
      fndm imbi12d adantr adantrl 3adant3 exlimiv sylbi reximia ssiun sseqtrrdi
      mpbird syl ) ECMZNZABEOZGHPZIPZUEWEAQABJPZOZWEQJWERSWFWDTWDWGUADTUBJWERUF
      IUGHUHZGPZMZUCZWAWBEWJNZGWHUDZWCWKQZWBEWKNWMWAWKEWAWHURZMWKCWOCABDUIWOFIJ
      ABHDUJUKULGWHUMUKZUNGEWHWJUOUPWMWCWJQZGWHUDWNWLWQGWHWIWHNWIKPZUEZWRAQZABL
      PZOZWRQZLWRRZSZXAWITWIXBUADTUBLWRRZUFZKUGZWLWQUQZXHGWHIJKLAWHBHGDWHUSUTVA
      XGXIKWSXEXIXFWSXDXIWTWSXDSXIEWRNZWCWRQZUQZXDXLWSXCXKLEWRXAEUBXBWCWRABXAEV
      BVCVDVEWSXIXLVFXDWSWLXJWQXKWSWJWREWRWIVIZVGWSWJWRWCXMVHVJVKVSVLVMVNVOVPGW
      HWJWCVQVTVOWPVR $.
  $}

  ${
    $d A w z $.  $d F w $.  $d R w $.
    wfrlem10OLD.1 $e |- R We A $.
    wfrlem10OLD.2 $e |- F = wrecs ( R , A , G ) $.
    $( Obsolete version as of 18-Nov-2024.  Lemma for well-ordered recursion.
       When ` z ` is an ` R ` minimal element of ` ( A \ dom F ) ` , then its
       predecessor class is equal to ` dom F ` .  (New usage is discouraged.)
       (Proof modification is discouraged.)  (Contributed by Scott Fenton,
       21-Apr-2011.) $)
    wfrlem10OLD $p |- ( ( z e. ( A \ dom F ) /\
                     Pred ( R , ( A \ dom F ) , z ) = (/) ) ->
                   Pred ( R , A , z ) = dom F ) $=
      ( vw cdm cdif cv cpred c0 wceq wcel wss wa wbr wn wb wfrlem8OLD a1i simpr
      biimpi predss weq eldifn eleq1w notbid syl5ibrcom con2d imp con3d syl5com
      ssel wfrdmclOLD impel eldifi elpredg sylan mtbid w3o wfrdmssOLD sseli wor
      ancoms wwe weso ax-mp solin mpan syl2anr ecase23d cvv vex elpred sylanbrc
      elv eqelssd sylan9eqr ) BDIZJZCAKZLMNZWCWBOZBCWCLZWACWCLZWAWDWFWGNBCDEWCG
      UAUDWEHWGWAWGWAPWEWACWCUEUBWEHKZWAOZQZWIWHWCCRZWHWGOZWEWIUCWJWKHAUFZWCWHC
      RZWEWIWMSWEWMWIWEWISWMWCWAOZSZWCBWAUGZWMWIWOHAWAUHUIUJUKULWJWCBCWHLZOZWNW
      EWRWAPZWSSZWIWEWPWTXAWQWTWSWOWRWAWCUOUMUNBCDEWHGUPUQWEWCBOZWIWSWNTZWCBWAU
      RZWIXBXCBWACWHWCUSVFUTVAWIWHBOZXBWKWMWNVBZWEWABWHBCDEGVCVDXDBCVEZXEXBQXFB
      CVGXGFBCVHVIBWHWCCVJVKVLVMWLWIWKQTAWAVNCWCWHHVOVPVRVQVSVT $.
  $}

  ${
    $d A f g h u v x y z $.  $d F u v x z $.  $d G f g h x y z $.
    $d R f g h x y z $.
    wfrfunOLD.1 $e |- R We A $.
    wfrfunOLD.2 $e |- R Se A $.
    wfrfunOLD.3 $e |- F = wrecs ( R , A , G ) $.
    $( Obsolete version of ~ wfrfun as of 17-Nov-2024.
       (New usage is discouraged.)  (Proof modification is discouraged.)
       (Contributed by Scott Fenton, 21-Apr-2011.)  (Revised by Mario Carneiro,
       26-Jun-2015.) $)
    wfrfunOLD $p |- Fun F $=
      ( vx vu vv vg vf vy vh cv wbr wa wex wcel df-br wfun weq wi wal wfrrelOLD
      wrel wfn wss cpred wral cfv cres wceq w3a cab cop cwrecs dfwrecsOLD eqtri
      cuni eleq2i eluni bitri anbi1i exbii 3bitr4i anbi12i exdistrv bitr4i eqid
      wfrlem5OLD impcom an4s exlimivv sylbi ax-gen gen2 dffun2 mpbir2an ) CUACU
      FHOZIOZCPZVTJOZCPZQZIJUBZUCZJUDZIUDHUDABCDGUEWHHIWGJWEVTWAKOZPZWILOZVTUGV
      TAUHABMOZUIZVTUHMVTUJQWLWKUKWKWMULDUKUMMVTUJUNHRLUOZSZQZVTWCNOZPZWQWNSZQZ
      QZNRKRZWFWEWPKRZWTNRZQXBWBXCWDXDVTWAUPZCSZXEWISZWOQZKRZWBXCXFXEWNUTZSXICX
      JXECABDUQXJGHMABLDURUSZVAKXEWNVBVCVTWACTWPXHKWJXGWOVTWAWITVDVEVFVTWCUPZCS
      ZXLWQSZWSQZNRZWDXDXMXLXJSXPCXJXLXKVANXLWNVBVCVTWCCTWTXONWRXNWSVTWCWQTVDVE
      VFVGWPWTKNVHVIXAWFKNWJWRWOWSWFWOWSQWJWRQWFHMJIAWNBLKNDEFWNVJVKVLVMVNVOVPV
      QHIJCVRVS $.

    ${
      $d F f $.
      $( Obsolete version as of 18-Nov-2024.  Lemma for well-ordered recursion.
         Here, we compute the value of the recursive definition generator.
         (New usage is discouraged.)  (Proof modification is discouraged.)
         (Contributed by Scott Fenton, 21-Apr-2011.)  (Revised by Mario
         Carneiro, 26-Jun-2015.) $)
      wfrlem12OLD $p |- ( y e. dom F ->
                    ( F ` y ) = ( G ` ( F |` Pred ( R , A , y ) ) ) ) $=
        ( vz vf vx cv cdm wcel wex cfv wceq wss wa wi cop cpred cres eldm2 wral
        vex wfn w3a cab cuni cwrecs dfwrecsOLD eqtri eluniab bitri abid elssuni
        eleq2i sseqtrrdi sylbir wel fnop rsp impcom fndm sseq2d anbi12d biimprd
        ex eleq2d expd wfun wfrfunOLD funssfv 3adant3l fun2ssres fveq2d eqeq12d
        3adant3r mp3an1 expcom com23 com34 sylcom adantl com14 syl7 exp4a syldc
        syl6com pm2.43d 3impd exlimdv mpdi imp exlimiv sylbi ) ALZDMNWRILZUAZDN
        ZIOWRDPZDBCWRUBZUCZEPZQZIWRDAUFUDXAXFIXAWTJLZNZXGKLZUGZXIBRZXCXIRZAXIUE
        ZSZWRXGPZXGXCUCZEPZQZAXIUEZUHZKOZSZJOZXFXAWTYAJUIZUJZNYCDYEWTDBCEUKYEHK
        ABCJEULUMZURYAJWTUNUOYBXFJXHYAXFXHYAXGDRZXFYAXGYDNZYGYAJUPYHXGYEDXGYDUQ
        YFUSUTXHXTYGXFTZKXHXJXNXSYIXJXHAKVAZXNXSYITTXJXHYJXIWRWSXGVBVIXJYJXSXNY
        IXJYJXSXNYITZTXJYJYJXSYKYJXSSXRXJYJYKXSYJXRXRAXIVCVDXNYJXRXJYIXMYJXRXJY
        ITTZTXKXMYJXLYLXLAXIVCYJXLXJXRYIYJXLXJXRYITZXLXJSYJXCXGMZRZWRYNNZSZYMXJ
        XLYJYQTXJXLYJYQXJYQXLYJSXJYOXLYPYJXJYNXIXCXIXGVEZVFXJYNXIWRYRVJVGVHVKVD
        YQYGXRXFYGYQXRXFTZDVLZYGYQYSBCDEFGHVMYTYGYQUHZXFXRUUAXBXOXEXQYTYGYPXBXO
        QYOWRDXGVNVOUUAXDXPEYTYGYOXDXPQYPXCDXGVPVSVQVRVHVTWAWBWJVKWCWDWEWFWGWHW
        KWCWIWLWMWNWOWPWQWPWQ $.
    $}
  $}

  ${
    $d A f x y z $.  $d F f x y z $.  $d G f x y $.  $d R f x y z $.
    $d C f x y $.
    wfrlem13OLD.1 $e |- R We A $.
    wfrlem13OLD.2 $e |- R Se A $.
    wfrlem13OLD.3 $e |- F = wrecs ( R , A , G ) $.
    wfrlem13OLD.4 $e |- C = ( F u.
                      { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } ) $.
    $( Obsolete version as of 18-Nov-2024.  Lemma for well-ordered recursion.
       From here through ~ wfrlem16OLD , we aim to prove that ` dom F = A ` .
       We do this by supposing that there is an element ` z ` of ` A ` that is
       not in ` dom F ` .  We then define ` C ` by extending ` dom F ` with the
       appropriate value at ` z ` .  We then show that ` z ` cannot be an ` R `
       minimal element of ` ( A \ dom F ) ` , meaning that ` ( A \ dom F ) `
       must be empty, so ` dom F = A ` .  Here, we show that ` C ` is a
       function extending the domain of ` F ` by one.
       (New usage is discouraged.)  (Proof modification is discouraged.)
       (Contributed by Scott Fenton, 21-Apr-2011.)  (Revised by Mario Carneiro,
       26-Jun-2015.) $)
    wfrlem13OLD $p |- ( z e. ( A \ dom F ) -> C Fn ( dom F u. { z } ) ) $=
      ( cdm wcel csn cun wfun wceq wfn wa cin c0 cv cdif cres cfv cop wfrfunOLD
      cpred vex funsn pm3.2i dmsnop ineq2i wn eldifn disjsn sylibr eqtrid funun
      fvex sylancr dmun uneq2i eqtri fneq1i df-fn bitri sylanblrc ) AUAZBEKZUBL
      ZEVHEBDVHUGUCZFUDZUEMZNZOZVNKZVIVHMZNZPZCVRQZVJEOZVMOZRVIVMKZSZTPVOWAWBBD
      EFGHIUFVHVLAUHVKFUSZUIUJVJWDVIVQSZTWCVQVIVHVLWEUKZULVJVHVILUMWFTPVHBVIUNV
      IVHUOUPUQEVMURUTVPVIWCNVREVMVAWCVQVIWGVBVCVTVNVRQVOVSRVRCVNJVDVNVRVEVFVG
      $.

    $( Obsolete version as of 18-Nov-2024.  Lemma for well-ordered recursion.
       Compute the value of ` C ` .  (New usage is discouraged.)
       (Proof modification is discouraged.)  (Contributed by Scott Fenton,
       21-Apr-2011.) $)
    wfrlem14OLD $p |- ( z e. ( A \ dom F ) ->
                     ( y e. ( dom F u. { z } ) ->
                       ( C ` y ) = ( G ` ( C |` Pred ( R , A , y ) ) ) ) ) $=
      ( cv wcel csn cun cfv cres wceq ax-mp c0 cdm cdif cpred wi wfrlem13OLD wo
      wfn elun velsn orbi2i bitri wa wfrlem12OLD wfun wb fnfun wss cop sseqtrri
      ssun1 funssfv wfrdmclOLD fun2ssres syl3an3 fveq2d eqeq12d sylan imbitrrid
      w3a mp3an2 ex pm2.43d vsnid elun2 reseq1i resundir wfr wwe wefr predfrirr
      wn ressnop0 mp2b uneq2i un0 eqtri 3eqtri fveq2i opeq2i opex elsn eleqtrri
      mpbir fnopfvb mpbiri mpan2 fveq2 predeq3 reseq2d syl5ibrcom jaod biimtrid
      syl ) BLZCFUAZUBMDXEXDNZOZUGZALZXGMZXIDPZDCEXIUCZQZGPZRZUDBCDEFGHIJKUEXJX
      IXEMZXIXDRZUFZXHXOXJXPXIXFMZUFXRXIXEXFUHXSXQXPAXDUIUJUKXHXPXOXQXHXPXOXHXP
      XPXOUDXPXOXHXPULXIFPZFXLQZGPZRZACEFGHIJUMXHDUNZXPXOYCUOZXGDUPYDFDUQZXPYEF
      FXDFCEXDUCZQZGPZURZNZOZDFYKUTKUSYDYFXPVIZXKXTXNYBXIDFVAYMXMYAGXPYDYFXLXEU
      QXMYARCEFGXIJVBXLDFVCVDVEVFVJVGVHVKVLXHXOXQXDDPZDYGQZGPZRZXHXDXGMZYQXDXFM
      YRBVMXDXFXEVNSXHYRULYQXDYPURZDMYSYLDYSYKMZYSYLMYTYSYJRYPYIXDYOYHGYOYLYGQY
      HYKYGQZOZYHDYLYGKVOFYKYGVPUUBYHTOYHUUATYHCEVQZXDYGMWAUUATRCEVRUUCHCEVSSCE
      XDVTXDYIYGWBWCWDYHWEWFWGWHWIYSYJXDYPWJWKWMYSYKFVNSKWLXGXDYPDWNWOWPXQXKYNX
      NYPXIXDDWQXQXMYOGXQXLYGDCEXIXDWRWSVEVFWTXAXBXC $.

    $( Obsolete version as of 18-Nov-2024.  Lemma for well-ordered recursion.
       When ` z ` is ` R ` minimal, ` C ` is an acceptable function.  This step
       is where the Axiom of Replacement becomes required.
       (New usage is discouraged.)  (Proof modification is discouraged.)
       (Contributed by Scott Fenton, 21-Apr-2011.) $)
    wfrlem15OLD $p |- ( ( z e. ( A \ dom F ) /\
                       Pred ( R , ( A \ dom F ) , z ) = (/) ) ->
                     C e. { f | E. x ( f Fn x /\
                     ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x ) /\
                     A. y e. x ( f ` y ) =
                     ( G ` ( f |` Pred ( R , A , y ) ) ) ) } ) $=
      ( cv wcel wceq wa wss wral cvv cdm cdif cpred c0 wfn cfv cres w3a wex csn
      cun wfrlem13OLD adantr wfrlem10OLD wse setlikespec sylancl eqeltrrd unexg
      eldifi snex mpan2 fnex sylan2 syl2anc wfrdmssOLD snssd biimpi sylancr weq
      unss wo elun velsn orbi2i bitri wi wfrdmclOLD ssun3 syl a1i ssun1 predeq3
      eqsstrdi sseq1d syl5ibrcom jaod biimtrid ralrimiv wfrlem14OLD fneq2 sseq1
      sseq2 raleqbi1dv anbi12d raleq 3anbi123d spcedv fneq1 fveq1 reseq1 fveq2d
      jca 3jca eqeq12d ralbidv 3anbi13d exbidv elabd ) CNZDHUAZUBZOZXLFXJUCUDPZ
      QZGNZANZUEZXQDRZDFBNZUCZXQRZBXQSZQZXTXPUFZXPYAUGZIUFZPZBXQSZUHZAUIEXQUEZY
      DXTEUFZEYAUGZIUFZPZBXQSZUHZAUIGETXOEXKXJUJZUKZUEZXKTOZETOZXMYTXNCDEFHIJKL
      MULUMZXODFXJUCZXKTCDFHIJLUNZXMUUDTOZXNXMXJDODFUOUUFXJDXKUTZKDFXJUPUQUMURZ
      UUAYTYSTOZUUBUUAYRTOZUUIXJVAZXKYRTTUSZVBYSTEVCVDVEXOYQYTYSDRZYAYSRZBYSSZQ
      ZYOBYSSZUHATYSXOUUAUUJUUIUUHUUKUULUQXOYTUUPUUQUUCXOUUMUUOXMUUMXNXMXKDRZYR
      DRZUUMDFHILVFXMXJDUUGVGUURUUSQUUMXKYRDVKVHVIUMXOUUNBYSXTYSOZXTXKOZBCVJZVL
      ZXOUUNUUTUVAXTYROZVLUVCXTXKYRVMUVDUVBUVABXJVNVOVPXOUVAUUNUVBUVAUUNVQXOUVA
      YAXKRUUNDFHIXTLVRYAXKYRVSVTWAXOUUNUVBUUDYSRXOUUDXKYSUUEXKYRWBWDUVBYAUUDYS
      DFXTXJWCWEWFWGWHWIXCXMUUQXNXMYOBYSBCDEFHIJKLMWJWIUMXDXQYSPZYKYTYDUUPYPUUQ
      XQYSEWKUVEXSUUMYCUUOXQYSDWLYBUUNBXQYSXQYSYAWMWNWOYOBXQYSWPWQWRXPEPZYJYQAU
      VFXRYKYIYPYDXQXPEWSUVFYHYOBXQUVFYEYLYGYNXTXPEWTUVFYFYMIXPEYAXAXBXEXFXGXHX
      I $.

    $( Obsolete version as of 18-Nov-2024.  Lemma for well-ordered recursion.
       If ` z ` is ` R ` minimal in ` ( A \ dom F ) ` , then ` C ` is
       acceptable and thus a subset of ` F ` , but ` dom C ` is bigger than
       ` dom F ` .  Thus, ` z ` cannot be minimal, so ` ( A \ dom F ) ` must be
       empty, and (due to ~ wfrdmssOLD ), ` dom F = A ` .
       (New usage is discouraged.)  (Proof modification is discouraged.)
       (Contributed by Scott Fenton, 21-Apr-2011.) $)
    wfrlem16OLD $p |- dom F = A $=
      ( vf vx vy cdm wss c0 wceq cv cpred wcel wfrdmssOLD cdif eldifn csn ssun2
      wrex cun vsnid sselii cres cfv cop dmeqi dmun fvex dmsnop uneq2i eleqtrri
      wa 3eqtri wfn wral w3a wex cab cuni wfrlem15OLD elssuni cwrecs dfwrecsOLD
      syl eqtri sseqtrrdi dmss sseld mpi mtand nrex wn df-ne difss tz6.26i mpan
      wne sylbir mt3 ssdif0 mpbir eqssi ) ENZBBDEFIUABWJOBWJUBZPQZWLWKDARZSPQZA
      WKUFZWNAWKWMWKTZWNWMWJTZWMBWJUCWPWNUSZWMCNZTWQWMWJWMUDZUGZWSWTXAWMWTWJUEA
      UHUIWSEWMEBDWMSUJZFUKZULUDZUGZNWJXDNZUGXACXEJUMEXDUNXFWTWJWMXCXBFUOUPUQUT
      URWRWSWJWMWRCEOWSWJOWRCKRZLRZVAXHBOBDMRZSZXHOMXHVBUSXIXGUKXGXJUJFUKQMXHVB
      VCLVDKVEZVFZEWRCXKTCXLOLMABCDKEFGHIJVGCXKVHVKEBDFVIXLILMBDKFVJVLVMCEVNVKV
      OVPVQVRWLVSWKPWDZWOWKPVTWKBOXMWOBWJWAABWKDGHWBWCWEWFBWJWGWHWI $.
  $}

  ${
    $d F f g x y z w $.  $d R f g x y z w $.  $d A f g x y z w $.
    $d G f g x y z w $.  $d X f g x y z w $.
    wfrlem17OLD.1 $e |- R We A $.
    wfrlem17OLD.2 $e |- R Se A $.
    wfrlem17OLD.3 $e |- F = wrecs ( R , A , G ) $.
    $( Obsolete version as of 18-Nov-2024.  Without using ~ ax-rep , show that
       all restrictions of ` wrecs ` are sets.  (New usage is discouraged.)
       (Proof modification is discouraged.)  (Contributed by Scott Fenton,
       31-Jul-2020.) $)
    wfrlem17OLD $p |- ( X e. dom F -> ( F |` Pred ( R , A , X ) ) e. _V ) $=
      ( vg vf vx vy vz vw wcel cfv cv wss wral wa cdm cop wfn cres wceq w3a wex
      cpred cab wfun wfrfunOLD funfvop mpan cuni cwrecs dfwrecsOLD eqtri eleq2i
      cvv eluni bitri sylib simprr wfrlem3OLDa 3simpa simprlr elssuni sseqtrrdi
      eqid vex syl predeq3 sseq1d simprrr adantl wbr simprll df-br fvex breldmg
      sylibr mp3an2 syldan simprrl fndmd eleqtrd rspcdva sseqtrrd mp3an2i resex
      fun2ssres eqeltrdi expr syl5 exlimdv mpd exlimddv ) ECUAZOZEECPZUBZIQZOZX
      BJQZKQZUCXEARABLQZUHZXERLXESTXFXDPXDXGUDDPUELXESUFKUGJUIZOZTZCABEUHZUDZUS
      OZIWSXACOZXJIUGZCUJZWSXNABCDFGHUKZECULUMXNXAXHUNZOXOCXRXACABDUOXRHKLABJDU
      PUQZURIXAXHUTVAVBWSXJTZXBMQZUCZYAARZABNQZUHZYARZNYASZTZYDXBPXBYEUDDPUENYA
      SZUFZMUGZXMXTXIYKWSXCXIVCKLMNAXHBJDXBXHVIIVJZVDVBXTYJXMMYJYBYHTZXTXMYBYHY
      IVEWSXJYMXMWSXJYMTZTZXLXBXKUDZUSXPYOXBCRZXKXBUAZRXLYPUEXQYOXIYQWSXCXIYMVF
      XIXBXRCXBXHVGXSVHVKYOXKYAYRYOYFXKYARNYAEYDEUEYEXKYAABYDEVLVMYNYGWSXJYBYCY
      GVNVOYOEYRYAWSYNEWTXBVPZEYROZYOXCYSWSXCXIYMVQEWTXBVRWAWSWTUSOYSYTECVSEWTW
      RUSXBVTWBWCYOYAXBWSXJYBYHWDWEZWFWGUUAWHXKCXBWKWIXBXKYLWJWLWMWNWOWPWQ $.
  $}

  ${
    $d R x $.  $d A x $.  $d X x $.  $d G x $.  $d F x $.
    wfr2aOLD.1 $e |- R We A $.
    wfr2aOLD.2 $e |- R Se A $.
    wfr2aOLD.3 $e |- F = wrecs ( R , A , G ) $.
    $( Obsolete version of ~ wfr2a as of 18-Nov-2024.
       (New usage is discouraged.)  (Proof modification is discouraged.)
       (Contributed by Scott Fenton, 30-Jul-2020.) $)
    wfr2aOLD $p |- ( X e. dom F ->
    ( F ` X ) = ( G ` ( F |` Pred ( R , A , X ) ) ) ) $=
      ( vx cv cfv cpred cres wceq cdm fveq2 predeq3 reseq2d fveq2d eqeq12d
      wfrlem12OLD vtoclga ) IJZCKZCABUCLZMZDKZNECKZCABELZMZDKZNIECOUCENZUDUHUGU
      KUCECPULUFUJDULUEUICABUCEQRSTIABCDFGHUAUB $.
  $}

  ${
    $d A z $.  $d F z $.  $d R z $.
    wfr1OLD.1 $e |- R We A $.
    wfr1OLD.2 $e |- R Se A $.
    wfr1OLD.3 $e |- F = wrecs ( R , A , G ) $.
    $( Obsolete version of ~ wfr1 as of 18-Nov-2024.
       (New usage is discouraged.)  (Proof modification is discouraged.)
       (Contributed by Scott Fenton, 22-Apr-2011.)  (Revised by Mario Carneiro,
       26-Jun-2015.) $)
    wfr1OLD $p |- F Fn A $=
      ( vz wfn wfun cdm wceq wfrfunOLD cv cpred cres cfv cop csn cun eqid df-fn
      wfrlem16OLD mpbir2an ) CAICJCKALABCDEFGMHACHNZCABUEOPDQRSTZBCDEFGUFUAUCCA
      UBUD $.
  $}

  ${
    $d A x $.  $d F x $.  $d G x $.  $d R x $.
    wfr2OLD.1 $e |- R We A $.
    wfr2OLD.2 $e |- R Se A $.
    wfr2OLD.3 $e |- F = wrecs ( R , A , G ) $.
    $( Obsolete version of ~ wfr2 as of 18-Nov-2024.
       (New usage is discouraged.)  (Proof modification is discouraged.)
       (Contributed by Scott Fenton, 18-Apr-2011.)  (Revised by Mario Carneiro,
       26-Jun-2015.) $)
    wfr2OLD $p |- ( X e. A ->
                  ( F ` X ) = ( G ` ( F |` Pred ( R , A , X ) ) ) ) $=
      ( vx wcel cdm cfv cpred cres wceq cv cop csn cun eqid wfrlem16OLD eleq2i
      wfr2aOLD sylbir ) EAJECKZJECLCABEMNDLOUEAEIACIPZCABUFMNDLQRSZBCDFGHUGTUAU
      BABCDEFGHUCUD $.
  $}

  ${
    wfrrel.1 $e |- F = wrecs ( R , A , G ) $.
    $( The well-ordered recursion generator generates a relation.  Avoids the
       axiom of replacement.  (Contributed by Scott Fenton, 8-Jun-2018.)
       (Proof shortened by Scott Fenton, 17-Nov-2024.) $)
    wfrrel $p |- Rel F $=
      ( c2nd ccom cwrecs cfrecs df-wrecs eqtri frrrel ) ABCDFGZCABDHABMIEABDJKL
      $.
    $( $j usage 'wfrrel' avoids 'ax-rep'; $)

    $( The domain of the well-ordered recursion generator is a subclass of
       ` A ` .  Avoids the axiom of replacement.  (Contributed by Scott Fenton,
       21-Apr-2011.)  (Proof shortened by Scott Fenton, 17-Nov-2024.) $)
    wfrdmss $p |- dom F C_ A $=
      ( c2nd ccom cwrecs cfrecs df-wrecs eqtri frrdmss ) ABCDFGZCABDHABMIEABDJK
      L $.
    $( $j usage 'wfrdmss' avoids 'ax-rep'; $)

    $( The predecessor class of an element of the well-ordered recursion
       generator's domain is a subset of its domain.  Avoids the axiom of
       replacement.  (Contributed by Scott Fenton, 21-Apr-2011.)  (Proof
       shortened by Scott Fenton, 17-Nov-2024.) $)
    wfrdmcl $p |- ( X e. dom F -> Pred ( R , A , X ) C_ dom F ) $=
      ( c2nd ccom cwrecs cfrecs df-wrecs eqtri frrdmcl ) ABCDGHZECABDIABNJFABDK
      LM $.
    $( $j usage 'wfrdmcl' avoids 'ax-rep'; $)
  $}

  ${
    wfrfun.1 $e |- F = wrecs ( R , A , G ) $.
    $( The "function" generated by the well-ordered recursion generator is
       indeed a function.  Avoids the axiom of replacement.  (Contributed by
       Scott Fenton, 21-Apr-2011.)  (Revised by Mario Carneiro, 26-Jun-2015.)
       (Revised by Scott Fenton, 17-Nov-2024.) $)
    wfrfun $p |- ( ( R We A /\ R Se A ) -> Fun F ) $=
      ( wwe wse wa wfr wpo wfun wefr adantr wor weso sopo syl simpr c2nd ccom
      cwrecs cfrecs df-wrecs eqtri fprfung syl3anc ) ABFZABGZHABIZABJZUHCKUGUIU
      HABLMUGUJUHUGABNUJABOABPQMUGUHRABCDSTZCABDUAABUKUBEABDUCUDUEUF $.
    $( $j usage 'wfrfun' avoids 'ax-rep'; $)

    $( Show without using the axiom of replacement that the restriction of the
       well-ordered recursion generator to a predecessor class is a set.
       (Contributed by Scott Fenton, 18-Nov-2024.) $)
    wfrresex $p |- ( ( ( R We A /\ R Se A ) /\ X e. dom F ) ->
    ( F |` Pred ( R , A , X ) ) e. _V ) $=
      ( wwe wse wa wfr wpo w3a cdm wcel cpred cres cvv wefr adantr wor weso syl
      sopo simpr 3jca c2nd ccom cwrecs cfrecs df-wrecs eqtri fprresex sylan ) A
      BGZABHZIZABJZABKZUOLECMNCABEOPQNUPUQURUOUNUQUOABRSUNURUOUNABTURABUAABUCUB
      SUNUOUDUEABCDUFUGZECABDUHABUSUIFABDUJUKULUM $.
    $( $j usage 'wfrresex' avoids 'ax-rep'; $)

    $( A weak version of ~ wfr2 which is useful for proofs that avoid the Axiom
       of Replacement.  (Contributed by Scott Fenton, 30-Jul-2020.)  (Proof
       shortened by Scott Fenton, 18-Nov-2024.) $)
    wfr2a $p |- ( ( ( R We A /\ R Se A ) /\ X e. dom F ) ->
    ( F ` X ) = ( G ` ( F |` Pred ( R , A , X ) ) ) ) $=
      ( wwe wse wa cdm wcel cfv cpred cres c2nd ccom co wfr adantr simpr cwrecs
      wpo w3a wceq wefr wor weso sopo syl cfrecs df-wrecs eqtri fpr2a sylan cvv
      3jca wfrresex opco2 eqtrd ) ABGZABHZIZECJZKZIZECLZECABEMNZDOPZQZVGDLVBABR
      ZABUBZVAUCVDVFVIUDVBVJVKVAUTVJVAABUESUTVKVAUTABUFVKABUGABUHUISUTVATUPABCV
      HECABDUAABVHUJFABDUKULUMUNVEEVGDVCUOVBVDTABCDEFUQURUS $.
    $( $j usage 'wfr2a' avoids 'ax-rep'; $)
  $}

  ${
    wfr1.1 $e |- F = wrecs ( R , A , G ) $.
    $( The Principle of Well-Ordered Recursion, part 1 of 3.  We start with an
       arbitrary function ` G ` .  Then, using a base class ` A ` and a
       set-like well-ordering ` R ` of ` A ` , we define a function ` F ` .
       This function is said to be defined by "well-ordered recursion".  The
       purpose of these three theorems is to demonstrate the properties of
       ` F ` .  We begin by showing that ` F ` is a function over ` A ` .
       (Contributed by Scott Fenton, 22-Apr-2011.)  (Revised by Mario Carneiro,
       26-Jun-2015.)  (Revised by Scott Fenton, 18-Nov-2024.) $)
    wfr1 $p |- ( ( R We A /\ R Se A ) -> F Fn A ) $=
      ( wwe wse wa wfr wpo wfn wefr adantr wor weso sopo syl simpr c2nd ccom
      cwrecs cfrecs df-wrecs eqtri fpr1 syl3anc ) ABFZABGZHABIZABJZUHCAKUGUIUHA
      BLMUGUJUHUGABNUJABOABPQMUGUHRABCDSTZCABDUAABUKUBEABDUCUDUEUF $.
  $}

  ${
    wfr2.1 $e |- F = wrecs ( R , A , G ) $.
    $( The Principle of Well-Ordered Recursion, part 2 of 3.  Next, we show
       that the value of ` F ` at any ` X e. A ` is ` G ` applied to all
       "previous" values of ` F ` .  (Contributed by Scott Fenton,
       18-Apr-2011.)  (Revised by Mario Carneiro, 26-Jun-2015.) $)
    wfr2 $p |- ( ( ( R We A /\ R Se A ) /\ X e. A ) ->
                  ( F ` X ) = ( G ` ( F |` Pred ( R , A , X ) ) ) ) $=
      ( wwe wse wa wcel cdm cfv cpred cres wceq wfr1 fndmd eleq2d biimpar wfr2a
      syldan ) ABGABHIZEAJZECKZJZECLCABEMNDLOUBUEUCUBUDAEUBACABCDFPQRSABCDEFTUA
      $.
  $}

  ${
    $d A z $.  $d F z $.  $d G z $.  $d H z $.  $d R z $.
    wfr3.3 $e |- F = wrecs ( R , A , G ) $.
    $( The principle of Well-Ordered Recursion, part 3 of 3.  Finally, we show
       that ` F ` is unique.  We do this by showing that any function ` H `
       with the same properties we proved of ` F ` in ~ wfr1 and ~ wfr2 is
       identical to ` F ` .  (Contributed by Scott Fenton, 18-Apr-2011.)
       (Revised by Mario Carneiro, 26-Jun-2015.)  (Revised by Scott Fenton,
       18-Nov-2024.) $)
    wfr3 $p |- ( ( ( R We A /\ R Se A ) /\ ( H Fn A /\
                    A. z e. A ( H ` z ) =
                            ( G ` ( H |` Pred ( R , A , z ) ) ) ) ) ->
                  F = H ) $=
      ( wwe wse wa wfn cv cfv cpred cres wceq wral simpl wfr1 wfr2 adantr simpr
      ralrimiva jca wfr3g syl3anc ) BCHBCIJZFBKALZFMFBCUHNZOEMPABQJZJUGDBKZUHDM
      DUIOEMPZABQZJZUJDFPUGUJRUGUNUJUGUKUMBCDEGSUGULABBCDEUHGTUCUDUAUGUJUBABCDF
      EUEUF $.
  $}

  ${
    $d A z $.  $d F z $.  $d G z $.  $d H z $.  $d R z $.
    wfr3OLD.1 $e |- R We A $.
    wfr3OLD.2 $e |- R Se A $.
    wfr3OLD.3 $e |- F = wrecs ( R , A , G ) $.
    $( Obsolete form of ~ wfr3 as of 18-Nov-2024.  (New usage is discouraged.)
       (Proof modification is discouraged.)  (Contributed by Scott Fenton,
       18-Apr-2011.)  (Revised by Mario Carneiro, 26-Jun-2015.) $)
    wfr3OLD $p |- ( ( H Fn A /\
                    A. z e. A ( H ` z ) =
                            ( G ` ( H |` Pred ( R , A , z ) ) ) ) ->
                  F = H ) $=
      ( wwe wse wa wfn cv cfv cpred cres wceq wral pm3.2i wfr1 mp2an wcel wfr3g
      wfr2 mpanl12 rgen mp3an12 ) BCJZBCKZLDBMZANZDODBCULPZQEORZABSZLFBMULFOFUM
      QEORABSLDFRUIUJGHTUKUOUIUJUKGHBCDEIUAUBUNABUIUJULBUCUNGHBCDEULIUEUFUGTABC
      DFEUDUH $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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 eqid rnmptss ssonuni imp syl2an eqeltrd ) BDEZCFEABGZHABCIZABCJ
      ZKZLZFUHUIULMUGABCFNOUGUKPEZUKFQZULFEZUHUGUJPEUMABCDRUJPSTABCFUJUJUAUBUMU
      NUOUKPUCUDUEUF $.
  $}

  ${
    $d x A $.
    $( The nonempty indexed intersection of a class of ordinal numbers
       ` B ( x ) ` is an ordinal number.  (Contributed by NM, 13-Oct-2003.)
       (Proof shortened by Mario Carneiro, 5-Dec-2016.) $)
    iinon $p |- ( ( A. x e. A B e. On /\ A =/= (/) ) ->
                |^|_ x e. A B e. On ) $=
      ( con0 wcel wral c0 wne wa ciin cmpt crn cint wceq dfiin3g adantr rnmptss
      wss eqid cdm dm0rn0 bitr3id necon3bid biimpar oninton syl2an2r eqeltrd
      dmmptg eqeq1d ) CDEABFZBGHZIABCJZABCKZLZMZDUJULUONUKABCDOPUJUNDRUKUNGHZUO
      DEABCDUMUMSQUJUPUKUJUNGBGUNGNUMTZGNUJBGNUMUAUJUQBGABCDUHUIUBUCUDUNUEUFUG
      $.
  $}

  ${
    $d x y S $.  $d x y F $.  $d x T $.
    onfununi.1 $e |- ( Lim y -> ( F ` y ) = U_ x e. y ( F ` x ) ) $.
    onfununi.2 $e |- ( ( x e. On /\ y e. On /\ x C_ y )
                  -> ( F ` x ) C_ ( F ` y ) ) $.
    $( A property of functions on ordinal numbers.  Generalization of Theorem
       Schema 8E of [Enderton] p. 218.  (Contributed by Eric Schmidt,
       26-May-2009.) $)
    onfununi $p |- ( ( S e. T /\ S C_ On /\ S =/= (/) )
                -> ( F ` U. S ) = U_ x e. S ( F ` x ) ) $=
      ( wcel con0 wss cfv ciun wa wceq word wi syl6 imp adantr fveq2 c0 wne w3a
      cuni cv wn wlim ssorduni ad2antrr nelneq wo elssuni adantl ssel ordsseleq
      eloni syl2an anabss1 mpbid ord con1d syl5 exp4b pm2.43d com23 ssrdv sylan
      wb ssn0 unissd orduniss syl eqssd df-lim syl3anbrc an32s 3adantl1 ssonuni
      limeq iuneq1 eqeq12d imbi12d vtoclg 3adant3 mpd wral eluni2 anim1d onelon
      wrex adantrd ordelss a1i syland 3jcad expd reximdvai biimtrid ssiun iunss
      ralrimiv cbviunv sseqtrdi 3ad2ant2 eqsstrd ex ssiun2s pm2.61d2 jca2 sseq2
      sylibr anbi2d sseq2d 3com12 3expib vtoclga sylsyld ) CDHZCIJZCUAUBZUCZCUD
      ZEKZACAUEZEKZLZYAYBCHZYCYFJZYAYGUFZYHYAYIMZYCAYBYELZYFYJYBUGZYCYKNZXSXTYI
      YLXRXSYIXTYLXSYIMZXTMYBOZYBUAUBZYBYBUDZNZYLXSYOYIXTCUHZUIYNCYBJXTYPYNACYB
      XSYIYDCHZYDYBHZPXSYTYIUUAXSYTYIUUAPXSYTYTYIUUAYTYIMYDYBNZUFXSYTMZUUAYDYBC
      UJUUCUUAUUBUUCUUAUUBUUCYDYBJZUUAUUBUKZYTUUDXSYDCULZUMXSYTUUDUUEVHZUUCYDOZ
      YOUUGXSXSYTUUHXSYTYDIHZUUHCIYDUNZYDUPQRYSYDYBUOUQURUSUTVAVBVCVDVERVFZCYBV
      IVGYNYRXTYNYBYQYNCYBUUKVJXSYQYBJZYIXSYOUULYSYBVKVLSVMSYBVNVOVPVQYAYLYMPZY
      IXRXSUUMXTXRXSUUMXRXSYBIHZUUMCDVRZBUEZUGZUUPEKZAUUPYELZNZPUUMBYBIUUPYBNZU
      UQYLUUTYMUUPYBVSUVAUURYCUUSYKUUPYBETZAUUPYBYEVTWAWBFWCQRWDSWEYAYKYFJZYIXS
      XRUVCXTXSYKBCUURLZYFXSYEUVDJZAYBWFYKUVDJXSUVEAYBXSUUAYEUURJZBCWJZUVEUUAYD
      UUPHZBCWJXSUVGBYDCWGXSUVHUVFBCXSUUPCHZUVHUVFXSUVIUVHMZUUIUUPIHZYDUUPJZUCU
      VFXSUVJUUIUVKUVLXSUVJUVKUVHMUUIXSUVIUVKUVHCIUUPUNZWHUUPYDWIQXSUVIUVKUVHUV
      MWKXSUVIUUPOZUVHUVLXSUVIUVKUVNUVMUUPUPQUVNUVHMUVLPXSUUPYDWLWMWNWOGQWPWQWR
      BCUURYEWSQXAAYBYEUVDWTXKBACUURYEUUPYDETXBXCXDSXEXFACYEYBYCYDYBETXGXHYAYEY
      CJZACWFYFYCJYAUVOACYAUUNYTUUIUUDMZUVOXRXSUUNXTXRXSUUNUUORWDYAYTUUIUUDXSXR
      YTUUIPXTUUJXDUUFXIUUIUVLMZUVFPUVPUVOPBYBIUVAUVQUVPUVFUVOUVAUVLUUDUUIUUPYB
      YDXJXLUVAUURYCYEUVBXMWBUVKUUIUVLUVFUUIUVKUVLUVFGXNXOXPXQXAACYEYCWTXKVM $.
  $}

  ${
    $d w x y z A $.  $d w x y z F $.  $d w x y z K $.  $d w x y L $.
    $d w x y z S $.  $d w T $.
    onovuni.1 $e |- ( Lim y -> ( A F y ) = U_ x e. y ( A F x ) ) $.
    onovuni.2 $e |- ( ( x e. On /\ y e. On /\ x C_ y )
                  -> ( A F x ) C_ ( A F y ) ) $.
    ${
      $d x T $.
      $( A variant of ~ onfununi for operations.  (Contributed by Eric Schmidt,
         26-May-2009.)  (Revised by Mario Carneiro, 11-Sep-2015.) $)
      onovuni $p |- ( ( S e. T /\ S C_ On /\ S =/= (/) )
                     -> ( A F U. S ) = U_ x e. S ( A F x ) ) $=
        ( vz wcel con0 cvv cv co cfv ciun wceq oveq2 ovex fvmpt wss c0 wne cuni
        w3a cmpt wlim eqid elv iuneq2i 3eqtr4g 3sstr4g onfununi uniexg 3ad2ant1
        a1i syl 3eqtr3d ) DEJZDKUAZDUBUCZUEZDUDZILCIMZFNZUFZOZADAMZVFOZPZCVCFNZ
        ADCVHFNZPZABDEVFBMZUGCVNFNZAVNVLPVNVFOZAVNVIPGVPVOQBIVNVEVOLVFVDVNCFRVF
        UHZCVNFSTUIZAVNVIVLVIVLQZVHVNJVSAIVHVEVLLVFVDVHCFRVQCVHFSTUIZUPUJUKVHKJ
        VNKJVHVNUAUEVLVOVIVPHVTVRULUMUSUTVGVKQZVAUSVCLJWADEUNIVCVEVKLVFVDVCCFRV
        QCVCFSTUQUOVJVMQVBADVIVLVSVHDJVTUPUJUPUR $.
    $}

    $( A variant of ~ onovuni with indexed unions.  (Contributed by Eric
       Schmidt, 26-May-2009.)  (Proof shortened by Mario Carneiro,
       5-Dec-2016.) $)
    onoviun $p |- ( ( K e. T /\ A. z e. K L e. On /\ K =/= (/) )
                -> ( A F U_ z e. K L ) = U_ z e. K ( A F L ) ) $=
      ( vw wcel con0 c0 wne ciun co wceq 3ad2ant2 cvv wral w3a cmpt crn cuni cv
      dfiun3g oveq2d wss simp1 mptexg rnexg 3syl simp2 eqid fmpt sylib frnd cdm
      wf dmmptg simp3 eqnetrd necon3bii onovuni syl3anc wrex wb oveq2 rexrnmptw
      dm0rn0 eleq2d eliun 3bitr4g eqrdv 3eqtrd ) GELZHMLCGUAZGNOZUBZDCGHPZFQDCG
      HUCZUDZUEZFQZAWCDAUFZFQZPZCGDHFQZPZVTWAWDDFVRVQWAWDRVSCGHMUGSUHVTWCTLZWCM
      UIWCNOZWEWHRVTVQWBTLWKVQVRVSUJCGHEUKWBTULUMVTGMWBVTVRGMWBUTVQVRVSUNCGMHWB
      WBUOZUPUQURVTWBUSZNOWLVTWNGNVRVQWNGRVSCGHMVASVQVRVSVBVCWNNWCNWBVKVDUQABDW
      CTFIJVEVFVTKWHWJVTKUFZWGLZAWCVGZWOWILZCGVGZWOWHLWOWJLVRVQWQWSVHVSWPWRCAGH
      WBMWMWFHRWGWIWOWFHDFVIVLVJSAWOWCWGVMCWOGWIVMVNVOVP $.
  $}

  ${
    $d w x y z F $.
    $( There are no length ` _om ` decreasing sequences in the ordinals.  See
       also ~ noinfep for a stronger version assuming Regularity.  (Contributed
       by Mario Carneiro, 19-May-2015.) $)
    onnseq $p |- ( ( F ` (/) ) e. On ->
      E. x e. _om -. ( F ` suc x ) e. ( F ` x ) ) $=
      ( vy vw vz c0 cfv con0 wcel cv csuc com wral wrex wceq fveq2 eleq1d sylib
      wne cvv wn crn cin wa cep wwe wss epweon wf simpl wi suceq fveq2d eleq12d
      cmpt rspcv onelon expcom syl6 adantld finds2 com12 ralrimiv eqid fmpt cdm
      frnd peano1 fdmd eleqtrrid ne0d dm0rn0 necon3bii wefrc mp3an2i fvex rgenw
      cbvmptv ineq2 eqeq1d rexrnmptw ax-mp peano2 adantl sylancl elrnmpt sylibr
      wb rspceeqv rspccva adantll inelcm syl2anc neneqd nrexdv pm2.65da rexnal
      ) FBGZHIZAJZKZBGZWTBGZIZALMZUAXDUAALNWSXECLCJZBGZUOZUBZDJZBGZUCZFOZDLNZWS
      XEUDZXIEJZUCZFOZEXINZXNHUEUFXOXIHUGXIFSZXSUHXOLHXHXOXGHIZCLMLHXHUIXOYACLX
      FLIXOYAYAWSXPBGZHIZXPKZBGZHIZXOCEXFFOXGWRHXFFBPQXFXPOXGYBHXFXPBPQXFYDOXGY
      EHXFYDBPQWSXEUJXPLIZXEYCYFUKZWSYGXEYEYBIZYHXDYIAXPLWTXPOZXBYEXCYBYJXAYDBW
      TXPULUMWTXPBPUNUPYCYIYFYBYEUQURUSUTVAVBVCCLHXGXHXHVDZVERZVGXOXHVFZFSXTXOY
      MFXOFLYMVHXOLHXHYLVIVJVKYMFXIFXHVLVMREHXIVNVOXKTIZDLMXSXNWHYNDLXJBVPVQXRX
      MDELXKXHTCDLXGXKXFXJBPVRXPXKOXQXLFXPXKXIVSVTWAWBRXOXMDLXOXJLIZUDZXLFYPXJK
      ZBGZXIIZYRXKIZXLFSYPYRXGOCLNZYSYPYQLIZYRYROUUAYOUUBXOXJWCWDYRVDCYQLXGYRYR
      XFYQBPWIWEYRTIYSUUAWHYQBVPCLXGYRXHTYKWFWBWGXEYOYTWSXDYTAXJLWTXJOZXBYRXCXK
      UUCXAYQBWTXJULUMWTXJBPUNWJWKYRXIXKWLWMWNWOWPXDALWQWG $.
  $}

  $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.)  Avoid ~ ax-13 .  (Revised
       by GG, 19-May-2023.) $)
    issmo $p |- Smo A $=
      ( wsmo cdm con0 wf word wel cv cfv wcel wral mpbir eleq2i wi wceq syl2anb
      feq2i wb ordeq ax-mp rgen2 df-smo mpbir3an ) CICJZKCLZUKMZABNAOZCPBOZCPQU
      AZBUKRAUKRULDKCLEUKDKCHUDSUMDMZFUKDUBUMUQUEHUKDUFUGSUPABUKUKUNUKQUNDQUODQ
      UPUOUKQUKDUNHTUKDUOHTGUCUHABCUIUJ $.
    $( $j usage 'issmo' avoids 'ax-13'; $)
  $}

  ${
    $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 wfn crn wss ax-mp
      wa cres wfun funfn 3imtr3i resss rnssi sstr mpan anim12i df-f 3imtr4i a1i
      funres ordelord expcom ordin ex syli wceq wb dmres ordeq imbitrrdi ssralv
      cin dmss ralimi inss1 eqsstri simpl sselid fvresd eleq12d imbi2d ralbidva
      syl simpr ralbiia sylibr 3anim123d df-smo 3imtr4g impcom ) BAEZFZAGZABUAZ
      GZWEWDHAIZWDJZCKZDKZFZWKALZWLALZFZMZDWDNZCWDNZOWGEZHWGIZWTJZWMWKWGLZWLWGL
      ZFZMZDWTNZCWTNZOWFWHWEWIXAWJXBWSXHWIXAMWEAWDPZAQZHRZTWGWTPZWGQZHRZTWIXAXI
      XLXKXNAUBWGUBXIXLBAUMAUCWGUCUDXMXJRXKXNWGAABUEZUFXMXJHUGUHUIWDHAUJWTHWGUJ
      UKULWEWJBWDVEZJZXBWJWEBJZXQWJWEXRWDBUNUOXRWJXQBWDUPUQURWTXPUSXBXQUTABVAZW
      TXPVBSVCWSXHMWEWSWQDWTNZCWTNZXHWSWRCWTNZYAWTWDRZWSYBMWGARYCXOWGAVFSZWRCWT
      WDVDSWRXTCWTYCWRXTMYDWQDWTWDVDSVGVPXGXTCWTWKWTFZXFWQDWTYEWLWTFZTZXEWPWMYG
      XCWNXDWOYGWKBAYGWTBWKWTXPBXSBWDVHVIZYEYFVJVKVLYGWLBAYGWTBWLYHYEYFVQVKVLVM
      VNVOVRVSULVTCDAWACDWGWAWBWC $.

    $( 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 elinel2 ordelss ancoms sylan 3adant1 resabs1 smoeq
      3syl mpbid ) ABDZEZCAFZBGZHZBIZJZUJCDZEZACDZEZUKUNURUOUNUKCUJFZHURVAUMCVA
      BULGUMABKBULLMNUJCOQPUPCBRZUQUSSURUTTUNUOVBUKUNCBHZUOVBCULBUAUOVCVBBCUBUC
      UDUEACBUFUQUSUGUHUI $.
  $}

  ${
    $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 dfsmo2 syl wceq
      adantr cres wfun simp1bi ffund funres funfnd cima df-ima imassrn eqsstrri
      wfn frnd sstrid df-f sylanbrc smodm cin ordin wb dmres ordeq ax-mp sylibr
      ancoms sylan resss simp3bi ssralv mpsyl wel wi ordtr1 inss1 eqsstri sseli
      dmss syl6 expcomd imp31 fvresd ad2antlr eleq12d ralbidva mpbird syl3anbrc
      ) BEZAFZGZBAUAZHZIWIJZWJFZCKZWILZDKZWILZMZCWONZDWJNZWIEWFWKWGWFWIWJUKZWIO
      ZIPWKWFBUBZWTWFBHZIBWFXCIBJZXCFZWMBLZWOBLZMZCWONZDXCNZDCBQZUCZUDXBWIABUEU
      FRWFXABOZIXABAUGXMBAUHBAUIUJWFXCIBXLULUMWJIWIUNUOTWFXEWGWLBUPWGXEWLWGXEGA
      XCUQZFZWLAXCURWJXNSWLXOUSBAUTZWJXNVAVBVCVDVEZWHWSXIDWJNZWFXRWGWJXCPZWFXJX
      RWIBPXSBAVFWIBVPVBWFXDXEXJXKVGXIDWJXCVHVITWHWRXIDWJWHWOWJMZGZWQXHCWOYACDV
      JZGZWNXFWPXGYCWMABWHXTYBWMAMZWHYBXTYDWHYBXTGZWMWJMZYDWHWLYEYFVKXQWMWOWJVL
      RWJAWMWJXNAXPAXCVMVNZVOVQVRVSVTXTWPXGSWHYBXTWOABWJAWOYGVOVTWAWBWCWCWDDCWI
      QWE $.
  $}

  $( 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.) $)
    smofvon2 $p |- ( Smo F -> ( F ` B ) e. On ) $=
      ( vy vx cdm wcel wsmo cfv con0 wi wf word cv wral dfsmo2 simp1bi ffvelcdm
      expcom syl5 wn c0 ndmfv 0elon eqeltrdi a1d pm2.61i ) ABEZFZBGZABHZIFZJUIU
      GIBKZUHUKUIULUGLCMBHDMZBHFCUMNDUGNDCBOPULUHUKUGIABQRSUHTZUKUIUNUJUAIABUBU
      CUDUEUF $.
  $}

  ${
    $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 null 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 wiso word con0 wss w3a wf cv wcel cfv wral wb wa eleq2d wbr
      cdm wi wsmo wf1o isof1o f1of syl ffdm simpld fss 3adant2 syl3an1 wceq fdm
      sylan eqcomd ordeq 4syl biimpa 3adant3 3syl isorel epel fvex epeli biimpd
      anbi12d 3bitr3g ex sylbid ralrimivv 3ad2ant1 df-smo syl3anbrc ) ABFFCGZAH
      ZBIJZKCUAZICLZVRHZDMZEMZNZWACOZWBCOZNZUBZEVRPDVRPZCUCVOABCLZVPVQVSVOABCUD
      ZWIABFFCUEZABCUFZUGWIVQVSVPWIVRBCLZVQVSWIWMVRAJABCUHUIVRBICUJUOUKULVOVPVT
      VQVOVPVTVOWJWIAVRUMVPVTQWKWLWIVRAABCUNZUPAVRUQURUSUTVOVPWHVQVOWGDEVRVRVOW
      AVRNZWBVRNZRZWAANZWBANZRZWGVOWJWIWQWTQWKWLWIWOWRWPWSWIVRAWAWNSWIVRAWBWNSV
      GVAVOWTWGVOWTRZWCWFXAWAWBFTWDWEFTWCWFABWAWBFFCVBEWAVCWDWEWBCVDVEVHVFVIVJV
      KVLDECVMVN $.
  $}

  $( A strictly monotone ordinal function preserves the membership 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 $.

  ${
    $d A w x y z $.  $d F w x y z $.
    $( A strictly monotone ordinal function is one-to-one.  (Contributed by
       Mario Carneiro, 28-Feb-2013.) $)
    smo11 $p |- ( ( F : A --> B /\ Smo F ) -> F : A -1-1-> B ) $=
      ( vz vw vy vx wa cv cfv weq wi wral wcel word ex w3a 3ad2ant1 fveq2 sylan
      wsmo wceq wf1 simpl wfn ffn smodm2 ordelord syl anim12d wel w3o ordtri3or
      wf simp1rr smoel2 ralrimivva adantr simp2 eleq2d raleqbi1dv eleq1d rspccv
      simp3 rspcv syl6 3imp eleq1 biimpac syl31anc wn con0 smofvon2 ordirr 3syl
      eloni ad2antlr pm2.21dd 3exp ax-1 simp1rl eleq2 3jaod syl5 mpdd ralrimivv
      a1i dff13 sylanbrc ) ABCUNZCUAZHWJDIZCJZEIZCJZUBZDEKZLZEAMDAMZABCUCWJWKUD
      WJCAUEZWKWSABCUFWTWKHZWRDEAAXAWLANZWNANZHZWLOZWNOZHZWRXAXBXEXCXFXAAOZXBXE
      LACUGZXHXBXEAWLUHPUIXAXHXCXFLXIXHXCXFAWNUHPUIUJXAXDXGWRLXGDEUKZWQEDUKZULX
      AXDHZWRWLWNUMXLXJWRWQXKXLXJWPWQXLXJWPQZWOWONZWQXMXCFIZCJZGIZCJZNZFXQMZGAM
      ZXJWPXNXBXCXAXJWPUOXLXJYAWPXAYAXDXAXSGFAXQAXQXOCUPUQURZRXLXJWPUSXLXJWPVDX
      CYAXJQWMWONZWPXNXCYAXJYCXCYAXPWONZFWNMZXJYCLXTYEGWNAXSYDFXQWNGEKXRWOXPXQW
      NCSUTVAVEYDYCFWLWNFDKXPWMWOXOWLCSVBVCVFVGWPYCXNWMWOWOVHVITVJXLXJXNVKZWPWK
      YFWTXDWKWOVLNWOOYFWNCVMWOVPWOVNVOVQZRVRVSWQWRLXLWQWPVTWGXLXKWPWQXLXKWPQZX
      NWQYHXBYAXKWPXNXBXCXAXKWPWAXLXKYAWPYBRXLXKWPUSXLXKWPVDXBYAXKQWOWMNZWPXNXB
      YAXKYIXBYAXPWMNZFWLMZXKYILXTYKGWLAXSYJFXQWLGDKXRWMXPXQWLCSUTVAVEYJYIFWNWL
      FEKXPWOWMXOWNCSVBVCVFVGWPYIXNWMWOWOWBVITVJXLXKYFWPYGRVRVSWCWDPWEWFTDEABCW
      HWI $.
  $}

  $( A strictly monotone ordinal function preserves strict ordering.
     (Contributed by Mario Carneiro, 4-Mar-2013.) $)
  smoord $p |- ( ( ( F Fn A /\ Smo F ) /\ ( C e. A /\ D e. A ) ) ->
    ( C e. D <-> ( F ` C ) e. ( F ` D ) ) ) $=
    ( wa wcel word cfv ordelord syl2an2r w3a simp3 smoel2 3expia ordirr 3adant3
    wi wn syl neleqtrd wfn wsmo wb smodm2 simprl simprr wceq w3o ordtri3or expr
    adantrl 3impia 2thd con0 smofvon2 ad2antlr eloni 3syl 2falsed ordn2lp pm3.2
    fveq2d 3ad2ant3 mtod adantrlr 3impb pm3.21 3jaod syl5 mp2and ) DAUAZDUBZEZB
    AFZCAFZEZEZBGZCGZBCFZBDHZCDHZFZUCZVMAGZVPVNVRADUDZVMVNVOUEABIJZVMWEVPVOVSWF
    VMVNVOUFACIJZVRVSEVTBCUGZCBFZUHVQWDBCUIVQVTWDWIWJVMVPVTWDVMVPVTKVTWCVMVPVTL
    VMVPVTWCVMVOVTWCQVNVMVOVTWCACBDMUJUKULUMNVMVPWIWDVMVPWIKZVTWCWKBCBVMVPBBFRZ
    WIVQVRWLWGBOSPVMVPWILZTWKWAWBWAVMVPWAWAFRZWIVQWAUNFZWAGZWNVLWOVKVPBDUOUPZWA
    UQZWAOURPWKBCDWMVBTUSNVMVPWJWDVMVPWJKZVTWCWSVTWJVTEZWSVSWTRVMVPVSWJWHPCBUTS
    WJVMVTWTQVPWJVTVAVCVDWSWCWCWBWAFZEZWSWPXBRVMVPWPWJVQWOWPWQWRSPWAWBUTSWSXAWC
    XBQVMVPWJXAVMVNWJXAVOABCDMVEVFXAWCVGSVDUSNVHVIVJ $.

  $( A strictly monotone ordinal function preserves weak ordering.
     (Contributed by Mario Carneiro, 4-Mar-2013.) $)
  smoword $p |- ( ( ( F Fn A /\ Smo F ) /\ ( C e. A /\ D e. A ) ) ->
    ( C C_ D <-> ( F ` C ) C_ ( F ` D ) ) ) $=
    ( wfn wa wcel wn cfv wss wb word ordelord syl2an2r ordtri1 syl2anc smofvon2
    con0 eloni 3syl smoord notbid ancom2s smodm2 simprl simprr simplr 3bitr4d
    wsmo ) DAEZDUIZFZBAGZCAGZFZFZCBGZHZCDIZBDIZGZHZBCJZUTUSJZULUNUMURVBKULUNUMF
    FUQVAACBDUAUBUCUPBLZCLZVCURKULALZUOUMVEADUDZULUMUNUEABMNULVGUOUNVFVHULUMUNU
    FACMNBCOPUPUTLZUSLZVDVBKUPUKUTRGVIUJUKUOUGZBDQUTSTUPUKUSRGVJVKCDQUSSTUTUSOP
    UH $.

  ${
    $d A y x $.  $d C x $.  $d F y x $.
    $( A strictly monotone ordinal function is greater than or equal to its
       argument.  Exercise 1 in [TakeutiZaring] p. 50.  (Contributed by Andrew
       Salmon, 23-Nov-2011.)  (Revised by Mario Carneiro, 28-Feb-2013.) $)
    smogt $p |- ( ( F Fn A /\ Smo F /\ C e. A ) -> C C_ ( F ` C ) ) $=
      ( vx vy wcel cfv wss wa cv wi id fveq2 sseq12d con0 w3a word syl2anc wral
      imp wfn wsmo wceq imbi2d smodm2 3adant3 simp3 ordelord elon sylibr eleq1w
      vex weq 3anbi3d imbi12d simpl1 simpl2 ordtr1 expcomd sylc pm2.27 ralimdva
      wel syl3anc simp31 simp32 smofvon2 3ad2ant2 eloni syl simp33 smoel2 3impa
      3adantr3 ordtr2 3expia 3expd 3impia dfss3 imbitrrdi syldc a1i tfis2 mpcom
      syl22anc com12 vtoclga ) CAUAZCUBZBAFZBBCGZHZWJWHWIIZWLWMDJZWNCGZHZKWMWLK
      DBAWNBUCZWPWLWMWQWNBWOWKWQLWNBCMNUDWMWNAFZWPWHWIWRWPWNOFZWHWIWRPZWPWTWNQZ
      WSWTAQZWRXAWHWIXBWRACUEZUFZWHWIWRUGZAWNUHZRWNDULUIUJWTWPKZWHWIEJZAFZPZXHX
      HCGZHZKZDEDEUMZWTXJWPXLXNWRXIWHWIDEAUKUNXNWNXHWOXKXNLWNXHCMNUOXMEWNSZXGKW
      SWTXOXLEWNSZWPWTXMXLEWNWTEDVCZIWHWIXIXMXLKWHWIWRXQUPWHWIWRXQUQWTXQXIWTXBW
      RXQXIKXDXEXBXQWRXIXHWNAURUSUTTXJXLVAVDVBWTXPXHWOFZEWNSWPWTXLXREWNWTXQXLXR
      KZWHWIWRXQXSKWMWRXQXLXRWHWIWRXQXLPZXRWHWIXTPZXHQZWOQZXLXKWOFZXRYAXAXQYBYA
      XBWRXAWHWIXBXTXCUFWHWIWRXQXLVEXFRWHWIWRXQXLVFWNXHUHRYAWOOFZYCWIWHYEXTWNCV
      GVHWOVIVJWHWIWRXQXLVKWHWIXTYDWMWRXQYDXLAWNXHCVLVNVMYBYCIXLYDIXRXHXKWOVOTW
      EVPVQVRTVBEWNWOVSVTWAWBWCWDVPWFWGWFVR $.
  $}

  ${
    $d A x $.  $d B x $.  $d F x $.
    $( The codomain of a strictly monotone ordinal function dominates the
       domain.  (Contributed by Mario Carneiro, 13-Mar-2013.) $)
    smocdmdom $p |- ( ( F : A --> B /\ Smo F /\ Ord B ) -> A C_ B ) $=
      ( vx wf wsmo word w3a cv wcel wa cfv wss wfn simpl1 simpl2 smodm2 syl2anc
      ffnd ordelord sylancom simpl3 simpr syl3anc ffvelcdm 3ad2antl1 ordtr2 imp
      smogt syl22anc ex ssrdv ) ABCEZCFZBGZHZDABUPDIZAJZUQBJZUPURKZUQGZUOUQUQCL
      ZMZVBBJZUSUPURAGZVAUTCANZUNVEUTABCUMUNUOUROSZUMUNUOURPZACQRAUQTUAUMUNUOUR
      UBUTVFUNURVCVGVHUPURUCAUQCUIUDUMUNURVDUOABUQCUEUFVAUOKVCVDKUSUQVBBUGUHUJU
      KUL $.
  $}

  ${
    $d A x y $.  $d B x y $.  $d F x y $.
    $( The strictly monotone ordinal functions are also isomorphisms of
       subclasses of ` On ` equipped with the membership relation.
       (Contributed by Mario Carneiro, 20-Mar-2013.) $)
    smoiso2 $p |- ( ( Ord A /\ B C_ On ) ->
      ( ( F : A -onto-> B /\ Smo F ) <-> F Isom _E , _E ( A , B ) ) ) $=
      ( vx vy word con0 wss wa wfo wsmo cep cv wbr cfv wral sylan sylanbrc wcel
      adantl wiso wf1o wb wf1 fof smo11 simpl df-f1o wfn fofn smoord epel epeli
      wf 3bitr4g ralrimivva df-isom ex w3a isof1o f1ofo syl 3ad2ant1 smoiso jca
      fvex 3expib com12 impbid ) AFZBGHZIZABCJZCKZIZABLLCUAZVLVOVPVLVOIABCUBZDM
      ZEMZLNZVRCOZVSCOZLNZUCZEAPDAPZVPVOVQVLVOABCUDZVMVQVMABCUNVNWFABCUEABCUFQV
      MVNUGABCUHRTVOWEVLVMCAUIZVNWEABCUJWGVNIZWDDEAAWHVRASVSASIIVRVSSWAWBSVTWCA
      VRVSCUKEVRULWAWBVSCVFUMUOUPQTDEABLLCUQRURVPVLVOVPVJVKVOVPVJVKUSVMVNVPVJVM
      VKVPVQVMABLLCUTABCVAVBVCABCVDVEVGVHVI $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  "Strong" transfinite recursion
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c recs $.

  $( Notation for a function defined by strong transfinite recursion. $)
  crecs $a class recs ( F ) $.

  $( 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-rdg for more details on why this
     definition is desirable.  Unlike ~ df-rdg 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 ~ recsfnon and ~ recsval for the
     primary contract of this definition.  (Contributed by Stefan O'Rear,
     18-Jan-2015.)  (Revised by Scott Fenton, 3-Aug-2020.) $)
  df-recs $a |- recs ( F ) = wrecs ( _E , On , F ) $.

  ${
    $d F f x y $.
    $( The old definition of transfinite recursion.  This version is preferred
       for development, as it demonstrates the properties of transfinite
       recursion without relying on well-ordered recursion.  (Contributed by
       Scott Fenton, 3-Aug-2020.)  (Proof revised by Scott Fenton,
       18-Nov-2024.) $)
    dfrecs3 $p |- recs ( F ) = U. { f | E. x e. On ( f Fn x /\
        A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) } $=
      ( con0 cep cv cfv cres wceq wral wa cab cuni wss co wex wcel vex wwe c2nd
      crecs cwrecs ccom cfrecs wrex df-recs df-wrecs cpred df-frecs 3anass word
      wfn w3a wtr elon ordsson jca epweon wess mpi anim1ci df-ord sylibr impbii
      ordtr dftr3 wel wb ssel2 predon sseq1d syl ralbidva bitr4id 3bitri anbi1i
      pm5.32i onelon reseq2d oveq2d cvv id resex a1i opco2 adantl eqeq2d bitr3i
      eqtrd anbi2i an12 exbii df-rex bitr4i abbii unieqi eqtri 3eqtri ) DUBEFDU
      CEFDUAUDZUEZCGZAGZUMZBGZXBHZXBXEIZDHZJZBXCKZLZAEUFZCMZNZDUGEFDUHXAXDXCEOZ
      EFXEUIZXCOZBXCKZLZXFXEXBXPIZWTPZJZBXCKZUNZAQZCMZNXNABEFCWTUJYFXMYEXLCYEXC
      ERZXKLZAQXLYDYHAYDXDXSYCLZLXDYGXJLZLYHXDXSYCUKYIYJXDYIYGYCLYJYGXSYCYGXCUL
      ZXOXCUOZLZXSXCASUPYKYMYKXOYLXCUQXCVFURYMYLXCFTZLYKXOYNYLXOEFTYNUSXCEFUTVA
      VBXCVCVDVEXOYLXRXOYLXEXCOZBXCKXRBXCVGXOXQYOBXCXOBAVHZLXEERZXQYOVIXCEXEVJY
      QXPXEXCXEVKZVLVMVNVOVRVPVQYGYCXJYGYBXIBXCYGYPLZYAXHXFYSYAXEXGWTPZXHYSXTXG
      XEWTYSXPXEXBYSYQXPXEJXCXEVSYRVMVTWAYPYTXHJYGYPXEXGDXCWBYPWCXGWBRYPXBXECSW
      DWEWFWGWJWHVNVRWIWKXDYGXJWLVPWMXKAEWNWOWPWQWRWS $.
  $}

  ${
    $d F f x y $.
    $( Obsolete version of ~ dfrecs3 as of 18-Nov-2024.
       (New usage is discouraged.)  (Proof modification is discouraged.)
       (Contributed by Scott Fenton, 3-Aug-2020.) $)
    dfrecs3OLD $p |- recs ( F ) = U. { f | E. x e. On ( f Fn x /\
        A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) } $=
      ( con0 cep cv wss wral wa cfv cres wceq wex cab cuni wcel wwe wb syl wrex
      crecs cwrecs wfn cpred w3a df-recs dfwrecsOLD 3anass word wtr vex ordsson
      elon ordtr jca epweon mpi anim1ci df-ord sylibr impbii dftr3 ssel2 predon
      wess wel sseq1d bitr4id pm5.32i 3bitri anbi1i onelon fveq2d eqeq2d bitr3i
      ralbidva reseq2d anbi2i an12 exbii df-rex bitr4i abbii unieqi 3eqtri ) DU
      BEFDUCCGZAGZUDZWHEHZEFBGZUEZWHHZBWHIZJZWKWGKZWGWLLZDKZMZBWHIZUFZANZCOZPWI
      WPWGWKLZDKZMZBWHIZJZAEUAZCOZPDUGABEFCDUHXCXJXBXICXBWHEQZXHJZANXIXAXLAXAWI
      WOWTJZJWIXKXGJZJXLWIWOWTUIXMXNWIXMXKWTJXNXKWOWTXKWHUJZWJWHUKZJZWOWHAULUNX
      OXQXOWJXPWHUMWHUOUPXQXPWHFRZJXOWJXRXPWJEFRXRUQWHEFVFURUSWHUTVAVBWJXPWNWJX
      PWKWHHZBWHIWNBWHVCWJWMXSBWHWJBAVGZJWKEQZWMXSSWHEWKVDYAWLWKWHWKVEZVHTVQVIV
      JVKVLXKWTXGXKWSXFBWHXKXTJYAWSXFSWHWKVMYAWRXEWPYAWQXDDYAWLWKWGYBVRVNVOTVQV
      JVPVSWIXKXGVTVKWAXHAEWBWCWDWEWF $.
  $}

  $( Equality theorem for ` recs ` .  (Contributed by Stefan O'Rear,
     18-Jan-2015.) $)
  recseq $p |- ( F = G -> recs ( F ) = recs ( G ) ) $=
    ( wceq con0 cep cwrecs crecs wrecseq3 df-recs 3eqtr4g ) ABCDEAFDEBFAGBGDEAB
    HAIBIJ $.

  ${
    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 ) $=
      ( crecs con0 cep cwrecs df-recs nfcv nfwrecs nfcxfr ) ABDEFBGBHAEFBAFIAEI
      CJK $.
  $}

  ${
    $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 vw wss cv cfv wceq wral wi wa vy ssid con0 wcel sseq1 imbi12d imbi2d
      raleq r19.21v cres cdm wfn wfun ad4antr simpld funfnd word eloni ad3antlr
      vu ordelss sylan simplr sstrd simprd fnssres syl2anc fveq2 eqeq12d adantr
      simp-4r rspcdva simpr fvres adantl 3eqtr4d fveq2d reseq2 sselda ralrimiva
      mpd eqfnfvd cbvralvw sylibr exp31 expcom a2d biimtrid tfis3 mpcom mpi ) A
      CCNZBOZEPZWMFPZQZBCRZCUBCUCUDAWLWQSZGAUAOZCNZWPBWSRZSZSZALOZCNZWPBXDRZSZS
      ZAWRSUALCWSXDQZXBXGAXIWTXEXAXFWSXDCUEWPBWSXDUHUFUGWSCQZXBWRAXJWTWLXAWQWSC
      CUEWPBWSCUHUFUGXHLWSRAXGLWSRZSWSUCUDZXCAXGLWSUIXLAXKXBAXLXKXBSAXLTZXKWTXA
      XMXKTZWTTZMOZEPZXPFPZQZMWSRXAXOXSMWSXOXPWSUDZTZEXPUJZDPZFXPUJZDPZXQXRYAYB
      YDDYAUTXPYBYDYAEEUKZULXPYFNYBXPULYAEYAEUMZCYFNZAYGYHTXLXKWTXTHUNZUOUPYAXP
      CYFYAXPWSCXOWSUQZXTXPWSNXLYJAXKWTWSURUSWSXPVAVBXNWTXTVCVDZYAYGYHYIVEVDYFX
      PEVFVGYAFFUKZULXPYLNYDXPULYAFYAFUMZCYLNZAYMYNTXLXKWTXTIUNZUOUPYAXPCYLYKYA
      YMYNYOVEVDYLXPFVFVGYAUTOZXPUDZTZYPEPZYPFPZYPYBPZYPYDPZYRWPYSYTQBXPYPWMYPQ
      WNYSWOYTWMYPEVHWMYPFVHVIYRXPCNZWPBXPRZYAUUCYQYKVJYRXGUUCUUDSLWSXPXDXPQXEU
      UCXFUUDXDXPCUEWPBXDXPUHUFXMXKWTXTYQVKXOXTYQVCVLWAYAYQVMVLYQUUAYSQYAYPXPEV
      NVOYQUUBYTQYAYPXPFVNVOVPWBVQYAWNEWMUJZDPZQZXQYCQBCXPWMXPQZWNXQUUFYCWMXPEV
      HZUUHUUEYBDWMXPEVRVQVIAUUGBCRXLXKWTXTJUNXOWSCXPXNWTVMVSZVLYAWOFWMUJZDPZQZ
      XRYEQBCXPUUHWOXRUULYEWMXPFVHZUUHUUKYDDWMXPFVRVQVIAUUMBCRXLXKWTXTKUNUUJVLV
      PVTWPXSBMWSUUHWNXQWOXRUUIUUNVIWCWDWEWFWGWHWIWJWK $.
  $}

  ${
    $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 ) ) ) } $.
    ${
      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 fneq12 simpll fveq12d reseq12d
        simpr fveq2d eqeq12d simplr cbvraldva2 anbi12d cbvrexdva elab2 ) FKZAKZ
        LZBKZULMZULUONZGMZOZBUMPZQZARSHCKZLZDKZHMZHVDNZGMZOZDVBPZQZCRSFHEJULHOZ
        VAVJACRVKUMVBOZQZUNVCUTVIUMVBULHTVMUSVHBDUMVBVMUOVDOZQZUPVEURVGVOUOVDUL
        HVKVLVNUAZVMVNUDZUBVOUQVFGVOULHUOVDVPVQUCUEUFVKVLVNUGUHUIUJIUK $.
    $}

    $( 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 eqabi ) GJZCJZKDJZU
      BLUBUDMHLNDUCOPCQRGEABCDEFHUBIGSTUA $.
  $}

  ${
    $d f g x y z B $.  $d f x y z C $.  $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 eqabri 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 cdm simp2ll fnfun inss1 fndmd
      wfun wss syl sseqtrrid simp2rl inss2 simp2lr ssralv mpsyl simp2rr tfrlem1
      jca simp3l fnbr syl2anc elind rspcdva funbrfv sylc 3eqtr3d 3exp rexlimivv
      simp3r sylbir syl2anb ) GNZEOWEKNZUAZLNZWEPZWEWHUBIPQZLWFRZSZKTUCZHNZMNZU
      AZWHWNPZWNWHUBIPQZLWORZSZMTUCZANZDNZWEUGZXBCNZWNUGZSZXCXEQZUDZWNEOABKLEFI
      WEJGUHUEABMLEFIWNJHUHUEWMXASWLWTSZMTUCKTUCXIWLWTKMTTUFXJXIKMTTWFTOWOTOSZX
      JXGXHXKXJXGUIZXBWEPZXBWNPZXCXEXLWIWQQXMXNQLWFWOUJZXBWHXBQWIXMWQXNWHXBWEUK
      WHXBWNUKULXLLXOIWEWNXKXJXOTOXGWFWOUMUNXLWEUTZXOWEUOZVAXLWGXPWGWKWTXKXGUPZ
      WFWEUQVBZXLWFXOXQWFWOURZXLWFWEXRUSVCVKXLWNUTZXOWNUOZVAXLWPYAWPWSWLXKXGVDZ
      WOWNUQVBZXLWOXOYBWFWOVEZXLWOWNYCUSVCVKXOWFVAXLWKWJLXORXTWGWKWTXKXGVFWJLXO
      WFVGVHXOWOVAXLWSWRLXORYEWPWSWLXKXGVIWRLXOWOVGVHVJXLWFWOXBXLWGXDXBWFOXRXKX
      JXDXFVLZWFXBXCWEVMVNXLWPXFXBWOOYCXKXJXDXFWBZWOXBXEWNVMVNVOVPXLXPXDXMXCQXS
      YFXBXCWEVQVRXLYAXFXNXEQYDYGXBXEWNVQVRVSVTWAWCWD $.

    $( 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 dfrecs3 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 weq tfrlem6 cuni recsfval exdistrv bitr4i wbr df-br tfrlem5 sylanbr
      wi impcom an4s exlimivv sylbi ax-gen gen2 dffun4 mpbir2an ) EUCZUAVCUBAKZ
      GKZLZVCMZVDHKZLZVCMZNZGHUDZUNZHOZGOAOABCDEFUEVNAGVMHVKVFIKZMZVOCMZNZVIJKZ
      MZVSCMZNZNZJPIPZVLVKVRIPZWBJPZNWDVGWEVJWFVGVFCUFZMWEVCWGVFABCDEFUGZQIVFCR
      SVJVIWGMWFVCWGVIWHQJVICRSTVRWBIJUHUIWCVLIJVPVTVQWAVLVPVTNVDVEVOUJZVDVHVSU
      JZNZVQWANZVLWIVPWJVTVDVEVOUKVDVHVSUKTWLWKVLABHGCDIJEFULUOUMUPUQURUSUTAGHV
      CVAVB $.

    $( Lemma for transfinite recursion.  The domain of ` recs ` is an 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 eqabri 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 dfrecs3 eleq2i eluniab bitri wss eqabri 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 $.

    $( Lemma for transfinite recursion.  Without using ~ ax-rep , show that all
       the restrictions of ` recs ` are sets.  (Contributed by Mario Carneiro,
       16-Nov-2014.) $)
    tfrlem9a $p |- ( B e. dom recs ( F ) -> ( recs ( F ) |` B ) e. _V ) $=
      ( vg vz va cdm wcel cfv cv wa cres cvv wceq con0 wss cop wex wfun tfrlem7
      crecs funfvop mpan cuni recsfval eleq2i eluni bitri sylib wfn wral simprr
      wrex vex tfrlem3a a1i simplrr elssuni sseqtrrdi word fndm ad2antll simprl
      syl eqeltrd eloni wbr simpll fvexd simplrl sylibr breldmg syl3anc ordelss
      df-br syl2anc fun2ssres resex expr adantrd rexlimdva mpd exlimddv ) DFUEZ
      KZLZDDWHMZUAZHNZLZWMCLZOZWHDPZQLZHWJWLWHLZWPHUBZWHUCZWJWSABCEFGUDZDWHUFUG
      WSWLCUHZLWTWHXCWLABCEFGUIZUJHWLCUKULUMWJWPOZWMINZUNZJNZWMMWMXHPFMRJXFUOZO
      ZISUQZWRXEWOXKWJWNWOUPABIJCEFWMGHURZUSUMXEXJWRISXEXFSLZOXGWRXIXEXMXGWRXEX
      MXGOZOZWQWMDPZQXOXAWMWHTDWMKZTZWQXPRXAXOXBUTXOWMXCWHXOWOWMXCTWJWNWOXNVAWM
      CVBVHXDVCXOXQVDZDXQLZXRXOXQSLXSXOXQXFSXGXQXFRXEXMXFWMVEVFXEXMXGVGVIXQVJVH
      XOWJWKQLDWKWMVKZXTWJWPXNVLXODWHVMXOWNYAWJWNWOXNVNDWKWMVSVODWKWIQWMVPVQXQD
      VRVTDWHWMWAVQXPQLXOWMDXLWBUTVIWCWDWEWFWG $.

    ${
      tfrlem.3 $e |- C = ( recs ( F ) u. { <. dom recs ( F ) ,
        ( F ` recs ( F ) ) >. } ) $.
      $( Lemma for transfinite recursion.  We define class ` C ` by extending
         ` recs ` with one ordered pair.  We will assume, falsely, that domain
         of ` recs ` is a member of, and thus not equal to, ` On ` .  Using
         this assumption we will prove facts about ` C ` that will lead to a
         contradiction in ~ tfrlem14 , thus showing the domain of ` recs ` does
         in fact equal ` On ` .  Here we show (under the false assumption) that
         ` C ` is a function extending the domain of ` recs ( F ) ` by one.
         (Contributed by NM, 14-Aug-1994.)  (Revised by Mario Carneiro,
         9-May-2015.) $)
      tfrlem10 $p |- ( dom recs ( F ) e. On -> C Fn suc dom recs ( F ) ) $=
        ( crecs cdm con0 wcel csn cun wfn wfun wceq cin c0 cvv cfv csuc wa fvex
        cop funsng mpan2 tfrlem7 jctil dmsnop ineq2i word tfrlem8 orddisj ax-mp
        eqtri funun sylancl uneq2i df-suc 3eqtr4i df-fn sylanblrc fneq1i sylibr
        dmun ) FIZJZKLZVGVHVGFUAZUEMZNZVHUBZOZDVMOVIVLPZVLJZVMQVNVIVGPZVKPZUCVH
        VKJZRZSQVOVIVRVQVIVJTLVRVGFUDZVHVJKTUFUGABCEFGUHUIVTVHVHMZRZSVSWBVHVHVJ
        WAUJZUKVHULWCSQABCEFGUMVHUNUOUPVGVKUQURVHVSNVHWBNVPVMVSWBVHWDUSVGVKVFVH
        UTVAVLVMVBVCVMDVLHVDVE $.

      $( Lemma for transfinite recursion.  Compute the value of ` C ` .
         (Contributed by NM, 18-Aug-1994.)  (Revised by Mario Carneiro,
         9-May-2015.) $)
      tfrlem11 $p |- ( dom recs ( F ) e. On ->
         ( B e. suc dom recs ( F ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) ) $=
        ( wcel wceq con0 cfv cres wi wa syl wss cop csn cdm csuc wo elsuci wfun
        crecs wfn tfrlem10 fnfun cun ssun1 tfrlem9 funssfv 3expa adantrl onelss
        sseqtrri fun2ssres fveq2d sylan2 eqeq12d imbitrrid mpanl2 sylan pm2.43i
        imp exp32 pm2.43d opex opeq1 adantl eqimss mp3an2 syl2an reseq2 tfrlem6
        snid wrel resdm ax-mp eqtrdi eqtrd sneqd eleqtrid elun2 eleqtrrdi simpr
        opeq2d wb sucidg adantr eqeltrd fnopfvb syl2an2r mpbird ex jaod syl5 )
        DGUFZUAZUBZJZDWTJZDWTKZUCWTLJZDEMZEDNZGMZKZDWTUDXEXCXIXDXEXCXIXEXCXCXIO
        ZOXEXEXCXJXEEUEZXEXCPZXJXEEXAUGZXKABCEFGHIUHZXAEUIQZXKWSERZXLXJWSWSWTWS
        GMZSZTZUJZEWSXSUKIUQZXCXIXKXPPZXLPZDWSMZWSDNZGMZKABCDFGHULYCXFYDXHYFYBX
        CXFYDKZXEXKXPXCYGDEWSUMUNUOXLYBDWTRZXHYFKXEXCYHWTDUPVFYBYHPXGYEGXKXPYHX
        GYEKZDEWSURZUNUSUTVAVBVCVDVGVEVHXEXDXIXEXDPZXIDXHSZEJZYKYLXTEYKYLXSJYLX
        TJYKYLYLTXSYLDXHVIVQYKYLXRYKYLWTXHSZXRXDYLYNKXEDWTXHVJVKYKXHXQWTYKXGWSG
        YKXGYEWSXEXKYHYIXDXODWTVLXKXPYHYIYAYJVMVNXDYEWSKXEXDYEWSWTNZWSDWTWSVOWS
        VRYOWSKABCFGHVPWSVSVTWAVKWBUSWHWBWCWDYLXSWSWEQIWFXEXMXDXBXIYMWIXNYKDWTX
        AXEXDWGXEWTXAJXDWTLWJWKWLXADXHEWMWNWOWPWQWR $.

      $( Lemma for transfinite recursion.  Show ` C ` is an acceptable
         function.  (Contributed by NM, 15-Aug-1994.)  (Revised by Mario
         Carneiro, 9-May-2015.) $)
      tfrlem12 $p |- ( recs ( F ) e. _V -> C e. A ) $=
        ( vz cvv wcel cv wfn cfv cres wceq wral wa con0 wrex crecs word tfrlem8
        dmexg elon2 sylanbrc csuc onsuc tfrlem10 tfrlem11 ralrimiv fveq2 reseq2
        cdm a1i fveq2d eqeq12d cbvralvw sylib fneq2 anbi12d rspcev syl12anc syl
        raleq cop csn cun snex unexg mpan2 eqeltrid fneq1 fveq1 ralbidv rexbidv
        wb reseq1 elab2g mpbird ) FUAZJKZDCKZDALZMZBLZDNZDWFOZFNZPZBWDQZRZASTZW
        BWAUNZSKZWMWBWNUBZWNJKWOWPWBABCEFGUCUOWAJUDWNUEUFWOWNUGZSKDWQMZWJBWQQZW
        MWNUHABCDEFGHUIWOILZDNZDWTOZFNZPZIWQQWSWOXDIWQABCWTDEFGHUJUKXDWJIBWQWTW
        FPZXAWGXCWIWTWFDULXEXBWHFWTWFDUMUPUQURUSWLWRWSRAWQSWDWQPWEWRWKWSWDWQDUT
        WJBWDWQVEVAVBVCVDWBDJKWCWMVQWBDWAWNWAFNVFZVGZVHZJHWBXGJKXHJKXFVIWAXGJJV
        JVKVLELZWDMZWFXINZXIWFOZFNZPZBWDQZRZASTWMEDCJXIDPZXPWLASXQXJWEXOWKWDXID
        VMXQXNWJBWDXQXKWGXMWIWFXIDVNXQXLWHFXIDWFVRUPUQVOVAVPGVSVDVT $.
    $}

    $( Lemma for transfinite recursion.  If ` recs ` is a set function, then
       ` C ` is acceptable, and thus a subset of ` recs ` , but ` dom C ` is
       bigger than ` dom recs ` .  This is a contradiction, so ` recs ` must be
       a proper class function.  (Contributed by NM, 14-Aug-1994.)  (Revised by
       Mario Carneiro, 14-Nov-2014.) $)
    tfrlem13 $p |- -. recs ( F ) e. _V $=
      ( crecs cvv wcel cdm word wn tfrlem8 ordirr ax-mp cfv cop wss 3syl con0
      csn cun eqid tfrlem12 cuni elssuni recsfval sseqtrrdi dmss csuc a1i dmexg
      elon2 sylanbrc sucidg syl wfn wceq tfrlem10 fndm eleqtrrd sseldd mto ) EG
      ZHIZVDJZVFIZVFKZVGLABCDEFMZVFNOVEVDVFVDEPQUAUBZJZVFVFVEVJCIZVJVDRVKVFRABC
      VJDEFVJUCZUDVLVJCUEVDVJCUFABCDEFUGUHVJVDUISVEVFVFUJZVKVEVFTIZVFVNIVEVHVFH
      IVOVHVEVIUKVDHULVFUMUNZVFTUOUPVEVOVJVNUQVKVNURVPABCVJDEFVMUSVNVJUTSVAVBVC
      $.

    $( Lemma for transfinite recursion.  Assuming ~ ax-rep ,
       ` dom recs e. _V <-> recs e. _V ` , so since ` dom recs ` is an ordinal,
       it must be equal to ` On ` .  (Contributed by NM, 14-Aug-1994.)
       (Revised by Mario Carneiro, 9-May-2015.) $)
    tfrlem14 $p |- dom recs ( F ) = On $=
      ( crecs cdm con0 wcel wceq cvv tfrlem13 wfun tfrlem7 funex mpan mto word
      wo tfrlem8 ordeleqon mpbi mtpor ) EGZHZIJZUFIKZUGUELJZABCDEFMUENUGUIABCDE
      FOIUEPQRUFSUGUHTABCDEFUAUFUBUCUD $.

    $( Lemma for transfinite recursion.  Without assuming ~ ax-rep , we can
       show that all proper initial subsets of ` recs ` are sets, while nothing
       larger is a set.  (Contributed by Mario Carneiro, 14-Nov-2014.) $)
    tfrlem15 $p |- ( B e. On ->
        ( B e. dom recs ( F ) <-> ( recs ( F ) |` B ) e. _V ) ) $=
      ( con0 wcel crecs cdm cres cvv tfrlem9a adantl wa wss wn tfrlem13 word wb
      simpr resss a1i wrel tfrlem6 resdm ax-mp ssres2 eqsstrrid eqssd syl5ibcom
      wceq eleq1d tfrlem8 eloni adantr ordtri1 con2bid sylancr mpbird impbida
      mtoi ) DHIZDFJZKZIZVEDLZMIZVGVIVDABCDEFGNOVDVIPZVGVFDQZRZVJVKVEMIZABCEFGS
      VJVIVKVMVDVIUBVKVHVEMVKVHVEVHVEQVKVEDUCUDVKVEVEVFLZVHVEUEVNVEUMABCEFGUFVE
      UGUHVFDVEUIUJUKUNULVCVJVFTZDTZVGVLUAABCEFGUOVDVPVIDUPUQVOVPPVKVGVFDURUSUT
      VAVB $.

    $( Lemma for finite recursion.  Without assuming ~ ax-rep , we can show
       that the domain of the constructed function is a limit ordinal, and
       hence contains all the finite ordinals.  (Contributed by Mario Carneiro,
       14-Nov-2014.) $)
    tfrlem16 $p |- Lim dom recs ( F ) $=
      ( vz crecs cdm c0 wceq cv con0 wcel cres cvv ax-mp pm2.21i csn cun ordzsl
      csuc wrex wlim w3o word tfrlem8 mpbi res0 eqeltri wb 0elon tfrlem15 mpbir
      n0ii tfrlem13 wa simpr df-suc eqtrdi reseq2d wrel tfrlem6 resundi 3eqtr3g
      0ex resdm vex sucid eleqtrrid tfrlem9a syl cfv cop snex wfun wss funressn
      tfrlem7 ssexi unexg sylancl eqeltrd rexlimiva mto id 3jaoi ) EHZIZJKZWIGL
      ZUBZKZGMUCZWIUDZUEZWOWIUFWPABCDEFUGGWIUAUHWJWOWNWOWJWOJWIJWINZWHJOZPNZWRJ
      PWHUIVFUJJMNWQWSUKULABCJDEFUMQUNUORWNWOWNWHPNZABCDEFUPWMWTGMWKMNZWMUQZWHW
      HWKOZWHWKSZOZTZPXBWHWIOZWHWKXDTZOWHXFXBWIXHWHXBWIWLXHXAWMURZWKUSUTVAWHVBX
      GWHKABCDEFVCWHVGQWHWKXDVDVEXBXCPNZXEPNXFPNXBWKWINXJXBWKWLWIWKGVHVIXIVJABC
      WKDEFVKVLXEWKWKWHVMVNZSZXKVOWHVPXEXLVQABCDEFVSWKWHVRQVTXCXEPPWAWBWCWDWERW
      OWFWGQ $.
  $}

  ${
    $d f x y A $.  $d x y B $.  $d x y F $.  $d f x y G $.
    tfr.1 $e |- F = recs ( G ) $.
    $( A weak version of ~ tfr1 which is useful for proofs that avoid the Axiom
       of Replacement.  (Contributed by Mario Carneiro, 24-Jun-2015.) $)
    tfr1a $p |- ( Fun F /\ Lim dom F ) $=
      ( vx vy vf wfun cdm wlim crecs cv wfn cfv cres wceq wral con0 wrex mpbir
      wa cab eqid tfrlem7 funeqi tfrlem16 wb dmeqi limeq ax-mp pm3.2i ) AGZAHZI
      ZUKBJZGDEFKZDKZLEKZUOMUOUQNBMOEUPPTDQRFUAZFBURUBZUCAUNCUDSUMUNHZIZDEURFBU
      SUEULUTOUMVAUFAUNCUGULUTUHUISUJ $.

    $( A weak version of ~ tfr2 which is useful for proofs that avoid the Axiom
       of Replacement.  (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 $.

    $( Without assuming ~ ax-rep , we can show that all proper initial subsets
       of ` recs ` are sets, while nothing larger is a set.  (Contributed by
       Mario Carneiro, 24-Jun-2015.) $)
    tfr2b $p |- ( Ord A -> ( A e. dom F <-> ( F |` A ) e. _V ) ) $=
      ( vx vy vf word con0 wcel wceq wo cdm cres cvv wb ordeleqon cv cfv mtbiri
      crecs wfn wral wrex cab eqid tfrlem15 dmeqi eleq2i reseq1i eleq1i 3bitr4g
      wa onprc elex mto eleq1 tfrlem13 eqneltri reseq2 wrel wss wfun wlim tfr1a
      simpli funrel ax-mp simpri limord ordsson mp2b relssres mp2an eqtrdi jaoi
      eleq1d 2falsed sylbi ) AHAIJZAIKZLABMZJZBANZOJZPZAQVTWFWAVTACUAZMZJWGANZO
      JWCWEEFGRZERZUBFRZWJSWJWLNCSKFWKUCUMEIUDGUEZAGCWMUFZUGWBWHABWGDUHUIWDWIOB
      WGADUJUKULWAWCWEWAWCIWBJZWOIOJUNIWBUOUPAIWBUQTWAWEBOJBWGODEFWMGCWNURUSWAW
      DBOWAWDBINZBAIBUTBVAZWBIVBZWPBKBVCZWQWSWBVDZBCDVEZVFBVGVHWTWBHWRWSWTXAVIW
      BVJWBVKVLBIVMVNVOVQTVRVPVS $.

    $( Principle of Transfinite Recursion, part 1 of 3.  Theorem 7.41(1) of
       [TakeutiZaring] p. 47.  We start with an arbitrary class ` G ` ,
       normally a function, and 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 NM,
       17-Aug-1994.)  (Revised by Mario Carneiro, 18-Jan-2015.) $)
    tfr1 $p |- F Fn On $=
      ( vx vy vf con0 wfn crecs wfun cdm wceq cv cfv cres wral wa wrex cab eqid
      tfrlem7 tfrlem14 df-fn mpbir2an fneq1i mpbir ) AGHBIZGHZUHUGJUGKGLDEFMZDM
      ZHEMZUINUIUKOBNLEUJPQDGRFSZFBULTZUADEULFBUMUBUGGUCUDGAUGCUEUF $.

    $( Principle of Transfinite Recursion, part 2 of 3.  Theorem 7.41(2) of
       [TakeutiZaring] p. 47.  Here we show that the function ` F ` has the
       property that for any function ` G ` whatsoever, the "next" value of
       ` F ` is ` G ` recursively applied to all "previous" values of ` F ` .
       (Contributed by NM, 9-Apr-1995.)  (Revised by Stefan O'Rear,
       18-Jan-2015.) $)
    tfr2 $p |- ( A e. On -> ( F ` A ) = ( G ` ( F |` A ) ) ) $=
      ( con0 wcel cdm cfv cres wceq tfr1 fndmi eleq2i tfr2a sylbir ) AEFABGZFAB
      HBAICHJPEAEBBCDKLMABCDNO $.

    $( Principle of Transfinite Recursion, part 3 of 3.  Theorem 7.41(3) of
       [TakeutiZaring] p. 47.  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
       NM, 18-Aug-1994.)  (Revised by Mario Carneiro, 9-May-2015.) $)
    tfr3 $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 nfra1 wi fveq2 wb imp nfan rsp
      wcel nfim eqeq12d imbi2d r19.21v wss onss fvreseq mpanl2 biimtrrdi sylan2
      tfr1 ancoms adantr tfr2 jctr jcab sylibr eqeq12 adantl mpbird exp43 com4t
      syl6 exp4a pm2.43d syl com3l impd a2d biimtrid tfis2f com12 ralrimi mpan2
      eqfnfv biimpar syldan ) BGHZAIZBJZBWBKZDJZLZAGMZWCWBCJZLZAGMZBCLZWAWGNZWI
      AGWAWGAWAAOWFAGPUAZWBGUCZWLWIWLWIQZWLFIZBJZWPCJZLZQZAFWLWSAWMWSAOUDWBWPLZ
      WIWSWLXAWCWQWHWRWBWPBRWBWPCRUEUFWTFWBMWLWSFWBMZQWNWOWLWSFWBUGWNWLXBWIWNWA
      WGXBWIQZWGWNWAXCWGWNWFQZWNWAXCQZQWFAGUBXDWNXEXDWNWNWAXCWNWANZXBXDWNWIXFXB
      XDWNWIXFXBNZXDWNNZNWIWECWBKZDJZLZXGXKXHXFXBXKWAWNXBXKQZWNWAWBGUHZXLWBUIWA
      XMNXBWDXILZXKWACGHZXMXNXBSCDEUNZFGWBBCUJUKWDXIDRULUMUOTUPXHWIXKSZXGXDWNXQ
      XDWNWFWHXJLZNZXQXDXDWNXRQZNWNXSQXDXTWBCDEUQURWNWFXRUSUTWCWEWHXJVAVFTVBVCV
      DVEVGVHVIVJVKVLVMVNVOVPWAWKWJWAXOWKWJSXPAGBCVRVQVSVT $.
  $}

  ${
    tfrALT.1 $e |- F = recs ( G ) $.
    $( Alternate proof of ~ tfr1 using well-ordered recursion.  (Contributed by
       Scott Fenton, 3-Aug-2020.)  (New usage is discouraged.)
       (Proof modification is discouraged.) $)
    tfr1ALT $p |- F Fn On $=
      ( con0 cep wwe wse wfn epweon epse crecs cwrecs df-recs eqtri wfr1 mp2an
      ) DEFDEGADHIDJDEABABKDEBLCBMNOP $.

    $( Alternate proof of ~ tfr2 using well-ordered recursion.  (Contributed by
       Scott Fenton, 3-Aug-2020.)  (New usage is discouraged.)
       (Proof modification is discouraged.) $)
    tfr2ALT $p |- ( A e. On -> ( F ` A ) = ( G ` ( F |` A ) ) ) $=
      ( con0 wcel cfv cep cpred cres wwe wceq epweon crecs cwrecs df-recs eqtri
      wse epse wfr2 mpanl12 predon reseq2d fveq2d eqtrd ) AEFZABGZBEHAIZJZCGZBA
      JZCGEHKEHRUFUGUJLMESEHBCABCNEHCODCPQTUAUFUIUKCUFUHABAUBUCUDUE $.

    $d B x $.  $d G x $.  $d F x $.
    $( Alternate proof of ~ tfr3 using well-ordered recursion.  (Contributed by
       Scott Fenton, 3-Aug-2020.)  (New usage is discouraged.)
       (Proof modification is discouraged.) $)
    tfr3ALT $p |- ( ( B Fn On /\
             A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> B = F ) $=
      ( con0 wfn cv cfv cres wceq wral wa cep wcel predon reseq2d fveq2d eqeq2d
      cpred ralbiia wwe wse epweon epse crecs cwrecs df-recs eqtri wfr3 mpanl12
      sylan2br eqcomd ) BFGZAHZBIZBUOJZDIZKZAFLZMCBUTUNUPBFNUOTZJZDIZKZAFLZCBKZ
      VDUSAFUOFOZVCURUPVGVBUQDVGVAUOBUOPQRSUAFNUBFNUCUNVEMVFUDFUEAFNCDBCDUFFNDU
      GEDUHUIUJUKULUM $.
  $}

  $( Strong transfinite recursion defines a function on ordinals.  (Contributed
     by Stefan O'Rear, 18-Jan-2015.) $)
  recsfnon $p |- recs ( F ) Fn On $=
    ( crecs eqid tfr1 ) ABZAECD $.

  $( Strong transfinite recursion in terms of all previous values.
     (Contributed by Stefan O'Rear, 18-Jan-2015.) $)
  recsval $p |- ( A e. On -> ( recs ( F ) ` A ) =
        ( F ` ( recs ( F ) |` A ) ) ) $=
    ( crecs eqid tfr2 ) ABCZBFDE $.

  ${
    $d x y $.  $d y A $.  $d y H $.
    tz7.44lem1.1 $e |- G = { <. x , y >. | ( ( x = (/) /\ y = A ) \/
            ( -. ( x = (/) \/ Lim dom x ) /\ y = ( H ` ( x ` U. dom x ) ) )
                \/ ( Lim dom x /\ y = U. ran x ) ) } $.
    $( The ordered pair abstraction ` G ` defined in the hypothesis is a
       function.  This was a lemma for ~ tz7.44-1 , ~ tz7.44-2 , and ~ tz7.44-3
       when they used that definition of ` G ` .  Now, they use the maps-to
       ~ df-mpt idiom so this lemma is not needed anymore, but is kept in case
       other applications (for instance in intuitionistic set theory) need it.
       (Contributed by NM, 23-Apr-1995.)  (Revised by David Abernethy,
       19-Jun-2012.) $)
    tz7.44lem1 $p |- Fun G $=
      ( wfun cv c0 wceq wa cdm wlim wo cuni cfv cvv wcel wb limeq crn w3o copab
      wn wmo funopab fvex vex rnexg uniexg mp2b nlim0 dm0 ax-mp dmeq syl biimpa
      mtbir mto moeq3 mpgbir funeqi mpbir ) DGAHZIJZBHZCJKVEVDLZMZNUDVFVGOVDPZE
      PZJKVHVFVDUAZOZJKUBZABUCZGZVOVMBUEAVMABUFVEVHBCVJVLVIEUGVDQRVKQRVLQRAUHVD
      QUIVKQUJUKVEVHKILZMZVQIMZULVPIJVQVRSUMVPITUNURVEVHVQVEVGVPJVHVQSVDIUOVGVP
      TUPUQUSUTVADVNFVBVC $.
  $}

  ${
    $d x A $.  $d x y B $.  $d x y F $.  $d y G $.  $d x H $.  $d y X $.
    tz7.44.1 $e |- G = ( x e. _V |-> if ( x = (/) , A ,
                if ( Lim dom x , U. ran x , ( H ` ( x ` U. dom x ) ) ) ) ) $.
    tz7.44.2 $e |- ( y e. X -> ( F ` y ) = ( G ` ( F |` y ) ) ) $.
    ${
      tz7.44-1.3 $e |- A e. _V $.
      $( The value of ` F ` at ` (/) ` .  Part 1 of Theorem 7.44 of
         [TakeutiZaring] p. 49.  (Contributed by NM, 23-Apr-1995.)  (Revised by
         Mario Carneiro, 14-Nov-2014.) $)
      tz7.44-1 $p |- ( (/) e. X -> ( F ` (/) ) = A ) $=
        ( c0 wcel cfv cv cres wceq eqtrdi cvv cuni cif fveq2 reseq2 res0 fveq2d
        eqeq12d vtoclga 0ex cdm wlim crn iftrue fvmpt ax-mp ) KGLKDMZKEMZCBNZDM
        ZDUPOZEMZPUNUOPBKGUPKPZUQUNUSUOUPKDUAUTURKEUTURDKOKUPKDUBDUCQUDUEIUFKRL
        UOCPUGAKANZKPZCVAUHZUIVAUJSVCSVAMFMTZTCREVBCVDUKHJULUMQ $.
    $}

    tz7.44.3 $e |- ( y e. X -> ( F |` y ) e. _V ) $.
    tz7.44.4 $e |- F Fn X $.
    tz7.44.5 $e |- Ord X $.
    $( The value of ` F ` at a successor ordinal.  Part 2 of Theorem 7.44 of
       [TakeutiZaring] p. 49.  (Contributed by NM, 23-Apr-1995.)  Remove
       unnecessary distinct variable conditions.  (Revised by David Abernethy,
       19-Jun-2012.)  (Revised by Mario Carneiro, 14-Nov-2014.) $)
    tz7.44-2 $p |- ( suc B e. X -> ( F ` suc B ) = ( H ` ( F ` B ) ) ) $=
      ( wcel cfv c0 wceq cuni fveq2d cvv csuc cres cdm wlim crn cv fveq2 reseq2
      cif eqeq12d vtoclga eqeq1 wb dmeq limeq rneq unieqd fveq1 eqtrd ifbieq12d
      syl ifbieq2d eleq1d noel dm0 eqtrdi con0 word wss ordsson ax-mp wtr ordtr
      trsuc mpan sselid sucidg dmres ordelss fndmi sseqtrrdi dfss2 sylib eqtrid
      cin eleqtrrd eleq2 syl5ibcom syl5 mtoi iffalsed nlimsucg mtbird ordunisuc
      wn eloni 3syl fvresd 3eqtrd fvex eqeltrdi fvmptd3 ) DUAZHNZXCEOZEXCUBZFOZ
      XFPQZCXFUCZUDZXFUEZRZXIRZXFOZGOZUIZUIZDEOZGOZBUFZEOZEXTUBZFOZQXEXGQBXCHXT
      XCQZYAXEYCXGXTXCEUGYDYBXFFXTXCEUHZSUJJUKXDAXFAUFZPQZCYFUCZUDZYFUEZRZYHRZY
      FOZGOZUIZUIXQTFTIYFXFQZYGXHYOXPCYFXFPULYPYIXJYKYNXLXOYPYHXIQYIXJUMYFXFUNZ
      YHXIUOVAYPYJXKYFXFUPUQYPYMXNGYPYMYLXFOXNYLYFXFURYPYLXMXFYPYHXIYQUQSUSSUTV
      BYBTNXFTNBXCHYDYBXFTYEVCKUKXDXQXSTXDXQXPXOXSXDXHCXPXDXHDPNZDVDXHXIPQZXDYR
      XHXIPUCPXFPUNVEVFXDDXINYSYRXDDXCXIXDDVGNZDXCNXDHVGDHVHZHVGVIMHVJVKHVLZXDD
      HNUUAUUBMHVMVKHDVNVOVPZDVGVQVAZXDXIXCEUCZWEZXCEXCVRXDXCUUEVIUUFXCQXDXCHUU
      EUUAXDXCHVIMHXCVSVOHELVTWAXCUUEWBWCWDZWFXIPDWGWHWIWJWKXDXJXLXOXDXJXCUDZXD
      YTUUHWOUUCDVGWLVAXDXIXCQXJUUHUMUUGXIXCUOVAWMWKXDXNXRGXDXNDXFOXRXDXMDXFXDX
      MXCRZDXDXIXCUUGUQXDYTDVHUUIDQUUCDWPDWNWQUSSXDDXCEUUDWRUSSWSZXRGWTXAXBUUJW
      S $.

    $( The value of ` F ` at a limit ordinal.  Part 3 of Theorem 7.44 of
       [TakeutiZaring] p. 49.  (Contributed by NM, 23-Apr-1995.)  (Revised by
       David Abernethy, 19-Jun-2012.) $)
    tz7.44-3 $p |- ( ( B e. X /\ Lim B ) -> ( F ` B ) = U. ( F " B ) ) $=
      ( wcel wlim cfv cuni wceq c0 cvv wa cres crn cima cv fveq2 reseq2 eqeq12d
      fveq2d vtoclga adantr cdm cif eleq1d simpr wn nlim0 wb cin dmres wss word
      ordelss mpan wfn fndm ax-mp sseqtrrdi dfss2 sylib eqtrid eqtrdi sylan9req
      dmeq dm0 limeq syl mtbiri mt2d iffalsed mpbird iftrued eqtrd rnexg uniexg
      ex 3syl eqeltrd eqeq1 rneq unieqd fveq1 ifbieq12d ifbieq2d fvmptg syl2anc
      df-ima unieqi eqtr4di ) DHNZDOZUAZDEPZEDUBZUCZQZEDUDZQXBXCXDFPZXFWTXCXHRZ
      XABUEZEPZEXJUBZFPZRXIBDHXJDRZXKXCXMXHXJDEUFXNXLXDFXJDEUGZUIUHJUJUKXBXHXDS
      RZCXDULZOZXFXQQZXDPZGPZUMZUMZXFXBXDTNZYCTNXHYCRWTYDXAXLTNYDBDHXNXLXDTXOUN
      KUJUKZXBYCXFTXBYCYBXFXBXPCYBXBXPXAWTXAUOZXBXPXAUPXBXPUAZXASOZUQYGDSRXAYHU
      RXBXPDXQSXBXQDEULZUSZDEDUTXBDYIVAYJDRXBDHYIWTDHVAZXAHVBWTYKMHDVCVDUKEHVEY
      IHRLHEVFVGVHDYIVIVJVKZXPXQSULSXDSVNVOVLVMDSVPVQVRWFVSVTXBXRXFYAXBXRXAYFXB
      XQDRXRXAURYLXQDVPVQWAWBWCZXBYDXETNXFTNYEXDTWDXETWEWGWHAXDAUEZSRZCYNULZOZY
      NUCZQZYPQZYNPZGPZUMZUMYCTTFYNXDRZYOXPUUCYBCYNXDSWIUUDYQXRYSUUBXFYAUUDYPXQ
      RYQXRURYNXDVNZYPXQVPVQUUDYRXEYNXDWJWKUUDUUAXTGUUDUUAYTXDPXTYTYNXDWLUUDYTX
      SXDUUDYPXQUUEWKUIWCUIWMWNIWOWPYMWCWCXGXEEDWQWRWS $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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 g F $.  $d g I $.
    $( Define a recursive definition generator on ` On ` (the class of ordinal
       numbers) with characteristic function ` F ` and initial value ` I ` .
       This combines functions ` F ` in ~ tfr1 and ` G ` in ~ tz7.44-1 into one
       definition.  This rather amazing operation allows 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, as in for example ~ oav , from which
       we prove the recursive textbook definition as Theorems ~ oa0 , ~ oasuc ,
       and ~ oalim (with the help of Theorems ~ rdg0 , ~ rdgsuc , and
       ~ rdglim2a ).  We can also restrict the ` rec ` operation to define
       otherwise recursive functions on the natural numbers ` _om ` ; see
       ~ fr0g and ~ frsuc .  Our ` rec ` operation apparently does not appear
       in published literature, although closely related is Definition 25.2 of
       [Quine] p. 177, which he uses to "turn...a recursion into a genuine or
       direct definition" (p. 174).  Note that the ` if ` operations (see
       ~ df-if ) select cases based on whether the domain of ` g ` is zero, a
       successor, or a limit ordinal.

       An important use of this definition is in the recursive sequence
       generator ~ df-seq on the natural numbers (as a subset of the complex
       numbers), allowing us to define, with direct definitions, recursive
       infinite sequences such as the factorial function ~ df-fac and integer
       powers ~ df-exp .

       _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 NM, 9-Apr-1995.)  (Revised by Mario Carneiro, 9-May-2015.) $)
    df-rdg $a |- rec ( F , I ) = recs ( ( g e. _V |-> if ( g = (/) , I ,
                if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) ) ) $.
  $}

  ${
    $d g F $.  $d g G $.  $d g A $.  $d 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 wceq cvv cv cdm wlim crn cuni cfv cif cmpt crecs fveq1 ifeq2d df-rdg
      c0 crdg mpteq2dv recseq syl 3eqtr4g ) BCEZDFDGZSEZAUFHZIZUFJKZUHKUFLZBLZM
      ZMZNZOZDFUGAUIUJUKCLZMZMZNZOZBATCATUEUOUTEUPVAEUEDFUNUSUEUGUMURAUEUIULUQU
      JUKBCPQQUAUOUTUBUCDBARDCARUD $.

    $( 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 wceq cvv cv cdm wlim crn cuni cfv cif cmpt crecs crdg ifeq1 mpteq2dv
      c0 df-rdg recseq syl 3eqtr4g ) ABEZDFDGZSEZAUEHZIUEJKUGKUELCLMZMZNZOZDFUF
      BUHMZNZOZCAPCBPUDUJUMEUKUNEUDDFUIULUFABUHQRUJUMUAUBDCATDCBTUC $.
  $}

  $( Equality theorem for the recursive definition generator.  (Contributed by
     Scott Fenton, 28-Apr-2012.) $)
  rdgeq12 $p |- ( ( F = G /\ A = B ) -> rec ( F , A ) = rec ( G , B ) ) $=
    ( wceq crdg rdgeq2 rdgeq1 sylan9eqr ) ABECDECAFCBFDBFABCGBCDHI $.

  ${
    $d g F $.  $d g A $.  $d x g $.
    nfrdg.1 $e |- F/_ x F $.
    nfrdg.2 $e |- F/_ x A $.
    $( Bound-variable hypothesis builder for the recursive definition
       generator.  (Contributed by NM, 14-Sep-2003.)  (Revised by Mario
       Carneiro, 8-Sep-2013.) $)
    nfrdg $p |- F/_ x rec ( F , A ) $=
      ( vg crdg cvv cv c0 wceq cdm wlim crn cuni cfv cif nfcv nfv nfif df-rdg
      cmpt crecs nffv nfmpt nfrecs nfcxfr ) ACBGFHFIZJKZBUHLZMZUHNOZUJOUHPZCPZQ
      ZQZUBZUCFCBUAAUQAFHUPAHRUIABUOUIASEUKAULUNUKASAULRAUMCDAUMRUDTTUEUFUG $.
  $}

  ${
    $d x y f g v $.  $d x y z g $.  $d z f $.  $d f g G $.  $d x z G $.
    $d y z w g v G $.
    $( Lemma used with the recursive definition generator.  This is a trivial
       lemma that just changes bound variables for later use.  (Contributed by
       NM, 9-Apr-1995.) $)
    rdglem1 $p |- { f | E. x e. On ( f Fn x /\
                A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }
                = { g | E. z e. On ( g Fn z /\
                A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) } $=
      ( vv cv wfn cfv cres wceq wral wa con0 wrex cab eqid tfrlem3 fveq2 reseq2
      fveq2d eqeq12d cbvralvw anbi2i rexbii abbii eqtri ) EIZAIZJBIZUJKUJULLGKM
      BUKNOAPQERZFIZCIZJZHIZUNKZUNUQLZGKZMZHUONZOZCPQZFRUPDIZUNKZUNVELZGKZMZDUO
      NZOZCPQZFRABCHUMEFGUMSTVDVLFVCVKCPVBVJUPVAVIHDUOUQVEMZURVFUTVHUQVEUNUAVMU
      SVGGUQVEUNUBUCUDUEUFUGUHUI $.
  $}

  ${
    $d g F $.  $d g A $.
    $( The recursive definition generator is a function.  (Contributed by Mario
       Carneiro, 16-Nov-2014.) $)
    rdgfun $p |- Fun rec ( F , A ) $=
      ( vg crdg wfun cdm wlim cvv cv c0 wceq crn cuni cfv cif cmpt df-rdg tfr1a
      simpli ) BADZETFGTCHCIZJKAUAFZGUALMUBMUANBNOOPCBAQRS $.

    $( The domain of the recursive definition generator is a limit ordinal.
       (Contributed by NM, 16-Nov-2014.) $)
    rdgdmlim $p |- Lim dom rec ( F , A ) $=
      ( vg crdg wfun cdm wlim cvv cv c0 wceq crn cuni cfv cif cmpt df-rdg tfr1a
      simpri ) BADZETFGTCHCIZJKAUAFZGUALMUBMUANBNOOPCBAQRS $.

    $( The recursive definition generator is a function on ordinal numbers.
       (Contributed by NM, 9-Apr-1995.)  (Revised by Mario Carneiro,
       9-May-2015.) $)
    rdgfnon $p |- rec ( F , A ) Fn On $=
      ( vg crdg cvv cv c0 wceq cdm wlim crn cuni cfv cif cmpt df-rdg tfr1 ) BAD
      CECFZGHARIZJRKLSLRMBMNNOCBAPQ $.
  $}

  ${
    $d g F $.  $d g A $.
    $( Value of the recursive definition generator.  (Contributed by NM,
       9-Apr-1995.)  (Revised by Mario Carneiro, 8-Sep-2013.) $)
    rdgvalg $p |- ( B e. dom rec ( F , A ) ->
      ( rec ( F , A ) ` B ) = ( ( g e. _V |->
  if ( g = (/) , A , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) )
                   ` ( rec ( F , A ) |` B ) ) ) $=
      ( crdg cvv cv c0 wceq cdm wlim crn cuni cfv cif cmpt df-rdg tfr2a ) BDAEC
      FCGZHIASJZKSLMTMSNDNOOPCDAQR $.

    $( Value of the recursive definition generator.  (Contributed by NM,
       9-Apr-1995.)  (Revised by Mario Carneiro, 8-Sep-2013.) $)
    rdgval $p |- ( B e. On -> ( rec ( F , A ) ` B ) = ( ( g e. _V |->
  if ( g = (/) , A , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) )
                   ` ( rec ( F , A ) |` B ) ) ) $=
      ( crdg cvv cv c0 wceq cdm wlim crn cuni cfv cif cmpt df-rdg tfr2 ) BDAECF
      CGZHIASJZKSLMTMSNDNOOPCDAQR $.
  $}

  ${
    $d x y f g v w F $.  $d x y f g v w A $.  $d f x y B $.
    ${
      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 $=
        ( vx vy crdg cdm wcel cfv wceq com wlim wss rdgdmlim limomss ax-mp cuni
        c0 cv cif peano1 sselii cvv crn cmpt eqid rdgvalg tz7.44-1 ) RBAFZGZHRU
        IIAJKUJRUJLKUJMABNUJOPUAUBDEAUIDUCDSZRJAUKGZLUKUDQULQUKIBITTUEZBUJUMUFA
        ESDBUGCUHP $.
    $}

    $( The initial segments of the recursive definition generator are sets.
       (Contributed by Mario Carneiro, 16-Nov-2014.) $)
    rdgseg $p |- ( B e. dom rec ( F , A ) ->
                 ( rec ( F , A ) |` B ) e. _V ) $=
      ( vg vx vy vw vv vf crdg cdm wcel cres cvv cv c0 wceq cuni cfv cif df-rdg
      wlim crn cmpt crecs reseq1i wfn wral con0 wrex cab rdglem1 tfrlem9a dmeqi
      wa eleq2s eqeltrid ) BCAJZKZLURBMDNDOZPQAUTKZUBUTUCRVARUTSCSTTUDZUEZBMZNU
      RVCBDCAUAZUFVDNLBVCKUSEFGOZFOZUGHOZVFSVFVHMVBSQHVGUHUOFUIUJGUKBIVBFHEFGIV
      BULUMURVCVEUNUPUQ $.

    $( The value of the recursive definition generator at a successor.
       (Contributed by NM, 16-Nov-2014.) $)
    rdgsucg $p |- ( B e. dom rec ( F , A ) ->
                 ( rec ( F , A ) ` suc B ) = ( F ` ( rec ( F , A ) ` B ) ) ) $=
      ( vx vy crdg cdm wcel csuc cfv wceq wlim wb rdgdmlim limsuc ax-mp cv cuni
      cvv cif c0 crn cmpt eqid rdgvalg rdgseg wfun wfn rdgfun funfn mpbi limord
      word tz7.44-2 sylbi ) BCAFZGZHZBIZUQHZUSUPJBUPJCJKUQLZURUTMACNZUQBOPDEABU
      PDSDQZUAKAVCGZLVCUBRVDRVCJCJTTUCZCUQVEUDAEQZDCUEAVFCUFUPUGUPUQUHACUIUPUJU
      KVAUQUMVBUQULPUNUO $.

    $( The value of the recursive definition generator at a successor.
       (Contributed by NM, 23-Apr-1995.)  (Revised by Mario Carneiro,
       14-Nov-2014.) $)
    rdgsuc $p |- ( B e. On ->
                 ( rec ( F , A ) ` suc B ) = ( F ` ( rec ( F , A ) ` B ) ) ) $=
      ( con0 wcel crdg cdm csuc cfv wceq rdgfnon fndmi eleq2i rdgsucg sylbir )
      BDEBCAFZGZEBHPIBPICIJQDBDPACKLMABCNO $.

    $( The value of the recursive definition generator at a limit ordinal.
       (Contributed by NM, 16-Nov-2014.) $)
    rdglimg $p |- ( ( B e. dom rec ( F , A ) /\ Lim B ) ->
                   ( rec ( F , A ) ` B ) = U. ( rec ( F , A ) " B ) ) $=
      ( vx vy crdg cvv cv c0 wceq cdm wlim crn cuni cfv cif cmpt rdgvalg rdgseg
      eqid wfun wfn rdgfun funfn mpbi word rdgdmlim limord ax-mp tz7.44-3 ) DEA
      BCAFZDGDHZIJAULKZLULMNUMNULOCOPPQZCUKKZUNTAEHZDCRAUPCSUKUAUKUOUBACUCUKUDU
      EUOLUOUFACUGUOUHUIUJ $.

    $( The value of the recursive definition generator at a limit ordinal.
       (Contributed by NM, 23-Apr-1995.)  (Revised by Mario Carneiro,
       14-Nov-2014.) $)
    rdglim $p |- ( ( B e. C /\ Lim B ) ->
                   ( rec ( F , A ) ` B ) = U. ( rec ( F , A ) " B ) ) $=
      ( wcel wlim crdg cdm cfv cima cuni wceq wa con0 limelon wfn rdgfnon ax-mp
      fndm eleqtrrdi rdglimg sylancom ) BCEZBFZBDAGZHZEBUEIUEBJKLUCUDMBNUFBCOUE
      NPUFNLADQNUESRTABDUAUB $.
  $}

  ${
    $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 $.
  $}

  ${
    rdgsucmptf.1 $e |- F/_ x A $.
    rdgsucmptf.2 $e |- F/_ x B $.
    rdgsucmptf.3 $e |- F/_ x D $.
    rdgsucmptf.4 $e |- F = rec ( ( x e. _V |-> C ) , A ) $.
    rdgsucmptf.5 $e |- ( x = ( F ` B ) -> C = D ) $.
    $( The value of the recursive definition generator at a successor (special
       case where the characteristic function uses the map operation).
       (Contributed by NM, 22-Oct-2003.)  (Revised by Mario Carneiro,
       15-Oct-2016.) $)
    rdgsucmptf $p |- ( ( B e. On /\ D e. V ) -> ( F ` suc B ) = D ) $=
      ( con0 wcel csuc cfv cvv cmpt crdg fveq1i rdgsuc fveq2i 3eqtr4g wceq fvex
      nfmpt1 nfrdg nfcxfr nffv eqid fvmptf mpan sylan9eq ) CMNZEGNZCOZFPZCFPZAQ
      DRZPZEUNUPUSBSZPCVAPZUSPUQUTBCUSUAUPFVAKTURVBUSCFVAKTUBUCURQNUOUTEUDCFUEA
      URDEQUSGACFAFVAKABUSAQDUFHUGUHIUIJLUSUJUKULUM $.

    $( The value of the recursive definition generator at a successor (special
       case where the characteristic function is an ordered-pair class
       abstraction and where the mapping class ` D ` is a proper class).  This
       is a technical lemma that can be used together with ~ rdgsucmptf to help
       eliminate redundant sethood antecedents.  (Contributed by NM,
       22-Oct-2003.)  (Revised by Mario Carneiro, 15-Oct-2016.) $)
    rdgsucmptnf $p |- ( -. D e. _V -> ( F ` suc B ) = (/) ) $=
      ( cvv wcel wn csuc cfv cmpt crdg c0 fveq1i cdm wceq rdgdmlim limsuc ax-mp
      wlim wb rdgsucg fveq2i eqtr4di nfmpt1 nfrdg nfcxfr nffv fvmptnf sylan9eqr
      eqid ex biimtrrid ndmfv pm2.61d1 eqtrid ) ELMNZCOZFPVDALDQZBRZPZSVDFVFJTV
      CVDVFUAZMZVGSUBZVICVHMZVCVJVHUFVKVIUGBVEUCVHCUDUEVCVKVJVKVCVGCFPZVEPZSVKV
      GCVFPZVEPVMBCVEUHVLVNVECFVFJTUIUJAVLDELVEACFAFVFJABVEALDUKGULUMHUNIKVEUQU
      OUPURUSVDVFUTVAVB $.
  $}

  ${
    $d x y $.  $d y A $.  $d y B $.  $d y C $.  $d y D $.  $d x E $.
    rdgsucmpt2.1 $e |- F = rec ( ( x e. _V |-> C ) , A ) $.
    rdgsucmpt2.2 $e |- ( y = x -> E = C ) $.
    rdgsucmpt2.3 $e |- ( y = ( F ` B ) -> E = D ) $.
    $( This version of ~ rdgsucmpt avoids the not-free hypothesis of
       ~ rdgsucmptf by using two substitutions instead of one.  (Contributed by
       Mario Carneiro, 11-Sep-2015.) $)
    rdgsucmpt2 $p |- ( ( B e. On /\ D e. V ) -> ( F ` suc B ) = D ) $=
      ( nfcv cvv cmpt crdg wceq cbvmptv rdgeq1 ax-mp eqtr4i rdgsucmptf ) BCDGFH
      IBCMBDMBFMHANEOZCPZBNGOZCPZJUEUCQUFUDQBANGEKRCUEUCSTUALUB $.
  $}

  ${
    $d x A $.  $d x B $.  $d x D $.
    rdgsucmpt.1 $e |- F = rec ( ( x e. _V |-> C ) , A ) $.
    rdgsucmpt.2 $e |- ( x = ( F ` B ) -> C = D ) $.
    $( The value of the recursive definition generator at a successor (special
       case where the characteristic function uses the map operation).
       (Contributed by Mario Carneiro, 9-Sep-2013.) $)
    rdgsucmpt $p |- ( ( B e. On /\ D e. V ) -> ( F ` suc B ) = D ) $=
      ( nfcv rdgsucmptf ) ABCDEFGABJACJAEJHIK $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y F $.
    $( The value of the recursive definition generator at a limit ordinal, in
       terms of the union of all smaller values.  (Contributed by NM,
       23-Apr-1995.) $)
    rdglim2 $p |- ( ( B e. C /\ Lim B ) -> ( rec ( F , A ) ` B ) =
                  U. { y | E. x e. B y = ( rec ( F , A ) ` x ) } ) $=
      ( wcel wlim wa crdg cfv cima cuni cv wceq cab wex con0 wb word rdglim cop
      wrex dfima3 df-rex wi limord ordelord ex vex elon imbitrrdi syl eqcom wfn
      rdgfnon fnopfvb bitrid pm5.32d exbidv bitr2id abbidv eqtrid unieqd adantl
      mpan syl6 eqtrd ) DEGZDHZIDFCJZKVKDLZMZBNZANZVKKZOZADUCZBPZMZCDEFUAVJVMVT
      OVIVJVLVSVJVLVODGZVOVNUBVKGZIZAQZBPVSABVKDUDVJWDVRBVRWAVQIZAQVJWDVQADUEVJ
      WEWCAVJWAVQWBVJWAVORGZVQWBSVJDTZWAWFUFDUGWGWAVOTZWFWGWAWHDVOUHUIVOAUJUKUL
      UMVQVPVNOZWFWBVNVPUNVKRUOWFWIWBSCFUPRVOVNVKUQVFURVGUSUTVAVBVCVDVEVH $.

    $( The value of the recursive definition generator at a limit ordinal, in
       terms of indexed union of all smaller values.  (Contributed by NM,
       28-Jun-1998.) $)
    rdglim2a $p |- ( ( B e. C /\ Lim B ) -> ( rec ( F , A ) ` B ) =
                  U_ x e. B ( rec ( F , A ) ` x ) ) $=
      ( vy wcel wlim wa crdg cfv cv wceq wrex cab cuni ciun rdglim2 fvex dfiun2
      eqtr4di ) CDGCHICEBJZKFLALZUBKZMACNFOPACUDQAFBCDERAFCUDUCUBSTUA $.
  $}

  ${
    $d F g $.  $d A g $.
    $( If ` A ` is a proper class, then the recursive function generator at
       ` (/) ` is the empty set.  (Contributed by Scott Fenton,
       31-Oct-2024.) $)
    rdg0n $p |- ( -. A e. _V -> ( rec ( F , A ) ` (/) ) = (/) ) $=
      ( vg cvv wcel wn c0 crdg cfv cv wceq cdm wlim crn cuni cif cmpt cres con0
      0elon df-rdg tfr2 ax-mp res0 fveq2i eqtri iftrue eqid fvmptn eqtrid ) ADE
      FGBAHZIZGCDCJZGKZAUMLZMUMNOUOOUMIBIPZPZQZIZGULUKGRZURIZUSGSEULVAKTGUKURCB
      AUAUBUCUTGURUKUDUEUFCDUQAGURUNAUPUGURUHUIUJ $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Finite recursion
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( We use the next 3 theorems for finite recursion.  Note that we take
     advantage of what we have done for transfinite recursion, which allows
     to explicitly specify the function and moreover not require the
     existence of omega. $)

  $( The function generated by finite recursive definition generation is a
     function on omega.  (Contributed by NM, 15-Oct-1996.)  (Revised by Mario
     Carneiro, 14-Nov-2014.) $)
  frfnom $p |- ( rec ( F , A ) |` _om ) Fn _om $=
    ( crdg com cres wfn wfun cdm wceq rdgfun funres ax-mp cin wss wlim rdgdmlim
    dmres limomss dfss2 mpbi eqtri df-fn mpbir2an ) BACZDEZDFUEGZUEHZDIUDGUFABJ
    DUDKLUGDUDHZMZDUDDQDUHNZUIDIUHOUJABPUHRLDUHSTUAUEDUBUC $.

  $( The initial value resulting from finite recursive definition generation.
     (Contributed by NM, 15-Oct-1996.) $)
  fr0g $p |- ( A e. B -> ( ( rec ( F , A ) |` _om ) ` (/) ) = A ) $=
    ( wcel c0 crdg com cres cfv wceq peano1 fvres ax-mp rdg0g eqtrid ) ABDECAFZ
    GHIZEPIZAEGDQRJKEGPLMABCNO $.

  $( The successor value resulting from finite recursive definition generation.
     (Contributed by NM, 15-Oct-1996.)  (Revised by Mario Carneiro,
     16-Nov-2014.) $)
  frsuc $p |- ( B e. _om -> ( ( rec ( F , A ) |` _om ) ` suc B ) =
              ( F ` ( ( rec ( F , A ) |` _om ) ` B ) ) ) $=
    ( com wcel csuc crdg cfv cres cdm wceq wlim wss limomss ax-mp sseli rdgsucg
    rdgdmlim syl fvres peano2b sylbi fveq2d 3eqtr4d ) BDEZBFZCAGZHZBUGHZCHZUFUG
    DIZHZBUKHZCHUEBUGJZEUHUJKDUNBUNLDUNMACRUNNOPABCQSUEUFDEULUHKBUAUFDUGTUBUEUM
    UICBDUGTUCUD $.

  ${
    frsucmpt.1 $e |- F/_ x A $.
    frsucmpt.2 $e |- F/_ x B $.
    frsucmpt.3 $e |- F/_ x D $.
    frsucmpt.4 $e |- F = ( rec ( ( x e. _V |-> C ) , A ) |` _om ) $.
    frsucmpt.5 $e |- ( x = ( F ` B ) -> C = D ) $.
    $( The successor value resulting from finite recursive definition
       generation (special case where the generation function is expressed in
       maps-to notation).  (Contributed by NM, 14-Sep-2003.)  (Revised by Scott
       Fenton, 2-Nov-2011.) $)
    frsucmpt $p |- ( ( B e. _om /\ D e. V ) -> ( F ` suc B ) = D ) $=
      ( com wcel csuc cfv cvv cmpt crdg fveq1i cres frsuc fveq2i 3eqtr4g nfmpt1
      wceq fvex nfrdg nfcv nfres nfcxfr nffv eqid fvmptf mpan sylan9eq ) CMNZEG
      NZCOZFPZCFPZAQDRZPZEUQUSVBBSZMUAZPCVEPZVBPUTVCBCVBUBUSFVEKTVAVFVBCFVEKTUC
      UDVAQNURVCEUFCFUGAVADEQVBGACFAFVEKAVDMABVBAQDUEHUHAMUIUJUKIULJLVBUMUNUOUP
      $.

    $( The value of the finite recursive definition generator at a successor
       (special case where the characteristic function is a mapping abstraction
       and where the mapping class ` D ` is a proper class).  This is a
       technical lemma that can be used together with ~ frsucmpt to help
       eliminate redundant sethood antecedents.  (Contributed by Scott Fenton,
       19-Feb-2011.)  (Revised by Mario Carneiro, 11-Sep-2015.) $)
    frsucmptn $p |- ( -. D e. _V -> ( F ` suc B ) = (/) ) $=
      ( cvv wcel wn csuc cfv com c0 fveq1i wceq cmpt crdg cres cdm frfnom ax-mp
      wfn fndm eleq2i peano2b frsuc fveq2i eqtr4di nfmpt1 nfrdg nfcv nfres nffv
      nfcxfr eqid fvmptnf sylan9eqr ex biimtrrid biimtrid ndmfv pm2.61d1 eqtrid
      ) ELMNZCOZFPVJALDUAZBUBZQUCZPZRVJFVMJSVIVJVMUDZMZVNRTZVPVJQMZVIVQVOQVJVMQ
      UGVOQTBVKUEQVMUHUFUIVRCQMZVIVQCUJVIVSVQVSVIVNCFPZVKPZRVSVNCVMPZVKPWABCVKU
      KVTWBVKCFVMJSULUMAVTDELVKACFAFVMJAVLQABVKALDUNGUOAQUPUQUSHURIKVKUTVAVBVCV
      DVEVJVMVFVGVH $.
  $}

  ${
    $d x y $.  $d y A $.  $d y B $.  $d y C $.  $d y D $.  $d x E $.
    frsucmpt2.1 $e |- F = ( rec ( ( x e. _V |-> C ) , A ) |` _om ) $.
    frsucmpt2.2 $e |- ( y = x -> E = C ) $.
    frsucmpt2.3 $e |- ( y = ( F ` B ) -> E = D ) $.
    $( The successor value resulting from finite recursive definition
       generation (special case where the generation function is expressed in
       maps-to notation), using double-substitution instead of a bound variable
       condition.  (Contributed by Mario Carneiro, 11-Sep-2015.) $)
    frsucmpt2 $p |- ( ( B e. _om /\ D e. V ) -> ( F ` suc B ) = D ) $=
      ( nfcv cvv cmpt crdg com cres wceq cbvmptv rdgeq1 reseq1i eqtr4i frsucmpt
      ax-mp ) BCDGFHIBCMBDMBFMHANEOZCPZQRBNGOZCPZQRJUIUGQUHUFSUIUGSBANGEKTCUHUF
      UAUEUBUCLUD $.
    $( $j usage 'frsucmpt2' avoids 'ax-13'; $)
  $}

  ${
    $d y z w A $.  $d x y z w F $.
    tz7.48.1 $e |- F Fn On $.
    ${
      $d x A $.
      $( A way of showing an ordinal function is one-to-one.  (Contributed by
         NM, 9-Feb-1997.) $)
      tz7.48lem $p |- ( ( A C_ On /\ A. x e. A A. y e. x
                    -. ( F ` x ) = ( F ` y ) ) -> Fun `' ( F |` A ) ) $=
        ( vw vz con0 cv cfv wceq wn wral wa weq wi wel wcel wal r2al cres simpl
        ccnv wfun anim1i imim1i 2alimi sylbi sylibr elequ1 fveq2 eqeq2d imbi12d
        expd notbid cbvralvw ralbii elequ2 fveqeq2 ralbidv 3bitri ralcom biimpi
        wss bitri ancri r19.26-2 syl wb fvres eqeqan12d ad2antrl anim12d pm3.48
        ssel wo oridm eqcom notbii orbi1i bitr3i imbitrrdi word ordtri3 biimprd
        con2d eloni syl2an syl9r syl6 imp32 exp32 a2d 2alimdv 3imtr4g imdistani
        sylbid syl5 wfn fnssres mpan cvv wf dffn2 wf1 dff13 df-f1 simprbi sylan
        sylanb ) CHVDZAIZDJZBIZDJZKZLZBXLMACMZNXKXLDCUAZJZXNXSJZKZABOZPZBCMACMZ
        NXSUCUDZXKXRYEXRABQZXOXMKZLZPZBAQZXQPZNZBCMACMZXKYEXRYLBCMZACMZYNXRXLCR
        ZXNCRZNZYLPZBSASZYPXRYQYKNZXQPZBSASUUAXQABCXLTUUCYTABUUCYSYKXQYSYKNUUBX
        QYSYQYKYQYRUBUEUFUNUGUHYLABCCTUIYPYJBCMACMZYPNYNYPUUDYPYJACMZBCMZUUDYPF
        AQZXMFIZDJZKZLZPZFCMZACMFGQZGIZDJZUUIKZLZPZFCMZGCMZUUFYOUUMACYLUULBFCBF
        OZYKUUGXQUUKBFAUJUVBXPUUJUVBXOUUIXMXNUUHDUKULUOUMUPUQUUMUUTAGCAGOZUULUU
        SFCUVCUUGUUNUUKUURAGFURUVCUUJUUQXLUUOUUIDUSUOUMUTUPUVAAGQZUUPXMKZLZPZAC
        MZGCMUUFUUTUVHGCUUSUVGFACFAOZUUNUVDUURUVFFAGUJUVIUUQUVEUVIUUIXMUUPUUHXL
        DUKULUOUMUPUQUVHUUEGBCGBOZUVGYJACUVJUVDYGUVFYIGBAURUVJUVEYHUUOXNXMDUSUO
        UMUTUPVEVAUUFUUDYJBACCVBVCUHVFYJYLABCCVGUIVHXKYSYMPZBSASYSYDPZBSASYNYEX
        KUVKUVLABXKYSYMYDXKYSYMYDXKYSYMNNYBXPYCYSYBXPVIXKYMYQYRXTXMYAXOXLCDVJXN
        CDVJVKVLXKYSYMXPYCPZXKYSXLHRZXNHRZNZYMUVMPXKYQUVNYRUVOCHXLVOCHXNVOVMYMX
        PYGYKVPZLZUVPYCYMUVQXPYMUVQYIXQVPZXQYGYIYKXQVNXQXQXQVPUVSXQVQXQYIXQXPYH
        XMXOVRVSVTWAWBWFUVNXLWCZXNWCZUVRYCPUVOXLWGXNWGUVTUWANYCUVRXLXNWDWEWHWIW
        JWKWQWLWMWNYMABCCTYDABCCTWOWRWPXKXSCWSZYEYFDHWSXKUWBEHCDWTXAUWBCXBXSXCZ
        YEYFCXSXDUWCYENZUWCYFUWDCXBXSXEUWCYFNABCXBXSXFCXBXSXGWAXHXJXIVH $.
    $}

    $( Proposition 7.48(2) of [TakeutiZaring] p. 51.  (Contributed by NM,
       9-Feb-1997.)  (Revised by David Abernethy, 5-May-2013.) $)
    tz7.48-2 $p |- ( A. x e. On ( F ` x ) e. ( A \ ( F " x ) ) -> Fun `' F ) $=
      ( vy cv cfv cima cdif wcel con0 wral cres ccnv wfun wss wceq wa wi ax-mp
      wn ssid wel onelon ancoms cdm fndmi eleq2i wfn fnfun funfvima mpan impcom
      eleq1a eldifn nsyli syl sylan2br syldan com12 ralrimiv ralimiaa tz7.48lem
      expimpd sylancr wrel fnrel eqimssi relssres mp2an cnveqi funeqi sylib ) A
      FZCGZBCVNHZIJZAKLZCKMZNZOZCNZOVRKKPVOEFZCGZQZUAZEVNLZAKLWAKUBVQWGAKVNKJZV
      QRZWFEVNEAUCZWIWFWJWHVQWFWJWHWCKJZVQWFSZWHWJWKVNWCUDUEWKWJWCCUFZJZWLWMKWC
      KCDUGZUHWJWNRWDVPJZWLWNWJWPCOZWNWJWPSCKUIZWQDKCUJTVNWCCUKULUMWPWEVOVPJVQW
      DVPVOUNVOBVPUOUPUQURUSVDUTVAVBAEKCDVCVEVTWBVSCCVFZWMKPVSCQWRWSDKCVGTWMKWO
      VHCKVIVJVKVLVM $.

    $d x A $.
    $( Proposition 7.48(1) of [TakeutiZaring] p. 51.  (Contributed by NM,
       9-Feb-1997.) $)
    tz7.48-1 $p |- ( A. x e. On ( F ` x ) e. ( A \ ( F " x ) ) ->
                   ran F C_ A ) $=
      ( vy cv cfv cima cdif wcel con0 wral crn wceq wa wex cop vex elrn2 wi cdm
      opeldm fndmi eleqtrdi ancri wfn wb fnopfvb mpan pm5.32i eximi sylbi nfra1
      sylibr nfv rsp eldifi eleq1 syl5ibcom imim2i impd syl exlimd syl5 ssrdv )
      AFZCGZBCVFHZIJZAKLZECMZBEFZVKJZVFKJZVGVLNZOZAPZVJVLBJZVMVFVLQCJZAPVQAVLCE
      RZSVSVPAVSVNVSOVPVSVNVSVFCUAKVFVLCARVTUBKCDUCUDUEVNVOVSCKUFVNVOVSUGDKVFVL
      CUHUIUJUNUKULVJVPVRAVIAKUMVRAUOVJVNVITZVPVRTVIAKUPWAVNVOVRVIVOVRTVNVIVGBJ
      VOVRVGBVHUQVGVLBURUSUTVAVBVCVDVE $.

    $( Proposition 7.48(3) of [TakeutiZaring] p. 51.  (Contributed by NM,
       9-Feb-1997.) $)
    tz7.48-3 $p |- ( A. x e. On ( F ` x ) e. ( A \ ( F " x ) ) ->
                    -. A e. _V ) $=
      ( cv cfv cima cdif wcel con0 wral cvv crn cdm fndmi onprc eqneltri eleq1i
      wi syl ccnv wfun tz7.48-2 funrnex com12 df-rn dfdm4 3imtr4g mtoi tz7.48-1
      wss ssexg ex mtod ) AEZCFBCUOGHIAJKZBLIZCMZLIZUPUSCNZLIZUTJLJCDOPQUPCUAZU
      BZUSVASABCDUCVCVBNZLIZVBMZLIZUSVAVEVCVGLVBUDUEURVDLCUFRUTVFLCUGRUHTUIUPUR
      BUKZUQUSSABCDUJVHUQUSURBLULUMTUN $.
  $}

  ${
    tz7.49.1 $e |- F Fn On $.
    $d x y z A $.  $d x y z F $.  $d ph y z $.
    tz7.49.2 $e |- ( ph <-> A. x e. On ( ( A \ ( F " x ) ) =/= (/) ->
                               ( F ` x ) e. ( A \ ( F " x ) ) ) ) $.
    $( Proposition 7.49 of [TakeutiZaring] p. 51.  (Contributed by NM,
       10-Feb-1997.)  (Revised by Mario Carneiro, 10-Jan-2013.) $)
    tz7.49 $p |- ( ( A e. B /\ ph ) ->
                     E. x e. On ( A. y e. x ( A \ ( F " y ) ) =/= (/) /\
                     ( F " x ) = A /\ Fun `' ( F |` x ) ) ) $=
      ( vz wcel c0 wral wceq con0 wrex wa wn wi wss nfv cima cdif wne cres ccnv
      wfun w3a cfv df-ne ralbii ralim sylbi biimtrrid tz7.48-3 elex nsyl3 nsyli
      cv cvv dfrex2 imbitrrdi imaeq2 difeq2d eqeq1d onminex anbi2i rexbii nfra1
      syl6 nfxfr simpllr wfn fnfun ax-mp fvelima mpan rsp adantld onelon neeq1d
      nfan fveq2 eleq12d imbi12d rspcv biimtrid com23 syl sylcom expcomd eldifi
      com3r imp eleq1 syl5ibcom syl8 com34 rexlimd ssrdv ssdif0 biimpri anim12i
      syl5 eqss onss ssel cdm fndmi sseqtrrdi funfvima2 sylancr com12 a1i syl10
      sylibr imp4a eldifn eleq1a con3d syl5com syldd com4r ralrimi ex tz7.48lem
      imp31 ancld syl56 ancoms adantr 3jca exp41 reximdai syld impcom ) ADEJZDF
      CURZUAZUBZKUCZCBURZLZFUUAUAZDMZFUUAUDUEUFZUGZBNOZAYPDUUCUBZKMZUUBPZBNOZUU
      GAYPUUIYSKMZQZCUUALZPZBNOZUUKAYPUUIBNOZUUPAYPUUIQZBNLZQUUQAUUSUUAFUHZUUHJ
      ZBNLZYPUUSUUHKUCZBNLZAUVBUVCUURBNUUHKUIUJAUVCUVARZBNLZUVDUVBRHUVCUVABNUKU
      LUMUVBDUSJYPBDFGUNDEUOUPUQUUIBNUTVAUUIUULBCUUAYQMZUUHYSKUVGUUCYRDUUAYQFVB
      VCZVDVEVIUUJUUOBNUUBUUNUUIYTUUMCUUAYSKUIUJVFVGVAAUUJUUFBNAUVFBHUVEBNVHVJA
      UUANJZUUIUUBUUFAUVIUUBUUIUUFAUUBUVIUUIUUFRAUUBUVIUUIUUFAUUBPZUVIPZUUIPZUU
      BUUDUUEAUUBUVIUUIVKUVLUUCDSZDUUCSZPUUDUVKUVMUUIUVNUVKIUUCDUVJUVIIURZUUCJZ
      UVODJZRUVJUVPUVIUVQUVPYQFUHZUVOMZCUUAOZUVJUVIUVQRZFUFZUVPUVTFNVLUWBGNFVMV
      NZCUVOUUAFVOVPUVJUVSUWACUUAAUUBCACTZYTCUUAVHZWAUWACTUVJYQUUAJZUVIUVSUVQUV
      JUWFUVIUVRYSJZUVSUVQRUVJUVIUWFUWGAUUBUVIUWFPZUWGRUUBUWHAUWGUUBUWHYTAUWGRZ
      UUBUWFYTUVIYTCUUAVQZVRUWHYQNJZYTUWIRUUAYQVSUWKAYTUWGAUVFUWKYTUWGRZHUVEUWL
      BYQNUVGUVCYTUVAUWGUVGUUHYSKUVHVTUVGUUTUVRUUHYSUUAYQFWBUVHWCWDWEWFWGZWHWIW
      LWMWJUWGUVRDJUVSUVQUVRDYRWKUVRUVODWNWOWPWQWRXCWGWMWSUVNUUIDUUCWTXAXBUUCDX
      DXOUVKUUEUUIUVJUVIUUEUUBAUVIUUERUVIUUANSZUUBAPZUWNUVRUVOFUHZMZQZIYQLZCUUA
      LZPUUEUUAXEUWOUWNUWTUWOUWNUWTUWOUWNPZUWSCUUAUWOUWNCUUBACUWEUWDWAUWNCTWAUX
      AUWFUWSUXAUWFPZUWRIYQUXBITUWOUWNUWFUVOYQJZUWRRUWNUWFUXCUWOUWRUWNUWFUXCUWP
      YRJZUWOUWRRUWNUWFUWKUXCUXDRZUUANYQXFZUWKUWBYQFXGZSUXEUWCUWKYQNUXGYQXENFGX
      HXIYQUVOFXJXKVIUWNUWFUWOUXDUWRUWNUWFUWOUWGUXDUWRRUWNUWFUUBAUWGUWNUWFUWKUU
      BYTUWIUXFUWFUUBYTRRUWNUUBUWFYTUWJXLXMUWMXNXPUWGUVRYRJZQUXDUWRUVRDYRXQUXDU
      WQUXHUWPYRUVRXRXSXTWPWQYAYBYFYCYDYCYDYGCIUUAFGYEYHYIWMYJYKYLWGWQXPYMYNYO
      $.
  $}

  ${
    $d A x y $.  $d F x y $.
    tz7.49c.1 $e |- F Fn On $.
    $( Corollary of Proposition 7.49 of [TakeutiZaring] p. 51.  (Contributed by
       NM, 10-Feb-1997.)  (Revised by Mario Carneiro, 19-Jan-2013.) $)
    tz7.49c $p |- ( ( A e. B /\ A. x e. On ( ( A \ ( F " x ) ) =/= (/) ->
                 ( F ` x ) e. ( A \ ( F " x ) ) ) ) ->
                 E. x e. On ( F |` x ) : x -1-1-onto-> A ) $=
      ( vy wcel cv cima cdif c0 wne cfv con0 wral wa wceq w3a wrex wfn cres crn
      wi ccnv wfun wf1o biid tz7.49 3simpc fnssres sylancr df-ima eqeq1i biimpi
      wss onss anim12i anim1i dff1o2 3anan32 bitri sylibr expl syl5 reximia syl
      ) BCGBDAHZIZJZKLVGDMVIGUCANOZPBDFHIJKLFVGOZVHBQZDVGUAZUDUEZRZANSVGBVMUFZA
      NSVJAFBCDEVJUGUHVOVPANVOVLVNPVGNGZVPVKVLVNUIVQVLVNVPVQVLPZVNPVMVGTZVMUBZB
      QZPZVNPZVPVRWBVNVQVSVLWAVQDNTVGNUOVSEVGUPNVGDUJUKVLWAVHVTBDVGULUMUNUQURVP
      VSVNWARWCVGBVMUSVSVNWAUTVAVBVCVDVEVF $.
  $}

  $c seqom $.

  $( Extend class notation to include index-aware recursive definitions. $)
  cseqom $a class seqom ( F , I ) $.

  ${
    $d F i v $.  $d I i v $.

    $( Index-aware recursive definitions over ` _om ` .  A mashup of ~ df-rdg
       and ~ df-seq , this allows for recursive definitions that use an index
       in the recursion in cases where Infinity is not admitted.  (Contributed
       by Stefan O'Rear, 1-Nov-2014.) $)
    df-seqom $a |- seqom ( F , I ) = ( rec ( ( i e. _om , v e. _V |->
        <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) " _om ) $.
  $}

  ${
    $d F a b c d $.  $d I a b c d $.

    $( Lemma for ` seqom ` .  Change bound variables.  (Contributed by Stefan
       O'Rear, 1-Nov-2014.) $)
    seqomlem0 $p |- rec ( ( a e. _om , b e. _V |-> <. suc a , ( a F b ) >. ) ,
      <. (/) , ( _I ` I ) >. ) = rec ( ( c e. _om , d e. _V |->
      <. suc c , ( c F d ) >. ) , <. (/) , ( _I ` I ) >. ) $=
      ( com cvv cv csuc co cop cmpo wceq c0 cid cfv crdg weq suceq oveq1 opeq2d
      opeq12d oveq2 cbvmpov rdgeq1 ax-mp ) CDGHCIZJZUHDIZAKZLZMZEFGHEIZJZUNFIZA
      KZLZMZNUMOBPQLZRUSUTRNCDEFGHULURUOUNUJAKZLCESUIUOUKVAUHUNTUHUNUJAUAUCDFSV
      AUQUOUJUPUNAUDUBUEUTUMUSUFUG $.
  $}

  ${
    $d Q a b c i v $.  $d A a b i v $.  $d F a b i v $.  $d I a b $.
    seqomlem.a $e |- Q = rec ( ( i e. _om , v e. _V |->
        <. suc i , ( i F v ) >. ) , <. (/) , ( _I ` I ) >. ) $.
    $( Lemma for ` seqom ` .  The underlying recursion generates a sequence of
       pairs with the expected first values.  (Contributed by Stefan O'Rear,
       1-Nov-2014.)  (Revised by Mario Carneiro, 23-Jun-2015.) $)
    seqomlem1 $p |- ( A e. _om -> ( Q ` A ) =
      <. A , ( 2nd ` ( Q ` A ) ) >. ) $=
      ( cv cfv c2nd cop wceq c0 fveq2 id 2fveq3 opeq12d eqeq12d com opeq2d csuc
      va vb cid cvv co cmpo crdg fveq1i opex rdg0 eqtri 0ex op2nd eqcomi opeq2i
      fvex 3eqtr4a ax-mp df-ov suceq oveq1 oveq2 ovmpo mpan2 eqtr3id syl5ibrcom
      wcel eqid fveqeq2 vex sucex ovex a1i syld cres frsuc peano2 eqtr4di fvres
      fvresd fveq2d 3eqtr3d sylibrd finds ) UBHZCIZWFWGJIZKZLMCIZMWJJIZKZLZUCHZ
      CIZWNWOJIZKZLZWNUAZCIZWSWTJIZKZLZBCIZBXDJIZKZLUBUCBWFMLZWGWJWIWLWFMCNXGWF
      MWHWKXGOWFMJCPQRWFWNLZWGWOWIWQWFWNCNXHWFWNWHWPXHOWFWNJCPQRWFWSLZWGWTWIXBW
      FWSCNXIWFWSWHXAXIOWFWSJCPQRWFBLZWGXDWIXFWFBCNXJWFBWHXEXJOWFBJCPQRWJMFUDIZ
      KZLZWMWJMDASUEDHZUAZXNAHZEUFZKZUGZXLUHZIXLMCXTGUIXLXSMXKUJUKULXMXLMXLJIZK
      WJWLXKYAMYAXKMXKUMFUDUQUNUOUPXMOXMWKYAMWJXLJNTURUSWNSVHZWRWOXSIZWSYCJIZKZ
      LZXCYBWRYCWSWNWPEUFZKZLZYFYBYIWRWQXSIZYHLYBYJWNWPXSUFZYHWNWPXSUTYBWPUEVHY
      KYHLWOJUQDAWNWPSUEXRYHXSWSWNXPEUFZKXNWNLXOWSXQYLXNWNVAXNWNXPEVBQXPWPLYLYG
      WSXPWPWNEVCTXSVIWSYGUJVDVEVFWOWQYHXSVJVGYBYFYIYHWSYHJIZKZLYBYGYMWSYGYMLYB
      YMYGWSYGWNUCVKVLWNWPEVMUNUOVNTYIYCYHYEYNYIOYIYDYMWSYCYHJNTRVGVOYBWTYCXBYE
      YBWSXTSVPZIZWNYOIZXSIWTYCXLWNXSVQYBYPWSXTIWTYBWSSXTWNVRWAWSCXTGUIVSYBYQWO
      XSYBYQWNXTIWOWNSXTVTWNCXTGUIVSWBWCZYBXAYDWSYBWTYCJYRWBTRWDWE $.

    $( Lemma for ` seqom ` .  (Contributed by Stefan O'Rear, 1-Nov-2014.)
       (Revised by Mario Carneiro, 23-Jun-2015.) $)
    seqomlem2 $p |- ( Q " _om ) Fn _om $=
      ( va vb vc com wfn cvv wf cv cfv wcel cop mpbir c2nd wceq crn cxp wss wbr
      cima cres weu wral csuc co cmpo c0 cid crdg frfnom fneq1i fvres seqomlem1
      reseq1i eqtrd fvex opelxpi mpan2 eqeltrd rgen ffnfv mpbir2an frn ax-mp wb
      wal wex wrex df-br fvelrnb eqeq1d rexbiia 3bitri wa adantl opth1 biimtrdi
      fveqeq2 biimpd syli fveq2 op2nd eqtr2di syl6 rexlimdva rspcev mpdan opeq2
      eqeq2d rexbidv syl5ibrcom impbid bitrid alrimiv eqeq2 bibi2d albidv spcev
      vex syl eu6 sylibr dff3 df-ima feq1i dffn2 ) BJUEZJKJLXLMZXMJLBJUFZUAZMZX
      PXOJLUBZUCZGNZHNZXOUDZHUGZGJUHJXQXNMZXRYCXNJKZXTXNOZXQPZHJUHYDCAJLCNZUIYG
      ANDUJQUKZULEUMOQZUNZJUFZJKYIYHUOJXNYKBYJJFUSUPRZYFHJXTJPZYEXTXTBOZSOZQZXQ
      YMYEYNYPXTJBUQAXTBCDEFURUTYMYOLPYPXQPYNSVAXTYOJLVBVCVDVEHJXQXNVFVGJXQXNVH
      VIYBGJXSJPZYAXTINZTZVJZHVKZIVLZYBYQYAXTXSBOZSOZTZVJZHVKZUUBYQUUFHYAYRBOZX
      SXTQZTZIJVMZYQUUEYAUUIXOPZYRXNOZUUITZIJVMZUUKXSXTXOVNYDUULUUOVJYLIJUUIXNV
      OVIUUNUUJIJYRJPZUUMUUHUUIYRJBUQVPVQVRYQUUKUUEYQUUJUUEIJYQUUPVSZUUJUUCUUIT
      ZUUEUUJUUQYRXSTZUURUUQUUJYRUUHSOZQZUUITUUSUUQUUHUVAUUIUUPUUHUVATYQAYRBCDE
      FURVTVPYRUUTXSXTIXDUUHSVAWAWBUUSUUJUURYRXSUUIBWCWDWEUURUUDUUISOXTUUCUUISW
      FXSXTGXDHXDWGWHWIWJYQUUKUUEUUHXSUUDQZTZIJVMZYQUUCUVBTZUVDAXSBCDEFURUVCUVE
      IXSJYRXSUVBBWCWKWLUUEUUJUVCIJUUEUUIUVBUUHXTUUDXSWMWNWOWPWQWRWSUUAUUGIUUDU
      UCSVAYRUUDTZYTUUFHUVFYSUUEYAYRUUDXTWTXAXBXCXEYAHIXFXGVEGHJLXOXHVGJLXLXOBJ
      XIXJRJXLXKR $.

    $( Lemma for ` seqom ` .  (Contributed by Stefan O'Rear, 1-Nov-2014.) $)
    seqomlem3 $p |- ( ( Q " _om ) ` (/) ) = ( _I ` I ) $=
      ( c0 com cima cfv cid wceq cop wcel cres cv peano1 wfn mpbir mp2an wbr co
      crn cvv csuc cmpo crdg fvres ax-mp fveq1i opex rdg0 3eqtri frfnom reseq1i
      fneq1i fnfvelrn eqeltrri df-ima eleqtrri df-br wb seqomlem2 fnbrfvb ) GBH
      IZJEKJZLZGVFVEUAZVHGVFMZVENVIBHOZUCZVEGVJJZVIVKVLGBJZGCAHUDCPZUEVNAPDUBMU
      FZVIUGZJVIGHNZVLVMLQGHBUHUIGBVPFUJVIVOGVFUKULUMVJHRZVQVLVKNVRVPHOZHRVIVOU
      NHVJVSBVPHFUOUPSQHGVJUQTURBHUSUTGVFVEVASVEHRVQVGVHVBABCDEFVCQHGVFVEVDTS
      $.

    $( Lemma for ` seqom ` .  (Contributed by Stefan O'Rear, 1-Nov-2014.)
       (Revised by Mario Carneiro, 23-Jun-2015.) $)
    seqomlem4 $p |- ( A e. _om -> ( ( Q " _om ) ` suc A ) =
      ( A F ( ( Q " _om ) ` A ) ) ) $=
      ( com wcel csuc cfv co wceq wbr cop cres fvresd cvv eqtrd wfn cima peano2
      crn cv cmpo c2nd c0 cid crdg frsuc fveq1i eqtr4di fvres 3eqtr3d seqomlem1
      fveq2d df-ov fvex suceq oveq1 opeq12d oveq2 opeq2d eqid opex ovmpo frfnom
      mpan2 reseq1i fneq1i mpbir fnfvelrn mpan eqeltrrd df-ima eleqtrrdi sylibr
      df-br wb seqomlem2 fnbrfvb mpbird eqcomd oveq2d eqtr3id 3eqtrd sylancr )
      BHIZBJZCHUAZKBBWJKZELZMZWIWLWJNZWHWIWLOZWJIWNWHWOCHPZUCZWJWHWIWPKZWOWQWHW
      RWICKZWOWHWIHCBUBZQWHWSBCKZDAHRDUDZJZXBAUDZELZOZUEZKZBXAUFKZOZXGKZWOWHWIX
      GUGFUHKOZUIZHPZKZBXNKZXGKWSXHXLBXGUJWHXOWIXMKWSWHWIHXMWTQWICXMGUKULWHXPXA
      XGWHXPBXMKXABHXMUMBCXMGUKULUPUNWHXAXJXGABCDEFGUOZUPWHXKBXIXGLZWOBXIXGUQWH
      XRWIBXIELZOZWOWHXIRIXRXTMXAUFURDABXIHRXFXTXGWIBXDELZOXBBMXCWIXEYAXBBUSXBB
      XDEUTVAXDXIMYAXSWIXDXIBEVBVCXGVDWIXSVEVFVHWHXSWLWIWHXIWKBEWHWKXIWHWKXIMZB
      XIWJNZWHXJWJIYCWHXJWQWJWHBWPKZXJWQWHYDXAXJBHCUMXQSWPHTZWHYDWQIYEXNHTXLXGV
      GHWPXNCXMHGVIVJVKZHBWPVLVMVNCHVOZVPBXIWJVRVQWJHTZWHYBYCVSACDEFGVTZHBXIWJW
      AVMWBWCWDVCSWEWFSWHYEWIHIZWRWQIYFWTHWIWPVLWGVNYGVPWIWLWJVRVQWHYHYJWMWNVSY
      IWTHWIWLWJWAWGWB $.
  $}

  ${
    $d A a b $.  $d B a b $.  $d C a b $.  $d D a b $.
    $( Equality theorem for ` seqom ` .  (Contributed by Stefan O'Rear,
       1-Nov-2014.) $)
    seqomeq12 $p |- ( ( A = B /\ C = D ) -> seqom ( A , C ) =
      seqom ( B , D ) ) $=
      ( va vb wceq com cvv cv co cop cmpo c0 cid cfv crdg cima cseqom opeq2d wa
      csuc oveq mpoeq3dv fveq2 rdgeq12 syl2an imaeq1d df-seqom 3eqtr4g ) ABGZCD
      GZUAZEFHIEJZUBZUNFJZAKZLZMZNCOPZLZQZHREFHIUOUNUPBKZLZMZNDOPZLZQZHRACSBDSU
      MVBVHHUKUSVEGVAVGGVBVHGULUKEFHIURVDUKUQVCUOUNUPABUCTUDULUTVFNCDOUETVAVGUS
      VEUFUGUHFEACUIFEBDUIUJ $.
  $}

  ${
    $d F a b c d $.  $d I a b c d $.  $d G a b c d $.  $d A a b c d $.
    seqom.a $e |- G = seqom ( F , I ) $.
    $( An index-aware recursive definition defines a function on the natural
       numbers.  (Contributed by Stefan O'Rear, 1-Nov-2014.) $)
    fnseqom $p |- G Fn _om $=
      ( va vb vd vc com wfn cvv cv csuc co cop cmpo c0 cid cfv crdg cima cseqom
      seqomlem0 seqomlem2 df-seqom eqtri fneq1i mpbir ) BIJEFIKELZMUIFLANOPQCRS
      OTZIUAZIJGUJHACACEFHGUCUDIBUKBACUBUKDFEACUEUFUGUH $.

    $( Value of an index-aware recursive definition at 0.  (Contributed by
       Stefan O'Rear, 1-Nov-2014.)  (Revised by AV, 17-Sep-2021.) $)
    seqom0g $p |- ( I e. V -> ( G ` (/) ) = I ) $=
      ( va vb vd vc wcel c0 cfv cid com cvv cv csuc co cop eqtri cmpo crdg cima
      cseqom df-seqom fveq1i seqomlem0 seqomlem3 fvi eqtrid ) CDJKBLZCMLZCUKKFG
      NOFPZQUMGPARSUAKULSUBZNUCZLULKBUOBACUDUOEGFACUETUFHUNIACACFGIHUGUHTCDUIUJ
      $.

    $( Value of an index-aware recursive definition at a successor.
       (Contributed by Stefan O'Rear, 1-Nov-2014.) $)
    seqomsuc $p |- ( A e. _om -> ( G ` suc A ) = ( A F ( G ` A ) ) ) $=
      ( va vb vd vc com wcel csuc cvv cv co cop cmpo c0 cfv fveq1i cid df-seqom
      crdg cima seqomlem0 seqomlem4 cseqom eqtri oveq2i 3eqtr4g ) AJKALZFGJMFNZ
      LULGNBOPQRDUASPUCZJUDZSAAUNSZBOUKCSAACSZBOHAUMIBDBDFGIHUEUFUKCUNCBDUGUNEG
      FBDUBUHZTUPUOABACUNUQTUIUJ $.
  $}

  $( Membership is inherited by successors for natural numbers.  (Contributed
     by AV, 15-Sep-2023.) $)
  omsucelsucb $p |- ( N e. _om <-> suc N e. suc _om ) $=
    ( com word wcel csuc wb ordom ordsucelsuc ax-mp ) BCABDAEBEDFGABHI $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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 ^o $.  $( Ordinal exponentiation operation (caret subscript o) $)

  $( 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. $)
  coe $a class ^o $.

  $( Define the ordinal number 1.  Definition 2.1 of [Schloeder] p. 4.
     (Contributed by NM, 29-Oct-1995.) $)
  df-1o $a |- 1o = suc (/) $.

  $( Define the ordinal number 2.  Lemma 3.17 of [Schloeder] p. 10.
     (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.  (Contributed by NM,
       30-Dec-2004.) $)
    df-oexp $a |- ^o = ( x e. On , y e. On |->
                   if ( x = (/) , ( 1o \ y ) ,
                 ( rec ( ( z e. _V |-> ( z .o x ) ) , 1o ) ` y ) ) ) $.
  $}

  $( Expanded value of the ordinal number 1.  Definition 2.1 of [Schloeder]
     p. 4.  (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 $.

  $( Ordinal 1 is a set.  (Contributed by BJ, 6-Apr-2019.)  (Proof shortened by
     AV, 1-Jul-2022.)  Remove dependency on ~ ax-10 , ~ ax-11 , ~ ax-12 ,
     ~ ax-un .  (Revised by Zhi Wang, 19-Sep-2024.) $)
  1oex $p |- 1o e. _V $=
    ( c1o c0 csn cvv df1o2 snex eqeltri ) ABCDEBFG $.
  $( $j usage '1oex' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-un'; $)

  $( ` 2o ` is a set.  (Contributed by BJ, 6-Apr-2019.)  Remove dependency on
     ~ ax-10 , ~ ax-11 , ~ ax-12 , ~ ax-un .  (Proof shortened by Zhi Wang,
     19-Sep-2024.) $)
  2oex $p |- 2o e. _V $=
    ( c2o c0 c1o cpr cvv df2o3 prex eqeltri ) ABCDEFBCGH $.
  $( $j usage '2oex' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-un'; $)

  $( Ordinal 1 is an ordinal number.  (Contributed by NM, 29-Oct-1995.)  Avoid
     ~ ax-un .  (Revised by BTernaryTau, 30-Nov-2024.) $)
  1on $p |- 1o e. On $=
    ( c1o csuc con0 df-1o wcel cvv 0elon 1oex eqeltrri sucexeloni mp2an eqeltri
    c0 ) AMBZCDMCENFENCEGANFDHIMFJKL $.
  $( $j usage '1on' avoids 'ax-un'; $)

  $( Obsolete version of ~ 1on as of 30-Nov-2024.  (Contributed by NM,
     29-Oct-1995.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  1onOLD $p |- 1o e. On $=
    ( c1o c0 csuc con0 df-1o 0elon onsuci eqeltri ) ABCDEBFGH $.

  $( Ordinal 2 is an ordinal number.  (Contributed by NM, 18-Feb-2004.)  (Proof
     shortened by Andrew Salmon, 12-Aug-2011.)  Avoid ~ ax-un .  (Revised by
     BTernaryTau, 30-Nov-2024.) $)
  2on $p |- 2o e. On $=
    ( c2o c1o csuc con0 df-2o wcel cvv 2oex eqeltrri sucexeloni mp2an eqeltri
    1on ) ABCZDEBDFNGFNDFMANGEHIBGJKL $.
  $( $j usage '2on' avoids 'ax-un'; $)

  $( Obsolete version of ~ 2on as of 30-Nov-2024.  (Contributed by NM,
     18-Feb-2004.)  (Proof shortened by Andrew Salmon, 12-Aug-2011.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  2onOLD $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 nsuceq0 eqnetri ) ABCDEBFG $.

  $( Ordinal 3 is an ordinal class.  (Contributed by BTernaryTau,
     6-Jan-2025.) $)
  ord3 $p |- Ord 3o $=
    ( c3o word c2o csuc con0 wcel 2on eloni ordsuci mp2b wceq df-3o ordeq ax-mp
    wb mpbir ) ABZCDZBZCEFCBSGCHCIJARKQSOLARMNP $.
  $( $j usage 'ord3' avoids 'ax-un'; $)

  $( 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 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 $.

  $( 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 $.

  $( 1 is not a limit ordinal.  (Contributed by BTernaryTau, 1-Dec-2024.) $)
  nlim1 $p |- -. Lim 1o $=
    ( c1o wlim word c0 wne cuni wceq w3a csn 1n0 0ex unisn neeqtrri unieqi neii
    df1o2 simp3 mto df-lim mtbir ) ABACZADEZAAFZGZHZUEUDAUCADIZFZUCADUGJDKLMAUF
    PNMOUAUBUDQRAST $.
  $( $j usage 'nlim1' avoids 'ax-un'; $)

  $( 2 is not a limit ordinal.  (Contributed by BTernaryTau, 1-Dec-2024.) $)
  nlim2 $p |- -. Lim 2o $=
    ( c2o wlim word c0 wne cuni wceq w3a c1o wcel cpr 1oex prid2 df2o3 eleqtrri
    1on onirri eleq2 mtbiri mt2 neir cun unieqi 0ex unipr 3eqtri neeqtrri simp3
    0un neii mto df-lim mtbir ) ABACZADEZAAFZGZHZURUQAUPAIUPAIAIGZIAJZIDIKZADIL
    MNOUSUTIIJIPQAIIRSTUAUPVAFDIUBIAVANUCDIUDLUEIUIUFUGUJUNUOUQUHUKAULUM $.
  $( $j usage 'nlim2' avoids 'ax-un'; $)

  $( 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 NM,
     21-Dec-2004.) $)
  ordge1n0 $p |- ( Ord A -> ( 1o C_ A <-> A =/= (/) ) ) $=
    ( word c0 wcel c1o wss wne ordgt0ge1 ord0eln0 bitr3d ) ABCADEAFACGAHAIJ $.

  $( 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
    $.

  $( An ordinal that is not 0 or 1 contains 1.  (Contributed by BTernaryTau,
     1-Dec-2024.) $)
  ord1eln01 $p |- ( Ord A -> ( 1o e. A <-> ( A =/= (/) /\ A =/= 1o ) ) ) $=
    ( word c1o wcel c0 wne wa ne0i wceq 1on onirri eleq2 mtbiri necon2ai jca wo
    wn el1o biimpi necon3ai nesym anim12ci pm4.56 onordi ordtri2 mpan imbitrrid
    sylib wb impbid2 ) ABZCADZAEFZACFZGZULUMUNACHULACACIULCCDCJKACCLMNOUOULUKCA
    IZACDZPQZUOUPQZUQQZGURUMUTUNUSUQAEUQAEIARSTUNUSACUASUBUPUQUCUHCBUKULURUICJU
    DCAUEUFUGUJ $.
  $( $j usage 'ord1eln01' avoids 'ax-un'; $)

  $( An ordinal that is not 0, 1, or 2 contains 2.  (Contributed by
     BTernaryTau, 1-Dec-2024.) $)
  ord2eln012 $p |- ( Ord A ->
      ( 2o e. A <-> ( A =/= (/) /\ A =/= 1o /\ A =/= 2o ) ) ) $=
    ( word c2o wcel c0 wne c1o w3a ne0i wceq 2on0 nemtbir eleq2 mtbiri necon2ai
    el1o 2on onirri 3jca wn wo wa nesym biimpi 3ad2ant3 cpr simp1 nelprd eleq2i
    simp2 df2o3 sylnibr jca pm4.56 sylib onordi ordtri2 mpan imbitrrid impbid2
    wb ) ABZCADZAEFZAGFZACFZHZVCVDVEVFACIVCAGAGJVCCGDZVHCEKCPLAGCMNOVCACACJVCCC
    DCQRACCMNOSVGVCVBCAJZACDZUATZVGVITZVJTZUBVKVGVLVMVFVDVLVEVFVLACUCUDUEVGAEGU
    FZDVJVGAEGVDVEVFUGVDVEVFUJUHCVNAUKUIULUMVIVJUNUOCBVBVCVKVACQUPCAUQURUSUT $.
  $( $j usage 'ord2eln012' avoids 'ax-un'; $)

  $( A limit ordinal contains 1.  (Contributed by BTernaryTau, 1-Dec-2024.) $)
  1ellim $p |- ( Lim A -> 1o e. A ) $=
    ( wlim c1o wcel c0 wceq nlim0 limeq mtbiri necon2ai nlim1 word wa wb limord
    wne ord1eln01 syl mpbir2and ) ABZCADZAEPZACPZTAEAEFTEBGAEHIJTACACFTCBKACHIJ
    TALUAUBUCMNAOAQRS $.
  $( $j usage '1ellim' avoids 'ax-un'; $)

  $( A limit ordinal contains 2.  (Contributed by BTernaryTau, 1-Dec-2024.) $)
  2ellim $p |- ( Lim A -> 2o e. A ) $=
    ( wlim c2o wcel c0 wne c1o wceq nlim0 limeq mtbiri necon2ai nlim1 nlim2 w3a
    word wb limord ord2eln012 syl mpbir3and ) ABZCADZAEFZAGFZACFZUBAEAEHUBEBIAE
    JKLUBAGAGHUBGBMAGJKLUBACACHUBCBNACJKLUBAPUCUDUEUFOQARASTUA $.
  $( $j usage '2ellim' avoids 'ax-un'; $)

  $( 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 $.

  $( Two ways to say that ` A ` is a nonzero ordinal number.  Lemma 1.10 of
     [Schloeder] p. 2.  (Contributed by Mario Carneiro, 21-May-2015.) $)
  ondif1 $p |- ( A e. ( On \ 1o ) <-> ( A e. On /\ (/) e. A ) ) $=
    ( con0 c1o cdif wcel c0 wne wa dif1o on0eln0 pm5.32i bitr4i ) ABCDEABEZAFGZ
    HMFAEZHABIMONAJKL $.

  $( Two ways to say that ` A ` is an ordinal greater than one.  (Contributed
     by Mario Carneiro, 21-May-2015.) $)
  ondif2 $p |- ( A e. ( On \ 2o ) <-> ( A e. On /\ 1o e. A ) ) $=
    ( con0 c2o cdif wcel wn wa c1o eldif wb 1on wss ontri1 onsssuc df-2o eleq2i
    csuc bitr4di bitr3d mpan2 con1bid pm5.32i bitri ) ABCDEABEZACEZFZGUDHAEZGAB
    CIUDUFUGUDUGUEUDHBEZUGFZUEJKUDUHGZAHLZUIUEAHMUJUKAHQZEUEAHNCULAOPRSTUAUBUC
    $.

  $( 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 $.

  $( An ordinal greater than one is greater than zero.  (Contributed by Mario
     Carneiro, 25-May-2015.) $)
  dif20el $p |- ( A e. ( On \ 2o ) -> (/) e. A ) $=
    ( con0 c2o cdif wcel c1o c0 ondif2 simprbi 0lt1o wa wi eldifi ontr1 syl mpd
    mpani ) ABCDEZFAEZGAEZRABEZSAHIRGFEZSTJRUAUBSKTLABCMGFANOQP $.

  $( The empty set is a well-ordering of ordinal one.  (Contributed by Mario
     Carneiro, 9-Feb-2015.) $)
  0we1 $p |- (/) We 1o $=
    ( c1o c0 wwe csn wbr wn br0 wrel wb rel0 wesn ax-mp mpbir wceq df1o2 weeq2
    ) ABCZBDZBCZSBBBEFZBBGBHSTIJBBKLMARNQSIOARBPLM $.

  ${
    brwitnlem.r $e |- R = ( `' O " ( _V \ 1o ) ) $.
    brwitnlem.o $e |- O Fn X $.
    $( Lemma for relations which assert the existence of a witness in a
       two-parameter set.  (Contributed by Stefan O'Rear, 25-Jan-2015.)
       (Revised by Mario Carneiro, 23-Aug-2015.) $)
    brwitnlem $p |- ( A R B <-> ( A O B ) =/= (/) ) $=
      ( cop ccnv cvv c1o cdif cima wcel cfv c0 wne wbr wa 3bitr4i co fvex dif1o
      mpbiran anbi2i wfn wb elpreima ax-mp cdm ndmfv necon1ai eleqtrdi pm4.71ri
      fndmi breqi df-br bitri df-ov neeq1i ) ABHZDIJKLZMZNZVADOZPQZABCRZABDUAZP
      QVAENZVEVBNZSZVIVFSVDVFVJVFVIVJVEJNVFVADUBVEJUCUDUEDEUFVDVKUGGEVAVBDUHUIV
      FVIVFVADUJZEVAVLNVEPVADUKULEDGUOUMUNTVGABVCRVDABCVCFUPABVCUQURVHVEPABDUSU
      TT $.
  $}

  ${
    $d x y z $.
    $( Functionality and domain of ordinal addition.  (Contributed by NM,
       26-Aug-1995.)  (Revised by Mario Carneiro, 8-Sep-2013.) $)
    fnoa $p |- +o Fn ( On X. On ) $=
      ( vx vy vz con0 cv cvv csuc cmpt crdg cfv coa df-oadd fvex fnmpoi ) ABDDB
      EZCFCEGHAEIZJKABCLOPMN $.

    $( Functionality and domain of ordinal multiplication.  (Contributed by NM,
       26-Aug-1995.)  (Revised by Mario Carneiro, 8-Sep-2013.) $)
    fnom $p |- .o Fn ( On X. On ) $=
      ( vx vy vz con0 cv cvv coa co cmpt c0 crdg cfv comu df-omul fvex fnmpoi )
      ABDDBEZCFCEAEGHIJKZLMABCNQROP $.

    $( Functionality and domain of ordinal exponentiation.  (Contributed by
       Mario Carneiro, 29-May-2015.) $)
    fnoe $p |- ^o Fn ( On X. On ) $=
      ( vx vy vz con0 cv c0 wceq c1o cdif cvv comu co cmpt crdg cfv cif df-oexp
      coe wcel 1on difexg ax-mp fvex ifex fnmpoi ) ABDDAEZFGZHBEZIZUHCJCEUFKLMH
      NZOZPRABCQUGUIUKHDSUIJSTHUHDUAUBUHUJUCUDUE $.
  $}

  ${
    $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 cv cvv csuc cmpt crdg cfv coa wceq rdgeq2 fveq2 df-oadd fvex
      fveq1d ovmpo ) DEBCFFEGZAHAGIJZDGZKZLCUBBKZLMUAUELUCBNUAUDUEUCBUBOSUACUEP
      DEAQCUERT $.

    $( 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 cv cvv coa co cmpt c0 crdg cfv comu wceq mpteq2dv rdgeq1 syl
      oveq2 fveq1d fveq2 df-omul fvex ovmpo ) DEBCFFEGZAHAGZDGZIJZKZLMZNCAHUGBI
      JZKZLMZNOUFUNNUHBPZUFUKUNUOUJUMPUKUNPUOAHUIULUHBUGITQLUJUMRSUAUFCUNUBDEAU
      CCUNUDUE $.
  $}

  ${
    oe0lem.1 $e |- ( ( ph /\ A = (/) ) -> ps ) $.
    oe0lem.2 $e |- ( ( ( A e. On /\ ph ) /\ (/) e. A ) -> ps ) $.
    $( A helper lemma for ~ oe0 and others.  (Contributed by NM,
       6-Jan-2005.) $)
    oe0lem $p |- ( ( A e. On /\ ph ) -> ps ) $=
      ( con0 wcel wa c0 wceq wi ex adantl wne on0eln0 adantr sylbird pm2.61dne
      wb ) CFGZAHZBCIACIJZBKTAUBBDLMUACINZICGZBTUDUCSACOPUAUDBELQR $.
  $}

  ${
    $d x y z A $.  $d y z B $.
    $( Value of ordinal exponentiation.  (Contributed by NM, 30-Dec-2004.)
       (Revised by Mario Carneiro, 23-Aug-2014.) $)
    oev $p |- ( ( A e. On /\ B e. On ) -> ( A ^o B ) = if ( A = (/) ,
      ( 1o \ B ) , ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) ) ) $=
      ( vy vz con0 cv c0 wceq c1o cdif cvv comu co cmpt crdg cfv cif coe eqeq1
      oveq2 mpteq2dv rdgeq1 fveq1d ifbieq2d difeq2 fveq2 ifeq12d df-oexp difexi
      syl 1oex fvex ifex ovmpo ) DEBCFFDGZHIZJEGZKZURALAGZUPMNZOZJPZQZRBHIZJCKZ
      CALUTBMNZOZJPZQZRSVEUSURVIQZRUPBIZUQVEVDVKUSUPBHTVLURVCVIVLVBVHIVCVIIVLAL
      VAVGUPBUTMUAUBJVBVHUCUKUDUEURCIVEUSVFVKVJURCJUFURCVIUGUHDEAUIVEVFVJJCULUJ
      CVIUMUNUO $.

    $( Value of ordinal exponentiation at a nonzero base.  (Contributed by NM,
       31-Dec-2004.)  (Revised by Mario Carneiro, 8-Sep-2013.) $)
    oevn0 $p |- ( ( ( A e. On /\ B e. On ) /\ (/) e. A ) -> ( A ^o B ) =
                ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) ) $=
      ( con0 wcel wa c0 coe co cvv cv comu cmpt c1o crdg cfv wceq wn wb wne cif
      on0eln0 df-ne bitrdi adantr cdif oev iffalse sylan9eq ex sylbid imp ) BDE
      ZCDEZFZGBEZBCHIZCAJAKBLIMNOPZQZUOUPBGQZRZUSUMUPVASUNUMUPBGTVABUBBGUCUDUEU
      OVAUSUOVAUQUTNCUFZURUAURABCUGUTVBURUHUIUJUKUL $.
  $}

  ${
    $d x A $.
    $( Addition with zero.  Proposition 8.3 of [TakeutiZaring] p. 57.
       Definition 2.3 of [Schloeder] p. 4.  (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(a) of [TakeutiZaring]
       p. 62.  Definition 2.5 of [Schloeder] p. 4.  See ~ om0x for a way to
       remove the antecedent ` A e. On ` .  (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 $.

    $( Value of zero raised to an ordinal.  (Contributed by NM, 31-Dec-2004.)
       (Revised by Mario Carneiro, 8-Sep-2013.) $)
    oe0m $p |- ( A e. On -> ( (/) ^o A ) = ( 1o \ A ) ) $=
      ( vx con0 wcel c0 coe co wceq c1o cdif cvv cv comu cmpt cfv cif 0elon oev
      crdg mpan eqid iftruei eqtrdi ) ACDZEAFGZEEHZIAJZABKBLEMGNISOZPZUGECDUDUE
      UIHQBEARTUFUGUHEUAUBUC $.
  $}

  $( Ordinal multiplication with zero.  Definition 8.15 of [TakeutiZaring]
     p. 62.  Unlike ~ om0 , this version works whether or not ` A ` is an
     ordinal.  However, since it is an artifact of our particular function
     value definition outside the domain, we will not use it in order to be
     conventional and present it only as a curiosity.  (Contributed by NM,
     1-Feb-1996.)  (New usage is discouraged.) $)
  om0x $p |- ( A .o (/) ) = (/) $=
    ( con0 wcel c0 wa comu co wceq om0 adantr cxp fnom fndmi ndmov pm2.61i ) AB
    CZDBCZEADFGDHZPRQAIJADBFBBKFLMNO $.

  $( Ordinal exponentiation with zero base and zero exponent.  Proposition 8.31
     of [TakeutiZaring] p. 67.  (Contributed by NM, 31-Dec-2004.) $)
  oe0m0 $p |- ( (/) ^o (/) ) = 1o $=
    ( c0 coe co c1o cdif con0 wcel wceq 0elon oe0m ax-mp dif0 eqtri ) AABCZDAEZ
    DAFGNOHIAJKDLM $.

  $( Ordinal exponentiation with zero base and nonzero exponent.  Proposition
     8.31(2) of [TakeutiZaring] p. 67 and its converse.  Definition 2.6 of
     [Schloeder] p. 4.  (Contributed by NM, 5-Jan-2005.) $)
  oe0m1 $p |- ( A e. On -> ( (/) e. A <-> ( (/) ^o A ) = (/) ) ) $=
    ( con0 wcel c0 c1o wss coe co wceq word wb eloni ordgt0ge1 cdif ssdif0 oe0m
    syl eqeq1d bitr4id bitrd ) ABCZDACZEAFZDAGHZDIZUAAJUBUCKALAMQUAUCEANZDIUEEA
    OUAUDUFDAPRST $.

  ${
    $d x A $.
    $( Ordinal exponentiation with zero exponent.  Definition 8.30 of
       [TakeutiZaring] p. 67.  Definition 2.6 of [Schloeder] p. 4.
       (Contributed by NM, 31-Dec-2004.)  (Revised by Mario Carneiro,
       8-Sep-2013.) $)
    oe0 $p |- ( A e. On -> ( A ^o (/) ) = 1o ) $=
      ( vx con0 wcel c0 coe co c1o wceq oveq1 oe0m0 eqtrdi adantl cvv comu cmpt
      wa cv crdg cfv 0elon oevn0 mpanl2 1oex rdg0 adantll oe0lem anidms ) ACDZA
      EFGZHIZUIUKAAEIZUKUIULUJEEFGHAEEFJKLMUIEADZUKUIUIUMQUJEBNBRAOGPZHSTZHUIEC
      DUMUJUOIUABAEUBUCHUNUDUELUFUGUH $.

    $( Alternate value of ordinal exponentiation.  Compare ~ oev .
       (Contributed by NM, 2-Jan-2004.)  (Revised by Mario Carneiro,
       8-Sep-2013.) $)
    oev2 $p |- ( ( A e. On /\ B e. On ) -> ( A ^o B ) = ( ( rec (
          ( x e. _V |-> ( x .o A ) ) , 1o ) ` B )
                  i^i ( ( _V \ |^| A ) u. |^| B ) ) ) $=
      ( con0 wcel coe co cvv c1o cint cdif cun cin wceq c0 eqtrdi eqtr4d ineq2d
      wa adantl cv comu cmpt crdg cfv oveq12 oe0m0 fveq2 1oex rdg0 int0 ineq12d
      inteq inv1 a1i sylan9eqr oveq1 oe0m1 biimpa an32s int0el in0 oe0lem difid
      difeq2d uneq2d uncom un0 3eqtr3g oevn0 dif0 unv eqtr2di eqtrd ) CDEZBCFGZ
      CAHAUABUBGUCZIUDZUEZHBJZKZCJZLZMZNBVOBONZSZVPVSWBMZWDWEVPWGNCWECONZSZVPIW
      GWIVPOOFGIBOCOFUFUGPWHWEWGIHMZIWHVSIWBHWHVSOVRUEICOVRUHIVQUIUJPWHWBOJZHCO
      UMUKPULWJINWEIUNUOUPQWFOCEZSVPOWGVOWLWEVPONWEVOWLSVPOCFGZOBOCFUQVOWLWMONC
      URUSUPUTWLWGONWFWLWGVSOMOWLWBOVSCVARVSVBPTQVCWFWCWBVSWEWCWBNVOWEWBWALZWBO
      LWCWBWEWAOWBWEWAHHKOWEVTHHWEVTWKHBOUMUKPVEHVDPVFWBWAVGZWBVHVITRQBDEVOSZOB
      EZSZVPVSWDABCVJWRWDVSHMVSWRWCHVSWQWCHNWPWQWNWBHLWCHWQWAHWBWQWAHOKHWQVTOHB
      VAVEHVKPVFWOWBVLVITRVSUNVMVNVC $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Addition with successor.  Definition 8.1 of [TakeutiZaring] p. 56.
       Definition 2.3 of [Schloeder] p. 4.  (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 cvv cv cmpt crdg cfv coa co wceq rdgsuc adantl oav
      onsuc sylan2 ovex suceq eqid sucex fvmpt ax-mp fveq2d eqtr3id 3eqtr4d ) A
      DEZBDEZFZBGZCHCIZGZJZAKZLZBUQLZUPLZAUMMNZABMNZGZUKURUTOUJABUPPQUKUJUMDEVA
      UROBSCAUMRTULVCVBUPLZUTVBHEVDVCOABMUAZCVBUOVCHUPUNVBUBUPUCVBVEUDUEUFULVBU
      SUPCABRUGUHUI $.
  $}

  ${
    $d x A $.  $d x B $.
    oesuclem.1 $e |- Lim X $.
    oesuclem.2 $e |- ( B e. X ->
      ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` suc B ) =
        ( ( x e. _V |-> ( x .o A ) ) `
          ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) ) ) $.
    $( Lemma for ~ oesuc .  (Contributed by NM, 31-Dec-2004.)  (Revised by
       Mario Carneiro, 15-Nov-2014.) $)
    oesuclem $p |- ( ( A e. On /\ B e. X ) ->
                ( A ^o suc B ) = ( ( A ^o B ) .o A ) ) $=
      ( wcel coe co comu wceq c0 wa oveq1 ax-mp mpan syl wb con0 cfv csuc oe0m1
      word wlim limord ordelord 0elsuc ordelon sylbi mpbid sylan9eqr id oveq12d
      limsuc oveq2 c1o oe0m0 1on eqeltri eqeltrdi adantl biimpa adantll mpancom
      0elon oe0lem om0 eqtr4d cvv cv cmpt crdg ad2antlr oevn0 sylanl2 ovex eqid
      fvmpt fveq2d eqtr3id 3eqtr4d ) CDGZBCUAZHIZBCHIZBJIZKBWBBLKZMWDLWFWGWBWDL
      WCHIZLBLWCHNWBLWCGZWHLKZWBCUCZWIDUCZWBWKDUDZWLEDUEOZDCUFPCUGQWBWCDGZWIWJR
      ZWMWBWOREDCUNOZWOWCSGZWPWLWOWRWNDWCUHPZWCUBQUIUJUKWGWBWFLCHIZLJIZLWGWEWTB
      LJBLCHNWGULUMWBWTSGZXALKCSGZWBXBWLWBXCWNDCUHPZWBXBCCLKZXBWBXEWTLLHIZSCLLH
      UOXFUPSUQURUSUTVAWBLCGZXBXCWBXGMWTLSWBXGWTLKZWBXCXGXHRXDCUBQVBVEUTVCVFVDW
      TVGQUKVHBSGZWBMLBGZMZWCAVIAVJZBJIZVKZUPVLZTZCXOTZXNTZWDWFWBXPXRKXIXJFVMWB
      XIWRXJWDXPKWBWOWRWQWSUIABWCVNVOXKWFWEXNTZXRWEVIGXSWFKBCHVPAWEXMWFVIXNXLWE
      BJNXNVQWEBJVPVROXKWEXQXNWBXIXCXJWEXQKXDABCVNVOVSVTWAVF $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Multiplication with successor.  Definition 8.15 of [TakeutiZaring]
       p. 62.  Definition 2.5 of [Schloeder] p. 4.  (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 cvv cv coa cmpt crdg cfv comu wceq rdgsuc omv ovex
      co c0 adantl onsuc sylan2 oveq1 eqid fvmpt ax-mp fveq2d eqtr3id 3eqtr4d )
      ADEZBDEZFZBGZCHCIZAJSZKZTLZMZBURMZUQMZAUNNSZABNSZAJSZULUSVAOUKTBUQPUAULUK
      UNDEVBUSOBUBCAUNQUCUMVDVCUQMZVAVCHEVEVDOABNRCVCUPVDHUQUOVCAJUDUQUEVCAJRUF
      UGUMVCUTUQCABQUHUIUJ $.

    $( Ordinal exponentiation with a successor exponent.  Definition 8.30 of
       [TakeutiZaring] p. 67.  Definition 2.6 of [Schloeder] p. 4.
       (Contributed by NM, 31-Dec-2004.)  (Revised by Mario Carneiro,
       8-Sep-2013.) $)
    oesuc $p |- ( ( A e. On /\ B e. On ) ->
                ( A ^o suc B ) = ( ( A ^o B ) .o A ) ) $=
      ( vx con0 limon c1o cvv cv comu co cmpt rdgsuc oesuclem ) CABDEFBCGCHAIJK
      LM $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Addition with successor.  Theorem 4I(A2) of [Enderton] p. 79.  (Note
       that this version of ~ oasuc does not need Replacement.)  (Contributed
       by Mario Carneiro, 16-Nov-2014.) $)
    onasuc $p |- ( ( A e. On /\ B e. _om ) ->
                  ( A +o suc B ) = suc ( A +o B ) ) $=
      ( vx con0 wcel com wa csuc cvv cv cmpt crdg cfv coa co wceq adantl fveq2d
      oav sylan2 cres frsuc peano2 fvres 3eqtr3d nnon onsuc syl ovex suceq eqid
      fvresd sucex fvmpt ax-mp eqtr3id 3eqtr4d ) ADEZBFEZGZBHZCICJZHZKZALZMZBVE
      MZVDMZAVANOZABNOZHZUTVAVEFUAZMZBVLMZVDMZVFVHUSVMVOPURABVDUBQUTVAFVEUSVAFE
      URBUCQULUTVNVGVDUSVNVGPURBFVEUDQRUEUSURVADEZVIVFPUSBDEZVPBUFZBUGUHCAVASTU
      TVKVJVDMZVHVJIEVSVKPABNUIZCVJVCVKIVDVBVJUJVDUKVJVTUMUNUOUTVJVGVDUSURVQVJV
      GPVRCABSTRUPUQ $.

    $( 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 ) ) $=
      ( vx con0 wcel com wa comu cvv coa cfv wceq nnon omv sylan2 adantl eqtr4d
      co c0 ovex csuc cv cmpt crdg cres peano2 syl oveq1 eqid fvmpt ax-mp fvres
      fvresd fveq2d eqtr3id frsuc ) ADEZBFEZGZABUAZHRZUTCICUBZAJRZUCZSUDZFUEZKZ
      ABHRZAJRZUSVAUTVEKZVGURUQUTDEZVAVJLURUTFEZVKBUFZUTMUGCAUTNOUSUTFVEURVLUQV
      MPUMQUSVIBVFKZVDKZVGUSVIVHVDKZVOVHIEVPVILABHTCVHVCVIIVDVBVHAJUHVDUIVHAJTU
      JUKUSVHVNVDUSVHBVEKZVNURUQBDEVHVQLBMCABNOURVNVQLUQBFVEULPQUNUOURVGVOLUQSB
      VDUPPQQ $.

    $( Exponentiation with a successor exponent.  Definition 8.30 of
       [TakeutiZaring] p. 67.  (Contributed by Mario Carneiro, 14-Nov-2014.) $)
    onesuc $p |- ( ( A e. On /\ B e. _om ) ->
                ( A ^o suc B ) = ( ( A ^o B ) .o A ) ) $=
      ( vx com limom wcel csuc cvv cv comu cmpt c1o crdg cres cfv peano2 fvresd
      co frsuc fvres fveq2d 3eqtr3d oesuclem ) CABDEBDFZBGZCHCIAJRKZLMZDNZOBUHO
      ZUFOUEUGOBUGOZUFOLBUFSUDUEDUGBPQUDUIUJUFBDUGTUAUBUC $.
  $}

  $( Addition with 1 is same as successor.  Proposition 4.34(a) of [Mendelson]
     p. 266.  Remark 2.4 of [Schloeder] p. 4.  (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 $.

  ${
    $d x y A $.  $d x B $.
    $( Ordinal addition with a limit ordinal.  Definition 8.1 of
       [TakeutiZaring] p. 56.  Definition 2.3 of [Schloeder] p. 4.
       (Contributed by NM, 3-Aug-2004.)  (Revised by Mario Carneiro,
       8-Sep-2013.) $)
    oalim $p |- ( ( A e. On /\ ( B e. C /\ Lim B ) ) ->
                ( A +o B ) = U_ x e. B ( A +o x ) ) $=
      ( vy wcel wlim wa con0 coa co ciun wceq limelon simpr jca cfv oav sylan2
      cv cvv csuc cmpt crdg rdglim2a adantl wb anassrs iuneq2dv eqeq12d adantrr
      onelon mpbird ) CDFZCGZHZBIFZCIFZUOHZBCJKZACBATZJKZLZMZUPURUOCDNUNUOOPUQU
      SHVDCEUAETUBUCZBUDZQZACVAVFQZLZMZUSVJUQABCIVEUEUFUQURVDVJUGUOUQURHZUTVGVC
      VIEBCRVKACVBVHUQURVACFZVBVHMZURVLHUQVAIFVMCVAULEBVARSUHUIUJUKUMS $.

    $( Ordinal multiplication with a limit ordinal.  Definition 8.15 of
       [TakeutiZaring] p. 62.  Definition 2.5 of [Schloeder] p. 4.
       (Contributed by NM, 3-Aug-2004.)  (Revised by Mario Carneiro,
       8-Sep-2013.) $)
    omlim $p |- ( ( A e. On /\ ( B e. C /\ Lim B ) ) ->
                ( A .o B ) = U_ x e. B ( A .o x ) ) $=
      ( vy wcel wlim wa con0 comu co cv ciun wceq limelon simpr cfv omv sylan2
      c0 jca cvv coa cmpt crdg rdglim2a adantl anassrs iuneq2dv eqeq12d adantrr
      wb onelon mpbird ) CDFZCGZHZBIFZCIFZUPHZBCJKZACBALZJKZMZNZUQUSUPCDOUOUPPU
      AURUTHVECEUBELBUCKUDZTUEZQZACVBVGQZMZNZUTVKURATCIVFUFUGURUSVEVKULUPURUSHZ
      VAVHVDVJEBCRVLACVCVIURUSVBCFZVCVINZUSVMHURVBIFVNCVBUMEBVBRSUHUIUJUKUNS $.

    $( Ordinal exponentiation with a limit exponent and nonzero base.
       Definition 8.30 of [TakeutiZaring] p. 67.  Definition 2.6 of [Schloeder]
       p. 4.  (Contributed by NM, 1-Jan-2005.)  (Revised by Mario Carneiro,
       8-Sep-2013.) $)
    oelim $p |- ( ( ( A e. On /\ ( B e. C /\ Lim B ) ) /\ (/) e. A ) ->
                ( A ^o B ) = U_ x e. B ( A ^o x ) ) $=
      ( vy wcel wlim wa con0 c0 coe co ciun wceq limelon c1o cfv oevn0 sylanl2
      cv simpr jca comu cmpt crdg rdglim2a ad2antlr wb onelon exp42 com34 imp41
      cvv iuneq2dv eqeq12d adantlrr mpbird ) CDFZCGZHZBIFZCIFZUSHZJBFZBCKLZACBA
      TZKLZMZNZUTVBUSCDOURUSUAUBVAVCHVDHVICEUMETBUCLUDZPUEZQZACVFVKQZMZNZVCVOVA
      VDAPCIVJUFUGVAVBVDVIVOUHUSVAVBHVDHZVEVLVHVNEBCRVPACVGVMVAVBVDVFCFZVGVMNZV
      AVBVQVDVRVAVBVQVDVRVBVQHVAVFIFVDVRCVFUIEBVFRSUJUKULUNUOUPUQS $.
  $}

  ${
    $d x y A $.  $d x B $.
    $( Closure law for ordinal addition.  Proposition 8.2 of [TakeutiZaring]
       p. 57.  Remark 2.8 of [Schloeder] p. 5.  (Contributed by NM,
       5-May-1995.) $)
    oacl $p |- ( ( A e. On /\ B e. On ) -> ( A +o B ) e. On ) $=
      ( vx vy con0 wcel coa co cv c0 csuc wceq oveq2 eleq1d oa0 wi wa imbitrrid
      expcom cvv ibir onsuc oasuc wlim wral ciun vex iunon oalim mpanr1 tfinds3
      mpan impcom ) BEFAEFZABGHZEFZACIZGHZEFZAJGHZEFZADIZGHZEFZAVBKZGHZEFZUPUNC
      DBUQJLURUTEUQJAGMNUQVBLURVCEUQVBAGMNUQVELURVFEUQVEAGMNUQBLURUOEUQBAGMNUNV
      AUNUTAEAONUAUNVBEFZVDVGPVDVGUNVHQZVCKZEFVCUBVIVFVJEAVBUCNRSUNUQUDZVDDUQUE
      ZUSPVLUSUNVKQZDUQVCUFZEFZUQTFZVLVOCUGZDUQVCTUHULVMURVNEUNVPVKURVNLVQDAUQT
      UIUJNRSUKUM $.

    $( Closure law for ordinal multiplication.  Proposition 8.16 of
       [TakeutiZaring] p. 57.  Remark 2.8 of [Schloeder] p. 5.  (Contributed by
       NM, 3-Aug-2004.)  (Proof shortened by Andrew Salmon, 22-Oct-2011.) $)
    omcl $p |- ( ( A e. On /\ B e. On ) -> ( A .o B ) e. On ) $=
      ( vx vy con0 wcel comu co cv c0 csuc wceq oveq2 eleq1d 0elon wi wa expcom
      om0 cvv eqeltrdi coa oacl adantr omsuc sylibrd wlim wral ciun iunon omlim
      vex mpan mpanr1 imbitrrid tfinds3 impcom ) BEFAEFZABGHZEFZACIZGHZEFZAJGHZ
      EFADIZGHZEFZAVEKZGHZEFZUTURCDBVAJLVBVDEVAJAGMNVAVELVBVFEVAVEAGMNVAVHLVBVI
      EVAVHAGMNVABLVBUSEVABAGMNURVDJEASOUAURVEEFZVGVJPURVKQZVGVFAUBHZEFZVJURVGV
      NPVKVGURVNVFAUCRUDVLVIVMEAVEUENUFRURVAUGZVGDVAUHZVCPVPVCURVOQZDVAVFUIZEFZ
      VATFZVPVSCULZDVAVFTUJUMVQVBVREURVTVOVBVRLWADAVATUKUNNUORUPUQ $.

    $( Closure law for ordinal exponentiation.  Remark 2.8 of [Schloeder] p. 5.
       (Contributed by NM, 1-Jan-2005.)  (Proof shortened by Andrew Salmon,
       22-Oct-2011.) $)
    oecl $p |- ( ( A e. On /\ B e. On ) -> ( A ^o B ) e. On ) $=
      ( vx vy con0 wcel coe co c0 wceq c1o 1on eqeltrdi oe0lem eleq1d imbitrrid
      oveq2 wa wi cvv oe0m0 eqeltri adantl oe0m1 biimpa 0elon adantll anidms cv
      oveq1 impcom csuc adantr comu omcl expcom oesuc sylibrd adantrd wlim wral
      oe0 ciun vex iunon mpan oelim mpanlr1 anasss an12s ex tfinds3 com12 imp31
      expd ) BEFZABGHZEFZAAIJZVPVRVPVRVSIBGHZEFZVPWAVPWABBIJZWAVPWBVTIIGHZEBIIG
      QWCKEUALUBMUCVPIBFZWAVPVPWDRVTIEVPWDVTIJBUDUEUFMUGNUHVSVQVTEAIBGUJOPUKAEF
      ZVPIAFZVRVPWEWFVRSVPWEWFVRACUIZGHZEFZAIGHZEFZADUIZGHZEFZAWLULZGHZEFZVRWEW
      FRZCDBWGIJWHWJEWGIAGQOWGWLJWHWMEWGWLAGQOWGWOJWHWPEWGWOAGQOWGBJWHVQEWGBAGQ
      OWEWKWFWEWJKEAVBLMUMWLEFZWEWNWQSZWFWEWSWTWEWSRZWNWMAUNHZEFZWQWEWNXCSWSWNW
      EXCWMAUOUPUMXAWPXBEAWLUQOURUPUSWGUTZWRWNDWGVAZWISXEWIXDWRRZDWGWMVCZEFZWGT
      FZXEXHCVDZDWGWMTVEVFXFWHXGEWEXDWFWHXGJZWEXDWFXKWEXIXDWFXKXJDAWGTVGVHVIVJO
      PVKVLVOVMVNN $.
  $}

  ${
    $d x y $.  $d x A $.
    $( Ordinal addition with zero.  Proposition 8.3 of [TakeutiZaring] p. 57.
       Lemma 2.14 of [Schloeder] p. 5.  (Contributed by NM, 5-May-1995.) $)
    oa0r $p |- ( A e. On -> ( (/) +o A ) = A ) $=
      ( vx vy c0 cv coa co wceq csuc oveq2 id eqeq12d con0 wcel 0elon oa0 ax-mp
      mpan ciun cvv oasuc suceq sylan9eq ex wral wlim iuneq2 uniiun eqtr4di vex
      cuni wa oalim limuni imbitrrid tfinds ) DBEZFGZUQHZDDFGZDHZDCEZFGZVBHZDVB
      IZFGZVEHZDAFGZAHBCAUQDHZURUTUQDUQDDFJVIKLUQVBHZURVCUQVBUQVBDFJVJKLUQVEHZU
      RVFUQVEUQVEDFJVKKLUQAHZURVHUQAUQADFJVLKLDMNZVAODPQVBMNZVDVGVNVDVFVCIZVEVM
      VNVFVOHODVBUARVCVBUBUCUDVDCUQUEZUSUQUFZCUQVCSZUQUKZHVPVRCUQVBSVSCUQVCVBUG
      CUQUHUIVQURVRUQVSUQTNZVQURVRHZBUJVMVTVQULWAOCDUQTUMRRUQUNLUOUP $.
  $}

  ${
    $d x y A $.
    $( Ordinal multiplication with zero.  Proposition 8.18(1) of
       [TakeutiZaring] p. 63.  (Contributed by NM, 3-Aug-2004.) $)
    om0r $p |- ( A e. On -> ( (/) .o A ) = (/) ) $=
      ( vx vy c0 cv comu wceq csuc oveq2 eqeq1d con0 wcel 0elon ax-mp imbitrrid
      co coa mpan ciun cvv om0 oveq1 omsuc oa0 eqcomi a1i wral wlim iuneq2 iun0
      eqeq12d eqtrdi vex wa omlim tfinds ) DBEZFPZDGZDDFPZDGZDCEZFPZDGZDVBHZFPZ
      DGZDAFPZDGBCAUQDGURUTDUQDDFIJUQVBGURVCDUQVBDFIJUQVEGURVFDUQVEDFIJUQAGURVH
      DUQADFIJDKLZVAMDUANVDVGVBKLZVCDQPZDDQPZGVCDDQUBVJVFVKDVLVIVJVFVKGMDVBUCRD
      VLGVJVLDVIVLDGMDUDNUEUFUKOVDCUQUGZUSUQUHZCUQVCSZDGVMVOCUQDSDCUQVCDUICUQUJ
      ULVNURVODUQTLZVNURVOGZBUMVIVPVNUNVQMCDUQTUORRJOUP $.
  $}

  $( 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 $.

  $( 2 + 2 = 4 for ordinal numbers.  Ordinal numbers are modeled as Von Neumann
     ordinals; see ~ df-suc .  For the usual proof using complex numbers, see
     ~ 2p2e4 .  (Contributed by NM, 18-Aug-2021.)  Avoid ~ ax-rep , from a
     comment by Sophie.  (Revised by SN, 23-Mar-2024.) $)
  o2p2e4 $p |- ( 2o +o 2o ) = 4o $=
    ( c2o coa co c3o csuc c4o c1o con0 wcel com wceq 2on c0 df-1o peano1 peano2
    ax-mp eqeltri onasuc eqtr4i mp2an df-2o oveq2i df-3o oa1suc suceq 3eqtr4i
    df-4o ) AABCZDEZFAGEZBCZAGBCZEZUIUJAHIZGJIULUNKLGMEZJNMJIUPJIOMPQRAGSUAAUKA
    BUBUCDUMKUJUNKDAEZUMUDUOUMUQKLAUEQTDUMUFQUGUHT $.
  $( $j usage 'o2p2e4' avoids 'ax-rep'; $)

  $( Ordinal multiplication with 1.  Proposition 8.18(2) of [TakeutiZaring]
     p. 63.  Lemma 2.15 of [Schloeder] p. 5.  (Contributed by NM,
     29-Oct-1995.) $)
  om1 $p |- ( A e. On -> ( A .o 1o ) = A ) $=
    ( con0 wcel c1o comu co coa csuc df-1o oveq2i com wceq peano1 onmsuc eqtrid
    c0 mpan2 om0 oveq1d oa0r 3eqtrd ) ABCZADEFZAPEFZAGFZPAGFAUBUCAPHZEFZUEDUFAE
    IJUBPKCUGUELMAPNQOUBUDPAGARSATUA $.

  ${
    $d x y $.  $d x A $.
    $( Ordinal multiplication with 1.  Proposition 8.18(2) of [TakeutiZaring]
       p. 63.  Lemma 2.15 of [Schloeder] p. 5.  (Contributed by NM,
       3-Aug-2004.) $)
    om1r $p |- ( A e. On -> ( 1o .o A ) = A ) $=
      ( vx vy c1o cv comu co wceq c0 oveq2 id eqeq12d con0 wcel 1on wa coa mpan
      ciun cvv csuc ax-mp omsuc oveq1 sylan9eq oa1suc adantr eqtrd ex wral wlim
      om0 cuni iuneq2 uniiun eqtr4di vex omlim limuni imbitrrid tfinds ) DBEZFG
      ZVBHZDIFGZIHZDCEZFGZVGHZDVGUAZFGZVJHZDAFGZAHBCAVBIHZVCVEVBIVBIDFJVNKLVBVG
      HZVCVHVBVGVBVGDFJVOKLVBVJHZVCVKVBVJVBVJDFJVPKLVBAHZVCVMVBAVBADFJVQKLDMNZV
      FODULUBVGMNZVIVLVSVIPVKVGDQGZVJVSVIVKVHDQGZVTVRVSVKWAHODVGUCRVHVGDQUDUEVS
      VTVJHVIVGUFUGUHUIVICVBUJZVDVBUKZCVBVHSZVBUMZHWBWDCVBVGSWECVBVHVGUNCVBUOUP
      WCVCWDVBWEVBTNZWCVCWDHZBUQVRWFWCPWGOCDVBTURRRVBUSLUTVA $.
  $}

  $( Ordinal exponentiation with an exponent of 1.  Lemma 2.16 of [Schloeder]
     p. 6.  (Contributed by NM, 2-Jan-2005.) $)
  oe1 $p |- ( A e. On -> ( A ^o 1o ) = A ) $=
    ( con0 wcel c1o coe co comu csuc df-1o oveq2i com wceq peano1 onesuc eqtrid
    c0 mpan2 oe0 oveq1d om1r 3eqtrd ) ABCZADEFZAPEFZAGFZDAGFAUBUCAPHZEFZUEDUFAE
    IJUBPKCUGUELMAPNQOUBUDDAGARSATUA $.

  ${
    $d x y $.  $d x A $.
    $( Ordinal exponentiation with a base of 1.  Proposition 8.31(3) of
       [TakeutiZaring] p. 67.  Lemma 2.17 of [Schloeder] p. 6.  (Contributed by
       NM, 2-Jan-2005.) $)
    oe1m $p |- ( A e. On -> ( 1o ^o A ) = 1o ) $=
      ( vx vy c1o cv coe co wceq c0 oveq2 eqeq1d con0 wcel ax-mp comu mpan ciun
      1on cvv wa csuc oe0 oesuc oveq1 eqtrdi sylan9eq ex wral wlim iuneq2 0lt1o
      om1 vex oelim mpan2 wne 0ellim ne0i iunconst 3syl eqeq2d bitr4d imbitrrid
      tfinds ) DBEZFGZDHZDIFGZDHZDCEZFGZDHZDVJUAZFGZDHZDAFGZDHBCAVEIHVFVHDVEIDF
      JKVEVJHVFVKDVEVJDFJKVEVMHVFVNDVEVMDFJKVEAHVFVPDVEADFJKDLMZVIRDUBNVJLMZVLV
      OVRVLVNVKDOGZDVQVRVNVSHRDVJUCPVLVSDDOGZDVKDDOUDVQVTDHRDULNUEUFUGVLCVEUHVG
      VEUIZCVEVKQZCVEDQZHZCVEVKDUJWAVGWBDHWDWAVFWBDVESMZWAVFWBHZBUMVQWEWATZWFRV
      QWGTIDMWFUKCDVESUNUOPPKWAWCDWBWAIVEMVEIUPWCDHVEUQVEIURCVEDUSUTVAVBVCVD $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.
    $( Ordering property of ordinal addition.  Proposition 8.4 of
       [TakeutiZaring] p. 58.  (Contributed by NM, 5-Dec-2004.) $)
    oaordi $p |- ( ( B e. On /\ C e. On ) ->
                 ( A e. B -> ( C +o A ) e. ( C +o B ) ) ) $=
      ( vx vy con0 wcel coa co wi wa csuc wss oveq2 sseq2d imbi2d ancoms ex cvv
      wceq onelon adantll word eloni ordsucss ad2antlr onsucb ssid 2a1i sssucid
      syl cv sstr2 mpi oasuc imbitrrid ad2antrr wlim wral sucssel sylbir limsuc
      a2d ciun biimpd sylan9r imp ssiun2s adantr vex oalim adantlr sseqtrrd a1d
      mpanr1 tfindsg exp31 biimtrid com4r imp31 ovex ax-mp biimtrdi 3syld an32s
      sseq1d mpdan ) CFGZBFGZABGZCAHIZCBHIZGZJWHWIKZWJWMWNWJKAFGZWMWIWJWOWHBAUA
      UBWNWOWJWMWNWOKZWJWMWPWJALZBMZCWQHIZWLMZWMWIWJWRJZWHWOWIBUCXABUDABUEUKUFW
      HWIWOWRWTJWIWOWRWHWTWOWQFGZWIWRWHWTJZJAUGZWIXBWRXCWHWSCDULZHIZMZJZWHWSWSM
      ZJWHWSCEULZHIZMZJZWHWSCXJLZHIZMZJXCDEBWQXEWQTZXGXIWHXQXFWSWSXEWQCHNOPXEXJ
      TZXGXLWHXRXFXKWSXEXJCHNOPXEXNTZXGXPWHXSXFXOWSXEXNCHNOPXEBTZXGWTWHXTXFWLWS
      XEBCHNOPXIXBWHWSUHUIXJFGZXBKWQXJMZKWHXLXPYAWHXLXPJZJXBYBYAWHYCXLXPYAWHKZW
      SXKLZMZXLXKYEMYFXKUJWSXKYEUMUNYDXOYEWSWHYAXOYETCXJUOQOUPRUQVCXEURZXBKZWQX
      EMZKZXHYBXMJEXEUSYJWHXGYJWHKWSEXEXKVDZXFYJWSYKMZWHYJWQXEGZYLYHYIYMXBYIAXE
      GZYGYMXBWOYIYNJXDAXEFUTVAYGYNYMXEAVBVEVFVGEXEXKWQWSXJWQCHNVHUKVIYHWHXFYKT
      ZYIYGWHYOXBWHYGYOWHXESGYGYODVJECXESVKVOQVLVLVMRVNVPVQVRVSVTWHWOWTWMJWIWHW
      OKZWTWKLZWLMZWMYPWSYQWLCAUOWFWKSGYRWMJCAHWAWKWLSUTWBWCVLWDVGWEWGRQ $.
  $}

  $( Ordering property of ordinal addition.  Proposition 8.4 of [TakeutiZaring]
     p. 58 and its converse.  (Contributed by NM, 5-Dec-2004.) $)
  oaord $p |- ( ( A e. On /\ B e. On /\ C e. On ) ->
                 ( A e. B <-> ( C +o A ) e. ( C +o B ) ) ) $=
    ( con0 wcel coa co wi oaordi wceq wo wn word wa wb oacl syl anim12i ordtri2
    eloni w3a 3adant1 oveq2 a1i 3adant2 orim12d con3d df-3an ancom anandi sylbi
    3bitri 3adant3 3imtr4d impbid ) ADEZBDEZCDEZUAZABEZCAFGZCBFGZEZUQURUTVCHUPA
    BCIUBUSVAVBJZVBVAEZKZLZABJZBAEZKZLZVCUTUSVJVFUSVHVDVIVEVHVDHUSABCFUCUDUPURV
    IVEHUQBACIUEUFUGUSVAMZVBMZNZVCVGOUSURUPNZURUQNZNZVNUSUPUQNZURNURVRNVQUPUQUR
    UHVRURUIURUPUQUJULVOVLVPVMVOVADEVLCAPVATQVPVBDEVMCBPVBTQRUKVAVBSQUSAMZBMZNZ
    UTVKOUPUQWAURUPVSUQVTATBTRUMABSQUNUO $.

  $( Left cancellation law for ordinal addition.  Corollary 8.5 of
     [TakeutiZaring] p. 58.  (Contributed by NM, 5-Dec-2004.) $)
  oacan $p |- ( ( A e. On /\ B e. On /\ C e. On ) ->
                   ( ( A +o B ) = ( A +o C ) <-> B = C ) ) $=
    ( con0 wcel w3a wo wn coa co wceq wb oaord word eloni ordtri3 syl2an wa syl
    oacl 3comr 3com13 orbi12d notbid 3adant1 3impdi 3bitr4rd ) ADEZBDEZCDEZFZBC
    EZCBEZGZHZABIJZACIJZEZUQUPEZGZHZBCKZUPUQKZUKUNUTUKULURUMUSUIUJUHULURLBCAMUA
    UJUIUHUMUSLCBAMUBUCUDUIUJVBUOLZUHUIBNCNVDUJBOCOBCPQUEUHUIUJVCVALZUHUIRZUPNZ
    UQNZVEUHUJRZVFUPDEVGABTUPOSVIUQDEVHACTUQOSUPUQPQUFUG $.

  $( Weak ordering property of ordinal addition.  (Contributed by NM,
     6-Dec-2004.)  (Proof shortened by Andrew Salmon, 22-Oct-2011.) $)
  oaword $p |- ( ( A e. On /\ B e. On /\ C e. On ) ->
                 ( A C_ B <-> ( C +o A ) C_ ( C +o B ) ) ) $=
    ( con0 wcel w3a wn coa co wb oaord 3com12 notbid ontri1 3adant3 oacl ancoms
    wss 3adant2 3adant1 syl2anc 3bitr4d ) ADEZBDEZCDEZFZBAEZGZCBHIZCAHIZEZGZABR
    ZUJUIRZUFUGUKUDUCUEUGUKJBACKLMUCUDUMUHJUEABNOUFUJDEZUIDEZUNULJUCUEUOUDUEUCU
    OCAPQSUDUEUPUCUEUDUPCBPQTUJUINUAUB $.

  ${
    $d x y A $.  $d x y B $.  $d x y C $.
    $( Weak ordering property of ordinal addition.  Proposition 8.7 of
       [TakeutiZaring] p. 59.  (Contributed by NM, 7-Dec-2004.) $)
    oawordri $p |- ( ( A e. On /\ B e. On /\ C e. On ) ->
                 ( A C_ B -> ( A +o C ) C_ ( B +o C ) ) ) $=
      ( vx vy con0 wcel wss coa co wi cv c0 csuc wa wceq oveq2 sseq12d oa0 cvv
      adantr adantl biimpar wb word eloni syl ordsucsssuc syl2an anandirs oasuc
      oacl adantlr adantll bitr4d biimpd expcom wlim wral vex ciun ss2iun oalim
      adantrd imbitrrid mpanr1 tfinds3 exp4c 3imp231 ) CFGZAFGZBFGZABHZACIJZBCI
      JZHZKVJVKVLVMVPADLZIJZBVQIJZHZAMIJZBMIJZHZAELZIJZBWDIJZHZAWDNZIJZBWHIJZHZ
      VPVKVLOZVMODECVQMPVRWAVSWBVQMAIQVQMBIQRVQWDPVRWEVSWFVQWDAIQVQWDBIQRVQWHPV
      RWIVSWJVQWHAIQVQWHBIQRVQCPVRVNVSVOVQCAIQVQCBIQRWLWCVMWLWAAWBBVKWAAPVLASUA
      VLWBBPVKBSUBRUCWDFGZWLWGWKKZVMWLWMWNWLWMOZWGWKWOWGWENZWFNZHZWKVKVLWMWGWRU
      DZVKWMOZWEUEZWFUEZWSVLWMOZWTWEFGXAAWDULWEUFUGXCWFFGXBBWDULWFUFUGWEWFUHUIU
      JWOWIWPWJWQVKWMWIWPPVLAWDUKUMVLWMWJWQPVKBWDUKUNRUOUPUQVDVQURZWLWGEVQUSZVT
      KZVMWLXDXFWLVQTGZXDXFDUTXEVTWLXGXDOZOZEVQWEVAZEVQWFVAZHEVQWEWFVBXIVRXJVSX
      KVKXHVRXJPVLEAVQTVCUMVLXHVSXKPVKEBVQTVCUNRVEVFUQVDVGVHVI $.
  $}

  $( An ordinal is less than its sum with a nonzero ordinal.  Theorem 18 of
     [Suppes] p. 209 and its converse.  (Contributed by NM, 6-Dec-2004.) $)
  oaord1 $p |- ( ( A e. On /\ B e. On ) ->
               ( (/) e. B <-> A e. ( A +o B ) ) ) $=
    ( con0 wcel c0 coa co wb wa 0elon oaord mp3an1 wceq oa0 adantl eleq1d bitrd
    ancoms ) BCDZACDZEBDZAABFGZDZHSTIZUAAEFGZUBDZUCECDSTUAUFHJEBAKLUDUEAUBTUEAM
    SANOPQR $.

  $( An ordinal is less than or equal to its sum with another.  Part of
     Exercise 5 of [TakeutiZaring] p. 62.  Lemma 3.2 of [Schloeder] p. 7.  (For
     the other part see ~ oaord1 .)  (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 0ss 0elon oaword 3com13 mp3an3
    wb mpbii eqsstrrd ) ACDZBCDZEZAAFGHZABGHZUAUDAIUBAJKUCFBLZUDUELZBMUAUBFCDZU
    FUGRZNUHUBUAUIFBAOPQST $.

  $( An ordinal is less than or equal to its sum with another.  Theorem 21 of
     [Suppes] p. 209.  Lemma 3.3 of [Schloeder] p. 7.  (Contributed by NM,
     7-Dec-2004.) $)
  oaword2 $p |- ( ( A e. On /\ B e. On ) -> A C_ ( B +o A ) ) $=
    ( con0 wcel coa co wss wa c0 0ss wi oawordri mp3an1 wceq oa0r adantl sseq1d
    0elon sylibd mpi ancoms ) BCDZACDZABAEFZGZUBUCHZIBGZUEBJUFUGIAEFZUDGZUEICDU
    BUCUGUIKRIBALMUFUHAUDUCUHANUBAOPQSTUA $.

  ${
    $d x y z A $.  $d x y z B $.  $d x z S $.
    oawordeulem.1 $e |- A e. On $.
    oawordeulem.2 $e |- B e. On $.
    oawordeulem.3 $e |- S = { y e. On | B C_ ( A +o y ) } $.
    $( Lemma for ~ oawordex .  (Contributed by NM, 11-Dec-2004.) $)
    oawordeulem $p |- ( A C_ B -> E! x e. On ( A +o x ) = B ) $=
      ( vz wss cv coa co wceq con0 wa wcel c0 oveq2 ax-mp wrex wi wral wreu wne
      cint ssrab3 oaword2 mp2an sseq2d mpbir2an ne0ii oninton csuc cvv wlim w3o
      elrab2 onzsl mpbi oa0 eqtrdi biimprd oasuc mpan sylan9eqr vex sucid eleq2
      sseq1d mpbiri wn crab inteqi eleq2i onnminsb biimtrid oacl ontri1 sylancr
      oneli wb con2bid sylibrd mpcom word onordi ordsucss 3syl adantl rexlimiva
      eqsstrd a1d ciun oalim iunss onelssi mprgbir eqsstrdi 3jaoi rspcev nfrab1
      nfcv nfint nfov nfss onminsb oveq2i sseqtrri sylanblrc eqeq1d eqtr3 oacan
      syl eqss mp3an1 imbitrid rgen2 reu4 ) CDJZCAKZLMZDNZAOUAZYCCBKZLMZDNZPZYA
      YENZUBZBOUCAOUCYCAOUDXTEUFZOQZCYKLMZDNZYDEOJERUEYLDYFJZBOEHUGDEDEQDOQZDCD
      LMZJZGYPCOQZYRGFDCUHUIZYOYRBDOEYEDNYFYQDYEDCLSUJZHURUKULEUMUIZXTYMDJZDYMJ
      YNYKRNZYKIKZUNZNZIOUAZYKUOQYKUPPZUQZXTUUCUBZYLUUJUUBIYKUSUTUUDUUKUUHUUIUU
      DUUCXTUUDYMCDUUDYMCRLMZCYKRCLSYSUULCNFCVATVBVJVCUUHUUCXTUUGUUCIOUUEOQZUUG
      PYMCUUELMZUNZDUUGUUMYMCUUFLMZUUOYKUUFCLSYSUUMUUPUUONFCUUEVDVEVFUUGUUODJZU
      UMUUGUUEYKQZUUNDQZUUQUUGUURUUEUUFQUUEIVGVHYKUUFUUEVIVKUUMUURUUSYKUUEUUBWA
      UUMUURDUUNJZVLZUUSUURUUEYOBOVMZUFZQUUMUVAYKUVCUUEEUVBHVNZVOYOUUTBUUEYEUUE
      NYFUUNDYEUUECLSUJVPVQUUMUUTUUSUUMYPUUNOQZUUTUUSVLWBGYSUUMUVEFCUUEVRVEDUUN
      VSVTWCWDWEZDWFUUSUUQUBDGWGUUNDWHTWIWJWLWKWMUUIUUCXTUUIYMIYKUUNWNZDYSUUIYM
      UVGNFICYKUOWOVEUVGDJUUNDJZIYKIYKUUNDWPUURUUSUVHUVFDUUNGWQXNWRWSWMWTTDCUVC
      LMZYMYOBOUAZDUVIJZYPYRUVJGYTYOYRBDOUUAXAUIYOUVKBBDUVIBDXCBCUVCLBCXCBLXCBU
      VBYOBOXBXDXEXFYEUVCNYFUVIDYEUVCCLSUJXGTYKUVCCLUVDXHXIYMDXOXJYCYNAYKOYAYKN
      YBYMDYAYKCLSXKXAVTYJABOOYHYBYFNZYAOQZYEOQZPYIYBYFDXLYSUVMUVNUVLYIWBFCYAYE
      XMXPXQXRYCYGABOYIYBYFDYAYECLSXKXSXJ $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( Existence theorem for weak ordering of ordinal sum.  Proposition 8.8 of
       [TakeutiZaring] p. 59.  (Contributed by NM, 11-Dec-2004.) $)
    oawordeu $p |- ( ( ( A e. On /\ B e. On ) /\ A C_ B ) ->
                  E! x e. On ( A +o x ) = B ) $=
      ( vy con0 wcel wa wss cv coa co wceq wreu wi c0 cif reubidv imbi12d 0elon
      elimel sseq1 oveq1 eqeq1d sseq2 eqeq2 crab eqid oawordeulem dedth2h imp )
      BEFZCEFZGBCHZBAIZJKZCLZAEMZUKULUMUQNUKBOPZCHZURUNJKZCLZAEMZNURULCOPZHZUTV
      CLZAEMZNBCOOBURLZUMUSUQVBBURCUAVGUPVAAEVGUOUTCBURUNJUBUCQRCVCLZUSVDVBVFCV
      CURUDVHVAVEAECVCUTUEQRADURVCVCURDIJKHDEUFZBOESTCOESTVIUGUHUIUJ $.

    $( Existence theorem for weak ordering of ordinal sum.  (Contributed by NM,
       12-Dec-2004.) $)
    oawordexr $p |- ( ( A e. On /\ E. x e. On ( A +o x ) = B ) -> A C_ B ) $=
      ( con0 wcel cv coa co wceq wrex wss oaword1 sseq2 syl5ibcom rexlimdva imp
      wa ) BDEZBAFZGHZCIZADJBCKZRUAUBADRSDEQBTKUAUBBSLTCBMNOP $.

    $( Existence theorem for weak ordering of ordinal sum.  Proposition 8.8 of
       [TakeutiZaring] p. 59 and its converse.  See ~ oawordeu for uniqueness.
       (Contributed by NM, 12-Dec-2004.) $)
    oawordex $p |- ( ( A e. On /\ B e. On ) ->
                  ( A C_ B <-> E. x e. On ( A +o x ) = B ) ) $=
      ( con0 wcel wa wss cv co wceq wrex wreu oawordeu ex reurex syl6 oawordexr
      coa wi adantr impbid ) BDEZCDEZFZBCGZBAHRICJZADKZUDUEUFADLZUGUDUEUHABCMNU
      FADOPUBUGUESUCUBUGUEABCQNTUA $.

    $( Existence theorem for ordering of ordinal sum.  Similar to Proposition
       4.34(f) of [Mendelson] p. 266 and its converse.  (Contributed by NM,
       12-Dec-2004.) $)
    oaordex $p |- ( ( A e. On /\ B e. On ) ->
                  ( A e. B <-> E. x e. On ( (/) e. x /\ ( A +o x ) = B ) ) ) $=
      ( con0 wcel wa c0 cv coa co wceq wss onelss adantl oawordex sylibd oaord1
      wrex wi adantr eleq2 sylan9bb exp4c com12 imp4b simpr jca2 expd reximdvai
      biimprcd ex mpdd biimpd exp31 com34 imp4a rexlimdv impbid ) BDEZCDEZFZBCE
      ZGAHZEZBVCIJZCKZFZADRZVAVBVFADRZVHVAVBBCLZVIUTVBVJSUSCBMNABCOPUSVBVIVHSZS
      UTUSVBVKUSVBFZVFVGADVLVCDEZVFVGVLVMVFFVDVFUSVBVMVFVDVBUSVMVFVDSSVBUSVMVFV
      DUSVMFZVFFZVDVBVNVDBVEEVFVBBVCQVECBUAUBZUJUCUDUEVMVFUFUGUHUIUKTULUSVHVBSU
      TUSVGVBADUSVMVDVFVBUSVMVFVDVBUSVMVFVDVBSVOVDVBVPUMUNUOUPUQTUR $.
  $}

  $( An ordinal sum is zero iff both of its arguments are zero.  Lemma 3.10 of
     [Schloeder] p. 8.  (Contributed by NM, 6-Dec-2004.) $)
  oa00 $p |- ( ( A e. On /\ B e. On ) ->
             ( ( A +o B ) = (/) <-> ( A = (/) /\ B = (/) ) ) ) $=
    ( con0 wcel wa coa co c0 wceq wne on0eln0 adantr oaword1 sseld sylbird ne0i
    wb syl6 necon4d 0elon adantl oaord mp3an1 ancoms bitr3d biimtrdi oveq12 oa0
    jcad ax-mp eqtrdi impbid1 ) ACDZBCDZEZABFGZHIZAHIZBHIZEZUOUQURUSUOAHUPHUOAH
    JZHUPDZUPHJZUOVAHADZVBUMVDVAQUNAKLUOAUPHABMNOUPHPRSUOBHUPHUOBHJZAHFGZUPDZVC
    UOHBDZVEVGUNVHVEQUMBKUAUNUMVHVGQZHCDZUNUMVITHBAUBUCUDUEUPVFPUFSUIUTUPHHFGZH
    AHBHFUGVJVKHITHUHUJUKUL $.

  ${
    $d x y A $.  $d x y B $.  $d x y C $.
    $( The ordinal sum with a limit ordinal is a limit ordinal.  Proposition
       8.11 of [TakeutiZaring] p. 60.  Lemma 3.4 of [Schloeder] p. 7.
       (Contributed by NM, 8-Dec-2004.) $)
    oalimcl $p |- ( ( A e. On /\ ( B e. C /\ Lim B ) ) -> Lim ( A +o B ) ) $=
      ( vy vx con0 wcel wlim wa coa co c0 wceq csuc wn sylan2 wi mpd adantl wb
      word cv wrex wo limelon oacl eloni syl 0ellim n0i ad2antll simpr biimtrdi
      oa00 con3d vex sucid oalim eqeq1 imbitrid imp eleqtrid eliun sylib onelon
      sylan onnbtwn imnan sylibr com12 ad2antrl ordsucelsuc oasuc eleq2d bitr4d
      ciun eleq1 bicomd sylan9bbr adantr onsucb oaord 3expa ancoms biimpd exp32
      3syl com4l imp32 mtod exp44 rexlimdv expcom pm2.01d nrexdv ioran sylanbrc
      jca dflim3 ) AFGZBCGZBHZIZIZABJKZUAZXELMZXEDUBZNZMZDFUCZUDOZXEHXCWTBFGZXF
      BCUEZWTXMIZXEFGXFABUFXEUGUHPXDXGOZXKOXLXDBLMZOZXPXBXRWTXAXBLBGXRBUIBLUJUH
      UKXCWTXMXRXPQXNXOXGXQXOXGALMZXQIXQABUNXSXQULUMUOPRXDXJDFXDXJOZXHFGXDXJXJX
      DXTXJXDIZXHAEUBZJKZGZEBUCZXTYAXHEBYCVPZGYEYAXHXIYFXHDUPUQXJXDXIYFMZXDXEYF
      MXJYGEABCURXEXIYFUSUTVAVBEXHBYCVCVDXDYEXTQXJXDYDXTEBWTXCYBBGZYDXTQQWTXCYH
      YDXTWTXCYHIZYDIIXJBYBNZGZYIYKOZWTYDYIYBFGZYLXCXMYHYMXNBYBVEVFZYHYMYLQXCYM
      YHYLYMYHYKIOYHYLQYBBVGYHYKVHVIVJSRVKWTYIYDXJYKQXJWTYIYDYKXJWTYIYDYKQXJWTY
      IIZIZYDYKYPYDXEAYJJKZGZYKYOYDXIYQGZXJYRYIWTYMYDYSTYNWTYMIZYDXIYCNZGZYSYTY
      CFGYCUAYDUUBTAYBUFYCUGXHYCVLWGYTYQUUAXIAYBVMVNVOPXJYRYSXEXIYQVQVRVSYOYKYR
      TZXJYIWTUUCYIXMYJFGZIWTUUCYIXMUUDXCXMYHXNVTYIYMUUDYNYBWAVDWRXMUUDWTUUCBYJ
      AWBWCVFWDSVOWEWFWHWIWJWKVAWLSRWMWNVTWOXGXKWPWQDXEWSWQ $.
  $}

  ${
    $d x y z w A $.  $d x y z w B $.  $d x C $.
    $( Ordinal addition is associative.  Theorem 25 of [Suppes] p. 211.
       Theorem 4.2 of [Schloeder] p. 11.  (Contributed by NM, 10-Dec-2004.) $)
    oaass $p |- ( ( A e. On /\ B e. On /\ C e. On ) ->
                 ( ( A +o B ) +o C ) = ( A +o ( B +o C ) ) ) $=
      ( vy vz con0 wcel coa co wceq wa oveq2 oveq2d eqeq12d adantl cvv wss wrex
      c0 wi vx vw cv csuc oa0 syl eqtr4d suceq oasuc sylan sylan2 eqtrd anassrs
      oacl imbitrrid expcom wlim wral iuneq2 oalim mpanr1 ancoms adantr oalimcl
      ciun vex ovex limelon mpan onelon ex adantld 0ellim onelss sseq2d sylibrd
      rspcev syl2an expr adantrl adantrr wb oawordex ad2ant2l oaord 3expb eleq1
      imp sylan9bb an32s biimpar eqimss2 ad3antlr jca anasss reximdv2 sylbid wo
      word eloni ordtri2or syl2anr mpjaod exp45 imp32 simplrr exp32 com12 imp31
      ad4ant24 simpll ad2antlr oaword syl3anc rexbidva mpbid mpd exp4a ralrimiv
      mpdd iunss2 oaordi anim1d eleq2d rexlimdv eliun 3imtr4g ssrdv eqssd an12s
      syl6 expd 3eqtr4d exp31 tfinds3 3impia ) AFGZBFGZCFGZABHIZCHIZABCHIZHIZJZ
      YSYQYRKZUUDYTUAUCZHIZABUUFHIZHIZJZYTSHIZABSHIZHIZJYTDUCZHIZABUUNHIZHIZJZY
      TUUNUDZHIZABUUSHIZHIZJZUUDUUEUADCUUFSJZUUGUUKUUIUUMUUFSYTHLUVDUUHUULAHUUF
      SBHLMNUUFUUNJZUUGUUOUUIUUQUUFUUNYTHLUVEUUHUUPAHUUFUUNBHLMNUUFUUSJZUUGUUTU
      UIUVBUUFUUSYTHLUVFUUHUVAAHUUFUUSBHLMNUUFCJZUUGUUAUUIUUCUUFCYTHLUVGUUHUUBA
      HUUFCBHLMNUUEUUKYTUUMUUEYTFGZUUKYTJABUNZYTUEUFYRUUMYTJYQYRUULBAHBUEZMOUGU
      UEUUNFGZUURUVCTUURUVCUUEUVKKZUUOUDZUUQUDZJUUOUUQUHUVLUUTUVMUVBUVNUUEUVHUV
      KUUTUVMJUVIYTUUNUIUJYQYRUVKUVBUVNJYQYRUVKKZKUVBAUUPUDZHIZUVNUVOUVBUVQJYQU
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      UUFURZUUJUVTUUEKZUWAKDUUFUUOVEZDUUFUUQVEZUUGUUIUWAUWCUWDJUWBDUUFUUOUUQUSO
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      UPUULUWKUUNSBHLVOVQVRVSVTWAUXOUXQUXLTUVTUXOUXQUUPUWKJZDFRZUXLYRUXGUXQUYAW
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      KBXAXBWDOXCXDWHVLXEUXJUXKUWRDUUFUXJUYDKUXGUVRYQUXKUWRWBUXFUXBUXGUYDXFUXEU
      YDUVRUVTUXIUXAYRUYDUVRYRUXAUYDUVRTYRUXAUYDUVRUXAUYDKYRUVKUVRUUFUUNVJUVSUK
      XGXHXIXJUXIYQUXFUYDYQUWTUXGXKXLUWKUUPAXMXNXOXPXGXTXGXQXRXIXSEDUUHUUFUWLUU
      QYAUFVBUWIUWDUWMQZYQUVTUXAYRUYIUXDUXEUBUWDUWMUXEUBUCZUUQGZDUUFRUYJUWLGZEU
      UHRZUYJUWDGUYJUWMGUXEUYKUYMDUUFUXEUYDUYKUYMUXEUYDUYKKUYGUYKKUYMUXEUYDUYGU
      YKUUNUUFBYBYCUYLUYKEUUPUUHUWKUUPJUWLUUQUYJUWKUUPAHLYDVQYKYLYEDUYJUUFUUQYF
      EUYJUUHUWLYFYGYHUJOYIULYJVCYMYNYOXHYP $.
  $}

  ${
    $d x y z w A $.  $d x y z B $.
    $( Recursive definition of ordinal addition.  Exercise 25 of [Enderton]
       p. 240.  (Contributed by NM, 26-Dec-2004.)  (Revised by Mario Carneiro,
       30-May-2015.) $)
    oarec $p |- ( ( A e. On /\ B e. On ) ->
                ( A +o B ) = ( A u. ran ( x e. B |-> ( A +o x ) ) ) ) $=
      ( vw vy wcel coa co cv cmpt crn wceq c0 oveq2 mpteq1 eqtrdi rneqd wa wrex
      cun vz con0 csuc rn0 uneq2d eqeq12d oa0 un0 eqtr4di wi csn uneq1 unass wo
      mpt0 rexun df-suc rexeqi wb cvv elrnmpt elv velsn vex eqeq2d rexsn bitr4i
      eqid orbi12i 3bitr4i ovex elun eqriv uneq2i eqtr4i oasuc eqeq1d imbitrrid
      elrnmpti expcom wlim wral oalim mpanr1 ancoms adantr iuneq2 adantl 0ellim
      ciun iunun wne ne0i iunconst 3syl cuni df-rex bitri rexbii eluni2 r19.41v
      anbi1i exbii rexcom4 rexeqdv bitr4id eliun 3bitr4g eqrdv uneq12d ad2antrr
      wex limuni eqtrid 3eqtrd exp31 tfinds3 impcom ) CUBFBUBFZBCGHZBACBAIZGHZJ
      ZKZTZLZBUAIZGHZBAYGYBJZKZTZLZBMGHZBMTZLBDIZGHZBAYOYBJZKZTZLZBYOUCZGHZBAUU
      AYBJZKZTZLZYFXSUADCYGMLZYHYMYKYNYGMBGNUUGYJMBUUGYJMKMUUGYIMUUGYIAMYBJMAYG
      MYBOAYBUOPQUDPUEUFYGYOLZYHYPYKYSYGYOBGNUUHYJYRBUUHYIYQAYGYOYBOQUEUFYGUUAL
      ZYHUUBYKUUEYGUUABGNUUIYJUUDBUUIYIUUCAYGUUAYBOQUEUFYGCLZYHXTYKYEYGCBGNUUJY
      JYDBUUJYIYCAYGCYBOQUEUFXSYMBYNBUGBUHUIXSYOUBFZYTUUFUJYTUUFXSUUKRZYPYPUKZT
      ZUUELYTUUNYSUUMTZUUEYPYSUUMULUUOBYRUUMTZTUUEBYRUUMUMUUDUUPBEUUDUUPEIZYBLZ
      AUUASZUUQYRFZUUQUUMFZUNZUUQUUDFUUQUUPFUURAYOYOUKZTZSUURAYOSZUURAUVCSZUNUU
      SUVBUURAYOUVCUPUURAUUAUVDYOUQURUUTUVEUVAUVFUUTUVEUSEAYOYBUUQYQUTYQVHVAVBZ
      UVAUUQYPLZUVFEYPVCUURUVHAYODVDYAYOLYBYPUUQYAYOBGNVEVFVGVIVJAUUAYBUUQUUCUU
      CVHBYAGVKZVSUUQYRUUMVLVJVMVNVOPUULUUBUUNUUEUULUUBYPUCUUNBYOVPYPUQPVQVRVTY
      GWAZXSYTDYGWBZYLUVJXSRZUVKRYHDYGYPWJZDYGYSWJZYKUVLYHUVMLZUVKXSUVJUVOXSYGU
      TFUVJUVOUAVDDBYGUTWCWDWEWFUVKUVMUVNLUVLDYGYPYSWGWHUVJUVNYKLXSUVKUVJUVNDYG
      BWJZDYGYRWJZTYKDYGBYRWKUVJUVPBUVQYJUVJMYGFYGMWLUVPBLYGWIYGMWMDYGBWNWOUVJE
      UVQYJUVJUUTDYGSZUURAYGSZUUQUVQFUUQYJFUVJUVRUURAYGWPZSZUVSUVRYAYOFZUURRZAX
      LZDYGSZUWAUUTUWDDYGUUTUVEUWDUVGUURAYOWQWRWSYAUVTFZUURRZAXLUWCDYGSZAXLUWAU
      WEUWGUWHAUWGUWBDYGSZUURRUWHUWFUWIUURDYAYGWTXBUWBUURDYGXAVGXCUURAUVTWQUWCD
      AYGXDVJVGUVJUURAYGUVTYGXMXEXFDUUQYGYRXGAYGYBUUQYIYIVHUVIVSXHXIXJXNXKXOXPX
      QXR $.
  $}

  ${
    $d x y A $.
    $( Left addition by a constant is a bijection from ordinals to ordinals
       greater than the constant.  (Contributed by Mario Carneiro,
       30-May-2015.) $)
    oaf1o $p |- ( A e. On ->
        ( x e. On |-> ( A +o x ) ) : On -1-1-onto-> ( On \ A ) ) $=
      ( vy con0 wcel cv coa co cdif wral wceq wa wss wn ontri1 syldan ralrimiva
      wreu wb adantl cmpt wf1o oacl oaword1 mpbid eldifd eldifi eldifn oawordeu
      simpl mpbird syl21anc eqcom reubii sylib eqid f1ompt sylanbrc ) BDEZBAFZG
      HZDBIZEZADJCFZVAKZADRZCVBJDVBADVAUAZUBUSVCADUSUTDEZLZVADBBUTUCZVIBVAMZVAB
      ENZBUTUDUSVHVADEVKVLSVJBVAOPUEUFQUSVFCVBUSVDVBEZLZVAVDKZADRZVFVNUSVDDEZBV
      DMZVPUSVMUJVMVQUSVDDBUGTZVNVRVDBENZVMVTUSVDDBUHTUSVMVQVRVTSVSBVDOPUKABVDU
      IULVOVEADVAVDUMUNUOQACDVBVAVGVGUPUQUR $.
  $}

  ${
    $d x A $.  $d x B $.
    oacomf1olem.1 $e |- F = ( x e. A |-> ( B +o x ) ) $.
    $( Lemma for ~ oacomf1o .  (Contributed by Mario Carneiro, 30-May-2015.) $)
    oacomf1olem $p |- ( ( A e. On /\ B e. On ) ->
      ( F : A -1-1-onto-> ran F /\ ( ran F i^i B ) = (/) ) ) $=
      ( con0 wcel wa crn wf1o cin c0 wceq cdif wf1 cv cmpt wss syl wb coa oaf1o
      cres adantl f1of1 onss adantr f1ssres syl2anc resmptd eqtr4di f1eq1 mpbid
      co f1f1orn wf f1f frn 3syl difss2d reldisj mpbird jca ) BFGZCFGZHZBDIZDJZ
      VGCKLMZVFBFCNZDOZVHVFBVJAFCAPUAUNZQZBUCZOZVKVFFVJVMOZBFRZVOVFFVJVMJZVPVEV
      RVDACUBUDFVJVMUESVDVQVEBUFUGZFVJBVMUHUIVFVNDMVOVKTVFVNABVLQDVFAFBVLVSUJEU
      KBVJVNDULSUMZBVJDUOSVFVIVGVJRZVFVKBVJDUPWAVTBVJDUQBVJDURUSZVFVGFRVIWATVFV
      GFCWBUTVGCFVASVBVC $.
  $}

  ${
    $d x A $.  $d x B $.
    oacomf1o.1 $e |- F = ( ( x e. A |-> ( B +o x ) ) u.
      `' ( x e. B |-> ( A +o x ) ) ) $.
    $( Define a bijection from ` A +o B ` to ` B +o A ` .  Thus, the two are
       equinumerous even if they are not equal (which sometimes occurs, e.g.,
       ~ oancom ).  (Contributed by Mario Carneiro, 30-May-2015.) $)
    oacomf1o $p |- ( ( A e. On /\ B e. On ) ->
      F : ( A +o B ) -1-1-onto-> ( B +o A ) ) $=
      ( con0 wcel wa coa co wf1o cmpt crn cun cin wceq eqid oacomf1olem simpld
      c0 cv ccnv ancoms f1ocnv syl incom simprd eqtrid f1oun syl22anc wb f1oeq1
      ax-mp sylibr oarec f1oeq2d mpbird uncom eqtrdi f1oeq3d ) BFGZCFGZHZBCIJZC
      BIJZDKVDABCAUAZIJLZMZCNZDKZVCVJBACBVFIJLZMZNZVIDKZVCVMVIVGVKUBZNZKZVNVCBV
      HVGKZVLCVOKZBVLOZTPVHCOTPZVQVCVRWAABCVGVGQRZSVCCVLVKKZVSVCWCVLBOZTPZVBVAW
      CWEHACBVKVKQRUCZSCVLVKUDUEVCVTWDTBVLUFVCWCWEWFUGUHVCVRWAWBUGBVHVLCVGVOUIU
      JDVPPVNVQUKEVMVIDVPULUMUNVCVDVMVIDABCUOUPUQVCVEVIVDDVCVECVHNZVIVBVAVEWGPA
      CBUOUCCVHURUSUTUQ $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.
    $( Ordering property of ordinal multiplication.  Half of Proposition 8.19
       of [TakeutiZaring] p. 63.  Lemma 3.15 of [Schloeder] p. 9.  (Contributed
       by NM, 14-Dec-2004.) $)
    omordi $p |- ( ( ( B e. On /\ C e. On ) /\ (/) e. C ) ->
                 ( A e. B -> ( C .o A ) e. ( C .o B ) ) ) $=
      ( vx vy con0 wcel c0 comu co wi wa wceq eleq2 oveq2 eleq2d imbi12d biimpa
      sylan adantr onelon ex cv csuc noel pm2.21i a1i coa elsuci omcl simpl jca
      wo oaword1 sseld imim2d imp adantrl oaord1 eleq1d syl5ibrcom adantrr jaod
      syl5 wb omsuc sylibrd exp43 com12 adantld impd wlim wral id ad2ant2r ciun
      wss limsuc ssiun2s syl adantll cvv omlim mpanr1 sseqtrrd anabss1 eleqtrrd
      sseldd exp53 com13 imp4c a1dd tfinds3 com23 exp4a mpdd com34 com24 imp31
      vex ) BFGZCFGZHCGZABGZCAIJZCBIJZGZKZXAXDXCXBXGXAXDXBXCXGXAXDAFGZXBXCXGKZK
      XAXDXIBAUAUBXAXDXIXBXJXAXDXIXBLZXCXGXAXKXCLZXDXGADUCZGZXECXMIJZGZKZAHGZXE
      CHIJZGZKZAEUCZGZXECYBIJZGZKZAYBUDZGZXECYGIJZGZKZXHXLDEBXMHMZXNXRXPXTXMHAN
      YLXOXSXEXMHCIOPQXMYBMZXNYCXPYEXMYBANYMXOYDXEXMYBCIOPQXMYGMZXNYHXPYJXMYGAN
      YNXOYIXEXMYGCIOPQXMBMZXNXDXPXGXMBANYOXOXFXEXMBCIOPQYAXLXRXTAUEUFUGYBFGZXK
      XCYFYKKZYPXBXCYQKZXIXBYPYRXBYPXCYFYKXBYPLZXCYFLZLZYHXEYDCUHJZGZYJYHYCAYBM
      ZUMZUUAUUCAYBUIYSYDFGZXBLZYTUUEUUCKYSUUFXBCYBUJXBYPUKULUUGYTLYCUUCUUDUUGY
      FYCUUCKZXCUUGYFUUHUUGYEUUCYCUUGYDUUBXEYDCUNUOUPUQURUUGXCUUDUUCKYFUUGXCLUU
      CUUDYDUUBGZUUGXCUUIYDCUSRUUDXEYDUUBAYBCIOUTVAVBVCSVDYSYJUUCVEYTYSYIUUBXEC
      YBVFPTVGVHVIVJVKXMVLZXLXQYFEXMVMUUJXIXBXCXQXBXIUUJXCXQKXBXIUUJXCXNXPXBXIL
      ZUUJXCLLZXNLCAUDZIJZXOXEUULXBUUJLZXNUUNXOVQXBUUJUUOXIXCUUOVNVOUUOXNLUUNEX
      MYDVPZXOUUJXNUUNUUPVQZXBUUJXNLUUMXMGZUUQUUJXNUURXMAVRREXMYDUUMUUNYBUUMCIO
      VSVTWAUUOXOUUPMZXNXBXMWBGUUJUUSDWTECXMWBWCWDTWESUULXEUUNGZXNUUKXCUUTUUJUU
      KXCLXEXECUHJZUUNUUKXCXEUVAGZXBXIXCUVBVEZUUKXEFGXBUVCCAUJXECUSSWFRUUKUUNUV
      AMXCCAVFTWGURTWHWIWJWKWLWMWNWOWOWPWQWRWS $.
  $}

  $( Ordering property of ordinal multiplication.  (Contributed by NM,
     25-Dec-2004.) $)
  omord2 $p |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ (/) e. C ) ->
               ( A e. B <-> ( C .o A ) e. ( C .o B ) ) ) $=
    ( con0 wcel wa comu co wi omordi wceq wo wn omcl word ordtri2 syl2an adantr
    wb eloni w3a 3adantl1 oveq2 a1i 3adantl2 orim12d con3d anandis ancoms 3impa
    c0 3adant3 3imtr4d impbid ) ADEZBDEZCDEZUAZUKCEZFZABEZCAGHZCBGHZEZUPUQUSVAV
    DIUOABCJUBUTVBVCKZVCVBEZLZMZABKZBAEZLZMZVDVAUTVKVGUTVIVEVJVFVIVEIUTABCGUCUD
    UOUQUSVJVFIUPBACJUEUFUGURVDVHSZUSUOUPUQVMUQUOUPFVMUQUOUPVMUQUOFVBDEZVCDEZVM
    UQUPFCANCBNVNVBOVCOVMVOVBTVCTVBVCPQQUHUIUJRURVAVLSZUSUOUPVPUQUOAOBOVPUPATBT
    ABPQULRUMUN $.

  $( Ordering property of ordinal multiplication.  Proposition 8.19 of
     [TakeutiZaring] p. 63.  Theorem 3.16 of [Schloeder] p. 9.  (Contributed by
     NM, 14-Dec-2004.) $)
  omord $p |- ( ( A e. On /\ B e. On /\ C e. On ) ->
               ( ( A e. B /\ (/) e. C ) <-> ( C .o A ) e. ( C .o B ) ) ) $=
    ( con0 wcel w3a c0 wa comu co wb omord2 ex pm5.32rd simpl wi ne0i wceq om0r
    wne eqeq1d syl5ibrcom necon3d adantr on0eln0 adantl sylibrd 3adant1 impbid2
    oveq1 syl5 ancld bitrd ) ADEZBDEZCDEZFZABEZGCEZHCAIJZCBIJZEZUSHZVBUQUSURVBU
    QUSURVBKABCLMNUQVCVBVBUSOUQVBUSUOUPVBUSPUNUOUPHVBCGTZUSUOVBVDPUPVBVAGTUOVDV
    AUTQUOCGVAGUOVAGRCGRZGBIJZGRBSVEVAVFGCGBIUJUAUBUCUKUDUPUSVDKUOCUEUFUGUHULUI
    UM $.

  $( Left cancellation law for ordinal multiplication.  Proposition 8.20 of
     [TakeutiZaring] p. 63 and its converse.  (Contributed by NM,
     14-Dec-2004.) $)
  omcan $p |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ (/) e. A ) ->
                   ( ( A .o B ) = ( A .o C ) <-> B = C ) ) $=
    ( con0 wcel wa comu co wceq wo wn wi omordi ex ancoms imp word omcl eloni
    wb w3a c0 3adant2 3adant3 orim12d syl ordtri3 syl2an 3impdi 3adant1 3imtr4d
    con3d adantr oveq2 impbid1 ) ADEZBDEZCDEZUAZUBAEZFZABGHZACGHZIZBCIZVAVBVCEZ
    VCVBEZJZKZBCEZCBEZJZKZVDVEVAVLVHVAVJVFVKVGUSUTVJVFLZUPURUTVNLZUQURUPVOURUPF
    UTVNBCAMNOUCPUSUTVKVGLZUPUQUTVPLZURUQUPVQUQUPFUTVPCBAMNOUDPUEULUSVDVITZUTUP
    UQURVRUPUQFZVBQZVCQZVRUPURFZVSVBDEVTABRVBSUFWBVCDEWAACRVCSUFVBVCUGUHUIUMUSV
    EVMTZUTUQURWCUPUQBQCQWCURBSCSBCUGUHUJUMUKBCAGUNUO $.

  $( Weak ordering property of ordinal multiplication.  (Contributed by NM,
     21-Dec-2004.) $)
  omword $p |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ (/) e. C ) ->
                   ( A C_ B <-> ( C .o A ) C_ ( C .o B ) ) ) $=
    ( con0 wcel w3a c0 wa wceq wo co wss omord2 wb 3anrot omcan onsseleq adantr
    comu omcl sylanbr bicomd orbi12d 3adant3 anim12dan ancoms 3impa syl 3bitr4d
    ) ADEZBDEZCDEZFZGCEZHZABEZABIZJZCASKZCBSKZEZUSUTIZJZABLZUSUTLZUOUPVAUQVBABC
    MUOVBUQUMULUJUKFUNVBUQNULUJUKOCABPUAUBUCUMVDURNZUNUJUKVFULABQUDRUMVEVCNZUNU
    MUSDEZUTDEZHZVGUJUKULVJULUJUKHVJULUJVHUKVICATCBTUEUFUGUSUTQUHRUI $.

  $( Weak ordering property of ordinal multiplication.  (Contributed by NM,
     21-Dec-2004.) $)
  omwordi $p |- ( ( A e. On /\ B e. On /\ C e. On ) ->
                   ( A C_ B -> ( C .o A ) C_ ( C .o B ) ) ) $=
    ( con0 wcel w3a c0 wss comu co wi wa omword biimpd ex wn wceq sseq12d oveq1
    om0r wb word ord0eln0 necon2bbid syl 3ad2ant3 ssid adantr adantl syl5ibrcom
    eloni mpbiri 3adant3 sylbird a1dd pm2.61d ) ADEZBDEZCDEZFZGCEZABHZCAIJZCBIJ
    ZHZKZUTVAVFUTVALVBVEABCMNOUTVAPZVEVBUTVGCGQZVEUSUQVHVGUAZURUSCUBZVICUKVJVAC
    GCUCUDUEUFUQURVHVEKUSUQURLZVEVHGAIJZGBIJZHZVKVNGGHGUGVKVLGVMGUQVLGQURATUHUR
    VMGQUQBTUIRULVHVCVLVDVMCGAISCGBISRUJUMUNUOUP $.

  ${
    $d x y A $.  $d x y B $.  $d x y C $.
    $( Weak ordering property of ordinal multiplication.  Proposition 8.21 of
       [TakeutiZaring] p. 63.  (Contributed by NM, 20-Dec-2004.) $)
    omwordri $p |- ( ( A e. On /\ B e. On /\ C e. On ) ->
        ( A C_ B -> ( A .o C ) C_ ( B .o C ) ) ) $=
      ( vx vy con0 wcel wss comu co wi wa cv c0 wceq oveq2 sseq12d coa omcl cvv
      csuc om0 0ss eqsstrdi ad2antrr w3a 3adant2 3adant1 simp1 oawordri syl3anc
      imp adantrl wb oaword syld3an3 biimpa adantrr sstrd adantr 3sstr4d exp520
      omsuc com3r imp4c wlim wral vex ss2iun omlim ad2ant2rl imbitrrid anandirs
      ciun adantl mpanr1 expcom adantrd tfinds3 expd 3impib 3coml ) CFGZAFGZBFG
      ZABHZACIJZBCIJZHZKZWCWDWEWJWCWDWELZWFWIADMZIJZBWLIJZHZANIJZBNIJZHZAEMZIJZ
      BWSIJZHZAWSUAZIJZBXCIJZHZWIWKWFLDECWLNOWMWPWNWQWLNAIPWLNBIPQWLWSOWMWTWNXA
      WLWSAIPWLWSBIPQWLXCOWMXDWNXEWLXCAIPWLXCBIPQWLCOWMWGWNWHWLCAIPWLCBIPQWDWRW
      EWFWDWPNWQAUBWQUCUDUEWSFGZWDWEWFXBXFKZWDWEXGWFXHKWDWEXGWFXBXFWDWEXGUFZWFX
      BLZLZWTARJZXABRJZXDXEXKXLXAARJZXMXIXBXLXNHZWFXIXBXOXIWTFGZXAFGZWDXBXOKWDX
      GXPWEAWSSUGWEXGXQWDBWSSUHZWDWEXGUIWTXAAUJUKULUMXIWFXNXMHZXBXIWFXSWDWEXGXQ
      WFXSUNXRABXAUOUPUQURUSXIXDXLOZXJWDXGXTWEAWSVCUGUTXIXEXMOZXJWEXGYAWDBWSVCU
      HUTVAVBVDVEWLVFZWKXBEWLVGZWOKZWFWKYBYDWKWLTGZYBYDDVHWDWEYEYBLZYDYCWOWDYFL
      ZWEYFLZLZEWLWTVNZEWLXAVNZHEWLWTXAVIYIWMYJWNYKWDYFWMYJOYFWEEAWLTVJVKYHWNYK
      OYGEBWLTVJVOQVLVMVPVQVRVSVTWAWB $.
  $}

  $( An ordinal is less than or equal to its product with another.  Lemma 3.11
     of [Schloeder] p. 8.  (Contributed by NM, 21-Dec-2004.) $)
  omword1 $p |- ( ( ( A e. On /\ B e. On ) /\ (/) e. B ) ->
                 A C_ ( A .o B ) ) $=
    ( con0 wcel wa c0 comu co wss c1o wb word eloni ordgt0ge1 syl adantl wi 1on
    omwordi mp3an1 ancoms wceq om1 adantr sseq1d sylibd sylbid imp ) ACDZBCDZEZ
    FBDZAABGHZIZUKULJBIZUNUJULUOKZUIUJBLUPBMBNOPUKUOAJGHZUMIZUNUJUIUOURQZJCDUJU
    IUSRJBASTUAUKUQAUMUIUQAUBUJAUCUDUEUFUGUH $.

  $( An ordinal is less than or equal to its product with another.  Lemma 3.12
     of [Schloeder] p. 9.  (Contributed by NM, 21-Dec-2004.) $)
  omword2 $p |- ( ( ( A e. On /\ B e. On ) /\ (/) e. B ) ->
                 A C_ ( B .o A ) ) $=
    ( con0 wcel wa c0 c1o comu co wceq om1r ad2antrr wss eloni ordgt0ge1 biimpa
    word sylan adantll wi 1on omwordri mp3an1 ancoms adantr mpd eqsstrrd ) ACDZ
    BCDZEZFBDZEZAGAHIZBAHIZUHUMAJUIUKAKLULGBMZUMUNMZUIUKUOUHUIBQZUKUOBNUQUKUOBO
    PRSUJUOUPTZUKUIUHURGCDUIUHURUAGBAUBUCUDUEUFUG $.

  $( The product of two ordinal numbers is zero iff at least one of them is
     zero.  Proposition 8.22 of [TakeutiZaring] p. 64.  (Contributed by NM,
     21-Dec-2004.) $)
  om00 $p |- ( ( A e. On /\ B e. On ) -> ( ( A .o B ) = (/) <->
              ( A = (/) \/ B = (/) ) ) ) $=
    ( con0 wcel wa comu co c0 wceq c1o wss wne wi word wb eloni ordge1n0 adantr
    adantl ex wo neanior biimprd on0eln0 omword1 sylbird anim12d sstr biimtrrid
    wn syl syl6 omcl 3syl sylibd necon4bd oveq1 om0r sylan9eqr oveq2 om0 impbid
    jaod ) ACDZBCDZEZABFGZHIZAHIZBHIZUAZVFVKVGHVFVKUJZJVGKZVGHLZVLAHLZBHLZEZVFV
    MAHBHUBVFVQJAKZAVGKZEVMVFVOVRVPVSVDVOVRMVEVDVRVOVDANVRVOOAPAQUKUCRVFVPHBDZV
    SVEVTVPOVDBUDSVFVTVSABUETUFUGJAVGUHULUIVFVGCDVGNVMVNOABUMVGPVGQUNUOUPVFVIVH
    VJVEVIVHMVDVEVIVHVIVEVGHBFGHAHBFUQBURUSTSVDVJVHMVEVDVJVHVJVDVGAHFGHBHAFUTAV
    AUSTRVCVB $.

  $( The product of two nonzero ordinal numbers is nonzero.  (Contributed by
     NM, 28-Dec-2004.) $)
  om00el $p |- ( ( A e. On /\ B e. On ) -> ( (/) e. ( A .o B ) <->
              ( (/) e. A /\ (/) e. B ) ) ) $=
    ( con0 wcel wa comu co c0 wne wceq wo om00 necon3abid omcl on0eln0 bi2anan9
    wn wb syl neanior bitrdi 3bitr4d ) ACDZBCDZEZABFGZHIZAHJBHJKZQZHUFDZHADZHBD
    ZEZUEUHUFHABLMUEUFCDUJUGRABNUFOSUEUMAHIZBHIZEUIUCUKUNUDULUOAOBOPAHBHTUAUB
    $.

  ${
    $d x y z w v A $.  $d x y z w v B $.  $d x y C $.
    $( Ordering involving the product of a limit ordinal.  Proposition 8.23 of
       [TakeutiZaring] p. 64.  (Contributed by NM, 25-Dec-2004.) $)
    omordlim $p |- ( ( ( A e. On /\ ( B e. D /\ Lim B ) ) /\ C e. ( A .o B ) )
                -> E. x e. B C e. ( A .o x ) ) $=
      ( con0 wcel wlim wa comu co cv wrex ciun omlim eleq2d eliun bitrdi biimpa
      ) BFGCEGCHIIZDBCJKZGZDBALJKZGACMZTUBDACUCNZGUDTUAUEDABCEOPADCUCQRS $.

    $( The product of any nonzero ordinal with a limit ordinal is a limit
       ordinal.  Proposition 8.24 of [TakeutiZaring] p. 64.  (Contributed by
       NM, 25-Dec-2004.) $)
    omlimcl $p |- ( ( ( A e. On /\ ( B e. C /\ Lim B ) ) /\ (/) e. A ) ->
                  Lim ( A .o B ) ) $=
      ( vy vx con0 wcel wa c0 comu co word wceq wn syl adantr wb sylan wi c1o
      wlim cv csuc wrex wo limelon omcl sylan2 0ellim n0i anim12ci adantll om00
      eloni notbid ioran bitrdi mpbird ciun vex sucid omlim eqeq1 biimpac eliun
      eleqtrid sylib adantlr onelon onnbtwn imnan sylibr com12 adantl mpd simpl
      jca anim2i anassrs coa ordsucelsuc oa1suc eleq2d bitr4d wss ordgt0ge1 1on
      ad5ant24 oaword mp3an1 syldan bitrd biimpa omsuc sseld eleq1 biimprd syl9
      sseqtrrd sylbid com23 adantlrl omord biimtrrdi syl3an2b 3comr 3expb syl6d
      onsucb w3a an32s imp mtod rexlimdva2 pm2.01da nrexdv sylanbrc dflim3 ) AF
      GZBCGZBUAZHZHZIAGZHZABJKZLZYFIMZYFDUBZUCZMZDFUDZUENZYFUAYCYGYDYBXSBFGZYGB
      CUFZXSYNHZYFFGYGABUGYFUNOUHPYEYHNZYLNYMYEYQAIMZNZBIMZNZHZYBYDUUBXSYAYDUUB
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      CUUFXIYNUWDXSXJUVGUUQYDHUUQBUUPAXCUUQYDVPXDXEXFXGPXHRXKXLXMXNPVOXOPXPYHYL
      UPXQDYFXRXQ $.

    $( Distributive law for ordinal arithmetic (left-distributivity).
       Proposition 8.25 of [TakeutiZaring] p. 64.  Theorem 4.3 of [Schloeder]
       p. 12.  (Contributed by NM, 26-Dec-2004.) $)
    odi $p |- ( ( A e. On /\ B e. On /\ C e. On ) ->
                 ( A .o ( B +o C ) ) = ( ( A .o B ) +o ( A .o C ) ) ) $=
      ( vz vw vv con0 wcel coa co comu wceq c0 wa oveq2 oveq2d syl wi cvv wss
      vx vy cv csuc eqeq12d omcl oa0 om0 adantr adantl 3eqtr4rd w3a oveq1 oasuc
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      3exp pm2.43a com4r pm2.43i 3imp eqtr4d imbitrrid com3r impd wlim wral vex
      limelon mpan om0r oveqan12d 0elon ax-mp eqtr2di ancoms expd imp a1dd ciun
      oveq12d wo wn wb simplr onelon sylan ontri1 oawordex bitr3d syl2anc oaord
      wrex 3expb eleq1 sylan9bb iba an32s biimpcd reximdvai sylbid orrd sylanl1
      exp4c adantlrl adantlr 0ellim om00el biimprd sylan2i a1d adantlrr ancom1s
      imp31 omordi onelss sseq2d sylibrd syld adantll jcad rspcev mpd ralrimiva
      syl6 adantrr iunss2 mpanr1 oaordi an12s ovex adantrd rspccv eqeq1 eqimss2
      imbitrid imim2i ad2antll rexlimdvw jaod omordlim ex limord ordelon eleq2d
      word rspccva rexlimdva adantrl eqssd oalimcl anim2i ad2antlr jctl anim1ci
      omlim omlimcl jctil oalim 3eqtr4d exp43 com3l oe0lem tfinds3 expdcom
      com12 ) AGHZBGHZCGHZABCIJZKJZABKJZACKJZIJZLZUVRUVPUVQUWDABUAUCZIJZKJZUWAA
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      YFUYJVXGUVPUYJVUBVUCUVCUVDAUWESUVFWPYAAUWEKYTUVGEUWAUWHSUVHWTYOUVIUVJUVKW
      GUVLUVOUVMUVNVJ $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x C $.
    $( Multiplication of ordinal numbers is associative.  Theorem 8.26 of
       [TakeutiZaring] p. 65.  Theorem 4.4 of [Schloeder] p. 13.  (Contributed
       by NM, 28-Dec-2004.) $)
    omass $p |- ( ( A e. On /\ B e. On /\ C e. On ) ->
                 ( ( A .o B ) .o C ) = ( A .o ( B .o C ) ) ) $=
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      wi cvv wss vx csuc omcl om0 syl eqtr4d w3a coa oveq1 omsuc stoic3 3adant1
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      ex com23 imp reximdvai exp31 imp4c mpcom ralrimiva adantr omlimcl mpanlr1
      eqssd ovex wn word eloni ord0eln0 necon2bbid oveq1d om0r anassrs sylan9eq
      wb adantll sylbird a1dd pm2.61d com3l tfinds3 ) AFGZBFGZCFGZABHIZCHIZABCH
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      UIUYDPNUVBXTVKUFWSQYBYCYDYEXCYFVBYGUOYFUR $.
  $}

  $( If an ordinal number is even, its successor is odd.  (Contributed by NM,
     26-Jan-2006.) $)
  oneo $p |- ( ( A e. On /\ B e. On /\ C = ( 2o .o A ) ) ->
              -. suc C = ( 2o .o B ) ) $=
    ( con0 wcel c2o comu co wceq csuc wa wb sucid mpbii 2on mp3an3 c1o coa mpan
    omord w3a wn onnbtwn 3ad2ant1 suceq eqeq1d 3ad2ant3 wi ovex eleq2 biimtrrdi
    c0 simpl syl5 simpr omcl oa1suc syl df-2o eleqtrri 1on oaord mp3an12i omsuc
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    jcad 3adant3 sylbid mtod ) ADEZBDEZCFAGHZIZUAZCJZFBGHZIZABEZBAJZEZKZVTWAWKU
    BWCABUCUDWDWGWBJZWFIZWKWCVTWGWMLWAWCWEWLWFCWBUEUFUGVTWAWMWKUHWCVTWAKZWMWHWJ
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    IWAXJWRXIOBWIFTPVJVKVLVMVNVOVPVQVRVS $.

  ${
    $d A w x z $.  $d A x y z $.  $d B w x z $.  $d B x y z $.
    $( Lemma for ~ omeu : existence part.  (Contributed by Mario Carneiro,
       28-Feb-2013.) $)
    omeulem1 $p |- ( ( A e. On /\ B e. On /\ A =/= (/) ) ->
      E. x e. On E. y e. A ( ( A .o x ) +o y ) = B ) $=
      ( vz vw con0 wcel c0 w3a cv csuc comu co wrex wceq eleq2d syl2anc wa wi
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      mpbird oawordex mpbid 3adantr1 simp3r simp21 simp11 omsuc eleqtrd eqeltrd
      simp23 simp3l oaord jca reximdv2 mpd expcom com13 ) CGHZDGHZCIUAZJZDCAKZL
      ZMNZHZAGOZCUUFMNZBKZUBNZDPZBCOZAGOZUUEUUCDCDLZMNZHZUUJUUBUUCUUDUCZUUEUUQU
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      UUNSZUULCHZUUNSUUEUVNUXSJZUXTUUNUYAUXTUUMUUKCUBNZHZUYAUUMDUYBUUEUVNUXRUUN
      YHZUYADUUHUYBUUEUUIUVJUVMUXSYIUYAUUBUVMUUHUYBPUUBUUCUUDUVNUXSYJZUUEUUIUVJ
      UVMUXSYNZCUUFYKRYLYMUYAUXRUUBUXNUXTUYCXRUUEUVNUXRUUNYOUYEUYAUUBUVMUXNUYEU
      YFUXORUULCUUKYPXFYDUYDYQWFYRYSYTWFUUAXHUOYS $.
  $}

  $( Lemma for ~ omeu : uniqueness part.  (Contributed by Mario Carneiro,
     28-Feb-2013.)  (Revised by Mario Carneiro, 29-May-2015.) $)
  omeulem2 $p |- ( ( ( A e. On /\ A =/= (/) ) /\
      ( B e. On /\ C e. A ) /\ ( D e. On /\ E e. A ) ) ->
    ( ( B e. D \/ ( B = D /\ C e. E ) ) ->
      ( ( A .o B ) +o C ) e. ( ( A .o D ) +o E ) ) ) $=
    ( con0 wcel c0 wa w3a comu co coa wceq wss wi wb syl syl31anc syl2anc eloni
    wne csuc word simp3l ordsucss 3syl simp2l onsuc simp1l simp1r mpbird omword
    on0eln0 sylibd omcl simp3r onelon oaword1 sstr expcom simp2r oaord eleqtrrd
    syld biimpa omsuc syl6ci simpr imbitrid oveq1d adantr eleq2d mpbidi syl3anc
    ssel oveq2 jaod ) AFGZAHUBZIZBFGZCAGZIZDFGZEAGZIZJZBDGZABKLZCMLZADKLZEMLZGZ
    BDNZCEGZIZWHWIABUCZKLZWMOZWKWSGWNWHWIWSWLOZWTWHWIWRDOZXAWHWEDUDWIXBPWAWDWEW
    FUEZDUABDUFUGWHWRFGZWEVSHAGZXBXAQWHWBXDWAWBWCWGUHZBUIRXCVSVTWDWGUJZWHXEVTVS
    VTWDWGUKWHVSXEVTQXGAUNRULWRDAUMSUOWHWLFGZEFGZXAWTPZWHVSWEXHXGXCADUPTWHVSWFX
    IXGWAWDWEWFUQAEURTZXHXIIWLWMOZXJWLEUSXAXLWTWSWLWMUTVARTVEWHWKWJAMLZWSWHCFGZ
    VSWJFGZWCWKXMGZWHVSWCXNXGWAWBWCWGVBZACURTZXGWHVSWBXOXGXFABUPTZXQXNVSXOJWCXP
    CAWJVCVFSWHVSWBWSXMNXGXFABVGTVDWSWMWKVPVHWHXNXIXOWQWNPXRXKXSWQWKWJEMLZGZWNX
    NXIXOJZWQWPYBYAWOWPVICEWJVCVJWQXTWMWKWOXTWMNWPWOWJWLEMBDAKVQVKVLVMVNVOVR $.

  $( An ordered pair-like theorem for ordinal multiplication.  (Contributed by
     Mario Carneiro, 29-May-2015.) $)
  omopth2 $p |- ( ( ( A e. On /\ A =/= (/) ) /\
      ( B e. On /\ C e. A ) /\ ( D e. On /\ E e. A ) ) ->
    ( ( ( A .o B ) +o C ) = ( ( A .o D ) +o E ) <-> ( B = D /\ C = E ) ) ) $=
    ( con0 wcel wa comu coa wceq w3o word eloni syl syl2anc syl5 mtod pm2.21d
    co wne w3a simpl2l simpl3l ordtri3or simpr simpl1l omcl simpl3r onelon oacl
    c0 wn ordirr 3syl eqneltrd wo orc omeulem2 adantr neleqtrrd simpl1r simpl2r
    wi idd syl222anc 3jaod mpd olc mpand eqcomd jca ex oveq2 oveqan12d impbid1
    id ) AFGZAULUAZHZBFGZCAGZHZDFGZEAGZHZUBZABITZCJTZADITZEJTZKZBDKZCEKZHZWGWLW
    OWGWLHZWMWNWPBDGZWMDBGZLZWMWPBMZDMZWSWPWAWTWAWBVTWFWLUCZBNOWPWDXAWDWEVTWCWL
    UDZDNOBDUEPWPWQWMWMWRWPWQWMWPWQWIWKGZWPWIWKWKWGWLUFZWPWKFGZWKMWKWKGUMWPWJFG
    ZEFGZXFWPVRWDXGVRVSWCWFWLUGZXCADUHPWPVRWEXHXIWDWEVTWCWLUIZAEUJZPWJEUKPWKNWK
    UNUOZUPZWQWQWMCEGZHZUQZWPXDWQXOURWGXPXDVDWLABCDEUSUTZQRSWPWMVEWPWRWMWPWRWKW
    IGZWPWIWKWKXLXEVAZWRWRDBKZECGZHZUQZWPXRWRYBURWPVRVSWDWEWAWBYCXRVDXIVRVSWCWF
    WLVBXCXJXBWAWBVTWFWLVCZADEBCUSVFZQRSVGVHZWPXNWNYALZWNWPCMZEMZYGWPVRWBYHXIYD
    VRWBHCFGYHACUJCNOPWPVRWEYIXIXJVRWEHXHYIXKENOPCEUEPWPXNWNWNYAWPXNWNWPXNXDXMW
    PWMXNXDYFXOXPWPXDXOWQVIXQQVJRSWPWNVEWPYAWNWPYAXRXSWPXTYAXRWPBDYFVKYBYCWPXRY
    BWRVIYEQVJRSVGVHVLVMWMWNWHWJCEJBDAIVNWNVQVOVP $.

  ${
    $d A r s t x y z $.  $d B r s t x y z $.
    $( The division algorithm for ordinal multiplication.  (Contributed by
       Mario Carneiro, 28-Feb-2013.) $)
    omeu $p |- ( ( A e. On /\ B e. On /\ A =/= (/) ) ->
        E! z E. x e. On E. y e. A
          ( z = <. x , y >. /\ ( ( A .o x ) +o y ) = B ) ) $=
      ( vt vr vs con0 wcel cv cop wceq comu co coa wa wrex wex wi omeulem1 opex
      wne w3a wal weu isseti 19.41v mpbiran rexbii rexcom4 bitr3i bitri simp2rl
      c0 sylib simp3rl simp2rr simp3rr eqtr4d wb simp11 simp2ll simp2lr simp3ll
      simp13 simp3lr omopth2 syl222anc mpbid opeq12 3expia exp4b expd rexlimdvv
      syl imp expimpd alrimivv opeq1 eqeq2d oveq2 oveq1d eqeq1d opeq2 cbvrex2vw
      anbi12d eqeq1 anbi1d 2rexbidv bitrid eu4 sylanbrc ) DIJZEIJZDUOUCZUDZCKZA
      KZBKZLZMZDWSNOZWTPOZEMZQZBDRZAIRZCSZXHFKZGKZHKZLZMZDXKNOZXLPOZEMZQZHDRGIR
      ZQWRXJMZTZFUECUEXHCUFWQXEBDRZAIRZXIABDEUAYCXGCSZAIRXIYBYDAIYBXFCSZBDRYDYE
      XEBDYEXBCSXECXAWSWTUBUGXBXECUHUIUJXFBCDUKULUJXGACIUKUMUPWQYACFWQXHXSXTWQX
      HQXRXTGHIDWQXHXKIJZXLDJZQZXRXTTTZWQXFYIABIDWQWSIJZWTDJZQZXFYIWQYLXFQZYHXR
      XTWQYMYHXRQZXTWQYMYNUDZWRXAXJXBXEYLWQYNUNYOXJXMXAXNXQYHWQYMUQYOWSXKMZWTXL
      MZQZXAXMMYOXDXPMZYRYOXDEXPXBXEYLWQYNURXNXQYHWQYMUSUTYOWNWPYJYKYFYGYSYRVAW
      NWOWPYMYNVBWNWOWPYMYNVFYJYKXFWQYNVCYJYKXFWQYNVDYFYGXRWQYMVEYFYGXRWQYMVGDW
      SWTXKXLVHVIVJWSWTXKXLVKVPUTUTVLVMVNVOVQVOVRVSXHXSCFXHWRXMMZXQQZHDRGIRXTXS
      XFUUAWRXKWTLZMZXOWTPOZEMZQABGHIDYPXBUUCXEUUEYPXAUUBWRWSXKWTVTWAYPXDUUDEYP
      XCXOWTPWSXKDNWBWCWDWGYQUUCYTUUEXQYQUUBXMWRWTXLXKWEWAYQUUDXPEWTXLXOPWBWDWG
      WFXTUUAXRGHIDXTYTXNXQWRXJXMWHWIWJWKWLWM $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( Ordinal exponentiation with a nonzero base is nonzero.  Proposition 8.32
       of [TakeutiZaring] p. 67.  (Contributed by NM, 4-Jan-2005.) $)
    oen0 $p |- ( ( ( A e. On /\ B e. On ) /\ (/) e. A ) ->
               (/) e. ( A ^o B ) ) $=
      ( vx vy con0 wcel c0 coe co wi cv wa wceq oveq2 eleq2d c1o adantr wss cvv
      wb csuc oe0 eleqtrrid comu oecl omordi om0 eleq1d ad2antlr sylibd syldanl
      0lt1o oesuc sylibrd exp31 com12 com34 impd wlim wral ciun wrex 0ellim syl
      eqimss2 sseq2d rspcev syl2an ssiun adantrr vex oelim mpanlr1 anasss an12s
      sseqtrrd word limelon mpan ancoms sylan eloni ordgt0ge1 mpbird ex tfinds3
      3syl a1dd expd imp31 ) AEFZBEFZGAFZGABHIZFZWLWKWMWOJWLWKWMWOGACKZHIZFZGAG
      HIZFZGADKZHIZFZGAXAUAZHIZFZWOWKWMLZCDBWPGMWQWSGWPGAHNOWPXAMWQXBGWPXAAHNOW
      PXDMWQXEGWPXDAHNOWPBMWQWNGWPBAHNOWKWTWMWKGPWSULAUBZUCQXAEFZWKWMXCXFJXIWKX
      CWMXFWKXIXCWMXFJZJWKXIXCXJWKXILZXCLWMGXBAUDIZFZXFWKXIXBEFZXCWMXMJAXAUEWKX
      NLXCLWMXBGUDIZXLFZXMGAXBUFXNXPXMTWKXCXNXOGXLXBUGUHUIUJUKXKXFXMTXCXKXEXLGA
      XAUMOQUNUOUPUQURWPUSZXGWRXCDWPUTXQXGWRXQXGLZWRPWQRZXRPDWPXBVAZWQXQWKPXTRZ
      WMXQWKLZPXBRZDWPVBZYAXQGWPFPWSRZYDWKWPVCWKWSPMYEXHPWSVEVDYCYEDGWPXAGMXBWS
      PXAGAHNVFVGVHDWPXBPVIVDVJWKXQWMWQXTMZWKXQWMYFWKWPSFZXQWMYFCVKZDAWPSVLVMVN
      VOVPXQWKWRXSTZWMYBWQEFZWQVQYIXQWPEFZWKYJYGXQYKYHWPSVRVSWKYKYJAWPUEVTWAWQW
      BWQWCWGVJWDWEWHWFWIUPWJ $.
  $}

  ${
    $d x y A $.  $d x B $.  $d x y C $.
    $( Ordering law for ordinal exponentiation.  Proposition 8.33 of
       [TakeutiZaring] p. 67.  (Contributed by NM, 5-Jan-2005.)  (Revised by
       Mario Carneiro, 24-May-2015.) $)
    oeordi $p |- ( ( B e. On /\ C e. ( On \ 2o ) ) ->
                 ( A e. B -> ( C ^o A ) e. ( C ^o B ) ) ) $=
      ( vy con0 wcel coe co wi wceq oveq2 eleq2d imbi2d wa comu adantr syl21anc
      c1o syl cvv vx c2o cdif cv csuc eldifi oecl sylan ondif2 simprbi c0 simpr
      om1 dif20el oen0 omordi eqeltrrd oesuc eleqtrrd expcom onsuc syl2an ontr1
      mpd mpan2d a2d wral wlim bi2.04 ralbii r19.21v bitri limsuc biimpa sucexb
      ciun elex sucidg sylbir eleq2 imbi12d rspcv mpid anc2li eliuni syl6 simpl
      adantl vex oelim mpanlr1 adantlr sylibrd ex biimtrid tfindsg2 impancom )
      BEFABFCEUBUCFZCAGHZCBGHZFZWRWSCUAUDZGHZFZIZWRWSCAUEZGHZFZIWRWSCDUDZGHZFZI
      ZWRWSCXIUEZGHZFZIWRXAIUADBAXBXFJZXDXHWRXPXCXGWSXBXFCGKLMXBXIJZXDXKWRXQXCX
      JWSXBXICGKLMXBXMJZXDXOWRXRXCXNWSXBXMCGKLMXBBJZXDXAWRXSXCWTWSXBBCGKLMWRAEF
      ZXHWRXTNZWSWSCOHZXGYAWSROHZWSYBYAWSEFZYCWSJWRCEFZXTYDCEUBUFZCAUGUHZWSUMSY
      ARCFZYCYBFZWRYHXTWRYEYHCUIUJZPYAYEYDUKWSFZYHYIIWRYEXTYFPZYGYAYEXTUKCFZYKY
      LWRXTULWRYMXTCUNZPCAUOQRCWSUPQVDUQWRYEXTXGYBJYFCAURUHUSUTXIEFZAXIFZNWRXKX
      OYOWRXKXOIZIYPWRYOYQWRYONZXKXJXNFZXOYRXJXJCOHZXNYRXJROHZXJYTYRXJEFZUUAXJJ
      WRYEYOUUBYFCXIUGUHZXJUMSYRYHUUAYTFZWRYHYOYJPYRYEUUBUKXJFZYHUUDIWRYEYOYFPZ
      UUCYRYEYOYMUUEUUFWRYOULWRYMYOYNPCXIUOQRCXJUPQVDUQWRYEYOXNYTJYFCXIURUHUSYR
      XNEFZXKYSNXOIWRYEXMEFUUGYOYFXIVACXMUGVBWSXJXNVCSVEUTPVFYPXLIZDXBVGZWRYPXK
      IZDXBVGZIZXBVHZAXBFZNZXEUUIWRUUJIZDXBVGUULUUHUUPDXBYPWRXKVIVJWRUUJDXBVKVL
      UUOWRUUKXDUUOWRUUKXDIUUOWRNZUUKWSDXBXJVPZFZXDUUOUUKUUSIZWRUUOXFXBFZUUTUUM
      UUNUVAXBAVMVNUVAUUKUVAXHNUUSUVAUUKXHUVAUUKAXFFZXHUVAXFTFZUVBXFXBVQUVCATFU
      VBAVOATVRVSSUUJUVBXHIDXFXBXIXFJZYPUVBXKXHXIXFAVTUVDXJXGWSXIXFCGKZLWAWBWCW
      DDXFXJXGXBWSUVEWEWFSPUUQXCUURWSUUMWRXCUURJZUUNUUMWRNYEUUMYMUVFWRYEUUMYFWH
      UUMWRWGWRYMUUMYNWHYEXBTFUUMYMUVFUAWIDCXBTWJWKQWLLWMWNVFWOWPWQ $.
  $}

  $( Ordering property of ordinal exponentiation.  Corollary 8.34 of
     [TakeutiZaring] p. 68 and its converse.  (Contributed by NM, 6-Jan-2005.)
     (Revised by Mario Carneiro, 24-May-2015.) $)
  oeord $p |- ( ( A e. On /\ B e. On /\ C e. ( On \ 2o ) ) ->
               ( A e. B <-> ( C ^o A ) e. ( C ^o B ) ) ) $=
    ( con0 wcel c2o coe co wi oeordi wceq wo wn oecl syl2anc word eloni ordtri2
    wb syl2an w3a 3adant1 oveq2 a1i 3adant2 orim12d con3d eldifi 3ad2ant3 simp1
    cdif simp2 3adant3 3imtr4d impbid ) ADEZBDEZCDFUKEZUAZABEZCAGHZCBGHZEZUQURU
    TVCIUPABCJUBUSVAVBKZVBVAEZLZMZABKZBAEZLZMZVCUTUSVJVFUSVHVDVIVEVHVDIUSABCGUC
    UDUPURVIVEIUQBACJUEUFUGUSVADEZVBDEZVCVGSZUSCDEZUPVLURUPVOUQCDFUHUIZUPUQURUJ
    CANOUSVOUQVMVPUPUQURULCBNOVLVAPVBPVNVMVAQVBQVAVBRTOUPUQUTVKSZURUPAPBPVQUQAQ
    BQABRTUMUNUO $.

  $( Left cancellation law for ordinal exponentiation.  (Contributed by NM,
     6-Jan-2005.)  (Revised by Mario Carneiro, 24-May-2015.) $)
  oecan $p |- ( ( A e. ( On \ 2o ) /\ B e. On /\ C e. On ) ->
                   ( ( A ^o B ) = ( A ^o C ) <-> B = C ) ) $=
    ( con0 c2o wcel co wceq wo wn wi oeordi ancoms wb oecl syl2anc word ordtri3
    coe eloni cdif w3a 3adant2 3adant3 con3d eldifi 3ad2ant1 simp2 simp3 syl2an
    orim12d 3adant1 3imtr4d oveq2 impbid1 ) ADEUAFZBDFZCDFZUBZABSGZACSGZHZBCHZU
    SUTVAFZVAUTFZIZJZBCFZCBFZIZJZVBVCUSVJVFUSVHVDVIVEUPURVHVDKZUQURUPVLBCALMUCU
    PUQVIVEKZURUQUPVMCBALMUDUKUEUSUTDFZVADFZVBVGNZUSADFZUQVNUPUQVQURADEUFUGZUPU
    QURUHABOPUSVQURVOVRUPUQURUIACOPVNUTQVAQVPVOUTTVATUTVARUJPUQURVCVKNZUPUQBQCQ
    VSURBTCTBCRUJULUMBCASUNUO $.

  $( Weak ordering property of ordinal exponentiation.  (Contributed by NM,
     6-Jan-2005.)  (Revised by Mario Carneiro, 24-May-2015.) $)
  oeword $p |- ( ( A e. On /\ B e. On /\ C e. ( On \ 2o ) ) ->
                   ( A C_ B <-> ( C ^o A ) C_ ( C ^o B ) ) ) $=
    ( con0 wcel c2o cdif w3a wceq wo coe co wss oeord oecan 3coml onsseleq oecl
    wb wa bicomd orbi12d 3adant3 eldifi id anim12dan syl syl2anr 3impa 3bitr4d
    ) ADEZBDEZCDFGEZHZABEZABIZJZCAKLZCBKLZEZURUSIZJZABMZURUSMZUNUOUTUPVAABCNUNV
    AUPUMUKULVAUPSCABOPUAUBUKULVCUQSUMABQUCUKULUMVDVBSZUMCDEZUKULTZVEVGCDFUDVGU
    EVFVGTURDEZUSDEZTVEVFUKVHULVICARCBRUFURUSQUGUHUIUJ $.

  $( Weak ordering property of ordinal exponentiation.  Lemma 3.19 of
     [Schloeder] p. 10.  (Contributed by NM, 6-Jan-2005.) $)
  oewordi $p |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ (/) e. C ) ->
                   ( A C_ B -> ( C ^o A ) C_ ( C ^o B ) ) ) $=
    ( con0 wcel w3a c0 wss coe co wi c1o wceq wo wb word syl wa oe1m oveq1 mpan
    eloni ordgt0ge1 1on onsseleq bitrd 3ad2ant3 c2o ondif2 oeword biimpd 3expia
    cdif biimtrrid 3impia adantr adantl eqtr4d eqimss sseq12d syl5ibcom 3adant3
    expd a1dd jaod sylbid imp ) ADEZBDEZCDEZFZGCEZABHZCAIJZCBIJZHZKZVKVLLCEZLCM
    ZNZVQVJVHVLVTOVIVJVLLCHZVTVJCPVLWAOCUBCUCQLDEVJWAVTOUDLCUEUAUFUGVKVRVQVSVHV
    IVJVRVQKVHVIRZVJVRVQVJVRRCDUHUMEZWBVQCUIVHVIWCVQVHVIWCFVMVPABCUJUKULUNVCUOV
    KVSVPVMVHVIVSVPKVJWBLAIJZLBIJZHZVSVPWBWDWEMWFWBWDLWEVHWDLMVIASUPVIWELMVHBSU
    QURWDWEUSQVSWDVNWEVOLCAITLCBITUTVAVBVDVEVFVG $.

  ${
    $d x y A $.  $d x y B $.  $d x y C $.
    $( Weak ordering property of ordinal exponentiation.  Proposition 8.35 of
       [TakeutiZaring] p. 68.  (Contributed by NM, 6-Jan-2005.) $)
    oewordri $p |- ( ( B e. On /\ C e. On ) ->
        ( A e. B -> ( A ^o C ) C_ ( B ^o C ) ) ) $=
      ( vx vy con0 wcel coe co wss wi c0 wceq oveq2 sseq12d c1o adantr imp cvv
      wa csuc onelon oe0 syl eqtr4d eqimss simpl onelss jca31 comu oecl 3adant2
      w3a 3adant1 simp1 omwordri syl3anc adantrl omwordi syld3an3 adantrr sstrd
      cv oesuc 3sstr4d exp520 com3r imp4c syl5 wlim wral vex limelon mpan oe0m1
      0ellim biimpa 0ss eqsstrdi oveq1 sseq1d imbitrrid adantl a1dd ciun ss2iun
      syl2anc oelim mpanlr1 adantllr anim1i wne ne0i on0eln0 ad4ant24 ex oe0lem
      an32s ancri syl11 tfinds3 expd impcom ) CFGZBFGZABGZACHIZBCHIZJZKXDXEXFXI
      ADVCZHIZBXJHIZJZALHIZBLHIZJZAEVCZHIZBXQHIZJZAXQUAZHIZBYAHIZJZXIXEXFTZDECX
      JLMXKXNXLXOXJLAHNXJLBHNOXJXQMXKXRXLXSXJXQAHNXJXQBHNOXJYAMXKYBXLYCXJYAAHNX
      JYABHNOXJCMXKXGXLXHXJCAHNXJCBHNOYEXNXOMXPYEXNPXOYEAFGZXNPMBAUBZAUCUDXEXOP
      MXFBUCQUEXNXOUFUDYEYFXETABJZTXQFGZXTYDKZYEYFXEYHYGXEXFUGZXEXFYHBAUHRUIYIY
      FXEYHYJYFXEYIYHYJKYFXEYIYHXTYDYFXEYIUMZYHXTTZTZXRAUJIZXSBUJIZYBYCYNYOXSAU
      JIZYPYLXTYOYQJZYHYLXTYRYLXRFGZXSFGZYFXTYRKYFYIYSXEAXQUKULXEYIYTYFBXQUKUNZ
      YFXEYIUOXRXSAUPUQRURYLYHYQYPJZXTYLYHUUBYFXEYIYTYHUUBKUUAABXSUSUTRVAVBYLYB
      YOMZYMYFYIUUCXEAXQVDULQYLYCYPMZYMXEYIUUDYFBXQVDUNQVEVFVGVHVIYFYETZXJVJZXT
      EXJVKZXMKZYEYEUUFUUHKAYEALMZTUUFXMUUGUUIUUFXMKYEUUFXMUUILXJHIZXLJUUFUUJLX
      LUUFXJFGZLXJGZUUJLMZXJSGZUUFUUKDVLZXJSVMVNXJVPUUKUULUUMXJVOVQWGXLVRVSUUIX
      KUUJXLALXJHVTWAWBWCWDUUELAGZTZUUFUUHUUGXMUUQUUFTZEXJXRWEZEXJXSWEZJEXJXRXS
      WFUURXKUUSXLUUTYFUUPUUFXKUUSMZYEYFUUFUUPUVAYFUUNUUFUUPUVAUUOEAXJSWHWIWRWJ
      YEUUFXLUUTMZYFUUPYEUUFTXEUUFTLBGZUVBYEXEUUFYKWKYEUVCUUFXEXFUVCXFUVCXEBLWL
      BAWMBWNWBRQXEUUNUUFUVCUVBUUOEBXJSWHWIWGWOOWBWPWQYEYFYGWSWTXAXBXC $.

    $( Ordinal exponentiation compared to its exponent.  Proposition 8.37 of
       [TakeutiZaring] p. 68.  Lemma 3.20 of [Schloeder] p. 10.  (Contributed
       by NM, 7-Jan-2005.)  (Revised by Mario Carneiro, 24-May-2015.) $)
    oeworde $p |- ( ( A e. ( On \ 2o ) /\ B e. On ) -> B C_ ( A ^o B ) ) $=
      ( vx vy con0 wcel c2o coe co wss cv c0 csuc wceq id oveq2 sseq12d wa word
      eloni cdif 0ss a1i wi wb eldifi oecl sylan syl ordsucsssuc syl2an2 syl2an
      onsuc vex sucid oeordi mpi syl2anr ordsucss sylc sstr2 sylbid expcom wlim
      syl5com wral dif20el jca ciun ss2iun cuni limuni uniiun eqtrdi adantr cvv
      oelim mpanlr1 anasss an12s imbitrrid ex syl5 tfinds3 impcom ) BEFAEGUAFZB
      ABHIZJZCKZAWIHIZJZLALHIZJZDKZAWNHIZJZWNMZAWQHIZJZWHWFCDBWILNZWILWJWLWTOWI
      LAHPQWIWNNZWIWNWJWOXAOWIWNAHPQWIWQNZWIWQWJWRXBOWIWQAHPQWIBNZWIBWJWGXCOWIB
      AHPQWMWFWLUBUCWFWNEFZWPWSUDWFXDRZWPWQWOMZJZWSXDWNSWFWOSZWPXGUEWNTXEWOEFZX
      HWFAEFZXDXIAEGUFZAWNUGUHWOTUIWNWOUJUKXEXFWRJZXGWSXEWRSZWOWRFZXLXEWREFZXMW
      FXJWQEFZXOXDXKWNUMZAWQUGULWRTUIXDXPWFXNWFXQWFOXPWFRWNWQFXNWNDUNUOWNWQAUPU
      QURWOWRUSUTWQXFWRVAVEVBVCWFXJLAFZRZWIVDZWPDWIVFZWKUDZWFXJXRXKAVGVHXTXSYBY
      AWKXTXSRZDWIWNVIZDWIWOVIZJDWIWNWOVJYCWIYDWJYEXTWIYDNXSXTWIWIVKYDWIVLDWIVM
      VNVOXJXTXRWJYENZXJXTXRYFXJWIVPFXTXRYFCUNDAWIVPVQVRVSVTQWAWBWCWDWE $.
  $}

  $( Ordering property of ordinal exponentiation with a successor exponent.
     Corollary 8.36 of [TakeutiZaring] p. 68.  (Contributed by NM,
     7-Jan-2005.) $)
  oeordsuc $p |- ( ( B e. On /\ C e. On ) ->
      ( A e. B -> ( A ^o suc C ) e. ( B ^o suc C ) ) ) $=
    ( con0 wcel wa coe co wi ex adantr comu wss 3adant1 oecl 3adant2 syld oesuc
    wceq c0 csuc onelon w3a oewordri simp1 omwordri syl3anc sseq1d sylibrd ne0i
    wne on0eln0 imbitrrid oen0 omordi com23 mpdd eleq2d jcad 3expa onsucb ontr2
    syldanl syl2an anandirs sylan2b exp31 com4l imp ) BDEZCDEZFZABEZADEZACUAZGH
    ZBVOGHZEZVJVMVNIVKVJVMVNBAUBJKVJVKVMVNVRIIVNVJVKVMVRVNVJVKVMVRIVNVJFZVKFVMV
    PBCGHZALHZMZWAVQEZFZVRVNVJVKVMWDIVNVJVKUCZVMWBWCWEVMACGHZALHZWAMZWBWEVMWFVT
    MZWHVJVKVMWIIVNABCUDNWEWFDEZVTDEZVNWIWHIVNVKWJVJACOPVJVKWKVNBCOZNVNVJVKUEWF
    VTAUFUGQWEVPWGWAVNVKVPWGSVJACRPUHUIWEVMWAVTBLHZEZWCVJVKVMWNIZVNVLVMTVTEZWNV
    LVMTBEZWPVJVMWQIVKVMWQVJBTUKBAUJBULUMKVLWQWPBCUNJQVLWPVMWNVLWPWOVJVKWKWPWOW
    LABVTUOVCJUPUQNWEVQWMWAVJVKVQWMSVNBCRNURUIUSUTVKVSVODEZWDVRIZCVAVNVJWRWSVNW
    RFVPDEVQDEWSVJWRFAVOOBVOOVPWAVQVBVDVEVFQVGVHVIUQ $.

  ${
    $d x y A $.  $d x y B $.  $d y C $.
    $( Ordinal exponentiation with a limit exponent.  Part of Exercise 4.36 of
       [Mendelson] p. 250.  (Contributed by NM, 6-Jan-2005.) $)
    oelim2 $p |- ( ( A e. On /\ ( B e. C /\ Lim B ) ) ->
                ( A ^o B ) = U_ x e. ( B \ 1o ) ( A ^o x ) ) $=
      ( vy wcel wa coe co c1o cv ciun wceq c0 con0 adantl wi sylan wne wss wlim
      cdif limelon 0ellim oe0m1 biimpa syl2anc word limord ordelon on0eln0 el1o
      wn eldif necon3bbii bitr4di biimpd sylbird syl impr sylan2b iuneq2dv csuc
      df-1o limsuc mpbid eqeltrid 1on onirri sylanblrc ne0i iunconst 3syl eqtrd
      eqtr4d oveq1 iuneq2d eqeq12d imbitrrid impcom oelim wrex nsuceq0 dif1o ex
      wral ad2antlr sssucid w3a onsuc jccir 3expa ancoms sylan2 anassrs oewordi
      id an32s jcad oveq2 sseq2d rspcev syl6 ralrimiv iunss2 difss iunss1 ax-mp
      mpi cbviunv sseqtri a1i eqssd adantlrl oe0lem ) CDFZCUAZGZBCHIZACJUBZBAKZ
      HIZLZMZBBNMZXRYDXRYDYENCHIZAXTNYAHIZLZMXRYFNYHXRCOFZNCFZYFNMZCDUCXQYJXPCU
      DZPYIYJYKCUEUFUGXQYHNMXPXQYHAXTNLZNXQAXTYGNYAXTFXQYACFZYAJFZUMZGYGNMZYACJ
      UNXQYNYPYQXQYNGYAOFZYPYQQXQCUHZYNYRCUIZCYAUJRYRYPNYAFZYQYRUUAYANSYPYAUKYO
      YANYAULUOUPYRUUAYQYAUEUQURUSUTVAVBXQJXTFZXTNSYMNMXQJCFJJFUMUUBXQJNVCZCVDX
      QYJUUCCFYLCNVEVFVGJVHVIJCJUNVJXTJVKAXTNVLVMVNPVOYEXSYFYCYHBNCHVPYEAXTYBYG
      BNYAHVPVQVRVSVTBOFZXRGNBFZGXSECBEKZHIZLZYCEBCDWAUUDXQUUEUUHYCMXPUUDXQGZUU
      EGZUUHYCUUJUUGYBTZAXTWBZECWFUUHYCTUUJUULECUUJUUFCFZUUFVCZXTFZUUGBUUNHIZTZ
      GUULUUJUUMUUOUUQXQUUMUUOQUUDUUEXQUUMUUOXQUUMGZUUNCFZUUNNSUUOXQUUMUUSCUUFV
      EUFUUFWCUUNCWDVJWEWGUUJUUMUUQUUJUUMGUUFUUNTZUUQUUFWHUUIUUMUUEUUTUUQQZUUIU
      UMGUUFOFZUUNOFZUUDWIZUUEUVAUUDXQUUMUVDUURUUDUVBUVCGZUVDUURUVBUVCXQYSUUMUV
      BYTCUUFUJRUUFWJWKUVEUUDUVDUVBUVCUUDUVDUVDWQWLWMWNWOUUFUUNBWPRWRXIWEWSUUKU
      UQAUUNXTYAUUNMYBUUPUUGYAUUNBHWTXAXBXCXDEACXTUUGYBXEUSYCUUHTUUJYCACYBLZUUH
      XTCTYCUVFTCJXFAXTCYBXGXHAECYBUUGYAUUFBHWTXJXKXLXMXNVNXO $.
  $}

  ${
    $d x C $.  $d x y z w A $.  $d x y z w B $.
    oeoalem.1 $e |- A e. On $.
    oeoalem.2 $e |- (/) e. A $.
    oeoalem.3 $e |- B e. On $.
    $( Lemma for ~ oeoa .  (Contributed by Eric Schmidt, 26-May-2009.) $)
    oeoalem $p |- ( C e. On
               -> ( A ^o ( B +o C ) ) = ( ( A ^o B ) .o ( A ^o C ) ) ) $=
      ( vy coa co coe comu wceq c0 oveq2 oveq2d con0 wcel wa mpan cvv vx vz c1o
      vw csuc eqeq12d oecl mp2an om1 ax-mp oe0 oveq2i 3eqtr4ri oasuc oacl oesuc
      cv oa0 sylancr eqtrd oveq1 sylan9eq omass mp3an13 eqtr4d adantr wlim wral
      syl ex ciun vex oalim wne word limord ordelon sylan ralrimiva 0ellim ne0d
      oelim mpan2 wss wi w3a oewordi mp3an3 3impia onoviun mp3an2i iuneq2 omlim
      omwordi tfinds ) ABUAUQZHIZJIZABJIZAWPJIZKIZLZABMHIZJIZWSAMJIZKIZLABGUQZH
      IZJIZWSAXGJIZKIZLZABXGUEZHIZJIZWSAXMJIZKIZLZABCHIZJIZWSACJIZKIZLUAGCWPMLZ
      WRXDXAXFYCWQXCAJWPMBHNOYCWTXEWSKWPMAJNOUFWPXGLZWRXIXAXKYDWQXHAJWPXGBHNOYD
      WTXJWSKWPXGAJNOUFWPXMLZWRXOXAXQYEWQXNAJWPXMBHNOYEWTXPWSKWPXMAJNOUFWPCLZWR
      XTXAYBYFWQXSAJWPCBHNOYFWTYAWSKWPCAJNOUFWSUCKIZWSXFXDWSPQZYGWSLAPQZBPQZYHD
      FABUGUHZWSUIUJXEUCWSKYIXEUCLDAUKUJULXCBAJYJXCBLFBURUJULUMXGPQZXLXRYLXLRXO
      XKAKIZXQYLXLXOXIAKIZYMYJYLXOYNLFYJYLRZXOAXHUEZJIZYNYOXNYPAJBXGUNOYOYIXHPQ
      ZYQYNLDBXGUOZAXHUPUSUTSXIXKAKVAVBYLYMXQLZXLYIYLYTDYIYLRZYMWSXJAKIZKIZXQUU
      AXJPQZYMUUCLZAXGUGZYHUUDYIUUEYKDWSXJAVCVDVIUUAXPUUBWSKAXGUPOVESVFUTVJWPVG
      ZXLGWPVHZXBUUGUUHRWRGWPXKVKZXAUUGUUHWRGWPXIVKZUUIUUGWRAGWPXHVKZJIZUUJUUGW
      QUUKAJWPTQZUUGWQUUKLZUAVLZYJUUMUUGRZUUNFGBWPTVMSSOUUMUUGYRGWPVHWPMVNZUULU
      UJLUUOUUGYRGWPUUGXGWPQZRZYJYLYRFUUGWPVOUURYLWPVPWPXGVQVRZYSUSVSUUGWPMWPVT
      WAZUBUDGATJWPXHUDUQZTQZUVBVGZAUVBJIZUBUVBAUBUQZJIZVKLZUDVLZYIUVCUVDRZUVHD
      YIUVJRMAQZUVHEUBAUVBTWBWCSSUVFPQZUVBPQZUVFUVBWDZUVGUVEWDZUVLUVMYIUVNUVOWE
      ZDUVLUVMYIWFUVKUVPEUVFUVBAWGWCWHWIWJWKUTGWPXIXKWLVBUUGXAUUILUUHUUGXAWSGWP
      XJVKZKIZUUIUUGWTUVQWSKUUMUUGWTUVQLZUUOYIUUPUVSDYIUUPRUVKUVSEGAWPTWBWCSSOU
      UMUUGUUDGWPVHUUQUVRUUILUUOUUGUUDGWPUUSYIYLUUDDUUTUUFUSVSUVAUBUDGWSTKWPXJU
      VCUVDWSUVBKIZUBUVBWSUVFKIZVKLZUVIYHUVJUWBYKUBWSUVBTWMSSUVLUVMUVNUWAUVTWDZ
      UVLUVMYHUVNUWCWEYKUVFUVBWSWNWHWIWJWKUTVFVEVJWO $.
  $}

  $( Sum of exponents law for ordinal exponentiation.  Theorem 8R of [Enderton]
     p. 238.  Also Proposition 8.41 of [TakeutiZaring] p. 69.  Theorem 4.7 of
     [Schloeder] p. 14.  (Contributed by Eric Schmidt, 26-May-2009.) $)
  oeoa $p |- ( ( A e. On /\ B e. On /\ C e. On )
          -> ( A ^o ( B +o C ) ) = ( ( A ^o B ) .o ( A ^o C ) ) ) $=
    ( con0 wcel coa co coe comu wceq wa c0 oveq2d c1o oveq2 0elon oecl wb oveq1
    syl oa00 biimpar oe0m0 eqtrdi oveqan12d mp2an om1 adantl eqtr4d wn wne oacl
    ax-mp on0eln0 oe0m1 necon3abid 3bitr3d adantr orbi12d neorian bitrdi biimpa
    wo oveq1d mpan om0r sylan9eq an32s om0 sylan9eqr anassrs jaodan sylbird imp
    ex pm2.61dan oveq12d eqeq12d imbitrrid impcom wi imbi2d eleq1 eleq2 anbi12d
    cif 1on 0lt1o pm3.2i elimhyp simpli simpri elimel oeoalem impr oe0lem 3impb
    dedth2h ) ADEZBDEZCDEZABCFGZHGZABHGZACHGZIGZJZWTXAKZXGAALJZXHXGXHXGXILXBHGZ
    LBHGZLCHGZIGZJZXHBLJZCLJZKZXNXHXQKZXJLLHGZXMXRXBLLHXHXBLJXQBCUAZUBMXQXMXSJX
    HXQXMXSNIGZXSXOXPXKXSXLNIBLLHOXPXLXSNCLLHOUCUDUEXSDEZYAXSJLDEZYCYBPPLLQUFXS
    UGUMUDUHUIXHXQUJZKXJLXMXHXJLJZYDXHLXBEZXBLUKZYEYDXHXBDEZYFYGRBCULZXBUNTXHYH
    YFYERYIXBUOTXHXQXBLXTUPUQUBXHYDXMLJZXHYDLBEZLCEZVCZYJXHYMBLUKZCLUKZVCYDXHYK
    YNYLYOWTYKYNRXABUNURXAYLYORWTCUNUHUSBLCLUTVAXHYMYJXHYKYJYLWTYKXAYJWTYKKZXAX
    MLXLIGZLYPXKLXLIWTYKXKLJBUOVBVDXAXLDEZYQLJYCXAYRPLCQVEXLVFTVGVHWTXAYLYJXAYL
    KZWTXMXKLIGZLYSXLLXKIXAYLXLLJCUOVBMWTXKDEZYTLJYCWTUUAPLBQVEXKVITVJVKVLVOVMV
    NUIVPXIXCXJXFXMALXBHSXIXDXKXEXLIALBHSALCHSVQVRVSVTWSLAEZXHXGWSUUBKZWTXAXGUU
    CWTXAXGWAXAUUCANWFZXBHGZUUDBHGZUUDCHGZIGZJZWAXAUUDWTBNWFZCFGZHGZUUDUUJHGZUU
    GIGZJZWAABNNAUUDJZXGUUIXAUUPXCUUEXFUUHAUUDXBHSUUPXDUUFXEUUGIAUUDBHSAUUDCHSV
    QVRWBBUUJJZUUIUUOXAUUQUUEUULUUHUUNUUQXBUUKUUDHBUUJCFSMUUQUUFUUMUUGIBUUJUUDH
    OVDVRWBUUDUUJCUUDDEZLUUDEZUUCUURUUSKNDEZLNEZKANUUPWSUURUUBUUSAUUDDWCAUUDLWD
    WENUUDJUUTUURUVAUUSNUUDDWCNUUDLWDWEUUTUVAWGWHWIWJZWKUURUUSUVBWLBNDWGWMWNWRW
    OVHWPWQ $.

  ${
    $d x C $.  $d x y z w A $.  $d x y z w B $.
    oeoelem.1 $e |- A e. On $.
    oeoelem.2 $e |- (/) e. A $.
    $( Lemma for ~ oeoe .  (Contributed by Eric Schmidt, 26-May-2009.) $)
    oeoelem $p |- ( ( B e. On /\ C e. On )
               -> ( ( A ^o B ) ^o C ) = ( A ^o ( B .o C ) ) ) $=
      ( vy vz con0 wcel coe co comu wceq cv c0 oveq2 oveq2d eqeq12d wa cvv csuc
      vx vw c1o oecl mpan oe0 syl om0 ax-mp eqtrdi eqtr4d oveq1 oesuc sylan coa
      wi omsuc omcl oeoa mp3an1 anabss1 eqtrd imbitrrid expcom wlim wral iuneq2
      ciun vex oen0 mpan2 oelim sylanl1 mpidan mpanl1 mpanr1 omlim word ordelon
      wne limord sylan2 anassrs ralrimiva 0ellim ne0d adantl wss oewordi mp3an3
      w3a 3impia onoviun mp3an2i tfinds3 impcom ) CHIBHIZABJKZCJKZABCLKZJKZMZWS
      UBNZJKZABXDLKZJKZMZWSOJKZABOLKZJKZMWSFNZJKZABXLLKZJKZMZWSXLUAZJKZABXQLKZJ
      KZMZXCWRUBFCXDOMZXEXIXGXKXDOWSJPYBXFXJAJXDOBLPQRXDXLMZXEXMXGXOXDXLWSJPYCX
      FXNAJXDXLBLPQRXDXQMZXEXRXGXTXDXQWSJPYDXFXSAJXDXQBLPQRXDCMZXEWTXGXBXDCWSJP
      YEXFXAAJXDCBLPQRWRXIUDXKWRWSHIZXIUDMAHIZWRYFDABUEZUFZWSUGUHWRXKAOJKZUDWRX
      JOAJBUIQYGYJUDMDAUGUJUKULWRXLHIZXPYAUQXPYAWRYKSZXMWSLKZXOWSLKZMXMXOWSLUMY
      LXRYMXTYNWRYFYKXRYMMYIWSXLUNUOYLXTAXNBUPKZJKZYNYLXSYOAJBXLURQWRYKYPYNMZYL
      XNHIZWRYQBXLUSZYGYRWRYQDAXNBUTVAUOVBVCRVDVEWRXDVFZXPFXDVGZXHUQUUAXHWRYTSZ
      FXDXMVIZFXDXOVIZMFXDXMXOVHUUBXEUUCXGUUDWRXDTIZYTXEUUCMZUBVJZYGWRUUEYTSZUU
      FDYGWRSZUUHOWSIZUUFUUIOAIZUUJEABVKVLUUIYFUUHUUJUUFYHFWSXDTVMVNVOVPVQUUBXG
      AFXDXNVIZJKZUUDUUBXFUULAJWRUUEYTXFUULMUUGFBXDTVRVQQUUEUUBYRFXDVGXDOWAZUUM
      UUDMUUGUUBYRFXDWRYTXLXDIZYRYTUUOSWRYKYRYTXDVSUUOYKXDWBXDXLVTUOYSWCWDWEYTU
      UNWRYTXDOXDWFWGWHGUCFATJXDXNUCNZTIZUUPVFZAUUPJKZGUUPAGNZJKZVIMZUCVJYGUUQU
      URSZUVBDYGUVCSUUKUVBEGAUUPTVMVLUFUFUUTHIZUUPHIZUUTUUPWIZUVAUUSWIZUVDUVEYG
      UVFUVGUQZDUVDUVEYGWLUUKUVHEUUTUUPAWJVLWKWMWNWOVCRVDVEWPWQ $.
  $}

  $( Product of exponents law for ordinal exponentiation.  Theorem 8S of
     [Enderton] p. 238.  Also Proposition 8.42 of [TakeutiZaring] p. 70.
     (Contributed by Eric Schmidt, 26-May-2009.) $)
  oeoe $p |- ( ( A e. On /\ B e. On /\ C e. On )
          -> ( ( A ^o B ) ^o C ) = ( A ^o ( B .o C ) ) ) $=
    ( con0 wcel coe co wceq wa c0 c1o oveq2 oe0m0 eqtrdi oveq1d sylan9eqr oe0m1
    syl biimpa oveq1 comu wo oe1m adantll 0elon oecl oe0 adantlr jaodan biimpar
    mpan om00 oveq2d eqtr4d wne on0eln0 bi2anan9 neanior bitrdi sylan9eq om00el
    wn an4s wb omcl bitr3d ex sylbird imp pm2.61dan eqeq12d imbitrrid impcom wi
    cif imbi2d eleq1 eleq2 anbi12d 1on 0lt1o pm3.2i elimhyp simpli simpri dedth
    oeoelem an32s oe0lem 3impb ) ADEZBDEZCDEZABFGZCFGZABCUAGZFGZHZWLWMIZWRAAJHZ
    WSWRWSWRWTJBFGZCFGZJWPFGZHZWSBJHZCJHZUBZXDWSXGIZXBKXCWSXEXBKHZXFWMXEXIWLXEW
    MXBKCFGKXEXAKCFXEXAJJFGZKBJJFLMNOCUCPUDWLXFXIWMXFWLXBXAJFGZKCJXAFLWLXADEZXK
    KHJDEWLXLUEJBUFUKXAUGRPUHUIXHXCXJKXHWPJJFWSWPJHXGBCULUJUMMNUNWSXGVBZXDWSXMJ
    BEZJCEZIZXDWSXPBJUOZCJUOZIXMWLXNXQWMXOXRBUPCUPUQBJCJURUSWSXPXDWSXPIXBJXCWLX
    NWMXOXBJHWLXNIZWMXOIXBJCFGZJXSXAJCFWLXNXAJHBQSOWMXOXTJHCQSUTVCWSXPXCJHZWSJW
    PEZXPYABCVAWSWPDEYBYAVDBCVEWPQRVFSUNVGVHVIVJWTWOXBWQXCWTWNXACFAJBFTOAJWPFTV
    KVLVMWKJAEZWSWRWKYCIZWSWRYDWSWRVNWSYDAKVOZBFGZCFGZYEWPFGZHZVNAKAYEHZWRYIWSY
    JWOYGWQYHYJWNYFCFAYEBFTOAYEWPFTVKVPYEBCYEDEZJYEEZYDYKYLIKDEZJKEZIAKYJWKYKYC
    YLAYEDVQAYEJVRVSKYEHYMYKYNYLKYEDVQKYEJVRVSYMYNVTWAWBWCZWDYKYLYOWEWGWFVIWHWI
    WJ $.

  ${
    $d x y A $.  $d x y B $.  $d x y C $.
    $( The ordinal exponential with a limit ordinal is a limit ordinal.
       (Contributed by Mario Carneiro, 29-May-2015.) $)
    oelimcl $p |- ( ( A e. ( On \ 2o ) /\ ( B e. C /\ Lim B ) ) ->
               Lim ( A ^o B ) ) $=
      ( vx vy con0 c2o cdif wcel wlim wa coe co word c0 cv syl adantr syl2anc
      wi csuc wral eldifi limelon oecl syl2an eloni adantl dif20el syl21anc c1o
      oen0 ciun wceq oelim2 sylan eleq2d wrex wss simprl onelon simprr ordsucss
      eliun sylc simpll oeordi mpd onsuc ontr2 mp2and sylan2 rexlimdva biimtrid
      expr sylbid ralrimiv dflim4 syl3anbrc ) AFGHIZBCIBJKZKZABLMZNZOWCIZDPZUAZ
      WCIZDWCUBWCJWBWCFIZWDVTAFIZBFIZWIWAAFGUCZBCUDZABUEUFZWCUGQWBWJWKOAIZWEVTW
      JWAWLRZWAWKVTWMUHZVTWOWAAUIRABULUJWBWHDWCWBWFWCIWFEBUKHZAEPZLMZUMZIZWHWBW
      CXAWFVTWJWAWCXAUNWLEABCUOUPUQXBWFWTIZEWRURWBWHEWFWRWTVDWBXCWHEWRWSWRIWBWS
      BIZXCWHTWSBUKUCWBXDXCWHWBXDXCKZKZWGWTUSZWTWCIZWHXFWTNZXCXGXFWTFIZXIXFWJWS
      FIZXJWBWJXEWPRXFWKXDXKWBWKXEWQRZWBXDXCUTZBWSVASAWSUESZWTUGQWBXDXCVBZWFWTV
      CVEXFXDXHXMXFWKVTXDXHTXLVTWAXEVFWSBAVGSVHXFWGFIZWIXGXHKWHTXFWFFIZXPXFXJXC
      XQXNXOWTWFVASWFVIQWBWIXEWNRWGWTWCVJSVKVOVLVMVNVPVQDWCVRVS $.
  $}

  ${
    $d a d e w x y z A $.  $d a d e w x y z B $.  $d a d e D $.  $d a d e E $.
    $d a d e w y z X $.
    oeeu.1 $e |- X = U. |^| { x e. On | B e. ( A ^o x ) } $.
    $( Lemma for ~ oeeu .  (Contributed by Mario Carneiro, 28-Feb-2013.) $)
    oeeulem $p |- ( ( A e. ( On \ 2o ) /\ B e. ( On \ 1o ) ) -> ( X e. On /\
        ( A ^o X ) C_ B /\ B e. ( A ^o suc X ) ) ) $=
      ( vy con0 wcel c1o wa coe co wss syl wceq oveq2 eleq2d wn c0 simprbi cdif
      c2o csuc cv crab cint cuni wrex eldifi adantl onsuc oeworde syldan sucidg
      sseldd rspcev syl2anc onintrab2 sylib onuni eqeltrid suceq ax-mp wlim wne
      dif1o ssrab2 rabn0 sylibr onint sylancr eleq1 syl5ibcom elrab adantr el1o
      wo oe0 bitrdi imbitrid syld necon3ad ciun limuni eqtr4di eqeltrrd cbvrabv
      mpd elrab2 ad2antrr wb limeq anim12i dif20el oelim syl21anc eleqtrd eliun
      ibi onss sselda biimpar onnminsb sylc nrexdv pm2.65da ioran sylanbrc word
      eloni unizlim 3syl mtbird orduniorsuc eqtr4id inteqi eleqtrdi oecl ontri1
      ord mpbird eqeltrd 3jca ) BGUBUAHZCGIUAHZJZDGHZBDKLZCMZCBDUCZKLZHZYFDCBAU
      DZKLZHZAGUEZUFZUGZGEYFYQGHZYRGHYFYOAGUHZYSYFCUCZGHZCBUUAKLZHZYTYFCGHZUUBY
      EUUEYDCGIUIUJZCUKNZYFUUAUUCCYDYEUUBUUAUUCMUUGBUUAULUMYFUUECUUAHUUFCGUNNUO
      YOUUDAUUAGYMUUAOYNUUCCYMUUABKPQUPUQZYOAURUSZYQUTNVAZYFYICYHHZRZYFYGDCBFUD
      ZKLZHZFGUEZUFZHUULUUJYFDYQUUQYFDYJYQYFYGDYJHUUJDGUNNYFYJYRUCZYQDYROYJUURO
      EDYRVBVCYFYQYROZRYQUUROZYFUUSYQSOZYQVDZVQZYFUVARZUVBRUVCRYFCSVEZUVDYEUVEY
      DYEUUEUVECGVFTUJYFUVACSYFUVASYPHZCSOZYFYQYPHZUVAUVFYFYPGMYPSVEZUVHYOAGVGY
      FYTUVIUUHYOAGVHVIYPVJVKZYQSYPVLVMUVFCBSKLZHZYFUVGUVFSGHUVLYOUVLASGYMSOYNU
      VKCYMSBKPQVNTYFUVLCIHUVGYFUVKICYFBGHZUVKIOYDUVMYEBGUBUIZVOZBVRNQCVPVSVTWA
      WBWHYFUVBUUOFDUHZYFUVBJZCFDUUNWCZHUVPUVQCYHUVRUVQDYPHZUUKUVQYQDYPUVBYQDOZ
      YFUVBYQYRDYQWDEWEZUJZYFUVHUVBUVJVOWFUVSYGUUKUUOUUKFDGYPUUMDOUUNYHCUUMDBKP
      QZYOUUOAFGYMUUMOYNUUNCYMUUMBKPQZWGZWITNUVQUVMYGDVDZJSBHZYHUVROYDUVMYEUVBU
      VNWJYFYGUVBUWFUUJUVBUWFUVBUVTUVBUWFWKUWAYQDWLNWSWMYDUWGYEUVBBWNWJFBDGWOWP
      WQFCDUUNWRUSUVQUUOFDUVQUUMDHZJUUMGHUUMYQHZUUORUVQDGUUMUVQYGDGMYFYGUVBUUJV
      ODWTNXAUVQUWIUWHUVQYQDUUMUWBQXBYOUUOAUUMUWDXCXDXEXFUVAUVBXGXHYFYSYQXIZUUS
      UVCWKUUIYQXJZYQXKXLXMYFUUSUUTYFYSUWJUUSUUTVQUUIUWKYQXNXLXTWHXOZWQYPUUPUWE
      XPXQUUOUUKFDUWCXCXDYFYHGHZUUEYIUULWKYFUVMYGUWMUVOUUJBDXRUQUUFYHCXSUQYAYFY
      JYPHZYLYFYJYQYPUWLUVJYBUWNYJGHYLUUOYLFYJGYPUUMYJOUUNYKCUUMYJBKPQUWEWITNYC
      $.

    oeeu.2 $e |- P = ( iota w E. y e. On E. z e. ( A ^o X )
      ( w = <. y , z >. /\ ( ( ( A ^o X ) .o y ) +o z ) = B ) ) $.
    oeeu.3 $e |- Y = ( 1st ` P ) $.
    oeeu.4 $e |- Z = ( 2nd ` P ) $.
    $( The division algorithm for ordinal exponentiation.  (This version of
       ~ oeeu gives an explicit expression for the unique solution of the
       equation, in terms of the solution ` P ` to ~ omeu .)  (Contributed by
       Mario Carneiro, 25-May-2015.) $)
    oeeui $p |- ( ( A e. ( On \ 2o ) /\ B e. ( On \ 1o ) ) ->
        ( ( ( C e. On /\ D e. ( A \ 1o ) /\ E e. ( A ^o C ) ) /\
          ( ( ( A ^o C ) .o D ) +o E ) = B ) <->
          ( C = X /\ D = Y /\ E = Z ) ) ) $=
      ( con0 wcel co va vd ve c2o cdif c1o wa coe comu coa wceq w3a wb wss csuc
      eldifi adantr ad2antrr simprl oecl syl2anc om1 syl c0 csn df1o2 wne dif1o
      simprbi ad2antll onelon on0eln0 mpbird snssd eqsstrid 1on omwordi syl3anc
      wi a1i eqsstrrd omcl simplrl oaword1 simplrr sseqtrd sstrd oeeulem simp3d
      mpd simp1d onsuc ontr2 mp2and simplll oeord onsssuc simp2d eloni ordsucss
      word sylc dif20el oen0 syl21anc omword syl31anc mpbid eqeltrrd odi oa1suc
      oaord oveq2d 3eqtr3d eleqtrrd sseldd oesuc eqssd jca eqeltrd simprr suceq
      ad2antrl eqtr3d eleqtrd omord2 wn eqsstrd oveq1d eleq1d 3anass eqeq1d cop
      oveq2 cv wrex cio weq eqeq2d anbi12d adantl ontri1 oa0r syl5ibrcom sylibd
      om0 eqtrd necon3bd sylanbrc impbida pm5.32rd anass bitrdi eleq2d 3anbi23d
      ex bitr3id ne0d weu opeq1 opeq2 cbvrex2vw eqeq1 anbi1d 2rexbidv cbviotavw
      omeu bitrid eqtri opiota sylan9bbr pm5.32da bitrd 3an4anass 3bitr4g ) ERU
      DUESZFRUFUESZUGZGRSZHEUFUESZUGZJEGUHTZSZUWBHUITZJUJTZFUKZUGZUGZGKUKZHLUKZ
      JMUKZUGZUGZUVSUVTUWCULUWFUGUWIUWJUWKULUVRUWHUWIHRSZUWGUGZUGZUWMUVRUWHUWIU
      WNUGZUWGUGUWPUVRUWGUWAUWQUVRUWGUWAUWQUMUVRUWGUGZUWAUWQUWRUWAUGZUWIUWNUWSG
      KUWSGKUNZGKUOZSZUWSUXBUWBEUXAUHTZSZUWSUWBFUNZFUXCSZUXDUWSUWBUWDFUWSUWBUWB
      UFUITZUWDUWSUWBRSZUXGUWBUKUWSERSZUVSUXHUVRUXIUWGUWAUVPUXIUVQERUDUPUQZURZU
      WRUVSUVTUSZEGUTZVAZUWBVBVCZUWSUFHUNZUXGUWDUNZUWSUFVDVEHVFUWSVDHUWSVDHSZHV
      DVGZUVTUXSUWRUVSUVTHESZUXSHEVHZVIVJUWSUWNUXRUXSUMUWSUXIUXTUWNUXKUVTUXTUWR
      UVSHEUFUPVJZEHVKVAZHVLVCVMVNVOUWSUFRSZUWNUXHUXPUXQVSUYDUWSVPVTZUYCUXNUFHU
      WBVQVRWJWAUWSUWDUWEFUWSUWDRSZJRSZUWDUWEUNZUWSUXHUWNUYFUXNUYCUWBHWBZVAZUWS
      UXHUWCUYGUXNUVRUWCUWFUWAWCZUWBJVKZVAZUWDJWDZVAUVRUWCUWFUWAWEZWFWGUVRUXFUW
      GUWAUVRKRSZEKUHTZFUNZUXFAEFKNWHZWIZURUWSUXHUXCRSZUXEUXFUGUXDVSUXNUWSUXIUX
      ARSZVUAUXKUWSUYPVUBUVRUYPUWGUWAUVRUYPUYRUXFUYSWKZURZKWLVCZEUXAUTVAUWBFUXC
      WMVAWNUWSUVSVUBUVPUXBUXDUMUXLVUEUVPUVQUWGUWAWOZGUXAEWPVRVMUWSUVSUYPUWTUXB
      UMUXLVUDGKWQVAVMUWSKGUNZKGUOZSZUWSVUIUYQEVUHUHTZSZUWSUYRFVUJSZVUKUVRUYRUW
      GUWAUVRUYPUYRUXFUYSWRZURUWSFUWBEUITZVUJUWSUWBHUOZUITZVUNFUWSVUOEUNZVUPVUN
      UNZUWSEXAZUXTVUQUWSUXIVUSUXKEWSVCUYBHEWTXBUWSVUORSZUXIUXHVDUWBSZVUQVURUMU
      WSUWNVUTUYCHWLVCUXKUXNUWSUXIUVSVDESZVVAUXKUXLUWSUVPVVBVUFEXCZVCEGXDZXEVUO
      EUWBXFXGXHUWSFUWDUWBUJTZVUPUWSUWEFVVEUYOUWSUWCUWEVVESZUYKUWSUYGUXHUYFUWCV
      VFUMUYMUXNUYJJUWBUWDXLVRXHXIUWSUWBHUFUJTZUITZUWDUXGUJTZVUPVVEUWSUXHUWNUYD
      VVHVVIUKUXNUYCUYEUWBHUFXJVRUWSVVGVUOUWBUIUWSUWNVVGVUOUKUYCHXKVCXMUWSUXGUW
      BUWDUJUXOXMXNXOXPUWSUXIUVSVUJVUNUKZUXKUXLEGXQZVAXOUWSUYQRSZVUJRSZUYRVULUG
      VUKVSUWSUXIUYPVVLUXKVUDEKUTZVAUWSUXIVUHRSZVVMUXKUVSVVOUWRUVTGWLYCZEVUHUTV
      AUYQFVUJWMVAWNUWSUYPVVOUVPVUIVUKUMVUDVVPVUFKVUHEWPVRVMUWSUYPUVSVUGVUIUMVU
      DUXLKGWQVAVMXRUYCXSUWRUWQUGZUVSUVTVVQGKRUWRUWIUWNUSZUVRUYPUWGUWQVUCURXTZV
      VQUXTUXSUVTVVQUXTUWDVUNSZVVQUWDFUNZFVUNSZVVTVVQUWDUWEFVVQUYFUYGUYHVVQUXHU
      WNUYFVVQUXIUVSUXHUVRUXIUWGUWQUXJURZVVSUXMVAZUWRUWIUWNYAZUYIVAZVVQUXHUWCUY
      GVWDUVRUWCUWFUWQWCZUYLVAZUYNVAUVRUWCUWFUWQWEZWFVVQFUXCVUNUVRUXFUWGUWQUYTU
      RVVQVUJUXCVUNVVQVUHUXAEUHUWIVUHUXAUKUWRUWNGKYBYCXMVVQUXIUVSVVJVWCVVSVVKVA
      YDYEVVQUYFVUNRSZVWAVWBUGVVTVSVWFVVQUXHUXIVWJVWDVWCUWBEWBVAUWDFVUNWMVAWNVV
      QUWNUXIUXHVVAUXTVVTUMVWEVWCVWDVVQUXIUVSVVBVVAVWCVVSUVRVVBUWGUWQUVPVVBUVQV
      VCUQZURVVDXEHEUWBYFXGVMVVQFUWBSZYGZUXSVVQUXEVWMVVQUWBUYQFVVQGKEUHVVRXMUVR
      UYRUWGUWQVUMURYHVVQUXHFRSZUXEVWMUMVWDUVRVWNUWGUWQUVQVWNUVPFRUFUPUUAZURUWB
      FUUBVAXHVVQVWLHVDVVQHVDUKZUWEUWBSZVWLVVQVWQVWPUWBVDUITZJUJTZUWBSVVQVWSJUW
      BVVQVWSVDJUJTZJVVQVWRVDJUJVVQUXHVWRVDUKVWDUWBUUFVCYIVVQUYGVWTJUKVWHJUUCVC
      UUGVWGXTVWPUWEVWSUWBVWPUWDVWRJUJHVDUWBUIYNYIYJUUDVVQUWEFUWBVWIYJUUEUUHWJU
      YAUUIXSUUJUUPUUKUWIUWNUWGUULUUMUVRUWIUWOUWLUWIUWOUWNJUYQSZUYQHUITZJUJTZFU
      KZULZUVRUWLUWOUWNUWCUWFULUWIVXEUWNUWCUWFYKUWIUWCVXAUWFVXDUWNUWIUWBUYQJGKE
      UHYNZUUNUWIUWEVXCFUWIUWDVXBJUJUWIUWBUYQHUIVXFYIYIYLUUOUUQUVRVVLVWNUYQVDVG
      ZVXEUWLUMZUVRUXIUYPVVLUXJVUCVVNVAVWOUVRUYQVDUVRUXIUYPVVBVDUYQSUXJVUCVWKEK
      XDXEUURVVLVWNVXGULUAYOZUBYOZUCYOZYMZUKZUYQVXJUITZVXKUJTZFUKZUGZUCUYQYPUBR
      YPZUAUUSVXHUBUCUAUYQFUVGVXPVXBVXKUJTZFUKVXDUBUCUARUYQHJILMIDYOZBYOZCYOZYM
      ZUKZUYQVYAUITZVYBUJTZFUKZUGZCUYQYPBRYPZDYQVXRUAYQOVYIVXRDUAVYIVXTVXLUKZVX
      PUGZUCUYQYPUBRYPDUAYRZVXRVYHVYKVXTVXJVYBYMZUKZVXNVYBUJTZFUKZUGBCUBUCRUYQB
      UBYRZVYDVYNVYGVYPVYQVYCVYMVXTVYAVXJVYBUUTYSVYQVYFVYOFVYQVYEVXNVYBUJVYAVXJ
      UYQUIYNYIYLYTCUCYRZVYNVYJVYPVXPVYRVYMVXLVXTVYBVXKVXJUVAYSVYRVYOVXOFVYBVXK
      VXNUJYNYLYTUVBVYLVYKVXQUBUCRUYQVYLVYJVXMVXPVXTVXIVXLUVCUVDUVEUVHUVFUVIPQV
      XJHUKZVXOVXSFVYSVXNVXBVXKUJVXJHUYQUIYNYIYLVXKJUKVXSVXCFVXKJVXBUJYNYLUVJVC
      VRUVKUVLUVMUVSUVTUWCUWFUVNUWIUWJUWKYKUVO $.
  $}

  ${
    $d a b c d w x y z A $.  $d a b c d w x y z B $.
    $( The division algorithm for ordinal exponentiation.  (Contributed by
       Mario Carneiro, 25-May-2015.) $)
    oeeu $p |- ( ( A e. ( On \ 2o ) /\ B e. ( On \ 1o ) ) ->
        E! w E. x e. On E. y e. ( A \ 1o ) E. z e. ( A ^o x )
          ( w = <. x , y , z >. /\ ( ( ( A ^o x ) .o y ) +o z ) = B ) ) $=
      ( va vd vb vc con0 wcel wa cv wceq coe co wex wrex eqid c2o cdif c1o cotp
      w3a comu coa weu crab cint cuni cop cio c1st cfv c2nd cvv wss csuc simp1d
      oeeulem fvexd oeeui euotd df-3an biancomi anbi1i anbi2i an12 anass 3bitri
      exbii df-rex r19.42v 3bitr2i 2exbii r2ex bitr4i eubii sylib ) EKUAUBLFKUC
      UBLMZDNANZBNZCNZUDOZWBKLZWCEUCUBZLZWDEWBPQZLZUEZWIWCUFQWDUGQFOZMZMZCRZBRA
      RZDUHWEWLMZCWISZBWGSAKSZDUHWAWMDFEGNPQLGKUIUJUKZHNINZJNZULOEWTPQZXAUFQXBU
      GQFOMJXCSIKSHUMZUNUOZXDUPUOZKUQUQABCWAWTKLXCFURFEWTUSPQLGEFWTWTTZVAUTWAXD
      UNVBWAXDUPVBGIJHEFWBWCXDWDWTXEXFXGXDTXETXFTVCVDWPWSDWPWFWHMZWRMZBRARWSWOX
      IABWOWJXHWQMZMZCRXJCWISXIWNXKCWNWEWJXHMZWLMZMXLWQMXKWMXMWEWKXLWLWKWJXHWFW
      HWJVEVFVGVHWEXLWLVIWJXHWQVJVKVLXJCWIVMXHWQCWIVNVOVPWRABKWGVQVRVSVT $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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 $.

  $( Exponentiation with a successor exponent.  Definition 8.30 of
     [TakeutiZaring] p. 67.  (Contributed by Mario Carneiro, 14-Nov-2014.) $)
  nnesuc $p |- ( ( A e. _om /\ B e. _om ) ->
              ( A ^o suc B ) = ( ( A ^o B ) .o A ) ) $=
    ( com wcel con0 csuc coe co comu wceq nnon onesuc sylan ) ACDAEDBCDABFGHABG
    HAIHJAKABLM $.

  ${
    $d x y A $.  $d x B $.
    $( Addition to zero.  Remark in proof of Theorem 4K(2) of [Enderton] p. 81.
       Note:  In this and later theorems, we deliberately avoid the more
       general ordinal versions of these theorems (in this case ~ oa0r ) so
       that we can avoid ~ ax-rep , which is not needed for finite recursive
       definitions.  (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.  Theorem 2.20 of [Schloeder] p. 6.  (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.  Theorem 2.20 of [Schloeder] p. 6.  (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 $.

    $( Closure of exponentiation of natural numbers.  Proposition 8.17 of
       [TakeutiZaring] p. 63.  Theorem 2.20 of [Schloeder] p. 6.  (Contributed
       by NM, 24-Mar-2007.)  (Proof shortened by Andrew Salmon,
       22-Oct-2011.) $)
    nnecl $p |- ( ( A e. _om /\ B e. _om ) -> ( A ^o B ) e. _om ) $=
      ( vx vy com wcel coe co cv wi wceq oveq2 eleq1d imbi2d csuc c1o con0 nnon
      c0 expcom oe0 syl df-1o peano1 peano2 ax-mp eqeltri eqeltrdi nnmcl adantr
      wa comu nnesuc sylibrd finds2 vtoclga impcom ) BEFAEFZABGHZEFZURACIZGHZEF
      ZJURUTJCBEVABKZVCUTURVDVBUSEVABAGLMNVCASGHZEFADIZGHZEFZAVFOZGHZEFZURCDVAS
      KVBVEEVASAGLMVAVFKVBVGEVAVFAGLMVAVIKVBVJEVAVIAGLMURVEPEURAQFVEPKARAUAUBPS
      OZEUCSEFVLEFUDSUEUFUGUHURVFEFZVHVKJURVMUKZVHVGAULHZEFZVKURVHVPJVMVHURVPVG
      AUITUJVNVJVOEAVFUMMUNTUOUPUQ $.
  $}

  ${
    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 $.
  $}

  $( Reverse closure law for addition of natural numbers.  Exercise 1 of
     [TakeutiZaring] p. 62 and its converse.  (Contributed by NM,
     12-Dec-2004.) $)
  nnarcl $p |- ( ( A e. On /\ B e. On ) ->
               ( ( A +o B ) e. _om <-> ( A e. _om /\ B e. _om ) ) ) $=
    ( con0 wcel wa coa co com wi oaword1 word eloni ordom ordtr2 sylancl adantr
    wss expd mpd oaword2 ancoms adantl jcad nnacl impbid1 ) ACDZBCDZEZABFGZHDZA
    HDZBHDZEUHUJUKULUHAUIQZUJUKIZABJUFUMUNIUGUFUMUJUKUFAKHKZUMUJEUKIALMAUIHNORP
    SUHBUIQZUJULIZUGUFUPBATUAUGUPUQIUFUGUPUJULUGBKUOUPUJEULIBLMBUIHNORUBSUCABUD
    UE $.

  ${
    $d x y A $.  $d x y z B $.  $d x y z C $.
    $( 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 $.

    $( 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 wi wa ancoms csuc wss cv wceq oveq2 sseq2d imbi2d
      nnasuc elnn adantll word nnord ordsucss syl ad2antlr peano2b ssid sssucid
      2a1i sstr2 imbitrrid ex ad2antrr findsg exp31 biimtrid com4r imp31 sseq1d
      mpi a2d cvv ovex sucssel ax-mp biimtrdi adantlr 3syld imp an32s mpdan ) C
      FGZBFGZABGZCAHIZCBHIZGZJVNVOKZVPVSVTVPKAFGZVSVOVPWAVNVPVOWAABUALUBVTWAVPV
      SVTWAKZVPVSWBVPAMZBNZCWCHIZVRNZVSVOVPWDJZVNWAVOBUCWGBUDABUEUFUGVNVOWAWDWF
      JVOWAWDVNWFWAWCFGZVOWDVNWFJZJAUHVOWHWDWIVNWECDOZHIZNZJVNWEWENZJVNWECEOZHI
      ZNZJVNWECWNMZHIZNZJWIDEBWCWJWCPZWLWMVNWTWKWEWEWJWCCHQRSWJWNPZWLWPVNXAWKWO
      WEWJWNCHQRSWJWQPZWLWSVNXBWKWRWEWJWQCHQRSWJBPZWLWFVNXCWKVRWEWJBCHQRSWMWHVN
      WEUIUKWNFGZWHKWCWNNZKVNWPWSXDVNWPWSJZJWHXEXDVNXFWPWSXDVNKZWEWOMZNZWPWOXHN
      XIWOUJWEWOXHULVBXGWRXHWEVNXDWRXHPCWNTLRUMUNUOVCUPUQURUSUTVNWAWFVSJVOVNWAK
      ZWFVQMZVRNZVSXJWEXKVRCATVAVQVDGXLVSJCAHVEVQVRVDVFVGVHVIVJVKVLVMUNL $.
  $}

  $( 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 coa co wi nnaordi wceq wo wn word wa nnacl nnord anim12i ordtri2
    wb syl w3a 3adant1 oveq2 a1i 3adant2 con3d df-3an ancom anandi 3bitri sylbi
    orim12d 3adant3 3imtr4d impbid ) ADEZBDEZCDEZUAZABEZCAFGZCBFGZEZUQURUTVCHUP
    ABCIUBUSVAVBJZVBVAEZKZLZABJZBAEZKZLZVCUTUSVJVFUSVHVDVIVEVHVDHUSABCFUCUDUPUR
    VIVEHUQBACIUEULUFUSVAMZVBMZNZVCVGSUSURUPNZURUQNZNZVNUSUPUQNZURNURVRNVQUPUQU
    RUGVRURUHURUPUQUIUJVOVLVPVMVOVADEVLCAOVAPTVPVBDEVMCBOVBPTQUKVAVBRTUSAMZBMZN
    ZUTVKSUPUQWAURUPVSUQVTAPBPQUMABRTUNUO $.

  $( 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 y $.
    $( 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 ) ) ) $=
      ( vx vy com wcel wss coa co wi wa c0 wceq sseq12d imbi2d con0 nnon ancoms
      oveq2 cv csuc weq oa0 adantr adantl biimprd syl2an wb nnacl adantrr eloni
      word adantrl ordsucsssuc syl2anc biimpa nnasuc mpbird ex imim2d a2d finds
      3syl com12 3impia ) AFGZBFGZCFGZABHZACIJZBCIJZHZKZVIVGVHLZVNVOVJADUAZIJZB
      VPIJZHZKZKVOVJAMIJZBMIJZHZKZKVOVJAEUAZIJZBWEIJZHZKZKVOVJAWEUBZIJZBWJIJZHZ
      KZKVOVNKDECVPMNZVTWDVOWOVSWCVJWOVQWAVRWBVPMAITVPMBITOPPDEUCZVTWIVOWPVSWHV
      JWPVQWFVRWGVPWEAITVPWEBITOPPVPWJNZVTWNVOWQVSWMVJWQVQWKVRWLVPWJAITVPWJBITO
      PPVPCNZVTVNVOWRVSVMVJWRVQVKVRVLVPCAITVPCBITOPPVGAQGZBQGZWDVHARBRWSWTLZWCV
      JXAWAAWBBWSWAANWTAUDUEWTWBBNWSBUDUFOUGUHWEFGZVOWIWNXBVOWIWNKXBVOLZWHWMVJX
      CWHWMXCWHLWMWFUBZWGUBZHZXCWHXFXCWFUMZWGUMZWHXFUIXCWFFGZWFQGXGXBVGXIVHVGXB
      XIAWEUJSUKWFRWFULVDXCWGFGZWGQGXHXBVHXJVGVHXBXJBWEUJSUNWGRWGULVDWFWGUOUPUQ
      XCWMXFUIWHXCWKXDWLXEXBVGWKXDNZVHVGXBXKAWEURSUKXBVHWLXENZVGVHXBXLBWEURSUNO
      UEUSUTVAUTVBVCVEVF $.
  $}

  ${
    $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 stoic3 3adant1 nndi
      syl3an2 3exp expd com34 pm2.43d 3imp eqtrd imbitrrid com3r finds2 vtoclga
      impd expdcom ) AFGZBFGZCFGZABHIZCHIZABCHIZHIZJZVGVEVFVLVEVFKZVHDLZHIZABVN
      HIZHIZJZMVMVLMDCFVNCJZVRVLVMVSVOVIVQVKVNCVHHNVSVPVJAHVNCBHNOPUAVRVHQHIZAB
      QHIZHIZJVHELZHIZABWCHIZHIZJZVHWCUBZHIZABWHHIZHIZJZVMDEVNQJZVOVTVQWBVNQVHH
      NWMVPWAAHVNQBHNOPVNWCJZVOWDVQWFVNWCVHHNWNVPWEAHVNWCBHNOPVNWHJZVOWIVQWKVNW
      HVHHNWOVPWJAHVNWHBHNOPVMVTQWBVMVHFGZVTQJABRZVHSUCVFVEWBAQHIQVFWAQAHBSOASU
      DUEWCFGZVEVFWGWLMZVEVFWRWSVEVFWRWSWGWLVEVFWRUFZWDVHTIZWFVHTIZJWDWFVHTUGWT
      WIXAWKXBVEVFWPWRWIXAJWQVHWCUHUIWTWKAWEBTIZHIZXBWTWJXCAHVFWRWJXCJVEBWCUHUJ
      OVEVFWRXDXBJZVEVFWRXEMVEVFWRVFXEVEVFWRVFXEMVEVFWRKZVFXEXFVEWEFGVFXEBWCRAW
      EBUKULUMUNUOUPUQURPUSUMUTVCVAVBVDUQ $.

    $( 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 $.
  $}

  $( 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 ) ) ) $=
    ( com wcel w3a wn coa co wss nnaord 3com12 notbid word nnord ordtri1 syl2an
    wb nnacl ancoms 3adant3 3adant2 3adant1 syl2anc 3bitr4d ) ADEZBDEZCDEZFZBAE
    ZGZCBHIZCAHIZEZGZABJZUMULJZUIUJUNUGUFUHUJUNRBACKLMUFUGUPUKRZUHUFANBNURUGAOB
    OABPQUAUIUMDEZULDEZUQUORZUFUHUSUGUHUFUSCASTUBUGUHUTUFUHUGUTCBSTUCUSUMNULNVA
    UTUMOULOUMULPQUDUE $.

  $( 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 wa c0 coa co wceq adantr wss 0ss wb peano1 nnaword 3com13 mp3an3
    nna0 mpbii eqsstrrd ) ACDZBCDZEZAAFGHZABGHZUAUDAIUBARJUCFBKZUDUEKZBLUAUBFCD
    ZUFUGMZNUHUBUAUIFBAOPQST $.

  $( 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 $.

  ${
    $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 3adant1 wne wceq wb nnord wo syl2anc
    word w3a impcomd ne0i nnm0r oveq1 eqeq1d syl5ibrcom necon3d adantr ord0eln0
    syl5 syl adantl sylibrd wn oveq2 3adantl2 orim12d con3d simpl3 simpl1 nnmcl
    a1i simpl2 ordtri2 syl2an 3imtr4d com23 mpdd jcad impbid ) ADEZBDEZCDEZUAZA
    BEZFCEZGZCAHIZCBHIZEZVMVNVRWAJVLVMVNGZVQVPWAWBVQVPWAJABCKLUBMVOWAVPVQVOWAVQ
    VPVMVNWAVQJVLWBWACFNZVQVMWAWCJVNWAVTFNVMWCVTVSUCVMCFVTFVMVTFOCFOZFBHIZFOBUD
    WDVTWEFCFBHUEUFUGUHUKUIVNVQWCPZVMVNCTWFCQCUJULUMUNMZVOVQWAVPVOVQWAVPJVOVQGZ
    VSVTOZVTVSEZRZUOZABOZBAEZRZUOZWAVPWHWOWKWHWMWIWNWJWMWIJWHABCHUPVCVLVNVQWNWJ
    JVMBACKUQURUSWHVSDEZVTDEZWAWLPZWHVNVLWQVLVMVNVQUTZVLVMVNVQVAZCAVBSWHVNVMWRW
    TVLVMVNVQVDZCBVBSWQVSTVTTWSWRVSQVTQVSVTVEVFSWHVLVMVPWPPZXAXBVLATBTXCVMAQBQA
    BVEVFSVGLVHVIWGVJVK $.

  $( 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 co wss iba con0 nnon syl ontri1 syl2anc nnmcl
    wb nnmord 3com12 sylan9bbr notbid simpl1 simpl2 simpl3 3bitr4d ) ADEZBDEZCD
    EZFZGCEZHZBAEZIZCBJKZCAJKZEZIZABLZURUQLZUNUOUSUMUOUOUMHZULUSUMUOMUJUIUKVCUS
    TBACUAUBUCUDUNANEZBNEZVAUPTUNUIVDUIUJUKUMUEZAOPUNUJVEUIUJUKUMUFZBOPABQRUNUR
    NEZUQNEZVBUTTUNURDEZVHUNUKUIVJUIUJUKUMUGZVFCASRUROPUNUQDEZVIUNUKUJVLVKVGCBS
    RUQOPURUQQRUH $.

  $( 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 $.

  $( Weak ordering property of multiplication.  (Contributed by Mario Carneiro,
     17-Nov-2014.) $)
  nnmwordi $p |- ( ( A e. _om /\ B e. _om /\ C e. _om ) ->
                  ( A C_ B -> ( C .o A ) C_ ( C .o B ) ) ) $=
    ( com wcel w3a c0 wss comu co wi wa nnmword biimpd ex wn wceq nnm0r sseq12d
    oveq1 word nnord ord0eln0 necon2bbid syl 3ad2ant3 ssid adantr adantl mpbiri
    wb syl5ibrcom 3adant3 sylbird a1dd pm2.61d ) ADEZBDEZCDEZFZGCEZABHZCAIJZCBI
    JZHZKZUTVAVFUTVALVBVEABCMNOUTVAPZVEVBUTVGCGQZVEUSUQVHVGUKZURUSCUAZVICUBVJVA
    CGCUCUDUEUFUQURVHVEKUSUQURLZVEVHGAIJZGBIJZHZVKVNGGHGUGVKVLGVMGUQVLGQURARUHU
    RVMGQUQBRUISUJVHVCVLVDVMCGAITCGBITSULUMUNUOUP $.

  $( Weak ordering property of ordinal multiplication.  Proposition 8.21 of
     [TakeutiZaring] p. 63, limited to natural numbers.  (Contributed by Mario
     Carneiro, 17-Nov-2014.) $)
  nnmwordri $p |- ( ( A e. _om /\ B e. _om /\ C e. _om ) ->
                    ( A C_ B -> ( A .o C ) C_ ( B .o C ) ) ) $=
    ( com wcel w3a comu co nnmwordi wceq nnmcom 3adant2 3adant1 sseq12d sylibrd
    wss ) ADEZBDEZCDEZFZABPCAGHZCBGHZPACGHZBCGHZPABCITUCUAUDUBQSUCUAJRACKLRSUDU
    BJQBCKMNO $.

  ${
    $d x y A $.  $d x y 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 ) ) $=
      ( vy com wcel wa wss coa co wceq con0 oveq2 sseq2d syl2anc word c0 adantr
      syl nfcv cv wrex crab cint simplr simpll nnaword2 elrabd intss1 wi ssrab2
      nnon wne ne0d oninton sylancr eloni ordom ordtr2 sylancl mp2and csuc nna0
      ad2antrr simpr eqsstrd sseq1d syl5ibrcom simprr oveq2d simprl eqtrd nnord
      nnasuc wn vex sucid eleqtrrid onnminsb sylc adantl ordtri1 con2bid mpbird
      wb nnacl ordsucss rexlimdvaa nn0suc mpjaod onint nfrab1 nfint nfov elrabf
      wo nfss simprbi eqssd eqeq1d rspcev ex nnaword1 sseq2 syl5ibcom rexlimdva
      adantlr impbid ) BEFZCEFZGZBCHZBAUAZIJZCKZAEUBZXKXLXPXKXLGZCBDUAZIJZHZDLU
      CZUDZEFZBYBIJZCKZXPXQYBCHZXJYCXQCYAFYFXQXTCBCIJZHZDCLXRCKXSYGCXRCBIMNXQXJ
      CLFXIXJXLUEZCULSXQXJXIYHYIXIXJXLUFZCBUGOUHZCYAUISYIXQYBPZEPYFXJGYCUJXQYBL
      FZYLXQYALHZYAQUMZYMXTDLUKZXQYACYKUNZYAUOUPYBUQSURYBCEUSUTVAZXQYDCXQYBQKZY
      DCHZYBXMVBZKZAEUBZXQYTYSBQIJZCHXQUUDBCXIUUDBKXJXLBVCVDXKXLVEVFYSYDUUDCYBQ
      BIMVGVHXQUUBYTAEXQXMEFZUUBGZGZYDXNVBZCUUGYDBUUAIJZUUHUUGYBUUABIXQUUEUUBVI
      VJUUGXIUUEUUIUUHKXQXIUUFYJRZXQUUEUUBVKZBXMVNOVLUUGCPZXNCFZUUHCHXQUULUUFXQ
      XJUULYICVMSRZUUGUUMCXNHZVOZUUFUUPXQUUFXMLFZXMYBFUUPUUEUUQUUBXMULRUUFXMUUA
      YBXMAVPVQUUEUUBVEVRXTUUODXMXRXMKXSXNCXRXMBIMNVSVTWAUUGUUOUUMUUGUULXNPZUUO
      UUMVOWEUUNUUGXNEFZUURUUGXIUUEUUSUUJUUKBXMWFOXNVMSCXNWBOWCWDXNCWGVTVFWHXQY
      CYSUUCWPYRAYBWISWJXQYBYAFZCYDHZXQYNYOUUTYPYQYAWKUPUUTYMUVAXTUVADYBLDYAXTD
      LWLWMZDLTDCYDDCTDBYBIDBTDITUVBWNWQXRYBKXSYDCXRYBBIMNWOWRSWSXOYEAYBEXMYBKX
      NYDCXMYBBIMWTXAOXBXKXOXLAEXKUUEGBXNHZXOXLXIUUEUVCXJBXMXCXGXNCBXDXEXFXH $.

    $( 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 ) ) ) $=
      ( com wcel wa c0 cv coa co wceq wrex con0 wi nnon adantl onelss wb adantr
      wss nnawordex sylibd simplr syl5ibrcom peano1 nnaord mp3an1 ancoms eleq1d
      syl eleq2 nna0 bitrd adantlr sylibrd ancrd reximdva mpdd biimpa syl5ibcom
      ex expimpd rexlimdva impbid ) BDEZCDEZFZBCEZGAHZEZBVIIJZCKZFZADLZVGVHVLAD
      LZVNVGVHBCTZVOVGCMEZVHVPNVFVQVECOPCBQUJABCUAUBVEVHVOVNNZNVFVEVHVRVEVHFZVL
      VMADVSVIDEZFZVLVJWAVLBVKEZVJWAWBVLVHVEVHVTUCVKCBUKZUDVEVTVJWBRVHVEVTFZVJB
      GIJZVKEZWBVTVEVJWFRZGDEVTVEWGUEGVIBUFUGUHWDWEBVKVEWEBKVTBULSUIUMZUNUOUPUQ
      VASURVEVNVHNVFVEVMVHADWDVJVLVHWDVJFWBVLVHWDVJWBWHUSWCUTVBVCSVD $.
  $}

  ${
    $d A x y $.  $d B x y $.
    $( Equivalence for ordering.  (Contributed by Scott Fenton,
       18-Apr-2025.) $)
    nnaordex2 $p |- ( ( A e. _om /\ B e. _om ) -> ( A e. B <->
                   E. x e. _om ( A +o suc x ) = B ) ) $=
      ( vy com wcel wa c0 cv coa co wceq wrex csuc nnaordex nn0suc ad2antrl syl
      wo rexlimdvaa wn simprrl n0i orcnd simprrr oveq2 eqeq1d syl5ibcom reximdv
      mpd peano2 nnord 0elsuc simprr eleq2 anbi12d rspcev syl12anc impbid bitrd
      word ) BEFCEFGZBCFHDIZFZBVCJKZCLZGZDEMZBAIZNZJKZCLZAEMZDBCOVBVHVMVBVGVMDE
      VBVCEFZVGGGZVCVJLZAEMZVMVOVCHLZVQVNVRVQSVBVGAVCPQVOVDVRUAVBVNVDVFUBVCHUCR
      UDVOVPVLAEVOVFVPVLVBVNVDVFUEVPVEVKCVCVJBJUFUGZUHUIUJTVBVLVHAEVBVIEFZVLGGZ
      VJEFZHVJFZVLVHVTWBVBVLVIUKQWAVIVAZWCVTWDVBVLVIULQVIUMRVBVTVLUNVGWCVLGDVJE
      VPVDWCVFVLVCVJHUOVSUPUQURTUSUT $.
  $}

  $( The ordinal 1 is a natural number.  For a shorter proof using Peano's
     postulates that depends on ~ ax-un , see ~ 1onnALT .  Lemma 2.2 of
     [Schloeder] p. 4.  (Contributed by NM, 29-Oct-1995.)  Avoid ~ ax-un .
     (Revised by BTernaryTau, 1-Dec-2024.) $)
  1onn $p |- 1o e. _om $=
    ( vx c1o com wcel con0 cv wlim wi wal 1on 1ellim ax-gen elom mpbir2an ) BCD
    BEDAFZGBODHZAIJPAOKLABMN $.
  $( $j usage '1onn' avoids 'ax-un'; $)

  $( Shorter proof of ~ 1onn using Peano's postulates that depends on ~ ax-un .
     (Contributed by NM, 29-Oct-1995.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  1onnALT $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.  For a shorter proof using Peano's
     postulates that depends on ~ ax-un , see ~ 2onnALT .  (Contributed by NM,
     28-Sep-2004.)  Avoid ~ ax-un .  (Revised by BTernaryTau, 1-Dec-2024.) $)
  2onn $p |- 2o e. _om $=
    ( vx c2o com wcel con0 cv wlim wi wal 2on 2ellim ax-gen elom mpbir2an ) BCD
    BEDAFZGBODHZAIJPAOKLABMN $.
  $( $j usage '2onn' avoids 'ax-un'; $)

  $( Shorter proof of ~ 2onn using Peano's postulates that depends on ~ ax-un .
     (Contributed by NM, 28-Sep-2004.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  2onnALT $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
    $.

  $( Ordinal one is not ordinal two.  Analogous to ~ 1ne2 .  (Contributed by
     AV, 17-Oct-2023.) $)
  1one2o $p |- 1o =/= 2o $=
    ( c1o csuc c2o com wcel wne 1onn omsucne ax-mp df-2o neeqtrri ) AABZCADEALF
    GAHIJK $.

  ${
    $d x y z A $.  $d x y z B $.  $d x y z C $.
    $( Lemma for ~ oaabs .  (Contributed by NM, 9-Dec-2004.) $)
    oaabslem $p |- ( ( _om e. On /\ A e. _om ) -> ( A +o _om ) = _om ) $=
      ( vx com con0 wcel wa coa co wss cv ciun wlim wceq nnon limom jctr syl2an
      oalim wral word ordom nnacl ordelss sylancr ralrimiva iunss sylibr adantr
      eqsstrd ancoms oaword2 sylan2 eqssd ) CDEZACEZFACGHZCUOUNUPCIUOUNFUPBCABJ
      ZGHZKZCUOADEZUNCLZFUPUSMUNANZUNVAOPBACDRQUOUSCIZUNUOURCIZBCSVCUOVDBCUOUQC
      EFCTURCEVDUAAUQUBCURUCUDUEBCURCUFUGUHUIUJUOUNUTCUPIVBCAUKULUM $.

    $( Ordinal addition absorbs a natural number added to the left of a
       transfinite number.  Proposition 8.10 of [TakeutiZaring] p. 59.
       (Contributed by NM, 9-Dec-2004.)  (Proof shortened by Mario Carneiro,
       29-May-2015.) $)
    oaabs $p |- ( ( ( A e. _om /\ B e. On ) /\ _om C_ B )
        -> ( A +o B ) = B ) $=
      ( vx com wcel con0 wa wss cv coa wceq wrex wreu cvv ssexg omelon2 syl6com
      co ex adantr imp adantll simplr jca oawordeu sylancom syl ad3antrrr simpr
      reurex nnon oaass syl3anc simpll oaabslem syl2anc oveq1d oveq2 id eqeq12d
      eqtr3d syl5ibcom rexlimdva mpd ) ADEZBFEZGZDBHZGZDCIZJRZBKZCFLZABJRZBKZVI
      VLCFMZVMVGVHDFEZVFGVPVIVQVFVFVHVQVEVFVHVQVHVFDNEZVQVHVFVRDBFOSPQUAUBZVEVF
      VHUCUDCDBUEUFVLCFUJUGVIVLVOCFVIVJFEZGZAVKJRZVKKVLVOWAADJRZVJJRZWBVKWAAFEZ
      VQVTWDWBKVEWEVFVHVTAUKUHVIVQVTVSTVIVTUIADVJULUMWAWCDVJJVIWCDKZVTVIVQVEWFV
      SVEVFVHUNAUOUPTUQVAVLWBVNVKBVKBAJURVLUSUTVBVCVD $.

    $( The absorption law ~ oaabs is also a property of higher powers of
       ` _om ` .  (Contributed by Mario Carneiro, 29-May-2015.) $)
    oaabs2 $p |- ( ( ( A e. ( _om ^o C ) /\ B e. On ) /\ ( _om ^o C ) C_ B )
        -> ( A +o B ) = B ) $=
      ( vx vy com coe co wcel con0 wa wss coa wceq c0 syl syl2anc adantr c1o wi
      vz cv wrex wreu n0i cxp wfn cdm fnoe fndm ax-mp ndmov nsyl2 ad2antrr oecl
      simplr simpr oawordeu syl21anc reurex simpll onelon syl3anc csuc wlim w3o
      oaass word simprd eloni ordzsl sylib simplll simpld oe0 sylan9eqr eleqtrd
      oveq2 el1o oveq1d oa0r eqtrd ex simprl limom jctir simprr oveq2d omordlim
      comu oesuc nnon ad2antrl ordelss oawordri mpd odi oaabslem eqtr3d 3eqtr4d
      omcl sseqtrd rexlimddv oaword2 eqssd rexlimdvaa cdif ciun oelim2 syl12anc
      eliun eldifi c2o 1onn ondif2 sylanblrc oelimcl oalim wb eleq2d bitrdi cun
      simplrl simplrr ssun1 ad3antrrr ordunel peano1 a1i oewordi syl31anc sstrd
      wral mpi ordsucss 1on onsuc expr sylan2 rexlimdva ssun2 oaword mpbid mp1i
      ordom mp2 omwordi omsuc eqtr2d 3sstr4d sylc sylbid ralrimiv iunss eqsstrd
      om1 sylibr 3jaod id eqeq12d syl5ibcom ) AFCGHZIZBJIZKZUVBBLZKZUVBDUBZMHZB
      NZDJUCZABMHZBNZUVGUVJDJUDZUVKUVGUVBJIZUVDUVFUVNUVGFJIZCJIZKZUVOUVCUVRUVDU
      VFUVCUVBONUVRUVBAUEFCJGGJJUFZUGGUHUVSNUIUVSGUJUKULUMUNZFCUOPZUVCUVDUVFUPU
      VEUVFUQDUVBBURUSUVJDJUTPUVGUVJUVMDJUVGUVHJIZKZAUVIMHZUVINUVJUVMUWCAUVBMHZ
      UVHMHZUWDUVIUWCAJIZUVOUWBUWFUWDNUVGUWGUWBUVGUVOUVCUWGUWAUVCUVDUVFVAUVBAVB
      QZRUVGUVOUWBUWARUVGUWBUQAUVBUVHVGVCUWCUWEUVBUVHMUVGUWEUVBNZUWBUVGCONZCUVH
      VDZNZDJUCZCVEZVFZUWIUVGCVHZUWOUVGUVQUWPUVGUVPUVQUVTVIZCVJPZDCVKVLUVGUWJUW
      IUWMUWNUVGUWJUWIUVGUWJKZUWEOUVBMHZUVBUWSAOUVBMUWSASIAONUWSAUVBSUVCUVDUVFU
      WJVMUWJUVGUVBFOGHZSCOFGVRUVGUVPUXASNUVGUVPUVQUVTVNZFVOPVPVQAVSVLVTUWSUVOU
      WTUVBNUVGUVOUWJUWARUVBWAPWBWCUVGUWLUWIDJUVGUWBUWLKZKZUWEUVBUXDAFUVHGHZEUB
      ZWJHZIZUWEUVBLZEFUXDUXEJIZUVPFVEZKAUXEFWJHZIUXHEFUCUXDUVPUWBUXJUVGUVPUXCU
      XBRZUVGUWBUWLWDZFUVHUOZQZUXDUVPUXKUXMWEWFUXDAUVBUXLUVCUVDUVFUXCVMUXDUVBFU
      WKGHZUXLUXDCUWKFGUVGUWBUWLWGWHUXDUVPUWBUXQUXLNUXMUXNFUVHWKQWBZVQEUXEFAJWI
      USUXDUXFFIZUXHKZKZUWEUXGUVBMHZUVBUYAAUXGLZUWEUYBLZUYAUXGVHZUXHUYCUYAUXGJI
      ZUYEUYAUXJUXFJIZUYFUXDUXJUXTUXPRZUXSUYGUXDUXHUXFWLWMZUXEUXFXAQZUXGVJPUXDU
      XSUXHWGUXGAWNQUYAUWGUYFUVOUYCUYDTUVGUWGUXCUXTUWHUNUYJUVGUVOUXCUXTUWAUNAUX
      GUVBWOVCWPUYAUXGUXLMHZUXLUYBUVBUYAUXEUXFFMHZWJHZUYKUXLUYAUXJUYGUVPUYMUYKN
      UYHUYIUXDUVPUXTUXMRZUXEUXFFWQVCUYAUYLFUXEWJUYAUVPUXSUYLFNUYNUXDUXSUXHWDUX
      FWRQWHWSUYAUVBUXLUXGMUXDUVBUXLNUXTUXRRZWHUYOWTXBXCUVGUVBUWELZUXCUVGUVOUWG
      UYPUWAUWHUVBAXDQZRXEXFUVGUWNUWIUVGUWNKZUWEUVBUYRAUXEIZDCSXGZUCZUXIUYRADUY
      TUXEXHZIVUAUYRAUVBVUBUVCUVDUVFUWNVMUYRUVPUVQUWNUVBVUBNUVGUVPUWNUXBRZUVGUV
      QUWNUWQRZUVGUWNUQZDFCJXIXJVQDAUYTUXEXKVLUYRUYSUXIDUYTUVHUYTIUYRUVHCIZUYSU
      XITUVHCSXLUYRVUFUYSUXIUYRVUFUYSKZKZUWEEUVBAUXFMHZXHZUVBVUHUWGUVOUVBVEZUWE
      VUJNUVGUWGUWNVUGUWHUNZUVGUVOUWNVUGUWAUNVUHFJXMXGIZUVQUWNVUKVUHUVPSFIZVUMU
      YRUVPVUGVUCRZXNFXOXPUYRUVQVUGVUDRZUVGUWNVUGUPFCJXQXJEAUVBJXRXJVUHVUIUVBLZ
      EUVBYMVUJUVBLVUHVUQEUVBVUHUXFUVBIZUXFFUAUBZGHZIZUAUYTUCZVUQUYRVURVVBXSVUG
      UYRVURUXFUAUYTVUTXHZIVVBUYRUVBVVCUXFUYRUVPUVQUWNUVBVVCNVUCVUDVUEUAFCJXIXJ
      XTUAUXFUYTVUTXKYARVUHVVAVUQUAUYTVUSUYTIVUHVUSCIZVVAVUQTVUSCSXLVUHVVDVVAVU
      QVUHVVDVVAKZKZVUIFUVHVUSYBZVDZGHZUVBVVFVUIFVVGGHZVVJMHZVVIVVFVUIVVJUXFMHZ
      VVKVVFAVVJLZVUIVVLLZVVFAUXEVVJVVFUXEVHZUYSAUXELVVFUXJVVOVVFUVPUWBUXJVUHUV
      PVVEVUORZVVFUVQVUFUWBVUHUVQVVEVUPRZUYRVUFUYSVVEYCZCUVHVBQZUXOQUXEVJPUYRVU
      FUYSVVEYDUXEAWNQVVFUVHVVGLZUXEVVJLZUVHVUSYEVVFUWBVVGJIZUVPOFIZVVTVWATVVSV
      VFUVQVVGCIZVWBVVQVVFUWPVUFVVDVWDUVGUWPUWNVUGVVEUWRYFZVVRVUHVVDVVAWDZCUVHV
      USYGVCZCVVGVBQZVVPVWCVVFYHYIZUVHVVGFYJYKYNYLVVFUWGVVJJIZUYGVVMVVNTVUHUWGV
      VEVULRVVFUVPVWBVWJVVPVWHFVVGUOQZVVFVUTJIZVVAUYGVVFUVPVUSJIZVWLVVPVVFUVQVV
      DVWMVVQVWFCVUSVBQZFVUSUOQZVUHVVDVVAWGZVUTUXFVBQZAVVJUXFWOVCWPVVFUXFVVJLZV
      VLVVKLZVVFUXFVUTVVJVVFVUTVHZVVAUXFVUTLVVFVWLVWTVWOVUTVJPVWPVUTUXFWNQVVFVU
      SVVGLZVUTVVJLZVUSUVHUUAVVFVWMVWBUVPVWCVXAVXBTVWNVWHVVPVWIVUSVVGFYJYKYNYLV
      VFUYGVWJVWJVWRVWSXSVWQVWKVWKUXFVVJVVJUUBVCUUCYLVVFVVJSVDZWJHZVVJFWJHZVVKV
      VIVVFVXCFLZVXDVXELZFVHVUNVXFUUEXNSFYOUUFVVFVXCJIZUVPVWJVXFVXGTSJIZVXHVVFY
      PSYQUUDVVPVWKVXCFVVJUUGVCYNVVFVXDVVJSWJHZVVJMHZVVKVVFVWJVXIVXDVXKNVWKVXIV
      VFYPYIVVJSUUHQVVFVXJVVJVVJMVVFVWJVXJVVJNVWKVVJUUPPVTUUIVVFUVPVWBVVIVXENVV
      PVWHFVVGWKQUUJYLVVFVVHCLZVVIUVBLZVVFUWPVWDVXLVWEVWGVVGCYOUUKVVFVVHJIZUVQU
      VPVWCVXLVXMTVVFVWBVXNVWHVVGYQPVVQVVPVWIVVHCFYJYKWPYLYRYSYTUULUUMEUVBVUIUV
      BUUNUUQUUOYRYSYTWPUVGUYPUWNUYQRXEWCUURWPRVTWSUVJUWDUVLUVIBUVIBAMVRUVJUUSU
      UTUVAYTWP $.

    $( Lemma for ~ omabs .  (Contributed by Mario Carneiro, 30-May-2015.) $)
    omabslem $p |- ( ( _om e. On /\ A e. _om /\ (/) e. A ) ->
        ( A .o _om ) = _om ) $=
      ( vx com con0 wcel c0 w3a comu co wa ciun wlim wceq nnon limom jctr omlim
      wss cv syl2an wral word ordom nnmcl ordelss ralrimiva iunss sylibr adantr
      sylancr eqsstrd ancoms 3adant3 omword2 3impa syl3an2 eqssd ) CDEZACEZFAEZ
      GACHIZCURUSVACRZUTUSURVBUSURJVABCABSZHIZKZCUSADEZURCLZJVAVEMURANZURVGOPBA
      CDQTUSVECRZURUSVDCRZBCUAVIUSVJBCUSVCCEJCUBVDCEVJUCAVCUDCVDUEUJUFBCVDCUGUH
      UIUKULUMUSURVFUTCVARZVHURVFUTVKCAUNUOUPUQ $.

    $( Ordinal multiplication is also absorbed by powers of ` _om ` .
       (Contributed by Mario Carneiro, 30-May-2015.) $)
    omabs $p |- ( ( ( A e. _om /\ (/) e. A ) /\ ( B e. On /\ (/) e. B ) ) ->
        ( A .o ( _om ^o B ) ) = ( _om ^o B ) ) $=
      ( vy vz com wcel c0 wa con0 coe co comu wi oveq2d eqeq12d adantr ad2antrr
      wceq syl syl2anc vx cv csuc eleq2 oveq2 imbi12d pm2.21i a1i simprl simpll
      noel simplr omabslem syl3anc c1o suceq df-1o eqtr4di oe1 ad2antrl 3eqtr4d
      sylan9eqr a1dd oveq1 oesuc adantl nnon oecl omass eqtr4d imbitrrid imim2d
      ex com23 wo simprr on0eqel mpjaod anassrs expcom wlim wral ciun ad3antrrr
      cvv vex jctil limelon c2o cdif 1onn ondif2 sylanblrc oelimcl syl12anc wss
      omlim wrex simplrl oelim2 eleq2d eliun bitrdi wb anass wne onelon on0eln0
      pm5.32da dif1o bitr4di anbi1d rexbidv2 bitr4d r19.29 id imp anim1i anasss
      bitr3id word eloni omord2 syl31anc mpbid eleqtrd oeord ontr1 ordelss syl5
      mp2and rexlimdva expdimp sylbid ralrimiv sylibr eqsstrd simpllr omword2
      iunss syl21anc eqssd tfinds3 com12 adantrr imp32 wn nnm0 cxp wfn cdm fnoe
      an32s fndm ax-mp ndmov pm2.61dan ) AEFZGAFZHZBIFZGBFZHZHZEIFZUVAHZAEBJKZL
      KZUVGRZUUTUVFUVCUVIUUTUVFHUVAUVBUVIUUTUVEUVAUVBUVIMZMUVAUVAUUTUVEHZUVJGUA
      UBZFZAEUVLJKZLKZUVNRZMZGGFZAEGJKZLKZUVSRZMZGCUBZFZAEUWCJKZLKZUWERZMZGUWCU
      CZFZAEUWIJKZLKZUWKRZMZUVJUVKUACBUVLGRZUVMUVRUVPUWAUVLGGUDUWOUVOUVTUVNUVSU
      WOUVNUVSALUVLGEJUEZNUWPOUFUVLUWCRZUVMUWDUVPUWGUVLUWCGUDUWQUVOUWFUVNUWEUWQ
      UVNUWEALUVLUWCEJUEZNUWROUFUVLUWIRZUVMUWJUVPUWMUVLUWIGUDUWSUVOUWLUVNUWKUWS
      UVNUWKALUVLUWIEJUEZNUWTOUFUVLBRZUVMUVBUVPUVIUVLBGUDUXAUVOUVHUVNUVGUXAUVNU
      VGALUVLBEJUEZNUXBOUFUWBUVKUVRUWAGUKUGUHUVKUWCIFZUWHUWNMZUUTUVEUXCUXDUUTUV
      EUXCHZHZUWHUWMUWJUXFUWCGRZUWHUWMMUWDUXFUXGUWMUWHUXFUXGUWMUXFUXGHZAELKZEUW
      LUWKUXFUXIERZUXGUXFUVEUURUUSUXJUUTUVEUXCUIZUURUUSUXEUJUURUUSUXEULAUMUNPUX
      HUWKEALUXGUXFUWKEUOJKZEUXGUWIUOEJUXGUWIGUCUOUWCGUPUQURNUVEUXLERUUTUXCEUSU
      TVBZNUXMVAVMVCUXFUWHUWDUWMUXFUWGUWMUWDUWGUWMUXFUWFELKZUWEELKZRUWFUWEELVDU
      XFUWLUXNUWKUXOUXFUWLAUXOLKZUXNUXFUWKUXOALUXEUWKUXORUUTEUWCVEVFZNUXFAIFZUW
      EIFZUVEUXNUXPRUURUXRUUSUXEAVGZQUXEUXSUUTEUWCVHZVFUXKAUWEEVIUNVJUXQOVKVLVN
      UXFUXCUXGUWDVOUUTUVEUXCVPUWCVQSVRVCVSVTUVKUVLWAZUWHCUVLWBZUVQMUVKUYBHUYCU
      VPUVMUUTUVEUYBUYCUVPMUUTUVEUYBHZHZUYCUVPUYEUYCHZUVOUVNUYFUVODUVNADUBZLKZW
      CZUVNUYFUXRUVNIFZUVNWAZUVOUYIRUURUXRUUSUYDUYCUXTWDZUYEUYJUYCUYEUVEUVLIFZU
      YJUUTUVEUYBUIZUYEUVLWEFZUYBHZUYMUYEUYBUYOUUTUVEUYBVPUAWFWGZUVLWEWHSZEUVLV
      HTZPZUYFEIWIWJFZUYPUYKUYEVUAUYCUYEUVEUOEFVUAUYNWKEWLWMZPUYEUYPUYCUYQPZEUV
      LWEWNTDAUVNIWQWOUYFUYHUVNWPZDUVNWBUYIUVNWPUYFVUDDUVNUYFUYGUVNFZUWDUYGUWEF
      ZHZCUVLWRZVUDUYFVUEVUFCUVLUOWJZWRZVUHUYFVUEUYGCVUIUWEWCZFVUJUYFUVNVUKUYGU
      YFUVEUYPUVNVUKRUUTUVEUYBUYCWSVUCCEUVLWEWTTXACUYGVUIUWEXBXCUYFUYMVUHVUJXDU
      YEUYMUYCUYRPUYMVUGVUFCUVLVUIUWCUVLFZVUGHVULUWDHZVUFHUYMUWCVUIFZVUFHVULUWD
      VUFXEUYMVUMVUNVUFUYMVUMVULUWCGXFZHVUNUYMVULUWDVUOUYMVULHUXCUWDVUOXDUVLUWC
      XGZUWCXHSXIUWCUVLXJXKXLXTXMSXNUYEUYCVUHVUDUYCVUHHUWHVUGHZCUVLWRUYEVUDUWHV
      UGCUVLXOUYEVUQVUDCUVLVUQUWGVUFHZUYEVULHZVUDUWHUWDVUFVURUWHUWDHUWGVUFUWHUW
      DUWGUWHXPXQXRXSVUSVURVUDVUSVURHZUVNYAZUYHUVNFZVUDVUTUYJVVAUYEUYJVULVURUYS
      QZUVNYBSVUTUYHUWEFZUWEUVNFZVVBVUTUYHUWFUWEVUTVUFUYHUWFFZVUSUWGVUFVPZVUTUY
      GIFZUXSUXRUUSVUFVVFXDVUTUXSVUFVVHVUTUVEUXCUXSUYEUVEVULVURUYNQVUTUYMVULUXC
      UYEUYMVULVURUYRQZUYEVULVURULZVUPTZUYATZVVGUWEUYGXGTVVLUYEUXRVULVURUURUXRU
      USUYDUXTQQUYEUUSVULVURUURUUSUYDULQUYGUWEAYCYDYEVUSUWGVUFUIYFVUTVULVVEVVJV
      UTUXCUYMVUAVULVVEXDVVKVVIUYEVUAVULVURVUBQUWCUVLEYGUNYEVUTUYJVVDVVEHVVBMVV
      CUYHUWEUVNYHSYKUVNUYHYITVMYJYLYJYMYNYODUVNUYHUVNYTYPYQUYFUYJUXRUUSUVNUVOW
      PUYTUYLUURUUSUYDUYCYRUVNAYSUUAUUBVMVSVCVTUUCUUDUUEUUFUUMUVDUVFUUGZHZAGLKZ
      GUVHUVGUURVVOGRUUSUVCVVMAUUHWDVVNUVGGALVVMUVGGRUVDEBIJJIIUUIZUUJJUUKVVPRU
      ULVVPJUUNUUOUUPVFZNVVQVAUUQ $.
  $}

  $( 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 $.

  $( If a natural number is even, its successor is odd.  (Contributed by Mario
     Carneiro, 16-Nov-2014.) $)
  nnneo $p |- ( ( A e. _om /\ B e. _om /\ C = ( 2o .o A ) ) ->
              -. suc C = ( 2o .o B ) ) $=
    ( com wcel comu co wceq csuc wa con0 nnon wb sucid mpbii 2onn nnmord mp3an3
    c2o c1o w3a wn onnbtwn syl 3ad2ant1 suceq eqeq1d 3ad2ant3 wi eleq2 c0 simpl
    ovex biimtrrdi syl5 simpr nnmcl mpan oa1suc 3syl 1oex df-2o eleqtrri nnaord
    1onn mp3an12i nnmsuc eleqtrrd eqeltrrd ad2antrr peano2 sylan2 ancoms adantr
    coa mpbird simpld ex jcad 3adant3 sylbid mtod ) ADEZBDEZCSAFGZHZUAZCIZSBFGZ
    HZABEZBAIZEZJZWCWDWNUBZWFWCAKEWOALABUCUDUEWGWJWEIZWIHZWNWFWCWJWQMWDWFWHWPWI
    CWEUFUGUHWCWDWQWNUIWFWCWDJZWQWKWMWQWEWIEZWRWKWQWEWPEWSWESAFUMNWPWIWEUJOWRWS
    WKUKSEZJZWKWCWDSDEZXAWSMPABSQRWKWTULUNUOWRWQWMWRWQJZWMWTXCWMWTJZWISWLFGZEZX
    CWPWIXEWRWQUPWCWPXEEWDWQWCWETVOGZWPXEWCWEDEZWEKEXGWPHXBWCXHPSAUQURZWELWEUSU
    TWCXGWESVOGZXEWCTSEZXGXJEZTTISTVANVBVCTDEXBWCXHXKXLMVEPXITSWEVDVFOXBWCXEXJH
    PSAVGURVHVIVJVIWRXDXFMZWQWDWCXMWCWDWLDEZXMAVKWDXNXBXMPBWLSQRVLVMVNVPVQVRVSV
    TWAWB $.

  ${
    $d x y z A $.
    $( A natural number is even iff its successor is odd.  (Contributed by NM,
       26-Jan-2006.)  (Revised by Mario Carneiro, 15-Nov-2014.) $)
    nneob $p |- ( A e. _om -> ( E. x e. _om A = ( 2o .o x )
          <-> -. E. x e. _om suc A = ( 2o .o x ) ) ) $=
      ( vy com wcel comu co wceq wrex csuc wn wi c0 suceq eqeq1d rexbidv notbid
      c2o eqeq1 imbi12d vz cv oveq2 eqeq2d cbvrexvw wa nnneo 3com23 3expa sylbi
      nrexdv rexlimiva peano1 eqid con0 2on om0 ax-mp eqtrdi rspceeqv mp2an a1i
      peano2 coa c1o 2onn nnmsuc mpan df-2o oveq2i nnmcl nnasuc sylancl eqtr2id
      1onn nnon oa1suc 4syl 3eqtr2rd syl2anc syl5ibrcom rexlimiv biimtrid con3d
      syl con1 syl9 finds impbid2 ) BDEBRAUBZFGZHZADIZBJZWKHZADIZKZWMBRCUBZFGZH
      ZCDIWQWLWTACDWJWRHZWKWSBWJWRRFUCZUDUEWTWQCDWRDEZWTUFWOADXCWTWJDEZWOKZXCXD
      WTXEWRWJBUGUHUIUKULUJWRJZWKHZADIZKZWRWKHZADIZLMJZWKHZADIZKZMWKHZADIZLUAUB
      ZJZWKHZADIZKZXRWKHZADIZLZXSJZWKHZADIZKZYALWQWMLCUABWRMHZXIXOXKXQYJXHXNYJX
      GXMADYJXFXLWKWRMNOPQYJXJXPADWRMWKSPTWRXRHZXIYBXKYDYKXHYAYKXGXTADYKXFXSWKW
      RXRNOPQYKXJYCADWRXRWKSPTWRXSHZXIYIXKYAYLXHYHYLXGYGADYLXFYFWKWRXSNOPQYLXJX
      TADWRXSWKSPTWRBHZXIWQXKWMYMXHWPYMXGWOADYMXFWNWKWRBNOPQYMXJWLADWRBWKSPTXQX
      OMDEMMHXQUMMUNAMDWKMMWJMHWKRMFGZMWJMRFUCRUOEYNMHUPRUQURUSUTVAVBXRDEZYIYDK
      YEYAYOYDYHYDXRWSHZCDIZYOYHYCYPACDXAWKWSXRXBUDUEYQYHLYOYPYHCDXCYHYPWSJZJZW
      KHZADIZXCXFDEYSRXFFGZHUUAWRVCXCUUBWSRVDGZWSVEVDGZJZYSRDEZXCUUBUUCHVFRWRVG
      VHXCUUCWSVEJZVDGZUUERUUGWSVDVIVJXCWSDEZVEDEUUHUUEHUUFXCUUIVFRWRVKVHZVOWSV
      EVLVMVNXCUUIWSUOEUUDYRHUUEYSHUUJWSVPWSVQUUDYRNVRVSAXFDWKUUBYSWJXFRFUCUTVT
      YPYGYTADYPYFYSWKYPXSYRHYFYSHXRWSNXSYRNWEOPWAWBVBWCWDYAYDWFWGWHWI $.
  $}

  ${
    $d y z w A $.  $d x y z w F $.
    $( Lemma for ~ omsmo .  (Contributed by NM, 30-Nov-2003.)  (Revised by
       David Abernethy, 1-Jan-2014.) $)
    omsmolem $p |- ( y e. _om -> ( ( ( A C_ On /\ F : _om --> A ) /\
        A. x e. _om ( F ` x ) e. ( F ` suc x ) ) ->
       ( z e. y -> ( F ` z ) e. ( F ` y ) ) ) ) $=
      ( vw cv wcel cfv wi c0 csuc con0 com wa eleq2 fveq2 eleq2d imbi12d weq wf
      wss wral wceq noel pm2.21i a1i wo vex elsuc suceq eleq12d rspccva adantll
      fveq2d peano2b ffvelcdm sylan2b ssel ontr1 expcomd syl56 impl adantlr mpd
      imim2d imp eleq1d syl5ibrcom ad4ant23 jaod biimtrid exp31 com12 finds2 )
      CGZBGZHZVPEIZVQEIZHZJVPKHZVSKEIZHZJZVPFGZHZVSWFEIZHZJZVPWFLZHZVSWKEIZHZJZ
      DMUBZNDEUAZOZAGZEIZWSLZEIZHZANUCZOZBFVQKUDZVRWBWAWDVQKVPPXFVTWCVSVQKEQRSB
      FTZVRWGWAWIVQWFVPPXGVTWHVSVQWFEQRSVQWKUDZVRWLWAWNVQWKVPPXHVTWMVSVQWKEQRSW
      EXEWBWDVPUEUFUGXEWFNHZWJWOJXEXIWJWOWLWGCFTZUHXEXIOZWJOZWNVPWFCUIUJXLWGWNX
      JXKWJWGWNJXKWIWNWGXKWHWMHZWIWNJZXDXIXMWRXCXMAWFNAFTZWTWHXBWMWSWFEQXOXAWKE
      WSWFUKUOULUMZUNWRXIXMXNJZXDWPWQXIXQWQXIOWMDHZWPWMMHZXQXIWQWKNHXRWFUPNDWKE
      UQURDMWMUSXSWIXMWNVSWHWMUTVAVBVCVDVEVFVGXDXIXJWNJWRWJXDXIOWNXJXMXPXJVSWHW
      MVPWFEQVHVIVJVKVLVMVNVO $.

    $( A strictly monotonic ordinal function on the set of natural numbers is
       one-to-one.  (Contributed by NM, 30-Nov-2003.)  (Revised by David
       Abernethy, 1-Jan-2014.) $)
    omsmo $p |- ( ( ( A C_ On /\ F : _om --> A ) /\
          A. x e. _om ( F ` x ) e. ( F ` suc x ) ) -> F : _om -1-1-> A ) $=
      ( vy vz con0 com wa cv cfv wcel wral wceq wi wo omsmolem adantl imp word
      wn wss wf csuc wf1 simplr adantr orim12d ancoms con3d wb ffvelcdm expdimp
      ssel syl5 eloni syl6 anim12d ordtri3 syl adantlr nnord 3imtr4d ralrimivva
      syl2an dff13 sylanbrc ) BFUAZGBCUBZHZAIZCJVJUCCJKAGLZHZVHDIZCJZEIZCJZMZVM
      VOMZNZEGLDGLGBCUDVGVHVKUEVLVSDEGGVLVMGKZVOGKZHZHZVNVPKZVPVNKZOZTZVMVOKZVO
      VMKZOZTZVQVRWCWJWFWBVLWJWFNWBVLHWHWDWIWEWBVLWHWDNZWAVLWLNVTAEDBCPQRWBVLWI
      WENZVTVLWMNWAADEBCPUFRUGUHUIVIWBVQWGUJZVKVIWBHVNSZVPSZHZWNVIWBWQVIVTWOWAW
      PVIVTVNFKZWOVGVHVTWRVHVTHVNBKVGWRGBVMCUKBFVNUMUNULVNUOUPVIWAVPFKZWPVGVHWA
      WSVHWAHVPBKVGWSGBVOCUKBFVPUMUNULVPUOUPUQRVNVPURUSUTWBVRWKUJZVLVTVMSVOSWTW
      AVMVAVOVAVMVOURVDQVBVCDEGBCVEVF $.
  $}

  ${
    omopthlem1.1 $e |- A e. _om $.
    omopthlem1.2 $e |- C e. _om $.
    $( Lemma for ~ omopthi .  (Contributed by Scott Fenton, 18-Apr-2012.)
       (Revised by Mario Carneiro, 17-Nov-2014.) $)
    omopthlem1 $p |- ( A e. C ->
                ( ( A .o A ) +o ( A .o 2o ) ) e. ( C .o C ) ) $=
      ( csuc wss comu co wcel c2o coa wi ax-mp mp3an nnoni onsucssi nnmcli wceq
      com mp2an peano2 nnmwordi nnmwordri sstrd 2onn nnacli nnasuc nnmsuc eqtri
      nnaass nnmcom oveq1i nnm2 oveq2i 3eqtr4ri suceq sseq1i bitri 3imtr4i ) AE
      ZBFZUTUTGHZBBGHZFZABIAAGHZAJGHZKHZVCIZVAVBUTBGHZVCUTSIZBSIZVJVAVBVIFLASIZ
      VJCAUAMZDVMUTBUTUBNVJVKVKVAVIVCFLVMDDUTBBUCNUDABACOBDOPVHVGEZVCFVDVGVCVGV
      EVFAACCQZAJCUEQUFOVCBBDDQOPVNVBVCUTAGHZUTKHZVPAKHZEZVBVNVPSIVLVQVSRUTAVMC
      QCVPAUGTVJVLVBVQRVMCUTAUHTVGVRRVNVSRVEAKHZAKHZVEAAKHZKHZVRVGVESIVLVLWAWCR
      VOCCVEAAUJNVPVTAKVPAUTGHZVTVJVLVPWDRVMCUTAUKTVLVLWDVTRCCAAUHTUIULVFWBVEKV
      LVFWBRCAUMMUNUOVGVRUPMUOUQURUS $.
  $}

  ${
    omopthlem2.1 $e |- A e. _om $.
    omopthlem2.2 $e |- B e. _om $.
    omopthlem2.3 $e |- C e. _om $.
    omopthlem2.4 $e |- D e. _om $.
    $( Lemma for ~ omopthi .  (Contributed by Scott Fenton, 16-Apr-2012.)
       (Revised by Mario Carneiro, 17-Nov-2014.) $)
    omopthlem2 $p |- ( ( A +o B ) e. C ->
     -. ( ( C .o C ) +o D ) = ( ( ( A +o B ) .o ( A +o B ) ) +o B ) ) $=
      ( comu co coa wceq wcel nnmcli nnacli nnoni com wss mp2an nnacom nnaword1
      onirri eleq1 mtbii c2o sseqtri nnaass mp3an nnm2 ax-mp eqtr4i wi nnawordi
      2onn 3sstr4i omopthlem1 con0 wa ontr2 sylancr sselid nsyl3 ) CCIJZDKJZABK
      JZVEIJZBKJZLZVGVDMZVECMZVHVDVDMVIVDVDVCDCCGGNZHOPUBVDVGVDUCUDVJVCVDVGVCQM
      DQMVCVDRVKHVCDUASVJVGVFVEUEIJZKJZRZVMVCMZVGVCMZBVFKJZVLVFKJZVGVMBVLRZVQVR
      RZBVEAKJZBKJZVLBBWAKJZWBBQMZWAQMZBWCRFVEAABEFOZEOZBWAUASWDWEWCWBLFWGBWATS
      UFWBVEVEKJZVLVEQMZAQMWDWBWHLWFEFVEABUGUHWIVLWHLWFVEUIUJUKUFWDVLQMZVFQMZVS
      VTULFVEUEWFUNNZVEVEWFWFNZBVLVFUMUHUJWKWDVGVQLWMFVFBTSWKWJVMVRLWMWLVFVLTSU
      OVECWFGUPVGUQMVCUQMVNVOURVPULVGVFBWMFOPVCVKPVGVMVCUSSUTVAVB $.
  $}

  ${
    omopth.1 $e |- A e. _om $.
    omopth.2 $e |- B e. _om $.
    omopth.3 $e |- C e. _om $.
    omopth.4 $e |- D e. _om $.
    $( An ordered pair theorem for ` _om ` .  Theorem 17.3 of [Quine] p. 124.
       This proof is adapted from ~ nn0opthi .  (Contributed by Scott Fenton,
       16-Apr-2012.)  (Revised by Mario Carneiro, 17-Nov-2014.) $)
    omopthi $p |- ( ( ( ( A +o B ) .o ( A +o B ) ) +o B ) =
                     ( ( ( C +o D ) .o ( C +o D ) ) +o D ) <->
                     ( A = C /\ B = D ) ) $=
      ( coa co comu wceq wn wcel word wb nnacli mp2an oveq12d com wa wo ordtri3
      nnoni onordi con2bii omopthlem2 eqcom sylnib jaoi sylbir id oveq1d eqtr4d
      con4i nnmcli nnacan mp3an sylib oveq2d nnacom 3eqtr4g oveq12 simpr impbii
      jca ) ABIJZVGKJZBIJZCDIJZVJKJZDIJZLZACLZBDLZUAZVMVNVOVMBAIJZBCIJZLZVNVMVG
      CBIJZVQVRVMVGVJVTVGVJLZVMWAMVGVJNZVJVGNZUBZVMMZWAWDVGOVJOWAWDMPVGVGABEFQZ
      UDUEVJVJCDGHQZUDUEVGVJUCRUFWBWEWCWBVLVILVMABVJDEFWGHUGVLVIUHUICDVGBGHWFFU
      GUJUKUOZVMBDCIVMVIVHDIJZLZVOVMVIVLWIVMULVMVHVKDIVMVGVJVGVJKWHWHSUMUNVHTNB
      TNZDTNWJVOPVGVGWFWFUPFHVHBDUQURUSZUTUNWKATNZVQVGLFEBAVARWKCTNZVRVTLFGBCVA
      RVBWKWMWNVSVNPFEGBACUQURUSWLVFVPVHVKBDIVPVGVJVGVJKACBDIVCZWOSVNVOVDSVE $.
  $}

  $( An ordered pair theorem for finite integers.  Analogous to ~ nn0opthi .
     (Contributed by Scott Fenton, 1-May-2012.) $)
  omopth $p |- ( ( ( A e. _om /\ B e. _om ) /\ ( C e. _om /\ D e. _om ) ) ->
                 ( ( ( ( A +o B ) .o ( A +o B ) ) +o B ) =
                     ( ( ( C +o D ) .o ( C +o D ) ) +o D ) <->
                     ( A = C /\ B = D ) ) ) $=
    ( com wcel coa co comu wceq wa wb c0 cif oveq1 oveq12d oveq1d peano1 elimel
    bibi12d eqeq1d eqeq1 anbi1d oveq2 id anbi2d eqeq2d eqeq2 omopthi dedth4h )
    AEFZBEFZCEFZDEFZABGHZUOIHZBGHZCDGHZURIHZDGHZJZACJZBDJZKZLUKAMNZBGHZVFIHZBGH
    ZUTJZVECJZVCKZLVEULBMNZGHZVMIHZVLGHZUTJZVJVLDJZKZLVOUMCMNZDGHZVTIHZDGHZJZVE
    VSJZVQKZLVOVSUNDMNZGHZWGIHZWFGHZJZWDVLWFJZKZLABCDMMMMAVEJZVAVIVDVKWMUQVHUTW
    MUPVGBGWMUOVFUOVFIAVEBGOZWNPQUAWMVBVJVCAVECUBUCTBVLJZVIVPVKVRWOVHVOUTWOVGVN
    BVLGWOVFVMVFVMIBVLVEGUDZWPPWOUEPUAWOVCVQVJBVLDUBUFTCVSJZVPWCVRWEWQUTWBVOWQU
    SWADGWQURVTURVTICVSDGOZWRPQUGWQVJWDVQCVSVEUHUCTDWFJZWCWJWEWLWSWBWIVOWSWAWHD
    WFGWSVTWGVTWGIDWFVSGUDZWTPWSUEPUGWSVQWKWDDWFVLUHUFTVEVLVSWFAMERSBMERSCMERSD
    MERSUIUJ $.

  ${
    $d A x y $.  $d B x y $.
    $( There is at most one left additive inverse for natural number addition.
       (Contributed by Scott Fenton, 17-Oct-2024.) $)
    nnasmo $p |- ( A e. _om -> E* x e. _om ( A +o x ) = B ) $=
      ( vy com wcel cv coa co wceq wa weq wi wral wrmo w3a eqtr3 imbitrid 3expb
      nnacan ralrimivva oveq2 eqeq1d rmo4 sylibr ) BEFZBAGZHIZCJZBDGZHIZCJZKZAD
      LZMZDENAENUIAEOUFUOADEEUFUGEFZUJEFZUOUMUHUKJUFUPUQPUNUHUKCQBUGUJTRSUAUIUL
      ADEUNUHUKCUGUJBHUBUCUDUE $.
  $}

  ${
    $d A x y $.  $d B x y $.
    $( Condition for membership in the difference of ` _om ` and a nonzero
       finite ordinal.  (Contributed by Scott Fenton, 24-Oct-2024.) $)
    eldifsucnn $p |- ( A e. _om -> ( B e. ( _om \ suc A ) <->
     E. x e. ( _om \ A ) B = suc x ) ) $=
      ( vy com wcel csuc wss wa cv wceq wrex cdif coa co wb peano2 nnord syl2an
      word nnawordex sylan wi nnacl nnaword1 nnasuc ancoms nnacom suceq 3eqtr4d
      syl sseq2 eqeq2d anbi12d rspcev syl12anc eqeq1 anbi2d syl5ibcom rexlimdva
      rexbidv adantr sylbid ad2antlr ordsucsssuc biimpa eleq1 syl5ibrcom impbid
      expimpd jca wn eldif ordtri1 pm5.32da bitr4id anbi1i anass rexbii2 anbi1d
      bitri rexbidva 3bitr4d ) BEFZCEFZBGZCHZIZBAJZHZCWIGZKZIZAELZCEWFMFZWLAEBM
      ZLZWDWHWNWDWEWGWNWDWEIWGWFDJZNOZCKZDELZWNWDWFEFZWEWGXAPBQZDWFCUAUBWDXAWNU
      CWEWDWTWNDEWDWREFZIZWJWSWKKZIZAELZWTWNXEBWRNOZEFBXIHZWSXIGZKZXHBWRUDBWRUE
      XEWRWFNOZWRBNOZGZWSXKXDWDXMXOKWRBUFUGWDXBXDWSXMKXCWFWRUHUBXEXIXNKXKXOKBWR
      UHXIXNUIUKUJXGXJXLIAXIEWIXIKZWJXJXFXLWIXIBULXPWKXKWSWIXIUIUMUNUOUPWTXGWMA
      EWTXFWLWJWSCWKUQURVAUSUTVBVCVJWDWMWHAEWDWIEFZIZWJWLWHXRWJIZWHWLWKEFZWFWKH
      ZIXSXTYAXQXTWDWJWIQVDXRWJYAWDBTZWITZWJYAPXQBRZWIRZBWIVESVFVKWLWEXTWGYACWK
      EVGCWKWFULUNVHVJUTVIWDWOWECWFFVLZIWHCEWFVMWDWEWGYFWDWFTZCTWGYFPWEWDXBYGXC
      WFRUKCRWFCVNSVOVPWDWQWIBFVLZWLIZAELWNWLYIAWPEWIWPFZWLIXQYHIZWLIXQYIIYJYKW
      LWIEBVMVQXQYHWLVRWAVSWDWMYIAEXRWJYHWLWDYBYCWJYHPXQYDYEBWIVNSVTWBVPWC $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Natural addition
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare new constants $)
  $c +no $.

  $( Declare the syntax for natural ordinal addition.  See ~ df-nadd . $)
  cnadd $a class +no $.

  ${
    $d x y a z w $.
    $( Define natural ordinal addition.  This is a commutative form of addition
       over the ordinals.  (Contributed by Scott Fenton, 26-Aug-2024.) $)
    df-nadd $a |- +no =
     frecs ( { <. x , y >. | ( x e. ( On X. On ) /\ y e. ( On X. On )
     /\ ( ( ( 1st ` x ) _E ( 1st ` y ) \/ ( 1st ` x ) = ( 1st ` y ) ) /\
          ( ( 2nd ` x ) _E ( 2nd ` y ) \/ ( 2nd ` x ) = ( 2nd ` y ) ) /\
          x =/= y ) ) } , ( On X. On ) ,
          ( z e. _V , a e. _V |->
          |^| { w e. On | ( ( a " ( { ( 1st ` z ) } X. ( 2nd ` z ) ) ) C_ w /\
          ( a " ( ( 1st ` z ) X. { ( 2nd ` z ) } ) ) C_ w ) } ) ) $.
  $}

  ${
    on2recs.1 $e |- F = frecs ( { <. x , y >. |
     ( x e. ( On X. On ) /\ y e. ( On X. On )
     /\ ( ( ( 1st ` x ) _E ( 1st ` y ) \/ ( 1st ` x ) = ( 1st ` y ) ) /\
          ( ( 2nd ` x ) _E ( 2nd ` y ) \/ ( 2nd ` x ) = ( 2nd ` y ) ) /\
          x =/= y ) ) } , ( On X. On ) , G ) $.

    ${
      $d x y $.
      $( Show that double recursion over ordinals yields a function over pairs
         of ordinals.  (Contributed by Scott Fenton, 3-Sep-2024.) $)
      on2recsfn $p |- F Fn ( On X. On ) $=
        ( con0 cv wcel c1st cfv cep wbr wceq wo c2nd w3a wfr wtru a1i mptru cxp
        wne copab wpo wse wfn eqid onfr frxp2 wwe wor weso sopo mp2b poxp2 epse
        epweon sexp2 fpr1 mp3an ) FFUAZAGZVAHBGZVAHVBIJZVCIJZKLVDVEMNVBOJZVCOJZ
        KLVFVGMNVBVCUBPPABUCZQZVAVHUDZVAVHUEZCVAUFVIRABFFKKVHVHUGZFKQRUHSZVMUIT
        VJRABFFKKVHVLFKUDZRFKUJFKUKVNUQFKULFKUMUNSZVOUOTVKRABFFKKVHVLFKUERFUPSZ
        VPURTVAVHCDEUSUT $.
    $}

    ${
      $d x y $.
      $( Calculate the value of the double ordinal recursion operator.
         (Contributed by Scott Fenton, 3-Sep-2024.) $)
      on2recsov $p |- ( ( A e. On /\ B e. On ) ->
     ( A F B ) = ( <. A , B >. G
     ( F |` ( ( suc A X. suc B ) \ { <. A , B >. } ) ) ) ) $=
        ( con0 wcel co cxp cfv cep wceq w3a cpred csn wtru a1i cun cop c1st wbr
        wa cv c2nd wne copab cres csuc cdif df-ov opelxp wfr wpo wse eqid frxp2
        wo onfr mptru wwe wor weso sopo mp2b poxp2 epse sexp2 3pm3.2i fpr2 mpan
        epweon sylbir eqtrid predon adantr uneq1d df-suc eqtr4di adantl xpeq12d
        xpord2pred difeq1d eqtrd reseq2d oveq2d ) CHIZDHIZUDZCDEJZCDUAZEHHKZAUE
        ZWMIBUEZWMIWNUBLZWOUBLZMUCWPWQNUSWNUFLZWOUFLZMUCWRWSNUSWNWOUGOOABUHZWLP
        ZUIZFJZWLECUJZDUJZKZWLQZUKZUIZFJWJWKWLELZXCCDEULWJWLWMIZXJXCNZCDHHUMWMW
        TUNZWMWTUOZWMWTUPZOXKXLXMXNXOXMRABHHMMWTWTUQZHMUNRUTSZXQURVAXNRABHHMMWT
        XPHMUOZRHMVBHMVCXRVMHMVDHMVEVFSZXSVGVAXORABHHMMWTXPHMUPRHVHSZXTVIVAVJWM
        WTEFWLGVKVLVNVOWJXBXIWLFWJXAXHEWJXAHMCPZCQZTZHMDPZDQZTZKZXGUKXHABHHMMWT
        CDXPWCWJYGXFXGWJYCXDYFXEWJYCCYBTXDWJYACYBWHYACNWICVPVQVRCVSVTWJYFDYETXE
        WJYDDYEWIYDDNWHDVPWAVRDVSVTWBWDWEWFWGWE $.
    $}

  $}

  ${
    $d a b $.  $d a c $.  $d a d $.  $d a ps $.  $d a ta $.  $d b c $.
    $d b ch $.  $d b d $.  $d b et $.  $d c d $.  $d c ph $.  $d c th $.
    $d d ps $.  $d X a $.  $d X b $.  $d Y b $.
    on2ind.1 $e |- ( a = c -> ( ph <-> ps ) ) $.
    on2ind.2 $e |- ( b = d -> ( ps <-> ch ) ) $.
    on2ind.3 $e |- ( a = c -> ( th <-> ch ) ) $.
    on2ind.4 $e |- ( a = X -> ( ph <-> ta ) ) $.
    on2ind.5 $e |- ( b = Y -> ( ta <-> et ) ) $.
    on2ind.i $e |- ( ( a e. On /\ b e. On ) ->
       ( ( A. c e. a A. d e. b ch /\
           A. c e. a ps /\
           A. d e. b th ) -> ph ) ) $.
    $( Double induction over ordinal numbers.  (Contributed by Scott Fenton,
       26-Aug-2024.) $)
    on2ind $p |- ( ( X e. On /\ Y e. On ) -> et ) $=
      ( con0 cep onfr wwe wor wpo epweon weso sopo mp2b epse cv wcel cpred wral
      wa wceq predon adantr adantl raleqdv raleqbidv 3anbi123d sylbid xpord2ind
      w3a ) ABCDEFSSTTGHIJKLUASTUBSTUCSTUDUESTUFSTUGUHZSUIZUAVEVFMNOPQIUJZSUKZJ
      UJZSUKZUNZCLSTVIULZUMZKSTVGULZUMZBKVNUMZDLVLUMZVDCLVIUMZKVGUMZBKVGUMZDLVI
      UMZVDAVKVOVSVPVTVQWAVKVMVRKVNVGVHVNVGUOVJVGUPUQZVKCLVLVIVJVLVIUOVHVIUPURZ
      USUTVKBKVNVGWBUSVKDLVLVIWCUSVARVBVC $.
  $}

  ${
    $d X a b c d e f $.  $d Y a b c d e f $.  $d Z a b c d e f $.  $d ps a $.
    $d rh a $.  $d th a b c $.  $d ch b f $.  $d mu b $.  $d la c $.
    $d ph d $.  $d ta d $.  $d et e $.  $d ps e $.  $d ze e $.  $d si f $.
    $d a b c d e f $.
    on3ind.1 $e |- ( a = d -> ( ph <-> ps ) ) $.
    on3ind.2 $e |- ( b = e -> ( ps <-> ch ) ) $.
    on3ind.3 $e |- ( c = f -> ( ch <-> th ) ) $.
    on3ind.4 $e |- ( a = d -> ( ta <-> th ) ) $.
    on3ind.5 $e |- ( b = e -> ( et <-> ta ) ) $.
    on3ind.6 $e |- ( b = e -> ( ze <-> th ) ) $.
    on3ind.7 $e |- ( c = f -> ( si <-> ta ) ) $.
    on3ind.8 $e |- ( a = X -> ( ph <-> rh ) ) $.
    on3ind.9 $e |- ( b = Y -> ( rh <-> mu ) ) $.
    on3ind.10 $e |- ( c = Z -> ( mu <-> la ) ) $.
    on3ind.i $e |- ( ( a e. On /\ b e. On /\ c e. On ) ->
       ( ( ( A. d e. a A. e e. b A. f e. c th /\
             A. d e. a A. e e. b ch /\
             A. d e. a A. f e. c ze )
             /\
             ( A. d e. a ps /\
               A. e e. b A. f e. c ta /\
               A. e e. b si ) /\
             A. f e. c et )
           -> ph ) ) $.
    $( Triple induction over ordinals.  (Contributed by Scott Fenton,
       4-Sep-2024.) $)
    on3ind $p |- ( ( X e. On /\ Y e. On /\ Z e. On ) -> la ) $=
      ( con0 cep onfr wwe wor wpo epweon weso sopo mp2b epse cv wcel cpred wral
      wceq predon 3ad2ant1 3ad2ant2 3ad2ant3 raleqdv raleqbidv 3anbi123d sylbid
      w3a xpord3ind ) ABCDEFGHIJKULULULUMUMUMLMNOPQRSTUNULUMUOULUMUPULUMUQURULU
      MUSULUMUTVAZULVBZUNVRVSUNVRVSUAUBUCUDUEUFUGUHUIUJQVCZULVDZRVCZULVDZSVCZUL
      VDZVPZDMULUMWDVEZVFZLULUMWBVEZVFZTULUMVTVEZVFZCLWIVFZTWKVFZGMWGVFZTWKVFZV
      PZBTWKVFZEMWGVFZLWIVFZHLWIVFZVPZFMWGVFZVPDMWDVFZLWBVFZTVTVFZCLWBVFZTVTVFZ
      GMWDVFZTVTVFZVPZBTVTVFZEMWDVFZLWBVFZHLWBVFZVPZFMWDVFZVPAWFWQXKXBXPXCXQWFW
      LXFWNXHWPXJWFWJXETWKVTWAWCWKVTVGWEVTVHVIZWFWHXDLWIWBWCWAWIWBVGWEWBVHVJZWF
      DMWGWDWEWAWGWDVGWCWDVHVKZVLVMVMWFWMXGTWKVTXRWFCLWIWBXSVLVMWFWOXITWKVTXRWF
      GMWGWDXTVLVMVNWFWRXLWTXNXAXOWFBTWKVTXRVLWFWSXMLWIWBXSWFEMWGWDXTVLVMWFHLWI
      WBXSVLVNWFFMWGWDXTVLVNUKVOVQ $.
  $}

  ${
    $d A b c $.  $d A x $.  $d B b x y $.  $d B z $.  $d C b $.  $d C w z $.
    $d a b c ph $.  $d a x y $.  $d b w z $.  $d c w $.
    coflton.1 $e |- ( ph -> A C_ On ) $.
    coflton.2 $e |- ( ph -> B C_ On ) $.
    coflton.3 $e |- ( ph -> C C_ On ) $.
    coflton.4 $e |- ( ph -> A. x e. A E. y e. B x C_ y ) $.
    coflton.5 $e |- ( ph -> A. z e. B A. w e. C z e. w ) $.
    $( Cofinality theorem for ordinals.  If ` A ` is cofinal with ` B ` and
       ` B ` precedes ` C ` , then ` A ` precedes ` C ` .  Compare ~ cofsslt
       for surreals.  (Contributed by Scott Fenton, 20-Jan-2025.) $)
    coflton $p |- ( ph -> A. a e. A A. c e. C a e. c ) $=
      ( vb wel cv wcel wa wss wrex weq sseq1 rexbidv wral simpr rspcdva adantrr
      adantr sseq2 cbvrexvw sylib simplrr ad2antrr elequ1 rspc2va syl21anc con0
      elequ2 wi sselda adantrl ontr2 syl2an2r mpan2d rexlimdva mpd ralrimivva )
      AIJQZIJFHAIRZFSZJRZHSZTZTZVKPRZUAZPGUBZVJVPVKCRZUAZCGUBZVSAVLWBVNAVLTBRZV
      TUAZCGUBZWBBFVKBIUCWDWACGWCVKVTUDUEAWEBFUFVLNUJAVLUGUHUIWAVRCPGVTVQVKUKUL
      UMVPVRVJPGVPVQGSZTZVRPJQZVJWGWFVNDEQZEHUFDGUFZWHVPWFUGAVLVNWFUNAWJVOWFOUO
      WIWHPEQDEVQVMGHDPEUPEJPUTUQURVPVKUSSZWFVMUSSZVRWHTVJVAAVLWKVNAFUSVKKVBUIV
      PWLWFAVNWLVLAHUSVMMVBVCUJVKVQVMVDVEVFVGVHVI $.
  $}

  ${
    $d A a b $.  $d A w $.  $d A x $.  $d A z $.  $d B a b $.  $d B w $.
    $d B a b x y $.  $d B a b z $.  $d a b ph $.  $d ph z $.  $d w z $.
    cofon1.1 $e |- ( ph -> A e. ~P On ) $.
    cofon1.2 $e |- ( ph -> A. x e. A E. y e. B x C_ y ) $.
    cofon1.3 $e |- ( ph -> B C_ |^| { z e. On | A C_ z } ) $.
    $( Cofinality theorem for ordinals.  If ` A ` is cofinal with ` B ` and the
       upper bound of ` A ` dominates ` B ` , then their upper bounds are
       equal.  Compare with ~ cofcut1 for surreals.  (Contributed by Scott
       Fenton, 20-Jan-2025.) $)
    cofon1 $p |- ( ph ->
    |^| { z e. On | A C_ z } = |^| { w e. On | B C_ w } ) $=
      ( va vb cv wss con0 sseq2 wcel wa wrex syl crab cbvrabv wel sseq1 rexbidv
      cint weq wral ad2antrr simprr rspcdva cbvrexvw sylib simprl sselda elpwid
      ad3antrrr simplrr sseldd simpllr ontr2 syl2anc mpan2d rexlimdva mpd ssrdv
      wi expr ex ss2rabdv eqsstrid intss cuni csuc word ssorduni ordsuc cvv cpw
      uniexd sucexg elong mpbird onsucuni rspcev onintrab2 elrabd intss1 eqssd
      wb ) AFDMZNZDOUAZUFZGEMZNZEOUAZUFZAWQWMNWNWRNAWQGWKNZDOUAWMWPWSEDOWOWKGPU
      BAWSWLDOAWKOQZRZWSWLXAWSRKFWKXAWSKMZFQZKDUCZXAWSXCRZRZXBLMZNZLGSZXDXFXBCM
      ZNZCGSZXIXFBMZXJNZCGSZXLBFXBBKUGXNXKCGXMXBXJUDUEAXOBFUHWTXEIUIXAWSXCUJUKX
      KXHCLGXJXGXBPULUMXFXHXDLGXFXGGQZRZXHLDUCZXDXFGWKXGXAWSXCUNUOXQXBOQWTXHXRR
      XDVGXQFOXBAFONZWTXEXPAFOHUPZUQXAWSXCXPURUSAWTXEXPUTXBXGWKVAVBVCVDVEVHVFVI
      VJVKWQWMVLTAWNWQQWRWNNAWPGWNNEWNOWOWNGPAWLDOSZWNOQAFVMZVNZOQZFYCNZYAAYDYC
      VOZAYBVOZYFAXSYGXTFVPTYBVQUMAYCVRQZYDYFWJAYBVRQYHAFOVSHVTYBVRWATYCVRWBTWC
      AXSYEXTFWDTWLYEDYCOWKYCFPWEVBWLDWFUMJWGWNWQWHTWI $.
  $}

  ${
    $d A a b c $.  $d A w z c $.  $d A x $.  $d B a b $.  $d B c $.  $d B x $.
    $d B y $.  $d B z $.  $d a b c ph $.  $d b c w z $.  $d x y $.
    cofon2.1 $e |- ( ph -> A e. ~P On ) $.
    cofon2.2 $e |- ( ph -> B e. ~P On ) $.
    cofon2.3 $e |- ( ph -> A. x e. A E. y e. B x C_ y ) $.
    cofon2.4 $e |- ( ph -> A. z e. B E. w e. A z C_ w ) $.
    $( Cofinality theorem for ordinals.  If ` A ` and ` B ` are mutually
       cofinal, then their upper bounds agree.  Compare ~ cofcut2 for surreals.
       (Contributed by Scott Fenton, 20-Jan-2025.) $)
    cofon2 $p |- ( ph ->
    |^| { a e. On | A C_ a } = |^| { b e. On | B C_ b } ) $=
      ( vc cv wss con0 wcel wrex cvv crab cint wa weq sseq1 rexbidv wral adantr
      simpr rspcdva sseq2 cbvrexvw ssintub a1i sselda wi elpwid ad2antrr simplr
      sylib sseldd cuni csuc word ssorduni syl ordsuc wb cpw uniexd sucexg 3syl
      elong mpbird onsucuni rspcev syl2anc onintrab2 ontr2 mpan2d rexlimdva mpd
      ex ssrdv cofon1 ) ABCHIFGJLAIGFHOZPZHQUAUBZAIOZGRZWIWHRZAWJUCZWINOZPZNFSZ
      WKWLWIEOZPZEFSZWOWLDOZWPPZEFSZWRDGWIDIUDWTWQEFWSWIWPUEUFAXADGUGWJMUHAWJUI
      UJWQWNENFWPWMWIUKULUTWLWNWKNFWLWMFRZUCZWNWMWHRZWKWLFWHWMFWHPWLHFQUMUNUOXC
      WIQRWHQRZWNXDUCWKUPXCGQWIAGQPWJXBAGQKUQURAWJXBUSVAAXEWJXBAWGHQSZXEAFVBZVC
      ZQRZFXHPZXFAXIXHVDZAXGVDZXKAFQPZXLAFQJUQZFVEVFXGVGUTAXGTRXHTRXIXKVHAFQVIJ
      VJXGTVKXHTVMVLVNAXMXJXNFVOVFWGXJHXHQWFXHFUKVPVQWGHVRUTURWIWMWHVSVQVTWAWBW
      CWDWE $.
  $}

  ${
    $d A x $.  $d A z $.  $d X x $.  $d X z $.  $d ph x y $.  $d ph z $.
    $d y z $.
    cofonr.1 $e |- ( ph -> A e. On ) $.
    cofonr.2 $e |- ( ph -> A = |^| { x e. On | X C_ x } ) $.
    $( Inverse cofinality law for ordinals.  Contrast with ~ cofcutr for
       surreals.  (Contributed by Scott Fenton, 20-Jan-2025.) $)
    cofonr $p |- ( ph -> A. y e. A E. z e. X y C_ z ) $=
      ( cv wss wrex wcel wa wel wn con0 onss syl sselda adantr wral word ordirr
      eloni 3syl crab cint wceq simpr sseq2 elrab sylanbrc intss1 simplr sseldd
      eqsstrd mtand dfss3 sylnib wb eqeltrrd onintrab2 sylibr wi sstr rexlimdva
      adantl expcom mpd ontri1 syl2an2r rexbidva rexnal bitrdi mpbird ralrimiva
      ) ACIZDIZJZDFKZCEAVQELZMZVTDCNZDFUAZOZWBFVQJZWDWBWFCCNZWBVQPLZVQUBWGOAEPV
      QAEPLEPJGEQRSZVQUDVQUCUEWBWFMZEVQVQWJEFBIZJZBPUFZUGZVQWBEWNUHZWFAWOWAHTTW
      JVQWMLZWNVQJWJWHWFWPWBWHWFWITWBWFUIWLWFBVQPWKVQFUJUKULVQWMUMRUPAWAWFUNUOU
      QDFVQURUSWBVTWCOZDFKWEWBVSWQDFWBWHVRFLVRPLVSWQUTWIWBFPVRWBWLBPKZFPJZAWRWA
      AWNPLWRAEWNPHGVAWLBVBVCTWBWLWSBPWBWKPLZMWKPJZWLWSVDWTXAWBWKQVGWLXAWSFWKPV
      EVHRVFVISVQVRVJVKVLWCDFVMVNVOVP $.
  $}

  ${
    $d a w x y z $.
    $( Natural addition is a function over pairs of ordinals.  (Contributed by
       Scott Fenton, 26-Aug-2024.) $)
    naddfn $p |- +no Fn ( On X. On ) $=
      ( vx vy vz va vw cnadd cvv cv c1st cfv csn c2nd cxp cima wss wa con0 crab
      cint cmpo df-nadd on2recsfn ) ABFCDGGDHZCHZIJZKUDLJZMNEHZOUCUEUFKMNUGOPEQ
      RSTABCEDUAUB $.
  $}

  ${
    $d A a b c d f t p q x $.  $d B b c d f t p q x $.
    $( Lemma for ordinal addition closure.  (Contributed by Scott Fenton,
       26-Aug-2024.) $)
    naddcllem $p |- ( ( A e. On /\ B e. On ) ->
     ( ( A +no B ) e. On /\
       ( A +no B ) = |^| { x e. On | ( ( +no " ( { A } X. B ) ) C_ x /\
       ( +no " ( A X. { B } ) ) C_ x ) } ) ) $=
      ( vc vd vt cnadd co con0 wcel cxp cima wss wa wceq imaeq2d sseq1d anbi12d
      wral cvv va vb vf vp vq cv csn crab cint oveq1 eleq1d sneq xpeq1d rabbidv
      weq xpeq1 inteqd eqeq12d oveq2 xpeq2 xpeq2d w3a simpl ralimi 3ad2ant2 jca
      3ad2ant3 cop csuc cdif cres c1st cfv c2nd cmpo on2recsov adantr opex wfun
      df-nadd wfn naddfn fnfun ax-mp vex sucex difexi resfunexg mp2an wn wel wo
      xpex word eloni ad2antlr ordirr olcd ianor opelxp xchnxbir sylibr sssucid
      sucid snssi xpss12 ssdifsn mpbiran resima2 ad2antrr orcd wrex cuni simprr
      syl cun ralbidv ralsn cdm onss syl2an sseqtrrdi funimassov sylancr mpbird
      wb fndmi simprl ralbii unssd ssorduni vsnex funimaexg sylib sseq2 eqeltrd
      unex sneqd xpeq12d imaeq1 uniex elon onsucb onsucuni unssad unssbd rspcev
      syl12anc onintrab2 op1std op2ndd eqid ovmpog mp3an12i 3eqtrd syl5 on2ind
      ex ) UAUFZUBUFZGHZIJZUVAGUUSUGZUUTKZLZAUFZMZGUUSUUTUGZKZLZUVFMZNZAIUHZUIZ
      OZNZDUFZUUTGHZIJZUVRGUVQUGZUUTKZLZUVFMZGUVQUVHKZLZUVFMZNZAIUHZUIZOZNZUVQE
      UFZGHZIJZUWMGUVTUWLKZLZUVFMZGUVQUWLUGZKZLZUVFMZNZAIUHZUIZOZNZUUSUWLGHZIJZ
      UXGGUVCUWLKZLZUVFMZGUUSUWRKZLZUVFMZNZAIUHZUIZOZNZBUUTGHZIJZUXTGBUGZUUTKZL
      ZUVFMZGBUVHKZLZUVFMZNZAIUHZUIZOZNBCGHZIJZUYMGUYBCKZLZUVFMZGBCUGZKZLZUVFMZ
      NZAIUHZUIZOZNBCUAUBDEUADUOZUVBUVSUVOUWJVUFUVAUVRIUUSUVQUUTGUJZUKVUFUVAUVR
      UVNUWIVUGVUFUVMUWHVUFUVLUWGAIVUFUVGUWCUVKUWFVUFUVEUWBUVFVUFUVDUWAGVUFUVCU
      VTUUTUUSUVQULZUMPQVUFUVJUWEUVFVUFUVIUWDGUUSUVQUVHUPPQRUNUQURRUBEUOZUVSUWN
      UWJUXEVUIUVRUWMIUUTUWLUVQGUSZUKVUIUVRUWMUWIUXDVUJVUIUWHUXCVUIUWGUXBAIVUIU
      WCUWQUWFUXAVUIUWBUWPUVFVUIUWAUWOGUUTUWLUVTUTPQVUIUWEUWTUVFVUIUWDUWSGVUIUV
      HUWRUVQUUTUWLULVAPQRUNUQURRVUFUXHUWNUXRUXEVUFUXGUWMIUUSUVQUWLGUJZUKVUFUXG
      UWMUXQUXDVUKVUFUXPUXCVUFUXOUXBAIVUFUXKUWQUXNUXAVUFUXJUWPUVFVUFUXIUWOGVUFU
      VCUVTUWLVUHUMPQVUFUXMUWTUVFVUFUXLUWSGUUSUVQUWRUPPQRUNUQURRUUSBOZUVBUYAUVO
      UYLVULUVAUXTIUUSBUUTGUJZUKVULUVAUXTUVNUYKVUMVULUVMUYJVULUVLUYIAIVULUVGUYE
      UVKUYHVULUVEUYDUVFVULUVDUYCGVULUVCUYBUUTUUSBULUMPQVULUVJUYGUVFVULUVIUYFGU
      USBUVHUPPQRUNUQURRUUTCOZUYAUYNUYLVUEVUNUXTUYMIUUTCBGUSZUKVUNUXTUYMUYKVUDV
      UOVUNUYJVUCVUNUYIVUBAIVUNUYEUYQUYHVUAVUNUYDUYPUVFVUNUYCUYOGUUTCUYBUTPQVUN
      UYGUYTUVFVUNUYFUYSGVUNUVHUYRBUUTCULVAPQRUNUQURRUXFEUUTSDUUSSZUWKDUUSSZUXS
      EUUTSZVBZUVSDUUSSZUXHEUUTSZNZUUSIJZUUTIJZNZUVPVUSVUTVVAVUQVUPVUTVURUWKUVS
      DUUSUVSUWJVCVDVEVURVUPVVAVUQUXSUXHEUUTUXHUXRVCVDVGVFVVEVVBUVPVVEVVBNZUVBU
      VOVVFUVAUVNIVVFUVAUUSUUTVHZGUUSVIZUUTVIZKZVVGUGZVJZVKZFUCTTUCUFZFUFZVLVMZ
      UGZVVOVNVMZKZLZUVFMZVVNVVPVVRUGZKZLZUVFMZNZAIUHZUIZVOZHZVVMUVDLZUVFMZVVMU
      VILZUVFMZNZAIUHZUIZUVNVVEUVAVWJOVVBUDUEUUSUUTGVWIUDUEFAUCVTVPVQVVGTJVVMTJ
      ZVVFVWQIJVWJVWQOUUSUUTVRGVSZVVLTJVWRGIIKZWAVWSWBVWTGWCWDZVVJVVKVVHVVIUUSU
      AWEZWFUUTUBWEZWFWMWGGVVLTWHWIVVFVWQUVNIVVFVWPUVMVVFVWOUVLAIVVFVWLUVGVWNUV
      KVVFVWKUVEUVFVVFUVDVVLMZVWKUVEOVVFVVGUVDJZWJZVXDVVFUUSUVCJZWJZUBUBWKZWJZW
      LZVXFVVFVXJVXHVVFUUTWNZVXJVVDVXLVVCVVBUUTWOWPUUTWQXOWRVXGVXINVXKVXEVXGVXI
      WSUUSUUTUVCUUTWTXAXBVXDUVDVVJMZVXFUVCVVHMZUUTVVIMVXMUUSVVHJVXNUUSVXBXDUUS
      VVHXEWDUUTXCUVCVVHUUTVVIXFWIUVDVVJVVGXGXHXBGUVDVVLXIXOQVVFVWMUVJUVFVVFUVI
      VVLMZVWMUVJOVVFVVGUVIJZWJZVXOVVFUAUAWKZWJZUUTUVHJZWJZWLZVXQVVFVXSVYAVVFUU
      SWNZVXSVVCVYCVVDVVBUUSWOXJUUSWQXOXKVXRVXTNVYBVXPVXRVXTWSUUSUUTUUSUVHWTXAX
      BVXOUVIVVJMZVXQUUSVVHMUVHVVIMZVYDUUSXCUUTVVIJVYEUUTVXCXDUUTVVIXEWDUUSVVHU
      VHVVIXFWIUVIVVJVVGXGXHXBGUVIVVLXIXOQRUNUQZVVFUVLAIXLZUVNIJVVFUVEUVJXPZXMZ
      VIZIJZUVEVYJMZUVJVYJMZVYGVVFVYIIJZVYKVVFVYIWNZVYNVVFVYHIMZVYOVVFUVEUVJIVV
      FUVEIMZVVOUWLGHZIJZEUUTSZFUVCSZVVFVVAWUAVVEVUTVVAXNVYTVVAFUUSVXBFUAUOZVYS
      UXHEUUTWUBVYRUXGIVVOUUSUWLGUJUKXQXRXBVVFVWSUVDGXSZMVYQWUAYFVXAVVFUVDVWTWU
      CVVEUVDVWTMZVVBVVCUVCIMUUTIMWUDVVDUUSIXEUUTXTUVCIUUTIXFYAVQVWTGWBYGZYBFEU
      VCUUTIGYCYDYEVVFUVJIMZUVQVVOGHZIJZFUVHSZDUUSSZVVFVUTWUJVVEVUTVVAYHWUIUVSD
      UUSWUHUVSFUUTVXCFUBUOWUGUVRIVVOUUTUVQGUSUKXRYIXBVVFVWSUVIWUCMWUFWUJYFVXAV
      VFUVIVWTWUCVVEUVIVWTMZVVBVVCUUSIMUVHIMWUKVVDUUSXTUUTIXEUUSIUVHIXFYAVQWUEY
      BDFUUSUVHIGYCYDYEYJZVYHYKXOVYIVYHUVEUVJVWSUVDTJUVETJVXAUVCUUTUAYLVXCWMGUV
      DTYMWIVWSUVITJUVJTJVXAUUSUVHVXBUBYLWMGUVITYMWIYQUUAUUBXBVYIUUCYNVVFUVEUVJ
      VYJVVFVYPVYHVYJMWULVYHUUDXOZUUEVVFUVEUVJVYJWUMUUFUVLVYLVYMNAVYJIUVFVYJOUV
      GVYLUVKVYMUVFVYJUVEYOUVFVYJUVJYORUUGUUHUVLAUUIYNZYPFUCVVGVVMTTVWHVWQVWIVV
      NUVDLZUVFMZVVNUVILZUVFMZNZAIUHZUIIVVOVVGOZVWGWUTWVAVWFWUSAIWVAVWAWUPVWEWU
      RWVAVVTWUOUVFWVAVVSUVDVVNWVAVVQUVCVVRUUTWVAVVPUUSUUSUUTVVOVXBVXCUUJZYRUUS
      UUTVVOVXBVXCUUKZYSPQWVAVWDWUQUVFWVAVWCUVIVVNWVAVVPUUSVWBUVHWVBWVAVVRUUTWV
      CYRYSPQRUNUQVVNVVMOZWUTVWPWVDWUSVWOAIWVDWUPVWLWURVWNWVDWUOVWKUVFVVNVVMUVD
      YTQWVDWUQVWMUVFVVNVVMUVIYTQRUNUQVWIUULUUMUUNVYFUUOZWUNYPWVEVFUURUUPUUQ $.
  $}

  ${
    $d A x $.  $d B x $.
    $( Closure law for natural addition.  (Contributed by Scott Fenton,
       26-Aug-2024.) $)
    naddcl $p |- ( ( A e. On /\ B e. On ) -> ( A +no B ) e. On ) $=
      ( vx con0 wcel wa cnadd co csn cxp cima cv wss crab cint naddcllem simpld
      wceq ) ADEBDEFABGHZDESGAIBJKCLZMGABIJKTMFCDNORCABPQ $.

    $( The value of natural addition.  (Contributed by Scott Fenton,
       26-Aug-2024.) $)
    naddov $p |- ( ( A e. On /\ B e. On ) -> ( A +no B ) =
     |^| { x e. On | ( ( +no " ( { A } X. B ) ) C_ x /\
                       ( +no " ( A X. { B } ) ) C_ x ) } ) $=
      ( con0 wcel wa cnadd co csn cxp cima wss crab cint wceq naddcllem simprd
      cv ) BDECDEFBCGHZDESGBICJKARZLGBCIJKTLFADMNOABCPQ $.

    $d A t y $.  $d A z $.  $d B t y $.  $d B z $.  $d t x y $.  $d t x z $.
    $( Alternate expression for natural addition.  (Contributed by Scott
       Fenton, 26-Aug-2024.) $)
    naddov2 $p |- ( ( A e. On /\ B e. On ) -> ( A +no B ) =
     |^| { x e. On | ( A. y e. B ( A +no y ) e. x /\
     A. z e. A ( z +no B ) e. x ) } ) $=
      ( vt con0 wcel wa cnadd co csn cxp cima cv wss crab cint wral wb cdm onss
      naddov snssi xpss12 syl2an fndmi sseqtrrdi wfun wfn fnfun funimassov mpan
      naddfn ax-mp wceq oveq1 eleq1d ralbidv ralsng adantr bitrd adantl anbi12d
      syl oveq2 rabbidv inteqd eqtrd ) DGHZEGHZIZDEJKJDLZEMZNAOZPZJDELZMZNVOPZI
      ZAGQZRDBOZJKZVOHZBESZCOZEJKZVOHZCDSZIZAGQZRADEUCVLWAWKVLVTWJAGVLVPWEVSWIV
      LVPFOZWBJKZVOHZBESZFVMSZWEVLVNJUAZPZVPWPTZVLVNGGMZWQVJVMGPEGPVNWTPVKDGUDE
      UBVMGEGUEUFWTJUNUGZUHJUIZWRWSJWTUJXBUNWTJUKUOZFBVMEVOJULUMVEVJWPWETVKWOWE
      FDGWLDUPZWNWDBEXDWMWCVOWLDWBJUQURUSUTVAVBVLVSWFWLJKZVOHZFVQSZCDSZWIVLVRWQ
      PZVSXHTZVLVRWTWQVJDGPVQGPVRWTPVKDUBEGUDDGVQGUEUFXAUHXBXIXJXCCFDVQVOJULUMV
      EVKXHWITVJVKXGWHCDXFWHFEGWLEUPXEWGVOWLEWFJVFURUTUSVCVBVDVGVHVI $.
  $}

  ${
    $d A x $.  $d B x $.
    $( Alternate expression for natural addition.  (Contributed by Scott
       Fenton, 20-Jan-2025.) $)
    naddov3 $p |- ( ( A e. On /\ B e. On ) -> ( A +no B ) =
    |^| { x e. On | ( ( +no " ( { A } X. B ) ) u. ( +no " ( A X. { B } ) ) )
    C_ x } ) $=
      ( con0 wcel wa cnadd co csn cxp cima wss crab cint cun naddov unss rabbii
      cv inteqi eqtrdi ) BDECDEFBCGHGBICJKZASZLGBCIJKZUCLFZADMZNUBUDOUCLZADMZNA
      BCPUFUHUEUGADUBUDUCQRTUA $.
  $}

  ${
    $d x y z $.
    $( Function statement for natural addition.  (Contributed by Scott Fenton,
       20-Jan-2025.) $)
    naddf $p |- +no : ( On X. On ) --> On $=
      ( vx vy vz con0 cxp cnadd wf wfn cv cfv wcel wral naddfn naddcl rgen2 cop
      co wceq fveq2 df-ov eqtr4di eleq1d ralxp mpbir ffnfv mpbir2an ) DDEZDFGFU
      GHAIZFJZDKZAUGLZMUKBIZCIZFQZDKZCDLBDLUOBCDDULUMNOUJUOABCDDUHULUMPZRZUIUND
      UQUIUPFJUNUHUPFSULUMFTUAUBUCUDAUGDFUEUF $.
  $}

  ${
    $d A a b c d x $.  $d B b $.
    $( Natural addition commutes.  (Contributed by Scott Fenton,
       26-Aug-2024.) $)
    naddcom $p |- ( ( A e. On /\ B e. On ) -> ( A +no B ) = ( B +no A ) ) $=
      ( va vb vc vd vx cv cnadd co wceq weq oveq1 oveq2 eqeq12d con0 wcel wa wb
      wral w3a crab cint eleq1 ralimi 3ad2ant3 adantl 3ad2ant2 anbi12d biancomd
      ralbi syl rabbidv inteqd naddov2 adantr ancoms 3eqtr4d ex on2ind ) CHZDHZ
      IJZVBVAIJZKZEHZVBIJZVBVFIJZKZVFFHZIJZVJVFIJZKZVAVJIJZVJVAIJZKZAVBIJZVBAIJ
      ZKABIJZBAIJZKABCDEFCELZVCVGVDVHVAVFVBIMVAVFVBINODFLVGVKVHVLVBVJVFINVBVJVF
      IMOWAVNVKVOVLVAVFVJIMVAVFVJINOVAAKVCVQVDVRVAAVBIMVAAVBINOVBBKVQVSVRVTVBBA
      INVBBAIMOVAPQZVBPQZRZVMFVBTEVATZVIEVATZVPFVBTZUAZVEWDWHRZVNGHZQZFVBTZVGWJ
      QZEVATZRZGPUBZUCZVHWJQZEVATZVOWJQZFVBTZRZGPUBZUCZVCVDWIWPXCWIWOXBGPWIWOWS
      XAWIWLXAWNWSWHWLXASZWDWGWEXEWFWGWKWTSZFVBTXEVPXFFVBVNVOWJUDUEWKWTFVBUKULU
      FUGWHWNWSSZWDWFWEXGWGWFWMWRSZEVATXGVIXHEVAVGVHWJUDUEWMWREVAUKULUHUGUIUJUM
      UNWDVCWQKWHGFEVAVBUOUPWDVDXDKZWHWCWBXIGEFVBVAUOUQUPURUSUT $.
  $}

  ${
    $d A a b c x $.
    $( Ordinal zero is the additive identity for natural addition.
       (Contributed by Scott Fenton, 26-Aug-2024.) $)
    naddrid $p |- ( A e. On -> ( A +no (/) ) = A ) $=
      ( va vb vc vx cv c0 cnadd co wceq oveq1 id eqeq12d con0 wcel wral wa crab
      cint adantr weq wss 0elon naddov2 mpan2 ral0 biantrur wel wb eleq1 ralimi
      ralbi syl adantl dfss3 bitr4di bitr3id rabbidv inteqd intmin 3eqtrd tfis3
      ex ) BFZGHIZVDJZCFZGHIZVGJZAGHIZAJBCABCUAZVEVHVDVGVDVGGHKVKLMVDAJZVEVJVDA
      VDAGHKVLLMVDNOZVICVDPZVFVMVNQZVEVDDFHIEFZOZDGPZVHVPOZCVDPZQZENRZSZVDVPUBZ
      ENRZSZVDVMVEWCJZVNVMGNOWGUCEDCVDGUDUETVOWBWEVOWAWDENWAVTVOWDVRVTVQDUFUGVO
      VTCEUHZCVDPZWDVNVTWIUIZVMVNVSWHUIZCVDPWJVIWKCVDVHVGVPUJUKVSWHCVDULUMUNCVD
      VPUOUPUQURUSVMWFVDJVNEVDNUTTVAVCVB $.
  $}

  $( Ordinal zero is the additive identity for natural addition.  (Contributed
     by Scott Fenton, 20-Feb-2025.) $)
  naddlid $p |- ( A e. On -> ( (/) +no A ) = A ) $=
    ( con0 wcel c0 cnadd co wceq 0elon naddcom mpan2 naddrid eqtr3d ) ABCZADEFZ
    DAEFZAMDBCNOGHADIJAKL $.

  ${
    $d A c d w x y z $.  $d B c d w x y z $.  $d C c $.
    $( Ordinal less-than-or-equal is preserved by natural addition.
       (Contributed by Scott Fenton, 7-Sep-2024.) $)
    naddssim $p |- ( ( A e. On /\ B e. On /\ C e. On ) ->
     ( A C_ B -> ( A +no C ) C_ ( B +no C ) ) ) $=
      ( vc vd vw vx vz vy con0 wcel wss cnadd co wi wa cv oveq2 imbi2d wral weq
      sseq12d wceq r19.21v imbi2i crab cint wel rspccva ad4ant24 simprrl eleq1d
      bitri sylan simplrl adantr simp-4l onelon naddcl syl2anc mp2and ralrimiva
      ontr2 simpllr simprrr ssralv sylc jca expr ss2rabdv intss simplll naddov2
      syl simplrr 3sstr4d exp31 a2d ex biimtrid tfis3 com12 3impia ) AJKZBJKZCJ
      KZABLZACMNZBCMNZLZOZWFWDWEPZWKWLWGADQZMNZBWMMNZLZOZOZWLWGAEQZMNZBWSMNZLZO
      ZOZWLWKODECDEUAZWQXCWLXEWPXBWGXEWNWTWOXAWMWSAMRWMWSBMRUBSSWMCUCZWQWKWLXFW
      PWJWGXFWNWHWOWIWMCAMRWMCBMRUBSSXDEWMTZWLWGXBEWMTZOZOZWMJKZWRXGWLXCEWMTZOX
      JWLXCEWMUDXLXIWLWGXBEWMUDUEUMXKWLXIWQXKWLXIWQOXKWLPZWGXHWPXMWGXHWPXMWGPZX
      HPZAFQZMNZGQZKZFWMTZHQWMMNXRKZHATZPZGJUFZUGZBIQZMNZXRKZIWMTZYAHBTZPZGJUFZ
      UGZWNWOXOYLYDLYEYMLXOYKYCGJXOXRJKZYKYCXOYNYKPZPZXTYBYPXSFWMYPFDUHZPZXQBXP
      MNZLZYSXRKZXSXHYQYTXNYOXBYTEXPWMEFUAWTXQXAYSWSXPAMRWSXPBMRUBUIUJYPYIYQUUA
      XOYNYIYJUKYHUUAIXPWMIFUAYGYSXRYFXPBMRULUIUNYRXQJKZYNYTUUAPXSOYRWDXPJKZUUB
      YPWDYQXOWDYOXNWDXHXKWDWEWGUOUPZUPUPYPXKYQUUCXKWLWGXHYOUQWMXPURUNAXPUSUTXO
      YNYKYQUOXQYSXRVCUTVAVBYPWGYJYBXMWGXHYOVDXOYNYIYJVEYAHABVFVGVHVIVJYLYDVKVN
      XOWDXKWNYEUCUUDXKWLWGXHVLZGFHAWMVMUTXOWEXKWOYMUCXNWEXHXKWDWEWGVOUPUUEGIHB
      WMVMUTVPVQVRVSVRVTWAWBWC $.
  $}

  ${
    $d A b c x $.  $d B b c x $.  $d C b c x $.
    $( Ordinal less-than is preserved by natural addition.  (Contributed by
       Scott Fenton, 9-Sep-2024.) $)
    naddelim $p |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( A e. B ->
    ( A +no C ) e. ( B +no C ) ) ) $=
      ( vc vx vb con0 wcel w3a cnadd co wa cv wral crab cint wceq oveq1 eleq1d
      wi ad2antlr adantld ralrimiva ovex elintrab sylibr naddov2 3adant1 adantr
      rspcv eleqtrrd ex ) AGHZBGHZCGHZIZABHZACJKZBCJKZHUPUQLZURBDMJKEMZHDCNZFMZ
      CJKZVAHZFBNZLZEGOPZUSUTVGURVAHZTZEGNURVHHUTVJEGUTVAGHZLVFVIVBUQVFVITUPVKV
      EVIFABVCAQVDURVAVCACJRSUJUAUBUCVGEURGACJUDUEUFUPUSVHQZUQUNUOVLUMEDFBCUGUH
      UIUKUL $.
  $}

  $( Ordinal less-than is not affected by natural addition.  (Contributed by
     Scott Fenton, 9-Sep-2024.) $)
  naddel1 $p |- ( ( A e. On /\ B e. On /\ C e. On ) ->
  ( A e. B <-> ( A +no C ) e. ( B +no C ) ) ) $=
    ( con0 wcel w3a cnadd co naddelim wss naddssim 3com12 ontri1 ancoms 3adant3
    wn wi wb naddcl 3adant1 3imp3i2an 3imtr3d impcon4bid ) ADEZBDEZCDEZFZABEZAC
    GHZBCGHZEZABCIUGBAJZUJUIJZUHPZUKPZUEUDUFULUMQBACKLUDUEULUNRZUFUEUDUPBAMNOUD
    UEUFUJDEZUIDEUMUORUEUFUQUDBCSTACSUJUIMUAUBUC $.

  $( Ordinal less-than is not affected by natural addition.  (Contributed by
     Scott Fenton, 9-Sep-2024.) $)
  naddel2 $p |- ( ( A e. On /\ B e. On /\ C e. On ) ->
  ( A e. B <-> ( C +no A ) e. ( C +no B ) ) ) $=
    ( con0 wcel w3a cnadd co naddel1 wceq naddcom 3adant2 3adant1 eleq12d bitrd
    ) ADEZBDEZCDEZFZABEACGHZBCGHZECAGHZCBGHZEABCISTUBUAUCPRTUBJQACKLQRUAUCJPBCK
    MNO $.

  $( Ordinal less-than-or-equal is not affected by natural addition.
     (Contributed by Scott Fenton, 9-Sep-2024.) $)
  naddss1 $p |- ( ( A e. On /\ B e. On /\ C e. On ) ->
  ( A C_ B <-> ( A +no C ) C_ ( B +no C ) ) ) $=
    ( con0 wcel w3a wn cnadd co wss naddel1 3com12 notbid ontri1 3adant3 naddcl
    wb 3adant2 3adant1 syl2anc 3bitr4d ) ADEZBDEZCDEZFZBAEZGZBCHIZACHIZEZGZABJZ
    UIUHJZUEUFUJUCUBUDUFUJQBACKLMUBUCULUGQUDABNOUEUIDEZUHDEZUMUKQUBUDUNUCACPRUC
    UDUOUBBCPSUIUHNTUA $.

  $( Ordinal less-than-or-equal is not affected by natural addition.
     (Contributed by Scott Fenton, 9-Sep-2024.) $)
  naddss2 $p |- ( ( A e. On /\ B e. On /\ C e. On ) ->
  ( A C_ B <-> ( C +no A ) C_ ( C +no B ) ) ) $=
    ( con0 wcel w3a cnadd co naddss1 wceq naddcom 3adant2 3adant1 sseq12d bitrd
    wss ) ADEZBDEZCDEZFZABPACGHZBCGHZPCAGHZCBGHZPABCITUAUCUBUDQSUAUCJRACKLRSUBU
    DJQBCKMNO $.

  $( Weak-ordering principle for natural addition.  (Contributed by Scott
     Fenton, 21-Jan-2025.) $)
  naddword1 $p |- ( ( A e. On /\ B e. On ) -> A C_ ( A +no B ) ) $=
    ( con0 wcel wa c0 cnadd co wceq naddrid adantr wss 0ss 0elon naddss2 mp3an1
    wb ancoms mpbii eqsstrrd ) ACDZBCDZEZAAFGHZABGHZUAUDAIUBAJKUCFBLZUDUELZBMUB
    UAUFUGQZFCDUBUAUHNFBAOPRST $.

  $( Weak-ordering principle for natural addition.  (Contributed by Scott
     Fenton, 15-Feb-2025.) $)
  naddword2 $p |- ( ( A e. On /\ B e. On ) -> A C_ ( B +no A ) ) $=
    ( con0 wcel wa cnadd co naddword1 naddcom sseqtrd ) ACDBCDEAABFGBAFGABHABIJ
    $.

  ${
    $d A a b c d p q r s w x z $.  $d B a b c d p q r s w z $.  $d B y $.
    $d X a c d p q r s w x z $.  $d Y a c d p q r s w z $.  $d Y y $.
    $d ph p q r s w x $.  $d ph q y z $.
    naddunif.1 $e |- ( ph -> A e. On ) $.
    naddunif.2 $e |- ( ph -> B e. On ) $.
    naddunif.3 $e |- ( ph -> A = |^| { x e. On | X C_ x } ) $.
    naddunif.4 $e |- ( ph -> B = |^| { y e. On | Y C_ y } ) $.
    $( Uniformity theorem for natural addition.  If ` A ` is the upper bound of
       ` X ` and ` B ` is the upper bound of ` Y ` , then ` ( A +no B ) ` can
       be expressed in terms of ` X ` and ` Y ` .  (Contributed by Scott
       Fenton, 20-Jan-2025.) $)
    naddunif $p |- ( ph -> ( A +no B ) = |^| { z e. On |
     ( ( +no " ( X X. { B } ) ) u. ( +no " ( { A } X. Y ) ) ) C_ z } ) $=
      ( vq vp cnadd wss con0 wcel wrex wral vw vc vd va vb vs vr co csn cxp cun
      cima cv crab cint wceq naddov3 syl2anc cpw cvv wfun wfn naddfn fnfun snex
      ax-mp xpexg sylancr funimaexg crn imassrn wf naddf frn sstri elpwd pwuncl
      a1i sylancl eqeltrrd onintrab2 sylibr vex ssex rexlimivw syl wo cofonr wa
      wb onss sselda adantr wi adantl sstr expcom rexlimdva mpd naddss2 syl3anc
      ad2antrr rexbidva mpbid snssd xpss12 sseq2 imaeqexov oveq1 sseq2d rexbidv
      ralbidva rexsng bitrd ralbidv mpbird olc ralimi rexun ralbii sseq1 sseq1d
      imaeqalov ralsng ssintub eqsstrrd sstrid naddss1 oveq2 sylanbrc sseqtrrid
      orc ralunb ssid rspcev ralrimiva bitrdi 3bitr4d cofon2 eqtrd ) AEFOUHZOEU
      IZFUJZULZOEFUIZUJZULZUKZUAUMPUAQUNUOZOGUUEUJZULZOUUBHUJZULZUKZDUMPDQUNUOA
      EQRZFQRZUUAUUIUPIJUAEFUQURAUBUCUDUEUUHUUNUADAUUDQUSZRUUGUUQRUUHUUQRAUUDQU
      TAOVAZUUCUTRZUUDUTROQQUJZVBZUURVCUUTOVDVFZAUUBUTRZUUPUUSEVEZJUUBFUTQVGVHO
      UUCUTVIVHUUDQPAUUDOVJZQOUUCVKUUTQOVLUVEQPVMUUTQOVNVFZVOVRVPAUUGQUTAUURUUF
      UTRZUUGUTRUVBAUUOUUEUTRZUVGIFVEZEUUEQUTVGVSOUUFUTVIVHUUGQPAUUGUVEQOUUFVKU
      VFVOVRVPUUDUUGQVQURAUUKUUQRUUMUUQRUUNUUQRAUUKQUTAUURUUJUTRZUUKUTRUVBAGUTR
      ZUVHUVJAGBUMZPZBQSZUVKAUVMBQUNUOZQRUVNAEUVOQKIVTUVMBWAWBUVMUVKBQGUVLBWCWD
      WEWFUVIGUUEUTUTVGVSOUUJUTVIVHUUKQPAUUKUVEQOUUJVKUVFVOVRVPAUUMQUTAUURUULUT
      RZUUMUTRUVBAUVCHUTRZUVPUVDAHCUMZPZCQSZUVQAUVSCQUNUOZQRUVTAFUWAQLJVTUVSCWA
      WBZUVSUVQCQHUVRCWCWDWEWFUUBHUTUTVGVHOUULUTVIVHUUMQPAUUMUVEQOUULVKUVFVOVRV
      PUUKUUMQVQURAUBUMZUCUMZPZUCUUNSZUBUUDTZUWFUBUUGTZUWFUBUUHTAUWGEMUMZOUHZUW
      DPZUCUUNSZMFTZAUWKUCUUKSZUWKUCUUMSZWGZMFTZUWMAUWOMFTZUWQAUWRUWJEUFUMZOUHZ
      PZUFHSZMFTZAUWIUWSPZUFHSZMFTUXCACMUFFHJLWHAUXEUXBMFAUWIFRZWIZUXDUXAUFHUXG
      UWSHRZWIUWIQRZUWSQRZUUOUXDUXAWJZUXGUXIUXHAFQUWIAUUPFQPZJFWKWFZWLWMUXGHQUW
      SAHQPZUXFAUVTUXNUWBAUVSUXNCQAUVRQRZWIUVRQPZUVSUXNWNUXOUXPAUVRWKWOUVSUXPUX
      NHUVRQWPWQWFWRWSZWMWLAUUOUXFUXHIXBUWIUWSEWTZXAXCXLXDAUWOUXBMFAUWOUWJUGUMZ
      UWSOUHZPZUFHSZUGUUBSZUXBAUVAUULUUTPZUWOUYCWJVCAUUBQPZUXNUYDAEQIXEZUXQUUBQ
      HQXFURZUWKUYAUCUGUFUUTUUBHOUWDUXTUWJXGXHVHAUUOUYCUXBWJIUYBUXBUGEQUXSEUPZU
      YAUXAUFHUYHUXTUWTUWJUXSEUWSOXIXJZXKXMWFXNXOXPUWOUWPMFUWOUWNXQXRWFUWLUWPMF
      UWKUCUUKUUMXSXTWBAUWGNUMZUWIOUHZUWDPZUCUUNSZMFTZNUUBTZUWMAUVAUUCUUTPZUWGU
      YOWJVCAUYEUXLUYPUYFUXMUUBQFQXFURZUWFUYMUBNMUUTUUBFOUWCUYKUPUWEUYLUCUUNUWC
      UYKUWDYAXKZYCVHAUUOUYOUWMWJIUYNUWMNEQUYJEUPZUYMUWLMFUYSUYLUWKUCUUNUYSUYKU
      WJUWDUYJEUWIOXIZYBXKXOYDWFXNXPAUWHUYMMUUETZNETZAVUBUYJFOUHZUWDPZUCUUNSZNE
      TZAVUDUCUUKSZVUDUCUUMSZWGZNETZVUFAVUGNETZVUJAVUKVUCUXSFOUHZPZUGGSZNETZAUY
      JUXSPZUGGSZNETVUOABNUGEGIKWHAVUQVUNNEAUYJERZWIZVUPVUMUGGVUSUXSGRZWIUYJQRZ
      UXSQRZUUPVUPVUMWJZVUSVVAVUTAEQUYJAUUOEQPZIEWKWFZWLWMVUSGQUXSAGQPZVURAGUVO
      QBGQYEZAUVOEQKVVEYFYGZWMWLAUUPVURVUTJXBUYJUXSFYHZXAXCXLXDAVUGVUNNEAVUGVUC
      UXTPZUFUUESZUGGSZVUNAUVAUUJUUTPZVUGVVLWJVCAVVFUUEQPZVVMVVHAFQJXEZGQUUEQXF
      URZVUDVVJUCUGUFUUTGUUEOUWDUXTVUCXGXHVHAVVKVUMUGGAUUPVVKVUMWJJVVJVUMUFFQUW
      SFUPUXTVULVUCUWSFUXSOYIXJXMWFZXKXNXOXPVUGVUINEVUGVUHYLXRWFVUEVUINEVUDUCUU
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      VWTAGEUYJAUVOGEVVGKYKWLUYJYNVUPVXCUGUYJEUXSUYJUYJXGYOVSYPAVWTVWRNGVXBVUPV
      UMUGEVXBUXSERZWIVVAVVBUUPVVCVXBVVAVXDAGQUYJVVHWLWMVXBEQUXSAVVDVXAVVEWMWLA
      UUPVXAVXDJXBVVIXAXCXLXDAVWNVWRNGAVWNVVKUGESZVWRAUVAVVTVWNVXEWJVCVWAVWLVVJ
      UEUGUFUUTEUUEOVWCUXTVUCXGXHVHAVVKVUMUGEVVQXKXNXOXPVWNVWONGVWNVWMXQXRWFAVW
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      YKVUCVWCVVSYBXKYDWFWMVWLUEUUDUUGXSYQXLXPAUVAVVMVWFVWKWJVCVVPVWEVWIUDNMUUT
      GUUEOVWBUYKUPVWDVWHUEUUHVWBUYKVWCYAXKZYCVHXPAVWGVWIMHTZNUUBTZAVXJUWJVWCPZ
      UEUUHSZMHTZAVXKUEUUDSZVXKUEUUGSZWGZMHTZVXMAVXNMHTZVXQAVXRUXDUFFSZMHTAVXSM
      HAUWIHRZWIZUXFUWIUWIPZVXSAHFUWIAUWAHFCHQYELYKWLUWIYNUXDVYBUFUWIFUWSUWIUWI
      XGYOVSYPAVXNVXSMHVYAUYAUFFSZUGUUBSZUXAUFFSZVXNVXSAVYDVYEWJZVXTAUUOVYFIVYC
      VYEUGEQUYHUYAUXAUFFUYIXKXMWFWMAVXNVYDWJZVXTAUVAUYPVYGVCUYQVXKUYAUEUGUFUUT
      UUBFOVWCUXTUWJXGXHVHWMVYAUXDUXAUFFVYAUWSFRZWIUXIUXJUUOUXKVYAUXIVYHAHQUWIU
      XQWLWMVYAFQUWSAUXLVXTUXMWMWLAUUOVXTVYHIXBUXRXAXCYRXLXPVXNVXPMHVXNVXOYLXRW
      FVXLVXPMHVXKUEUUDUUGXSXTWBAUUOVXJVXMWJIVXIVXMNEQUYSVWIVXLMHUYSVWHVXKUEUUH
      UYSUYKUWJVWCUYTYBXKXOYDWFXPAUVAUYDVWGVXJWJVCUYGVWEVWIUDNMUUTUUBHOVXHYCVHX
      PVWEUDUUKUUMYMYJYSYT $.
  $}

  ${
    $d A a b c p x $.  $d B a b c p x $.  $d C a b c p x $.
    $( Lemma for ~ naddass .  Expand out the expression for natural addition of
       three arguments.  (Contributed by Scott Fenton, 20-Jan-2025.) $)
    naddasslem1 $p |- ( ( A e. On /\ B e. On /\ C e. On ) ->
     ( ( A +no B ) +no C ) = |^| { x e. On |
     ( A. a e. A ( ( a +no B ) +no C ) e. x /\
     A. b e. B ( ( A +no b ) +no C ) e. x /\
     A. c e. C ( ( A +no B ) +no c ) e. x ) } ) $=
      ( vp con0 wcel cnadd co cxp cima cv wss wral wceq wb sylancr w3a csn crab
      cun cint naddcl 3adant3 naddov3 intmin eqcomd 3ad2ant3 naddunif wa df-3an
      simp3 unss ancom xpundir imaeq2i imaundi eqtri sseq1i 3bitr4i anbi1i wfun
      3bitrri cdm wfn naddfn fnfun ax-mp crn imassrn wf naddf frn simpl3 xpss12
      sstri snssd fndmi sseqtrrdi funimassov eleq1d ralsng syl ralbidv 3ad2ant1
      oveq2 adantr simpl2 syl2anc oveq1 imaeqalov oveq1d 3bitrd simpl1 3ad2ant2
      onss bitrd ovex ralsn bitrdi 3anbi123d bitrid rabbidva inteqd eqtrd ) BIJ
      ZCIJZDIJZUAZBCKLZDKLKKBUBZCMZNZKBCUBZMZNZUDZDUBZMZNZKXMUBZDMZNZUDAOZPZAIU
      CZUEEOZCKLZDKLZYGJZEBQZBFOZKLZDKLZYGJZFCQZXMGOZKLZYGJZGDQZUAZAIUCZUEXLEGA
      XMDXTDXIXJXMIJZXKBCUFUGZXIXJXKUOXIXJXMXTYJPEIUCUERXKEBCUHUGXKXIDDYTPGIUCU
      EZRXJXKUUHDGDIUIUJUKULXLYIUUEXLYHUUDAIYHKXSYAMZNZYGPZKXPYAMZNZYGPZYFYGPZU
      AZXLYGIJZUMZUUDUUPUUKUUNUMZUUOUMYCYGPZUUOUMYHUUKUUNUUOUNUUSUUTUUOUUNUUKUM
      UUMUUJUDZYGPUUSUUTUUMUUJYGUPUUKUUNUQYCUVAYGYCKUULUUIUDZNUVAYBUVBKXPXSYAUR
      USKUULUUIUTVAVBVCVDYCYFYGUPVFUURUUKYNUUNYSUUOUUCUURUUKHOZYTKLZYGJZGYAQZHX
      SQZUVCDKLZYGJZHXSQZYNUURKVEZUUIKVGZPUUKUVGSKIIMZVHZUVKVIUVMKVJVKZUURUUIUV
      MUVLUURXSIPYAIPZUUIUVMPXSKVLZIKXRVMUVMIKVNUVQIPVOUVMIKVPVKZVSUURDIXIXJXKU
      UQVQZVTZXSIYAIVRTUVMKVIWAZWBHGXSYAYGKWCTUURUVFUVIHXSUURXKUVFUVISUVSUVEUVI
      GDIYTDRUVDUVHYGYTDUVCKWIWDWEWFZWGUURUVJYJYOKLZDKLZYGJZFXQQZEBQZYNUURUVNXR
      UVMPZUVJUWGSVIUURBIPZXQIPUWHXLUWIUUQXIXJUWIXKBWSWHWJUURCIXIXJXKUUQWKZVTBI
      XQIVRWLUVIUWEHEFUVMBXQKUVCUWCRUVHUWDYGUVCUWCDKWMWDZWNTUURUWFYMEBUURXJUWFY
      MSUWJUWEYMFCIYOCRZUWDYLYGUWLUWCYKDKYOCYJKWIWOWDWEWFWGWTWPUURUUNUVFHXPQZUV
      IHXPQZYSUURUVKUULUVLPUUNUWMSUVOUURUULUVMUVLUURXPIPUVPUULUVMPXPUVQIKXOVMUV
      RVSUVTXPIYAIVRTUWAWBHGXPYAYGKWCTUURUVFUVIHXPUWBWGUURUWNUWEFCQZEXNQZYSUURU
      VNXOUVMPZUWNUWPSVIUURXNIPCIPZUWQUURBIXIXJXKUUQWQZVTXLUWRUUQXJXIUWRXKCWSWR
      WJXNICIVRWLUVIUWEHEFUVMXNCKUWKWNTUURXIUWPYSSUWSUWOYSEBIYJBRZUWEYRFCUWTUWD
      YQYGUWTUWCYPDKYJBYOKWMWOWDWGWEWFWTWPUURUUOYJYTKLZYGJZGDQZEYDQZUUCUURUVKYE
      UVLPUUOUXDSUVOUURYEUVMUVLUURYDIPDIPZYEUVMPUURXMIXLUUFUUQUUGWJVTXLUXEUUQXK
      XIUXEXJDWSUKWJYDIDIVRWLUWAWBEGYDDYGKWCTUXCUUCEXMBCKXAYJXMRZUXBUUBGDUXFUXA
      UUAYGYJXMYTKWMWDWGXBXCXDXEXFXGXH $.

    $( Lemma for ~ naddass .  Expand out the expression for natural addition of
       three arguments.  (Contributed by Scott Fenton, 20-Jan-2025.) $)
    naddasslem2 $p |- ( ( A e. On /\ B e. On /\ C e. On ) ->
     ( A +no ( B +no C ) ) = |^| { x e. On |
     ( A. a e. A ( a +no ( B +no C ) ) e. x /\
     A. b e. B ( A +no ( b +no C ) ) e. x /\
     A. c e. C ( A +no ( B +no c ) ) e. x ) } ) $=
      ( vp con0 wcel cnadd co cxp cima cv wss wral wceq wa wb w3a csn crab cint
      simp1 naddcl 3adant1 intmin eqcomd 3ad2ant1 naddov3 naddunif 3anass ancom
      cun unss xpundi imaeq2i imaundi eqtri sseq1i 3bitr4i 3bitrri wfun cdm wfn
      anbi2i naddfn fnfun ax-mp onss adantr snssd syl2an2r sseqtrrdi funimassov
      xpss12 fndmi sylancr ovex oveq2 eleq1d ralsn ralbii bitrdi simpl1 imassrn
      crn wf naddf sstri sylancl oveq1 ralbidv ralsng 3ad2ant2 simpl3 imaeqalov
      frn oveq2d bitrd 3bitrd simpl2 3ad2ant3 syl2anc 3anbi123d bitrid rabbidva
      syl inteqd eqtrd ) BIJZCIJZDIJZUAZBCDKLZKLKBXPUBZMZNZKBUBZKCUBZDMZNZKCDUB
      ZMZNZUOZMZNZUOAOZPZAIUCZUDEOZXPKLZYJJZEBQZBFOZDKLZKLZYJJZFCQZBCGOZKLZKLZY
      JJZGDQZUAZAIUCZUDXOEHABXPBYGXLXMXNUEXMXNXPIJZXLCDUFUGZXLXMBBYMPEIUCUDZRXN
      XLUUKBEBIUHUIUJXMXNXPYGHOZPHIUCUDRXLHCDUKUGULXOYLUUHXOYKUUGAIYKXSYJPZKXTY
      FMZNZYJPZKXTYCMZNZYJPZUAZXOYJIJZSZUUGUUTUUMUUPUUSSZSUUMYIYJPZSYKUUMUUPUUS
      UMUVCUVDUUMUUSUUPSUURUUOUOZYJPUVCUVDUURUUOYJUPUUPUUSUNYIUVEYJYIKUUQUUNUOZ
      NUVEYHUVFKXTYCYFUQURKUUQUUNUSUTVAVBVGXSYIYJUPVCUVBUUMYPUUPUUAUUSUUFUVBUUM
      YMUULKLZYJJZHXQQZEBQZYPUVBKVDZXRKVEZPUUMUVJTKIIMZVFZUVKVHUVMKVIVJZUVBXRUV
      MUVLXOBIPZUVAXQIPXRUVMPXLXMUVPXNBVKUJUVBXPIXOUUIUVAUUJVLVMBIXQIVQVNUVMKVH
      VRZVOEHBXQYJKVPVSUVIYOEBUVHYOHXPCDKVTUULXPRUVGYNYJUULXPYMKWAWBWCWDWEUVBUU
      PUVHHYFQZEXTQZBUULKLZYJJZHYFQZUUAUVBUVKUUNUVLPUUPUVSTUVOUVBUUNUVMUVLUVBXT
      IPZYFIPUUNUVMPUVBBIXLXMXNUVAWFZVMZYFKWHZIKYEWGUVMIKWIUWFIPWJUVMIKWSVJZWKX
      TIYFIVQWLUVQVOEHXTYFYJKVPVSUVBXLUVSUWBTUWDUVRUWBEBIYMBRZUVHUWAHYFUWHUVGUV
      TYJYMBUULKWMWBZWNWOXIUVBUWBBYQUUBKLZKLZYJJZGYDQZFCQZUUAUVBUVNYEUVMPZUWBUW
      NTVHXOCIPZUVAYDIPUWOXMXLUWPXNCVKWPUVBDIXLXMXNUVAWQZVMCIYDIVQVNUWAUWLHFGUV
      MCYDKUULUWJRUVTUWKYJUULUWJBKWAWBZWRVSUVBUWMYTFCUVBXNUWMYTTUWQUWLYTGDIUUBD
      RZUWKYSYJUWSUWJYRBKUUBDYQKWAWTWBWOXIWNXAXBUVBUUSUVHHYCQZEXTQZUWAHYCQZUUFU
      VBUVKUUQUVLPUUSUXATUVOUVBUUQUVMUVLUVBUWCYCIPUUQUVMPUWEYCUWFIKYBWGUWGWKXTI
      YCIVQWLUVQVOEHXTYCYJKVPVSUVBXLUXAUXBTUWDUWTUXBEBIUWHUVHUWAHYCUWIWNWOXIUVB
      UXBUWLGDQZFYAQZUUFUVBUVNYBUVMPZUXBUXDTVHUVBYAIPDIPZUXEUVBCIXLXMXNUVAXCZVM
      XOUXFUVAXNXLUXFXMDVKXDVLYAIDIVQXEUWAUWLHFGUVMYADKUWRWRVSUVBXMUXDUUFTUXGUX
      CUUFFCIYQCRZUWLUUEGDUXHUWKUUDYJUXHUWJUUCBKYQCUUBKWMWTWBWNWOXIXAXBXFXGXHXJ
      XK $.
  $}

  ${
    $d A a b c x y z w $.  $d B a b c x y z w $.  $d C a b c x y z w $.
    $( Natural ordinal addition is associative.  (Contributed by Scott Fenton,
       20-Jan-2025.) $)
    naddass $p |- ( ( A e. On /\ B e. On /\ C e. On ) ->
    ( ( A +no B ) +no C ) = ( A +no ( B +no C ) ) ) $=
      ( vx vy vz vw cv cnadd co wceq oveq1 oveq1d eqeq12d oveq2 oveq2d con0 w3a
      wcel wral va vb vc weq wa crab cint wb simpr21 eleq1 ralimi ralbi simpr23
      simpr3 3anbi123d rabbidv inteqd naddasslem1 adantr naddasslem2 3eqtr4d ex
      3syl on3ind ) UAHZUBHZIJZUCHZIJZVEVFVHIJZIJZKZDHZVFIJZVHIJZVMVJIJZKZVMEHZ
      IJZVHIJZVMVRVHIJZIJZKZVSFHZIJZVMVRWDIJZIJZKZVEVRIJZWDIJZVEWFIJZKZVGWDIJZV
      EVFWDIJZIJZKZVNWDIJZVMWNIJZKZWIVHIJZVEWAIJZKZAVFIJZVHIJZAVJIJZKABIJZVHIJZ
      ABVHIJZIJZKXFCIJZABCIJZIJZKEFABCUAUBUCDUADUDZVIVOVKVPXMVGVNVHIVEVMVFILMVE
      VMVJILNUBEUDZVOVTVPWBXNVNVSVHIVFVRVMIOZMXNVJWAVMIVFVRVHILPNUCFUDZVTWEWBWG
      VHWDVSIOXPWAWFVMIVHWDVRIOZPNXMWJWEWKWGXMWIVSWDIVEVMVRILMVEVMWFILNXNWMWJWO
      WKXNVGWIWDIVFVRVEIOMXNWNWFVEIVFVRWDILZPNXNWQWEWRWGXNVNVSWDIXOMXNWNWFVMIXR
      PNXPWTWJXAWKVHWDWIIOXPWAWFVEIXQPNVEAKZVIXDVKXEXSVGXCVHIVEAVFILMVEAVJILNVF
      BKZXDXGXEXIXTXCXFVHIVFBAIOMXTVJXHAIVFBVHILPNVHCKZXGXJXIXLVHCXFIOYAXHXKAIV
      HCBIOPNVEQSVFQSVHQSRZWHFVHTEVFTDVETWCEVFTDVETWSFVHTDVETRZVQDVETZWLFVHTEVF
      TZXBEVFTZRZWPFVHTZRZVLYBYIUEZVOGHZSZDVETZWTYKSZEVFTZWMYKSZFVHTZRZGQUFZUGZ
      VPYKSZDVETZXAYKSZEVFTZWOYKSZFVHTZRZGQUFZUGZVIVKYJYSUUHYJYRUUGGQYJYMUUBYOU
      UDYQUUFYJYDYLUUAUHZDVETYMUUBUHYDYEYFYCYHYBUIVQUUJDVEVOVPYKUJUKYLUUADVEULV
      CYJYFYNUUCUHZEVFTYOUUDUHYDYEYFYCYHYBUMXBUUKEVFWTXAYKUJUKYNUUCEVFULVCYJYHY
      PUUEUHZFVHTYQUUFUHYBYCYGYHUNWPUULFVHWMWOYKUJUKYPUUEFVHULVCUOUPUQYBVIYTKYI
      GVEVFVHDEFURUSYBVKUUIKYIGVEVFVHDEFUTUSVAVBVD $.
  $}

  $( Commutative/associative law that swaps the last two terms in a triple sum.
     (Contributed by Scott Fenton, 20-Jan-2025.) $)
  nadd32 $p |- ( ( A e. On /\ B e. On /\ C e. On ) ->
  ( ( A +no B ) +no C ) = ( ( A +no C ) +no B ) ) $=
    ( con0 wcel w3a cnadd co wceq naddcom 3adant1 oveq2d naddass 3com23 3eqtr4d
    ) ADEZBDEZCDEZFZABCGHZGHACBGHZGHZABGHCGHACGHBGHZSTUAAGQRTUAIPBCJKLABCMPRQUC
    UBIACBMNO $.

  $( Rearragement of terms in a quadruple sum.  (Contributed by Scott Fenton,
     5-Feb-2025.) $)
  nadd4 $p |- ( ( ( A e. On /\ B e. On ) /\ ( C e. On /\ D e. On ) ) ->
  ( ( A +no B ) +no ( C +no D ) ) = ( ( A +no C ) +no ( B +no D ) ) ) $=
    ( con0 wcel wa cnadd wceq nadd32 adantrr oveq1d naddcl adantr simprl simprr
    co 3expa naddass syl3anc ad2ant2r simplr 3eqtr3d ) AEFZBEFZGZCEFZDEFZGZGZAB
    HQZCHQZDHQZACHQZBHQZDHQZUKCDHQHQZUNBDHQHQZUJULUODHUFUGULUOIZUHUDUEUGUSABCJR
    KLUJUKEFZUGUHUMUQIUFUTUIABMNUFUGUHOUFUGUHPZUKCDSTUJUNEFZUEUHUPURIUDUGVBUEUH
    ACMUAUDUEUIUBVAUNBDSTUC $.

  $( Rearragement of terms in a quadruple sum.  (Contributed by Scott Fenton,
     5-Feb-2025.) $)
  nadd42 $p |- ( ( ( A e. On /\ B e. On ) /\ ( C e. On /\ D e. On ) ) ->
  ( ( A +no B ) +no ( C +no D ) ) = ( ( A +no C ) +no ( D +no B ) ) ) $=
    ( con0 wcel wa cnadd co nadd4 wceq naddcom ad2ant2l oveq2d eqtrd ) AEFZBEFZ
    GCEFZDEFZGGZABHICDHIHIACHIZBDHIZHIUADBHIZHIABCDJTUBUCUAHQSUBUCKPRBDLMNO $.

  $( Natural addition to both sides of ordinal less-than.  (Contributed by
     Scott Fenton, 7-Feb-2025.) $)
  naddel12 $p |- ( ( C e. On /\ D e. On ) ->
  ( ( A e. C /\ B e. D ) -> ( A +no B ) e. ( C +no D ) ) ) $=
    ( con0 wcel wa cnadd simprr onelon ad2ant2l simplr ad2ant2r naddel2 syl3anc
    co wb mpbid simprl simpll naddel1 wi naddcl adantr ontr1 syl mp2and ex ) CE
    FZDEFZGZACFZBDFZGZABHPZCDHPZFZUKUNGZUOADHPZFZUSUPFZUQURUMUTUKULUMIURBEFZUJA
    EFZUMUTQUJUMVBUIULDBJKUIUJUNLZUIULVCUJUMCAJMZBDANORURULVAUKULUMSURVCUIUJULV
    AQVEUIUJUNTVDACDUAORURUPEFZUTVAGUQUBUKVFUNCDUCUDUOUSUPUEUFUGUH $.


  ${
    $d A a b $.  $d B b $.  $d a b c d x $.
    $( Natural addition with successor.  (Contributed by RP, 1-Jan-2025.) $)
    naddsuc2 $p |- ( ( A e. On /\ B e. On ) ->
                     ( A +no suc B ) = suc ( A +no B ) ) $=
      ( va vb vc vd vx cv csuc cnadd co wceq oveq1 suceq eqeq12d con0 wcel wral
      syl wa weq oveq2d oveq2 w3a wi simp2 a1i crab cint csn cun df-suc raleqdv
      cvv vex eleq1d ralunsn biancomd bitrd nfv nfra1 nfan wel r19.21bi ralbida
      simplr anbi12d anass simpll3 simpr simpll2 onelon syl2anc simpll1 naddel2
      wb syl3anc mpbid jca ontr1 sylc ralrimiva cuni 3jca naddelim elunii eloni
      word ordsucuniel ad4ant124 pm4.71d bitr4id rabbidva inteqd naddov2 sylan2
      ex onsuc adantr naddcl onsucmin 3eqtr4d syld on2ind ) CHZDHZIZJKZXEXFJKZI
      ZLZEHZXGJKZXLXFJKZIZLZXLFHZIZJKZXLXQJKZIZLZXEXRJKZXEXQJKZIZLZAXGJKZAXFJKZ
      IZLABIZJKZABJKZIZLABCDEFCEUAZXHXMXJXOXEXLXGJMYNXIXNLXJXOLXEXLXFJMXIXNNSOD
      FUAZXMXSXOYAYOXGXRXLJXFXQNUBYOXNXTLXOYALXFXQXLJUCXNXTNSOYNYCXSYEYAXEXLXRJ
      MYNYDXTLYEYALXEXLXQJMYDXTNSOXEALZXHYGXJYIXEAXGJMYPXIYHLXJYILXEAXFJMXIYHNS
      OXFBLZYGYKYIYMYQXGYJAJXFBNUBYQYHYLLYIYMLXFBAJUCYHYLNSOXEPQZXFPQZTZYBFXFRE
      XERZXPEXERZYFFXFRZUDZUUBXKUUDUUBUEYTUUAUUBUUCUFUGYTUUBXKYTUUBTZYDGHZQZFXG
      RZXMUUFQZEXERZTZGPUHZUIZXIUUFQZGPUHZUIZXHXJUUEUULUUOUUEUUKUUNGPUUEUUFPQZT
      ZUUKUUNUUGFXFRZTZXOUUFQZEXERZTZUUNUURUUHUUTUUJUVBUURUUHUUGFXFXFUJUKZRZUUT
      UURUUGFXGUVDXGUVDLUURXFULUGUMUURUVEUUNUUSUURXFUNQZUVEUUSUUNTVPUVFUURDUOUG
      UUGUUNFXFXFUNFDUAYDXIUUFXQXFXEJUCUPUQSURUSUURUUIUVAEXEUUEUUQEYTUUBEYTEUTX
      PEXEVAVBUUQEUTVBUURECVCZTXMXOUUFUURXPEXEYTUUBUUQVFVDUPVEVGUURUVCUUNUUSUVB
      TZTUUNUUNUUSUVBVHUURUUNUVHYRYSUUQUUNUVHUEUUBYRYSUUQUDZUUNUVHUVIUUNTZUUSUV
      BUVJUUGFXFUVJFDVCZTZUUQYDXIQZUUNTUUGYRYSUUQUUNUVKVIUVLUVMUUNUVLUVKUVMUVJU
      VKVJZUVLXQPQZYSYRUVKUVMVPUVLYSUVKUVOYRYSUUQUUNUVKVKZUVNXFXQVLVMUVPYRYSUUQ
      UUNUVKVNXQXFXEVOVQVRUVIUUNUVKVFVSYDXIUUFVTWAWBUVJUVAEXEUVJUVGTZXNUUFWCQZU
      VAUVQXNXIQZUUNUVRUVQXLPQZYRYSUDUVGUVSUVQUVTYRYSUVQYRUVGUVTYRYSUUQUUNUVGVN
      ZUVJUVGVJZXEXLVLVMUWAYRYSUUQUUNUVGVKWDUWBXLXEXFWEWAUVIUUNUVGVFXNXIUUFWFVM
      UVQUUFWHZUVRUVAVPUVQUUQUWCYRYSUUQUUNUVGVIUUFWGSXNUUFWISVRWBVSWQWJWKWLUSWM
      WNYTXHUUMLZUUBYSYRXGPQUWDXFWRGFEXEXGWOWPWSYTXJUUPLZUUBYTXIPQUWEXEXFWTGXIX
      ASWSXBWQXCXD $.
  $}

  ${
    $d A x y $.  $d B x y $.
    $( Natural addition of a natural is the same as regular addition.
       (Contributed by Scott Fenton, 20-Aug-2025.) $)
    naddoa $p |- ( ( A e. On /\ B e. _om ) -> ( A +no B ) = ( A +o B ) ) $=
      ( vy vx com wcel con0 cnadd co coa wceq cv wi c0 csuc oveq2 imbi2d ancoms
      eqeq12d 3adant3 weq naddrid oa0 eqtr4d w3a suceq 3ad2ant3 naddsuc2 sylan2
      nnon onasuc 3eqtr4d 3exp a2d finds impcom ) BEFAGFZABHIZABJIZKZUQACLZHIZA
      VAJIZKZMUQANHIZANJIZKZMUQADLZHIZAVHJIZKZMUQAVHOZHIZAVLJIZKZMUQUTMCDBVANKZ
      VDVGUQVPVBVEVCVFVANAHPVANAJPSQCDUAZVDVKUQVQVBVIVCVJVAVHAHPVAVHAJPSQVAVLKZ
      VDVOUQVRVBVMVCVNVAVLAHPVAVLAJPSQVABKZVDUTUQVSVBURVCUSVABAHPVABAJPSQUQVEAV
      FAUBAUCUDVHEFZUQVKVOVTUQVKVOVTUQVKUEVIOZVJOZVMVNVKVTWAWBKUQVIVJUFUGVTUQVM
      WAKZVKUQVTWCVTUQVHGFWCVHUJAVHUHUIRTVTUQVNWBKZVKUQVTWDAVHUKRTULUMUNUOUP $.
  $}

  $( The naturals are closed under natural addition.  (Contributed by Scott
     Fenton, 20-Aug-2025.) $)
  omnaddcl $p |- ( ( A e. _om /\ B e. _om ) -> ( A +no B ) e. _om ) $=
    ( com wcel wa cnadd co coa con0 wceq nnon naddoa sylan nnacl eqeltrd ) ACDZ
    BCDZEABFGZABHGZCPAIDQRSJAKABLMABNO $.

$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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 $.

  $( A nonempty equivalence class implies the representative is a set.
     (Contributed by Mario Carneiro, 9-Jul-2014.) $)
  ecexr $p |- ( A e. [ B ] R -> B e. _V ) $=
    ( cvv wcel csn cima cec c0 wceq n0i wn snprc imaeq2 sylbi ima0 eqtrdi nsyl2
    df-ec eleq2s ) BDEZACBFZGZBCHAUCEUCIJUAUCAKUALZUCCIGZIUDUBIJUCUEJBMUBICNOCP
    QRBCST $.

  ${
    $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 brrelex1 syl2an wceq wer errel
      wrel breq2 breq1 anbi12d spcedv wb simpl brrelex2 brcog syl2anc mpbird ex
      ccnv cun wss cdm df-er simp3bi unssbd ssbrd syld ) ABCEIZCDEIZJZBDEEKZIZB
      DEIAVFVHAVFJZVHBHLZEIZVJDEIZJZHMZVIVMVFHNCAEUCZVECNOVFAFEUAZVOGFEUBPZVDVE
      QZCDERSAVFQVJCTVKVDVLVEVJCBEUDVJCDEUEUFUGVIBNOZDNOZVHVNUHAVOVDVSVFVQVDVEU
      IBCERSAVOVEVTVFVQVRCDEUJSHBDEENNUKULUMUNAVGEBDAEUOZVGEAVPWAVGUPEUQZGVPVOE
      URFTWBFEUSUTPVAVBVC $.

    ${
      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 sqxpexg ssexg syl2an ex ) ABDZACEZBFEZNBAAGZH
    QFEPOABIACJBQFKLM $.

  $( 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 $.
  $}

  ${
    $d x y z R $.  $d x A $.
    iseri.1 $e |- Rel R $.
    iseri.2 $e |- ( x R y -> y R x ) $.
    iseri.3 $e |- ( ( x R y /\ y R z ) -> x R z ) $.
    iseri.4 $e |- ( x e. A <-> x R x ) $.
    $( A reflexive, symmetric, transitive relation is an equivalence relation
       on its domain.  Inference version of ~ iserd , which avoids the need to
       provide a "dummy antecedent" ` ph ` if there is no natural one to
       choose.  (Contributed by AV, 30-Apr-2021.) $)
    iseri $p |- R Er A $=
      ( wer wtru wrel a1i cv wbr adantl wa wcel wb iserd mptru ) DEJKABCDEELKFM
      ANZBNZEOZUCUBEOKGPUDUCCNZEOQUBUEEOKHPUBDRUBUBEOSKIMTUA $.

    $( Alternate proof of ~ iseri , avoiding the usage of ~ mptru and ` T. ` as
       antecedent by using ~ ax-mp and one of the hypotheses as antecedent.
       This results, however, in a slightly longer proof.  (Contributed by AV,
       30-Apr-2021.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    iseriALT $p |- R Er A $=
      ( wrel wer id cv wbr adantl wa wcel wb a1i iserd ax-mp ) EJZDEKFUBABCDEUB
      LAMZBMZENZUDUCENUBGOUEUDCMZENPUCUFENUBHOUCDQUCUCENRUBISTUA $.
  $}

  ${
    $d V x y z $.  $d .~ x y z $.
    brinxper.r $e |- ( x e. V -> x .~ x ) $.
    brinxper.s $e |- ( x e. V -> ( x .~ y -> y .~ x ) ) $.
    brinxper.t $e |- ( x e. V -> ( ( x .~ y /\ y .~ z ) -> x .~ z ) ) $.
    $( Conditions for a reflexive, symmetric and transitive binary relation to
       be an equivalence relation over a class ` V ` .  (Contributed by AV,
       11-Jun-2025.) $)
    brinxper $p |- ( .~ i^i ( V X. V ) ) Er V $=
      ( cv wbr wa wcel wi brxp adantr sylbi impcom brin com12 adantl cxp sylbb2
      cin relinxp ancom jctird 3imtr4i expd simplr simprl anim12ci exp31 anbi2i
      imp jca 3imtr4g sylibr id sylanbrc simplbi impbii bitr4i iseri ) ABCEDEEU
      AZUCZEEDUDAIZBIZDJZVFVGVDJZKZVGVFDJZVGVFVDJZKZVFVGVEJZVGVFVEJVIVHVMVIVFEL
      ZVGELZKZVHVMMVFVGEENZVQVHVKVLVOVHVKMVPGOVQVPVOKVLVOVPUEVGVFEENUBUFPQVFVGD
      VDRZVGVFDVDRUGVNVGCIZVEJZKVFVTDJZVFVTVDJZKZVFVTVEJVNWAWDVNVJWAWDMVSWAVJWD
      WAVGVTDJZVGVTVDJZKZVJWDMVGVTDVDRWGVHVQKZWBVOVTELZKZKZVJWDWFWEWHWKMZWFVPWI
      KZWEWLMVGVTEENWMWEWHWKWMWEKZWHKWBWJWNWHWBWEWHWBMWMWHWEWBVQVHWEWBMZVOVHWOM
      VPVOVHWEWBHUHOQSTUNWNWIWHVOVPWIWEUIVHVOVPUJUKUOULPQVIVQVHVRUMWCWJWBVFVTEE
      NUMUPPSPUNVFVTDVDRUQVOVFVFDJZVFVFVDJZKZVFVFVEJVOWRVOWPWQFVOVOVOWQVOURZWSV
      FVFEENZUSUOWQVOWPWQVOVOWTUTTVAVFVFDVDRVBVC $.
  $}

  ${
    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 $.  $d x y Y $.
    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 $.
    $}

    swoso.4 $e |- ( ph -> Y C_ X ) $.
    swoso.5 $e |- ( ( ph /\ ( x e. Y /\ y e. Y /\ x R y ) ) -> x = y ) $.
    $( If the incomparability relation is equivalent to equality in a subset,
       then the partial order strictly orders the subset.  (Contributed by
       Mario Carneiro, 30-Dec-2014.) $)
    swoso $p |- ( ph -> .< Or Y ) $=
      ( wpo cv wcel wa wbr wo w3o wss poss sylc weq wn sselda anim12dan brdifun
      swopo wb syl w3a df-3an sylan2br expr sylbird 3orcomb df-3or bitri sylibr
      orrd issod ) ABCHFAHGUAGFNHFNLABCDGFJKUIHGFUBUCABOZHPZCOZHPZQZQZVCVEFRZVE
      VCFRZSZBCUDZSZVIVLVJTZVHVKVLVHVKUEZVCVEERZVLVHVCGPZVEGPZQVPVOUJAVDVQVFVRA
      HGVCLUFAHGVELUFUGVCVEEFGIUHUKAVGVPVLVGVPQAVDVFVPULVLVDVFVPUMMUNUOUPVAVNVI
      VJVLTVMVIVLVJUQVIVJVLURUSUTVB $.
  $}

  ${
    $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 wbr wceq wsbc csb wb cvv nfcsb1v weq sbciegf elv brabsb nfeq nfv vex
      csbie csbeq1 eqtr3id eqeq2d csbeq1a eqeq1d bitrid bitri ) CJZDJZGKEFLZBUN
      MZAUMMZAUMENZAUNENZLZUOABUMUNGIUAUQUTOCUPUTAUMPAURUSAUMEQAUNEQUBUPEUSLZAC
      RZUTUPVAODUOVABUNPVABUCBDRZFUSEVCFABJZENUSAVDEFBUDHUEAVDUNEUFUGUHSTVBEURU
      SAUMEUIUJUKSTUL $.

    $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.)  (Proof shortened by AV, 1-May-2021.) $)
    eqer $p |- R Er _V $=
      ( vz vw vv cvv wceq relopabiv cv csb wbr id eqerlem 3imtr4i wa eqcomd vex
      eqtr anbi12i wcel eqid mpbir 2th iseri ) HIJKECDLABEGMAHNZCOZAINZCOZLZUMU
      KLUJULEPZULUJEPUNUKUMUNQUAABHICDEFGRZABIHCDEFGRSUNUMAJNZCOZLZTUKURLUOULUQ
      EPZTUJUQEPUKUMURUCUOUNUTUSUPABIJCDEFGRUDABHJCDEFGRSUJKUEUJUJEPZHUBVAUKUKL
      UKUFABHHCDEFGRUGUHUI $.
  $}

  ${
    $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 weq id df-id eqer ) ABACBCDABEFABGH $.

    $( The empty set is an equivalence relation on the empty set.  (Contributed
       by Mario Carneiro, 5-Sep-2015.)  (Proof shortened by AV, 1-May-2021.) $)
    0er $p |- (/) Er (/) $=
      ( vx vy vz c0 rel0 cv wbr cop wcel df-br noel pm2.21i sylbi adantr 2false
      bitr4i iseri ) ABCDDEAFZBFZDGZRSHZDIZSRDGZRSDJZUBUCUAKZLMTRCFZDGZSUFDGTUB
      UGUDUBUGUELMNRDIZRRHZDIZRRDGUHUJRKUIKORRDJPQ $.
  $}

  $( 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 $.

  $( All elements are in their own equivalence class.  (Contributed by Thierry
     Arnoux, 14-Feb-2025.) $)
  ecref $p |- ( ( R Er X /\ A e. X ) -> A e. [ A ] R ) $=
    ( wer wcel wa cec wbr simpl simpr erref wb elecg sylancom mpbird ) CBDZACEZ
    FZAABGEZAABHZRABCPQIPQJZKPQQSTLUAAABCCMNO $.

  ${
    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 a nonempty equivalence class belongs to the domain
       of the equivalence relation.  (Contributed by NM, 15-Feb-1996.)
       (Revised by Mario Carneiro, 9-Jul-2014.) $)
    ecdmn0 $p |- ( A e. dom R <-> [ A ] R =/= (/) ) $=
      ( vx cdm wcel cvv cec c0 wne elex cv wex n0 ecexr exlimiv sylbi wbr elecg
      wb vex mpan exbidv a1i eldmg 3bitr4rd pm5.21nii ) ABDZEZAFEZABGZHIZAUGJUK
      CKZUJEZCLZUICUJMZUMUICULABNOPUIUNAULBQZCLUKUHUIUMUPCULFEUIUMUPSCTULABFFRU
      AUBUKUNSUIUOUCCABFUDUEUF $.
  $}

  ${
    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 ) ) $=
      ( cdm wcel cec c0 wne neeq1d ecdmn0 3bitr4g wer wceq erdm syl eleq2d
      3bitr3d ) ABDHZIZCUBIZBEICEIABDJZKLCDJZKLUCUDAUEUFKGMBDNCDNOAUBEBAEDPUBEQ
      FEDRSZTAUBECUGTUA $.
  $}

  ${
    $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 impbida cvv wb adantr elecg sylancr ersymb
      wceq biimpa syldanl vex wrel wer errel brrelex2 sylan 3bitr4d eqrdv erref
      cv syl syl2anc mpbird simpr eleqtrd ereldm mpbid ersym ) ABCDIZBDJZCDJZUB
      ZAVCKZHVDVEVGBHUNZDIZCVHDIZVHVDLZVHVELZVGVIVJAVCCBDIZVIVJAVCVMABCDEFUAUCA
      VMVIVJACBVHDEFMNUDAVCVJVIABCVHDEFMNOVGVHPLZBELZVKVIQHUEZAVOVCGRVHBDPESTVG
      VNCPLZVLVJQVPADUFZVCVQAEDUGZVRFEDUHUOBCDUIUJVHCDPPSTUKULAVFKZCBDEAVSVFFRZ
      VTBVELZVMVTBVDVEVTBVDLZBBDIZAWDVFABDEFGUMRVTVOVOWCWDQAVOVFGRZWEBBDEESUPUQ
      AVFURZUSVTVOCELZWBVMQWEVTVOWGWEVTBCDEWAWFUTVABCDEESUPVAVBO $.
  $}

  ${
    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 $.
  $}

  ${
    $d x A $.  $d x B $.  $d x R $.  $d x X $.
    $( Equivalence classes do not overlap.  In other words, two equivalence
       classes are either equal or disjoint.  Theorem 74 of [Suppes] p. 83.
       (Contributed by NM, 15-Jun-2004.)  (Revised by Mario Carneiro,
       9-Jul-2014.) $)
    erdisj $p |- ( R Er X ->
                ( [ A ] R = [ B ] R \/ ( [ A ] R i^i [ B ] R ) = (/) ) ) $=
      ( vx wer cec cin c0 wceq wcel wbr adantl cvv wb ecexr elecg sylancr mpbid
      syl wn cv wex neq0 wa simpl elinel1 elinel2 ertr4d erthi exlimdv biimtrid
      vex ex orrd orcomd ) DCFZACGZBCGZHZIJZURUSJZUQVAVBVAUAEUBZUTKZEUCUQVBEUTU
      DUQVDVBEUQVDVBUQVDUEZABCDUQVDUFZVEAVCBCDVFVEVCURKZAVCCLZVDVGUQVCURUSUGMZV
      EVCNKZANKZVGVHOEUMZVEVGVKVIVCACPTVCACNNQRSVEVCUSKZBVCCLZVDVMUQVCURUSUHMZV
      EVJBNKZVMVNOVLVEVMVPVOVCBCPTVCBCNNQRSUIUJUNUKULUOUP $.
  $}

  $( 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 $.
  $}

  ${
    qseq2i.1 $e |- A = B $.
    $( Equality theorem for quotient set, inference form.  (Contributed by
       Peter Mazsa, 3-Jun-2021.) $)
    qseq2i $p |- ( C /. A ) = ( C /. B ) $=
      ( wceq cqs qseq2 ax-mp ) ABECAFCBFEDABCGH $.
  $}

  ${
    qseq1d.1 $e |- ( ph -> A = B ) $.
    $( Equality theorem for quotient set, deduction form.  (Contributed by
       Peter Mazsa, 27-May-2021.) $)
    qseq1d $p |- ( ph -> ( A /. C ) = ( B /. C ) ) $=
      ( wceq cqs qseq1 syl ) ABCFBDGCDGFEBCDHI $.

    $( Equality theorem for quotient set, deduction form.  (Contributed by
       Peter Mazsa, 27-May-2021.) $)
    qseq2d $p |- ( ph -> ( C /. A ) = ( C /. B ) ) $=
      ( wceq cqs qseq2 syl ) ABCFDBGDCGFEBCDHI $.
  $}

  $( Equality theorem for quotient set.  (Contributed by Peter Mazsa,
     17-Apr-2019.) $)
  qseq12 $p |- ( ( A = B /\ C = D ) -> ( A /. C ) = ( B /. D ) ) $=
    ( wceq cqs qseq1 qseq2 sylan9eq ) ABECDEACFBCFBDFABCGCDBHI $.

  ${
    $d R x y $.
    $( Quotient set with the empty set.  (Contributed by Peter Mazsa,
       14-Sep-2019.) $)
    0qs $p |- ( (/) /. R ) = (/) $=
      ( vy vx c0 cqs cv cec wceq wrex cab df-qs rex0 abf eqtri ) DAEBFCFAGHZCDI
      ZBJDCBDAKPBOCLMN $.
  $}

  ${
    $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 .~ x y $.  $d B x $.  $d W x $.  $d X x $.
    $( Membership in a quotient set by an equivalence class according to
       ` .~ ` .  (Contributed by Alexander van der Vekens, 12-Apr-2018.)
       (Revised by AV, 30-Apr-2021.) $)
    elqsecl $p |- ( B e. X -> ( B e. ( W /. .~ )
                                <-> E. x e. W B = { y | x .~ y } ) ) $=
      ( wcel cqs cv cec wceq wrex wbr cab elqsg cvv vex dfec2 mp1i eqeq2d bitrd
      rexbidv ) CFGZCEDHGCAIZDJZKZAELCUDBIDMBNZKZAELAECDFOUCUFUHAEUCUEUGCUDPGUE
      UGKUCAQBUDDPRSTUBUA $.
  $}

  ${
    $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 rspceeqv mpan2 cvv wb ecexg syl
      elqsg biimpar sylan2 ) BAFZCDFZBCGZEHZCGZIEAJZUEACKFZUCUEUEIUHUELEBAUGUEU
      EUFBCMNOUDUIUHUDUEPFUIUHQBDCREAUECPTSUAUB $.
  $}

  ${
    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 coset under the converse membership relation.
       (Note: the converse membership relation 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 epeli 3bitri eqriv ) CADEZFZ
      ACGZQHARPIRADIRAHRAPCJZBKARDBSLRABMNO $.
  $}

  ${
    $d x y A $.
    $( A set is equal to its quotient set modulo the converse membership
       relation.  (Note: the converse membership relation 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 ) $=
        ( cv cec wceq wrex wcel wa wb eqcoms syl5ibcom rexlimdva elqsi eleq2s
        cqs impel ) CEDLZGMZNZDFOZBEHPCUHBDFCUFFPQAUHBKABRUGEJSTUAUIEFGUDHDFEGU
        BIUCUE $.
    $}

    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 R $.  $d x A $.  $d x B $.
    $( A quotient set does not contain the empty set.  (Contributed by NM,
       24-Aug-1995.) $)
    elqsn0 $p |- ( ( dom R = A /\ B e. ( A /. R ) ) -> B =/= (/) ) $=
      ( vx cv cec c0 wne cdm wceq eqid neeq1 wcel wa eleq2 biimpar ecdmn0 sylib
      cqs ectocld ) DEZCFZGHZBGHCIZAJZDBACACSZUFKUBBGLUEUAAMZNUAUDMZUCUEUHUGUDA
      UAOPUACQRT $.
  $}

  $( 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 ) $=
    ( cdm wceq cec cqs wcel wa c0 wne elqsn0 ecdmn0 sylibr simpl eleqtrd ) CDZA
    EZBCFZACGHZIZBQAUASJKBQHASCLBCMNRTOP $.

  $( A Cartesian square is an equivalence relation (in general, it is 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 u v w x A $.  $d u v w x B $.  $d u v w R $.
    $( The intersection of a nonempty family of equivalence relations is an
       equivalence relation.  (Contributed by Mario Carneiro, 27-Sep-2015.) $)
    iiner $p |- ( ( A =/= (/) /\ A. x e. A R Er B ) ->
                  |^|_ x e. A R Er B ) $=
      ( vu wral wa cvv wrex cv wbr cop wcel wi df-br wb opex eliin ax-mp bitri
      vv vw c0 wne wer ciin cxp wss wrel r19.2z errel df-rel sylib reximi iinss
      sylibr id ersymb biimpd 3imtr3g ral2imi adantl 3imtr4g imp r19.26 anbi12i
      3syl ertr biimtrrid simpl simpr erref expcom ralimdv com12 cdm vex opeldm
      erdm eleq2d biimpa sylan2 rexlimivw syl ex expdimp impbid bitr4di iserd )
      BUCUDZCDUEZABFZGZEUAUBCABDUFZWMWNHHUGZUHZWNUIWMWKABIDWOUHZABIWPWKABUJWKWQ
      ABWKDUIWQCDUKDULUMUNABDWOUOVGWNULUPWMEJZUAJZWNKZWSWRWNKZWMWRWSLZDMZABFZWS
      WRLZDMZABFZWTXAWLXDXGNWJWKXCXFABWKWRWSDKZWSWRDKZXCXFWKXHXIWKWRWSDCWKUQZUR
      USWRWSDOZWSWRDOUTVAVBWTXBWNMZXDWRWSWNOXBHMXLXDPWRWSQAXBBDHRSTZXAXEWNMZXGW
      SWRWNOXEHMXNXGPWSWRQAXEBDHRSTVCVDWMWTWSUBJZWNKZGZWRXOWNKZWMXDWSXOLZDMZABF
      ZGZWRXOLZDMZABFZXQXRYBXCXTGZABFZWMYEXCXTABVEWLYGYENWJWKYFYDABWKXHWSXODKZG
      WRXODKYFYDWKWRWSXODCXJVHXHXCYHXTXKWSXODOVFWRXODOUTVAVBVIWTXDXPYAXMXPXSWNM
      ZYAWSXOWNOXSHMYIYAPWSXOQAXSBDHRSTVFXRYCWNMZYEWRXOWNOYCHMYJYEPWRXOQAYCBDHR
      STVCVDWMWRCMZWRWRLZDMZABFZWRWRWNKZWMYKYNWLYKYNNWJYKWLYNYKWKYMABWKYKYMWKYK
      GZWRWRDKYMYPWRDCWKYKVJWKYKVKVLWRWRDOUMVMVNVOVBWJWLYNYKWLYNGWKYMGZABFZWJYK
      WKYMABVEWJYRYKWJYRGYQABIYKYQABUJYQYKABYMWKWRDVPZMZYKWRWRDEVQZUUAVRWKYTYKW
      KYSCWRCDVSVTWAWBWCWDWEVIWFWGYOYLWNMZYNWRWRWNOYLHMUUBYNPWRWRQAYLBDHRSTWHWI
      $.

    $( The relative intersection of a family of equivalence relations is an
       equivalence relation.  (Contributed by Mario Carneiro, 27-Sep-2015.) $)
    riiner $p |- ( A. x e. A R Er B ->
      ( ( B X. B ) i^i |^|_ x e. A R ) Er B ) $=
      ( wer wral cxp ciin cin c0 wa xpider wb riin0 adantl ereq1 syl mpbiri wne
      wceq iiner ancoms wss erssxp ralimi riinn0 sylan mpbird pm2.61dane ) CDEZ
      ABFZCCCGZABDHZIZEZBJUKBJTZKZUOCULEZCLUQUNULTZUOURMUPUSUKAULDBNOCUNULPQRUK
      BJSZKZUOCUMEZUTUKVBABCDUAUBVAUNUMTZUOVBMUKDULUCZABFUTVCUJVDABCDUDUEAULDBU
      FUGCUNUMPQUHUI $.
  $}

  ${
    $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 cv wbr wa wcel brinxp2 sylib simplrd adantr syl21anbrc adantrr
      simprd cxp cin wrel relinxp simpr simplld ersym simprr ertrd sselda erref
      a1i wer ex pm4.71rd brin brxp anidm bitri anbi2i bitr4di iserd ) AGHICDCC
      UAZUBZVDUCACCDUDULAGJZHJZVDKZLZVFCMZVECMZVFVEDKVFVEVDKVHVJVIVEVFDKZVHVGVJ
      VILZVKLAVGUECCVEVFDNOZPVHVJVIVKVMUFZVHVEVFDBABDUMZVGEQVHVLVKVMTZUGCCVFVED
      NRAVGVFIJZVDKZLZLZVJVQCMZVEVQDKVEVQVDKAVGVJVRVNSVTVIWAVFVQDKZVTVRVIWALZWB
      LAVGVRUHCCVFVQDNOZPVTVEVFVQDBAVOVSEQAVGVKVRVPSVTWCWBWDTUICCVEVQDNRAVJVEVE
      DKZVJLZVEVEVDKZAVJWEAVJWEAVJLVEDBAVOVJEQACBVEFUJUKUNUOWGWEVEVEVCKZLWFVEVE
      DVCUPWHVJWEWHVJVJLVJVEVECCUQVJURUSUTUSVAVB $.
  $}

  $( 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 dfss2 sylib imaeq2d ineq1d
    wa imass2 df-ec syl simpl sstrd eqtr2d imainrect eqtr4di 3eqtr4g ) CADZAEZB
    AFZRZCBGZDZCAAHIZULDZBCJBUNJUKUMCULAIZDZAIZUOUKURUMAIZUMUKUQUMAUKUPULCUKULA
    EZUPULKUKBAUIUJLMZULANOPQUKUMAEUSUMKUKUMUHAUKUTUMUHEVAULACSUAUIUJUBUCUMANOU
    DAACULUEUFBCTBUNTUG $.

  ${
    $d x y A $.  $d x B $.  $d x y C $.  $d x y ph $.  $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 $.

    qsdisj.1 $e |- ( ph -> R Er X ) $.
    qsdisj.2 $e |- ( ph -> B e. ( A /. R ) ) $.
    qsdisj.3 $e |- ( ph -> C e. ( A /. R ) ) $.
    $( Members of a quotient set do not overlap.  (Contributed by Rodolfo
       Medina, 12-Oct-2010.)  (Revised by Mario Carneiro, 11-Jul-2014.) $)
    qsdisj $p |- ( ph -> ( B = C \/ ( B i^i C ) = (/) ) ) $=
      ( vx vy wcel wceq cin c0 wo cv cec eqeq1d orbi12d cqs eqid eqeq1 ineq1 wa
      eqeq2 ineq2 wer ad2antrr erdisj syl ectocld mpidan mpdan ) ACBEUAZLCDMZCD
      NZOMZPZHJQZERZDMZVADNZOMZPZUSAJCBEUOUOUBZVACMZVBUPVDURVACDUCVGVCUQOVACDUD
      STAUTBLZDUOLVEIVAKQZERZMZVAVJNZOMZPZVEAVHUEZKDBEUOVFVJDMZVKVBVMVDVJDVAUFV
      PVLVCOVJDVAUGSTVOVIBLZUEFEUHZVNAVRVHVQGUIUTVIEFUJUKULUMULUN $.
  $}

  ${
    $d x y A $.  $d x y X $.  $d x y R $.
    $( A quotient set is a disjoint set.  (Contributed by Mario Carneiro,
       10-Dec-2016.) $)
    qsdisj2 $p |- ( R Er X -> Disj_ x e. ( A /. R ) x ) $=
      ( vy wer cv wceq cin c0 wo cqs wral wdisj wcel simpl simprl simprr qsdisj
      wa ralrimivva id disjor sylibr ) DCFZAGZEGZHZUFUGIJHKZEBCLZMAUJMAUJUFNUEU
      IAEUJUJUEUFUJOZUGUJOZTZTBUFUGCDUEUMPUEUKULQUEUKULRSUAUJUFUGAEUHUBUCUD $.
  $}

  ${
    $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 wb elecg
      cvv elvd ibi simpll simpr erthi ex syl5 ectocld 3impia ) EDGZBADHZICBIZBC
      DJZKZCFLZDJZIZURUOKZMUNUPMULFBADUMUMNURBKUSUNUTUPURBCOURBUOPQUSUQCDRZULUQ
      AIZSZUTUSVAUSUSVATFCUQDURUBUAUCUDVCVAUTVCVASUQCDEULVBVAUEVCVAUFUGUHUIUJUK
      $.
  $}

  ${
    $d b c x A $.  $d b c x B $.  $d b c x C $.  $d b c x R $.
    uniinqs.1 $e |- R Er X $.
    $( Class union distributes over the intersection of two subclasses of a
       quotient space.  Compare ~ uniin .  (Contributed by FL, 25-May-2007.)
       (Proof shortened by Mario Carneiro, 11-Jul-2014.) $)
    uniinqs $p |- ( ( B C_ ( A /. R ) /\ C C_ ( A /. R ) ) ->
      U. ( B i^i C ) = ( U. B i^i U. C ) ) $=
      ( vx vb vc wss wa cin cuni a1i cv wcel wrex eluni2 c0 wceq cqs uniin elin
      anbi12i reeanv 3bitr4i simp3l simp2l inelcm 3ad2ant3 simp1l sseldd simp1r
      w3a wne wer simp2r qsdisj ord mpd eqeltrd elind elunii syl2anc rexlimdvva
      necon1ad 3expia biimtrid ssrdv eqssd ) BADUAZJZCVKJZKZBCLZMZBMZCMZLZVPVSJ
      VNBCUBNVNGVSVPGOZVSPZVTHOZPZVTIOZPZKZICQHBQZVNVTVPPZVTVQPZVTVRPZKWCHBQZWE
      ICQZKWAWGWIWKWJWLHVTBRIVTCRUDVTVQVRUCWCWEHIBCUEUFVNWFWHHIBCVNWBBPZWDCPZKZ
      WFWHVNWOWFUNZWCWBVOPWHVNWOWCWEUGWPBCWBVNWMWNWFUHZWPWBWDCWPWBWDLZSUOZWBWDT
      ZWFVNWSWOVTWBWDUIUJWPWTWRSWPWTWRSTWPAWBWDDEEDUPWPFNWPBVKWBVLVMWOWFUKWQULW
      PCVKWDVLVMWOWFUMVNWMWNWFUQZULURUSVFUTXAVAVBVTWBVOVCVDVGVEVHVIVJ $.
  $}

  ${
    $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 ) $.
    $( Lemma for theorems about a function lift.  (Contributed by Mario
       Carneiro, 23-Dec-2016.)  (Revised by AV, 3-Aug-2024.) $)
    qliftlem $p |- ( ( ph /\ x e. X ) -> [ x ] R e. ( X /. R ) ) $=
      ( cvv wcel cv cec cqs wer erex sylc ecelqsg sylan ) ADMNZBOZGNUDDPGDQNAGD
      RGFNUCKLGDFSTGUDDMUAUB $.

    $( ` F ` , a function lift, is a subset of ` R X. S ` .  (Contributed by
       Mario Carneiro, 23-Dec-2016.)  (Revised by AV, 3-Aug-2024.) $)
    qliftrel $p |- ( ph -> F C_ ( ( X /. R ) X. Y ) ) $=
      ( cv cec cqs qliftlem fliftrel ) ABBMDNCGDOHEGIABCDEFGHIJKLPJQ $.

    $( Elementhood in the relation ` F ` .  (Contributed by Mario Carneiro,
       23-Dec-2016.)  (Revised by AV, 3-Aug-2024.) $)
    qliftel $p |- ( ph -> ( [ C ] R F D <-> E. x e. X ( C R x /\ D = A ) ) ) $=
      ( cec wbr cv wceq wa wrex qliftlem fliftel wcel adantr simpr erth2 anbi1d
      cqs wer rexbidva bitr4d ) ADFOZEGPULBQZFOZRZECRZSZBITDUMFPZUPSZBITABUNCUL
      EIFUHJGIKABCFGHIJKLMNUALUBAUSUQBIAUMIUCZSZURUOUPVADUMFIAIFUIUTMUDAUTUEUFU
      GUJUK $.

    $( Elementhood in the relation ` F ` .  (Contributed by Mario Carneiro,
       23-Dec-2016.)  (Revised by AV, 3-Aug-2024.) $)
    qliftel1 $p |- ( ( ph /\ x e. X ) -> [ x ] R F A ) $=
      ( cv cec cqs qliftlem fliftel1 ) ABBMDNCGDOHEGIABCDEFGHIJKLPJQ $.

    ${
      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.)  (Revised by AV,
         3-Aug-2024.) $)
      qliftfun $p |- ( ph -> ( Fun F <-> A. x A. y ( x R y -> A = B ) ) ) $=
        ( cv cec wi wal wa wfun wceq 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 ) AGUABPZFQZCPZFQZUBZDEU
        BZRZCIUCBIUCZVIVKFUDZVNRZCSBSZABCVJDVLEIFUEJGIKABDFGHIJKLMNUFLVIVKFUJOU
        GAVSVIIUHZVKIUHZTZVORZCSBSVPAVRWCBCAVRWBVMTZVNRWCAVQWDVNAVQWBVQTWDAVQWB
        AVQWBAVQTZVTWAWEVIVKFIAIFUKZVQMUIZAVQULZUMWEVIVKFIWGWHUNUOUPUQAWBVQVMAW
        BTVIVKFIAWFWBMUIAVTWAURVBUSUTVAWBVMVNVCVDVEVOBCIIVFVGVH $.

      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.)  (Revised by AV,
         3-Aug-2024.) $)
      qliftfund $p |- ( ph -> Fun F ) $=
        ( wfun cv wbr wal wceq wi ex alrimivv qliftfun mpbird ) AGQBRCRFSZDEUAZ
        UBZCTBTAUIBCAUGUHPUCUDABCDEFGHIJKLMNOUEUF $.
    $}

    $( 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.)  (Revised by AV,
       3-Aug-2024.) $)
    qliftfuns $p |- ( ph -> ( Fun F <->
      A. y A. z ( y R z -> [_ y / x ]_ A = [_ z / x ]_ A ) ) ) $=
      ( cv csb cec cop cmpt wcel nfcv nfcsb1v nfop eceq1 csbeq1a opeq12d cbvmpt
      wceq rneqi eqtri wral ralrimiva nfel1 eleq1d rspc mpan9 csbeq1 qliftfun
      crn ) ACDBCOZEPZBDOZEPFGHIJGBIBOZFQZERZSZUSCIUTFQZVARZSZUSKVFVIBCIVEVHCVE
      UABVGVABVGUABUTEUBZUCVCUTUHZVDVGEVAVCUTFUDBUTEUEZUFUGUIUJAEJTZBIUKUTITVAJ
      TZAVMBILULVMVNBUTIBVAJVJUMVKEVAJVLUNUOUPMNBUTVBEUQUR $.

    $( The domain and codomain of the function ` F ` .  (Contributed by Mario
       Carneiro, 23-Dec-2016.)  (Revised by AV, 3-Aug-2024.) $)
    qliftf $p |- ( ph -> ( Fun F <-> F : ( X /. R ) --> Y ) ) $=
      ( vy wfun cv cec cmpt crn wf wceq cqs qliftlem fliftf wrex cab df-qs eqid
      rnmpt eqtr4i a1i feq2d bitr4d ) AENBGBODPZQZRZHESGDUAZHESABUMCUPHEGIABCDE
      FGHIJKLUBJUCAUPUOHEUPUOTAUPMOUMTBGUDMUEUOBMGDUFBMGUMUNUNUGUHUIUJUKUL $.

    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.)  (Revised by AV, 3-Aug-2024.) $)
    qliftval $p |- ( ( ph /\ C e. X ) -> ( F ` [ C ] R ) = B ) $=
      ( cv cec cqs qliftlem eceq1 fliftval ) ABBQZFRCEFRDIFSJGIEKABCFGHIJKLMNTL
      UCEFUAOPUB $.
  $}

  ${
    $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 $.
  $}

  ${
    brecop2.1 $e |- dom .~ = ( G X. G ) $.
    brecop2.2 $e |- H = ( ( G X. G ) /. .~ ) $.
    brecop2.3 $e |- R C_ ( H X. H ) $.
    brecop2.4 $e |- .<_ C_ ( G X. G ) $.
    brecop2.5 $e |- -. (/) e. G $.
    brecop2.6 $e |- dom .+ = ( G X. G ) $.
    brecop2.7 $e |- ( ( ( A e. G /\ B e. G ) /\ ( C e. G /\ D e. G ) ) ->
                    ( [ <. A , B >. ] .~ R [ <. C , D >. ] .~ <->
                    ( A .+ D ) .<_ ( B .+ C ) ) ) $.
    $( Binary relation on a quotient set.  Lemma for real number construction.
       Eliminates antecedent from last hypothesis.  (Contributed by NM,
       13-Feb-1996.)  (Revised by AV, 12-Jul-2022.) $)
    brecop2 $p |- ( [ <. A , B >. ] .~ R [ <. C , D >. ] .~ <->
                    ( A .+ D ) .<_ ( B .+ C ) ) $=
      ( wcel wa sylib cop cec wbr co brel cxp cqs cdm wceq ecelqsdm mpan eleq2s
      opelxp anim12i syl ndmovrcl an42 pm5.21nii ) ABUAZFUBZCDUAZFUBZGUCZAHRZBH
      RZSZCHRZDHRZSZSZADEUDZBCEUDZJUCZVCUTIRZVBIRZSVJUTVBIIGMUEVNVFVOVIVNUSHHUF
      ZRZVFVQUTVPFUGZIFUHVPUIZUTVRRVQKVPUSFUJUKLULABHHUMTVOVAVPRZVIVTVBVRIVSVBV
      RRVTKVPVAFUJUKLULCDHHUMTUNUOVMVDVHSZVEVGSZSZVJVMVKHRZVLHRZSWCVKVLHHJNUEWD
      WAWEWBADHEPOUPBCHEPOUPUNUOVDVHVEVGUQTQUR $.
  $}

  ${
    $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 x y z P $.  $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 x y z Q $.  $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 ~ erov and ~ 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 cbvrex2vw anbi12i bitr4i wbr wer wss sseldd erth
      simprll simprrl cxp wf fovcdmd 3imtr3d eqeq2 biimprcd syl6 eqeq1 bi2anan9
      impd wb imbi12d syl5ibrcom anassrs rexlimdvva biimtrrid alrimivv sylanbrc
      eu4 ) AQMUPZRNUPZUQZUQZQUCURZIUSZUTZRUBURZJUSZUTZUQZBURZYJYMHVAZKUSZUTZUQ
      ZUBFVBZUCEVBZBVCZUUCYPUOURZYSUTZUQZUBFVBUCEVBZUQZBUOVDZVKZUOVEBVEUUCBVFYI
      YPYTBVCZUQZUBFVBZUCEVBZUUDYIYPUBFVBUCEVBZUUOYIYLUCEVBZYOUBFVBZUQZUUPYHUUS
      AYFUUQYGUURUUQQEIVGMUCEQIVHUDVIUURRFJVGNUBFRJVHUEVIVJVLYLYOUCUBEFVMVNYIYP
      UUMUCUBEFYIUULYPYIKSUPZYSWMUPUULAUUTYHUFVOYRSKVPBYSWMVQVRVSVTWAUUOUUBBVCZ
      UCEVBUUDUUNUVAUCEUUNUUABVCZUBFVBUVAUUMUVBUBFUVBUUMYPYTBWBWCWDUUAUBBFWEWFW
      DUUBUCBEWEWFWGYIUUKBUOAUUKYHUUIQUAURZIUSZUTZRDURZJUSZUTZUQZYQUVCUVFHVAZKU
      SZUTZUQZDFVBZQTURZIUSZUTZRCURZJUSZUTZUQZUUEUVOUVRHVAZKUSZUTZUQZCFVBZUQZTE
      VBUAEVBZAUUJUWHUVNUAEVBZUWFTEVBZUQUUIUVNUWFUATEEVMUUCUWIUUHUWJUUAUVMUVEYO
      UQZYQUVCYMHVAZKUSZUTZUQUCUBUADEFUCUAVDZYPUWKYTUWNUWOYLUVEYOUWOYKUVDQYJUVC
      IWHWIWJUWOYSUWMYQUWOYRUWLKYJUVCYMHWKWLWIWNUBDVDZUWKUVIUWNUVLUWPYOUVHUVEUW
      PYNUVGRYMUVFJWHWIWOUWPUWMUVKYQUWPUWLUVJKYMUVFUVCHWPWLWIWNWQUUGUWEUVQYOUQZ
      UUEUVOYMHVAZKUSZUTZUQUCUBTCEFUCTVDZYPUWQUUFUWTUXAYLUVQYOUXAYKUVPQYJUVOIWH
      WIWJUXAYSUWSUUEUXAYRUWRKYJUVOYMHWKWLWIWNUBCVDZUWQUWAUWTUWDUXBYOUVTUVQUXBY
      NUVSRYMUVRJWHWIWOUXBUWSUWCUUEUXBUWRUWBKYMUVRUVOHWPWLWIWNWQWRWSAUWGUUJUATE
      EUWGUVMUWEUQZCFVBDFVBAUVCEUPZUVOEUPZUQZUQZUUJUVMUWEDCFFVMUXGUXCUUJDCFFAUX
      FUVFFUPZUVRFUPZUQZUXCUUJVKAUXFUXJUQZUQZUVMUWEUUJUXLUWEUUJVKUVMUVDUVPUTZUV
      GUVSUTZUQZUWDUQZUVKUUEUTZVKUXLUXOUWDUXQUXLUXOUVKUWCUTZUWDUXQVKUXLUVCUVOIW
      TZUVFUVRJWTZUQUVJUWBKWTUXOUXRUNUXLUXSUXMUXTUXNUXLUVCUVOILALIXAUXKUGVOUXLE
      LUVCAELXBUXKUJVOAUXDUXEUXJXEZXCXDUXLUVFUVRJOAOJXAUXKUHVOUXLFOUVFAFOXBUXKU
      KVOAUXFUXHUXIXFZXCXDWNUXLUVJUWBKPAPKXAUXKUIVOUXLGPUVJAGPXBUXKULVOUXLUVCUV
      FGEFHAEFXGGHXHUXKUMVOUYAUYBXIXCXDXJUWDUXQUXRUUEUWCUVKXKXLXMXPUVMUWEUXPUUJ
      UXQUVIUWEUXPXQUVLUVIUWAUXOUWDUVEUVQUXMUVHUVTUXNQUVDUVPXNRUVGUVSXNXOWJVOUV
      LUUJUXQXQUVIYQUVKUUEXNVLXRXSXPXTYAYBYAYBVOYCUUCUUHBUOUUJUUAUUGUCUBEFUUJYT
      UUFYPYQUUEYSXNWOVTYEYD $.

    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 ~ erov and ~ 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 ) $.
    $( The value of an operation defined on equivalence classes.  (Contributed
       by Jeff Madsen, 10-Jun-2010.)  (Revised by Mario Carneiro,
       30-Dec-2014.) $)
    erov $p |- ( ( ph /\ P e. A /\ Q e. B )
                             -> ( [ P ] R .+^ [ Q ] S ) = [ ( P .+ Q ) ] T ) $=
      ( wcel w3a cec co cv wceq wa wrex cio cmpo erovlem 3ad2ant1 simprl eqeq1d
      cvv simprr anbi12d anbi1d 2rexbidv iotabidv cqs ecelqsg eleqtrrdi 3adant3
      sylan 3adant2 iotaex ovmpod pm3.2i eceq1 eqeq2d oveq1 eceq1d anbi2d oveq2
      a1i eqid rspc2ev mp3an3 3adant1 ecexg syl weu simp1 eroveu syl12anc simpr
      iota2d mpbid eqtrd ) AJGVCZMHVCZVDZJNVEZMOVEZLVFXPUHVGZNVEZVHZXQUGVGZOVEZ
      VHZVIZDVGZXRYAKVFZPVEZVHZVIZUGHVJUHGVJZDVKZJMKVFZPVEZXOBCXPXQRSBVGZXSVHZC
      VGZYBVHZVIZYHVIZUGHVJUHGVJZDVKZYKLVQAXMLBCRSUUAVLVHXNABCDEFGHIKLNOPQRSTUA
      UDUEUFUGUHUIUJUKULUMUNUOUPUQURUSUTVMVNXOYNXPVHZYPXQVHZVIVIZYTYJDUUDYSYIUH
      UGGHUUDYRYDYHUUDYOXTYQYCUUDYNXPXSXOUUBUUCVOVPUUDYPXQYBXOUUBUUCVRVPVSVTWAW
      BAXMXPRVCZXNANUBVCZXMUUEVAUUFXMVIXPGNWCRGJNUBWDUIWEWGWFZAXNXQSVCZXMAOUCVC
      ZXNUUHVBUUIXNVIXQHOWCSHMOUCWDUJWEWGWHZYKVQVCXOYJDWIWRWJXOYDYMYGVHZVIZUGHV
      JUHGVJZYKYMVHXMXNUUMAXMXNXPXPVHZXQXQVHZVIZYMYMVHZVIZUUMUUPUUQUUNUUOXPWSXQ
      WSWKYMWSWKUULUURUUNYCVIZYMJYAKVFZPVEZVHZVIUHUGJMGHXRJVHZYDUUSUUKUVBUVCXTU
      UNYCUVCXSXPXPXRJNWLWMVTUVCYGUVAYMUVCYFUUTPXRJYAKWNWOWMVSYAMVHZUUSUUPUVBUU
      QUVDYCUUOUUNUVDYBXQXQYAMOWLWMWPUVDUVAYMYMUVDUUTYLPYAMJKWQWOWMVSWTXAXBXOYJ
      UUMDYMVQAXMYMVQVCZXNAPUDVCUVEUKYLUDPXCXDVNXOAUUEUUHYJDXEAXMXNXFUUGUUJADEF
      GHIKNOPQRSTUAXPXQUDUEUFUGUHUIUJUKULUMUNUOUPUQURUSXGXHXOYEYMVHZVIZYIUULUHU
      GGHUVGYHUUKYDUVGYEYMYGXOUVFXIVPWPWAXJXKXL $.

    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 x y z P $.  $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 $.  $d p q r s t u x y z Q $.
    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 ) ) ) $.
    $( The value of an operation defined on equivalence classes.  (Contributed
       by Jeff Madsen, 10-Jun-2010.) $)
    erov2 $p |- ( ( ph /\ P e. A /\ Q e. A )
                          -> ( [ P ] .~ .+^ [ Q ] .~ ) = [ ( P .+ Q ) ] .~ ) $=
      ( erov ) ABCDEFGGGHIJKLLLMNNMMOOOPQRSTTUBUCUCUCUDUDUDUEUFUAUBUBUG $.

    $( 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
          eqid anbi12d imbi1d anbi2d wb ecopoveq 3adant3 3adant1 oveq12 caov411
          breq2 caov4 eqtr3i 3eqtr4g biimtrdi caovcl ovex caovcan syl2an 3com12
          vex 3impb 3adant3l 3adant1r syld 3adant2 sylibrd 3optocl mpcom ) GLLU
          GZRZHWORZIWORZUHZGHKUIZHIKUIZSZGIKUIZXBWPWQWRSZSWSWTWPXAXDWTWPWQGHWOW
          OKKAUJZWORBUJZWORSXECUJZDUJZUKULXFEUJZFUJZUKULSXGXJJTXHXIJTULSFUOEUOD
          UOCUOZSABUMWOWOUGMXKABWOWOUNUPZUQURHIWOWOKXLUQUSWPWQWRUTVAUAUJZUBUJZU
          KZUCUJZUDUJZUKZKUIZXRUEUJZUFUJZUKZKUIZSZXOYBKUIZVBGXRKUIZYCSZGYBKUIZV
          BWTHYBKUIZSZYHVBXBXCVBUAUBUCUDUEUFGHILWOLWOVFXOGULZYDYGYEYHYKXSYFYCXO
          GXRKVCVDXOGYBKVCVEXRHULZYGYJYHYLYFWTYCYIXRHGKVPXRHYBKVCVGVHYBIULZYJXB
          YHXCYMYIXAWTYBIHKVPVIYBIGKVPVEXMLRZXNLRZSZXPLRXQLRSZXTLRZYALRZSZUHZYD
          XMYAJTZXNXTJTZULZYEUUAYDXPXQJTZUUBJTZUUEUUCJTZULZUUDUUAYDXMXQJTZXNXPJ
          TZULZXPYAJTZXQXTJTZULZSZUUHUUAXSUUKYCUUNYPYQXSUUKVJYTABCDEFXMXNXPXQJK
          LMVKVLYQYTYCUUNVJYPABCDEFXPXQXTYAJKLMVKVMVGUUOUUIUULJTUUJUUMJTZUUFUUG
          UUIUUJUULUUMJVNABCXPXQXMYAJUCWFZUDWFZUAWFNPUFWFVOXNXQJTXPXTJTJTUUGUUP
          ABCXNXQXPXTJUBWFZUURUUQNPUEWFZVOABCXNXQXPXTJUUSUURUUQNPUUTVQVRVSVTYNY
          QYTUUHUUDVBZYOYNYQYSUVAYRYQYNYSUVAYQYNYSUVAYQUUELRUUBLRUVAYNYSSABXPXQ
          LJOWAABXMYALJOWAABCUUEUUBUUCLJXNXTJWBQWCWDWGWEWHWIWJYPYTYEUUDVJYQABCD
          EFXMXNXTYAJKLMVKWKWLWMWN $.

        $( 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.)  (Proof shortened by AV, 1-May-2021.) $)
        ecopover $p |- .~ Er ( S X. S ) $=
          ( vf vg vh cv wcel wa cxp cop wceq relopabiv ecopovsym ecopovtrn wral
          co wex wbr caovcom ecopoveq mpbiri anidms rgen2 wb breq12 ralxp mpbir
          vex rspec copab opabssxp eqsstri ssbri brxp simplbi syl impbii iseri
          ) OPQIIUAZHARZVKSBRZVKSTVLCRZDRZUBUCVMERZFRZUBUCTVNVQGUHVOVPGUHUCTFUI
          EUIDUICUIZTZABHJUDABCDEFORZPRZGHIJKUEABCDEFVTWAQRZGHIJKLMNUFVTVKSZVTV
          THUJZWDOVKWDOVKUGWAWBUBZWEHUJZQIUGPIUGWFPQIIWAISWBISTZWFWGWGTWFWAWBGU
          HWBWAGUHUCABWAWBGPUTQUTKUKABCDEFWAWBWAWBGHIJULUMUNUOWDWFOPQIIVTWEUCWD
          WFUPVTWEVTWEHUQUNURUSVAWDVTVTVKVKUAZUJZWCHWHVTVTHVSABVBWHJVRABVKVKVCV
          DVEWIWCWCVTVTVKVKVFVGVHVIVJ $.
      $}
    $}
  $}

  ${
    $d x y .+ $.  $d x y S $.  $d x y A $.  $d x y B $.  $d x y C $.
    $d x y D $.
    eceqoveq.5 $e |- .~ Er ( S X. S ) $.
    eceqoveq.7 $e |- dom .+ = ( S X. S ) $.
    eceqoveq.8 $e |- -. (/) e. S $.
    eceqoveq.9 $e |- ( ( x e. S /\ y e. S ) -> ( x .+ y ) e. S ) $.
    eceqoveq.10 $e |- ( ( ( A e. S /\ B e. S ) /\
                     ( C e. S /\ D e. S ) ) ->
          ( <. A , B >. .~ <. C , D >. <-> ( A .+ D ) = ( B .+ C ) ) ) $.
    $( Equality of equivalence relation in terms of an operation.  (Contributed
       by NM, 15-Feb-1996.)  (Proof shortened by Mario Carneiro,
       12-Aug-2015.) $)
    eceqoveq $p |- ( ( A e. S /\ C e. S ) ->
                      ( [ <. A , B >. ] .~ = [ <. C , D >. ] .~ <->
                      ( A .+ D ) = ( B .+ C ) ) ) $=
      ( wcel wa wceq wi wn c0 cop cec co wb cxp opelxpi ad2antrr wer a1i ereldm
      simpr mpbid opelxp2 syl ex caovcl eleq1 imbitrrid ndmovrcl simprd syl6com
      adantll wbr adantr erth bitr3d expr pm5.21ndd an32s eqcom wne erdm eleq2i
      ax-mp ecdmn0 opelxp 3bitr3i simplbi2 ad2antlr necon2bd ndmov nsyl5 mtbiri
      cdm syl6 simprbi syl5 necon1bd impbid bitrid necon1bi adantl eqeq1d simpl
      eqeq2d 3bitr4d pm2.61dan ) CIOZEIOZPZDIOZCDUAZHUBZEFUAZHUBZQZCFGUCZDEGUCZ
      QZUDZWRXAWSXJWRXAPZWSPZFIOZXFXIXLXFXMXLXFPZXDIIUEZOZXMXNXBXOOZXPXKXQWSXFC
      DIIUFZUGXNXBXDHXOXOHUHZXNJUIXLXFUKUJULEFIIUMUNUOXAWSXIXMRWRXIXAWSPZXGIOZX
      MXTYAXIXHIOABDEIGMUPXGXHIUQURYAWRXMCFIGKLUSUTVAVBXKWSXMXJXKWSXMPZPZXBXDHV
      CXFXIYCXBXDHXOXSYCJUIXKXQYBXRVDVENVFVGVHVIWTXASZPZTXEQZXGTQZXFXIYFXETQZYE
      YGTXEVJYEYHYGYEYHXMSYGYEXMXETWSXMXETVKZRWRYDYIWSXMXDHWDZOXPYIYBYJXOXDXSYJ
      XOQJXOHVLVNZVMXDHVOEFIIVPVQZVRVSVTWRXMPXMYGWRXMUKCFIGKWAWBWEYGYASYEYHYGYA
      TIOLXGTIUQWCYEYAXETYIXMYEYAYIWSXMYLWFWRXMYARWSYDWRXMYAABCFIGMUPUOUGWGWHWG
      WIWJYEXCTXEYDXCTQWTXAXCTXCTVKZWRXAXBYJOXQYMXKYJXOXBYKVMXBHVOCDIIVPVQWFWKW
      LWMYEXHTXGYDXHTQZWTXTXAYNXAWSWNDEIGKWAWBWLWOWPWQ $.
  $}

  ${
    $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.
       (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 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.
       (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 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.
       (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 $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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 } ) $.
  $}

  $( Note:  an alternate way to express partial functions is as follows,
     which would be added after df-fun above.  The symbol would be _pfun.gif.
     This definition would be needed only if we need proper class partial
     functions, which seems unlikely.  While nice-looking, for ordinary usage
     it would inconveniently require a new set of equality theorems, etc. $)
  $( Define a partial function, which is a function from a subset of the
     domain ` A ` to the codomain ` B ` .  The notation of a stroke through
     the arrow is used by the Z language: see, for example,
     ~ http://staff.washington.edu/jon/z/dcs.html . $)
  $(
  df-fp $a |- ( F : A -|-> B <-> ( Fun F /\ F C_ ( X X. Y ) ) $.
  $)

  ${
    $d f A $.  $d f 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 } = (/) ) $=
      ( cvv wcel cv wf cab c0 wne wex abn0 cdm fdm dmex eqeltrrdi exlimiv sylbi
      vex necon1bi ) ADEZABCFZGZCHZIUDIJUCCKUAUCCLUCUACUCAUBMDABUBNUBCSOPQRT $.

    $( 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
      $.

    $( Obsolete version of ~ mapex as of 17-Jun-2025.  (Contributed by Raph
       Levien, 4-Dec-2003.)  (New usage is discouraged.)
       (Proof modification is discouraged.) $)
    mapexOLD $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 pwexd
      wf ssexg sylancr ) ACFBDFGZABEHZTZEIZABJZKZLUHMFUFMFUFUDUGLZEIUHUEUIEABUD
      NOEUGPQUCUGMABCDRSUFUHMUAUB $.
  $}

  ${
    $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 mapex el2v fnmpoi ) ABDDBEZAEZC
      EFCGZHABCIPDJBANODDCKLM $.

    $( 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 wfun cxp cpw crab cvv cpm wceq elex pweqd rabeqdv wss
      co ssrab2 xpexg ancoms pwexd ssexg xpeq2 xpeq1 df-pm ovmpog 3expia syl2an
      sylancr wi mpd ) ACHZBDHZIZEJKZEBALZMZNZOHZABPUBVCQZUSVCVBUAVBOHVDUTEVBUC
      USVAOURUQVAOHBADCUDUEUFVCVBOUGUNUQAOHZBOHZVDVEUOURACRBDRVFVGVDVEFGABOOUTE
      GJZFJZLZMZNVCPUTEVHALZMZNOVIAQZUTEVKVMVNVJVLVIAVHUHSTVHBQZUTEVMVBVOVLVAVH
      BAUISTFGEUJUKULUMUP $.
  $}

  ${
    $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 $.
  $}

  ${
    elmapdd.a $e |- ( ph -> A e. V ) $.
    elmapdd.b $e |- ( ph -> B e. W ) $.
    elmapdd.c $e |- ( ph -> C : B --> A ) $.
    $( Deduction associated with ~ elmapd .  (Contributed by SN,
       29-Jul-2024.) $)
    elmapdd $p |- ( ph -> C e. ( A ^m B ) ) $=
      ( cmap co wcel wf elmapd mpbird ) ADBCJKLCBDMIABCDEFGHNO $.
  $}

  ${
    $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 wb elmapg mpan2 f0bi bitrdi velsn
      0ex bitr4di eqrdv ) ABDZCAEFGZEHZUBCIZUCDZUEEJZUEUDDUBUFEAUEKZUGUBELDUFUH
      MSAEUEBLNOUEAPQCERTUA $.
  $}

  ${
    $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 pmvalg cvv wi a1i funeq elex
      eleq2d elrab bitrdi biancomd xpexg ancoms ssexg expcom wb elpwg pm5.21ndd
      syl anbi2d bitrd ) ADGZBEGZHZCABIJZGZCKZCBALZMZGZHVBCVCNZHUSVAVBVEUSVACFO
      ZKZFVDPZGVEVBHUSUTVICABDEFQUCVHVBFCVDVGCUAUDUEUFUSVEVFVBUSCRGZVEVFVEVJSUS
      CVDUBTUSVCRGZVFVJSURUQVKBAEDUGUHVFVKVJCVCRUIUJUNVJVEVFUKSUSCVCRULTUMUOUP
      $.
  $}

  $( 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 $.

  $( 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 ) ) $=
    ( cpm co wcel cdm wf wss wa cvv wb c0 wceq n0i cxp fnpm fndmi ndmov nsyl2
    elpm2g syl ibi ) CABDEZFZCGZACHUFBIJZUEAKFBKFJZUEUGLUEUDMNUHUDCOABKDKKPDQRS
    TABCKKUAUBUC $.

  $( 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 $.

  $( 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 ) ) $=
    ( cmap co wcel c0 wceq cvv wa n0i cxp fnmap fndmi ndmov nsyl2 ) ABCDEZFQGHB
    IFCIFJQAKBCIDIILDMNOP $.

  $( 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 (which does not require ~ ax-rep ) 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.) $)
    mapssfset $p |- ( B ^m A ) C_ { f | f : A --> B } $=
      ( cvv wcel cmap co cv wf cab wss wceq mapfset eqimss2 syl reldmmap ovprc1
      wn c0 0ss eqsstrdi pm2.61i ) BDEZBAFGZABCHICJZKZUCUEUDLUFABCDMUDUENOUCRUD
      SUEBAFPQUETUAUB $.

    $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 functions between two classes exists if the codomain exists.
       Generalization of ~ mapex to arbitrary domains.  (Contributed by AV,
       14-Aug-2024.) $)
    fsetex $p |- ( B e. V -> { f | f : A --> B } e. _V ) $=
      ( wcel cv wf cab cmap co cvv mapfset ovex eqeltrdi ) BDEABCFGCHBAIJKABCDL
      BAIMN $.

    $( The set of injections between two classes exists if the codomain exists.
       (Contributed by AV, 14-Aug-2024.) $)
    f1setex $p |- ( B e. V -> { f | f : A -1-1-> B } e. _V ) $=
      ( wcel cv wf1 cab wf cvv fsetex wss ccnv wfun df-f1 abbii abanssl eqsstri
      wa a1i ssexd ) BDEZABCFZGZCHZABUCIZCHZJABCDKUEUGLUBUEUFUCMNZSZCHUGUDUICAB
      UCOPUFUHCQRTUA $.

    $( The set of surjections between two classes exists (without any
       precondition).  (Contributed by AV, 8-Aug-2024.) $)
    fosetex $p |- { f | f : A -onto-> B } e. _V $=
      ( cv wfo cab cmap co ovex mapfoss ssexi ) ABCDECFBAGHBAGIABCJK $.

    $( The set of bijections between two classes exists.  (Contributed by AV,
       30-Mar-2024.)  (Revised by AV, 8-Aug-2024.)  (Proof shortened by SN,
       22-Aug-2024.) $)
    f1osetex $p |- { f | f : A -1-1-onto-> B } e. _V $=
      ( cv wf1o cab wfo fosetex f1ofo ss2abi ssexi ) ABCDZEZCFABLGZCFABCHMNCABL
      IJK $.
  $}

  ${
    $d A f g $.  $d B f g $.  $d F g $.  $d X g $.
    fsetfocdm.f $e |- F = { f | f : A --> B } $.
    fsetfocdm.s $e |- S = ( g e. F |-> ( g ` X ) ) $.
    $( The class of functions with a given domain and a given codomain is
       mapped, through evaluation at a point of the domain, into the codomain.
       (Contributed by AV, 15-Sep-2024.) $)
    fsetfcdm $p |- ( X e. A -> S : F --> B ) $=
      ( wcel cv cfv wf vex feq1 elab2 ffvelcdm expcom biimtrid imp fmptd ) GAJZ
      EFGEKZLZBCUBUCFJZUDBJZUEABUCMZUBUFABDKZMUGDUCFENABUHUCOHPUGUBUFABGUCQRSTI
      UA $.

    $d A f h x $.  $d B h x $.  $d F f h $.  $d S g h $.  $d V f g h x $.
    $d X f h x $.
    $( The class of functions with a given domain that is a set and a given
       codomain is mapped, through evaluation at a point of the domain, onto
       the codomain.  (Contributed by AV, 15-Sep-2024.) $)
    fsetfocdm $p |- ( ( A e. V /\ X e. A ) -> S : F -onto-> B ) $=
      ( vh vx wcel wa wf cv cfv wceq cmpt cvv wfo fsetfcdm adantl simplr fmpttd
      wrex wral wb simpll mptexd feq1 elab2g mpbird fveq1 cbvmptv eqtri fvmptd3
      syl fvexd eqidd eqid vex fvmpt ad2antlr eqtrd fveq2 sylan9req rspcedeq2vd
      eqcomd ralrimiva dffo3 sylanbrc ) AGMZHAMZNZFBCOZEPZKPZCQZRKFUFZEBUGFBCUA
      VNVPVMABCDEFHIJUBUCVOVTEBVOVQBMZNZKLAVQSZFVQVSWBWCFMZABWCOZWBLAVQBVOWALPZ
      AMUDUEWBWCTMWDWEUHWBLAVQGVMVNWAUIUJABDPZOWEDWCFTABWGWCUKIULURUMZWBVRWCRZV
      QWCCQZVSWBWJHWCQZVQWBDWCHWGQZWKFCTCEFHVQQZSDFWLSJEDFWMWLHVQWGUNUOUPHWGWCU
      NWHWBHWCUSUQVNWKVQRVMWALHVQVQAWCWFHRVQUTWCVAEVBVCVDVEWIVSWJVRWCCVFVIVGVHV
      JKEFBCVKVL $.
  $}

  ${
    $d A a f g m n $.  $d B a f g m n $.  $d V a m n $.
    $( The class of all functions from a nonempty set ` A ` into a proper class
       ` B ` is not a set.  If one of the preconditions is not fufilled, then
       ` { f | f : A --> B } ` is a set, see ~ fsetdmprc0 for ` A e/ _V ` ,
       ~ fset0 for ` A = (/) ` , and ~ fsetex for ` B e. _V ` , see also
       ~ fsetexb .  (Contributed by AV, 14-Sep-2024.)  (Proof shortened by BJ,
       15-Sep-2024.) $)
    fsetprcnex $p |- ( ( ( A e. V /\ A =/= (/) ) /\ B e/ _V )
                       -> { f | f : A --> B } e/ _V ) $=
      ( va vg vm vn wcel c0 wne wa cvv wnel cv wf cab wi wex cfv n0 cmpt cbvabv
      wfo fveq1 cbvmptv fsetfocdm focdmex syl5com nelcon3d expcom exlimiv sylbi
      feq1 impcom imp ) ADIZAJKZLBMNZABCOZPZCQZMNZURUQUSVCRZUREOZAIZESUQVDRZEAU
      AVFVGEUQVFVDUQVFLZVBMBMVHVBBFVBVEFOZTZUBZUDVBMIBMIABVKGHVBDVEVAABGOZPCGAB
      UTVLUNUCFHVBVJVEHOZTVEVIVMUEUFUGVBBMVKUHUIUJUKULUMUOUP $.
  $}

  ${
    $d A f $.  $d B f $.
    $( The class of all functions from a nonempty set ` A ` into a class ` B `
       is a set iff ` B ` is a set .  (Contributed by AV, 15-Sep-2024.) $)
    fsetcdmex $p |- ( ( A e. V /\ A =/= (/) )
                      -> ( B e. _V <-> { f | f : A --> B } e. _V ) ) $=
      ( wcel c0 wne wa cvv cv wf cab fsetex wnel wn fsetprcnex ex 3imtr3g con4d
      df-nel impbid2 ) ADEAFGHZBIEZABCJKCLZIEZABCIMUBUCUEUBBINZUDINZUCOUEOUBUFU
      GABCDPQBITUDITRSUA $.

    $( The class of all functions from a class ` A ` into a class ` B ` is a
       set iff ` B ` is a set or ` A ` is not a set or ` A ` is empty.
       (Contributed by AV, 15-Sep-2024.) $)
    fsetexb $p |- ( { f | f : A --> B } e. _V
                    <-> ( A e/ _V \/ A = (/) \/ B e. _V ) ) $=
      ( cv wf cab cvv wcel wnel c0 w3o wo wn wa ioran df-nel wne sylbi eqeltrdi
      wceq wi nnel df-ne bicomi anbi12i bitri fsetprcnex ex biimtrrid imp sylib
      con4i df-3or sylibr wfn fsetdmprc0 wss ffn ss2abi sseq0 mpan 0ex syl feq2
      csn abbidv fset0 eqtrdi p0ex fsetex 3jaoi impbii ) ABCDZEZCFZGHZAGIZAJTZB
      GHZKZVPVQVRLZVSLZVTWBVPWBMZVOGIZVPMWCWAMZVSMZNWDWAVSOWEWFWDWFBGIZWEWDBGPW
      EAGHZAJQZNZWGWDUAWEVQMZVRMZNWJVQVROWKWHWLWIAGUBWIWLAJUCUDUEUFWJWGWDABCGUG
      UHRUIUJRVOGPUKULVQVRVSUMUNVQVPVRVSVQVMAUOZCFZJTZVPACUPWOVOJGVOWNUQWOVOJTV
      NWMCABVMURUSVOWNUTVAVBSVCVRVOJVEZGVRVOJBVMEZCFWPVRVNWQCAJBVMVDVFBCVGVHVIS
      ABCGVJVKVL $.
  $}

  $( 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 elmapi ffnd ) ABCDEFCBAABCGH $.

  $( 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 elmapi ffund ) ABCDEFCBAABCGH $.

  $( 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 $.
  $}

  $( 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 ) ) $=
    ( wcel cpm co wa cvv cdm cin cres wf c0 wceq simpld adantl wss 3syl n0i cxp
    fnpm fndmi ndmov nsyl2 simpl elpmi inss1 fssres wfun ffun resres wrel resdm
    sylancl funrel reseq1 eqtr3id feq1d mpbid inss2 elpm2r mpanr2 syl21anc ) BE
    FZDACGHZFZIZAJFZVFDKZBLZADBMZNZVMABGHFZVHVJVFVHVJCJFZVHVGOPVJVPIVGDUAACJGJJ
    UBGUCUDUEUFQRVFVHUGVIVLADVLMZNZVNVIVKADNZVLVKSVRVHVSVFVHVSVKCSACDUHQRZVKBUI
    VKAVLDUJUPVIVLAVQVMVIVSDUKZVQVMPVTVKADULWAVQDVKMZBMZVMDVKBUMWADUNWBDPWCVMPD
    UQDUOWBDBURTUSTUTVAVJVFIVNVLBSVOVKBVBABVLVMJEVCVDVE $.

  ${
    $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 biancomi elin vex
      elmap velpw fneq1 elab anbi12i bitri 3bitr4i eqriv ) FABGHZBAIZJZCKZBLZCM
      ZNZBAFKZOZURULPZURBLZQZURUKRURUQRZUSUTVABAURSTABURDEUCVCURUMRZURUPRZQVBUR
      UMUPUAVDUTVEVAFULUDUOVACURFUBBUNURUEUFUGUHUIUJ $.

    $( 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 $.
    $( 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 cpm co cxp cpw cv wcel wss wfun wa cvv wb wceq n0i fnpm fndmi ndmov
      c0 nsyl2 elpmg syl ibi simprd velpw sylibr ssriv ) CABDEZBAFZGZCHZUIIZULU
      JJZULUKIUMULKZUNUMUOUNLZUMAMIBMILZUMUPNUMUITOUQUIULPABMDMMFDQRSUAABULMMUB
      UCUDUECUJUFUGUH $.

    $( 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 $.
  $}

  ${
    mapfvd.m $e |- M = ( A ^m B ) $.
    mapfvd.f $e |- ( ph -> F e. M ) $.
    mapfvd.x $e |- ( ph -> X e. B ) $.
    $( The value of a function that maps from ` B ` to ` A ` .  (Contributed by
       AV, 2-Feb-2023.) $)
    mapfvd $p |- ( ph -> ( F ` X ) e. A ) $=
      ( wcel cfv wi cmap co wf elmapi ffvelcdm expcom syl2imc eleq2s mpcom ) DE
      JAFDKBJZHAUBLDBCMNZEAFCJZDUCJCBDOZUBIDBCPUEUDUBCBFDQRSGTUA $.
  $}

  $( ~ fresaun transposed to mappings.  (Contributed by Stefan O'Rear,
     9-Oct-2014.)  (Revised by Stefan O'Rear, 6-May-2015.) $)
  elmapresaun $p |- ( ( F e. ( C ^m A ) /\ G e. ( C ^m B ) /\
    ( F |` ( A i^i B ) ) = ( G |` ( A i^i B ) ) ) ->
      ( F u. G ) e. ( C ^m ( A u. B ) ) ) $=
    ( cmap co wcel cin cres wceq w3a cun wf elmapi fresaun cvv elmapex simprd
    id syl3an simpld 3ad2ant1 unexg syl2an 3adant3 elmapd mpbird ) DCAFGHZECBFG
    HZDABIZJEUKJKZLZDEMZCABMZFGHUOCUNNZUIACDNUJBCENULULUPDCAOECBOULTABCDEPUAUMC
    UOUNQQUIUJCQHZULUIUQAQHZDCARZUBUCUIUJUOQHZULUIURBQHZUTUJUIUQURUSSUJUQVAECBR
    SABQQUDUEUFUGUH $.

  ${
    $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 $.
  $}

  $( 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.)  (Proof shortened
     by AV, 14-Jul-2022.) $)
  map0e $p |- ( A e. V -> ( A ^m (/) ) = 1o ) $=
    ( wcel c0 cmap co csn c1o mapdm0 df1o2 eqtr4di ) ABCADEFDGHABIJK $.

  ${
    $d f A $.  $d f B $.  $d f V $.  $d f W $.
    $( 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 n0 wf elmapg imbitrrid ne0i syl6
      csn cxp fconst6g exlimdv biimtrid necon4d feq2 mpbiri necon2d oveq1 map0b
      f0 jcad sylan9eq impbid1 ) ACFBDFGZABHIZJKZAJKZBJLZGUPURUSUTUPAJUQJAJLEMZ
      AFZENUPUQJLZEAOUPVBVCEUPVBBVAUAUBZUQFZVCVBVEUPBAVDPBVAAUCABVDCDQRUQVDSTUD
      UEUFUPBJUQJUPBJKZJUQFZVCVFVGUPBAJPZVFVHJAJPAULBJAJUGUHABJCDQRUQJSTUIUMUSU
      TUQJBHIJAJBHUJBUKUNUO $.
  $}

  $( The set of mappings of the empty set to the empty set is the singleton
     containing the empty set.  (Contributed by AV, 31-Mar-2024.) $)
  0map0sn0 $p |- ( (/) ^m (/) ) = { (/) } $=
    ( vf c0 cv wf cab wceq cmap co csn f0bi abbii 0ex mapval df-sn 3eqtr4i ) BB
    ACZDZAEPBFZAEBBGHBIQRAPBJKBBALLMABNO $.

  ${
    $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 wa wex syl wb mpbid cop wrex cmap co a1i elmapd
      snex crn wbr weu wfn ffn snidg fneu syl2anr cab euabsn cima wrel relimasn
      frel cdm imaeq2d imadmrn eqtr3di eqtr3d eqeq1d exbidv bitrid adantl sseld
      fdm frn vsnid eleq2 mpbiri impel adantll ffrn feq3 syl5ibcom imp ad2antrr
      vex fsng sylancl jca ex eximdv mpd df-rex sylibr w3a f1osng adantr f1oeq1
      wf1o bicomd f1of 3adant2 wss snssi 3ad2ant2 fssd rexlimdv3a impbid eqabdv
      bitrd ) AEJZDBJZUAKZLZBCUBZECDKZUCUDZAXIXOMXNCXINZXMACXNXIFOHXNOMADUGUEUF
      AXPXMAXPXMAXPPZXJCMZXLPZBQZXMXQXIUHZXJKZLZBQZXTXQDXJXIUIZBUJZYDXPXIXNUKDX
      NMZYFAXNCXIULADGMZYGIDGUMRBXNDXIUNUOXPYFYDSAYFYEBUPZYBLZBQXPYDYEBUQXPYJYC
      BXPYIYAYBXPXIXNURZYIYAXPXIUSYKYILXNCXIVABDXIUTRXPXIXIVBZURYKYAXPYLXNXIXNC
      XIVLVCXIVDVEVFVGVHVIVJTXQYCXSBXQYCXSXQYCPZXRXLXPYCXRAXPXJYAMZXRYCXPYACXJX
      NCXIVMVKYCYNXJYBMBVNYAYBXJVOVPVQVRYMXNYBXINZXLXPYCYOAXPYCYOXPXNYAXINYCYOX
      NCXIVSYAYBXNXIVTWAWBVRYMYHXJOMZYOXLSAYHXPYCIWCBWDZDXJGOXIWEWFTWGWHWIWJXLB
      CWKWLWHAXLXPBCAXRXLWMXNYBCXIAXLYOXRAXLPZXNYBXIWQZYOYRXNYBXKWQZYSAYTXLAYHY
      PYTIYQDXJGOWNWFWOXLYTYSSAXLYSYTXNYBXIXKWPWRVJTXNYBXIWSRWTXRAYBCXAXLXJCXBX
      CXDXEXFXHXG $.
  $}

  ${
    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 $.

    $d f y A $.  $d f y B $.
    $( The value of set exponentiation with a singleton exponent.  Theorem 98
       of [Suppes] p. 89.  (Contributed by NM, 10-Dec-2003.)  (Proof shortened
       by AV, 17-Jul-2022.) $)
    mapsn $p |- ( A ^m { B } ) = { f | E. y e. A f = { <. B , y >. } } $=
      ( cvv wcel csn cmap co cv cop wceq wrex cab id a1i mapsnd ax-mp ) BGHZBCI
      JKDLCALMINABODPNEUAABCDGGUAQCGHUAFRST $.
  $}

  ${
    $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 cv csn cxp cvv wceq sneq xpeq2d simpr snex xpexg mpan2 fvmptd3
      adantr ) DEHZFBHZIAFDAJZKZLDFKZLZBCMGUEFNUFUGDUEFOPUCUDQUCUHMHZUDUCUGMHUI
      FRDUGEMSTUBUA $.
  $}

  ${
    $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 ) } ) ) $=
      ( cfv csn cxp wceq cmap co wcel wf snex elmap cop fsn2 simprbi fvex sylbi
      xpeq1i xpsn eqtr2i eqtrdi oveq2i eleq2s ) CBDCHZIZJZKZCADIZLMZABLMCUNNUMA
      COZULAUMCFDPQUOCDUIRIZUKUOUIANCUPKDACGSTUKUMUJJUPBUMUJEUCDUIGDCUAUDUEUFUB
      BUMALEUGUH $.

    ${
      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
        WDFWSOZPZQWIXHXJWIWOCWSUCXHWOWDCURCWOWSHFUSUTVAWIXIWDXFXIWDPWIXGWOWDFBV
        QVBVCVDUJWTWQXHWFXJWAWSWPUPWTWEXIWDFWAWSVGUMUNUOUQVEWCWQWFWBWPWADWOCLGV
        FVHVIWLWTWIWKWSWADWOWJGVJVKVLVMVNWNWGABRZUAWHEXKEAWBWEUBXKJABWBWEVOVRVS
        WGABVPVRBACWKVOVT $.

      $( 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 ccnv cv cfv fnmpti csn snex fvex cxp eqeltri xpex
        cvv mapsncnv dff1o4 mpbir2an ) BCKLZBDMDUINDOZBNAUIEAPZQDEUKUAIRJBCJPZS
        ZUBUJCUMCESUEFETUCULTUDAJBCDEFGHIUFRUIBDUGUH $.
    $}

    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 $.
  $}

  ${
    $d ph g y $.  $d ps f $.  $d J f g y $.  $d S f g y $.  $d T f g y $.
    ralxpmap.j $e |- ( f = ( g u. { <. J , y >. } ) -> ( ph <-> ps ) ) $.
    $( Quantification over functions in terms of quantification over values and
       punctured functions.  (Contributed by Stefan O'Rear, 27-Feb-2015.)
       (Revised by Stefan O'Rear, 5-May-2015.) $)
    ralxpmap $p |- ( J e. T -> ( A. f e. ( S ^m T ) ph <->
          A. y e. S A. g e. ( S ^m ( T \ { J } ) ) ps ) ) $=
      ( wcel cv cop csn cun wceq wa wf cvv adantl wss cmap co cdif vex snex cfv
      unex wrex cres simpr wb elmapex elmapg mpbid simpl ffvelcdmd difss fssres
      syl sylancl simpld simprd difexd elmapd mpbird wfn ffnd fnsnsplit syl2anc
      opeq2 sneqd uneq2d eqeq2d uneq1 rspc2ev syl3anc ex cin c0 elmapi ad2antll
      wf1o f1osng f1of elvd adantr disjdifr a1i fun syl21anc snssd undifr sylib
      feq2d ssidd snssi ad2antrl unssd fssd ssun1 unexg eqeltrrid ssexg sylancr
      undif1 eleq1 syl5ibrcom rexlimdvva impbid ralxpxfr2d ) HEJZABFCGGKZHCKZLZ
      MZNZDEUAUBZDDEHMZUCZUAUBZXLXOGUDXNUEUGXKFKZXQJZYAXPOZGXTUHCDUHZXKYBYDXKYB
      PZHYAUFZDJYAXSUIZXTJZYAYGHYFLZMZNZOZYDYEEDHYAYEYBEDYAQZXKYBUJYEDRJZERJZPZ
      YBYMUKYBYPXKYADEULZSZDEYARRUMUSUNZXKYBUOZUPYEYHXSDYGQZYEYMXSETUUAYSEXRUQE
      DXSYAURUTYEDXSYGRRYBYNXKYBYNYOYQVASYEEXRRYEYNYOYRVBVCVDVEYEYAEVFXKYLYEEDY
      AYSVGYTEYAHVHVIYCYLYAXLYJNZOCGYFYGDXTXMYFOZXPUUBYAUUCXOYJXLUUCXNYIXMYFHVJ
      VKVLVMXLYGOUUBYKYAXLYGYJVNVMVOVPVQXKYCYBCGDXTXKXMDJZXLXTJZPZPZYBYCXPXQJZU
      UGUUHEDXPQUUGEDXMMZNZDXPUUGXSXRNZUUJXPQZEUUJXPQUUGXSDXLQZXRUUIXOQZXSXRVRV
      SOZUULUUEUUMXKUUDXLDXSVTWAXKUUNUUFXKUUNCXKXMRJPXRUUIXOWBUUNHXMERWCXRUUIXO
      WDUSWEWFUUOUUGXREWGWHXSXRDUUIXLXOWIWJUUGUUKEUUJXPUUGXRETUUKEOUUGHEXKUUFUO
      WKXREWLWMWNUNUUGDUUIDUUGDWOUUDUUIDTXKUUEXMDWPWQWRWSUUGDEXPRRUUGYNXSRJZUUE
      YNUUPPXKUUDXLDXSULWAZVAUUGEEXRNZTUURRJYOEXRWTUUGUURUUKREXRXEUUGUUPXRRJUUK
      RJUUGYNUUPUUQVBHUEXSXRRRXAUTXBEUURRXCXDVDVEYAXPXQXFXGXHXIYCABUKXKISXJ $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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 sbcel12 csbfv2g
      csb csbvarg fveq2d eqtrd eleq1d bitrid bitrd anbi2d abbidv eqtrid ) EDFZA
      EGZBHCIZUKJZAIZULKZBFZAUKLZMZCNUMEULKZAEBTZFZMZCNAUKBCOUJURVBCUJUQVAUMUJU
      QUPAEPZVAUPAEDQVCAEUOTZUTFUJVAAEUOBRUJVDUSUTUJVDAEUNTZULKUSAEUNDULSUJVEEU
      LAEDUAUBUCUDUEUFUGUHUI $.
  $}

  ${
    $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 wceq wa cv wf cab elixpconst eqabi mapvalg eqtr4id
      vex ancoms ) CEGZBDGZABCHZCBIJZKUBUCLUDBCFMZNZFOUEUGFUDABCUFFTPQCBEDFRSUA
      $.

    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 eleq12d abbii
      cbvralw anbi2i 3eqtr4i ) IJZCKZAJZUGLZDMZACNZOZIPUHBJZUGLZEMZBCNZOZIPACDQ
      BCEQUMURIULUQUHUKUPABCBUJDFRAUOEGRABSUJUODEUIUNUGUAHUBUDUEUCACDITBCEITUF
      $.
  $}

  ${
    $d A x y z $.  $d B y z $.  $d C x z $.
    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 $=
      ( vz cv wfn cfv wcel wral wa cab cixp weq fveq2 eleq12d cbvralvw dfixp
      anbi2i abbii 3eqtr4i ) GHZCIZAHZUDJZDKZACLZMZGNUEBHZUDJZEKZBCLZMZGNACDOBC
      EOUJUOGUIUNUEUHUMABCABPUGULDEUFUKUDQFRSUAUBACDGTBCEGTUC $.
    $( $j usage 'cbvixpv' avoids 'ax-10' 'ax-11' 'ax-12' 'ax-13'; $)
  $}

  ${
    $d x y z $.  $d A z $.  $d B z $.
    nfixpw.1 $e |- F/_ y A $.
    nfixpw.2 $e |- F/_ y B $.
    $( Bound-variable hypothesis builder for indexed Cartesian product.
       Version of ~ nfixp with a disjoint variable condition, which does not
       require ~ ax-13 .  (Contributed by Mario Carneiro, 15-Oct-2016.)  Avoid
       ~ ax-13 .  (Revised by GG, 26-Jan-2024.) $)
    nfixpw $p |- F/_ y X_ x e. A B $=
      ( vz cixp cv wcel cab wfn cfv nfcv wnfc wtru nfab a1i mptru wnf wa df-ixp
      wral nfel nffn wal df-ral nftru nffvd nfeld nfimd nfald nfxfr nfan nfcxfr
      wi ) BACDHGIZAIZCJZAKZLZURUQMZDJZACUCZUAZGKACDGUBVEBGVAVDBBUTUQBUQNZBUTOZ
      VGPUSBABURCBURNZEUDZQRSUEVDUSVCUPZAUFZBVCACUGVKBTPVJBAAUHPUSVCBUSBTPVIRPB
      VBDPBURUQBUQOPVFRBUROPVHRUIBDOPFRUJUKULSUMUNQUO $.
    $( $j usage 'nfixpw' avoids 'ax-13'; $)
  $}

  ${
    $d x z $.  $d 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.  Usage
       of this theorem is discouraged because it depends on ~ ax-13 .  Use the
       weaker ~ nfixpw when possible.  (Contributed by Mario Carneiro,
       15-Oct-2016.)  (New usage is discouraged.) $)
    nfixp $p |- F/_ y X_ x e. A B $=
      ( vz cixp cv wcel cab wfn cfv wa wnfc wtru wal a1i nfeld mptru wral nftru
      df-ixp nfcv weq wn nfcvf adantl nfabd2 nffn df-ral wnf nffvd nfimd nfald2
      wi nfxfr nfan nfab nfcxfr ) BACDHGIZAIZCJZAKZLZVBVAMZDJZACUAZNZGKACDGUCVI
      BGVEVHBBVDVABVAUDZBVDOPVCBAAUBZPBAUEBQUFZNZBVBCVLBVBOPBAUGUHZBCOVMERSZUIT
      UJVHVCVGUPZAQZBVGACUKVQBULPVPBAVKVMVCVGBVOVMBVFDVMBVBVABVAOVMVJRVNUMBDOVM
      FRSUNUOTUQURUSUT $.
  $}

  ${
    $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 cixp c0 wceq cvv wcel wn cv wex neq0 wfn cdm fndm vex dmex eqeltrrdi
      ixpfn syl exlimiv sylbi con1i ) ABCEZFGZBHIZUFJDKZUEIZDLUGDUEMUIUGDUIUHBN
      ZUGABCUHTUJBUHOHBUHPUHDQRSUAUBUCUD $.
  $}

  ${
    $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 Mario Carneiro,
       25-Jan-2015.) $)
    ixpexg $p |- ( A. x e. A B e. V -> X_ x e. A B e. _V ) $=
      ( cvv wcel wral cixp cuni ciun cxp wss uniixp iunexg xpexg syldan sylancr
      wa ssexg uniexb sylibr wn c0 ixpprc 0ex eqeltrdi adantr pm2.61ian ) BEFZC
      DFABGZABCHZEFZUIUJRZUKIZEFZULUMUNBABCJZKZLUQEFZUOABCMUIUJUPEFURABCEDNBUPE
      EOPUNUQESQUKTUAUIUBZULUJUSUKUCEABCUDUEUFUGUH $.

    $( 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 f C $.
    $( The indexed intersection of a collection of infinite Cartesian products.
       (Contributed by Mario Carneiro, 6-Feb-2015.) $)
    ixpiin $p |- ( B =/= (/) ->
      X_ x e. A |^|_ y e. B C = |^|_ y e. B X_ x e. A C ) $=
      ( vf c0 wne cixp ciin cv wcel wral wa wb cvv eliin elixp ralbii bitri wfn
      cfv r19.28zv elv vex fvex ax-mp ralcom anbi2i 3bitr4g eqrdv eqcomd ) DGHZ
      BDACEIZJZACBDEJZIZUMFUOUQUMFKZCUAZAKZURUBZELZACMZNZBDMZUSVCBDMZNZURUOLZUR
      UQLZUSVCBDUCVHURUNLZBDMZVEVHVKOFBURDUNPQUDVJVDBDACEURFUEZRSTVIUSVAUPLZACM
      ZNVGACUPURVLRVNVFUSVNVBBDMZACMVFVMVOACVAPLVMVOOUTURUFBVADEPQUGSVBABCDUHTU
      ITUJUKUL $.

    $( The intersection of a collection of infinite Cartesian products.
       (Contributed by Mario Carneiro, 3-Feb-2015.) $)
    ixpint $p |- ( B =/= (/) -> X_ x e. A |^| B = |^|_ y e. B X_ x e. A y ) $=
      ( c0 wne cint cixp cv ciin wceq ixpeq2 wcel intiin a1i mprg ixpiin eqtrid
      ) DEFACDGZHZACBDBIZJZHZBDACUAHJSUBKZTUCKACACSUBLUDAICMBDNOPABCDUAQR $.
  $}

  ${
    $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 eqabi eqtr4i ) ADBECFZDGZAFUCHBIZADJZKZCLDMZADBCNUGCUHUCUHIU
      CDOUDUGCDPUCQUFUDUEASRTUAUB $.
  $}

  ${
    $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-rep .  (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 c0 wceq n0i nsyl2
      ixpprc elmapg sylan2 mpbird ex ssrdv ) ABCFZDGZEABCHZUHBIJZUIEKZUJGZULUKG
      ZUIUMLUNBUHULMZUMUOUIABCULNOUMUIBPGZUNUOQUMUJRSUPUJULTABCUBUAUHBULDPUCUDU
      EUFUG $.

    $( 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 c0 wceq n0i ixpprc nsyl2
      id iunexg syl2anr ixpssmap2g syl simpr sseldd ex ssrdv ) CDFABGZEABCHZABC
      IZBJKZUJELZUKFZUNUMFUJUOMZUKUMUNUPULNFZUKUMOUOBNFZUJUQUJUOUKPQURUKUNRABCS
      TUJUAABCNDUBUCABCNUDUEUJUOUFUGUHUI $.
  $}

  $( 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 B $.  $d f x $.
    $( The infinite Cartesian product of a family ` B ( x ) ` with an empty
       member is empty.  The converse of this theorem is equivalent to the
       Axiom of Choice, see ~ ac9 .  (Contributed by Mario Carneiro,
       22-Jun-2016.) $)
    ixpn0 $p |- ( X_ x e. A B =/= (/) -> A. x e. A B =/= (/) ) $=
      ( vf cixp c0 wne cv wcel wex wral n0 cab wfn wa df-ixp eqabri ne0i ralimi
      cfv simplbiim exlimiv sylbi ) ABCEZFGDHZUDIZDJCFGZABKZDUDLUFUHDUFUEAHZBIA
      MNZUIUETZCIZABKZUHUJUMODUDABCDPQULUGABCUKRSUAUBUC $.

    $( The infinite Cartesian product of a family ` B ( x ) ` with an empty
       member is empty.  The converse of this theorem is equivalent to the
       Axiom of Choice, see ~ ac9 .  (Contributed by NM, 1-Oct-2006.)  (Proof
       shortened by Mario Carneiro, 22-Jun-2016.) $)
    ixp0 $p |- ( E. x e. A B = (/) -> X_ x e. A B = (/) ) $=
      ( c0 wceq wrex wne wral wn cixp rexbii rexnal bitr3i ixpn0 necon1bi sylbi
      nne ) CDEZABFZCDGZABHZIZABCJZDESTIZABFUBUDRABCDQKTABLMUAUCDABCNOP $.
  $}

  ${
    $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 A x $.  $d B x $.  $d F x $.  $d G x $.
    $( Union of two projections of a cartesian product.  (Contributed by FL,
       7-Nov-2011.) $)
    undifixp $p |-
    ( ( F e. X_ x e. B C /\ G e. X_ x e. ( A \ B ) C /\ B C_ A )
        -> ( F u. G ) e. X_ x e. A C ) $=
      ( cixp wcel cun wfn wral wi wa c0 wceq sylancl syl2imc wn eqcoms syl cdif
      wss w3a cvv cfv unexg 3adant3 ixpfn cin 3simpa ancomd disjdif fnun biimpi
      cv eqcomd 3ad2ant3 fneq2d mpbird 3exp 3imp elixp2 simp3bi cdm fndm elndif
      undif wb eleq2 notbid ndmfv biimtrdi syl2im ralrimiv uneq2 un0 eqtr eleq1
      biimpd com12 ral2imi impcom eldifn uneq1 uncom imp ralunb sylanbrc imbi2d
      ex raleq imbitrrid sylbi 3imp231 wfun df-fn simpl anim12i ineq12 ad2ant2l
      eqtrdi fvun syl2anc eleq1d ralbidv syl3anbrc ) EACDGZHZFABCUAZDGZHZCBUBZU
      CZEFIZUDHZXNBJZAUOZXNUEZDHZABKZXNABDGHXHXKXOXLEFXGXJUFUGXHXKXLXPXKFXIJZXH
      ECJZXLXPLAXIDFUHZACDEUHZYAYBXLXPYAYBXLUCZXPXNCXIIZJZYEYBYAMCXIUIZNOYGYEYA
      YBYAYBXLUJUKCBULZCXIEFUMPYEBYFXNXLYABYFOZYBXLYFBXLYFBOZCBVGZUNUPUQURUSUTQ
      VAXMXTXQEUEZXQFUEZIZDHZABKZXLXHXKYQXLYKXHXKYQLZLZYLYSBYFXHYRYJXKYPAYFKZLX
      HXKYTXHXKMYPACKZYPAXIKZYTXKXHUUAXKYAXHUUALYCXHYMDHZACKZYAYNNOZACKUUAXHEUD
      HYBUUDACDEVBVCYAUUEACYAFVDZXIOZXQCHZXQXIHZRZUUEXIFVEXQCBVFUUGUUJXQUUFHZRZ
      UUEUUJUULVHXIUUFXIUUFOUUIUUKXIUUFXQVIVJSXQFVKVLVMVNUUCUUEYPACUUEUUCYPUUEY
      OYMNIZOZUUMYMOZUUCYPLZYNNYMVOYMVPUUNUUOMYOYMOUUPYOUUMYMVQUUPYMYOYMYOOUUCY
      PYMYODVRVSSTPVTWAQTWBXHXKUUBXHYBXKUUBLYDXKYNDHZAXIKZYBYMNOZAXIKUUBXKFUDHY
      AUURAXIDFVBVCYBUUSAXIYBEVDZCOZUUIUUHRZUUSCEVEXQBCWCUVBUUSLCUUTCUUTOZUVBXQ
      UUTHZRUUSUVCUUHUVDCUUTXQVIVJXQEVKVLSVMVNUUQUUSYPAXIUUSUUQYPUUSYONYNIZOZUV
      EYNNIZOZUUQYPLZYMNYNWDNYNWEUVFUVHMYOUVGOZUVGYNOZUVIYOUVEUVGVQYNVPUVJUVKMY
      OYNOUVIYOUVGYNVQUVIYNYOYNYOOUUQYPYNYODVRVSSTPPVTWAQTWFYPACXIWGWHWJYJYQYTX
      KYPABYFWKWIWLSWMWNXHXKXLXTYQVHZXKYAXHYBXLUVLLZYCYDYAFWOZUUGMZYBUVMLFXIWPY
      BUVOUVMYBEWOZUVAMZUVOUVMLECWPUVQUVOXLUVLUVQUVOXLUCZXSYPABUVRXRYODUVRUVPUV
      NMZUUTUUFUIZNOZXRYOOUVQUVOUVSXLUVQUVPUVOUVNUVPUVAWQUVNUUGWQWRUGUVQUVOUWAX
      LUVAUUGUWAUVPUVNUVAUUGMUVTYHNUUTCUUFXIWSYIXAWTUGXQEFXBXCXDXEUTWMVTWMQVAUS
      ABDXNVBXF $.
  $}

  ${
    $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
      nfv nffvmpt1 nfel weq fveq2 eleq12d cbvralw anbi2i bitri mptexg biantrurd
      wf bitr2id bitrid ) BEGBHGZABCIZABDJZGZCDGZABKZLBEMWCWAHGZWABUAZFNZWAOZAW
      HDUBZGZFBKZPZPZVTWEWCWAFBWJJZGWFWGWLUCWNWBWOWAAFBDWJFDUDAWHDUEZAWHDUFZUGU
      HFBWJWAUIWFWGWLUJUNWEWMVTWNWEWGANZWAOZDGZABKZPZWMWEXBWEWGXAABCWADWAUKZULW
      DWTABWRBGZWDPWSCDABCDWAXCQXDWDUMUORUPWGXAWEWGBHWAVQZXAWESZBWAUQXECHGZABKZ
      XFABHCWAXCURXHWTWDSZABKXFXGXIABXDXGPZWTWDXJWSCDABCHWAXCQUSUTRWTWDABVATVBV
      CVDVEXAWLWGWTWKAFBWTFVFAWIWJABCWHVGWPVHAFVIWSWIDWJWRWHWAVJWQVKVLVMVNVTWFW
      MABCHVOVPVRVST $.
  $}

  ${
    $d f g h x y z A $.  $d f g h x y z B $.  $d f g h y C $.  $d g h y F $.
    resixpfo.1 $e |- F = ( f e. X_ x e. A C |-> ( f |` B ) ) $.
    $( Restriction of elements of an infinite Cartesian product creates a
       surjection, if the original Cartesian product is nonempty.  (Contributed
       by Mario Carneiro, 27-Aug-2015.) $)
    resixpfo $p |- ( ( B C_ A /\ X_ x e. A C =/= (/) ) ->
        F : X_ x e. A C -onto-> X_ x e. B C ) $=
      ( vh vy vg vz wa cv cfv wceq wral wcel cmpt wi cvv wss cixp c0 wne wf wfo
      wrex cres resixp fmptd adantr wex n0 cif eleq1w ifbid fveq12d cbvmptv wfn
      id vex elixp simprbi fveq1 eleq1d simpl wn simplrr ifbothda exp32 ralimi2
      imp syl adantl ralim impel wb n0i ixpprc mptelixpg mpbird eqeltrid reseq1
      nsyl2 iftrue fveq1d mpteq2ia resmpt ad2antrr ixpfn ad2antlr dffn5 3eqtr4a
      sylib eqeltrdi fvmptd3 eqtr2d fveq2 rspceeqv syl2anc ex ralrimdva exlimdv
      biimtrid dffo3 sylanbrc ) CBUAZABDUBZUCUDZLXHACDUBZFUEZHMZIMZFNZOIXHUGZHX
      JPZXHXJFUFXGXKXIXGEXHEMZCUHZXJFABCDXQUIGUJUKXGXIXPXIJMZXHQZJULXGXPJXHUMXG
      XTXPJXGXTXOHXJXGXLXJQZLZXTXOYBXTLZKBKMZYDCQZXLXSUNZNZRZXHQXLYHFNZOXOYCYHA
      BAMZYJCQZXLXSUNZNZRZXHKABYGYMYDYJOZYDYJYFYLYOYEYKXLXSKACUOUPYOUTUQURYCYNX
      HQZYMDQZABPZYBYJXSNZDQZABPZYRXTYBYTYQSZABPZUUAYRSYAUUCXGYAYJXLNZDQZACPZUU
      CYAXLCUSZUUFACDXLHVAZVBVCUUEUUBACBYKUUESZYJBQZYTYQYKUUEYTYQUUIUUJYTLZLZXL
      XSXLYLOUUDYMDYJXLYLVDVEXSYLOYSYMDYJXSYLVDVEUULYKUUEUUIUUKVFVLUUIUUJYTYKVG
      VHVIVJVKVMVNYTYQABVOVMXTXSBUSUUAABDXSJVAVBVCVPYCBTQZYPYRVQXTUUMYBXTXHUCOU
      UMXHXSVRABDVSWDVNABYMDTVTVMWAWBZYCYIYHCUHZXLYCEYHXRUUOXHFTGXQYHCWCUUNYCUU
      OXLTYCKCYGRZKCYDXLNZRZUUOXLKCYGUUQYEYDYFXLYEXLXSWEWFWGXGUUOUUPOYAXTKBCYGW
      HWIYCUUGXLUUROYAUUGXGXTACDXLWJWKKCXLWLWNWMZUUHWOWPUUSWQIYHXHXNYIXLXMYHFWR
      WSWTXAXBXCXDVLIHXHXJFXEXF $.
  $}

  ${
    $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 cfv wa vex eleq1d sneq ixpeq1d
      eleq2d opeq1 sneqd eqeq2d elex snex eleq1 mpbiri rexlimivw eqeq1 wfn wral
      rexbidv elixp fveq2 ralsn anbi2i simpl biimpri adantl ffnfv sylanbrc fsn2
      wf sylib opeq2 rspceeqv syl fvsn id eqeltrid jctil fneq1 fveq1 syl5ibrcom
      fnsn anbi12d rexlimiv impbii 3bitri vtoclbg pm5.21nii ) EAGIZJZDKZLZEWEBI
      ZMZJZNZBDOZEACJZDKZLECWIMZJZNZBDOGCFWECNZWGWOEWSAWFWNDWECUAUBUCWSWLWRBDWS
      WKWQEWSWJWPWECWIUDUEUFUOWHEPLZWMEWGUGWLWTBDWLWTWKPLWJUHEWKPUIUJUKHIZWGLZX
      AWKNZBDOZWHWMHEPXAEWGUIXAENXCWLBDXAEWKULUOXBXAWFUMZAIZXAQZDLZAWFUNZRXEWEX
      AQZDLZRZXDAWFDXAHSUPXIXKXEXHXKAWEGSZXFWENXGXJDXFWEXAUQTURUSXLXDXLXKXAWEXJ
      MZJZNRZXDXLWFDXAVFZXPXLXEWIXAQZDLZBWFUNZXQXEXKUTXKXTXEXTXKXSXKBWEXMWIWENX
      RXJDWIWEXAUQTURVAVBBWFDXAVCVDWEDXAXMVEVGBXJDWKXOXAWIXJNWJXNWIXJWEVHUEVIVJ
      XCXLBDWIDLZXLXCWKWFUMZWEWKQZDLZRYAYDYBYAYCWIDWEWIXMBSZVKYAVLVMWEWIXMYEVRV
      NXCXEYBXKYDWFXAWKVOXCXJYCDWEXAWKVPTVSVQVTWAWBWCWDWC $.
  $}

  ${
    $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 wa vex wceq eqeq2d cixp cxp snex xpex
      a1i rnex uniex cop wrex sneq xpeq1d anbi2d wb elixpsn elv ixpeq1d bitr3id
      eleq2d anbi1d xpsn eqeq2i anbi2i eqid weq opeq2 sneqd mpan2 op2nda eqcomi
      rspceeqv jctir eqeq1 rexbidv unieqd anbi12d syl5ibrcom imp wi eqidd ancli
      rneq eleq1w biimtrid imbi12d rexlimiv impbii bitri vtoclbg f1od ) EFKZAHC
      BELZCUAZWKAMZLZUBZHMZNZOZDPPGWOPKWJWMCKZQWKWNEUCWMUCUDUEWRPKWJWPWLKZQWQWP
      HRUFUGUEWSWPIMZLZWNUBZSZQZWPXAJMZUHZLZSZJCUIZWMWRSZQZWSWPWOSZQWTXKQIEFXAE
      SZXDXMWSXNXCWOWPXNXBWKWNXAEUJZUKTULXNXJWTXKXJWPBXBCUAZKZXNWTXQXJUMIBJXACW
      PPUNUOXNXPWLWPXNBXBWKCXOUPURUQUSXEWSWPXAWMUHZLZSZQZXLXDXTWSXCXSWPXAWMIRZA
      RZUTVAVBYAXLWSXTXLWSXLXTXSXHSZJCUIZWMXSNZOZSZQWSYEYHWSXSXSSYEXSVCJWMCXHXS
      XSJAVDXGXRXFWMXAVEVFVJVGYGWMXAWMYBYCVHVIVKXTXJYEXKYHXTXIYDJCWPXSXHVLVMXTW
      RYGWMXTWQYFWPXSWAVNTVOVPVQXJXKYAXIXKYAVRZJCXFCKZYIXIWMXHNZOZSZWSXHXSSZQZV
      RYMAJVDZYJYOYLXFWMXAXFYBJRVHVAYJYOYPYJXHXHSZQYJYQYJXHVSVTYPWSYJYNYQAJCWBY
      PXSXHXHYPXRXGWMXFXAVEVFTVOVPWCXIXKYMYAYOXIWRYLWMXIWQYKWPXHWAVNTXIXTYNWSWP
      XHXSVLULWDVPWEVQWFWGWHWI $.

    $( 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 cvv snex f1oeq3d
      ixpconstg eqcomd mpan adantr mpbird ) BEIZDFIZJZBBDKZLMZCNBHUJBOZCNZUHUMU
      GAHBCDFGPQUIUKULBCUGUKULRZUHUJSIZUGUNDTUOUGJULUKHUJBSEUBUCUDUEUAUF $.
  $}

  ${
    $d A y z $.  $d B y z $.  $d I x y z $.
    $( A rectangular subset of a rectangular set can be recovered as the
       relative intersection of single-axis restrictions.  (Contributed by
       Stefan O'Rear, 22-Feb-2015.) $)
    boxriin $p |- ( A. x e. I A C_ B -> X_ x e. I A = ( X_ x e. I B i^i
            |^|_ y e. I X_ x e. I if ( x = y , A , B ) ) ) $=
      ( vz wral cixp weq cv wcel wa ral2imi adantr impr eleq2 jca elixp bitri
      wi wss cif ciin cin wfn simprl ssel simplr wn ssel2 ifbothda ex ralrimivw
      cfv jca31 simprll simpr ralimi ralcom wceq iftrue equcoms eleq2d ralimiaa
      rspcva sylbi syl ad2antll impbida vex wb cvv eliin ralbii anbi12i 3bitr4g
      elin elv eqrdv ) CDUAZAEGZFAECHZAEDHZBEAEABIZCDUBZHZUCZUDZWAFJZEUEZAJZWIU
      NZCKZAEGZLZWJWLDKZAEGZLZWJWLWEKZAEGZLZBEGZLZWIWBKWIWHKZWAWOXCWAWOLZWJWQXB
      WAWJWNUFZWAWJWNWQWAWNWQTWJVTWMWPAECDWLUGMNOXEXABEXEWJWTXFWAWJWNWTWAWNWTTW
      JVTWMWSAEVTWMWSWDWMWPWSVTWMLZCDCWEWLPDWEWLPVTWMWDUHXGWPWDUICDWLUJNUKULMNO
      QUMUOWAXCLWJWNWAWJWQXBUPXBWNWAWRXBWTBEGZWNXAWTBEWJWTUQURXHWSBEGZAEGWNWSBA
      EEUSXIWMAEWSWMBWKEBAIWECWLWECUTABWDCDVAVBVCVEVDVFVGVHQVIAECWIFVJZRXDWIWCK
      ZWIWGKZLXCWIWCWGVQXKWRXLXBAEDWIXJRXLWIWFKZBEGZXBXLXNVKFBWIEWFVLVMVRXMXABE
      AEWEWIXJRVNSVOSVPVS $.
  $}

  ${
    $d A k l m z $.  $d B l m z $.  $d C l m z $.  $d X k l m z $.
    $( The relative complement of a box set restricted on one axis.
       (Contributed by Stefan O'Rear, 22-Feb-2015.) $)
    boxcutc $p |- ( ( X e. A /\ A. k e. A C C_ B ) ->
                    ( X_ k e. A B \ X_ k e. A if ( k = X , C , B ) ) =
                                    X_ k e. A if ( k = X , ( B \ C ) , B ) ) $=
      ( vl vm wcel wss wral wa cv cif cdif wn cfv csb eleq2 nfv eleq12d vz cixp
      wceq eldifi adantl sseq1 difss ssid keephyp rgenw ss2ixp sselda wfn elixp
      mp1i vex ixpfn biantrurd bitr4id notbid wrex rexnal simpl simprbi nfcsb1v
      nfel2 fveq2 csbeq1a cbvralw csbeq1 rspcva syl2an neldif adantr syl5ibrcom
      sylib sylan ad2antlr r19.21bi ifbothda ralrimiva dfral2 ex con4d difeq12d
      weq imp simpll iftrue eqtrd eldifbd rspcev syl2an2r impbida nfn ifbieq12d
      nfif eqeq1 cbvrexw nfdif 3bitr4g bitr3id bitrd ibar 3bitr3d eldif eqrdav
      wb ) EAHZCBIDAJZKZUADABUBZDADLZEUCZCBMZUBZNZDAXNBCNZBMZUBZXLUALZXQHZYAXLH
      ZXKYAXLXPUDUEXKXTXLYAXSBIZDAJXTXLIXKYDDAXNXRBIBBIYDXRBXRXSBUFBXSBUFBCUGBU
      HUIUJDAXSBUKUOULXKYCKZYCYAXPHZOZKZYAAUMZXMYAPZXSHZDAJZKZYBYAXTHYEYGYLYHYM
      YEYGYJXOHZDAJZOZYLYEYFYOYEYFYIYOKYODAXOYAUAUPZUNYEYIYOYCYIXKDABYAUQUEZURU
      SUTYPYNOZDAVAZYEYLYNDAVBYEFLZYAPZUUAEUCZDUUACQZDUUABQZMZHZOZFAVAZGLZYAPZU
      UJEUCZDUUJBQZDUUJCQZNZUUMMZHZGAJZYTYLYEUUIUURYEUUIKZUUQGAUULUUKUUOHZUUKUU
      MHZUUQUUSUUJAHZKZUUOUUMUUOUUPUUKRUUMUUPUUKRUVCUULUUTUVCUUTUULEYAPZDEBQZDE
      CQZNZHZUUSUVHUVBYEUUIUVHYEUVHUUIYEUVHOZUUIOZYEUVIKZUUGFAJUVJUVKUUGFAUUCUU
      BUUDHZUUBUUEHZUUGUVKUUAAHZKZUUDUUEUUDUUFUUBRUUEUUFUUBRUVOUUCUVLUVOUVLUUCU
      VDUVFHZUVKUVPUVNYEUVDUVEHZUVIUVPXKXIUVMFAJZUVQYCXIXJVCYCYJBHZDAJZUVRYCYIU
      VTDABYAYQUNVDZUVSUVMDFAUVSFSDUUBUUEDUUABVEZVFDFWFZYJUUBBUUEXMUUAYAVGZDUUA
      BVHZTVIVPZUVMUVQFEAUUCUUBUVDUUEUVEUUAEYAVGZDUUAEBVJTVKVLUVDUVEUVFVMVQVNUU
      CUUBUVDUUDUVFUWGDUUAECVJZTVOWGUVOUVMUUCOUVKUVMFAYCUVRXKUVIUWFVRVSVNVTWAUU
      GFAWBVPWCWDWGVNUULUUKUVDUUOUVGUUJEYAVGZUULUUMUVEUUNUVFDUUJEBVJDUUJECVJWEZ
      TVOWGUVCUVAUULOUUSUVAGAYCUVAGAJZXKUUIYCUVTUWKUWAUVSUVADGAUVSGSDUUKUUMDUUJ
      BVEZVFDGWFZYJUUKBUUMXMUUJYAVGZDUUJBVHZTVIVPVRVSVNVTWAYEXIUURUVPOZUUIXIXJY
      CWHZYEUURKUVDUVEUVFYEXIUURUVHUWQUUQUVHGEAUULUUKUVDUUPUVGUWIUULUUPUUOUVGUU
      LUUOUUMWIUWJWJTVKVQWKUUHUWPFEAUUCUUGUVPUUCUUBUVDUUFUVFUWGUUCUUFUUDUVFUUCU
      UDUUEWIUWHWJTUTWLWMWNYSUUHDFAYSFSUUGDDUUBUUFUUCDUUDUUEUUCDSDUUACVEUWBWQVF
      WOUWCYNUUGUWCYJUUBXOUUFUWDUWCXNUUCCBUUDUUEXMUUAEWRDUUACVHUWEWPTUTWSYKUUQD
      GAYKGSDUUKUUPUULDUUOUUMUULDSDUUMUUNUWLDUUJCVEWTUWLWQVFUWMYJUUKXSUUPUWNUWM
      XNUULXRBUUOUUMXMUUJEWRUWMBUUMCUUNUWODUUJCVHWEUWOWPTVIXAXBXCYCYGYHXHXKYCYG
      XDUEYEYIYLYRURXEYAXLXPXFDAXSYAYQUNXAXG $.
  $}

$(
  @{
    xpixp.1 @e |- P = X_ x e. { (/) , 1o } if ( x = (/) , A , B ) @.
    xpixp.1 @e |- P = X_ x e. 2o if ( x = (/) , A , B ) @.
    xpixp.1 @e |- P =
               X_ x e. { (/) , 1o } ( { <. (/) , A >. , <. 1o , B >. } ` x ) @.
    @( A Cartesian product in terms of an infinite Cartesian product. @)
    xpixp @p |- ( A X. B ) =
                ( ran ( U. P |` { (/) } ) X. ran ( U. P |` { 1o } ) ) @=
?@.
      @( [ ?] @) @( [29-Sep-2006] @)
  @}

  @{
    ixpxp.1 @e |- P = ( A X. B ) @.
    @( An infinite Cartesian product in terms of a Cartesian product. @)
    ixpxp @p |- X_ x e. { (/) , 1o } if ( x = (/) , A , B ) =
                { ( { (/) } X. dom P ) , ( { 1o } X. ran P ) } @=
?@.
      @( [ ?] @) @( [29-Sep-2006] @)
  @}
$)


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Equinumerosity
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Introduce new constant symbols. $)
  $c ~~ $.  $( Equinumerosity relation ("approximately equals" symbol) $)
  $c ~<_ $.  $( Dominance relation (curly "less than or equal to") $)
  $c ~< $.  $( Strict dominance relation (curly less-than) $)
  $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 to") $)
  cdom $a class ~<_ $.

  $( Extend class definition to include the strict dominance relation (curly
     less-than) $)
  csdm $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.  For an alternate definition see
       ~ dfdom2 .  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 strict dominance relation.  Alternate possible definitions
       are derived as ~ brsdom and ~ brsdom2 .  Definition 3 of [Suppes] p. 97.
       (Contributed by NM, 31-Mar-1998.) $)
    df-sdom $a |- ~< = ( ~<_ \ ~~ ) $.

    $( 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 .  If we accept Infinity, we can
       also express ` A e. Fin ` by ` A ~< _om ` (Theorem ~ isfinite .)
       (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 relopabiv ) ADBDCDECFABGABCHI $.

    $( Dominance is a relation.  (Contributed by NM, 28-Mar-1998.) $)
    reldom $p |- Rel ~<_ $=
      ( vx vy vf cv wf1 wex cdom df-dom relopabiv ) ADBDCDECFABGABCHI $.
  $}

  $( Strict dominance is a relation.  (Contributed by NM, 31-Mar-1998.) $)
  relsdom $p |- Rel ~< $=
    ( cdom wrel csdm reldom cen cdif reldif df-sdom releqi sylibr ax-mp ) ABZCB
    ZDLAEFZBMAEGCNHIJK $.

  $( 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 relen brrelex12i ) ABCDE $.

  ${
    $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'; $)

    $( Equinumerosity relation.  (Contributed by NM, 15-Jun-1998.)  Extract
       ~ breng as an intermediate result.  (Revised by BTernaryTau,
       23-Sep-2024.) $)
    bren $p |- ( A ~~ B <-> E. f f : A -1-1-onto-> B ) $=
      ( cen wbr cvv wcel wa wf1o wex encv wfn f1ofn cdm fndm vex dmex eqeltrrdi
      cv syl crn wfo wceq f1ofo forn rnex jca exlimiv breng pm5.21nii ) ABDEAFG
      ZBFGZHZABCSZIZCJABKUOUMCUOUKULUOUNALZUKABUNMUPAUNNFAUNOUNCPZQRTUOBUNUAZFU
      OABUNUBURBUCABUNUDABUNUETUNUQUFRUGUHABCFFUIUJ $.

    $( 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.)  Extract
       ~ brdom2g as an intermediate result.  (Revised by BTernaryTau,
       29-Nov-2024.) $)
    brdomg $p |- ( B e. C -> ( A ~<_ B <-> E. f f : A -1-1-> B ) ) $=
      ( cvv wcel cdom wbr cv wf1 wex wb wi brdom2g ex wn reldom brrelex1i f1f
      wf cdm fdm vex dmex eqeltrrdi syl exlimiv pm5.21ni a1d pm2.61i ) AEFZBCFZ
      ABGHZABDIZJZDKZLZMUKULUQABDECNOUKPUQULUMUKUPABGQRUOUKDUOABUNTZUKABUNSURAU
      NUAEABUNUBUNDUCUDUEUFUGUHUIUJ $.

    $( Obsolete version of ~ brdomg as of 29-Nov-2024.  (Contributed by NM,
       15-Jun-1998.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    brdomgOLD $p |- ( B e. C -> ( A ~<_ B <-> E. f f : A -1-1-> B ) ) $=
      ( vx vy cvv wcel cdom wbr cv wf1 wex wb wi wceq f1eq2 exbidv f1eq3 df-dom
      brabg ex reldom brrelex1i f1f cdm fdm vex dmex eqeltrrdi exlimiv pm5.21ni
      wn wf syl a1d pm2.61i ) AGHZBCHZABIJZABDKZLZDMZNZOURUSVDEKZFKZVALZDMAVFVA
      LZDMVCEFABGCIVEAPVGVHDVEAVFVAQRVFBPVHVBDVFBAVASREFDTUAUBURUMVDUSUTURVCABI
      UCUDVBURDVBABVAUNZURABVAUEVIAVAUFGABVAUGVADUHUIUJUOUKULUPUQ $.

    $( Dominance relation.  (Contributed by Mario Carneiro, 26-Apr-2015.)
       Avoid ~ ax-un .  (Revised by BTernaryTau, 29-Nov-2024.) $)
    brdomi $p |- ( A ~<_ B -> E. f f : A -1-1-> B ) $=
      ( cdom wbr cv wf1 wex cvv wcel wa wb reldom brrelex12i brdom2g syl ibi )
      ABDEZABCFGCHZRAIJBIJKRSLABDMNABCIIOPQ $.
    $( $j usage 'brdomi' avoids 'ax-un'; $)

    $( Obsolete version of ~ brdomi as of 29-Nov-2024.  (Contributed by Mario
       Carneiro, 26-Apr-2015.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    brdomiOLD $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 countable set 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
       Replacement nor the Axiom of Power Sets 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-pow' 'ax-rep' '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 Replacement nor the Axiom of Power Sets 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-pow' 'ax-rep' 'ax-un'; $)

    $( The domain and range of a one-to-one, onto set function are
       equinumerous.  This variation of ~ f1oeng does not require the Axiom of
       Replacement nor the Axiom of Power Sets.  (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-pow' 'ax-rep'; $)

    $( 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 Replacement nor the Axiom of Power Sets.  (Contributed by
       BTernaryTau, 9-Sep-2024.) $)
    f1dom3g $p |- ( ( F e. V /\ B e. W /\ F : A -1-1-> B ) -> A ~<_ B ) $=
      ( vf wcel wf1 w3a cdom wbr cv wex f1eq1 spcegv 3adant2 wb brdomg 3ad2ant2
      imp mpbird ) CDGZBEGZABCHZIABJKZABFLZHZFMZUBUDUHUCUBUDUHUGUDFCDABUFCNOTPU
      CUBUEUHQUDABEFRSUA $.
    $( $j usage 'f1dom3g' avoids 'ax-pow' 'ax-rep'; $)

    $( 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-rep'; $)

    $( 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.)  (Proof shortened by
       BTernaryTau, 25-Sep-2024.) $)
    f1dom2g $p |- ( ( A e. V /\ B e. W /\ F : A -1-1-> B ) -> A ~<_ B ) $=
      ( cvv wcel wf1 cdom wbr wf f1f fex2 syl3an1 3coml f1dom3g syld3an1 ) CFGZ
      BEGZADGZABCHZABIJUATSRUAABCKTSRABCLABCDEMNOABCFEPQ $.
    $( $j usage 'f1dom2g' avoids 'ax-rep'; $)

    $( 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 focdmex f1ofo impel f1oen2g 3com23 mpd3an3 )
      ACEZABDFZBGEZABHIZQABDJSRABCDKABDLMQSRTABDCGNOP $.

    $( 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 wf f1f f1dmex fex syl2an2r expcom f1eq1
      spcegv syli brdomg sylibrd ) BCFZABDGZABEHZGZEIZABJKUEUDDLFZUHUEUDUIUEABD
      MUDALFUIABDNABCDOABLDPQRUGUEEDLABUFDSTUAABCEUBUC $.
  $}

  ${
    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 $.
  $}

  $( Strict dominance relation, meaning " ` B ` is strictly greater in size
     than ` A ` ".  Definition of [Mendelson] p. 255.  (Contributed by NM,
     25-Jun-1998.) $)
  brsdom $p |- ( A ~< B <-> ( A ~<_ B /\ -. A ~~ B ) ) $=
    ( cop csdm wcel cdom cen cdif wbr wn wa df-sdom eleq2i df-br notbii anbi12i
    eldif bitr4i 3bitr4i ) ABCZDETFGHZEZABDIABFIZABGIZJZKZDUATLMABDNUFTFEZTGEZJ
    ZKUBUCUGUEUIABFNUDUHABGNOPTFGQRS $.

  ${
    $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
      UGBCAKLUFUICBQUHBQEAIBUDGMNOUCBRUEUHAIUCBUDGSTUAUB $.

    $( 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 opabidw
      cv 3imtr4i df-en eleq2i df-dom relssi ) ABDEFAPZBPZGZUBUCCPZHZCIZABJZKZUD
      UBUCUELZCIZABJZKZUDDKUDEKUGUKUIUMUFUJCUBUCUEMNUGABOUKABOQDUHUDABCRSEULUDA
      BCTSQUA $.
  $}

  $( Alternate definition of dominance.  (Contributed by NM, 17-Jun-1998.) $)
  dfdom2 $p |- ~<_ = ( ~< u. ~~ ) $=
    ( cen csdm cun cdom cdif df-sdom uncom wss wceq enssdom undif mpbi 3eqtr3ri
    uneq2i ) ABCADAEZCZBACDBOAFNABGADHPDIJADKLM $.

  $( 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 $.

  $( Strict dominance implies dominance.  (Contributed by NM, 10-Jun-1998.) $)
  sdomdom $p |- ( A ~< B -> A ~<_ B ) $=
    ( csdm wbr cdom cen wn brsdom simplbi ) ABCDABEDABFDGABHI $.

  $( Strict dominance implies non-equinumerosity.  (Contributed by NM,
     10-Jun-1998.) $)
  sdomnen $p |- ( A ~< B -> -. A ~~ B ) $=
    ( csdm wbr cdom cen wn brsdom simprbi ) ABCDABEDABFDGABHI $.

  $( Dominance in terms of strict dominance and equinumerosity.  Theorem 22(iv)
     of [Suppes] p. 97.  (Contributed by NM, 17-Jun-1998.) $)
  brdom2 $p |- ( A ~<_ B <-> ( A ~< B \/ A ~~ B ) ) $=
    ( cop cdom wcel csdm cen cun wbr wo dfdom2 eleq2i df-br orbi12i elun bitr4i
    3bitr4i ) ABCZDERFGHZEZABDIABFIZABGIZJZDSRKLABDMUCRFEZRGEZJTUAUDUBUEABFMABG
    MNRFGOPQ $.

  $( Equinumerosity expressed in terms of dominance and strict dominance.
     (Contributed by NM, 23-Oct-2004.) $)
  bren2 $p |- ( A ~~ B <-> ( A ~<_ B /\ -. A ~< B ) ) $=
    ( cen wbr cdom csdm wn endom sdomnen con2i jca brdom2 biimpi orcanai impbii
    wa wo ) ABCDZABEDZABFDZGZPRSUAABHTRABIJKSTRSTRQABLMNO $.

  $( 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
      fmpt sylib fvmpt2 adantll mpdan adantrr nffvmpt1 nfeq1 nfim eleq1w imbi1d
      wf1 nfv anbi1d anidm bitrdi fveq2 adantr biimparc sylbird pm5.74d chvarfv
      ex bitrd adantrl biimpd sylbid ralrimivva nfmpt1 nfcv dff13f sylanbrc ) A
      DEBDFUAZUBZBJZVQKZCJZVQKZLZVSWALZMZCDNBDNDEVQUPAFEOZBDNVRAWFBDHUCBDEFVQVQ
      UDZUEUFAWEBCDDAVSDOZWADOZPZPZWCFGLZWDWKVTFWBGAWHVTFLZWIAWHPZWFWMAWHWFHQWH
      WFWMABDFEVQWGUGUHUIZUJAWIWBGLZWHWNWMMZAWIPZWPMZBCWRWPBWRBUQBWBGBDFWAUKULU
      MWDWQWRWMMWSWDWNWRWMWDWHWIABCDUNZRUOWDWRWMWPWDWRWKWMWPSZWDWJWIAWDWJWIWIPW
      IWDWHWIWIWTURWIUSUTRWDWKXAWDWKPVTWBFGWDWCWKVSWAVQVAVBWKWLWDAWJWLWDSIQZVCT
      VGVDVEVHWOVFVITWKWLWDXBVJVKVLBCDEVQBDFVMCVQVNVOVP $.

    $( 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 spcedv wb brdomg mpbird ) ADEOPZDENUAZQZNUBZAUPDEBDFUCZQZNRURADEURUIZD
      HSEISZURRSAUSUTABCDEFGJKUDZDEURUETLMDEURHIUFUGVBDEUOURUHUJAVAUNUQUKMDEINU
      LTUM $.
  $}

  ${
    $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 wi cvv eqeng elv sylbi df-br
      ideq 3imtr3i relssi ) ABCDEAFZBFZCGZUAUBDGZUAUBHZCIUEDIUCABJZUDUAUBBKRUFU
      DLAUAUBMNOPUAUBCQUAUBDQST $.
  $}

  ${
    $d A f $.  $d B f $.  $d C f $.
    $( If ` A ` is a subset of ` B ` and ` C ` dominates ` B ` , then ` C `
       also dominates ` A ` .  (Contributed by BTernaryTau, 7-Dec-2024.) $)
    domssl $p |- ( ( A C_ B /\ B ~<_ C ) -> A ~<_ C ) $=
      ( vf wss cdom wbr wa cvv wcel simpr reldom brrelex12i simpl ssexg adantrr
      simprr jca32 sylan2 wf1 cv wex wi brdomi cres f1ssres vex f1dom4g mp3anl1
      resex ancoms sylan expl exlimiv syl sylc ) ABEZBCFGZHURUQAIJZCIJZHZHZACFG
      ZUQURKURUQBIJZUTHZVBBCFLMUQVEHUQUSUTUQVENUQVDUSUTABIOPUQVDUTQRSURBCDUAZTZ
      DUBVBVCUCZBCDUDVGVHDVGUQVAVCVGUQHACVFAUEZTZVAVCBCAVFUFVAVJVCVIIJUSUTVJVCV
      FADUGUJACVIIIIUHUIUKULUMUNUOUP $.
    $( $j usage 'domssl' avoids 'ax-pow' 'ax-un'; $)
  $}

  ${
    $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 $.
    $( $j usage 'domssr' avoids 'ax-pow' 'ax-un'; $)
  $}

  $( 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 mpbir funres11 sylanblrc
    funeqi df-f1 adantr f1dom2g syl3anc expcom ) ABDZBCEZABFGZUSUTHAIEUTABJAKZL
    ZVAABCMUSUTNUSVCUTUSABVBOZVBPQZVCAAVBOZUSVDAAVBTZVFVGVEAAVBRVGVEHAUAAAVBUBU
    CUDAAVBUESAABVBUFUGJPZQZVEVIJQUHVHJUIUMUJAJUKSABVBUNULUOABVBICUPUQUR $.

  ${
    $d f g x y z $.
    $( Equinumerosity is an equivalence relation.  (Contributed by NM,
       19-Mar-1998.)  (Revised by Mario Carneiro, 15-Nov-2014.)  (Proof
       shortened by AV, 1-May-2021.) $)
    ener $p |- ~~ Er _V $=
      ( vx vy vz vf vg cvv cen relen cv wbr wf1o wex bren wcel ccnv vex f1oen2g
      f1ocnv mp3an12i wa sylbi exdistrv ccom f1oco ancoms exlimivv sylbir enref
      exlimiv syl2anb 2th iseri ) ABCFGHAIZBIZGJZUMUNDIZKZDLUNUMGJZUMUNDMUQURDU
      NFNUMFNZUQUNUMUPOZKURBPAPZUMUNUPRUNUMUTFFQSUIUAUOUMUNEIZKZELZUNCIZUPKZDLZ
      UMVEGJZUNVEGJUMUNEMUNVEDMVDVGTVCVFTZDLELVHVCVFEDUBVIVHEDUSVEFNVIUMVEUPVBU
      CZKZVHVACPVFVCVKUMUNVEUPVBUDUEUMVEVJFFQSUFUGUJUSUMUMGJVAUMVAUHUKUL $.
  $}

  $( 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 exdistrv 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 $.

  $( If a function is one-to-one, then the image of a subset of its domain
     under it 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 $.

  $( If a function is one-to-one, then the image of a subset of its domain
     under it is equinumerous to the subset.  (This version of ~ f1imaeng does
     not need ~ ax-rep .)  (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 fimass
    syl ad2antrr ssexd f1ores ad2ant2r f1oen2g syl3anc ensymd ) ABDFZBEGZHZCAIZ
    CEGZHZHZCDCJZUOUMUPKGCUPDCLZMZCUPNOUKULUMPUOUPBEUIUJUNQUIUPBIZUJUNUIABDRUSA
    BDSABDCTUAUBUCUIULURUJUMABCDUDUECUPUQEKUFUGUH $.
  $( $j usage 'f1imaen2g' avoids 'ax-rep'; $)

  $( If a set function is one-to-one, then a subset of its domain is
     equinumerous to the image of that subset.  (This version of ~ f1imaeng
     does not need ~ ax-rep nor ~ ax-pow .)  (Contributed by BTernaryTau,
     13-Jan-2025.) $)
  f1imaen3g $p |- ( ( F : A -1-1-> B /\ C C_ A /\ F e. V ) ->
    C ~~ ( F " C ) ) $=
    ( wf1 wss wcel w3a cres cvv cima wf1o cen wbr resexg f1ores 3adant3 f1oen3g
    3ad2ant3 syl2anc ) ABDFZCAGZDEHZIDCJZKHZCDCLZUEMZCUGNOUDUBUFUCDCEPTUBUCUHUD
    ABCDQRCUGUEKSUA $.
  $( $j usage 'f1imaen3g' avoids 'ax-pow' 'ax-rep'; $)

  ${
    f1imaen.1 $e |- C e. _V $.
    $( If a function is one-to-one, then the image of a subset of its domain
       under it 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 A f $.
    $( The empty set is equinumerous only to itself.  Exercise 1 of
       [TakeutiZaring] p. 88.  (Contributed by NM, 27-May-1998.)  Avoid
       ~ ax-pow , ~ ax-un .  (Revised by BTernaryTau, 23-Sep-2024.) $)
    en0 $p |- ( A ~~ (/) <-> A = (/) ) $=
      ( vf c0 cen wbr wceq cv wf1o wex cvv wcel wa wb encv breng syl ibi f1ocnv
      ccnv 0ex f1o00 simprbi exlimiv f1oeq1 ceqsexv2d mp2an mpbir mpbiri impbii
      f1o0 breq1 ) ACDEZACFZULACBGZHZBIZUMULUPULAJKCJKZLULUPMACNACBJJOPQUOUMBUO
      CAUNSZHZUMACUNRUSURCFUMAURUAUBPUCPUMULCCDEZUTCCUNHZBIZVACCCHBCTCCUNCUDUJU
      EUQUQUTVBMTTCCBJJOUFUGACCDUKUHUI $.
    $( $j usage 'en0' avoids 'ax-pow' 'ax-un'; $)
  $}

  ${
    $d f A $.
    $( Shorter proof of ~ en0 , depending on ~ ax-pow and ~ ax-un .
       (Contributed by NM, 27-May-1998.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    en0ALT $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 A f $.
    $( The empty set is equinumerous only to itself.  (Contributed by
       BTernaryTau, 29-Nov-2024.) $)
    en0r $p |- ( (/) ~~ A <-> A = (/) ) $=
      ( vf c0 cen wbr wceq cv wf1o wex cvv wcel wa encv breng syl f1o00 simprbi
      wb ibi 0ex exlimiv f1oeq1 f1o0 ceqsexv2d mp2an mpbir breq2 mpbiri impbii
      ) CADEZACFZUJCABGZHZBIZUKUJUNUJCJKZAJKLUJUNRCAMCABJJNOSUMUKBUMULCFUKAULPQ
      UAOUKUJCCDEZUPCCULHZBIZUQCCCHBCTCCULCUBUCUDUOUOUPURRTTCCBJJNUEUFACCDUGUHU
      I $.
    $( $j usage 'en0r' avoids 'ax-pow' 'ax-un'; $)
  $}

  ${
    $d A f $.
    ensn1.1 $e |- A e. _V $.
    $( A singleton is equinumerous to ordinal one.  (Contributed by NM,
       4-Nov-2002.)  Avoid ~ ax-un .  (Revised by BTernaryTau, 23-Sep-2024.) $)
    ensn1 $p |- { A } ~~ 1o $=
      ( vf csn c0 c1o cen wbr wf1o wex cop snex f1oeq1 0ex f1osn ceqsexv2d wcel
      cv cvv wb breng mp2an mpbir df1o2 breqtrri ) ADZEDZFGUFUGGHZUFUGCRZIZCJZU
      JUFUGAEKZDZICUMULLUFUGUIUMMAEBNOPUFSQUGSQUHUKTALELUFUGCSSUAUBUCUDUE $.
    $( $j usage 'ensn1' avoids 'ax-un'; $)
  $}

  ${
    $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.)  Avoid ~ ax-un .  (Revised by
       BTernaryTau, 23-Sep-2024.) $)
    en1 $p |- ( A ~~ 1o <-> E. x A = { x } ) $=
      ( vf c1o cen wbr cv csn wceq wex c0 wf1o df1o2 breq2i cvv syl crn exlimiv
      wcel 0ex wa wb encv breng ibi sylbi ccnv cfv f1ocnv wfo f1ofo forn cop wf
      f1of fsn2 simprbi rneqd rnsnop eqtrdi eqtr3d fvex sneq eqeq2d spcev ensn1
      3syl vex breq1 mpbiri impbii ) BDEFZBAGZHZIZAJZVLBKHZCGZLZCJZVPVLBVQEFZVT
      DVQBEMNWAVTWABOSVQOSUAWAVTUBBVQUCBVQCOOUDPUEUFVSVPCVSVQBVRUGZLZBKWBUHZHZI
      ZVPBVQVRUIWCWBQZBWEWCVQBWBUJWGBIVQBWBUKVQBWBULPWCWGKWDUMHZQWEWCWBWHWCVQBW
      BUNZWBWHIZVQBWBUOWIWDBSWJKBWBTUPUQPURKWDTUSUTVAVOWFAWDKWBVBVMWDIVNWEBVMWD
      VCVDVEVGRPVOVLAVOVLVNDEFVMAVHVFBVNDEVIVJRVK $.
    $( $j usage 'en1' avoids 'ax-un'; $)

    $( A set is equinumerous to ordinal one iff it is a singleton.
       (Contributed by Mario Carneiro, 17-Jan-2015.)  Avoid ~ ax-un .  (Revised
       by BTernaryTau, 24-Sep-2024.) $)
    en1b $p |- ( A ~~ 1o <-> A = { U. A } ) $=
      ( vx c1o cen wbr cuni csn wceq cv wex id unieq unisnv eqtrdi sneqd eqtr4d
      en1 exlimiv sylbi cvv wcel eqsnuniex ensn1g syl eqbrtrd impbii ) ACDEZAAF
      ZGZHZUGABIZGZHZBJUJBAQUMUJBUMAULUIUMKUMUHUKUMUHULFUKAULLBMNOPRSUJAUICDUJK
      UJUHTUAUICDEAUBUHTUCUDUEUF $.
    $( $j usage 'en1b' avoids 'ax-un'; $)

    $( 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.)  Avoid ~ ax-un .  (Revised by BTernaryTau, 24-Sep-2024.) $)
  en1uniel $p |- ( S ~~ 1o -> U. S e. S ) $=
    ( c1o cen wbr cuni csn wcel wceq en1b eqsnuniex sylbi snidg biimpi eleqtrrd
    cvv syl ) ABCDZAEZRFZAQROGZRSGQASHZTAIZAJKROLPQUAUBMN $.
  $( $j usage 'en1uniel' avoids 'ax-un'; $)

  ${
    $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 wn wrex cfv wcel wceq ffvelcdm notbid
      sylancl wex weq df2o2 breq1i brdomi sylbi f1f prid1 snex prid2 0nep0 neii
      wf 0ex wb f1fveq mpanr12 mtbiri eqeq1 eqeq2 rspc2ev syl3anc exlimiv syl )
      ECFGZHHIZJZCDKZLZDUAZABUBZMZBCNACNZVEVGCFGVJEVGCFUCUDVGCDUEUFVIVMDVIHVHOZ
      CPZVFVHOZCPZVNVPQZMZVMVIVGCVHUMZHVGPZVOVGCVHUGZHVFUNUHZVGCHVHRTVIVTVFVGPZ
      VQWBHVFHUIUJZVGCVFVHRTVIVRHVFQZHVFUKULVIWAWDVRWFUOWCWEVGCHVFVHUPUQURVLVSV
      NBKZQZMABVNVPCCAKZVNQVKWHWIVNWGUSSWGVPQWHVRWGVPVNUTSVAVBVCVD $.
  $}

  ${
    $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 fvex eqtr2di impbid1 en3d ) AUAZCDAUBZACGZWFAUCZHZ
      DGZIZIZJJWEJKWDABUDLAJKWDBLWDWFWEKZWHAKWFAUEMWDAUFZWIAKZWKWEKZNAUGZWMWNWO
      AWIUHMUIWDWLWNOZWFWKPZWIWHPZQWDWQOWRWSWDWNWRWSNZWLWDWNOZWIEGZFGZHZPZFRERZ
      WTXAWIJJUJZKXFWDAXGWIWDWMAXGVEWPAUKULUMEFWIUNULXAXEWTEFXAXEWRWSXAXEWROZOZ
      WIWFXCHZWHXHWIXJPZXAWRXEXKWRXEOZXJXDPZXKWRXEXMWRXEWFXBPZXMXEXNWRWKXBPXEWK
      XDIZIXBXEWJXOWIXDUOUPXBXCESFSUQURWFWKXBUSUTWFXBXCVAVBVFXEXMXKNWRXMXKXEXJX
      DWIVCVDTVGZVHTXIWGXCWFWDWNXEWRWGXCPZWDWRXEWNXQWDWRXEWNXQNWDXLOWNXJAKZXQXL
      WNXRQWDXLWIXJAXPVITWDXRXQNXLWFXCAVJVKVLVMVNVQVOVPVMVRVGVSWSWKWHIZIWFWSWJX
      SWIWHUOUPWFWGCSWFAVTUQWAWBMWC $.
  $}

  ${
    $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 countable, 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 cfv cvv wcel a1i wceq wa wb co ovexd wi fvexd snex
      cop 2a1i wex wrex mapsnd eqabrd anbi1d r19.41v bicomi df-rex 3bitrd fveq1
      fvsng sylancl sylan9eqr eqeq2d equcom bitrdi pm5.32da anass ancom 3bitr2d
      vex anbi2d exbidv eleq1w opeq2 sneqd anbi12d equsexvw en2d ) AHIBCKZLUAZB
      CHMZNZCIMZUFZKZODOOABVQLUBFVSVRPZVTOPUCAWDCVSUDQWCOPAWABPZWBUEUGAWDWAVTRZ
      SZJMZBPZVSCWHUFZKZRZWFSZSZJUHZWHWARZWIWLSZSZJUHZWEVSWCRZSZAWGWLJBUIZWFSZW
      MJBUIZWOAWDXBWFAXBHVRAJBCHDEFGUJUKULXCXDTAXDXCWLWFJBUMUNQXDWOTAWMJBUOQUPA
      WNWRJAWNWIWLWPSZSZWQWPSZWRAWMXEWIAWLWFWPAWLSZWFWAWHRWPXHVTWHWAWLAVTCWKNZW
      HCVSWKUQACEPWHOPXIWHRGJVHCWHEOURUSUTVAIJVBVCVDVIXGXFTAWIWLWPVEQXGWRTAWQWP
      VFQVGVJWSXATAWQXAJIWPWIWEWLWTJIBVKWPWKWCVSWPWJWBWHWACVLVMVAVNVOQUPVP $.
  $}

  ${
    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.)  (Proof shortened by AV,
       17-Jul-2022.) $)
    mapsnen $p |- ( A ^m { B } ) ~~ A $=
      ( cvv wcel csn cmap co cen wbr id a1i mapsnend ax-mp ) AEFZABGHIAJKCPABEE
      PLBEFPDMNO $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y V $.  $d x y W $.
    $( Set exponentiation: a singleton to any set is equinumerous to that
       singleton.  (Contributed by NM, 17-Dec-2003.)  (Revised by AV,
       17-Jul-2022.) $)
    snmapen $p |- ( ( A e. V /\ B e. W ) -> ( { A } ^m B ) ~~ { A } ) $=
      ( vx vy wcel wa csn cmap co cxp cvv ovexd snex a1i cv a1d wceq wb anim1ci
      simpl xpexg syl velsn elmapg sylan fconst2g adantr bitr2d anbi12d bitr2di
      wf ancom en2d ) ACGZBDGZHZEFAIZBJKZUSABUSLZMMCMURUSBJNUSMGZURAOZPURUPEQZU
      TGZUPUQUBRURVAMGZFQZUSGZURUQVBHVFUPVBUQVBUPVCPZUABUSDMUCUDRURVHVDVASZHVGA
      SZVEHVEVKHURVHVKVJVEVHVKTURFAUEPURVEBUSVDUMZVJUPVBUQVEVLTVIUSBVDMDUFUGUPV
      LVJTUQBACVDUHUIUJUKVKVEUNULUO $.
  $}

  $( Set exponentiation: a singleton to any set is equinumerous to ordinal 1.
     (Proposed by BJ, 17-Jul-2022.)  (Contributed by AV, 17-Jul-2022.) $)
  snmapen1 $p |- ( ( A e. V /\ B e. W ) -> ( { A } ^m B ) ~~ 1o ) $=
    ( wcel wa csn cmap co cen wbr c1o snmapen ensn1g adantr entr syl2anc ) ACEZ
    BDEZFAGZBHIZTJKTLJKZUALJKABCDMRUBSACNOUATLPQ $.

  $( 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.)
     (Proof shortened by AV, 17-Jul-2022.) $)
  map1 $p |- ( A e. V -> ( 1o ^m A ) ~~ 1o ) $=
    ( wcel c1o cmap co csn cen df1o2 oveq1i cvv wbr 0ex snmapen1 mpan eqbrtrid
    c0 ) ABCZDAEFQGZAEFZDHDSAEIJQKCRTDHLMQAKBNOP $.

  ${
    $d A f $.  $d B f $.
    $( Two singletons are equinumerous.  (Contributed by NM, 9-Nov-2003.)
       Avoid ~ ax-pow .  (Revised by BTernaryTau, 31-Jul-2024.)  Avoid
       ~ ax-un .  (Revised by BTernaryTau, 25-Sep-2024.) $)
    en2sn $p |- ( ( A e. C /\ B e. D ) -> { A } ~~ { B } ) $=
      ( vf wcel wa csn wf1o wex cen wbr cop cvv snex f1osng f1oeq1 spcegv mpsyl
      cv wb breng mp2an sylibr ) ACFBDFGZAHZBHZETZIZEJZUFUGKLZABMZHZNFUEUFUGUMI
      ZUJULOABCDPUIUNEUMNUFUGUHUMQRSUFNFUGNFUKUJUAAOBOUFUGENNUBUCUD $.
    $( $j usage 'en2sn' avoids 'ax-pow' 'ax-un'; $)
  $}

  $( The empty set is finite.  (Contributed by FL, 14-Jul-2008.)  Avoid
     ~ ax-10 , ~ ax-un .  (Revised by BTernaryTau, 13-Jan-2025.) $)
  0fi $p |- (/) e. Fin $=
    ( vx c0 cfn wcel cv cen wbr com wrex peano1 wceq eqid en0 mpbir breq2 mp2an
    rspcev isfi ) BCDBAEZFGZAHIZBHDBBFGZUAJUBBBKBLBMNTUBABHSBBFOQPABRN $.
  $( $j usage '0fi' avoids 'ax-10' 'ax-un'; $)

  ${
    $d A x $.
    $( A singleton is finite.  (Contributed by NM, 4-Nov-2002.)  (Proof
       shortened by BTernaryTau, 13-Jan-2025.) $)
    snfi $p |- { A } e. Fin $=
      ( vx cvv wcel csn cfn cv cen wbr com wrex c1o ensn1g breq2 rspcev sylancr
      1onn isfi sylibr c0 wn wceq snprc 0fi eleq1 mpbiri sylbi pm2.61i ) ACDZAE
      ZFDZUIUJBGZHIZBJKZUKUILJDUJLHIZUNQACMUMUOBLJULLUJHNOPBUJRSUIUAUJTUBZUKAUC
      UPUKTFDUDUJTFUEUFUGUH $.
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.  $d x y D $.
    $( Obsolete version of ~ snfi as of 13-Jan-2025.  (Contributed by NM,
       4-Nov-2002.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    snfiOLD $p |- { A } e. Fin $=
      ( vx csn cfn wcel cv cen wbr com wrex cvv c1o ensn1g breq2 rspcev sylancr
      1onn wn c0 wceq snprc en0 peano1 mpan sylbir sylbi pm2.61i isfi mpbir ) A
      CZDEUJBFZGHZBIJZAKEZUMUNLIEUJLGHZUMQAKMULUOBLIUKLUJGNOPUNRUJSTZUMAUAUPUJS
      GHZUMUJUBSIEUQUMUCULUQBSIUKSUJGNOUDUEUFUGBUJUHUI $.

    $( 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 snfi eleq1 mpbiri
      wn exlimiv abssi df-nel ssexg mpan con3i 3imtr4i ax-mp ) ACZBCZDZEZBFZAGZ
      HIZJHIZABKUKHLZQJHLZQULUMUOUNUKJMUOUNUJAJUIUFJLZBUIUPUHJLUGNUFUHJOPRSUKJH
      UAUBUCUKHTJHTUDUE $.

    $( 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 wcel f1oun
      exdistrv unex f1oen3g sylancr ex exlimivv sylbir syl2anb imp ) ABGHZCDGHZ
      IACJKLBDJKLIZACMZBDMZGHZULABENZOZEPZCDFNZOZFPZUNUQQZUMABERCDFRUTVCIUSVBIZ
      FPEPVDUSVBEFUCVEVDEFVEUNUQVEUNIURVAMZSUAUOUPVFOUQURVAETFTUDABCDURVAUBUOUP
      VFSUEUFUGUHUIUJUK $.
  $}

  ${
    $d x y $.  $d A x $.
    $( Equinumerosity is reflexive for finite ordinals, proved without using
       the Axiom of Power Sets (unlike ~ enrefg ).  (Contributed by
       BTernaryTau, 31-Jul-2024.) $)
    enrefnn $p |- ( A e. _om -> A ~~ A ) $=
      ( vx vy cv cen wbr c0 csuc wceq id breq12d eqid en0 mpbir com wcel wa csn
      cun cvv cin en2sn el2v jctr word nnord orddisj syl syl2anr df-suc 3brtr4g
      jca unen ex finds ) BDZUPEFGGEFZCDZUREFZURHZUTEFZAAEFBCAUPGIZUPGUPGEVBJZV
      CKUPURIZUPURUPUREVDJZVEKUPUTIZUPUTUPUTEVFJZVGKUPAIZUPAUPAEVHJZVIKUQGGIGLG
      MNUROPZUSVAVJUSQURURRZSZVLUTUTEUSUSVKVKEFZQURVKUAGIZVNQVLVLEFVJUSVMVMCCUR
      URTTUBUCUDVJVNVNVJURUEVNURUFURUGUHZVOULURURVKVKUMUIURUJZVPUKUNUO $.
    $( $j usage 'enrefnn' avoids 'ax-pow'; $)
  $}

  ${
    $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 unordered pairs are equinumerous.  (Contributed by BTernaryTau,
       23-Dec-2024.) $)
    en2prd $p |- ( ph -> { A , B } ~~ { C , D } ) $=
      ( vf cpr cvv wcel prex cv wf1o wex cen wbr cop wne f1oprg syl22anc mp2and
      wa wi f1oeq1 spcegv mpsyl wb breng mp2an sylibr ) ABCQZDEQZPUAZUBZPUCZUTV
      AUDUEZBDUFZCEUFZQZRSAUTVAVHUBZVDVFVGTABCUGZDEUGZVINOABFSDHSCGSEISVJVKUKVI
      ULJLKMBDCEFHGIUHUIUJVCVIPVHRUTVAVBVHUMUNUOUTRSVARSVEVDUPBCTDETUTVAPRRUQUR
      US $.
    $( $j usage 'en2prd' avoids 'ax-pow' 'ax-un'; $)
  $}

  ${
    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.)  Avoid ~ ax-un .  (Revised
       by BTernaryTau, 23-Dec-2024.) $)
    enpr2d $p |- ( ph -> { A , B } ~~ 2o ) $=
      ( cpr c0 c1o c2o cen cvv wcel 0ex a1i 1oex neqned wne necomi en2prd df2o3
      1n0 breqtrrdi ) ABCIJKILMABCJKDENNFGJNOAPQKNOARQABCHSJKTAKJUDUAQUBUCUE $.
    $( $j usage 'enpr2d' avoids 'ax-pow' 'ax-un'; $)
  $}

  ${
    enpr2dOLD.1 $e |- ( ph -> A e. C ) $.
    enpr2dOLD.2 $e |- ( ph -> B e. D ) $.
    enpr2dOLD.3 $e |- ( ph -> -. A = B ) $.
    $( Obsolete version of ~ enpr2d as of 23-Dec-2024.  (Contributed by Rohan
       Ridenour, 3-Aug-2023.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    enpr2dOLD $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 $.
  $}

  $( Any subset of a countable set is countable.  (Contributed by Thierry
     Arnoux, 31-Jan-2017.)  Avoid ~ ax-pow , ~ ax-un .  (Revised by
     BTernaryTau, 7-Dec-2024.) $)
  ssct $p |- ( ( A C_ B /\ B ~<_ _om ) -> A ~<_ _om ) $=
    ( com domssl ) ABCD $.
  $( $j usage 'ssct' avoids 'ax-pow' 'ax-un'; $)

  $( Obsolete version of ~ ssct as of 7-Dec-2024.  (Contributed by Thierry
     Arnoux, 31-Jan-2017.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  ssctOLD $p |- ( ( A C_ B /\ B ~<_ _om ) -> A ~<_ _om ) $=
    ( wss com cdom wbr cvv wcel wi ctex ssdomg syl impcom domtr sylancom ) ABCZ
    BDEFZABEFZADEFQPRQBGHPRIBJABGKLMABDNO $.

  $( All decrements of a set are equinumerous.  (Contributed by Stefan O'Rear,
     19-Feb-2015.) $)
  difsnen $p |- ( ( X e. V /\ A e. X /\ B e. X ) ->
      ( X \ { A } ) ~~ ( X \ { B } ) ) $=
    ( wcel csn cdif cen wbr wceq cvv difexg enrefg syl wne cun cin disjdifr a1i
    c0 w3a 3ad2ant1 sneq difeq2d breq2d syl5ibcom wa simpl1 4syl dif32 breqtrdi
    imp simpl3 simpl2 en2sn syl2anc unen syl22anc simpr necomd eldifsn sylanbrc
    difsnid 3brtr3d pm2.61dane ) DCEZADEZBDEZUAZDAFZGZDBFZGZHIZABVIABJZVNVIVKVK
    HIZVOVNVFVGVPVHVFVKKEZVPDVJCLZVKKMNUBVOVKVMVKHVOVJVLDABUCUDUEUFULVIABOZUGZV
    KVLGZVLPZVMVJGZVJPZVKVMHVTWAWCHIVLVJHIZWAVLQTJZWCVJQTJZWBWDHIVTWAWAWCHVTVFV
    QWAKEWAWAHIVFVGVHVSUHVRVKVLKLWAKMUIDVJVLUJUKVTVHVGWEVFVGVHVSUMZVFVGVHVSUNZB
    ADDUOUPWFVTVLVKRSWGVTVJVMRSWAWCVLVJUQURVTBVKEZWBVKJVTVHBAOWJWHVTABVIVSUSZUT
    BDAVAVBVKBVCNVTAVMEZWDVMJVTVGVSWLWIWKADBVAVBVMAVCNVDVE $.

  ${
    $d A f x $.  $d B f x $.  $d C f x $.
    $( Dominance over a set with one element removed.  (Contributed by Stefan
       O'Rear, 19-Feb-2015.)  (Revised by Mario Carneiro, 24-Jun-2015.) $)
    domdifsn $p |- ( A ~< B -> A ~<_ ( B \ { C } ) ) $=
      ( vf vx csdm wbr wcel csn cdif wa cv wf1 cvv wb adantr adantl wn ad2antrr
      cdom wex sdomdom relsdom brrelex2i brdomg syl crn c0 wne wss f1f frnd cen
      mpbid sdomnen wi wceq wf1o vex dff1o5 biimpri f1oen3g sylancr ex necon3bd
      mpd pssdifn0 syl2anc n0 brrelex1i difexd cin eldifn disjsn sylibr reldisj
      sylib f1ssr syldan f1dom2g syl3anc eldifi ad2antll simplr difsnen domentr
      expr exlimdv exlimddv difsn breq2d mpbird pm2.61dan ) ABFGZCBHZABCIJZTGZW
      NWOKZABDLZMZWQDWNWTDUAZWOWNABTGZXAABUBZWNBNHZXBXAOABFUCUDZABNDUEUFUNPWRWT
      KZELZBWSUGZJZHZEUAZWQXFXIUHUIZXKXFXHBUJZXHBUIZXLWTXMWRWTABWSABWSUKULZQXFA
      BUMGZRZXNWNXQWOWTABUOSWTXQXNUPWRWTXPXHBWTXHBUQZXPWTXRKZWSNHABWSURZXPDUSXT
      XSABWSUTVAABWSNVBVCVDVEQVFXHBVGVHEXIVIVQXFXJWQEWRWTXJWQWRWTXJKZKZABXGIZJZ
      TGZYDWPUMGZWQYBANHZYDNHAYDWSMZYEWNYGWOYAABFUCVJSYBBYCNWNXDWOYAXESZVKYAYHW
      RWTXJXHYDUJZYHYAXHYCVLUHUQZYJXJYKWTXJXGXHHRYKXGBXHVMXHXGVNVOQYAXMYKYJOWTX
      MXJXOPXHYCBVPUFUNABYDWSVRVSQAYDWSNNVTWAYBXDXGBHZWOYFYIXJYLWRWTXGBXHWBWCWN
      WOYAWDXGCNBWEWAAYDWPWFVHWGWHVFWIWNWORZKWQXBWNXBYMXCPYMWQXBOWNYMWPBATCBWJW
      KQWLWM $.
  $}

  ${
    $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 ) EFABUAZUBZAEHZIZIZFHZBJZAVRCBUCUDCVTVSKZVTWCGHZJZ
      LZWCAKZWFVRKZTZTZGMZFMZWBNKZFGVTAVRUEZWLWOFGWHWOWKWHWBWCNWHWBWGIZIWCWHWAW
      QVTWGOPWCWFFQZGQRUFWRUGSUHUIWDNKWIWCBUJUMWEWCWBLZTVTWBBJZLZWBAKZTZWSTZVTW
      DLZWITZWIXETWSWEXCWSWOWEXCUKFWBULWEWSXFTZFMZWOXCWEWNXHWPWMXGFWMWFBLZWHWIT
      ZTZGMXFXGWLXKGXJWJTWJXJTWLXKXJWJUNWHWIWJUOWJXIXJGBUPUQURUSXJXFGBDXIWHXEWI
      XIWGWDVTWFBWCVDUTVAVBXFWSXETZWITXGXEXLWIXEWSXEWBWDIZIWCXEWAXMVTWDOPWCBWRD
      RVEZVCUQWSXEWIUOVFVGUSVFXFXCFWBNWSXEXAWIXBWSWDWTVTWCWBBVHUTWCWBAVNVIZVJVK
      VOVLXFXFWSTXDXFWSXEWSWIXNSVMWSXFXCXOVLVPXEWIUNVGVQ $.
  $}

  ${
    $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 two 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 1oex pm3.2i
      0ex xp01disj p0ex snex bi2anan9 ineq12 eqeq1d anbi12d spc2ev mp2an ) CGHZ
      IZCJKZDLHZIZDJKZMZUNUQNZGOZAPZCJKZBPZDJKZMZVBVDNZGOZMZBQAQUOURCGEUCRDLFUA
      RUBCDUDVIUSVAMABUNUQCUMEUESDUPFLUFSVBUNOZVDUQOZMZVFUSVHVAVJVCUOVKVEURVBUN
      CJTVDUQDJTUGVLVGUTGVBUNVDUQUHUIUJUKUL $.
  $}

  ${
    $d A x y $.  $d B x y $.  $d C x y $.  $d D x y $.
    $( Dominance law for union.  Proposition 4.24(a) of [Mendelson] p. 257.
       (Contributed by NM, 3-Sep-2004.)  (Revised by Mario Carneiro,
       26-Apr-2015.)  Avoid ~ ax-pow .  (Revised by BTernaryTau,
       4-Dec-2024.) $)
    undom $p |- ( ( ( A ~<_ B /\ C ~<_ D ) /\ ( B i^i D ) = (/) ) ->
                ( A u. C ) ~<_ ( B u. D ) ) $=
      ( vx vy cdom wbr wa cin wceq cun cvv wcel reldom brrelex2i syl2an wf1 wex
      c0 cdif undif2 unexg adantr wi brdomi exdistrv cres disjdif difss f1ssres
      cv wss mpan2 f1un sylanl2 mpanr1 vex resex unex f1dom3g mp3an1 expcom syl
      ex exlimivv sylbir imp mpd eqbrtrrid ) ABGHZCDGHZIZBDJTKZIZACLACAUAZLZBDL
      ZGACUBVOVRMNZVQVRGHZVMVSVNVKBMNDMNVSVLABGOPCDGOPBDMMUCQUDVMVNVSVTUEZVKABE
      ULZRZESZCDFULZRZFSZVNWAUEZVLABEUFCDFUFWDWGIWCWFIZFSESWHWCWFEFUGWIWHEFWIVN
      WAWIVNIVQVRWBWEVPUHZLZRZWAWIAVPJTKZVNWLACUIWFWCVPDWJRZWMVNIWLWFVPCUMWNCAU
      JCDVPWEUKUNABVPDWBWJUOUPUQVSWLVTWKMNVSWLVTWBWJEURWEVPFURUSUTVQVRWKMMVAVBV
      CVDVEVFVGQVHVIVJ $.
    $( $j usage 'undom' avoids 'ax-pow'; $)
  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.  $d x y D $.
    $( Obsolete version of ~ undom as of 4-Dec-2024.  (Contributed by NM,
       3-Sep-2004.)  (Revised by Mario Carneiro, 26-Apr-2015.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    undomOLD $p |- ( ( ( A ~<_ B /\ C ~<_ D ) /\ ( B i^i D ) = (/) ) ->
                   ( A u. C ) ~<_ ( B u. D ) ) $=
      ( vx vy cdom wbr wa cin c0 wceq cen wss wex cun cvv wcel reldom brrelex2i
      cv cdif wb domeng syl brrelex1i difss ssdomg mpisyl domtr mpancom anim12i
      ibi adantr exdistrv simprll simprrl disjdif a1i ss2in adantl simplr sseq0
      ad2ant2l syl2anc undif2 unen eqbrtrrid syl22anc ad3antrrr ad3antlr unss12
      unexg sylc endomtr ex exlimdvv biimtrrid mpd ) ABGHZCDGHZIZBDJZKLZIZAEUAZ
      MHZWFBNZIZEOZCAUBZFUAZMHZWLDNZIZFOZIZACPZBDPZGHZWBWQWDVTWJWAWPVTWJVTBQRZV
      TWJUCABGSTZEABQUDUEUMWAWKDGHZWPWKCGHZWAXCWACQRWKCNXDCDGSUFCAUGWKCQUHUIWKC
      DUJUKXCWPXCDQRZXCWPUCWKDGSTFWKDQUDUEUMUEULUNWQWIWOIZFOEOWEWTWIWOEFUOWEXFW
      TEFWEXFWTWEXFIZWRWFWLPZMHZXHWSGHZWTXGWGWMAWKJKLZWFWLJZKLZXIWEWGWHWOUPWEWI
      WMWNUQXKXGACURUSXGXLWCNZWDXMXFXNWEWHWNXNWGWMWFBWLDUTVDVAWBWDXFVBXLWCVCVEW
      GWMIXKXMIIWRAWKPXHMACVFAWFWKWLVGVHVIXGWSQRZXHWSNZXJXGXAXEXOVTXAWAWDXFXBVJ
      WAXEVTWDXFCDGSTVKBDQQVMVEXFXPWEWHWNXPWGWMWFBWLDVLVDVAXHWSQUHVNWRXHWSVOVEV
      PVQVRVS $.
  $}

  ${
    $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 nfcv copab cop ccom cmpo cfv wf1o wfun
      cxp cdm wb xpcomf1o f1ofun funbrfv2b mp2b ancom eqcom f1odm ax-mp anbi12i
      eleq2i 3bitri anbi1i anass bitri exbii breq1 anbi2d ceqsexv nfmpo2 nfcxfr
      fvex elxp nfbr 19.41 nfmpo1 fveq2 csn ccnv cuni sneq cnveqd unieqd opswap
      opelxpi eqtrdi opex fvmpt syl sylan9eq breq1d coprab df-br eqtri oprabidw
      df-mpo baib ancoms adantl bitrd pm5.32da pm5.32i 3bitr2i opabbii dfoprab2
      bitr3i df-co 3eqtr4i ) KNZLNZGOZXIMNZHOZPZLQZKMUAXHCNZBNZUBZRZXODSZXPESZP
      ZXKFRZPZPZBQZCQZKMUAZHGUCCBDEFUDZXNYFKMXNXIXHGUEZRZXHDEUHZSZXLPZPZLQYLYIX
      KHOZPZYFXMYNLXMYJYLPZXLPYNXJYQXLXJXHGUIZSZYIXIRZPZYTYSPYQYKEDUHZGUFZGUGXJ
      UUAUJADEGIUKZYKUUBGULXHXIGUMUNYSYTUOYTYJYSYLYIXIUPYRYKXHUUCYRYKRUUDYKUUBG
      UQURUTUSVAVBYJYLXLVCVDVEYMYPLYIXHGVKYJXLYOYLXIYIXKHVFVGVHYPXRYAPZBQZCQZYO
      PUUFYOPZCQYFYLUUGYOCBXHDEVLVBUUFYOCCYIXKHCYITCHBCEDFUDZJBCEDFVIVJCXKTVMVN
      UUHYECUUHUUEYOPZBQYEUUEYOBBYIXKHBYITBHUUIJBCEDFVOVJBXKTVMVNUUJYDBUUJXRYAY
      OPZPYDXRYAYOVCXRUUKYCXRYAYOYBUUEYOXPXOUBZXKHOZYBUUEYIUULXKHXRYAYIXQGUEZUU
      LXHXQGVPYAXQYKSUUNUULRXOXPDEWDAXQANZVQZVRZVSZUULYKGUUOXQRZUURXQVQZVRZVSUU
      LUUSUUQUVAUUSUUPUUTUUOXQVTWAWBXOXPWCWEIXPXOWFWGWHWIWJYAUUMYBUJZXRXTXSUVBU
      UMXTXSPZYBUUMUULXKUBZHSUVDUVCYBPZBCMWKZSUVEUULXKHWLHUVFUVDHUUIUVFJBCMEDFW
      OWMUTUVEBCMWNVAWPWQWRWSWTXAVDVEXEVEXBVAXCKMLHGXFYHYCCBMWKYGCBMDEFWOYCCBMK
      XDWMXG $.
  $}

  ${
    $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 snex xpcomeng mpan xpsneng ancoms entr syl2an2 )
    BDEZAFZBGZBSGZHIZACEZUABHIZTBHISJERUBAKSBJDLMRUCUDBADCNOTUABPQ $.

  ${
    $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 wceq wa wex unieqd vx vy cxp xpex
      cvv opex a1i sneq dmeqd sneqd vex 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 eqeq2d anbi1d anbi2d
      opeq1 ceqsexv 3bitri 3exbii anbi2i 19.42vv bitri bitr3i exrot3 opeq2 en2i
      cv ) UAUBABUCZCUCZABCUCZUCZUAWKZKZLZMZKZLZMZWTNZMZWQNZMZOZOZUBWKZKZLZMZXJ
      NZMZKZLZMZOZXONZMZOZWLCABDEUDFUDAWNDBCEFUDUDXHUEPWPWMPZXBXGUFUGYAUEPXIWOP
      ZXRXTUFUGWPGWKZHWKZOZIWKZOZQZYDAPZYEBPZRZYGCPZRZRZISZHSGSZXIXHQZRZXIYDYEY
      GOZOZQZYJYKYMRZRZRZISZHSZGSZWPYAQZRZYBYRRYCUUIRYPYRRZHSGSUUFUUIRZHSGSYSUU
      JUUKUULGHYOYRRZISUUEUUIRZISUUKUULUUMUUNIYIYRRZYNRUUBUUIRZUUDRUUMUUNUUOUUP
      YNUUDWPXIYHUUAXHYAYIYDXBYTXGYIXBYHKZLZMZKZLZMZYDYIXAUVAYIWTUUTYIWSUUSYIWR
      UURYIWQUUQWPYHUHZUITUJZUITUVBYFKZLZMYDUVAUVFUUTUVEUUSYFYFYGYDYEUFZIUKZULU
      MZUNUOYDYEGUKZHUKZULUPUQYIYEXDYGXFYIXDUUTNZMZYEYIXCUVLYIWTUUTUVDURTUVMUVE
      NZMYEUVLUVNUUTUVEUVIUSUOYDYEUVJUVKUTUPUQYIXFUUQNZMYGYIXEUVOYIWQUUQUVCURTY
      FYGUVGUVHUTUQVAVAUUBYFXRYGXTUUBYDXLYEXQUUBXLUUAKZLZMYDUUBXKUVQUUBXJUVPXIU
      UAUHZUITYDYTUVJYEYGUFZULUQUUBXQUVPNZMZKZLZMZYEUUBXPUWCUUBXOUWBUUBXNUWAUUB
      XMUVTUUBXJUVPUVRURTUJZUITUWDYTKZLZMYEUWCUWGUWBUWFUWAYTYDYTUVJUVSUTUMZUNUO
      YEYGUVKUVHULUPUQVAUUBXTUWBNZMZYGUUBXSUWIUUBXOUWBUWEURTUWJUWFNZMYGUWIUWKUW
      BUWFUWHUSUOYEYGUVKUVHUTUPUQVAVBYJYKYMVCVDYIYNYRVEUUBUUDUUIVEVFVGYOYRIVHUU
      EUUIIVHVIVJYPYRGHVKUUFUUIGHVKVIYBYQYRYBWPJWKZYGOZQZUWLWLPZYMRRZISJSUWPJSI
      SZYQJIWPWLCVLUWPJIVMUWQUWLYFQZYLRZUWNYMRZRZHSGSZJSISUXAJSZISHSGSYQUWPUXBI
      JUWOUWTRUWSHSGSZUWTRUWPUXBUWOUXDUWTGHUWLABVLVNUWNUWOYMVOUWSUWTGHVKVFVJUXA
      IJGHVPUXCYOGHIUXCUWRYLUWTRZRZJSYLYIYMRZRZYOUXAUXFJUWRYLUWTVCVGUXEUXHJYFUV
      GUWRUWTUXGYLUWRUWNYIYMUWRUWMYHWPUWLYFYGVTVQVRVSWAYLYIYMVOWBWCWBWBVNYCUUHU
      UIYCXIYDUWLOZQZYJUWLWNPZRRZJSZGSUUHGJXIAWNVLUXMUUGGUXMUWLYTQZUXJUUDRZRZIS
      HSZJSUXPJSZISHSUUGUXLUXQJUXJYJRZUXKRUXSUXNUUCRZISHSZRZUXLUXQUXKUYAUXSHIUW
      LBCVLWDUXJYJUXKVCUYBUXSUXTRZISHSUXQUXSUXTHIWEUYCUXPHIUYCUXNUXSUUCRZRUXPUX
      SUXNUUCVOUYDUXOUXNUXJYJUUCVCWDWFVJWGVIVGUXPJHIWHUXRUUEHIUXOUUEJYTUVSUXNUX
      JUUBUUDUXNUXIUUAXIUWLYTYDWIVQVRWAVJWBVGWFVNVFWJ $.
  $}

  ${
    $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 ) ) $=
      ( vz vw vv vu cdom cv wa csn cdm cuni cop cvv wcel wi wceq wb wbr wf1 cxp
      vf vx vy brdomi crn cfv wf f1f ffvelcdm syl anim2d adantld opelxp 3imtr4g
      ex elxp4 adantl wrex elxp2 fvex opth f1fveq ancoms anbi2d bitrid ad2ant2l
      vex adantlr 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 ) ABIUAZABUDJZUBZCAUCZCBUCZIUAUDABUDUGXAXCKUEUFXDXEUEJZLZMZNZXGUHZN
      ZXBUIZOZUFJZLZMZNZXOUHZNZXBUIZOZPPXCXFXDQZXMXEQZRXAXCXFXIXKOSZXICQZXKAQZK
      ZKYEXLBQZKZYBYCXCYGYIYDXCYFYHYEXCABXBUJZYFYHRABXBUKYJYFYHABXKXBULURUMUNUO
      XFCAUSXIXLCBUPUQUTXCYBXNXDQZKZXMYASZXFXNSZTZRXAYLXCYOYBXFEJZFJZOZSZFAVAEC
      VAZXNGJZHJZOZSZHAVAGCVAZXCYORZYKEFXFCAVBGHXNCAVBYTUUEUUFYTUUDUUFGHCAYSUUA
      CQZUUBAQZKZUUDUUFRZREFCAYPCQZYQAQZKZUUIYSUUJUUMUUIYSUUDXCYOUUMUUIKZYSUUDK
      ZKXCKYPYQXBUIZOZUUAUUBXBUIZOZSZYPUUASZYQUUBSZKZYMYNUUNXCUUTUVCTZUUOUUNXCU
      VDUULUUHXCUVDRUUKUUGUULUUHKZXCUVDUUTUVAUUPUURSZKUVEXCKZUVCYPUUPUUAUUREVJZ
      YQXBVCVDUVGUVFUVBUVAXCUVEUVFUVBTABYQUUBXBVEVFVGVHURVIWAVKUUOYMUUTTUUNXCYS
      UUDXMUUQYAUUSYSXIYPXLUUPYSXIYRLZMZNYPYSXHUVJYSXGUVIXFYRVLZVMVNYPYQUVHFVJZ
      VOVPYSXKYQXBYSXKUVIUHZNYQYSXJUVMYSXGUVIUVKVQVNYPYQUVHUVLVRVPVSWBUUDXQUUAX
      TUURUUDXQUUCLZMZNUUAUUDXPUVOUUDXOUVNXNUUCVLZVMVNUUAUUBGVJZHVJZVOVPUUDXSUU
      BXBUUDXSUVNUHZNUUBUUDXRUVSUUDXOUVNUVPVQVNUUAUUBUVQUVRVRVPVSWBVTWKUUOYNUVC
      TUUNXCUUOYNYRUUCSUVCXFYRXNUUCWCYPYQUUAUUBUVHUVLVDWDWKWEWFWGWHWIWAWJWLUTXA
      XDPQZXCXACPQZAPQUVTDABIWMWNCAPPWOWPWQXAXEPQZXCXAUWABPQUWBDABIWMWRCBPPWOWP
      WQWTWS $.
  $}

  ${
    $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 a nonempty set.
       Exercise 6 of [Suppes] p. 98.  (Contributed by NM, 27-Jul-2004.)
       (Revised by Mario Carneiro, 29-Apr-2015.) $)
    xpdom3 $p |- ( ( A e. V /\ B e. W /\ B =/= (/) ) -> A ~<_ ( A X. B ) ) $=
      ( vx wcel c0 wne cxp cdom wbr cv wex wa n0 w3a csn cen cvv wss ensymd syl
      xpsneng 3adant2 xpexg 3adant3 simp3 ssdomg endomtr syl2anc 3expia exlimdv
      snssd xpss2 sylc biimtrid 3impia ) ACFZBDFZBGHZAABIZJKZUTELZBFZEMURUSNZVB
      EBOVEVDVBEURUSVDVBURUSVDPZAAVCQZIZRKVHVAJKZVBVFVHAURVDVHARKUSAVCCBUCUDUAV
      FVASFZVHVATZVIURUSVJVDABCDUEUFVFVGBTVKVFVCBURUSVDUGUMVGBAUNUBVHVASUHUOAVH
      VAUIUJUKULUPUQ $.
  $}

  ${
    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 f A $.  $d f B $.  $d f X $.  $d f Y $.
    domunsncan.a $e |- A e. _V $.
    domunsncan.b $e |- B e. _V $.
    $( A singleton cancellation law for dominance.  (Contributed by Stefan
       O'Rear, 19-Feb-2015.)  (Revised by Stefan O'Rear, 5-May-2015.) $)
    domunsncan $p |- ( ( -. A e. X /\ -. B e. Y ) ->
        ( ( { A } u. X ) ~<_ ( { B } u. Y ) <-> X ~<_ Y ) ) $=
      ( vf wcel wa csn cun cdom wbr cvv wss sylancr cdif cima wceq ad2antll wf1
      wn ssun2 reldom brrelex2i adantl ssexg wex brdomi cen cres wf1o vex resex
      cv simprr difss f1ores sylancl f1oen3g wf ccnv df-f1 imadif simplbiim cfv
      wfun snex simprl unexg difexd f1f fimass syl ssdifd f1fn snid elun1 ax-mp
      fnsnfv difeq2d sseqtrrd ssdomg sylc ffvelcdm mp1i difsnen syl3anc domentr
      wfn syl2anc eqbrtrd endomtr uncom difeq1i difun2 eqtrid ad2antrr ad2antlr
      eqtri difsn 3brtr3d expr exlimdv impancom mpd cin en2sn mp2an endom simpr
      syl5 c0 incom disjsn biimpri undom syl21anc impbida ) ACHUBZBDHUBZIZAJZCK
      ZBJZDKZLMZCDLMZYBYGIZDNHZYHYIDYFOYFNHZYJDYEUCYGYKYBYDYFLUDUEUFDYFNUGPYBYJ
      YGYHYGYDYFGUOZUAZGUHYBYJIZYHYDYFGUIYNYMYHGYBYJYMYHYBYJYMIZIZYDYCQZYFYEQZC
      DLYPYQYLYQRZUJMZYSYRLMYQYRLMYPYLYQUKZNHYQYSUUAULZYTYLYQGUMUNYPYMYQYDOUUBY
      BYJYMUPYDYCUQYDYFYQYLURUSYQYSUUANUTPYPYSYLYDRZYLYCRZQZYRLYMYSUUESZYBYJYMY
      DYFYLVAZYLVBVGUUFYDYFYLVCYDYCYLVDVETYPUUEYFAYLVFZJZQZLMZUUJYRUJMZUUEYRLMY
      PUUJNHUUEUUJOUUKYPYFUUINYPYENHYJYKBVHYBYJYMVIYEDNNVJPZVKYPUUEYFUUDQUUJYPU
      UCYFUUDYMUUCYFOZYBYJYMUUGUUNYDYFYLVLZYDYFYLYDVMVNTVOYPUUIUUDYFYPYLYDWJZAY
      DHZUUIUUDSYMUUPYBYJYDYFYLVPTAYCHUUQAEVQAYCCVRVSZYDAYLVTUSWAWBUUEUUJNWCWDY
      PYKUUHYFHZBYFHZUULUUMYMUUSYBYJYMUUGUUQUUSUUOUURYDYFAYLWEUSTBYEHUUTYPBFVQB
      YEDVRWFUUHBNYFWGWHUUEUUJYRWIWKWLYQYSYRWMWKXTYQCSYAYOXTYQCYCQZCYQCYCKZYCQU
      VAYDUVBYCYCCWNWOCYCWPWTACXAWQWRYAYRDSXTYOYAYRDYEQZDYRDYEKZYEQUVCYFUVDYEYE
      DWNWODYEWPWTBDXAWQWSXBXCXDXLXEXFYBYHIZYCYELMZYHYEDXGZXMSZYGYCYEUJMZUVFUVE
      ANHBNHUVIEFABNNXHXIYCYEXJWFYBYHXKYAUVHXTYHYAUVGDYEXGZXMYEDXNUVJXMSYADBXOX
      PWQWSYCYECDXQXRXS $.
  $}

  ${
    $d A m n x y $.  $d B m n x y $.  $d F m n $.
    omxpenlem.1 $e |- F = ( x e. B , y e. A |-> ( ( A .o x ) +o y ) ) $.
    $( Lemma for ~ omxpen .  (Contributed by Mario Carneiro, 3-Mar-2013.)
       (Revised by Mario Carneiro, 25-May-2015.) $)
    omxpenlem $p |- ( ( A e. On /\ B e. On ) ->
        F : ( B X. A ) -1-1-onto-> ( A .o B ) ) $=
      ( vm vn con0 wcel wa comu co cv wss wi wb syl2anc wceq c0 cxp wfn ccnv wf
      wf1o wral csuc word eloni ad2antlr simprl ordsucss onelon ad2ant2lr onsuc
      coa sylc syl simplr simpll omwordi syl3anc mpd ad2ant2rl omcl oaord mpbid
      simprr omsuc eleqtrrd sseldd ralrimivva fmpo sylib ffnd cres wbr weu wrex
      cop wne sylan noel oveq1 om0r sylan9eqr eleq2d mtbiri necon2ad adantl imp
      wn omeu vex brcnv wex eleq1 biimpac simplll simpr simplrr oaword1 simplrl
      ex ad2antrr ontr2 mp2and simpllr ne0d on0eln0 mpbird om00el simpld omord2
      syl31anc impbid expr sylan2 eqcom anbi2i bitrdi anbi2d an12 2exbidv copab
      pm5.32rd coprab df-mpo dfoprab2 3eqtri breqi df-br opabidw 3bitri 3bitr4g
      cmpo r2ex bitrid eubidv ralrimiva fnres sylibr wrel cdm relcnv df-rn frnd
      crn eqsstrrid relssres sylancr fneq1d dff1o4 sylanbrc ) CIJZDIJZKZEDCUAZU
      BEUCZCDLMZUBZUURUUTEUEUUQUURUUTEUUQCANZLMZBNZUPMZUUTJZBCUFADUFUURUUTEUDUU
      QUVFABDCUUQUVBDJZUVDCJZKZKZCUVBUGZLMZUUTUVEUVJUVKDOZUVLUUTOZUVJDUHZUVGUVM
      UUPUVOUUOUVIDUIUJUUQUVGUVHUKUVBDULUQUVJUVKIJZUUPUUOUVMUVNPUVJUVBIJZUVPUUP
      UVGUVQUUOUVHDUVBUMZUNZUVBUOURUUOUUPUVIUSUUOUUPUVIUTZUVKDCVAVBVCUVJUVEUVCC
      UPMZUVLUVJUVHUVEUWAJZUUQUVGUVHVHUVJUVDIJZUUOUVCIJZUVHUWBQUUOUVHUWCUUPUVGC
      UVDUMZVDUVTUVJUUOUVQUWDUVTUVSCUVBVEZRUVDCUVCVFVBVGUVJUUOUVQUVLUWASUVTUVSC
      UVBVIRVJVKVLABDCUVEUUTEFVMVNZVOUUQUUSUUTVPZUUTUBZUVAUUQGNZHNZUUSVQZHVRZGU
      UTUFUWIUUQUWMGUUTUUQUWJUUTJZKZUWMUWKUVBUVDVTSZUVEUWJSZKZBCVSAIVSZHVRZUWOU
      UOUWJIJZCTWAZUWTUUOUUPUWNUTUUQUUTIJZUWNUXACDVEZUUTUWJUMWBUUQUWNUXBUUPUWNU
      XBPUUOUUPUWNCTUUPCTSZUWNWLUUPUXEKZUWNUWJTJUWJWCUXFUUTTUWJUXEUUPUUTTDLMTCT
      DLWDDWEWFWGWHXDWIWJWKABHCUWJWMVBUWOUWLUWSHUWLUWKUWJEVQZUWOUWSUWJUWKEGWNHW
      NWOUWOUWPUVIUWJUVESZKZKZBWPAWPZUVQUVHKZUWRKZBWPAWPUXGUWSUWOUXJUXMABUWOUXJ
      UWPUXLUWQKZKUXMUWOUXIUXNUWPUWOUXIUXLUXHKUXNUWOUXHUVIUXLUUQUWNUXHUVIUXLQZU
      WNUXHKUUQUVFUXOUXHUWNUVFUWJUVEUUTWQWRUUQUVFKUVHUVGUVQUUQUVFUVHUVGUVQQUUQU
      VFUVHKZKZUVGUVQUUPUVGUVQPUUOUXPUUPUVGUVQUVRXDUJUXQUVQUVGUXQUVQKZUVGUVCUUT
      JZUXRUVCUVEOZUVFUXSUXRUWDUWCUXTUXRUUOUVQUWDUUOUUPUXPUVQWSZUXQUVQWTZUWFRZU
      XRUUOUVHUWCUYAUUQUVFUVHUVQXAUWERUVCUVDXBRUUQUVFUVHUVQXCZUXRUWDUXCUXTUVFKU
      XSPUYCUUQUXCUXPUVQUXDXEZUVCUVEUUTXFRXGUXRUVQUUPUUOTCJZUVGUXSQUYBUUOUUPUXP
      UVQXHUYAUXRUYFTDJZUXRTUUTJZUYFUYGKZUXRUYHUUTTWAZUXRUUTUVEUYDXIUXRUXCUYHUY
      JQUYEUUTXJURXKUUQUYHUYIQUXPUVQCDXLXEVGXMUVBDCXNXOXKXDXPXQYFXRXQYFUXHUWQUX
      LUWJUVEXSXTYAYBUWPUXLUWQYCYAYDUXGUWKUWJUXKHGYEZVQUWKUWJVTUYKJUXKUWKUWJEUY
      KEABDCUVEYPUXIABGYGUYKFABGDCUVEYHUXIABGHYIYJYKUWKUWJUYKYLUXKHGYMYNUWRABIC
      YQYOYRYSXKYTGHUUTUUSUUAUUBUUQUUTUWHUUSUUQUUSUUCUUSUUDZUUTOUWHUUSSEUUEUUQU
      YLEUUHUUTEUUFUUQUURUUTEUWGUUGUUIUUSUUTUUJUUKUULVGUURUUTEUUMUUN $.
  $}

  ${
    $d A x y $.  $d B x y $.
    $( The cardinal and ordinal products are always equinumerous.  Exercise 10
       of [TakeutiZaring] p. 89.  (Contributed by Mario Carneiro,
       3-Mar-2013.) $)
    omxpen $p |- ( ( A e. On /\ B e. On ) -> ( A .o B ) ~~ ( A X. B ) ) $=
      ( vx vy con0 wcel wa cxp comu co cen wbr xpcomeng cvv coa cmpo wf1o xpexg
      cv ancoms omcl eqid omxpenlem f1oen2g syl3anc entr syl2anc ensymd ) AEFZB
      EFZGZABHZABIJZUKULBAHZKLUNUMKLZULUMKLABEEMUKUNNFZUMEFUNUMCDBAACSIJDSOJPZQ
      UOUJUIUPBAEERTABUACDABUQUQUBUCUNUMUQNEUDUEULUNUMUFUGUH $.
  $}

  ${
    $d x y z A $.  $d x y z B $.
    omf1o.1 $e |- F = ( x e. B , y e. A |-> ( ( A .o x ) +o y ) ) $.
    omf1o.2 $e |- G = ( x e. B , y e. A |-> ( ( B .o y ) +o x ) ) $.
    $( Construct an explicit bijection from ` A .o B ` to ` B .o A ` .
       (Contributed by Mario Carneiro, 30-May-2015.) $)
    omf1o $p |- ( ( A e. On /\ B e. On ) ->
        ( G o. `' F ) : ( A .o B ) -1-1-onto-> ( B .o A ) ) $=
      ( vz con0 wcel cxp comu co wf1o ccnv ccom cv cmpo eqid coa cuni omxpenlem
      wa cmpt ancoms xpcomf1o f1oco sylancl wceq wb xpcomco eqtr4i f1oeq1 ax-mp
      csn sylibr f1ocnv syl syl2anc ) CJKZDJKZUDZDCLZDCMNZFOZCDMNZVDEPZOZVGVEFV
      HQOVCVDVEBACDDBRMNARUANZSZIVDIRUPPUBUEZQZOZVFVCCDLZVEVKOZVDVOVLOVNVBVAVPB
      ADCVKVKTZUCUFIDCVLVLTZUGVDVOVEVKVLUHUIFVMUJVFVNUKFABDCVJSVMHIBADCVJVLVKVR
      VQULUMVDVEFVMUNUOUQVCVDVGEOVIABCDEGUCVDVGEURUSVGVDVEFVHUHUT $.
  $}

  ${
    $d x y z A $.  $d x y z B $.  $d x y z C $.  $d x y ph $.  $d x y z S $.
    $d x y G $.
    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 ) $.
    $( Lemma for ~ pw2f1o .  (Contributed by Mario Carneiro, 6-Oct-2014.) $)
    pw2f1olem $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 vx wcel wceq wa syl wb wss cv cif cmpt cpr wf ccnv csn cima cpw cmap
      prid2g prid1g ifcld adantr fmpttd simprr feq1d mpbird cfv iftrue ad2antrr
      co wn iffalse neeq1d syl5ibrcom necon4bd impbid2 simplrr fveq1d id eleq1w
      wne ifbid eqid fvmptg syl2anr eqtrd eqeq1d bitr4d pm5.32da sseld pm4.71rd
      wfn ffn fniniseg 3bitr4d eqrdv jca cdm cnvimass ad2antrl sseqtrid eqsstrd
      simprl fdm dffn5 sylib eleq2d baibd bitrd biimpa adantl eqtr4d ffvelcdmda
      wo fvex elpr ord sylibrd con1d pm2.61dan mpteq2dva impbida elpw2g cbvmptv
      imp a1i eqeq2d anbi12d cvv prex elmapg sylancr anbi1d ) AFCUAZGNCNUBZFPZE
      DUCZUDZQZRZCDEUEZGUFZFGUGEUHZUIZQZRZFCUJPZGBCBUBZFPZEDUCZUDZQZRGYNCUKVCPZ
      YRRAYMYSAYMRZYOYRUUGYOCYNYKUFZAUUHYMANCYJYNAYJYNPYHCPZAYIEDYNAEIPEYNPLDEI
      ULSZADIPDYNPKDEIUMSZUNUOUPUOUUGCYNGYKAYGYLUQURUSZUUGOFYQUUGOUBZCPZUUMFPZR
      UUNUUMGUTZEQZRZUUOUUMYQPZUUGUUNUUOUUQUUGUUNRZUUOUUOEDUCZEQZUUQUUTUUOUVBUU
      OEDVAUUTUUOUVAEUUTUVAEVNUUOVDZDEVNZAUVDYMUUNMVBUVCUVADEUUOEDVEVFVGVHVIUUT
      UUPUVAEUUTUUPUUMYKUTZUVAUUTUUMGYKAYGYLUUNVJVKUUNUUNUVAYNPZUVEUVAQUUGUUNVL
      AUVFYMAUUOEDYNUUJUUKUNUONUUMYJUVACYNYKYHUUMQYIUUOEDNOFVMVOYKVPVQVRVSVTWAW
      BUUGUUOUUNUUGFCUUMAYGYLWPWCWDUUGGCWEZUUSUURTUUGYOUVGUULCYNGWFZSCEUUMGWGSW
      HWIWJAYSRZYGYLUVIFYQCAYOYRUQUVIGWKZYQCGYPWLYOUVJCQAYRCYNGWQWMWNWOUVIGNCYH
      GUTZUDZYKUVIUVGGUVLQYOUVGAYRUVHWMZNCGWRWSUVINCUVKYJUVIUUIRZYIUVKYJQUVNYIR
      UVKEYJUVNYIUVKEQZUVNYIYHYQPZUVOUVNFYQYHAYOYRUUIVJWTUVIUVPUUIUVOUVIUVGUVPU
      UIUVORTUVMCEYHGWGSXAXBZXCYIYJEQUVNYIEDVAXDXEUVNYIVDZRUVKDYJUVNUVRUVKDQZUV
      NUVSYIUVNUVSVDUVOYIUVNUVSUVOUVNUVKYNPUVSUVOXGUVICYNYHGAYOYRWPXFUVKDEYHGXH
      XIWSXJUVQXKXLXRUVRYJDQUVNYIEDVEXDXEXMXNVSWJXOAYTYGUUEYLACHPZYTYGTJFCHXPSA
      UUDYKGUUDYKQABNCUUCYJUUAYHQUUBYIEDBNFVMVOXQXSXTYAAUUFYOYRAYNYBPUVTUUFYOTD
      EYCJYNCGYBHYDYEYFWH $.

    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.) $)
    pw2f1o $p |- ( ph -> F : ~P A -1-1-onto-> ( { B , C } ^m A ) ) $=
      ( vy cv wcel ccnv cima wa cpw cpr cmap co cif cmpt csn cvv wceq pw2f1olem
      eqid biimpa mpanr2 simpld vex cnvex imaex a1i f1od ) ABODUAZEFUBDUCUDZCDC
      PBPZQFEUEUFZOPZRZFUGZSZGVAUHNAVBUTQZTVCVAQZVBVCRVFSUIZAVHVCVCUIZVIVJTZVCU
      KAVHVKTVLACDEFVBVCHIJKLMUJULUMUNVGUHQAVDVAQTVEVFVDOUOUPUQURACDEFVBVDHIJKL
      MUJUS $.
  $}

  ${
    $d x z A $.  $d x V $.
    $( The power set of a set is equinumerous to set exponentiation with a base
       of ordinal ` 2o ` .  (Contributed by FL, 22-Feb-2011.)  (Revised by
       Mario Carneiro, 1-Jul-2015.) $)
    pw2eng $p |- ( A e. V -> ~P A ~~ ( 2o ^m A ) ) $=
      ( vx vz wcel cpw c0 csn cpr cmap co c2o cen cvv wel cif cmpt wf1o wbr a1i
      pwexg ovexd id 0ex p0ex 0nep0 eqid pw2f1o f1oen2g syl3anc df2o2 breqtrrdi
      wne oveq1i ) ABEZAFZGGHZIZAJKZLAJKMUOUPNEUSNEUPUSCUPDADCOUQGPQQZRUPUSMSAB
      UAUOURAJUBUOCDAGUQUTBNUOUCGNEUOUDTUQNEUOUETGUQUMUOUFTUTUGUHUPUSUTNNUIUJLU
      RAJUKUNUL $.

    pw2en.1 $e |- A e. _V $.
    $( The power set of a set is equinumerous to set exponentiation with a base
       of ordinal 2.  Proposition 10.44 of [TakeutiZaring] p. 96.  This is
       Metamath 100 proof #52.  (Contributed by NM, 29-Jan-2004.)  (Proof
       shortened by Mario Carneiro, 1-Jul-2015.) $)
    pw2en $p |- ~P A ~~ ( 2o ^m A ) $=
      ( cvv wcel cpw c2o cmap co cen wbr pw2eng ax-mp ) ACDAEFAGHIJBACKL $.
  $}

  ${
    $d A a b $.  $d B a b $.  $d F a b $.  $d V a b $.
    $( Covering implies injection on power sets.  (Contributed by Stefan
       O'Rear, 6-Nov-2014.)  (Revised by Mario Carneiro, 24-Jun-2015.)
       (Revised by AV, 18-Sep-2021.) $)
    fopwdom $p |- ( ( F e. V /\ F : A -onto-> B ) -> ~P B ~<_ ~P A ) $=
      ( va vb wcel wa cpw cv cima cvv wss adantl wb wceq imaeq2 elpwid foimacnv
      crn wfo ccnv imassrn cdm dfdm4 fdmd eqtr3id sseqtrid cnvexg adantr imaexg
      fof elpwg 3syl mpbird a1d simpllr simplrl syl2anc simplrr 3eqtr3d impbid1
      weq ex rnexg forn eleq1d syl5ibcom imp pwexd wf dmfex sylan2 dom3d ) CDGZ
      ABCUAZHZEFBIZAIZCUBZEJZKZVTFJZKZLLVQWBVSGZWAVRGZVQWEWBAMZVPWGVOVPVTTZWBAV
      TWAUCVPWHCUDACUEVPABCABCULZUFUGUHNVQVTLGZWBLGWEWGOVOWJVPCDUIUJVTWALUKWBAL
      UMUNUOUPVQWFWCVRGZHZWBWDPZEFVCZOVQWLHZWMWNWOWMWNWOWMHZCWBKZCWDKZWAWCWMWQW
      RPWOWBWDCQNWPVPWABMWQWAPVOVPWLWMUQZWPWABVQWFWKWMURRABWACSUSWPVPWCBMWRWCPW
      SWPWCBVQWFWKWMUTRABWCCSUSVAVDWAWCVTQVBVDVQBLVOVPBLGZVOCTZLGVPWTCDVEVPXABL
      ABCVFVGVHVIVJVQALVPVOABCVKALGWIABDCVLVMVJVN $.
  $}

  ${
    $d f g h A $.  $d f g h B $.  $d f g h X $.  $d f g h Y $.
    $( Given two equipollent sets, a bijection can always be chosen which fixes
       a single point.  (Contributed by Stefan O'Rear, 9-Jul-2015.) $)
    enfixsn $p |- ( ( A e. X /\ B e. Y /\ X ~~ Y ) ->
        E. f ( f : X -1-1-onto-> Y /\ ( f ` A ) = B ) ) $=
      ( vg vh wcel cen cv wf1o cfv wceq wa wex csn cvv cun a1i syl2anc wbr bren
      simp3 sylib cdif relen brrelex2i 3ad2ant3 adantr wf f1of adantl ffvelcdmd
      w3a simpl1 simpl2 difsnen syl3anc cop ccom cin fvex f1osng simprr disjdif
      c0 f1oun syl22anc wb ad2antrl uncom difsnid eqtrid syl mpbid simprl f1oco
      f1oeq23 wfn f1ofn fvco2 ad2antll snid fvun1 syl112anc fvsng snex vex unex
      3eqtrd coex f1oeq1 fveq1 eqeq1d anbi12d spcev expr exlimdv mpd exlimddv )
      ADHZBEHZDEIUAZUNZDEFJZKZDECJZKZAXGLZBMZNZCOZFXDXCXFFOXAXBXCUCDEFUBUDXDXFN
      ZEAXELZPZUEZEBPZUEZGJZKZGOZXLXMXPXRIUAZYAXMEQHZXNEHZXBYBXDYCXFXCXAYCXBDEI
      UFUGUHUIXMDEAXEXFDEXEUJZXDDEXEUKZULXAXBXCXFUOUMXAXBXCXFUPXNBQEUQURXPXRGUB
      UDXMXTXLGXDXFXTXLXDXFXTNZNZDEXNBUSZPZXSRZXEUTZKZAYLLZBMZXLYHEEYKKZXFYMYHX
      OXPRZXQXRRZYKKZYPYHXOXQYJKZXTXOXPVAVFMZXQXRVAVFMZYSYHXNQHZXBYTUUCYHAXEVBZ
      SZXAXBXCYGUPZXNBQEVCTZXDXFXTVDUUAYHXOEVESZUUBYHXQEVESXOXQXPXRYJXSVGVHYHYQ
      EMZYREMZYSYPVIYHYDUUIYHDEAXEXFYEXDXTYFVJXAXBXCYGUOZUMYDYQXPXOREXOXPVKEXNV
      LVMVNYHXBUUJUUFXBYRXRXQREXQXRVKEBVLVMVNYQEYREYKVRTVOXDXFXTVPDEEYKXEVQTYHY
      NXNYKLZXNYJLZBYHXEDVSZXAYNUULMXFUUNXDXTDEXEVTVJUUKDYKXEAWATYHYJXOVSZXSXPV
      SZUUAXNXOHZUULUUMMYHYTUUOUUGXOXQYJVTVNXTUUPXDXFXPXRXSVTWBUUHUUQYHXNUUDWCS
      XOXPYJXSXNWDWEYHUUCXBUUMBMUUEUUFXNBQEWFTWJXKYMYONCYLYKXEYJXSYIWGGWHWIFWHW
      KXGYLMZXHYMXJYODEXGYLWLUURXIYNBAXGYLWMWNWOWPTWQWRWSWT $.
  $}

  ${
    $d f w A $.  $d f w B $.
    $( Obsolete version of ~ sucdom2 as of 4-Dec-2024.  (Contributed by Mario
       Carneiro, 12-Jan-2013.)  (Proof shortened by Mario Carneiro,
       27-Apr-2015.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    sucdom2OLD $p |- ( A ~< B -> suc A ~<_ B ) $=
      ( vf vw csdm wbr cv wf1 cdom wex syl csn cun wceq cvv wcel adantr a1i wss
      wa csuc sdomdom brdomi crn cdif cin c0 relsdom brrelex1i vex rnex f1f1orn
      wf1o adantl f1of1 f1dom2g syl3anc wn cen sdomnen ssdif0 simplr frnd simpr
      f1f eqssd dff1o5 sylanbrc f1oen3g sylancr biimtrrid mtod neq0 sylib snssi
      ex en2sn sylancl brrelex2i difexg ssdomg 3syl endomtr syl6an syl5 exlimdv
      mpd disjdif undom syl21anc df-suc undif2 ssequn1 eqtr2id 3brtr4d exlimddv
      wi ) ABEFZABCGZHZAUAZBIFCWRABIFWTCJABUBABCUCKWRWTTZAALZMZWSUDZBXEUEZMZXAB
      IXBAXEIFZXCXFIFZXEXFUFUGNZXDXGIFXBAOPZXEOPZAXEWSHZXHWRXKWTABEUHUIQZXLXBWS
      CUJZUKRXBAXEWSUMZXMWTXPWRABWSULUNAXEWSUOKAXEWSOOUPUQXBDGZXFPZDJZXIXBXFUGN
      ZURXSXBXTABUSFZWRYAURWTABUTQXTBXESZXBYABXEVAXBYBYAXBYBTZWSOPABWSUMZYAXOYC
      WTXEBNYDWRWTYBVBZYCXEBYCWTXEBSZYEWTABWSABWSVEVCZKXBYBVDVFABWSVGVHABWSOVIV
      JVPVKVLDXFVMVNXBXRXIDXRXQLZXFSZXBXIXQXFVOXBXCYHUSFZYIYHXFIFZXIXBXKXQOPYJX
      NDUJAXQOOVQVRXBBOPZXFOPYIYKWQWRYLWTABEUHVSQBXEOVTYHXFOWAWBXCYHXFWCWDWEWFW
      GXJXBXEBWHRAXEXCXFWIWJXAXDNXBAWKRXBXGXEBMZBXEBWLXBYFYMBNWTYFWRYGUNXEBWMVN
      WNWOWP $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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 ~ sbth .  (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 eqabri imass2 sscon 3syl wi difss2
      ssconb exbiri syl5 pm2.43d imp sylbi elssuni sstrd mprgbir ) DIZBFJZCEJZU
      LKZLZKZLZMAJZURMADADURNUSDOZUSBUMCUNUSKZLZKZLZURUTUSBMZVCBUSLMZPZUSVDMZVG
      ADHQVEVFVHVEVFVHVFVCBMZVEVFVHUAVCBUSUBVEVIVHVFUSVCBUCUDUEUFUGUHUTUPVBMZUQ
      VCMVDURMUTUSULMVAUOMVJUSDUIUSULUNRVAUOCSTUPVBUMRUQVCBSTUJUK $.

    $( Lemma for ~ sbth .  (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 wi imassrn sstr2 ax-mp difss sylancl mpbiri jctil difexi sseq1 difeq2d
      wceq imaeq2 imaeq2d difeq2 sseq12d anbi12d elab sylibr eleqtrrdi elssuni
      syl ) FIZJZBKZBVFCEIZDLZMZNZMZNZDOVNVJKVHVNAIZBKZVFCVIVOMZNZMZBVONZKZPZAQ
      ZDVHVNBKZVFCVIVNMZNZMZBVNNZKZPZVNWCOVHWIWDVHWIVNBWGNKZWFVLKZWGVMKWKVJVNKV
      KWEKWLABCDEFGHUAVJVNVIRVKWECSTWFVLVFRWGVMBSTVHWGBKZWDWIWKUCWGVGKVHWMUDVFW
      FUEWGVGBUFUGBVMUHZWGVNBUBUIUJWNUKWBWJAVNBVMGULVOVNUOZVPWDWAWIVOVNBUMWOVSW
      GVTWHWOVRWFVFWOVQWECVOVNVIUPUNUQVOVNBURUSUTVAVBHVCVNDVDVE $.

    $( Lemma for ~ sbth .  (Contributed by NM, 22-Mar-1998.) $)
    sbthlem3 $p |- ( ran g C_ A ->
                  ( g " ( B \ ( f " U. D ) ) ) = ( A \ U. D ) ) $=
      ( cv crn wss cuni cdif cima wa wceq sbthlem2 sbthlem1 jctil eqss wi sstr2
      sylibr difeq2d imassrn ax-mp dfss4 sylib eqtr2d ) FIZJZBKZBDLZMBBUJCEIUMN
      MZNZMZMZUOULUMUPBULUMUPKZUPUMKZOUMUPPULUSURABCDEFGHQABCDEFGHRSUMUPTUCUDUL
      UOBKZUQUOPUOUKKULUTUAUJUNUEUOUKBUBUFUOBUGUHUI $.

    $( Lemma for ~ sbth .  (Contributed by NM, 27-Mar-1998.) $)
    sbthlem4 $p |- ( ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ->
                  ( `' g " ( A \ U. D ) ) = ( B \ ( f " U. D ) ) ) $=
      ( cv cdm wceq crn wss wa ccnv wfun cuni cdif cima cres df-ima difss sseq2
      ssdmres sylib dfdm4 eqtr3di funcnvres sbthlem3 reseq2d sylan9eqr sylan9eq
      mpbiri rneqd anassrs eqtr4id ) FIZJZCKZUQLBMZNUQOZPZNVABDQZRZSVAVDTZLZCEI
      VCSZRZVAVDUAUSUTVBVHVFKUSUTVBNZVHUQVHTZOZLZVFUSVJJZVHVLUSVHURMZVMVHKUSVNV
      HCMCVGUBURCVHUCUMVHUQUDUEVJUFUGVIVKVEVBUTVKVAUQVHSZTVEVHUQUHUTVOVDVAABCDE
      FGHUIUJUKUNULUOUP $.

    ${
      $d x H $.
      sbthlem.3 $e |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) ) $.
      $( Lemma for ~ sbth .  (Contributed by NM, 22-Mar-1998.) $)
      sbthlem5 $p |- ( ( dom f = A /\ ran g C_ A ) -> dom H = A ) $=
        ( cv cdm wceq wss cdif cun cin cres dmres eqtri wa cuni ccnv dmeqi dmun
        df-rn eqcomi ineq2i uneq12i cima sbthlem1 difss sstri sseq2 mpbiri dfss
        crn sylib uneq1d sbthlem3 imassrn eqsstrrdi sylan9eq eqtr4id undif mpbi
        uneq2d eqtrdi ) EKZLZBMZFKZUQZBNZUAZGLZDUBZBVQOZPZBVOVPVQVJQZVRVMQZPZVS
        VPVIVQRZVLUCZVRRZPZLZWBGWFJUDWGWCLZWELZPWBWCWEUEWHVTWIWAVIVQSWIVRWDLZQW
        AWDVRSWJVMVRVMWJVLUFUGUHTUITTVKVNVSVTVRPWBVKVQVTVRVKVQVJNZVQVTMVKWKVQBN
        ZVQBVLCVIVQUJOZUJZOBABCDEFHIUKBWNULUMZVJBVQUNUOVQVJUPURUSVNVRWAVTVNVRVM
        NVRWAMVNVRWNVMABCDEFHIUTVLWMVAVBVRVMUPURVGVCVDWLVSBMWOVQBVEVFVH $.

      $( Lemma for ~ sbth .  (Contributed by NM, 27-Mar-1998.) $)
      sbthlem6 $p |- ( ( ran f C_ B /\
               ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> ran H = B ) $=
        ( cv wceq crn wss wa cima cdif cun cres df-ima cdm ccnv wfun cuni rneqi
        rnun uneq1i 3eqtr4i sbthlem4 eqtr3di uneq2d eqtr4id imassrn sstr2 ax-mp
        wi undif sylib sylan9eqr ) FKZUACLUTMBNOUTUBZUCOZEKZMZCNZGMZVCDUDZPZCVH
        QZRZCVBVFVHVABVGQZSZMZRZVJVCVGSZVLRZMVOMZVMRVFVNVOVLUFGVPJUEVHVQVMVCVGT
        UGUHVBVIVMVHVBVAVKPVIVMABCDEFHIUIVAVKTUJUKULVEVHCNZVJCLVHVDNVEVRUPVCVGU
        MVHVDCUNUOVHCUQURUS $.

      $( Lemma for ~ sbth .  (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 ~ sbth .  (Contributed by NM, 27-Mar-1998.) $)
      sbthlem8 $p |- ( ( Fun `' f /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A )
                     /\ Fun `' g ) ) -> Fun `' H ) $=
        ( cv ccnv wfun cdm wceq wa crn cres cdif cun wss cin funres11 funcnvcnv
        cuni c0 syl ad3antrrr anim12i cima df-ima df-rn eqtr2i sbthlem4 eqtr3id
        eqtri ineq12 sylancr disjdif eqtrdi adantlll adantl funun syl2anc cnvun
        cnveqi funeqi sylibr ) EKZLMZFKZMZVKNCOZPVKQBUAZPVKLZMZPZPZVIDUEZRZLZVO
        BVSSZRZLZTZMZGLZMVRWAMZWDMZPWANZWDNZUBZUFOZWFVJWHVQWIVSVIUCVLWIVMVNVPVL
        VOLMWIVKUDWBVOUCUGUHUIVQWMVJVMVNVPWMVLVMVNPVPPZWLVIVSUJZCWOSZUBZUFWNWJW
        OOWKWPOWLWQOWOVTQWJVIVSUKVTULUMWNWKVOWBUJZWPWRWCQWKVOWBUKWCULUPABCDEFHI
        UNUOWJWOWKWPUQURWOCUSUTVAVBWAWDVCVDWGWEWGVTWCTZLWEGWSJVFVTWCVEUPVGVH $.

      $( Lemma for ~ sbth .  (Contributed by NM, 28-Mar-1998.) $)
      sbthlem9 $p |- ( ( f : A -1-1-> B /\ g : B -1-1-> A ) ->
                    H : A -1-1-onto-> B ) $=
        ( wfun cdm wceq wa crn ccnv wfn df-fn anbi1i bitri cv wss wf1o sbthlem7
        wf1 sbthlem5 adantrl anim12i an42s adantlr sbthlem8 adantll simpr df-rn
        anim1i sbthlem6 eqtr3id sylanr1 jca32 df-f1 df-f anbi12i dff1o4 3imtr4i
        wf ) EUAZKZVFLBMZNZVFOCUBZNZVFPKZNZFUAZKZVNLCMZNZVNOBUBZNZVNPKZNZNZGKZG
        LBMZNZGPZKZWFLZCMZNZNZBCVFUEZCBVNUEZNBCGUCZWBWEWGWIVKWAWEVLVIWAWEVJVGVT
        VHVSWEVGVTNWCVHVSNWDABCDEFGHIJUDVHVRWDVQABCDEFGHIJUFUGUHUIUJUJVLWAWGVKA
        BCDEFGHIJUKULVKWAWIVLVJWAWIVIVSVJVPVRNZVTWIVQVPVRVOVPUMUOVJWOVTNNWHGOCG
        UNABCDEFGHIJUPUQURULUJUSWLVMWMWAWLBCVFVEZVLNVMBCVFUTWPVKVLWPVFBQZVJNVKB
        CVFVAWQVIVJVFBRSTSTWMCBVNVEZVTNWACBVNUTWRVSVTWRVNCQZVRNVSCBVNVAWSVQVRVN
        CRSTSTVBWNGBQZWFCQZNWKBCGVCWTWEXAWJGBRWFCRVBTVD $.

      ${
        $d f g A $.  $d f g B $.
        sbthlem.4 $e |- B e. _V $.
        $( Lemma for ~ sbth .  (Contributed by NM, 28-Mar-1998.) $)
        sbthlem10 $p |- ( ( A ~<_ B /\ B ~<_ A ) -> A ~~ B ) $=
          ( cdom wbr wa cv wf1 wex brdom cvv cres cen exdistrv bitr4i wcel wf1o
          anbi12i cuni ccnv cdif cun vex resex eqeltri sbthlem9 f1oen3g sylancr
          cnvex unex exlimivv sylbi ) BCLMZCBLMZNZBCEOZPZCBFOZPZNZFQEQZBCUAMZVC
          VEEQZVGFQZNVIVAVKVBVLBCEKRCBFHRUFVEVGEFUBUCVHVJEFVHGSUDBCGUEVJGVDDUGZ
          TZVFUHZBVMUIZTZUJSJVNVQVDVMEUKULVOVPVFFUKUQULURUMABCDEFGHIJUNBCGSUOUP
          USUT $.
      $}
    $}
  $}

  ${
    $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.  The theorem can
       also be proved from the axiom of choice and the linear order of the
       cardinal numbers, but our development does not provide the linear order
       of cardinal numbers until much later and in ways that depend on
       Schroeder-Bernstein.

       The main proof consists of lemmas ~ sbthlem1 through ~ sbthlem10 ; this
       final piece mainly changes bound variables to eliminate the hypotheses
       of ~ sbthlem10 .  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.  In the Intuitionistic Logic Explorer (ILE) the
       Schroeder-Bernstein Theorem has been proven equivalent to the law of the
       excluded middle (LEM), and in ILE the LEM is not accepted as necessarily
       true; see ~ https://us.metamath.org/ileuni/exmidsbth.html .  This is
       Metamath 100 proof #25.  (Contributed by NM, 8-Jun-1998.) $)
    sbth $p |- ( ( A ~<_ B /\ B ~<_ A ) -> A ~~ B ) $=
      ( vz vw vx vy cdom wbr wa cen cvv wi cv wceq breq1 breq2 anbi12d wss cima
      cdif vg vf wcel reldom brrelex1i imbi12d cab cuni cres ccnv cun vex sseq1
      imaeq2 difeq2d imaeq2d difeq2 sseq12d cbvabv eqid vtocl2g syl2an pm2.43i
      sbthlem10 ) ABGHZBAGHZIZABJHZVEAKUCBKUCVGVHLZVFABGUDUEBAGUDUECMZDMZGHZVKV
      JGHZIZVJVKJHZLAVKGHZVKAGHZIZAVKJHZLVICDABKKVJANZVNVRVOVSVTVLVPVMVQVJAVKGO
      VJAVKGPQVJAVKJOUFVKBNZVRVGVSVHWAVPVEVQVFVKBAGPVKBAGOQVKBAJPUFEVJVKFMZVJRZ
      UAMZVKUBMZWBSZTZSZVJWBTZRZIZFUGZUBUAWEWLUHZUIWDUJVJWMTUIUKZCULWKEMZVJRZWD
      VKWEWOSZTZSZVJWOTZRZIFEWBWONZWCWPWJXAWBWOVJUMXBWHWSWIWTXBWGWRWDXBWFWQVKWB
      WOWEUNUOUPWBWOVJUQURQUSWNUTDULVDVAVBVC $.
    $( $j usage 'sbth' avoids 'ax-inf' 'ax-inf2' 'ax-ac' 'ax-ac2'; $)
  $}

  $( Schroeder-Bernstein Theorem and its converse.  (Contributed by NM,
     8-Jun-1998.) $)
  sbthb $p |- ( ( A ~<_ B /\ B ~<_ A ) <-> A ~~ B ) $=
    ( cdom wbr wa cen sbth endom ensym syl jca impbii ) ABCDZBACDZEABFDZABGOMNA
    BHOBAFDNABIBAHJKL $.

  ${
    $d x y $.
    $( Schroeder-Bernstein Theorem in class form.  (Contributed by NM,
       28-Mar-1998.) $)
    sbthcl $p |- ~~ = ( ~<_ i^i `' ~<_ ) $=
      ( vx vy cen cdom ccnv cin relen wss wrel inss1 reldom relss mp2 cv wbr wa
      brin vex brcnv anbi2i sbthb 3bitrri eqbrriv ) ABCDDEZFZGUEDHDIUEIDUDJKUED
      LMANZBNZUEOUFUGDOZUFUGUDOZPUHUGUFDOZPUFUGCOUFUGDUDQUIUJUHUFUGDARBRSTUFUGU
      AUBUC $.
  $}

  $( Alternate definition of strict dominance.  Compare Definition 3 of
     [Suppes] p. 97.  (Contributed by NM, 31-Mar-1998.) $)
  dfsdom2 $p |- ~< = ( ~<_ \ `' ~<_ ) $=
    ( csdm cdom cen cdif ccnv cin df-sdom sbthcl difeq2i difin 3eqtri ) ABCDBBB
    EZFZDBLDGCMBHIBLJK $.

  ${
    brsdom2.1 $e |- A e. _V $.
    brsdom2.2 $e |- B e. _V $.
    $( Alternate definition of strict dominance.  Definition 3 of [Suppes]
       p. 97.  (Contributed by NM, 27-Jul-2004.) $)
    brsdom2 $p |- ( A ~< B <-> ( A ~<_ B /\ -. B ~<_ A ) ) $=
      ( cop csdm wcel cdom ccnv cdif wbr wn dfsdom2 eleq2i df-br opelcnv bitr4i
      wa notbii anbi12i eldif 3bitr4i ) ABEZFGUCHHIZJZGZABFKABHKZBAHKZLZRZFUEUC
      MNABFOUJUCHGZUCUDGZLZRUFUGUKUIUMABHOUHULUHBAEHGULBAHOABHCDPQSTUCHUDUAQUB
      $.
  $}

  $( Strict dominance is asymmetric.  Theorem 21(ii) of [Suppes] p. 97.
     (Contributed by NM, 8-Jun-1998.) $)
  sdomnsym $p |- ( A ~< B -> -. B ~< A ) $=
    ( csdm wbr cen sdomnen cdom sdomdom sbth syl2an mtand ) ABCDZBACDZABEDZABFL
    ABGDBAGDNMABHBAHABIJK $.

  $( Theorem 22(i) of [Suppes] p. 97.  (Contributed by NM, 10-Jun-1998.) $)
  domnsym $p |- ( A ~<_ B -> -. B ~< A ) $=
    ( cdom wbr csdm cen wo wn brdom2 sdomnsym sdomnen ensym nsyl3 jaoi sylbi )
    ABCDABEDZABFDZGBAEDZHZABIPSQABJRBAFDQBAKABLMNO $.

  ${
    $d A f $.
    $( Any set dominates the empty set.  (Contributed by NM, 26-Oct-2003.)
       (Revised by Mario Carneiro, 26-Apr-2015.)  Avoid ~ ax-pow , ~ ax-un .
       (Revised by BTernaryTau, 29-Nov-2024.) $)
    0domg $p |- ( A e. V -> (/) ~<_ A ) $=
      ( vf wcel c0 cdom wbr cv wf1 wex 0ex f1eq1 f10 ceqsexv2d cvv brdom2g mpan
      wb mpbiri ) ABDZEAFGZEACHZIZCJZUCEAEICEKEAUBELAMNEODTUAUDRKEACOBPQS $.
    $( $j usage '0domg' avoids 'ax-pow' 'ax-un'; $)
  $}

  $( Obsolete version of ~ 0domg as of 29-Nov-2024.  (Contributed by NM,
     26-Oct-2003.)  (Revised by Mario Carneiro, 26-Apr-2015.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  0domgOLD $p |- ( A e. V -> (/) ~<_ A ) $=
    ( wcel c0 wss cdom wbr 0ss ssdomg mpi ) ABCDAEDAFGAHDABIJ $.

  ${
    $d A f $.
    $( A set dominated by the empty set is empty.  (Contributed by NM,
       22-Nov-2004.)  Avoid ~ ax-pow , ~ ax-un .  (Revised by BTernaryTau,
       29-Nov-2024.) $)
    dom0 $p |- ( A ~<_ (/) <-> A = (/) ) $=
      ( vf c0 cdom wbr cv wf1 wex brdomi wf f1f f00 simprbi syl exlimiv cen en0
      wceq endom sylbir impbii ) ACDEZACRZUBACBFZGZBHUCACBIUEUCBUEACUDJZUCACUDK
      UFUDCRUCAUDLMNONUCACPEUBAQACSTUA $.
    $( $j usage 'dom0' avoids 'ax-pow' 'ax-un'; $)
  $}

  $( Obsolete version of ~ dom0 as of 29-Nov-2024.  (Contributed by NM,
     22-Nov-2004.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  dom0OLD $p |- ( A ~<_ (/) <-> A = (/) ) $=
    ( c0 cdom wbr wa cen wceq cvv wcel reldom brrelex1i 0domg syl pm4.71i sbthb
    en0 3bitri ) ABCDZRBACDZEABFDABGRSRAHISABCJKAHLMNABOAPQ $.

  $( A set strictly dominates the empty set iff it is not empty.  (Contributed
     by NM, 23-Mar-2006.)  Avoid ~ ax-pow , ~ ax-un .  (Revised by BTernaryTau,
     29-Nov-2024.) $)
  0sdomg $p |- ( A e. V -> ( (/) ~< A <-> A =/= (/) ) ) $=
    ( wcel c0 csdm wbr cen wn wne cdom wb 0domg brsdom baib syl en0r necon3bbii
    bitrdi ) ABCZDAEFZDAGFZHZADISDAJFZTUBKABLTUCUBDAMNOUAADAPQR $.
  $( $j usage '0sdomg' avoids 'ax-pow' 'ax-un'; $)

  $( Obsolete version of ~ 0sdomg as of 29-Nov-2024.  (Contributed by NM,
     23-Mar-2006.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  0sdomgOLD $p |- ( A e. V -> ( (/) ~< A <-> A =/= (/) ) ) $=
    ( wcel c0 csdm wbr cen wn wne cdom wb 0domg brsdom baib syl wceq ensymb en0
    bitri necon3bbii bitrdi ) ABCZDAEFZDAGFZHZADIUBDAJFZUCUEKABLUCUFUEDAMNOUDAD
    UDADGFADPDAQARSTUA $.

  ${
    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 $.

    $( A set strictly dominates the empty set iff it is not empty.
       (Contributed by NM, 29-Jul-2004.) $)
    0sdom $p |- ( (/) ~< A <-> A =/= (/) ) $=
      ( cvv wcel c0 csdm wbr wne wb 0sdomg ax-mp ) ACDEAFGAEHIBACJK $.
  $}

  $( The empty set does not strictly dominate any set.  (Contributed by NM,
     26-Oct-2003.)  Avoid ~ ax-pow , ~ ax-un .  (Revised by BTernaryTau,
     29-Nov-2024.) $)
  sdom0 $p |- -. A ~< (/) $=
    ( c0 csdm wbr cdom cen wn wa wi wceq dom0 en0 sylbb2 iman mpbi brsdom mtbir
    ) ABCDABEDZABFDZGHZRSITGRABJSAKALMRSNOABPQ $.
  $( $j usage 'sdom0' avoids 'ax-pow' 'ax-un'; $)

  $( Obsolete version of ~ sdom0 as of 29-Nov-2024.  (Contributed by NM,
     26-Oct-2003.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  sdom0OLD $p |- -. A ~< (/) $=
    ( csdm wbr cdom cvv wcel relsdom brrelex1i 0domg syl domnsym con2i pm2.65i
    c0 ) ANBCZNADCZOAEFPANBGHAEIJPONAKLM $.

  $( Transitivity of strict dominance and dominance.  Theorem 22(iii) of
     [Suppes] p. 97.  (Contributed by NM, 26-Oct-2003.)  (Revised by Mario
     Carneiro, 26-Apr-2015.) $)
  sdomdomtr $p |- ( ( A ~< B /\ B ~<_ C ) -> A ~< C ) $=
    ( csdm wbr cdom wa cen sdomdom domtr sylan simpl simpr ensym domentr syl2an
    wn domnsym syl ex mt2d brsdom sylanbrc ) ABDEZBCFEZGZACFEZACHEZQACDEUDABFEU
    EUGABIABCJKUFUHUDUDUELUFUHUDQZUFUHGBAFEZUIUFUECAHEUJUHUDUEMACNBCAOPBARSTUAA
    CUBUC $.

  $( Transitivity of strict dominance and equinumerosity.  Exercise 11 of
     [Suppes] p. 98.  (Contributed by NM, 26-Oct-2003.) $)
  sdomentr $p |- ( ( A ~< B /\ B ~~ C ) -> A ~< C ) $=
    ( cen wbr csdm cdom endom sdomdomtr sylan2 ) BCDEABFEBCGEACFEBCHABCIJ $.

  $( Transitivity of dominance and strict dominance.  Theorem 22(ii) of
     [Suppes] p. 97.  (Contributed by NM, 10-Jun-1998.)  (Revised by Mario
     Carneiro, 26-Apr-2015.) $)
  domsdomtr $p |- ( ( A ~<_ B /\ B ~< C ) -> A ~< C ) $=
    ( cdom wbr wa cen wn sdomdom domtr sylan2 simpr ensym simpl endomtr syl2anr
    csdm domnsym syl ex mt2d brsdom sylanbrc ) ABDEZBCQEZFZACDEZACGEZHACQEUEUDB
    CDEUGBCIABCJKUFUHUEUDUELUFUHUEHZUFUHFCBDEZUIUHCAGEUDUJUFACMUDUENCABOPCBRSTU
    AACUBUC $.

  $( Transitivity of equinumerosity and strict dominance.  (Contributed by NM,
     26-Oct-2003.)  (Revised by Mario Carneiro, 26-Apr-2015.) $)
  ensdomtr $p |- ( ( A ~~ B /\ B ~< C ) -> A ~< C ) $=
    ( cen wbr cdom csdm endom domsdomtr sylan ) ABDEABFEBCGEACGEABHABCIJ $.

  $( Strict dominance is irreflexive.  Theorem 21(i) of [Suppes] p. 97.
     (Contributed by NM, 4-Jun-1998.) $)
  sdomirr $p |- -. A ~< A $=
    ( cvv wcel csdm wbr wn sdomnen enrefg nsyl3 relsdom brrelex1i con3i pm2.61i
    cen ) ABCZAADEZFPAANEOAAGABHIPOAADJKLM $.

  $( Strict dominance is transitive.  Theorem 21(iii) of [Suppes] p. 97.
     (Contributed by NM, 9-Jun-1998.) $)
  sdomtr $p |- ( ( A ~< B /\ B ~< C ) -> A ~< C ) $=
    ( csdm wbr cdom sdomdom domsdomtr sylan ) ABDEABFEBCDEACDEABGABCHI $.

  $( Strict dominance has no 2-cycle loops.  (Contributed by NM,
     6-May-2008.) $)
  sdomn2lp $p |- -. ( A ~< B /\ B ~< A ) $=
    ( csdm wbr wa sdomirr sdomtr mto ) ABCDBACDEAACDAFABAGH $.

  $( 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 $.

  $( Equality-like theorem for equinumerosity and strict dominance.
     (Contributed by NM, 8-Nov-2003.) $)
  sdomen1 $p |- ( A ~~ B -> ( A ~< C <-> B ~< C ) ) $=
    ( cen wbr csdm ensym ensdomtr sylan impbida ) ABDEZACFEZBCFEZKBADELMABGBACH
    IABCHJ $.

  $( Equality-like theorem for equinumerosity and strict dominance.
     (Contributed by NM, 8-Nov-2003.) $)
  sdomen2 $p |- ( A ~~ B -> ( C ~< A <-> C ~< B ) ) $=
    ( cen wbr csdm sdomentr ancoms ensym sylan impbida ) ABDEZCAFEZCBFEZMLNCABG
    HLBADEZNMABINOMCBAGHJK $.

  $( Dominance is trichotomous in the restricted case of ordinal numbers.
     (Contributed by Jeff Hankins, 24-Oct-2009.) $)
  domtriord $p |- ( ( A e. On /\ B e. On ) -> ( A ~<_ B <-> -. B ~< A ) ) $=
    ( con0 wcel wa cdom wbr csdm wn cen wi sbth expcom a1i iman brsdom xchbinxr
    wss ssdomg syld imbitrdi onelss adantl con3d wb ontri1 ancoms sylibrd ensym
    adantr endom syl con3i jca2 imbitrrdi con1d impbid ) ACDZBCDZEZABFGZBAHGZIZ
    UTVABAFGZBAJGZKZVCVAVFKUTVDVAVEBALMNVFVDVEIZEZVBVDVEOBAPZQUAUTVAVBUTVAIZVHV
    BUTVJVDVGUTVJBARZVDUTVJABDZIZVKUTVLVAUSVLVAKURUSVLABRVABAUBABCSTUCUDUSURVKV
    MUEBAUFUGUHURVKVDKUSBACSUJTVEVAVEABJGVABAUIABUKULUMUNVIUOUPUQ $.

  $( For ordinals, strict dominance implies membership.  (Contributed by Mario
     Carneiro, 13-Jan-2013.) $)
  sdomel $p |- ( ( A e. On /\ B e. On ) -> ( A ~< B -> A e. B ) ) $=
    ( con0 wcel csdm wbr wi wa wss cdom wn ssdomg adantl ontri1 domtriord con4d
    3imtr3d ancoms ) BCDZACDZABEFZABDZGSTHZUBUAUCBAIZBAJFZUBKUAKTUDUEGSBACLMBAN
    BAOQPR $.

  $( The difference of a set from a smaller set cannot be empty.  (Contributed
     by Mario Carneiro, 5-Feb-2013.) $)
  sdomdif $p |- ( A ~< B -> ( B \ A ) =/= (/) ) $=
    ( csdm wbr cdif c0 wceq wn cvv wcel wi relsdom brrelex1i ssdif0 cdom ssdomg
    wss domnsym syl6 biimtrrid syl con2d pm2.43i neqned ) ABCDZBAEZFUEUFFGZHUEU
    GUEUEAIJZUGUEHZKABCLMUGBAQZUHUIBANUHUJBAODUIBAIPBARSTUAUBUCUD $.

  $( An ordinal with more elements of some type is larger.  (Contributed by
     Stefan O'Rear, 2-Nov-2014.) $)
  onsdominel $p |- ( ( A e. On /\ B e. On /\ ( A i^i C ) ~< ( B i^i C ) ) ->
      A e. B ) $=
    ( con0 wcel cin csdm wbr wa wn wb ontri1 ancoms wi cdom inex1g ssrin ssdomg
    wss cvv syl2im domnsym syl6 adantr sylbird con4d 3impia ) ADEZBDEZACFZBCFZG
    HZABEZUHUIIZUMULUNUMJZBASZULJZUIUHUPUOKBALMUHUPUQNUIUHUPUKUJOHZUQUHUJTEUPUK
    UJSURACDPBACQUKUJTRUAUKUJUBUCUDUEUFUG $.

  ${
    $d f g z A $.  $d f g z B $.  $d z C $.
    $( Dominance over a set with one element added.  (Contributed by Mario
       Carneiro, 18-May-2015.) $)
    domunsn $p |- ( A ~< B -> ( A u. { C } ) ~<_ B ) $=
      ( vz csdm wbr cv wcel csn cun cdom c0 wceq wn wex sdom0 breq2 sylib cvv
      wa mtbiri con2i neq0 cdif domdifsn adantr cen en2sn elvd endom syl biimpi
      snprc vsnex eqbrtrdi pm2.61i cin disjdifr undom mpan2 sylancl uncom simpr
      0dom wss snssd undif eqtrid breqtrd exlimddv ) ABEFZDGZBHZACIZJZBKFDVKBLM
      ZNVMDOVPVKVPVKALEFAPBLAEQUAUBDBUCRVKVMTZVOBVLIZUDZVRJZBKVQAVSKFZVNVRKFZVO
      VTKFZVKWAVMABVLUEUFCSHZWBWDVNVRUGFZWBWDWEDCVLSSUHUIVNVRUJUKWDNZVNLVRKWFVN
      LMCUMULVRDUNVDUOUPWAWBTVSVRUQLMWCVRBURAVSVNVRUSUTVAVQVTVRVSJZBVSVRVBVQVRB
      VEWGBMVQVLBVKVMVCVFVRBVGRVHVIVJ $.

    $( There exists a mapping from a set onto any (nonempty) set that it
       dominates.  (Contributed by NM, 23-Mar-2006.) $)
    fodomr $p |- ( ( (/) ~< B /\ B ~<_ A ) -> E. f f : A -onto-> B ) $=
      ( vz vg c0 cdom wa cvv wcel cv wex adantl wne crn cun sylib wceq wfun cdm
      wbr wf1 wfo reldom brrelex2i wb brrelex1i 0sdomg n0 bitrdi biimpac brdomi
      csdm syl wi ccnv csn cxp difexg vsnex xpexg sylancl vex cnvex jctil unexb
      cdif wfn cin df-f1 simprbi fconst ffun ax-mp jctir df-rn eqcomi snnz dmxp
      ineq12i disjdif eqtri funun dmun uneq1i uneq2i 3eqtr2i wss f1f frnd undif
      wf eqtrid df-fn sylanbrc rnun dfdm4 eqtr3id uneq1d xpeq1 0xp eqtrdi rneqd
      f1dm rn0 0ss eqsstrdi a1d rnxp adantr eqsstrd pm2.61ine ssequn2 sylan9eqr
      snssi ex df-fo foeq1 spcegv syl2im expdimp exlimdv syl3c ) FBUMUAZBAGUAZH
      AIJZDKZBJZDLZBAEKZUBZELZABCKZUCZCLZYEYFYDBAGUDUEMYEYDYIYEBIJZYDYIUFBAGUDU
      GYPYDBFNYIBIUHDBUIUJUNUKYEYLYDBAEULMYFYHYLYOUOZDYFYHYQYFYHHYKYOEYFYHYKYOY
      FYJUPZAYJOZVGZYGUQZURZPZIJZYHYKHZABUUCUCZYOYFYRIJZUUBIJZHUUDYFUUHUUGYFYTI
      JUUAIJUUHAYSIUSDUTYTUUAIIVAVBYJEVCVDVEYRUUBVFQUUEUUCAVHZUUCOZBRUUFUUEUUCS
      ZUUCTZARZUUIYKUUKYHYKYRSZUUBSZHYRTZUUBTZVIZFRUUKYKUUNUUOYKBAYJWLUUNBAYJVJ
      VKYTUUAUUBWLUUOYTYGDVCZVLYTUUAUUBVMVNVOUURYSYTVIFUUPYSUUQYTYSUUPYJVPZVQUU
      AFNUUQYTRYGUUSVRYTUUAVSVNZVTYSAWAWBYRUUBWCVBMYKUUMYHYKUULYSYTPZAUULUUPUUQ
      PYSUUQPUVBYRUUBWDYSUUPUUQUUTWEUUQYTYSUVAWFWGYKYSAWHUVBARYKBAYJBAYJWIWJYSA
      WKQWMMUUCAWNWOUUEUUJYROZUUBOZPZBYRUUBWPYKYHUVEBUVDPZBYKUVCBUVDYKUVCYJTBYJ
      WQBAYJXDWRWSYHUVDBWHZUVFBRYHUVGUOYTFYTFRZUVGYHUVHUVDFBUVHUVDFOFUVHUUBFUVH
      UUBFUUAURFYTFUUAWTUUAXAXBXCXEXBBXFXGXHYTFNZYHUVGUVIYHHUVDUUABUVIUVDUUARYH
      YTUUAXIXJYHUUABWHUVIYGBXOMXKXPXLUVDBXMQXNWMABUUCXQWOYNUUFCUUCIABYMUUCXRXS
      XTYAYBXPYBYC $.
  $}

  ${
    $d f A $.  $d f B $.
    $( Injection of sets implies injection on power sets.  (Contributed by
       Mario Carneiro, 9-Apr-2015.) $)
    pwdom $p |- ( A ~<_ B -> ~P A ~<_ ~P B ) $=
      ( vf cdom wbr cpw c0 wceq pweq breq1d wne wa wfo wex csdm cvv wcel reldom
      cv syl brrelex1i 0sdomg biimpar simpl fodomr syl2anc fopwdom mpan exlimiv
      wb vex wss brrelex2i pwexd 0ss sspwi ssdomg mpisyl pm2.61ne ) ABDEZAFZBFZ
      DEZGFZVBDEZAGAGHVAVDVBDAGIJUTAGKZLZBACSZMZCNZVCVGGAOEZUTVJUTVKVFUTAPQVKVF
      UJABDRUAAPUBTUCUTVFUDBACUEUFVIVCCVHPQVIVCCUKBAVHPUGUHUITUTVBPQVDVBULVEUTB
      PABDRUMUNGBBUOUPVDVBPUQURUS $.
  $}

  ${
    $d x y f A $.
    canth2.1 $e |- A e. _V $.
    $( Cantor's Theorem.  No set is equinumerous to its power set.
       Specifically, any set has a cardinality (size) strictly less than the
       cardinality of its power set.  For example, the cardinality of real
       numbers is the same as the cardinality of the power set of integers, so
       real numbers cannot be put into a one-to-one correspondence with
       integers.  Theorem 23 of [Suppes] p. 97.  For the function version, see
       ~ canth .  This is Metamath 100 proof #63.  (Contributed by NM,
       7-Aug-1994.) $)
    canth2 $p |- A ~< ~P A $=
      ( vx vy vf cpw csdm wbr cdom cen wn cvv wcel pwex cv csn snelpwi wceq wb
      wa vex sneqr sneq impbii a1i dom3 mp2an wf1o wex wfo canth f1ofo mto bren
      nex mtbir brsdom mpbir2an ) AAFZGHAUSIHZAUSJHZKALMUSLMUTBABNCDAUSCOZPZDOZ
      PZLLVBAQVCVERZVBVDRZSVBAMVDAMTVFVGVBVDCUAUBVBVDUCUDUEUFUGVAAUSEOZUHZEUIVI
      EVIAUSVHUJAVHBUKAUSVHULUMUOAUSEUNUPAUSUQUR $.
  $}

  ${
    $d x A $.
    $( Cantor's theorem with the sethood requirement expressed as an
       antecedent.  Theorem 23 of [Suppes] p. 97.  (Contributed by NM,
       7-Nov-2003.) $)
    canth2g $p |- ( A e. V -> A ~< ~P A ) $=
      ( vx cv cpw csdm wbr wceq wb pweq breq12 mpdan vex canth2 vtoclg ) CDZPEZ
      FGZAAEZFGZCABPAHQSHRTIPAJPAQSFKLPCMNO $.
  $}

  $( 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 NM, 27-Jun-2008.) $)
  2pwuninel $p |- -. ~P ~P U. A e. A $=
    ( cuni cvv wcel cpw wn csdm wbr sdomirr wss elssuni cdom ssdomg pwexb sylbi
    canth2g sdomtr syl2anc domsdomtr ex syl6ci syl5 bitri sylibr con3i pm2.61i
    mtoi elex ) ABZCDZUIEZEZADZFUJUMULULGHZULIUMULUIJZUJUNULAKUJUOULUILHZUIULGH
    ZUNULUICMUJUIUKGHUKULGHZUQUICPUJUKCDZURUINZUKCPOUIUKULQRUPUQUNULUIULSTUAUBU
    GUMUJUMULCDZUJULAUHUJUSVAUTUKNUCUDUEUF $.

  $( No set equals the power set of its power set.  (Contributed by NM,
     17-Nov-2008.) $)
  2pwne $p |- ( A e. V -> ~P ~P A =/= A ) $=
    ( wcel cpw wceq csdm wbr sdomirr canth2g cvv pwexg syl sdomtr syl2anc breq1
    syl5ibrcom mtoi neqned ) ABCZADZDZASUAAEZUAUAFGZUAHSUCUBAUAFGZSATFGTUAFGZUD
    ABISTJCUEABKTJILATUAMNUAAUAFOPQR $.

  ${
    $d x A $.  $d x B $.  $d x V $.  $d x W $.
    $( A stronger form of ~ pwuninel .  We can use ~ pwuninel , ~ 2pwuninel to
       create one or two sets disjoint from a given set ` A ` , but here we
       show that in fact such constructions exist for arbitrarily large
       disjoint extensions, which is to say that for any set ` B ` we can
       construct a set ` x ` that is equinumerous to it and disjoint from
       ` A ` .  (Contributed by Mario Carneiro, 7-Feb-2015.) $)
    disjen $p |- ( ( A e. V /\ B e. W ) ->
      ( ( A i^i ( B X. { ~P U. ran A } ) ) = (/) /\
          ( B X. { ~P U. ran A } ) ~~ B ) ) $=
      ( vx wcel crn cuni cpw csn cxp wceq c2nd cfv c1st ad2antll fvex syl cvv
      wa cin c0 cen wbr cop 1st2nd2 simprl eqeltrrd opelrn pwuninel xp2nd elsni
      eleq1d mtbiri pm2.65da elin sylnibr eq0rdv simpr rnexg adantr uniexg 3syl
      cv pwexg xpsneng syl2anc jca ) ACFZBDFZTZABAGZHZIZJZKZUAZUBLVPBUCUDZVKEVQ
      VKEVDZAFZVSVPFZTZVSVQFVKWBVSMNZVLFZVKWBTZVSONZWCUEZAFWDWEVSWGAWAVSWGLVKVT
      VSBVOUFPVKVTWAUGUHWFWCAVSOQVSMQUIRWEWDVNVLFVLUJWEWCVNVLWEWCVOFZWCVNLWAWHV
      KVTVSBVOUKPWCVNULRUMUNUOVSAVPUPUQURVKVJVNSFZVRVIVJUSVKVLSFZVMSFWIVIWJVJAC
      UTVAVLSVBVMSVEVCBVNDSVFVGVH $.

    $( Existence version of ~ disjen .  (Contributed by Mario Carneiro,
       7-Feb-2015.) $)
    disjenex $p |- ( ( A e. V /\ B e. W ) ->
      E. x ( ( A i^i x ) = (/) /\ x ~~ B ) ) $=
      ( wcel wa cv cin c0 wceq cen wbr crn cuni cpw csn cxp cvv simpr sylancl
      snex xpexg disjen ineq2 eqeq1d breq1 anbi12d spcedv ) BDFZCEFZGZBAHZIZJKZ
      UMCLMZGBCBNOPZQZRZIZJKZUSCLMZGASUSULUKURSFUSSFUJUKTUQUBCURESUCUABCDEUDUMU
      SKZUOVAUPVBVCUNUTJUMUSBUEUFUMUSCLUGUHUI $.
  $}

  ${
    domss2.1 $e |- G = `' ( F u.
      ( 1st |` ( ( B \ ran F ) X. { ~P U. ran A } ) ) ) $.
    $( A corollary of ~ disjenex .  If ` F ` is an injection from ` A ` to
       ` B ` then ` G ` is a right inverse of ` F ` from ` B ` to a superset of
       ` A ` .  (Contributed by Mario Carneiro, 7-Feb-2015.)  (Revised by Mario
       Carneiro, 24-Jun-2015.) $)
    domss2 $p |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) ->
     ( G : B -1-1-onto-> ran G /\ A C_ ran G /\ ( G o. F ) = ( _I |` A ) ) ) $=
      ( wcel crn wf1o wss ccom cres wceq cun ccnv cin c0 cvv ax-mp wf1 w3a cdif
      cid cuni cpw csn cxp c1st f1f1orn 3ad2ant1 simp2 rnexg syl pwexg 1stconst
      uniexg cen wbr wa difexg 3ad2ant3 disjen syl2anc simpld disjdif a1i f1oun
      3syl syl22anc undif2 wf frnd ssequn1 sylib eqtrid f1oeq3d mpbid f1ocnv wb
      f1f f1oeq1 sylibr wfo f1ofo forn mpbird ssun1 sseqtrrid cores dmres f1odm
      ssid ineq2d eqtrdi wrel relres reldm0 uneq2d cnvun eqtri reseq1i resundir
      cdm df-rn reseq2i relcnv resdm uneq1i 3eqtrri un0 3eqtr3g coeq1d f1cocnv1
      eqtrd eqtr3id 3jca ) ABCUAZAEHZBFHZUBZBDIZDJZAYBKDCLZUDAMZNYAYCBABCIZUCZA
      IZUEZUFZUGUHZOZDJZYABYLCUIYKMZOZPZJZYMYAYLBYOJZYQYAYLYFYGOZYOJZYRYAAYFCJZ
      YKYGYNJZAYKQRNZYFYGQZRNZYTXRXSUUAXTABCUJUKYAYJSHZUUBYAYHSHZYISHUUFYAXSUUG
      XRXSXTULZAEUMUNYHSUQYISUOVIYGYJSUPUNZYAUUCYKYGURUSZYAXSYGSHZUUCUUJUTUUHXT
      XRUUKXSBYFFVAVBAYGESVCVDVEUUEYAYFBVFZVGAYFYKYGCYNVHVJYAYSBYLYOYAYSYFBOZBY
      FBVKYAYFBKUUMBNYAABCXRXSABCVLXTABCWAUKVMYFBVNVOVPVQVRYLBYOVSUNDYPNYMYQVTG
      BYLDYPWBTWCZYAYBYLBDYAYMBYLDWDYBYLNUUNBYLDWEBYLDWFVIZVQWGYAYLAYBAYKWHUUOW
      IYAYDDYFMZCLZYEYFYFKUUQYDNYFWMDCYFWJTYAUUQCPZCLZYEYAUUPUURCYAUURYNPZYFMZO
      ZUURROUUPUURYAUVARUURYAUVAXDZRNZUVARNZYAUVCYFUUTXDZQZRUUTYFWKYAUVGUUDRYAU
      VFYGYFYAUUBYGYKUUTJUVFYGNUUIYKYGYNVSYGYKUUTWLVIWNUULWOVPUVAWPUVEUVDVTUUTY
      FWQUVAWRTWCWSUUPUURUUTOZYFMUURYFMZUVAOUVBDUVHYFDYPUVHGCYNWTXAXBUURUUTYFXC
      UVIUURUVAUVIUURUURXDZMZUURYFUVJUURCXEXFUURWPUVKUURNCXGUURXHTXAXIXJUURXKXL
      XMXRXSUUSYENXTABCXNUKXOXPXQ $.
  $}

  ${
    $d f g x A $.  $d f g x B $.  $d g F $.  $d f W $.
    $( A corollary of ~ disjenex .  If ` F ` is an injection from ` A ` to
       ` B ` then there is a right inverse ` g ` of ` F ` from ` B ` to a
       superset of ` A ` .  (Contributed by Mario Carneiro, 7-Feb-2015.)
       (Revised by Mario Carneiro, 24-Jun-2015.) $)
    domssex2 $p |- ( ( F : A -1-1-> B /\ A e. V /\ B e. W ) ->
      E. g ( g : B -1-1-> _V /\ ( g o. F ) = ( _I |` A ) ) ) $=
      ( wf1 wcel w3a cvv ccom cres wceq wa crn wf fex2 sylancl syl wss cid c1st
      cv cdif cuni cpw csn cxp cun ccnv f1f syl3an1 f1stres 3ad2ant3 snex xpexg
      difexg mp3an2i unexg cnvexg wf1o eqid domss2 simp1d f1of1 ssv f1ss simp3d
      syl2anc jca f1eq1 coeq1 eqeq1d anbi12d spcedv ) ABDGZAEHZBFHZIZBJCUCZGZVT
      DKZUAALZMZNBJDUBBDOZUDZAOUEUFZUGZUHZLZUIZUJZGZWLDKZWCMZNCJWLVSWKJHZWLJHVS
      DJHZWJJHZWPVPABDPVQVRWQABDUKABDEFQULWIWFWJPVSWIJHZWFJHZWRWFWHUMVSWTWHJHWS
      VRVPWTVQBWEFUQUNZWGUOWFWHJJUPRXAWIWFWJJJQURDWJJJUSVIWKJUTSVSWMWOVSBWLOZWL
      GZXBJTWMVSBXBWLVAZXCVSXDAXBTZWOABDWLEFWLVBVCZVDBXBWLVESXBVFBXBJWLVGRVSXDX
      EWOXFVHVJVTWLMZWAWMWDWOBJVTWLVKXGWBWNWCVTWLDVLVMVNVO $.

    $( Weakening of ~ domssex2 to forget the functions in favor of dominance
       and equinumerosity.  (Contributed by Mario Carneiro, 7-Feb-2015.)
       (Revised by Mario Carneiro, 24-Jun-2015.) $)
    domssex $p |- ( A ~<_ B -> E. x ( A C_ x /\ B ~~ x ) ) $=
      ( vf cdom wbr cv wf1 wex cvv wcel wss cen brdomi reldom crn cres syl wceq
      wa brrelex2i wi c1st cdif cuni cpw csn cxp cun ccnv vex wf f1stres difexg
      adantl snex xpexg sylancl fex2 mp3an2i unexg sylancr cnvexg wf1o ccom cid
      rnexg w3a simpl cdm f1dm dmex eqeltrrdi adantr eqid domss2 syl3anc simp2d
      simpr simp1d f1oen3g syl2anc jca sseq2 breq2 anbi12d spcedv exlimiv sylc
      ex ) BCEFBCDGZHZDICJKZBAGZLZCWNMFZTZAIZBCDNBCEOUAWLWMWRUBDWLWMWRWLWMTZWQB
      WKUCCWKPZUDZBPUEUFZUGZUHZQZUIZUJZPZLZCXHMFZTAJXHWSXGJKZXHJKWSXFJKZXKWSWKJ
      KXEJKZXLDUKZXDXAXEULWSXDJKZXAJKZXMXAXCUMWSXPXCJKXOWMXPWLCWTJUNUOZXBUPXAXC
      JJUQURXQXDXAXEJJUSUTWKXEJJVAVBXFJVCRZXGJVGRWSXIXJWSCXHXGVDZXIXGWKVEVFBQSZ
      WSWLBJKZWMXSXIXTVHWLWMVIWLYAWMWLBWKVJJBCWKVKWKXNVLVMVNWLWMVSBCWKXGJJXGVOV
      PVQZVRWSXKXSXJXRWSXSXIXTYBVTCXHXGJWAWBWCWNXHSWOXIWPXJWNXHBWDWNXHCMWEWFWGW
      JWHWI $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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 opelxpd ralrimiva weq wb wrex simprd r19.21bi reu6
      xp2nd anim12dan reeanv pm4.38 ralimdv impcom reximi sylbir syl vex op1std
      ex com12 csbeq1d eqeq2d op2ndd anbi12d eqeq1 bibi12d ralxp nfv nfcv nfeq2
      nfan nfbi nfralw anbi2d opeq2 eqeq1d cbvralw anbi1d ralbidv bitrid bitr4i
      opeq1 eqeq2 bitrdi bibi2d 2ralbidv rexxp sylibr ralrimivva reubidv opeq12
      opth nfop syl2an cbvmpo opeq12d mpompt eqtr4i sylanbrc ) ABLUJZUBUCZHUDZC
      YFUEUCZIUDZUFZEGUGZQZLDFUGZRUAUJZYKSZLYNUHZUAYLRZYNYLBCDFHIUFZUIZUKAYMLYN
      AYFYNQZTZYHYJEGUUBYGDQZHEQZBDRZYHEQZUUAUUCAYFDFULUMAUUEUUAAUUEMUJZHSZBDUH
      ZMERZADEBDHUNZUKUUEUUJTJBMDEHUUKUUKUOUPUQZURUSUUDUUFBYGDBYHEBYGHUTVBBUJZY
      GSHYHEBYGHVAVCVDVEUUBYIFQZIGQZCFRZYJGQZUUAUUNAYFDFVNUMAUUPUUAAUUPNUJZISZC
      FUHZNGRZAFGCFIUNZUKUUPUVATKCNFGIUVBUVBUOUPUQZURUSUUOUUQCYIFCYJGCYIIUTVBCU
      JZYISIYJGCYIIVAVCVDVEVFVGAUUGYHSZUURYJSZTZLYNUHZNGRMERYRAUVHMNEGAUUGEQZUU
      RGQZTTZUVGLUAVHZVIZLYNRZUAYNVJZUVHUVKUUHUUSTZBOVHZCPVHZTZVIZCFRZBDRZPFVJZ
      ODVJZUVOUVKUUHUVQVIZBDRZODVJZUUSUVRVIZCFRZPFVJZTZUWDAUVIUWGUVJUWJAUVITUUI
      UWGAUUIMEAUUEUUJUULVKVLUUHBODVMUQAUVJTUUTUWJAUUTNGAUUPUVAUVCVKVLUUSCPFVMU
      QVOUWKUWFUWITZPFVJZODVJUWDUWFUWIOPDFVPUWMUWCODUWLUWBPFUWIUWFUWBUWIUWEUWAB
      DUWEUWIUWAUWEUWHUVTCFUWEUWHUVTUUHUUSUVQUVRVQWEVRWFVRVSVTVTWAWBUVNUWBUAOPD
      FUVNUVPUUMUVDUFZYOSZVIZCFRZBDRZYOOUJZPUJZUFZSZUWBUVNUUGBUWSHUDZSZUURCUWTI
      UDZSZTZUXAYOSZVIZPFRZODRUWRUVMUXILOPDFYFUXASZUVGUXGUVLUXHUXKUVEUXDUVFUXFU
      XKYHUXCUUGUXKBYGUWSHUWSUWTYFOWCZPWCZWDWGWHUXKYJUXEUURUXKCYIUWTIUWSUWTYFUX
      LUXMWIWGWHWJYFUXAYOWKWLWMUWQUXJBODUWQOWNUXIBPFBFWOUXGUXHBUXDUXFBBUUGUXCBU
      WSHUTWPUXFBWNWQUXHBWNWRWSUWQUUHUXFTZUUMUWTUFZYOSZVIZPFRUVQUXJUWPUXQCPFUWP
      PWNUXNUXPCUUHUXFCUUHCWNCUURUXECUWTIUTWPWQUXPCWNWRUVRUVPUXNUWOUXPUVRUUSUXF
      UUHUVRIUXEUURCUWTIVAWHWTUVRUWNUXOYOUVDUWTUUMXAXBWLXCUVQUXQUXIPFUVQUXNUXGU
      XPUXHUVQUUHUXDUXFUVQHUXCUUGBUWSHVAWHXDUVQUXOUXAYOUUMUWSUWTXHXBWLXEXFXCXGU
      XBUWPUVTBCDFUXBUWOUVSUVPUXBUWOUWNUXASUVSYOUXAUWNXIUUMUVDUWSUWTBWCCWCXRXJX
      KXLXFXMXNUVGLUAYNVMXNXOYQUVHUAMNEGYOUUGUURUFZSZYPUVGLYNUXSYPUXRYKSUVGYOUX
      RYKWKUUGUURYHYJMWCZNWCZXRXJXPWMXNLUAYNYLYKYTYTMNDFBUUGHUDZCUURIUDZUFZUILY
      NYKUNBCMNDFYSUYDMYSWONYSWOBUYBUYCBUUGHUTBUYCWOXSCUYBUYCCUYBWOCUURIUTXSBMV
      HHUYBSIUYCSYSUYDSCNVHBUUGHVACUURIVAHIUYBUYCXQXTYAMNLDFYKUYDYFUXRSZYHUYBYJ
      UYCUYEBYGUUGHUUGUURYFUXTUYAWDWGUYECYIUURIUUGUURYFUXTUYAWIWGYBYCYDUPYE $.
  $}

  $( Equinumerosity law for Cartesian product.  Proposition 4.22(b) of
     [Mendelson] p. 254.  (Contributed by NM, 24-Jul-2004.)  (Proof shortened
     by Mario Carneiro, 26-Apr-2015.) $)
  xpen $p |- ( ( A ~~ B /\ C ~~ D ) -> ( A X. C ) ~~ ( B X. D ) ) $=
    ( cen wbr cxp cdom cvv wcel relen brrelex1i endom xpdom1g syl2anr brrelex2i
    xpdom2g syl2an domtr syl2anc wa ensym syl sbth ) ABEFZCDEFZUAZACGZBDGZHFZUI
    UHHFZUHUIEFUGUHBCGZHFZULUIHFZUJUFCIJABHFUMUECDEKLABMABCINOUEBIJCDHFUNUFABEK
    PCDMCDBIQRUHULUISTUGUIADGZHFZUOUHHFZUKUFDIJBAHFZUPUECDEKPUEBAEFURABUBBAMUCB
    ADINOUEAIJDCHFZUQUFABEKLUFDCEFUSCDUBDCMUCDCAIQRUIUOUHSTUHUIUDT $.

  ${
    $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 wf f1of adantr adantl syl wceq
      vx vy cen wbr co bren exdistrv ccnv ovexd elmapi fco sylan f1ocnv fcod ex
      syl5 crn wfo f1ofo forn vex eqeltrrdi elmapd sylibrd id syl2anr cdm f1odm
      rnex dmex wb coass cid cres f1ococnv2 ad2antrr coeq1d adantrl fcoi2 eqtrd
      eqtr3id eqeq2d f1ococnv1 ad2antlr coeq2d fcoi1 eqtrid eqcom bitrdi bitr4d
      adantrr wf1 f1of1 simprl cocan1 syl3anc wfn ad2antll ffnd 3bitr3d syl2ani
      ffn cocan2 en3d exlimivv sylbir syl2anb ) ABUCUDABEGZHZEIZCDFGZHZFIZACJUE
      ZBDJUEZUCUDZCDUCUDABEUFCDFUFXJXMKXIXLKZFIEIXPXIXLEFUGXQXPEFXQUAUBXNXOXHUA
      GZLZXKUHZLZXHUHZUBGZXKLZLZMMXQACJUIXQBDJUIXQXRXNNZDBYAOZYAXONYFCAXROZXQYG
      XRACUJZXQYHYGXQYHKDCBXSXTXQABXHOZYHCBXSOZXIYJXLABXHPQCABXHXRUKULZXQDCXTOZ
      YHXQDCXTHZYMXLYNXICDXKUMRDCXTPSQUNZUOUPXQBDYAMMXQBXHUQZMXQABXHURZYPBTXIYQ
      XLABXHUSQABXHUTSXHEVAZVIVBXQDXKUQZMXQCDXKURZYSDTXLYTXICDXKUSRZCDXKUTSXKFV
      AZVIVBVCVDXQYCXONZCAYEOZYEXNNUUCDBYCOZXQUUDYCBDUJZXQUUEUUDXQUUEKCBAYBYDXQ
      BAYBOZUUEXQBAYBHZUUGXIUUHXLABXHUMQBAYBPSQUUEUUECDXKOZCBYDOZXQUUEVEXLUUIXI
      CDXKPRCDBYCXKUKVFZUNZUOUPXQACYEMMXQAXHVGZMXIUUMATXLABXHVHQXHYRVJVBXQCXKVG
      ZMXLUUNCTXICDXKVHRXKUUBVJVBVCVDYFXQYHUUEXRYETZYCYATZVKZUUCYIUUFXQYHUUEKZU
      UQXQUURKZXSXHYELZTZYDYAXKLZTZUUOUUPUUSUVAXSYDTZUVCUUSUUTYDXSUUSUUTXHYBLZY
      DLZYDXHYBYDVLUUSUVFVMBVNZYDLZYDUUSUVEUVGYDXIUVEUVGTXLUURABXHVOVPVQUUSUUJU
      VHYDTXQUUEUUJYHUUKVRCBYDVSSVTWAWBUUSUVCYDXSTUVDUUSUVBXSYDUUSUVBXSXTXKLZLZ
      XSXSXTXKVLUUSUVJXSVMCVNZLZXSUUSUVIUVKXSXLUVIUVKTXIUURCDXKWCWDWEUUSYKUVLXS
      TXQYHYKUUEYLWKCBXSWFSVTWGWBYDXSWHWIWJUUSABXHWLZYHUUDUVAUUOVKXIUVMXLUURABX
      HWMVPXQYHUUEWNXQUUEUUDYHUULVRCABXHXRYEWOWPUUSYTYCDWQZYADWQUVCUUPVKXQYTUUR
      UUAQUUEUVNXQYHDBYCXBWRUUSDBYAXQYHYGUUEYOWKWSCDXKYCYAXCWPWTUOXAXDXEXFXG $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Order-preserving property of set exponentiation.  Theorem 6L(c) of
       [Enderton] p. 149.  (Contributed by NM, 27-Jul-2004.)  (Revised by Mario
       Carneiro, 9-Mar-2013.) $)
    mapdom1 $p |- ( A ~<_ B -> ( A ^m C ) ~<_ ( B ^m C ) ) $=
      ( vx cdom wbr cvv wcel cmap co wa cv cen wss wex wb reldom adantl syl2anc
      c0 brrelex2i domeng syl ibi adantr simpl enrefg mapen syl2anr ovex simprr
      ad2antrr mapss ssdomg mpsyl endomtr exlimddv wn wceq elmapex simprd con3i
      eq0rdv 0dom eqbrtrdi pm2.61dan ) ABEFZCGHZACIJZBCIJZEFZVGVHKZADLZMFZVMBNZ
      KZVKDVGVPDOZVHVGVQVGBGHZVGVQPABEQUAZDABGUBUCUDUEVLVPKZVIVMCIJZMFZWAVJEFZV
      KVPVNCCMFZWBVLVNVOUFVHWDVGCGUGRAVMCCUHUIVJGHVTWAVJNZWCBCIUJZVTVRVOWEVGVRV
      HVPVSULVLVNVOUKVMBCGUMSWAVJGUNUOVIWAVJUPSUQVGVHURZKVITVJEWGVITUSVGWGDVIVM
      VIHZVHWHAGHVHVMACUTVAVBVCRVJWFVDVEVF $.
  $}

  ${
    $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 ovexd elmapi ffvelcdmda an32s ralrimiva fmpo sylib simp1 xpexg 3adant1
      syl elmapg syl2anc imbitrrid adantl fovcdm 3expa sylanl1 3adant3 ad2antrr
      wb fmpttd mpbird ovex simp3 sylancr sylibrd wfn elmapfn ad2antll adantlrl
      ex fnov 3adant2 simp1l2 simp1l1 syl3anc fvmpt2 fveq1d simp2 sylancl eqtrd
      fex2 mpoeq3dva eqtr4d nfmpt1 nfmpt nfeq2 fveq1 a1d ralrimi jctil mpoeq123
      nfcv eqeq2d syl5ibrcom feqmptd simprl sylan mpteq2dva nfmpo2 eqidd nfmpo1
      ad2antrl fvex ovmpt4g mp3an3 eqeq1d expcomd ralrimd mpteq12 syl6an impbid
      nfv oveq en3d ) ADIZBEIZCFIZUBZUAUEABJKZCJKZABCUCZJKZGHBCGLZHLZUALZMZMZUD
      ZHCGBYJYKUELZKZNZNZOOYEYFCJUFYEAYHJUFYLYGIZYOYIIZYEYHAYOPZYTYNAIZHCQZGBQU
      UBYTUUDGBYTYJBIZRUUCHCYTYKCIZUUEUUCYTUUFRZBAYJYMUUGYMYFIBAYMPZYTCYFYKYLYL
      YFCUGZUHYMABUGUPZUHUIUJUJGHBCYNAYOYOSZUKULYEYBYHOIZUUAUUBVFYBYCYDUMYCYDUU
      LYBBCEFUNUOAYHYODOUQURUSYEYPYIIZCYFYSPZYSYGIZYEUUMUUNYEUUMRZHCYRYFUUPUUFR
      ZYRYFIZBAYRPZUUQGBYQAUUPYHAYPPZUUFUUEYQAIZUUMUUTYEYPAYHUGUTUUTUUEUUFUVAUU
      TUUEUUFUVAYJYKABCYPVAVBUIVCVGZYEUURUUSVFZUUMUUFYBYCUVCYDABYRDEUQVDVEVHVGV
      QYEYFOIYDUUOUUNVFABJVIYBYCYDVJYFCYSOFUQVKVLYEYTUUMRZYLYSTZYPYOTZVFYEUVDRZ
      UVEUVFUVGUVFUVEYPGHBCYJYKYSMZMZUDZTUVGYPGHBCYQUDZUVJUVGYPYHVMZYPUVKTUUMUV
      LYEYTYPAYHVNVOGHBCYPVRULUVGGHBCUVIYQUVGUUEUUFUBZUVIYJYRMZYQUVMYJUVHYRUVMU
      UFYROIZUVHYRTUVGUUEUUFVJUVMUUSYCYBUVOUVGUUFUUSUUEYEUUMUUFUUSYTUVBVPVSYBYC
      YDUVDUUEUUFVTYBYCYDUVDUUEUUFWABAYREDWHWBHCYROYSYSSWCURWDUVMUUEYQOIUVNYQTU
      VGUUEUUFWEYJYKYPVIGBYQOYRYRSWCWFWGWIWJUVEYOUVJYPUVEBBTZCCTZYNUVITZHCQZRZG
      BQYOUVJTBSUVEUVTGBGYLYSGHCYRGCWSGBYQWKWLWMUVEUVTUUEUVEUVSUVQUVEUVRHCHYLYS
      HCYRWKWMUVEUVRUUFUVEYJYMUVHYKYLYSWNWDWOWPCSZWQWOWPGHBCYNBCUVIWRVKWTXAUVGU
      VEUVFYLHCGBYNNZNZTUVGYLHCYMNUWCUVGHCYFYLYTCYFYLPYEUUMUUIXIXBUVGHCYMUWBUVG
      UUFRGBAYMUVGYTUUFUUHYEYTUUMXCUUJXDXBXEWGUVFYSUWCYLUVFUVQYRUWBTZHCQYSUWCTU
      WAUVFUWDHCHYPYOGHBCYNXFWMUVFUVPUUFYQYNTZGBQUWDUVFBXGUVFUUFUWEGBGYPYOGHBCY
      NXHWMUUFGXSUVFUUEUUFUWEUUEUUFRUWEUVFYJYKYOKZYNTZUUEUUFYNOIUWGYJYMXJGHBCYN
      YOOUUKXKXLUVFYQUWFYNYJYKYPYOXTXMUSXNXOGBYQBYNXPXQWPHCYRCUWBXPVKWTXAXRVQYA
      $.
  $}

  ${
    $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 ) ) $=
        ( wcel cfv c1st wa wceq cxp cmap co cop ovex xpex cv wf ffvelcdm sylanb
        elmap xp1st fmptd sylibr c2nd xp2nd opelxpd ffvelcdmda 1st2nd2 ad2antlr
        syl sylib cmpt feqmptd simplr fveq1d cvv opex fvmpt2 mpan2 adantl eqtrd
        fveq2d fvex op1st eqtrdi mpteq2dva eqtrid eqtr4d op2nd opeq12d ad2antrr
        simpll simpr op1stg syl2anc sylan9eq op2ndg impbida en3i ) ABDEUAZFUBUC
        ZDFUBUCZEFUBUCZUAZGHUDZIWKFUBUEWMWNDFUBUEEFUBUEUFAUGZWLPZGHWMWNWRFDGUHG
        WMPZWRCFCUGZWQQZRQZDGWRWTFPZSZXAWKPZXBDPWRFWKWQUHZXCXEWKFWQDEJKUFZLUKZF
        WKWTWQUIUJZXADEULVAMUMDFGJLUKUNZWRFEHUHHWNPZWRCFXAUOQZEHXDXEXLEPXIXADEU
        PVANUMEFHKLUKUNZUQBUGZWOPZFWKIUHIWLPXOCFWTXNRQZQZWTXNUOQZQZUDZWKIXOXCSX
        QXSDEXOFDWTXPXOXPWMPFDXPUHXNWMWNULDFXPJLUKVBZURXOFEWTXRXOXRWNPFEXRUHXNW
        MWNUPEFXRKLUKVBZURUQOUMWKFIXGLUKUNWRXOSZWQITZXNWPTZYCYDSZXNXPXRUDZWPXOX
        NYGTWRYDXNWMWNUSUTYFXPGXRHYFXPCFXQVCZGXOXPYHTWRYDXOCFDXPYAVDUTYFGCFXBVC
        YHMYFCFXBXQYFXCSZXBXTRQXQYIXAXTRYIXAWTIQZXTYIWTWQIYCYDXCVEVFXCYJXTTZYFX
        CXTVGPYKXQXSVHCFXTVGIOVIVJVKVLZVMXQXSWTXPVNZWTXRVNZVOVPVQVRVSYFXRCFXSVC
        ZHXOXRYOTWRYDXOCFEXRYBVDUTYFHCFXLVCYONYFCFXLXSYIXLXTUOQXSYIXAXTUOYLVMXQ
        XSYMYNVTVPVQVRVSWAVLYCYESZWQCFXAVCZIYPCFWKWQYPWRXFWRXOYEWCXHVBZVDYPICFX
        TVCYQOYPCFXTXAYPXCSZXTXBXLUDZXAYSXQXBXSXLYPXCXQWTGQZXBYPWTXPGYPXPWPRQZG
        YPXNWPRYCYEWDZVMYPWSXKUUBGTWRWSXOYEXJWBZWRXKXOYEXMWBZGHWMWNWEWFVLVFXCXB
        VGPUUAXBTXARVNCFXBVGGMVIVJWGYPXCXSWTHQZXLYPWTXRHYPXRWPUOQZHYPXNWPUOUUCV
        MYPWSXKUUGHTUUDUUEGHWMWNWHWFVLVFXCXLVGPUUFXLTXAUOVNCFXLVGHNVIVJWGWAYSXE
        XAYTTYPFWKWTWQYRURXADEUSVAVSVQVRVSWIWJ $.
    $}

    $( 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 2fveq3 cbvmptv weq fveq2 opeq12d
      xpmapenlem ) GHIABCJCJKZGKZLZMLZNJCUEOLZNJCUCHKZMLZLZUCUHOLZLZPZNDEFJICUF
      IKZUDLZMLUCUNMUDQRJICUGUOOLUCUNOUDQRJICUMUNUILZUNUKLZPJISUJUPULUQUCUNUITU
      CUNUKTUARUB $.
  $}

  ${
    $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 c0 wceq wa cun cmap co cres cop cvv ovexd wf elmapi w3a cin cxp
      vy cv c1st cfv c2nd xpexd ssun1 fssres sylancl ssun2 opelxp simpl3 simpl1
      wss jca elmapd simpl2 anbi12d bitrid imbitrrid xp1st adantl syl simplr ex
      xp2nd fun2d unexg syl2anc sylibrd wb 1st2nd2 ad2antll adantrl res0 eqtr4i
      reseq2d 3eqtr4a fresaunres1 syl3anc fresaunres2 opeq12d eqtr4d syl5ibrcom
      reseq1 eqeq2d wfn ffn fnresdm ad2antrl eqcomd resex op1std op2ndd uneq12d
      3syl vex resundi eqtr4di impbid en3d ) ADHZBEHZCFHZUAZABUBZIJZKZGUDCABLZM
      NZCAMNZCBMNZUCZGUEZAOZXQBOZPZUDUEZUFUGZYAUHUGZLZQQXKCXLMRXKXNXOQQXKCAMRXK
      CBMRUIXQXMHZXTXPHZXKACXRSZBCXSSZKZYEYGYHYEXLCXQSZAXLUQYGXQCXLTZABUJXLCAXQ
      UKULYEYJBXLUQYHYKBAUMXLCBXQUKULURYFXRXNHZXSXOHZKXKYIXRXSXNXOUNXKYLYGYMYHX
      KCAXRFDXEXFXGXJUOZXEXFXGXJUPZUSXKCBXSFEYNXEXFXGXJUTZUSVAVBVCXKYAXPHZXLCYD
      SZYDXMHXKYQYRXKYQKZABCYBYCYSYBXNHZACYBSZYQYTXKYAXNXOVDVEYBCATVFZYSYCXOHZB
      CYCSZYQUUCXKYAXNXOVIVEYCCBTVFZXHXJYQVGVJVHXKCXLYDFQYNXKXEXFXLQHYOYPABDEVK
      VLUSVMXKYEYQKZXQYDJZYAXTJZVNXKUUFKZUUGUUHUUIUUHUUGYAYDAOZYDBOZPZJUUIYAYBY
      CPZUULYQYAUUMJXKYEYAXNXOVOVPUUIUUJYBUUKYCUUIUUAUUDYBXIOZYCXIOZJZUUJYBJXKY
      QUUAYEUUBVQZXKYQUUDYEUUEVQZUUIYBIOZYCIOZUUNUUOUUSIUUTYBVRYCVRVSUUIXIIYBXH
      XJUUFVGZVTUUIXIIYCUVAVTWAZABCYBYCWBWCUUIUUAUUDUUPUUKYCJUUQUURUVBABCYBYCWD
      WCWEWFUUGXTUULYAUUGXRUUJXSUUKXQYDAWHXQYDBWHWEWIWGUUIUUGUUHXQXQXLOZJUUIUVC
      XQYEUVCXQJZXKYQYEYJXQXLWJUVDYKXLCXQWKXLXQWLWSWMWNUUHYDUVCXQUUHYDXRXSLUVCU
      UHYBXRYCXSXRXSYAXQAGWTZWOZXQBUVEWOZWPXRXSYAUVFUVGWQWRXQABXAXBWIWGXCVHXD
      $.
  $}

  $( A cardinal power with exponent 2 is equivalent to a Cartesian product with
     itself.  (Contributed by Mario Carneiro, 31-May-2015.)  (Proof shortened
     by AV, 17-Jul-2022.) $)
  map2xp $p |- ( A e. V -> ( A ^m 2o ) ~~ ( A X. A ) ) $=
    ( wcel c2o cmap co c0 csn c1o cxp cen wbr cun cpr cvv wceq snex a1i syl2anc
    mapsnend df2o3 df-pr eqtri oveq2i cin id wn 1n0 neii elsni mto disjsn mpbir
    mapunen syl31anc eqbrtrid 0ex 1oex xpen entr ) ABCZADEFZAGHZEFZAIHZEFZJZKLV
    GAAJZKLZVBVHKLVAVBAVCVEMZEFZVGKDVJAEDGINVJUAGIUBUCUDVAVCOCZVEOCZVAVCVEUEGPZ
    VKVGKLVLVAGQRVMVAIQRVAUFZVNVAVNIVCCZUGVPIGPIGUHUIIGUJUKVCIULUMRVCVEAOOBUNUO
    UPVAVDAKLVFAKLVIVAAGBOVOGOCVAUQRTVAAIBOVOIOCVAURRTVDAVFAUSSVBVGVHUTS $.

  ${
    $d x A $.  $d x B $.  $d x C $.  $d x V $.  $d x W $.
    $( Order-preserving property of set exponentiation.  Theorem 6L(d) of
       [Enderton] p. 149.  (Contributed by NM, 23-Sep-2004.)  (Revised by Mario
       Carneiro, 30-Apr-2015.) $)
    mapdom2 $p |- ( ( A ~<_ B /\ -. ( A = (/) /\ C = (/) ) ) ->
                  ( C ^m A ) ~<_ ( C ^m B ) ) $=
      ( vx cdom wbr c0 wceq wa wn cvv wcel cmap wne simplr syl eqbrtrdi syl2anc
      co cen simpr oveq1d idd jctird mtod neqned map0b eqtrd ovex cv wss simpll
      0dom wex wb reldom brrelex2i ad2antrr domeng mpbid ad2antlr simprrl mapen
      enrefg cxp ovexd simprl wi difexd map0g simpl biimtrdi necon3d mpd xpdom3
      cdif syl3anc cun cin vex a1i disjdif mapunen syl31anc simprrr undif sylib
      ensymd oveq2d breqtrd domentr endomtr exlimdv adantlr pm2.61dane an32s ex
      expr reldmmap ovprc1 pm2.61d1 ) ABEFZAGHZCGHZIZJZIZCKLZCAMSZCBMSZEFZXGXHX
      KXBXHXFXKXBXHIZXFIZXKCGXMXDIZXIGXJEXNXIGAMSZGXNCGAMXMXDUAZUBXNAGNXOGHXNAG
      XNXCXEXLXFXDOXNXCXCXDXNXCUCXPUDUEUFAUGPUHXJCBMUIUMZQXLCGNZXKXFXLXRIZADUJZ
      TFZXTBUKZIZDUNZXKXSXBYDXBXHXRULXSBKLZXBYDUOXBYEXHXRABEUPUQZURDABKUSPUTXSY
      CXKDXLXRYCXKXLXRYCIZIZXICXTMSZTFZYIXJEFZXKYHCCTFZYAYJXHYLXBYGCKVDVAXLXRYA
      YBVBCCAXTVCRYHYIYICBXTVPZMSZVEZEFZYOXJTFYKYHYIKLYNKLYNGNZYPYHCXTMVFYHCYMM
      VFYHXRYQXLXRYCVGYHXHYMKLZXRYQVHXBXHYGOZYHBXTKXBYEXHYGYFURVIZXHYRIZYNGCGUU
      AYNGHXDYMGNZIXDCYMKKVJXDUUBVKVLVMRVNYIYNKKVOVQYHYOCXTYMVRZMSZXJTYHUUDYOYH
      XTKLZYRXHXTYMVSGHZUUDYOTFUUEYHDVTWAYTYSUUFYHXTBWBWAXTYMCKKKWCWDWHYHUUCBCM
      YHYBUUCBHXLXRYAYBWEXTBWFWGWIWJYIYOXJWKRXIYIXJWLRWRWMVNWNWOWPWQXHJXIGXJECA
      MWSWTXQQXA $.

    $( Set exponentiation dominates the base.  (Contributed by Mario Carneiro,
       30-Apr-2015.)  (Proof shortened by AV, 17-Jul-2022.) $)
    mapdom3 $p |- ( ( A e. V /\ B e. W /\ B =/= (/) ) -> A ~<_ ( A ^m B ) ) $=
      ( vx wcel c0 wne cmap co cdom wbr cv wex wa n0 w3a csn cen wceq ensymd wn
      simp1 simp3 mapsnend wss simp2 snssd ssdomg sylc vex simpl necon3ai ax-mp
      snnz mapdom2 sylancl endomtr syl2anc 3expia exlimdv biimtrid 3impia ) ACF
      ZBDFZBGHZAABIJZKLZVFEMZBFZENVDVEOZVHEBPVKVJVHEVDVEVJVHVDVEVJQZAAVIRZIJZSL
      VNVGKLZVHVLVNAVLAVICBVDVEVJUCVDVEVJUDZUEUAVLVMBKLZVMGTZAGTZOZUBZVOVLVEVMB
      UFVQVDVEVJUGVLVIBVPUHVMBDUIUJVMGHWAVIEUKUOVTVMGVRVSULUMUNVMBAUPUQAVNVGURU
      SUTVAVBVC $.

    $( If two sets are equinumerous, then their power sets are equinumerous.
       Proposition 10.15 of [TakeutiZaring] p. 87.  (Contributed by NM,
       29-Jan-2004.)  (Revised by Mario Carneiro, 9-Apr-2015.) $)
    pwen $p |- ( A ~~ B -> ~P A ~~ ~P B ) $=
      ( cen wbr cpw c2o cmap cvv wcel relen brrelex1i pw2eng syl com 2onn elexi
      co enref entr syl2anc mapen mpan brrelex2i ensym 3syl ) ABCDZAEZFAGQZCDZU
      HBEZCDZUGUJCDUFAHIUIABCJKAHLMUFUHFBGQZCDZULUJCDZUKFFCDUFUMFFNOPRFFABUAUBU
      FBHIUJULCDUNABCJUCBHLUJULUDUEUHULUJSTUGUHUJST $.
  $}

  ${
    $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 $.
  $}

  ${
    $d x y A $.  $d x y V $.
    limenpsi.1 $e |- Lim A $.
    $( A limit ordinal is equinumerous to a proper subset of itself.
       (Contributed by NM, 30-Oct-2003.)  (Revised by Mario Carneiro,
       16-Nov-2014.) $)
    limenpsi $p |- ( A e. V -> A ~~ ( A \ { (/) } ) ) $=
      ( vx vy wcel c0 csn cdif cdom wbr cv csuc wb ax-mp con0 wceq ordelon mpan
      cvv cen difexg wne wlim limsuc biimpi nsuceq0 sylanblrc word limord suc11
      eldifsn syl2an dom3 mpdan wss difss ssdomg mpi sbth syl2anc ) ABFZAAGHZIZ
      JKZVDAJKZAVDUAKVBVDTFVEAVCBUBDEAVDDLZMZELZMZBTVGAFZVHAFZVHGUCVHVDFVKVLAUD
      ZVKVLNCAVGUEOUFVGUGVHAGULUHVKVGPFZVIPFZVHVJQVGVIQNVIAFZAUIZVKVNVMVQCAUJOZ
      AVGRSVQVPVOVRAVIRSVGVIUKUMUNUOVBVDAUPVFAVCUQVDABURUSAVDUTVA $.
  $}

  ${
    limensuci.1 $e |- Lim A $.
    $( A limit ordinal is equinumerous to its successor.  (Contributed by NM,
       30-Oct-2003.) $)
    limensuci $p |- ( A e. V -> A ~~ suc A ) $=
      ( wcel csn cdif cun csuc cen wbr limenpsi ensymd cvv 0ex en2sn wceq ax-mp
      c0 cin mpbi mpan wa disjdifr wn word wlim limord ordirr disjsn mpbir unen
      mpanr12 syl2anc wss 0ellim snss undif uncom eqtr3i df-suc 3brtr4g ) ABDZA
      REZFZVCGZAAEZGZAAHIVBVDAIJZVCVFIJZVEVGIJZVBAVDABCKLRMDVBVINRAMBOUAVHVIUBV
      DVCSRPAVFSRPZVJVCAUCVKAADUDZAUEZVLAUFZVMCAUGQAUHQAAUIUJVDAVCVFUKULUMVCVDG
      ZAVEVCAUNZVOAPRADZVPVNVQCAUOQRANUPTVCAUQTVCVDURUSAUTVA $.
  $}

  $( A limit ordinal is equinumerous to its successor.  (Contributed by NM,
     30-Oct-2003.) $)
  limensuc $p |- ( ( A e. V /\ Lim A ) -> A ~~ suc A ) $=
    ( wlim wcel csuc cen wbr wi con0 cif wceq eleq1 suceq breq12d imbi12d limeq
    id limon elimhyp limensuci dedth impcom ) ACZABDZAAEZFGZUCUDUFHUCAIJZBDZUGU
    GEZFGZHAIAUGKZUDUHUFUJAUGBLUKAUGUEUIFUKQAUGMNOUGBUCUGCICAIAUGPIUGPRSTUAUB
    $.

  ${
    $d x y A $.
    $( Any infinite ordinal is equinumerous to its successor.  Exercise 7 of
       [TakeutiZaring] p. 88.  Proved without the Axiom of Infinity.
       (Contributed by NM, 30-Oct-2003.)  (Revised by Mario Carneiro,
       13-Jan-2013.) $)
    infensuc $p |- ( ( A e. On /\ _om C_ A ) -> A ~~ suc A ) $=
      ( vx vy con0 wcel com wss wa csuc cen wbr wceq cvv wi cv id suceq breq12d
      csn vex onprc eleq1 mtbiri ssexg nsyl3 omon ori nsyl2 weq limom limensuci
      ancoms sucex en2sn mp2an cin c0 wn word eloni ordirr disjsn sylibr onsucb
      wel syl 3imtr4i jca unen df-suc 3brtr4g ex syl5 mpan2 com12 ad2antrr wlim
      cun wral limensuc mpan a1d tfindsg exp31 com23 imp mpd ) ADEZFAGZHZFDEZAA
      IZJKZWJFDLZWKWNFMEZWJWNWODMEUAFDMUBUCWIWHWOFADUDULUEWKWNUFUGUHWHWIWKWMNWH
      WKWIWMWHWKWIWMBOZWPIZJKZFFIZJKCOZWTIZJKZXAXAIZJKZWMBCAFWPFLZWPFWQWSJXEPWP
      FQRBCUIZWPWTWQXAJXFPWPWTQRWPXALZWPXAWQXCJXGPWPXAQRWPALZWPAWQWLJXHPWPAQRFD
      UJUKWTDEZXBXDNWKFWTGZXBXIXDXBWTSZXASZJKZXIXDNWTMEXAMEXMCTZWTXNUMWTXAMMUNU
      OXIWTXKUPUQLZXAXLUPUQLZHZXBXMHZXDXIXOXPXICCVEURZXOXIWTUSXSWTUTWTVAVFWTWTV
      BVCXADEZXAXAEURZXIXPXTXAUSYAXAUTXAVAVFWTVDXAXAVBVGVHXRXQXDXRXQHWTXKVRXAXL
      VRXAXCJWTXAXKXLVIWTVJXAVJVKVLVMVNVOVPWPVQZWKHFWPGZHWRXJXBNCWPVSYBWRWKYCWP
      MEYBWRBTWPMVTWAVPWBWCWDWEWFWG $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Finite sets
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Lemma for ~ rexdif1en and ~ dif1en .  (Contributed by BTernaryTau,
     18-Aug-2024.)  Generalize to all ordinals and add a sethood requirement to
     avoid ~ ax-un .  (Revised by BTernaryTau, 5-Jan-2025.) $)
  dif1enlem $p |- ( ( ( F e. V /\ A e. W /\ M e. On ) /\
      F : A -1-1-onto-> suc M ) -> ( A \ { ( `' F ` M ) } ) ~~ M ) $=
    ( wcel con0 wf1o cfv csn cdif cres wa wfo adantr wceq wb mpbird c0 cvv csuc
    w3a ccnv cen wbr sucidg wfun dff1o3 simprbi f1ofo wfn f1ofn fnresdm 3syl wf
    foeq1 wne f1ocnvdm f1ocnvfv2 snidg adantl eqeltrd fressnfv biimp3ar syl3anc
    cin disjsn con2bii sylib fnresdisj syl neqned foconst syl2anc resdif sylan2
    wn mtbid word eloni orddif f1oeq3d ancoms 3ad2antl3 difexg f1oen4g syl3anl1
    resexg syl3anl2 syldan ) BDFZAEFZCGFZUBACUAZBHZACBUCZIZJZKZCBWSLZHZWSCUDUEZ
    WMWKWOXAWLWOWMXAWOWMMXAWSWNCJZKZWTHZWMWOCWNFZXECGUFWOXFMZWPUGZAWNBALZNZWRXC
    BWRLZNZXEWOXHXFWOAWNBNZXHAWNBUHUIOWOXJXFWOXJXMAWNBUJWOBAUKZXIBPXJXMQAWNBULZ
    ABUMAWNXIBUPUNROXGWRXCXKUOZXKSUQXLXGXNWQAFZWQBIZXCFZXPWOXNXFXOOAWNCBURZXGXR
    CXCAWNCBUSXFCXCFWOCWNUTVAVBXNXQXPXSAWQXCBVCVDVEXGXKSXGAWRVFSPZXKSPZXGXQYAVQ
    XTYAXQAWQVGVHVIWOYAYBQZXFWOXNYCXOAWRBVJVKOVRVLWRCXKVMVNAWRWNXCBVOVEVPWMXAXE
    QWOWMCXDWSWTWMCVSCXDPCVTCWAVKWBVARWCWDWLWKWSTFZWMXAXBAWREWEWKWTTFYDWMXAXBBW
    SDWHWSCWTTTGWFWGWIWJ $.
  $( $j usage 'dif1enlem' avoids 'ax-pow' 'ax-un'; $)

  $( Obsolete version of ~ dif1enlem as of 5-Jan-2025.  (Contributed by
     BTernaryTau, 18-Aug-2024.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  dif1enlemOLD $p |- ( ( F e. V /\ M e. _om /\ F : A -1-1-onto-> suc M ) ->
      ( A \ { ( `' F ` M ) } ) ~~ M ) $=
    ( wcel com wf1o cfv csn cdif cres wa wfo adantr wb mpbird c0 adantl syl3anc
    wceq csuc w3a ccnv cen wbr simp1 sucidg wfun dff1o3 simprbi f1ofo wfn f1ofn
    fnresdm foeq1 wf wne f1ocnvdm f1ocnvfv2 snidg eqeltrd fressnfv biimp3ar cin
    3syl disjsn con2bii sylib fnresdisj syl mtbid neqned foconst syl2anc resdif
    wn sylan2 word nnord orddif f1oeq3d ancoms 3adant1 cvv resexg f1oen3g sylan
    ) BDEZCFEZACUAZBGZUBWHACBUCZHZIZJZCBWOKZGZWOCUDUEZWHWIWKUFWIWKWQWHWKWIWQWKW
    ILWQWOWJCIZJZWPGZWIWKCWJEZXACFUGWKXBLZWLUHZAWJBAKZMZWNWSBWNKZMZXAWKXDXBWKAW
    JBMZXDAWJBUIUJNWKXFXBWKXFXIAWJBUKWKBAULZXEBTXFXIOAWJBUMZABUNAWJXEBUOVEPNXCW
    NWSXGUPZXGQUQXHXCXJWMAEZWMBHZWSEZXLWKXJXBXKNAWJCBURZXCXNCWSAWJCBUSXBCWSEWKC
    WJUTRVAXJXMXLXOAWMWSBVBVCSXCXGQXCAWNVDQTZXGQTZXCXMXQVPXPXQXMAWMVFVGVHWKXQXR
    OZXBWKXJXSXKAWNBVIVJNVKVLWNCXGVMVNAWNWJWSBVOSVQWIWQXAOWKWICWTWOWPWICVRCWTTC
    VSCVTVJWARPWBWCWHWPWDEWQWRBWODWEWOCWPWDWFWGVN $.

  ${
    $d A f x $.  $d M f x $.
    $( If a set is equinumerous to a nonzero ordinal, then there exists an
       element in that set such that removing it leaves the set equinumerous to
       the predecessor of that ordinal.  (Contributed by BTernaryTau,
       26-Aug-2024.)  Generalize to all ordinals and avoid ~ ax-un .  (Revised
       by BTernaryTau, 5-Jan-2025.) $)
    rexdif1en $p |- ( ( M e. On /\ A ~~ suc M ) ->
        E. x e. A ( A \ { x } ) ~~ M ) $=
      ( vf con0 wcel csuc cen wbr cv csn cdif wrex wi cvv encv simpld wa ancoms
      sylan wf1o wex wb breng syl ibi cfv sucidg f1ocnvdm adantll vex dif1enlem
      ccnv mp3anl1 wceq sneq difeq2d breq1d rspcev syl2anc exlimdv syl5 syldbl2
      ex ) CEFZBCGZHIZBAJZKZLZCHIZABMZVGVEVGVLNZVGBOFZVEVMVGVNVFOFZBVFPZQVGBVFD
      JZUAZDUBZVNVERZVLVGVSVGVNVORVGVSUCVPBVFDOOUDUEUFVTVRVLDVTVRVLVTVRRCVQUMUG
      ZBFZBWAKZLZCHIZVLVEVRWBVNVECVFFZVRWBCEUHVRWFWBBVFCVQUISTUJVQOFVNVEVRWEDUK
      BVQCOOULUNVKWEAWABVHWAUOZVJWDCHWGVIWCBVHWAUPUQURUSUTVDVAVBTSVC $.
    $( $j usage 'rexdif1en' avoids 'ax-pow' 'ax-un'; $)
  $}

  ${
    $d A f x $.  $d M f x $.
    $( Obsolete version of ~ rexdif1en as of 5-Jan-2025.  (Contributed by
       BTernaryTau, 26-Aug-2024.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    rexdif1enOLD $p |- ( ( M e. _om /\ A ~~ suc M ) ->
        E. x e. A ( A \ { x } ) ~~ M ) $=
      ( vf csuc cen wbr com wcel cv wf1o wex csn cdif wrex bren 19.42v ccnv cvv
      wa cfv sucidg f1ocnvdm ancoms sylan dif1enlemOLD mp3an1 wceq sneq difeq2d
      vex breq1d rspcev syl2anc exlimiv sylbir sylan2b ) BCEZFGCHIZBURDJZKZDLZB
      AJZMZNZCFGZABOZBURDPUSVBTUSVATZDLVGUSVADQVHVGDVHCUTRUAZBIZBVIMZNZCFGZVGUS
      CURIZVAVJCHUBVAVNVJBURCUTUCUDUEUTSIUSVAVMDUKBUTCSUFUGVFVMAVIBVCVIUHZVEVLC
      FVOVDVKBVCVIUIUJULUMUNUOUPUQ $.
  $}

  ${
    $d A f $.  $d M f $.  $d X f $.
    $( If a set ` A ` is equinumerous to the successor of an ordinal ` 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.)  Avoid ~ ax-pow .  (Revised by BTernaryTau, 26-Aug-2024.)
       Generalize to all ordinals.  (Revised by BTernaryTau, 6-Jan-2025.) $)
    dif1en $p |- ( ( M e. On /\ A ~~ suc M /\ X e. A ) ->
        ( A \ { X } ) ~~ M ) $=
      ( vf cen wbr wcel con0 csn cdif w3a cvv wf1o wi wa ccnv cfv cpr cop jca
      csuc simp1 encv simpld 3anim1i cv wex bren cres wceq wfun sucidg f1ocnvdm
      cun 3adant2 f1ofvswap syld3an3 wb f1ocnvfv2 opeq2d preq1d f1oeq1d syl3an3
      uneq2d mpbid 3adant3r1 f1ofun opex prid1 elun2 ax-mp funopfv mpi f1ocnvfv
      simpr2 syl2anc mpd sneqd difeq2d simpr1 3simpc anim2i 3anass sylibr simpl
      3syl simpr3 simpr vex prex unex dif1enlem mp3anl1 sylan2 eqbrtrrd exlimiv
      resex ex sylbi sylc 3comr ) ABUAZEFZCAGZBHGZACIZJZBEFZXCXDXEKXCALGZXDXEKZ
      XHXCXDXEUBXCXIXDXEXCXIXBLGAXBUCUDUEXCAXBDUFZMZDUGXJXHNZAXBDUHXLXMDXLXJXHX
      LXJOZABXKACBXKPQZRJZUIZCBSZXOCXKQSZRZUNZPQZIZJZXGBEXNYCXFAXNYBCXNCYAQBUJZ
      YBCUJZXNAXBYAMZYAUKZYEXLXDXEYGXIXEXLXDBXBGZYGBHULXLXDYIKAXBXQCXOXKQZSZXSR
      ZUNZMZYGXLXDYIXOAGZYNXLYIYOXDAXBBXKUMUOAXBXKCXOUPUQXLYIYNYGURXDXLYIOZAXBY
      MYAYPYLXTXQYPYKXRXSYPYJBCAXBBXKUSUTVAVDVBUOVEVCZVFZAXBYAVGYHXRYAGZYEXRXTG
      YSXRXSCBVHVIXRXTXQVJVKCBYAVLVMWFXNYGXDYEYFNYRXLXIXDXEVOAXBCBYAVNVPVQVRVSX
      NXIXLXDXEKZOZXIXEOZYTOYDBEFZXNXIYTXLXIXDXEVTXNXLXDXEOZOYTXJUUDXLXIXDXEWAW
      BXLXDXEWCWDTUUAUUBYTUUAXIXEXIYTWEXIXLXDXEWGTXIYTWHTYTUUBYGUUCYQYALGXIXEYG
      UUCXQXTXKXPDWIWQXRXSWJWKAYABLLWLWMWNWFWOWRWPWSWTXA $.
    $( $j usage 'dif1en' avoids 'ax-pow'; $)
  $}

  $( If a set ` A ` is equinumerous to the successor of a natural number
     ` M ` , then ` A ` with an element removed is equinumerous to ` M ` .  See
     also ~ dif1ennnALT .  (Contributed by BTernaryTau, 6-Jan-2025.) $)
  dif1ennn $p |- ( ( M e. _om /\ A ~~ suc M /\ X e. A ) ->
      ( A \ { X } ) ~~ M ) $=
    ( com wcel con0 csuc cen wbr csn cdif nnon dif1en syl3an1 ) BDEBFEABGHICAEA
    CJKBHIBLABCMN $.
  $( $j usage 'dif1ennn' avoids 'ax-pow'; $)

  ${
    $d A f $.  $d M f $.  $d X f $.
    $( Obsolete version of ~ dif1en as of 6-Jan-2025.  (Contributed by Jeff
       Madsen, 2-Sep-2009.)  (Revised by Stefan O'Rear, 16-Aug-2015.)  Avoid
       ~ ax-pow .  (Revised by BTernaryTau, 26-Aug-2024.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    dif1enOLD $p |- ( ( M e. _om /\ A ~~ suc M /\ X e. A ) ->
        ( A \ { X } ) ~~ M ) $=
      ( vf cen wbr wcel com csn cdif wf1o wex w3a 3anass ccnv cfv cpr cop cun
      wa csuc bren 19.41v exbii 3bitr4i cres wceq sucidg wfun 3adant2 f1ofvswap
      cv f1ocnvdm syld3an3 wb f1ocnvfv2 opeq2d preq1d uneq2d f1oeq1d mpbid opex
      f1ofun prid1 elun2 ax-mp funopfv mpi 3syl wi f1ocnvfv syl2anc mpd syl3an3
      simp2 sneqd difeq2d cvv vex resex prex unex dif1enlemOLD mp3an2i eqbrtrrd
      simp3 exlimiv sylbir syl3an1b 3comr ) ABUAZEFZCAGZBHGZACIZJZBEFZWLAWKDULZ
      KZDLZWMWNWQAWKDUBWTWMWNMZWSWMWNMZDLZWQWSWMWNTZTZDLWTXDTXCXAWSXDDUCXBXEDWS
      WMWNNUDWTWMWNNUEXBWQDXBABWRACBWROPZQJZUFZCBRZXFCWRPRZQZSZOPZIZJZWPBEXBXNW
      OAXBXMCWNWSWMBWKGZXMCUGZBHUHZWSWMXPMZCXLPBUGZXQXSAWKXLKZXLUIZXTXSAWKXHCXF
      WRPZRZXJQZSZKZYAWSWMXPXFAGZYGWSXPYHWMAWKBWRUMUJAWKWRCXFUKUNWSXPYGYAUOWMWS
      XPTZAWKYFXLYIYEXKXHYIYDXIXJYIYCBCAWKBWRUPUQURUSUTUJVAZAWKXLVCYBXIXLGZXTXI
      XKGYKXIXJCBVBVDXIXKXHVEVFCBXLVGVHVIXSYAWMXTXQVJYJWSWMXPVOAWKCBXLVKVLVMVNV
      PVQXLVRGXBWNYAXOBEFXHXKWRXGDVSVTXIXJWAWBWSWMWNWFWNWSWMXPYAXRYJVNAXLBVRWCW
      DWEWGWHWIWJ $.
  $}

  ${
    $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 dif1ennn 3expa vex breq1 imbi12d spcv syl5com ralrimdva imp
      difexi an32s 3impa sylc alrimdv cbvalvw imbitrrdi finds1 19.21bi rexlimiv
      3exp vtoclga ) AEFIUBMFUCZUBQWIPUCZRSZPTUDAPWIUEWKAPTWJTQWKAUFZFWLFUGWIUL
      RSZAUFZFUGWIUAUCZRSZAUFZFUGZWIWOUHZRSZAUFZFUGZPUAWJULUMZWLWNFXCWKWMAWJULW
      IRUIUJUKWJWOUMZWLWQFXDWKWPAWJWOWIRUIUJUKWJWSUMZWLXAFXEWKWTAWJWSWIRUIUJUKW
      NFWMWIULUMZAWIUNXFABNJUOUPUQWOTQZWRGUCZWSRSZDUFZGUGXBXGWRXJGXGWRXIDXGWRXI
      URXHUBQZCHXHVEZDXGXIXKWRXGXIUSZXHWJRSZPTUDZXKXGWSTQXIXOWOUTXNXIPWSTWJWSXH
      RUIVAVBPXHUEVCVDXGWRXIXLXGXIWRXLXMWRXLXMWRCHXHXMHUCZXHQZUSXHXPVFZVGZWORSZ
      WRCXGXIXQXTXHWOXPVHVIWQXTCUFFXSXHXRGVJVQWIXSUMWPXTACWIXSWORVKKVLVMVNVOVPV
      RVSOVTWGWAXAXJFGWIXHUMWTXIADWIXHWSRVKLVLWBWCWDWEWFUPWH $.
    $( $j usage 'findcard' avoids 'ax-pow'; $)
  $}

  ${
    $d ch x $.  $d th x $.  $d ta x $.  $d A x $.  $d ph v w y z $.
    $d v w x 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.)  Avoid ~ ax-pow .  (Revised by
       BTernaryTau, 26-Aug-2024.) $)
    findcard2 $p |- ( A e. Fin -> ta ) $=
      ( vw cen wbr wi wceq vv cfn cv wcel com wrex isfi wal breq2 imbi1d albidv
      c0 csuc en0 mpbiri sylbi ax-gen wsbc wa csn cdif con0 rexdif1en sylan wss
      cun snssi uncom undif biimpi eqtrid vex difexi breq1 anbi2d uneq1 sbceq1d
      nnon imbi2d imbi12d spvv rspe sylibr pm2.27 sylsyld syl5 vsnex unex sbcie
      adantl imbitrrdi vtocl dfsbcq imbitrid 3syl expd com12 rexlimdv adantr ex
      mpd com23 alrimdv nfv nfsbc1v nfim sbceq1a cbvalv1 finds1 19.21bi vtoclga
      rexlimiv ) AEFIUBMFUCZUBUDXMPUCZQRZPUEUFAPXMUGXOAPUEXNUEUDXOASZFXPFUHXMUL
      QRZASZFUHXMUAUCZQRZASZFUHZXMXSUMZQRZASZFUHZPUAXNULTZXPXRFYGXOXQAXNULXMQUI
      UJUKXNXSTZXPYAFYHXOXTAXNXSXMQUIUJUKXNYCTZXPYEFYIXOYDAXNYCXMQUIUJUKXRFXQXM
      ULTZAXMUNYJABNJUOUPUQXSUEUDZYBXNYCQRZAFXNURZSZPUHYFYKYBYNPYKYLYBYMYKYLYBY
      MSZYKYLUSXNHUCZUTZVAZXSQRZHXNUFZYOYKXSVBUDYLYTXSVRHXNXSVCVDYKYTYOSYLYKYSY
      OHXNYPXNUDZYKYSYOSUUAYKYSYOUUAYQXNVEZYRYQVFZXNTZYKYSUSZYOSYPXNVGUUBUUCYQY
      RVFZXNYRYQVHUUBUUFXNTYQXNVIVJVKUUEYBAFUUCURZSZUUDYOYKGUCZXSQRZUSZYBAFUUIY
      QVFZURZSZSUUEUUHSGYRXNYQPVLVMUUIYRTZUUKUUEUUNUUHUUOUUJYSYKUUIYRXSQVNVOUUO
      UUMUUGYBUUOAFUULUUCUUIYRYQVPVQVSVTUUKYBDUUMYBUUJCSZUUKDYAUUPFGXMUUITXTUUJ
      ACXMUUIXSQVNKVTWAUUKUUIUBUDZUUPCDUUKUUJUAUEUFUUQUUJUAUEWBUAUUIUGWCUUJUUPC
      SYKUUJCWDWJOWEWFADFUULUUIYQGVLHWGWHLWIWKWLUUDUUGYMYBAFUUCXNWMVSWNWOWPWQWR
      WSXAWTXBXCYEYNFPYEPXDYLYMFYLFXDAFXNXEXFXMXNTYDYLAYMXMXNYCQVNAFXNXGVTXHWKX
      IXJXLUPXK $.
    $( $j usage 'findcard2' avoids 'ax-pow'; $)
  $}

  ${
    $d A x y z $.  $d ch x $.  $d ph y z $.  $d ps x $.  $d ta x $.  $d th x $.
    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 ) $=
      ( cv wcel wceq cun sylib cfn wi wn ex csn wa wex wb snssi ssequn1 eqtr3di
      wss uncom vex eqvinc bicomd sylan9bb exlimiv biimpd pm2.61d2 findcard2
      syl ) ABCDEFGHIJKLMNGPZUAQZHPZVCQZCDUBZVDVFUCVGOUDVFCDVFFPZVCRZVHVCVEUEZS
      ZRZUFZFUGZCDUHZVFVCVKRVNVFVJVCSZVCVKVFVJVCULVPVCRVEVCUIVJVCUJTVJVCUMUKFVC
      VKGUNUOTVMVOFVICAVLDVIACKUPLUQURVBUSUTVA $.
  $}

  ${
    $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 .  (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 $.
    $( Natural numbers are finite sets.  (Contributed by Stefan O'Rear,
       21-Mar-2015.)  Avoid ~ ax-pow .  (Revised by BTernaryTau,
       23-Sep-2024.) $)
    nnfi $p |- ( A e. _om -> A e. Fin ) $=
      ( vx com wcel cv cen wbr wrex cfn enrefnn breq2 rspcev mpdan isfi sylibr
      ) ACDZABEZFGZBCHZAIDPAAFGZSAJRTBACQAAFKLMBANO $.
    $( $j usage 'nnfi' avoids 'ax-pow'; $)
  $}

  ${
    $d B w x $.  $d w x y z $.  $d A w x z $.
    $( A proper subset of a natural number is equinumerous to some smaller
       number.  Lemma 6F of [Enderton] p. 137.  (Contributed by NM,
       22-Jun-1998.)  (Revised by Mario Carneiro, 16-Nov-2014.)  Avoid
       ~ ax-pow .  (Revised by BTernaryTau, 31-Jul-2024.) $)
    pssnn $p |- ( ( A e. _om /\ B C. A ) -> E. x e. A B ~~ x ) $=
      ( vw vz vy com wcel wpss wa cen wbr wrex wss wi wal c0 wceq imbi12d wn cv
      cvv pssss ssexg sylan ancoms psseq2 rexeq albidv weq npss0 pm2.21i ax-gen
      csuc nfv nfa1 wel elequ1 biimpcd con3d adantl sseld elsuci ord con1d syl6
      imp impancom ssrdv anim1i dfpss2 sylibr elelsuc reximi2 imim12i exp4c sps
      syld com4t csn cdif anidm ssdif word orddif syl sseq2d imbitrrid syl5 wex
      nnord pssnel eleq2 eldifi biimtrrdi eleq1a sylan9r adantr pm2.61d expimpd
      ex exlimdv im2anan9r biimtrrid imbitrrdi psseq1 breq1 rexbidv cbvalvw vex
      difexi spcv sylbi sylan9 ordsucelsuc biimpd adantrd elnn cun cin cop wf1o
      snex f1osn f1oen3g mp2an jctr orddisj disjdifr unen syl2an difsnid eqcomd
      jctil df-suc a1i breq12d sylan2i breq2 rspcev exp4d com24 df-rex cbvrexvw
      imp4b jcad 3imtr4g eleq1w pm5.32i eqelsuc enrefnn sylbir pm2.43d pm2.61ii
      expl 2a1d alrimd finds spcgv com3l mpd ) BGHZCBIZJCUBHZCAUAZKLZABMZUVCUVB
      UVDUVCCBNUVBUVDCBUCCBGUDUEUFUVBUVCUVDUVGOUVDUVBUVCUVGUVBDUAZBIZUVHUVEKLZA
      BMZOZDPZUVDUVCUVGOZUVHEUAZIZUVJAUVOMZOZDPUVHQIZUVJAQMZOZDPUVHFUAZIZUVJAUW
      BMZOZDPZUVHUWBUNZIZUVJAUWGMZOZDPUVMEFBUVOQRZUVRUWADUWKUVPUVSUVQUVTUVOQUVH
      UGUVJAUVOQUHSUIEFUJZUVRUWEDUWLUVPUWCUVQUWDUVOUWBUVHUGUVJAUVOUWBUHSUIUVOUW
      GRZUVRUWJDUWMUVPUWHUVQUWIUVOUWGUVHUGUVJAUVOUWGUHSUIUVOBRZUVRUVLDUWNUVPUVI
      UVQUVKUVOBUVHUGUVJAUVOBUHSUIUWADUVSUVTUVHUKULUMUWBGHZUWFUWJDUWODUOUWEDUPU
      WOUWFUWJFDUQZDFUJZUWOUWFJZUWJOZUWRUWHUWPTZUWQTZUWIUWFUWHUWTUXAUWIOOOZUWOU
      WEUXBDUWEUWHUWTUXAUWIUWHUWTJZUXAJZUWCUWDUWIUXDUVHUWBNZUXAJUWCUXCUXEUXAUXC
      EUVHUWBUWHEDUQZUWTEFUQZUWHUXFJUWTUWLTZUXGUXFUWTUXHOUWHUXFUWLUWPUWLUXFUWPE
      FDURUSUTVAUWHUXFUXHUXGOZUWHUXFUVOUWGHZUXIUWHUVHUWGUVOUVHUWGUCZVBUXJUXGUWL
      UXJUXGUWLUVOUWBVCVDZVEVFVGVRVHVIVJUVHUWBVKVLUVJUVJAUWBUWGAFUQZUVEUWGHUVJU
      VEUWBVMVJVNVOVPVQVAVSUWPUWOUWFUWJUWPUWOJZUWFJUWHUVHUWBVTZWAZUVEKLZAUWBMZU
      WIUXNUWHUXPUWBIZUWFUXRUXNUWHUXPUWBNZUXPUWBRZTZJZUXSUWHUWHUWHJUXNUYCUWHWBU
      WOUWHUXTUWPUWHUYBUWHUVHUWGNZUWOUXTUXKUYDUXTUWOUXPUWGUXOWAZNUVHUWGUXOWCUWO
      UWBUYEUXPUWOUWBWDZUWBUYERUWBWKZUWBWEWFWGWHWIUWHUXJUXFTZJZEWJUWPUYBEUVHUWG
      WLUWPUYIUYBEUWPUXJUYHUYBUWPUXJJZUYAUXFUYJUYAUXFUYJUYAJUXGUXFUYAUXGUXFOUYJ
      UYAUXGUVOUXPHUXFUXPUWBUVOWMUVOUVHUXOWNWOVAUYJUXGTZUXFOUYAUXJUYKUWLUWPUXFU
      XLUWBUVHUVOWPWQWRWSXAUTWTXBWIXCXDUXPUWBVKXEUWFUVOUWBIZUVOUVEKLZAUWBMZOZEP
      UXSUXROZUWEUYODEDEUJZUWCUYLUWDUYNUVHUVOUWBXFUYQUVJUYMAUWBUVHUVOUVEKXGXHSX
      IUYOUYPEUXPUVHUXODXJZXKUVOUXPRZUYLUXSUYNUXRUVOUXPUWBXFUYSUYMUXQAUWBUVOUXP
      UVEKXGXHSXLXMXNUXNUXRUWIOUWFUXNUXMUXQJZAWJUVHUVOKLZEUWGMZUXRUWIUXNUYTVUBA
      UXNUYTUVEUNZUWGHZUVHVUCKLZJVUBUXNUYTVUDVUEUXNUXMVUDUXQUWOUXMVUDOZUWPUWOUY
      FVUFUYGUYFUXMVUDUVEUWBXOXPWFVAXQUWPUWOUXMUXQVUEUWPUXQUXMUWOVUEUWPUXQUXMUW
      OVUEUXMUWOJUWPUXQUVEGHZVUEUVEUWBXRUXQVUGJVUEUWPUXPUXOXSZUVEUVEVTZXSZKLZUX
      QUXQUXOVUIKLZJUXPUXOXTQRZUVEVUIXTQRZJVUKVUGUXQVULUWBUVEYAZVTZUBHUXOVUIVUP
      YBVULVUOYCUWBUVEFXJAXJYDUXOVUIVUPUBYEYFYGVUGVUNVUMVUGUVEWDVUNUVEWKUVEYHWF
      UXOUVHYIYNUXPUVEUXOVUIYJYKUWPUVHVUHVUCVUJKUWPVUHUVHUVHUWBYLYMVUCVUJRUWPUV
      EYOYPYQWHYRUUAUUBUUEUUFVUAVUEEVUCUWGUVOVUCUVHKYSYTVFXBUXQAUWBUUCUVJVUAAEU
      WGUVEUVOUVHKYSUUDUUGWRVRUUOUWQUWRUWJUWQUWOUWSUWFUWQUWOUWSUWQUWOJUWQUVHGHZ
      JZUWSUWQVUQUWODFGUUHUUIVURUWIUWRUWHUWQUVHUWGHUVHUVHKLZUWIVUQUVHUWBUYRUUJU
      VHUUKUVJVUSAUVHUWGUVEUVHUVHKYSYTYKUUPUULXAXQUUMUUNXAUUQUURUVLUVNDCUBUVHCR
      ZUVIUVCUVKUVGUVHCBXFVUTUVJUVFABUVHCUVEKXGXHSUUSWIUUTVGUVA $.
    $( $j usage 'pssnn' avoids 'ax-pow'; $)
  $}

  ${
    $d A x $.  $d B x $.
    $( A subset of a natural number is finite.  (Contributed by NM,
       24-Jun-1998.)  (Proof shortened by BTernaryTau, 23-Sep-2024.) $)
    ssnnfi $p |- ( ( A e. _om /\ B C_ A ) -> B e. Fin ) $=
      ( vx wss com wcel wpss wceq wo cfn sspss wa cv cen wbr wrex pssnn wi elnn
      expcom anim1d reximdv2 adantr mpd isfi sylibr eleq1 biimparc nnfi sylan2b
      syl jaodan ) BADAEFZBAGZBAHZIBJFZBAKUMUNUPUOUMUNLZBCMZNOZCEPZUPUQUSCAPZUT
      CABQUMVAUTRUNUMUSUSCAEUMURAFZUREFZUSVBUMVCURASTUAUBUCUDCBUEUFUMUOLBEFZUPU
      OVDUMBAEUGUHBUIUKULUJ $.
    $( $j usage 'ssnnfi' avoids 'ax-pow'; $)
  $}

  $( Obsolete version of ~ 0fi as of 13-Jan-2025.  (Contributed by FL,
     14-Jul-2008.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  0finOLD $p |- (/) e. Fin $=
    ( c0 com wcel wss cfn peano1 ssid ssnnfi mp2an ) ABCAADAECFAGAAHI $.

  ${
    $d A x $.  $d B x y z $.  $d A v w y z $.
    $( The union of two finite sets is finite.  Part of Corollary 6K of
       [Enderton] p. 144.  (Contributed by NM, 16-Nov-2002.)  Avoid ~ ax-pow .
       (Revised by BTernaryTau, 7-Aug-2024.) $)
    unfi $p |- ( ( A e. Fin /\ B e. Fin ) -> ( A u. B ) e. Fin ) $=
      ( vz vw vv cfn wcel cun cv wi c0 wceq uneq2 eleq1d imbi2d wn cen wbr com
      wa vx csn un0 eleq1i biimpri wss snssi ssequn2 biimpi uneq2d un12 3eqtr4g
      uncom syl biimprd adantld wrex isfi r19.41v cin disjsn elun notbii pm4.56
      vy wo bitr4i sylbbr word nnord orddisj cvv en2sn el2v unen mpanl2 sylanr2
      sylanr1 3impb 3comr 3expb unass csuc df-suc peano2 eqeltrrid breq2 rspcev
      sylan sylibr syldan rexlimiva sylbir ancoms expl pm2.61i ex imim2d adantl
      findcard2s impcom ) BFGAFGZABHZFGZXBAUAIZHZFGZJXBAKHZFGZJXBAVEIZHZFGZJZXB
      AXJCIZUBZHZHZFGZJZXBXDJUAVECBXEKLZXGXIXBXTXFXHFXEKAMNOXEXJLZXGXLXBYAXFXKF
      XEXJAMNOXEXPLZXGXRXBYBXFXQFXEXPAMNOXEBLZXGXDXBYCXFXCFXEBAMNOXIXBXHAFAUCUD
      UEXNXJGZPZXMXSJXJFGYEXLXRXBYEXLXRXNAGZYEXLTXRJYFXLXRYEYFXRXLYFXQXKFYFXOAU
      FZXQXKLXNAUGYGXJAXOHZHXJAHXQXKYGYHAXJYGYHALXOAUHUIUJAXJXOUKAXJUMULUNNUOUP
      YFPZYEXLXRXLYIYETZXRXLXKDIZQRZDSUQZYJXRXLYMDXKURUIYMYJTYLYJTZDSUQXRYLYJDS
      USYNXRDSYKSGZYNXKXOHZYKYKUBZHZQRZXRYOYLYJYSYLYJYOYSYLYJYOYSYJYLXKXOUTKLZY
      OYSYTXNXKGZPZYJXKXNVAUUBYFYDVFZPYJUUAUUCXNAXJVBVCYFYDVDVGVHYOYLYTYKYQUTKL
      ZYSYOYKVIUUDYKVJYKVKUNYLXOYQQRZYTUUDTYSUUECDXNYKVLVLVMVNXKYKXOYQVOVPVQVRV
      SVTWAYOYSTZXQYPFAXJXOWBUUFYPEIZQRZESUQZYPFGYOYRSGYSUUIYOYRYKWCSYKWDYKWEWF
      UUHYSEYRSUUGYRYPQWGWHWIEYPURWJWFWKWLWMWIWNWOWPWQWRWSWTXA $.
    $( $j usage 'unfi' avoids 'ax-pow'; $)
  $}

  ${
    unfid.1 $e |- ( ph -> A e. Fin ) $.
    unfid.2 $e |- ( ph -> B e. Fin ) $.
    $( The union of two finite sets is finite.  (Contributed by Glauco
       Siliprandi, 5-Feb-2022.) $)
    unfid $p |- ( ph -> ( A u. B ) e. Fin ) $=
      ( cfn wcel cun unfi syl2anc ) ABFGCFGBCHFGDEBCIJ $.
  $}

  ${
    $d B b $.  $d x y z $.  $d A b x $.  $d b c y z $.
    $( A subset of a finite set is finite.  Corollary 6G of [Enderton] p. 138.
       For a shorter proof using ~ ax-pow , see ~ ssfiALT .  (Contributed by
       NM, 24-Jun-1998.)  Avoid ~ ax-pow .  (Revised by BTernaryTau,
       12-Aug-2024.) $)
    ssfi $p |- ( ( A e. Fin /\ B C_ A ) -> B e. Fin ) $=
      ( vb vc wcel cfn wss wa cv wi sseq1 eleq1 imbi12d wal c0 cun sseq2 imbi1d
      wceq albidv vx vy vz cvv ancoms imbi2d csn ss0 0fi eqeltrdi ax-gen eleq1w
      ssexg wn cbvalvw w3a simp1 snssi undif sylib uncom eqtr3di sseq2i ssundif
      cdif sylbb anim12ci 3adant1 3anass sylanbrc vex difexi spcv imp snfi unfi
      sylancl biimparc stoic3 syl 3expib alrimiv sylbi cin disjsn sylbir biimpa
      wb disjssun sylan2b imim1i alimi wo exmid jctl andir syl5 alanimi syl2anc
      pm3.44 a1i findcard2 19.21bi vtoclg impd mpcom ) BUDEZAFEZBAGZHBFEZXIXHXG
      BAFUMUEXGXHXIXJXHCIZAGZXKFEZJZJXHXIXJJZJCBUDXKBSZXNXOXHXPXLXIXMXJXKBAKXKB
      FLMUFXHXNCXKUAIZGZXMJZCNXKOGZXMJZCNXKUBIZGZXMJZCNZXKYBUCIZUGZPZGZXMJZCNZX
      NCNUAUBUCAXQOSZXSYACYLXRXTXMXQOXKQRTXQYBSZXSYDCYMXRYCXMXQYBXKQRTXQYHSZXSY
      JCYNXRYIXMXQYHXKQRTXQASZXSXNCYOXRXLXMXQAXKQRTYACXTXKOFXKUHUIUJUKYEYKJYBFE
      YEYFXKEZYIHZXMJZCNZYPUNZYIHZXMJZCNYKYEDIZYBGZUUCFEZJZDNZYSYDUUFCDXKUUCSYC
      UUDXMUUEXKUUCYBKCDFULMUOUUGYRCUUGYPYIXMUUGYPYIUPZUUGXKYGVEZYBGZXKUUIYGPZS
      ZUPZXMUUHUUGUUJUULHZUUMUUGYPYIUQYPYIUUNUUGYPUULYIUUJYPYGUUIPZXKUUKYPYGXKG
      UUOXKSYFXKURYGXKUSUTYGUUIVAVBYIXKYGYBPZGZUUJYHUUPXKYBYGVAVCZXKYGYBVDVFVGV
      HUUGUUJUULVIVJUUGUUJUUKFEZUULXMUUGUUJHUUIFEZYGFEUUSUUGUUJUUTUUFUUJUUTJDUU
      IXKYGCVKVLUUCUUISUUDUUJUUEUUTUUCUUIYBKUUCUUIFLMVMVNYFVOUUIYGVPVQUULXMUUSX
      KUUKFLVRVSVTWAWBWCYDUUBCUUAYCXMYIYTUUQYCUURYTUUQYCYTXKYGWDOSUUQYCWHXKYFWE
      XKYGYBWIWFWGWJWKWLYRUUBYJCYIYQUUAWMZYRUUBHXMYIYPYTWMZYIHUVAYIUVBYPWNWOYPY
      TYIWPUTXMYQUUAWTWQWRWSXAXBXCXDXEXF $.
    $( $j usage 'ssfi' avoids 'ax-pow'; $)
  $}

  ${
    $d A x y z $.  $d B x y z $.
    $( Shorter proof of ~ ssfi using ~ ax-pow .  (Contributed by NM,
       24-Jun-1998.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    ssfiALT $p |- ( ( A e. Fin /\ B C_ A ) -> B e. Fin ) $=
      ( vx vz vy cfn wcel wss cv cen wbr com wrex wi isfi wf1o wex bren sylan
      wa cima wfo f1ofo crn imassrn forn sseqtrid syl ssnnfi sylib adantrr cres
      sylan2 wf1 f1of1 f1ores vex resex f1oeq1 sylibr entr ex reximdv imbitrrdi
      spcev adantl mpd exp32 exlimdv biimtrid rexlimiv sylbi imp ) AFGZBAHZBFGZ
      VNACIZJKZCLMVOVPNZCAOVRVSCLVRAVQDIZPZDQVQLGZVSAVQDRWBWAVSDWBWAVOVPWBWAVOT
      ZTVTBUAZEIZJKZELMZVPWBWAWGVOWAWBWDVQHZWGWAAVQVTUBZWHAVQVTUCWIVTUDWDVQVTBU
      EAVQVTUFUGUHWBWHTWDFGWGVQWDUIEWDOUJUMUKWCWGVPNWBWCWGBWEJKZELMVPWCWFWJELWC
      WFWJWCBWDVTBULZPZWFWJWAAVQVTUNVOWLAVQVTUOAVQBVTUPSWLBWDJKZWFWJWLBWDVQPZCQ
      WMWNWLCWKVTBDUQURBWDVQWKUSVEBWDCRUTBWDWEVASSVBVCEBOVDVFVGVHVIVJVKVLVM $.
  $}

  $( If ` A ` is finite, ` ( A \ B ) ` is finite.  (Contributed by FL,
     3-Aug-2009.) $)
  diffi $p |- ( A e. Fin -> ( A \ B ) e. Fin ) $=
    ( cfn wcel cdif wss difss ssfi mpan2 ) ACDABEZAFJCDABGAJHI $.

  ${
    $d A x $.  $d x y z $.  $d u v z $.
    $( If a set is finite, its converse is as well.  (Contributed by Mario
       Carneiro, 28-Dec-2014.)  Avoid ~ ax-pow .  (Revised by BTernaryTau,
       9-Sep-2024.) $)
    cnvfi $p |- ( A e. Fin -> `' A e. Fin ) $=
      ( vx vy vz vu vv cv ccnv cfn wcel c0 csn cun cnveq eleq1d 0fi cvv cop wex
      wceq cnv0 eqeltri wi cnvun cxp elvv sneq vex eqtrdi syl eqeltrdi exlimivv
      cnvsn snfi sylbi wn crn cdm dfdm4 dmsnn0 biimpri necon1bi eqtr3id wrel wb
      wne relcnv relrn0 ax-mp sylibr pm2.61i unfi mpan2 eqeltrid a1i findcard2
      ) BGZHZIJKHZIJCGZHZIJZVTDGZLZMZHZIJZAHZIJBCDAVQKTVRVSIVQKNOVQVTTVRWAIVQVT
      NOVQWETVRWFIVQWENOVQATVRWHIVQANOVSKIUAPUBWBWGUCVTIJWBWFWAWDHZMZIVTWDUDWBW
      IIJZWJIJWCQQUEJZWKWLWCEGZFGZRZTZFSESWKEFWCUFWPWKEFWPWIWNWMRZLZIWPWDWOLZTZ
      WIWRTWCWOUGWTWIWSHWRWDWSNWMWNEUHFUHUMUIUJWQUNUKULUOWLUPZWIKIXAWIUQZKTZWIK
      TZXAXBWDURZKWDUSWLXEKWLXEKVFWCUTVAVBVCWIVDXDXCVEWDVGWIVHVIVJPUKVKWAWIVLVM
      VNVOVP $.
    $( $j usage 'cnvfi' avoids 'ax-pow'; $)
  $}

  ${
    $d A x $.
    $( Every element of the power set of ` A ` is finite if and only if ` A `
       is finite.  (Contributed by Glauco Siliprandi, 17-Aug-2020.) $)
    pwssfi $p |- ( A e. V -> ( A e. Fin <-> ~P A C_ Fin ) ) $=
      ( vx wcel cfn cpw wss wral elpwi ssfi sylan2 ralrimiva dfss3 sylibr pwidg
      cv biimpi eleq1 rspcva syl2an ex impbid2 ) ABDZAEDZAFZEGZUDCPZEDZCUEHZUFU
      DUHCUEUGUEDUDUGAGUHUGAIAUGJKLCUEEMZNUCUFUDUCAUEDUIUDUFABOUFUIUJQUHUDCAUEU
      GAERSTUAUB $.
  $}

  ${
    $d A x y z $.  $d F x y z $.
    $( A version of ~ fnex for finite sets that does not require Replacement or
       Power 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 ) $=
      ( vx vy vz cfn wcel wa cres wceq adantr cv wi c0 csn reseq2 eleq1d imbi2d
      cun a1i wfn fnresdm res0 0fi eqeltri resundi cfv snfi wfun fnfun funressn
      cop wss syl ssfi sylancr unfi sylan2 eqeltrid expcom a2i anabsi7 eqeltrrd
      findcard2 ) BAUAZAFGZHZBAIZBFVEVHBJVFABUBKVEVFVHFGZVGBCLZIZFGZMVGBNIZFGZM
      VGBDLZIZFGZMZVGBVOELZOZSZIZFGZMZVGVIMCDEAVJNJZVLVNVGWEVKVMFVJNBPQRVJVOJZV
      LVQVGWFVKVPFVJVOBPQRVJWAJZVLWCVGWGVKWBFVJWABPQRVJAJZVLVIVGWHVKVHFVJABPQRV
      NVGVMNFBUCUDUETVRWDMVOFGVGVQWCVQVGWCVQVGHWBVPBVTIZSZFBVOVTUFVGVQWIFGZWJFG
      VGVSVSBUGULZOZFGWIWMUMZWKWLUHVEWNVFVEBUIWNABUJVSBUKUNKWMWIUOUPVPWIUQURUSU
      TVATVDVBVC $.
    $( $j usage 'fnfi' avoids 'ax-pow' 'ax-rep'; $)
  $}

  $( If the domain of a one-to-one, onto function is finite, then the domain
     and range of the function are equinumerous.  This theorem is proved
     without using the Axiom of Replacement or the Axiom of Power Sets (unlike
     ~ f1oeng ).  (Contributed by BTernaryTau, 8-Sep-2024.) $)
  f1oenfi $p |- ( ( A e. Fin /\ F : A -1-1-onto-> B ) -> A ~~ B ) $=
    ( cfn wcel wf1o cen wbr wfn f1ofn fnfi sylan ancoms f1oen3g sylancom ) ADEZ
    ABCFZCDEZABGHQPRQCAIPRABCJACKLMABCDNO $.
  $( $j usage 'f1oenfi' avoids 'ax-pow' 'ax-rep'; $)

  $( If the range of a one-to-one, onto function is finite, then the domain and
     range of the function are equinumerous.  (Contributed by BTernaryTau,
     9-Sep-2024.) $)
  f1oenfirn $p |- ( ( B e. Fin /\ F : A -1-1-onto-> B ) -> A ~~ B ) $=
    ( cfn wcel wf1o cen wbr ccnv f1ocnv f1ofn fnfi sylan ancoms cnvfi wrel wceq
    wfn f1orel sylancom dfrel2 sylib eleq1d biimpac f1oen3g ) BDEZABCFZCDEZABGH
    UFUGCIZDEZUHUGUFUJUGBAUIFZUFUJABCJUKUIBRUFUJBAUIKBUILMMNUJUIIZDEZUGUHUIOUGU
    MUHUGULCDUGCPULCQABCSCUAUBUCUDMTABCDUET $.
  $( $j usage 'f1oenfirn' avoids 'ax-pow' 'ax-rep'; $)

  $( If the codomain of a one-to-one function is finite, then the function's
     domain is dominated by its codomain.  This theorem is proved without using
     the Axiom of Replacement or the Axiom of Power Sets (unlike ~ f1domg ).
     (Contributed by BTernaryTau, 25-Sep-2024.) $)
  f1domfi $p |- ( ( B e. Fin /\ F : A -1-1-> B ) -> A ~<_ B ) $=
    ( cfn wcel wf1 wa cdom wbr ccnv crn wf1o f1cnv wss f1f frnd ssfi sylan2 wfn
    sylan f1ofn fnfi syl2an2 cnvfi wrel wceq f1rel dfrel2 sylib eleq1d sylancom
    biimpac f1dom3g 3expib mpcom ) CDEZBDEZABCFZGABHIZUQURCJZDEZUPURCKZAUTLZUQV
    BDEZVAABCMURUQVBBNVDURABCABCOPBVBQRVCUTVBSVDVAVBAUTUAVBUTUBTUCVAUTJZDEZURUP
    UTUDURVFUPURVECDURCUEVECUFABCUGCUHUIUJULTUKUPUQURUSABCDDUMUNUO $.
  $( $j usage 'f1domfi' avoids 'ax-pow' 'ax-rep'; $)

  $( If the domain of a one-to-one function is finite, then the function's
     domain is dominated by its codomain when the latter is a set.  This
     theorem is proved without using the Axiom of Power Sets (unlike
     ~ f1dom2g ).  (Contributed by BTernaryTau, 24-Nov-2024.) $)
  f1domfi2 $p |- ( ( A e. Fin /\ B e. V /\ F : A -1-1-> B ) -> A ~<_ B ) $=
    ( cfn wcel wf1 cdom wbr wfn f1fn fnfi sylan ancoms 3adant2 f1dom3g syld3an1
    ) CEFZBDFZAEFZABCGZABHITUARSUATRUACAJTRABCKACLMNOABCEDPQ $.
  $( $j usage 'f1domfi2' avoids 'ax-pow' 'ax-rep'; $)

  $( Equinumerosity is reflexive for finite sets, proved without using the
     Axiom of Power Sets (unlike ~ enrefg ).  (Contributed by BTernaryTau,
     8-Sep-2024.) $)
  enreffi $p |- ( A e. Fin -> A ~~ A ) $=
    ( cfn wcel cid cres wf1o cen wbr f1oi f1oenfi mpan2 ) ABCAADAEZFAAGHAIAALJK
    $.
  $( $j usage 'enreffi' avoids 'ax-pow'; $)

  ${
    $d A f $.  $d A g $.  $d B f $.  $d B g $.
    $( Symmetry of equinumerosity for finite sets, proved without using the
       Axiom of Power Sets (unlike ~ ensymb ).  (Contributed by BTernaryTau,
       9-Sep-2024.) $)
    ensymfib $p |- ( A e. Fin -> ( A ~~ B <-> B ~~ A ) ) $=
      ( vf vg cfn wcel cen wbr cv wf1o wex bren wa 19.42v f1ocnv sylan2 exlimiv
      ccnv sylbir sylan2b f1oenfirn f1oenfi impbida ) AEFZABGHZBAGHZUEUDABCIZJZ
      CKZUFABCLUDUIMUDUHMZCKUFUDUHCNUJUFCUHUDBAUGRZJUFABUGOBAUKUAPQSTUFUDBADIZJ
      ZDKZUEBADLUDUNMUDUMMZDKUEUDUMDNUOUEDUMUDABULRZJUEBAULOABUPUBPQSTUC $.
    $( $j usage 'ensymfib' avoids 'ax-pow'; $)
  $}

  ${
    $d A f g $.  $d B f g $.  $d C f g $.
    $( Transitivity of equinumerosity for finite sets, proved without using the
       Axiom of Power Sets (unlike ~ entr ).  (Contributed by BTernaryTau,
       10-Sep-2024.) $)
    entrfil $p |- ( ( A e. Fin /\ A ~~ B /\ B ~~ C ) -> A ~~ C ) $=
      ( vf vg cen wbr cfn wcel cv wf1o wex bren wa exdistrv 19.42vv ccom ancoms
      f1oco f1oenfi sylan2 exlimivv sylbir sylan2br 3impb syl3an2b syl3an3b ) B
      CFGAHIZABFGZBCDJZKZDLZACFGZBCDMUIUHABEJZKZELZULUMABEMUHUPULUMUPULNUHUOUKN
      ZDLELZUMUOUKEDOUHURNUHUQNZDLELUMUHUQEDPUSUMEDUQUHACUJUNQZKZUMUKUOVAABCUJU
      NSRACUTTUAUBUCUDUEUFUG $.
    $( $j usage 'entrfil' avoids 'ax-pow'; $)
  $}

  ${
    $d A x $.  $d B x $.
    $( A set equinumerous to a finite set is finite.  (Contributed by Mario
       Carneiro, 12-Mar-2015.)  Avoid ~ ax-pow .  (Revised by BTernaryTau,
       23-Sep-2024.) $)
    enfii $p |- ( ( B e. Fin /\ A ~~ B ) -> A e. Fin ) $=
      ( vx cen wbr cfn wcel wa com wrex wex isfi df-rex sylbb ensymfib biimparc
      cv 19.41v w3a sylibr simp1 nnfi biimpar 3adant3 entrfil syld3an2 3ad2ant1
      wb mpbid syl3an1 jca 3expa eximi sylbir syl2an2 ancoms ) ABDEZBFGZAFGZUQU
      RHZACQZDEZCIJZUSUTVAIGZVBHZCKZVCURVDBVADEZHZCKZUQBADEZVFURVGCIJVICBLVGCIM
      NURVJUQBAOPVIVJHVHVJHZCKVFVHVJCRVKVECVDVGVJVEVDVGVJSVDVBVDVGVJUAVDVAFGZVG
      VJVBVAUBVLVGVJSVAADEZVBVLVABDEZVGVJVMVLVGVNVJVLVNVGVABOUCUDVABAUEUFVLVGVM
      VBUHVJVAAOUGUIUJUKULUMUNUOVBCIMTCALTUP $.
    $( $j usage 'enfii' avoids 'ax-pow'; $)
  $}

  $( Equinumerous sets have the same finiteness.  For a shorter proof using
     ~ ax-pow , see ~ enfiALT .  (Contributed by NM, 22-Aug-2008.)  Avoid
     ~ ax-pow .  (Revised by BTernaryTau, 23-Sep-2024.) $)
  enfi $p |- ( A ~~ B -> ( A e. Fin <-> B e. Fin ) ) $=
    ( cen wbr cfn wcel wa ensymfib pm5.32i enfii sylbi expcom impbid ) ABCDZAEF
    ZBEFZONPONGOBACDZGPONQABHIBAJKLPNOABJLM $.
  $( $j usage 'enfi' avoids 'ax-pow'; $)

  ${
    $d A x $.  $d B x $.
    $( Shorter proof of ~ enfi using ~ ax-pow .  (Contributed by NM,
       22-Aug-2008.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    enfiALT $p |- ( A ~~ B -> ( A e. Fin <-> B e. Fin ) ) $=
      ( vx cen wbr cv com wrex cfn wcel enen1 rexbidv isfi 3bitr4g ) ABDEZACFZD
      EZCGHBPDEZCGHAIJBIJOQRCGABPKLCAMCBMN $.
  $}

  ${
    $d A x $.  $d B x $.
    $( A set dominated by a finite set is finite.  (Contributed by NM,
       23-Mar-2006.)  (Revised by Mario Carneiro, 12-Mar-2015.) $)
    domfi $p |- ( ( A e. Fin /\ B ~<_ A ) -> B e. Fin ) $=
      ( vx cfn wcel cdom wbr cv cen wss wa domeng ssfi adantrl adantrr sylancom
      wex enfii ex exlimdv sylbid imp ) ADEZBAFGZBDEZUCUDBCHZIGZUFAJZKZCQUECBAD
      LUCUIUECUCUIUEUCUIUFDEZUEUCUHUJUGAUFMNUJUGUEUHBUFROPSTUAUB $.
  $}

  $( Transitivity of equinumerosity for finite sets, proved without using the
     Axiom of Power Sets (unlike ~ entr ).  (Contributed by BTernaryTau,
     23-Sep-2024.) $)
  entrfi $p |- ( ( B e. Fin /\ A ~~ B /\ B ~~ C ) -> A ~~ C ) $=
    ( cfn wcel cen wbr enfii 3adant3 entrfil syld3an1 ) ADEZABFGZBDEZBCFGZACFGN
    MLOABHIABCJK $.
  $( $j usage 'entrfi' avoids 'ax-pow'; $)

  $( Transitivity of equinumerosity for finite sets, proved without using the
     Axiom of Power Sets (unlike ~ entr ).  (Contributed by BTernaryTau,
     23-Sep-2024.) $)
  entrfir $p |- ( ( C e. Fin /\ A ~~ B /\ B ~~ C ) -> A ~~ C ) $=
    ( cfn wcel cen wbr enfii 3adant2 entrfi syld3an1 ) BDEZABFGZCDEZBCFGZACFGNO
    LMBCHIABCJK $.
  $( $j usage 'entrfir' avoids 'ax-pow'; $)

  ${
    $d A f g $.  $d B f g $.  $d C f g $.
    $( Transitivity of dominance relation when ` A ` is finite, proved without
       using the Axiom of Power Sets (unlike ~ domtr ).  (Contributed by
       BTernaryTau, 24-Nov-2024.) $)
    domtrfil $p |- ( ( A e. Fin /\ A ~<_ B /\ B ~<_ C ) -> A ~<_ C ) $=
      ( vg vf cfn wcel cvv wa wbr reldom brrelex2i anim2i 3adant2 cv wf1 brdomi
      cdom wex exdistrv 19.42vv ccom f1co ancoms f1domfi2 3expa sylan2 exlimivv
      sylbir sylan2br 3impb syl3an3 syl3an2 syld3an1 ) AFGZCHGZIZABRJZUOBCRJZAC
      RJZUOUSUQURUSUPUOBCRKLMNURUQABDOZPZDSZUSUTABDQUSUQVCBCEOZPZESZUTBCEQUQVCV
      FUTVCVFIUQVBVEIZESDSZUTVBVEDETUQVHIUQVGIZESDSUTUQVGDEUAVIUTDEVGUQACVDVAUB
      ZPZUTVEVBVKABCVDVAUCUDUOUPVKUTACVJHUEUFUGUHUIUJUKULUMUN $.
    $( $j usage 'domtrfil' avoids 'ax-pow'; $)
  $}

  $( Transitivity of dominance relation when ` B ` is finite, proved without
     using the Axiom of Power Sets (unlike ~ domtr ).  (Contributed by
     BTernaryTau, 24-Nov-2024.) $)
  domtrfi $p |- ( ( B e. Fin /\ A ~<_ B /\ B ~<_ C ) -> A ~<_ C ) $=
    ( cfn wcel cdom wbr domfi 3adant3 domtrfil syld3an1 ) ADEZABFGZBDEZBCFGZACF
    GNMLOBAHIABCJK $.
  $( $j usage 'domtrfi' avoids 'ax-pow'; $)

  $( Transitivity of dominance relation for finite sets, proved without using
     the Axiom of Power Sets (unlike ~ domtr ).  (Contributed by BTernaryTau,
     24-Nov-2024.) $)
  domtrfir $p |- ( ( C e. Fin /\ A ~<_ B /\ B ~<_ C ) -> A ~<_ C ) $=
    ( cfn wcel cdom wbr domfi 3adant2 domtrfi syld3an1 ) BDEZABFGZCDEZBCFGZACFG
    NOLMCBHIABCJK $.
  $( $j usage 'domtrfir' avoids 'ax-pow'; $)

  $( If a function is one-to-one, then the image of a finite subset of its
     domain under it is equinumerous to the subset.  This theorem is proved
     without using the Axiom of Replacement or the Axiom of Power Sets (unlike
     ~ f1imaeng ).  (Contributed by BTernaryTau, 29-Sep-2024.) $)
  f1imaenfi $p |- ( ( F : A -1-1-> B /\ C C_ A /\ C e. Fin ) ->
      ( F " C ) ~~ C ) $=
    ( cfn wcel wf1 wss cima cen wbr wa cres wf1o f1ores f1oenfi ensymfib adantr
    wb mpbid sylan2 3impb 3coml ) CEFZABDGZCAHZDCIZCJKZUDUEUFUHUEUFLUDCUGDCMZNZ
    UHABCDOUDUJLCUGJKZUHCUGUIPUDUKUHSUJCUGQRTUAUBUC $.
  $( $j usage 'f1imaenfi' avoids 'ax-pow' 'ax-rep'; $)

  $( A finite set dominates its subsets, proved without using the Axiom of
     Power Sets (unlike ~ ssdomg ).  (Contributed by BTernaryTau,
     12-Nov-2024.) $)
  ssdomfi $p |- ( B e. Fin -> ( A C_ B -> A ~<_ B ) ) $=
    ( cfn wcel wss cdom wbr cid cres wf1 wf1o f1oi f1of1 ax-mp f1ss mpan sylan2
    f1domfi ex ) BCDZABEZABFGZUATABHAIZJZUBAAUCJZUAUDAAUCKUEALAAUCMNAABUCOPABUC
    RQS $.
  $( $j usage 'ssdomfi' avoids 'ax-pow'; $)

  $( A set dominates its finite subsets, proved without using the Axiom of
     Power Sets (unlike ~ ssdomg ).  (Contributed by BTernaryTau,
     24-Nov-2024.) $)
  ssdomfi2 $p |- ( ( A e. Fin /\ B e. V /\ A C_ B ) -> A ~<_ B ) $=
    ( wss cfn wcel cid cres wf1 cdom wbr wf1o f1oi f1of1 ax-mp f1domfi2 syl3an3
    f1ss mpan ) ABDZAEFBCFABGAHZIZABJKAAUAIZTUBAAUALUCAMAAUANOAABUARSABUACPQ $.
  $( $j usage 'ssdomfi2' avoids 'ax-pow'; $)

  ${
    $d D x $.  $d H x $.  $d A f g x $.  $d B f g x $.
    sbthfilem.1 $e |- A e. _V $.
    sbthfilem.2 $e |- D = { x |
                   ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) } $.
    sbthfilem.3 $e |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) ) $.
    sbthfilem.4 $e |- B e. _V $.
    $( Lemma for ~ sbthfi .  (Contributed by BTernaryTau, 4-Nov-2024.) $)
    sbthfilem $p |- ( ( B e. Fin /\ A ~<_ B /\ B ~<_ A ) -> A ~~ B ) $=
      ( cfn wcel cdom wbr w3a cv wex wa cvv wf1 cen 19.42vv 3anass 2exbii brdom
      anbi12i exdistrv bitr4i anbi2i 3bitr4ri wf1o wfn f1fn cuni cres ccnv cdif
      bitri cun vex resex fnfi cnvfi resexg 3syl sylancr eqeltrid ancoms sylan2
      unexg 3adant2 sbthlem9 3adant1 f1oen3g syl2anc exlimivv sylbi ) CLMZBCNOZ
      CBNOZPZVSBCEQZUAZCBFQZUAZPZFRERZBCUBOZVSWDWFSZSZFRERVSWJFRERZSZWHWBVSWJEF
      UCWGWKEFVSWDWFUDUEWBVSVTWASZSWMVSVTWAUDWNWLVSWNWDERZWFFRZSWLVTWOWAWPBCEKU
      FCBFHUFUGWDWFEFUHUIUJUSUKWGWIEFWGGTMZBCGULZWIVSWFWQWDWFVSWECUMZWQCBWEUNWS
      VSWQWSVSSZGWCDUOZUPZWEUQZBXAURZUPZUTZTJWTXBTMXETMZXFTMWCXAEVAVBWTWELMXCLM
      XGCWEVCWEVDXCXDLVEVFXBXETTVKVGVHVIVJVLWDWFWRVSABCDEFGHIJVMVNBCGTVOVPVQVR
      $.
    $( $j usage 'sbthfilem' avoids 'ax-pow'; $)
  $}

  ${
    $d B w $.  $d A w z $.  $d f g w x y z $.
    $( Schroeder-Bernstein Theorem for finite sets, proved without using the
       Axiom of Power Sets (unlike ~ sbth ).  (Contributed by BTernaryTau,
       4-Nov-2024.) $)
    sbthfi $p |- ( ( B e. Fin /\ A ~<_ B /\ B ~<_ A ) -> A ~~ B ) $=
      ( vw vz vx vy cfn wcel cdom wbr w3a cen wi cvv wceq breq1 wss cima cdif
      cv vg vf reldom brrelex1i breq2 3anbi23d imbi12d eleq1 3anbi123d cab cuni
      wa cres ccnv cun vex imaeq2 difeq2d imaeq2d difeq2 sseq12d anbi12d cbvabv
      sseq1 eqid sbthfilem vtocl2g syl2an 3adant1 pm2.43i ) BGHZABIJZBAIJZKZABL
      JZVLVMVNVOMZVKVLANHBNHVPVMABIUCUDBAIUCUDCTZGHZDTZVQIJZVQVSIJZKZVSVQLJZMVR
      AVQIJZVQAIJZKZAVQLJZMVPDCABNNVSAOZWBWFWCWGWHVTWDWAWEVRVSAVQIPVSAVQIUEUFVS
      AVQLPUGVQBOZWFVNWGVOWIVRVKWDVLWEVMVQBGUHVQBAIUEVQBAIPUIVQBALUEUGEVSVQFTZV
      SQZUATZVQUBTZWJRZSZRZVSWJSZQZULZFUJZUBUAWMWTUKZUMWLUNVSXASUMUOZDUPWSETZVS
      QZWLVQWMXCRZSZRZVSXCSZQZULFEWJXCOZWKXDWRXIWJXCVSVDXJWPXGWQXHXJWOXFWLXJWNX
      EVQWJXCWMUQURUSWJXCVSUTVAVBVCXBVECUPVFVGVHVIVJ $.
    $( $j usage 'sbthfi' avoids 'ax-pow'; $)
  $}

  $( If a set dominates a finite set, it cannot also be strictly dominated by
     the finite set.  This theorem is proved without using the Axiom of Power
     Sets (unlike ~ domnsym ).  (Contributed by BTernaryTau, 22-Nov-2024.) $)
  domnsymfi $p |- ( ( A e. Fin /\ A ~<_ B ) -> -. B ~< A ) $=
    ( cdom wbr cfn wcel csdm cen wo wn brdom2 sdomnen adantl sdomdom w3a sbthfi
    wa wb ensymfib 3ad2ant1 mpbird syl3an2 syl3an3 3com23 biimpa jaodan sylan2b
    3expa mtand nsyl3 ) ABCDZAEFZABGDZABHDZIBAGDZJZABKULUMUPUNULUMQUOUNUMUNJULA
    BLMULUMUOUNULUOUMUNUMULUOUKUNABNUOULBACDZUKUNBANULUQUKOUNBAHDZBAPULUQUNURRU
    KABSZTUAUBUCUDUHUIUOURULUNQBALULUNURUSUEUJUFUG $.
  $( $j usage 'domnsymfi' avoids 'ax-pow'; $)

  $( Transitivity of strict dominance and dominance when ` A ` is finite,
     proved without using the Axiom of Power Sets (unlike ~ sdomdomtr ).
     (Contributed by BTernaryTau, 25-Nov-2024.) $)
  sdomdomtrfi $p |- ( ( A e. Fin /\ A ~< B /\ B ~<_ C ) -> A ~< C ) $=
    ( cfn wcel csdm wbr cdom w3a cen wn sdomdom domtrfil syl3an2 simp1 ensymfib
    wa biimpa 3adant2 endom domtrfir syl3an3 syld3an3 domnsymfi sylancom 3expia
    domfi syl2anc con2d 3impia 3com23 brsdom sylanbrc ) ADEZABFGZBCHGZIACHGZACJ
    GZKZACFGUOUNABHGUPUQABLABCMNUNUPUOUSUNUPUOUSUNUPQURUOUNUPURUOKZUNUPURIUNBAH
    GZUTUNUPUROUNUPURCAJGZVAUNURVBUPUNURVBACPRSVBUNUPCAHGVACATBCAUAUBUCUNVABDEU
    TABUGBAUDUEUHUFUIUJUKACULUM $.
  $( $j usage 'sdomdomtrfi' avoids 'ax-pow'; $)

  $( Transitivity of dominance and strict dominance when ` A ` is finite,
     proved without using the Axiom of Power Sets (unlike ~ domsdomtr ).
     (Contributed by BTernaryTau, 25-Nov-2024.) $)
  domsdomtrfi $p |- ( ( A e. Fin /\ A ~<_ B /\ B ~< C ) -> A ~< C ) $=
    ( cfn wcel cdom wbr csdm w3a wn sdomdom domtrfil syl3an3 wa ensymfib biimpa
    cen 3adant3 enfii endom domtrfi syl3an2 jca syld3an2 domnsymfi 3com23 con2d
    syl 3expia 3impia brsdom sylanbrc ) ADEZABFGZBCHGZIACFGZACQGZJZACHGUOUMUNBC
    FGUPBCKABCLMUMUNUOURUMUNNUQUOUMUNUQUOJZUMUQUNUSUMUQUNICDEZCBFGZNZUSUMCAQGZU
    QUNVBUMUQVCUNUMUQVCACOPRUMVCUNIUTVAUMVCUTUNCASRVCUMCAFGUNVACATCABUAUBUCUDCB
    UEUHUFUIUGUJACUKUL $.
  $( $j usage 'domsdomtrfi' avoids 'ax-pow'; $)

  ${
    $d A f w $.  $d B f w $.
    $( Strict dominance of a set over another set implies dominance over its
       successor.  (Contributed by Mario Carneiro, 12-Jan-2013.)  (Proof
       shortened by Mario Carneiro, 27-Apr-2015.)  Avoid ~ ax-pow .  (Revised
       by BTernaryTau, 4-Dec-2024.) $)
    sucdom2 $p |- ( A ~< B -> suc A ~<_ B ) $=
      ( vf vw csdm wbr cv wf1 cdom wex syl wa csn cun wceq cvv wcel adantr wss
      c0 csuc sdomdom brdomi crn cdif cin vex rnex f1f1orn adantl f1of1 f1dom3g
      wf1o mp3an12i wn cen sdomnen ssdif0 simplr f1f frnd simpr dff1o5 sylanbrc
      eqssd f1oen3g sylancr biimtrrid mtod neq0 sylib relsdom brrelex1i sylancl
      ex snssi en2sn wi brrelex2i difexg cfn snfi ssdomfi2 mp3an1 endom domtrfi
      3syl sylan syl6an syl5 exlimdv mpd disjdif syl21anc df-suc undif2 ssequn1
      a1i undom eqtr2id 3brtr4d exlimddv ) ABEFZABCGZHZAUAZBIFCXCABIFXECJABUBAB
      CUCKXCXELZAAMZNZXDUDZBXJUEZNZXFBIXGAXJIFZXHXKIFZXJXKUFTOZXIXLIFXDPQZXJPQX
      GAXJXDHZXMCUGZXDXRUHXGAXJXDUMZXQXEXSXCABXDUIUJAXJXDUKKAXJXDPPULUNXGDGZXKQ
      ZDJZXNXGXKTOZUOYBXGYCABUPFZXCYDUOXEABUQRYCBXJSZXGYDBXJURXGYEYDXGYELZXPABX
      DUMZYDXRYFXEXJBOYGXCXEYEUSZYFXJBYFXEXJBSZYHXEABXDABXDUTVAZKXGYEVBVEABXDVC
      VDABXDPVFVGVOVHVIDXKVJVKXGYAXNDYAXTMZXKSZXGXNXTXKVPXGXHYKUPFZYLYKXKIFZXNX
      GAPQZXTPQYMXCYOXEABEVLVMRDUGAXTPPVQVNXGBPQZXKPQZYLYNVRXCYPXEABEVLVSRBXJPV
      TYQYLYNYKWAQZYQYLYNXTWBZYKXKPWCWDVOWGYMXHYKIFZYNXNXHYKWEYRYTYNXNYSXHYKXKW
      FWDWHWIWJWKWLXOXGXJBWMWRAXJXHXKWSWNXFXIOXGAWOWRXGXLXJBNZBXJBWPXGYIUUABOXE
      YIXCYJUJXJBWQVKWTXAXB $.
    $( $j usage 'sucdom2' avoids 'ax-pow'; $)
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Pigeonhole Principle
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Lemma for Pigeonhole Principle.  A natural number is equinumerous to its
     successor minus any element of the successor.  (Contributed by NM,
     26-May-1998.)  Avoid ~ ax-pow .  (Revised by BTernaryTau, 23-Sep-2024.) $)
  phplem1 $p |- ( ( A e. _om /\ B e. suc A ) -> A ~~ ( suc A \ { B } ) ) $=
    ( com wcel csuc wa csn cdif cen wbr simpl peano2 enrefnn syl simpr dif1ennn
    adantr syl3anc cfn wb nnfi ensymfib 3syl mpbird ) ACDZBAEZDZFZAUFBGHZIJZUIA
    IJZUHUEUFUFIJZUGUKUEUGKZUEULUGUEUFCDULALUFMNQUEUGOUFABPRUHUEASDUJUKTUMAUAAU
    IUBUCUD $.
  $( $j usage 'phplem1' avoids 'ax-pow'; $)

  ${
    $d A f $.  $d B f $.
    phplem2.1 $e |- A e. _V $.
    $( 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.)  Avoid
       ~ ax-pow .  (Revised by BTernaryTau, 4-Nov-2024.) $)
    phplem2 $p |- ( ( A e. _om /\ B e. _om ) ->
        ( suc A ~~ suc B -> A ~~ B ) ) $=
      ( vf csuc cen wbr com wcel wa csn cdif cima cfn nnfi wb ensymfib syl wceq
      adantr cv wf1o wex bren cfv wf1 f1of1 wss sssucid f1imaenfi mp3an2 mpbird
      syl2anr word nnord orddif imaeq2d wfn f1ofn sucid sylancl difeq2d crn cdm
      fnsnfv imadmrn eqcomi f1ofo forn f1odm 3eqtr3a difeq1d ccnv dff1o3 imadif
      wfo wfun simplbiim 3eqtr4rd sylan9eq breqtrd fnfvelrn eleq2d mpbid sylan2
      phplem1 entrfil syl3an1 syl3an3 3expa syldanl anandirs exlimdv biimtrid
      ex ) AEZBEZFGWPWQDUAZUBZDUCAHIZBHIZJZABFGZWPWQDUDXBWSXCDXBWSXCWTXAWSXCWTW
      SAWQAWRUEZKZLZFGZXAWSJZXCWTWSJZAWRAMZXFFXIAXJFGZXJAFGZWSWPWQWRUFZANIZXLWT
      WPWQWRUGAOZXMAWPUHXNXLAUIWPWQAWRUJUKUMWTXKXLPZWSWTXNXPXOAXJQRTULWTWSXJWRW
      PAKZLZMZXFWTAXRWRWTAUNAXRSAUOAUPRUQWSWRWPMZXELXTWRXQMZLZXFXSWSXEYAXTWSWRW
      PURZAWPIZXEYASWPWQWRUSZACUTZWPAWRVEVAVBWSWQXTXEWSWRVCZWRWRVDZMZWQXTYIYGWR
      VFVGWSWPWQWRVPZYGWQSWPWQWRVHZWPWQWRVIZRWSYHWPWRWPWQWRVJUQVKVLWSYJWRVMVQXS
      YBSWPWQWRVNWPXQWRVOVRVSVTWAWTXGXHXCXHWTXGXFBFGZXCXHBXFFGZYMWSXAXDWQIZYNWS
      XDYGIZYOWSYCYDYPYEYFWPAWRWBVAWSYJYPYOPYKYJYGWQXDYLWCRWDBXDWFWEXAYNYMPZWSX
      ABNIYQBOBXFQRTWDWTXNXGYMXCXOAXFBWGWHWIWJWKWLWOWMWN $.
    $( $j usage 'phplem2' avoids 'ax-pow'; $)
  $}

  ${
    $d x y $.  $d B z $.  $d w y z $.  $d A x z $.
    $( 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.)  Avoid ~ ax-pow .
       (Revised by BTernaryTau, 11-Nov-2024.) $)
    nneneq $p |- ( ( A e. _om /\ B e. _om ) -> ( A ~~ B <-> A = B ) ) $=
      ( vz vx vy vw com wcel cen wbr wceq cv wi wral weq c0 breq1 eqeq1 imbi12d
      breq2 wa csuc ralbidv cfn 0fi ensymfib ax-mp eqcom bitri sylbb rgenw wrex
      wb en0 wo nn0suc eqeq2 bibi12d mpbiri biimpd a1i nfv nfra1 phplem2 imim1d
      nfan vex ex a2d impel suceq syl8 biimprcd syl6 rexlimd jaod syl7 ralrimdv
      rsp cbvralvw imbitrdi finds rspcv mpan9 enrefnn syl5ibcom adantr impbid )
      AGHZBGHZUAABIJZABKZWIACLZIJZAWMKZMZCGNZWJWKWLMZDLZWMIJZDCOZMZCGNPWMIJZPWM
      KZMZCGNELZWMIJZECOZMZCGNZXFUBZWMIJZXKWMKZMZCGNZWQDEAWSPKZXBXECGXPWTXCXAXD
      WSPWMIQWSPWMRSUCDEOZXBXICGXQWTXGXAXHWSXFWMIQWSXFWMRSUCWSXKKZXBXNCGXRWTXLX
      AXMWSXKWMIQWSXKWMRSUCWSAKZXBWPCGXSWTWNXAWOWSAWMIQWSAWMRSUCXECGXCWMPIJZXDP
      UDHXCXTUMUEPWMUFUGXTWMPKXDWMUNWMPUHUIUJUKXFGHZXJXKFLZIJZXKYBKZMZFGNXOYAXJ
      YEFGYBGHYBPKZYBWMUBZKZCGULZUOZYAXJYECYBUPYAXJYJYEMYAXJUAZYFYEYIYFYEMYKYFY
      CYDYFYCYDUMXKPIJZXKPKZUMXKUNYFYCYLYDYMYBPXKITYBPXKUQURUSUTVAYKYHYECGYAXJC
      YACVBXICGVCVFYECVBYKWMGHZXKYGIJZXKYGKZMZYHYEMYKYNYOXHYPYAYNXIMYNYOXHMZMXJ
      YAYNXIYRYAYNXIYRMYAYNUAYOXGXHXFWMEVGVDVEVHVIXICGVSVJXFWMVKVLYHYEYQYHYCYOY
      DYPYBYGXKITYBYGXKUQSVMVNVOVPVHVQVRYEXNFCGFCOYCXLYDXMYBWMXKITYBWMXKUQSVTWA
      WBWPWRCBGWMBKWNWKWOWLWMBAITWMBAUQSWCWDWIWLWKMWJWIAAIJWLWKAWEABAITWFWGWH
      $.
    $( $j usage 'nneneq' avoids 'ax-pow'; $)
  $}

  ${
    $d A x $.  $d B x y $.
    $( Pigeonhole Principle.  A natural number is not equinumerous to a proper
       subset of itself.  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
       ~ phplem1 , ~ phplem2 , ~ nneneq , and this final piece of the proof.
       (Contributed by NM, 29-May-1998.)  Avoid ~ ax-pow .  (Revised by
       BTernaryTau, 18-Nov-2024.) $)
    php $p |- ( ( A e. _om /\ B C. A ) -> -. A ~~ B ) $=
      ( vx vy com wcel wpss cen wbr wn wa cv wceq wi c0 wss cdom wb cfn syl 0ss
      csuc wrex sspsstr mpan wne df-ne bitri sylib nn0suc orcanai sylan2 pssnel
      0pss wex csn cdif pssss ssdif cin disjsn disj3 sseq1 sylbi imbitrrid syl5
      bitr3i peano2 nnfi diffi ssdomfi 4syl sylan9 3impia 3com23 3expa ad2antrl
      adantrr simpl simpr phplem1 ensymfib mpbid endom domtrfir syl3an3 syl3anc
      adantr sylancom exp43 com4r imp exlimiv mpcom w3a syl3an2 syl3an1 sssucid
      simp1 mpisyl sbthfi mpd3an3 syl2anc 3expia peano2b nnord vex sucid nordeq
      sylancl nneneq sylanb anidms necon3bbid mpbird nsyli expcom pm2.43d com12
      word syli psseq2 breq1 notbid imbi12d syl5ibrcom rexlimiv syldbl2 ) AEFZB
      AGZABHIZJZYIYJKACLZUBZMZCEUCZYJYLNZYJYIAOMZJZYPYJOAGZYSOBPYJYTBUAOBAUDUEY
      TAOUFYSAUNAOUGUHUIYIYRYPCAUJUKULYOYQCEYMEFZYQYOBYNGZYNBHIZJZNUUBUUAUUDUUA
      UUBBYMQIZUUDDLZYNFZUUFBFJZKZDUOUUBUUAUUENZDBYNUMUUIUUBUUJNZDUUGUUHUUKUUHU
      UBUUAUUGUUEUUHUUBUUAUUGUUEUUHUUBKZUUAUUGKZBYNUUFUPZUQZQIZUUEUULUUAUUPUUGU
      UHUUBUUAUUPUUHUUAUUBUUPUUHUUAUUBUUPUUHUUBBUUOPZUUAUUPUUBBYNPZUUHUUQBYNURU
      URUUQUUHBUUNUQZUUOPZBYNUUNUSUUHBUUSMZUUQUUTRUUHBUUNUTOMUVABUUFVABUUNVBVGB
      UUSUUOVCVDVEVFUUAYNEFZYNSFZUUOSFUUQUUPNYMVHZYNVIZYNUUNVJBUUOVKVLVMVNVOVPV
      RUUPUUMKYMSFZUUPUUMUUEUUAUVFUUPUUGYMVIZVQUUPUUMVSUUPUUMVTUUMUVFUUPUUOYMQI
      ZUUEUUMUUOYMHIZUVHUUMYMUUOHIZUVIYMUUFWAUUAUVJUVIRZUUGUUAUVFUVKUVGYMUUOWBT
      WHWCUUOYMWDTBUUOYMWEWFWGWIWJWKWLWMWNUUEUUAUUDUUAUUEUUAUUDNUUAUUEKUUCYNYMH
      IZUUAUUAUUEUUCUVLUUAUUCUUEUVLUUAUUCUUEWOUUAYNYMQIZUVLUUAUUCUUEWSUUAUVFUUC
      UUEUVMUVGUUCUVFYNBQIUUEUVMYNBWDYNBYMWEWPWQUUAUVMYMYNQIZUVLUUAUVNUVMUUAUVB
      UVNUVDUVBUVCYMYNPUVNUVEYMWRYMYNVKWTTWHUUAUVFUVMUVNUVLUVGYNYMXAWQXBXCVOXDU
      UAUVLJYNYMUFZUUAYNXTZYMYNFUVOUUAUVBUVPYMXEZYNXFVDYMCXGXHYNYMXIXJUUAUVLYNY
      MUUAUVLYNYMMRZUUAUVBUUAUVRUVQYNYMXKXLXMXNXOXPXQXRYAXSYOYJUUBYLUUDAYNBYBYO
      YKUUCAYNBHYCYDYEYFYGTYH $.
    $( $j usage 'php' avoids 'ax-pow'; $)
  $}

  $( Corollary of Pigeonhole Principle.  (Contributed by NM, 31-May-1998.)
     Avoid ~ ax-pow .  (Revised by BTernaryTau, 20-Nov-2024.) $)
  php2 $p |- ( ( A e. _om /\ B C. A ) -> B ~< A ) $=
    ( com wcel wpss wa cdom wbr cen wn csdm cfn wss nnfi ssdomfi imp syl2an php
    pssss wi ensymfib biimprd syl adantr mtod brsdom sylanbrc ) ACDZBAEZFZBAGHZ
    BAIHZJBAKHUHALDZBAMZUKUIANZBASUMUNUKBAOPQUJULABIHZABRUHULUPTZUIUHUMUQUOUMUP
    ULABUAUBUCUDUEBAUFUG $.
  $( $j usage 'php2' avoids 'ax-pow'; $)

  ${
    $d A f x y $.  $d B f x y $.
    $( Corollary of Pigeonhole Principle.  If ` A ` is finite and ` B ` is a
       proper subset of ` A ` , the ` B ` is strictly less numerous than
       ` A ` .  Stronger version of Corollary 6C of [Enderton] p. 135.
       (Contributed by NM, 22-Aug-2008.)  Avoid ~ ax-pow .  (Revised by
       BTernaryTau, 26-Nov-2024.) $)
    php3 $p |- ( ( A e. Fin /\ B C. A ) -> B ~< A ) $=
      ( vx vf vy cfn wcel wpss csdm wbr cv cen com wi wf1o wex cima sylan2 cdom
      wss wrex isfi bren wa wn pssss imass2 syl adantl cdif c0 pssnel eldif cfv
      wceq f1ofn difss fnfvima 3expia sylancl ccnv wfun dff1o3 imadif simplbiim
      wfn wfo eleq2d sylibd n0i syl6 exlimdv imp ssdif0 sylnibr dfpss3 sylanbrc
      biimtrrid cdm crn imadmrn f1odm imaeq2d f1ofo forn 3eqtr3a psseq2d adantr
      wb mpbid php2 nnfi cres wf1 f1of1 f1ores syl2an resex f1oeq1 spcev sylibr
      endom sdomdom domfi 3adant2 3adant3 domsdomtrfi syld3an1 syl3an2 biimtrid
      vex mpd exp32 ensymfib biimp3ar sdomdomtrfi syl3an3 syld3an3 syl3an1 3exp
      3com23 syldd rexlimiv sylbi ) AFGZBAHZBAIJZYEACKZLJZCMUAYFYGNZCAUBYIYJCMY
      HMGZYIYFBYHIJZYGYIAYHDKZOZDPYKYFYLNZAYHDUCYKYNYODYKYNYFYLYKYNYFUDZUDYMBQZ
      YHIJZYLYPYKYQYHHZYRYPYQYMAQZHZYSYPYQYTTZYTYQTZUEUUAYFUUBYNYFBATZUUBBAUFZB
      AYMUGUHUIYPYTYQUJZUKUOZUUCYFYNEKZAGUUHBGUEUDZEPZUUGUEZEBAULYNUUJUUKYNUUIU
      UKEUUIUUHABUJZGZYNUUKUUHABUMYNUUMUUHYMUNZUUFGZUUKYNUUMUUNYMUULQZGZUUOYNYM
      AVFZUULATZUUMUUQNAYHYMUPABUQUURUUSUUMUUQAUULYMUUHURUSUTYNUUPUUFUUNYNAYHYM
      VGZYMVAVBUUPUUFUOAYHYMVCABYMVDVEVHVIUUFUUNVJVKVRVLVMRYTYQVNVOYQYTVPVQYNUU
      AYSWIYFYNYTYHYQYNYMYMVSZQYMVTZYTYHYMWAYNUVAAYMAYHYMWBWCYNUUTUVBYHUOAYHYMW
      DAYHYMWEUHWFWGWHWJYHYQWKRYKYHFGZBYQLJZYRYLNYPYHWLZYPBYQYMBWMZOZUVDYNAYHYM
      WNUUDUVGYFAYHYMWOUUEAYHBYMWPWQUVGBYQUUHOZEPUVDUVHUVGEUVFYMBDXKWRBYQUUHUVF
      WSWTBYQEUCXAUHUVCUVDYRYLUVDUVCBYQSJZYRYLBYQXBYQFGZUVIUVCYRYLUVCYRUVJUVIYR
      UVCYQYHSJUVJYQYHXCYHYQXDRXEBFGZUVIUVJYRYLUVJUVIUVKYRYQBXDXFBYQYHXGXHXHXIU
      SWQXLXMVLXJYKYIYLYGYKYLYIYGYKUVCYLYIYGUVEUVCYLYIYHALJZYGUVCYLUVLYIUVCUVLY
      IWIYLYHAXNWHXOUVLUVCYLYHASJZYGYHAXBUVKYLUVCUVMYGUVCYLUVKUVMYLUVCBYHSJUVKB
      YHXCYHBXDRXFBYHAXPXHXQXRXSYAXTYBYCYDVM $.
    $( $j usage 'php3' avoids 'ax-pow'; $)
  $}

  $( Corollary of the Pigeonhole Principle ~ php : a natural number is strictly
     dominated by its successor.  (Contributed by NM, 26-Jul-2004.) $)
  php4 $p |- ( A e. _om -> A ~< suc A ) $=
    ( com wcel csuc wpss csdm wbr sucidg word wb nnord ordsuc ordelpss syl2anc2
    biimpi mpbid peano2b php2 sylanb mpdan ) ABCZAADZEZAUBFGZUAAUBCZUCABHUAAIZU
    BIZUEUCJAKUFUGALOAUBMNPUAUBBCUCUDAQUBARST $.

  $( Corollary of the Pigeonhole Principle ~ php : 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 ) $=
    ( com wcel csuc csdm wbr cen wn php4 sdomnen syl ) ABCAADZEFALGFHAIALJK $.

  ${
    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
       ~ php expressed without negation.  (Contributed by Rohan Ridenour,
       3-Aug-2023.)  Avoid ~ ax-pow .  (Revised by BTernaryTau,
       28-Nov-2024.) $)
    phpeqd $p |- ( ph -> A = B ) $=
      ( cen wbr wceq wn csdm cfn wcel wpss wa wss adantr simpr neqcomd dfpss2
      sylanbrc php3 syl2an2r sdomnen ensymfib notbid biimpar syl2an syldan mt4d
      ex ) ABCGHZBCIZFAUMJZULJZAUNCBKHZUOABLMZUNCBNZUPDAUNOZCBPZCBIJURAUTUNEQUS
      BCAUNRSCBTUABCUBUCAUQCBGHZJZUOUPDCBUDUQUOVBUQULVABCUEUFUGUHUIUKUJ $.
    $( $j usage 'phpeqd' avoids 'ax-pow'; $)
  $}

  $( Cardinal ordering agrees with ordinal number ordering when the smaller
     number is a natural number.  Compare with ~ nndomo when both are natural
     numbers.  (Contributed by NM, 17-Jun-1998.)  Generalize from ~ nndomo .
     (Revised by RP, 5-Nov-2023.)  Avoid ~ ax-pow .  (Revised by BTernaryTau,
     29-Nov-2024.) $)
  nndomog $p |- ( ( A e. _om /\ B e. On ) -> ( A ~<_ B <-> A C_ B ) ) $=
    ( com wcel con0 wa cdom wbr wss wpss wn wi csdm cfn nnfi domnsymfi sylan ex
    word wb php2 nsyld adantr nnord eloni ordtri1 ordelpss ancoms notbid syl2an
    bitrd sylibrd ssdomfi2 3expia impbid ) ACDZBEDZFZABGHZABIZURUSBAJZKZUTUPUSV
    BLUQUPUSBAMHZVAUPUSVCKZUPANDZUSVDAOZABPQRUPVAVCABUARUBUCUPASZBSZUTVBTUQAUDB
    UEVGVHFZUTBADZKVBABUFVIVJVAVHVGVJVATBAUGUHUIUKUJULUPVEUQUTUSLVFVEUQUTUSABEU
    MUNQUO $.
  $( $j usage 'nndomog' avoids 'ax-pow'; $)

  $( Obsolete lemma for ~ php as of 22-Nov-2024.  (Contributed by NM,
     25-May-1998.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  phplem1OLD $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 $.
    phplem2OLD.1 $e |- A e. _V $.
    phplem2OLD.2 $e |- B e. _V $.
    $( Obsolete lemma for ~ php as of 22-Nov-2024.  (Contributed by NM,
       11-Jun-1998.)  (Revised by Mario Carneiro, 16-Nov-2014.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    phplem2OLD $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
      wf1o snex f1osn f1oen3g mp2an difexi enref pm3.2i incom difss ssrin ax-mp
      cop word nnord orddisj sseqtrid disjdif jctil unen sylancr adantr difsnid
      ss0 uncom adantl phplem1OLD 3brtr3d ) AEFZBAFZGBHZAVKIZJZAHZVLJZAAKVKILVI
      VMVOLMZVJVIVKVNLMZVLVLLMZGVKVLNOPZVNVLNZOPZGVPVQVRBAUMZHZQFVKVNWCUAVQWBUB
      BADCUCVKVNWCQUDUEVLAVKCUFUGUHVIWAVSVIVTVLVNNZOVNVLUIVIWDORWDOPVIAVNNZWDOV
      LARWDWERAVKUJVLAVNUKULVIAUNWEOPAUOAUPSUQWDVDSTVKAURUSVKVNVLVLUTVAVBVJVMAP
      VIVJVMVLVKJAVKVLVEABVCTVFABVGVH $.

    $( Obsolete version of ~ phplem1 as of 18-Nov-2024.  (Contributed by NM,
       26-May-1998.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    phplem3OLD $p |- ( ( A e. _om /\ B e. suc A )
                    -> A ~~ ( suc A \ { B } ) ) $=
      ( csuc wcel com wceq wo csn cdif cen wbr elsuci phplem2OLD wa enref nnord
      word orddif syl sneq difeq2d eqcoms sylan9eq breqtrid jaodan sylan2 ) BAE
      ZFAGFZBAFZBAHZIAUIBJZKZLMZBANUJUKUOULABCDOUJULPAAUNLACQUJULAUIAJZKZUNUJAS
      AUQHARATUAUQUNHABABHUPUMUIABUBUCUDUEUFUGUH $.
  $}

  ${
    $d x y A $.  $d x y B $.
    $( Obsolete version of ~ php as of 18-Nov-2024.  (Contributed by NM,
       29-May-1998.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    phpOLD $p |- ( ( A e. _om /\ B C. A ) -> -. A ~~ B ) $=
      ( vx vy com wcel wpss cen wbr wn wi wa cv wceq c0 wss wne cdom cvv imp wb
      csuc wrex 0ss sspsstr mpan df-ne bitri sylib nn0suc orcanai sylan2 pssnel
      0pss wex csn cdif pssss ssdif cin disjsn disj3 bitr3i sseq1 imbitrrid vex
      sylbi sucex difexi ssdomg ax-mp syl56 ensymd domentr syl2an exp43 exlimiv
      phplem3OLD com4r mpcom endomtr sssucid mp2 sbth sylancl expcom word nnord
      peano2b sucid nordeq nneneq sylanb anidms mpbird nsyli com12 psseq2 breq1
      necon3bbid syli notbid imbi12d syl5ibrcom rexlimiv syl ex pm2.43d ) AEFZB
      AGZABHIZJZXIXJXLXIXJXJXLKZXIXJLACMZUBZNZCEUCZXMXJXIAONZJZXQXJOAGZXSOBPXJX
      TBUDOBAUEUFXTAOQXSAUNAOUGUHUIXIXRXQCAUJUKULXPXMCEXNEFZXMXPBXOGZXOBHIZJZKY
      BYAYDYAYBBXNRIZYDDMZXOFZYFBFJZLZDUOYBYAYEKZDBXOUMYIYBYJKZDYGYHYKYHYBYAYGY
      EYHYBYAYGYEYHYBLBXOYFUPZUQZRIZYMXNHIYEYAYGLZYHYBYNYBBXOPZYHBYMPZYNBXOURYP
      YQYHBYLUQZYMPZBXOYLUSYHBYRNZYQYSUAYHBYLUTONYTBYFVABYLVBVCBYRYMVDVGVEYMSFY
      QYNKXOYLXNCVFZVHZVIBYMSVJVKVLTYOXNYMXNYFUUADVFVRVMBYMXNVNVOVPVSTVQVTYEYCX
      OXNHIZYAYCYEUUCYCYELXOXNRIXNXORIZUUCXOBXNWAXOSFXNXOPUUDUUBXNWBXNXOSVJWCXO
      XNWDWEWFYAUUCJXOXNQZYAXOWGZXNXOFUUEYAXOEFZUUFXNWIZXOWHVGXNUUAWJXOXNWKWEYA
      UUCXOXNYAUUCXOXNNUAZYAUUGYAUUIUUHXOXNWLWMWNWTWOWPXAWQXPXJYBXLYDAXOBWRXPXK
      YCAXOBHWSXBXCXDXEXFXGXHT $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Obsolete version of ~ php2 as of 20-Nov-2024.  (Contributed by NM,
       31-May-1998.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    php2OLD $p |- ( ( A e. _om /\ B C. A ) -> B ~< A ) $=
      ( vx com wcel wpss csdm wbr cv wa wceq eleq1 psseq2 anbi12d breq2 imbi12d
      wi cdom cen cvv wss vex pssss ssdomg mpsyl adantl php ensym nsyl sylanbrc
      wn brsdom vtoclg anabsi5 ) ADEZBAFZBAGHZCIZDEZBURFZJZBURGHZQUOUPJZUQQCADU
      RAKZVAVCVBUQVDUSUOUTUPURADLURABMNURABGOPVABURRHZBURSHZUKVBUTVEUSURTEUTBUR
      UAVECUBBURUCBURTUDUEUFVAURBSHVFURBUGBURUHUIBURULUJUMUN $.
  $}

  ${
    $d f x y A $.  $d f x y B $.
    $( Obsolete version of ~ php3 as of 26-Nov-2024.  (Contributed by NM,
       22-Aug-2008.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    php3OLD $p |- ( ( A e. Fin /\ B C. A ) -> B ~< A ) $=
      ( vx vf vy wcel wpss wbr cv cen com wi wa wn wss imp wf1o wex cima syl c0
      cfn csdm wrex isfi cvv relen brrelex1i pssss ssdomg syl2an adantll imass2
      cdom bren adantl cdif pssnel eldif cfv f1ofn difss fnfvima 3expia sylancl
      wceq wfn wfo ccnv wfun dff1o3 imadif simplbiim eleq2d sylibd syl6 exlimdv
      n0i biimtrrid sylan2 ssdif0 sylnibr dfpss3 sylanbrc cdm crn imadmrn f1odm
      imaeq2d f1ofo forn 3eqtr3a psseq2d adantr mpbid php cres wf1 f1of1 f1ores
      wb vex resex f1oeq1 spcev sylibr entr expcom mtod exp32 biimtrid imp31 ex
      ensym syl6com ad2antlr brsdom exp31 rexlimiv sylbi ) AUBFZBAGZBAUCHZYAACI
      ZJHZCKUDYBYCLZCAUEYEYFCKYDKFZYEYBYCYGYEMYBMZBAUNHZBAJHZNYCYEYBYIYGYEAUFFZ
      BAOZYIYBAYDJUGUHBAUIZYKYLYIBAUFUJPUKULYHYJYDBJHZYGYEYBYNNZYEAYDDIZQZDRYGY
      BYOLZAYDDUOYGYQYRDYGYQYBYOYGYQYBMZMYNYDYPBSZJHZYSYGYTYDGZUUANYSYTYPASZGZU
      UBYSYTUUCOZUUCYTOZNUUDYBUUEYQYBYLUUEYMBAYPUMTUPYSUUCYTUQZUAVFZUUFYBYQEIZA
      FUUIBFNMZERZUUHNZEBAURYQUUKUULYQUUJUULEUUJUUIABUQZFZYQUULUUIABUSYQUUNUUIY
      PUTZUUGFZUULYQUUNUUOYPUUMSZFZUUPYQYPAVGZUUMAOZUUNUURLAYDYPVAABVBUUSUUTUUN
      UURAUUMYPUUIVCVDVEYQUUQUUGUUOYQAYDYPVHZYPVIVJUUQUUGVFAYDYPVKABYPVLVMVNVOU
      UGUUOVRVPVSVQPVTUUCYTWAWBYTUUCWCWDYQUUDUUBXAYBYQUUCYDYTYQYPYPWEZSYPWFZUUC
      YDYPWGYQUVBAYPAYDYPWHWIYQUVAUVCYDVFAYDYPWJAYDYPWKTWLWMWNWOYDYTWPVTYSYNUUA
      LZYGYSBYTJHZUVDYSBYTYPBWQZQZUVEYQAYDYPWRYLUVGYBAYDYPWSYMAYDBYPWTUKUVGBYTU
      UIQZERUVEUVHUVGEUVFYPBDXBXCBYTUUIUVFXDXEBYTEUOXFTYNUVEUUAYDBYTXGXHTUPXIXJ
      VQXKXLYEYJYNLYGYBYJYEBYDJHZYNYJYEUVIBAYDXGXMBYDXNXOXPXIBAXQWDXRXSXTP $.
  $}

  ${
    phpeqdOLD.1 $e |- ( ph -> A e. Fin ) $.
    phpeqdOLD.2 $e |- ( ph -> B C_ A ) $.
    phpeqdOLD.3 $e |- ( ph -> A ~~ B ) $.
    $( Obsolete version of ~ phpeqd as of 28-Nov-2024.  (Contributed by Rohan
       Ridenour, 3-Aug-2023.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    phpeqdOLD $p |- ( ph -> A = B ) $=
      ( cen wbr wceq wn wa csdm cfn wcel wpss wss adantr simpr neqcomd dfpss2
      sylanbrc php3 syl2an2r sdomnen ensym nsyl syl ex mt4d ) ABCGHZBCIZFAUKJZU
      JJZAULKZCBLHZUMABMNULCBOZUODUNCBPZCBIJUPAUQULEQUNBCAULRSCBTUABCUBUCUOCBGH
      UJCBUDBCUEUFUGUHUI $.
  $}

  $( Obsolete version of ~ nndomog as of 29-Nov-2024.  (Contributed by NM,
     17-Jun-1998.)  Generalize from ~ nndomo .  (Revised by RP, 5-Nov-2023.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  nndomogOLD $p |- ( ( A e. _om /\ B e. On ) -> ( A ~<_ B <-> A C_ B ) ) $=
    ( com wcel con0 wa cdom wbr wss wpss wn wi csdm php2 ex domnsym adantr word
    nsyli wb ordtri1 ordelpss ancoms notbid syl2an sylibrd ssdomg adantl impbid
    nnord eloni bitrd ) ACDZBEDZFZABGHZABIZUOUPBAJZKZUQUMUPUSLUNUMURBAMHZUPUMUR
    UTABNOABPSQUMARZBRZUQUSTUNAUJBUKVAVBFZUQBADZKUSABUAVCVDURVBVAVDURTBAUBUCUDU
    LUEUFUNUQUPLUMABEUGUHUI $.

  $( Obsolete version of ~ snnen2o as of 18-Nov-2024.  (Contributed by AV,
     6-Aug-2019.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  snnen2oOLD $p |- -. { A } ~~ 2o $=
    ( cvv wcel csn c2o cen wbr wn c1o csuc com 1onn php5 ax-mp ensn1g wi ensymb
    df-2o c0 wceq eqcomi breq2i entr ex sylbi con3rr3 sylnbi mpsyl 2on0 nemtbir
    en0 bitri snprc biimpi breq1d mtbiri pm2.61i ) ABCZADZEFGZHZIIJZFGZHZURUSIF
    GZVAIKCVDLIMNABOVCIEFGZVEVAPVBEIFEVBRUAUBVEUTVFVEIUSFGZUTVFPUSIQVGUTVFIUSEU
    CUDUEUFUGUHURHZUTSEFGZVIESUIVIESFGESTSEQEUKULUJVHUSSEFVHUSSTAUMUNUOUPUQ $.


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Finite sets (cont.)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( An ordinal number equinumerous to a natural number is equal to it.
     Proposition 10.22 of [TakeutiZaring] p. 90 and its converse.  (Contributed
     by NM, 26-Jul-2004.)  Avoid ~ ax-pow .  (Revised by BTernaryTau,
     2-Dec-2024.) $)
  onomeneq $p |- ( ( A e. On /\ B e. _om ) -> ( A ~~ B <-> A = B ) ) $=
    ( con0 wcel com wa cen wbr wss cdom endom wn wi cfn adantl syl5 word ancoms
    ex wb wceq w3a wpss nnfi domfi simpr jca csdm domnsymfi nsyld expimpd mpand
    php3 eloni nnord ordtri1 ordelpss notbid syl2an sylibrd 3impia ensymfib syl
    bitrd biimtrrdi 3adant1 nndomog biimp3a syld3an3 eqssd 3expia enrefnn breq1
    imp syl5ibrcom impbid ) ACDZBEDZFZABGHZABUAZVQVRVTWAVQVRVTUBABVQVRVTABIZVTA
    BJHZVSWBABKVSWCBAUCZLZWBVRWCWEMZVQVRBNDZWCWEBUDZWGWCFZANDZWCFVRWEWIWJWCBAUE
    WGWCUFUGVRWJWCWEWJWFVRWJWCBAUHHZWDWJWCWKLABUISWJWDWKABUMSUJOUKPULOVQAQZBQZW
    BWETVRAUNBUOWLWMFZWBBADZLWEABUPWNWOWDWMWLWOWDTBAUQRURVDUSUTPVAVQVRVTBAJHZBA
    IZVRVTWPVQVRVTWPVRVTBAGHZWPVRWGWRVTTWHBAVBVCBAKVEVNVFVQVRWPWQVRVQWPWQTBAVGR
    VHVIVJVKVRWAVTMVQVRVTWABBGHBVLABBGVMVOOVP $.
  $( $j usage 'onomeneq' avoids 'ax-pow'; $)

  $( Obsolete version of ~ onomeneq as of 29-Nov-2024.  (Contributed by NM,
     26-Jul-2004.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  onomeneqOLD $p |- ( ( A e. On /\ B e. _om ) -> ( A ~~ B <-> A = B ) ) $=
    ( con0 wcel com wa cen wbr wceq wss adantl cdom ssdomg sylan2 ancoms adantr
    wn wb wi word csuc php5 ad2antlr enen1 mtbird peano2 sssucid mpisyl endomtr
    a1d adantll ssel com12 eloni ordelsuc sylibd syld jcad sbth syl6 mtod ordom
    ordtri1 sylancr con2bid ad2antrr mpbird simplr nneneq biimpa sylancom eqeng
    jca ex impbid ) ACDZBEDZFZABGHZABIZVRVSVTVRVSAEDZVQFZVTVRVSFZWAVQWCWAEAJZQZ
    WCWDABUAZGHZWCWGBWFGHZVQWHQVPVSBUBUCVSWGWHRVRABWFUDKUEWCWDAWFLHZWFALHZFWGWC
    WDWIWJVQVSWDWISVPVQVSFWIWDVSVQWIVQVSBWFLHZWIVQWFEDBWFJWKBUFBUGBWFEMUHABWFUI
    NOUJUKVRWDWJSZVSVQVPWLVQVPFZWDWFAJZWJWMWDBADZWNVQWDWOSVPWDVQWOEABULUMPVPVQA
    TZWOWNRAUNZBAEUONUPVPWNWJSVQWFACMKUQOPURAWFUSUTVAVPWAWERVQVSVPWDWAVPETWPWDW
    AQRVBWQEAVCVDVEVFVGVPVQVSVHVMWBVSVTABVIVJVKVNVPVTVSSVQABCVLPVO $.

  ${
    $d x A $.
    $( An ordinal number is finite iff it is a natural number.  Proposition
       10.32 of [TakeutiZaring] p. 92.  (Contributed by NM, 26-Jul-2004.) $)
    onfin $p |- ( A e. On -> ( A e. Fin <-> A e. _om ) ) $=
      ( vx cfn wcel cv cen wbr wrex con0 isfi wa wceq onomeneq wi eleq1a adantl
      com sylbid rexlimdva enrefnn breq2 rspcev mpdan impbid1 bitrid ) ACDABEZF
      GZBQHZAIDZAQDZBAJUIUHUJUIUGUJBQUIUFQDZKUGAUFLZUJAUFMUKULUJNUIUFQAOPRSUJAA
      FGZUHATUGUMBAQUFAAFUAUBUCUDUE $.
    $( $j usage 'onfin' avoids 'ax-pow'; $)
  $}

  $( A set is a natural number iff it is a finite ordinal.  (Contributed by
     Mario Carneiro, 22-Jan-2013.) $)
  onfin2 $p |- _om = ( On i^i Fin ) $=
    ( vx com con0 cfn cin cv wcel nnon onfin biimprcd jcai biimpa impbii bitr4i
    wa elin eqriv ) ABCDEZAFZBGZSCGZSDGZOZSRGTUCTUAUBSHUAUBTSIZJKUAUBTUDLMSCDPN
    Q $.

  $( 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 con0 cdom wbr wss wb nnon nndomog sylan2 ) BCDACDBEDABFGABHIBJAB
    KL $.

  $( Cardinal ordering agrees with natural number ordering.  (Contributed by
     NM, 17-Jun-1998.) $)
  nnsdomo $p |- ( ( A e. _om /\ B e. _om ) -> ( A ~< B <-> A C. B ) ) $=
    ( com wcel wa cdom wbr cen wn wceq csdm nndomo nneneq notbid anbi12d brsdom
    wss wpss dfpss2 3bitr4g ) ACDBCDEZABFGZABHGZIZEABQZABJZIZEABKGABRUAUBUEUDUG
    ABLUAUCUFABMNOABPABST $.

  $( Strict dominance of a set over a natural number is the same as dominance
     over its successor.  (Contributed by Mario Carneiro, 12-Jan-2013.)  Avoid
     ~ ax-pow .  (Revised by BTernaryTau, 4-Dec-2024.)  (Proof shortened by BJ,
     11-Jan-2025.) $)
  sucdom $p |- ( A e. _om -> ( A ~< B <-> suc A ~<_ B ) ) $=
    ( com wcel csdm wbr csuc cdom sucdom2 nnfi php4 sdomdomtrfi syl2anc impbid2
    cfn wi 3expia ) ACDZABEFZAGZBHFZABIRAODZATEFZUASPAJAKUBUCUASATBLQMN $.
  $( $j usage 'sucdom' avoids 'ax-pow'; $)

  $( Obsolete version of ~ sucdom as of 4-Dec-2024.  (Contributed by Mario
     Carneiro, 12-Jan-2013.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  sucdomOLD $p |- ( A e. _om -> ( A ~< B <-> suc A ~<_ B ) ) $=
    ( com wcel csdm wbr csuc cdom sucdom2 wi php4 sdomdomtr ex syl impbid2 ) AC
    DZABEFZAGZBHFZABIPAREFZSQJAKTSQARBLMNO $.

  ${
    $d A f $.
    $( A singleton ` { A } ` is never equinumerous with the ordinal number 2.
       This holds for proper singletons ( ` A e. _V ` ) as well as for
       singletons being the empty set ( ` A e/ _V ` ).  (Contributed by AV,
       6-Aug-2019.)  Avoid ~ ax-pow , ~ ax-un .  (Revised by BTernaryTau,
       1-Dec-2024.) $)
    snnen2o $p |- -. { A } ~~ 2o $=
      ( vf vx csn c2o cen wbr cv wf1o wex ccnv wf1 wceq c0 c1o cvv wcel wne nex
      mto cpr df2o3 0ex 1oex 1n0 necomi prnesn mp3an eqnetri neii f1cdmsn mpan2
      2on0 f1ocnv f1of1 syl wb snex 2oex breng mp2an mtbir ) ADZEFGZVCEBHZIZBJZ
      VFBVFEVCVEKZLZVIECHZDZMZCJZVLCEVKENOUAZVKUBNPQOPQNORVNVKRUCUDONUEUFNOVJPP
      UGUHUIUJSVIENRVMUMCEAVHUKULTVFEVCVHIVIVCEVEUNEVCVHUOUPTSVCPQEPQVDVGUQAURU
      SVCEBPPUTVAVB $.
    $( $j usage 'snnen2o' avoids 'ax-pow' 'ax-un'; $)
  $}

  ${
    $d A x $.
    $( Strict dominance over 0 is the same as dominance over 1.  For a shorter
       proof requiring ~ ax-un , see 0sdom1domALT .  (Contributed by NM,
       28-Sep-2004.)  Avoid ~ ax-un .  (Revised by BTernaryTau, 7-Dec-2024.) $)
    0sdom1dom $p |- ( (/) ~< A <-> 1o ~<_ A ) $=
      ( vx c0 csdm wbr cvv wcel c1o cdom relsdom brrelex2i reldom wne 0sdomg cv
      wex n0 csn wss cen snssi df1o2 0ex en2sn mp2an eqbrtri endom ax-mp domssr
      vex mp3an3 ex syl5 exlimdv biimtrid wceq 1n0 dom0 nemtbir mtbiri necon2ai
      breq2 impbid1 bitrd pm5.21nii ) CADEZAFGZHAIEZCADJKHAILKVGVFACMZVHAFNVGVI
      VHVIBOZAGZBPVGVHBAQVGVKVHBVKVJRZASZVGVHVJAUAVGVMVHVGVMHVLIEZVHHVLTEVNHCRZ
      VLTUBCFGVJFGVOVLTEUCBUJCVJFFUDUEUFHVLUGUHHVLAFUIUKULUMUNUOVHACACUPVHHCIEZ
      VPHCUQHURUSACHIVBUTVAVCVDVE $.
    $( $j usage '0sdom1dom' avoids 'ax-pow' 'ax-un'; $)
  $}

  $( Alternate proof of ~ 0sdom1dom , shorter but requiring ~ ax-un .
     (Contributed by NM, 28-Sep-2004.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  0sdom1domALT $p |- ( (/) ~< A <-> 1o ~<_ A ) $=
    ( c0 csdm wbr csuc cdom c1o wcel wb peano1 sucdom ax-mp df-1o breq1i bitr4i
    com ) BACDZBEZAFDZGAFDBPHQSIJBAKLGRAFMNO $.
  $( $j usage '0sdom1domALT' avoids 'ax-pow'; $)

  $( Ordinal 1 is strictly dominated by ordinal 2.  For a shorter proof
     requiring ~ ax-un , see ~ 1sdom2ALT .  (Contributed by NM, 4-Apr-2007.)
     Avoid ~ ax-un .  (Revised by BTernaryTau, 8-Dec-2024.) $)
  1sdom2 $p |- 1o ~< 2o $=
    ( c1o c2o csdm wbr cdom cen wn wne 2on0 2oex 0sdom mpbir 0sdom1dom mpbi csn
    c0 snnen2o df1o2 breq1i mtbir brsdom mpbir2an ) ABCDABEDZABFDZGPBCDZUCUEBPH
    IBJKLBMNUDPOZBFDPQAUFBFRSTABUAUB $.
  $( $j usage '1sdom2' avoids 'ax-pow' 'ax-un'; $)

  $( Alternate proof of ~ 1sdom2 , shorter but requiring ~ ax-un .
     (Contributed by NM, 4-Apr-2007.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  1sdom2ALT $p |- 1o ~< 2o $=
    ( c1o csuc c2o csdm com wcel wbr 1onn php4 ax-mp df-2o breqtrri ) AABZCDAEF
    AMDGHAIJKL $.
  $( $j usage '1sdom2ALT' avoids 'ax-pow'; $)

  ${
    $d A x $.  $d A f $.
    $( A set has less than one member iff it is empty.  (Contributed by Stefan
       O'Rear, 28-Oct-2014.)  Avoid ~ ax-pow , ~ ax-un .  (Revised by
       BTernaryTau, 12-Dec-2024.) $)
    sdom1 $p |- ( A ~< 1o <-> A = (/) ) $=
      ( vf vx c1o csdm wbr c0 wceq wne cdom cen wn wa wi csn df1o2 cv wex breq1
      mpbiri breq2i wf1 brdomi f1cdmsn vex exlimiv expcom exlimdv syl5 biimtrid
      ensn1 syl iman sylib brsdom sylnibr necon4ai 1n0 1oex 0sdom mpbir impbii
      ) ADEFZAGHZVCAGAGIZADJFZADKFZLMZVCVEVFVGNVHLVFAGOZJFZVEVGDVIAJPUAVJAVIBQZ
      UBZBRVEVGAVIBUCVEVLVGBVLVEVGVLVEMACQZOZHZCRVGCAGVKUDVOVGCVOVGVNDKFVMCUEUK
      AVNDKSTUFULUGUHUIUJVFVGUMUNADUOUPUQVDVCGDEFZVPDGIURDUSUTVAAGDESTVB $.
    $( $j usage 'sdom1' avoids 'ax-pow' 'ax-un'; $)
  $}

  $( Obsolete version of ~ sdom1 as of 12-Dec-2024.  (Contributed by Stefan
     O'Rear, 28-Oct-2014.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  sdom1OLD $p |- ( A ~< 1o <-> A = (/) ) $=
    ( c1o csdm wbr c0 wceq wn cdom domnsym con2i 0sdom1dom sylnibr wcel relsdom
    cvv wb brrelex1i 0sdomg necon2bbid syl mpbird 1oex 0sdom mpbir breq1 mpbiri
    wne 1n0 impbii ) ABCDZAEFZUJUKEACDZGZUJBAHDZULUNUJBAIJAKLUJAOMZUKUMPABCNQUO
    ULAEAORSTUAUKUJEBCDZUPBEUGUHBUBUCUDAEBCUEUFUI $.

  $( Two ways to express "at most one".  (Contributed by Stefan O'Rear,
     28-Oct-2014.) $)
  modom $p |- ( E* x ph <-> { x | ph } ~<_ 1o ) $=
    ( wmo wex weu wi wn wo cab c1o cdom wbr moeu imor csdm cen wceq abn0 bitr4i
    c0 necon1bbii sdom1 euen1 orbi12i brdom2 3bitri ) ABCABDZABEZFUGGZUHHZABIZJ
    KLZABMUGUHNUJUKJOLZUKJPLZHULUIUMUHUNUIUKTQUMUGUKTABRUAUKUBSABUCUDUKJUESUF
    $.

  ${
    $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 $.
    $( $j usage 'rex2dom' avoids 'ax-pow' 'ax-un'; $)
  $}

  ${
    $d A x y $.  $d A f x $.
    $( Strict dominance over 1 is the same as dominance over 2.  (Contributed
       by BTernaryTau, 23-Dec-2024.) $)
    1sdom2dom $p |- ( 1o ~< A <-> 2o ~<_ A ) $=
      ( vx vy vf c1o csdm wbr c2o cdom cvv wcel cv wne csn wceq wex c0 syl wtru
      cen wrex relsdom brrelex2i wo sdomdom 0sdom1dom sylibr wb mpbid n0snor2el
      0sdomg sdomnen df1o2 0ex vex en2sn mp2an eqbrtri breq2 exlimiv nsyl olcnd
      mpbiri rex2dom syl2anc wn wss cpr snsspr1 3sstr4i domssl mpan wf1 snnen2o
      df2o3 a1i 1oex 1n0 nesymi enpr2d mptru breq1 mpbii mto 2on0 f1cdmsn mpan2
      nex brdomi mtbiri con2i nexdv wf1o reldom breng wi ccnv f1of1 f1eq3 ax-mp
      f1ocnv sylib sylan expcom exlimdv sylbid mtod brsdom sylanbrc impbii ) EA
      FGZHAIGZXKAJKZBLZCLZMCAUABAUAZXLEAFUBUCZXKXPAXNNZOZBPZXKAQMZXPXTUDXKQAFGZ
      YAXKEAIGZYBEAUEAUFZUGXKXMYBYAUHZXQAJUKZRUIBCBAUJRXKEATGZXTEAULXSYGBXSYGEX
      RTGEQNZXRTUMQJKZXNJKYHXRTGUNBUOQXNJJUPUQURAXRETUSVCUTVAVBBCAJVDVEXLYCYGVF
      XKEHVGXLYCYHQEVHZEHQEVIUMVOVJEHAVKVLZXLYGXTXLXSBXSXLXSXLHXRIGZYLHXRDLZVMZ
      DPYNDYNHXONZOZCPZYPCYPYOHTGZXOVNYPHHTGYRHYJHTVOYJHTGSQEJJYISUNVPEJKZSVQVP
      QEOVFSEQVRVSVPVTWAURHYOHTWBWCWDWHYNHQMYQWECHXNYMWFWGWDWHHXRDWIWDAXRHIUSWJ
      WKWLXLYGEAYMWMZDPZXTXLXMYGUUAUHZHAIWNUCZYSXMUUBVQEADJJWOVLRXLYAUUAXTWPXLY
      BYAXLYCYBYKYDUGXLXMYEUUCYFRUIYAYTXTDYTYAXTYTAYHYMWQZVMZYAXTYTAEUUDWMZUUEE
      AYMXAUUFAEUUDVMZUUEAEUUDWREYHOUUGUUEUHUMEYHAUUDWSWTXBRBAQUUDWFXCXDXERXFXG
      EAXHXIXJ $.
    $( $j usage '1sdom2dom' avoids 'ax-pow' 'ax-un'; $)
  $}

  ${
    $d A x y $.
    $( A set that strictly dominates ordinal 1 has at least 2 different
       members.  (Closely related to ~ 2dom .)  (Contributed by Mario Carneiro,
       12-Jan-2013.)  Avoid ~ ax-un .  (Revised by BTernaryTau,
       30-Dec-2024.) $)
    1sdom $p |- ( A e. V -> ( 1o ~< A <-> E. x e. A E. y e. A -. x = y ) ) $=
      ( c1o csdm wbr c2o cdom wcel weq wn wrex 1sdom2dom 2dom wne df-ne 2rexbii
      cv rex2dom sylan2br ex impbid2 bitrid ) ECFGHCIGZCDJZABKLZBCMACMZCNUFUEUH
      ABCOUFUHUEUHUFASZBSZPZBCMACMUEUKUGABCCUIUJQRABCDTUAUBUCUD $.
    $( $j usage '1sdom' avoids 'ax-pow' 'ax-un'; $)
  $}

  ${
    $d a f x y A $.
    $( Obsolete version of ~ 1sdom as of 30-Dec-2024.  (Contributed by Mario
       Carneiro, 12-Jan-2013.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    1sdomOLD $p |- ( A e. V ->
        ( 1o ~< A <-> E. x e. A E. y e. A -. x = y ) ) $=
      ( va vf c1o cv csdm wbr wrex cdom c2o c0 cpr wf1 cop vex ccnv csn wn csuc
      weq breq2 rexeq rexeqbi1dv com wcel wb 1onn sucdom ax-mp df-2o 2dom df2o3
      breq1i wel wa wex wss wne wfun 0ex 1oex funpr df-ne wf 1n0 necomi mpbiran
      fpr df-f1 cun cnvsn uneq12i df-pr cnveqi cnvun eqtri 3eqtr4i funeqi prssi
      bitr2i 3imtr3i f1ss syl2an prex f1eq1 spcev brdom sylibr expcom rexlimivv
      syl eqbrtrid impbii 3bitr2i vtoclbg ) GEHZIJZABUCUAZBWSKZAWSKZGCIJXABCKZA
      CKECDWSCGIUDXBXDAWSCXABWSCUEUFWTGUBZWSLJZMWSLJZXCGUGUHWTXFUIUJGWSUKULMXEW
      SLUMUPXGXCABWSUNXCMNGOZWSLUOXAXHWSLJZABWSWSXAAEUQBEUQURZXIXAXJURZXHWSFHZP
      ZFUSZXIXKXHWSNAHZQZGBHZQZOZPZXNXAXHXOXQOZXSPZYAWSUTXTXJXOXQVAXONQZXQGQZOZ
      VBZXAYBXOXQNGARZBRZVCVDVEXOXQVFYBXSSZVBZYFYBXHYAXSVGZYJNGVAYKGNVHVINGXOXQ
      VCVDYGYHVKULXHYAXSVLVJYIYEXPTZSZXRTZSZVMZYCTZYDTZVMYIYEYMYQYOYRNXOVCYGVNG
      XQVDYHVNVOYIYLYNVMZSYPXSYSXPXRVPVQYLYNVRVSYCYDVPVTWAWCWDXOXQWSWBXHYAWSXSW
      EWFXMXTFXSXPXRWGXHWSXLXSWHWIWNXHWSFERWJWKWLWMWOWPWQWR $.
  $}

  ${
    $d F w z $.  $d a b m n s t w x z $.
    unxpdomlem1.1 $e |- F = ( x e. ( a u. b ) |-> G ) $.
    unxpdomlem1.2 $e |- G = if ( x e. a , <. x , if ( x = m , t , s ) >. ,
                                          <. if ( x = t , n , m ) , x >. ) $.
    $( Lemma for ~ unxpdom .  (Trivial substitution proof.)  (Contributed by
       Mario Carneiro, 13-Jan-2013.) $)
    unxpdomlem1 $p |- ( z e. ( a u. b ) -> ( F ` z ) =
       if ( z e. a , <. z , if ( z = m , t , s ) >. ,
                     <. if ( z = t , n , m ) , z >. ) ) $=
      ( cv wel weq cif cop equequ1 ifbid eqtrd elequ1 opeq1 opeq2d opeq1d opeq2
      cun ifbieq12d eqtrid opex ifex fvmpt ) ABMZGBINZULBDOZCMZHMZPZQZBCOZEMZDM
      ZPZULQZPZIMJMUFFABOZGAINZAMZADOZUOUPPZQZACOZUTVAPZVGQZPVDLVEVFUMVJVMURVCA
      BIUAVEVJULVIQURVGULVIUBVEVIUQULVEVHUNUOUPABDRSUCTVEVMVBVGQVCVEVLVBVGVEVKU
      SUTVAABCRSUDVGULVBUETUGUHKUMURVCULUQUIVBULUIUJUK $.

    ${
      unxpdomlem2.1 $e |- ( ph -> w e. ( a u. b ) ) $.
      unxpdomlem2.2 $e |- ( ph -> -. m = n ) $.
      unxpdomlem2.3 $e |- ( ph -> -. s = t ) $.
      $( Lemma for ~ unxpdom .  (Contributed by Mario Carneiro,
         13-Jan-2013.) $)
      unxpdomlem2 $p |- ( ( ph /\ ( z e. a /\ -. w e. a ) ) ->
        -. ( F ` z ) = ( F ` w ) ) $=
        ( cv wceq weq wel wn cfv adantr cif cop wcel elun1 ad2antrl unxpdomlem1
        wa cun syl iftrue eqtrd iffalse ad2antll eqeq12d biimpa ifex opth sylib
        vex simprd eqeq1d imbitrid simpld imbitrrid ad2antrr equequ1 syl5ibrcom
        syld notbid pm2.65d iffalsed mt3d 3eqtr3d mtand ) ACKUAZDKUAZUBZUKZUKZC
        RZHUCZDRZHUCZSZJETZAWIUBWBQUDWCWHUKZCFTZERZJRZUEZWFWMWLWJWDDETZGRZFRZUE
        ZSZWNWFSZWJWDWNUFZWRWFUFZSZWSWTUKWCWHXCWCWEXAWGXBWCWEVSXACETWPWQUEWDUFZ
        UEZXAWCWDKRZLRZULZUGZWEXESVSXIAWAWDXFXGUHUIBCEFGHIJKLMNUJUMVSXEXASAWAVS
        XAXDUNUIUOWCWGVTWFDFTWLWMUEUFZXBUEZXBWCWFXHUGZWGXKSAXLWBOUDBDEFGHIJKLMN
        UJUMWAXKXBSAVSVTXJXBUPUQUOURUSWDWNWRWFCVCWKWLWMEVCJVCUTVAVBZVDZWJWKWLWM
        WJWKCGTZWJWKWOXOWKWNWLSWJWOWKWLWMUNWJWNWFWLXNVEVFWOXOWJWRWPSWOWPWQUNWJW
        DWRWPWJWSWTXMVGZVEVHVLWJXOUBWKFGTZUBZAXRWBWHPVIWKXOXQCFGVJVMVKVNZVOWJWO
        WKXSWOUBWKWJWRWQSWOWPWQUPWJWDWRWQXPVEVHVPVQVR $.
    $}

    $( Lemma for ~ unxpdom .  (Contributed by Mario Carneiro, 13-Jan-2013.)
       (Revised by Mario Carneiro, 16-Nov-2014.) $)
    unxpdomlem3 $p |- ( ( 1o ~< a /\ 1o ~< b )
                        -> ( a u. b ) ~<_ ( a X. b ) ) $=
      ( vz vw cv weq wrex wa wel wceq cif c1o csdm wbr wn cun cxp cdom wb 1sdom
      cvv elv reeanv wcel w3a wf1 vex unex xpex wf cfv wi wral cop simpr simp2r
      simp1r ifcld ad2antrr opelxpd simp2l simp1l wo elun sylib ifclda eqeltrid
      orcanai fmptd unxpdomlem1 ad2antrl iftrue adantr sylan9eq ad2antll adantl
      eqeq12d ifex opth1 biimtrdi simprr simpll simplr unxpdomlem2 eqcom simprl
      pm2.21d ancom2s biimtrid iffalse opth 4casesdan ralrimivva 3ad2ant3 dff13
      simprbi sylanbrc f1dom2g mp3an12i rexlimdvva biimtrrid rexlimivv syl2anb
      3expia sylbir ) UAHNZUBUCZCDOUDZDXOPZCXOPZGBOUDZBINZPZGYAPZXOYAUEZXOYAUFZ
      UGUCZUAYAUBUCZXPXSUHHCDXOUJUIUKYGYCUHIGBYAUJUIUKXSYCQXRYBQZGYAPCXOPYFXRYB
      CGXOYAULYHYFCGXOYAYHXQXTQZBYAPDXOPCHRZGIRZQZYFXQXTDBXOYAULYLYIYFDBXOYAYLD
      HRZBIRZQZYIYFYDUJUMYEUJUMYLYOYIUNZYDYEEUOZYFXOYAHUPZIUPZUQXOYAYRYSURYPYDY
      EEUSLNZEUTZMNZEUTZSZLMOZVAZMYDVBLYDVBZYQYPAYDFYEEYPANZYDUMZQZFAHRZUUHACOZ
      BNZGNZTZVCZABOZDNZCNZTZUUHVCZTYEKUUJUUKUUPUVAYEUUJUUKQUUHUUOXOYAUUJUUKVDY
      PUUOYAUMUUIUUKYPUULUUMUUNYAYLYMYNYIVEYJYKYOYIVFVGVHVIUUJUUKUDZQUUTUUHXOYA
      YPUUTXOUMUUIUVBYPUUQUURUUSXOYLYMYNYIVJYJYKYOYIVKVGVHUUJUUKAIRZUUJUUIUUKUV
      CVLYPUUIVDUUHXOYAVMVNVQVIVOVPJVRYIYLUUGYOYIUUFLMYDYDYIYTYDUMZUUBYDUMZQZQZ
      LHRZMHRZUUFUVGUVHUVIQZQZUUDYTLCOZUUMUUNTZVCZUUBMCOUUMUUNTZVCZSUUEUVKUUAUV
      NUUCUVPUVGUVJUUAUVHUVNLBOZUURUUSTZYTVCZTZUVNUVDUUAUVTSYIUVEALBCDEFGHIJKVS
      VTZUVHUVTUVNSUVIUVHUVNUVSWAWBWCUVGUVJUUCUVIUVPMBOUURUUSTZUUBVCZTZUVPUVEUU
      CUWDSYIUVDAMBCDEFGHIJKVSWDZUVIUWDUVPSUVHUVIUVPUWCWAWEWCWFYTUVMUUBUVOLUPZU
      VLUUMUUNBUPGUPWGWHWIUVGUVHUVIUDZQQUUDUUEUVGALMBCDEFGHIJKYIUVDUVEWJXQXTUVF
      WKZXQXTUVFWLZWMWPUUDUUCUUASZUVGUVHUDZUVIQQZUUEUUAUUCWNUWLUWJUUEUVGUVIUWKU
      WJUDUVGAMLBCDEFGHIJKYIUVDUVEWOUWHUWIWMWQWPWRUVGUWKUWGQZQZUUDUVSUWCSZUUEUW
      NUUAUVSUUCUWCUVGUWMUUAUVTUVSUWAUWKUVTUVSSUWGUVHUVNUVSWSWBWCUVGUWMUUCUWDUW
      CUWEUWGUWDUWCSUWKUVIUVPUWCWSWEWCWFUWOUVRUWBSUUEUVRYTUWBUUBUVQUURUUSDUPCUP
      WGUWFWTXEWIXAXBXCLMYDYEEXDXFYDYEEUJUJXGXHXMXIXJXKXNXL $.
  $}

  ${
    $d A x y $.  $d B y $.  $d t u v w x y z $.
    $( Cartesian product dominates union for sets with cardinality greater than
       1.  Proposition 10.36 of [TakeutiZaring] p. 93.  (Contributed by Mario
       Carneiro, 13-Jan-2013.)  (Proof shortened by Mario Carneiro,
       27-Apr-2015.) $)
    unxpdom $p |- ( ( 1o ~< A /\ 1o ~< B ) -> ( A u. B ) ~<_ ( A X. B ) ) $=
      ( vx vy vz vw vv vu cvv wcel wa c1o csdm wbr cun cxp cdom cv wi cif breq2
      vt relsdom brrelex2i anim12i wceq anbi1d uneq1 xpeq1 breq12d anbi2d uneq2
      imbi12d xpeq2 weq cop cmpt eqid unxpdomlem3 vtocl2g mpcom ) AIJZBIJZKLAMN
      ZLBMNZKZABOZABPZQNZVDVBVEVCLAMUCUDLBMUCUDUELCRZMNZLDRZMNZKZVJVLOZVJVLPZQN
      ZSVDVMKZAVLOZAVLPZQNZSVFVISCDABIIVJAUFZVNVRVQWAWBVKVDVMVJALMUAUGWBVOVSVPV
      TQVJAVLUHVJAVLUIUJUMVLBUFZVRVFWAVIWCVMVEVDVLBLMUAUKWCVSVGVTVHQVLBAULVLBAU
      NUJUMEFGHEVOERZVJJWDEGUOFRUBRTUPEFUOHRGRTWDUPTZUQZWEUBCDWFURWEURUSUTVA $.
  $}

  $( Corollary of ~ unxpdom .  (Contributed by NM, 16-Sep-2004.) $)
  unxpdom2 $p |- ( ( 1o ~< A /\ B ~<_ A ) -> ( A u. B ) ~<_ ( A X. A ) ) $=
    ( c1o csdm wbr cdom cun csn cxp c0 cen cvv wcel com xpsneng sylancl domentr
    ensymd syl2anc sdomentr wa cin wceq relsdom brrelex2i adantr 1onn endom syl
    simpr 0ex wne 1n0 xpsndisj mp1i undom syl21anc syldan unxpdom xpen domtr )
    CADEZBAFEZUAZABGZACHIZAJHIZGZFEZVHAAIZFEZVEVJFEVDAVFFEZBVGFEZVFVGUBJUCZVIVD
    AVFKEZVLVDVFAVDALMZCNMVFAKEZVBVPVCCADUDUEUFZUGACLNOPZRZAVFUHUIVDVCAVGKEZVMV
    BVCUJVDVGAVDVPJLMVGAKEZVRUKAJLLOPZRZBAVGQSCJULVNVDUMACAJUNUOAVFBVGUPUQVDVHV
    FVGIZFEZWEVJKEZVKVDCVFDEZCVGDEZWFVBVCVOWHVTCAVFTURVBVCWAWIWDCAVGTURVFVGUSSV
    DVQWBWGVSWCVFAVGAUTSVHWEVJQSVEVHVJVAS $.

  $( Cartesian product dominates successor for set with cardinality greater
     than 1.  Proposition 10.38 of [TakeutiZaring] p. 93 (but generalized to
     arbitrary sets, not just ordinals).  (Contributed by NM, 3-Sep-2004.)
     (Proof shortened by Mario Carneiro, 27-Apr-2015.) $)
  sucxpdom $p |- ( 1o ~< A -> suc A ~<_ ( A X. A ) ) $=
    ( c1o csdm wbr csn cun cxp cdom c0 cen cvv wcel con0 xpsneng sylancl ensymd
    syl sdomentr syl2anc mpdan csuc df-suc cin wceq relsdom brrelex2i 1on endom
    ensn1g ensdomtr mpancom 0ex sdomdom wne 1n0 xpsndisj undom syl21anc unxpdom
    mp1i domtr xpen domentr eqbrtrid ) BACDZAUAAAEZFZAAGZHAUBVEVGABEGZAIEGZGZHD
    ZVKVHJDZVGVHHDVEVGVIVJFZHDZVNVKHDZVLVEAVIHDZVFVJHDZVIVJUCIUDZVOVEAVIJDZVQVE
    VIAVEAKLZBMLVIAJDZBACUEUFZUGABKMNOZPZAVIUHQVEVFVJCDZVRVEVFACDZAVJJDZWFVFBJD
    ZVEWGVEWAWIWCAKUIQVFBAUJUKVEVJAVEWAIKLVJAJDZWCULAIKKNOZPZVFAVJRSVFVJUMQBIUN
    VSVEUOABAIUPUTAVIVFVJUQURVEBVICDZBVJCDZVPVEVTWMWEBAVIRTVEWHWNWLBAVJRTVIVJUS
    SVGVNVKVASVEWBWJVMWDWKVIAVJAVBSVGVKVHVCSVD $.

  $( A set equinumerous to a proper subset of itself is infinite.  Corollary
     6D(a) of [Enderton] p. 136.  (Contributed by NM, 2-Jun-1998.) $)
  pssinf $p |- ( ( A C. B /\ A ~~ B ) -> -. B e. Fin ) $=
    ( wpss cen wbr cfn wcel wn csdm php3 ex sdomnen syl6com con2d imp ) ABCZABD
    EZBFGZHPRQRPABIEZQHRPSBAJKABLMNO $.

  $( 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 ) $=
    ( cen wbr wss cfn wcel wceq wne wpss df-pss pssinf expcom biimtrrid expdimp
    wa wn necon4ad 3impia 3com13 ) ABCDZABEZBFGZABHZUAUBUCUDUAUBPUCABUAUBABIZUC
    QZUBUEPABJZUAUFABKUGUAUFABLMNORST $.

  $( The set of natural numbers is infinite.  Corollary 6D(b) of [Enderton]
     p. 136.  (Contributed by NM, 2-Jun-1998.)  Avoid ~ ax-pow .  (Revised by
     BTernaryTau, 2-Jan-2025.) $)
  ominf $p |- -. _om e. Fin $=
    ( vx com cfn wcel wn wi cv cen wbr wrex isfi wpss wss wne wa wb nnord ordom
    word ordelssne sylancl ibi df-pss sylibr nnfi ensymfib syl biimpar syl2an2r
    pssinf rexlimiva sylbi pm2.01 ax-mp ) BCDZUOEZFUPUOBAGZHIZABJUPABKURUPABUQB
    DZUQBLZURUQBHIZUPUSUQBMUQBNOZUTUSVBUSUQSBSUSVBPUQQRUQBTUAUBUQBUCUDUSVAURUSU
    QCDVAURPUQUEUQBUFUGUHUQBUJUIUKULUOUMUN $.
  $( $j usage 'ominf' avoids 'ax-pow'; $)

  $( Obsolete version of ~ ominf as of 2-Jan-2025.  (Contributed by NM,
     2-Jun-1998.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  ominfOLD $p |- -. _om e. Fin $=
    ( vx com cfn wcel wn wi cv cen wbr wrex isfi wpss wss wne wa wb nnord ordom
    word ordelssne ibi df-pss sylibr ensym pssinf syl2an rexlimiva sylbi pm2.01
    sylancl ax-mp ) BCDZULEZFUMULBAGZHIZABJUMABKUOUMABUNBDZUNBLZUNBHIUMUOUPUNBM
    UNBNOZUQUPURUPUNSBSUPURPUNQRUNBTUJUAUNBUBUCBUNUDUNBUEUFUGUHULUIUK $.

  ${
    $d A m n x y $.  $d f g m x z $.  $d A f m x y z $.
    $( Any set that is not finite is literally infinite, in the sense that it
       contains subsets of arbitrarily large finite cardinality.  (It cannot be
       proven that the set has _countably infinite_ subsets unless AC is
       invoked.)  The proof does not require the Axiom of Infinity.
       (Contributed by Mario Carneiro, 15-Jan-2013.)  Avoid ~ ax-pow .
       (Revised by BTernaryTau, 2-Jan-2025.) $)
    isinf $p |- ( -. A e. Fin -> A. n e. _om E. x ( x C_ A /\ x ~~ n ) ) $=
      ( vm vy vf vz vg wcel cv wss cen wbr wa wex com c0 wceq wi wf1o cfn breq2
      wn anbi2d exbidv wb sseq1 adantl breq1 sylan9bbr anbi12d cbvexdvaw peano1
      csuc 0ss enrefnn ax-mp 0ex spcev mp2an a1i w3a cdif ssdif0 eqss wrex rspe
      biimpa sylan2 isfi sylibr sylanbr ex sylan2br 3impd com12 con3d bren neq0
      expcom csn cun eldifi snssd unss biimpi ad2ant2r cop cin vex f1osn eldifn
      jctr disjsn word nnord orddisj syl anim12i syl2an df-suc f1oeq3 snex unex
      f1oun f1oeq1 sylbir adantll syl2anc exlimiv sylbi com13 3imp21 syld com3l
      3expia finds2 ralrimiv ) BUAIZUCZAJZBKZYACJZLMZNZAOZCPYCPIXTYFYFYBYAQLMZN
      ZAOZYBYADJZLMZNZAOZEJZBKZYNYJUNZLMZNZEOZXTCDYCQRZYEYHAYTYDYGYBYCQYALUBUDU
      EYCYJRZYEYLAUUAYDYKYBYCYJYALUBUDUEYCYPRZYEYRAEUUBYAYNRZNYBYOYDYQUUCYBYOUF
      UUBYAYNBUGUHUUCYDYNYCLMUUBYQYAYNYCLUIYCYPYNLUBUJUKULYIXTQBKZQQLMZYIBUOQPI
      UUEUMQUPUQYHUUDUUENAQURYAQRYBUUDYGUUEYAQBUGYAQQLUIUKUSUTVAYMYJPIZXTYSYLUU
      FXTYSSZSAYBYKUUFUUGYBYKUUFVBZXTBYAVCZQRZUCZYSUUHUUJXSUUJUUHXSUUJYBYKUUFXS
      YBUUJYKUUFXSSZSZUUJYBBYAKZUUMBYAVDYBUUNNZYKUULUUOYABRZYKUULYABVEUUFUUPYKN
      ZXSUUFUUQNBYJLMZDPVFZXSUUQUUFUURUUSUUPYKUURYABYJLUIVHUURDPVGVIDBVJVKVTVLV
      MVNVTVOVPVQYKYBUUFUUKYSSZYKYAYJFJZTZFOYBUUFUUTSZSZYAYJFVRUVBUVDFYBUVBUVCU
      UKUUFYBUVBNZYSUUKGJZUUIIZGOUUFUVEYSSZSZGUUIVSUVGUVIGUVGUUFUVHUVEUVGUUFNZY
      SUVEUVJNYAUVFWAZWBZBKZUVLYPLMZYSYBUVGUVMUVBUUFUVGYBUVKBKZUVMUVGUVFBUVFBYA
      WCWDYBUVONUVMYAUVKBWEWFVIWGUVBUVJUVNYBUVBUVJNUVLYJYJWAZWBZUVAUVFYJWHZWAZW
      BZTZUVNUVBUVBUVKUVPUVSTZNYAUVKWIQRZYJUVPWIQRZNUWAUVJUVBUWBUVFYJGWJDWJWKWM
      UVGUWCUUFUWDUVGUVFYAIUCUWCUVFBYAWLYAUVFWNVKUUFYJWOUWDYJWPYJWQWRWSYAYJUVKU
      VPUVAUVSXEWTUWAUVLYPUVTTZUVNYPUVQRUWEUWAUFYJXAYPUVQUVLUVTXBUQUWEUVLYPHJZT
      ZHOUVNUWGUWEHUVTUVAUVSFWJUVRXCXDUVLYPUWFUVTXFUSUVLYPHVRVKXGWRXHYRUVMUVNNE
      UVLYAUVKAWJUVFXCXDYNUVLRYOUVMYQUVNYNUVLBUGYNUVLYPLUIUKUSXIVTVMXJXKXLVTXJX
      KXMXNXPXJXOXQVPXR $.
    $( $j usage 'isinf' avoids 'ax-pow'; $)
  $}

  ${
    $d A f m x y z $.  $d A m n x y $.  $d f g m x y z $.
    $( Obsolete version of ~ isinf as of 2-Jan-2025.  (Contributed by Mario
       Carneiro, 15-Jan-2013.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    isinfOLD $p |- ( -. A e. Fin -> A. n e. _om E. x ( x C_ A /\ x ~~ n ) ) $=
      ( vm vy vf vz vg wcel cv wss cen wbr wa wex com c0 wceq wi wf1o cfn breq2
      wn csuc anbi2d exbidv wb sseq1 adantl sylan9bbr anbi12d cbvexdvaw 0ss 0ex
      breq1 enref spcev mp2an a1i cdif ssdif0 eqss wrex biimpa rspe sylan2 isfi
      w3a sylibr expcom sylanbr ex sylan2br 3impd com12 con3d bren neq0 csn cun
      eldifi snssd unss biimpi ad2ant2r cop f1osn jctr eldifn disjsn word nnord
      cin vex orddisj anim12i f1oun syl2an df-suc f1oeq3 ax-mp snex unex f1oeq1
      syl sylbir adantll syl2anc exlimiv sylbi com13 3imp21 3expia com3l finds2
      syld ralrimiv ) BUAIZUCZAJZBKZXTCJZLMZNZAOZCPYBPIXSYEYEYAXTQLMZNZAOZYAXTD
      JZLMZNZAOZEJZBKZYMYIUDZLMZNZEOZXSCDYBQRZYDYGAYSYCYFYAYBQXTLUBUEUFYBYIRZYD
      YKAYTYCYJYAYBYIXTLUBUEUFYBYORZYDYQAEUUAXTYMRZNYAYNYCYPUUBYAYNUGUUAXTYMBUH
      UIUUBYCYMYBLMUUAYPXTYMYBLUOYBYOYMLUBUJUKULYHXSQBKZQQLMZYHBUMQUNUPYGUUCUUD
      NAQUNXTQRYAUUCYFUUDXTQBUHXTQQLUOUKUQURUSYLYIPIZXSYRYKUUEXSYRSZSAYAYJUUEUU
      FYAYJUUEVHZXSBXTUTZQRZUCZYRUUGUUIXRUUIUUGXRUUIYAYJUUEXRYAUUIYJUUEXRSZSZUU
      IYABXTKZUULBXTVAYAUUMNZYJUUKUUNXTBRZYJUUKXTBVBUUEUUOYJNZXRUUEUUPNBYILMZDP
      VCZXRUUPUUEUUQUURUUOYJUUQXTBYILUOVDUUQDPVEVFDBVGVIVJVKVLVMVJVNVOVPYJYAUUE
      UUJYRSZYJXTYIFJZTZFOYAUUEUUSSZSZXTYIFVQUVAUVCFYAUVAUVBUUJUUEYAUVANZYRUUJG
      JZUUHIZGOUUEUVDYRSZSZGUUHVRUVFUVHGUVFUUEUVGUVDUVFUUENZYRUVDUVINXTUVEVSZVT
      ZBKZUVKYOLMZYRYAUVFUVLUVAUUEUVFYAUVJBKZUVLUVFUVEBUVEBXTWAWBYAUVNNUVLXTUVJ
      BWCWDVFWEUVAUVIUVMYAUVAUVINUVKYIYIVSZVTZUUTUVEYIWFZVSZVTZTZUVMUVAUVAUVJUV
      OUVRTZNXTUVJWMQRZYIUVOWMQRZNUVTUVIUVAUWAUVEYIGWNDWNWGWHUVFUWBUUEUWCUVFUVE
      XTIUCUWBUVEBXTWIXTUVEWJVIUUEYIWKUWCYIWLYIWOXEWPXTYIUVJUVOUUTUVRWQWRUVTUVK
      YOUVSTZUVMYOUVPRUWDUVTUGYIWSYOUVPUVKUVSWTXAUWDUVKYOHJZTZHOUVMUWFUWDHUVSUU
      TUVRFWNUVQXBXCUVKYOUWEUVSXDUQUVKYOHVQVIXFXEXGYQUVLUVMNEUVKXTUVJAWNUVEXBXC
      YMUVKRYNUVLYPUVMYMUVKBUHYMUVKYOLUOUKUQXHVJVLXIXJXKVJXIXJXLXPXMXIXNXOVOXQ
      $.
  $}

  ${
    $d b c d e A $.  $d b c e V $.
    $( Lemma for ~ fineqv .  (Contributed by Mario Carneiro, 20-Jan-2013.)
       (Proof shortened by Stefan O'Rear, 3-Nov-2014.)  (Revised by Mario
       Carneiro, 17-May-2015.) $)
    fineqvlem $p |- ( ( A e. V /\ -. A e. Fin ) -> _om ~<_ ~P ~P A ) $=
      ( vb vc vd ve wcel wa cpw cvv com wbr cv cen crab wss wb syl wceq wi cdom
      cfn wn pwexg adantr pwexd ssrab2 elpw2g mpbiri a1d wex r19.21bi ad2ant2lr
      isinf velpw biimpri anim1i breq1 elrab sylibr eximi eleq2 biimpcd simprbi
      adantl ensym entr sylan syl2an ex nneneq biimpd ad2antlr exlimddv rabbidv
      3syld breq2 impbid1 dom2d mpd ) ABGZAUBGUCZHZAIZIZJGKWEUALWCWDJWAWDJGZWBA
      BUDUEZUFWCCDKWEEMZCMZNLZEWDOZWHDMZNLZEWDOZJWCWKWEGZWIKGZWCWOWKWDPZWJEWDUG
      WCWFWOWQQWGWKWDJUHRUIUJWCWPWLKGZHZWKWNSZWIWLSZQWCWSHZWTXAXBFMZWKGZWTXATFX
      BXCAPZXCWINLZHZFUKZXDFUKWBWPXHWAWRWBXHCKFACUNULUMXGXDFXGXCWDGZXFHXDXEXIXF
      XIXEFAUOUPUQWJXFEXCWDWHXCWINURUSZUTVARXBXDHWTXCWNGZWIWLNLZXAXDWTXKTXBWTXD
      XKWKWNXCVBVCVEXDXKXLTXBXDXKXLXDXFXCWLNLZXLXKXDXIXFXJVDXKXIXMWMXMEXCWDWHXC
      WLNURUSVDXFWIXCNLXMXLXCWIVFWIXCWLVGVHVIVJVEWSXLXATWCXDWSXLXAWIWLVKVLVMVPV
      NXAWJWMEWDWIWLWHNVQVOVRVJVSVT $.
  $}

  $( If the Axiom of Infinity is denied, then all sets are finite (which
     implies the Axiom of Choice).  (Contributed by Mario Carneiro,
     20-Jan-2013.)  (Revised by Mario Carneiro, 3-Jan-2015.) $)
  fineqv $p |- ( -. _om e. _V <-> Fin = _V ) $=
    ( va com cvv wcel wn cfn wceq wss ssv a1i cpw cdom wbr vex fineqvlem reldom
    cv mpan brrelex1i syl con1i a1d ssrdv eqssd ominf eleq2 mtbii impbii ) BCDZ
    EZFCGZUJFCFCHUJFIJUJACFUJAQZFDZULCDZUMUIUMEZBULKKZLMZUIUNUOUQANULCORBUPLPST
    UAUBUCUDUKBFDUIUEFCBUFUGUH $.
  $( $j usage 'fineqv' avoids 'ax-inf' 'ax-inf2'; $)

  ${
    $( The components of a nonempty finite Cartesian product are finite.
       (Contributed by Paul Chapman, 11-Apr-2009.)  (Proof shortened by Mario
       Carneiro, 29-Apr-2015.) $)
    xpfir $p |- ( ( ( A X. B ) e. Fin /\ ( A X. B ) =/= (/) ) ->
                 ( A e. Fin /\ B e. Fin ) ) $=
      ( cxp cfn wcel c0 wne wa cdom wbr cvv xpexr2 simpld simprd xpdom3 syl3anc
      simpr domfi syldan syl2anc xpnz sylibr cen xpcomeng domentr jca ) ABCZDEZ
      UGFGZHZADEZBDEZUHUIAUGIJZUKUJAKEZBKEZBFGZUMUJUNUOABDLZMZUJUNUOUQNZUJAFGZU
      PUJUIUTUPHUHUIQABUAUBZNABKKOPUGARSUHUIBUGIJZULUJBBACZIJZVCUGUCJZVBUJUOUNU
      TVDUSURUJUTUPVAMBAKKOPUJUOUNVEUSURBAKKUDTBVCUGUETUGBRSUF $.
  $}

  ${
    ssfid.1 $e |- ( ph -> A e. Fin ) $.
    ssfid.2 $e |- ( ph -> B C_ A ) $.
    $( A subset of a finite set is finite, deduction version of ~ ssfi .
       (Contributed by Glauco Siliprandi, 21-Nov-2020.) $)
    ssfid $p |- ( ph -> B e. Fin ) $=
      ( cfn wcel wss ssfi syl2anc ) ABFGCBHCFGDEBCIJ $.
  $}

  $( The intersection of two sets is finite if one of them is.  (Contributed by
     Thierry Arnoux, 14-Feb-2017.) $)
  infi $p |- ( A e. Fin -> ( A i^i B ) e. Fin ) $=
    ( cfn wcel cin wss inss1 ssfi mpan2 ) ACDABEZAFJCDABGAJHI $.

  ${
    $d x A $.
    $( A restricted class built from a finite set is finite.  (Contributed by
       Thierry Arnoux, 14-Feb-2017.) $)
    rabfi $p |- ( A e. Fin -> { x e. A | ph } e. Fin ) $=
      ( cfn wcel crab cab cin dfrab3 infi eqeltrid ) CDEABCFCABGZHDABCICLJK $.
  $}

  $( The restriction of a finite set is finite.  (Contributed by Alexander van
     der Vekens, 3-Jan-2018.) $)
  finresfin $p |- ( E e. Fin -> ( E |` B ) e. Fin ) $=
    ( cfn wcel cres wss resss ssfi mpan2 ) BCDBAEZBFJCDBAGBJHI $.

  $( Any injection from one finite set to another of equal size must be a
     bijection.  (Contributed by Jeff Madsen, 5-Jun-2010.)  (Revised by Mario
     Carneiro, 27-Feb-2014.)  Avoid ~ ax-pow .  (Revised by BTernaryTau,
     4-Jan-2025.) $)
  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 wfo simpr wfn crn wceq wf csdm wi sylanbrc
    ex f1f adantl ffnd simpll wne wpss wn wss wb frnd df-pss baib php3 ad2antlr
    enfii ancoms f1f1orn f1oenfi syl2an cdom endom domsdomtrfi syl3an2 syl2an2r
    syl 3expia syld sdomnen syl6 sylbird necon4ad df-fo df-f1o f1of1 impbid1
    mpd ) ABDEZBFGZHZABCIZABCJZVSVTWAVSVTHZVTABCKZWAVSVTLWBCAMCNZBOZWCWBABCVTAB
    CPVSABCUAUBZUCWBVQWEVQVRVTUDWBVQWDBWBWDBUEZWDBUFZVQUGZWBWDBUHZWHWGUIWBABCWF
    UJWHWJWGWDBUKULVEWBWHABQEZWIWBWHWDBQEZWKVRWHWLRVQVTVRWHWLBWDUMTUNVSAFGZVTAW
    DDEZWLWKRVRVQWMABUOUPZVSWMAWDCJWNVTWOABCUQAWDCURUSWMWNWLWKWNWMAWDUTEWLWKAWD
    VAAWDBVBVCVFVDVGABVHVIVJVKVPABCVLSABCVMSTABCVNVO $.
  $( $j usage 'f1finf1o' avoids 'ax-pow'; $)

  $( Obsolete version of ~ f1finf1o as of 4-Jan-2025.  (Contributed by Jeff
     Madsen, 5-Jun-2010.)  (Revised by Mario Carneiro, 27-Feb-2014.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  f1finf1oOLD $p |- ( ( A ~~ B /\ B e. Fin ) ->
    ( F : A -1-1-> B <-> F : A -1-1-onto-> B ) ) $=
    ( cen wbr cfn wcel wa wf1 wf1o wfo simpr wfn crn adantl syl cvv ex sylanbrc
    csdm wceq wf f1f ffnd simpll wne wpss wn wb frnd df-pss baib cmap co simplr
    wss brrelex1i elmapd mpbird f1f1orn f1oen3g syl2anc wi php3 ensdomtr syl6an
    relen sdomnen syl6 sylbird necon4ad mpd df-fo df-f1o f1of1 impbid1 ) ABDEZB
    FGZHZABCIZABCJZVSVTWAVSVTHZVTABCKZWAVSVTLWBCAMCNZBUAZWCWBABCVTABCUBZVSABCUC
    OZUDWBVQWEVQVRVTUEZWBVQWDBWBWDBUFZWDBUGZVQUHZWBWDBUPZWJWIUIWBABCWGUJWJWLWIW
    DBUKULPWBWJABTEZWKWBAWDDEZWJWDBTEZWMWBCBAUMUNZGZAWDCJZWNWBWQWFWGWBBACFQVQVR
    VTUOZWBVQAQGWHABDVGUQPURUSVTWRVSABCUTOAWDCWPVAVBWBVRWJWOVCWSVRWJWOBWDVDRPAW
    DBVEVFABVHVIVJVKVLABCVMSABCVNSRABCVOVP $.

  ${
    $d A x $.
    $( If a class is not finite, then it contains at least one element.
       (Contributed by Alexander van der Vekens, 12-Jan-2018.) $)
    nfielex $p |- ( -. A e. Fin -> E. x x e. A ) $=
      ( cfn wcel wn c0 wceq cv wex 0fi eleq1 mpbiri con3i neq0 sylib ) BCDZEBFG
      ZEAHBDAIQPQPFCDJBFCKLMABNO $.
  $}

  ${
    $d A x $.  $d B x $.
    $( A set with one element is a singleton.  (Contributed by FL,
       18-Aug-2008.)  Avoid ~ ax-pow , ~ ax-un .  (Revised by BTernaryTau,
       4-Jan-2025.) $)
    en1eqsn $p |- ( ( A e. B /\ B ~~ 1o ) -> B = { A } ) $=
      ( vx c1o cen wbr wcel csn wceq cv wex wi eleq2 elsni sneqd biimtrdi eqtr3
      en1 imp syldan ex exlimiv sylbi impcom ) BDEFZABGZBAHZIZUEBCJZHZIZCKUFUHL
      ZCBRUKULCUKUFUHUKUFUGUJIZUHUKUFUMUKUFAUJGZUMBUJAMUNAUIAUINOPSBUGUJQTUAUBU
      CUD $.
    $( $j usage 'en1eqsn' avoids 'ax-pow' 'ax-un'; $)
  $}

  $( Obsolete version of ~ en1eqsn as of 4-Jan-2025.  (Contributed by FL,
     18-Aug-2008.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  en1eqsnOLD $p |- ( ( A e. B /\ B ~~ 1o ) -> B = { A } ) $=
    ( wcel c1o cen wbr csn cfn wss wceq com 1onn ssid ssnnfi mp2an enfii adantl
    wa mpan snssi adantr ensn1g ensym entr syl2an fisseneq syl3anc eqcomd ) ABC
    ZBDEFZRZAGZBUKBHCZULBIZULBEFZULBJUJUMUIDHCZUJUMDKCDDIUPLDMDDNOBDPSQUIUNUJAB
    TUAUIULDEFDBEFUOUJABUBBDUCULDBUDUEULBUFUGUH $.

  $( A set containing an element has exactly one element iff it is a singleton.
     Formerly part of proof for ~ rngen1zr .  (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 A x $.  $d X x $.  $d M x $.
    $( Alternate proof of ~ dif1ennn using ~ ax-pow .  (Contributed by Jeff
       Madsen, 2-Sep-2009.)  (Revised by Stefan O'Rear, 16-Aug-2015.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    dif1ennnALT $p |- ( ( M e. _om /\ A ~~ suc M /\ X e. A ) ->
        ( A \ { X } ) ~~ M ) $=
      ( vx com wcel csuc cen wbr csn wrex wa cfn peano2 breq2 isfi wceq wi cin
      c0 w3a cdif cv rspcev sylibr sylan diffi sylib syl 3adant3 cun en2sn elvd
      cvv word nnord orddisj incom disjdif eqtri unen an4s mpanl2 expcom syl2an
      3ad2antl3 wb difsnid df-suc eqcomi a1i breq12d 3ad2ant3 adantr ensym entr
      nneneq sylan2 imbitrid expd syl5 imp an32s sylbid peano4 biimpd 3ad2antl1
      3adantl3 3syld biimprcd sylcom rexlimdva mpd ) BEFZABGZHIZCAFZUAZACJZUBZD
      UCZHIZDEKZWTBHIZWNWPXCWQWNWPLAMFZXCWNWOEFZWPXEBNZXFWPLAXAHIZDEKXEXHWPDWOE
      XAWOAHOUDDAPUEUFXEWTMFXCAWSUGDWTPUHUIUJWRXBXDDEWRXAEFZLZXBBXAQZXDXJXBWTWS
      UKZXAXAJZUKZHIZWOXAGZQZXKWQWNXIXBXORZWPWQWSXMHIZXAXMSTQZXRXIWQXSDCXAAUNUL
      UMXIXAUOXTXAUPXAUQUIXBXSXTLZXOXBWTWSSZTQZYAXOYBWSWTSTWTWSURWSAUSUTXBXSYCX
      TXOWTXAWSXMVAVBVCVDVEVFXJXOAXPHIZXQWRXOYDVGZXIWQWNYEWPWQXLAXNXPHACVHXNXPQ
      WQXPXNXAVIVJVKVLVMVNWNWPXIYDXQRZWQWNXIWPYFWNXILZWPYFWNXFXIWPYFRXGWPWOAHIZ
      XFXILZYFAWOVOYIYHYDXQYHYDLWOXPHIZYIXQWOAXPVPXIXFXPEFYJXQVGXANWOXPVQVRVSVT
      WAUFWBWCWHWDWNWPXIXQXKRWQYGXQXKBXAWEWFWGWIXKXDXBBXAWTHOWJWKWLWM $.
  $}

  ${
    enp1ilem.1 $e |- T = ( { x } u. S ) $.
    $( Lemma for uses of ~ enp1i .  (Contributed by Mario Carneiro,
       5-Jan-2016.) $)
    enp1ilem $p |- ( x e. A -> ( ( A \ { x } ) = S -> A = T ) ) $=
      ( cv csn cdif wceq cun wcel uneq1 undif1 uncom eqtr4i 3eqtr3g wss ssequn2
      snssi sylib eqeq1d imbitrid ) BAFZGZHZCIZBUDJZDIUCBKZBDIUFUEUDJCUDJZUGDUE
      CUDLBUDMUIUDCJDCUDNEOPUHUGBDUHUDBQUGBIUCBSUDBRTUAUB $.
  $}

  ${
    $d A x $.  $d M x $.
    enp1i.1 $e |- Ord M $.
    enp1i.2 $e |- N = suc M $.
    enp1i.3 $e |- ( ( A \ { x } ) ~~ M -> ph ) $.
    enp1i.4 $e |- ( x e. A -> ( ph -> ps ) ) $.
    $( Proof induction for ~ en2 and related theorems.  (Contributed by Mario
       Carneiro, 5-Jan-2016.)  Generalize to all ordinals and avoid ~ ax-pow ,
       ~ ax-un .  (Revised by BTernaryTau, 6-Jan-2025.) $)
    enp1i $p |- ( A ~~ N -> E. x ps ) $=
      ( cen wbr csuc wex breq2i wrex wcel cvv 3syl sylbi cv cdif con0 word encv
      csn wb simprd sssucid ssexg mpan elong mpbiri rexdif1en mpancom reximi wa
      wss df-rex imp eximi ) DFKLDEMZKLZBCNZFVBDKHOVCDCUAZUFUBEKLZCDPZACDPZVDEU
      CQZVCVGVCVIEUDZGVCVBRQZERQZVIVJUGVCDRQVKDVBUEUHEVBURVKVLEUIEVBRUJUKERULSU
      MCDEUNUOVFACDIUPVHVEDQZAUQZCNVDACDUSVNBCVMABJUTVATST $.
    $( $j usage 'enp1i' avoids 'ax-pow' 'ax-un'; $)
  $}

  ${
    $d x A $.  $d x N $.
    enp1iOLD.1 $e |- M e. _om $.
    enp1iOLD.2 $e |- N = suc M $.
    enp1iOLD.3 $e |- ( ( A \ { x } ) ~~ M -> ph ) $.
    enp1iOLD.4 $e |- ( x e. A -> ( ph -> ps ) ) $.
    $( Obsolete version of ~ enp1i as of 6-Jan-2025.  (Contributed by Mario
       Carneiro, 5-Jan-2016.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    enp1iOLD $p |- ( A ~~ N -> E. x ps ) $=
      ( cen wbr cv wcel wex c0 wceq wn csuc sylib wne nsuceq0 breq1 en0 eqtr3id
      ensym biimtrdi necon3ad mpi con2i neq0 wi breq2i wa csn cdif com dif1ennn
      mp3an1 syl ex sylbi sylcom eximdv mpd ) DFKLZCMZDNZCOZBCOVFDPQZRVIVJVFVJE
      SZPUAVFREUBVJVFVKPVJVFPFKLZVKPQDPFKUCVLVKFPHVLFPKLFPQPFUFFUDTUEUGUHUIUJCD
      UKTVFVHBCVFVHABVFDVKKLZVHAULFVKDKHUMVMVHAVMVHUNDVGUOUPEKLZAEUQNVMVHVNGDEV
      GURUSIUTVAVBJVCVDVE $.
  $}

  ${
    $d x y A $.
    $( A set equinumerous to ordinal 2 is a pair.  (Contributed by Mario
       Carneiro, 5-Jan-2016.) $)
    en2 $p |- ( A ~~ 2o -> E. x E. y A = { x , y } ) $=
      ( csn cdif wceq wex cpr c1o c2o 1on onordi df-2o cen wbr en1 biimpi df-pr
      cv wcel enp1ilem eximdv enp1i ) CASZDEZBSZDZFZBGZCUDUFHZFZBGACIJIKLMUEINO
      UIBUEPQUDCTUHUKBACUGUJUDUFRUAUBUC $.
    $( $j usage 'en2' avoids 'ax-pow' 'ax-un'; $)
  $}

  ${
    $d x y z A $.
    $( A set equinumerous to ordinal 3 is a triple.  (Contributed by Mario
       Carneiro, 5-Jan-2016.) $)
    en3 $p |- ( A ~~ 3o -> E. x E. y E. z A = { x , y , z } ) $=
      ( cv csn cdif cpr wceq wex ctp c2o c3o 2on onordi df-3o en2 wcel enp1ilem
      tpass 2eximdv enp1i ) DAEZFGZBEZCEZHZIZCJBJDUCUEUFKZIZCJBJADLMLNOPBCUDQUC
      DRUHUJBCADUGUIUCUEUFTSUAUB $.
    $( $j usage 'en3' avoids 'ax-pow' 'ax-un'; $)
  $}

  ${
    $d w x y z A $.
    $( A set equinumerous to ordinal 4 is a quadruple.  (Contributed by Mario
       Carneiro, 5-Jan-2016.) $)
    en4 $p |- ( A ~~ 4o ->
      E. x E. y E. z E. w A = ( { x , y } u. { z , w } ) ) $=
      ( cv csn cdif ctp wceq wex cpr cun c3o c4o ord3 df-4o en3 wcel qdassr
      enp1ilem eximdv 2eximdv enp1i ) EAFZGHZBFZCFZDFZIZJZDKZCKBKEUEUGLUHUILMZJ
      ZDKZCKBKAENOPQBCDUFRUEESZULUOBCUPUKUNDAEUJUMUEUGUHUITUAUBUCUD $.
    $( $j usage 'en4' avoids 'ax-pow' 'ax-un'; $)
  $}

  ${
    $d ch x $.  $d ta w x $.  $d x y z $.  $d A w x $.  $d ph w y z $.
    findcard3.1 $e |- ( x = y -> ( ph <-> ch ) ) $.
    findcard3.2 $e |- ( x = A -> ( ph <-> ta ) ) $.
    findcard3.3 $e |- ( y e. Fin -> ( A. x ( x C. y -> ph ) -> ch ) ) $.
    $( Schema for strong induction on the cardinality of a finite set.  The
       inductive hypothesis is that the result is true on any proper subset.
       The result is then proven to be true for all finite sets.  (Contributed
       by Mario Carneiro, 13-Dec-2013.)  Avoid ~ ax-pow .  (Revised by
       BTernaryTau, 7-Jan-2025.) $)
    findcard3 $p |- ( A e. Fin -> ta ) $=
      ( vw vz cfn wcel cen wbr wi wal wa com wrex cv isfi wral con0 nnon eleq1w
      breq2 imbi1d albidv imbi12d wpss rspe sylibr 19.21v ralbii ralcom4 bitr3i
      weq wss pssss ssfi syl2an csdm simprl wb nnfi ensymfib syl biimpar adantl
      sylib simplll php3 sylan adantr simpllr cdom endom 3adant2 sdomdom sylan2
      3adant3 syld3an1 sdomdomtrfi syl3an3 syl3anc syl3an1 syl3an2 nnsdomo word
      domfi domsdomtrfi nnord ordelpss bitr4d syl2anc mpbid simpr jca2 reximdv2
      ex mpd r19.29 expcom ordom ordelss mpan ad2antrr sseld impd syl6 rexlimdv
      pm2.27 syld com23 alimdv biimtrid sylsyld alrimiv breq1 cbvalvw imbitrrdi
      impancom a1i tfis2 mpcom rgen sylbi wceq spcgv rexlimdvw ) FLMZDUAZJUAZNO
      ZAPZDQZFYNNOZRZJSTZCYLYRJSTZYTJFUBYQJSUCUUAYTYQJSYNUDMZYNSMZYQYNUEUUCYQPZ
      KUAZSMZYMUUENOZAPZDQZPZJKJKURZUUCUUFYQUUIJKSUFUUKYPUUHDUUKYOUUGAYNUUEYMNU
      GUHUIUJUUJKYNUCZUUDPUUBUULUUCEUAZYNNOZBPZEQZYQUUCUULUUPUUCUULRUUOEUUCUUNU
      ULBUUCUUNRZUUMLMZUULYMUUMUKZAPZDQZBUUQUUNJSTUURUUNJSULJUUMUBUMZUULUUFUUHP
      ZKYNUCZDQZUUQUVAUULUVCDQZKYNUCUVEUVFUUJKYNUUFUUHDUNUOUVCKDYNUPUQUUQUVDUUT
      DUUQUUSUVDAUUQUUSUVDAPUUQUUSRZUVDUVCUUGRZKYNTZAUVGUUGKYNTZUVDUVIPUVGUUGKS
      TZUVJUUQUURYMUUMUSZUVKUUSUVBYMUUMUTUURUVLRYMLMZUVKUUMYMVAKYMUBVKVBUVGUUGU
      UGKSYNUVGUUFUUGRZUUEYNMZUUGUVGUVNUVOUVGUVNRZUUEYNVCOZUVOUVPUUFUUEYMNOZYMY
      NVCOZUVQUVGUUFUUGVDZUVNUVRUVGUUFUVRUUGUUFUUELMZUVRUUGVEUUEVFZUUEYMVGVHVIV
      JUVPUUCYMUUMVCOZUUNUVSUUCUUNUUSUVNVLZUVGUWCUVNUUQUURUUSUWCUVBUUMYMVMVNVOU
      UCUUNUUSUVNVPUUNUUCUWCUUMYNVQOZUVSUUMYNVRUVMUWCUUCUWEUVSUURUWCUUCUWEUVMUU
      CUWEUURUWCUUCYNLMUWEUURYNVFYNUUMWKVNVSUURUWCUVMUWEUWCUURYMUUMVQOUVMYMUUMV
      TUUMYMWKWAWBWCYMUUMYNWDWCWEWFUVRUUFUUEYMVQOZUVSUVQUUEYMVRUUFUWAUWFUVSUVQU
      WBUUEYMYNWLWGWHWFUVPUUFUUCUVQUVOVEUVTUWDUUFUUCRUVQUUEYNUKZUVOUUEYNWIUUFUU
      EWJYNWJUVOUWGVEUUCUUEWMYNWMUUEYNWNVBWOWPWQXAUUFUUGWRWSWTXBUVDUVJUVIUVCUUG
      KYNXCXDVHUVGUVHAKYNUVGUVOUUFUVHAPUVGYNSUUEUUCYNSUSZUUNUUSSWJUUCUWHXESYNXF
      XGXHXIUUFUVCUUGAUUFUUHXMXJXKXLXNXAXOXPXQIXRYCXSXDYPUUODEDEURYOUUNABYMUUMY
      NNXTGUJYAYBYDYEYFYGYQYRJSXCXGYHYLYSCJSYLYQYRCYPYRCPDFLYMFYIYOYRACYMFYNNXT
      HUJYJXJYKXB $.
    $( $j usage 'findcard3' avoids 'ax-pow'; $)
  $}

  ${
    $d w x y z $.  $d w y z ph $.  $d w x A $.  $d w x ta $.  $d x ch $.
    findcard3OLD.1 $e |- ( x = y -> ( ph <-> ch ) ) $.
    findcard3OLD.2 $e |- ( x = A -> ( ph <-> ta ) ) $.
    findcard3OLD.3 $e |- ( y e. Fin -> ( A. x ( x C. y -> ph ) -> ch ) ) $.
    $( Obsolete version of ~ findcard3 as of 7-Jan-2025.  (Contributed by Mario
       Carneiro, 13-Dec-2013.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    findcard3OLD $p |- ( A e. Fin -> ta ) $=
      ( vw vz cfn wcel cen wbr wi wal wa com wrex cv isfi wral con0 nnon eleq1w
      wceq breq2 imbi1d albidv imbi12d wpss sylibr 19.21v ralbii ralcom4 bitr3i
      rspe wss pssss ssfi sylib csdm ensym ad2antll php3 sylan simpllr sdomentr
      syl2an syl2an2r ensdomtr syl2anc ad2antrl ad3antrrr sdomel mpd simpr jca2
      ex reximdv2 r19.29 expcom syl word ordom ordelss mpan ad2antrr sseld impd
      pm2.27 syl6 rexlimdv com23 alimdv biimtrid sylsyld impancom alrimiv breq1
      syld cbvalvw imbitrrdi a1i tfis2 mpcom rgen sylbi spcgv rexlimdvw ) FLMZD
      UAZJUAZNOZAPZDQZFXNNOZRZJSTZCXLXRJSTZXTJFUBXQJSUCYAXTXQJSXNUDMZXNSMZXQXNU
      EZYCXQPZKUAZSMZXMYFNOZAPZDQZPZJKXNYFUGZYCYGXQYJJKSUFYLXPYIDYLXOYHAXNYFXMN
      UHUIUJUKYKKXNUCZYEPYBYMYCEUAZXNNOZBPZEQZXQYCYMYQYCYMRYPEYCYOYMBYCYORZYNLM
      ZYMXMYNULZAPZDQZBYRYOJSTYSYOJSURJYNUBUMZYMYGYIPZKXNUCZDQZYRUUBYMUUDDQZKXN
      UCUUFUUGYKKXNYGYIDUNUOUUDKDXNUPUQYRUUEUUADYRYTUUEAYRYTUUEAPYRYTRZUUEUUDYH
      RZKXNTZAUUHYHKXNTZUUEUUJPUUHYHKSTZUUKYRYSXMYNUSZUULYTUUCXMYNUTYSUUMRXMLMU
      ULYNXMVAKXMUBVBVJUUHYHYHKSXNUUHYGYHRZYFXNMZYHUUHUUNUUOUUHUUNRZYFXNVCOZUUO
      UUPYFXMNOZXMXNVCOZUUQYHUURUUHYGXMYFVDVEUUHXMYNVCOZUUNYOUUSYRYSYTUUTUUCYNX
      MVFVGYCYOYTUUNVHXMYNXNVIVKYFXMXNVLVMUUPYFUDMZYBUUQUUOPYGUVAUUHYHYFUEVNYCY
      BYOYTUUNYDVOYFXNVPVMVQVTYGYHVRVSWAVQUUEUUKUUJUUDYHKXNWBWCWDUUHUUIAKXNUUHU
      UOYGUUIAPUUHXNSYFYCXNSUSZYOYTSWEYCUVBWFSXNWGWHWIWJYGUUDYHAYGYIWLWKWMWNXBV
      TWOWPWQIWRWSWTWCXPYPDEXMYNUGXOYOABXMYNXNNXAGUKXCXDXEXFXGXHXQXRJSWBWHXIXLX
      SCJSXLXQXRCXPXRCPDFLXMFUGXOXRACXMFXNNXAHUKXJWKXKVQ $.
  $}

  ${
    $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 ) ) $.
    $( A version of ~ ac6s 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 ) ) $=
      ( vz vg wral cv wf wa wex wi c0 wceq raleq anbi12d vu vw cfn wcel csn cun
      wrex cfv wsbc feq2 exbidv imbi12d feq1 fvex sbcie sbceq1d bitr3id ralbidv
      weq fveq1 cbvexvw bitrdi f0 0ex ral0 biantru spcev mp1i wel wn wss ssralv
      ssun1 ax-mp imim1i ssun2 wb cvv ralsnsg elv sbcrex bitri nfv nfcv nfsbc1v
      sylib nfralw nfan nfex nfim w3a cop simprl wf1o vex f1osn f1of snssd fssd
      simpl2 cin simpl1 disjsn sylibr simprr wne eleq1a necon3bd impcom bitr2di
      fun2d fvunsn dfsbcq 3syl ralbidva syl mpbid simpl3 wfun ffun vsnid dmsnop
      cdm eleqtrri funssfv mp3an23 eqtr2di elsni fveq2d eqeq2d biimparc sbceq1a
      fvsn ralun syl2anc snex unex ex exlimdv syl5 3exp rexlimd a2d findcard2s
      adantl imp ) EUCUDADFUGZCEKZEFGLZMZBCEKZNZGOZUUGCUALZKZUUNFUUIMZBCUUNKZNZ
      GOZPUUGCQKZQFUUIMZBCQKZNZGOZPUUGCUBLZKZUVEFUUIMZBCUVEKZNZGOZPZUUGCUVEILZU
      EZUFZKZUVNFJLZMZADCLZUVPUHZUIZCUVNKZNZJOZPZUUHUUMPUAUBIEUUNQRZUUOUUTUUSUV
      DUUGCUUNQSUWEUURUVCGUWEUUPUVAUUQUVBUUNQFUUIUJBCUUNQSTUKULUAUBUSZUUOUVFUUS
      UVJUUGCUUNUVESUWFUURUVIGUWFUUPUVGUUQUVHUUNUVEFUUIUJBCUUNUVESTUKULUUNUVNRZ
      UUOUVOUUSUWCUUGCUUNUVNSUWGUUSUVNFUUIMZBCUVNKZNZGOUWCUWGUURUWJGUWGUUPUWHUU
      QUWIUUNUVNFUUIUJBCUUNUVNSTUKUWJUWBGJGJUSZUWHUVQUWIUWAUVNFUUIUVPUMUWKBUVTC
      UVNBADUVRUUIUHZUIZUWKUVTABDUWLUVRUUIUNHUOZUWKADUWLUVSUVRUUIUVPUTUPUQURTVA
      VBULUUNERZUUOUUHUUSUUMUUGCUUNESUWOUURUULGUWOUUPUUJUUQUUKUUNEFUUIUJBCUUNES
      TUKULQFQMZUVDUUTFVCUVCUWPGQVDUVCUVAUUIQRUWPUVBUVABCVEVFQFUUIQUMUQVGVHIUBV
      IZVJZUVKUWDPUVEUCUDUVKUVOUVJPUWRUWDUVOUVFUVJUVEUVNVKUVOUVFPUVEUVMVMUUGCUV
      EUVNVLVNVOUWRUVOUVJUWCUVOACUVLUIZDFUGZUWRUVJUWCPZUVOUUGCUVMKZUWTUVMUVNVKU
      VOUXBPUVMUVEVPUUGCUVMUVNVLVNUXBUUGCUVLUIZUWTUXBUXCVQIUUGCUVLVRVSVTACDUVLF
      WAWBWFUWRUWSUXADFUWRDWCUVJUWCDUVJDWCUWBDJUVQUWADUVQDWCUVTDCUVNDUVNWDADUVS
      WEWGWHWIWJUWRDLZFUDZUWSUXAUWRUXEUWSWKZUVIUWCGUXFUVIUWCUXFUVINZUVNFUUIUVLU
      XDWLZUEZUFZMZADUVRUXJUHZUIZCUVNKZUWCUXGUVEUVMFUUIUXIUXFUVGUVHWMUXGUVMUXDU
      EZFUXIUVMUXOUXIWNUVMUXOUXIMUXGUVLUXDIWOZDWOZWPUVMUXOUXIWQVHUXGUXDFUWRUXEU
      WSUVIWTWRWSUXGUWRUVEUVMXAQRUWRUXEUWSUVIXBZUVEUVLXCXDXKZUXGUXMCUVEKZUXMCUV
      MKZUXNUXGUVHUXTUXFUVGUVHXEUXGUWRUVHUXTVQUXRUWRBUXMCUVEUWRCUBVIZNUVLUVRXFZ
      UXLUWLRZBUXMVQUYBUWRUYCUYBUWQUVLUVRUVRUVEUVLXGXHXIUUIUVLUXDUVRXLUYDUXMUWM
      BADUXLUWLXMUWNXJXNXOXPXQUXGUWSUYAUWRUXEUWSUVIXRUXGUXDUVLUXJUHZRZUWSUYAVQU
      XGUYEUVLUXIUHZUXDUXGUXKUXJXSZUYEUYGRZUXSUVNFUXJXTUYHUXIUXJVKUVLUXIYCZUDUY
      IUXIUUIVPUVLUVMUYJIYAUVLUXDUXQYBYDUVLUXJUXIYEYFXNUVLUXDUXPUXQYMYGUWSACUVM
      KZUYFUYAUYKUWSVQIACUVLVRVSVTUYFAUXMCUVMUYFUVRUVMUDZNUXDUXLRZAUXMVQUYLUYMU
      YFUYLUXLUYEUXDUYLUVRUVLUXJUVRUVLYHYIYJYKADUXLYLXPXOUQXPXQUXMCUVEUVMYNYOUW
      BUXKUXNNJUXJUUIUXIGWOUXHYPYQUVPUXJRZUVQUXKUWAUXNUVNFUVPUXJUMUYNUVTUXMCUVN
      UYNADUVSUXLUVRUVPUXJUTUPURTVGYOYRYSUUAUUBYTUUCYTUUEUUDUUF $.
  $}

  ${
    $d R u v w x y z $.  $d A w x y $.
    $( A partial order is well-founded on a finite set.  (Contributed by Jeff
       Madsen, 18-Jun-2010.)  (Proof shortened by Mario Carneiro,
       29-Jan-2014.) $)
    frfi $p |- ( ( R Po A /\ A e. Fin ) -> R Fr A ) $=
      ( vx vw vv vu vz wcel wpo wfr cv wi c0 imbi12d wss wa wbr wn wral wrex vy
      cfn csn cun wceq poeq2 freq2 weq fr0 a1i ssun1 poss ax-mp imim1i wne cdif
      wal uncom sseq2i ssundif bitri anbi1i wel cbvrexvw simpllr simplrl impcom
      breq1 w3a sylan2br ad2ant2r simpr1 simpr2 poirr 3ad2antr3 syl2anc eldifsn
      nbrne2 sylanbrc sylan ne0d difss cvv vex difexi fri mpanl1 mpsyl syl12anc
      ssrexv notbid rspcv syl adantr simplr2 simpll simplr1 simpr potr syl13anc
      simplr3 mpand con3d ralsn imbitrrdi syld ex sylcom difsnid raleqdv sylibd
      ralun wb reximdva mpd 3exp2 rexlimdv biimtrid ralnex breq2 ralbidv rspcev
      expcom sylbir difsn expr neeq1 raleq rexbidv syl5ibcom com23 adantll impr
      pm2.61d1 syl5 pm2.61d alrimiv df-fr sylibr findcard2 ) AUBHABIZABJZCKZBIZ
      UUCBJZLMBIZMBJZLUAKZBIZUUHBJZLZUUHDKZUCZUDZBIZUUNBJZLZUUAUUBLCUADAUUCMUEU
      UDUUFUUEUUGUUCMBUFUUCMBUGNCUAUHUUDUUIUUEUUJUUCUUHBUFUUCUUHBUGNUUCUUNUEUUD
      UUOUUEUUPUUCUUNBUFUUCUUNBUGNUUCAUEUUDUUAUUEUUBUUCABUFUUCABUGNUUGUUFBUIUJU
      UKUUQLUUHUBHUUKUUOUUJUUPUUOUUIUUJUUHUUNOUUOUUILUUHUUMUKUUHUUNBULUMUNUUOUU
      JUUPUUOUUJPZUUCUUNOZUUCMUOZPZEKZFKZBQZRZEUUCSZFUUCTZLZCUQUUPUURUVHCUVAUUC
      UUMUPZUUHOZUUTPZUURUVGUUSUVJUUTUUSUUCUUMUUHUDZOUVJUUNUVLUUCUUHUUMURUSUUCU
      UMUUHUTVAZVBUURUVKUVGUURUVKPZDCVCZUVGUVNUVBUULBQZEUUCTZUVOUVGLZUVQGKZUULB
      QZGUUCTUVNUVRUVPUVTEGUUCUVBUVSUULBVHVDUVNUVTUVRGUUCUVNGCVCZUVTUVOUVGUVNUW
      AUVTUVOVIZPZUVEEUVISZFUUCTZUVGUWCUUJUVJUVIMUOZUWEUUOUUJUVKUWBVEUURUVJUUTU
      WBVFUWCUVIUVSUVNUUDUWBUVSUVIHZUUOUVJUUDUUJUUTUVJUUOUUSUUDUVMUUSUUOUUDUUCU
      UNBULVGVJVKZUUDUWBPZUWAUVSUULUOZUWGUUDUWAUVTUVOVLUWIUVTUULUULBQRZUWJUUDUW
      AUVTUVOVMUUDUWAUVOUWKUVTUUCUULBVNVOUVSUULUULBVRVPUVSUUCUULVQVSZVTWAUVIUUC
      OUUJUVJUWFPZPUWDFUVITZUWEUUCUUMWBUVIWCHUUJUWMUWNUUCUUMCWDWEFEUUHUVIWCBWFW
      GUWDFUVIUUCWJWHZWIUVNUUDUWBUWEUVGLUWHUWIUWDUVFFUUCUWIFCVCZPZUWDUVEEUVIUUM
      UDZSZUVFUWQUWDUVEEUUMSZUWSUWQUWDUVSUVCBQZRZUWTUWIUWDUXBLZUWPUWIUWGUXCUWLU
      VEUXBEUVSUVIEGUHUVDUXAUVBUVSUVCBVHWKWLWMWNUWQUXBUULUVCBQZRZUWTUWQUXDUXAUW
      QUVTUXDUXAUWAUVTUVOUUDUWPWOUWQUUDUWAUVOUWPUVTUXDPUXALUUDUWBUWPWPUWAUVTUVO
      UUDUWPWQUWAUVTUVOUUDUWPXAZUWIUWPWRUUCUVSUULUVCBWSWTXBXCUVEUXEEUULDWDEDUHU
      VDUXDUVBUULUVCBVHWKXDXEXFUWDUWTUWSUVEEUVIUUMXLXGXHUWQUVOUWSUVFXMUXFUVOUVE
      EUWRUUCUUCUULXIXJWMXKXNVTXOXPXQXRUVQRUVPRZEUUCSZUVRUVPEUUCXSUVOUXHUVGUVFU
      XHFUULUUCFDUHZUVEUXGEUUCUXIUVDUVPUVCUULUVBBXTWKYAYBYCYDYNUVORUVIUUCUEZUVN
      UVGUULUUCYEUURUVJUUTUXJUVGLZUUJUVJUUTUXKLUUOUUJUVJPZUXJUUTUVGUXLUWFUWELUX
      JUUTUVGLUUJUVJUWFUWEUWOYFUXJUWFUUTUWEUVGUVIUUCMYGUXJUWDUVFFUUCUVEEUVIUUCY
      HYINYJYKYLYMYOYPXGXRYQCFEUUNBYRYSXGXHUJYTVG $.
  $}

  ${
    $d R x y z $.  $d A x y z $.
    $( A finite set has a maximum under a total order.  (Contributed by Jeff
       Madsen, 18-Jun-2010.)  (Proof shortened by Mario Carneiro,
       29-Jan-2014.) $)
    fimax2g $p |- ( ( R Or A /\ A e. Fin /\ A =/= (/) ) ->
                                        E. x e. A A. y e. A -. x R y ) $=
      ( wor cfn wcel c0 wne w3a ccnv cv wbr wn wral wrex wpo sylib wa vex cnvpo
      wfr sopo frfi sylan 3adant3 wss ssid fri mpanr1 an32s brcnv notbii ralbii
      wi rexbii ex 3adant1 mpd ) CDEZCFGZCHIZJCDKZUBZALZBLZDMZNZBCOZACPZUTVAVDV
      BUTCVCQZVAVDUTCDQVKCDUCCDUARCVCUDUEUFVAVBVDVJUOUTVAVBSZVDVJVLVDSVFVEVCMZN
      ZBCOZACPZVJVAVDVBVPVAVDSCCUGVBVPCUHABCCFVCUIUJUKVOVIACVNVHBCVMVGVFVEDBTAT
      ULUMUNUPRUQURUS $.

    $( A finite set has a maximum under a total order.  (Contributed by Jeff
       Madsen, 2-Sep-2009.)  (Proof shortened by Mario Carneiro,
       29-Jan-2014.) $)
    fimaxg $p |- ( ( R Or A /\ A e. Fin /\ A =/= (/) ) ->
                            E. x e. A A. y e. A ( x =/= y -> y R x ) ) $=
      ( wor cfn wcel c0 wne w3a cv wbr wi wral wrex wn fimax2g wb wa weq imbi1i
      df-ne pm4.64 bitri sotric con2bid bitrid anassrs ralbidva rexbidva mpbird
      wo 3ad2ant1 ) CDEZCFGZCHIZJAKZBKZIZURUQDLZMZBCNZACOZUQURDLZPZBCNZACOZABCD
      QUNUOVCVGRUPUNVBVFACUNUQCGZSVAVEBCUNVHURCGZVAVERVAABTZUTULZUNVHVISSZVEVAV
      JPZUTMVKUSVMUTUQURUBUAVJUTUCUDVLVDVKCUQURDUEUFUGUHUIUJUMUK $.

    $( Lemma showing existence and closure of supremum of a finite set.
       (Contributed by Jeff Madsen, 2-Sep-2009.) $)
    fisupg $p |- ( ( R Or A /\ A e. Fin /\ A =/= (/) ) -> E. x e. A ( A. y e. A
            -. x R y /\ A. y e. A ( y R x -> E. z e. A y R z ) ) ) $=
      ( wor cfn wcel c0 wne w3a cv wbr wi wral wrex wn wa ex anassrs fimaxg weq
      sotrieq2 simprbda a1dd pm2.27 so2nr pm3.21 con3d syl9r pm2.61dne ralimdva
      syl5com breq2 rspcev ralrimivw adantl jctird reximdva 3ad2ant1 mpd ) DEFZ
      DGHZDIJZKALZBLZJZVFVEEMZNZBDOZADPZVEVFEMZQZBDOZVHVFCLZEMZCDPZNZBDOZRZADPZ
      ABDEUAVBVCVKWANVDVBVJVTADVBVEDHZRZVJVNVSWCVIVMBDWCVFDHZRZVIVMNVEVFWEABUBZ
      VMVIVBWBWDWFVMNVBWBWDRRZWFVMWGWFVMVHQDVEVFEUCUDSTUEVGVIVHWEVMVGVHUFVBWBWD
      VHVMNWGVLVHRZQVHVMDVEVFEUGVHVLWHVHVLUHUIUMTUJUKULWBVSVBWBVRBDWBVHVQVPVHCV
      EDVOVEVFEUNUOSUPUQURUSUTVA $.
  $}

  $( A total order on a finite set is a well-order.  (Contributed by Jeff
     Madsen, 18-Jun-2010.)  (Proof shortened by Mario Carneiro,
     29-Jan-2014.) $)
  wofi $p |- ( ( R Or A /\ A e. Fin ) -> R We A ) $=
    ( wor cfn wcel wa wfr wwe wpo sopo frfi sylan simpl df-we sylanbrc ) ABCZAD
    EZFABGZPABHPABIQRABJABKLPQMABNO $.

  ${
    $d A x y $.
    $( The maximum of a finite collection of ordinals is in the set.
       (Contributed by Mario Carneiro, 28-May-2013.)  (Revised by Mario
       Carneiro, 29-Jan-2014.) $)
    ordunifi $p |- ( ( A C_ On /\ A e. Fin /\ A =/= (/) ) -> U. A e. A ) $=
      ( vx vy con0 wss cfn wcel c0 wne w3a cuni cv wrex cep wn wral wb wa ssel2
      wor wbr wwe epweon weso ax-mp soss mpi fimax2g syl3an1 adantr epel notbii
      adantlr ontri1 bitr4id syl2anc ralbidva bitr4di rexbidva 3ad2ant1 elssuni
      unissb mpbid wceq eqss eleq1 biimpcd biimtrrid mpand rexlimiv syl ) ADEZA
      FGZAHIZJZAKZBLZEZBAMZVPAGZVOVQCLZNUAZOZCAPZBAMZVSVLANTZVMVNWEVLDNTZWFDNUB
      WGUCDNUDUEADNUFUGBCANUHUIVLVMWEVSQVNVLWDVRBAVLVQAGZRZWDWAVQEZCAPVRWIWCWJC
      AWIWAAGZRWADGZVQDGZWCWJQVLWKWLWHADWASUMWIWMWKADVQSUJWLWMRWCVQWAGZOWJWBWNC
      VQUKULWAVQUNUOUPUQCAVQVBURUSUTVCVRVTBAWHVQVPEZVRVTVQAVAWOVRRVQVPVDZWHVTVQ
      VPVEWPWHVTVQVPAVFVGVHVIVJVK $.
  $}

  $( The union (supremum) of a finite set of finite ordinals is a finite
     ordinal.  (Contributed by Stefan O'Rear, 5-Nov-2014.) $)
  nnunifi $p |- ( ( S C_ _om /\ S e. Fin ) -> U. S e. _om ) $=
    ( com wss cfn wcel wa cuni c0 wceq unieq peano1 eqeltri eqeltrdi adantl wne
    uni0 simpll con0 omsson sstrdi simplr ordunifi syl3anc sseldd pm2.61dane
    simpr ) ABCZADEZFZAGZBEZAHAHIZUKUIULUJHGZBAHJUMHBPKLMNUIAHOZFZABUJUGUHUNQZU
    OARCUHUNUJAEUOABRUPSTUGUHUNUAUIUNUFAUBUCUDUE $.

  ${
    $d x y A $.  $d x y B $.
    $( Lemma for ~ unbnn .  After removing the successor of an element from an
       unbounded set of natural numbers, the intersection of the result belongs
       to the original unbounded set.  (Contributed by NM, 3-Dec-2003.) $)
    unblem1 $p |- ( ( ( B C_ _om /\ A. x e. _om E. y e. B x e. y ) /\
                 A e. B ) -> |^| ( B \ suc A ) e. B ) $=
      ( com wss cv wcel wrex wral wa csuc cdif cint con0 c0 ssel wn wi syl6 wne
      omsson sstr mpan2 ssdifssd ad2antrr peano2b imbitrdi wceq rexbidv rspccva
      eleq1 word nnord ordn2lp imnan sylibr con2d syl imdistand eldif ne0i expd
      sylbir rexlimdv syl5 sylan2d impl onint syl2anc eldifad ) DEFZAGZBGZHZBDI
      ZAEJZKCDHZKZDCLZMZNZDVTVSWAOFZWAPUAZWBWAHVLWCVQVRVLDOVTVLEOFDOFUBDEOUCUDU
      EUFVLVQVRWDVLVRVTEHZVQWDVLVRCEHWEDECQCUGUHVQWEKVTVNHZBDIZVLWDVPWGAVTEVMVT
      UIVOWFBDVMVTVNULUJUKVLWFWDBDVLVNDHZWFWDVLWHWFKWHVNVTHZRZKZWDVLWHWFWJVLWHV
      NEHZWFWJSZDEVNQWLVNUMZWMVNUNWNWIWFWNWIWFKRWIWFRSVNVTUOWIWFUPUQURUSTUTWKVN
      WAHWDVNDVTVAWAVNVBVDTVCVEVFVGVHWAVIVJVK $.
  $}

  ${
    $d u v w x y z A $.  $d u v w y z F $.
    unblem.2 $e |- F =
          ( rec ( ( x e. _V |-> |^| ( A \ suc x ) ) , |^| A ) |` _om ) $.
    $( Lemma for ~ unbnn .  The value of the function ` F ` belongs to the
       unbounded set of natural numbers ` A ` .  (Contributed by NM,
       3-Dec-2003.) $)
    unblem2 $p |- ( ( A C_ _om /\ A. w e. _om E. v e. A w e. v )
        -> ( z e. _om -> ( F ` z ) e. A ) ) $=
      ( cv com wcel wss wa cfv c0 csuc wceq fveq2 eleq1d cint con0 vu wrex wral
      vy wne omsson sstr mpan2 wi peano1 eleq1 rexbidv rspcv ax-mp df-rex sylib
      wex exsimpl syl n0 sylibr onint syl2an cdif cmpt crdg cres fveq1i eqtr2id
      cvv fr0g ibi unblem1 wb suceq difeq2d inteqd frsucmpt2 eqcomd ex ibd syl5
      expd finds2 com12 ) BHZIJEIKZCHZDHZJZDEUBZCIUCZLZWFFMZEJZWONFMZEJZUAHZFMZ
      EJZWROZFMZEJZWMBUAWFNPWNWPEWFNFQRWFWRPWNWSEWFWRFQRWFXAPWNXBEWFXAFQRWMESZE
      JZWQWGETKZENUEZXEWLWGITKXFUFEITUGUHWLWIEJZDUQZXGWLXHNWIJZLDUQZXIWLXJDEUBZ
      XKNIJWLXLUIUJWKXLCNIWHNPWJXJDEWHNWIUKULUMUNXJDEUOUPXHXJDURUSDEUTVAEVBVCXE
      WQXEXDWPEXEWPNAVJEAHZOZVDZSZVEZXDVFIVGZMXDNFXRGVHXDEXQVKVIRVLUSWRIJZWMWTX
      CWMWTLEWSOZVDZSZEJZXSXCCDWSEVMXSYCXCXSYCYCXCVNXSYCLZYBXBEYDXBYBAUDXDWRXPY
      BEUDHZOZVDZSFEGYEXMPZYGXOYHYFXNEYEXMVOVPVQYEWSPZYGYAYIYFXTEYEWSVOVPVQVRVS
      RVTWAWBWCWDWE $.

    $( Lemma for ~ unbnn .  The value of the function ` F ` is less than its
       value at a successor.  (Contributed by NM, 3-Dec-2003.) $)
    unblem3 $p |- ( ( A C_ _om /\ A. w e. _om E. v e. A w e. v )
        -> ( z e. _om -> ( F ` z ) e. ( F ` suc z ) ) ) $=
      ( vy com wss cv wcel wa cfv csuc cdif cint con0 wceq ex wrex wral unblem2
      imp omsson sstr mpan2 ssel anc2li syl ad2antrr mpd onmindif unblem1 suceq
      wi syld difeq2d inteqd frsucmpt2 sylcom eleqtrrd ) EIJZCKDKLDEUACIUBZMZBK
      ZILZVFFNZVFOFNZLVEVGMZVHEVHOZPZQZVIVJERJZVHRLZMZVHVMLVJVHELZVPVEVGVQABCDE
      FGUCZUDVCVQVPUPZVDVGVCVNVSVCIRJVNUEEIRUFUGVNVQVOERVHUHUIUJUKULEVHUMUJVEVG
      VIVMSZVEVGVMELZVTVEVGVQWAVRVEVQWACDVHEUNTUQVGWAVTAHEQVFEAKZOZPZQVMEHKZOZP
      ZQFEGWEWBSZWGWDWHWFWCEWEWBUOURUSWEVHSZWGVLWIWFVKEWEVHUOURUSUTTVAUDVBT $.

    $( Lemma for ~ unbnn .  The function ` F ` maps the set of natural numbers
       one-to-one to the set of unbounded natural numbers ` A ` .  (Contributed
       by NM, 3-Dec-2003.) $)
    unblem4 $p |- ( ( A C_ _om /\ A. w e. _om E. v e. A w e. v ) ->
                 F : _om -1-1-> A ) $=
      ( vz com wss cv wcel wrex wral wa con0 cfv csuc wfn cint ralrimiv wf sstr
      wf1 omsson mpan2 adantr cvv cdif cmpt crdg cres frfnom fneq1i mpbir ffnfv
      unblem2 biimpri sylancr unblem3 omsmo syl21anc ) DHIZBJCJKCDLBHMZNZDOIZHD
      EUAZGJZEPZVGQEPKZGHMHDEUCVBVEVCVBHOIVEUDDHOUBUEUFVDEHRZVHDKZGHMZVFVJAUGDA
      JQUHSUIZDSZUJHUKZHRVNVMULHEVOFUMUNVDVKGHAGBCDEFUPTVFVJVLNGHDEUOUQURVDVIGH
      AGBCDEFUSTGDEUTVA $.
  $}

  ${
    $d x y z A $.
    $( Any unbounded subset of natural numbers is equinumerous to the set of
       all natural numbers.  Part of the proof of Theorem 42 of [Suppes]
       p. 151.  See ~ unbnn3 for a stronger version without the first
       assumption.  (Contributed by NM, 3-Dec-2003.) $)
    unbnn $p |- ( ( _om e. _V /\ A C_ _om /\ A. x e. _om E. y e. A x e. y ) ->
      A ~~ _om ) $=
      ( vz com cvv wcel wss wrex wral w3a cdom wbr cen ssdomg 3adant3 csuc cint
      cv imp cdif cmpt crdg cres wf1 simp1 ssexg ancoms unblem4 3adant1 f1dom2g
      eqid syl3anc sbth syl2anc ) EFGZCEHZASBSGBCIAEJZKZCELMZECLMZCENMUPUQUTURU
      PUQUTCEFOTPUSUPCFGZECDFCDSQUARUBCRUCEUDZUEZVAUPUQURUFUPUQVBURUQUPVBCEFUGU
      HPUQURVDUPDABCVCVCULUIUJECVCFFUKUMCEUNUO $.

    $( Version of ~ unbnn that does not require a strict upper bound.
       (Contributed by NM, 24-Apr-2004.) $)
    unbnn2 $p |- ( ( _om e. _V /\ A C_ _om /\ A. x e. _om E. y e. A x C_ y ) ->
      A ~~ _om ) $=
      ( vz cv wss wrex com wral cvv wcel cen wbr csuc peano2 wceq sseq1 rexbidv
      rspcv wi sucssel elv reximi syl6com syl5 ralrimiv unbnn syl3an3 ) AEZBEZF
      ZBCGZAHIZHJKCHFDEZUJKZBCGZDHICHLMUMUPDHUNHKUNNZHKZUMUPUNOURUMUQUJFZBCGZUP
      ULUTAUQHUIUQPUKUSBCUIUQUJQRSUSUOBCUSUOTDUNUJJUAUBUCUDUEUFDBCUGUH $.
  $}

  ${
    $d y z w A $.
    $( Any set strictly dominated by the class of natural numbers is finite.
       Sufficiency part of Theorem 42 of [Suppes] p. 151.  This theorem does
       not require the Axiom of Infinity.  (Contributed by NM, 24-Apr-2004.) $)
    isfinite2 $p |- ( A ~< _om -> A e. Fin ) $=
      ( vy vz vw com cvv wcel csdm wbr cfn cv cen wss wa wrex wral syl2anc con0
      wn wi relsdom brrelex2i cdom sdomdom domeng imbitrid ensym ad2antrl simpl
      wex ensdomtr sdomnen syl simpr 3expia syl2an mtod rexnal csuc word omsson
      unbnn sstr mpan2 nnord cuni ssel2 elon sylib ordtri1 sylan an32s ralbidva
      vex unissb ralnex bicomi 3bitr4g ordunisssuc bitr3d peano2b ssnnfi sylanb
      wb ex adantl sylbid rexlimdva biimtrrid ad2antll mpd simprl enfii exlimdv
      sylcom mpcom ) EFGZAEHIZAJGZAEHUAUBZWQWRABKZLIZXAEMZNZBUJZWSWRAEUCIWQXEAE
      UDBAEFUEUFWRXDWSBWRXDWSWRXDNZXAJGZXBWSXFCKZDKZGZDXAOZCEPZSZXGXFXLXAELIZXF
      XAEHIZXNSXFXAALIZWRXOXBXPWRXCAXAUGUHWRXDUIXAAEUKQXAEULUMWRWQXCXLXNTXDWTXB
      XCUNWQXCXLXNCDXAVBUOUPUQXCXMXGTWRXBXMXKSZCEOXCXGXKCEURXCXQXGCEXCXHEGZNXQX
      AXHUSZMZXGXCXARMZXHUTZXQXTWDXRXCERMYAVAXAERVCVDXHVEYAYBNZXAVFXHMZXQXTYCXI
      XHMZDXAPXJSZDXAPZYDXQYCYEYFDXAYAXIXAGZYBYEYFWDZYAYHNZXIUTZYBYIYJXIRGYKXAR
      XIVGXIDVNVHVIXIXHVJVKVLVMDXAXHVOYGXQXJDXAVPVQVRXAXHVSVTUPXRXTXGTXCXRXTXGX
      RXSEGXTXGXHWAXSXAWBWCWEWFWGWHWIWJWKWRXBXCWLAXAWMQWEWNWOWP $.
  $}

  ${
    $d A x $.
    $( Omega strictly dominates a natural number.  Example 3 of [Enderton]
       p. 146.  In order to avoid the Axiom of Infinity, we include it as part
       of the antecedent.  See ~ nnsdom for the version without this sethood
       requirement.  (Contributed by NM, 15-Jun-1998.)  Avoid ~ ax-pow .
       (Revised by BTernaryTau, 7-Jan-2025.) $)
    nnsdomg $p |- ( ( _om e. _V /\ A e. _om ) -> A ~< _om ) $=
      ( vx com cvv wcel wa cdom wbr cen csdm wss word ordom ordelss mpan adantr
      wn cfn nnfi ssdomfi2 syl3an1 mpd3an3 ancoms ominf wb ensymfib syl cv wrex
      breq2 rspcev isfi sylibr ex sylbid mtoi adantl brsdom sylanbrc ) CDEZACEZ
      FACGHZACIHZQZACJHVAUTVBVAUTACKZVBVAVEUTCLVAVEMCANOPVAAREZUTVEVBASZACDTUAU
      BUCVAVDUTVAVCCREZUDVAVCCAIHZVHVAVFVCVIUEVGACUFUGVAVIVHVAVIFCBUHZIHZBCUIVH
      VKVIBACVJACIUJUKBCULUMUNUOUPUQACURUS $.
    $( $j usage 'nnsdomg' avoids 'ax-pow'; $)

    $( Obsolete version of ~ nnsdomg as of 7-Jan-2025.  (Contributed by NM,
       15-Jun-1998.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    nnsdomgOLD $p |- ( ( _om e. _V /\ A e. _om ) -> A ~< _om ) $=
      ( vx com cvv wcel wa cdom wbr cen csdm wss ssdomg word ordom ordelss mpan
      wn impel cfn ominf ensym wrex breq2 rspcev isfi sylibr syl5 adantl brsdom
      cv ex mtoi sylanbrc ) CDEZACEZFACGHZACIHZQZACJHUNACKZUPUOACDLCMUOUSNCAOPR
      UOURUNUOUQCSEZTUQCAIHZUOUTACUAUOVAUTUOVAFCBUJZIHZBCUBUTVCVABACVBACIUCUDBC
      UEUFUKUGULUHACUIUM $.

    $( A set is finite iff it is strictly dominated by the class of natural
       number.  Theorem 42 of [Suppes] p. 151.  In order to avoid the Axiom of
       infinity, we include it as a hypothesis.  (Contributed by NM,
       3-Nov-2002.)  (Revised by Mario Carneiro, 27-Apr-2015.) $)
    isfiniteg $p |- ( _om e. _V -> ( A e. Fin <-> A ~< _om ) ) $=
      ( vx com cvv wcel cfn csdm wbr cv wrex isfi wa nnsdomg sdomen1 syl5ibrcom
      cen rexlimdva biimtrid isfinite2 impbid1 ) CDEZAFEZACGHZUBABIZPHZBCJUAUCB
      AKUAUEUCBCUAUDCELUCUEUDCGHUDMAUDCNOQRAST $.
  $}

  $( An infinite set strictly dominates a natural number.  (Contributed by NM,
     22-Nov-2004.)  (Revised by Mario Carneiro, 27-Apr-2015.)  Avoid ~ ax-pow .
     (Revised by BTernaryTau, 7-Jan-2025.) $)
  infsdomnn $p |- ( ( _om ~<_ A /\ B e. _om ) -> B ~< A ) $=
    ( com cdom wbr wcel cfn csdm nnfi adantl cvv reldom brrelex1i nnsdomg sylan
    wa simpl sdomdomtrfi syl3anc ) CADEZBCFZPBGFZBCHEZTBAHEUAUBTBIJTCKFUAUCCADL
    MBNOTUAQBCARS $.
  $( $j usage 'infsdomnn' avoids 'ax-pow'; $)

  $( Obsolete version of ~ infsdomnn as of 7-Jan-2025.  (Contributed by NM,
     22-Nov-2004.)  (Revised by Mario Carneiro, 27-Apr-2015.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  infsdomnnOLD $p |- ( ( _om ~<_ A /\ B e. _om ) -> B ~< A ) $=
    ( com cdom wbr wcel csdm cvv reldom brrelex1i nnsdomg sylan simpl sdomdomtr
    wa syl2anc ) CADEZBCFZOBCGEZQBAGEQCHFRSCADIJBKLQRMBCANP $.

  ${
    $d A f $.
    $( An infinite set is not empty.  For a shorter proof using ~ ax-un , see
       ~ infn0ALT .  (Contributed by NM, 23-Oct-2004.)  Avoid ~ ax-un .
       (Revised by BTernaryTau, 8-Jan-2025.) $)
    infn0 $p |- ( _om ~<_ A -> A =/= (/) ) $=
      ( vf com cdom wbr cv wf1 wex c0 wne brdomi wceq wcel peano1 wf1o ccnv crn
      wa wn f1f1orn adantr wss frnd sseq0 sylan f1oeq3d mpbid f1ocnv noel f1o00
      f1f simprbi eleq2d mtbiri 3syl mt2 imnani neqned exlimiv syl ) CADECABFZG
      ZBHAIJZCABKVBVCBVBAIVBAILZVBVDRZICMZNVECIVAOZICVAPZOZVFSVECVAQZVAOZVGVBVK
      VDCAVATUAVEVJICVAVBVJAUBVDVJILVBCAVACAVAUKUCVJAUDUEUFUGCIVAUHVIVFIIMIUIVI
      CIIVIVHILCILCVHUJULUMUNUOUPUQURUSUT $.
    $( $j usage 'infn0' avoids 'ax-pow' 'ax-un'; $)
  $}

  $( Shorter proof of ~ infn0 using ~ ax-un .  (Contributed by NM,
     23-Oct-2004.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  infn0ALT $p |- ( _om ~<_ A -> A =/= (/) ) $=
    ( com cdom wbr c0 csdm wne wcel peano1 infsdomnn mpan2 cvv reldom brrelex2i
    wb 0sdomg syl mpbid ) BACDZEAFDZAEGZSEBHTIAEJKSALHTUAOBACMNALPQR $.

  $( This (useless) theorem, which was proved without the Axiom of Infinity,
     demonstrates an artifact of our definition of binary relation, which is
     meaningful only when its arguments exist.  In particular, the antecedent
     cannot be satisfied unless ` _om ` exists.  (Contributed by NM,
     13-Nov-2003.) $)
  fin2inf $p |- ( A ~< _om -> _om e. _V ) $=
    ( com csdm relsdom brrelex2i ) ABCDE $.

  ${
    $d x y A $.  $d w x y z B $.  $d w y z F $.
    unfilem1.1 $e |- A e. _om $.
    unfilem1.2 $e |- B e. _om $.
    unfilem1.3 $e |- F = ( x e. B |-> ( A +o x ) ) $.
    $( Lemma for proving that the union of two finite sets is finite.
       (Contributed by NM, 10-Nov-2002.)  (Revised by Mario Carneiro,
       31-Aug-2015.) $)
    unfilem1 $p |- ran F = ( ( A +o B ) \ A ) $=
      ( vy coa co cv wceq wrex wcel wa com wb word nnord eleq1 crn cdif wn elnn
      mpan2 nnaord mp3an23 syl ibi wss nnaword1 nnacl ordtri1 mpbid sylancr jca
      syl2an2r notbid anbi12d biimparc rexlimiva nnacli syl2an nnawordex bitr3d
      sylan sylan9bb biimprcd eqcom biimpi adantl jca2 reximdv2 sylbid imp ovex
      impbii elrnmpti eldif 3bitr4i eqriv ) HDUAZBCIJZBUBZHKZBAKZIJZLZACMZWEWCN
      ZWEBNZUCZOZWEWBNWEWDNWIWMWHWMACWFCNZWGWCNZWGBNZUCZOZWHWMWNWOWQWNWOWNWFPNZ
      WNWOQZWNCPNZWSFWFCUDUEZWSXABPNZWTFEWFCBUFUGZUHUIWNXCWSWQEXBXCWSOZBWGUJZWQ
      BWFUKXCBRZWSWGRZXFWQQBSZXEWGPNXHBWFULWGSUHBWGUMUQUNUOUPWHWMWRWHWJWOWLWQWE
      WGWCTWHWKWPWEWGBTURUSUTVFVAWJWLWIWJWLWGWELZAPMZWIWJXCWEPNZWLXKQEWJWCPNXLB
      CEFVBWEWCUDUEXCXLOBWEUJZWLXKXCXGWERXMWLQXLXIWESBWEUMVCABWEVDVEUOWJXJWHAPC
      WJWSXJOZWNWHXNWNWJWSWNWOXJWJXDWGWEWCTVGVHXJWHWSXJWHWGWEVIVJVKVLVMVNVOVQAC
      WGWEDGBWFIVPVRWEWCBVSVTWA $.

    $( Lemma for proving that the union of two finite sets is finite.
       (Contributed by NM, 10-Nov-2002.)  (Revised by Mario Carneiro,
       31-Aug-2015.) $)
    unfilem2 $p |- F : B -1-1-onto-> ( ( A +o B ) \ A ) $=
      ( vz vw coa co cv cfv wceq wral ovex mpbir2an wcel oveq2 com cdif wf1 wfo
      wf1o wf wi wfn crn fnmpti unfilem1 df-fo ax-mp wa fvmpt eqeqan12d wb elnn
      fof mpan2 nnacan mp3an3an bitrd biimpd rgen2 dff13 df-f1o ) CBCJKBUAZDUDC
      VGDUBZCVGDUCZVHCVGDUEZHLZDMZILZDMZNZVKVMNZUFZICOHCOVIVJVIDCUGDUHVGNACBALZ
      JKZDBVRJPGUIABCDEFGUJCVGDUKQZCVGDURULVQHICCVKCRZVMCRZUMZVOVPWCVOBVKJKZBVM
      JKZNZVPWAWBVLWDVNWEAVKVSWDCDVRVKBJSGBVKJPUNAVMVSWECDVRVMBJSGBVMJPUNUOBTRW
      AVKTRZWBVMTRZWFVPUPEWACTRZWGFVKCUQUSWBWIWHFVMCUQUSBVKVMUTVAVBVCVDHICVGDVE
      QVTCVGDVFQ $.
  $}

  ${
    $d x A $.  $d x B $.
    $( Lemma for proving that the union of two finite sets is finite.
       (Contributed by NM, 16-Nov-2002.)  (Revised by Mario Carneiro,
       31-Aug-2015.) $)
    unfilem3 $p |- ( ( A e. _om /\ B e. _om ) -> B ~~ ( ( A +o B ) \ A ) ) $=
      ( vx com wcel coa co cdif cen wbr c0 wceq oveq1 id difeq12d breq2d peano1
      cif cvv elimel oveq2 difeq1d breq12d cv cmpt wf1o difexi unfilem2 f1oen2g
      ovex eqid mp3an dedth2h ) ADEZBDEZBABFGZAHZIJBUNAKRZBFGZURHZIJUOBKRZURVAF
      GZURHZIJZABKKAURLZUQUTBIVEUPUSAURAURBFMVENOPBVALZBVAUTVCIVFNVFUSVBURBVAUR
      FUAUBUCVADEVCSEVAVCCVAURCUDFGUEZUFVDBKDQTZVBURURVAFUJUGCURVAVGAKDQTVHVGUK
      UHVAVCVGDSUIULUM $.
  $}

  $( If a union is finite, the operands are finite.  Converse of ~ unfi .
     (Contributed by FL, 3-Aug-2009.) $)
  unfir $p |- ( ( A u. B ) e. Fin -> ( A e. Fin /\ B e. Fin ) ) $=
    ( cun cfn wcel wss ssun1 ssfi mpan2 ssun2 jca ) ABCZDEZADEZBDEZMALFNABGLAHI
    MBLFOBAJLBHIK $.

  $( A union is finite if and only if the operands are finite.  (Contributed by
     AV, 10-May-2025.) $)
  unfib $p |- ( ( A u. B ) e. Fin <-> ( A e. Fin /\ B e. Fin ) ) $=
    ( cun cfn wcel wa unfir unfi impbii ) ABCDEADEBDEFABGABHI $.

  $( The union of two finite sets is finite.  Part of Corollary 6K of
     [Enderton] p. 144.  This version of ~ unfi is useful only if we assume the
     Axiom of Infinity (see comments in ~ fin2inf ).  (Contributed by NM,
     22-Oct-2004.)  (Revised by Mario Carneiro, 27-Apr-2015.) $)
  unfi2 $p |- ( ( A ~< _om /\ B ~< _om ) -> ( A u. B ) ~< _om ) $=
    ( com csdm wbr wa cun cfn wcel isfinite2 syl2an wb fin2inf adantr isfiniteg
    unfi cvv syl mpbid ) ACDEZBCDEZFZABGZHIZUCCDEZTAHIBHIUDUAAJBJABPKUBCQIZUDUE
    LTUFUAAMNUCORS $.

  $( An infinite set ` A ` minus a finite set is infinite.  (Contributed by FL,
     3-Aug-2009.) $)
  difinf $p |- ( ( -. A e. Fin /\ B e. Fin ) -> -. ( A \ B ) e. Fin ) $=
    ( cfn wcel wn cdif wa cun undif1 eleq1i unfir simpld sylbi syl expcom con3d
    unfi impcom ) BCDZACDZEABFZCDZESUBTUBSTUBSGUABHZCDZTUABQUDABHZCDZTUCUECABIJ
    UFTSABKLMNOPR $.

  ${
    $d A x y z $.  $d F x y z $.
    $( An onto function implies dominance of domain over range, for finite
       sets.  Unlike ~ fodomg for arbitrary sets, this theorem does not require
       the Axiom of Replacement nor the Axiom of Power Sets nor the Axiom of
       Choice for its proof.  (Contributed by NM, 23-Mar-2006.)  (Proof
       shortened by Mario Carneiro, 16-Nov-2014.)  Avoid ~ ax-pow .  (Revised
       by BTernaryTau, 20-Jun-2025.) $)
    fodomfi $p |- ( ( A e. Fin /\ F : A -onto-> B ) -> B ~<_ A ) $=
      ( vz cfn wcel wa cima cdom wceq wbr cv wi c0 csn imaeq2 id breq12d imbi2d
      cvv vx wfo foima adantl wfn cun ima0 eqtrdi 0ex 0dom a1i cid cfv wss cres
      vy wn crn cop wfun fnfun ad2antrl funressn rnss 3syl df-ima rnsnop eqcomi
      vex 3sstr4g snfi ssexg sylancl fvi syl uneq2d imaundi eqtr4di cin ssdomfi
      simprr mpsyl fvex en2sn mp2an domtrfi mp3an1 sylan2 eqbrtrd simplr disjsn
      cen endom sylibr undom syl21anc eqbrtrrd exp32 a2d findcard2s fofn impel
      ) AEFZABCUBZGCAHZBAIXDXEBJXCABCUCUDXCCAUEZXEAIKZXDXFCUALZHZXHIKZMXFNNIKZM
      XFCUPLZHZXLIKZMXFCXLDLZOZUFZHZXQIKZMXFXGMUAUPDAXHNJZXJXKXFXTXINXHNIXTXICN
      HNXHNCPCUGUHXTQRSXHXLJZXJXNXFYAXIXMXHXLIXHXLCPYAQRSXHXQJZXJXSXFYBXIXRXHXQ
      IXHXQCPYBQRSXHAJZXJXGXFYCXIXEXHAIXHACPYCQRSXKXFNUIUJUKXLEFZXOXLFUQZGZXFXN
      XSYFXFXNXSYFXFXNGZGZXMCXPHZULUMZUFZXRXQIYHYKXMYIUFXRYHYJYIXMYHYITFZYJYIJY
      HYIXOCUMZOZUNZYNEFZYLYHCXPUOZURZXOYMUSOZURZYIYNYHCUTZYQYSUNYRYTUNXFUUAYFX
      NACVAVBXOCVCYQYSVDVECXPVFYTYNXOYMDVIZVGVHVJZYMVKZYIYNEVLVMYITVNVOZVPCXLXP
      VQVRYHXNYJXPIKXLXPVSNJZYKXQIKYFXFXNWAYHYJYIXPIUUEYHYIYNIKZYNXPWLKZYIXPIKZ
      YPYHYOUUGUUDUUCYIYNVTWBYMTFXOTFUUHXOCWCUUBYMXOTTWDWEUUHUUGYNXPIKZUUIYNXPW
      MYPUUGUUJUUIUUDYIYNXPWFWGWHVMWIYHYEUUFYDYEYGWJXLXOWKWNXMXLYJXPWOWPWQWRWSW
      TABCXAXBWQ $.
    $( $j usage 'fodomfi' avoids 'ax-rep' 'ax-pow' 'ax-ac' 'ax-ac2'; $)
  $}

  ${
    $( If an onto function has a finite domain, its codomain/range is finite.
       Theorem 37 of [Suppes] p. 104.  (Contributed by NM, 25-Mar-2007.) $)
    fofi $p |- ( ( A e. Fin /\ F : A -onto-> B ) -> B e. Fin ) $=
      ( cfn wcel wfo cdom wbr fodomfi domfi syldan ) ADEABCFBAGHBDEABCIABJK $.

    $( If a 1-to-1 function has a finite codomain its domain is finite.
       (Contributed by FL, 31-Jul-2009.)  (Revised by Mario Carneiro,
       24-Jun-2015.) $)
    f1fi $p |- ( ( B e. Fin /\ F : A -1-1-> B ) -> A e. Fin ) $=
      ( cfn wcel wf1 wa crn ccnv wfo wss frnd ssfi sylan2 f1f1orn adantl f1ocnv
      f1f wf1o f1ofo 3syl fofi syl2anc ) BDEZABCFZGZCHZDEZUGACIZJZADEUEUDUGBKUH
      UEABCABCRLBUGMNUFAUGCSZUGAUISUJUEUKUDABCOPAUGCQUGAUITUAUGAUIUBUC $.
  $}

  $( Images of finite sets are finite.  (Contributed by Stefan O'Rear,
     22-Feb-2015.) $)
  imafi $p |- ( ( Fun F /\ X e. Fin ) -> ( F " X ) e. Fin ) $=
    ( wfun cfn wcel wa cima cres cdm imadmres wfo wss simpr dmres inss1 eqsstri
    cin ssfi sylancl resss dmss mp1i fores syldan fofi syl2anc eqeltrrid ) ACZB
    DEZFZABGAABHZIZGZDABJUJULDEZULUMAULHZKZUMDEUJUIULBLUNUHUIMULBAIZQBABNBUQOPB
    ULRSUHUIULUQLZUPUKALURUJABTUKAUAUBULAUCUDULUMUOUEUFUG $.
  $( $j usage 'imafi' avoids 'ax-pow'; $)

  ${
    $d X x $.  $d F x y z $.
    $( Obsolete version of ~ imafi as of 25-Jun-2025.  (Contributed by Stefan
       O'Rear, 22-Feb-2015.)  Avoid ~ ax-pow .  (Revised by BTernaryTau,
       7-Sep-2024.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    imafiOLD $p |- ( ( Fun F /\ X e. Fin ) -> ( F " X ) e. Fin ) $=
      ( vx vy vz cfn wcel cima cv wi c0 csn cun imaeq2 eleq1d imbi2d 0fi a1i wa
      wceq wfun ima0 eqeltri cdm cfv wfn fnsnfv sylanb snfi eqeltrrdi wn ndmima
      funfn eqeltrdi adantl pm2.61dan unfi eqeltrid sylan2 expcom a2i findcard2
      imaundi impcom ) BFGAUAZABHZFGZVEACIZHZFGZJVEAKHZFGZJVEADIZHZFGZJZVEAVMEI
      ZLZMZHZFGZJZVEVGJCDEBVHKTZVJVLVEWCVIVKFVHKANOPVHVMTZVJVOVEWDVIVNFVHVMANOP
      VHVSTZVJWAVEWEVIVTFVHVSANOPVHBTZVJVGVEWFVIVFFVHBANOPVLVEVKKFAUBQUCRVPWBJV
      MFGVEVOWAVOVEWAVEVOAVRHZFGZWAVEVQAUDZGZWHVEWJSWGVQAUEZLZFVEAWIUFWJWLWGTAU
      MWIVQAUGUHWKUIUJWJUKZWHVEWMWGKFVQAULQUNUOUPVOWHSVTVNWGMFAVMVRVCVNWGUQURUS
      UTVARVBVD $.
  $}

  ${
    $d B x y $.
    $( If the power set of a set is finite, then the set itself is finite.
       (Contributed by BTernaryTau, 7-Sep-2024.) $)
    pwfir $p |- ( ~P B e. Fin -> B e. Fin ) $=
      ( vx vy cpw cfn wcel cv csn wceq wa cima cres crn df-ima wrel cdm wss cab
      copab 3eqtri relopab dmopabss relssres mp2an rneqi wex eleq1 biimparc vex
      rnopab snelpw sylibr exlimiv wrex snelpwi eqid eqeq2 rspcev sylancl sylib
      df-rex impbii abbii abid2 wfun wmo funopab mosneq moani mpgbir imafi mpan
      eqeltrrid ) ADZEFZABGZVNFZCGZHZVPIZJZBCSZVNKZEWCWBVNLZMWBMZAWBVNNWDWBWBOW
      BPVNQWDWBIWABCUAVTBCVNUBWBVNUCUDUEWEWABUFZCRVRAFZCRAWABCUJWFWGCWFWGWAWGBW
      AVSVNFZWGVTWHVQVSVPVNUGUHVRACUIUKULUMWGVTBVNUNZWFWGWHVSVSIZWIVRAUOVSUPVTW
      JBVSVNVPVSVSUQURUSVTBVNVAUTVBVCCAVDTTWBVEZVOWCEFWKWACVFBWABCVGVTVQCCVPVHV
      IVJWBVNVKVLVM $.
    $( $j usage 'pwfir' avoids 'ax-pow'; $)
  $}

  ${
    $d F d $.  $d b c d $.  $d c d x $.
    pwfilem.1 $e |- F = ( c e. ~P b |-> ( c u. { x } ) ) $.
    $( Lemma for ~ pwfi .  (Contributed by NM, 26-Mar-2007.)  Avoid ~ ax-pow .
       (Revised by BTernaryTau, 7-Sep-2024.) $)
    pwfilem $p |- ( ~P b e. Fin -> ~P ( b u. { x } ) e. Fin ) $=
      ( vd cv cpw cfn wcel csn cun cdif pwundif crn wss wfun cima wceq sylib wa
      funmpt2 cdm vsnex unex dmmpti imaeq2i imadmrn eqtr3i imafi eqeltrrid mpan
      wrex eldifi elpwun undif1 elpwunsn snssd ssequn2 eqtr2id rspceeqv syl2anc
      vex uneq1 elrnmptd ssriv ssfi sylancl unfi mpancom eqeltrid ) CGZHZIJZVLA
      GZKZLHZVQVMMZVMLZIVLVPNVRIJZVNVSIJVNBOZIJZVRWAPVTBQZVNWBDVMDGZVPLZBEUBWCV
      NUAWABVMRZIBBUCZRWFWAWGVMBDVMWEBWDVPDVCAUDZUEEUFUGBUHUIBVMUJUKULFVRWAFGZV
      RJZDVMWEWIBVQEWJWIVPMZVMJZWIWKVPLZSWIWESDVMUMWJWIVQJWLWIVQVMUNZWIVLVPWHUO
      TWJWMWIVPLZWIWIVPUPWJVPWIPWOWISWJVOWIWIVLVOUQURVPWIUSTUTDWKVMWEWMWIWDWKVP
      VDVAVBWNVEVFWAVRVGVHVRVMVIVJVK $.
    $( $j usage 'pwfilem' avoids 'ax-pow'; $)
  $}

  ${
    $d A x $.  $d x y z $.  $d c y z $.
    $( The power set of a finite set is finite and vice-versa.  Theorem 38 of
       [Suppes] p. 104 and its converse, Theorem 40 of [Suppes] p. 105.
       (Contributed by NM, 26-Mar-2007.)  Avoid ~ ax-pow .  (Revised by
       BTernaryTau, 7-Sep-2024.) $)
    pwfi $p |- ( A e. Fin <-> ~P A e. Fin ) $=
      ( vx vy vz vc cfn wcel cpw cv c0 csn cun wceq pweq eleq1d snfi eqeltri wi
      pw0 cmpt eqid pwfilem a1i findcard2 pwfir impbii ) AFGAHZFGZBIZHZFGJHZFGC
      IZHZFGZULDIKZLZHZFGZUHBCDAUIJMUJUKFUIJNOUIULMUJUMFUIULNOUIUPMUJUQFUIUPNOU
      IAMUJUGFUIANOUKJKFSJPQUNURRULFGDEUMEIUOLTZCEUSUAUBUCUDAUEUF $.
    $( $j usage 'pwfi' avoids 'ax-pow'; $)
  $}

  $( The Cartesian product of two finite sets is finite.  (Contributed by Jeff
     Madsen, 2-Sep-2009.)  (Revised by Mario Carneiro, 12-Mar-2015.)  Avoid
     ~ ax-pow .  (Revised by BTernaryTau, 10-Jan-2025.) $)
  xpfi $p |- ( ( A e. Fin /\ B e. Fin ) -> ( A X. B ) e. Fin ) $=
    ( cfn wcel wa cun cpw cxp wss unfi pwfi bitri sylib xpsspw ssfi sylancl ) A
    CDBCDEZABFZGZGZCDZABHZTIUBCDQRCDZUAABJUCSCDUARKSKLMABNTUBOP $.
  $( $j usage 'xpfi' avoids 'ax-pow'; $)

  ${
    $d A w x y z $.  $d B w x y z $.
    $( Obsolete version of ~ xpfi as of 10-Jan-2025.  (Contributed by Jeff
       Madsen, 2-Sep-2009.)  (Revised by Mario Carneiro, 12-Mar-2015.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    xpfiOLD $p |- ( ( A e. Fin /\ B e. Fin ) -> ( A X. B ) e. Fin ) $=
      ( vx vy vz vw cfn wcel cxp cv wi c0 csn cdif wceq xpeq1 eleq1d imbi2d cvv
      wa 0xp 0fi eqeltri a1i wral wn wex neq0 sneq difeq2d xpeq1d adantl pm2.27
      rspcv ad2antlr cen wbr c2nd cres wf1o snex xpexg mpan id 2ndconst f1oen2g
      vex mp1i syl3anc enfii mpdan cun unfi xpundir difsnid eqtr3id biimpd syl5
      mpan2d 3syld ex exlimdv biimtrid eqeltrdi a1d pm2.61d2 com23 findcard imp
      ) AGHBGHZABIZGHZWJCJZBIZGHZKWJLBIZGHZKWJDJZEJZMZNZBIZGHZKZWJWRBIZGHZKWJWL
      KCDEAWMLOZWOWQWJXGWNWPGWMLBPQRWMXAOZWOXCWJXHWNXBGWMXABPQRWMWROZWOXFWJXIWN
      XEGWMWRBPQRWMAOZWOWLWJXJWNWKGWMABPQRWQWJWPLGBUAUBUCZUDWRGHZWJXDEWRUEZXFXL
      WJXMXFKZXLWJTZWRLOZXNXPUFFJZWRHZFUGXOXNFWRUHXOXRXNFXOXRXNXOXRTZXMWJWRXQMZ
      NZBIZGHZKZYCXFXRXMYDKXOXDYDEXQWRWSXQOZXCYCWJYEXBYBGYEXAYABYEWTXTWRWSXQUIU
      JUKQRUNULWJYDYCKXLXRWJYCUMUOXSYCXTBIZGHZXFWJYGXLXRWJYFBUPUQZYGWJYFSHZWJYF
      BURYFUSZUTZYHXTSHWJYIXQVAXTBSGVBVCWJVDXQSHYKWJFVGXQBSVEVHYFBYJSGVFVIYFBVJ
      VKUOYCYGTYBYFVLZGHZXSXFYBYFVMXRYMXFKXOXRYMXFXRYLXEGXRYLYAXTVLZBIXEYAXTBVN
      XRYNWRBWRXQVOUKVPQVQULVRVSVTWAWBWCXPXFXMXPXEWPGWRLBPXKWDWEWFWAWGWHWI $.
  $}

  $( 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 a b c f $.  $d B a b c f $.  $d X a b c f $.  $d Y a b c f $.
    $( A finite set union cancellation law for dominance.  (Contributed by
       Stefan O'Rear, 19-Feb-2015.)  (Revised by Stefan O'Rear, 5-May-2015.) $)
    domunfican $p |- ( ( ( A e. Fin /\ B ~~ A ) /\
          ( ( A i^i X ) = (/) /\ ( B i^i Y ) = (/) ) ) ->
        ( ( A u. X ) ~<_ ( B u. Y ) <-> X ~<_ Y ) ) $=
      ( wcel wbr c0 wceq wa cun cdom wb wi cima anbi1d uneq1d breq12d bibi1d wn
      wss vf va vb vc cfn cen cin cv wf1o wex ensym bren sylib ssid sseq1 uneq1
      w3a csn imaeq2 imbi12d uncom un0 eqtri ima0 breq12i a1i ssun1 sstr2 ax-mp
      uneq1i anim1i imim1i cfv unass imaundi wfn simprlr f1ofn syl ssun2 sylibr
      vex snss adantr ad2antrl fnsnfv syl2anc eqcomd uneq2d eqtrid eqtrdi incom
      simplr simprrl minel wo ioran elun xchnxbir sylanbrc f1of1 adantl f1elima
      wf1 syl3anc biimpd mtod adantrr wf f1of ffvelcdmd simprrr fvex domunsncan
      bitrd bitr ex syl5 findcard2s expd mpani 3imp f1ofo foima breq2d 3ad2ant2
      a2d wfo mpbid 3exp exlimdv imp31 ) AUEEZBAUFFZACUGZGHZBDUGZGHZIZACJZBDJZK
      FZCDKFZLZYNABUAUHZUIZUAUJZYMYSUUDMZYNABUFFUUGBAUKABUAULUMYMUUFUUHUAYMUUFY
      SUUDYMUUFYSUQYTUUEANZDJZKFZUUCLZUUDYMUUFYSUULYMAATZUUFYSUULMAUNYMUUMUUFIZ
      YSUULUBUHZATZUUFIZYSIZUUOCJZUUEUUONZDJZKFZUUCLZMGATZUUFIZYSIZGCJZUUEGNZDJ
      ZKFZUUCLZMUCUHZATZUUFIZYSIZUVLCJZUUEUVLNZDJZKFZUUCLZMZUVLUDUHZURZJZATZUUF
      IZYSIZUWDCJZUUEUWDNZDJZKFZUUCLZMZUUNYSIZUULMUBUCUDAUUOGHZUURUVFUVCUVKUWOU
      UQUVEYSUWOUUPUVDUUFUUOGAUOOOUWOUVBUVJUUCUWOUUSUVGUVAUVIKUUOGCUPUWOUUTUVHD
      UUOGUUEUSPQRUTUUOUVLHZUURUVOUVCUVTUWPUUQUVNYSUWPUUPUVMUUFUUOUVLAUOOOUWPUV
      BUVSUUCUWPUUSUVPUVAUVRKUUOUVLCUPUWPUUTUVQDUUOUVLUUEUSPQRUTUUOUWDHZUURUWGU
      VCUWLUWQUUQUWFYSUWQUUPUWEUUFUUOUWDAUOOOUWQUVBUWKUUCUWQUUSUWHUVAUWJKUUOUWD
      CUPUWQUUTUWIDUUOUWDUUEUSPQRUTUUOAHZUURUWNUVCUULUWRUUQUUNYSUWRUUPUUMUUFUUO
      AAUOOOUWRUVBUUKUUCUWRUUSYTUVAUUJKUUOACUPUWRUUTUUIDUUOAUUEUSPQRUTUVKUVFUVG
      CUVIDKUVGCGJCGCVACVBVCUVIGDJZDUVHGDUUEVDVJUWSDGJDGDVADVBVCVCVEVFUWAUWGUVT
      MUVLUEEZUWBUVLEZSZIZUWMUWGUVOUVTUWFUVNYSUWEUVMUUFUVLUWDTUWEUVMMUVLUWCVGUV
      LUWDAVHVIZVKVKVLUXCUWGUVTUWLUXCUWGUVTUWLMZUXCUWGIZUWKUVSLZUXEUXFUWKUWCUVP
      JZUWBUUEVMZURZUVRJZKFZUVSUXFUWHUXHUWJUXKKUWHUXHHUXFUWHUWCUVLJZCJUXHUWDUXM
      CUVLUWCVAVJUWCUVLCVNVCVFUXFUWJUVQUXJJZDJZUXKUXFUWIUXNDUXFUWIUVQUUEUWCNZJU
      XNUUEUVLUWCVOUXFUXPUXJUVQUXFUXJUXPUXFUUEAVPZUWBAEZUXJUXPHUXFUUFUXQUXCUWEU
      UFYSVQZABUUEVRVSUWFUXRUXCYSUWEUXRUUFUWEUWCATZUXRUWCUWDTUWEUXTMUWCUVLVTUWC
      UWDAVHVIUWBAUDWBZWCWAWDZWEZAUWBUUEWFWGWHWIWJPUXOUXJUVQJZDJUXKUXNUYDDUVQUX
      JVAVJUXJUVQDVNVCWKQUXFUWBUVPEZSZUXIUVREZSZUXLUVSLUXFUXBUWBCEZSZUYFUWTUXBU
      WGWMUXFUXRCAUGZGHUYJUYCUXFUYKYOGCAWLUXCUWFYPYRWNWJUWBACWOWGUXAUYIWPUXBUYJ
      IUYEUXAUYIWQUWBUVLCWRWSWTUXFUXIUVQEZSZUXIDEZSZUYHUXCUWFUYMYSUXCUWFIUYLUXA
      UWTUXBUWFWMUWFUYLUXAMUXCUWFUYLUXAUWFABUUEXDZUXRUVMUYLUXALUUFUYPUWEABUUEXA
      XBUYBUWEUVMUUFUXDWDABUUEUWBUVLXCXEXFXBXGXHUXFUXIBEDBUGZGHUYOUXFABUWBUUEUX
      FUUFABUUEXIUXSABUUEXJVSUYCXKUXFUYQYQGDBWLUXCUWFYPYRXLWJUXIBDWOWGUYLUYNWPU
      YMUYOIUYGUYLUYNWQUXIUVQDWRWSWTUWBUXIUVPUVRUYAUWBUUEXMXNWGXOUXGUVTUWLUWKUV
      SUUCXPXQVSXQYGXRXSXTYAYBUUFYMUULUUDLYSUUFUUKUUBUUCUUFUUJUUAYTKUUFUUIBDUUF
      ABUUEYHUUIBHABUUEYCABUUEYDVSPYERYFYIYJYKXRYL $.
  $}

  ${
    $d x A $.
    infcntss.1 $e |- A e. _V $.
    $( Every infinite set has a denumerable subset.  Similar to Exercise 8 of
       [TakeutiZaring] p. 91.  (However, we need neither AC nor the Axiom of
       Infinity because of the way we express "infinite" in the antecedent.)
       (Contributed by NM, 23-Oct-2004.) $)
    infcntss $p |- ( _om ~<_ A -> E. x ( x C_ A /\ x ~~ _om ) ) $=
      ( com cdom wbr cv cen wss wa wex domen ensym anim1ci eximi sylbi ) DBEFDA
      GZHFZQBIZJZAKSQDHFZJZAKADBCLTUBARUASDQMNOP $.
  $}

  ${
    $d A x $.  $d B x $.
    $( An unordered pair is finite.  For a shorter proof using ~ ax-un , see
       ~ prfiALT .  (Contributed by NM, 22-Aug-2008.)  Avoid ~ ax-11 ,
       ~ ax-un .  (Revised by BTernaryTau, 13-Jan-2025.) $)
    prfi $p |- { A , B } e. Fin $=
      ( vx cvv wcel cpr cfn wn csn prprc1 snfi eqeltrdi prprc2 wceq w3a cen wbr
      wa com c2o wrex 2onn simp1 simp2 simp3 enpr2d breq2 rspcev sylancr sylibr
      cv isfi 3expia dfsn2 preq2 eqtr2id pm2.61d2 ecase ) ADEZBDEZABFZGEZUSHVAB
      IGABJBKLUTHVAAIZGABMAKZLUSUTRABNZVBUSUTVEHZVBUSUTVFOZVACUKZPQZCSUAZVBVGTS
      EVATPQZVJUBVGABDDUSUTVFUCUSUTVFUDUSUTVFUEUFVIVKCTSVHTVAPUGUHUICVAULUJUMVE
      VAVCGVEVCAAFVAAUNABAUOUPVDLUQUR $.
    $( $j usage 'prfi' avoids 'ax-11' 'ax-un'; $)
  $}

  $( Shorter proof of ~ prfi using ~ ax-un .  (Contributed by NM, 22-Aug-2008.)
     (Proof modification is discouraged.)  (New usage is discouraged.) $)
  prfiALT $p |- { A , B } e. Fin $=
    ( cpr csn cun cfn df-pr wcel snfi unfi mp2an eqeltri ) ABCADZBDZEZFABGMFHNF
    HOFHAIBIMNJKL $.

  $( An unordered triple is finite.  (Contributed by Mario Carneiro,
     28-Sep-2013.) $)
  tpfi $p |- { A , B , C } e. Fin $=
    ( ctp cpr csn cun cfn df-tp wcel prfi snfi unfi mp2an eqeltri ) ABCDABEZCFZ
    GZHABCIPHJQHJRHJABKCLPQMNO $.

  ${
    $d A v x y $.  $d A u v y $.  $d A t u v w z $.  $d A f t u v z $.
    $( Equivalent ways of stating the finite intersection property.  We show
       two ways of saying, "the intersection of elements in every finite
       nonempty subcollection of ` A ` is in ` A ` ".  This theorem is
       applicable to a topology, which (among other axioms) is closed under
       finite intersections.  Some texts use the left-hand version of this
       axiom and others the right-hand version, but as our proof here shows,
       their "intuitively obvious" equivalence can be non-trivial to establish
       formally.  (Contributed by NM, 22-Sep-2002.)  Use a separate setvar for
       the right-hand side and avoid ~ ax-pow .  (Revised by BTernaryTau,
       14-Jan-2025.) $)
    fiint $p |- ( A. x e. A A. y e. A ( x i^i y ) e. A <->
        A. z ( ( z C_ A /\ z =/= (/) /\ z e. Fin ) -> |^| z e. A ) ) $=
      ( vv vu vw cv cin wcel wss c0 wa wi wal cen wbr wceq cvv eleq1d vt vf wne
      wral cfn cint w3a com wrex isfi wb nnfi ensymfib csuc breq1 anbi2d imbi1d
      syl albidv weq en0r biimpi anim1i ancoms adantll wn df-ne pm2.21i sylan2b
      pm3.24 ax-gen a1i nfv nfa1 wf1o wex bren cima ssel wf f1of sucid ffvelcdm
      cfv vex sylancl impel adantr crn imassrn wfn ccnv dff1o2 simp3bi sseqtrid
      sstr2 anim1d wf1 f1of1 sssucid f1imaen3g mp3an23 jctird imaex sseq1 neeq1
      wfun anbi12d breq2 inteq imbi12d sylan9 ineq1 ineq2 rspc2v ex com4r exp5c
      spcv syl6 com14 biimtrrid int0 eqtrdi ineq1d ssv sseqin2 biimprd pm2.61d2
      imp43 mpbi mpd csn cun fvex intunsn f1ofn fnsnfv com13 biantru uneq2d wfo
      df-suc imaeq2i imaundi eqtr2i f1ofo foima inteqd eqtr3id ad2antlr exlimdv
      eqtrd mpbid exp43 biimtrid imp adantlr alrimd finds2 exp4a com24 rexlimiv
      sylbird sylbi impd alrimiv cpr zfpair2 eleq1 prss prnz prfi 3bitrri intpr
      sp eleq1i 3imtr3g ralrimivv impbii cbvral2vw df-3an imbi1i albii 3bitr4i
      ) EHZFHZIZDJZFDUDEDUDZCHZDKZUWKLUCZMZUWKUEJZMZUWKUFZDJZNZCOZAHZBHZIZDJZBD
      UDADUDUWLUWMUWOUGZUWRNZCOUWJUWTUWJUWSCUWJUWNUWOUWRUWOUWNUWJUWRUWOUWKGHZPQ
      ZGUHUIUWNUWJUWRNZNZGUWKUJUXHUXJGUHUXGUHJZUXHUXGUWKPQZUXJUXKUXGUEJUXLUXHUK
      UXGULUXGUWKUMURUXKUWJUWNUXLUWRUXKUWJUWNUXLUWRUXKUWJUWNUXLMZUWRNZCOZUXNUXO
      UWNLUWKPQZMZUWRNZCOZUWNUAHZUWKPQZMZUWRNZCOZUWNUXTUNZUWKPQZMZUWRNZCOZUWJGU
      AUXGLRZUXNUXRCUYJUXMUXQUWRUYJUXLUXPUWNUXGLUWKPUOUPUQUSGUAUTZUXNUYCCUYKUXM
      UYBUWRUYKUXLUYAUWNUXGUXTUWKPUOUPUQUSUXGUYERZUXNUYHCUYLUXMUYGUWRUYLUXLUYFU
      WNUXGUYEUWKPUOUPUQUSUXSUWJUXRCUXQUWKLRZUWMMZUWRUWMUXPUYNUWLUXPUWMUYNUXPUY
      MUWMUXPUYMUWKVAVBVCVDVEUWMUYMUYMVFZUWRUWKLVGUYMUYOMUWRUYMVJVHVIURVKVLUWJU
      YDUYINNUXTUHJUWJUYDUYHCUWJCVMUYCCVNUYGUYDUWJUWRUWLUYFUYDUXINZUWMUWLUYFUYP
      UYFUYEUWKUBHZVOZUBVPUWLUYPUYEUWKUBVQUWLUYRUYPUBUWLUYRUYDUWJUWRUWLUYRMZUYD
      UWJMZMZUYQUXTVRZUFZUXTUYQWDZIZDJZUWRVUAVUDDJZVUFUYSVUGUYTUWLVUDUWKJZVUGUY
      RUWKDVUDVSUYRUYEUWKUYQVTUXTUYEJZVUHUYEUWKUYQWAUXTUAWEWBZUYEUWKUXTUYQWCWFW
      GWHVUAVUBLRZVUGVUFNZVUKVFVUBLUCZVUAVULVUBLVGUWLUYRUYDUWJVUMVULNZUWJUYRUYD
      UWLVUNUWJUYRUYDUWLVUMVULUYRUYDMZUWLVUMMZVUGUWJVUFVUOVUPVUCDJZVUGUWJVUFNZN
      UYRVUPVUBDKZVUMMZUXTVUBPQZMZUYDVUQUYRVUPVUTVVAUYRUWLVUSVUMUYRVUBUWKKUWLVU
      SNUYRUYQWIZVUBUWKUYQUXTWJUYRUYQUYEWKZUYQWLXGVVCUWKRUYEUWKUYQWMWNWOVUBUWKD
      WPURWQUYRUYEUWKUYQWRZVVAUYEUWKUYQWSVVEUXTUYEKUYQSJVVAUXTWTUBWEZUYEUWKUXTU
      YQSXAXBURXCUYCVVBVUQNCVUBUYQUXTVVFXDUWKVUBRZUYBVVBUWRVUQVVGUWNVUTUYAVVAVV
      GUWLVUSUWMVUMUWKVUBDXEUWKVUBLXFXHUWKVUBUXTPXIXHVVGUWQVUCDUWKVUBXJTXKXSXLV
      UQVUGVURUWIVUFVUCUWGIZDJEFVUCVUDDDUWFVUCRUWHVVHDUWFVUCUWGXMTUWGVUDRVVHVUE
      DUWGVUDVUCXNTXOXPXTXQXRYAYJYBVUKVUFVUGVUKVUEVUDDVUKVUESVUDIZVUDVUKVUCSVUD
      VUKVUCLUFSVUBLXJYCYDYEVUDSKVVIVUDRVUDYFVUDSYGYKYDTYHYIYLUYRVUFUWRUKUWLUYT
      UYRVUEUWQDUYRVUEVUBVUDYMZYNZUFUWQVUBVUDUXTUYQYOYPUYRVVKUWKUYRVVKUYQUYEVRZ
      UWKUYRVVKVUBUYQUXTYMZVRZYNZVVLUYRVVJVVNVUBUYRVVDVUIVVJVVNRUYEUWKUYQYQVUJU
      YEUXTUYQYRWFUUAVVLUYQUXTVVMYNZVRVVOUYEVVPUYQUXTUUCUUDUYQUXTVVMUUEUUFYDUYR
      UYEUWKUYQUUBVVLUWKRUYEUWKUYQUUGUYEUWKUYQUUHURUUMUUIUUJTUUKUUNUUOUULUUPUUQ
      UURYSUUSVLUUTUXNCUVPXTUVAUVBUVDUVCUVEYSUVFUVGUWTUWIEFDDUWTUWFUWGUVHZDKZVV
      QLUCZMZVVQUEJZMZVVQUFZDJZUWFDJUWGDJMZUWIUWSVWBVWDNCVVQEFUVIUWKVVQRZUWPVWB
      UWRVWDVWFUWNVVTUWOVWAVWFUWLVVRUWMVVSUWKVVQDXEUWKVVQLXFXHUWKVVQUEUVJXHVWFU
      WQVWCDUWKVVQXJTXKXSVWEVVRVVTVWBUWFUWGDEWEZFWEZUVKVVSVVRUWFUWGVWGUVLYTVWAV
      VTUWFUWGUVMYTUVNVWCUWHDUWFUWGVWGVWHUVOUVQUVRUVSUVTUXDUWIUWFUXBIZDJABEFDDA
      EUTUXCVWIDUXAUWFUXBXMTBFUTVWIUWHDUXBUWGUWFXNTUWAUXFUWSCUXEUWPUWRUWLUWMUWO
      UWBUWCUWDUWE $.
    $( $j usage 'fiint' avoids 'ax-pow'; $)
  $}

  ${
    $d x y z w v f A $.
    $( Obsolete version of ~ fiint as of 14-Jan-2025.  (Contributed by NM,
       22-Sep-2002.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    fiintOLD $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 cint wi wal cen wbr wceq cvv eleq1d wral
      wne cfn w3a com wrex isfi ensym csuc breq1 anbi2d imbi1d albidv weq sylib
      vv en0 anim1i ancoms adantll wn pm3.24 pm2.21i sylan2b syl ax-gen a1i nfv
      df-ne nfa1 wf1o wex bren cima cfv ssel wf f1of vex sucid ffvelcdm sylancl
      impel adantr crn imassrn wfn ccnv wfun dff1o2 simp3bi sseqtrid anim1d wf1
      sstr2 f1of1 sssucid mpanr12 ensymd jctird imaex sseq1 neeq1 anbi12d breq2
      f1imaen2g inteq imbi12d spcv sylan9 ineq1 ineq2 ex syl6 com4r exp5c com14
      rspc2v biimtrrid int0 eqtrdi ineq1d ssv sseqin2 mpbi biimprd pm2.61d2 mpd
      imp43 wb csn cun fvex intunsn f1ofn fnsnfv uneq2d df-suc com13 biantru sp
      imaeq2i imaundi eqtr2i wfo f1ofo foima eqtrd eqtr3id ad2antlr mpbid exp43
      inteqd exlimdv biimtrid imp alrimd finds2 exp4a com24 syl5 rexlimiv sylbi
      adantlr impd alrimiv zfpair2 eleq1 prss prnz 3bitrri intpr eleq1i 3imtr3g
      cpr prfi ralrimivv impbii cbvral2vw df-3an imbi1i albii 3bitr4i ) DGZEGZH
      ZCIZECUADCUAZAGZCJZUWIKUBZLZUWIUCIZLZUWIMZCIZNZAOZUWIBGZHZCIZBCUAACUAUWJU
      WKUWMUDZUWPNZAOUWHUWRUWHUWQAUWHUWLUWMUWPUWMUWLUWHUWPUWMUWIUWSPQZBUEUFUWLU
      WHUWPNZNZBUWIUGUXDUXFBUEUXDUWSUWIPQZUWSUEIZUXFUWIUWSUHUXHUWHUWLUXGUWPUXHU
      WHUWLUXGUWPUXHUWHUWLUXGLZUWPNZAOZUXJUXKUWLKUWIPQZLZUWPNZAOZUWLUPGZUWIPQZL
      ZUWPNZAOZUWLUXPUIZUWIPQZLZUWPNZAOZUWHBUPUWSKRZUXJUXNAUYFUXIUXMUWPUYFUXGUX
      LUWLUWSKUWIPUJUKULUMBUPUNZUXJUXSAUYGUXIUXRUWPUYGUXGUXQUWLUWSUXPUWIPUJUKUL
      UMUWSUYARZUXJUYDAUYHUXIUYCUWPUYHUXGUYBUWLUWSUYAUWIPUJUKULUMUXOUWHUXNAUXMU
      WIKRZUWKLZUWPUWKUXLUYJUWJUXLUWKUYJUXLUYIUWKUXLUWIKPQUYIKUWIUHUWIUQUOURUSU
      TUWKUYIUYIVAZUWPUWIKVIUYIUYKLUWPUYIVBVCVDVEVFVGUWHUXTUYENNUXPUEIUWHUXTUYD
      AUWHAVHUXSAVJUYCUXTUWHUWPUWJUYBUXTUXENZUWKUWJUYBUYLUYBUYAUWIFGZVKZFVLUWJU
      YLUYAUWIFVMUWJUYNUYLFUWJUYNUXTUWHUWPUWJUYNLZUXTUWHLZLZUYMUXPVNZMZUXPUYMVO
      ZHZCIZUWPUYQUYTCIZVUBUYOVUCUYPUWJUYTUWIIZVUCUYNUWICUYTVPUYNUYAUWIUYMVQUXP
      UYAIZVUDUYAUWIUYMVRUXPUPVSZVTZUYAUWIUXPUYMWAWBWCWDUYQUYRKRZVUCVUBNZVUHVAU
      YRKUBZUYQVUIUYRKVIUWJUYNUXTUWHVUJVUINZUWHUYNUXTUWJVUKUWHUYNUXTUWJVUJVUIUY
      NUXTLZUWJVUJLZVUCUWHVUBVULVUMUYSCIZVUCUWHVUBNZNUYNVUMUYRCJZVUJLZUXPUYRPQZ
      LZUXTVUNUYNVUMVUQVURUYNUWJVUPVUJUYNUYRUWIJUWJVUPNUYNUYMWEZUYRUWIUYMUXPWFU
      YNUYMUYAWGZUYMWHWIVUTUWIRUYAUWIUYMWJWKWLUYRUWICWOVEWMUYNUYRUXPUYNUYAUWIUY
      MWNZUWISIZUYRUXPPQZUYAUWIUYMWPAVSVVBVVCLUXPUYAJUXPSIVVDUXPWQVUFUYAUWIUXPU
      YMSXFWRWBWSWTUXSVUSVUNNAUYRUYMUXPFVSXAUWIUYRRZUXRVUSUWPVUNVVEUWLVUQUXQVUR
      VVEUWJVUPUWKVUJUWIUYRCXBUWIUYRKXCXDUWIUYRUXPPXEXDVVEUWOUYSCUWIUYRXGTXHXIX
      JVUNVUCVUOUWGVUBUYSUWEHZCIDEUYSUYTCCUWDUYSRUWFVVFCUWDUYSUWEXKTUWEUYTRVVFV
      UACUWEUYTUYSXLTXRXMXNXOXPXQYIXSVUHVUBVUCVUHVUAUYTCVUHVUASUYTHZUYTVUHUYSSU
      YTVUHUYSKMSUYRKXGXTYAYBUYTSJVVGUYTRUYTYCUYTSYDYEYATYFYGYHUYNVUBUWPYJUWJUY
      PUYNVUAUWOCUYNVUAUYRUYTYKZYLZMUWOUYRUYTUXPUYMYMYNUYNVVIUWIUYNVVIUYMUYAVNZ
      UWIUYNVVIUYRUYMUXPYKZVNZYLZVVJUYNVVHVVLUYRUYNVVAVUEVVHVVLRUYAUWIUYMYOVUGU
      YAUXPUYMYPWBYQVVJUYMUXPVVKYLZVNVVMUYAVVNUYMUXPYRUUBUYMUXPVVKUUCUUDYAUYNUY
      AUWIUYMUUEVVJUWIRUYAUWIUYMUUFUYAUWIUYMUUGVEUUHUUMUUITUUJUUKUULUUNUUOUUPUV
      DYSUUQVGUURUXJAUUAXNUUSUUTUVAUVBUVCYSUVEUVFUWRUWGDECCUWRUWDUWEUVOZCJZVVOK
      UBZLZVVOUCIZLZVVOMZCIZUWDCIUWECILZUWGUWQVVTVWBNAVVODEUVGUWIVVORZUWNVVTUWP
      VWBVWDUWLVVRUWMVVSVWDUWJVVPUWKVVQUWIVVOCXBUWIVVOKXCXDUWIVVOUCUVHXDVWDUWOV
      WACUWIVVOXGTXHXIVWCVVPVVRVVTUWDUWECDVSZEVSZUVIVVQVVPUWDUWEVWEUVJYTVVSVVRU
      WDUWEUVPYTUVKVWAUWFCUWDUWEVWEVWFUVLUVMUVNUVQUVRUXAUWGUWDUWSHZCIABDECCADUN
      UWTVWGCUWIUWDUWSXKTBEUNVWGUWFCUWSUWEUWDXLTUVSUXCUWQAUXBUWNUWPUWJUWKUWMUVT
      UWAUWBUWC $.
  $}

  ${
    $d A f g z $.  $d B f g z $.
    $( There exists a mapping from a finite set onto any nonempty set that it
       dominates, proved without using the Axiom of Power Sets (unlike
       ~ fodomr ).  (Contributed by BTernaryTau, 23-Jun-2025.) $)
    fodomfir $p |- ( ( A e. Fin /\ (/) ~< B /\ B ~<_ A ) ->
        E. f f : A -onto-> B ) $=
      ( vz vg cfn wcel c0 cv wex wa wne wi crn cun wceq wfun cdm adantl wss wbr
      cdom csdm wfo cvv wb relsdom brrelex2i 0sdomg n0 bitrdi syl ibi domfi wf1
      simpl brdomi wfn f1fn fnfi sylan ex ccnv cdif csn cxp cnvfi diffi sylancl
      snfi xpfi unfi syl2an cin df-f1 simprbi vex fconst ffun ax-mp jctir df-rn
      wf eqcomi snnz dmxp ineq12i disjdif eqtri funun uneq1i uneq2i 3eqtr2i f1f
      dmun frnd undif sylib eqtrid df-fn sylanbrc rnun dfdm4 f1dm eqtr3id xpeq1
      uneq1d 0xp eqtrdi rneqd rn0 0ss eqsstrdi a1d rnxp snssi eqsstrd pm2.61ine
      adantr ssequn2 sylan9eqr df-fo foeq1 spcegv syl2im expcomd syland exlimiv
      com12 mp2and exlimdv syl5 3impia 3com23 ) AFGZBAUBUAZHBUCUAZABCIZUDZCJZYO
      YPYQYTYQDIZBGZDJZYOYPKZYTYQUUCYQBUEGZYQUUCUFHBUCUGUHUUEYQBHLUUCBUEUIDBUJU
      KULUMUUDUUBYTDUUDBFGZYOUUBYTMZABUNYOYPUPYPUUFYOKUUGMZYOYPBAEIZUOZEJUUHBAE
      UQUUJUUHEUUJUUFUUIFGZYOUUGUUJUUFUUKUUJUUIBURUUFUUKBAUUIUSBUUIUTVAVBUUKYOK
      ZUUJUUGUULUUBUUJYTUULUUIVCZAUUINZVDZUUAVEZVFZOZFGZUUBUUJKZABUURUDZYTUUKUU
      MFGUUQFGZUUSYOUUIVGYOUUOFGUUPFGUVBAUUNVHUUAVJUUOUUPVKVIUUMUUQVLVMUUTUURAU
      RZUURNZBPUVAUUTUURQZUURRZAPZUVCUUJUVEUUBUUJUUMQZUUQQZKUUMRZUUQRZVNZHPUVEU
      UJUVHUVIUUJBAUUIWCUVHBAUUIVOVPUUOUUPUUQWCUVIUUOUUADVQZVRUUOUUPUUQVSVTWAUV
      LUUNUUOVNHUVJUUNUVKUUOUUNUVJUUIWBZWDUUPHLUVKUUOPUUAUVMWEUUOUUPWFVTZWGUUNA
      WHWIUUMUUQWJVISUUJUVGUUBUUJUVFUUNUUOOZAUVFUVJUVKOUUNUVKOUVPUUMUUQWOUUNUVJ
      UVKUVNWKUVKUUOUUNUVOWLWMUUJUUNATUVPAPUUJBAUUIBAUUIWNWPUUNAWQWRWSSUURAWTXA
      UUTUVDUUMNZUUQNZOZBUUMUUQXBUUJUUBUVSBUVROZBUUJUVQBUVRUUJUVQUUIRBUUIXCBAUU
      IXDXEXGUUBUVRBTZUVTBPUUBUWAMUUOHUUOHPZUWAUUBUWBUVRHBUWBUVRHNHUWBUUQHUWBUU
      QHUUPVFHUUOHUUPXFUUPXHXIXJXKXIBXLXMXNUUOHLZUUBUWAUWCUUBKUVRUUPBUWCUVRUUPP
      UUBUUOUUPXOXSUUBUUPBTUWCUUABXPSXQVBXRUVRBXTWRYAWSABUURYBXAYSUVACUURFABYRU
      URYCYDYEYFYIYGYHULSYJYKYLYMYN $.
    $( $j usage 'fodomfir' avoids 'ax-pow'; $)
  $}

  ${
    $d A f $.  $d B f $.
    $( Equivalence of an onto mapping and dominance for a nonempty finite set.
       Unlike ~ fodomb for arbitrary sets, this theorem does not require the
       Axiom of Replacement nor the Axiom of Power Sets nor the Axiom of Choice
       for its proof.  (Contributed by NM, 23-Mar-2006.)  Avoid ~ ax-pow .
       (Revised by BTernaryTau, 23-Jun-2025.) $)
    fodomfib $p |- ( A e. Fin ->
    ( ( A =/= (/) /\ E. f f : A -onto-> B ) <-> ( (/) ~< B /\ B ~<_ A ) ) ) $=
      ( cfn wcel c0 wne cv wfo wex wa csdm wbr cdom wceq eqeq1d cvv 0sdomg jcad
      ex cdm fof fdmd crn dm0rn0 bitrid bitr3d necon3bid biimpac adantll wb vex
      forn rnex eqeltrrdi adantl syl adantlr mpbird fodomfi exlimdv expimpd 0fi
      wi adantr sdomdomtrfi mp3an1 imbitrid fodomfir 3expib impbid ) ADEZAFGZAB
      CHZIZCJZKFBLMZBANMZKZVLVMVPVSVLVMKZVOVSCVTVOVQVRVTVOVQVTVOKVQBFGZVMVOWAVL
      VOVMWAVOAFBFVOVNUAZFOZAFOBFOZVOWBAFVOABVNABVNUBUCPWCVNUDZFOVOWDVNUEVOWEBF
      ABVNUMZPUFUGUHUIUJVLVOVQWAUKZVMVLVOKBQEZWGVOWHVLVOBWEQWFVNCULUNUOUPBQRUQU
      RUSTVLVOVRVDVMVLVOVRABVNUTTVESVAVBVLVSVMVPVSFALMZVLVMFDEVQVRWIVCFBAVFVGAD
      RVHVLVQVRVPABCVIVJSVK $.
    $( $j usage 'fodomfib' avoids 'ax-rep' 'ax-pow' 'ax-ac' 'ax-ac2'; $)
  $}

  ${
    $d f x y z A $.  $d f B $.  $d x y z F $.
    $( Obsolete version of ~ fodomfi as of 20-Jun-2025.  (Contributed by NM,
       23-Mar-2006.)  (Proof shortened by Mario Carneiro, 16-Nov-2014.)
       (Proof modification is discouraged.)  (New usage is discouraged.) $)
    fodomfiOLD $p |- ( ( A e. Fin /\ F : A -onto-> B ) -> B ~<_ A ) $=
      ( vz wcel wa cima cdom wceq wbr cv wi c0 csn cun imaeq2 id breq12d imbi2d
      cvv vx cfn wfo foima adantl wfn ima0 eqtrdi 0ex 0dom a1i cid cfv wss cres
      vy wn crn cop wfun fnfun ad2antrl funressn rnss 3syl df-ima rnsnop eqcomi
      vex 3sstr4g snex sylancl fvi syl uneq2d imaundi eqtr4di cin simprr ssdomg
      ssexg cen mpsyl c1o fvex ensn1 entr4i domentr eqbrtrd simplr disjsn undom
      sylibr syl21anc eqbrtrrd exp32 a2d findcard2s fofn impel ) AUBEZABCUCZFCA
      GZBAHXBXCBIXAABCUDUEXACAUFZXCAHJZXBXDCUAKZGZXFHJZLXDMMHJZLXDCUPKZGZXJHJZL
      XDCXJDKZNZOZGZXOHJZLXDXELUAUPDAXFMIZXHXIXDXRXGMXFMHXRXGCMGMXFMCPCUGUHXRQR
      SXFXJIZXHXLXDXSXGXKXFXJHXFXJCPXSQRSXFXOIZXHXQXDXTXGXPXFXOHXFXOCPXTQRSXFAI
      ZXHXEXDYAXGXCXFAHXFACPYAQRSXIXDMUIUJUKXJUBEZXMXJEUQZFZXDXLXQYDXDXLXQYDXDX
      LFZFZXKCXNGZULUMZOZXPXOHYFYIXKYGOXPYFYHYGXKYFYGTEZYHYGIYFYGXMCUMZNZUNZYLT
      EZYJYFCXNUOZURZXMYKUSNZURZYGYLYFCUTZYOYQUNYPYRUNXDYSYDXLACVAVBXMCVCYOYQVD
      VECXNVFYRYLXMYKDVIZVGVHVJZYKVKZYGYLTWAVLYGTVMVNZVOCXJXNVPVQYFXLYHXNHJXJXN
      VRMIZYIXOHJYDXDXLVSYFYHYGXNHUUCYFYGYLHJZYLXNWBJYGXNHJYNYFYMUUEUUBUUAYGYLT
      VTWCYLWDXNYKXMCWEWFXMYTWFWGYGYLXNWHVLWIYFYCUUDYBYCYEWJXJXMWKWMXKXJYHXNWLW
      NWOWPWQWRABCWSWTWO $.

    $( Obsolete version of ~ fodomfib as of 23-Jun-2025.  (Contributed by NM,
       23-Mar-2006.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    fodomfibOLD $p |- ( A e. Fin ->
    ( ( A =/= (/) /\ E. f f : A -onto-> B ) <-> ( (/) ~< B /\ B ~<_ A ) ) ) $=
      ( cfn wcel c0 wne cv wfo wex wa csdm wbr cdom cdm wceq eqeq1d cvv 0sdomg
      ex fof fdmd dm0rn0 forn bitrid bitr3d necon3bid biimpac adantll eqeltrrdi
      crn wb vex rnex adantl syl adantlr mpbird wi fodomfi jcad exlimdv expimpd
      adantr sdomdomtr imbitrid fodomr jca2 impbid ) ADEZAFGZABCHZIZCJZKFBLMZBA
      NMZKZVJVKVNVQVJVKKZVMVQCVRVMVOVPVRVMVOVRVMKVOBFGZVKVMVSVJVMVKVSVMAFBFVMVL
      OZFPZAFPBFPZVMVTAFVMABVLABVLUAUBQWAVLUKZFPVMWBVLUCVMWCBFABVLUDZQUEUFUGUHU
      IVJVMVOVSULZVKVJVMKBREZWEVMWFVJVMBWCRWDVLCUMUNUJUOBRSUPUQURTVJVMVPUSVKVJV
      MVPABVLUTTVDVAVBVCVJVQVKVNVQFALMVJVKFBAVEADSVFABCVGVHVI $.
  $}

  ${
    $d u w x y z A $.  $d u w x y z B $.  $d u w x y z F $.
    $( Any surjection from one finite set to another of equal size must be a
       bijection.  (Contributed by Mario Carneiro, 19-Aug-2014.) $)
    fofinf1o $p |- ( ( F : A -onto-> B /\ A ~~ B /\ B e. Fin ) ->
                     F : A -1-1-onto-> B ) $=
      ( vx vy vz vw vu wbr cfn wcel cv cfv wceq wa syl2anc sylanbrc adantr wrex
      wne wfo cen w3a wf1 wf1o wf weq wi wral simp1 fof syl csn cdif wn domnsym
      cdom csdm wpss simp3 simp2 enfii ad2antrr difssd simplrr neldifsn sylancl
      wss nelne1 necomd df-pss php3 sdomentr nsyl3 cres difss ssfi fssres sylan
      foelrn simprll simprrr eldifsn simprrl eqcomd rspceeqv fveqeq2 syl5ibrcom
      fveq2 rexbidv imp eqid mpan2 sylbir adantll pm2.61dane fvres eqeq2d eqeq1
      rexbiia bitrid rexlimdva syldan ralrimiva dffo3 fodomfi expr necon1bd mpd
      anassrs ex ralrimivva dff13 df-f1o ) ABCUAZABUBIZBJKZUCZABCUDZXOABCUEXRAB
      CUFZDLZCMZELZCMZNZDEUGZUHZEAUIDAUIXSXRXOXTXOXPXQUJZABCUKULZXRYGDEAAXRYAAK
      ZYCAKZOZOZYEYFYMYEOZBAYCUMZUNZUQIZUOYFYQYPBURIZYNBYPUPYNYPAURIZXPYRYNAJKZ
      YPAUSZYSXRYTYLYEXRXQXPYTXOXPXQUTXOXPXQVAZABVBPZVCYNYPAVHZYPATUUAYNAYOVDYN
      AYPYNYKYCYPKUOAYPTXRYJYKYEVEYCAVFYCAYPVIVGVJYPAVKQAYPVLPXRXPYLYEUUBVCYPAB
      VMPVNYNYQYAYCYMYEYAYCTZYQXRYLYEUUEOZYQXRYLUUFOZOZYPJKZYPBCYPVOZUAZYQUUHYT
      UUDUUIXRYTUUGUUCRAYOVPZAYPVQVGUUHYPBUUJUFZFLZGLZUUJMZNZGYPSZFBUIUUKUUHXTU
      UDUUMXRXTUUGYIRUULABYPCVRVGUUHUURFBUUHUUNBKZUUNHLZCMZNZHASZUURUUHXOUUSUVC
      XRXOUUGYHRHABUUNCVTVSUUHUVCUURUUHUVBUURHAUUHUUTAKZOZUURUVBUVAUUOCMZNZGYPS
      ZUVEUVHUUTYCUVEHEUGZUVHUUHUVIUVHUHUVDUUHUVHUVIYDUVFNZGYPSZUUHYAYPKZYDYBNU
      VKUUHYJUUEUVLXRYJYKUUFWAXRYLYEUUEWBYAAYCWCQUUHYBYDXRYLYEUUEWDWEGYAYPUVFYB
      YDUUOYACWIWFPUVIUVGUVJGYPUUTYCUVFCWGWJWHRWKUVDUUTYCTZUVHUUHUVDUVMOUUTYPKZ
      UVHUUTAYCWCUVNUVAUVANUVHUVAWLGUUTYPUVFUVAUVAUUOUUTCWIWFWMWNWOWPUURUUNUVFN
      ZGYPSUVBUVHUUQUVOGYPUUOYPKUUPUVFUUNUUOYPCWQWRWTUVBUVOUVGGYPUUNUVAUVFWSWJX
      AWHXBWKXCXDGFYPBUUJXEQYPBUUJXFPXJXGXHXIXKXLDEABCXMQYHABCXNQ $.
  $}

  $( Any function from a finite set onto the same set must be a bijection.
     (Contributed by AV, 5-Jul-2021.) $)
  rneqdmfinf1o $p |- ( ( A e. Fin /\ F Fn A /\ ran F = A )
                       -> F : A -1-1-onto-> A ) $=
    ( cfn wcel wfn crn wceq w3a wfo cen wbr wf1o dffn4 biimpi 3ad2ant2 wb foeq3
    3ad2ant3 mpbid enrefg 3ad2ant1 simp1 fofinf1o syl3anc ) ACDZBAEZBFZAGZHZAAB
    IZAAJKZUEAABLUIAUGBIZUJUFUEULUHUFULABMNOUHUEULUJPUFUGAABQRSUEUFUKUHACTUAUEU
    FUHUBAABUCUD $.

  ${
    $d x F $.
    $( Any finite set dominates its domain.  (Contributed by Mario Carneiro,
       22-Sep-2013.)  (Revised by Mario Carneiro, 16-Nov-2014.) $)
    fidomdm $p |- ( F e. Fin -> dom F ~<_ F ) $=
      ( vx cfn wcel cdm cvv cres cdom dmresv wbr cv c1st cfv cmpt wfo finresfin
      crn wfn fvex eqid fnmpti dffn4 mpbi wrel wceq wb relres reldm foeq3 mpbir
      mp2b fodomfi sylancl wss resss ssdomg mpi domtr syl2anc eqbrtrrid ) ACDZA
      EAFGZEZAHAIVAVCVBHJZVBAHJZVCAHJVAVBCDVBVCBVBBKZLMZNZOZVDFAPVIVBVHQZVHOZVH
      VBRVKBVBVGVHVFLSVHTUAVBVHUBUCVBUDVCVJUEVIVKUFAFUGBVBUHVCVJVBVHUIUKUJVBVCV
      HULUMVAVBAUNVEAFUOVBACUPUQVCVBAURUSUT $.
  $}

  $( The domain of a finite set is finite.  (Contributed by Mario Carneiro,
     24-Sep-2013.) $)
  dmfi $p |- ( A e. Fin -> dom A e. Fin ) $=
    ( cfn wcel cdm cdom wbr fidomdm domfi mpdan ) ABCADZAEFJBCAGAJHI $.

  $( 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 dmfi wfn funfn fnfi sylanb ex impbid2 ) ABZACDZAEZCDZAF
    MPNMAOGPNAHOAIJKL $.

  $( The restriction of a function to a finite set is finite.  (Contributed by
     Alexander van der Vekens, 3-Feb-2018.) $)
  resfnfinfin $p |- ( ( F Fn A /\ B e. Fin ) -> ( F |` B ) e. Fin ) $=
    ( wfn cfn wcel wa cres cdm cin wrel wceq fnrel adantr resindm eqcomd syl wi
    fnfun funfnd fnresin2 infi fnfi sylan2 ex 3syl imp eqeltrd ) CADZBEFZGZCBHZ
    CBCIZJZHZEUKCKZULUOLUIUPUJACMNUPUOULCBOPQUIUJUOEFZUICUMDUOUNDZUJUQRUICACSTU
    MBCUAURUJUQUJURUNEFUQBUMUBUNUOUCUDUEUFUGUH $.

  $( 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 dmfi eqeltrrid wfn wfun funi funfn mpbi resfnfinfin
    dmresi mpan impbii ) BACZDEZADEZSARFDAORGHBBFZIZTSBJUBKBLMUAABNPQ $.

  $( Shorter proof of ~ cnvfi using ~ ax-pow .  (Contributed by Mario Carneiro,
     28-Dec-2014.)  (Proof modification is discouraged.)
     (New usage is discouraged.) $)
  cnvfiALT $p |- ( A e. Fin -> `' A e. Fin ) $=
    ( cfn wcel ccnv cen wbr wss cnvcnvss ssfi mpan2 relcnv cnvexg cnven sylancr
    wrel cvv enfii syl2anc ) ABCZADZDZBCZTUAEFZTBCSUAAGUBAHAUAIJSTOTPCUCAKABLTP
    MNTUAQR $.

  $( The range of a finite set is finite.  (Contributed by Mario Carneiro,
     28-Dec-2014.) $)
  rnfi $p |- ( A e. Fin -> ran A e. Fin ) $=
    ( cfn wcel crn ccnv cdm df-rn cnvfi dmfi syl eqeltrid ) ABCZADAEZFZBAGLMBCN
    BCAHMIJK $.

  $( 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 rnfi cdm cen wbr simpr wf1o wceq wi f1dm wb syl eleq1
    f1f1orn f1oeq2 anbi12d eqcoms biimpd expcomd sylc impcom adantr f1oeng wfun
    enfii syl2anc f1fun ad2antlr fundmfibi mpbird ex impbid2 ) ADEZABCFZGZCHEZC
    IZHEZCJVCVFVDVCVFGZVDCKZHEZVGVFVHVELMZVIVCVFNVGVHDEZVHVECOZGZVJVCVMVFVBVAVM
    VBVHAPZAVECOZVAVMQABCRABCUBVNVAVOVMVNVAVOGZVMVPVMSAVHAVHPVAVKVOVLAVHDUAAVHV
    ECUCUDUEUFUGUHUIUJVHVEDCUKTVHVEUMUNVGCULZVDVISVBVQVAVFABCUOUPCUQTURUSUT $.

  $( 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 w x y z A $.  $d w y z B $.
    $( The finite union of finite sets is finite.  Exercise 13 of [Enderton]
       p. 144.  This is the indexed union version of ~ unifi .  Note that ` B `
       depends on ` x ` , i.e. can be thought of as ` B ( x ) ` .  (Contributed
       by NM, 23-Mar-2006.)  (Proof shortened by Mario Carneiro,
       31-Aug-2015.) $)
    iunfi $p |- ( ( A e. Fin /\ A. x e. A B e. Fin )
        -> U_ x e. A B e. Fin ) $=
      ( vw vy vz cfn wcel wral ciun cv wi c0 cun wceq iuneq1 eleq1d imbi12d a1i
      raleq csn 0iun eqtrdi 0fi wss ssun1 ssralv ax-mp imim1i wa iunxun nfcsb1v
      csb nfcv csbeq1a cbviun csbeq1 iunxsn eqtri ssun2 vsnid sselii nfel1 rspc
      vex eqeltrid unfi sylan2 expcom sylcom findcard2 imp ) BGHCGHZABIZABCJZGH
      ZVMADKZIZAVQCJZGHZLVMAMIZMGHZLVMAEKZIZAWCCJZGHZLZVMAWCFKZUAZNZIZAWJCJZGHZ
      LZVNVPLDEFBVQMOZVRWAVTWBVMAVQMTWOVSMGWOVSAMCJMAVQMCPACUBUCQRVQWCOZVRWDVTW
      FVMAVQWCTWPVSWEGAVQWCCPQRVQWJOZVRWKVTWMVMAVQWJTWQVSWLGAVQWJCPQRVQBOZVRVNV
      TVPVMAVQBTWRVSVOGAVQBCPQRWBWAUDSWGWNLWCGHWGWKWFWMWKWDWFWCWJUEWKWDLWCWIUFV
      MAWCWJUGUHUIWFWKWMWFWKUJWLWEAWICJZNZGAWCWICUKWKWFWSGHWTGHWKWSAWHCUMZGWSEW
      IAWCCUMZJXAAEWICXBECUNAWCCULAWCCUOUPEWHXBXAFVEAWCWHCUQURUSWHWJHWKXAGHZLWI
      WJWHWIWCUTFVAVBVMXCAWHWJAXAGAWHCULVCAKWHOCXAGAWHCUOQVDUHVFWEWSVGVHVFVIVJS
      VKVL $.

    $( The finite union of finite sets is finite.  Exercise 13 of [Enderton]
       p. 144.  (Contributed by NM, 22-Aug-2008.)  (Revised by Mario Carneiro,
       31-Aug-2015.) $)
    unifi $p |- ( ( A e. Fin /\ A C_ Fin ) -> U. A e. Fin ) $=
      ( vx cfn wss wcel cv wral cuni dfss3 ciun uniiun iunfi eqeltrid sylan2b
      wa ) ACDACEZBFZCEBAGZAHZCEBACIPROSBAQJCBAKBAQLMN $.

    $( The finite union of finite sets is finite.  Exercise 13 of [Enderton]
       p. 144.  This version of ~ unifi is useful only if we assume the Axiom
       of Infinity (see comments in ~ fin2inf ).  (Contributed by NM,
       11-Mar-2006.) $)
    unifi2 $p |- ( ( A ~< _om /\ A. x e. A x ~< _om ) -> U. A ~< _om ) $=
      ( com csdm wbr cv wral wa cuni cfn wcel wss isfinite2 ralimi dfss3 sylibr
      unifi syl2an cvv wb fin2inf adantr isfiniteg syl mpbid ) BCDEZAFZCDEZABGZ
      HZBIZJKZUKCDEZUFBJKBJLZULUIBMUIUGJKZABGUNUHUOABUGMNABJOPBQRUJCSKZULUMTUFU
      PUIBUAUBUKUCUDUE $.
  $}

  ${
    $d A x $.  $d B x $.
    $( If an infinite set ` A ` is included in the underlying set of a finite
       cover ` B ` , then there exists a set of the cover that contains an
       infinite number of element of ` A ` .  (Contributed by FL,
       2-Aug-2009.) $)
    infssuni $p |- ( ( -. A e. Fin /\ B e. Fin /\ A C_ U. B )
        -> E. x e. B -. ( A i^i x ) e. Fin ) $=
      ( cfn wcel wn cuni wss w3a cv cin wrex wral dfral2 wi wa ciun iunfi sylbi
      eleq1i iunin2 uniiun eqcomi ineq2i wceq dfss2 eleq1 pm2.24 biimtrdi com12
      syl ex com24 3imp21 biimtrrid pm2.18d ) BDEZFZCDEZBCGZHZIZBAJZKZDEZFACLZV
      FFVEACMZVBVFVEACNUSURVAVGVFOUSVGVAURVFUSVGVAURVFOZOZUSVGPACVDQZDEZVIACVDR
      VKBACVCQZKZDEZVIVJVMDACBVCUATVNBUTKZDEZVIVMVODVLUTBUTVLACUBUCUDTVAVPVHVAV
      OBUEZVPVHOBUTUFVQVPUQVHVOBDUGUQVFUHUISUJSSUKULUMUNUOUP $.
  $}

  ${
    unirnffid.1 $e |- ( ph -> F : T --> Fin ) $.
    unirnffid.2 $e |- ( ph -> T e. Fin ) $.
    $( The union of the range of a function from a finite set into the class of
       finite sets is finite.  Deduction form.  (Contributed by David Moews,
       1-May-2017.) $)
    unirnffid $p |- ( ph -> U. ran F e. Fin ) $=
      ( crn cfn wcel wss cuni wfn ffnd fnfi syl2anc rnfi syl frnd unifi ) ACFZG
      HZSGISJGHACGHZTACBKBGHUAABGCDLEBCMNCOPABGCDQSRN $.
  $}

  $( 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 ) $=
    ( cfn wcel wa cxp cpw cmap wss xpfi ancoms pwfi sylib mapsspw ssfi sylancl
    co ) ACDZBCDZEZBAFZGZCDZABHQZUBIUDCDTUACDZUCSRUEBAJKUALMABNUBUDOP $.

  ${
    $d A x $.
    $( A Cartesian product of finitely many finite sets is finite.
       (Contributed by Jeff Madsen, 19-Jun-2011.) $)
    ixpfi $p |- ( ( A e. Fin /\ A. x e. A B e. Fin ) -> X_ x e. A B e. Fin ) $=
      ( cfn wcel wral wa ciun cmap co cixp iunfi simpl mapfi syl2anc ixpssmap2g
      wss syl ssfid ) BDEZCDEABFZGZABCHZBIJZABCKZUBUCDEZTUDDEABCLZTUAMUCBNOUBUF
      UEUDQUGABCDPRS $.
  $}

  ${
    $d f g x A $.  $d f g B $.  $d f g x C $.  $d f g x ph $.
    ixpfi2.1 $e |- ( ph -> C e. Fin ) $.
    ixpfi2.2 $e |- ( ( ph /\ x e. A ) -> B e. Fin ) $.
    ixpfi2.3 $e |- ( ( ph /\ x e. ( A \ C ) ) -> B C_ { D } ) $.
    $( A Cartesian product of finite sets such that all but finitely many are
       singletons is finite.  (Note that ` B ( x ) ` and ` D ( x ) ` are both
       possibly dependent on ` x ` .)  (Contributed by Mario Carneiro,
       25-Jan-2015.) $)
    ixpfi2 $p |- ( ph -> X_ x e. A B e. Fin ) $=
      ( vf vg cfn wcel cv wral syl2anc wa wceq cfv wfn cin cixp cres cmpt inss2
      wf1 wss ssfi sylancl inss1 ralrimiva ssralv mpsyl ixpfi wi resixp mpan wb
      a1i cdif simprl vex elixp sylib simprd simprr difss ax-mp csn sseld elsni
      r19.26 syl6 anim12d eqtr3 ralimdva adantr biimtrrid mp2and biantrud fvres
      syl5 eqeq12d ralbiia inundif raleqi ralunb bitr3i 3bitr4g fnssres 3bitr4d
      cun simpld eqfnfv ex dom2lem f1fi ) ABCEUAZDUBZLMZBCDUBZWSJXAJNZWRUCZUDZU
      FXALMAWRLMZDLMZBWROZWTAELMWREUGXEGCEUEEWRUHUIWRCUGZAXFBCOXGCEUJZAXFBCHUKX
      FBWRCULUMBWRDUNPAJKXAWSXCKNZWRUCZXBXAMZXCWSMZUOAXHXLXMXIBCWRDXBUPUQUSAXLX
      JXAMZQZXCXKRZXBXJRZURAXOQZBNZXCSZXSXKSZRZBWROZXSXBSZXSXJSZRZBCOZXPXQXRYFB
      WROZYHYFBCEUTZOZQZYCYGXRYJYHXRYDDMZBCOZYEDMZBCOZYJXRXBCTZYMXRXLYPYMQAXLXN
      VABCDXBJVBVCVDZVEXRXJCTZYOXRXNYRYOQAXLXNVFBCDXJKVBVCVDZVEYMYOQYLYNQZBCOZX
      RYJYLYNBCVLUUAYTBYIOZXRYJYICUGUUAUUBUOCEVGYTBYICULVHAUUBYJUOXOAYTYFBYIAXS
      YIMQZYTYDFRZYEFRZQYFUUCYLUUDYNUUEUUCYLYDFVIZMUUDUUCDUUFYDIVJYDFVKVMUUCYNY
      EUUFMUUEUUCDUUFYEIVJYEFVKVMVNYDYEFVOVMVPVQWBVRVSVTYBYFBWRXSWRMXTYDYAYEXSW
      RXBWAXSWRXJWAWCWDYGYFBWRYIWLZOYKYFBUUGCCEWEWFYFBWRYIWGWHWIXRXCWRTZXKWRTZX
      PYCURXRYPXHUUHXRYPYMYQWMZXICWRXBWJUIXRYRXHUUIXRYRYOYSWMZXICWRXJWJUIBWRXCX
      KWNPXRYPYRXQYGURUUJUUKBCXBXJWNPWKWOWPXAWSXDWQP $.
  $}

  ${
    $d A x y $.  $d B y $.
    $( A finite mapping set is finite.  (Contributed by Mario Carneiro,
       31-Aug-2015.) $)
    mptfi $p |- ( A e. Fin -> ( x e. A |-> B ) e. Fin ) $=
      ( cfn wcel cmpt cdm wfn wfun funmpt funfn mpbi wss eqid dmmptss ssfi fnfi
      mpan2 sylancr ) BDEZABCFZUAGZHZUBDEZUADEUAIUCABCJUAKLTUBBMUDABCUAUANOBUBP
      RUBUAQS $.

    $( An image set from a finite set is finite.  (Contributed by Mario
       Carneiro, 13-Feb-2014.) $)
    abrexfi $p |- ( A e. Fin -> { y | E. x e. A y = B } e. Fin ) $=
      ( cfn wcel cv wceq wrex cab cmpt crn eqid rnmpt mptfi rnfi syl eqeltrrid
      ) CEFZBGDHACIBJACDKZLZEABCDTTMNSTEFUAEFACDOTPQR $.
  $}

  ${
    $d p N $.
    cnvimamptfin.n $e |- ( ph -> N e. Fin ) $.
    $( A preimage of a mapping with a finite domain under any class is finite.
       In contrast to ~ fisuppfi , the range of the mapping needs not to be
       known.  (Contributed by AV, 21-Dec-2018.) $)
    cnvimamptfin $p |- ( ph -> ( `' ( p e. N |-> X ) " Y ) e. Fin ) $=
      ( cfn wcel cmpt ccnv cima wss cnvimass eqid dmmptss sstri ssfi sylancl
      cdm ) ABGHEBCIZJDKZBLUAGHFUATSBTDMEBCTTNOPBUAQR $.
  $}

  $( 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 b A $.  $d a b B $.  $d a b F $.
    $( A set is the union of its finite subsets.  (Contributed by Stefan
       O'Rear, 2-Apr-2015.) $)
    unifpw $p |- U. ( ~P A i^i Fin ) = A $=
      ( va cpw cfn cin cuni cv inss1 unissi unipw sseqtri sseli csn wss snelpwi
      wcel snfi a1i elind elssuni syl snidg sseldd impbii eqriv ) BACZDEZFZABGZ
      UHPUIAPZUHAUIUHUFFAUGUFUFDHIAJKLUJUIMZUHUIUJUKUGPUKUHNUJUFDUKUIAOUKDPUJUI
      QRSUKUGTUAUIAUBUCUDUE $.

    $( A one-to-one mapping induces a one-to-one mapping on finite subsets.
       (Contributed by Mario Carneiro, 25-Jan-2015.) $)
    f1opwfi $p |- ( F : A -1-1-onto-> B ->
      ( b e. ( ~P A i^i Fin ) |-> ( F " b ) ) :
        ( ~P A i^i Fin ) -1-1-onto-> ( ~P B i^i Fin ) ) $=
      ( va wf1o cpw cfn cin cv cima wcel wa cres wfo wss syl adantl wceq adantr
      ccnv cmpt eqid simpr elin2d cdm f1ofun elinel1 elpwi f1odm sseqtrrd fores
      wfun syl2an2r fofi syl2anc imassrn f1ofo forn sseqtrid elpwd elind dff1o3
      crn simprbi f1ocnv dfdm4 eqtr3id wb anim12i foimacnv syl2an eqcomd imaeq2
      eqeq2d syl5ibrcom wf1 f1of1 f1imacnv impbid sylan2 f1o2d ) ABCFZDEAGZHIZB
      GZHIZCDJZKZCUAZEJZKZDWEWIUBZWMUCWCWHWELZMZWFHWIWOWIBHWOWHHLWHWICWHNZOZWIH
      LWOWDHWHWCWNUDUEWCCUMWNWHCUFZPWQABCUGWOWHAWRWNWHAPZWCWNWHWDLZWSWHWDHUHZWH
      AUIZQRWCWRASWNABCUJZTUKWHCULUNWHWIWPUOUPZWCWIBPWNWCCVDZWIBCWHUQWCABCOZXEB
      SABCURZABCUSQUTTVAXDVBWCWKWGLZMZWDHWLXIWLAHXIWKHLWKWLWJWKNZOZWLHLXIWFHWKW
      CXHUDUEWCWJUMZXHWKWJUFZPXKWCXFXLABCVCVEXIWKBXMXIWKWFLZWKBPZXHXNWCWKWFHUHZ
      RWKBUIZQXIBAWJFZXMBSWCXRXHABCVFTBAWJUJQUKWKWJULUNWKWLXJUOUPZWCWLAPXHWCWJV
      DZWLAWJWKUQWCXTWRACVGXCVHUTTVAXSVBWNXHMWCWTXNMZWHWLSZWKWISZVIWNWTXHXNXAXP
      VJWCYAMZYBYCYDYCYBWKCWLKZSYDYEWKWCXFXOYEWKSYAXGXNXOWTXQRABWKCVKVLVMYBWIYE
      WKWHWLCVNVOVPYDYBYCWHWJWIKZSYDYFWHWCABCVQWSYFWHSYAABCVRWTWSXNXBTABWHCVSVL
      VMYCWLYFWHWKWIWJVNVOVPVTWAWB $.
  $}

  ${
    $d A c f $.  $d A x $.  $d B c f $.  $d B x z $.  $d f x z $.
    $( A finite subset of a union is covered by finitely many elements.
       (Contributed by Stefan O'Rear, 2-Apr-2015.) $)
    fissuni $p |- ( ( A C_ U. B /\ A e. Fin ) ->
        E. c e. ( ~P B i^i Fin ) A C_ U. c ) $=
      ( vf vx vz cuni wss cfn wcel wa cv wral simpr dfss3 adantr syl2anc sylibr
      wrex ad2antrl cfv cpw cin wel wex eluni2 ralbii sylbb eleq2 ac6sfi fimass
      wf cima vex imaex elpw wfun ffun simplr imafi elind wfn ffn ssidd fnfvima
      syl3anc elssuni syl sseld ralimdva imp adantl wceq sseq2d rspcev exlimddv
      unieq ) ABGZHZAIJZKZABDLZULZELZWDWBUAZJZEAMZKZACLZGZHZCBUBZIUCZSZDWAVTEFU
      DZFBSZEAMZWHDUEVSVTNVSWQVTVSWDVRJZEAMWQEAVROWRWPEAFWDBUFUGUHPWOWFEFABDFLW
      EWDUIUJQWAWHKZWBAUMZWMJAWTGZHZWNWSWLIWTWCWTWLJZWAWGWCWTBHXCABWBAUKWTBWBAD
      UNUOUPRTWSWBUQZVTWTIJWCXDWAWGABWBURTVSVTWHUSWBAUTQVAWHXBWAWHWDXAJZEAMZXBW
      CWGXFWCWFXEEAWCWDAJZKZWEXAWDXHWEWTJZWEXAHXHWBAVBZAAHXGXIWCXJXGABWBVCPXHAV
      DWCXGNAAWBWDVEVFWEWTVGVHVIVJVKEAXAORVLWKXBCWTWMWIWTVMWJXAAWIWTVQVNVOQVP
      $.
  $}

  ${
    $d A c f $.  $d A x $.  $d B c f $.  $d B x y $.  $d F c f $.  $d F x y $.
    $d f x y $.
    $( Given a finite subset ` A ` of the range of a function, there exists a
       finite subset of the domain whose image is ` A ` .  (Contributed by Jeff
       Madsen, 2-Sep-2009.)  (Revised by Stefan O'Rear, 22-Feb-2015.) $)
    fipreima $p |- ( ( F Fn B /\ A C_ ran F /\ A e. Fin ) ->
        E. c e. ( ~P B i^i Fin ) ( F " c ) = A ) $=
      ( vf vx vy wfn crn wss cfn wcel cv cfv wceq wral wa cima syl2anc ad2antrl
      w3a wf cpw cin wrex wex simp3 dfss3 fvelrnb ralbidv bitrid biimpa 3adant3
      fveqeq2 ac6sfi fimass imaex elpw sylibr wfun ffun simpl3 imafi elind ccom
      vex cid cres fvco3 fvresi adantl eqeq12d ralbidva biimprd impr simpl1 ffn
      wi wb frn fnco syl3anc fnresi eqfnfv sylancl mpbird imaeq1d imaco resiima
      ssid ax-mp 3eqtr3g imaeq2 eqeq1d rspcev exlimddv ) CBHZACIZJZAKLZUAZABEMZ
      UBZFMZXBNZCNZXDOZFAPZQZCDMZRZAOZDBUCZKUDZUEZEXAWTGMZCNXDOZGBUEZFAPZXIEUFW
      QWSWTUGWQWSXSWTWQWSXSWSXDWRLZFAPWQXSFAWRUHWQXTXRFAGBXDCUIUJUKULUMXQXGFGAB
      EXPXEXDCUNUOSXAXIQZXBARZXNLCYBRZAOZXOYAXMKYBXCYBXMLZXAXHXCYBBJYEABXBAUPYB
      BXBAEVFUQURUSTYAXBUTZWTYBKLXCYFXAXHABXBVATWQWSWTXIVBXBAVCSVDYACXBVEZARVGA
      VHZARZYCAYAYGYHAYAYGYHOZXDYGNZXDYHNZOZFAPZXAXCXHYNXCXHYNVRXAXCYNXHXCYMXGF
      AXCXDALZQYKXFYLXDABXDCXBVIYOYLXDOXCAXDVJVKVLVMVNVKVOYAYGAHZYHAHYJYNVSYAWQ
      XBAHZXBIBJZYPWQWSWTXIVPXCYQXAXHABXBVQTXCYRXAXHABXBVTTBACXBWAWBAWCFAYGYHWD
      WEWFWGCXBAWHAAJYIAOAWJAAWIWKWLXLYDDYBXNXJYBOXKYCAXJYBCWMWNWOSWP $.
  $}

  ${
    $d a b c u w z A $.  $d a b c z B $.
    $( A finite subset of the union of a superset chain is a subset of some
       element of the chain.  A useful preliminary result for ~ alexsub and
       others.  (Contributed by Jeff Hankins, 25-Jan-2010.)  (Proof shortened
       by Mario Carneiro, 11-Feb-2015.)  (Revised by Mario Carneiro,
       18-May-2015.) $)
    finsschain $p |- ( ( ( A =/= (/) /\ [C.] Or A ) /\
        ( B e. Fin /\ B C_ U. A ) ) -> E. z e. A B C_ z ) $=
      ( va vb vc vw vu c0 wa wcel wss cv wrex wceq sseq1 rexbidv imbi12d imbi2d
      wi wne crpss wor cfn cuni csn cun wral 0ss rgenw r19.2z adantr a1d unssad
      mpan2 id imim1i sseq2 cbvrexvw simpr unssbd vex snss sylibr eluni2 reeanv
      sylib simpllr simprlr simprll simprrr simprrl snssd unss12 syl2anc rspcev
      sorpssun syl12anc expr rexlimdvva biimtrrid mpand biimtrid ex a2d a2i a1i
      syl5 findcard2 com12 imp32 ) BIUAZBUBUCZJZCUDKZCBUEZLZCAMZLZABNZWOWNWQWTT
      ZWNDMZWPLZXBWRLZABNZTZTWNIWPLZIWRLZABNZTZTWNEMZWPLZXKWRLZABNZTZTZWNXKFMZU
      FZUGZWPLZXSWRLZABNZTZTZWNXATDEFCXBIOZXFXJWNYEXCXGXEXIXBIWPPYEXDXHABXBIWRP
      QRSXBXKOZXFXOWNYFXCXLXEXNXBXKWPPYFXDXMABXBXKWRPQRSXBXSOZXFYCWNYGXCXTXEYBX
      BXSWPPYGXDYAABXBXSWRPQRSXBCOZXFXAWNYHXCWQXEWTXBCWPPYHXDWSABXBCWRPQRSWNXIX
      GWLXIWMWLXHABUHXIXHABWRUIUJXHABUKUOULUMXPYDTXKUDKWNXOYCXOXTXNTWNYCXTXLXNX
      TXKXRWPXTUPUNUQWNXTXNYBWNXTXNYBTXNXKGMZLZGBNZWNXTJZYBXMYJAGBWRYIXKURUSYLX
      QHMZKZHBNZYKYBYLXQWPKZYOYLXRWPLYPYLXKXRWPWNXTUTVAXQWPFVBVCVDHXQBVEVGYOYKJ
      YNYJJZGBNHBNYLYBYNYJHGBBVFYLYQYBHGBBYLYMBKZYIBKZJZYQYBYLYTYQJZJZYIYMUGZBK
      ZXSUUCLZYBUUBWMYSYRUUDWLWMXTUUAVHYLYRYSYQVIYLYRYSYQVJBYIYMVQVRUUBYJXRYMLU
      UEYLYTYNYJVKUUBXQYMYLYTYNYJVLVMXKYIXRYMVNVOYAUUEAUUCBWRUUCXSURVPVOVSVTWAW
      BWCWDWEWHWFWGWIWJWK $.
  $}

  ${
    $d c f w x y z A $.  $d c f w x y z B $.  $d c f w z ph $.
    $( If for every element of a finite indexing set ` A ` there exists a
       corresponding element of another set ` B ` , then there exists a finite
       subset of ` B ` consisting only of those elements which are indexed by
       ` A ` .  Proven without the Axiom of Choice, unlike ~ indexdom .
       (Contributed by Jeff Madsen, 2-Sep-2009.)  (Proof shortened by Mario
       Carneiro, 12-Sep-2015.) $)
    indexfi $p |- ( ( A e. Fin /\ B e. M /\ A. x e. A E. y e. B ph ) ->
                E. c e. Fin ( c C_ B /\ A. x e. A E. y e. c ph /\
                                                A. y e. c E. x e. A ph ) ) $=
      ( vf vz vw cfn wcel wrex wral cv wss w3a wa wsbc nfv wf cfv ralbii dfsbcq
      wex nfsbc1v sbceq1a cbvrexw ac6sfi sylan2b crn wfo wfn ffn ad2antrl dffn4
      simpll sylib fofi syl2anc frn wi fnfvelrn sylan rspesbca syl ralimdva imp
      ex adantl simpr simprr weq fveq2 sbceq1d bitrd cbvralw r19.21bi ralrimiva
      wb wceq rexbidv ralrn mpbird nfcv nfrexw sylibr sseq1 rexeq ralbidv raleq
      3anbi123d rspcev syl13anc exlimddv 3adant2 ) DKLZACEMZBDNZGOZEPZACWTMZBDN
      ZABDMZCWTNZQZGKMZEFLWQWSRZDEHOZUAZACBOZXIUBZSZBDNZRZXGHWSWQACIOZSZIEMZBDN
      XOHUEWRXRBDAXQCIEAITACXPUFZACXPUGZUHUCXQXMBIDEHACXPXLUDUIUJXHXORZXIUKZKLZ
      YBEPZACYBMZBDNZXDCYBNZXGYAWQDYBXIULZYCWQWSXOUQYAXIDUMZYHXJYIXHXNDEXIUNZUO
      ZDXIUPURDYBXIUSUTXJYDXHXNDEXIVAUOXOYFXHXJXNYFXJXMYEBDXJXKDLZRXLYBLZXMYEVB
      XJYIYLYMYJDXKXIVCVDYMXMYEACXLYBVEVIVFVGVHVJYAXQBDMZIYBNZYGYAYOACJOZXIUBZS
      ZBDMZJDNZYAYSJDYAYPDLZRUUAYRBYPSZYSYAUUAVKYAUUBJDYAXNUUBJDNXHXJXNVLXMUUBB
      JDXMJTYRBYPUFBJVMZXMYRUUBUUCACXLYQXKYPXIVNVOYRBYPUGVPVQURVRYRBYPDVEUTVSYA
      YIYOYTVTYKYNYSIJDXIXPYQWAXQYRBDACXPYQUDWBWCVFWDXDYNCIYBXDITXQCBDCDWEXSWFC
      IVMAXQBDXTWBVQWGXFYDYFYGQGYBKWTYBWAZXAYDXCYFXEYGWTYBEWHUUDXBYEBDACWTYBWIW
      JXDCWTYBWKWLWMWNWOWP $.
    $( $j usage 'indexfi' avoids 'ax-ac' 'ax-ac2'; $)
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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 $.
  $}

  ${
    fisuppfi.1 $e |- ( ph -> A e. Fin ) $.
    fisuppfi.2 $e |- ( ph -> F : A --> B ) $.
    $( A function on a finite set is finitely supported.  (Contributed by Mario
       Carneiro, 20-Jun-2015.) $)
    fisuppfi $p |- ( ph -> ( `' F " C ) e. Fin ) $=
      ( ccnv cima cnvimass fssdm ssfid ) ABEHDIZFABCMEEDJGKL $.
  $}

  ${
    fidmfisupp.1 $e |- ( ph -> F : D --> R ) $.
    fidmfisupp.2 $e |- ( ph -> D e. Fin ) $.
    fidmfisupp.3 $e |- ( ph -> Z e. V ) $.
    $( A function with a finite domain is finitely supported.  (Contributed by
       Glauco Siliprandi, 24-Dec-2020.) $)
    fidmfisupp $p |- ( ph -> F finSupp Z ) $=
      ( cfsupp wbr csupp co cfn wcel ccnv cvv csn cdif cima suppimacnv fisuppfi
      wceq fexd syl2anc eqeltrd wfun wb ffund funisfsupp syl3anc mpbird ) ADFJK
      ZDFLMZNOZAUNDPQFRSZTZNADQOZFEOZUNUQUCABCNDGHUDZIDQEFUAUEABCUPDHGUBUFADUGU
      RUSUMUOUHABCDGUIUTIDQEFUJUKUL $.
  $}

  ${
    $d ph x $.  $d A x $.  $d F x $.  $d Z x $.
    finnzfsuppd.1 $e |- ( ph -> F e. V ) $.
    finnzfsuppd.2 $e |- ( ph -> F Fn D ) $.
    finnzfsuppd.3 $e |- ( ph -> Z e. U ) $.
    finnzfsuppd.4 $e |- ( ph -> A e. Fin ) $.
    finnzfsuppd.5 $e |- ( ( ph /\ x e. D ) -> ( x e. A \/ ( F ` x ) = Z ) ) $.
    $( If a function is zero outside of a finite set, it has finite support.
       (Contributed by Rohan Ridenour, 13-May-2024.) $)
    finnzfsuppd $p |- ( ph -> F finSupp Z ) $=
      ( cfsupp wbr wcel wa cvv wb syl3anc csupp co cfn cfv wceq wne wfn fndmexd
      cv wo elsuppfn biimpa simpld syldan simprd neneqd olcnd ssrdv ssfid fnfun
      ex wfun syl funisfsupp mpbird ) AFHNOZFHUAUBZUCPZACVGLABVGCABUIZVGPZVICPZ
      AVJQZVKVIFUDZHUEZAVJVIDPZVKVNUJVLVOVMHUFZAVJVOVPQZAFDUGZDRPHEPZVJVQSJADFG
      IJUHKVIFREDHUKTULZUMMUNVLVMHVLVOVPVTUOUPUQVAURUSAFVBZFGPVSVFVHSAVRWAJDFUT
      VCIKFGEHVDTVE $.
  $}

  ${
    fdmfisuppfi.f $e |- ( ph -> F : D --> R ) $.
    fdmfisuppfi.d $e |- ( ph -> D e. Fin ) $.
    fdmfisuppfi.z $e |- ( ph -> Z e. V ) $.
    $( The support of a function with a finite domain is always finite.
       (Contributed by AV, 27-Apr-2019.) $)
    fdmfisuppfi $p |- ( ph -> ( F supp Z ) e. Fin ) $=
      ( csupp co ccnv cvv csn cdif cima cfn wcel wceq fexd suppimacnv fisuppfi
      syl2anc eqeltrd ) ADFJKZDLMFNOZPZQADMRFERUEUGSABCQDGHTIDMEFUAUCABCUFDHGUB
      UD $.

    $( A function with a finite domain is always finitely supported.
       (Contributed by AV, 25-May-2019.) $)
    fdmfifsupp $p |- ( ph -> F finSupp Z ) $=
      ( cfsupp wbr wfun csupp co cfn wcel ffund fdmfisuppfi cvv syl2anc wa ffnd
      wb wfn fnex isfsupp mpbir2and ) ADFJKZDLZDFMNOPZABCDGQABCDEFGHIRADSPZFEPU
      HUIUJUAUCADBUDBOPUKABCDGUBHBODUETIDSEFUFTUG $.
  $}

  ${
    $d A x $.  $d V x $.  $d ph x $.
    fsuppmptdm.f $e |- F = ( x e. A |-> Y ) $.
    fsuppmptdm.a $e |- ( ph -> A e. Fin ) $.
    fsuppmptdm.y $e |- ( ( ph /\ x e. A ) -> Y e. V ) $.
    fsuppmptdm.z $e |- ( ph -> Z e. W ) $.
    $( A mapping with a finite domain is finitely supported.  (Contributed by
       AV, 7-Jun-2019.) $)
    fsuppmptdm $p |- ( ph -> F finSupp Z ) $=
      ( fmptd fdmfifsupp ) ACEDFHABCGEDKIMJLN $.
  $}

  ${
    fndmfisuppfi.f $e |- ( ph -> F Fn D ) $.
    fndmfisuppfi.d $e |- ( ph -> D e. Fin ) $.
    fndmfisuppfi.z $e |- ( ph -> Z e. V ) $.
    $( The support of a function with a finite domain is always finite.
       (Contributed by AV, 25-May-2019.) $)
    fndmfisuppfi $p |- ( ph -> ( F supp Z ) e. Fin ) $=
      ( crn wfn wf dffn3 sylib fdmfisuppfi ) ABCIZCDEACBJBOCKFBCLMGHN $.

    $( A function with a finite domain is always finitely supported.
       (Contributed by AV, 25-May-2019.) $)
    fndmfifsupp $p |- ( ph -> F finSupp Z ) $=
      ( crn wfn wf dffn3 sylib fdmfifsupp ) ABCIZCDEACBJBOCKFBCLMGHN $.
  $}

  $( 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 ) ) ) $=
    ( cvv wcel wa csupp co cfsupp wbr wb wi cfn simpl funisfsupp syl3anc adantr
    wfun simprlr simprll simpr adantl impcom eleq1 bicomd sylan9bb bitr4d exp31
    wceq ex wn relfsupp brrelex2i pm5.21ni 2a1d pm2.61i ) EFGZBAGZBTZHZCDGZCTZH
    ZHZBEIJZCEIJZUKZBEKLZCEKLZMZNNUSVFVIVLUSVFHZVIHVJVGOGZVKVMVJVNMZVIVMVAUTUSV
    OUSUTVAVEUAUSUTVAVEUBUSVFPBAFEQRSVMVKVHOGZVIVNVFUSVKVPMZVEUSVQNVBVEUSVQVEUS
    HVDVCUSVQVEVDUSVCVDUCSVEVCUSVCVDPSVEUSUCCDFEQRULUDUEVIVNVPVGVHOUFUGUHUIUJUS
    UMVLVFVIVJUSVKBEKUNUOCEKUNUOUPUQUR $.

  $( If the support of a function is a subset of a finite set, the function is
     finitely supported.  (Contributed by AV, 15-Jul-2019.) $)
  suppssfifsupp $p |- ( ( ( G e. V /\ Fun G /\ Z e. W )
                          /\ ( F e. Fin /\ ( G supp Z ) C_ F ) )
                        -> G finSupp Z ) $=
    ( wcel wfun w3a cfn csupp co wss wa cfsupp wbr ssfi adantl 3ancoma biimpi
    wb adantr funisfsupp syl mpbird ) BCFZBGZEDFZHZAIFBEJKZALMZMZBENOZUIIFZUJUM
    UHAUIPQUKUFUEUGHZULUMTUHUNUJUHUNUEUFUGRSUABCDEUBUCUD $.

  $( If the support of a function is a subset of the support of a finitely
     supported function, the function is finitely supported.  (Contributed by
     AV, 2-Jul-2019.)  (Proof shortened by AV, 15-Jul-2019.) $)
  fsuppsssupp $p |- ( ( ( G e. V /\ Fun G )
                        /\ ( F finSupp Z /\ ( G supp Z ) C_ ( F supp Z ) ) )
                      -> G finSupp Z ) $=
    ( wcel wfun wa cfsupp wbr csupp co wss cvv simpll simplr relfsupp brrelex2i
    cfn ad2antrl id fsuppimpd anim1i adantl suppssfifsupp syl31anc ) BCEZBFZGZA
    DHIZBDJKADJKZLZGZGUFUGDMEZUJREZUKGZBDHIUFUGULNUFUGULOUIUMUHUKADHPQSULUOUHUI
    UNUKUIADUITUAUBUCUJBCMDUDUE $.

  ${
    fsuppsssuppgd.g $e |- ( ph -> G e. V ) $.
    fsuppsssuppgd.z $e |- ( ph -> Z e. W ) $.
    fsuppsssuppgd.1 $e |- ( ph -> Fun G ) $.
    fsuppsssuppgd.2 $e |- ( ph -> F finSupp O ) $.
    fsuppsssuppgd.3 $e |- ( ph -> ( G supp Z ) C_ ( F supp O ) ) $.
    $( If the support of a function is a subset of a finite support, it is
       finite.  Deduction associated with ~ fsuppsssupp .  (Contributed by SN,
       6-Mar-2025.) $)
    fsuppsssuppgd $p |- ( ph -> G finSupp Z ) $=
      ( wcel wfun csupp co cfn wss cfsupp wbr fsuppimpd suppssfifsupp syl32anc
      ) ACEMCNGFMBDOPZQMCGOPUDRCGSTHJIABDKUALUDCEFGUBUC $.
  $}

  ${
    fsuppss.1 $e |- ( ph -> F C_ G ) $.
    fsuppss.2 $e |- ( ph -> G finSupp Z ) $.
    $( A subset of a finitely supported function is a finitely supported
       function.  (Contributed by SN, 8-Mar-2025.) $)
    fsuppss $p |- ( ph -> F finSupp Z ) $=
      ( cvv cfsupp wrel wbr wcel relfsupp brrelex1 sylancr ssexd brrelex2 csupp
      wss wfun co fsuppfund funss sylc funsssuppss syl3anc fsuppsssuppgd ) ACBD
      GGDABCGAHIZCDHJZCGKZLFCDHMNZEOAUGUHDGKLFCDHPNABCRZCSZBSEACDFUAZBCUBUCFAUL
      UKUIBDQTCDQTRUMEUJBCGDUDUEUF $.
  $}

  ${
    $d ph v $.  $d ph x $.  $d v B $.  $d x D $.  $d v O $.  $d v R $.
    $d v Y $.  $d x Y $.  $d v Z $.  $d x Z $.  $d x V $.
    fsuppssov1.s $e |- ( ph -> ( x e. D |-> A ) finSupp Y ) $.
    fsuppssov1.o $e |- ( ( ph /\ v e. R ) -> ( Y O v ) = Z ) $.
    fsuppssov1.a $e |- ( ( ph /\ x e. D ) -> A e. V ) $.
    fsuppssov1.b $e |- ( ( ph /\ x e. D ) -> B e. R ) $.
    fsuppssov1.z $e |- ( ph -> Z e. W ) $.
    $( Formula building theorem for finite support: operator with left
       annihilator.  Finite support version of ~ suppssov1 .  (Contributed by
       SN, 26-Apr-2025.) $)
    fsuppssov1 $p |- ( ph -> ( x e. D |-> ( A O B ) ) finSupp Z ) $=
      ( cvv wcel cfsupp cmpt co wbr relfsupp brrelex1i syl fmpttd dmfex syl2anc
      wf mptexd wfun funmpt a1i csupp ssidd brrelex2i suppssov1 fsuppsssuppgd )
      ABFDUAZBFDEHUBZUAZKRJLABFVARAUTRSZFIUTUJFRSAUTKTUCZVCMUTKTUDUEUFABFDIOUGF
      IRUTUHUIUKQVBULABFVAUMUNMABCDEFGUTKUOUBZHIRKLAVEUPNOPAVDKRSMUTKTUDUQUFURU
      S $.
  $}

  $( 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 cfsupp wbr wfun csupp co cfn wcel wfn fnconstg fnfun cvv c0 0fi
      3syl fczsupp0 eqeltri a1i wa snex xpexg sylancl isfsupp syl2anc mpbir2and
      wb ) ABEHZIZEJKZUPLZUPEMNZOPZAEDPZUPBQURGBEDRBUPSUCUTAUSUAOBEUDUBUEUFAUPT
      PZVAUQURUTUGUNABCPUOTPVBFEUHBUOCTUIUJGUPTDEUKULUM $.
  $}

  ${
    fsuppun.f $e |- ( ph -> F finSupp Z ) $.
    fsuppun.g $e |- ( ph -> G finSupp Z ) $.
    $( The union of two finitely supported functions is finitely supported (but
       not necessarily a function!).  (Contributed by AV, 3-Jun-2019.) $)
    fsuppun $p |- ( ph -> ( ( F u. G ) supp Z ) e. Fin ) $=
      ( cun cvv wcel wa csupp cfn ccnv cima wceq sylbir suppimacnv sylan adantr
      co wi csn cdif cnvun imaeq1i imaundir eqtri unexb eqcomd fsuppimpd adantl
      simpl eqeltrd simpr unfi syl2anc eqeltrid wb eleq1d mpbird ex wn supp0prc
      c0 0fi eqeltrdi a1d pm2.61i ) BCGZHIZDHIZJZAVIDKTZLIZUAVLAVNVLAJZVNVIMZHD
      UBUCZNZLIZVOVRBMZVQNZCMZVQNZGZLVRVTWBGZVQNWDVPWEVQBCUDUEVTWBVQUFUGVOWALIW
      CLIWDLIVOWABDKTZLVLWAWFOAVLWFWAVJBHIZVKWFWAOVJWGCHIZJZWGBCUHZWGWHULPBHHDQ
      RUISAWFLIVLABDEUJUKUMVOWCCDKTZLVLWCWKOZAVJWHVKWLVJWIWHWJWGWHUNPWHVKJWKWCC
      HHDQUIRSAWKLIVLACDFUJUKUMWAWCUOUPUQVLVNVSURAVLVMVRLVIHHDQUSSUTVAVLVBZVNAW
      MVMVDLVIDVCVEVFVGVH $.

    $( The union of the support of two finitely supported functions is finite.
       (Contributed by AV, 1-Jul-2019.) $)
    fsuppunfi $p |- ( ph -> ( ( F supp Z ) u. ( G supp Z ) ) e. Fin ) $=
      ( cfsupp wbr csupp co cun cfn wcel wfun wi fsuppimp wa unfi expcom adantl
      3syl com12 simpl2im mpcom ) BDGHZABDIJZCDIJZKLMZEUEBNUFLMZAUHOBDPAUIUHACD
      GHCNZUGLMZQUIUHOZFCDPUKULUJUIUKUHUFUGRSTUAUBUCUD $.
  $}

  ${
    fsuppunbi.u $e |- ( ph -> Fun ( F u. G ) ) $.
    $( If the union of two classes/functions is a function, this union is
       finitely supported iff the two functions are finitely supported.
       (Contributed by AV, 18-Jun-2019.) $)
    fsuppunbi $p |- ( ph -> ( ( F u. G ) finSupp Z
                              <-> ( F finSupp Z /\ G finSupp Z ) ) ) $=
      ( cfsupp wbr wa cvv wcel relfsupp csupp co cfn simpr adantr wb funisfsupp
      wfun adantl cun wi brrelex12i unexb simprlr suppun ssfid fununfun simprll
      simpld syl3anc mpbird uncom oveq1i eleq1i biimpi simprd ex fsuppimp syl11
      jca a1d sylanbr mpcom com12 simpl brrelex1i unexg syl2an brrelex2i impbid
      fsuppun ) ABCUAZDFGZBDFGZCDFGZHZVNAVQVMIJZDIJZHVNAVQUBZVMDFKUCVRBIJZCIJZH
      ZVSVNVTUBBCUDVMSZVMDLMZNJZHZWCVSHZVTVNWGWHVTWGWHHZVQAWIVOVPWIVOBDLMZNJZWI
      WEWJWGWFWHWDWFOPWIBCIDWGWAWBVSUEZUFUGWIBSZWAVSVOWKQWGWMWHWDWMWFWDWMCSZBCU
      HZUJPPWGWAWBVSUIZWHVSWGWCVSOTZBIIDRUKULWIVPCDLMZNJZWICBUAZDLMZWRWGXANJZWH
      WFXBWDWFXBWEXANVMWTDLBCUMUNUOUPTPWICBIDWPUFUGWIWNWBVSVPWSQWGWNWHWDWNWFWDW
      MWNWOUQPPWLWQCIIDRUKULVAVBURVMDUSUTVCVDVEAVQVNAVQHZVNWFVQWFAVQBCDVOVPVFVO
      VPOVLTXCWDVRVSVNWFQAWDVQEPVQVRAVOWAWBVRVPBDFKVGCDFKVGBCIIVHVITVQVSAVOVSVP
      BDFKVJPTVMIIDRUKULURVK $.
  $}

  $( 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
    $.

  $( A singleton containing an ordered pair is a finitely supported function.
     (Contributed by AV, 19-Jul-2019.) $)
  snopfsupp $p |- ( ( X e. V /\ Y e. W /\ Z e. U )
                    -> { <. X , Y >. } finSupp Z ) $=
    ( wcel w3a cop csn cfsupp wbr csupp co cfn wss wa snfi snopsuppss cvv simp3
    pm3.2i ssfi mp1i wfun wb funsng 3adant3 snex a1i funisfsupp syl3anc mpbird
    ) DBGZECGZFAGZHZDEIZJZFKLZUSFMNZOGZDJZOGZVAVCPZQVBUQVDVEDRDEFSUBVCVAUCUDUQU
    SUEZUSTGZUPUTVBUFUNUOVFUPDEBCUGUHVGUQURUIUJUNUOUPUAUSTAFUKULUM $.

  $( Finite support for a function extended by a singleton.  (Contributed by
     Stefan O'Rear, 27-Feb-2015.)  (Revised by AV, 19-Jul-2019.) $)
  funsnfsupp $p |- ( ( ( X e. V /\ Y e. W ) /\ ( Fun F /\ X e/ dom F ) )
                 -> ( ( F u. { <. X , Y >. } ) finSupp Z <-> F finSupp Z ) ) $=
    ( cvv wcel wa wfun cdm csn cfsupp wbr simpl syl cin c0 wceq adantl wnel cop
    cun wb wi w3a anim2i ancomd df-3an sylibr snopfsupp funsng anim12ci dmsnopg
    ineq2d df-nel disjsn sylbb2 sylan9eq jca funun fsuppunbi mpbiran2d relfsupp
    wn ex brrelex2i pm5.21ni a1d pm2.61i ) FGHZDBHZECHZIZAJZDAKZUAZIZIZADEUBLZU
    CZFMNZAFMNZUDZUEVKVSWDVKVSIZWBWCVTFMNZWEVLVMVKUFZWFWEVNVKIWGWEVKVNVSVNVKVNV
    ROUGUHVLVMVKUIUJGBCDEFUKPWEAVTFWEVOVTJZIZVPVTKZQZRSZIZWAJVSWMVKVSWIWLVNWHVR
    VODEBCULVOVQOUMVNVRWKVPDLZQZRVNWJWNVPVMWJWNSVLDECUNTUOVQWORSZVOVQDVPHVEWPDV
    PUPVPDUQURTUSUTTAVTVAPVBVCVFVKVEWDVSWBVKWCWAFMVDVGAFMVDVGVHVIVJ $.

  ${
    fsuppres.s $e |- ( ph -> F finSupp Z ) $.
    fsuppres.z $e |- ( ph -> Z e. V ) $.
    $( The restriction of a finitely supported function is finitely supported.
       (Contributed by AV, 14-Jul-2019.) $)
    fsuppres $p |- ( ph -> ( F |` X ) finSupp Z ) $=
      ( cfsupp wbr csupp co cfn wcel wfun wa wi cvv syl adantr 3syl cres expcom
      fsuppimp wss relprcnfsupp con4i jca ressuppss ssfi com23 imp mpcom funres
      wb resexg funisfsupp syl3anc mpbird ) ABDUAZEHIZUSEJKZLMZBEHIZAVBFVCBNZBE
      JKZLMZOZAVBPZBEUCZVDVFVHVDAVFVBAVDVFVBPZAVDOBQMZECMZOZVAVEUDZVJAVMVDAVKVL
      AVCVKFVKVCBEUEUFZRGUGSDBQCEUHVFVNVBVEVAUIUBTUBUJUKRULAUSNZUSQMZVLUTVBUNAV
      CVGVPFVIVDVPVFDBUMSTAVCVKVQFVOBDQUOTGUSQCEUPUQUR $.
  $}

  ${
    $d A x $.  $d C x $.
    fmptssfisupp.1 $e |- ( ph -> ( x e. A |-> B ) finSupp Z ) $.
    fmptssfisupp.2 $e |- ( ph -> C C_ A ) $.
    fmptssfisupp.3 $e |- ( ph -> Z e. V ) $.
    $( The restriction of a mapping function has finite support if that
       function has finite support.  (Contributed by Thierry Arnoux,
       21-Jan-2024.) $)
    fmptssfisupp $p |- ( ph -> ( x e. C |-> B ) finSupp Z ) $=
      ( cmpt cres cfsupp resmptd fsuppres eqbrtrrd ) ABCDKZELBEDKGMABCEDINAQFEG
      HJOP $.
  $}

  ${
    ressuppfi.b $e |- ( ph -> ( dom F \ B ) e. Fin ) $.
    ressuppfi.f $e |- ( ph -> F e. W ) $.
    ressuppfi.g $e |- ( ph -> G = ( F |` B ) ) $.
    ressuppfi.s $e |- ( ph -> ( G supp Z ) e. Fin ) $.
    ressuppfi.z $e |- ( ph -> Z e. V ) $.
    $( If the support of the restriction of a function by a set which,
       subtracted from the domain of the function so that its difference is
       finite, the support of the function itself is finite.  (Contributed by
       AV, 22-Apr-2019.) $)
    ressuppfi $p |- ( ph -> ( F supp Z ) e. Fin ) $=
      ( cres csupp co cdm cdif cfn wcel syl2anc cun eqcomd eqeltrd ressuppssdif
      oveq1d unfi wss ssfid ) ACBMZGNOZCPBQZUAZCGNOZAUJRSUKRSULRSAUJDGNORAUIDGN
      ADUIJUBUEKUCHUJUKUFTACFSGESUMULUGILBCFEGUDTUH $.
  $}

  ${
    resfsupp.b $e |- ( ph -> ( dom F \ B ) e. Fin ) $.
    resfsupp.e $e |- ( ph -> F e. W ) $.
    resfsupp.f $e |- ( ph -> Fun F ) $.
    resfsupp.g $e |- ( ph -> G = ( F |` B ) ) $.
    resfsupp.s $e |- ( ph -> G finSupp Z ) $.
    resfsupp.z $e |- ( ph -> Z e. V ) $.
    $( If the restriction of a function by a set which, subtracted from the
       domain of the function so that its difference is finitely supported, the
       function itself is finitely supported.  (Contributed by AV,
       27-May-2019.) $)
    resfsupp $p |- ( ph -> F finSupp Z ) $=
      ( cfsupp wbr csupp co cfn wcel fsuppimpd ressuppfi wfun funisfsupp mpbird
      wb syl3anc ) ACGNOZCGPQRSZABCDEFGHIKADGLTMUAACUBCFSGESUGUHUEJIMCFEGUCUFUD
      $.
  $}

  ${
    resfifsupp.f $e |- ( ph -> Fun F ) $.
    resfifsupp.x $e |- ( ph -> X e. Fin ) $.
    resfifsupp.z $e |- ( ph -> Z e. V ) $.
    $( The restriction of a function to a finite set is finitely supported.
       (Contributed by AV, 12-Dec-2019.) $)
    resfifsupp $p |- ( ph -> ( F |` X ) finSupp Z ) $=
      ( cdm cin cres cfsupp wrel wceq wfun funrel syl wfn cfn wcel resindm infi
      funfnd fnresin2 fndmfifsupp eqbrtrrd ) ABDBIZJZKZBDKZELABMZUIUJNABOUKFBPQ
      BDUAQAUHUICEABUGRUIUHRABFUCUGDBUDQADSTUHSTGDUGUBQHUEUF $.
  $}

  $( 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 $.

  ${
    $d A k $.  $d F k $.  $d Z k $.  $d ph k $.
    fsuppmptif.f $e |- ( ph -> F : A --> B ) $.
    fsuppmptif.a $e |- ( ph -> A e. V ) $.
    fsuppmptif.z $e |- ( ph -> Z e. W ) $.
    fsuppmptif.s $e |- ( ph -> F finSupp Z ) $.
    $( A function mapping an argument to either a value of a finitely supported
       function or zero is finitely supported.  (Contributed by AV,
       6-Jun-2019.) $)
    fsuppmptif $p |- ( ph
                 -> ( k e. A |-> if ( k e. D , ( F ` k ) , Z ) ) finSupp Z ) $=
      ( cv wcel cif csupp co cvv wa cfv cmpt cfsupp wbr wfun cfn adantr sylancr
      fvex ifexg fmpttd ffund fsuppimpd cdif suppssr ifeq1d ifid eqtrdi suppss2
      ssidd ssfid wb mptexd isfsupp syl2anc mpbir2and ) AEBENZDOZVGFUAZIPZUBZIU
      CUDZVKUEZVKIQRZUFOZABSVKAEBVJSAVGBOZTVISOIHOZVJSOVGFUIAVQVPLUGVHVIISHUJUH
      UKULAFIQRZVNAFIMUMABVJEGVRIAVGBVRUNOTZVJVHIIPIVSVHVIIIABCHFGVRVGIJAVRUTKL
      UOUPVHIUQURKUSVAAVKSOVQVLVMVOTVBAEBVJGKVCLVKSHIVDVEVF $.
  $}

  ${
    $d I x $.  $d X x $.  $d .0. x $.  $d ph x $.
    sniffsupp.i $e |- ( ph -> I e. V ) $.
    sniffsupp.0 $e |- ( ph -> .0. e. W ) $.
    sniffsupp.f $e |- F = ( x e. I |-> if ( x = X , A , .0. ) ) $.
    $( A function mapping all but one arguments to zero is finitely supported.
       (Contributed by AV, 8-Jul-2019.) $)
    sniffsupp $p |- ( ph -> F finSupp .0. ) $=
      ( cv wceq cif cmpt cfsupp cfn wcel cvv wbr csupp co csn wss snfi cdif wne
      wa eldifsni adantl neneqd iffalsed suppss2 ssfi sylancr funmpt funisfsupp
      wfun wb mptexd mp3an2i mpbird eqbrtrid ) ADBEBMZHNZCIOZPZIQLAVHIQUAZVHIUB
      UCZRSZAHUDZRSVJVLUEVKHUFAEVGBFVLIAVEEVLUGSZUIZVFCIVNVEHVMVEHUHAVEEHUJUKUL
      UMJUNVLVJUOUPVHUSAVHTSIGSVIVKUTBEVGUQABEVGFJVAKVHTGIURVBVCVD $.
  $}

  ${
    fsuppcolem.f $e |- ( ph -> ( `' F " ( _V \ { Z } ) ) e. Fin ) $.
    fsuppcolem.g $e |- ( ph -> G : X -1-1-> Y ) $.
    $( Lemma for ~ fsuppco .  Formula building theorem for finite supports:
       rearranging the index set.  (Contributed by Stefan O'Rear,
       21-Mar-2015.) $)
    fsuppcolem $p |- ( ph -> ( `' ( F o. G ) " ( _V \ { Z } ) ) e. Fin ) $=
      ( ccom ccnv cvv csn cdif cima cfn cnvco imaeq1i imaco eqtri wcel wfun wf1
      wf df-f1 simprbi syl imafi syl2anc eqeltrid ) ABCIJZKFLMZNZCJZBJZUKNZNZOU
      LUMUNIZUKNUPUJUQUKBCPQUMUNUKRSAUMUAZUOOTUPOTADECUBZURHUSDECUCURDECUDUEUFG
      UMUOUGUHUI $.
  $}

  ${
    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 ) $.
    $( The composition of a 1-1 function with a finitely supported function is
       finitely supported.  (Contributed by AV, 28-May-2019.) $)
    fsuppco $p |- ( ph -> ( F o. G ) finSupp Z ) $=
      ( cfsupp cfn wcel ccnv cvv wfun syl syl2anc ccom wbr csupp cdif cima wceq
      co csn wf df-f1 simprbi cofunex2g suppimacnv fsuppimpd fsuppcolem eqeltrd
      wf1 eqeltrrd wb fsuppimp simpld f1fun funco funisfsupp syl3anc mpbird ) A
      BCUAZHMUBZVGHUCUGZNOZAVIVGPQHUHUDZUEZNAVGQOZHEOZVIVLUFABDOZCPRZVMLAFGCUQZ
      VPJVQFGCUIVPFGCUJUKSBCDULTZKVGQEHUMTABCFGHABHUCUGZBPVKUEZNAVOVNVSVTUFLKBD
      EHUMTABHIUNURJUOUPAVGRZVMVNVHVJUSABRZCRZWAABHMUBZWBIWDWBVSNOBHUTVASAVQWCJ
      FGCVBSBCVCTVRKVGQEHVDVEVF $.
  $}

  ${
    $d x Z $.  $d x F $.  $d x G $.  $d x ph $.
    fsuppco2.z $e |- ( ph -> Z e. W ) $.
    fsuppco2.f $e |- ( ph -> F : A --> B ) $.
    fsuppco2.g $e |- ( ph -> G : B --> B ) $.
    fsuppco2.a $e |- ( ph -> A e. U ) $.
    fsuppco2.b $e |- ( ph -> B e. V ) $.
    fsuppco2.n $e |- ( ph -> F finSupp Z ) $.
    fsuppco2.i $e |- ( ph -> ( G ` Z ) = Z ) $.
    $( The composition of a function which maps the zero to zero with a
       finitely supported function is finitely supported.  This is not only a
       special case of ~ fsuppcor because it does not require that the "zero"
       is an element of the range of the finitely supported function.
       (Contributed by AV, 6-Jun-2019.) $)
    fsuppco2 $p |- ( ph -> ( G o. F ) finSupp Z ) $=
      ( wcel syl2anc cfv cvv vx ccom cfsupp wfun csupp co ffund funco fsuppimpd
      wbr cfn wf fco cv cdif wa eldifi fvco3 syl2an ssidd suppssr fveq2d adantr
      wceq 3eqtrd suppss ssfid wb fexd coexg isfsupp mpbir2and ) AFEUBZIUCUJZVM
      UDZVMIUEUFZUKQZAFUDEUDVOACCFLUGABCEKUGFEUHRAEIUEUFZVPAEIOUIABCUAVMVRIACCF
      ULBCEULZBCVMULLKBCCFEUMRAUAUNZBVRUOQZUPZVTVMSZVTESZFSZIFSZIAVSVTBQWCWEVDW
      AKVTBVRUQBCVTFEURUSWBWDIFABCHEDVRVTIKAVRUTMJVAVBAWFIVDWAPVCVEVFVGAVMTQZIH
      QVNVOVQUPVHAFTQETQWGACCGFLNVIABCDEKMVIFETTVJRJVMTHIVKRVL $.
  $}

  ${
    $d x Z $.  $d x F $.  $d x G $.  $d x ph $.  $d x .0. $.
    fsuppcor.0 $e |- ( ph -> .0. e. W ) $.
    fsuppcor.z $e |- ( ph -> Z e. B ) $.
    fsuppcor.f $e |- ( ph -> F : A --> C ) $.
    fsuppcor.g $e |- ( ph -> G : B --> D ) $.
    fsuppcor.s $e |- ( ph -> C C_ B ) $.
    fsuppcor.a $e |- ( ph -> A e. U ) $.
    fsuppcor.b $e |- ( ph -> B e. V ) $.
    fsuppcor.n $e |- ( ph -> F finSupp Z ) $.
    fsuppcor.i $e |- ( ph -> ( G ` Z ) = .0. ) $.
    $( The composition of a function which maps the zero of the range of a
       finitely supported function to the zero of its range with this finitely
       supported function is finitely supported.  (Contributed by AV,
       6-Jun-2019.) $)
    fsuppcor $p |- ( ph -> ( G o. F ) finSupp .0. ) $=
      ( vx ccom cfsupp wbr wfun csupp co cfn wcel ffund funco syl2anc fsuppimpd
      cres wf fssresd fco2 cv cdif wa cfv wceq eldifi fvco3 syl2an ssidd fveq2d
      suppssr adantr 3eqtrd suppss ssfid cvv wb fexd coexg isfsupp mpbir2and )
      AHGUCZKUDUEZVTUFZVTKUGUHZUIUJZAHUFGUFWBACEHPUKABDGOUKHGULUMAGLUGUHZWCAGLT
      UNABEUBVTWEKADEHDUOUPBDGUPZBEVTUPACEDHPQUQOBDEHGURUMAUBUSZBWEUTUJZVAZWGVT
      VBZWGGVBZHVBZLHVBZKAWFWGBUJWJWLVCWHOWGBWEVDBDWGHGVEVFWIWKLHABDCGFWEWGLOAW
      EVGRNVIVHAWMKVCWHUAVJVKVLVMAVTVNUJZKJUJWAWBWDVAVOAHVNUJGVNUJWNACEIHPSVPAB
      DFGORVPHGVNVNVQUMMVTVNJKVRUMVS $.
  $}

  ${
    $d x A $.  $d x B $.  $d x C $.  $d f g x F $.  $d f g x G $.  $d f g ph $.
    $d x D $.  $d f g S $.  $d f g T $.  $d x W $.  $d x Z $.
    mapfien.s $e |- S = { x e. ( B ^m A ) | x finSupp Z } $.
    mapfien.t $e |- T = { x e. ( D ^m C ) | x finSupp W } $.
    mapfien.w $e |- W = ( G ` Z ) $.
    mapfien.f $e |- ( ph -> F : C -1-1-onto-> A ) $.
    mapfien.g $e |- ( ph -> G : B -1-1-onto-> D ) $.
    mapfien.a $e |- ( ph -> A e. U ) $.
    mapfien.b $e |- ( ph -> B e. V ) $.
    mapfien.c $e |- ( ph -> C e. X ) $.
    mapfien.d $e |- ( ph -> D e. Y ) $.
    mapfien.z $e |- ( ph -> Z e. B ) $.
    $( Lemma 1 for ~ mapfien .  (Contributed by AV, 3-Jul-2019.)  (Revised by
       AV, 28-Jul-2024.) $)
    mapfienlem1 $p |- ( ( ph /\ f e. S )
                          -> ( G o. ( f o. F ) ) finSupp W ) $=
      ( cv wcel wa ccom cvv fvexi a1i adantr cfsupp wbr cmap crab elrabi elmapi
      wf syl eleq2s wf1o f1of fco syl2anr ssidd breq1 elrab2 simprbi adantl wf1
      co f1of1 simpr fsuppco cfv wceq eqcomi fsuppcor ) AJUHZGUIZUJZEDDFOWCKUKZ
      LMULNQNULUIWENQLTUMUNAQDUIWDUGUOZWDCDWCVBZECKVBZEDWFVBAWHWCBUHZQUPUQZBDCU
      RVOZUSZGWCWMUIWCWLUIZWHWKBWCWLUTWCDCVAVCRVDAECKVEZWIUAECKVFVCECDWCKVGVHAD
      FLVBZWDADFLVEWPUBDFLVFVCUOWEDVIAEOUIWDUEUOADMUIWDUDUOWEWCKGDECQWDWCQUPUQZ
      AWDWNWQWKWQBWCWLGWJWCQUPVJRVKVLVMAECKVNZWDAWOWRUAECKVPVCUOWGAWDVQVRQLVSZN
      VTWENWSTWAUNWB $.

    $( Lemma 2 for ~ mapfien .  (Contributed by AV, 3-Jul-2019.)  (Revised by
       AV, 28-Jul-2024.) $)
    mapfienlem2 $p |- ( ( ph /\ g e. T )
                          -> ( ( `' G o. g ) o. `' F ) finSupp Z ) $=
      ( cv wcel wa ccnv ccom cvv adantr cfv wf1o wf f1of syl ffvelcdmd eqeltrid
      cfsupp wbr cmap crab elrabi elmapi eleq2s adantl f1ocnv ssidd breq1 elrab
      co 3syl simprbi wceq jca eqcomi jctir f1ocnvfv imp fsuppcor wf1 f1of1 fex
      cnvexg coexg sylan fsuppco ) AJUHZHUIZUJZLUKZWKULZKUKZUMDCEQWMEFFDOWKWNPD
      QNAQDUIZWLUGUNZANFUIWLANQLUOZFTADFQLADFLUPZDFLUQZUBDFLURUSZUGUTVAUNWLEFWK
      UQZAXCWKBUHZNVBVCZBFEVDVNZVEZHWKXGUIZWKXFUIZXCXEBWKXFVFWKFEVGUSSVHVIAFDWN
      UQZWLAWTFDWNUPXJUBDFLVJFDWNURVOUNWMFVKAEOUIWLUEUNAFPUIWLUFUNWLWKNVBVCZAXK
      WKXGHXHXIXKXEXKBWKXFXDWKNVBVLVMVPSVHVIWMWTWQUJZWSNVQZUJZNWNUOQVQZAXNWLAXL
      XMAWTWQUBUGVRNWSTVSVTUNXLXMXODFQNLWAWBUSWCACEWPWDZWLAECKUPCEWPUPXPUAECKVJ
      CEWPWEVOUNWRAWNUMUIZWLWOUMUIAXADMUIZUJLUMUIXQAXAXRXBUDVRDFMLWFLUMWGVOWNWK
      UMHWHWIWJ $.

    $( Lemma 3 for ~ mapfien .  (Contributed by AV, 3-Jul-2019.)  (Revised by
       AV, 28-Jul-2024.) $)
    mapfienlem3 $p |- ( ( ph /\ g e. T )
                           -> ( ( `' G o. g ) o. `' F ) e. S ) $=
      ( cv wcel wa ccnv ccom cmap co cfsupp wbr wf wf1o f1ocnv f1of 3syl adantr
      crab elrabi eleq2s adantl elmapi syl fcod elmapd mpbird mapfienlem2 breq1
      wb elrab2 sylanbrc ) AJUHZHUIZUJZLUKZVQULZKUKZULZDCUMUNZUIZWCQUOUPZWCGUIV
      SWECDWCUQZVSCEDWAWBVSEFDVTVQAFDVTUQZVRADFLURFDVTURWHUBDFLUSFDVTUTVAVBVSVQ
      FEUMUNZUIZEFVQUQVRWJAWJVQBUHZNUOUPZBWIVCHWLBVQWIVDSVEVFVQFEVGVHVIACEWBUQZ
      VRAECKURCEWBURWMUAECKUSCEWBUTVAVBVIAWEWGVNVRADCWCMIUDUCVJVBVKABCDEFGHIJKL
      MNOPQRSTUAUBUCUDUEUFUGVLWKQUOUPWFBWCWDGWKWCQUOVMRVOVP $.

    $( A bijection of the base sets induces a bijection on the set of finitely
       supported functions.  (Contributed by Mario Carneiro, 30-May-2015.)
       (Revised by AV, 3-Jul-2019.)  (Revised by AV, 28-Jul-2024.) $)
    mapfien $p |- ( ph ->
        ( f e. S |-> ( G o. ( f o. F ) ) ) : S -1-1-onto-> T ) $=
      ( vg cv ccom ccnv cmpt eqid wcel wa cmap co cfsupp wbr wf wf1o syl adantr
      f1of breq1 elrab2 simplbi adantl elmapi fcod wb elmapd mpbird mapfienlem1
      sylanbrc mapfienlem3 wceq coass cres f1ococnv1 coeq2d f1ocnv simpr simpld
      3syl sylib adantrl fcoi1 eqtrd eqtrid eqeq2d coeq1d adantrr fcoi2 eqtr3id
      cid eqcom bitrdi bitr4d wfo wfn f1ofo ffn cocan2 syl3anc wf1 f1of1 cocan1
      ffnd 3bitr3d f1o2d ) AJUHGHLJUIZKUJZUJZLUKZUHUIZUJZKUKZUJZJGXNULZXTUMAXLG
      UNZUOZXNFEUPUQZUNZXNNURUSZXNHUNYBYDEFXNUTZYBEDFLXMADFLUTZYAADFLVAZYGUBDFL
      VDVBVCYBECDXLKYBXLDCUPUQZUNZCDXLUTZYAYJAYAYJXLQURUSZBUIZQURUSYLBXLYIGYMXL
      QURVERVFVGVHZXLDCVIZVBAECKUTZYAAECKVAZYPUAECKVDVBVCVJZVJZAYDYFVKYAAFEXNPO
      UFUEVLVCVMABCDEFGHIJKLMNOPQRSTUAUBUCUDUEUFUGVNYMNURUSZYEBXNYCHYMXNNURVESV
      FVOABCDEFGHIUHKLMNOPQRSTUAUBUCUDUEUFUGVPAYAXPHUNZUOZUOZXMXSKUJZVQZXQXOXNU
      JZVQZXLXSVQZXPXNVQZUUCUUEXMXQVQZUUGUUCUUDXQXMUUCUUDXQXRKUJZUJZXQXQXRKVRUU
      CUULXQWPEVSZUJZXQUUCUUKUUMXQUUCYQUUKUUMVQAYQUUBUAVCZECKVTVBWAUUCEDXQUTZUU
      NXQVQAUUAUUPYAAUUAUOZEFDXOXPAFDXOUTZUUAAYHFDXOVAZUURUBDFLWBZFDXOVDWEVCUUQ
      XPYCUNZEFXPUTZUUQUVAXPNURUSZUUQUUAUVAUVCUOAUUAWCYTUVCBXPYCHYMXPNURVESVFWF
      WDXPFEVIVBZVJZWGEDXQWHVBWIWJWKUUCUUGXQXMVQUUJUUCUUFXMXQUUCUUFXOLUJZXMUJZX
      MXOLXMVRUUCUVGWPDVSZXMUJZXMUUCUVFUVHXMUUCYHUVFUVHVQAYHUUBUBVCDFLVTVBWLUUC
      EDXMUTZUVIXMVQAYAUVJUUAYRWMEDXMWNVBWIWOWKXQXMWQWRWSUUCECKWTZXLCXAZXSCXAZU
      UEUUHVKUUCYQUVKUUOECKXBVBAYAUVLUUAYBYJYKUVLYNYOCDXLXCWEWMAUUAUVMYAUUQCDXS
      UUQCEDXQXRUVEACEXRUTZUUAAYQCEXRVAUVNUAECKWBCEXRVDWEVCVJXIWGECKXLXSXDXEUUC
      FDXOXFZUVBYFUUGUUIVKUUCUUSUVOAUUSUUBAYHUUSUBUUTVBVCFDXOXGVBAUUAUVBYAUVDWG
      AYAYFUUAYSWMEFDXOXPXNXHXEXJXK $.
  $}

  ${
    $d w x y z $.  $d w x z A $.  $d w x z y B $.  $d w x z C $.
    $d w x z y D $.  $d w y z S $.  $d w x y z .0. $.  $d w y z ph $.
    $d x z y W $.  $d y z T $.
    mapfien2.s $e |- S = { x e. ( B ^m A ) | x finSupp .0. } $.
    mapfien2.t $e |- T = { x e. ( D ^m C ) | x finSupp W } $.
    mapfien2.ac $e |- ( ph -> A ~~ C ) $.
    mapfien2.bd $e |- ( ph -> B ~~ D ) $.
    mapfien2.z $e |- ( ph -> .0. e. B ) $.
    mapfien2.w $e |- ( ph -> W e. D ) $.
    $( Equinumerousity relation for sets of finitely supported functions.
       (Contributed by Stefan O'Rear, 9-Jul-2015.)  (Revised by AV,
       7-Jul-2019.) $)
    mapfien2 $p |- ( ph -> S ~~ T ) $=
      ( cen wbr wcel cvv vy vz vw cv wf1o cfv wceq wa wex enfixsn syl3anc sylib
      wi bren w3a cfsupp cmap co crab ccnv ccom cmpt eqid f1ocnv 3ad2ant2 simp3
      3ad2ant1 relen brrelex1i syl brrelex2i mapfien ovex rabex2 f1oen 3adant3r
      breq2 rabbidv eqtr4di adantl 3ad2ant3 breqtrd 3exp exlimdv mpd ) ADFUAUDZ
      UEZJWFUFZIUGZUHZUAUIZGHQRZAJDSZIFSDFQRZWKOPNJIUADFUJUKAWJWLUAACEUBUDZUEZU
      BUIZWJWLUMZACEQRZWQMCEUBUNULAWPWRUBAWPWJWLAWPWJUOGBUDZWHUPRZBFEUQURZUSZHQ
      AWPWGGXCQRZWIAWPWGUOZGXCUCGWFUCUDWOUTZVAVAVBZUEXDXEBCDEFGXCTUCXFWFTWHTTJK
      XCVCWHVCWPAECXFUEWGCEWOVDVEAWPWGVFXEWSCTSAWPWSWGMVGZCEQVHVIVJXEWNDTSAWPWN
      WGNVGZDFQVHVIVJXEWSETSXHCEQVHVKVJXEWNFTSXIDFQVHVKVJAWPWMWGOVGVLGXCXGWTJUP
      RBDCUQURGKDCUQVMVNVOVJVPWJAXCHUGZWPWIXJWGWIXCWTIUPRZBXBUSHWIXAXKBXBWHIWTU
      PVQVRLVSVTWAWBWCWDWEWDWE $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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 pweq wss sylibr
      ineq1d rexeqdv abbidv elex cuni wa simpr c0 elinel1 elpwid eqvisset intex
      intssuni2 syl2an eqsstrd velpw rexlimiva abssi uniexg pwexd ssexg sylancr
      wne fvmptd3 ) CDFZECBGZAGZHZIZAEGZJZKLZMZBNVIACJZKLZMZBNZOPOEABQVJCIZVMVP
      BVRVIAVLVOVRVKVNKVJCRUAUBUCCDUDVEVQCUEZJZSVTOFVQOFVPBVTVIVFVTFZAVOVGVOFZV
      IUFZVFVSSWAWCVFVHVSWBVIUGWBVGCSVGUHVCZVHVSSVIWBVGCVGVNKUIUJVIVHOFWDBVHUKV
      GULTVGCUMUNUOBVSUPTUQURVEVSOCDUSUTVQVTOVAVBVD $.

    $( 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 ) ) $=
      ( wcel cvv cfi cfv cv cint wceq cpw cfn c0 wrex wi a1i wa simpr ancom cin
      csn cdif elex wne eldifsni adantr intex sylib eqeltrd rexlimiva elfi vprc
      wb elsni inteqd eqtrdi eleq1d mtbiri simpll eqeltrrd nsyl3 biantrud eldif
      wn int0 bitr4di pm5.32da 3bitr4g rexbidv2 bitrd expcom pm5.21ndd ) CDEZBF
      EZBCGHZEZBAIZJZKZACLMUAZNUBZUCZOZVQVOPVNBVPUDQWDVOPVNVTVOAWCVRWCEZVTRZBVS
      FWEVTSWFVRNUEZVSFEZWEWGVTVRWANUFUGVRUHUIUJUKQVOVNVQWDUNVOVNRZVQVTAWAOWDAB
      CFDULWIVTVTAWAWCWIVTVRWAEZRVTWERWJVTRWFWIVTWJWEWIVTRZWJWJVRWBEZVEZRWEWKWM
      WJWLWHWKWLWHFFEUMWLVSFFWLVSNJFWLVRNVRNUOUPVFUQURUSWKBVSFWIVTSVOVNVTUTVAVB
      VCVRWAWBVDVGVHWJVTTWEVTTVIVJVKVLVM $.

    $( 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 wcel wss c0 wne cfn w3a wa cint cfi cfv cv wceq cpw cin wrex cvv imp
      simp1 elpw2g imbitrrid simpr3 elind inteq rspceeqv sylancl wb simp2 intex
      eqid sylib id elfi syl2anr mpbird ) BCEZABFZAGHZAIEZJZKZALZBMNEZVEDOZLZPD
      BQZIRZSZVDAVJEVEVEPVKVDVIIAUSVCAVIEZVCVLUSUTUTVAVBUBABCUCUDUAUSUTVAVBUEUF
      VEUMDAVJVHVEVEVGAUGUHUIVCVETEZUSVFVKUJUSVCVAVMUTVAVBUKAULUNUSUODVEBTCUPUQ
      UR $.

    $( Sufficient condition for the intersection of the range of a function to
       be in the set of finite intersections.  (Contributed by Mario Carneiro,
       30-Aug-2015.) $)
    intrnfi $p |- ( ( B e. V /\ ( F : A --> B /\ A =/= (/) /\ A e. Fin ) )
        -> |^| ran F e. ( fi ` B ) ) $=
      ( wcel wf c0 wne cfn w3a crn wss cint cfi cfv wa simpr1 frnd cdm sylib
      fdmd simpr2 eqnetrd dm0rn0 necon3bii wfo wfn ffnd dffn4 fofi syl2anc 3jca
      simpr3 elfir syldan ) BDEZABCFZAGHZAIEZJZCKZBLZVAGHZVAIEZJVAMBNOEUPUTPZVB
      VCVDVEABCUPUQURUSQZRVECSZGHVCVEVGAGVEABCVFUAUPUQURUSUBUCVGGVAGCUDUETVEUSA
      VACUFZVDUPUQURUSUMVECAUGVHVEABCVFUHACUITAVACUJUKULVABDUNUO $.
  $}

  ${
    $d x y A $.  $d y B $.  $d x C $.
    $( An indexed intersection of elements of ` C ` is an element of the finite
       intersections of ` C ` .  (Contributed by Mario Carneiro,
       30-Aug-2015.) $)
    iinfi $p |- ( ( C e. V /\ ( A. x e. A B e. C /\ A =/= (/) /\ A e. Fin ) )
        -> |^|_ x e. A B e. ( fi ` C ) ) $=
      ( vy wcel wral c0 wne cfn w3a wa ciin cmpt crn cint cfi cfv wceq wrex cab
      simpr1 dfiin2g syl eqid rnmpt inteqi eqtr4di fmpt 3anbi1i intrnfi sylan2b
      cv wf eqeltrd ) DEGZCDGABHZBIJZBKGZLZMZABCNZABCOZPZQZDRSZVBVCFUNCTABUAFUB
      ZQZVFVBURVCVITUQURUSUTUCAFBCDUDUEVEVHAFBCVDVDUFZUGUHUIVAUQBDVDUOZUSUTLVFV
      GGURVKUSUTABDCVDVJUJUKBDVDEULUMUP $.
  $}

  ${
    $d p A $.  $d p B $.  $d p V $.  $d p X $.
    $( The intersection of two sets is a finite intersection.  (Contributed by
       Thierry Arnoux, 6-Jan-2017.) $)
    inelfi $p |- ( ( X e. V /\ A e. X /\ B e. X )
      -> ( A i^i B ) e. ( fi ` X ) ) $=
      ( vp wcel w3a cin cfi cfv cint wceq cpw cfn wrex cpr 3adant1 syl2anc cvv
      cv prelpwi prfi a1i elind intprg eqcomd inteq rspceeqv wb inex1g 3ad2ant2
      simp1 elfi mpbird ) DCFZADFZBDFZGZABHZDIJFZUSETZKZLEDMZNHZOZURABPZVDFUSVF
      KZLVEURVCNVFUPUQVFVCFUOABDUAQVFNFURABUBUCUDURVGUSUPUQVGUSLUOABDDUEQUFEVFV
      DVBVGUSVAVFUGUHRURUSSFZUOUTVEUIUPUOVHUQABDUJUKUOUPUQULEUSDSCUMRUN $.
  $}

  ${
    $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 snfi elfir syl13anc eqeltrrid ex ssrdv ) ABDZCAAEFZUHCGZADZUJU
      IDUHUKHZUJUJIZJZUIUJCKZLULUHUMAMUMNOZUMPDZUNUIDUHUKQULUJAUHUKRTUPULUJUOUA
      SUQULUJUBSUMABUCUDUEUFUG $.

    $( 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 x y z A $.  $d x y z B $.  $d x y z C $.
    $( The elements of ` ( fi `` C ) ` are closed under finite intersection.
       (Contributed by Mario Carneiro, 24-Nov-2013.) $)
    fiin $p |- ( ( A e. ( fi ` C ) /\ B e. ( fi ` C ) ) ->
                 ( A i^i B ) e. ( fi ` C ) ) $=
      ( vz vx vy cfi wcel wa cin cv cint wceq cfn wrex cvv wb elfi adantr elin
      cfv cpw elfvex mpdan ibi simpr ancoms sylan mpbid cun pwuncl unfi anim12i
      an4s syl2anb sylibr ineq12 intun eqtr4di inteq rspceeqv syl2an rexlimdvaa
      wi rexlimiva sylc inex1g syl2anc mpbird ) ACGUAZHZBVJHZIZABJZVJHZVNDKZLZM
      DCUBZNJZOZVMAEKZLZMZEVSOZBFKZLZMZFVSOZVTVKWDVLVKWDVKCPHZVKWDQACGUCZEACVJP
      RUDUESVMVLWHVKVLUFVKWIVLVLWHQZWJVLWIWKFBCVJPRUGUHUIWCWHVTVDEVSWAVSHZWCIWG
      VTFVSWLWEVSHZWCWGVTWLWMIZWAWEUJZVSHZVNWOLZMVTWCWGIZWNWOVRHZWONHZIZWPWLWAV
      RHZWANHZIWEVRHZWENHZIXAWMWAVRNTWEVRNTXBXDXCXEXAXBXDIWSXCXEIWTWAWECUKWAWEU
      LUMUNUOWOVRNTUPWRVNWBWFJWQAWBBWFUQWAWEURUSDWOVSVQWQVNVPWOUTVAVBUNVCVEVFVK
      VOVTQZVLVKVNPHWIXFABVJVGWJDVNCPPRVHSVI $.
  $}

  ${
    $d t x y z A $.  $d x y z B $.  $d y z V $.
    $( The set of finite intersections is the smallest set that contains ` A `
       and is closed under pairwise intersection.  (Contributed by Mario
       Carneiro, 24-Nov-2013.) $)
    dffi2 $p |- ( A e. V -> ( fi ` A ) =
        |^| { z | ( A C_ z /\ A. x e. z A. y e. z ( x i^i y ) e. z ) } ) $=
      ( vt wcel cvv cfi cv wss cin wral wa cint wceq wi wal cfn w3a cfv cab cpw
      elex wrex wb vex elfi mpan biimpd wex df-rex c0 wne fiint elpwid 3ad2ant2
      elinel1 simp1 sstrd eqvisset intex sylibr elinel2 3jca 3expib pm2.27 syl6
      3ad2ant3 eleq1 biimprd adantl a1i syldd alimdv biimtrid imp 19.23v sylan9
      com23 sylib ssrdv alrimiv ssintab ssfii fiin rgen2 sseq2 eleq2 raleqbi1dv
      ex fvex anbi12d elab sylanblrc intss1 syl eqssd ) DEGDHGZDIUAZDCJZKZAJZBJ
      ZLZXAGZBXAMZAXAMZNZCUBZOZPDEUDWSWTXKWSXIWTXAKZQZCRWTXKKWSXMCWSXIXLWSXINFW
      TXAWSFJZWTGZXNXCOZPZADUCZSLZUEZXIXNXAGZWSXOXTXNHGWSXOXTUFFUGAXNDHHUHUIUJX
      TXCXSGZXQNZAUKZXIYAXQAXSULXIYCYAQZARZYDYAQXBXHYFXHXCXAKZXCUMUNZXCSGZTZXPX
      AGZQZARXBYFABAXAUOXBYLYEAXBYCYLYAXBYCYLYKYAXBYCYJYLYKQXBYBXQYJXBYBXQTZYGY
      HYIYMXCDXAYBXBXCDKXQYBXCDXCXRSURUPUQXBYBXQUSUTXQXBYHYBXQXPHGYHFXPVAXCVBVC
      VIYBXBYIXQXCXRSVDUQVEVFYJYKVGVHYCYKYAQZQXBXQYNYBXQYAYKXNXPXAVJVKVLVMVNVTV
      OVPVQYCYAAVRWAVPVSWBWKWCXICWTWDVCWSWTXJGZXKWTKWSDWTKZXEWTGZBWTMZAWTMZYODH
      WEYQABWTWTXCXDDWFWGXIYPYSNCWTDIWLXAWTPXBYPXHYSXAWTDWHXGYRAXAWTXFYQBXAWTXA
      WTXEWIWJWJWMWNWOWTXJWPWQWRWQ $.

    $( 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 ) ) $=
      ( vy vx vz wcel wss wa cv cin wral cab cint cfi cfv syl cvv wceq dffi2 wi
      sstr2 adantl anim1d ss2abdv intss ssexg ancoms adantr 3sstr4d ) BCGZABHZI
      ZADJZHZEJFJKUNGFUNLEUNLZIZDMZNZBUNHZUPIZDMZNZAOPZBOPZUMVBURHUSVCHUMVAUQDU
      MUTUOUPULUTUOUAUKABUNUBUCUDUEVBURUFQUMARGZVDUSSULUKVFABCUGUHEFDARTQUKVEVC
      SULEFDBCTUIUJ $.

    $( A set which is closed under pairwise intersection is closed under finite
       intersection.  (Contributed by Jeff Madsen, 2-Sep-2009.)  (Revised by
       Mario Carneiro, 24-Nov-2013.) $)
    inficl $p |- ( A e. V ->
        ( A. x e. A A. y e. A ( x i^i y ) e. A <-> ( fi ` A ) = A ) ) $=
      ( vz wcel cv cin wral cfi cfv wceq wss ssfii wa cab cint eleq2 raleqbi1dv
      eqimss2 biantrurd bitr3d elabg intss1 biimtrrdi dffi2 sseq1d sylibrd eqss
      simplbi2com sylsyld fiin rgen2 mpbii impbid1 ) CDFZAGZBGZHZCFZBCIZACIZCJK
      ZCLZUPCVCMZVBVCCMZVDCDNUPVBCEGZMZUSVGFZBVGIZAVGIZOZEPZQZCMZVFUPVBCVMFVOVL
      VBECDVGCLZVKVLVBVPVHVKCVGTUAVJVAAVGCVIUTBVGCVGCUSRSSUBUCCVMUDUEUPVCVNCABE
      CDUFUGUHVDVFVEVCCUIUJUKVDUSVCFZBVCIZAVCIVBVQABVCVCUQURCULUMVRVAAVCCVQUTBV
      CCVCCUSRSSUNUO $.

    $( The set of finite intersections of a set is contained in the powerset of
       the union of the elements of ` A ` .  (Contributed by Mario Carneiro,
       24-Nov-2013.)  (Proof shortened by Mario Carneiro, 21-Mar-2015.) $)
    fipwuni $p |- ( fi ` A ) C_ ~P U. A $=
      ( vx vy cvv wcel cfi cfv cuni cpw wss uniexg pwexd pwuni fiss sylancl cin
      cv wral wceq elpw ssinss1 vex inex1 3imtr4i adantr rgen2 inficl syl mpbii
      wb sseqtrd wn c0 fvprc 0ss eqsstrdi pm2.61i ) ADEZAFGZAHZIZJURUSVAFGZVAUR
      VADEZAVAJUSVBJURUTDADKLZAMAVADNOURBQZCQZPZVAEZCVARBVARZVBVASZVHBCVAVAVEVA
      EZVHVFVAEVEUTJVGUTJVKVHVEVFUTUAVEUTBUBZTVGUTVEVFVLUCTUDUEUFURVCVIVJUJVDBC
      VADUGUHUIUKURULUSUMVAAFUNVAUOUPUQ $.

    $( A singleton is closed under finite intersections.  (Contributed by Mario
       Carneiro, 3-Sep-2015.) $)
    fisn $p |- ( fi ` { A } ) = { A } $=
      ( vx vy cv cin csn wcel wral cfi cfv wceq wa elsni ineqan12d inidm eqtrdi
      vex inex1 elsn cvv sylibr rgen2 wb snex inficl ax-mp mpbi ) BDZCDZEZAFZGZ
      CUKHBUKHZUKIJUKKZULBCUKUKUHUKGZUIUKGZLZUJAKULUQUJAAEAUOUPUHAUIAUHAMUIAMNA
      OPUJAUHUIBQRSUAUBUKTGUMUNUCAUDBCUKTUEUFUG $.
  $}

  $( 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 ) ) $=
    ( wcel cuni cfi cfv ssfii unissd wss cpw fipwuni unissi unipw sseqtri eqssd
    a1i ) ABCZADZAEFZDZQASABGHTRIQTRJZDRSUAAKLRMNPO $.

  $( 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.) $)
  fipwss $p |- ( A C_ ~P X -> ( fi ` A ) C_ ~P X ) $=
    ( cvv wcel cpw wss cfi cuni fiuni sseq1d sspwuni 3bitr4g biimpa wn c0 fvprc
    cfv 0ss eqsstrdi adantr pm2.61ian ) ACDZABEZFZAGQZUCFZUBUDUFUBAHZBFUEHZBFUD
    UFUBUGUHBACIJABKUEBKLMUBNZUFUDUIUEOUCAGPUCRSTUA $.

  ${
    $d x y z A $.  $d x y z B $.  $d x y z C $.  $d x y z D $.  $d x y z K $.
    $( A finite intersection of elements taken from a union of collections.
       (Contributed by Jeff Hankins, 15-Nov-2009.)  (Proof shortened by Mario
       Carneiro, 26-Nov-2013.) $)
    elfiun $p |- ( ( B e. D /\ C e. K ) -> ( A e. ( fi ` ( B u. C ) ) <-> ( A
    e. ( fi ` B ) \/ A e. ( fi ` C ) \/ E. x e. ( fi ` B ) E. y e. ( fi ` C ) A
    = ( x i^i y ) ) ) ) $=
      ( vz wcel wa cin wceq wrex cvv wi 3expib cfn c0 wss cdif cun cfi cfv elex
      cv w3o w3a adantl simpll simplr 3jca 3anim1i inex1 eleq1 mpbiri rexlimivv
      vex a1i 3jaoi impcom cint cpw simp1 unexg 3adant1 elfi syl2anc wne simpl1
      wb intex bitr4di syl5ibcom simp22 inss2 simp1l simp3l elin2d ssfi sylancl
      inss1 elfir simp23 simp1r elinel1 elpwid dfss2 biimpi indi eqtr3di inteqd
      intun eqtrdi 3syl ineq1 eqeq2d ineq2 rspc2ev syl3anc 3mix3d reldisj mpbid
      syl13anc uncom difeq1i difun2 eqtri difss sstrdi simp3r 3mix2d pm2.61iine
      eqsstri 3mix1d eqeq1 2rexbidv 3orbi123d syl5ibrcom expr com23 mpdd sylbid
      rexlimdva ssun1 fiss sseld ssun2 anim12d fiin eleq1a syl6 rexlimdvv 3jaod
      syl impbid pm5.21nd ) DFIZEGIZJZCDEUAZUBUCZIZCDUBUCZIZCEUBUCZIZCAUEZBUEZK
      ZLZBUUEMAUUCMZUFZCNIZYQYRUGZYSUUBJUUMYQYRUUBUUMYSCUUAUDUHYQYRUUBUIYQYRUUB
      UJUKUULYSUUNUUDYSUUNOUUFUUKUUDYQYRUUNUUDUUMYQYRCUUCUDULPUUFYQYRUUNUUFUUMY
      QYRCUUEUDULPUUKYQYRUUNUUKUUMYQYRUUJUUMABUUCUUEUUJUUMOUUGUUCIZUUHUUEIZJZUU
      JUUMUUINIUUGUUHAUQUMCUUINUNUOURUPULPUSUTUUNUUBUULUUNUUBCHUEZVAZLZHYTVBZQK
      ZMZUULUUNUUMYTNIZUUBUVCVJUUMYQYRVCYQYRUVDUUMDEFGVDZVEHCYTNNVFVGUUNUUTUULH
      UVBUUNUURUVBIZJZUUTUURRVHZUULUVGUUMUUTUVHUUMYQYRUVFVIUUTUUMUUSNIUVHCUUSNU
      NUURVKVLVMUVGUVHUUTUULUUNUVFUVHUUTUULOUUNUVFUVHJZJZUULUUTUUSUUCIZUUSUUEIZ
      UUSUUILZBUUEMAUUCMZUFZUVJUVOOUURDKZUUREKZRRUVPRVHZUVQRVHZJZUUNUVIUVOUVTUU
      NUVIUGZUVNUVKUVLUWAUVPVAZUUCIZUVQVAZUUEIZUUSUWBUWDKZLZUVNUWAYQUVPDSZUVRUV
      PQIZUWCUVTUUMYQYRUVIVNUWHUWAUURDVOURUVRUVSUUNUVIVPUWAUURQIZUVPUURSUWIUWAU
      VAQUURUVTUUNUVFUVHVQZVRZUURDWAUURUVPVSVTUVPDFWBXCUWAYRUVQESZUVSUVQQIZUWEU
      VTUUMYQYRUVIWCUWMUWAUUREVOURUVRUVSUUNUVIWDUWAUWJUVQUURSUWNUWLUUREWAUURUVQ
      VSVTUVQEGWBXCUWAUVFUURYTSZUWGUWKUVFUURYTUURUVAQWEWFZUWOUUSUVPUVQUAZVAUWFU
      WOUURUWQUWOUURYTKZUURUWQUWOUWRUURLUURYTWGWHUURDEWIWJWKUVPUVQWLWMWNUVMUWGU
      USUWBUUHKZLABUWBUWDUUCUUEUUGUWBLUUIUWSUUSUUGUWBUUHWOWPUUHUWDLUWSUWFUUSUUH
      UWDUWBWQWPWRWSWTPUVPRLZUUNUVIUVOUWTUUNUVIUGZUVLUVKUVNUXAYRUURESUVHUWJUVLU
      WTUUMYQYRUVIWCUXAUURYTDTZEUXAUWTUURUXBSZUWTUUNUVIVCUXAUVFUWOUWTUXCVJUWTUU
      NUVFUVHVQZUWPUURDYTXAWNXBUXBEDTZEUXBEDUAZDTUXEYTUXFDDEXDXEEDXFXGEDXHXMXIU
      WTUUNUVFUVHXJUXAUVAQUURUXDVRUUREGWBXCXKPUVQRLZUUNUVIUVOUXGUUNUVIUGZUVKUVL
      UVNUXHYQUURDSUVHUWJUVKUXGUUMYQYRUVIVNUXHUURYTETZDUXHUXGUURUXISZUXGUUNUVIV
      CUXHUVFUWOUXGUXJVJUXGUUNUVFUVHVQZUWPUUREYTXAWNXBUXIDETDDEXFDEXHXMXIUXGUUN
      UVFUVHXJUXHUVAQUURUXKVRUURDFWBXCXNPXLUUTUUDUVKUUFUVLUUKUVNCUUSUUCUNCUUSUU
      EUNUUTUUJUVMABUUCUUECUUSUUIXOXPXQXRXSXTYAYCYBUUNUUDUUBUUFUUKUUNUUCUUACYQY
      RUUCUUASZUUMYSUVDDYTSUXLUVEDEYDDYTNYEVTVEZYFUUNUUEUUACYQYRUUEUUASZUUMYSUV
      DEYTSUXNUVEEDYGEYTNYEVTVEZYFUUNUUJUUBABUUCUUEUUNUUQUUGUUAIZUUHUUAIZJZUUJU
      UBOZUUNUUOUXPUUPUXQUUNUUCUUAUUGUXMYFUUNUUEUUAUUHUXOYFYHUXRUUIUUAIUXSUUGUU
      HYTYIUUIUUACYJYNYKYLYMYOYP $.
  $}

  ${
    $d a b c d m n v x y A $.  $d a b c d m n v x y R $.  $d c d m n x y V $.
    $d a b u v y z $.
    dffi3.1 $e |- R = ( u e. _V |->
      ran ( y e. u , z e. u |-> ( y i^i z ) ) ) $.
    $( The set of finite intersections can be "constructed" inductively by
       iterating binary intersection ` _om ` -many times.  (Contributed by
       Mario Carneiro, 21-Mar-2015.) $)
    dffi3 $p |- ( A e. V -> ( fi ` A ) = U. ( rec ( R , A ) " _om ) ) $=
      ( vx vn va vb vv wcel com cv wss wral wa wceq cvv vc vd cfi cfv crdg cres
      vm crn cuni cima cin cab cint dffi2 fr0g wfn frfnom peano1 fnfvelrn mp2an
      c0 eqeltrrdi elssuni syl wrex ciun reeanv eliun anbi12i wb fniunfv eleq2d
      anbi12d ax-mp 3bitr2i cun word ordom ordunel mp3an1 adantl simprl jca weq
      wi wo con0 nnon ad2antlr csuc fveq2 sseq1d sseq2d raleqbi1dv eqsstrd cmpo
      eqidd ineq1 eqeq2d ineq2 eqtrdi rspc2ev syl3anc eqid rnmpo cpw fvex uniex
      sylibr pwex cxp wf adantr sstrid vex inex1 elpw rgen2 fmpo mpbi frn ssexi
      nfcv cmpt mpoeq12 anidms rneqd frsucmpt sylancl sylib a1i ralrimiva sseq2
      ex imbi12d sseq1 rspc2v syl3c sseld simprr onsseleq syl2an2 rzal biantrud
      bitr3d ssfii csn id inidm eqabri ssriv simpl inss1 cbvmpov cbvmptv rdgeq1
      eqtri reseq1i sseqtrrid sstr2 syl5com ralimdv jctird df-suc raleqi ralunb
      ralsn imbitrrdi fiin ss2ralv mpi frnd jctild expimpd finds2 impcom simprd
      bitri a1d r19.21bi eqimss jaod sylbid ssun1 ssun2 peano2 ssiun2s 2rexbidv
      3syl eqeq1 elab eleqtrrdi eleqtrrd sseldd eleqtrdi syl2and rexlimdvva imp
      sylan2br ralrimivva simpld elpw2 fnfvrnss sylancr sspwuni ssexg mpbir2and
      eleq2 elabg intss1 eqssd df-ima unieqi eqtr4di ) DFMZDUCUDZEDUEZNUFZUHZUI
      ZUXQNUJZUIUXOUXPUXTUXOUXPDHOZPZUAOZUBOZUKZUYBMZUBUYBQZUAUYBQZRZHULZUMZUXT
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      UBUXTUXTUYDUXTMZUYEUXTMZRZUXOUYDUGOZUXRUDZMZUYEIOZUXRUDZMZRZINVEUGNVEZUYO
      VULVUGUGNVEZVUJINVEZRUYDUGNVUFVFZMZUYEINVUIVFZMZRZVUDVUGVUJUGINNVGVUPVUMV
      URVUNUGUYDNVUFVHIUYENVUIVHVIUYTVUSVUDVJVUAUYTVUPVUBVURVUCUYTVUOUXTUYDUGNU
      XRVKVLUYTVUQUXTUYEINUXRVKVLVMVNVOUXOVULUYOUXOVUKUYOUGINNUXOVUENMZVUHNMZRZ
      RZVUGUYDVUEVUHVPZUXRUDZMZVUJUYEVVEMZUYOVVCVUFVVEUYDVVCVVDNMZVUTRAOZUYBPZV
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      VVHVRNVUEVUHVSVTZWAZUXOVUTVVAWBWCUXOVVPVVBUXOVVOHNUXOUYBNMZRZVVNANVWBVVIN
      MZRZVVJVVIUYBMZAHWDZWFZVVMVWCVVIWGMVWBUYBWGMZVVJVWGVJVVIWHVWAVWHUXOVWCUYB
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      ZVVKVWRPZAVWQQZRZUXOHIUYBVASZVWIVWKVWLVXCVWJVWIVVMAUYBUUCUUDVXCVVLUYRUXPU
      YBVAUXRWKWLUUEHIWDZVWIVWMVWJVWOVXDVVLVUIUXPUYBVUHUXRWKZWLVVMVWNAUYBVUHVXD
      VVLVUIVVKVXEWMWNVMUYBVWQSZVWIVWSVWJVXAVXFVVLVWRUXPUYBVWQUXRWKZWLVVMVWTAUY
      BVWQVXFVVLVWRVVKVXGWMWNVMUXOUYRDUXPUYSDFUUFWOVVAVWPVXBWEUXOVVAVWMVWOVXBVV
      AVWMRZVWOVXAVWSVXHVWOVWTAVUHQZVWTAVUHUUGZQZRZVXAVXHVWOVXIVXKVXHVWNVWTAVUH
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  $}

  ${
    $d x y A $.  $d x F $.  $d x y V $.
    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 ) ) $=
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  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Hall's marriage theorem
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
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    $( Core induction for Philip Hall's marriage theorem.  (Contributed by
       Stefan O'Rear, 19-Feb-2015.) $)
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      JXXOXYEWYLWTZWYKXXOVXPWYLTXYEXYTTXXOVXPWYMWYLXYCXXDXCVXPWYLXAWAWYIWYKXBXC
      WIVXCVUAUWBXYEWYKVXCYLYMXEXYDVXCXYEYLYPVWSXYDWYIXYEVXLVXPUWCYNXXQXYIVUSXY
      FXXRXXKXYIVUSOZXXMXXPXUCYUAVWPXUCYUAVWSVUSUWFWJWBXFUWGYOVUSXYFXXRUWHWAXYF
      UWIVUSXYFKXXRUWJYMVVBXXTBXXRVWMVUSKVUJXXRVFVGYPYRYRYPUYKUWKYQUYOUYLUYMUYP
      UYQVWOVVDUDDVVFUDDULZVWLVVAVWNVVCYUBVWKVUFEVUTYUBVWJVUEVUCGVWIVUDVUCXPXRX
      SYUBVVBBVWMVULVWIVUDUOXTURXJRUYRUYSUWNUYT $.
  $}

  ${
    $d ph d f $.  $d A b c d f $.  $d B b c $.  $d C c d f $.
    marypha1.a $e |- ( ph -> A e. Fin ) $.
    marypha1.b $e |- ( ph -> B e. Fin ) $.
    marypha1.c $e |- ( ph -> C C_ ( A X. B ) ) $.
    marypha1.d $e |- ( ( ph /\ d C_ A ) -> d ~<_ ( C " d ) ) $.
    $( (Philip) Hall's marriage theorem, sufficiency: a finite relation
       contains an injection if there is no subset of its domain which would be
       forced to violate the pigeonhole principle.  (Contributed by Stefan
       O'Rear, 20-Feb-2015.) $)
    marypha1 $p |- ( ph -> E. f e. ~P C f : A -1-1-> B ) $=
      ( vc vb cvv cv cpw wral wcel wss wi cfn wf1 wrex cima wbr elpwi ralrimiva
      cdom sylan2 wceq imaeq1 breq2d ralbidv pweq rexeqdv imbi12d xpeq2 raleqdv
      cxp pweqd imbi2d marypha1lem com12 vtoclga sylc xpexd sselpwd rspcdva mpd
      wa crn sylan9ssr rnss syl rnxpss sstrdi f1ssr expcom reximdva ) ABMENZUAZ
      EDOZUBZBCVSUAZEWAUBAFNZDWDUCZUGUDZFBOZPZWBAWFFWGWDWGQAWDBRWFWDBUEJUHUFAWD
      KNZWDUCZUGUDZFWGPZVTEWIOZUBZSZWHWBSKBCURZOZDWIDUIZWLWHWNWBWRWKWFFWGWRWJWE
      WDUGWIDWDUJUKULWRVTEWMWAWIDUMUNUOACTQBTQZWOKWQPZHGWSWOKBLNZURZOZPZSWSWTSL
      CTXACUIZXDWTWSXEWOKXCWQXEXBWPXACBUPUSUQUTWSXATQXDBELKFVAVBVCVDADWPMABCTTG
      HVEIVFVGVHAVTWCEWAAVSWAQZVIZVSVJZCRZVTWCSXGXHWPVJZCXGVSWPRXHXJRXFAVSDWPVS
      DUEIVKVSWPVLVMBCVNVOVTXIWCBMCVSVPVQVMVRVH $.
  $}

  ${
    $d A x y z $.  $d F x y z $.  $d G x y $.  $d X x y $.
    marypha2lem.t $e |- T = U_ x e. A ( { x } X. ( F ` x ) ) $.
    $( Lemma for ~ marypha2 .  Properties of the used relation.  (Contributed
       by Stefan O'Rear, 20-Feb-2015.) $)
    marypha2lem1 $p |- T C_ ( A X. U. ran F ) $=
      ( csn cfv cxp ciun crn cuni wss iunss wcel snssi fvssunirn xpss12 sylancl
      cv mprgbir eqsstri ) CABASZFZUBDGZHZIZBDJKZHZEUFUHLUEUHLZABABUEUHMUBBNUCB
      LUDUGLUIUBBODUBPUCBUDUGQRTUA $.

    $( Lemma for ~ marypha2 .  Properties of the used relation.  (Contributed
       by Stefan O'Rear, 20-Feb-2015.) $)
    marypha2lem2 $p |- T = { <. x , y >. | ( x e. A /\ y e. ( F ` x ) ) } $=
      ( vz cv csn cfv cxp ciun wcel wa copab wceq fveq2 wrex bitri 3eqtri df-xp
      sneq xpeq12d cbviunv iuneq2i iunopab velsn equcom anbi1i rexbii ceqsrexbv
      a1i eleq2d opabbii ) DACAHZIZUOEJZKZLGCGHZIZUSEJZKZLZUOCMBHZUQMZNZABOZFAG
      CURVBUOUSPZUPUTUQVAUOUSUBUOUSEQUCUDVCGCUOUTMZVDVAMZNZABOZLVKGCRZABOVGGCVB
      VLVBVLPUSCMABUTVAUAULUEVKABGCUFVMVFABVMUSUOPZVJNZGCRVFVKVOGCVIVNVJVIVHVNA
      USUGAGUHSUIUJVJVEGUOCVNVAUQVDUSUOEQUMUKSUNTT $.

    $( Lemma for ~ marypha2 .  Properties of the used relation.  (Contributed
       by Stefan O'Rear, 20-Feb-2015.) $)
    marypha2lem3 $p |- ( ( F Fn A /\ G Fn A ) -> ( G C_ T <->
          A. x e. A ( G ` x ) e. ( F ` x ) ) ) $=
      ( vy wfn wa wss cv wcel cfv wceq wi wal wral copab bitrdi albii dffn5 a1i
      biimpi adantl df-mpt eqtrdi marypha2lem2 sseq12d ssopab2bw imdistan eleq1
      cmpt 19.21v fvex ceqsalv imbi2i 3bitr3i df-ral bitr4i ) DBHZEBHZIZECJZAKZ
      BLZGKZVDEMZNZIZVEVFVDDMZLZIZOZGPZAPZVGVJLZABQZVBVCVIAGRZVLAGRZJVOVBEVRCVS
      VBEABVGULZVRVAEVTNZUTVAWAABEUAUCUDAGBVGUEUFCVSNVBAGBCDFUGUBUHVIVLAGUISVOV
      EVPOZAPVQVNWBAVEVHVKOZOZGPVEWCGPZOVNWBVEWCGUMWDVMGVEVHVKUJTWEVPVEVKVPGVGV
      DEUNVFVGVJUKUOUPUQTVPABURUSS $.

    $( Lemma for ~ marypha2 .  Properties of the used relation.  (Contributed
       by Stefan O'Rear, 20-Feb-2015.) $)
    marypha2lem4 $p |- ( ( F Fn A /\ X C_ A ) -> ( T " X ) = U. ( F " X ) ) $=
      ( vy wfn wss wa cima cv cfv wcel copab crn wceq cab eqtrid eqtrd resopab2
      ciun cuni cres marypha2lem2 imaeq1i df-ima eqtri adantl rneqd rnopab wrex
      wex df-rex bicomi abbii df-iun eqtr4i a1i wfun fnfun adantr funiunfv syl
      ) DBHZEBIZJZCEKZAEALZDMZUBZDEKUCZVGVHVIBNGLVJNZJAGOZEUDZPZVKVHVNEKVPCVNEA
      GBCDFUEUFVNEUGUHVGVPVIENVMJZAGOZPZVKVGVOVRVFVOVRQVEVMAGEBUAUIUJVGVSVQAUMZ
      GRZVKVQAGUKWAVKQVGWAVMAEULZGRVKVTWBGWBVTVMAEUNUOUPAGEVJUQURUSSTSVGDUTZVKV
      LQVEWCVFBDVAVBAEDVCVDT $.
  $}

  ${
    $d ph d g x $.  $d A d g x $.  $d F d g x $.
    marypha2.a $e |- ( ph -> A e. Fin ) $.
    marypha2.b $e |- ( ph -> F : A --> Fin ) $.
    marypha2.c $e |- ( ( ph /\ d C_ A ) -> d ~<_ U. ( F " d ) ) $.
    $( Version of ~ marypha1 using a functional family of sets instead of a
       relation.  (Contributed by Stefan O'Rear, 20-Feb-2015.) $)
    marypha2 $p |- ( ph -> E. g ( g : A -1-1-> _V /\
          A. x e. A ( g ` x ) e. ( F ` x ) ) ) $=
      ( cuni cv wf1 cfv cxp cvv wcel wa wex wss cima crn csn ciun cpw wrex wral
      unirnffid eqid marypha2lem1 a1i cdom wfn wceq cfn ffnd marypha2lem4 sylan
      breqtrrd marypha1 df-rex ssv f1ss mpan2 ad2antll ad2antrl wb marypha2lem3
      elpwi f1fn syl2an2r mpbid jca ex eximdv biimtrid mpd ) ACEUAJZDKZLZDBCBKZ
      UBVTEMZNUCZUDZUEZCOVRLZVTVRMWAPBCUFZQZDRZACVQWBDFGACEHGUGWBCVQNSABCWBEWBU
      HZUIUJAFKZCSZQWJEWJTJZWBWJTZUKIAECULZWKWMWLUMACUNEHUOZBCWBEWJWIUPUQURUSWD
      VRWCPZVSQZDRAWHVSDWCUTAWQWGDAWQWGAWQQZWEWFVSWEAWPVSVQOSWEVQVACVQOVRVBVCVD
      WRVRWBSZWFWPWSAVSVRWBVHVEAWNWQVRCULZWSWFVFWOVSWTAWPCVQVRVIVDBCWBEVRWIVGVJ
      VKVLVMVNVOVP $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  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.  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 whose square
       is less than 2; in this case the supremum is defined as the square root
       of 2 per ~ sqrtval .  See ~ dfsup2 for alternate definition not
       requiring dummy variables.  (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 A x y z $.  $d B x y z $.  $d R x y z $.
    $( Quantifier-free definition of supremum.  (Contributed by Scott Fenton,
       19-Feb-2013.) $)
    dfsup2 $p |- sup ( B , A , R ) =
      U. ( A \ ( ( `' R " B ) u. ( R " ( A \ ( `' R " B ) ) ) ) ) $=
      ( vx vy vz cv wbr wn wral wrex wi wa cdif cvv wcel vex elima bitri eqtri
      csup crab cuni ccnv cima cun df-sup cab cin dfrab3 wceq wb eqabcb mpbiran
      eldif wo dfrex2 orbi12i ianor 3bitr4i con2bii notbii ralbii impexp imbi1i
      rexbii imbi2i con34b bitr3i ralbii2 anbi12i 3bitr2ri mpgbir ineq2i invdif
      elun brcnv unieqi ) BACUADGZEGZCHZIZEBJZVTVSCHZVTFGZCHZFBKZLZEAJZMZDAUBZU
      CACUDZBUEZCAWMNZUEZUFZNZUCDEFBACUGWKWQWKAWJDUHZUIZWQWJDAUJWSAOWPNZUIWQWRW
      TAWRWTUKWJVSWTPZULDWJDWTUMXAVSWPPZIZVTVSWLHZIZEBJZWDIZEWNJZMZWJXAVSOPXCDQ
      ZVSOWPUOUNXBXIVSWMPZVSWOPZUPXFIZXHIZUPXBXIIXKXMXLXNXKXDEBKXMEVSWLBXJRXDEB
      UQSXLWDEWNKXNEVSCWNXJRWDEWNUQSURVSWMWOVPXFXHUSUTVAXFWCXHWIXEWBEBXDWAVTVSC
      EQZXJVQVBVCXGWHEWNAVTAPZVTWMPZIZMZXGLXPXRXGLZLVTWNPZXGLXPWHLXPXRXGVDYAXSX
      GVTAWMUOVEWHXTXPWHWDXQLXTXQWGWDXQWEVTWLHZFBKWGFVTWLBXORYBWFFBWEVTCFQXOVQV
      FSVGWDXQVHVIVGUTVJVKVLVMVNAWPVOTTVRT $.
  $}

  ${
    $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 $.
  $}

  ${
    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 ) $=
      ( csup ccnv cima cdif cun cuni dfsup2 nfcnv nfima nfdif nfun nfuni nfcxfr
      ) ABCDHCDIZBJZDCUBKZJZLZKZMCBDNAUFACUEFAUBUDAUABADGOEPZADUCGACUBFUGQPRQST
      $.
  $}

  ${
    $d x y z w A $.  $d x y z w R $.  $d x y z w B $.  $d w C $.  $d w ph $.
    supmo.1 $e |- ( ph -> R Or A ) $.
    $( Any class ` B ` has at most one supremum in ` A ` (where ` R ` is
       interpreted as 'less than').  (Contributed by NM, 5-May-1999.)  (Revised
       by Mario Carneiro, 24-Dec-2016.) $)
    supmo $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 breq2 biimtrid wrmo wcel anbi2ci
      ancom an42 3bitr4i ralnex rexbidv imbi12d rspcva cbvrexvw imbitrrdi con3d
      wor expimpd ad2antrl ad2antll anim12d sotrieq2 sylibrd ralrimivva ralbidv
      syl notbid imbi1d anbi12d rmo4 sylibr ) ABJZCJZGKZLZCFMZVJVIGKZVJDJZGKZDF
      NZOZCEMZPZIJZVJGKZLZCFMZVJWAGKZVQOZCEMZPZPZBIQZOZIEMBEMZVTBEUAAEGUNZWLHWM
      WKBIEEWMVIEUBZWAEUBZPPZWIVIWAGKZLZWAVIGKZLZPZWJWIWGVMPZVSWDPZPZWPXAVMWDPZ
      WGVSPZPXFWDVMPZPWIXDXEXGXFVMWDUDUCVMVSWDWGUEWGVMVSWDUEUFWPXBWRXCWTWNXBWRO
      WMWOWNWGVMWRVMVKCFNZLWNWGPZWRVKCFUGXIWQXHXIWQVIVOGKZDFNZXHWFWQXKOCVIECBQZ
      WEWQVQXKVJVIWAGRXLVPXJDFVJVIVOGRUHUIUJVKXJCDFVJVOVIGSUKULUMTUOUPWOXCWTOWM
      WNWOVSWDWTWDWBCFNZLWOVSPZWTWBCFUGXNWSXMXNWSWAVOGKZDFNZXMVRWSXPOCWAECIQZVN
      WSVQXPVJWAVIGRXQVPXODFVJWAVOGRUHUIUJWBXOCDFVJVOWAGSUKULUMTUOUQURTEVIWAGUS
      UTVAVCVTWHBIEWJVMWDVSWGWJVLWCCFWJVKWBVIWAVJGRVDVBWJVRWFCEWJVNWEVQVIWAVJGS
      VEVBVFVGVH $.

    $( A supremum is a set.  (Contributed by NM, 22-May-1999.)  (Revised by
       Mario Carneiro, 24-Dec-2016.) $)
    supexd $p |- ( ph -> sup ( B , A , R ) e. _V ) $=
      ( vx vy vz csup cv wbr wn wral wrex wi wa crab cuni cvv wcel df-sup supmo
      wrmo rmorabex uniexg 3syl eqeltrid ) ACBDIFJZGJZDKLGCMUIUHDKUIHJDKHCNOGBM
      PZFBQZRZSFGHCBDUAAUJFBUCUKSTULSTAFGHBCDEUBUJFBUDUKSUEUFUG $.

    ${
      supeu.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 NM, 12-Oct-2004.) $)
      supeu $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 supmo reu5 sylanbrc ) ABJZCJZGKLC
        FMUCUBGKUCDJGKDFNOCEMPZBENUDBEQUDBERIABCDEFGHSUDBETUA $.
    $}

    $( Alternate expression for the supremum.  (Contributed by Mario Carneiro,
       24-Dec-2016.)  (Revised by Thierry Arnoux, 24-Sep-2017.) $)
    supval2 $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 ) ) ) ) $=
      ( wor csup cv wbr wn wral wrex wa wceq cuni syl c0 crio crab df-sup simpl
      wi wreu simpr supeu riotauni eqtr4id necon1bbii biimpi unieqd uni0 eqtrdi
      rabn0 eqtrid reurex riotaund nsyl5 eqtr4d adantl pm2.61dan ) AEGIZFEGJZBK
      ZCKZGLMCFNVGVFGLVGDKGLDFOUECENPZBEUAZQZHVDVHBEOZVJVDVKPZVEVHBEUBZRZVIBCDF
      EGUCZVLVHBEUFZVIVNQVLBCDEFGVDVKUDVDVKUGUHVHBEUISUJVKMZVJVDVQVETVIVQVEVNTV
      OVQVNTRTVQVMTVQVMTQVKVMTVHBEUPUKULUMUNUOUQVPVKVITQVHBEURVHBEUSUTVAVBVCS
      $.

    ${
      $d x y C $.
      $( Sufficient condition for an element to be equal to the supremum.
         (Contributed by Mario Carneiro, 21-Apr-2015.) $)
      eqsup $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 ralbidv syl2anc w3a wor adantr
        csup crio supval2 3simpc adantl wreu simpr1 breq1 notbid imbi1d anbi12d
        wb breq2 rspcev supeu riota2 mpbid eqtrd ex ) AFDJZFBKZGLZMZBENZVDFGLZV
        DCKGLCEOZPZBDNZUAZEDGUDZFQAVLRZVMIKZVDGLZMZBENZVDVOGLZVIPZBDNZRZIDUEZFV
        NIBCDEGADGUBVLHUCZUFVNVGVKRZWCFQZVLWEAVCVGVKUGUHZVNVCWBIDUIWEWFUOAVCVGV
        KUJZVNIBCDEGWDVNVCWEWBIDOWHWGWBWEIFDVOFQZVRVGWAVKWIVQVFBEWIVPVEVOFVDGUK
        ULSWIVTVJBDWIVSVHVIVOFVDGUPUMSUNZUQTURWBWEIDFWJUSTUTVAVB $.

      $d y ph $.
      eqsupd.2 $e |- ( ph -> C e. A ) $.
      eqsupd.3 $e |- ( ( ph /\ y e. B ) -> -. C R y ) $.
      eqsupd.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 Mario Carneiro, 21-Apr-2015.) $)
      eqsupd $p |- ( ph -> sup ( B , A , R ) = C ) $=
        ( wcel cv wbr wn wral wrex wi csup ralrimiva wceq expr eqsup mp3and ) A
        FDLFBMZGNOZBEPUEFGNZUECMGNCEQZRZBDPEDGSFUAIAUFBEJTAUIBDAUEDLUGUHKUBTABC
        DEFGHUCUD $.
    $}

    supcl.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 ~ supub
       and ~ suplub .  (Contributed by NM, 12-Oct-2004.) $)
    supcl $p |- ( ph -> sup ( B , A , R ) e. A ) $=
      ( csup cv wbr wn wral wrex wi wa crio supval2 wreu wcel supeu riotacl syl
      eqeltrd ) AFEGJBKZCKZGLMCFNUGUFGLUGDKGLDFOPCENQZBERZEABCDEFGHSAUHBETUIEUA
      ABCDEFGHIUBUHBEUCUDUE $.

    $( A supremum is an upper bound.  See also ~ supcl and ~ suplub .

       This proof demonstrates how to expand an iota-based definition
       ( ~ df-iota ) using ~ riotacl2 .

       (Contributed by NM, 12-Oct-2004.)  (Proof shortened by Mario Carneiro,
       24-Dec-2016.) $)
    supub $p |- ( ph -> ( C e. B -> -. sup ( B , A , R ) R C ) ) $=
      ( vw cv wbr wn wral wcel wi syl wceq notbid csup crab wrex wa a1i ss2rabi
      simpl crio supval2 wreu supeu riotacl2 eqeltrd sselid breq2 breq1 ralbidv
      cbvralvw bitrid elrab simprbi rspccv ) AFEHUAZKLZHMZNZKFOZGFPVCGHMZNZQAVC
      BLZCLZHMZNZCFOZBEUBZPZVGAVNVKVJHMVKDLHMDFUCQCEOZUDZBEUBZVOVCVRVNBEVRVNQVJ
      EPVNVQUGUEUFAVCVRBEUHZVSABCDEFHIUIAVRBEUJVTVSPABCDEFHIJUKVRBEULRUMUNVPVCE
      PVGVNVGBVCEVNVJVDHMZNZKFOVJVCSZVGVMWBCKFVKVDSVLWAVKVDVJHUOTURWCWBVFKFWCWA
      VEVJVCVDHUPTUQUSUTVARVFVIKGFVDGSVEVHVDGVCHUOTVBR $.

    $d z C $.
    $( A supremum is the least upper bound.  See also ~ supcl and ~ supub .
       (Contributed by NM, 13-Oct-2004.)  (Revised by Mario Carneiro,
       24-Dec-2016.) $)
    suplub $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 wceq breq1 syl csup wa crab rexbidv imbi12d
      simpr cbvralvw sylib a1i ss2rabi crio supval2 wreu supeu riotacl2 eqeltrd
      wn sselid breq2 imbi1d ralbidv elrab simprbi rspccv impd ) AKLZFEHUAZHMZV
      FDLZHMZDFNZOZKEPZGEQZGVGHMZUBGVIHMZDFNZOAVGVFBLZHMZVKOZKEPZBEUCZQZVMAVRCL
      ZHMUQCFPZWDVRHMZWDVIHMZDFNZOZCEPZUBZBEUCZWBVGWKWABEWKWAOVREQWKWJWAWEWJUFW
      IVTCKEWDVFRZWFVSWHVKWDVFVRHSWMWGVJDFWDVFVIHSUDUEUGUHUIUJAVGWKBEUKZWLABCDE
      FHIULAWKBEUMWNWLQABCDEFHIJUNWKBEUOTUPURWCVGEQVMWAVMBVGEVRVGRZVTVLKEWOVSVH
      VKVRVGVFHUSUTVAVBVCTVMVNVOVQVLVOVQOKGEVFGRZVHVOVKVQVFGVGHSWPVJVPDFVFGVIHS
      UDUEVDVET $.

    ${
      suplub2.3 $e |- ( ph -> B C_ A ) $.
      $( Bidirectional form of ~ suplub .  (Contributed by Mario Carneiro,
         6-Sep-2014.) $)
      suplub2 $p |- ( ( ph /\ C e. A ) ->
                    ( C R sup ( B , A , R ) <-> E. z e. B C R z ) ) $=
        ( vw wcel wa wbr cv wrex breq2 wi ad2antrr csup suplub expdimp cbvrexvw
        wceq biimprd a1i wor simplr wss adantr sselda supcl syl13anc expcomd wo
        sotr supub imp sotric syl12anc con2bid mpbird mpjaod rexlimdva biimtrid
        wn wb impbid ) AGEMZNZGFEHUAZHOZGDPZHOZDFQZAVJVMVPABCDEFGHIJUBUCVPGLPZH
        OZLFQVKVMVOVRDLFVNVQGHRUDVKVRVMLFVKVQFMZNZVLVQUEZVRVMSZVQVLHOZWAWBSVTWA
        VMVRVLVQGHRUFUGVTVRWCVMVTEHUHZVJVQEMZVLEMZVRWCNVMSAWDVJVSITZAVJVSUIVKFE
        VQAFEUJVJKUKULZAWFVJVSABCDEFHIJUMTZEGVQVLHUQUNUOVTWAWCUPZVLVQHOZVGZVKVS
        WLAVSWLSVJABCDEFVQHIJURUKUSVTWKWJVTWDWFWEWKWJVGVHWGWIWHEVLVQHUTVAVBVCVD
        VEVFVI $.
    $}

    $( An upper bound is not less than the supremum.  (Contributed by NM,
       13-Oct-2004.) $)
    supnub $p |- ( ph -> ( ( C e. A /\ A. z e. B -. C R z ) ->
                    -. C R sup ( B , A , R ) ) ) $=
      ( wcel cv wbr wn wral csup wa wrex suplub expdimp dfrex2 imbitrdi expimpd
      con2d ) AGEKZGDLHMZNDFOZGFEHPHMZNAUEQZUHUGUIUHUFDFRZUGNAUEUHUJABCDEFGHIJS
      TUFDFUAUBUDUC $.
  $}

  ${
    supex.1 $e |- R Or A $.
    $( A supremum is a set.  (Contributed by NM, 22-May-1999.) $)
    supex $p |- sup ( B , A , R ) e. _V $=
      ( wor csup cvv wcel id supexd ax-mp ) ACEZBACFGHDLABCLIJK $.
  $}

  ${
    $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 x y z $.  $d R x y z $.
    $( The supremum of an empty set is the smallest element of the base set.
       (Contributed by AV, 4-Sep-2020.) $)
    sup0riota $p |- ( R Or A -> sup ( (/) , A , R )
                                = ( iota_ x e. A A. y e. A -. y R x ) ) $=
      ( vz wor c0 csup cv wbr wn wral wrex wi wa crio id supval2 wb ral0 ralbii
      biantrur rex0 imnot ax-mp bitr3i a1i riotabidv eqtrd ) CDFZGCDHAIZBIZDJKZ
      BGLZULUKDJZULEIDJZEGMZNZBCLZOZACPUOKZBCLZACPUJABECGDUJQRUJUTVBACUTVBSUJUT
      USVBUNUSUMBTUBURVABCUQKURVASUPEUCUOUQUDUEUAUFUGUHUI $.

    $d X x y $.
    $( The supremum of an empty set under a base set which has a unique
       smallest element is the smallest element of the base set.  (Contributed
       by AV, 4-Sep-2020.) $)
    sup0 $p |- ( ( R Or A /\ ( X e. A /\ A. y e. A -. y R X )
                   /\ E! x e. A A. y e. A -. y R x )
                 -> sup ( (/) , A , R ) = X ) $=
      ( wor wcel cv wbr wn wral wa wreu w3a csup crio wceq sup0riota 3ad2ant1
      c0 simp2r wb simpl anim1i 3adant1 breq2 notbid ralbidv riota2 mpbid eqtrd
      syl ) CDFZECGZBHZEDIZJZBCKZLZUOAHZDIZJZBCKZACMZNZTCDOZVCACPZEUMUSVFVGQVDA
      BCDRSVEURVGEQZUMUNURVDUAVEUNVDLZURVHUBUSVDVIUMUSUNVDUNURUCUDUEVCURACEUTEQ
      ZVBUQBCVJVAUPUTEUODUFUGUHUIULUJUK $.
  $}

  ${
    $d A y z $.  $d B y z $.  $d C y z $.  $d R y z $.  $d y ph $.
    supmax.1 $e |- ( ph -> R Or A ) $.
    supmax.2 $e |- ( ph -> C e. A ) $.
    supmax.3 $e |- ( ph -> C e. B ) $.
    supmax.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 Jeff Hoffman, 17-Jun-2008.)  (Proof shortened by OpenAI,
       30-Mar-2020.) $)
    supmax $p |- ( ph -> sup ( B , A , R ) = C ) $=
      ( vz wcel cv wbr wa wrex simprr breq2 rspcev syl2an2r eqsupd ) ABKCDEFGHJ
      AEDLBMZCLZUBEFNZOUDUBKMZFNZKDPIAUCUDQUFUDKEDUEEUBFRSTUA $.
  $}

  ${
    $d A x y z $.  $d R x y z $.  $d B x y z $.
    $( A finite set satisfies the conditions to have a supremum.  (Contributed
       by Mario Carneiro, 28-Apr-2015.) $)
    fisup2g $p |- ( ( R Or A /\ ( B e. Fin /\ B =/= (/) /\ B C_ A ) ) ->
      E. x e. B ( A. y e. B -. x R y /\
        A. y e. A ( y R x -> E. z e. B y R z ) ) ) $=
      ( wor cfn wcel c0 wne wss w3a wa cv wbr wn wral wrex wi wreu simp1 fisupg
      soss supeu 3exp syl6 com4l 3imp2 reurex breq2 rspcev ralrimivw a1d anim2d
      ex reximia 3syl ) DFGZEHIZEJKZEDLZMNAOZBOZFPQBERZVDVCFPZVDCOZFPZCESZTZBER
      ZNZAEUAZVLAESVEVJBDRZNZAESUSUTVAVBVMVBUSUTVAVMVBUSEFGZUTVAVMTTEDFUDVPUTVA
      VMVPUTVAMABCEEFVPUTVAUBABCEFUCUEUFUGUHUIVLAEUJVLVOAEVCEIZVKVNVEVQVNVKVQVJ
      BDVQVFVIVHVFCVCEVGVCVDFUKULUPUMUNUOUQUR $.

    $( A nonempty finite set contains its supremum.  (Contributed by Jeff
       Madsen, 9-May-2011.) $)
    fisupcl $p |- ( ( R Or A /\ ( B e. Fin /\ B =/= (/) /\ B C_ A ) )
                                    -> sup ( B , A , R ) e. B ) $=
      ( vx vy vz wor cfn wcel c0 wa cv wbr wral wrex wi crio wreu sylc supeu wn
      wne wss w3a csup simpl supval2 wceq simpr3 rspcev ex ralrimivw a1d anim2d
      breq2 rgen a1i soss simpr1 simpr2 fisupg syl3anc ssrexv riotass2 syl22anc
      fisup2g riotacl syl eqeltrrd eqeltrd ) ACGZBHIZBJUBZBAUCZUDZKZBACUEDLZELZ
      CMUAEBNZVRVQCMZVRFLZCMZFBOZPZEANZKZDAQZBVPDEFABCVKVOUFZUGVPVSWDEBNZKZDBQZ
      WGBVPVNWJWFPZDBNZWJDBOZWFDARWKWGUHVKVLVMVNUIZWMVPWLDBVQBIZWIWEVSWPWEWIWPW
      DEAWPVTWCWBVTFVQBWAVQVRCUOUJUKULUMUNUPUQVPBCGZVLVMWNVPVNVKWQWOWHBACURSZVK
      VLVMVNUSVKVLVMVNUTDEFBCVAVBZVPDEFABCWHVPVNWFDBOWFDAOWODEFABCVFWFDBAVCSTWJ
      WFDBAVDVEVPWJDBRWKBIVPDEFBBCWRWSTWJDBVGVHVIVJ $.
  $}

  ${
    $d A x y z $.  $d B x y z $.  $d R x y z $.
    supgtoreq.1 $e |- ( ph -> R Or A ) $.
    supgtoreq.2 $e |- ( ph -> B C_ A ) $.
    supgtoreq.3 $e |- ( ph -> B e. Fin ) $.
    supgtoreq.4 $e |- ( ph -> C e. B ) $.
    supgtoreq.5 $e |- ( ph -> S = sup ( B , A , R ) ) $.
    $( The supremum of a finite set is greater than or equal to all the
       elements of the set.  (Contributed by AV, 1-Oct-2019.) $)
    supgtoreq $p |- ( ph -> ( C R S \/ C = S ) ) $=
      ( vx vy vz wbr wo wn wcel cv wrex wceq csup wss wral wi wa wor cfn c0 wne
      ne0d fisup2g syl13anc ssrexv supub mpd eqnbrtrd wb fisupcl sseldd eqeltrd
      sotric syl12anc orcom eqcom orbi2i bitri notbii bitr2di mtbird notnotrd
      sylc ) ADFEOZDFUAZPZAVOQZFDEOZAFCBEUBZDEKADCRVRDEOQJALMNBCDEGACBUCZLSZMSZ
      EOQMCUDWAVTEOWANSEONCTUEMBUDUFZLCTZWBLBTHABEUGZCUHRZCUIUJZVSWCGIACDJUKZHL
      MNBCEULUMWBLCBUNVLUOUPUQAVQFDUAZVMPZQZVPAWDFBRDBRVQWJURGAFVRBKACBVRHAWDWE
      WFVSVRCRGIWGHBCEUSUMUTVAACBDHJUTBFDEVBVCWIVOWIVMWHPVOWHVMVDWHVNVMFDVEVFVG
      VHVIVJVK $.
  $}

  ${
    $d y A $.  $d y B $.  $d y C $.  $d y R $.
    $( The supremum of a pair.  (Contributed by NM, 17-Jun-2007.)  (Proof
       shortened by Mario Carneiro, 24-Dec-2016.) $)
    suppr $p |- ( ( R Or A /\ B e. A /\ C e. A ) -> sup ( { B , C } , A , R ) =
                if ( C R B , B , C ) ) $=
      ( vy wor wcel w3a cpr wbr 3adant1 wn wceq breq1 notbid adantr ifbothda wa
      sonr breq2 cif simp1 ifcl ifpr cv wral 3adant3 simpr wi so2nr 3impb imnan
      3com23 sylibr imp 3adant2 wb ralprg mpbir2and r19.21bi supmax ) ADFZBAGZC
      AGZHZEABCIZCBDJZBCUAZDVBVCVDUBVCVDVHAGVBVGBCAUCKVCVDVHVFGVBVGBCAAUDKVEVHE
      UEZDJZLZEVFVEVKEVFUFZVHBDJZLZVHCDJZLZVGBBDJZLZVGLZVNVEBCBVHMZVQVMBVHBDNOC
      VHMZVGVMCVHBDNOVEVRVGVBVCVRVDABDSUGPVEVSUHQVGBCDJZLZCCDJZLZVPVEBCVTWBVOBV
      HCDNOWAWDVOCVHCDNOVEVGWCVEVGWBRLZVGWCUIVBVDVCWFVBVDVCWFACBDUJUKUMVGWBULUN
      UOVEWEVSVBVDWEVCACDSUPPQVCVDVLVNVPRUQVBVKVNVPEBCAAVIBMVJVMVIBVHDTOVICMVJV
      OVICVHDTOURKUSUTVA $.

    $( The supremum of a singleton.  (Contributed by NM, 2-Oct-2007.) $)
    supsn $p |- ( ( R Or A /\ B e. A ) -> sup ( { B } , A , R ) = B ) $=
      ( wor wcel wa csn csup wbr cif cpr dfsn2 supeq1i wceq suppr 3anidm23 ifid
      eqtrid eqtrdi ) ACDZBAEZFZBGZACHZBBCIZBBJZBUBUDBBKZACHZUFAUCUGCBLMTUAUHUF
      NABBCOPRUEBQS $.
  $}

  ${
    $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 ~ supiso .  (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 ~ supiso .  (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 $.

    supiso.4 $e |- ( ph -> R Or A ) $.
    $( Image of a supremum under an isomorphism.  (Contributed by Mario
       Carneiro, 24-Dec-2016.) $)
    supiso $p |- ( ph ->
      sup ( ( F " C ) , B , S ) = ( F ` sup ( C , A , R ) ) ) $=
      ( vw vv vu cv wbr wral cima csup cfv wor wiso isoso syl mpbid wf1o isof1o
      wb wf f1of 3syl supcl ffvelcdmd wn wrex wi wa supub wcel suplub supisolem
      ralrimiv expd mpdan mpbi2and simpld r19.21bi simprd impr eqsupd ) AOPFJGU
      AZGEHUBZJUCZIAEHUDZFIUDZNAEFHIJUEZVQVRUKKEFHIJUFUGUHAEFVOJAVSEFJUIEFJULKE
      FHIJUJEFJUMUNABCDEGHNMUOZUPAVPORZISUQZOVNAWBOVNTZWAVPISZWAPRISPVNURZUSZOF
      TZAVOQRZHSUQZQGTZWHVOHSZWHDRHSDGURZUSZQETZWCWGUTZAWIQGABCDEGWHHNMVAVEAWMQ
      EAWHEVBWKWLABCDEGWHHNMVCVFVEAVOEVBWJWNUTWOUKVTAQDOPEFGVOHIJKLVDVGVHZVIVJA
      WAFVBWDWEAWFOFAWCWGWPVKVJVLVM $.
  $}

  $( 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
      $.
  $}

  ${
    infexd.1 $e |- ( ph -> R Or A ) $.
    $( An infimum is a set.  (Contributed by AV, 2-Sep-2020.) $)
    infexd $p |- ( ph -> inf ( B , A , R ) e. _V ) $=
      ( cinf ccnv csup cvv df-inf wor cnvso sylib supexd eqeltrid ) ACBDFCBDGZH
      ICBDJABCPABDKBPKEBDLMNO $.

    $d A y z $.  $d B y z $.  $d R y z $.
    ${
      $d C y z $.
      $( Sufficient condition for an element to be equal to the infimum.
         (Contributed by AV, 2-Sep-2020.) $)
      eqinf $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 wi w3a wa wb cvv bicomd cinf wceq df-inf wor
        ccnv csup cnvso sylib eqsup brcnvg elvd notbid ralbidv vex mpan rexbidv
        brcnv a1i imbi12d anbi12d pm5.32i 3anass 3bitr4i biimpi impel eqtrid ex
        ) AFDIZBJZFGKZLZBEMZFVIGKZCJZVIGKZCENZOZBDMZPZEDGUAZFUBAVSQVTEDGUEZUFZF
        EDGUCAVHFVIWAKZLZBEMZVIFWAKZVIVNWAKZCENZOZBDMZPZWBFUBVSABCDEFWAADGUDDWA
        UDHDGUGUHUIVSWKVHVLVRQZQVHWEWJQZQVSWKVHWLWMVHVLWEVRWJVHVKWDBEVHVJWCVHVJ
        WCRBVHVISIZQWCVJFVIDSGUJTUKULUMVHVQWIBDVHVMWFVPWHVHWFVMWNVHWFVMRBUNZVIF
        SDGUJUOTVHVOWGCEVHWGVOWGVORVHVIVNGWOCUNUQURTUPUSUMUTVAVHVLVRVBVHWEWJVBV
        CVDVEVFVG $.

      $d ph y $.
      eqinfd.2 $e |- ( ph -> C e. A ) $.
      eqinfd.3 $e |- ( ( ph /\ y e. B ) -> -. y R C ) $.
      eqinfd.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 AV, 3-Sep-2020.) $)
      eqinfd $p |- ( ph -> inf ( B , A , R ) = C ) $=
        ( wcel cv wbr wn wral wrex wi cinf ralrimiva wceq expr eqinf mp3and ) A
        FDLBMZFGNOZBEPFUEGNZCMUEGNCEQZRZBDPEDGSFUAIAUFBEJTAUIBDAUEDLUGUHKUBTABC
        DEFGHUCUD $.
    $}

    $d A x $.  $d B x $.  $d R x $.  $d ph x y z $.
    $( Alternate expression for the infimum.  (Contributed by AV,
       2-Sep-2020.) $)
    infval $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 wi wa crio wb vex brcnv a1i cinf df-inf cnvso sylib
      ccnv csup supval2 notbid ralbidv rexbidv imbi12d anbi12d riotabidv eqtrid
      wor eqtrd ) AFEGUAFEGUEZUFZCIZBIZGJZKZCFLZUTUSGJZDIZUSGJZDFMZNZCELZOZBEPZ
      FEGUBAURUTUSUQJZKZCFLZUSUTUQJZUSVEUQJZDFMZNZCELZOZBEPVKABCDEFUQAEGUOEUQUO
      HEGUCUDUGAVTVJBEAVNVCVSVIAVMVBCFAVLVAVLVAQAUTUSGBRZCRZSTUHUIAVRVHCEAVOVDV
      QVGVOVDQAUSUTGWBWASTAVPVFDFVPVFQAUSVEGWBDRSTUJUKUIULUMUPUN $.
  $}

  ${
    $d A x y z $.  $d B x y z $.  $d R x y z $.
    infcl.1 $e |- ( ph -> R Or A ) $.
    infcl.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 ) ) ) $.
    $( Lemma for ~ infcl , ~ inflb , ~ infglb , etc.  (Contributed by AV,
       3-Sep-2020.) $)
    infcllem $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 vex brcnv bicomi ralbii notbii rexbii imbi12i
      ccnv anbi12i sylib ) ACJZBJZGKZLZCFMZUHUGGKZDJZUGGKZDFNZOZCEMZPZBENUHUGGU
      DZKZLZCFMZUGUHUSKZUGUMUSKZDFNZOZCEMZPZBENIURVHBEUKVBUQVGUJVACFUIUTUTUIUHU
      GGBQZCQZRSUATUPVFCEULVCUOVEVCULUGUHGVJVIRSUNVDDFVDUNUGUMGVJDQRSUBUCTUEUBU
      F $.

    $( An infimum belongs to its base class (closure law).  See also ~ inflb
       and ~ infglb .  (Contributed by AV, 3-Sep-2020.) $)
    infcl $p |- ( ph -> inf ( B , A , R ) e. A ) $=
      ( cinf ccnv csup df-inf wor cnvso sylib infcllem supcl eqeltrid ) AFEGJFE
      GKZLEFEGMABCDEFTAEGNETNHEGOPABCDEFGHIQRS $.

    $( An infimum is a lower bound.  See also ~ infcl and ~ infglb .
       (Contributed by AV, 3-Sep-2020.) $)
    inflb $p |- ( ph -> ( C e. B -> -. C R inf ( B , A , R ) ) ) $=
      ( wcel cinf wbr wn wa ccnv csup wor cnvso sylib infcllem supub imp df-inf
      wceq a1i breq2d wb supcl brcnvg bicomd sylan bitrd mtbird ex ) AGFKZGFEHL
      ZHMZNAUPOZURFEHPZQZGUTMZAUPVBNABCDEFGUTAEHREUTRIEHSTZABCDEFHIJUAZUBUCUSUR
      GVAHMZVBUSUQVAGHUQVAUEUSFEHUDUFUGAVAEKZUPVEVBUHABCDEFUTVCVDUIVFUPOVBVEVAG
      EFHUJUKULUMUNUO $.

    $d C z $.  $d ph z $.
    $( An infimum is the greatest lower bound.  See also ~ infcl and ~ inflb .
       (Contributed by AV, 3-Sep-2020.) $)
    infglb $p |- ( ph -> ( ( C e. A /\ inf ( B , A , R ) R C ) ->
                    E. z e. B z R C ) ) $=
      ( wcel cinf wbr cv wrex wa wb wor brcnvg cvv ccnv csup df-inf simpr cnvso
      breq1i sylib infcllem adantr bicomd syl2anc bitrid suplub expdimp sylancl
      supcl vex rexbidv sylibd sylbid expimpd ) AGEKZFEHLZGHMZDNZGHMZDFOZAVBPZV
      DGFEHUAZUBZVIMZVGVDVJGHMZVHVKVCVJGHFEHUCUFVHVBVJEKZVLVKQAVBUDZAVMVBABCDEF
      VIAEHREVIRIEHUEUGZABCDEFHIJUHZUPUIVBVMPVKVLGVJEEHSUJUKULVHVKGVEVIMZDFOZVG
      AVBVKVRABCDEFGVIVOVPUMUNVHVQVFDFVHVBVETKVQVFQVNDUQGVEETHSUOURUSUTVA $.

    ${
      infglbb.3 $e |- ( ph -> B C_ A ) $.
      $( Bidirectional form of ~ infglb .  (Contributed by AV, 3-Sep-2020.) $)
      infglbb $p |- ( ( ph /\ C e. A ) ->
                    ( inf ( B , A , R ) R C <-> E. z e. B z R C ) ) $=
        ( cinf wbr wcel wa wrex wb wor brcnvg cvv ccnv csup df-inf breq1i simpr
        cv cnvso sylib infcllem supcl adantr bicomd syl2anc suplub2 vex sylancl
        rexbidv 3bitrd bitrid ) FEHLZGHMFEHUAZUBZGHMZAGENZOZDUFZGHMZDFPZUTVBGHF
        EHUCUDVEVCGVBVAMZGVFVAMZDFPVHVEVDVBENZVCVIQAVDUEZAVKVDABCDEFVAAEHREVARI
        EHUGUHZABCDEFHIJUIZUJUKVDVKOVIVCGVBEEHSULUMABCDEFGVAVMVNKUNVEVJVGDFVEVD
        VFTNVJVGQVLDUOGVFETHSUPUQURUS $.
    $}

    $( A lower bound is not greater than the infimum.  (Contributed by AV,
       3-Sep-2020.) $)
    infnlb $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 infglb expdimp dfrex2 imbitrdi expimpd
      con2d ) AGEKZDLGHMZNDFOZFEHPGHMZNAUEQZUHUGUIUHUFDFRZUGNAUEUHUJABCDEFGHIJS
      TUFDFUAUBUDUC $.
  $}

  ${
    infex.1 $e |- R Or A $.
    $( An infimum is a set.  (Contributed by AV, 3-Sep-2020.) $)
    infex $p |- inf ( B , A , R ) e. _V $=
      ( wor cinf cvv wcel id infexd ax-mp ) ACEZBACFGHDLABCLIJK $.
  $}

  ${
    $d A y z $.  $d B y z $.  $d C y z $.  $d R y z $.  $d y z ph $.
    infmin.1 $e |- ( ph -> R Or A ) $.
    infmin.2 $e |- ( ph -> C e. A ) $.
    infmin.3 $e |- ( ph -> C e. B ) $.
    infmin.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 AV, 3-Sep-2020.) $)
    infmin $p |- ( ph -> inf ( B , A , R ) = C ) $=
      ( vz wcel cv wbr wa wrex simprr breq1 rspcev syl2an2r eqinfd ) ABKCDEFGHJ
      AEDLBMZCLZEUBFNZOUDKMZUBFNZKDPIAUCUDQUFUDKEDUEEUBFRSTUA $.
  $}

  ${
    $d x y z w A $.  $d x y z w R $.  $d x y z w B $.  $d w ph $.
    infmo.1 $e |- ( ph -> R Or A ) $.
    $( Any class ` B ` has at most one infimum in ` A ` (where ` R ` is
       interpreted as 'less than').  (Contributed by AV, 6-Oct-2020.) $)
    infmo $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 ) ) ) $=
      ( vw cv wbr wn wral wrex wi wa weq breq2 breq1 biimtrid wrmo wcel anbi2ci
      ancom an42 3bitr4i ralnex rexbidv imbi12d rspcva cbvrexvw imbitrrdi con3d
      wor expimpd ad2antrl ad2antll anim12d imp sotrieq2 sylibrd ralrimivva syl
      ancomd ex notbid ralbidv imbi1d anbi12d rmo4 sylibr ) ACJZBJZGKZLZCFMZVMV
      LGKZDJZVLGKZDFNZOZCEMZPZVLIJZGKZLZCFMZWDVLGKZVTOZCEMZPZPZBIQZOZIEMBEMZWCB
      EUAAEGUNZWOHWPWNBIEEWPVMEUBZWDEUBZPPZWLVMWDGKZLZWDVMGKZLZPZWMWLWJVPPZWBWG
      PZPZWSXDVPWGPZWJWBPZPXIWGVPPZPWLXGXHXJXIVPWGUDUCVPWBWGWJUEWJVPWBWGUEUFWSX
      GXDWSXGPXCXAWSXGXCXAPWSXEXCXFXAWQXEXCOWPWRWQWJVPXCVPVNCFNZLWQWJPZXCVNCFUG
      XLXBXKXLXBVRVMGKZDFNZXKWIXBXNOCVMECBQZWHXBVTXNVLVMWDGRXOVSXMDFVLVMVRGRUHU
      IUJVNXMCDFVLVRVMGSUKULUMTUOUPWRXFXAOWPWQWRWBWGXAWGWECFNZLWRWBPZXAWECFUGXQ
      WTXPXQWTVRWDGKZDFNZXPWAWTXSOCWDECIQZVQWTVTXSVLWDVMGRXTVSXRDFVLWDVRGRUHUIU
      JWEXRCDFVLVRWDGSUKULUMTUOUQURUSVDVETEVMWDGUTVAVBVCWCWKBIEWMVPWGWBWJWMVOWF
      CFWMVNWEVMWDVLGRVFVGWMWAWICEWMVQWHVTVMWDVLGSVHVGVIVJVK $.

    infeu.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 AV, 6-Oct-2020.) $)
    infeu $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 infmo reu5 sylanbrc ) ACJZBJZGKLCFM
      UCUBGKDJUBGKDFNOCEMPZBENUDBEQUDBERIABCDEFGHSUDBETUA $.
  $}

  ${
    $d R x y z $.  $d A x y z $.
    $( A finite set has a minimum under a total order.  (Contributed by AV,
       6-Oct-2020.) $)
    fimin2g $p |- ( ( R Or A /\ A e. Fin /\ A =/= (/) ) ->
                                        E. x e. A A. y e. A -. y R x ) $=
      ( wor cfn wcel c0 wne w3a wa wfr cv wbr wn wral wrex 3simpc wpo syl2anc
      sopo 3ad2ant1 simp2 frfi wss ssid fri mpanr1 an32s ) CDEZCFGZCHIZJZUKULKC
      DLZBMAMDNOBCPACQZUJUKULRUMCDSZUKUNUJUKUPULCDUAUBUJUKULUCCDUDTUKUNULUOUKUN
      KCCUEULUOCUFABCCFDUGUHUIT $.

    $( A finite set has a minimum under a total order.  (Contributed by AV,
       6-Oct-2020.) $)
    fiming $p |- ( ( R Or A /\ A e. Fin /\ A =/= (/) ) ->
                            E. x e. A A. y e. A ( x =/= y -> x R y ) ) $=
      ( wor cfn wcel c0 wne w3a cv wbr wi wral wrex wn fimin2g wb wa weq imbi1i
      wo nesym pm4.64 sotric ancom2s con2bid anassrs ralbidva rexbidva 3ad2ant1
      bitri bitrid mpbird ) CDEZCFGZCHIZJAKZBKZIZURUSDLZMZBCNZACOZUSURDLZPZBCNZ
      ACOZABCDQUOUPVDVHRUQUOVCVGACUOURCGZSVBVFBCUOVIUSCGZVBVFRVBBATZVAUBZUOVIVJ
      SSZVFVBVKPZVAMVLUTVNVAURUSUCUAVKVAUDULVMVEVLUOVJVIVEVLPRCUSURDUEUFUGUMUHU
      IUJUKUN $.

    $( Lemma showing existence and closure of infimum of a finite set.
       (Contributed by AV, 6-Oct-2020.) $)
    fiinfg $p |- ( ( R Or A /\ A e. Fin /\ A =/= (/) ) -> E. x e. A ( A. y e. A
            -. y R x /\ A. y e. A ( x R y -> E. z e. A z R y ) ) ) $=
      ( wor cfn wcel c0 wne cv wbr wi wral wrex wn wa weq ex anassrs w3a fiming
      equcom sotrieq2 ancom2s bitrid simprbda a1dd pm2.27 soasym syl9r ralimdva
      wb pm2.61dne breq1 rspcev ralrimivw adantl jctird reximdva 3ad2ant1 mpd )
      DEFZDGHZDIJZUAAKZBKZJZVFVGELZMZBDNZADOZVGVFELPZBDNZVICKZVGELZCDOZMZBDNZQZ
      ADOZABDEUBVCVDVLWAMVEVCVKVTADVCVFDHZQZVKVNVSWCVJVMBDWCVGDHZQZVJVMMVFVGWEA
      BRZVMVJVCWBWDWFVMMVCWBWDQQZWFVMWGWFVMVIPZWFBARZWGVMWHQZABUCVCWDWBWIWJUMDV
      GVFEUDUEUFUGSTUHVHVJVIWEVMVHVIUIVCWBWDVIVMMDEVFVGUJTUKUNULWBVSVCWBVRBDWBV
      IVQVPVICVFDVOVFVGEUOUPSUQURUSUTVAVB $.
  $}

  ${
    $d R x y z $.  $d B x y z $.
    $( A finite set satisfies the conditions to have an infimum.  (Contributed
       by AV, 6-Oct-2020.) $)
    fiinf2g $p |- ( ( R Or A /\ ( B e. Fin /\ B =/= (/) /\ B C_ A ) ) ->
      E. x e. B ( A. y e. B -. y R x /\
        A. y e. A ( x R y -> E. z e. B z R y ) ) ) $=
      ( wor cfn wcel c0 wne wss w3a wa cv wbr wn wral wrex wi wreu simp1 fiinfg
      soss infeu 3exp syl6 com4l 3imp2 reurex breq1 rspcev ralrimivw a1d anim2d
      ex reximia 3syl ) DFGZEHIZEJKZEDLZMNBOZAOZFPQBERZVDVCFPZCOZVCFPZCESZTZBER
      ZNZAEUAZVLAESVEVJBDRZNZAESUSUTVAVBVMVBUSUTVAVMVBUSEFGZUTVAVMTTEDFUDVPUTVA
      VMVPUTVAMABCEEFVPUTVAUBABCEFUCUEUFUGUHUIVLAEUJVLVOAEVDEIZVKVNVEVQVNVKVQVJ
      BDVQVFVIVHVFCVDEVGVDVCFUKULUPUMUNUOUQUR $.
  $}

  $( A nonempty finite set contains its infimum.  (Contributed by AV,
     3-Sep-2020.) $)
  fiinfcl $p |- ( ( R Or A /\ ( B e. Fin /\ B =/= (/) /\ B C_ A ) )
                                    -> inf ( B , A , R ) e. B ) $=
    ( wor cfn wcel c0 wne wss w3a wa cinf ccnv csup df-inf cnvso fisupcl sylanb
    eqeltrid ) ACDZBEFBGHBAIJZKBACLBACMZNZBBACOTAUBDUAUCBFACPABUBQRS $.

  ${
    infltoreq.1 $e |- ( ph -> R Or A ) $.
    infltoreq.2 $e |- ( ph -> B C_ A ) $.
    infltoreq.3 $e |- ( ph -> B e. Fin ) $.
    infltoreq.4 $e |- ( ph -> C e. B ) $.
    infltoreq.5 $e |- ( ph -> S = inf ( B , A , R ) ) $.
    $( The infimum of a finite set is less than or equal to all the elements of
       the set.  (Contributed by AV, 4-Sep-2020.) $)
    infltoreq $p |- ( ph -> ( S R C \/ C = S ) ) $=
      ( wbr wceq wo ccnv wor cnvso sylib cinf wcel csup df-inf eqtrdi supgtoreq
      wb cfn wne wss ne0d fiinfcl syl13anc eqeltrd brcnvg bicomd syl2anc orbi1d
      c0 wa mpbird ) AFDELZDFMZNDFEOZLZVANABCDVBFABEPZBVBPGBEQRHIJAFCBESZCBVBUA
      KCBEUBUCUDAUTVCVAADCTZFCTZUTVCUEJAFVECKAVDCUFTCUQUGCBUHVECTGIACDJUIHBCEUJ
      UKULVFVGURVCUTDFCCEUMUNUOUPUS $.
  $}

  ${
    $d A y $.  $d B y $.  $d C y $.  $d R y $.
    $( The infimum of a pair.  (Contributed by AV, 4-Sep-2020.) $)
    infpr $p |- ( ( R Or A /\ B e. A /\ C e. A ) -> inf ( { B , C } , A , R ) =
                  if ( B R C , B , C ) ) $=
      ( vy wor wcel w3a cpr wbr 3adant1 wn wceq breq2 notbid adantr ifbothda wa
      sonr breq1 cif simp1 ifcl ifpr cv wral 3adant3 simpr wi so2nr 3impb imnan
      sylibr imp 3adant2 wb ralprg mpbir2and r19.21bi infmin ) ADFZBAGZCAGZHZEA
      BCIZBCDJZBCUAZDVAVBVCUBVBVCVGAGVAVFBCAUCKVBVCVGVEGVAVFBCAAUDKVDEUEZVGDJZL
      ZEVEVDVJEVEUFZBVGDJZLZCVGDJZLZVFBBDJZLZVFLZVMVDBCBVGMZVPVLBVGBDNOCVGMZVFV
      LCVGBDNOVDVQVFVAVBVQVCABDSUGPVDVRUHQVFCBDJZLZCCDJZLZVOVDBCVSWAVNBVGCDNOVT
      WCVNCVGCDNOVDVFWBVDVFWARLZVFWBUIVAVBVCWEABCDUJUKVFWAULUMUNVDWDVRVAVCWDVBA
      CDSUOPQVBVCVKVMVORUPVAVJVMVOEBCAAVHBMVIVLVHBVGDTOVHCMVIVNVHCVGDTOUQKURUSU
      T $.
  $}

  $( The infimum of a proper pair is less than the supremum of this pair.
     (Contributed by AV, 13-Mar-2023.) $)
  infsupprpr $p |- ( ( R Or A /\ ( B e. A /\ C e. A /\ B =/= C ) )
             -> inf ( { B , C } , A , R ) R sup ( { B , C } , A , R ) ) $=
    ( wcel w3a wa wbr cif wceq 3adantr3 wi iftrue adantr wo wn wb sotric mpcom
    ex wor wne cpr cinf csup w3o solin biimpac ioran iffalse breqtrrd simplbiim
    simprl eqbrtrd 2a1d 3impd adantld pm3.22 3adant3 biimpd sylan2 impcom simpr
    eqneqall breqan12d mpbird a1d expimpd 3jaoi infpr suppr breq12d 3adant3r3 )
    ADUAZBAEZCAEZBCUBZFZGZBCUCZADUDZVTADUEZDHZBCDHZBCIZCBDHZBCIZDHZWDBCJZWFUFZV
    SWHVNVOVPWJVQABCDUGKWDVSWHLWIWFWDVSWHWDVSGZWEBWGDWDWEBJVSWDBCMNWIWFOPZWKBWG
    DHZVSWDWLVNVOVPWDWLQVQABCDRKUHWLWIPWFPZWKWMLWIWFUIWNWKWMWNWKGBCWGDWNWDVSUMW
    NWGCJWKWFBCUJNUKTULSUNTWIVRWHVNWIVOVPVQWHWIVQWHLVOVPWHBCVDUOUPUQWFVSWHCBJZW
    DOPZWFVSGZWHVSWFWPVRVNVPVOGZWFWPLVOVPWRVQVOVPURUSVNWRGWFWPACBDRUTVAVBWPWOPW
    DPZWQWHLWOWDUIWSWFVSWHWSWFGZWHVSWTWHWFWSWFVCWSWFWECWGBDWDBCUJWFBCMVEVFVGVHU
    LSTVISVNVOVPWCWHQVQVNVOVPFWAWEWBWGDABCDVJABCDVKVLVMVF $.

  $( The infimum of a singleton.  (Contributed by NM, 2-Oct-2007.) $)
  infsn $p |- ( ( R Or A /\ B e. A ) -> inf ( { B } , A , R ) = B ) $=
    ( wor wcel wa csn cinf wbr cif cpr dfsn2 infeq1i wceq infpr 3anidm23 eqtrid
    ifid eqtrdi ) ACDZBAEZFZBGZACHZBBCIZBBJZBUBUDBBKZACHZUFAUCUGCBLMTUAUHUFNABB
    COPQUEBRS $.

  $( 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 x y $.  $d R x y $.  $d X x y $.
    $( The infimum of an empty set under a base set which has a unique greatest
       element is the greatest element of the base set.  (Contributed by AV,
       4-Sep-2020.) $)
    infempty $p |- ( ( R Or A /\ ( X e. A /\ A. y e. A -. X R y )
                       /\ E! x e. A A. y e. A -. x R y )
                     -> inf ( (/) , A , R ) = X ) $=
      ( wor wcel cv wbr wn wral wa wreu c0 brcnvg ancoms bicomd notbid ralbidva
      wb cinf ccnv csup df-inf wceq cnvso pm5.32i reubiia sup0 syl3anb eqtrid
      w3a ) CDFZECGZEBHZDIZJZBCKZLZAHZUODIZJZBCKZACMZULNCDUANCDUBZUCZENCDUDUMCV
      EFUSUNUOEVEIZJZBCKZLVDUOUTVEIZJZBCKZACMVFEUECDUFUNURVIUNUQVHBCUNUOCGZLZUP
      VGVNVGUPVMUNVGUPTUOECCDOPQRSUGVCVLACUTCGZVBVKBCVOVMLZVAVJVPVJVAVMVOVJVATU
      OUTCCDOPQRSUHABCVEEUIUJUK $.
  $}

  ${
    $d A x y z $.  $d B x y z $.  $d C x y z $.  $d F x y z $.  $d R x y z $.
    $d S x y z $.  $d ph x y z $.
    infiso.1 $e |- ( ph -> F Isom R , S ( A , B ) ) $.
    infiso.2 $e |- ( ph -> C C_ A ) $.
    infiso.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 ) ) ) $.
    infiso.4 $e |- ( ph -> R Or A ) $.
    $( Image of an infimum under an isomorphism.  (Contributed by AV,
       4-Sep-2020.) $)
    infiso $p |- ( ph ->
                   inf ( ( F " C ) , B , S ) = ( F ` inf ( C , A , R ) ) ) $=
      ( ccnv csup cfv cinf wiso sylib cima isocnv2 infcllem cnvso supiso df-inf
      wor fveq2i 3eqtr4g ) AJGUAZFIOZPGEHOZPZJQUJFIRGEHRZJQABCDEFGULUKJAEFHIJSE
      FULUKJSKEFHIJUBTLABCDEGHNMUCAEHUGEULUGNEHUDTUEUJFIUFUNUMJGEHUFUHUI $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Ordinal isomorphism, Hartogs's theorem
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c OrdIso $.

  $( Extend class definition to include the canonical order isomorphism to an
     ordinal. $)
  coi $a class OrdIso ( R , A ) $.

  ${
    $d h j t u v w x z A $.  $d h j t u v w x z R $.
    $( Define the canonical order isomorphism from the well-order ` R ` on
       ` A ` to an ordinal.  (Contributed by Mario Carneiro, 23-May-2015.) $)
    df-oi $a |- OrdIso ( R , A ) = if ( ( R We A /\ R Se A ) ,
      ( recs ( ( h e. _V |->
      ( iota_ v e. { w e. A | A. j e. ran h j R w }
           A. u e. { w e. A | A. j e. ran h j R w } -. u R v ) ) ) |`
       { x e. On | E. t e. A A. z e. ( recs ( ( h e. _V |->
      ( iota_ v e. { w e. A | A. j e. ran h j R w }
           A. u e. { w e. A | A. j e. ran h j R w } -. u R v ) ) ) " x )
      z R t } ) , (/) ) $.
  $}

  ${
    $d h j t u v w x z A $.  $d u v C $.  $d z F $.  $d h j t u v w x z R $.
    dfoi.1 $e |- C = { w e. A | A. j e. ran h j R w } $.
    dfoi.2 $e |- G = ( h e. _V |-> ( iota_ v e. C A. u e. C -. u R v ) ) $.
    dfoi.3 $e |- F = recs ( G ) $.
    $( Rewrite ~ df-oi with abbreviations.  (Contributed by Mario Carneiro,
       24-Jun-2015.) $)
    dfoi $p |- OrdIso ( R , A ) = if ( ( R We A /\ R Se A ) , ( F |`
       { x e. On | E. t e. A A. z e. ( F " x ) z R t } ) , (/) ) $=
      ( cvv cv wral wceq coi wwe wse wa wbr crab crio cmpt crecs cima wrex con0
      wn crn cres c0 cif df-oi wcel a1i riotaeqbidv mpteq2ia eqtri recseq ax-mp
      raleqdv imaeq1i raleqi rexbii rabbii reseq12i ifeq1 eqtr4i ) GIUAGIUBGIUC
      UDZJQERDRIUEUMZEKRCRIUEKJRZUNSCGUFZSZDVQUGZUHZUIZBRFRIUEZBWAARZUJZSZFGUKZ
      AULUFZUOZUPUQZVNLWBBLWCUJZSZFGUKZAULUFZUOZUPUQZABCDEFGIJKURWNWHTWOWITLWAW
      MWGLMUIZWAPMVTTWPWATMJQVOEHSZDHUGZUHVTOJQWRVSVPQUSZWQVRDHVQHVQTWSNUTZWSVO
      EHVQWTVFVAVBVCMVTVDVEVCZWLWFAULWKWEFGWBBWJWDLWAWCXAVGVHVIVJVKVNWNWHUPVLVE
      VM $.
  $}

  ${
    $d h j t u v w x z A $.  $d h j t u v w x z B $.  $d h j t u v w x z R $.
    $d h j t u v w x z S $.
    $( Equality theorem for ordinal isomorphism.  (Contributed by Mario
       Carneiro, 23-May-2015.) $)
    oieq1 $p |- ( R = S -> OrdIso ( R , A ) = OrdIso ( S , A ) ) $=
      ( vh vu vv vj vw vz vt vx wceq cvv cv wbr wral crab con0 c0 breq wwe crio
      wse wa wn crn cmpt crecs cima wrex cres cif weeq1 anbi12d ralbidv rabbidv
      coi notbid raleqbidv riotaeqbidv mpteq2dv recseq imaeq1d rexbidv reseq12d
      seeq1 syl ifbieq1d df-oi 3eqtr4g ) BCLZABUAZABUCZUDZDMENZFNZBOZUEZEGNZHNZ
      BOZGDNUFZPZHAQZPZFWDUBZUGZUHZINZJNZBOZIWHKNZUIZPZJAUJZKRQZUKZSULACUAZACUC
      ZUDZDMVOVPCOZUEZEVSVTCOZGWBPZHAQZPZFXEUBZUGZUHZWIWJCOZIXIWLUIZPZJAUJZKRQZ
      UKZSULABUQACUQVKVNWTWQXOSVKVLWRVMWSABCUMABCVFUNVKWHXIWPXNVKWGXHLWHXILVKDM
      WFXGVKWEXFFWDXEVKWCXDHAVKWAXCGWBVSVTBCTUOUPZVKVRXBEWDXEXPVKVQXAVOVPBCTURU
      SUTVAWGXHVBVGZVKWOXMKRVKWNXLJAVKWKXJIWMXKVKWHXIWLXQVCWIWJBCTUSVDUPVEVHKIH
      FEJABDGVIKIHFEJACDGVIVJ $.

    $( Equality theorem for ordinal isomorphism.  (Contributed by Mario
       Carneiro, 23-May-2015.) $)
    oieq2 $p |- ( A = B -> OrdIso ( R , A ) = OrdIso ( R , B ) ) $=
      ( vh vu vv vj vw vz vt vx wceq wwe cvv cv wbr wral crab con0 c0 wse wa wn
      crn crio cmpt crecs cima wrex cif weeq2 seeq2 anbi12d raleqdv riotaeqbidv
      cres coi mpteq2dv recseq syl imaeq1d rexeqbi1dv rabbidv reseq12d ifbieq1d
      rabeq df-oi 3eqtr4g ) ABLZACMZACUAZUBZDNEOFOCPUCZEGOHOCPGDOUDQZHARZQZFVOU
      EZUFZUGZIOJOCPZIVSKOZUHZQZJAUIZKSRZUPZTUJBCMZBCUAZUBZDNVMEVNHBRZQZFWJUEZU
      FZUGZVTIWNWAUHZQZJBUIZKSRZUPZTUJACUQBCUQVIVLWIWFWSTVIVJWGVKWHABCUKABCULUM
      VIVSWNWEWRVIVRWMLVSWNLVIDNVQWLVIVPWKFVOWJVNHABVFZVIVMEVOWJWTUNUOURVRWMUSU
      TZVIWDWQKSWCWPJABVIVTIWBWOVIVSWNWAXAVAUNVBVCVDVEKIHFEJACDGVGKIHFEJBCDGVGV
      H $.
  $}

  ${
    $d a h j t u v w x z $.  $d a h j t u v w z A $.  $d a h j t u v w z R $.
    nfoi.1 $e |- F/_ x R $.
    nfoi.2 $e |- F/_ x A $.
    $( Hypothesis builder for ordinal isomorphism.  (Contributed by Mario
       Carneiro, 23-May-2015.)  (Revised by Mario Carneiro, 15-Oct-2016.) $)
    nfoi $p |- F/_ x OrdIso ( R , A ) $=
      ( vh vu vv vj vw vz vt va cvv cv wbr wral con0 c0 nfcv coi wwe wse wa crn
      wn crab crio cmpt crecs cima wrex cres df-oi nfwe nfse nfan nfralw nfrabw
      cif nfbr nfn nfriota nfmpt nfrecs nfima nfrexw nfres nfif nfcxfr ) ABCUAB
      CUBZBCUCZUDZFNGOZHOZCPZUFZGIOZJOZCPZIFOUEZQZJBUGZQZHWCUHZUIZUJZKOZLOZCPZK
      WGMOZUKZQZLBULZMRUGZUMZSUTMKJHGLBCFIUNVMAWPSVKVLAABCDEUOABCDEUPUQAWGWOAWF
      AFNWEANTWDAHWCVQAGWCWBAJBVTAIWAAWATAVRVSCAVRTDAVSTVAUREUSZVPAAVNVOCAVNTDA
      VOTVAVBURWQVCVDVEZWNAMRWMALBEWJAKWLAWGWKWRAWKTVFAWHWICAWHTDAWITVAURVGARTU
      SVHASTVIVJ $.
  $}

  ${
    $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 vw cep cv cfv wceq wral wcel wi fveq2 id eqeq12d wa wb syl wbr vz
      wiso word w3a con0 ordsson 3ad2ant2 eleq1w imbi12d imbi2d r19.21v ordelss
      sseld 3ad2antl2 sselda pm5.5 ralbidva ccnv wf1o 3ad2ant1 ad2antrr simpll3
      wss isof1o simpr f1of simplrl ffvelcdmd jca ordtr1 sylc f1ocnvfv2 syl2anc
      wf eqeltrd simpll1 f1ocnv 3syl isorel syl12anc epel epeli 3bitr3g simplrr
      fvex mpbird rspcv eqtr3d simprr rspccva sylan biimpri adantl simpl2 mpbid
      simprl sylib eqeltrrd impbida eqrdv expr sylbid ex com23 a2i a1i biimtrid
      tfis2 com3l mpdd ralrimiv adantll ffvelcdmda 3ad2antl1 adantlr crn simpl1
      wrex wfo f1ofo forn eleq2d f1ofn adantr fvelrnb bitr3d simpl simplr exp43
      wfn syldd imp rexlimdv impbid mpdan ) ABGGCUBZAUCZBUCZUDZDHZCIZYTJZDAKZAB
      JYSUUBDAYSYTALZYTUELZUUBYSAUEYTYQYPAUEVCYRAUFUGUMUUEYSUUDUUBYSUUDUUBMZMZY
      SEHZALZUUHCIZUUHJZMZMZDEYTUUHJZUUFUULYSUUNUUDUUIUUBUUKDEAUHUUNUUAUUJYTUUH
      YTUUHCNUUNOPUIUJUUMEYTKYSUULEYTKZMZUUEUUGYSUULEYTUKUUPUUGMUUEYSUUOUUFYSUU
      DUUOUUBYSUUDUUOUUBMYSUUDQZUUOUUKEYTKZUUBUUQUULUUKEYTUUQUUHYTLQUUIUULUUKRU
      UQYTAUUHYQYPUUDYTAVCZYRAYTULZUNUOUUIUUKUPSUQYSUUDUURUUBYSUUDUURQZQZUAUUAY
      TUVBUAHZUUALZUVCYTLZUVBUVDQZUVCUVCCURZIZYTUVFUVHCIZUVCUVHUVFABCUSZUVCBLZU
      VIUVCJYSUVJUVAUVDYPYQUVJYRABGGCVDZUTVAZUVFYRUVDUUABLZQUVKYPYQYRUVAUVDVBUV
      FUVDUVNUVBUVDVEZUVFABYTCYSABCVNZUVAUVDYPYQUVPYRYPUVJUVPUVLABCVFSZUTVAYSUU
      DUURUVDVGZVHVIUVCUUABVJVKZABUVCCVLVMZUVFUVHYTLZUURUVIUVHJZUVFUWAUVIUUALZU
      VFUVIUVCUUAUVTUVOVOUVFUVHYTGTZUVIUUAGTZUWAUWCUVFYPUVHALUUDUWDUWERYPYQYRUV
      AUVDVPUVFBAUVCUVGUVFUVJBAUVGUSBAUVGVNUVMABCVQBAUVGVFVRUVSVHUVRABUVHYTGGCV
      SVTDUVHWAUVIUUAYTCWEZWBWCWFZYSUUDUURUVDWDUUKUWBEUVHYTUUHUVHJZUUJUVIUUHUVH
      UUHUVHCNUWHOPWGVKWHUWGVOUVBUVEQZUVCCIZUVCUUAUVBUURUVEUWJUVCJZYSUUDUURWIUU
      KUWKEUVCYTUUHUVCJZUUJUWJUUHUVCUUHUVCCNUWLOPWJWKUWIUWJUUAGTZUWJUUALUWIUVCY
      TGTZUWMUVEUWNUVBUWNUVEDUVCWAWLWMUWIYPUVCALZUUDUWNUWMRYPYQYRUVAUVEVPUVBYTA
      UVCUVBYQUUDUUSYPYQYRUVAWNYSUUDUURWPUUTVMUOYSUUDUURUVEVGABUVCYTGGCVSVTWOUW
      JUUAUWFWBWQWRWSWTXAXBXCXDXEXFXGXHXIXJXKYSUUCQZUAABUWPUWOUVKUWPUWOUVKUWPUW
      OQUWJUVCBUUCUWOUWKYSUUBUWKDUVCAYTUVCJZUUAUWJYTUVCYTUVCCNUWQOPWJXLYSUWOUWJ
      BLZUUCYPYQUWOUWRYRYPABUVCCUVQXMXNXOWRXCUWPUVKFHZCIZUVCJZFAXRZUWOUWPUVCCXP
      ZLZUVKUXBUWPUXCBUVCUWPYPUXCBJZYPYQYRUUCXQYPUVJABCXSUXEUVLABCXTABCYAVRSYBU
      WPCAYJZUXDUXBRYSUXFUUCYPYQUXFYRYPUVJUXFUVLABCYCSUTYDFAUVCCYESYFUWPUXAUWOF
      AYSUUCUWSALZUXAUWOMZMYSUXGUUCUXHYSUXGUUCUWTUWSJZUXHUXGUUCUXIMMYSUUBUXIDUW
      SAYTUWSJZUUAUWTYTUWSYTUWSCNUXJOPWGXFYSUXGUXIUXAUWOYSUXGQZUXIUXAQZQUVCUWSA
      UXLUVCUWSJUXKUXLUWTUVCUWSUXIUXAVEUXIUXAYGWHWMYSUXGUXLYHVOYIYKXDYLYMXBYNWT
      YO $.

    $( 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 $.
  $}

  ${
    $d f r s u v C $.  $d a h j t u v w x z M $.  $d a b j u w N $.
    $d a b c f h i j m r s t u v w x y z R $.  $d a b c T $.
    $d a b c h j m r s t u v w x y z A $.  $d a b c m t u v x O $.
    $d a b m t x ph $.
    ordtypelem.1 $e |- F = recs ( G ) $.
    ordtypelem.2 $e |- C = { w e. A | A. j e. ran h j R w } $.
    ordtypelem.3 $e |- G =
      ( h e. _V |-> ( iota_ v e. C A. u e. C -. u R v ) ) $.
    $( Lemma for ~ ordtype .  (Contributed by Mario Carneiro, 26-Jun-2015.) $)
    ordtypecbv $p |- recs ( ( f e. _V |->
      ( iota_ s e. { y e. A | A. i e. ran f i R y }
           A. r e. { y e. A | A. i e. ran f i R y } -. r R s ) ) ) = F $=
      ( cv wral crecs cvv wbr crn crab crio cmpt wceq weq breq1 notbid cbvralvw
      breq2 ralbidv bitrid cbvriotavw cbvrabv eqtri rneq raleqdv rabbidv eqtrid
      wn riotaeqbidv cbvmptv recseq ax-mp eqtr2i ) LMUAZHUBOSZNSZGUCZVCZOJSZASZ
      GUCZJHSZUDZTZAEUEZTZNVTUFZUGZUAZPMWCUHVIWDUHMIUBDSZCSZGUCZVCZDFTZCFUFZUGW
      CRIHUBWJWBIHUIZWJVMOFTZNFUFWBWIWLCNFWIVJWFGUCZVCZOFTCNUIZWLWHWNDOFDOUIWGW
      MWEVJWFGUJUKULWOWNVMOFWOWMVLWFVKVJGUMUKUNUOUPWKWLWANFVTWKFVPJISZUDZTZAEUE
      ZVTFKSZBSZGUCZKWQTZBEUEWSQXCWRBAEXCVNXAGUCZJWQTBAUIZWRXBXDKJWQWTVNXAGUJUL
      XEXDVPJWQXAVOVNGUMUNUOUQURWKWRVSAEWKVPJWQVRWPVQUSUTVAVBZWKVMOFVTXFUTVDVBV
      EURMWCVFVGVH $.

    $d a b c h j t u v w x z F $.
    ordtypelem.5 $e |- T = { x e. On | E. t e. A A. z e. ( F " x ) z R t } $.
    ordtypelem.6 $e |- O = OrdIso ( R , A ) $.
    ordtypelem.7 $e |- ( ph -> R We A ) $.
    ordtypelem.8 $e |- ( ph -> R Se A ) $.
    $( Lemma for ~ ordtype .  (Contributed by Mario Carneiro, 24-Jun-2015.) $)
    ordtypelem1 $p |- ( ph -> O = ( F |` T ) ) $=
      ( wwe wse wa cv wbr cima wral wrex con0 crab cres cif wceq iftrue syl2anc
      c0 coi dfoi eqtri reseq2i 3eqtr4g ) AHJUDZHJUEZUFZNCUGGUGJUHCNBUGUIUJGHUK
      BULUMZUNZUSUOZVIPNKUNAVEVFVJVIUPUBUCVGVIUSUQURPHJUTVJUABCDEFGHIJLMNORSQVA
      VBKVHNTVCVD $.

    $( Lemma for ~ ordtype .  (Contributed by Mario Carneiro, 24-Jun-2015.) $)
    ordtypelem2 $p |- ( ph -> Ord T ) $=
      ( va wtr word cv wss wral wcel wa wbr cima wrex con0 crab ssrab3 a1i onss
      sselda syl eloni weq imaeq2 raleqdv rexbidv elrab2 simprbi adantl ordelss
      wi imass2 ssralv reximdv 3syl ralrimdva sylc sylanbrc sseqtrrdi ralrimiva
      ssrab dftr3 sylibr ordon trssord mp3an23 ) AKUEZKUFZAUDUGZKUHZUDKUIWGAWJU
      DKAWIKUJZUKZWICUGGUGJULZCNBUGZUMZUIZGHUNZBUOUPZKWLWIUOUHZWQBWIUIZWIWRUHWL
      WIUOUJZWSAKUOWIKUOUHZAWQBUOKTUQZURUTZWIUSVAWLWIUFZWMCNWIUMZUIZGHUNZWTWLXA
      XEXDWIVBVAWKXHAWKXAXHWQXHBWIUOKBUDVCZWPXGGHXIWMCWOXFWNWINVDVEVFTVGVHVIXEX
      HWQBWIXEWNWIUJUKWNWIUHWOXFUHZXHWQVKWIWNVJWNWINVLXJXGWPGHWMCWOXFVMVNVOVPVQ
      WQBUOWIWAVRTVSVTUDKWBWCWGXBUOUFWHXCWDKUOWEWFVA $.

    $( Lemma for ~ ordtype .  (Contributed by Mario Carneiro, 24-Jun-2015.) $)
    ordtypelem3 $p |- ( ( ph /\ M e. ( T i^i dom F ) ) ->
        ( F ` M ) e. { v e. { w e. A | A. j e. ( F " M ) j R w } |
          A. u e. { w e. A | A. j e. ( F " M ) j R w } -. u R v } ) $=
      ( cdm cin wcel wa cfv cv wbr wn cima wral crab crio cres wceq simpr tfr2a
      elin2d syl cvv word wb wlim wfun tfr1a simpri limord ax-mp ordelord tfr2b
      sylancr mpbid crn rneq df-ima eqtr4di raleqdv rabbidv riotaeqbidv riotaex
      eqtrid fvmpt eqtrd wreu wwe wse wss c0 wne adantr ssrab2 wrex elin1d con0
      imaeq2 rexbidv elrab2 simprbi breq1 cbvralvw breq2 bitrid cbvrexvw sylibr
      a1i ralbidv rabn0 wereu2 syl22anc riotacl2 eqeltrd ) APKNUEZUFUGZUHZPNUIZ
      FUJEUJJUKULZFMUJZDUJZJUKZMNPUMZUNZDHUOZUNZEYEUPZYFEYEUOZXQXRNPUQZOUIZYGXQ
      PXOUGZXRYJURXQKXOPAXPUSZVAZPNORUTVBXQYIVCUGZYJYGURXQYKYNYMXQPVDZYKYNVEXQX
      OVDZYKYOXOVFZYPNVGYQNORVHVIXOVJVKYMXOPVLVNPNORVMVBVOLYIXSFIUNZEIUPYGVCOLU
      JZYIURZYRYFEIYEYTIYBMYSVPZUNZDHUOYESYTUUBYDDHYTYBMUUAYCYTUUAYIVPYCYSYIVQN
      PVRVSVTWAWDZYTXSFIYEUUCVTWBTYFEYEWCWEVBWFXQYFEYEWGZYGYHUGXQHJWHZHJWIZYEHW
      JZYEWKWLZUUDAUUEXPUCWMAUUFXPUDWMUUGXQYDDHWNXHXQYDDHWOZUUHXQCUJZGUJZJUKZCY
      CUNZGHWOZUUIXQPKUGZUUNXQKXOPYLWPUUOPWQUGUUNUULCNBUJZUMZUNZGHWOUUNBPWQKUUP
      PURZUURUUMGHUUSUULCUUQYCUUPPNWRVTWSUAWTXAVBYDUUMDGHYDUUJYAJUKZCYCUNYAUUKU
      RZUUMYBUUTMCYCXTUUJYAJXBXCUVAUUTUULCYCYAUUKUUJJXDXIXEXFXGYDDHXJXGEFHYEJXK
      XLYFEYEXMVBXN $.

    $( Lemma for ~ ordtype .  (Contributed by Mario Carneiro, 24-Jun-2015.) $)
    ordtypelem4 $p |- ( ph -> O : ( T i^i dom F ) --> A ) $=
      ( va cdm cin wf cres wfn cfv wcel wral wfun wlim tfr1a simpli funres mp1i
      cv funfnd dmres fneq2i sylib wa simpr elin1d fvresd wbr cima ssrab2 sstri
      wn ordtypelem3 sselid eqeltrd ralrimiva ffnfv sylanbrc ordtypelem1 mpbird
      crab feq1d ) AKNUEZUFZHPUGWDHNKUHZUGZAWEWDUIZUDUSZWEUJZHUKZUDWDULWFAWEWEU
      EZUIWGAWENUMZWEUMAWLWCUNNOQUOUPKNUQURUTWKWDWENKVAVBVCAWJUDWDAWHWDUKZVDZWI
      WHNUJZHWNWHKNWNKWCWHAWMVEVFVGWNFUSEUSJVHVLFMUSDUSJVHMNWHVIULZDHWAZULZEWQW
      AZHWOWSWQHWREWQVJWPDHVJVKABCDEFGHIJKLMNOWHPQRSTUAUBUCVMVNVOVPUDWDHWEVQVRA
      WDHPWEABCDEFGHIJKLMNOPQRSTUAUBUCVSWBVT $.

    $( Lemma for ~ ordtype .  (Contributed by Mario Carneiro, 25-Jun-2015.) $)
    ordtypelem5 $p |- ( ph -> ( Ord dom O /\ O : dom O --> A ) ) $=
      ( cdm word wf cin ordtypelem2 wlim wfun tfr1a simpri limord ax-mp sylancl
      ordin wceq wb ordtypelem4 fdmd ordeq syl mpbird ffdmd jca ) APUDZUEZVFHPU
      FAVGKNUDZUGZUEZAKUEVHUEZVJABCDEFGHIJKLMNOPQRSTUAUBUCUHVHUIZVKNUJVLNOQUKUL
      VHUMUNKVHUPUOAVFVIUQVGVJURAVIHPABCDEFGHIJKLMNOPQRSTUAUBUCUSZUTVFVIVAVBVCA
      VIHPVMVDVE $.

    $( Lemma for ~ ordtype .  (Contributed by Mario Carneiro, 24-Jun-2015.) $)
    ordtypelem6 $p |- ( ( ph /\ M e. dom O )
            -> ( N e. M -> ( O ` N ) R ( O ` M ) ) ) $=
      ( va cdm wcel cfv wbr wa cv wceq fveq2 breq1d wral cima crab ssrab2 simpr
      wn cin ordtypelem4 adantr eleqtrd ordtypelem3 syldan sselid breq2 ralbidv
      fdmd elrab simprbi syl wfn wss wb wfun wlim tfr1a simpli mpbi word simpri
      funfn limord ax-mp inss2 sselda ordelss sylancr breq1 ralima mpbid simprr
      eqsstrdi adantrr rspcdva cres ordtypelem1 fveq1d ordtypelem2 inss1 sseldd
      syl2an2r fvresd eqtrd 3brtr4d expr ) APRUGZUHZQPUHZQRUIZPRUIZJUJAXKXLUKZU
      KZQNUIZPNUIZXMXNJXPUFULZNUIZXRJUJZXQXRJUJUFPQXSQUMXTXQXRJXSQNUNUOAXKYAUFP
      UPZXLAXKUKZMULZXRJUJZMNPUQZUPZYBYCXRYDDULZJUJZMYFUPZDHURZUHZYGYCFULEULJUJ
      VAFYKUPZEYKURZYKXRYMEYKUSAXKPKNUGZVBZUHXRYNUHYCPXJYPAXKUTAXJYPUMXKAYPHRAB
      CDEFGHIJKLMNORSTUAUBUCUDUEVCVKZVDVEABCDEFGHIJKLMNOPRSTUAUBUCUDUEVFVGVHYLX
      RHUHYGYJYGDXRHYHXRUMYIYEMYFYHXRYDJVIVJVLVMVNYCNYOVOZPYOVPZYGYBVQNVRZYRYTY
      OVSZNOSVTZWANWEWBYCYOWCZPYOUHYSUUAUUCYTUUAUUBWDYOWFWGAXJYOPAXJYPYOYQKYOWH
      WPWIYOPWJWKYEYAMUFYOPNYDXTXRJWLWMWKWNWQAXKXLWOZWRXPXMQNKWSZUIXQXPQRUUEARU
      UEUMXOABCDEFGHIJKLMNORSTUAUBUCUDUEWTVDZXAXPQKNXPPKQAKWCXOPKUHZPKVPABCDEFG
      HIJKLMNORSTUAUBUCUDUEXBAXKUUGXLAXJKPAXJYPKYQKYOXCWPWIWQZKPWJXEUUDXDXFXGXP
      XNPUUEUIXRXPPRUUEUUFXAXPPKNUUHXFXGXHXI $.

    $( Lemma for ~ ordtype . ` ran O ` is an initial segment of ` A ` under the
       well-order ` R ` .  (Contributed by Mario Carneiro, 25-Jun-2015.) $)
    ordtypelem7 $p |- ( ( ( ph /\ N e. A ) /\ M e. dom O )
            -> ( ( O ` M ) R N \/ N e. ran O ) ) $=
      ( va vb wcel wa cdm crn cfv wbr wn cdif eldif con0 cin ordtypelem4 adantr
      wi wf fdmd inss1 word wss ordtypelem2 ordsson syl sstrid eqsstrd sseld cv
      wceq eleq1 fveq2 breq1d imbi12d imbi2d wral r19.21v wb wlim simpri limord
      wfun tfr1a ax-mp ordin sylancl ordeq mpbird ordelss sylan sselda ralbidva
      pm5.5 eldifn ad2antlr wfn ad2antrr ffnd simprl eleqtrd fnfvelrn syl5ibcom
      syl2anc mtod cima crab breq1 notbid cres ordtypelem1 fveq1d elin1d fvresd
      eqtrd simpll ordtypelem3 eqeltrd breq2 ralbidv elrab eldifi simprr fssdmd
      simprbi adantrr sstrdi fveq1 ssel2 sylan9eq mpbid simpli funfn mpbi inss2
      anassrs ralima sylancr elrabd rspcdva wor wwe weso ffvelcdmd sotric ioran
      wo syl12anc bitrdi mpbir2and sylbid ex com23 a2i a1i biimtrid tfis3 com3l
      expr mpdd sylan2br impancom orrd orcomd ) AQHUHZUIZPRUJZUHZUIZQRUKZUHZPRU
      LZQJUMZUVLUVNUVPUVIUVNUNZUVKUVPAUVHUVQUVKUVPVAZUVHUVQUIAQHUVMUOUHZUVRQHUV
      MUPAUVSUIZUVKPUQUHZUVPUVTUVJUQPUVTUVJKNUJZURZUQUVTUWCHRAUWCHRVBZUVSABCDEF
      GHIJKLMNORSTUAUBUCUDUEUSZUTVCZUVTUWCKUQKUWBVDZUVTKVEZKUQVFAUWHUVSABCDEFGH
      IJKLMNORSTUAUBUCUDUEVGUTZKVHVIVJVKVLUWAUVTUVKUVPUVTUFVMZUVJUHZUWJRULZQJUM
      ZVAZVAZUVTUGVMZUVJUHZUWPRULZQJUMZVAZVAZUVTUVRVAUFUGPUWJUWPVNZUWNUWTUVTUXB
      UWKUWQUWMUWSUWJUWPUVJVOUXBUWLUWRQJUWJUWPRVPVQVRVSUWJPVNZUWNUVRUVTUXCUWKUV
      KUWMUVPUWJPUVJVOUXCUWLUVOQJUWJPRVPVQVRVSUXAUGUWJVTUVTUWTUGUWJVTZVAZUWJUQU
      HZUWOUVTUWTUGUWJWAUXEUWOVAUXFUVTUXDUWNUVTUWKUXDUWMUVTUWKUXDUWMVAUVTUWKUIZ
      UXDUWSUGUWJVTZUWMUXGUWTUWSUGUWJUXGUWPUWJUHZUIUWQUWTUWSWBUXGUWJUVJUWPUVTUV
      JVEZUWKUWJUVJVFZUVTUXJUWCVEZUVTUWHUWBVEZUXLUWIUWBWCZUXMNWFZUXNNOSWGZWDUWB
      WEWHKUWBWIWJUVTUVJUWCVNUXJUXLWBUWFUVJUWCWKVIWLUVJUWJWMWNZWOUWQUWSWQVIWPUV
      TUWKUXHUWMUVTUWKUXHUIZUIZUWMUWLQVNZUNZQUWLJUMZUNZUXSUXTUVNUVSUVQAUXRQHUVM
      WRWSUXSUWLUVMUHZUXTUVNUXSRUWCWTUWJUWCUHZUYDUXSUWCHRAUWDUVSUXRUWEXAZXBUXSU
      WJUVJUWCUVTUWKUXHXCUXSUWCHRUYFVCXDZUWCUWJRXEXGUWLQUVMVOXFXHUXSFVMZUWLJUMZ
      UNZUYCFMVMZDVMZJUMZMNUWJXIZVTZDHXJZQUYHQVNUYIUYBUYHQUWLJXKXLUXSUWLUYHEVMZ
      JUMZUNZFUYPVTZEUYPXJZUHZUYJFUYPVTZUXSUWLUWJNULZVUAUXSUWLUWJNKXMZULVUDUXSU
      WJRVUEARVUEVNZUVSUXRABCDEFGHIJKLMNORSTUAUBUCUDUEXNXAZXOUXSUWJKNUXSKUWBUWJ
      UYGXPXQXRUXSAUYEVUDVUAUHAUVSUXRXSUYGABCDEFGHIJKLMNOUWJRSTUAUBUCUDUEXTXGYA
      VUBUWLUYPUHVUCUYTVUCEUWLUYPUYQUWLVNZUYSUYJFUYPVUHUYRUYIUYQUWLUYHJYBXLYCYD
      YHVIUXSUYOUYKQJUMZMUYNVTZDQHUYLQVNUYMVUIMUYNUYLQUYKJYBYCUVSUVHAUXRQHUVMYE
      WSZUXSVUJUWPNULZQJUMZUGUWJVTZUXSUXHVUNUVTUWKUXHYFUXSVUFUWJKVFZUXHVUNWBVUG
      UXSUWJUWCKUXSUWCHUWJRUYFUVTUWKUXKUXHUXQYIYGZUWGYJVUFVUOUIZUWSVUMUGUWJVUQU
      XIUIUWRVULQJVUFVUOUXIUWRVULVNVUFVUOUXIUIZUWRUWPVUEULVULUWPRVUEYKVURUWPKNU
      WJKUWPYLXQYMYSVQWPXGYNUXSNUWBWTZUWJUWBVFVUJVUNWBUXOVUSUXOUXNUXPYONYPYQUXS
      UWJUWCUWBVUPKUWBYRYJVUIVUMMUGUWBUWJNUYKVULQJXKYTUUAWLUUBUUCUXSUWMUXTUYBUU
      JUNZUYAUYCUIUXSHJUUDZUWLHUHUVHUWMVUTWBAVVAUVSUXRAHJUUEVVAUDHJUUFVIXAUXSUW
      CHUWJRUYFUYGUUGVUKHUWLQJUUHUUKUXTUYBUUIUULUUMUVBUUNUUOUUPUUQUURUUSUUTUVAU
      VCUVDYSUVEUVFUVG $.

    $( Lemma for ~ ordtype .  (Contributed by Mario Carneiro, 25-Jun-2015.) $)
    ordtypelem8 $p |- ( ph -> O Isom _E , R ( dom O , ran O ) ) $=
      ( va vb cdm cep wor crn wpo wfo cv wbr cfv wral wiso con0 wss ordtypelem4
      wi cin fdmd inss1 word ordtypelem2 ordsson syl sstrid eqsstrd epweon weso
      wwe ax-mp soss mpisyl frnd wess sylc sopo 3syl wfun ffund funforn wcel wa
      sylib epel ordtypelem6 biimtrid ralrimiva ralrimivw soisoi syl22anc ) APU
      FZUGUHZPUIZJUJZWNWPPUKZUDULZUEULZUGUMZWSPUNWTPUNJUMZUTZUEWNUOZUDWNUOWNWPU
      GJPUPAWNUQURUQUGUHZWOAWNKNUFZVAZUQAXGHPABCDEFGHIJKLMNOPQRSTUAUBUCUSZVBAXG
      KUQKXFVCAKVDKUQURABCDEFGHIJKLMNOPQRSTUAUBUCVEKVFVGVHVIUQUGVLXEVJUQUGVKVMW
      NUQUGVNVOAWPJVLZWPJUHWQAWPHURHJVLXIAXGHPXHVPUBWPHJVQVRWPJVKWPJVSVTAPWAWRA
      XGHPXHWBPWCWFAXDUDWNAXCUEWNXAWSWTWDAWTWNWDWEXBUEWSWGABCDEFGHIJKLMNOWTWSPQ
      RSTUAUBUCWHWIWJWKUDUEWNWPUGJPWLWM $.

    ${
      ordtypelem9.1 $e |- ( ph -> O e. V ) $.
      $( Lemma for ~ ordtype .  Either the function ` OrdIso ` is an
         isomorphism onto all of ` A ` , or ` OrdIso ` is not a set, which by
         ~ oif implies that either ` ran O C_ A ` is a proper class or
         ` dom O = On ` .  (Contributed by Mario Carneiro, 25-Jun-2015.)
         (Revised by AV, 28-Jul-2024.) $)
      ordtypelem9 $p |- ( ph -> O Isom _E , R ( dom O , A ) ) $=
        ( vb vm vc va cdm crn cep wiso ordtypelem8 wceq wb cin ordtypelem4 frnd
        cv wcel wa cfv wbr wrex wral cima word ordtypelem2 ordirr syl con0 wlim
        wn wfun tfr1a simpri limord ax-mp cres ordtypelem1 elexd eqeltrrd tfr2b
        mpbird ordelon sylancr imaeq2 raleqdv rexbidv crab breq1 cbvralvw breq2
        ralbidv bitrid cbvrexvw cbvrabv eqtri elrab2 baib mtbid ralnex r19.21bi
        cvv sylibr rneqd df-ima eqtr4di adantr ffund funfnd ralrn bitr3d rexnal
        wfn ordtypelem7 ord rexlimdva mpd eqelssd isoeq5 mpbid ) APUJZPUKZULJPU
        MZYDHULJPUMZABCDEFGHIJKLMNOPRSTUAUBUCUDUNAYEHUOYFYGUPAUFYEHAKNUJZUQZHPA
        BCDEFGHIJKLMNOPRSTUAUBUCUDURZUSAUFUTZHVAZVBZUGUTZPVCZYKJVDZVNZUGYDVEZYK
        YEVAZYMYPUGYDVFZVNYRYMUHUTZYKJVDZUHNKVGZVFZYTAUUDVNZUFHAUUDUFHVEZVNUUEU
        FHVFAKKVAZUUFAKVHZUUGVNABCDEFGHIJKLMNOPRSTUAUBUCUDVIZKVJVKAKVLVAZUUGUUF
        UPAYHVHZKYHVAZUUJYHVMZUUKNVOUUMNORVPVQYHVRVSAUULNKVTZXEVAZAPUUNXEABCDEF
        GHIJKLMNOPRSTUAUBUCUDWAZAPQUEWBWCAUUHUULUUOUPUUIKNORWDVKWEYHKWFWGUUGUUJ
        UUFUUBUHNUIUTZVGZVFZUFHVEZUUFUIKVLKUUQKUOZUUSUUDUFHUVAUUBUHUURUUCUUQKNW
        HWIWJKCUTZGUTZJVDZCNBUTZVGZVFZGHVEZBVLWKUUTUIVLWKUAUVHUUTBUIVLUVHUUBUHU
        VFVFZUFHVEUVEUUQUOZUUTUVGUVIGUFHUVGUUAUVCJVDZUHUVFVFUVCYKUOZUVIUVDUVKCU
        HUVFUVBUUAUVCJWLWMUVLUVKUUBUHUVFUVCYKUUAJWNWOWPWQUVJUVIUUSUFHUVJUUBUHUV
        FUURUVEUUQNWHWIWJWPWRWSWTXAVKXBUUDUFHXCXFXDYMUUBUHYEVFZUUDYTYMUUBUHYEUU
        CAYEUUCUOYLAYEUUNUKUUCAPUUNUUPXGNKXHXIXJWIYMPYDXPZUVMYTUPAUVNYLAPAYIHPY
        JXKXLXJUUBYPUHUGYDPUUAYOYKJWLXMVKXNXBYPUGYDXOXFYMYQYSUGYDYMYNYDVAVBYPYS
        ABCDEFGHIJKLMNOYNYKPRSTUAUBUCUDXQXRXSXTYAYDYEHULJPYBVKYC $.
    $}

    $( Lemma for ~ ordtype .  Using ~ ax-rep , exclude the possibility that
       ` O ` is a proper class and does not enumerate all of ` A ` .
       (Contributed by Mario Carneiro, 25-Jun-2015.) $)
    ordtypelem10 $p |- ( ph -> O Isom _E , R ( dom O , A ) ) $=
      ( vb vc vm cdm crn cep wiso ordtypelem8 wceq wb cin ordtypelem4 frnd wcel
      cv wa wn simprl wf1o wfo cvv wwe adantr wse wfn ffund funfnd isof1o f1of1
      wf1 3syl wbr crab simpl seex syl2an wss wral rexnal ordtypelem7 rexlimdva
      cfv wrex ord biimtrrid con1d impr breq1 ralrn mpbird ssrab sylanbrc ssexd
      f1dmex syl2anc fnexd ordtypelem9 f1ofo forn 4syl eleqtrrd pm2.18d eqelssd
      syl expr isoeq5 mpbid ) APUGZPUHZUIJPUJZXKHUIJPUJZABCDEFGHIJKLMNOPQRSTUAU
      BUCUKAXLHULZXMXNUMAUDXLHAKNUGUNZHPABCDEFGHIJKLMNOPQRSTUAUBUCUOZUPZAUDURZH
      UQZUSZXSXLUQZAXTYBUTZYBAXTYCUSZUSZXSHXLAXTYCVAYEXNXKHPVBXKHPVCXOYEBCDEFGH
      IJKLMNOPVDQRSTUAAHJVEYDUBVFZAHJVGZYDUCVFZYEXKPVDAPXKVHZYDAPAXPHPXQVIVJVFZ
      YEXKXLPVMZXLVDUQXKVDUQYEXMXKXLPVBYKYEBCDEFGHIJKLMNOPQRSTUAYFYHUKXKXLUIJPV
      KXKXLPVLVNYEXLUEURZXSJVOZUEHVPZVDAYGXTYNVDUQYDUCXTYCVQUEHXSJVRVSYEXLHVTZY
      MUEXLWAZXLYNVTAYOYDXRVFYEYPUFURZPWEZXSJVOZUFXKWAZAXTYCYTYAYTYBYTUTYSUTZUF
      XKWFYAYBYSUFXKWBYAUUAYBUFXKYAYQXKUQUSYSYBABCDEFGHIJKLMNOYQXSPQRSTUAUBUCWC
      WGWDWHWIWJYEYIYPYTUMYJYMYSUEUFXKPYLYRXSJWKWLXGWMYMUEHXLWNWOWPXKXLVDPWQWRW
      SWTXKHUIJPVKXKHPXAXKHPXBXCXDXHXEXFXKXLHUIJPXIXGXJ $.
  $}

  ${
    $d t u v x F $.  $d h j t u v w x z M $.  $d f h i j r s t u v w x y z R $.
    $d f h j r s t u v w x y z A $.  $d j u w N $.
    oicl.1 $e |- F = OrdIso ( R , A ) $.
    $( Definition of the ordinal isomorphism when its arguments are not
       meaningful.  (Contributed by Mario Carneiro, 25-Jun-2015.) $)
    oi0 $p |- ( -. ( R We A /\ R Se A ) -> F = (/) ) $=
      ( vh vu vv vj vw vz vt vx wwe wse wn cv wbr wral crab c0 wa cvv crio cmpt
      crn crecs cima wrex con0 cres cif coi df-oi eqtri iffalse eqtrid ) ABMABN
      UAZOCUQEUBFPGPBQOFHPIPBQHEPUERIASZRGURUCUDUFZJPKPBQJUSLPUGRKAUHLUISUJZTUK
      ZTCABULVADLJIGFKABEHUMUNUQUTTUOUP $.

    $( The order type of the well-order ` R ` on ` A ` is an ordinal.
       (Contributed by Mario Carneiro, 23-May-2015.)  (Revised by Mario
       Carneiro, 25-Jun-2015.) $)
    oicl $p |- Ord dom F $=
      ( vx vz vw vv vu vt vj vh vf vr vs cv wbr wral eqid c0 vi vy wwe wse word
      wa cdm wf crn crab cvv wn crio cmpt crecs cima wrex con0 ordtypecbv simpl
      simpr ordtypelem5 simpld ord0 wceq oi0 dmeqd dm0 eqtrdi ordeq syl pm2.61i
      wb mpbiri ) ABUCZABUDZUFZCUGZUEZVQVSVRACUHVQEFGHIJAKPGPBQKLPUIRGAUJZBFPJP
      BQFMUKNPOPBQULNUAPUBPBQUAMPUIRUBAUJZROWAUMUNUOZEPUPRJAUQEURUJZLKWBLUKIPHP
      BQULIVTRHVTUMUNZCUBGHIAVTBMLUAKWDUOZWDONWESVTSZWDSZUSWFWGWCSDVOVPUTVOVPVA
      VBVCVQULZVSTUEZVDWHVRTVEVSWIVMWHVRTUGTWHCTABCDVFVGVHVIVRTVJVKVNVL $.

    $( The order isomorphism of the well-order ` R ` on ` A ` is a function.
       (Contributed by Mario Carneiro, 23-May-2015.) $)
    oif $p |- F : dom F --> A $=
      ( vx vz vw vv vu vt vj vh vf vr vs cv wbr wral eqid c0 vi vy wwe wse word
      wa cdm wf crn crab cvv wn crio cmpt crecs cima wrex con0 ordtypecbv simpl
      simpr ordtypelem5 simprd f0 oi0 dmeqd dm0 eqtrdi feq12d mpbiri pm2.61i )
      ABUCZABUDZUFZCUGZACUHZVNVOUEVPVNEFGHIJAKPGPBQKLPUIRGAUJZBFPJPBQFMUKNPOPBQ
      ULNUAPUBPBQUAMPUIRUBAUJZROVRUMUNUOZEPUPRJAUQEURUJZLKVSLUKIPHPBQULIVQRHVQU
      MUNZCUBGHIAVQBMLUAKWAUOZWAONWBSVQSZWASZUSWCWDVTSDVLVMUTVLVMVAVBVCVNULZVPT
      ATUHAVDWEVOTACTABCDVEZWEVOTUGTWECTWFVFVGVHVIVJVK $.

    $( The order isomorphism of the well-order ` R ` on ` A ` is an isomorphism
       onto ` ran O ` (which is a subset of ` A ` by ~ oif ).  (Contributed by
       Mario Carneiro, 25-Jun-2015.) $)
    oiiso2 $p |- ( ( R We A /\ R Se A ) -> F Isom _E , R ( dom F , ran F ) ) $=
      ( vx vz vw vv vu vt vj vh vf vr vs vi cv wbr wral eqid vy wwe wse wa crab
      crn cvv crio cmpt crecs cima wrex con0 ordtypecbv simpl simpr ordtypelem8
      wn ) ABUBZABUCZUDEFGHIJAKQGQBRKLQUFSGAUEZBFQJQBRFMUGNQOQBRURNPQUAQBRPMQUF
      SUAAUEZSOVBUHUIUJZEQUKSJAULEUMUEZLKVCLUGIQHQBRURIVASHVAUHUIZCUAGHIAVABMLP
      KVEUJZVEONVFTVATZVETZUNVGVHVDTDUSUTUOUSUTUPUQ $.

    $( For any set-like well-ordered class, there is an isomorphic ordinal
       number called its order type.  (Contributed by Jeff Hankins,
       17-Oct-2009.)  (Revised by Mario Carneiro, 25-Jun-2015.) $)
    ordtype $p |- ( ( R We A /\ R Se A ) -> F Isom _E , R ( dom F , A ) ) $=
      ( vx vz vw vv vu vt vj vh vf vr vs vi cv wbr wral eqid vy wwe wse wa crab
      crn wn crio cmpt crecs cima wrex con0 ordtypecbv simpl simpr ordtypelem10
      cvv ) ABUBZABUCZUDEFGHIJAKQGQBRKLQUFSGAUEZBFQJQBRFMURNQOQBRUGNPQUAQBRPMQU
      FSUAAUEZSOVBUHUIUJZEQUKSJAULEUMUEZLKVCLURIQHQBRUGIVASHVAUHUIZCUAGHIAVABML
      PKVEUJZVEONVFTVATZVETZUNVGVHVDTDUSUTUOUSUTUPUQ $.

    $( ` ran F ` is an initial segment of ` A ` under the well-order ` R ` .
       (Contributed by Mario Carneiro, 26-Jun-2015.) $)
    oiiniseg $p |- ( ( ( R We A /\ R Se A ) /\ ( N e. A /\ M e. dom F ) )
            -> ( ( F ` M ) R N \/ N e. ran F ) ) $=
      ( vx vz vw vv vu vt vj vh vf vr wbr cv wral eqid vs vi vy wwe wse wa wcel
      cdm cfv crn wo crab crio cmpt crecs cima wrex con0 ordtypecbv simpl simpr
      cvv wn ordtypelem7 anasss ) ABUDZABUEZUFZEAUGDCUHUGDCUIEBQECUJUGUKVHGHIJK
      LAMRIRBQMNRUJSIAULZBHRLRBQHOVBPRUARBQVCPUBRUCRBQUBORUJSUCAULZSUAVJUMUNUOZ
      GRUPSLAUQGURULZNMVKNVBKRJRBQVCKVISJVIUMUNZDECUCIJKAVIBONUBMVMUOZVMUAPVNTV
      ITZVMTZUSVOVPVLTFVFVGUTVFVGVAVDVE $.

    $( For any set-like well-ordered class, if the order isomorphism exists (is
       a set), then it maps some ordinal onto ` A ` isomorphically.  Otherwise,
       ` F ` is a proper class, which implies that either ` ran F C_ A ` is a
       proper class or ` dom F = On ` .  This weak version of ~ ordtype does
       not require the Axiom of Replacement.  (Contributed by Mario Carneiro,
       25-Jun-2015.) $)
    ordtype2 $p |- ( ( R We A /\ R Se A /\ F e. _V )
           -> F Isom _E , R ( dom F , A ) ) $=
      ( vx vz vw vv vu vt vj vh vf vr vs cvv cv wbr wral eqid vi vy wwe wse w3a
      wcel crn crab crio cmpt crecs cima wrex con0 ordtypecbv simp1 simp2 simp3
      wn ordtypelem9 ) ABUCZABUDZCPUFZUEEFGHIJAKQGQBRKLQUGSGAUHZBFQJQBRFMPNQOQB
      RUSNUAQUBQBRUAMQUGSUBAUHZSOVEUIUJUKZEQULSJAUMEUNUHZLKVFLPIQHQBRUSIVDSHVDU
      IUJZCPUBGHIAVDBMLUAKVHUKZVHONVITVDTZVHTZUOVJVKVGTDVAVBVCUPVAVBVCUQVAVBVCU
      RUT $.
    $( $j usage 'ordtype2' avoids 'ax-rep'; $)

    $( The order isomorphism on a set is a set.  (Contributed by Mario
       Carneiro, 25-Jun-2015.) $)
    oiexg $p |- ( A e. V -> F e. _V ) $=
      ( wcel wwe wse wa cvv cdm wf1 cep wiso wf1o ordtype isof1o f1of1 3syl wf
      f1f f1dmex fex syl2an2r expcom syl5 wn c0 oi0 0ex eqeltrdi pm2.61d1 ) ADF
      ZABGABHIZCJFZUNCKZACLZUMUOUNUPAMBCNUPACOUQABCEPUPAMBCQUPACRSUQUMUOUQUPACT
      UMUPJFUOUPACUAUPADCUBUPAJCUCUDUEUFUNUGCUHJABCEUIUJUKUL $.

    $( The order type of the well-order ` R ` on ` A ` is an ordinal.
       (Contributed by Stefan O'Rear, 11-Feb-2015.)  (Revised by Mario
       Carneiro, 23-May-2015.) $)
    oion $p |- ( A e. V -> dom F e. On ) $=
      ( wcel cdm con0 word oicl cvv wb oiexg dmexg elong 3syl mpbiri ) ADFZCGZH
      FZSIZABCEJRCKFSKFTUALABCDEMCKNSKOPQ $.

    $( The order isomorphism of the well-order ` R ` on ` A ` is an
       isomorphism.  (Contributed by Mario Carneiro, 23-May-2015.) $)
    oiiso $p |- ( ( A e. V /\ R We A ) -> F Isom _E , R ( dom F , A ) ) $=
      ( wcel wse wwe cdm cep wiso exse ordtype ancoms sylan ) ADFABGZABHZCIAJBC
      KZABDLQPRABCEMNO $.

    $( The order type of a well-ordered set is equinumerous to the set.
       (Contributed by Mario Carneiro, 23-May-2015.) $)
    oien $p |- ( ( A e. V /\ R We A ) -> dom F ~~ A ) $=
      ( wcel cvv wwe cdm wf1o cen wbr oiexg cep wiso oiiso isof1o syl f1oen3g
      wa syl2an2r ) ADFZCGFABHZCIZACJZUDAKLABCDEMUBUCTUDANBCOUEABCDEPUDANBCQRUD
      ACGSUA $.

    $( Uniqueness of the unique ordinal isomorphism.  (Contributed by Mario
       Carneiro, 23-May-2015.)  (Revised by Mario Carneiro, 25-Jun-2015.) $)
    oieu $p |- ( ( R We A /\ R Se A ) ->
      ( ( Ord B /\ G Isom _E , R ( B , A ) ) <-> ( B = dom F /\ G = F ) ) ) $=
      ( wwe wse wa word cep wiso cdm wceq ccnv adantr syl a1i wb isoeq4 ordtype
      ccom simprr isocnv isotr syl2anc oicl ordiso2 syl3anc ordwe ad2antrl epse
      simprl mpbird weisoeq syl22anc jca ex jctil ordeq isoeq1 sylan9bb anbi12d
      syl5ibrcom impbid ) ACGACHIZBJZBAKCELZIZBDMZNZEDNZIZVFVIVMVFVIIZVKVLVNBVJ
      KKDOZEUBZLZVGVJJZVKVNVHAVJCKVOLZVQVFVGVHUCZVNVJAKCDLZVSVFWAVIACDFUAZPZVJA
      KCDUDQBAVJKCKVOEUEUFVFVGVHUMVRVNACDFUGZRBVJVPUHUIZVNBKGZBKHZVHBAKCDLZVLVG
      WFVFVHBUJUKWGVNBULRVTVNWHWAWCVNVKWHWASWEBAVJKCDTQUNBAKCEDUOUPUQURVFVIVMVR
      WAIVFWAVRWBWDUSVMVGVRVHWAVKVGVRSVLBVJUTPVKVHVJAKCELVLWABAVJKCETVJAKCDEVAV
      BVCVDVE $.
  $}

  ${
    $d x y A $.  $d x y F $.
    oismo.1 $e |- F = OrdIso ( _E , A ) $.
    $( When ` A ` is a subclass of ` On ` , ` F ` is a strictly monotone
       ordinal functions, and it is also complete (it is an isomorphism onto
       all of ` A ` ).  The proof avoids ~ ax-rep (the second statement is
       trivial under ~ ax-rep ).  (Contributed by Mario Carneiro,
       26-Jun-2015.) $)
    oismo $p |- ( A C_ On -> ( Smo F /\ ran F = A ) ) $=
      ( vx vy con0 wss wfo wa cep wiso wwe wf sylancr a1i wcel cvv mp1i syl3anc
      cv wsmo crn wceq cdm wse epweon wess mpi epse oiiso2 sylancl word wb oicl
      oif frn ax-mp id sstrid smoiso2 mpbird simprd simprl wf1o adantr wfn wral
      wn cfv ffn wbr simplrr ad2antrr simplrl simpr oiiniseg syl22anc mt3d epel
      wo ord sylib ralrimiva ffnfv sylanbrc smogt ordelon simpll sseldd syl2anc
      wi ontr2 mp2and ssrdv ssexd fex2 ordtype2 isof1o f1ofo forn 4syl eleqtrrd
      ex expr pm2.18d eqelssd jca ) AFGZBUAZBUBZAUCZXHBUDZXJBHZXIXHXMXIIZXLXJJJ
      BKZXHAJLZAJUEZXOXHFJLXPUFAFJUGUHZAUIZAJBCUJUKXHXLULZXJFGXNXOUMAJBCUNZXHXJ
      AFXLABMZXJAGZAJBCUOZXLABUPUQZXHURUSXLXJBUTNVAVBZXHDXJAYCXHYEOXHDTZAPZIYGX
      JPZXHYHYIVHZYIXHYHYJIZIZYGAXJXHYHYJVCZYLXLAJJBKZXLABVDXLABHXKYLXPXQBQPZYN
      XHXPYKXRVEXQYLXSOYLXLYGBMZXLQPYHYOYLBXLVFZETZBVIZYGPZEXLVGYPYBYQYLYDXLABV
      JZRYLYTEXLYLYRXLPZIZYSYGJVKZYTUUCUUDYIXHYHYJUUBVLUUCUUDYIUUCXPXQYHUUBUUDY
      IVTXHXPYKUUBXRVMXQUUCXSOXHYHYJUUBVNZYLUUBVOZAJBYRYGCVPVQWAVRDYSVSWBZWCEXL
      YGBWDWEYLXLYGAYMYLEXLYGYLUUBYRYGPZUUCYRYSGZYTUUHUUCYQXIUUBUUIYBYQUUCYDUUA
      RXHXIYKUUBYFVMUUFXLYRBWFSUUGUUCYRFPZYGFPUUIYTIUUHWKUUCXTUUBUUJYAUUFXLYRWG
      NUUCAFYGXHYKUUBWHUUEWIYRYSYGWLWJWMXCWNWOYMXLYGBQAWPSAJBCWQSXLAJJBWRXLABWS
      XLABWTXAXBXDXEXFXG $.
  $}

  $( The order type of an ordinal under the ` e. ` order is itself, and the
     order isomorphism is the identity function.  (Contributed by Mario
     Carneiro, 26-Jun-2015.) $)
  oiid $p |- ( Ord A -> OrdIso ( _E , A ) = ( _I |` A ) ) $=
    ( word cep wwe wse coi wiso cid cres wceq ordwe epse a1i cdm crn eqid mpbid
    wb syl syl3anc oiiso2 sylancl wsmo con0 wss wa ordsson isoeq5 simpl2im oicl
    oismo id ordiso2 isoeq4 weniso ) ABZACDZACEZAACCACFZGZUSHAIJAKZURUPALZMUPUS
    NZACCUSGZUTUPVCUSOZCCUSGZVDUPUQURVFVAVBACUSUSPZUAUBUPUSUCZVEAJZVFVDRUPAUDUE
    VHVIUFAUGAUSVGUKSVCVEACCUSUHUIQZUPVCAJZVDUTRUPVDVCBZUPVKVJVLUPACUSVGUJMUPUL
    VCAUSUMTVCAACCUSUNSQACUSUOT $.

  ${
    $d f s t w y z $.  $d f r x y A $.  $d r x R $.  $d r y V $.
    hartogslem.2 $e |- F = { <. r , y >. |
       ( ( ( dom r C_ A /\ ( _I |` dom r ) C_ r /\ r C_ ( dom r X. dom r ) ) /\
         ( r \ _I ) We dom r ) /\ y = dom OrdIso ( ( r \ _I ) , dom r ) ) } $.
    hartogslem.3 $e |- R = { <. s , t >. | E. w e. y E. z e. y
       ( ( s = ( f ` w ) /\ t = ( f ` z ) ) /\ w _E z ) } $.
    $( Lemma for ~ hartogs .  (Contributed by Mario Carneiro, 14-Jan-2013.)
       (Revised by Mario Carneiro, 15-May-2015.) $)
    hartogslem1 $p |- ( dom F C_ ~P ( A X. A ) /\ Fun F /\
      ( A e. V -> ran F = { x e. On | x ~<_ A } ) ) $=
      ( wss wceq cid wa cep syl cdm cxp cpw wfun wcel crn cv cdom wbr con0 crab
      cres w3a cdif wwe coi wex cab copab dmeqi dmopab eqtri simp3 simp1 xpss12
      syl2anc sstrd velpw sylibr ad2antrr exlimiv abssi eqsstri funopab4 funeqi
      wi mpbir rneqi rnopab breq1 elrab wf1 cun cin f1f adantl frnd resss ssun2
      wf sstri idssxp ssini a1i inss2 3jca word eloni ordwe adantr wiso wb wf1o
      cfv wrex f1f1orn f1oiso sylancl isores2 sylib isowe mpbid c0 wal wor weso
      wn brel simpld sonr syl2an pm2.01da alrimiv intirr disj3 weeq1 isoeq3 cvv
      vex ax-mp eqid mpbiri dmss sseq12d indif1 anbi12d ex exlimdv impel cen id
      wse rnex exse oieu mpbi2and xpex inex2 sseq1 dmxpid sseqtrdi dmresi sseq2
      eqsstrrid eqssd sseq1d reseq2d sqxpeqd difeq1 difun2 ineq1i 3eqtr3i oieq1
      3anbi123d eqtrdi weeq12d oieq2 eqtrd dmeqd eqeq2d spcev brdomi simpr dmex
      syl21anc oion eqeltrdi simplr oien sylancr eqbrtrd ssdomg simpll1 endomtr
      syl2an2 jca impbid2 bitrid eqabdv eqtr4id 3pm3.2i ) IUAZFFUBZUCZOIUDZFJUE
      ZIUFZAUGZFUHUIZAUJUKZPVPUWLLUGZUAZFOZQUXBULZUXAOZUXAUXBUXBUBZOZUMZUXBUXAQ
      UNZUOZRZBUGZUXBUXIUPZUAZPZRZBUQZLURZUWNUWLUXPLBUSZUAUXRIUXSMUTUXPLBVAVBUX
      QLUWNUXPUXAUWNUEZBUXHUXTUXJUXOUXHUXAUWMOUXTUXHUXAUXFUWMUXCUXEUXGVCUXHUXCU
      XCUXFUWMOUXCUXEUXGVDZUYAUXBFUXBFVEVFVGLUWMVHVIVJVKVLVMUWOUXSUDUXKLBUXNVNI
      UXSMVOVQUWPUWQUXPLUQZBURZUWTUWQUXSUFUYCIUXSMVRUXPLBVSVBUWPUYBBUWTUXLUWTUE
      UXLUJUEZUXLFUHUIZRZUWPUYBUWSUYEAUXLUJUWRUXLFUHVTWAUWPUYFUYBUYDUXLFHUGZWBZ
      HUQUYBUYEUYDUYHUYBHUYDUYHUYBUYDUYHRZUYGUFZFOZQUYJULZGQWCZUYJUYJUBZWDZOZUY
      OUYNOZUMZUYJGUYNWDZQUNZUOZUXLUYJUYTUPZUAZPZUYBUYIUYKUYPUYQUYIUXLFUYGUYHUX
      LFUYGWJUYDUXLFUYGWEWFWGUYPUYIUYLUYMUYNUYLQUYMQUYJWHQGWIWKUYJWLWMZWNUYQUYI
      UYMUYNWOZWNWPUYIUYJUYSUOZVUAUYIUXLSUOZVUGUYDVUHUYHUYDUXLWQZVUHUXLWRZUXLWS
      TWTUYIUXLUYJSUYSUYGXAZVUHVUGXBUYIUXLUYJSGUYGXAZVUKUYIUXLUYJUYGXCZGKUGDUGZ
      UYGXDPEUGCUGZUYGXDPRVUNVUOSUIRCUXLXEDUXLXEKEUSPVULUYHVUMUYDUXLFUYGXFWFNDC
      KEUXLUYJSGUYGXGXHUXLUYJSGUYGXIXJZUXLUYJSUYSUYGXKTXLZUYIUYSUYTPZVUGVUAXBUY
      IUYSQWDXMPZVURUYIUWRUWRUYSUIZXQZAXNVUSUYIVVAAUYIVUTUYIUYJUYSXOZUWRUYJUEZV
      VAVUTUYIVUGVVBVUQUYJUYSXPTVUTVVCVVCUWRUWRUYJUYJUYSGUYNWOXRXSUYJUWRUYSXTYA
      YBYCAUYSYDVIUYSQYEXJZUYJUYSUYTYFTXLZUYIVUDUYGVUBPZUYIVUIUXLUYJSUYTUYGXAZV
      UDVVFRZUYDVUIUYHVUJWTUYIVUKVVGVUPUYIVURVUKVVGXBVVDUXLUYJSUYSUYTUYGYGTXLUY
      IVUAUYJUYTUUBZVUIVVGRVVHXBVVEUYJYHUEVVIUYGHYIUUCZUYJUYTYHUUDYJUYJUXLUYTVU
      BUYGVUBYKUUEXHUUFXSUXPUYRVUARZVUDRLUYOUYNUYMUYJUYJVVJVVJUUGUUHUXAUYOPZUXK
      VVKUXOVUDVVLUXHUYRUXJVUAVVLUXCUYKUXEUYPUXGUYQVVLUXBUYJFVVLUXBUYJVVLUXBUYN
      UAZUYJVVLUXAUYNOZUXBVVMOVVLVVNUYQVUFUXAUYOUYNUUIYLUXAUYNYMTUYJUUJUUKVVLUY
      JUYLUAZUXBUYJUULVVLUYLUXAOZVVOUXBOVVLVVPUYPVUEUXAUYOUYLUUMYLUYLUXAYMTUUNU
      UOZUUPVVLUXDUYLUXAUYOVVLUXBUYJQVVQUUQVVLUUAZYNVVLUXAUYOUXFUYNVVRVVLUXBUYJ
      VVQUURYNUVDVVLUXBUYJUXIUYTVVLUXIUYOQUNZUYTUXAUYOQUUSUYMQUNZUYNWDGQUNZUYNW
      DVVSUYTVVTVWAUYNGQUUTUVAUYMUYNQYOGUYNQYOUVBUVEZVVQUVFYPVVLUXNVUCUXLVVLUXM
      VUBVVLUXMUXBUYTUPZVUBVVLUXIUYTPUXMVWCPVWBUXBUXIUYTUVCTVVLUXBUYJPVWCVUBPVV
      QUXBUYJUYTUVGTUVHUVIUVJYPUVKUVOYQYRUXLFHUVLYSUWPUXPUYFLUWPUXPUYFUWPUXPRUY
      DUYEUXPUYDUWPUXPUXLUXNUJUXKUXOUVMZUXBYHUEZUXNUJUEUXALYIUVNZUXBUXIUXMYHUXM
      YKZUVPYJUVQWFUXPUXLUXBYTUIUWPUXBFUHUIZUYEUXPUXLUXNUXBYTVWDUXPVWEUXJUXNUXB
      YTUIVWFUXHUXJUXOUVRUXBUXIUXMYHVWGUVSUVTUWAUWPUXCVWHUXPUXBFJUWBUXCUXEUXGUX
      JUXOUWCYSUXLUXBFUWDUWEUWFYQYRUWGUWHUWIUWJUWK $.

    $( Lemma for ~ hartogs .  (Contributed by Mario Carneiro, 14-Jan-2013.) $)
    hartogslem2 $p |- ( A e. V -> { x e. On | x ~<_ A } e. _V ) $=
      ( wcel crn cv cdom cvv sylancr wbr con0 crab cdm cxp cpw wss wfun wceq wi
      hartogslem1 simp3i simp2i simp1i sqxpexg pwexd ssexg funex rnexg eqeltrrd
      syl ) FJOZIPZAQFRUAAUBUCZSIUDZFFUEZUFZUGZIUHZVBVCVDUIUJZABCDEFGHIJKLMNUKZ
      ULVBISOZVCSOVBVIVESOZVLVHVIVJVKUMVBVHVGSOVMVHVIVJVKUNVBVFSFJUOUPVEVGSUQTS
      IURTISUSVAUT $.
  $}

  ${
    $d g r s t w x y z $.  $d g r x y z A $.  $d r y V $.
    $( The class of ordinals dominated by a given set is an ordinal.  A shorter
       (when taking into account lemmas ~ hartogslem1 and ~ hartogslem2 ) proof
       can be given using the axiom of choice, see ~ ondomon .  As its label
       indicates, this result is used to justify the definition of the Hartogs
       function ~ df-har .  (Contributed by Jeff Hankins, 22-Oct-2009.)
       (Revised by Mario Carneiro, 15-May-2015.) $)
    hartogs $p |- ( A e. V -> { x e. On | x ~<_ A } e. On ) $=
      ( vy vz vw vt vs vg vr wcel cv cdom wbr con0 word wss wa cvv wceq crab wi
      wtr wal onelon vex onelss imp ssdomg mpsyl jca domtr anim2i anassrs sylan
      exp31 com12 impd breq1 elrab 3imtr4g gen2 dftr2 mpbir ordon trssord mp3an
      ssrab2 wb cfv cep wrex cdm cid cres cxp w3a cdif wwe coi eqid hartogslem2
      copab elong syl mpbiri ) BCKZALZBMNZAOUAZOKZWJPZWJUCZWJOQOPWLWMDLZELZKZWO
      WJKZRWNWJKZUBZEUDDUDWSDEWPWQWRWPWOOKZWOBMNZRWNOKZWNBMNZRZWQWRWPWTXAXDWTWP
      XAXDUBWTWPXAXDWTWPRZXBWNWOMNZRXAXDXEXBXFWOWNUEWOSKXEWNWOQZXFEUFWTWPXGWOWN
      UGUHWNWOSUIUJUKXBXFXAXDXFXARXCXBWNWOBULUMUNUOUPUQURWIXAAWOOWHWOBMUSUTWIXC
      AWNOWHWNBMUSUTVAUHVBDEWJVCVDWIAOVHVEWJOVFVGWGWJSKWKWLVIADEFGBHLFLZILZVJTG
      LWOXIVJTRXHWOVKNREWNVLFWNVLHGWCZIJLZVMZBQVNXLVOXKQXKXLXLVPQVQXLXKVNVRZVSR
      WNXLXMVTVMTRJDWCZCHJXNWAXJWAWBWJSWDWEWF $.
  $}

  ${
    $d y z A $.  $d y z R $.
    wofib.1 $e |- A e. _V $.
    $( The only sets which are well-ordered forwards and backwards are finite
       sets.  (Contributed by Mario Carneiro, 30-Jan-2014.)  (Revised by Mario
       Carneiro, 23-May-2015.) $)
    wofib $p |- ( ( R Or A /\ A e. Fin ) <-> ( R We A /\ `' R We A ) ) $=
      ( vz vy wor cfn wcel wa wwe ccnv wofi jca adantr com wss cep wbr wn cvv
      cnvso sylanb weso coi cdm cv wral wrex csuc peano2 sucidg wceq brcnv epel
      vex bitri eleq2 bitrid rspcev syl2anc dfrex2 sylib nrex word wb eqid oicl
      ordom ordtri1 mp2an wfr c0 wne con0 oion mp1i simpr ssexd wiso oiiso mpan
      isocnv2 wefr isofr biimpar syl2an c1o 1onn ne0i fri syl22anc ex biimtrrid
      mt3i ssid ssnnfi sylancl cen simpl oien sylancr enfi syl mpbid impbii ) A
      BFZAGHZIZABJZABKZJZIZXHXIXKABLXFAXJFXGXKABUAAXJLUBMXLXFXGXIXFXKABUCNXLABU
      DZUEZGHZXGXLXNOHZXNXNPXOXLXPDUFZEUFZQKZRZSDOUGZEOUHZYAEOXROHZXTDOUHZYASYC
      XRUIZOHXRYEHZYDXRUJXROUKXTYFDYEOXTXRXQHZXQYEULYFXTXRXQQRYGXQXRQDUOEUOUMDX
      RUNUPXQYEXRUQURUSUTXTDOVAVBVCXPSZOXNPZXLYBOVDXNVDYIYHVEVHABXMXMVFZVGOXNVI
      VJXLYIYBXLYIIZOTHXNXSVKZYIOVLVMZYBYKOXNVNATHZXNVNHYKCABXMTYJVOVPXLYIVQZVR
      XLYLYIXIXNAXSXJXMVSZAXJVKZYLXKXIXNAQBXMVSZYPYNXIYRCABXMTYJVTWAXNAQBXMWBVB
      AXJWCYPYLYQXNAXSXJXMWDWEWFNYOWGOHYMYKWHOWGWIVPEDXNOTXSWJWKWLWMWNXNWOXNXNW
      PWQXLXNAWRRZXOXGVEXLYNXIYSCXIXKWSABXMTYJWTXAXNAXBXCXDMXE $.
  $}

  ${
    $d a b c d x B $.  $d a b c d T $.  $d a b c d U $.  $d a b c w x y z X $.
    $d a b c d w x y z A $.  $d a b c d w x y z P $.  $d a b c d w x y z Q $.
    $d a b c d w x y z R $.  $d a b c d w x y z S $.  $d c d x Z $.
    wemapso.t $e |- T = { <. x , y >. | E. z e. A ( ( x ` z ) S ( y ` z ) /\
        A. w e. A ( w R z -> ( x ` w ) = ( y ` w ) ) ) } $.
    $( Value of the lexicographic order on a sequence space.  (Contributed by
       Stefan O'Rear, 18-Jan-2015.) $)
    wemaplem1 $p |- ( ( P e. V /\ Q e. W ) -> ( P T Q <->
        E. a e. A ( ( P ` a ) S ( Q ` a ) /\
          A. b e. A ( b R a -> ( P ` b ) = ( Q ` b ) ) ) ) ) $=
      ( cv cfv wbr wceq wi wral wrex breqan12d eqeqan12d imbi2d ralbidv anbi12d
      fveq1 rexbidv weq fveq2 breq12d breq2 imbi1d breq1 eqeq12d imbi12d bitrdi
      wa cbvralvw cbvrexvw brabga ) CPZAPZQZVCBPZQZIRZDPZVCHRZVIVDQZVIVFQZSZTZD
      EUAZUSZCEUBZMPZFQZVRGQZIRZNPZVRHRZWBFQZWBGQZSZTZNEUAZUSZMEUBZABFGJKLVDFSZ
      VFGSZUSZVQVCFQZVCGQZIRZVJVIFQZVIGQZSZTZDEUAZUSZCEUBWJWMVPXBCEWMVHWPVOXAWK
      WLVEWNVGWOIVCVDFUHVCVFGUHUCWMVNWTDEWMVMWSVJWKWLVKWQVLWRVIVDFUHVIVFGUHUDUE
      UFUGUIXBWICMECMUJZWPWAXAWHXCWNVSWOVTIVCVRFUKVCVRGUKULXCXAVIVRHRZWSTZDEUAW
      HXCWTXEDEXCVJXDWSVCVRVIHUMUNUFXEWGDNEDNUJZXDWCWSWFVIWBVRHUOXFWQWDWRWEVIWB
      FUKVIWBGUKUPUQUTURUGVAUROVB $.

    ${
      $d ph d $.
      wemaplem2.p $e |- ( ph -> P e. ( B ^m A ) ) $.
      wemaplem2.x $e |- ( ph -> X e. ( B ^m A ) ) $.
      wemaplem2.q $e |- ( ph -> Q e. ( B ^m A ) ) $.
      wemaplem2.r $e |- ( ph -> R Or A ) $.
      wemaplem2.s $e |- ( ph -> S Po B ) $.
      ${
        wemaplem2.px1 $e |- ( ph -> a e. A ) $.
        wemaplem2.px2 $e |- ( ph -> ( P ` a ) S ( X ` a ) ) $.
        wemaplem2.px3 $e |- ( ph -> A. c e. A ( c R a ->
            ( P ` c ) = ( X ` c ) ) ) $.
        wemaplem2.xq1 $e |- ( ph -> b e. A ) $.
        wemaplem2.xq2 $e |- ( ph -> ( X ` b ) S ( Q ` b ) ) $.
        wemaplem2.xq3 $e |- ( ph -> A. c e. A ( c R b ->
            ( X ` c ) = ( Q ` c ) ) ) $.
        $( Lemma for ~ wemapso .  Transitivity.  (Contributed by Stefan O'Rear,
           17-Jan-2015.)  (Revised by AV, 21-Jul-2024.) $)
        wemaplem2 $p |- ( ph -> P T Q ) $=
          ( vd wbr cv cfv wceq wi wral wrex cif wcel ifcld adantr breq1 eqeq12d
          fveq2 imbi12d rspcdva imp breqtrd iftrue fveq2d breq12d adantl mpbird
          wa wb wpo w3a cmap co wf elmapi syl ffvelcdmd 3jca syl5ibcom syl22anc
          potr ifeq1 ifid eqtrdi eqbrtrd wn wor sopo po2nr syl12anc nan iffalse
          mpbi solin mpjao3dan r19.26 sylanbrc anim12 eqtr ralimi simpl1 simpl2
          w3o simpr simpl3 soltmin syl13anc biimpd imim1d ralimdva syl2im breq2
          syl6 mpd imbi1d ralbidv anbi12d rspcev wemaplem1 syl2anc ) AHILUJZUIU
          KZHULZYGIULZKUJZPUKZYGJUJZYKHULZYKIULZUMZUNZPFUOZVMZUIFUPZANUKZOUKZJU
          JZYTUUAUQZFURUUCHULZUUCIULZKUJZYKUUCJUJZYOUNZPFUOZYSAUUBYTUUAFUCUFUSA
          UUBUUFYTUUAUMZUUAYTJUJZAUUBVMZUUFYTHULZYTIULZKUJZUULUUMYTMULZUUNKAUUM
          UUPKUJZUUBUDUTAUUBUUPUUNUMZAYKUUAJUJZYKMULZYNUMZUNZUUBUURUNPFYTYKYTUM
          ZUUSUUBUVAUURYKYTUUAJVAUVCUUTUUPYNUUNYKYTMVCYKYTIVCVBVDUHUCVEVFVGUUBU
          UFUUOVNAUUBUUDUUMUUEUUNKUUBUUCYTHUUBYTUUAVHZVIUUBUUCYTIUVDVIVJVKVLAUU
          JVMZUUFUUAHULZUUAIULZKUJZUVEGKVOZUVFGURZUUAMULZGURZUVGGURZVPZUVFUVKKU
          JZUVKUVGKUJZUVHAUVIUUJUBUTAUVNUUJAUVJUVLUVMAFGUUAHAHGFVQVRZURZFGHVSRH
          GFVTWAUFWBAFGUUAMAMUVQURFGMVSSMGFVTWAUFWBAFGUUAIAIUVQURZFGIVSTIGFVTWA
          UFWBWCUTAUUJUVOAUUQUUJUVOUDUUJUUMUVFUUPUVKKYTUUAHVCYTUUAMVCVJWDVFAUVP
          UUJUGUTUVIUVNVMUVOUVPVMUVHGUVFUVKUVGKWFVFWEUUJUUFUVHVNZAUUJUUDUVFUUEU
          VGKUUJUUCUUAHUUJUUCUUBUUAUUAUQUUAUUBYTUUAUUAWGUUBUUAWHWIZVIUUJUUCUUAI
          UWAVIVJVKVLAUUKVMZUUFUVHUWBUVFUVKUVGKAUUKUVFUVKUMZAYKYTJUJZYMUUTUMZUN
          ZUUKUWCUNPFUUAYKUUAUMZUWDUUKUWEUWCYKUUAYTJVAUWGYMUVFUUTUVKYKUUAHVCYKU
          UAMVCVBVDUEUFVEVFAUVPUUKUGUTWJUWBUUBWKZUVTAUUKUUBVMWKZUNUWBUWHUNAFJVO
          ZUUAFURZYTFURZUWIAFJWLZUWJUAFJWMWAUFUCFUUAYTJWNWOAUUKUUBWPWRUWHUUDUVF
          UUEUVGKUWHUUCUUAHUUBYTUUAWQZVIUWHUUCUUAIUWNVIVJWAVLAUWMUWLUWKUUBUUJUU
          KXHUAUCUFFYTUUAJWSWOWTAUWFUVBVMZPFUOZUUIAUWFPFUOUVBPFUOUWPUEUHUWFUVBP
          FXAXBAUWMUWLUWKVPZUWPUWDUUSVMZYOUNZPFUOUUIAUWMUWLUWKUAUCUFWCUWOUWSPFU
          WOUWRUWEUVAVMYOUWDUWEUUSUVAXCYMUUTYNXDXRXEUWQUWSUUHPFUWQYKFURZVMZUUGU
          WRYOUXAUUGUWRUXAUWMUWTUWLUWKUUGUWRVNUWMUWLUWKUWTXFUWQUWTXIUWMUWLUWKUW
          TXGUWMUWLUWKUWTXJYKYTUUAJFXKXLXMXNXOXPXSYRUUFUUIVMUIUUCFYGUUCUMZYJUUF
          YQUUIUXBYHUUDYIUUEKYGUUCHVCYGUUCIVCVJUXBYPUUHPFUXBYLUUGYOYGUUCYKJXQXT
          YAYBYCWOAUVRUVSYFYSVNRTBCDEFHIJKLUVQUVQUIPQYDYEVL $.
      $}

      $d ph a b c $.
      wemaplem3.px $e |- ( ph -> P T X ) $.
      wemaplem3.xq $e |- ( ph -> X T Q ) $.
      $( Lemma for ~ wemapso .  Transitivity.  (Contributed by Stefan O'Rear,
         17-Jan-2015.)  (Revised by AV, 21-Jul-2024.) $)
      wemaplem3 $p |- ( ph -> P T Q ) $=
        ( va vc vb cv cfv wbr wceq wi wral wa wrex cmap co wb wemaplem1 syl2anc
        wcel mpbid ad2antrr wor wpo simplrl simp2rl 3expa simprr simprl simprrl
        ad2antlr simprrr wemaplem2 rexlimdvaa mp2d ) AUBUEZHUFVNMUFKUGZUCUEZVNJ
        UGVPHUFVPMUFZUHUIUCFUJZUKZUBFULZUDUEZMUFWAIUFKUGZVPWAJUGVQVPIUFUHUIUCFU
        JZUKZUDFULZHILUGZAHMLUGZVTTAHGFUMUNZURZMWHURZWGVTUOOPBCDEFHMJKLWHWHUBUC
        NUPUQUSAMILUGZWEUAAWJIWHURZWKWEUOPQBCDEFMIJKLWHWHUDUCNUPUQUSAVSWEWFUIUB
        FAVNFURZVSUKZUKZWDWFUDFWOWAFURZWDUKZUKBCDEFGHIJKLMUBUDUCNAWIWNWQOUTAWJW
        NWQPUTAWLWNWQQUTAFJVAWNWQRUTAGKVBWNWQSUTAWMVSWQVCAWNWQVOVOVRWMAWQVDVEWN
        VRAWQWMVOVRVFVIWOWPWDVGWOWPWBWCVHWOWPWBWCVJVKVLVLVM $.
    $}

    $( Construct lexicographic order on a function space based on a
       well-ordering of the indices and a total ordering of the values.

       Without totality on the values or least differing indices, the best we
       can prove here is a partial order.  (Contributed by Stefan O'Rear,
       18-Jan-2015.)  (Revised by AV, 21-Jul-2024.) $)
    wemappo $p |- ( ( R Or A /\ S Po B ) -> T Po ( B ^m A ) ) $=
      ( va vb vc wa cv wcel cfv wbr simpllr cvv wor wpo cmap co wceq wi wral wn
      wrex wf elmapi adantl ffvelcdmda poirr syl2anc intnanrd wb wemaplem1 el2v
      nrexdv sylnibr simplr1 simplr2 simplr3 simplll simprl simprr wemaplem3 ex
      w3a ispod ) EGUAZFHUBZNZKLMFEUCUDZIVNKOZVOPZNZLOZVPQZVTHRZMOZVSGRWBVPQZWC
      UEUFMEUGZNZLEUIZVPVPIRZVRWELEVRVSEPZNZWAWDWIVMVTFPWAUHVLVMVQWHSVREFVSVPVQ
      EFVPUJVNVPFEUKULUMFVTHUNUOUPUTWGWFUQKKABCDEVPVPGHITTLMJURUSVAVNVQVSVOPZWB
      VOPZVJZNZVPVSIRZVSWBIRZNZVPWBIRWMWPNABCDEFVPWBGHIVSJVQWJWKVNWPVBVQWJWKVNW
      PVCVQWJWKVNWPVDVLVMWLWPVEVLVMWLWPSWMWNWOVFWMWNWOVGVHVIVK $.

    ${
      $d ph a b c d $.
      wemapsolem.1 $e |- U C_ ( B ^m A ) $.
      wemapsolem.2 $e |- ( ph -> R Or A ) $.
      wemapsolem.3 $e |- ( ph -> S Or B ) $.
      wemapsolem.4 $e |- ( ( ph /\ ( ( a e. U /\ b e. U ) /\ a =/= b ) ) ->
        E. c e. dom ( a \ b ) A. d e. dom ( a \ b ) -. d R c ) $.
      $( Lemma for ~ wemapso .  (Contributed by Stefan O'Rear, 18-Jan-2015.)
         (Revised by Mario Carneiro, 8-Feb-2015.)  (Revised by AV,
         21-Jul-2024.) $)
      wemapsolem $p |- ( ph -> T Or U ) $=
        ( cmap co wss wpo wor sopo syl wemappo syl2anc poss mpsyl wcel wceq wbr
        cv wa wo w3o wn wne df-ne cfv wi wral wrex cdif cdm crab wfn wf simprll
        sselid elmapi simprlr fndmdif eleq2d nesym eqeq12d notbid bitrid bitrdi
        ffnd fveq2 imbi1d impexp con34b imbi2i bitr4i ralbidv2 anbi12d rexbidv2
        elrab anass ad2antrr ffvelcdmda sotrieq con2bid biimprd syl12anc anim1d
        mpbid reximdva mpd cvv wemaplem1 el2v orbi12i r19.43 andir eqcom ralbii
        anbi2i orbi2i bitr2i rexbii 3bitr2i sylibr expr biimtrrid 3orrot 3orass
        wb orrd sylib issod ) ALMKJKGFUAUBZUCAYFJUDZKJUDQAFHUEGIUDZYGRAGIUEZYHS
        GIUFUGBCDEFGHIJPUHUIKYFJUJUKALUOZKULZMUOZKULZUPZUPZYJYLUMZYLYJJUNZYJYLJ
        UNZUQZUQZYRYPYQURZYOYPYSYPUSYJYLUTZYOYSYJYLVAAYNUUBYSAYNUUBUPZUPZNUOZYL
        VBZUUEYJVBZIUNZUUGUUFIUNZUQZOUOZUUEHUNZUUKYLVBZUUKYJVBZUMZVCZOFVDZUPZNF
        VEZYSUUDUUFUUGUMZUSZUUQUPZNFVEZUUSUUDUULUSZOYJYLVFVGZVDZNUVEVEUVCTUUDUV
        FUVBNUVEFUUDUUEUVEULZUVFUPUUEFULZUVAUPZUUQUPUVHUVBUPUUDUVGUVIUVFUUQUUDU
        VGUUEBUOZYJVBZUVJYLVBZUTZBFVHZULUVIUUDUVEUVNUUEUUDYJFVIYLFVIUVEUVNUMUUD
        FGYJUUDYJYFULFGYJVJUUDKYFYJQAYKYMUUBVKVLYJGFVMUGZWBUUDFGYLUUDYLYFULFGYL
        VJUUDKYFYLQAYKYMUUBVNVLYLGFVMUGZWBBFYJYLVOUIZVPUVMUVABUUEFUVMUVLUVKUMZU
        SZUVJUUEUMZUVAUVKUVLVQZUVTUVRUUTUVTUVLUUFUVKUUGUVJUUEYLWCUVJUUEYJWCVRVS
        VTWLWAUUDUVDUUPOUVEFUUDUUKUVEULZUVDVCUUKFULZUUOUSZUPZUVDVCZUWCUUPVCZUUD
        UWBUWEUVDUUDUWBUUKUVNULUWEUUDUVEUVNUUKUVQVPUVMUWDBUUKFUVMUVSUVJUUKUMZUW
        DUWAUWHUVRUUOUWHUVLUUMUVKUUNUVJUUKYLWCUVJUUKYJWCVRVSVTWLWAWDUWFUWCUWDUV
        DVCZVCUWGUWCUWDUVDWEUUPUWIUWCUULUUOWFWGWHWAWIWJUVHUVAUUQWMWAWKXAUUDUVBU
        URNFUUDUVHUPZUVAUUJUUQUWJYIUUFGULZUUGGULZUVAUUJVCAYIUUCUVHSWNUUDFGUUEYL
        UVPWOUUDFGUUEYJUVOWOYIUWKUWLUPUPZUUJUVAUWMUUTUUJGUUFUUGIWPWQWRWSWTXBXCY
        SUUHUUQUPZNFVEZUUIUULUUNUUMUMZVCZOFVDZUPZNFVEZUQUWNUWSUQZNFVEUUSYQUWOYR
        UWTYQUWOYBMLBCDEFYLYJHIJXDXDNOPXEXFYRUWTYBLMBCDEFYJYLHIJXDXDNOPXEXFXGUW
        NUWSNFXHUXAUURNFUURUWNUUIUUQUPZUQUXAUUHUUIUUQXIUXBUWSUWNUUQUWRUUIUUPUWQ
        OFUUOUWPUULUUMUUNXJWGXKXLXMXNXOXPXQXRXSYCUUAYPYQYRURYTYRYPYQXTYPYQYRYAX
        NYDYE $.
    $}

    $( Construct lexicographic order on a function space based on a
       well-ordering of the indices and a total ordering of the values.
       (Contributed by Stefan O'Rear, 18-Jan-2015.)  (Revised by Mario
       Carneiro, 8-Feb-2015.)  (Revised by AV, 21-Jul-2024.) $)
    wemapso $p |- ( ( R We A /\ S Or B ) -> T Or ( B ^m A ) ) $=
      ( va vb vc vd wor wa cv wcel wss c0 wwe cmap co ssid weso adantr wne cdif
      simpr cdm cvv wfr wbr wn wral wrex vex difexi dmex a1i wefr ad2antrr dmss
      difss ax-mp wf simprll elmapi syl fssdm simprr wceq wb simprlr fndmdifeq0
      wfn ffnd syl2anc necon3bid mpbird fri syl22anc wemapsolem ) EGUAZFHOZPZAB
      CDEFGHIFEUBUCZKLMNJWGUDWDEGOWEEGUEUFWDWEUIWFKQZWGRZLQZWGRZPZWHWJUGZPZPZWH
      WJUHZUJZUKRZEGULZWQESWQTUGZNQMQGUMUNNWQUOMWQUPWRWOWPWHWJKUQURUSUTWDWSWEWN
      EGVAVBWOEFWQWHWPWHSWQWHUJSWHWJVDWPWHVCVEWOWIEFWHVFWFWIWKWMVGWHFEVHVIZVJWO
      WTWMWFWLWMVKWOWQTWHWJWOWHEVPWJEVPWQTVLWHWJVLVMWOEFWHXAVQWOEFWJWOWKEFWJVFW
      FWIWKWMVNWJFEVHVIVQEWHWJVOVRVSVTMNEWQUKGWAWBWC $.

    $d V a b c d $.  $d W a b c d $.  $d Z a b $.
    wemapso2.u $e |- U = { x e. ( B ^m A ) | x finSupp Z } $.
    $( Lemma for ~ wemapso2 .  (Contributed by Mario Carneiro, 8-Feb-2015.)
       (Revised by AV, 1-Jul-2019.) $)
    wemapso2lem $p |- ( ( ( A e. V /\ R Or A /\ S Or B ) /\ Z e. W )
                        -> T Or U ) $=
      ( vc wcel wa cv cfsupp va vb vd wor w3a wbr cmap ssrab3 simpl2 simpl3 wne
      co cdif cdm cfn csupp cun wfr wss c0 wn wral simprll breq1 elrab2 simprbi
      wrex syl simprlr fsuppunfi cfv crab wceq wf sselid elmapi fndmdif syl2anc
      wfn ffnd wi wo neneor elun simpr cvv wb adantr 3ad2ant1 ad2antrr elsuppfn
      syl3anc mpbirand simpll1 orbi12d bitrid imbitrrid ralrimiva rabss eqsstrd
      elex sylibr ssfid wwe suppssdm fssdm unssd soss sylc wofi wefr fndmdifeq0
      simprr necon3bid mpbird fri syl22anc wemapsolem ) EKQZEGUDZFHUDZUEZMLQZRZ
      ABCDEFGHIJUAUBPUCNASZMTUFZAFEUGULZJOUHZXSXTYAYCUIZXSXTYAYCUJYDUASZJQZUBSZ
      JQZRZYJYLUKZRZRZYJYLUMUNZUOQYJMUPULZYLMUPULZUQZGURZYRUUAUSYRUTUKZUCSPSZGU
      FVAUCYRVBPYRVGYQUUAYRYQYJYLMYQYKYJMTUFZYDYKYMYOVCZYKYJYGQZUUEYFUUEAYJYGJY
      EYJMTVDOVEVFVHYQYMYLMTUFZYDYKYMYOVIZYMYLYGQZUUHYFUUHAYLYGJYEYLMTVDOVEVFVH
      VJZYQYRUUDYJVKZUUDYLVKZUKZPEVLZUUAYQYJEVSZYLEVSZYRUUOVMYQEFYJYQUUGEFYJVNY
      QJYGYJYHUUFVOYJFEVPVHZVTZYQEFYLYQUUJEFYLVNYQJYGYLYHUUIVOYLFEVPVHZVTZPEYJY
      LVQVRYQUUNUUDUUAQZWAZPEVBUUOUUAUSYQUVCPEUUNUVBYQUUDEQZRZUULMUKZUUMMUKZWBZ
      UULUUMMWCUVBUUDYSQZUUDYTQZWBUVEUVHUUDYSYTWDUVEUVIUVFUVJUVGUVEUVIUVDUVFYQU
      VDWEZUVEUUPEWFQZYCUVIUVDUVFRWGYQUUPUVDUUSWHYDUVLYPUVDYBUVLYCXSXTUVLYAEKXA
      WIWHWJYDYCYPUVDYBYCWEWJZUUDYJWFLEMWKWLWMUVEUVJUVDUVGUVKUVEUUQXSYCUVJUVDUV
      GRWGYQUUQUVDUVAWHYQXSUVDXSXTYAYCYPWNWHUVMUUDYLKLEMWKWLWMWOWPWQWRUUNPEUUAW
      SXBWTZXCYQUUAGXDZUUBYQUUAGUDZUUAUOQUVOYQUUAEUSXTUVPYQYSYTEYQEFYSYJYJMXEUU
      RXFYQEFYTYLYLMXEUUTXFXGYDXTYPYIWHUUAEGXHXIUUKUUAGXJVRUUAGXKVHUVNYQUUCYOYD
      YNYOXMYQYRUTYJYLYQUUPUUQYRUTVMYJYLVMWGUUSUVAEYJYLXLVRXNXOPUCUUAYRUOGXPXQX
      R $.

    $( An alternative to having a well-order on ` R ` in ~ wemapso is to
       restrict the function set to finitely-supported functions.  (Contributed
       by Mario Carneiro, 8-Feb-2015.)  (Revised by AV, 1-Jul-2019.) $)
    wemapso2 $p |- ( ( A e. V /\ R Or A /\ S Or B ) -> T Or U ) $=
      ( cvv wcel wor wn c0 wceq w3a wi wemapso2lem expcom so0 wb cv cfsupp cmap
      wbr co crab relfsupp brrelex2i con3i ralrimivw rabeq0 sylibr eqtrid soeq2
      wral syl mpbiri a1d pm2.61i ) LOPZEKPEGQFHQUAZJIQZUBVGVFVHABCDEFGHIJKOLMN
      UCUDVFRZVHVGVIVHSIQZIUEVIJSTVHVJUFVIJAUGZLUHUJZAFEUIUKZULZSNVIVLRZAVMVAVN
      STVIVOAVMVLVFVKLUHUMUNUOUPVLAVMUQURUSJSIUTVBVCVDVE $.
  $}

  ${
    $d A x y z $.
    $( The alternate definition of the cardinal of a set given in ~ cardval2
       always gives a set, and indeed an ordinal.  (Contributed by Mario
       Carneiro, 14-Jan-2013.) $)
    card2on $p |- { x e. On | x ~< A } e. On $=
      ( vy vz cvv wcel cv csdm wbr con0 crab word wss wa wi wal cdom imp breq1
      c0 wtr onelon vex onelss ssdomg mpsyl domsdomtr anassrs sylan exp31 com12
      jca anim2i impd elrab 3imtr4g gen2 dftr2 mpbir ssrab2 ordon trssord mp3an
      hartogs sdomdom a1i ss2rabi ssexg mpan elong 3syl mpbiri wceq 0elon eleq1
      wb wn wrex wne df-ne rabn0 bitr3i relsdom brrelex2i rexlimivw sylbi nsyl4
      pm2.61i ) BEFZAGZBHIZAJKZJFZWIWMWLLZWLUAZWLJMJLWNWOCGZDGZFZWQWLFZNWPWLFZO
      ZDPCPXACDWRWSWTWRWQJFZWQBHIZNWPJFZWPBHIZNZWSWTWRXBXCXFXBWRXCXFOXBWRXCXFXB
      WRNZXDWPWQQIZNXCXFXGXDXHWQWPUBWQEFXGWPWQMZXHDUCXBWRXIWQWPUDRWPWQEUEUFULXD
      XHXCXFXHXCNXEXDWPWQBUGUMUHUIUJUKUNWKXCAWQJWJWQBHSUOWKXEAWPJWJWPBHSUOUPRUQ
      CDWLURUSWKAJUTVAWLJVBVCWIWJBQIZAJKZJFZWLEFZWMWNVPABEVDWLXKMXLXMWKXJAJWKXJ
      OWJJFWJBVEVFVGWLXKJVHVIWLEVJVKVLWLTVMZWMWIXNWMTJFVNWLTJVOVLXNVQZWKAJVRZWI
      XOWLTVSXPWLTVTWKAJWAWBWKWIAJWJBHWCWDWEWFWGWH $.
  $}

  ${
    $d A x y n $.
    card2inf.1 $e |- A e. _V $.
    $( The alternate definition of the cardinal of a set given in ~ cardval2
       has the curious property that for non-numerable sets (for which ~ ndmfv
       yields ` (/) ` ), it still evaluates to a nonempty set, and indeed it
       contains ` _om ` .  (Contributed by Mario Carneiro, 15-Jan-2013.)
       (Revised by Mario Carneiro, 27-Apr-2015.) $)
    card2inf $p |- ( -. E. y e. On y ~~ A -> _om C_ { x e. On | x ~< A } ) $=
      ( vn cv cen wbr con0 wn csdm com wcel c0 breq1 rspcev cdom brsdom mpbiran
      wss wrex wral crab csuc 0elon mpan con3i 0dom wa sucdom2 ad2antll wi nnon
      sylibr onsuc ex 3syl con3dimp sylanbrc exp32 finds2 com12 ralrimiv omsson
      adantrr ssrab ) BFZCGHZBIUAZJZAFZCKHZALUBZLVLAIUCTZVJVLALVKLMVJVLVLNCKHZE
      FZCKHZVPUDZCKHZVJAEVKNCKOVKVPCKOVKVRCKOVJNCGHZJZVOVTVINIMVTVIUEVHVTBNIVGN
      CGOPUFUGVONCQHWACDUHNCRSUNVPLMZVJVQVSWBVJVQUIUIVRCQHZVRCGHZJZVSVQWCWBVJVP
      CUJUKWBVJWEVQWBWDVIWBVPIMVRIMZWDVIULVPUMVPUOWFWDVIVHWDBVRIVGVRCGOPUPUQURV
      EVRCRUSUTVAVBVCVNLITVMVDVLAILVFSUN $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Hartogs function
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c har $.

  $( Class symbol for the Hartogs function. $)
  char $a class har $.

  ${
    $d x y $.
    $( Define the Hartogs function as mapping a set to the class of ordinals it
       dominates.  That class is an ordinal by ~ hartogs , which is used in
       ~ harf .

       The _Hartogs number_ of a set is the least ordinal not dominated by that
       set.  Theorem ~ harval2 proves that the Hartogs function actually gives
       the Hartogs number for well-orderable sets.

       The Hartogs number of an ordinal is its cardinal successor.  This is
       proved for finite ordinal in ~ harsucnn .

       Traditionally, the Hartogs number of a set ` X ` is written
       ` aleph ( X ) ` , and its cardinal successor, ` X + ` ; we use
       functional notation for this, and cannot use the aleph symbol because it
       is taken for the enumerating function of the infinite initial ordinals
       ~ df-aleph .

       Some authors define the Hartogs number of a set to be the least
       *infinite* ordinal which does not inject into it, thus causing the range
       to consist only of alephs.  We use the simpler definition where the
       value can be any successor cardinal.  (Contributed by Stefan O'Rear,
       11-Feb-2015.) $)
    df-har $a |- har = ( x e. _V |-> { y e. On | y ~<_ x } ) $.
  $}

  ${
    $d x y X $.  $d x y Y $.
    $( Functionality of the Hartogs function.  (Contributed by Stefan O'Rear,
       11-Feb-2015.) $)
    harf $p |- har : _V --> On $=
      ( vx vy cvv con0 cv cdom wbr crab char df-har hartogs fmpti ) ACDBEAEZFGB
      DHIABJBMCKL $.

    $( Values of the Hartogs function are ordinals (closure of the Hartogs
       function in the ordinals).  (Contributed by Stefan O'Rear,
       11-Feb-2015.) $)
    harcl $p |- ( har ` X ) e. On $=
      ( cvv con0 char harf 0elon f0cli ) BCADEFG $.

    $( Function value of the Hartogs function.  (Contributed by Stefan O'Rear,
       11-Feb-2015.) $)
    harval $p |- ( X e. V -> ( har ` X ) = { y e. On | y ~<_ X } ) $=
      ( vx wcel cvv char cfv cdom wbr con0 crab wceq elex breq2 rabbidv hartogs
      cv df-har fvmpt3 syl ) CBECFECGHARZCIJZAKLZMCBNDCUBDRZIJZAKLUDFGKUECMUFUC
      AKUECUBIOPDASAUEFQTUA $.

    $( The Hartogs number of a set contains exactly the ordinals that set
       dominates.  Combined with ~ harcl , this implies that the Hartogs number
       of a set is greater than all ordinals that set dominates.  (Contributed
       by Stefan O'Rear, 11-Feb-2015.)  (Revised by Mario Carneiro,
       15-May-2015.) $)
    elharval $p |- ( Y e. ( har ` X ) <->
          ( Y e. On /\ Y ~<_ X ) ) $=
      ( vy char cfv wcel cvv con0 cdom wbr wa elfvex reldom brrelex2i adantl cv
      crab harval eleq2d breq1 elrab bitrdi pm5.21nii ) BADEZFZAGFZBHFZBAIJZKZB
      ADLUHUFUGBAIMNOUFUEBCPZAIJZCHQZFUIUFUDULBCGARSUKUHCBHUJBAITUAUBUC $.

    $( The Hartogs number of a set does not inject into that set.  (Contributed
       by Stefan O'Rear, 11-Feb-2015.)  (Revised by Mario Carneiro,
       15-May-2015.) $)
    harndom $p |- -. ( har ` X ) ~<_ X $=
      ( char cfv wcel cdom wbr harcl onirri con0 elharval mpbiran mtbi ) ABCZMD
      ZMAEFZMAGZHNMIDOPAMJKL $.

    $( Weak ordering property of the Hartogs function.  (Contributed by Stefan
       O'Rear, 14-Feb-2015.) $)
    harword $p |- ( X ~<_ Y -> ( har ` X ) C_ ( har ` Y ) ) $=
      ( vy cdom wbr cv con0 crab char cfv wi wcel expcom adantr cvv wceq reldom
      domtr harval syl ss2rabdv brrelex1i brrelex2i 3sstr4d ) ABDEZCFZADEZCGHZU
      FBDEZCGHZAIJZBIJZUEUGUICGUEUGUIKUFGLUGUEUIUFABRMNUAUEAOLUKUHPABDQUBCOASTU
      EBOLULUJPABDQUCCOBSTUD $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Weak dominance
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c ~<_* $.

  $( Class symbol for the weak dominance relation. $)
  cwdom $a class ~<_* $.

  ${  $d x y z $.
    $( A set is _weakly dominated_ by a "larger" set if the "larger" set can be
       mapped onto the "smaller" set or the smaller set is empty, or
       equivalently, if the smaller set can be placed into bijection with some
       partition of the larger set.  Dominance ( ~ df-dom ) implies weak
       dominance (over ZF).  The principle asserting the converse is known as
       the partition principle and is independent of ZF. Theorem ~ fodom proves
       that the axiom of choice implies the partition principle (over ZF).  It
       is not known whether the partition principle is equivalent to the axiom
       of choice (over ZF), although it is know to imply dependent choice.
       (Contributed by Stefan O'Rear, 11-Feb-2015.) $)
    df-wdom $a |- ~<_* = { <. x , y >. | ( x = (/) \/
            E. z z : y -onto-> x ) } $.
  $}

  ${
    $d X x y z w $.  $d Y x y z w $.  $d F x y z $.  $d Z x y z w $.
    $( Weak dominance is a relation.  (Contributed by Stefan O'Rear,
       11-Feb-2015.) $)
    relwdom $p |- Rel ~<_* $=
      ( vx vy vz cv c0 wceq wfo wex wo cwdom df-wdom relopabiv ) ADZEFBDMCDGCHI
      ABJABCKL $.

    $( Property of weak dominance (definitional form).  (Contributed by Stefan
       O'Rear, 11-Feb-2015.) $)
    brwdom $p |- ( Y e. V -> ( X ~<_* Y <->
          ( X = (/) \/ E. z z : Y -onto-> X ) ) ) $=
      ( vx vy wcel cvv cwdom wbr c0 wceq cv wfo wex wo wb wi a1i exbidv relwdom
      elex brrelex1i 0ex eleq1a ax-mp crn forn vex rnex eqeltrrdi exlimiv eqeq1
      jaoi foeq3 orbi12d foeq2 orbi2d df-wdom brabg expcom pm5.21ndd syl ) DBGD
      HGZCDIJZCKLZDCAMZNZAOZPZQZDBUBVDCHGZVEVJVEVLRVDCDIUAUCSVJVLRVDVFVLVIKHGVF
      VLRUDKHCUEUFVHVLAVHCVGUGHDCVGUHVGAUIUJUKULUNSVLVDVKEMZKLZFMZVMVGNZAOZPVFV
      OCVGNZAOZPVJEFCDHHIVMCLZVNVFVQVSVMCKUMVTVPVRAVMCVOVGUOTUPVODLZVSVIVFWAVRV
      HAVODCVGUQTUREFAUSUTVAVBVC $.

    $( Property of weak dominance, forward direction only.  (Contributed by
       Mario Carneiro, 5-May-2015.) $)
    brwdomi $p |- ( X ~<_* Y -> ( X = (/) \/ E. z z : Y -onto-> X ) ) $=
      ( cwdom wbr c0 wceq cv wfo wex wo cvv wb relwdom brrelex2i brwdom syl ibi
      wcel ) BCDEZBFGCBAHIAJKZTCLSTUAMBCDNOALBCPQR $.

    $( Weak dominance over nonempty sets.  (Contributed by Stefan O'Rear,
       11-Feb-2015.)  (Revised by Mario Carneiro, 5-May-2015.) $)
    brwdomn0 $p |- ( X =/= (/) -> ( X ~<_* Y <-> E. z z : Y -onto-> X ) ) $=
      ( c0 wne cvv wcel cwdom wbr cv wfo wex relwdom brrelex2i a1i cdm fof fdmd
      wi wb vex dmex eqeltrrdi exlimiv wo brwdom wn df-ne biorf sylbi sylan9bbr
      wceq bicomd ex pm5.21ndd ) BDEZCFGZBCHIZCBAJZKZALZURUQSUPBCHMNOVAUQSUPUTU
      QAUTCUSPFUTCBUSCBUSQRUSAUAUBUCUDOUPUQURVATUQURBDULZVAUEZUPVAAFBCUFUPVAVCU
      PVBUGVAVCTBDUHVBVAUIUJUMUKUNUO $.

    $( Any set weakly dominates the empty set.  (Contributed by Stefan O'Rear,
       11-Feb-2015.) $)
    0wdom $p |- ( X e. V -> (/) ~<_* X ) $=
      ( vz wcel c0 cwdom wbr wceq cv wfo wex wo eqid orci brwdom mpbiri ) BADEB
      FGEEHZBECIJCKZLQREMNCAEBOP $.

    $( An onto function implies weak dominance.  (Contributed by Stefan O'Rear,
       11-Feb-2015.) $)
    fowdom $p |- ( ( F e. V /\ F : Y -onto-> X ) -> X ~<_* Y ) $=
      ( vz wcel cvv wfo cwdom wbr elex wa c0 wceq cv wex wo foeq1 spcegv imp wb
      olcd wf fof dmfex sylan2 brwdom syl mpbird sylan ) ABFAGFZDCAHZCDIJZABKUK
      ULLZUMCMNZDCEOZHZEPZQZUNURUOUKULURUQULEAGDCUPARSTUBUNDGFZUMUSUAULUKDCAUCU
      TDCAUDDCGAUEUFEGCDUGUHUIUJ $.

    $( Reflexivity of weak dominance.  (Contributed by Stefan O'Rear,
       11-Feb-2015.) $)
    wdomref $p |- ( X e. V -> X ~<_* X ) $=
      ( wcel cid cres cvv wfo cwdom wbr resiexg wf1o f1ofo ax-mp fowdom sylancl
      f1oi ) BACDBEZFCBBQGZBBHIBAJBBQKRBPBBQLMQFBBNO $.

    $( Alternate characterization of the weak dominance predicate which does
       not require special treatment of the empty set.  (Contributed by Stefan
       O'Rear, 11-Feb-2015.) $)
    brwdom2 $p |- ( Y e. V -> ( X ~<_* Y <->
          E. y e. ~P Y E. z z : y -onto-> X ) ) $=
      ( vx vw wcel cvv cwdom cv wfo wex c0 wceq wa foeq1 adantl cun wfn wbr cpw
      wrex elex 0wdom breq1 syl5ibrcom imp 0elpw wf1o f1o0 f1ofo 0ex spcev mp2b
      wb foeq2 exbidv rspcev mp2an foeq3 rexbidv mpbiri 2thd wne brwdomn0 pwidg
      cbvexvw ad2antrr sylancom ex biimtrid n0 biimpi ad2antlr cdif csn cxp vex
      difexg vsnex xpexg sylancl unexg sylancr adantr crn fofn fnconstg disjdif
      mp1i cin fnund elpwi undif sylib ad2antrl fneq2d mpbid rnun forn ad2antll
      a1i wss uneq1d fconst6g frnd ssequn2 eqtrd eqtrid df-fo sylanbrc exlimddv
      spcedv expr exlimdv rexlimdva impbid bitrd pm2.61dane syl ) ECHEIHZDEJUAZ
      AKZDBKZLZBMZAEUBZUCZUPZECUDYBYJDNYBDNOZPYCYIYBYKYCYBYCYKNEJUAIEUEDNEJUFUG
      UHYKYIYBYKYIYDNYELZBMZAYHUCZNYHHNNYELZBMZYNEUINNNUJNNNLZYPUKNNNULYOYQBNUM
      NNYENQUNUOYMYPANYHYDNOYLYOBYDNNYEUQURUSUTYKYGYMAYHYKYFYLBDNYDYEVAURVBVCRV
      DYBDNVEZPZYCEDFKZLZFMZYIYRYCUUBUPYBFDEVFRYSUUBYIUUBEDYELZBMZYSYIUUAUUCFBE
      DYTYEQVHYSUUDYIYSUUDEYHHZYIYBUUEYRUUDEIVGVIYGUUDAEYHYDEOYFUUCBYDEDYEUQURU
      SVJVKVLYSYGUUBAYHYSYDYHHZPYFUUBBYSUUFYFUUBYSUUFYFPZPZGKZDHZUUBGYRUUJGMZYB
      UUGYRUUKGDVMVNVOUUHUUJPZUUAEDYEEYDVPZUUIVQZVRZSZLZFIUUPYSUUPIHZUUGUUJYBUU
      RYRYBYEIHUUOIHZUURBVSYBUUMIHUUNIHUUSEYDIVTGWAUUMUUNIIWBWCYEUUOIIWDWEWFVIU
      ULUUPETZUUPWGZDOUUQUULUUPYDUUMSZTUUTUULYDUUMYEUUOUUGYEYDTZYSUUJYFUVCUUFYD
      DYEWHRVOUUIIHUUOUUMTUULGVSUUMUUIIWIWKYDUUMWLNOUULYDEWJXCWMUULUVBEUUPUUHUV
      BEOZUUJUUFUVDYSYFUUFYDEXDUVDYDEWNYDEWOWPWQWFWRWSUULUVAYEWGZUUOWGZSZDYEUUO
      WTUULUVGDUVFSZDUULUVEDUVFUUHUVEDOZUUJYFUVIYSUUFYDDYEXAXBWFXEUULUVFDXDZUVH
      DOUUJUVJUUHUUJUUMDUUOUUMUUIDXFXGRUVFDXHWPXIXJEDUUPXKXLEDYTUUPQXNXMXOXPXQX
      RXSXTYA $.

    $( Weak dominance is implied by dominance in the usual sense.  (Contributed
       by Stefan O'Rear, 11-Feb-2015.) $)
    domwdom $p |- ( X ~<_ Y -> X ~<_* Y ) $=
      ( vy cdom wbr cwdom c0 wceq cv wfo wex wo wn wa wb cvv wcel reldom mpbird
      syl csdm wne neqne adantl brrelex1i 0sdomg adantr simpl fodomr syl2anc ex
      orrd brrelex2i brwdom ) ABDEZABFEZAGHZBACIJCKZLZUOUQURUOUQMZURUOUTNZGAUAE
      ZUOURVAVBAGUBZUTVCUOAGUCUDUOVBVCOZUTUOAPQVDABDRUEAPUFTUGSUOUTUHBACUIUJUKU
      LUOBPQUPUSOABDRUMCPABUNTS $.

    $( Transitivity of weak dominance.  (Contributed by Stefan O'Rear,
       11-Feb-2015.)  (Revised by Mario Carneiro, 5-May-2015.) $)
    wdomtr $p |- ( ( X ~<_* Y /\ Y ~<_* Z ) -> X ~<_* Z ) $=
      ( vz vy cwdom wbr wa c0 cvv wcel adantl syl wne cv wfo wex brwdomn0 mpbid
      wb wceq wi relwdom brrelex2i 0wdom breq1 syl5ibrcom simpll simpllr simplr
      cdm crn dm0rn0 necon3bii a1i fof fdmd neeq1d forn 3bitr3rd ccom coex foco
      vex fowdom sylancr expr exlimdv mpd exlimddv ex pm2.61dne ) ABFGZBCFGZHZA
      CFGZAIVOCJKZAIUAZVPUBVNVQVMBCFUCUDLVQVPVRICFGJCUEAICFUFUGMVOAINZVPVOVSHZB
      ADOZPZVPDVTVMWBDQZVMVNVSUHVSVMWCTVODABRLSVTWBHZCBEOZPZEQZVPWDVNWGVMVNVSWB
      UIWDBINZVNWGTWDVSWHVOVSWBUJWBVSWHTVTWBWAUKZINZWAULZINZWHVSWJWLTWBWIIWKIWA
      UMUNUOWBWIBIWBBAWABAWAUPUQURWBWKAIBAWAUSURUTLSEBCRMSWDWFVPEVTWBWFVPWBWFHZ
      VPVTWMWAWEVAZJKCAWNPVPWAWEDVDEVDVBCBAWAWEVCWNJACVEVFLVGVHVIVJVKVL $.

    $( Equality-like theorem for equinumerosity and weak dominance.
       (Contributed by Mario Carneiro, 18-May-2015.) $)
    wdomen1 $p |- ( A ~~ B -> ( A ~<_* C <-> B ~<_* C ) ) $=
      ( cen wbr cwdom cdom ensym endom domwdom 3syl wdomtr sylan syl impbida )
      ABDEZACFEZBCFEZPBAFEZQRPBADEBAGESABHBAIBAJKBACLMPABFEZRQPABGETABIABJNABCL
      MO $.

    $( Equality-like theorem for equinumerosity and weak dominance.
       (Contributed by Mario Carneiro, 18-May-2015.) $)
    wdomen2 $p |- ( A ~~ B -> ( C ~<_* A <-> C ~<_* B ) ) $=
      ( cen wbr cwdom cdom endom domwdom syl wdomtr syl2anr ensym 3syl impbida
      id ) ABDEZCAFEZCBFEZRRABFEZSQRPQABGETABHABIJCABKLSSBAFEZRQSPQBADEBAGEUAAB
      MBAHBAINCBAKLO $.

    $( Weak dominance strengthens to usual dominance on the power sets.
       (Contributed by Stefan O'Rear, 11-Feb-2015.)  (Revised by Mario
       Carneiro, 5-May-2015.) $)
    wdompwdom $p |- ( X ~<_* Y -> ~P X ~<_ ~P Y ) $=
      ( vz cwdom wbr cpw cdom wi c0 wceq cvv wcel relwdom brrelex2i pwexd sspwi
      wss 0ss ssdomg mpisyl pweq breq1d imbitrrid wne wfo wex brwdomn0 vex mpan
      cv fopwdom exlimiv biimtrdi pm2.61ine ) ABDEZAFZBFZGEZHAIUOURAIJZIFZUQGEZ
      UOUQKLUTUQQVAUOBKABDMNOIBBRPUTUQKSTUSUPUTUQGAIUAUBUCAIUDUOBACUJZUEZCUFURC
      ABUGVCURCVBKLVCURCUHBAVBKUKUIULUMUN $.
  $}

  ${
    $d x f A $.
    $( Cantor's Theorem, stated using weak dominance (this is actually a
       stronger statement than ~ canth2 , equivalent to ~ canth ).
       (Contributed by Mario Carneiro, 15-May-2015.) $)
    canthwdom $p |- -. ~P A ~<_* A $=
      ( vf vx cpw cwdom wbr cv wfo wex c0 wne wcel 0elpw ne0i mp1i syl cvv wceq
      wb wn brwdomn0 ibi relwdom brrelex2i foeq2 pweq foeq3 bitrd notbid vtoclg
      vex canth nexdv pm2.65i ) ADZAEFZAUOBGZHZBIZUPUSUPUOJKZUPUSSJUOLUTUPAMUOJ
      NOBUOAUAPUBUPURBUPAQLURTZUOAEUCUDCGZVBDZUQHZTVACAQVBARZVDURVEVDAVCUQHZURV
      BAVCUQUEVEVCUORVFURSVBAUFVCUOAUQUGPUHUIVBUQCUKULUJPUMUN $.
  $}

  ${
    $d A v w x y z $.  $d B v w x y z $.  $d X v w x z $.  $d ph w x y $.
    wdom2d.a $e |- ( ph -> A e. V ) $.
    wdom2d.b $e |- ( ph -> B e. W ) $.
    wdom2d.o $e |- ( ( ph /\ x e. A ) -> E. y e. B x = X ) $.
    $( Deduce weak dominance from an implicit onto function (stated in a way
       which avoids ~ ax-rep ).  (Contributed by Stefan O'Rear,
       13-Feb-2015.) $)
    wdom2d $p |- ( ph -> A ~<_* B ) $=
      ( vz vw vv cv wcel wbr cvv wceq wrex csb crab cwdom cmpt wfo rabexg xpexd
      cxp syl wf wss csbeq1 eleq1d elrab simprbi adantl fmpttd fssxp ssexd wral
      cfv wa wi eleq1 biimpcd ancrd reximdv mpd nfv nfcsb1v nfel1 nfeq2 csbeq1a
      nfan eqeq2d anbi12d cbvrexw sylib eqid fvmptg mpdan rexbiia rexrab sylibr
      bitri ralrimiva dffo3 sylanbrc fowdom syl2anc cdom ssrab2 ssdomg mpi 3syl
      domwdom wdomtr ) ADCLOZHUAZDPZLEUBZUCQZXAEUCQZDEUCQAMXACMOZHUAZUDZRPXADXF
      UEZXBAXFXADUHZRAXADRFAEGPZXARPJWTLEGUFUIIUGAXADXFUJZXFXHUKAMXAXEDXDXAPZXE
      DPZAXKXDEPXLWTXLLXDEWRXDSWSXEDCWRXDHULUMUNUOUPUQZXADXFURUIUSAXJBOZNOZXFVA
      ZSZNXATZBDUTXGXMAXRBDAXNDPZVBZCXOHUAZDPZXNYASZVBZNETZXRXTHDPZXNHSZVBZCETZ
      YEXTYGCETYIKXTYGYHCEXSYGYHVCAXSYGYFYGXSYFXNHDVDVEVFUPVGVHYHYDCNEYHNVIYBYC
      CCYADCXOHVJZVKCXNYAYJVLVNCOXOSZYFYBYGYCYKHYADCXOHVMZUMYKHYAXNYLVOVPVQVRXR
      YCNXATYEXQYCNXAXOXAPZXPYAXNYMYBXPYASYMXOEPYBWTYBLXOEWRXOSWSYADCWRXOHULUMZ
      UNUOMXOXEYAXADXFCXDXOHULXFVSVTWAVOWBWTYBYCNLEYNWCWEWDWFNBXADXFWGWHXFRDXAW
      IWJAXIXAEWKQZXCJXIXAEUKYOWTLEWLXAEGWMWNXAEWPWODXAEWQWJ $.
  $}

  ${
    $d A x y $.  $d B x y $.  $d ph x y $.  $d X x $.
    wdomd.b $e |- ( ph -> B e. W ) $.
    wdomd.o $e |- ( ( ph /\ x e. A ) -> E. y e. B x = X ) $.
    $( Deduce weak dominance from an implicit onto function.  (Contributed by
       Stefan O'Rear, 13-Feb-2015.) $)
    wdomd $p |- ( ph -> A ~<_* B ) $=
      ( cvv cv wceq wrex cab wcel abrexexg syl wi wal wss ex alrimiv ssab ssexd
      sylibr wdom2d ) ABCDEJFGADBKZGLCEMZBNZJAEFOUIJOHCBEGFPQAUGDOZUHRZBSDUITAU
      KBAUJUHIUAUBUHBDUCUEUDHIUF $.
  $}

  ${
    $d X f w x y z $.  $d Y f w x y z $.
    $( Condition for weak dominance with a condition reminiscent of ~ wdomd .
       (Contributed by Stefan O'Rear, 13-Feb-2015.)  (Revised by Mario
       Carneiro, 25-Jun-2015.) $)
    brwdom3 $p |- ( ( X e. V /\ Y e. W ) -> ( X ~<_* Y <->
          E. f A. x e. X E. y e. Y x = ( f ` y ) ) ) $=
      ( vz vw wcel cvv cv cfv wceq wrex wral wex wb wa adantl cwdom wbr wfo cpw
      elex brwdom2 wf dffo3 simprbi wi wss elpwi ssrexv syl ralimdv syl5 eximdv
      rexlimdva sylbid simpll simplr eqeq1 rexbidv fveq2 eqeq2d cbvrexvw bitrdi
      cbvralvw biimpi r19.21bi wdom2d ex exlimdv impbid syl2an ) FDJFKJZGKJZFGU
      AUBZALZBLZCLZMZNZBGOZAFPZCQZRGEJFDUEGEUEVPVQSZVRWFWGVRHLZFWAUCZCQZHGUDZOZ
      WFVQVRWLRVPHCKFGUFTWGWJWFHWKWGWHWKJZSZWIWECWIWCBWHOZAFPZWNWEWIWHFWAUGWPBA
      WHFWAUHUIWNWOWDAFWMWOWDUJZWGWMWHGUKWQWHGULWCBWHGUMUNTUOUPUQURUSWGWEVRCWGW
      EVRWGWESZHIFGKKILZWAMZVPVQWEUTVPVQWEVAWRWHWTNZIGOZHFWEXBHFPZWGWEXCWDXBAHF
      VSWHNZWDWHWBNZBGOXBXDWCXEBGVSWHWBVBVCXEXABIGVTWSNWBWTWHVTWSWAVDVEVFVGVHVI
      TVJVKVLVMVNVO $.

    $( Weak dominance implies existence of a covering function.  (Contributed
       by Stefan O'Rear, 13-Feb-2015.) $)
    brwdom3i $p |- ( X ~<_* Y -> E. f A. x e. X E. y e. Y x = ( f ` y ) ) $=
      ( cwdom wbr cv cfv wceq wrex wral wex wcel wb relwdom brrelex1i brrelex2i
      cvv brwdom3 syl2anc ibi ) DEFGZAHBHCHIJBEKADLCMZUCDSNESNUCUDODEFPQDEFPRAB
      CSSDETUAUB $.
  $}

  ${
    $d A a b f g y z $.  $d B a b f g y z $.  $d C a b f g y z $.
    $d D a b f g y z $.
    $( Weak dominance of a (disjoint) union.  (Contributed by Stefan O'Rear,
       13-Feb-2015.)  (Revised by Mario Carneiro, 25-Jun-2015.) $)
    unwdomg $p |- ( ( A ~<_* B /\ C ~<_* D /\ ( B i^i D ) = (/) ) ->
      ( A u. C ) ~<_* ( B u. D ) ) $=
      ( va vb vf vg vz cwdom wbr wceq cv cfv wrex wa cvv wcel relwdom eqeq2d vy
      cin c0 w3a wral cun wex brwdom3i 3ad2ant1 3ad2ant2 adantr brrelex1i unexg
      cif syl2an 3adant3 brrelex2i wi wo elun eqeq1 rexbidv rspcva cbvrexvw wss
      fveq2 ssun1 iftrue fveq1d biimprd reximia ssrexv mpsyl syl ancoms adantlr
      sylbi adantll bitrdi rspccva ssun2 wn minel iffalsed reximdva imp anassrs
      sylan2 adantlrl jaodan sylan2b expl 3ad2ant3 impl wdom2d expr exlimdv mpd
      exlimddv ) ABJKZCDJKZBDUBUCLZUDZEMZFMZGMZNZLZFBOZEAUEZACUFZBDUFZJKZGWTXAX
      JGUGXBEFGABUHUIXCXJPZXDXEHMZNZLZFDOZECUEZHUGZXMXCXTXJXAWTXTXBEFHCDUHUJUKX
      NXSXMHXCXJXSXMXCXJXSPZPUAIXKXLQQIMZYBBRZXFXOUNZNZXCXKQRZYAWTXAYFXBWTAQRCQ
      RYFXAABJSULCDJSULACQQUMUOUPUKXCXLQRZYAWTXAYGXBWTBQRDQRYGXAABJSUQCDJSUQBDQ
      QUMUOUPUKXCYAUAMZXKRZYHYELZIXLOZXBWTYAYIPYKURXAXBYAYIYKYIXBYAPZYHARZYHCRZ
      USYKYHACUTYLYMYKYNYAYMYKXBXJYMYKXSYMXJYKYMXJPYHXGLZFBOZYKXIYPEYHAXDYHLZXH
      YOFBXDYHXGVAVBVCYPYHYBXFNZLZIBOZYKYOYSFIBXEYBLZXGYRYHXEYBXFVFTVDBXLVEYTYJ
      IBOYKBDVGYSYJIBYCYJYSYCYEYRYHYCYBYDXFYCXFXOVHVITVJVKYJIBXLVLVMVQVNVOVPVRX
      BXSYNYKXJXBXSYNYKXSYNPXBYHYBXONZLZIDOZYKXRUUDEYHCYQXRYHXPLZFDOUUDYQXQUUEF
      DXDYHXPVAVBUUEUUCFIDUUAXPUUBYHXEYBXOVFTVDVSVTDXLVEXBUUDPYJIDOZYKDBWAXBUUD
      UUFXBUUCYJIDXBYBDRZPZYJUUCUUHYEUUBYHUUHYBYDXOUUHYCXFXOUUGXBYCWBYBDBWCVOWD
      VITVJWEWFYJIDXLVLVMWHWGWIWJWKWLWMWNWOWPWQWRWS $.
  $}

  ${
    $d A a b c f g x y $.  $d B a b c d f g x y $.  $d C a c d f g x y $.
    $d D a b c d f g x y $.
    $( Weak dominance of a Cartesian product.  (Contributed by Stefan O'Rear,
       13-Feb-2015.)  (Revised by Mario Carneiro, 25-Jun-2015.) $)
    xpwdomg $p |- ( ( A ~<_* B /\ C ~<_* D ) -> ( A X. C ) ~<_* ( B X. D ) ) $=
      ( va vb vf vc vd vg vx cwdom wa cv cfv wceq wrex wral cvv wcel vy wbr wex
      cxp brwdom3i adantr adantl wi c1st c2nd relwdom brrelex1i xpexg brrelex2i
      cop syl2an pm3.2 ralimdv com12 impcom reximdv 2ralimi syl vex opth bitrdi
      eqeq1 2rexbidv ralxp sylibr r19.21bi op1std fveq2d op2ndd opeq12d adantll
      eqeq2d rexxp wdom2d expr exlimdv ex mp2d ) ABLUBZCDLUBZMZENZFNZGNZOZPZFBQ
      ZEARZGUCZHNZINZJNZOZPZIDQZHCRZJUCZACUDZBDUDZLUBZWDWNWEEFGABUEUFWEXBWDHIJC
      DUEUGWFWMXBXEUHZGWFWMXFWFWMMXAXEJWFWMXAXEWFWMXAMZMKUAXCXDSSUANZUIOZWIOZXH
      UJOZWQOZUOZWFXCSTZXGWDASTCSTXNWEABLUKULCDLUKULACSSUMUPUFWFXDSTZXGWDBSTDST
      XOWEABLUKUNCDLUKUNBDSSUMUPUFXGKNZXCTZXPXMPZUAXDQZWFXGXQMXPWJWRUOZPZIDQFBQ
      ZXSXGYBKXCXGWKWSMZIDQZFBQZHCREARZYBKXCRXGWLWTMZHCRZEARZYFXAWMYIXAWLYHEAWL
      XAYHWLWTYGHCWLWTUQURUSURUTYGYEEHACWTWLYEWTWKYDFBWKWTYDWKWSYCIDWKWSUQVAUSV
      AUTVBVCYBYEKEHACXPWGWOUOZPZYAYCFIBDYKYAYJXTPYCXPYJXTVGWGWOWJWREVDHVDVEVFV
      HVIVJVKXRYAUAFIBDXHWHWPUOPZXMXTXPYLXJWJXLWRYLXIWHWIWHWPXHFVDZIVDZVLVMYLXK
      WPWQWHWPXHYMYNVNVMVOVQVRVJVPVSVTWAWBWAWC $.
  $}

  $( A set is weakly dominant over its image under any function.  This version
     of ~ wdomimag is stated so as to avoid ~ ax-rep .  (Contributed by Mario
     Carneiro, 25-Jun-2015.) $)
  wdomima2g $p |- ( ( Fun F /\ A e. V /\ ( F " A ) e. W ) ->
      ( F " A ) ~<_* A ) $=
    ( wfun wcel cima w3a crn cwdom df-ima cdm wbr cvv wfo wf funres syl syl2anc
    cres funforn sylib 3ad2ant1 fof wss dmres inss1 eqsstri simp2 ssexg sylancr
    cin simp3 eqeltrrid fex2 syl3anc fowdom cdom ssdomg domwdom 3ad2ant2 wdomtr
    mpi eqbrtrid ) BEZACFZBAGZDFZHZVGBATZIZAJBAKZVIVKVJLZJMZVMAJMZVKAJMVIVJNFZV
    MVKVJOZVNVIVMVKVJPZVMNFZVKDFVPVIVQVRVEVFVQVHVEVJEVQABQVJUAUBUCZVMVKVJUDRVIV
    MAUEZVFVSVMABLZULABAUFAWBUGUHZVEVFVHUIVMACUJUKVIVKVGDVLVEVFVHUMUNVMVKVJNDUO
    UPVTVJNVKVMUQSVFVEVOVHVFVMAURMZVOVFWAWDWCVMACUSVCVMAUTRVAVKVMAVBSVD $.

  $( A set is weakly dominant over its image under any function.  (Contributed
     by Stefan O'Rear, 14-Feb-2015.)  (Revised by Mario Carneiro,
     25-Jun-2015.) $)
  wdomimag $p |- ( ( Fun F /\ A e. V ) -> ( F " A ) ~<_* A ) $=
    ( wfun wcel cima cvv cwdom wbr funimaexg wdomima2g mpd3an3 ) BDACEBAFZGEMAH
    IBACJABCGKL $.

  ${
    $d f x y A $.  $d f x y B $.  $d f x y C $.
    $( Lemma for ~ unxpwdom .  (Contributed by Mario Carneiro, 15-May-2015.) $)
    unxpwdom2 $p |- ( ( A X. A ) ~~ ( B u. C ) -> ( A ~<_* B \/ A ~<_ C ) ) $=
      ( vf vx cen wbr cwdom cdom cv c1st wceq wss wa cvv syl sylancr adantr cfv
      wcel vy cxp cun ensym wf1o wex bren cres ccom cima cdif ssdif0 cdm dmxpid
      wo c0 crn wfo f1ofo forn vex eqeltrrdi dmexd eqeltrrid imassrn wf f1stres
      rnex f1of fco frnd sstrid ssexd simpr sylc domwdom wfun ffund ssun1 f1odm
      ssdomg dmex ssexg wdomima2g syl3anc wdomtr syl2anc orcd ex biimtrrid ccnv
      wne n0 csn ssun2 wb f1ofn elpreima wn wi elun df-or bitri eldifn ad2antlr
      wfn ad2antrr simprr sselid fvco3 eldifi adantl snssd simprl sseldd fvresd
      xpss1 xp1st eqeltrd elsni eqtrd ffnd a1i fnfvima eqeltrrd imim1d biimtrid
      expr mtod sylbid ssrdv wf1 f1ocnv f1of1 vsnex f1imaen2g syl22anc xpsnen2g
      impd xpexg entr domen1 mpbid olcd exlimdv pm2.61dne exlimiv sylbi ) AAUBZ
      BCUCZFGUUJUUIFGZABHGZACIGZUOZUUIUUJUDUUKUUJUUIDJZUEZDUFUUNUUJUUIDUGUUPUUN
      DUUPUUNAKUUIUHZUUOUIZBUJZUKZUPUUTUPLAUUSMZUUPUUNAUUSULUUPUVAUUNUUPUVANZUU
      LUUMUVBAUUSHGZUUSBHGZUULUVBAUUSIGZUVCUVBUUSOTZUVAUVEUUPUVFUVAUUPUUSAOUUPA
      UUIUMOAUNUUPUUIOUUPUUIUUOUQZOUUPUUJUUIUUOURUVGUUILUUJUUIUUOUSUUJUUIUUOUTP
      UUODVAZVHVBVCVDZUUPUUSUURUQAUURBVEUUPUUJAUURUUPUUIAUUQVFUUJUUIUUOVFZUUJAU
      URVFAAVGUUJUUIUUOVIZUUJUUIAUUQUUOVJQZVKVLVMZRUUPUVAVNAUUSOWAVOAUUSVPPUUPU
      VDUVAUUPUURVQBOTZUVFUVDUUPUUJAUURUVLVRUUPBUUJMZUUJOTZUVNBCVSZUUPUUJUUOUMO
      UUJUUIUUOVTUUOUVHWBVBZBUUJOWCQUVMBUUROOWDWERAUUSBWFWGWHWIWJUUTUPWLEJZUUTT
      ZEUFUUPUUNEUUTWMUUPUVTUUNEUUPUVTUUNUUPUVTNZUUMUULUWAUUOWKZUVSWNZAUBZUJZCI
      GZUUMUWACOTZUWECMUWFUUPUWGUVTUUPCUUJMUVPUWGCBWOUVRCUUJOWCQRUWAUAUWECUWAUA
      JZUWETZUWHUUJTZUWHUUOSZUWDTZNZUWHCTZUUPUWIUWMWPZUVTUUPUUOUUJXFUWOUUJUUIUU
      OWQUUJUWHUWDUUOWRPRUWAUWJUWLUWNUWJUWHBTZWSZUWNWTZUWAUWLUWNWTUWJUWPUWNUOUW
      RUWHBCXAUWPUWNXBXCUWAUWLUWQUWNUWAUWLUWQUWAUWLNUWPUVSUUSTZUVTUWSWSUUPUWLUV
      SAUUSXDXEUWAUWLUWPUWSUWAUWLUWPNZNZUWHUURSZUVSUUSUXAUXBUWKUUQSZUVSUXAUVJUW
      JUXBUXCLUUPUVJUVTUWTUVKXGUXABUUJUWHUVQUWAUWLUWPXHZXIUUJUUIUWHUUQUUOXJWGUX
      AUXCUWCTUXCUVSLUXAUXCUWKKSZUWCUXAUWKUUIKUXAUWDUUIUWKUWAUWDUUIMZUWTUWAUWCA
      MUXFUWAUVSAUVTUVSATUUPUVSAUUSXKXLXMUWCAAXQPZRUWAUWLUWPXNZXOXPUXAUWLUXEUWC
      TUXHUWKUWCAXRPXSUXCUVSXTPYAUXAUURUUJXFZUVOUWPUXBUUSTUUPUXIUVTUWTUUPUUJAUU
      RUVLYBXGUVOUXAUVQYCUXDUUJBUURUWHYDWEYEYHYIWIYFYGYSYJYKUWECOWAVOUWAUWEAFGZ
      UWFUUMWPUWAUWEUWDFGZUWDAFGZUXJUWAUUIUUJUWBYLZUVPUXFUWDOTZUXKUUPUXMUVTUUPU
      UIUUJUWBUEUXMUUJUUIUUOYMUUIUUJUWBYNPRUUPUVPUVTUVRRUXGUWAUWCOTAOTZUXNEYOUU
      PUXOUVTUVIRZUWCAOOYTQUUIUUJUWDUWBOYPYQUWAUVSOTUXOUXLEVAUXPUVSAOOYRQUWEUWD
      AUUAWGUWEACUUBPUUCUUDWIUUEYGUUFUUGUUHP $.

    $( If a Cartesian product is dominated by a union, then the base set is
       either weakly dominated by one factor of the union or dominated by the
       other.  Extracted from Lemma 2.3 of [KanamoriPincus] p. 420.
       (Contributed by Mario Carneiro, 15-May-2015.) $)
    unxpwdom $p |- ( ( A X. A ) ~<_ ( B u. C ) -> ( A ~<_* B \/ A ~<_ C ) ) $=
      ( vx cun cdom wbr cen wss wa cwdom wo cvv wcel syl wi ssexg sylancr inss2
      cin cxp cv wex reldom brrelex2i domeng ibi simprl indi simprr dfss2 sylib
      wceq eqtr3id breqtrrd unxpwdom2 ssun1 adantr ssdomg mpisyl domwdom wdomtr
      wb expcom ssun2 domtr orim12d mpd exlimddv ) AAUAZBCEZFGZVJDUBZHGZVMVKIZJ
      ZABKGZACFGZLZDVLVPDUCZVLVKMNZVLVTVCVJVKFUDUEZDVJVKMUFOUGVLVPJZAVMBTZKGZAV
      MCTZFGZLZVSWCVJWDWFEZHGWHWCVJVMWIHVLVNVOUHWCWIVMVKTZVMVMBCUIWCVOWJVMUMVLV
      NVOUJVMVKUKULUNUOAWDWFUPOWCWEVQWGVRWCWDBKGZWEVQPWCWDBFGZWKWCBMNZWDBIWLWCB
      VKIWAWMBCUQVLWAVPWBURZBVKMQRVMBSWDBMUSUTWDBVAOWEWKVQAWDBVBVDOWCWFCFGZWGVR
      PWCCMNZWFCIWOWCCVKIWAWPCBVEWNCVKMQRVMCSWFCMUSUTWGWOVRAWFCVFVDOVGVHVI $.
  $}

  ${
    $d f g k x y z A $.  $d f g k y z B $.  $d f z V $.  $d f z W $.
    $( Describe an onto function from the indexed cartesian product to the
       indexed union.  Together with ~ ixpssmapg this shows that
       ` U_ x e. A B ` and ` X_ x e. A B ` have closely linked cardinalities.
       (Contributed by Mario Carneiro, 27-Aug-2015.) $)
    ixpiunwdom $p |- ( ( A e. V /\ U_ x e. A B e. W /\ X_ x e. A B =/= (/) ) ->
        U_ x e. A B ~<_* ( X_ x e. A B X. A ) ) $=
      ( vf vy vz vg vk wcel cv cfv cvv wral wfn syl weq sylib wceq ciun cixp c0
      wne w3a cmpo cxp wfo cwdom wbr vex elixp simprbi ssiun2 sseld ralimia nfv
      wf nfiu1 nfel2 fveq2 eleq1d cbvralw adantl ralrimiva eqid fmpo ixpssmap2g
      cmap co wss 3ad2ant2 ovex ssex simp1 xpexd fexd crn ffnd dffn4 wb wrex wi
      cab wal wex n0 eliun nfixp1 nfrexw wa cif cmpt csb simplrr iftrue csbeq1a
      equcoms eqcomd eleq12d syl5ibrcom adantr nfcsb1v r19.21bi iffalse pm2.61d
      wn cdm ixpfn dmex eqeltrrdi mptelixpg mpbird nfcv cbvixp eleqtrrdi simprl
      fndmd fvmpt ad2antrl fveq1 rspc2ev syl3anc exp32 rexlimd biimtrid exlimiv
      eqeq2d sylbi 3ad2ant3 ssab sylibr rnmpo sseqtrrdi frnd eqssd foeq3 fowdom
      alrimiv syl2anc ) BDKZABCUAZEKZABCUBZUCUDZUEZFGUUDBGLZFLZMZUFZNKUUDBUGZUU
      BUUJUHZUUBUUKUIUJUUFUUKUUBNUUJUUFUUIUUBKZGBOZFUUDOUUKUUBUUJURUUFUUNFUUDUU
      HUUDKZUUNUUFUUOALZUUHMZUUBKZABOZUUNUUOUUQCKZABOZUUSUUOUUHBPUVAABCUUHFUKUL
      UMUUTUURABUUPBKZCUUBUUQABCUNUOUPQUURUUMAGBUURGUQAUUIUUBABCUSUTAGRUUQUUIUU
      BUUPUUGUUHVAVBVCSVDVEFGUUDBUUIUUBUUJUUJVFZVGSZUUFUUDBNDUUFUUDUUBBVIVJZVKZ
      UUDNKUUCUUAUVFUUEABCEVHVLUUDUVEUUBBVIVMVNQUUAUUCUUEVOVPVQUUFUULUUKUUJVRZU
      UJUHZUUFUUJUUKPUVHUUFUUKUUBUUJUVDVSUUKUUJVTSUUFUUBUVGTUULUVHWAUUFUUBUVGUU
      FUUBHLZUUITZGBWBZFUUDWBZHWDZUVGUUFUVIUUBKZUVLWCZHWEUUBUVMVKUUFUVOHUUEUUAU
      VOUUCUUEILZUUDKZIWFUVOIUUDWGUVQUVOIUVNUVICKZABWBUVQUVLAUVIBCWHUVQUVRUVLAB
      AUVPUUDABCWIZUTUVKAFUUDUVSUVKAUQWJUVQUVBUVRUVLUVQUVBUVRWKZWKZJBJARZUVIJLZ
      UVPMZWLZWMZUUDKUVBUVIUUPUWFMZTZUVLUWAUWFJBAUWCCWNZUBZUUDUWAUWFUWJKZUWEUWI
      KZJBOZUWAUWLJBUWAUWCBKZWKZUWBUWLUWOUWLUWBUVRUVQUVBUVRUWNWOUWBUWEUVIUWICUW
      BUVIUWDWPZUWBCUWICUWITAJAUWCCWQZWRWSWTXAUWOUWLUWBXGZUWDUWIKZUWAUWSJBUWAUU
      PUVPMZCKZABOZUWSJBOUVQUXBUVTUVQUVPBPZUXBABCUVPIUKZULUMXBUXAUWSAJBUXAJUQAU
      WDUWIAUWCCXCZUTAJRUWTUWDCUWIUUPUWCUVPVAUWQWTVCSXDUWRUWEUWDUWIUWBUVIUWDXEV
      BXAXFVEUWABNKUWKUWMWAUWABUVPXHNUWABUVPUVQUXCUVTABCUVPXIXBXRUVPUXDXJXKJBUW
      EUWINXLQXMAJBCUWIJCXNUXEUWQXOXPUVQUVBUVRXQUWAUWGUVIUVBUWGUVITUVQUVRJUUPUW
      EUVIBUWFUWPUWFVFHUKXSXTWSUVJUWHUVIUUGUWFMZTFGUWFUUPUUDBUUHUWFTUUIUXFUVIUU
      GUUHUWFYAYHGARUXFUWGUVIUUGUUPUWFVAYHYBYCYDYEYFYGYIYJYSUVLHUUBYKYLFGHUUDBU
      UIUUJUVCYMYNUUFUUKUUBUUJUVDYOYPUUBUVGUUKUUJYQQXMUUJNUUBUUKYRYT $.
  $}

  ${
    $d r y V $.  $d f r s t w x y z X $.
    $( The value of the Hartogs function at a set ` X ` is weakly dominated by
       ` ~P ( X X. X ) ` .  This follows from a more precise analysis of the
       bound used in ~ hartogs to prove that ` ( har `` X ) ` is an ordinal.
       (Contributed by Mario Carneiro, 15-May-2015.) $)
    harwdom $p |- ( X e. V -> ( har ` X ) ~<_* ~P ( X X. X ) ) $=
      ( vr vy vx vz vw vt vs vf wcel cfv cv cdm wss wa wceq cwdom wbr cvv copab
      char cid cres cxp w3a cdif wwe coi cpw wfo wfun crn cdom con0 crab wi cep
      wrex eqid hartogslem1 simp2i simp1i sqxpexg pwexd ssexg sylancr funex wfn
      funfn mpbi a1i simp3i harval eqtr4d sylanbrc fowdom syl2anc ssdomg mpisyl
      df-fo domwdom syl wdomtr ) BAKZBUBLZCMZNZBOUCWHUDWGOWGWHWHUEOUFWHWGUCUGZU
      HPDMZWHWIUINQPCDUAZNZRSZWLBBUEZUJZRSZWFWORSWEWKTKZWLWFWKUKZWMWEWKULZWLTKZ
      WQWLWOOZWSWEWKUMZEMBUNSEUOUPZQUQZEDFGHBIMGMZJMZLQHMFMZXFLQPXEXGURSPFWJUSG
      WJUSIHUAZJWKAICWKUTXHUTVAZVBZWEXAWOTKZWTXAWSXDXIVCZWEWNTBAVDVEZWLWOTVFVGT
      WKVHVGWEWKWLVIZXBWFQWRXNWEWSXNXJWKVJVKVLWEXBXCWFXAWSXDXIVMEABVNVOWLWFWKWA
      VPWKTWFWLVQVRWEWLWOUNSZWPWEXKXAXOXMXLWLWOTVSVTWLWOWBWCWFWLWOWDVR $.
  $}


$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
  ZF Set Theory - add the Axiom of Regularity
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Introduce the Axiom of Regularity
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x y z $.
    $( Axiom of Regularity.  An axiom of Zermelo-Fraenkel set theory.  Also
       called the Axiom of Foundation.  A rather non-intuitive axiom that
       denies more than it asserts, it states (in the form of ~ zfreg ) that
       every nonempty set contains a set disjoint from itself.  One consequence
       is that it denies the existence of a set containing itself ( ~ elirrv ).
       A stronger version that works for proper classes is proved as ~ zfregs .
       (Contributed by NM, 14-Aug-1993.) $)
    ax-reg $a |- ( E. y y e. x ->
                 E. y ( y e. x /\ A. z ( z e. y -> -. z e. x ) ) ) $.

    $( Axiom of Regularity expressed more compactly.  (Contributed by NM,
       14-Aug-2003.) $)
    axreg2 $p |- ( x e. y ->
                 E. x ( x e. y /\ A. z ( z e. x -> -. z e. y ) ) ) $=
      ( wel wn wi wal wa wex ax-reg 19.23bi ) ABDZLCADCBDEFCGHAIABACJK $.
  $}

  ${
    $d A x y z $.
    $( The Axiom of Regularity with class variables.  (Contributed by NM,
       5-Aug-1994.)  Replace sethood hypothesis with sethood antecedent.
       (Revised by BJ, 27-Apr-2021.) $)
    zfregcl $p |-
              ( A e. V -> ( E. x x e. A -> E. x e. A A. y e. x -. y e. A ) ) $=
      ( vz wel wex wn cv wral wrex wi wcel wceq eleq2 exbidv ralbidv rexeqbi1dv
      notbid imbi12d nfre1 wal axreg2 df-ral rexbii df-rex bitr2i exlimi vtoclg
      wa sylib ) AEFZAGZBEFZHZBAIZJZAEIZKZLUPCMZAGZBIZCMZHZBUPJZACKZLECDURCNZUM
      VAUSVFVGULUTAURCUPOPUQVEAURCVGUOVDBUPVGUNVCURCVBOSQRTULUSAUQAURUAULULBAFU
      OLBUBZUJAGZUSAEBUCUSVHAURKVIUQVHAURUOBUPUDUEVHAURUFUGUKUHUI $.
  $}

  ${
    $d A x y $.
    $( The Axiom of Regularity using abbreviations.  Axiom 6 of [TakeutiZaring]
       p. 21.  This is called the "weak form".  Axiom Reg of [BellMachover]
       p. 480.  There is also a "strong form", not requiring that ` A ` be a
       set, that can be proved with more difficulty (see ~ zfregs ).
       (Contributed by NM, 26-Nov-1995.)  Replace sethood hypothesis with
       sethood antecedent.  (Revised by BJ, 27-Apr-2021.) $)
    zfreg $p |- ( ( A e. V /\ A =/= (/) ) -> E. x e. A ( x i^i A ) = (/) ) $=
      ( vy wcel c0 wne wa cv wex wn wral wrex cin wceq n0 biimpi anim2i zfregcl
      imp disj rexbii biimpri 3syl ) BCEZBFGZHUEAIZBEAJZHDIBEKDUGLZABMZUGBNFOZA
      BMZUFUHUEUFUHABPQRUEUHUJADBCSTULUJUKUIABDUGBUAUBUCUD $.
  $}

  ${
    $d x y z $.
    $( The membership relation is irreflexive: no set is a member of itself.
       Theorem 105 of [Suppes] p. 54.  (This is trivial to prove from ~ zfregfr
       and ~ efrirr , but this proof is direct from the Axiom of Regularity.)
       (Contributed by NM, 19-Aug-1993.) $)
    elirrv $p |- -. x e. x $=
      ( vz vy cv wcel csn wn wral cvv wex vsnex eleq1w vsnid speivw zfregcl mp2
      wrex weq velsn wi ax9 equcoms com12 biimtrid notbid rspccv nsyli ralrimiv
      mt2i con2d ralnex sylib mt2 ) ADZUNEZBDUNFZEZGZBCDZHZCUPQZUPIEUSUPEZCJVAA
      KVBUNUPEZCACAUPLAMZNCBUPIOPUOUTGZCUPHVAGUOVECUPUOUTVBUOVBUNUSEZUTVBCARZUO
      VFCUNSVGUOVFUOVFTACACAUAUBUCUDUTVFVCVDURVCGBUNUSBARUQVCBAUPLUEUFUIUGUJUHU
      TCUPUKULUM $.
    $( $j usage 'elirrv' avoids 'ax-13'; $)
  $}

  ${
    $d A x $.
    $( No class is a member of itself.  Exercise 6 of [TakeutiZaring] p. 22.
       Theorem 1.9(i) of [Schloeder] p. 1.  (Contributed by NM, 7-Aug-1994.)
       (Proof shortened by Andrew Salmon, 9-Jul-2011.) $)
    elirr $p |- -. A e. A $=
      ( vx wcel wn wi cv wceq id eleq12d notbid elirrv vtoclg pm2.01 ax-mp ) AA
      CZODZEPBFZQCZDPBAAQAGZROSQAQASHZTIJBKLOMN $.
  $}

  $( A class is not equal to any of its elements.  (Contributed by AV,
     14-Jun-2022.) $)
  elneq $p |- ( A e. B -> A =/= B ) $=
    ( wcel wn wne wi elirr nelelne ax-mp ) BBCDABCABEFBGBBAHI $.

  $( A class is not an element of and equal to a class at the same time.
     Variant of ~ elneq analogously to ~ elnotel and ~ en2lp .  (Proposed by
     BJ, 18-Jun-2022.)  (Contributed by AV, 18-Jun-2022.) $)
  nelaneq $p |- -. ( A e. B /\ A = B ) $=
    ( wcel wceq wa wn wo wne wi elneq orc neneq olcd ja ax-mp ianor mpbir ) ABC
    ZABDZEFRFZSFZGZRABHZIUBABJRUCUBTUAKUCUATABLMNORSPQ $.

  ${
    $d x y $.
    $( The membership relation and the identity relation are disjoint.
       Variable-free version of ~ nelaneq .  (Proposed by BJ, 18-Jun-2022.)
       (Contributed by AV, 18-Jun-2022.) $)
    epinid0 $p |- ( _E i^i _I ) = (/) $=
      ( vx vy cep cid cin wel copab wa c0 df-eprel df-id ineq12i inopab wceq wn
      weq wal cv nelaneq gen2 opab0 mpbir 3eqtri ) CDEABFZABGZABPZABGZEUDUFHZAB
      GZICUEDUGABJABKLUDUFABMUIINUHOZBQAQUJABARBRSTUHABUAUBUC $.
  $}

  $( A class is equal to its successor iff it is a proper class (assuming the
     Axiom of Regularity).  (Contributed by NM, 9-Jul-2004.)  (Proof shortened
     by BJ, 16-Apr-2019.) $)
  sucprcreg $p |- ( -. A e. _V <-> suc A = A ) $=
    ( cvv wcel wn csuc sucprc wa elirr csn wss cun df-suc eqeq1i ssequn2 sylbb2
    wceq snidg ssel2 syl2an mto imnani impbii ) ABCZDAEZAPZAFUEUCUEUCGAACZAHUEA
    IZAJZAUGCUFUCUEAUGKZAPUHUDUIAALMUGANOABQUGAARSTUAUB $.

  $( 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 cv wnel cab wcel vex elirr nelir 2th eqabi eqcomi ) BACZMDZAENABMBFNA
    GMMMHIJKL $.

  $( Alternate proof of ~ ru , simplified using (indirectly) the Axiom of
     Regularity ~ ax-reg .  (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 vprc nelir wceq wb ruv neleq1 ax-mp mpbir ) ABZNCADZECZEE
    CZEEFGOEHPQIAJOEEKLM $.

  $( A class is disjoint from its singleton.  A consequence of regularity.
     (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)  (Revised by BJ,
     4-Apr-2019.) $)
  disjcsn $p |- ( A i^i { A } ) = (/) $=
    ( csn cin c0 wceq wcel wn elirr disjsn mpbir ) AABCDEAAFGAHAAIJ $.

  ${
    $d x y A $.
    $( The membership relation is well-founded on any class.  (Contributed by
       NM, 26-Nov-1995.) $)
    zfregfr $p |- _E Fr A $=
      ( vx vy cep wfr cv wss c0 wne wa cin wceq wrex dfepfr cvv wcel zfreg mpan
      wi vex incom eqeq1i rexbii sylib adantl mpgbir ) ADEBFZAGZUGHIZJUGCFZKZHL
      ZCUGMZSBBCANUIUMUHUIUJUGKZHLZCUGMZUMUGOPUIUPBTCUGOQRUOULCUGUNUKHUJUGUAUBU
      CUDUEUF $.
  $}

  $( No class has 2-cycle membership loops.  Theorem 7X(b) of [Enderton]
     p. 206.  (Contributed by NM, 16-Oct-1996.)  (Revised by Mario Carneiro,
     25-Jun-2015.) $)
  en2lp $p |- -. ( A e. B /\ B e. A ) $=
    ( cvv wcel wa wn cep wfr zfregfr efrn2lp mpan elex anim12i con3i pm2.61i )
    ACDZBCDZEZABDZBADZEZFZCGHRUBCICABJKUARSPTQABLBALMNO $.

  $( Two classes are not elements of each other simultaneously.  This is just a
     rewriting of ~ en2lp and serves as an example in the context of Godel
     codes, see ~ elnanelprv .  (Contributed by AV, 5-Nov-2023.)
     (New usage is discouraged.) $)
  elnanel $p |- ( A e. B -/\ B e. A ) $=
    ( wcel wnan wa wn en2lp df-nan mpbir ) ABCZBACZDJKEFABGJKHI $.

  ${
    $d x y $.
    $( The membership (epsilon) relation and its converse are disjoint, i.e.,
       ` _E ` is an asymmetric relation.  Variable-free version of ~ en2lp .
       (Proposed by BJ, 18-Jun-2022.)  (Contributed by AV, 19-Jun-2022.) $)
    cnvepnep $p |- ( `' _E i^i _E ) = (/) $=
      ( vy vx cep ccnv cin wel wa copab c0 df-eprel cnveqi cnvopab eqtri inopab
      ineq12i wceq wn wal cv en2lp gen2 opab0 mpbir ) CDZCEZABFZBAFZGZBAHZIUEUF
      BAHZUGBAHZEUIUDUJCUKUDUFABHZDUJCULABJKUFABLMBAJOUFUGBANMUIIPUHQZARBRUMBAA
      SBSTUAUHBAUBUCM $.

    $( The membership (epsilon) relation is not symmetric.  (Contributed by AV,
       18-Jun-2022.) $)
    epnsym $p |- `' _E =/= _E $=
      ( cep ccnv cin c0 wceq wa wb wne cnvepnep disjeq0 wi epn0 eqneqall adantl
      mpi a1i wn neqne a1d bija mp2b ) ABZACDEUBAEZUBDEZADEZFZGUBAHZIUBAJUCUFUG
      UFUGKUCUEUGUDUEADHUGLUGADMONPUCQUGUFQUBARSTUA $.
  $}

  $( A class cannot be an element of one of its elements.  (Contributed by AV,
     14-Jun-2022.) $)
  elnotel $p |- ( A e. B -> -. B e. A ) $=
    ( wcel en2lp imnani ) ABCBACABDE $.

  $( A class cannot be an element of one of its elements.  (Contributed by AV,
     14-Jun-2022.) $)
  elnel $p |- ( A e. B -> B e/ A ) $=
    ( wcel wn wnel elnotel df-nel sylibr ) ABCBACDBAEABFBAGH $.

  ${
    $d x A $.  $d x B $.  $d x C $.
    $( Lemma for ~ en3lp .  (Contributed by Alan Sare, 28-Oct-2011.) $)
    en3lplem1 $p |- ( ( A e. B /\ B e. C /\ C e. A ) ->
                      ( x = A -> ( x i^i { A , B , C } ) =/= (/) ) ) $=
      ( wcel w3a cv wceq ctp cin c0 wne simp3 syl5ibrcom tpid3g 3ad2ant3 inelcm
      eleq2 sylan2 expcom syld ) BCEZCDEZDBEZFZAGZBHZDUFEZUFBCDIZJKLZUEUHUGUDUB
      UCUDMUFBDRNUHUEUJUEUHDUIEZUJUDUBUKUCDBBCOPDUFUIQSTUA $.

    $( Lemma for ~ en3lp .  (Contributed by Alan Sare, 28-Oct-2011.) $)
    en3lplem2 $p |- ( ( A e. B /\ B e. C /\ C e. A ) -> ( x e. { A , B , C }
                      -> ( x i^i { A , B , C } ) =/= (/) ) ) $=
      ( wcel ctp cin c0 wne wceq wo wi en3lplem1 a1d tprot ineq2i neeq1i syl8ib
      imp imbitrdi w3a cv 3comr bicomi jao sylsyld 3coml w3o cab idd dftp2 abid
      wa eleq2i df-3or mpjaod ex ) BCEZCDEZDBEZUAZAUBZBCDFZEZVBVCGZHIZVAVDUMVBB
      JZVBCJZKZVFVBDJZVAVDVIVFLZVAVGVFLVDVHVFLVKABCDMVAVDVHVBCDBFZGZHIZVFVAVHVN
      LZVDUSUTURVOACDBMUCNVFVNVEVMHVCVLVBBCDOPQUDRVGVFVHUEUFSVAVDVJVFLVAVDVJVBD
      BCFZGZHIZVFVAVJVRLZVDUTURUSVSADBCMUGNVQVEHVPVCVBDBCOPQRSVAVDVIVJKZVAVDVGV
      HVJUHZVTVAVDVBWAAUIZEZWAVAVDVDWCVAVDUJVCWBVBABCDUKUNTWAAULTVGVHVJUOTSUPUQ
      $.

    $( No class has 3-cycle membership loops.  This proof was automatically
       generated from the virtual deduction proof ~ en3lpVD using a translation
       program.  (Contributed by Alan Sare, 24-Oct-2011.) $)
    en3lp $p |- -. ( A e. B /\ B e. C /\ C e. A ) $=
      ( vx wcel w3a wn ctp wceq noel eleq2 mtbiri tpid3g nsyl intn3an3d wne cin
      c0 cv cvv wrex tpex zfreg en3lplem2 com12 necon2bd rexlimiv syl pm2.61ine
      mpan ) ABEZBCEZCAEZFZGZABCHZRUPRIZUMUKULUQCUPEZUMUQURCRECJUPRCKLCAABMNOUP
      RPZDSZUPQZRIZDUPUAZUOUPTEUSVCABCUBDUPTUCUJVBUODUPUTUPEZUNVARUNVDVARPDABCU
      DUEUFUGUHUI $.
  $}

  $( Equality of two unordered pairs when one member of each pair contains the
     other member.  Closed form of ~ preleq .  (Contributed by AV,
     15-Jun-2022.) $)
  preleqg $p |- ( ( ( A e. B /\ B e. V /\ C e. D ) /\ { A , B } = { C , D } )
                  -> ( A = C /\ B = D ) ) $=
    ( wcel w3a cpr wa wo wne wb elneq 3ad2ant1 preq12nebg syld3an3 eleq12 com12
    wceq wi elnotel pm2.21d biimtrdi com3l a1d 3imp jao1i sylbid imp ) ABFZBEFZ
    CDFZGZABHCDHSZACSBDSIZUMUNUOADSBCSIZJZUOUJUKULABKZUNUQLUJUKURULABMNABCDBEOP
    UQUMUOUOUPUMUMUPUOUJUKULUPUOTZUJULUSTUKUPUJULUOUPUJDCFZULUOTADBCQUTULUODCUA
    UBUCUDUEUFRUGRUHUI $.

  ${
    preleq.b $e |- B e. _V $.
    $( Equality of two unordered pairs when one member of each pair contains
       the other member.  (Contributed by NM, 16-Oct-1996.)  (Revised by AV,
       15-Jun-2022.) $)
    preleq $p |- ( ( ( A e. B /\ C e. D ) /\ { A , B } = { C , D } ) ->
                   ( A = C /\ B = D ) ) $=
      ( wcel cvv cpr wceq wa preleqg mp3anl2 ) ABFBGFCDFABHCDHIACIBDIJEABCDGKL
      $.

    preleqALT.d $e |- D e. _V $.
    $( Alternate proof of ~ preleq , not based on ~ preleqg :  Equality of two
       unordered pairs when one member of each pair contains the other member.
       (Contributed by NM, 16-Oct-1996.)  (Proof modification is discouraged.)
       (New usage is discouraged.) $)
    preleqALT $p |- ( ( ( A e. B /\ C e. D ) /\ { A , B } = { C , D } ) ->
                   ( A = C /\ B = D ) ) $=
      ( wcel wa cpr wceq wi wn wo cvv wb jctr preq12bg syl2an biimpa ord eleq12
      en2lp anbi1d mtbiri syl6 con4d ex pm2.43a imp ) ABGZCDGZHZABICDIJZACJBDJH
      ZUMULUNULUMULUNKULUMHZUNULUOUNLADJBCJHZULLUOUNUPULUMUNUPMZUJUJBNGZHUKDNGZ
      HUMUQOUKUJUREPUKUSFPABCDBNDNQRSTUPULDCGZUKHDCUBUPUJUTUKADBCUAUCUDUEUFUGUH
      UI $.
  $}

  ${
    opthreg.1 $e |- A e. _V $.
    opthreg.2 $e |- B e. _V $.
    opthreg.3 $e |- C e. _V $.
    opthreg.4 $e |- D e. _V $.
    $( Theorem for alternate representation of ordered pairs, requiring the
       Axiom of Regularity ~ ax-reg (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.)  (Proof shortened
       by AV, 15-Jun-2022.) $)
    opthreg $p |- ( { A , { A , B } } = { C , { C , D } } <->
                    ( A = C /\ B = D ) ) $=
      ( cpr wceq wa wcel prid1 prex preleq mpanl12 preq1 eqeq1d preqr2 biimtrdi
      imdistani syl adantr preq12 preq2d eqtrd impbii ) AABIZIZCCDIZIZJZACJZBDJ
      ZKZULUMUHUJJZKZUOAUHLCUJLULUQABEMCDGMAUHCUJABNOPUMUPUNUMUPCBIZUJJUNUMUHUR
      UJACBQRBDCFHSTUAUBUOUICUHIZUKUMUIUSJUNACUHQUCUOUHUJCABCDUDUEUFUG $.
  $}

  $( The successor operation behaves like a one-to-one function (assuming the
     Axiom of Regularity).  Exercise 35 of [Enderton] p. 208 and its converse.
     (Contributed by NM, 25-Oct-2003.) $)
  suc11reg $p |- ( suc A = suc B <-> A = B ) $=
    ( csuc wceq cvv wcel wi wa wn wo sucidg eleq2 elsucg sylibd imp com23 eqcom
    ord ex sucexb en2lp ianor mpbi syl5ibcom syl5ibrcom imbitrdi notbii syl2anb
    jaao mpi nelneq ancoms biimtrid sucprc eqeqan12d biimpd 4cases suceq impbii
    pm2.21d ) ACZBCZDZABDZAEFZBEFZVCVDGZVEVFHABFZIZBAFZIZJZVGVHVJHIVLABUAVHVJUB
    UCVEVIVGVFVKVEVCVIVDVEVCVIVDGVEVCHVHVDVEVCVHVDJZVEVCAVBFZVMVEAVAFVCVNAEKVAV
    BALUDABEMNORSPVFVCVKVDVFVCVKVDGVFVCHZVKBADZVDVOVJVPVFVCVJVPJZVFVCBVAFZVQVFV
    RVCBVBFBEKVAVBBLUEBAEMNORBAQUFSPUIUJVEVFIZHVCVDVEVAEFZVBEFZIVCIVSATZVFWABTZ
    UGVAVBEUKUHUTVCVBVADZVEIZVFHZVDVAVBQWFWDVDVFWEWDIZVFWAVTIWGWEWCVEVTWBUGVBVA
    EUKUHULUTUMWEVSHVCVDWEVSVAAVBBAUNBUNUOUPUQABURUS $.

  ${
    $d x y A $.
    $( Assuming ~ ax-reg , an ordinal is a transitive class on which inclusion
       satisfies trichotomy.  (Contributed by Scott Fenton, 27-Oct-2010.) $)
    dford2 $p |- ( Ord A <-> ( Tr A /\
                    A. x e. A A. y e. A ( x e. y \/ x = y \/ y e. x ) ) ) $=
      ( word wtr cep wwe wa wel weq w3o wral df-ord wbr wfr zfregfr dfwe2 bitri
      cv epel mpbiran biid 3orbi123i 2ralbii anbi2i ) CDCEZCFGZHUFABIZABJZBAIZK
      ZBCLACLZHCMUGULUFUGASZBSZFNZUIUNUMFNZKZBCLACLZULUGCFOURCPABCFQUAUQUKABCCU
      OUHUIUIUPUJBUMTUIUBAUNTUCUDRUER $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Axiom of Infinity equivalents
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( First we prove the equivalence of our version of Infinity with
     some others. $)

  ${  $d x y z w v f $.
    $( Existence of ` _om ` implies our axiom of infinity ~ ax-inf .  The proof
       shows that the especially contrived
       class " ` ran ( rec ( ( v e. _V |-> suc v ) , x ) |`` _om ) ` " exists,
       is a subset of its union, and contains a given set ` x ` (and thus is
       nonempty).  Thus, it provides an example demonstrating that a set ` y `
       exists with the necessary properties demanded by ~ ax-inf .
       (Contributed by NM, 15-Oct-1996.)  Revised to closed form.  (Revised by
       BJ, 20-May-2024.) $)
    inf0 $p |- ( _om e. V -> E. y ( x e. y /\
                            A. z ( z e. y -> E. w ( z e. w /\ w e. y ) ) ) ) $=
      ( vv vf com wcel cv cvv csuc wel wa wex wi c0 cfv wceq eleq2 cmpt crn wal
      crdg cres fr0g elv wfn frfnom peano1 fnfvelrn mp2an eqeltrri wrex fvelrnb
      ax-mp fvex sucid sucex eqid suceq frsucmpt2 mpan2 eleqtrrid eleq1 peano2b
      wb imbitrid mpan sylbi jca2 anbi12d syl6com rexlimiv ax-gen cdm wfun fndm
      spcev id eqeltrid fnfun funrnex mpisyl anbi2d exbidv albidv spcegv mp2ani
      imbi12d syl ) HEIZAJZFKFJZLZUAZWMUDHUEZUBZIZCJZWRIZCDMZDJZWRIZNZDOZPZCUCZ
      ABMZCBMZXBDBMZNZDOZPZCUCZNZBOZQWQRZWMWRXRWMSAWMKWPUFUGWQHUHZQHIXRWRIWMWPU
      IZUJHQWQUKULUMXGCXAGJZWQRZWTSZGHUNZXFXSXAYDVGXTGHWTWQUOUPYCXFGHYCYAHIZWTY
      ALZWQRZIZYGWRIZNZXFYCYEYHYIYEYBYGIYCYHYEYBYBLZYGYBYAWQUQZURYEYKKIYGYKSYBY
      LUSFCWMYAWOYKWTLWQKWQUTWTWNVAWTYBVAVBVCVDYBWTYGVEVHYEYFHIZYIYAVFXSYMYIXTH
      YFWQUKVIVJVKXEYJDYGYFWQUQXCYGSXBYHXDYIXCYGWTTXCYGWRVEVLVSVMVNVJVOWLWRKIZW
      SXHNZXQPWLWQVPZEIWQVQZYNWLYPHEXSYPHSXTHWQVRUPWLVTWAXSYQXTHWQWBUPEWQWCWDXP
      YOBWRKBJZWRSZXIWSXOXHYRWRWMTYSXNXGCYSXJXAXMXFYRWRWTTYSXLXEDYSXKXDXBYRWRXC
      TWEWFWJWGVLWHWKWI $.
  $}

  ${
    inf1.1 $e |- E. x ( y e. x /\
                 A. y ( y e. x -> E. z ( y e. z /\ z e. x ) ) ) $.
    $( Variation of Axiom of Infinity (using ~ zfinf as a hypothesis).  Axiom
       of Infinity in [FreydScedrov] p. 283.  (Contributed by NM, 14-Oct-1996.)
       (Revised by David Abernethy, 1-Oct-2013.) $)
    inf1 $p |- E. x ( x =/= (/) /\
                 A. y ( y e. x -> E. z ( y e. z /\ z e. x ) ) ) $=
      ( wel wa wex wi wal cv c0 wne ne0i anim1i eximii ) BAEZPBCECAEFCGHBIZFAJZ
      KLZQFADPSQRBJMNO $.

    $d x y z $.
    $( Variation of Axiom of Infinity.  There exists a nonempty set that is a
       subset of its union (using ~ zfinf as a hypothesis).  Abbreviated
       version of the Axiom of Infinity in [FreydScedrov] p. 283.  (Contributed
       by NM, 28-Oct-1996.) $)
    inf2 $p |- E. x ( x =/= (/) /\ x C_ U. x ) $=
      ( cv c0 wne cuni wss wa wex wel wi wal inf1 wcel df-ss eluni imbi2i albii
      bitri anbi2i exbii mpbir ) AEZFGZUEUEHZIZJZAKUFBALZBCLCALJCKZMZBNZJZAKABC
      DOUIUNAUHUMUFUHUJBEZUGPZMZBNUMBUEUGQUQULBUPUKUJCUOUERSTUAUBUCUD $.
  $}

  ${
    $d x y w v u f $.  $d v u f G $.  $d v u f F $.  $d v u f A $.
    $d v u f B $.
    inf3lem.1 $e |- G = ( y e. _V |-> { w e. x | ( w i^i x ) C_ y } ) $.
    inf3lem.2 $e |- F = ( rec ( G , (/) ) |` _om ) $.
    inf3lem.3 $e |- A e. _V $.
    inf3lem.4 $e |- B e. _V $.
    $( Lemma for our Axiom of Infinity => standard Axiom of Infinity.  See
       ~ inf3 for detailed description.  (Contributed by NM, 28-Oct-1996.) $)
    inf3lema $p |- ( A e. ( G ` B ) <-> ( A e. x /\ ( A i^i x ) C_ B ) ) $=
      ( vf vv cv cin wss wceq ineq1 cvv crab cfv sseq1d wcel sseq2 rabbidv cmpt
      cbvrabv eqtrdi cbvmptv eqtri vex rabex fvmpt ax-mp elrab2 ) LNZANZOZEPZDU
      QOZEPLDUQEGUAZUPDQURUTEUPDUQRUBESUCVAUSLUQTZQKMEURMNZPZLUQTZVBSGVCEQVDUSL
      UQVCEURUDUEGBSCNZUQOZBNZPZCUQTZUFMSVEUFHBMSVJVEVHVCQZVJVGVCPZCUQTVEVKVIVL
      CUQVHVCVGUDUEVLVDCLUQVFUPQVGURVCVFUPUQRUBUGUHUIUJUSLUQAUKULUMUNUO $.

    $( Lemma for our Axiom of Infinity => standard Axiom of Infinity.  See
       ~ inf3 for detailed description.  (Contributed by NM, 28-Oct-1996.) $)
    inf3lemb $p |- ( F ` (/) ) = (/) $=
      ( c0 cfv crdg com cres fveq1i cvv wcel wceq 0ex fr0g ax-mp eqtri ) LFMLGL
      NOPZMZLLFUEIQLRSUFLTUALRGUBUCUD $.

    $( Lemma for our Axiom of Infinity => standard Axiom of Infinity.  See
       ~ inf3 for detailed description.  (Contributed by NM, 28-Oct-1996.) $)
    inf3lemc $p |- ( A e. _om -> ( F ` suc A ) = ( G ` ( F ` A ) ) ) $=
      ( com wcel csuc c0 crdg cres cfv frsuc fveq1i fveq2i 3eqtr4g ) DLMDNZGOPL
      QZRDUDRZGRUCFRDFRZGRODGSUCFUDITUFUEGDFUDITUAUB $.

    $( Lemma for our Axiom of Infinity => standard Axiom of Infinity.  See
       ~ inf3 for detailed description.  (Contributed by NM, 28-Oct-1996.) $)
    inf3lemd $p |- ( A e. _om -> ( F ` A ) C_ x ) $=
      ( vv vu com wcel cfv cv wss c0 wceq wi fveq2 inf3lemb eqtrdi 0ss eqsstrdi
      a1d wne wa csuc wrex nnsuc vex inf3lemc eleq2d cin fvex inf3lema biimtrdi
      simplbi ssrdv sseq1d syl5ibrcom rexlimiv syl expcom pm2.61ine ) DNOZDFPZA
      QZRZUADSDSTZVKVHVLVISVJVLVISFPSDSFUBABCDEFGHIJKUCUDVJUEUFUGVHDSUHZVKVHVMU
      IDLQZUJZTZLNUKVKLDULVPVKLNVNNOZVKVPVOFPZVJRVQMVRVJVQMQZVROVSVNFPZGPZOZVSV
      JOZVQVRWAVSABCVNEFGHILUMKUNUOWBWCVSVJUPVTRABCVSVTFGHIMUMVNFUQURUTUSVAVPVI
      VRVJDVOFUBVBVCVDVEVFVG $.

    $( Lemma for our Axiom of Infinity => standard Axiom of Infinity.  See
       ~ inf3 for detailed description.  (Contributed by NM, 28-Oct-1996.) $)
    inf3lem1 $p |- ( A e. _om -> ( F ` A ) C_ ( F ` suc A ) ) $=
      ( vv cfv csuc wss c0 wceq fveq2 suceq wcel vu fveq2d sseq12d inf3lemb 0ss
      cv eqsstri com wa wi cin sstr2 com12 anim2d inf3lemc eleq2d fvex inf3lema
      vex bitrdi peano2b sucex sylbi imbi12d imbitrrid imp ssrdv ex finds ) LUF
      ZFMZVJNZFMZOPFMZPNZFMZOUAUFZFMZVQNZFMZOZVTVSNZFMZOZDFMZDNZFMZOLUADVJPQZVK
      VNVMVPVJPFRWHVLVOFVJPSUBUCVJVQQZVKVRVMVTVJVQFRWIVLVSFVJVQSUBUCVJVSQZVKVTV
      MWCVJVSFRWJVLWBFVJVSSUBUCVJDQZVKWEVMWGVJDFRWKVLWFFVJDSUBUCVNPVPABCDDFGHIJ
      JUDVPUEUGVQUHTZWAWDWLWAUILVTWCWLWAVJVTTZVJWCTZUJZWAWOWLVJAUFZTZVJWPUKZVRO
      ZUIZWQWRVTOZUIZUJWAWSXAWQWSWAXAWRVRVTULUMUNWLWMWTWNXBWLWMVJVRGMZTWTWLVTXC
      VJABCVQDFGHIUAUSZJUOUPABCVJVRFGHILUSZVQFUQURUTWLWNVJVTGMZTXBWLWCXFVJWLVSU
      HTWCXFQVQVAABCVSDFGHIVQXDVBJUOVCUPABCVJVTFGHIXEVSFUQURUTVDVEVFVGVHVI $.

    $( Lemma for our Axiom of Infinity => standard Axiom of Infinity.  See
       ~ inf3 for detailed description.  (Contributed by NM, 28-Oct-1996.) $)
    inf3lem2 $p |- ( ( x =/= (/) /\ x C_ U. x ) ->
                    ( A e. _om -> ( F ` A ) =/= x ) ) $=
      ( vv vf wcel c0 wne wa cfv wi wceq vu com cv cuni wss fveq2 neeq1d imbi2d
      csuc inf3lemb eqeq1i eqcom necon3i adantr vex inf3lemd wn wex wpss df-pss
      sylbb pssnel sylbir ssel eluni imbitrdi biimparc inf3lemc eleq2d cin elin
      eleq2 inf3lema simprbi sseld biimtrrid biimtrdi com23 exp5c com34 exlimdv
      fvex syl5 impd sylan9r pm2.43d necon3bd syl6 sylani exp4b pm2.43a adantld
      id a2d finds com12 ) DUBNAUCZOPZWQWQUDZUEZQZDFRZWQPZXALUCZFRZWQPZSXAOFRZW
      QPZSXAUAUCZFRZWQPZSXAXIUIZFRZWQPZSXAXCSLUADXDOTZXFXHXAXOXEXGWQXDOFUFUGUHX
      DXITZXFXKXAXPXEXJWQXDXIFUFUGUHXDXLTZXFXNXAXQXEXMWQXDXLFUFUGUHXDDTZXFXCXAX
      RXEXBWQXDDFUFUGUHWRXHWTXGWQWQOXGWQTOWQTWQOTXGOWQABCDEFGHIJKUJUKOWQULVAUMU
      NXIUBNZXAXKXNXSWTXKXNSZWRWTXSXTXSWTXSXKXNXSXSWTQZXJWQUEZXKXNABCXIEFGHIUAU
      OZKUPYBXKQZXDWQNZXDXJNZUQZQZLURZYAXNYDXJWQUSYIXJWQUTLXJWQVBVCYAYHXNLYAYEY
      GXNYAYEXMWQTZYFSZYGXNSYAYEYKWTYEXDMUCZNZYLWQNZQZMURZXSYEYKSZWTYEXDWSNYPWQ
      WSXDVDMXDWQVEVFXSYOYQMXSYMYNYQXSYMYEYNYKXSYMYEYNYJYFXSYNYJQZYMYEQZYFYRYLX
      MNZXSYSYFSZYJYTYNXMWQYLVLVGXSYTYLXJGRZNZUUAXSXMUUBYLABCXIEFGHIYCKVHVIYSXD
      YLWQVJZNUUCYFXDYLWQVKUUCUUDXJXDUUCYNUUDXJUEABCYLXJFGHIMUOXIFWBVMVNVOVPVQW
      CVRVSVTWDWAWEWFYKYFXMWQYKWMWGWHWDWAWCWIWJWKWLWNWOWP $.

    $( Lemma for our Axiom of Infinity => standard Axiom of Infinity.  See
       ~ inf3 for detailed description.  In the proof, we invoke the Axiom of
       Regularity in the form of ~ zfreg .  (Contributed by NM,
       29-Oct-1996.) $)
    inf3lem3 $p |- ( ( x =/= (/) /\ x C_ U. x ) ->
                    ( A e. _om -> ( F ` A ) =/= ( F ` suc A ) ) ) $=
      ( vv wcel cv c0 wne wss wa cfv cin cuni cdif csuc inf3lemd inf3lem2 com12
      com pssdifn0 syl6an wceq wrex cvv vex difexi zfreg mpan wn eldifi biimpri
      inssdif0 anim12i fvex inf3lema sylibr eleq2d imbitrrid eldifn adantr jca2
      inf3lemc eleq2 biimprd iman sylib necon2ai syl6 expd rexlimdv syl5 syldc
      wi ) DUGMZANZOPWCWCUAQRZWCDFSZUBZOPZWEDUCFSZPZWBWEWCQWDWEWCPZWGABCDEFGHIJ
      KUDWDWBWJABCDEFGHIJKUEUFWEWCUHUIWGLNZWFTOUJZLWFUKZWBWIWFULMWGWMWCWEAUMUNL
      WFULUOUPWBWLWILWFWBWKWFMZWLWIWBWNWLRZWKWHMZWKWEMZUQZRZWIWBWOWPWRWOWPWBWKW
      EGSZMZWOWKWCMZWKWCTWEQZRXAWNXBWLXCWKWCWEURXCWLWKWCWEUTUSVAABCWKWEFGHILUMD
      FVBVCVDWBWHWTWKABCDEFGHIJKVJVEVFWNWRWLWKWCWEVGVHVIWSWEWHWEWHUJZWPWQWAWSUQ
      XDWQWPWEWHWKVKVLWPWQVMVNVOVPVQVRVSVT $.

    $( Lemma for our Axiom of Infinity => standard Axiom of Infinity.  See
       ~ inf3 for detailed description.  (Contributed by NM, 29-Oct-1996.) $)
    inf3lem4 $p |- ( ( x =/= (/) /\ x C_ U. x ) ->
                    ( A e. _om -> ( F ` A ) C. ( F ` suc A ) ) ) $=
      ( cv c0 wne cuni wss wa com wcel cfv csuc wpss inf3lem1 a1i inf3lem3 jcad
      wi df-pss imbitrrdi ) ALZMNUJUJOPQZDRSZDFTZDUAFTZPZUMUNNZQUMUNUBUKULUOUPU
      LUOUGUKABCDEFGHIJKUCUDABCDEFGHIJKUEUFUMUNUHUI $.

    $( Lemma for our Axiom of Infinity => standard Axiom of Infinity.  See
       ~ inf3 for detailed description.  (Contributed by NM, 29-Oct-1996.) $)
    inf3lem5 $p |- ( ( x =/= (/) /\ x C_ U. x ) ->
                    ( ( A e. _om /\ B e. A ) -> ( F ` B ) C. ( F ` A ) ) ) $=
      ( com wcel wa cfv wpss wi wceq fveq2 psseq2d vv vu cv wne cuni wss ancoms
      c0 elnn csuc word nnord ordsucss syl adantr peano2b imbi2d inf3lem4 com12
      sylbir vex psstr expcom syl6com a2d ad2antrr findsg sylan2b syld impancom
      ex mpd ) DLMZEDMZNZAUCZUHUDVPVPUEUFNZEFOZDFOZPZVOELMZVQVTQZVNVMWAEDUIUGVM
      WAVNWBVMWANVNEUJZDUFZWBVMVNWDQZWAVMDUKWEDULEDUMUNUOWAVMWCLMZWDWBQEUPZVMWF
      NWDWBVQVRUAUCZFOZPZQVQVRWCFOZPZQZVQVRUBUCZFOZPZQZVQVRWNUJZFOZPZQZWBUAUBDW
      CWHWCRZWJWLVQXBWIWKVRWHWCFSTUQWHWNRZWJWPVQXCWIWOVRWHWNFSTUQWHWRRZWJWTVQXD
      WIWSVRWHWRFSTUQWHDRZWJVTVQXEWIVSVRWHDFSTUQWFWAWMWGVQWAWLABCEEFGHIKKURUSUT
      WNLMZWQXAQWFWCWNUFXFVQWPWTVQXFWOWSPZWPWTQABCWNEFGHIUBVAKURWPXGWTVRWOWSVBV
      CVDVEVFVGVKVHVIVJVLUS $.

    $( Lemma for our Axiom of Infinity => standard Axiom of Infinity.  See
       ~ inf3 for detailed description.  (Contributed by NM, 29-Oct-1996.) $)
    inf3lem6 $p |- ( ( x =/= (/) /\ x C_ U. x ) -> F : _om -1-1-> ~P x ) $=
      ( vv vu cv c0 wss wa com wceq wcel wne cuni cpw wf cfv wi wral wo wn wpss
      wf1 vex inf3lem5 dfpss2 simprbi syl6 expdimp adantrl eqcom sylnib adantrr
      jaod con2d wb word nnord ordtri3 syl2an adantl sylibrd ralrimivva wfn crn
      crdg cres frfnom fneq1 mpbiri wrex fvelrnb inf3lemd fvex sylibr syl5ibcom
      elpw eleq1 rexlimiv biimtrdi ssrdv ancli mp2b df-f mpbir jctil dff13 ) AN
      ZOUAWPWPUBPQZRWPUCZFUDZLNZFUEZMNZFUEZSZWTXBSZUFZMRUGLRUGZQRWRFUKWQXGWSWQX
      FLMRRWQWTRTZXBRTZQZQZXDWTXBTZXBWTTZUHZUIZXEXKXNXDXKXLXDUIZXMWQXIXLXPUFXHW
      QXIXLXPWQXIXLQXAXCUJZXPABCXBWTFGHIMULZLULZUMXQXAXCPXPXAXCUNUOUPUQURWQXHXM
      XPUFXIWQXHXMXPWQXHXMQXCXAUJZXPABCWTXBFGHIXSXRUMXTXCXASZXDXTXCXAPYAUIXCXAU
      NUOXCXAUSUTUPUQVAVBVCXJXEXOVDZWQXHWTVEXBVEYBXIWTVFXBVFWTXBVGVHVIVJVKWSFRV
      LZFVMZWRPZQZFGOVNRVOZSZYCYFIYHYCYGRVLOGVPRFYGVQVRYCYEYCMYDWRYCXBYDTXAXBSZ
      LRVSXBWRTZLRXBFVTYIYJLRXHXAWRTZYIYJXHXAWPPYKABCWTEFGHIXSKWAXAWPWTFWBWEWCX
      AXBWRWFWDWGWHWIWJWKRWRFWLWMWNLMRWRFWOWC $.

    $( Lemma for our Axiom of Infinity => standard Axiom of Infinity.  See
       ~ inf3 for detailed description.  In the proof, we invoke the Axiom of
       Replacement in the form of ~ f1dmex .  (Contributed by NM, 29-Oct-1996.)
       (Proof shortened by Mario Carneiro, 19-Jan-2013.) $)
    inf3lem7 $p |- ( ( x =/= (/) /\ x C_ U. x ) -> _om e. _V ) $=
      ( cv c0 wne cuni wss wa com cvv wcel cpw inf3lem6 vpwex f1dmex sylancl
      wf1 ) ALZMNUGUGOPQRUGUAZFUFUHSTRSTABCDEFGHIJKUBAUCRUHSFUDUE $.
  $}

  ${
    $d x y w $.
    inf3.1 $e |- E. x ( x =/= (/) /\ x C_ U. x ) $.
    $( Note: the special <HTML> tag is handled in mmwtex.c.  All comment text
       between <HTML> and </HTML> is passed as-is to the web page, except
       that ` (symbols) ` , ~ (label) , and [(reference)] markups are still
       interpreted. $)
    $( Our Axiom of Infinity ~ ax-inf implies the standard Axiom of Infinity.
       The hypothesis is a variant of our Axiom of Infinity provided by
       ~ inf2 , and the conclusion is the version of the Axiom of Infinity
       shown as Axiom 7 in [TakeutiZaring] p. 43.  (Other standard versions are
       proved later as ~ axinf2 and ~ zfinf2 .)  The main proof is provided by
       ~ inf3lema through ~ inf3lem7 , and this final piece eliminates the
       auxiliary hypothesis of ~ inf3lem7 .  This proof is due to
       Ian Sutherland, Richard Heck, and Norman Megill and was posted
       on Usenet as shown below.  Although the result is not new, the authors
       were unable to find a published proof.
       <HTML>
       <PRE>
       (As posted to sci.logic on 30-Oct-1996, with annotations added.)

       Theorem:  The statement "There exists a nonempty set that is a subset
       of its union" implies the Axiom of Infinity.

       Proof:  Let X be a nonempty set which is a subset of its union; the
       latter
       property is equivalent to saying that for any y in X, there exists a z
       in X
       such that y is in z.

       Define by finite recursion a function F:omega-->(power X) such that
       F_0 = 0  (See ~ inf3lemb .)
       F_n+1 = {y&lt;X | y^X subset F_n}  (See ~ inf3lemc .)
       Note: ^ means intersect, &lt; means \in ("element of").
       (Finite recursion as typically done requires the existence of omega;
       to avoid this we can just use transfinite recursion restricted to omega.
       F is a class-term that is not necessarily a set at this point.)

       Lemma 1.  F_n subset F_n+1.  (See ~ inf3lem1 .)
       Proof:  By induction:  F_0 subset F_1.  If y &lt; F_n+1, then y^X subset
       F_n,
       so if F_n subset F_n+1, then y^X subset F_n+1, so y &lt; F_n+2.

       Lemma 2.  F_n =/= X.  (See ~ inf3lem2 .)
       Proof:  By induction:  F_0 =/= X because X is not empty.  Assume F_n =/=
       X.
       Then there is a y in X that is not in F_n.  By definition of X, there is
       a
       z in X that contains y.  Suppose F_n+1 = X.  Then z is in F_n+1, and z^X
       contains y, so z^X is not a subset of F_n, contrary to the definition of
       F_n+1.

       Lemma 3.  F_n =/= F_n+1.  (See ~ inf3lem3 .)
       Proof:  Using the identity y^X subset F_n &lt;-> y^(X-F_n) = 0, we have
       F_n+1 = {y&lt;X | y^(X-F_n) = 0}.  Let q = {y&lt;X-F_n | y^(X-F_n) = 0}.
       Then q subset F_n+1.  Since X-F_n is not empty by Lemma 2 and q is the
       set of \in-minimal elements of X-F_n, by Foundation q is not empty, so q
       and therefore F_n+1 have an element not in F_n.

       Lemma 4.  F_n proper_subset F_n+1.  (See ~ inf3lem4 .)
       Proof:  Lemmas 1 and 3.

       Lemma 5.  F_m proper_subset F_n, m &lt; n.  (See ~ inf3lem5 .)
       Proof:  Fix m and use induction on n > m.  Basis: F_m proper_subset
       F_m+1
       by Lemma 4.  Induction:  Assume F_m proper_subset F_n.  Then since F_n
       proper_subset F_n+1, F_m proper_subset F_n+1 by transitivity of proper
       subset.

       By Lemma 5, F_m =/= F_n for m =/= n, so F is 1-1.  (See ~ inf3lem6 .)
       Thus, the inverse of F is a function with range omega and domain a
       subset
       of power X, so omega exists by Replacement.  (See ~ inf3lem7 .)
       Q.E.D.
       </PRE> </HTML>  (Contributed by NM, 29-Oct-1996.) $)
    inf3 $p |- _om e. _V $=
      ( vy vw cv c0 wne cuni wss com cvv wcel cin crab cmpt crdg cres eqid vex
      wa inf3lem7 exlimiiv ) AEZFGUCUCHITJKLAACDUCUCCKDEUCMCEIDUCNOZFPJQZUDUDRU
      ERASZUFUABUB $.
  $}

  ${
    $d x y w $.
    $( Half of ~ infeq5 .  (Contributed by Mario Carneiro, 16-Nov-2014.) $)
    infeq5i $p |- ( _om e. _V -> E. x x C. U. x ) $=
      ( com cvv wcel c0 csn cdif cuni wpss cv wex difexg 0ex snid wn wceq disj4
      cin disj3 bitr3i peano1 eleq2 mpbii eldifbd sylbi mt4 unidif0 wlim limuni
      limom ax-mp eqtr4i psseq2i mpbir psseq1 unieq psseq2d bitrd spcegv mpisyl
      ) BCDBEFZGZCDVBVBHZIZAJZVEHZIZAKBVACLVDVBBIZEVADZVHEMNVHOZBVBPZVIOVJBVARE
      PVKBVAQBVASTVKEBVAVKEBDEVBDUABVBEUBUCUDUEUFVCBVBVCBHZBBUGBUHBVLPUJBUIUKUL
      UMUNVGVDAVBCVEVBPZVGVBVFIVDVEVBVFUOVMVFVCVBVEVBUPUQURUSUT $.

    $( The statement "there exists a set that is a proper subset of its union"
       is equivalent to the Axiom of Infinity (shown on the right-hand side in
       the form of ~ omex .)  The left-hand side provides us with a very short
       way to express the Axiom of Infinity using only elementary symbols.
       This proof of equivalence does not depend on the Axiom of Infinity.
       (Contributed by NM, 23-Mar-2004.)  (Revised by Mario Carneiro,
       16-Nov-2014.) $)
    infeq5 $p |- ( E. x x C. U. x <-> _om e. _V ) $=
      ( vy vw cv cuni wpss wex com cvv wcel c0 wne wa df-pss wceq unieq eqtr2di
      wss uni0 eqid eqtr mpdan necon3i anim1i ancoms sylbi eximi crab cmpt crdg
      cin cres vex inf3lem7 exlimiv syl infeq5i impbii ) ADZUSEZFZAGZHIJZVBUSKL
      ZUSUTRZMZAGVCVAVFAVAVEUSUTLZMVFUSUTNVGVEVFVGVDVEUSKUSUTUSKOZKUTOUSUTOVHUT
      KEKUSKPSQUSKUTUAUBUCUDUEUFUGVFVCAABCUSUSBICDUSUKBDRCUSUHUIZKUJHULZVIVITVJ
      TAUMZVKUNUOUPAUQUR $.
  $}


$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
  ZF Set Theory - add the Axiom of Infinity
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Introduce the Axiom of Infinity
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x y z w $.
    $( Axiom of Infinity.  An axiom of Zermelo-Fraenkel set theory.  This axiom
       is the gateway to "Cantor's paradise" (an expression coined by Hilbert).
       It asserts that given a starting set ` x ` , an infinite set ` y ` built
       from it exists.  Although our version is apparently not given in the
       literature, it is similar to, but slightly shorter than, the Axiom of
       Infinity in [FreydScedrov] p. 283 (see ~ inf1 and ~ inf2 ).  More
       standard versions, which essentially state that there exists a set
       containing all the natural numbers, are shown as ~ zfinf2 and ~ omex and
       are based on the (nontrivial) proof of ~ inf3 .  This version has the
       advantage that when expanded to primitives, it has fewer symbols than
       the standard version ~ ax-inf2 .  Theorem ~ inf0 shows the reverse
       derivation of our axiom from a standard one.  Theorem ~ inf5 shows a
       very short way to state this axiom.

       The standard version of Infinity ~ ax-inf2 requires this axiom along
       with Regularity ~ ax-reg for its derivation (as Theorem ~ axinf2 below).
       In order to more easily identify the normal uses of Regularity, we will
       usually reference ~ ax-inf2 instead of this one.  The derivation of this
       axiom from ~ ax-inf2 is shown by Theorem ~ axinf .

       Proofs should normally use the standard version ~ ax-inf2 instead of
       this axiom.  (New usage is discouraged.)  (Contributed by NM,
       16-Aug-1993.) $)
    ax-inf $a |- E. y ( x e. y /\
                 A. z ( z e. y -> E. w ( z e. w /\ w e. y ) ) ) $.

    $( Axiom of Infinity expressed with the fewest number of different
       variables.  (New usage is discouraged.)  (Contributed by NM,
       14-Aug-2003.) $)
    zfinf $p |- E. x ( y e. x /\
                 A. y ( y e. x -> E. z ( y e. z /\ z e. x ) ) ) $=
      ( vw wel wa wex wi wal ax-inf elequ1 anbi1d exbidv imbi12d cbvalvw anbi2i
      weq exbii mpbi ) BAEZDAEZDCEZCAEZFZCGZHZDIZFZAGTTBCEZUCFZCGZHZBIZFZAGBADC
      JUHUNAUGUMTUFULDBDBQZUATUEUKDBAKUOUDUJCUOUBUIUCDBCKLMNOPRS $.

    $( A standard version of Axiom of Infinity, expanded to primitives, derived
       from our version of Infinity ~ ax-inf and Regularity ~ ax-reg .

       This theorem should not be referenced in any proof.  Instead, use
       ~ ax-inf2 below so that the ordinary uses of Regularity can be more
       easily identified.  (New usage is discouraged.)  (Contributed by NM,
       3-Nov-1996.) $)
    axinf2 $p |- E. x ( E. y ( y e. x /\ A. z -. z e. y ) /\ A. y ( y e. x ->
          E. z ( z e. x /\ A. w ( w e. z <-> ( w e. y \/ w = y ) ) ) ) ) $=
      ( c0 cv wcel wel csuc wi wal wa wex wn wceq com eleq2 wrex df-rex bitri
      wo wb peano1 peano2 ax-gen zfinf inf2 inf3 imbi12d albidv spcev mp2an 0el
      anbi12d sucel imbi2i albii anbi12i exbii mpbi ) EAFZGZBAHZBFZIZVAGZJZBKZL
      ZAMZVCCBHNCKZLBMZVCCAHDCHDBHDFVDOUAUBDKZLCMZJZBKZLZAMEPGZVDPGZVEPGZJZBKZV
      JUCWABVDUDUEVIVRWBLAPAABCABCUFUGUHVAPOZVBVRVHWBVAPEQWCVGWABWCVCVSVFVTVAPV
      DQVAPVEQUIUJUNUKULVIVQAVBVLVHVPVBVKBVARVLBCVAUMVKBVASTVGVOBVFVNVCVFVMCVAR
      VNCDVDVAUOVMCVASTUPUQURUSUT $.

    $( A standard version of Axiom of Infinity of ZF set theory.  In English,
       it says: there exists a set that contains the empty set and the
       successors of all of its members.  Theorem ~ zfinf2 shows it converted
       to abbreviations.  This axiom was derived as Theorem ~ axinf2 above,
       using our version of Infinity ~ ax-inf and the Axiom of Regularity
       ~ ax-reg .  We will reference ~ ax-inf2 instead of ~ axinf2 so that the
       ordinary uses of Regularity can be more easily identified.  The reverse
       derivation of ~ ax-inf from ~ ax-inf2 is shown by Theorem ~ axinf .
       (Contributed by NM, 3-Nov-1996.) $)
    ax-inf2 $a |- E. x ( E. y ( y e. x /\ A. z -. z e. y ) /\ A. y ( y e. x ->
          E. z ( z e. x /\ A. w ( w e. z <-> ( w e. y \/ w = y ) ) ) ) ) $.
  $}

  ${
    $d x y z w $.
    $( A standard version of the Axiom of Infinity, using definitions to
       abbreviate.  Axiom Inf of [BellMachover] p. 472.  (See ~ ax-inf2 for the
       unabbreviated version.)  (Contributed by NM, 30-Aug-1993.) $)
    zfinf2 $p |- E. x ( (/) e. x /\ A. y e. x suc y e. x ) $=
      ( vz vw c0 cv wcel csuc wral wa wex wel wn wal weq wo wrex df-rex bitri
      wb wi ax-inf2 0el sucel ralbii df-ral anbi12i exbii mpbir ) EAFZGZBFZHUJG
      ZBUJIZJZAKBALZCBLMCNZJBKZUPCALDCLDBLDBOPTDNZJCKZUABNZJZAKABCDUBUOVBAUKURU
      NVAUKUQBUJQURBCUJUCUQBUJRSUNUTBUJIVAUMUTBUJUMUSCUJQUTCDULUJUDUSCUJRSUEUTB
      UJUFSUGUHUI $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Existence of omega (the set of natural numbers)
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  ${
    $d x y $.
    $( The existence of omega (the class of natural numbers).  Axiom 7 of
       [TakeutiZaring] p. 43.  Remark 1.21 of [Schloeder] p. 3.  This theorem
       is proved assuming the Axiom of Infinity and in fact is equivalent to
       it, as shown by the reverse derivation ~ inf0 .

       A finitist (someone who doesn't believe in infinity) could, without
       contradiction, replace the Axiom of Infinity by its denial
       ` -. _om e. _V ` ; this would lead to ` _om = On ` by ~ omon and
       ` Fin = _V ` (the universe of all sets) by ~ fineqv .  The finitist
       could still develop natural number, integer, and rational number
       arithmetic but would be denied the real numbers (as well as much of the
       rest of mathematics).  In deference to the finitist, much of our
       development is done, when possible, without invoking the Axiom of
       Infinity; an example is Peano's axioms ~ peano1 through ~ peano5 (which
       many textbooks prove more easily assuming Infinity).  (Contributed by
       NM, 6-Aug-1994.) $)
    omex $p |- _om e. _V $=
      ( vx vy com cv wss cvv wcel vex ssex c0 csuc wral zfinf2 wel ax-1 ralimi2
      wa wi peano5 sylan2 eximii exlimiiv ) CADZEZCFGACUCAHIJUCGZBDZKUCGZBUCLZQ
      UDAABMUHUEBANUGRZBCLUDUGUIBUCCUIUFCGOPBUCSTUAUB $.
  $}

  ${
    $d x y z w $.
    $( The first version of the Axiom of Infinity ~ ax-inf proved from the
       second version ~ ax-inf2 .  Note that we didn't use ~ ax-reg , unlike
       the other direction ~ axinf2 .  (Contributed by NM, 24-Apr-2009.) $)
    axinf $p |- E. y ( x e. y /\
                 A. z ( z e. y -> E. w ( z e. w /\ w e. y ) ) ) $=
      ( com cvv wcel wel wa wex wi wal omex inf0 ax-mp ) EFGABHCBHCDHDBHIDJKCLI
      BJMABCDFNO $.
  $}

  $( The statement "there exists a set that is a proper subset of its union" is
     equivalent to the Axiom of Infinity (see Theorem ~ infeq5 ).  This
     provides us with a very compact way to express the Axiom of Infinity using
     only elementary symbols.  (Contributed by NM, 3-Jun-2005.) $)
  inf5 $p |- E. x x C. U. x $=
    ( com cvv wcel cv cuni wpss wex omex infeq5i ax-mp ) BCDAEZLFGAHIAJK $.

  $( Omega is an ordinal number.  Theorem 1.22 of [Schloeder] p. 3.
     (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 $.

  ${
    $d x y z $.
    $( The class of natural numbers ` _om ` can be defined as the intersection
       of all inductive sets (which is the smallest inductive set, since
       inductive sets are closed under intersection), which is valid provided
       we assume the Axiom of Infinity.  Definition 6.3 of [Eisenberg] p. 82.
       Definition 1.20 of [Schloeder] p. 3.  (Contributed by NM,
       6-Aug-1994.) $)
    dfom3 $p |- _om = |^| { x | ( (/) e. x /\ A. y e. x suc y e. x ) } $=
      ( vz com c0 cv wcel csuc wral wa cab cint wss 0ex elintab simpl wal eleq2
      wi wceq mpgbir suceq eleq1d rspccv adantl a2i alimi vex sucex rgenw mp2an
      3imtr4i peano5 peano1 peano2 rgen raleqbi1dv anbi12d imbi12d sylbi mp2ani
      omex spcv ssriv eqssi ) DEAFZGZBFZHZVFGZBVFIZJZAKLZEVMGZCFZVMGZVOHZVMGZSZ
      CDIDVMMVNVLVGSAVLAENOVGVKPUAVSCDVLVOVFGZSZAQZVLVQVFGZSZAQVPVRWAWDAVLVTWCV
      KVTWCSVGVJWCBVOVFVHVOTVIVQVFVHVOUBUCUDUEUFUGVLAVOCUHZOZVLAVQVOWEUIOULUJCV
      MUMUKCVMDVPEDGZVIDGZBDIZVODGZUNWHBDVHUOUPVPWBWGWIJZWJSZWFWAWLADVBVFDTZVLW
      KVTWJWMVGWGVKWIVFDERVJWHBVFDVFDVIRUQURVFDVORUSVCUTVAVDVE $.
  $}

  ${
    $d A x $.
    $( A simplification of ~ elom assuming the Axiom of Infinity.  (Contributed
       by NM, 30-May-2003.) $)
    elom3 $p |- ( A e. _om <-> A. x ( Lim x -> A e. x ) ) $=
      ( com wcel con0 cv wlim wi wal wa elom limom omex wceq limeq imbi12d spcv
      eleq2 mpi nnon syl pm4.71ri bitr4i ) BCDZBEDZAFZGZBUFDZHZAIZJUJABKUJUEUJU
      DUEUJCGZUDLUIUKUDHACMUFCNUGUKUHUDUFCOUFCBRPQSBTUAUBUC $.
  $}

  ${
    $d x y $.
    $( A simplification of ~ df-om assuming the Axiom of Infinity.
       (Contributed by NM, 30-May-2003.) $)
    dfom4 $p |- _om = { x | A. y ( Lim y -> x e. y ) } $=
      ( cv wlim wel wi wal com elom3 eqabi ) BCDABEFBGAHBACIJ $.
  $}

  ${
    $d x y $.
    $( ` _om ` is the smallest limit ordinal and can be defined as such
       (although the Axiom of Infinity is needed to ensure that at least one
       limit ordinal exists).  Theorem 1.23 of [Schloeder] p. 4.  (Contributed
       by FL, 22-Feb-2011.)  (Revised by Mario Carneiro, 2-Feb-2013.) $)
    dfom5 $p |- _om = |^| { x | Lim x } $=
      ( vy com cv wlim cab cint wcel wel wi wal elom3 vex elintab bitr4i eqriv
      ) BCADEZAFGZBDZCHQBAIJAKSRHASLQASBMNOP $.
  $}

  $( Ordinal addition is not commutative.  This theorem shows a counterexample.
     Remark in [TakeutiZaring] p. 60.  (Contributed by NM, 10-Dec-2004.) $)
  oancom $p |- ( 1o +o _om ) =/= ( _om +o 1o ) $=
    ( c1o com coa wcel wne csuc omex sucid con0 wceq omelon 1onn oaabslem mp2an
    co oa1suc ax-mp 3eltr4i 1on oacl wss wa wb onelpss simprbi ) ABCOZBACOZDZUF
    UGEZBBFZUFUGBGHBIDZABDUFBJKLAMNUKUGUJJKBPQRUHUFUGUAZUIUFIDZUGIDZUHULUIUBUCA
    IDZUKUMSKABTNUKUOUNKSBATNUFUGUDNUEQ $.

  $( A set is finite iff it is strictly dominated by the class of natural
     number.  Theorem 42 of [Suppes] p. 151.  The Axiom of Infinity is used for
     the forward implication.  (Contributed by FL, 16-Apr-2011.) $)
  isfinite $p |- ( A e. Fin <-> A ~< _om ) $=
    ( com cvv wcel cfn csdm wbr wb omex isfiniteg ax-mp ) BCDAEDABFGHIAJK $.

  $( A finite set is countable (weaker version of ~ isfinite ).  (Contributed
     by Thierry Arnoux, 27-Mar-2018.) $)
  fict $p |- ( A e. Fin -> A ~<_ _om ) $=
    ( cfn wcel com csdm wbr cdom isfinite sdomdom sylbi ) ABCADEFADGFAHADIJ $.

  $( A natural number is strictly dominated by the set of natural numbers.
     Example 3 of [Enderton] p. 146.  (Contributed by NM, 28-Oct-2003.) $)
  nnsdom $p |- ( A e. _om -> A ~< _om ) $=
    ( com cvv wcel csdm wbr omex nnsdomg mpan ) BCDABDABEFGAHI $.

  $( Omega is equinumerous to a proper subset of itself.  Example 13.2(4) of
     [Eisenberg] p. 216.  (Contributed by NM, 30-Jul-2003.) $)
  omenps $p |- _om ~~ ( _om \ { (/) } ) $=
    ( com cvv wcel c0 csn cdif cen wbr omex limom limenpsi ax-mp ) ABCAADEFGHIA
    BJKL $.

  $( The set of natural numbers is equinumerous to its successor.  (Contributed
     by NM, 30-Oct-2003.) $)
  omensuc $p |- _om ~~ suc _om $=
    ( com cvv wcel csuc cen wbr omex limom limensuci ax-mp ) ABCAADEFGABHIJ $.

  ${
    $d f x y z A $.  $d f x y z B $.
    $( Removing a singleton from an infinite set does not change the
       cardinality of the set.  (Contributed by Mario Carneiro, 30-Apr-2015.)
       (Revised by Mario Carneiro, 16-May-2015.) $)
    infdifsn $p |- ( _om ~<_ A -> ( A \ { B } ) ~~ A ) $=
      ( vf com cdom wbr wcel csn cdif cen wa cvv adantl peano1 sylancl cun wceq
      c0 syl wss wf1 wex brdomi adantr cfv reldom brrelex2i ad2antrr simplr f1f
      cv wf ffvelcdm difsnen syl3anc crn cin vex f1f1orn f1oen3g sylancr ensymd
      wf1o brrelex1i limom limenpsi cima cres resex simpr difss f1ores ccnv wfn
      wfun f1orn simprbi imadif 3syl fnima fnsnfv eqcomd difeq12d eqtrd breqtrd
      f1fn entr syl2anc difexg enrefg a1i ssrin ax-mp sseq0 mp2an unen syl22anc
      disjdif frnd undif sylib uncom eldifn fnfvelrn nsyl3 disjsn sylibr undif4
      wn eqtrid difeq1d 3brtr3d exlimddv difsn eqbrtrd pm2.61dan ) DAEFZBAGZABH
      IZAJFZXQXRKZDACUKZUAZXTCXQYCCUBXRDACUCUDYAYCKZXSARYBUEZHZIZJFZYGAJFXTYDAL
      GZXRYEAGZYHXQYIXRYCDAEUFUGZUHZXQXRYCUIYDDAYBULZRDGZYJYCYMYADAYBUJMZNDARYB
      UMOBYELAUNUOYDAYGYDYBUPZAYPIZPZYPYFIZYQPZAYGJYDYPYSJFZYQYQJFZYPYQUQZRQZYS
      YQUQZRQZYRYTJFYDYPDJFDYSJFZUUAYDDYPYDYBLGDYPYBVCZDYPJFCURZYCUUHYADAYBUSMZ
      DYPYBLUTVAVBYDDDRHZIZJFZUULYSJFUUGYDDLGZUUMXQUUNXRYCDAEUFVDUHDLVEVFSYDUUL
      YBUULVGZYSJYDYBUULVHZLGUULUUOUUPVCZUULUUOJFYBUULUUIVIYDYCUULDTUUQYAYCVJDU
      UKVKDAUULYBVLOUULUUOUUPLUTVAYDUUOYBDVGZYBUUKVGZIZYSYDUUHYBVMVOZUUOUUTQUUJ
      UUHYBDVNZUVADYBVPVQDUUKYBVRVSYDUURYPUUSYFYDUVBUURYPQYCUVBYADAYBWFMZDYBVTS
      YDYFUUSYDUVBYNYFUUSQUVCNDRYBWAOWBWCWDWEDUULYSWGWHYPDYSWGWHYDYIYQLGUUBYLAY
      PLWIYQLWJVSUUDYDYPAWRZWKUUFYDUUEUUCTZUUDUUFYSYPTUVEYPYFVKYSYPYQWLWMUVDUUE
      UUCWNWOWKYPYSYQYQWPWQYDYPATYRAQYDDAYBYOWSYPAWTXAZYDYTYQYSPZYGYSYQXBYDUVGY
      QYPPZYFIZYGYDYQYFUQRQZUVGUVIQYDYEYQGZXIUVJUVKYEYPGZYDYEAYPXCYDUVBYNUVLUVC
      NDRYBXDOXEYQYEXFXGYQYPYFXHSYDUVHAYFYDUVHYRAYQYPXBUVFXJXKWDXJXLVBXSYGAWGWH
      XMXQXRXIZKXSAAJUVMXSAQXQBAXNMXQAAJFZUVMXQYIUVNYKALWJSUDXOXP $.

    $( Removing a finite set from an infinite set does not change the
       cardinality of the set.  (Contributed by Mario Carneiro,
       30-Apr-2015.) $)
    infdiffi $p |- ( ( _om ~<_ A /\ B e. Fin ) -> ( A \ B ) ~~ A ) $=
      ( vx vy vz cfn wcel com cdom wbr cdif cen cv wi wceq difeq2 eqtrdi breq1d
      c0 imbi2d csn dif0 cun difun1 cvv reldom brrelex2i enrefg syl wa biimparc
      domen2 infdifsn entr sylancom ex a2i a1i findcard2 impcom ) BFGHAIJZABKZA
      LJZVAACMZKZALJZNVAAALJZNVAADMZKZALJZNZVAVIEMZUAZKZALJZNZVAVCNCDEBVDSOZVFV
      GVAVQVEAALVQVEASKAVDSAPAUBQRTVDVHOZVFVJVAVRVEVIALVDVHAPRTVDVHVMUCZOZVFVOV
      AVTVEVNALVTVEAVSKVNVDVSAPAVHVMUDQRTVDBOZVFVCVAWAVEVBALVDBAPRTVAAUEGVGHAIU
      FUGAUEUHUIVKVPNVHFGVAVJVOVAVJVOVAVJVNVILJZVOVAVJUJHVIIJZWBVJWCVAVIAHULUKV
      IVLUMUIVNVIAUNUOUPUQURUSUT $.
  $}

  ${
    $d x y A $.
    $( Any unbounded subset of natural numbers is equinumerous to the set of
       all natural numbers.  This version of ~ unbnn eliminates its hypothesis
       by assuming the Axiom of Infinity.  (Contributed by NM, 4-May-2005.) $)
    unbnn3 $p |- ( ( A C_ _om /\ A. x e. _om E. y e. A x e. y )
        -> A ~~ _om ) $=
      ( com cvv wcel wss wel wrex wral cen wbr omex unbnn mp3an1 ) DEFCDGABHBCI
      ADJCDKLMABCNO $.
  $}

  ${
    $d w x y z F $.
    $( Using the Axiom of Regularity in the form ~ zfregfr , show that there
       are no infinite descending ` e. ` -chains.  Proposition 7.34 of
       [TakeutiZaring] p. 44.  (Contributed by NM, 26-Jan-2006.)  (Revised by
       Mario Carneiro, 22-Mar-2013.) $)
    noinfep $p |- E. x e. _om ( F ` suc x ) e/ ( F ` x ) $=
      ( vz vy vw cv cep wn com cfv wrex wnel cvv wcel c0 wne wceq fvex ax-mp wi
      wbr cmpt crn wral csuc wfr omex mptex rnex zfregfr ssid dmmptg fvexd mprg
      wss cdm peano1 ne0ii eqnetri dm0rn0 necon3bii mpbi fri wfn wb eqid fnmpti
      mp4an fvelrnb peano2 fveq2 fvmpt syl fnfvelrn sylancr eqeltrrd epel eleq1
      bitrid notbid df-nel bitr4di rspccv syl5com eqeq1 syl5ibcom neleq2 biimpd
      wel syl6 com23 syldc reximdvai biimtrid com12 rexlimiv ) CFZDFZGUAZHZCEIE
      FZBJZUBZUCZUDZDXDKZAFZUEZBJZXGBJZLZAIKZXDMNXDGUFXDXDUOXDOPZXFXCEIXBUGUHUI
      XDUJXDUKXCUPZOPXMXNIOXBMNXNIQEIEIXBMULXAINXABUMUNOIUQURUSXNOXDOXCUTVAVBDC
      XDXDMGVCVHXEXLDXDXEWRXDNZXLXOXGXCJZWRQZAIKZXEXLXCIVDZXOXRVEEIXBXCXABRXCVF
      ZVGZAIWRXCVISXEXQXKAIXGINZXEXIWRLZXQXKTYBXIXDNXEYCYBXHXCJZXIXDYBXHINZYDXI
      QXGVJZEXHXBXIIXCXAXHBVKXTXHBRVLVMYBXSYEYDXDNYAYFIXHXCVNVOVPWTYCCXIXDWQXIQ
      ZWTXIWRNZHYCYGWSYHWSCDWIYGYHDWQVQWQXIWRVRVSVTXIWRWAWBWCWDYBXQYCXKYBXQWRXJ
      QZYCXKTYBXPXJQXQYIEXGXBXJIXCXAXGBVKXTXGBRVLXPWRXJWEWFYIYCXKWRXJXIWGWHWJWK
      WLWMWNWOWPS $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Cantor normal form
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $c CNF $.

  $( Extend class notation with the Cantor normal form function. $)
  ccnf $a class CNF $.

  ${
    $d x y f g h k z $.
    $( Define the Cantor normal form function, which takes as input a finitely
       supported function from ` y ` to ` x ` and outputs the corresponding
       member of the ordinal exponential ` x ^o y ` .  The content of the
       original Cantor Normal Form theorem is that for ` x = _om ` this
       function is a bijection onto ` _om ^o y ` for any ordinal ` y ` (or,
       since the function restricts naturally to different ordinals, the
       statement that the composite function is a bijection to ` On ` ).  More
       can be said about the function, however, and in particular it is an
       order isomorphism for a certain easily defined well-ordering of the
       finitely supported functions, which gives an alternate definition
       ~ cantnffval2 of this function in terms of ~ df-oi .  (Contributed by
       Mario Carneiro, 25-May-2015.)  (Revised by AV, 28-Jun-2019.) $)
    df-cnf $a |- CNF = ( x e. On , y e. On |->
      ( f e. { g e. ( x ^m y ) | g finSupp (/) } |->
        [_ OrdIso ( _E , ( f supp (/) ) ) / h ]_
        ( seqom ( ( k e. _V , z e. _V |-> ( ( ( x ^o ( h ` k ) ) .o
          ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) ) ) $.
  $}

  ${
    $d f g h k x y z A $.  $d f g h k x y z B $.  $d f x y S $.
    cantnffval.s $e |- S = { g e. ( A ^m B ) | g finSupp (/) } $.
    cantnffval.a $e |- ( ph -> A e. On ) $.
    cantnffval.b $e |- ( ph -> B e. On ) $.
    $( The value of the Cantor normal form function.  (Contributed by Mario
       Carneiro, 25-May-2015.)  (Revised by AV, 28-Jun-2019.) $)
    cantnffval $p |- ( ph -> ( A CNF B ) =
      ( f e. S |-> [_ OrdIso ( _E , ( f supp (/) ) ) / h ]_
        ( seqom ( ( k e. _V , z e. _V |-> ( ( ( A ^o ( h ` k ) ) .o
          ( f ` ( h ` k ) ) ) +o z ) ) , (/) ) ` dom h ) ) ) $=
      ( con0 wcel co cv c0 cvv cfv wceq vx ccnf csupp cep coi cdm coe comu cmpo
      vy coa cseqom csb cmpt cfsupp wbr cmap crab wa oveq12 rabeqdv eqtr4di w3a
      simp1l oveq1d mpoeq3dva seqomeq12 fveq1d csbeq2dv mpteq12dv df-cnf rabex2
      eqid sylancl ovex mptex ovmpoa syl2anc ) ACMNDMNCDUBOFEHFPZQUCOUDUEZHPZUF
      ZIBRRCIPZWASZUGOZWDVSSZUHOZBPZUKOZUIZQULZSZUMZUNZTKLUAUJCDMMFGPQUOUPZGUAP
      ZUJPZUQOZURZHVTWBIBRRWPWDUGOZWFUHOZWHUKOZUIZQULZSZUMZUNWNUBWPCTZWQDTZUSZF
      WSXFEWMXIWSWOGCDUQOZUREXIWOGWRXJWPCWQDUQUTVAJVBXIHVTXEWLXIWBXDWKXIXCWJTQQ
      TXDWKTXIIBRRXBWIXIWCRNZWHRNZVCZXAWGWHUKXMWTWEWFUHXMWPCWDUGXGXHXKXLVDVEVEV
      EVFQVMXCWJQQVGVNVHVIVJUAUJBFGHIVKFEWMWOGXJEJCDUQVOVLVPVQVR $.

    $( The domain of the Cantor normal form function (in later lemmas we will
       use ` dom ( A CNF B ) ` to abbreviate "the set of finitely supported
       functions from ` B ` to ` A ` ").  (Contributed by Mario Carneiro,
       25-May-2015.)  (Revised by AV, 28-Jun-2019.) $)
    cantnfdm $p |- ( ph -> dom ( A CNF B ) = S ) $=
      ( vf vh vk vz ccnf co cdm cv c0 csupp cvv cfv cep coi coe comu coa cseqom
      cmpo cmpt cantnffval dmeqd wcel wral wceq csbex rgenw dmmptg ax-mp eqtrdi
      csb fvex ) ABCMNZOIDJIPZQRNUAUBZJPZOZKLSSBKPVDTZUCNVFVBTUDNLPUENUGQUFZTZU
      SZUHZOZDAVAVJALBCDIEJKFGHUIUJVISUKZIDULVKDUMVLIDJVCVHVEVGUTUNUOIDVISUPUQU
      R $.
  $}

  ${
    $d k z A $.  $d k z B $.  $d x y F $.
    cantnfvalf.f $e |- F =
      seqom ( ( k e. A , z e. B |-> ( C +o D ) ) , (/) ) $.
    $( Lemma for ~ cantnf .  The function appearing in ~ cantnfval is
       unconditionally a function.  (Contributed by Mario Carneiro,
       20-May-2015.) $)
    cantnfvalf $p |- F : _om --> On $=
      ( vx vy com con0 wf cfv wcel wral coa co c0 wceq wfn cv cmpo fnseqom csuc
      wrex wo nn0suc fveq2 cvv 0ex seqom0g ax-mp eqtrdi 0elon eqeltrdi seqomsuc
      cop df-ov cxp fnoa oacl rgen2 ffnov mpbir2an f0cli eqeltri eqid fmpo mpbi
      rgen2w eleq1d syl5ibrcom rexlimiv jaoi syl rgen ffnfv ) KLGMGKUAIUBZGNZLO
      ZIKPFABCDEQRZUCZGSHUDWAIKVSKOVSSTZVSJUBZUEZTZJKUFZUGWAJVSUHWDWAWHWDVTSLWD
      VTSGNZSVSSGUISUJOWISTUKWCGSUJHULUMUNUOUPWGWAJKWEKOZWAWGWFGNZLOWJWKWEWEGNZ
      URZWCNZLWJWKWEWLWCRWNWEWCGSHUQWEWLWCUSUNBCUTZLWMWCWBLOZACPFBPWOLWCMWPFABC
      WBDEURZQNLDEQUSLLUTZLWQQWRLQMQWRUAVSWEQRLOZJLPILPVAWSIJLLVSWEVBVCIJLLLQVD
      VEUOVFVGVKFABCWBLWCWCVHVIVJUOVFUPWGVTWKLVSWFGUIVLVMVNVOVPVQIKLGVRVE $.
  $}

  ${
    $d c f g h k n t u v w x y z B $.  $d a b c d g w x y z C $.  $d k n z D $.
    $d a b c d f g h k n t u v w x y z A $.  $d x y M $.  $d c f g k t u v T $.
    $d f g h k u v w x y z F $.  $d c f g k t u v x y z S $.  $d g t x y z Z $.
    $d c f h k t u v w x y z G $.  $d f h u v x y H $.  $d k t u v x z K $.
    $d k u w x y z O $.  $d f g k n t u v x y z ph $.  $d k t w x y z Y $.
    $d a b d k t u w x y z X $.
    cantnfs.s $e |- S = dom ( A CNF B ) $.
    cantnfs.a $e |- ( ph -> A e. On ) $.
    cantnfs.b $e |- ( ph -> B e. On ) $.
    $( Elementhood in the set of finitely supported functions from ` B ` to
       ` A ` .  (Contributed by Mario Carneiro, 25-May-2015.)  (Revised by AV,
       28-Jun-2019.) $)
    cantnfs $p |- ( ph -> ( F e. S <->
                               ( F : B --> A /\ F finSupp (/) ) ) ) $=
      ( vg wcel cmap co c0 cfsupp wbr wa wf cv crab con0 ccnf cdm eqid cantnfdm
      eqtrid eleq2d breq1 elrab bitrdi elmapd anbi1d bitrd ) AEDJZEBCKLZJZEMNOZ
      PZCBEQZUPPAUMEIRZMNOZIUNSZJUQADVAEADBCUALUBVAFABCVAIVAUCGHUDUEUFUTUPIEUNU
      SEMNUGUHUIAUOURUPABCETTGHUJUKUL $.

    ${
      cantnfcl.g $e |- G = OrdIso ( _E , ( F supp (/) ) ) $.
      cantnfcl.f $e |- ( ph -> F e. S ) $.
      $( Basic properties of the order isomorphism ` G ` used later.  The
         support of an ` F e. S ` is a finite subset of ` A ` , so it is
         well-ordered by ` _E ` and the order isomorphism has domain a finite
         ordinal.  (Contributed by Mario Carneiro, 25-May-2015.)  (Revised by
         AV, 28-Jun-2019.) $)
      cantnfcl $p |- ( ph -> ( _E We ( F supp (/) ) /\ dom G e. _om ) ) $=
        ( c0 csupp cep wwe com wcel con0 cfn cvv co cdm wss suppssdm cfsupp wbr
        wf wa cantnfs mpbid simpld fssdm onss syl sstrd epweon mpisyl cin ovexd
        wess oion cen simprd fsuppimpd syl2anc enfii elind onfin2 eleqtrrdi jca
        oien ) AELMUAZNOZFUBZPQAVLRUCRNOVMAVLCRACBVLEELUDACBEUGZELUEUFZAEDQVOVP
        UHKABCDEGHIUIUJZUKULACRQCRUCICUMUNUOUPVLRNUTUQZAVNRSURPARSVNAVLTQZVNRQA
        ELMUSZVLNFTJVAUNAVLSQVNVLVBUFZVNSQAELAVOVPVQVCVDAVSVMWAVTVRVLNFTJVKVEVN
        VLVFVEVGVHVIVJ $.

      cantnfval.h $e |- H = seqom ( ( k e. _V , z e. _V |->
        ( ( ( A ^o ( G ` k ) ) .o ( F ` ( G ` k ) ) ) +o z ) ) , (/) ) $.
      $( The value of the Cantor normal form function.  (Contributed by Mario
         Carneiro, 25-May-2015.)  (Revised by AV, 28-Jun-2019.) $)
      cantnfval $p |- ( ph -> ( ( A CNF B ) ` F ) = ( H ` dom G ) ) $=
        ( co cfv c0 cvv wceq vf vg vh ccnf cv cfsupp wbr cmap csupp cep coi cdm
        crab coe comu coa cmpo cseqom cmpt eqid cantnffval fveq1d wcel cantnfdm
        csb eqtrid eleqtrd ovex oiexg mp1i simpr oveq1 adantr oieq2 syl eqtr4di
        wa eqtrd oveq2d simpl fveq12d oveq12d oveq1d mpoeq3dv seqomeq12 sylancl
        dmeqd csbied fvex fvmpt ) AGCDUDPZQGUAUBUERUFUGUBCDUHPUMZUCUAUEZRUIPZUJ
        UKZUCUEZULZFBSSCFUEZWPQZUNPZWSWMQZUOPZBUEZUPPZUQZRURZQZVEZUSZQZHULZIQZA
        GWKXIABCDWLUAUBUCFWLUTZKLVAVBAGWLVCXJXLTAGEWLNAEWKULWLJACDWLUBXMKLVDVFV
        GUAGXHXLWLXIWMGTZUCWOXGXLSWNSVCWOSVCXNWMRUIVHWNUJWOSWOUTVIVJXNWPWOTZVQZ
        WQXKXFIXPXFFBSSCWRHQZUNPZXQGQZUOPZXCUPPZUQZRURZIXPXEYBTRRTXFYCTXPFBSSXD
        YAXPXBXTXCUPXPWTXRXAXSUOXPWSXQCUNXPWRWPHXPWPGRUIPZUJUKZHXPWPWOYEXNXOVKX
        PWNYDTZWOYETXNYFXOWMGRUIVLVMWNYDUJVNVOVRMVPZVBZVSXPWSXQWMGXNXOVTYHWAWBW
        CWDRUTXEYBRRWEWFOVPXPWPHYGWGWAWHXIUTXKIWIWJVOVR $.

      $( Alternate expression for the value of the Cantor normal form function.
         (Contributed by Mario Carneiro, 25-May-2015.)  (Revised by AV,
         28-Jun-2019.) $)
      cantnfval2 $p |- ( ph -> ( ( A CNF B ) ` F ) =
          ( seqom ( ( k e. dom G , z e. On |-> ( ( ( A ^o ( G ` k ) ) .o
            ( F ` ( G ` k ) ) ) +o z ) ) , (/) ) ` dom G ) ) $=
        ( co cfv c0 wceq wi vu ccnf cdm con0 coe comu coa cmpo cseqom cantnfval
        vv wss ssid com wcel csupp cep wwe cantnfcl simprd csuc sseq1 fveq2 cvv
        cv 0ex seqom0g ax-mp eqtrdi eqid eqeq12d imbi12d imbi2d 2a1i wa sssucid
        sstr mpan imim1i oveq2 seqomsuc ad2antrl cres resmpo mp2an oveqi simprr
        cxp ssv vex sucid a1i sseldd cantnfvalf ffvelcdmi ovres syl2anc eqtr3id
        eqtrd imbitrrid expr a2d syl5 expcom finds mpcom mpi ) AGCDUBPQHUCZIQZX
        HFBXHUDCFVEHQZUEPXJGQUFPZBVEZUGPZUHZRUIZQZABCDEFGHIJKLMNOUJAXHXHULZXIXP
        SZXHUMXHUNUOZAXQXRTZAGRUPPUQURXSACDEGHJKLMNUSUTAUAVEZXHULZYAIQZYAXOQZSZ
        TZTARXHULZRRSZTZTAUKVEZXHULZYJIQZYJXOQZSZTZTAYJVAZXHULZYPIQZYPXOQZSZTZT
        AXTTUAUKXHYARSZYFYIAUUBYBYGYEYHYARXHVBUUBYCRYDRUUBYCRIQZRYARIVCRVDUOZUU
        CRSVFFBVDVDXMUHZIRVDOVGVHVIUUBYDRXOQZRYARXOVCUUDUUFRSVFXNXORVDXOVJZVGVH
        VIVKVLVMYAYJSZYFYOAUUHYBYKYEYNYAYJXHVBUUHYCYLYDYMYAYJIVCYAYJXOVCVKVLVMY
        AYPSZYFUUAAUUIYBYQYEYTYAYPXHVBUUIYCYRYDYSYAYPIVCYAYPXOVCVKVLVMYAXHSZYFX
        TAUUJYBXQYEXRYAXHXHVBUUJYCXIYDXPYAXHIVCYAXHXOVCVKVLVMYHAYGRVJVNYJUNUOZA
        YOUUAAUUKYOUUATYOYQYNTAUUKVOZUUAYQYKYNYJYPULYQYKYJVPYJYPXHVQVRVSUULYQYN
        YTAUUKYQYNYTTYNYTAUUKYQVOVOZYJYLUUEPZYJYMUUEPZSYLYMYJUUEVTUUMYRUUNYSUUO
        UUKYRUUNSAYQYJUUEIROWAWBUUMYSYJYMXNPZUUOUUKYSUUPSAYQYJXNXORUUGWAWBUUMUU
        PYJYMUUEXHUDWHWCZPZUUOUUQXNYJYMXHVDULUDVDULUUQXNSXHWIUDWIFBVDVDXHUDXMWD
        WEWFUUMYJXHUOYMUDUOZUURUUOSUUMYPXHYJAUUKYQWGYJYPUOUUMYJUKWJWKWLWMUUKUUS
        AYQUNUDYJXOBXHUDXKXLFXOUUGWNWOWBYJYMXHUDUUEWPWQWRWSVKWTXAXBXCXDXBXEXFXG
        WS $.

      $( The value of the recursive function ` H ` at a successor.
         (Contributed by Mario Carneiro, 25-May-2015.)  (Revised by AV,
         28-Jun-2019.) $)
      cantnfsuc $p |- ( ( ph /\ K e. _om ) -> ( H ` suc K ) =
        ( ( ( A ^o ( G ` K ) ) .o ( F ` ( G ` K ) ) ) +o ( H ` K ) ) ) $=
        ( cfv cvv co coa vu vv com wcel wa csuc cv coe comu cmpo wceq c0 adantl
        seqomsuc elex fvex simpl fveq2d oveq2d oveq12d simpr fveq2 oveq1d oveq2
        cbvmpov ovex ovmpoa sylancl eqtrd ) AJUCUDZUEZJUFIQZJJIQZFBRRCFUGZHQZUH
        SZVOGQZUISZBUGZTSZUJZSZCJHQZUHSZWCGQZUISZVMTSZVJVLWBUKAJWAIULPUNUMVKJRU
        DZVMRUDWBWGUKVJWHAJUCUOUMJIUPUAUBJVMRRCUAUGZHQZUHSZWJGQZUISZUBUGZTSZWGW
        AWIJUKZWNVMUKZUEZWMWFWNVMTWRWKWDWLWEUIWRWJWCCUHWRWIJHWPWQUQURZUSWRWJWCG
        WSURUTWPWQVAUTFBUAUBRRVTWOWMVSTSVNWIUKZVRWMVSTWTVPWKVQWLUIWTVOWJCUHVNWI
        HVBZUSWTVOWJGXAURUTVCVSWNWMTVDVEWFVMTVFVGVHVI $.

      ${
        cantnfle.c $e |- ( ph -> C e. B ) $.
        $( A lower bound on the ` CNF ` function.  Since
           ` ( ( A CNF B ) `` F ) ` is defined as the sum of
           ` ( A ^o x ) .o ( F `` x ) ` over all ` x ` in the support of
           ` F ` , it is larger than any of these terms (and all other terms
           are zero, so we can extend the statement to all ` C e. B ` instead
           of just those ` C ` in the support).  (Contributed by Mario
           Carneiro, 28-May-2015.)  (Revised by AV, 28-Jun-2019.) $)
        cantnfle $p |- ( ph ->
            ( ( A ^o C ) .o ( F ` C ) ) C_ ( ( A CNF B ) ` F ) ) $=
          ( wss c0 wcel vx vy coe co cfv comu ccnf wceq oveq2 sseq1d wne wa cdm
          ccnv csupp wf1o wf cep wiso cvv wwe ovexd com cantnfcl simpld syl2anc
          oiiso isof1o syl adantr f1ocnv f1of 3syl anim1i wfn wb cfsupp cantnfs
          con0 wbr mpbid 0ex a1i elsuppfn syl3anc mpbird ffvelcdmd wi simprd cv
          ffnd eqimss biantrurd eleq2 bitr3d sseq2d imbi12d imbi2d csuc anbi12d
          fveq2 sseq1 noel pm2.21i adantl fvex elsuc sssucid sstr mpan ad2antrl
          wo simprr pm2.27 coa cantnfvalf ffvelcdmi ad2antlr ad3antrrr suppssdm
          fssdm simpr sucidg sseldd onelon oecl omcl oaword2 cantnfsuc ad4ant13
          sseqtrrd expcom adantrr syld expr fveq2d f1ocnvfv2 ad2antrr eqtr3d
          oif oveq2d oveq12d oaword1 eqsstrrd a1dd jaod biimtrid expimpd finds2
          com23 vtoclga mpcom mpd cantnfval om0 0ss eqsstrdi pm2.61ne ) ACEUCUD
          ZEHUEZUFUDZHCDUGUDUEZRUUSSUFUDZUVBRUUTSUUTSUHUVAUVCUVBUUTSUUSUFUIUJAU
          UTSUKZULZUVAIUMZJUEZUVBUVEEIUNZUEZUVFTZUVAUVGRZUVEHSUOUDZUVFEUVHUVEUV
          FUVLIUPZUVLUVFUVHUPUVLUVFUVHUQAUVMUVDAUVFUVLURURIUSZUVMAUVLUTTUVLURVA
          ZUVNAHSUOVBAUVOUVFVCTZACDFHIKLMNOVDZVEUVLURIUTNVGVFUVFUVLURURIVHVIVJZ
          UVFUVLIVKUVLUVFUVHVLVMUVEEUVLTZEDTZUVDULZAUVTUVDQVNUVEHDVODVSTZSUTTZU
          VSUWAVPUVEDCHADCHUQZUVDAUWDHSVQVTZAHFTUWDUWEULOACDFHKLMVRWAVEZVJWKAUW
          BUVDMVJUWCUVEWBWCEHVSUTDSWDWEWFZWGUVPUVEUVJUVKWHZAUVPUVDAUVOUVPUVQWIV
          JUVEUAWJZUVFRZUVIUWITZULZUVAUWIJUEZRZWHZWHUVEUWHWHUAUVFVCUWIUVFUHZUWO
          UWHUVEUWPUWLUVJUWNUVKUWPUWKUWLUVJUWPUWJUWKUWIUVFWLWMUWIUVFUVIWNWOUWPU
          WMUVGUVAUWIUVFJXAWPWQWRUWOSUVFRZUVISTZULZUVASJUEZRZWHZUBWJZUVFRZUVIUX
          CTZULZUVAUXCJUEZRZWHZUXCWSZUVFRZUVIUXJTZULZUVAUXJJUEZRZWHZUVEUAUBUWIS
          UHZUWLUWSUWNUXAUXQUWJUWQUWKUWRUWISUVFXBUWISUVIWNWTUXQUWMUWTUVAUWISJXA
          WPWQUWIUXCUHZUWLUXFUWNUXHUXRUWJUXDUWKUXEUWIUXCUVFXBUWIUXCUVIWNWTUXRUW
          MUXGUVAUWIUXCJXAWPWQUWIUXJUHZUWLUXMUWNUXOUXSUWJUXKUWKUXLUWIUXJUVFXBUW
          IUXJUVIWNWTUXSUWMUXNUVAUWIUXJJXAWPWQUXBUVEUWRUXAUWQUWRUXAUVIXCXDXEWCU
          VEUXCVCTZUXIUXPWHUVEUXTULZUXMUXIUXOUYAUXKUXLUXIUXOWHZUXLUXEUVIUXCUHZX
          LUYAUXKULZUYBUVIUXCEUVHXFXGUYDUXEUYBUYCUYAUXKUXEUYBUYAUXKUXEULULZUXIU
          XHUXOUYEUXDUXEUXIUXHWHUXKUXDUYAUXEUXCUXJRUXKUXDUXCXHUXCUXJUVFXIXJXKUY
          AUXKUXEXMUXFUXHXNVFUYAUXKUXHUXOWHZUXEUYDUXGUXNRZUYFUYDUXGCUXCIUEZUCUD
          ZUYHHUEZUFUDZUXGXOUDZUXNUYDUXGVSTZUYKVSTZUXGUYLRUXTUYMUVEUXKVCVSUXCJB
          UTUTCGWJIUEZUCUDUYOHUEUFUDBWJGJPXPXQXRZUYDUYIVSTZUYJVSTZUYNUYDCVSTZUY
          HVSTZUYQAUYSUVDUXTUXKLXSZUYDUWBUYHDTUYTAUWBUVDUXTUXKMXSUYDUVLDUYHAUVL
          DRUVDUXTUXKADCUVLHHSXTUWFYAXSUYDUXCUVFTUYHUVLTUYDUXJUVFUXCUYAUXKYBUXT
          UXCUXJTUVEUXKUXCVCYCXRYDUVFUVLUXCIUVLURINYTXQVIYDZDUYHYEVFCUYHYFVFUYD
          UYSUYJCTUYRVUAUYDDCUYHHAUWDUVDUXTUXKUWFXSVUBWGCUYJYEVFUYIUYJYGVFZUXGU
          YKYHVFAUXTUXNUYLUHZUVDUXKABCDFGHIJUXCKLMNOPYIZYJYKUXHUYGUXOUVAUXGUXNX
          IYLVIYMYNYOUYDUYCUXOUXIUYAUXKUYCUXOUYAUXKUYCULZULZUVAUYLUXNVUGUVAUYKU
          YLVUGUYIUUSUYJUUTUFVUGUYHECUCVUGUVIIUEZUYHEVUGUVIUXCIUYAUXKUYCXMYPUVE
          VUHEUHZUXTVUFUVEUVMUVSVUIUVRUWGUVFUVLEIYQVFYRYSZUUAVUGUYHEHVUJYPUUBUY
          AUXKUYKUYLRZUYCUYDUYNUYMVUKVUCUYPUYKUXGUUCVFYMUUDAUXTVUDUVDVUFVUEYJYK
          YOUUEUUFUUGUUHUUJYLUUIUUKUULUUMAUVBUVGUHUVDABCDFGHIJKLMNOPUUNVJYKAUVC
          SUVBAUUSVSTZUVCSUHAUYSEVSTZVULLAUWBUVTVUMMQDEYEVFCEYFVFUUSUUOVIUVBUUP
          UUQUUR $.
      $}

      cantnflt.a $e |- ( ph -> (/) e. A ) $.
      cantnflt.k $e |- ( ph -> K e. suc dom G ) $.
      cantnflt.c $e |- ( ph -> C e. On ) $.
      cantnflt.s $e |- ( ph -> ( G " K ) C_ C ) $.
      $( An upper bound on the partial sums of the ` CNF ` function.  Since
         each term dominates all previous terms, by induction we can bound the
         whole sum with any exponent ` A ^o C ` where ` C ` is larger than any
         exponent ` ( G `` x ) , x e. K ` which has been summed so far.
         (Contributed by Mario Carneiro, 28-May-2015.)  (Revised by AV,
         29-Jun-2019.) $)
      cantnflt $p |- ( ph -> ( H ` K ) e. ( A ^o C ) ) $=
        ( vx vy c0 wceq cfv coe co wcel cv csuc com wrex con0 oen0 syl21anc cvv
        fveq2 0ex comu coa cmpo seqom0g ax-mp eqtrdi eleq1d syl5ibrcom wss word
        wa adantr eloni syl cima cdm wfn csupp wf cep oif ffn mp1i wb oicl mpbi
        ordsuc ordelon sylancr ordsssuc sylancl mpbird vex sucid simprr fnfvima
        eleqtrrid syl3anc sseldd ordsucss sylc wi suppssdm cfsupp cantnfs mpbid
        wbr simpld fssdm onss sstrd eqeltrrd ordsucelsuc sylibr ffvelcdmi onsuc
        oewordi syl31anc fveq2d simprl simpl eleq1 suceq oveq2d eleq12d imbi12d
        mpd sselda sylan2 ffvelcdmd onelon syl2anc oecl cantnfsuc oesuc 3eltr4d
        omcl ad2antrr ad2antrl epeli peano2 omord2 peano1 a1i oveq2i oa0 eqtrid
        eqtrd wtr ordtr trsuc mpan imim1i omwordi sseqtrrd sucex mpbir wiso wwe
        ex ovexd cantnfcl oiiso isorel syl12anc mpbii sylib ad2antlr cantnfvalf
        fvex oaord omsuc exp32 a2d syl5 expcom finds2 syl3c rexlimdvaa simpl2im
        eqeltrd wo elnn nn0suc mpjaod ) AKUDUEZKJUFZCEUGUHZUIZKUBUJZUKZUEZUBULU
        MZAUWHUWEUDUWGUIZACUNUIZEUNUIZUDCUIZUWMMTRCEUOUPUWEUWFUDUWGUWEUWFUDJUFZ
        UDKUDJURUDUQUIUWQUDUEUSGBUQUQCGUJIUFZUGUHUWRHUFUTUHZBUJZVAUHVBJUDUQQVCV
        DZVEVFVGAUWKUWHUBULAUWIULUIZUWKVJZVJZCUWIIUFZUKZUGUHZUWGUWFUXDUXFEVHZUX
        GUWGVHZUXDEVIZUXEEUIUXHUXDUWOUXJAUWOUXCTVKZEVLVMUXDIKVNZEUXEAUXLEVHUXCU
        AVKUXDIIVOZVPZKUXMVHZUWIKUIUXEUXLUIUXMHUDVQUHZIVRUXNUXDUXPVSIOVTZUXMUXP
        IWAWBAUXOUXCAUXOKUXMUKZUIZSAKUNUIZUXMVIZUXOUXSWCAUXRVIZUXSUXTUYAUYBUXPV
        SIOWDZUXMWFWESUXRKWGWHUYCKUXMWIWJWKVKUXDUWIUWJKUWIUBWLWMAUXBUWKWNZWPUXM
        KIUWIWOWQWRUXEEWSWTUXDUXFUNUIZUWOUWNUWPUXHUXIXAUXDUXEUNUIUYEUXDUXPUNUXE
        AUXPUNVHZUXCAUXPDUNADCUXPHHUDXBADCHVRZHUDXCXFZAHFUIUYGUYHVJPACDFHLMNXDX
        EXGZXHZADUNUIDUNVHNDXIVMXJZVKUXDUWIUXMUIZUXEUXPUIUXDUWJUXRUIZUYLUXDKUWJ
        UXRUYDAUXSUXCSVKXKUYAUYLUYMWCUYCUWIUXMXLVDXMZUXMUXPUWIIUXQXNVMWRUXEXOVM
        UXKAUWNUXCMVKAUWPUXCRVKUXFECXPXQYFUXDUWFUWJJUFZUXGUXDKUWJJUYDXRUXDUXBAU
        YLUYOUXGUIZAUXBUWKXSAUXCXTUYNUYLUYPXAUDUXMUIZUDUKZJUFZCUDIUFZUKZUGUHZUI
        ZXAUCUJZUXMUIZVUDUKZJUFZCVUDIUFZUKZUGUHZUIZXAZVUFUXMUIZVUFUKZJUFZCVUFIU
        FZUKZUGUHZUIZXAZAUBUCUWIUDUEZUYLUYQUYPVUCUWIUDUXMYAVVAUYOUYSUXGVUBVVAUW
        JUYRJUWIUDYBXRVVAUXFVUACUGVVAUXEUYTUEUXFVUAUEUWIUDIURUXEUYTYBVMYCYDYEUW
        IVUDUEZUYLVUEUYPVUKUWIVUDUXMYAVVBUYOVUGUXGVUJVVBUWJVUFJUWIVUDYBXRVVBUXF
        VUICUGVVBUXEVUHUEUXFVUIUEUWIVUDIURUXEVUHYBVMYCYDYEUWIVUFUEZUYLVUMUYPVUS
        UWIVUFUXMYAVVCUYOVUOUXGVURVVCUWJVUNJUWIVUFYBXRVVCUXFVUQCUGVVCUXEVUPUEUX
        FVUQUEUWIVUFIURUXEVUPYBVMYCYDYEAUYQVUCAUYQVJZCUYTUGUHZUYTHUFZUTUHZVVECU
        TUHZUYSVUBVVDVVFCUIZVVGVVHUIZVVDDCUYTHAUYGUYQUYIVKUYQAUYTUXPUIZUYTDUIUX
        MUXPUDIUXQXNZAUXPDUYTUYJYGYHYIZVVDVVFUNUIZUWNVVEUNUIZUDVVEUIZVVIVVJWCVV
        DUWNVVIVVNAUWNUYQMVKZVVMCVVFYJYKZVVQVVDUWNUYTUNUIZVVOVVQUYQAVVKVVSVVLAU
        XPUNUYTUYKYGYHZCUYTYLYKZVVDUWNVVSUWPVVPVVQVVTAUWPUYQRVKCUYTUOUPVVFCVVEU
        UAXQXEVVDUYSVVGUWQVAUHZVVGUYQAUDULUIZUYSVWBUEVWCUYQUUBUUCABCDFGHIJUDLMN
        OPQYMYHVVDVWBVVGUDVAUHZVVGUWQUDVVGVAUXAUUDVVDVVGUNUIZVWDVVGUEVVDVVOVVNV
        WEVWAVVRVVEVVFYPYKVVGUUEVMUUFUUGVVDUWNVVSVUBVVHUEVVQVVTCUYTYNYKYOUUSAVU
        DULUIZVULVUTXAVULVUMVUKXAAVWFVJZVUTVUMVUEVUKUXMUUHZVUMVUEUYAVWHUYCUXMUU
        IVDUXMVUDUUJUUKZUULVWGVUMVUKVUSVWGVUMVUKVUSVWGVUMVUKVJZVJZCVUPUGUHZVUPH
        UFZUKZUTUHZVURVUOVWKVWOVWLCUTUHZVURVWKVWNCVHZVWOVWPVHZVWKCVIZVWMCUIZVWQ
        VWKUWNVWSAUWNVWFVWJMYQZCVLVMVWKDCVUPHAUYGVWFVWJUYIYQVWKUXPDVUPAUXPDVHVW
        FVWJUYJYQVUMVUPUXPUIVWGVUKUXMUXPVUFIUXQXNYRZWRYIZVWMCWSWTVWKVWNUNUIZUWN
        VWLUNUIZVWQVWRXAVWKVWMUNUIZVXDVWKUWNVWTVXFVXAVXCCVWMYJYKZVWMXOVMVXAVWKU
        WNVUPUNUIZVXEVXAVWKUXPUNVUPAUYFVWFVWJUYKYQZVXBWRZCVUPYLYKZVWNCVWLUUMWQY
        FVWKUWNVXHVURVWPUEVXAVXJCVUPYNYKUUNVWKVWLVWMUTUHZVUGVAUHZVXLVWLVAUHZVUO
        VWOVWKVUGVWLUIZVXMVXNUIZVWKVUJVWLVUGVWKVUIVUPVHZVUJVWLVHZVWKVUPVIZVUHVU
        PUIZVXQVWKVXHVXSVXJVUPVLVMVWKVUHVUPVSXFZVXTVWKVUDVUFVSXFZVYAVYBVUDVUFUI
        VUDUCWLZWMVUDVUFVUDVYCUUOYSUUPVWKUXMUXPVSVSIUUQZVUEVUMVYBVYAWCAVYDVWFVW
        JAUXPUQUIUXPVSUURZVYDAHUDVQUUTAVYEUXMULUIZACDFHILMNOPUVAZXGUXPVSIUQOUVB
        YKYQVUMVUEVWGVUKVWIYRZVWGVUMVUKXSUXMUXPVUDVUFVSVSIUVCUVDUVEVUHVUPVUFIUV
        IYSUVFVUHVUPWSWTVWKVUIUNUIZVXHUWNUWPVXQVXRXAVWKVUHUNUIVYIVWKUXPUNVUHVXI
        VWKVUEVUHUXPUIVYHUXMUXPVUDIUXQXNVMWRVUHXOVMVXJVXAAUWPVWFVWJRYQVUIVUPCXP
        XQYFVWGVUMVUKWNWRVWKVUGUNUIZVXEVXLUNUIZVXOVXPWCVWKVUFULUIZVYJVWFVYLAVWJ
        VUDYTZUVGULUNVUFJBUQUQUWSUWTGJQUVHXNVMVXKVWKVXEVXFVYKVXKVXGVWLVWMYPYKVU
        GVWLVXLUVJWQXEVWGVUOVXMUEZVWJVWFAVYLVYNVYMABCDFGHIJVUFLMNOPQYMYHVKVWKVX
        EVXFVWOVXNUEVXKVXGVWLVWMUVKYKYOWRUVLUVMUVNUVOUVPUVQUVTWRUVRAKULUIZUWEUW
        LUWAAUXSUXRULUIZVYOSAVYEVYFVYPVYGUXMYTUVSKUXRUWBYKUBKUWCVMUWD $.
    $}

    ${
      cantnflt2.f $e |- ( ph -> F e. S ) $.
      cantnflt2.a $e |- ( ph -> (/) e. A ) $.
      cantnflt2.c $e |- ( ph -> C e. On ) $.
      cantnflt2.s $e |- ( ph -> ( F supp (/) ) C_ C ) $.
      $( An upper bound on the ` CNF ` function.  (Contributed by Mario
         Carneiro, 28-May-2015.)  (Revised by AV, 29-Jun-2019.) $)
      cantnflt2 $p |- ( ph -> ( ( A CNF B ) ` F ) e. ( A ^o C ) ) $=
        ( vk vz co cfv cep cvv wcel ccnf csupp coi cdm coe comu coa cmpo cseqom
        c0 cv eqid cantnfval con0 csuc oion sucidg 3syl cima wiso wf1o wfo wceq
        ovexd wwe cantnfcl simpld oiiso syl2anc isof1o f1ofo foima 4syl eqsstrd
        com cantnflt eqeltrd ) AFBCUAPQFUJUBPZRUCZUDZNOSSBNUKVSQZUEPWAFQUFPOUKU
        GPUHUJUIZQBDUEPAOBCENFVSWBGHIVSULZJWBULZUMAOBCDENFVSWBVTGHIWCJWDKAVRSTZ
        VTUNTVTVTUOTAFUJUBVDZVRRVSSWCUPVTUNUQURLAVSVTUSZVRDAVTVRRRVSUTZVTVRVSVA
        VTVRVSVBWGVRVCAWEVRRVEZWHWFAWIVTVOTABCEFVSGHIWCJVFVGVRRVSSWCVHVIVTVRRRV
        SVJVTVRVSVKVTVRVSVLVMMVNVPVQ $.
    $}

    $( The ` CNF ` function is a function from finitely supported functions
       from ` B ` to ` A ` , to the ordinal exponential ` A ^o B ` .
       (Contributed by Mario Carneiro, 28-May-2015.) $)
    cantnff $p |- ( ph -> ( A CNF B ) : S --> ( A ^o B ) ) $=
      ( vf vk vz cv c0 co cvv cfv coe wcel wa wceq adantr vx vh vg cep coi comu
      csupp cdm coa cmpo cseqom csb ccnf fvex csbex a1i wbr cmap crab cmpt eqid
      cfsupp cantnffval cantnfdm eqtrid mpteq1d eqtr4d c1o con0 simpr cantnfval
      cen wwe ovex com cantnfcl simpld sylancr wss suppssdm wf cantnfs simprbda
      oien fssdm feq3 syl5ibcom imp f00 sylib simprd sseq0 syl2an2r breqtrd en0
      fveq2d 0ex seqom0g mp1i 3eqtrd el1o sylibr oveq2d oe0 syl eleqtrrd wne wb
      eqtrd on0eln0 biimpar cantnflt2 pm2.61dane fmpt2d ) AHUADUBHKZLUGMUDUEZUB
      KZUHZIJNNBIKZXQOZPMXTXOOUFMJKZUIMUJLUKZOZULZBCPMZBCUMMZNYDNQAXODQRUBXPYCX
      RYBUNUOUPAYFHUCKLVBUQUCBCURMUSZYDUTHDYDUTAJBCYGHUCUBIYGVAZFGVCAHDYGYDADYF
      UHYGEABCYGUCYHFGVDVEVFVGAUAKZDQZRZYIYFOZYEQBLYKBLSZRZYLVHYEYNYLLSYLVHQYNY
      LYILUGMZUDUEZUHZIJNNBXSYPOZPMYRYIOUFMYAUIMUJZLUKZOZLYTOZLYKYLUUASYMYKJBCD
      IYIYPYTEABVIQZYJFTZACVIQZYJGTZYPVAZAYJVJZYTVAZVKTYNYQLYTYNYQLVLUQYQLSYNYQ
      YOLVLYKYQYOVLUQZYMYKYONQYOUDVMZUUJYILUGVNYKUUKYQVOQYKBCDYIYPEUUDUUFUUGUUH
      VPVQYOUDYPNUUGWDVRTYKYOCVSZYMCLSZYOLSYKCBYOYIYILVTAYJCBYIWAZYILVBUQABCDYI
      EFGWBWCZWEZYNYILSZUUMYNCLYIWAZUUQUUMRYKYMUURYKUUNYMUURUUOBLCYIWFWGWHCYIWI
      WJWKZYOCWLWMWNYQWOWJWPLNQUUBLSYNWQYSYTLNUUIWRWSWTYLXAXBYNYEBLPMZVHYNCLBPU
      USXCYNUUCUUTVHSYKUUCYMUUDTBXDXEXIXFYKBLXGZRBCCDYIEYKUUCUVAUUDTYKUUEUVAUUF
      TZYKYJUVAUUHTYKLBQZUVAYKUUCUVCUVAXHUUDBXJXEXKUVBYKUULUVAUUPTXLXMXN $.

    ${
      cantnf0.a $e |- ( ph -> (/) e. A ) $.
      $( The value of the zero function.  (Contributed by Mario Carneiro,
         30-May-2015.) $)
      cantnf0 $p |- ( ph -> ( ( A CNF B ) ` ( B X. { (/) } ) ) = (/) ) $=
        ( vk vz c0 co cfv cep cvv eqid wcel con0 wceq 0ex csn cxp csupp coi cdm
        ccnf coe comu coa cmpo cseqom cfsupp wbr fconst6g syl fczfsuppd cantnfs
        cv wf mpbir2and cantnfval eqidd wfn fnconstg mp1i a1i fnsuppeq0 syl3anc
        wb mpbird oieq2 dmeqd cen wwe we0 oien mp2an mpbi eqtrdi fveq2d seqom0g
        en0 3eqtrd ) ACKUAUBZBCUFLMWDKUCLZNUDZUEZIJOOBIURWFMZUGLWHWDMUHLJURUILU
        JZKUKZMKWJMZKAJBCDIWDWFWJEFGWFPAWDDQCBWDUSZWDKULUMAKBQWLHCKBUNUOACRBKGH
        UPABCDWDEFGUQUTWJPZVAAWGKWJAWGKNUDZUEZKAWFWNAWEKSZWFWNSAWPWDWDSZAWDVBAW
        DCVCZCRQKOQZWPWQVIWSWRATCKOVDVEGWSATVFCWDORKVGVHVJWEKNVKUOVLWOKVMUMZWOK
        SWSKNVNWTTNVOKNWNOWNPVPVQWOWBVRVSVTWSWKKSATWIWJKOWMWAVEWC $.
    $}

    ${
      cantnfrescl.d $e |- ( ph -> D e. On ) $.
      cantnfrescl.b $e |- ( ph -> B C_ D ) $.
      cantnfrescl.x $e |- ( ( ph /\ n e. ( D \ B ) ) -> X = (/) ) $.
      cantnfrescl.a $e |- ( ph -> (/) e. A ) $.
      cantnfrescl.t $e |- T = dom ( A CNF D ) $.
      $( A function is finitely supported from ` B ` to ` A ` iff the extended
         function is finitely supported from ` D ` to ` A ` .  (Contributed by
         Mario Carneiro, 25-May-2015.) $)
      cantnfrescl $p |- ( ph -> ( ( n e. B |-> X ) e. S <->
                                     ( n e. D |-> X ) e. T ) ) $=
        ( c0 wa wcel cvv cmpt wf cfsupp wbr wral cdif adantr ralrimiva raldifeq
        cv eqeltrd eqid fmpt 3bitr3g wfun csupp co wceq mptexd funmpt a1i jctir
        wb con0 jca31 extmptsuppeq suppeqfsuppbi sylc anbi12d cantnfs 3bitr4d )
        ACBGCHUAZUBZVLQUCUDZRDBGDHUAZUBZVOQUCUDZRVLESVOFSAVMVPVNVQAHBSZGCUEVRGD
        UEVMVPAVRGCDMAVRGDCUFZAGUJVSSZRHQBNAQBSVTOUGUKUHUIGCBHVLVLULUMGDBHVOVOU
        LUMUNAVLTSZVLUOZRVOTSZVOUOZRZRVLQUPUQVOQUPUQURVNVQVCAWAWBWEAGCHVDKUSWBA
        GCHUTVAAWCWDAGDHVDLUSGDHUTVBVEACDGVDHQLMNVFTVLVOTQVGVHVIABCEVLIJKVJABDF
        VOPJLVJVK $.

      $d k z T $.
      cantnfres.m $e |- ( ph -> ( n e. B |-> X ) e. S ) $.
      $( The ` CNF ` function respects extensions of the domain to a larger
         ordinal.  (Contributed by Mario Carneiro, 25-May-2015.) $)
      cantnfres $p |- ( ph -> ( ( A CNF B ) ` ( n e. B |-> X ) ) =
                              ( ( A CNF D ) ` ( n e. D |-> X ) ) ) $=
        ( vk c0 con0 vz cmpt csupp co cep coi cdm cv cfv coe comu coa cmpo ccnf
        cseqom wceq wcel w3a extmptsuppeq oieq2 syl fveq1d 3ad2ant1 oveq2d cres
        wss suppssdm eqid dmmptss a1i sstrid oif 3ad2ant2 sseldd fvresd resmptd
        ffvelcdmi fveq2d 3eqtr3d oveq12d oveq1d mpoeq3dva dmeqd mpoeq12 sylancl
        eqtrd seqomeq12 fveq12d cvv cantnfval2 cantnfrescl mpbid 3eqtr4d ) AGCH
        UBZSUCUDZUEUFZUGZRUAWQTBRUHZWPUIZUJUDZWSWNUIZUKUDZUAUHZULUDZUMZSUOZUIGD
        HUBZSUCUDZUEUFZUGZRUAXJTBWRXIUIZUJUDZXKXGUIZUKUDZXCULUDZUMZSUOZUIWNBCUN
        UDUIXGBDUNUDUIAWQXJXFXQAXEXPUPSSUPXFXQUPAXERUAWQTXOUMZXPARUAWQTXDXOAWRW
        QUQZXCTUQZURZXBXNXCULYAWTXLXAXMUKYAWSXKBUJAXSWSXKUPXTAWRWPXIAWOXHUPWPXI
        UPACDGTHSLMNUSWOXHUEUTVAZVBVCZVDYAWSXGCVEZUIWSXGUIXAXMYAWSCXGYAWOCWSAXS
        WOCVFXTAWOWNUGZCWNSVGYECVFAGCHWNWNVHVIVJVKVCXSAWSWOUQXTWQWOWRWPWOUEWPWP
        VHZVLVQVMVNVOYAWSYDWNYAGDCHAXSCDVFXTMVCVPVBYAWSXKXGYCVRVSVTWAWBAWQXJUPT
        TUPXRXPUPAWPXIYBWCZTVHRUAWQTXJTXOWDWEWFSVHXEXPSSWGWEYGWHAUABCERWNWPRUAW
        IWIXDUMSUOZIJKYFQYHVHWJAUABDFRXGXIRUAWIWIXOUMSUOZPJLXIVHAWNEUQXGFUQQABC
        DEFGHIJKLMNOPWKWLYIVHWJWM $.
    $}

    ${
      cantnfp1.g $e |- ( ph -> G e. S ) $.
      cantnfp1.x $e |- ( ph -> X e. B ) $.
      cantnfp1.y $e |- ( ph -> Y e. A ) $.
      cantnfp1.s $e |- ( ph -> ( G supp (/) ) C_ X ) $.
      cantnfp1.f $e |- F = ( t e. B |-> if ( t = X , Y , ( G ` t ) ) ) $.
      $( Lemma for ~ cantnfp1 .  (Contributed by Mario Carneiro, 20-Jun-2015.)
         (Revised by AV, 30-Jun-2019.) $)
      cantnfp1lem1 $p |- ( ph -> F e. S ) $=
        ( wcel c0 cvv vk wf cfsupp wbr cv wceq cfv cif wa adantr cantnfs simpld
        mpbid ffvelcdmda ifcld fmptd csupp co cfn csn cun simprd fsuppimpd snfi
        unfi sylancl cdif eqeq1 fveq2 ifbieq2d eldifi adantl fvex ifexg fvmptd3
        wn eldifn velsn elun2 sylbir nsyl iffalsed ssun1 sscon ax-mp sseli con0
        wss ssidd 0ex suppssr sylan2 3eqtrd suppss ssfid wfun wb funmpt2 mptexg
        a1i cmpt eqeltrid syl funisfsupp mp3an2i mpbird mpbir2and ) AFERDCFUBFS
        UCUDZABDBUEZHUFZIXIGUGZUHZCFAXIDRZUIXJIXKCAICRZXMOUJADCXIGADCGUBZGSUCUD
        ZAGERXOXPUIMACDEGJKLUKUMZULZUNUOQUPZAXHFSUQURZUSRZAGSUQURZHUTZVAZXTAYBU
        SRYCUSRYDUSRAGSAXOXPXQVBVCHVDYBYCVEVFADCUAFYDSXSAUAUEZDYDVGZRZUIZYEFUGY
        EHUFZIYEGUGZUHZYJSYHBYEXLYKDFTQXIYEUFXJYIXKYJIXIYEHVHXIYEGVIVJYGYEDRAYE
        DYDVKVLYHXNYJTRYKTRAXNYGOUJYEGVMYIIYJCTVNVFVOYHYIIYJYHYEYDRZYIYGYLVPAYE
        DYDVQVLYIYEYCRYLUAHVRYEYCYBVSVTWAWBYGAYEDYBVGZRYJSUFYFYMYEYBYDWHYFYMWHY
        BYCWCYBYDDWDWEWFADCTGWGYBYESXRAYBWILSTRZAWJWTZWKWLWMWNWOFWPAFTRZYNXHYAW
        QBDXLFQWRADWGRZYPLYQFBDXLXATQBDXLWGWSXBXCYOFTTSXDXEXFACDEFJKLUKXG $.

      ${
        cantnfp1.e $e |- ( ph -> (/) e. Y ) $.
        cantnfp1.o $e |- O = OrdIso ( _E , ( F supp (/) ) ) $.
        $( Lemma for ~ cantnfp1 .  (Contributed by Mario Carneiro,
           28-May-2015.)  (Revised by AV, 30-Jun-2019.) $)
        cantnfp1lem2 $p |- ( ph -> dom O = suc U. dom O ) $=
          ( cdm cuni wceq wn csuc c0 wlim wo csupp co cfv wne cv iftrue fvmptd3
          wcel cif ne0d eqnetrd wfn con0 cvv wa wb adantr wf cfsupp wbr cantnfs
          mpbid simpld ffvelcdmda ifcld fmptd ffnd elexd elsuppfn mpbir2and n0i
          syl3anc syl cen cep wwe ovexd com cantnfp1lem1 cantnfcl syl2anc breq1
          oien ensymb en0 bitri bitrdi syl5ibcom mtod nnlim ioran sylanbrc word
          simprd nnord unizlim 3syl mtbird orduniorsuc ord mpd ) AHUAZXJUBZUCZU
          DXJXKUEUCZAXLXJUFUCZXJUGZUHZAXNUDXOUDZXPUDAXNFUFUIUJZUFUCZAIXRUPZXSUD
          AXTIDUPZIFUKZUFULZOAYBJUFABIBUMZIUCZJYDGUKZUQZJDFCRYEJYFUNOPUOAJUFSUR
          USAFDUTDVAUPUFVBUPXTYAYCVCVDADCFABDYGCFAYDDUPZVCYEJYFCAJCUPYHPVEADCYD
          GADCGVFZGUFVGVHZAGEUPYIYJVCNACDEGKLMVIVJVKVLVMRVNVOMAUFJSVPIFVAVBDUFV
          QVTVRXRIVSWAAXJXRWBVHZXNXSAXRVBUPXRWCWDZYKAFUFUIWEAYLXJWFUPZACDEFHKLM
          TABCDEFGIJKLMNOPQRWGWHZVKXRWCHVBTWKWIXNYKUFXRWBVHZXSXJUFXRWBWJYOXRUFW
          BVHXSUFXRWLXRWMWNWOWPWQAYMXQAYLYMYNXBZXJWRWAXNXOWSWTAYMXJXAZXLXPVDYPX
          JXCZXJXDXEXFAXLXMAYMYQXLXMUHYPYRXJXGXEXHXI $.

        cantnfp1.h $e |- H = seqom ( ( k e. _V , z e. _V |->
          ( ( ( A ^o ( O ` k ) ) .o ( F ` ( O ` k ) ) ) +o z ) ) , (/) ) $.
        cantnfp1.k $e |- K = OrdIso ( _E , ( G supp (/) ) ) $.
        cantnfp1.m $e |- M = seqom ( ( k e. _V , z e. _V |->
          ( ( ( A ^o ( K ` k ) ) .o ( G ` ( K ` k ) ) ) +o z ) ) , (/) ) $.
        $( Lemma for ~ cantnfp1 .  (Contributed by Mario Carneiro,
           28-May-2015.)  (Revised by AV, 1-Jul-2019.) $)
        cantnfp1lem3 $p |- ( ph -> ( ( A CNF B ) ` F ) =
            ( ( ( A ^o X ) .o Y ) +o ( ( A CNF B ) ` G ) ) ) $=
          ( vx vy co cfv cdm csuc coe comu coa cantnfval fveq2d wcel wceq csupp
          com c0 cep wwe cantnfcl simprd sylibr cantnfsuc wo wn wss wf1o wf cvv
          wiso simpld syl2anc syl 3syl wne cv cif fvmptd3 wfn con0 wa wb adantr
          wbr mpbid ffnd 0ex a1i elsuppfn syl3anc word sylancr fvex epeli bitrd
          eleq2d csn sseld wi suppssdm fssdm ffvelcdmi sseldd eloni syld ordirr
          oif syl5ibrcom cun cdif weq eqeq1 fveq2 ifbieq2d eldifi ifexg sylancl
          adantl iffalsed ax-mp ssidd suppssr 3eqtrd suppss sylib eqtrd oveq12d
          cima eqtr3d eleq1 eqeq12d imbi12d imbi2d ccnf cuni cantnfp1lem1 mpdan
          cantnfp1lem2 eqeltrrd ccnv ovexd oiiso isof1o f1ocnv f1of iftrue ne0d
          peano2b eqnetrd cfsupp cantnfs ffvelcdmda mpbir2and ffvelcdmd elssuni
          ifcld fmptd oicl ordelon nnon ontri1 sucidg eleqtrrd syl12anc 3bitr3g
          isorel f1ocnvfv2 mtbid sstrd ordn2lp imnan onelon elsni notbid eldifn
          onss velsn elun2 sylbir nsyl ssun1 sscon sseli sylan2 mpjaod sylanbrc
          elun ioran ordtri3 mpbird oveq2d cres nnord sseqtrrid wfo f1ofo foima
          sssucid ffn fnsnfv sneqd difeq12d disjsn difun2 eqtr4di df-suc eqtrdi
          disj3 difeq1d eqtr4d imaeq2d wfun dff1o3 simprbi imadif ssneldd dif1o
          cin c1o mpbiran bicomd anbi2d sylbird mpand biimtrrid fveqeq2 iffalse
          necon1bd mpd eqeq1d syl5ibcom pm2.61d reldisj uncom sseqtrdi 3eqtr4rd
          ssundif eqssd isores3 wse epse oieu mpbi2and cmpo seqom0g ordtr trsuc
          wtr sylan imim1d oveq2 elnn ancoms syl2an2r syldan fveq1d ordsucelsuc
          ex biimpa fvresd eleqtrd nelneq 3eqtr3rd oveq1d imbitrrid finds mpcom
          a2d a2i 3eqtr4d ) AHDEUUAUKZULMUMZJULVVSUUBZUNZJULZDNUOUKZOUPUKZIVVRU
          LZUQUKZABDEFGHMJPQRUEACDEFHINOPQRSTUAUBUCUUCZUFURAVVSVWAJACDEFHIMNOPQ
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          MUXGWSAVWINWVTUXHYPUXIAWWJMVVSWWLXQZYOZWWNAVVTWWSMAVVTVVTWWLXPZWWLXQZ
          WWSAVVTVVTWWLXQZWXBAVVTWWLUYEVDVAZVVTWXCVAAVVTVVTUTVLZWXDAWWFWXEWWIVV
          TXMVTVVTVVTUXJVIVVTWWLUXOYLVVTWWLUXKUXLAVVSWXAWWLAVVSVWAWXAVWHVVTUXMU
          XNUXPUXQUXRAVXMVXGUXSZWWTWWNVAVXPVXMWWPWXFVVSVWPMUXTUYAVVSWWLMUYBWAYM
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          AWUFVWPVMWXGWXHWIAEDGIVWPVDVYLAWVBEVWPXQUTZWHZWVGWVRWXMWVRWVGNIULZVDV
          AZAWXOWXLAWUPWXOWUTAWUOWXNVDWXNVDWBZWXNVPUYFXQUTZAWUOWXQWXNVPUTWXPNIW
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          IWXTWIAEDIVYLWMRVYNNIWGVPEVDWPWQAWXPWXQVXRAWXQWXPWXQWXPWIAWXRWOUYHUYI
          XBAWUFNNUBXEUYJUYKUYLUYOUYPWJWVBNVDIUYMXOWXMWVIVDVAWVGVLZWVRWXMWVFWVI
          VDWXMCWVBVYDWVIEHVPUCWVJWXLWVKAWVBEVWPYBYEWXMVYIWVLWVMAVYIWXLUAWJWVNW
          VOYDWEAEDOHWGVWPWVBVDVYMAVWPYHRUDYIYPWYAWVIWVHVDWVGOWVHUYNUYQUYRUYSYK
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          LZVDVAZXFZXFAUJWCZVVSUTZWYRJULZWYRLULZVAZXFZXFZAWYRUNZVVSUTZXUEJULZXU
          ELULZVAZXFZXFZAWYGXFUIUJVVTWYHVDVAZWYMWYQAXULWYIWYNWYLWYPWYHVDVVSYQXU
          LWYJWYOWYKVDWYHVDJXTXULWYKVDLULZVDWYHVDLXTVYGXUMVDVAWNGBVPVPDWVBKULZU
          OUKXUNIULUPUKBWCZUQUKVUKLVDVPUHVULYGUXNYRYSYTUIUJXRZWYMXUCAXUPWYIWYSW
          YLXUBWYHWYRVVSYQXUPWYJWYTWYKXUAWYHWYRJXTWYHWYRLXTYRYSYTWYHXUEVAZWYMXU
          JAXUQWYIXUFWYLXUIWYHXUEVVSYQXUQWYJXUGWYKXUHWYHXUEJXTWYHXUELXTYRYSYTWY
          HVVTVAZWYMWYGAXURWYIWUDWYLWYFWYHVVTVVSYQXURWYJVWMWYKWWAWYHVVTJXTWYHVV
          TLXTYRYSYTWYQAGBVPVPDWVBMULZUOUKXUSHULUPUKXUOUQUKVUKJVDVVSUFVULWOXUDX
          UKXFWYRVCUTZAXUCXUJAXUCXUFXUBXFXUJAXUFWYSXUBAXUFWYSAVVSVUOZXUFWYSAVYS
          XVAAVWRVYSVWTVVSUWTVTVVSVUMVTVVSWYRVUNVUPZVVEVUQAXUFXUBXUIAXUFXUBXUIX
          FXUBXUIAXUFWHZDWYRMULZUOUKZXVDHULZUPUKZWYTUQUKZXVGXUAUQUKZVAWYTXUAXVG
          UQVURXVCXUGXVHXUHXVIAXUFXUTXUGXVHVAAVWRXUFWYSXUTVWTXVBWYSVWRXUTWYRVVS
          VUSVUTVVAZABDEFGHMJWYRPQRUEVWGUFVJVVBXVCXUHDWYRKULZUOUKZXVKIULZUPUKZX
          UAUQUKZXVIAXUFXUTXUHXVOVAXVJABDEFGIKLWYRPQRUGSUHVJVVBXVCXVNXVGXUAUQXV
          CXVLXVEXVMXVFUPXVCXVKXVDDUOXVCWYRWWDULZXVKXVDAXVPXVKVAXUFAWYRWWDKAWWC
          WWEWYDVHVVCWJXVCWYRVVTMXVCWYRVVTUTZXUEVWAUTZAXUFXVRAVVSVWAXUEVWHXCVVF
          XVCWWFXVQXVRWIAWWFXUFWWIWJWYRVVTVVDVTUWQZVVGYPZUWRXVCXVKHULXVKNVAZOXV
          MWDZXVFXVMXVCCXVKVYDXWBEHVPUCVYAXVKVAVYBXWAVYCXVMOVYAXVKNXSVYAXVKIXTY
          AXVCWUFEXVKAWUFEVMXUFAEDWUFIIVDXGVYLXHWJXVCWYRWWBUTXVKWUFUTZXVCWYRVVT
          WWBXVSAWWCXUFWYEWJVVHWWBWUFWYRKWUFVEKUGXNXIVTZXJXVCVYIXVMVPUTXWBVPUTA
          VYIXUFUAWJXVKIWTXWAOXVMDVPYCYDWEXVCXVKXVDHXVTUSXVCXWAOXVMXVCXWCWXJXWA
          VLXWDAWXJXUFWXKWJXVKNWUFVVIVSYFVVJYNVVKYMYRVVLVVEVVOXLVVPWOVVMVVNUYPA
          BDEFGIKLPQRUGSUHURVVQYNYMYJ $.
      $}

      $( If ` F ` is created by adding a single term ` ( F `` X ) = Y ` to
         ` G ` , where ` X ` is larger than any element of the support of
         ` G ` , then ` F ` is also a finitely supported function and it is
         assigned the value ` ( ( A ^o X ) .o Y ) +o z ` where ` z ` is the
         value of ` G ` .  (Contributed by Mario Carneiro, 28-May-2015.)
         (Revised by AV, 1-Jul-2019.) $)
      cantnfp1 $p |- ( ph -> ( F e. S /\ ( ( A CNF B ) ` F ) =
          ( ( ( A ^o X ) .o Y ) +o ( ( A CNF B ) ` G ) ) ) ) $=
        ( wcel co c0 vz vk ccnf cfv coe comu coa wceq wa cv cmpt wn con0 onelon
        cif word syl2anc eloni ordirr 3syl wne cvv c1o cdif dif1o mpbiran csupp
        fvex wfn wb wf cfsupp wbr cantnfs mpbid simpld 0ex a1i elsuppfn syl3anc
        ffnd bicomi anbi2d bitrd sseld sylbird biimtrrid necon1bd mpd ad3antrrr
        mpand simpr fveq2d simpllr eqidd ifeqda mpteq2dva eqtrid feqmptd adantr
        3eqtr4rd eqtr4d eqeltrd oecl cantnff ffvelcdmd oa0r syl oveq2 sylan9eqr
        om0 oveq1d jca wss cantnfp1lem1 cep cseqom on0eln0 biimpar cantnfp1lem3
        coi cmpo eqid pm2.61dane ) AFERZFCDUCSZUDZCHUESZIUFSZGYFUDZUGSZUHZUIITA
        ITUHZUIZYEYLYNFGEYNFBDBUJZGUDZUKZGYNFBDYOHUHZIYPUOZUKYQQYNBDYSYPYNYODRZ
        UIZYRIYPYPUUAYRUIZHGUDZTYPIAUUCTUHZYMYTYRAHHRZULZUUDAHUMRZHUPUUFADUMRZH
        DRZUUGLNDHUNUQZHURHUSUTAUUEUUCTUUCTVAZUUCVBVCVDRZAUUEUULUUCVBRUUKHGVHUU
        CVBVEVFZAUUIUULUUENAUUIUULUIZHGTVGSZRZUUEAUUPUUIUUKUIZUUNAGDVIUUHTVBRZU
        UPUUQVJADCGADCGVKZGTVLVMZAGERZUUSUUTUIMACDEGJKLVNVOVPZWALUURAVQVRHGUMVB
        DTVSVTAUUKUULUUIUUKUULVJAUULUUKUUMWBVRWCWDAUUOHHPWEWFWKWGWHWIWJUUBYOHGU
        UAYRWLWMAYMYTYRWNXAUUAYRULUIYPWOWPWQWRAGYQUHYMABDCGUVBWSWTXBZAUVAYMMWTX
        CYNTYJUGSZYJYKYGYNYJUMRZUVDYJUHAUVEYMACDUESZUMRZYJUVFRUVEACUMRZUUHUVGKL
        CDXDUQAEUVFGYFACDEJKLXEMXFUVFYJUNUQWTYJXGXHYNYITYJUGYMAYIYHTUFSZTITYHUF
        XIAYHUMRZUVITUHAUVHUUGUVJKUUJCHXDUQYHXKXHXJXLYNFGYFUVCWMXAXMAITVAZUIZYE
        YLUVLBCDEFGHIJAUVHUVKKWTZAUUHUVKLWTZAUVAUVKMWTZAUUIUVKNWTZAICRZUVKOWTZA
        UUOHXNUVKPWTZQXOUVLUABCDEUBFGUBUAVBVBCUBUJZFTVGSXPYAZUDZUESUWBFUDUFSUAU
        JZUGSYBTXQZUUOXPYAZUBUAVBVBCUVTUWEUDZUESUWFGUDUFSUWCUGSYBTXQZUWAHIJUVMU
        VNUVOUVPUVRUVSQATIRZUVKAIUMRZUWHUVKVJAUVHUVQUWIKOCIUNUQIXRXHXSUWAYCUWDY
        CUWEYCUWGYCXTXMYD $.
    $}

    oemapval.t $e |- T = { <. x , y >. | E. z e. B ( ( x ` z ) e. ( y ` z )
      /\ A. w e. B ( z e. w -> ( x ` w ) = ( y ` w ) ) ) } $.
    $( The relation ` T ` is a strict order on ` S ` (a corollary of
       ~ wemapso2 ).  (Contributed by Mario Carneiro, 28-May-2015.) $)
    oemapso $p |- ( ph -> T Or S ) $=
      ( vg wor cv c0 wbr wcel cep cfsupp cmap co crab con0 ccnv wwe eloni ordwe
      word weso 4syl cnvso sylib cfv wceq wi wral wa wrex copab epeli vex brcnv
      fvex epel bitri imbi1i ralbii anbi12i rexbii opabbii eqtr4i breq1 cbvrabv
      wemapso2 syl3anc wb ccnf cdm eqid cantnfdm eqtrid soeq2 syl mpbird ) AHIO
      ZNPZQUARZNFGUBUCZUDZIOZAGUESZGTUFZOZFTOZWLLAGTOZWOAWMGUJGTUGWQLGUHGUIGTUK
      ULGTUMUNAFUESFUJFTUGWPKFUHFUIFTUKULBCDEGFWNTIWKUEQIDPZBPZUOZWRCPZUOZSZWRE
      PZSZXDWSUOXDXAUOUPZUQZEGURZUSZDGUTZBCVAWTXBTRZXDWRWNRZXFUQZEGURZUSZDGUTZB
      CVAMXPXJBCXOXIDGXKXCXNXHWTXBWRXAVEVBXMXGEGXLXEXFXLWRXDTRXEXDWRTEVCDVCVDEW
      RVFVGVHVIVJVKVLVMWIWSQUARNBWJWHWSQUAVNVOVPVQAHWKUPWGWLVRAHFGVSUCVTWKJAFGW
      KNWKWAKLWBWCHWKIWDWEWF $.

    ${
      oemapval.f $e |- ( ph -> F e. S ) $.
      oemapval.g $e |- ( ph -> G e. S ) $.
      $( Value of the relation ` T ` .  (Contributed by Mario Carneiro,
         28-May-2015.) $)
      oemapval $p |- ( ph -> ( F T G <-> E. z e. B ( ( F ` z ) e. ( G ` z )
        /\ A. w e. B ( z e. w -> ( F ` w ) = ( G ` w ) ) ) ) ) $=
        ( wcel cfv wceq wbr cv wi wral wa wrex wb fveq1 eleq12 syl2an eqeqan12d
        imbi2d ralbidv anbi12d rexbidv brabga syl2anc ) AJHRKHRJKIUADUBZJSZURKS
        ZRZUREUBZRZVBJSZVBKSZTZUCZEGUDZUEZDGUFZUGPQURBUBZSZURCUBZSZRZVCVBVKSZVB
        VMSZTZUCZEGUDZUEZDGUFVJBCJKIHHVKJTZVMKTZUEZWAVIDGWDVOVAVTVHWBVLUSTVNUTT
        VOVAUGWCURVKJUHURVMKUHVLUSVNUTUIUJWDVSVGEGWDVRVFVCWBWCVPVDVQVEVBVKJUHVB
        VMKUHUKULUMUNUOOUPUQ $.

      $d c F $.  $d c ph $.
      oemapvali.r $e |- ( ph -> F T G ) $.
      oemapvali.x $e |- X = U. { c e. B | ( F ` c ) e. ( G ` c ) } $.
      $( If ` F < G ` , then there is some ` z ` witnessing this, but we can
         say more and in fact there is a definable expression ` X ` that also
         witnesses ` F < G ` .  (Contributed by Mario Carneiro,
         25-May-2015.) $)
      oemapvali $p |- ( ph -> ( X e. B /\ ( F ` X ) e. ( G ` X ) /\
        A. w e. B ( X e. w -> ( F ` w ) = ( G ` w ) ) ) ) $=
        ( cv cfv wcel wceq wi wral w3a wbr wrex oemapval mpbid crab ssrab2 cuni
        wa con0 wss cfn c0 wne adantr onss syl sstrid cfsupp csupp co wf simprd
        cantnfs wfn 3ad2ant1 simp2 simpld ffnd ne0i 3ad2ant3 fvn0elsupp rabssdv
        syl22anc wfun fsuppimp ssfi ex simpl2im sylc weq eleq12d simprl simprrl
        fveq2 elrabd ne0d ordunifi syl3anc eqeltrid sselid cbvrabv elrab2 sylib
        simprrr wb word ffvelcdmd onelon syl2anc eloni ordirr 3syl nelneq eleq2
        wn eqeq12d imbi12d rspcdva mtod cvv ssexg sylancr ssonuni ontri1 mpbird
        elssuni sseqtrrdi eqssd eleq1 imbi1d ralbidv 3jca rexlimddv ) ADUBZJUCZ
        YLKUCZUDZYLEUBZUDZYPJUCZYPKUCZUEZUFZEGUGZUPZLGUDZLJUCZLKUCZUDZLYPUDZYTU
        FZEGUGZUHDGAJKIUIUUCDGUJTABCDEFGHIJKNOPQRSUKULAYLGUDZUUCUPZUPZUUDUUGUUJ
        UUMMUBZJUCZUUNKUCZUDZMGUMZGLUUQMGUNZUUMLUURUOZUURUAUUMUURUQURZUURUSUDZU
        URUTVAUUTUURUDUUMUURGUQUUSUUMGUQUDZGUQURAUVCUULPVBZGVCVDVEZUUMKUTVFUIZU
        URKUTVGVHZURZUVBAUVFUULAGFKVIZUVFAKHUDUVIUVFUPSAFGHKNOPVKULZVJVBAUVHUUL
        AUUQMGUVGAUUNGUDZUUQUHUVCUVKKGVLZUUPUTVAZUUNUVGUDAUVKUVCUUQPVMAUVKUUQVN
        AUVKUVLUUQAGFKAUVIUVFUVJVOZVPVMUUQAUVMUVKUUPUUOVQVRGKUQUUNVSWAVTVBUVFKW
        BUVGUSUDZUVHUVBUFKUTWCUVOUVHUVBUVGUURWDWEWFWGUUMUURYLUUMUUQYOMYLGMDWHUU
        OYMUUPYNUUNYLJWLUUNYLKWLWIAUUKUUCWJZAUUKYOUUBWKWMZWNUURWOWPWQZWRZUUMUUD
        UUGUUMLUURUDUUDUUGUPUVRBUBZJUCZUVTKUCZUDZUUGBLGUURUVTLUEUWAUUEUWBUUFUVT
        LJWLUVTLKWLWIUUQUWCMBGMBWHUUOUWAUUPUWBUUNUVTJWLUUNUVTKWLWIWSWTXAVJZUUMU
        UJUUBAUUKYOUUBXBZUUMLYLUEZUUJUUBXCUUMLYLUUMLYLURZYLLUDZXMZUUMUWHUUEUUFU
        EZUUMUUGUUFUUFUDXMZUWJXMUWDUUMUUFUQUDZUUFXDUWKUUMFUQUDZUUFFUDUWLAUWMUUL
        OVBUUMGFLKAUVIUULUVNVBUVSXEFUUFXFXGUUFXHUUFXIXJUUEUUFUUFXKXGUUMUUAUWHUW
        JUFEGLYPLUEZYQUWHYTUWJYPLYLXLUWNYRUUEYSUUFYPLJWLYPLKWLXNXOUWEUVSXPXQUUM
        LUQUDYLUQUDZUWGUWIXCUUMLUUTUQUAUUMUURXRUDZUVAUUTUQUDUUMUURGURUVCUWPUUSU
        VDUURGUQXSXTUVEUURXRYAWGWQUUMUVCUUKUWOUVDUVPGYLXFXGLYLYBXGYCUUMYLUURUDZ
        YLLURUVQUWQYLUUTLYLUURYDUAYEVDYFUWFUUIUUAEGUWFUUHYQYTLYLYPYGYHYIVDYCYJY
        K $.

      $( Lemma for ~ cantnf .  (Contributed by Mario Carneiro, 4-Jun-2015.)
         (Revised by AV, 2-Jul-2019.) $)
      cantnflem1a $p |- ( ph -> X e. ( G supp (/) ) ) $=
        ( c0 csupp co wcel cfv wne cv wceq wi wral oemapvali simp1d simp2d ne0d
        wfn con0 cvv wa wb wf cfsupp wbr cantnfs mpbid simpld ffnd 0ex elsuppfn
        a1i syl3anc mpbir2and ) ALKUBUCUDUEZLGUEZLKUFZUBUGZAVNLJUFZVOUEZLEUHZUE
        VSJUFVSKUFUIUJEGUKZABCDEFGHIJKLMNOPQRSTUAULZUMAVOVQAVNVRVTWAUNUOAKGUPGU
        QUEUBURUEZVMVNVPUSUTAGFKAGFKVAZKUBVBVCZAKHUEWCWDUSSAFGHKNOPVDVEVFVGPWBA
        VHVJLKUQURGUBVIVKVL $.

      cantnflem1.o $e |- O = OrdIso ( _E , ( G supp (/) ) ) $.
      $( Lemma for ~ cantnf .  (Contributed by Mario Carneiro, 4-Jun-2015.)
         (Revised by AV, 2-Jul-2019.) $)
      cantnflem1b $p |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) )
                             -> X C_ ( O ` u ) ) $=
        ( cv csuc cdm wcel ccnv cfv wss wa wn simprr con0 wb word c0 csupp oicl
        co cep wf1o wiso cvv wwe ovexd com cantnfcl simpld oiiso syl2anc isof1o
        wf syl f1ocnv f1of 3syl cantnflem1a ffvelcdmd ordelon sylancr a1i sylan
        onsucb sylibr adantrr ontri1 syl2an2r mpbid wbr adantr wtr ordtr simprl
        mp1i trsuc isorel syl12anc fvex epeli wceq f1ocnvfv2 eleq2d bitrd mtbid
        3bitr3g wi wral oemapvali simp1d onelon suppssdm cfsupp fssdm ffvelcdmi
        cantnfs oif sseldd mpbird ) AFUEZUFZMUGZUHZNMUIZUJZYAUKZULZULZNYAMUJZUK
        ZYJNUHZUMZYIYAYFUHZYLYIYGYNUMZAYDYGUNAYFUOUHZYHYAUOUHZYGYOUPAYCUQZYFYCU
        HZYPLURUSVAZVBMUDUTZAYTYCNYEAYCYTMVCZYTYCYEVCYTYCYEVNAYCYTVBVBMVDZUUBAY
        TVEUHYTVBVFZUUCALURUSVGAUUDYCVHUHAGHILMPQRUDUAVIVJYTVBMVEUDVKVLZYCYTVBV
        BMVMVOZYCYTMVPYTYCYEVQVRABCDEGHIJKLNOPQRSTUAUBUCVSZVTZYCYFWAWBAYDYQYGAY
        DULYBUOUHZYQAYRYDUUIYRAUUAWCYCYBWAWDYAWEWFWGYFYAWHWIWJYIYNYJYFMUJZUHZYL
        YIYAYFVBWKZYJUUJVBWKZYNUUKYIUUCYAYCUHZYSUULUUMUPAUUCYHUUEWLYIYCWMZYDUUN
        YRUUOYIUUAYCWNWPAYDYGWOYCYAWQVLZAYSYHUUHWLYCYTYAYFVBVBMWRWSYAYFNYEWTXAY
        JUUJYFMWTXAXGYIUUJNYJAUUJNXBZYHAUUBNYTUHUUQUUFUUGYCYTNMXCVLWLXDXEXFANUO
        UHZYHYJUOUHZYKYMUPAHUOUHZNHUHZUURRAUVANKUJNLUJUHNEUEZUHUVBKUJUVBLUJXBXH
        EHXIABCDEGHIJKLNOPQRSTUAUBUCXJXKHNXLVLAUUTYHYJHUHUUSRYIYTHYJAYTHUKYHAHG
        YTLLURXMAHGLVNZLURXNWKZALIUHUVCUVDULUAAGHILPQRXQWJVJXOWLYIUUNYJYTUHUUPY
        CYTYAMYTVBMUDXRXPVOXSHYJXLWINYJWHWIXT $.

      $( Lemma for ~ cantnf .  (Contributed by Mario Carneiro, 4-Jun-2015.)
         (Revised by AV, 2-Jul-2019.)  (Proof shortened by AV, 4-Apr-2020.) $)
      cantnflem1c $p |- ( ( ( ( ph /\ ( suc u e. dom O
                                           /\ ( `' O ` X ) C_ u ) ) /\ x e. B )
                               /\ ( ( F ` x ) =/= (/) /\ ( O ` u ) e. x ) ) ->
                            x e. ( G supp (/) ) ) $=
        ( cv csuc cdm wcel ccnv cfv wss wa c0 wne wfn csupp co ad3antrrr simplr
        con0 wf cfsupp wbr cantnfs simpld ffnd wceq cantnflem1b ad2antrr simprr
        mpbid wi wral oemapvali simp1d onelon syl2anc syl sselda ad4ant13 ontr2
        onss mp2and eleq2w fveq2 eqeq12d imbi12d simp3d rspcdva simprl eqnetrrd
        mpd fvn0elsupp syl22anc ) AFUEZUFMUGUHNMUIUJWOUKULZULZBUEZHUHZULZWRKUJZ
        UMUNZWOMUJZWRUHZULZULZHUTUHZWSLHUOZWRLUJZUMUNWRLUMUPUQUHAXGWPWSXERURWQW
        SXEUSZAXHWPWSXEAHGLAHGLVAZLUMVBVCZALIUHXKXLULUAAGHILPQRVDVKVEVFURXFXAXI
        UMXFNWRUHZXAXIVGZXFNXCUKZXDXMWQXOWSXEABCDEFGHIJKLMNOPQRSTUAUBUCUDVHVIWT
        XBXDVJXFNUTUHZWRUTUHZXOXDULXMVLAXPWPWSXEAXGNHUHZXPRAXRNKUJNLUJUHZNEUEZU
        HZXTKUJZXTLUJZVGZVLZEHVMZABCDEGHIJKLNOPQRSTUAUBUCVNZVOHNVPVQURAWSXQWPXE
        AHUTWRAXGHUTUKRHWBVRVSVTNXCWRWAVQWCXFYEXMXNVLEHWRXTWRVGZYAXMYDXNEBNWDYH
        YBXAYCXIXTWRKWEXTWRLWEWFWGAYFWPWSXEAXRXSYFYGWHURXJWIWLWTXBXDWJWKHLUTWRW
        MWN $.

      cantnflem1.h $e |- H = seqom ( ( k e. _V , z e. _V |->
        ( ( ( A ^o ( O ` k ) ) .o ( G ` ( O ` k ) ) ) +o z ) ) , (/) ) $.
      $( Lemma for ~ cantnf .  (Contributed by Mario Carneiro, 4-Jun-2015.)
         (Revised by AV, 2-Jul-2019.) $)
      cantnflem1d $p |- ( ph ->
         ( ( A CNF B ) ` ( x e. B |-> if ( x C_ X , ( F ` x ) , (/) ) ) ) e.
           ( H ` suc ( `' O ` X ) ) ) $=
        ( coe co cfv comu ccnv csuc cv wss c0 cif cmpt ccnf con0 wcel wceq wral
        coa wi oemapvali simp1d onelon syl2anc oecl wf cfsupp wbr cantnfs mpbid
        wa simpld ffvelcdmd omcl com cdm csupp wf1o cep wiso cvv ovexd cantnfcl
        wwe oiiso isof1o syl f1ocnv f1of 3syl cantnflem1a simprd elnn ffvelcdmi
        cantnfvalf oaword1 cantnfsuc mpdan f1ocnvfv2 oveq2d fveq2d oveq1d eqtrd
        oveq12d sseqtrrd wo sselda adantr onsseleq orcom bitrdi ifbid mpteq2dva
        onss ffvelcdmda wne ne0d on0eln0 mpbird ifcld fmpttd 0ex a1i fsuppmptif
        wb mpbir2and cdif wn eldifn adantl iffalsed suppss2 ifor syl3anc sseldd
        fveq2 ifeq1da eleq1w ifbieq1d eqid fvmpt ifeq2d eqtr3d eqtr4id mpteq2ia
        fvex ifex cantnfp1 omsuc word eloni simp2d ordsucss sylc onsuc eqsstrrd
        omwordi mpd cantnflt2 oaord eqeltrd ) AFOUGUHZOLUIZUJUHZONUKZUIZULMUIZB
        GBUMZOUNZUVLKUIZUOUPZUQZFGURUHZUIZAUVHUVHUVJMUIZVCUHZUVKAUVHUSUTZUVSUSU
        TZUVHUVTUNAUVFUSUTZUVGUSUTZUWAAFUSUTZOUSUTZUWCRAGUSUTZOGUTZUWFSAUWHOKUI
        ZUVGUTZOEUMZUTUWKKUIUWKLUIVAVDEGVBZABCDEFGHIKLOPQRSTUAUBUCUDVEZVFZGOVGV
        HZFOVIVHZAUWEUVGFUTUWDRAGFOLAGFLVJZLUOVKVLZALHUTUWQUWRVOUBAFGHLQRSVMVNV
        PUWNVQZFUVGVGVHZUVFUVGVRVHAUVJVSUTZUWBAUVJNVTZUTUXBVSUTZUXAALUOWAUHZUXB
        OUVIAUXBUXDNWBZUXDUXBUVIWBUXDUXBUVIVJAUXBUXDWCWCNWDZUXEAUXDWEUTUXDWCWHZ
        UXFALUOWAWFAUXGUXCAFGHLNQRSUEUBWGZVPUXDWCNWEUEWIVHUXBUXDWCWCNWJWKZUXBUX
        DNWLUXDUXBUVIWMWNABCDEFGHIKLOPQRSTUAUBUCUDWOZVQAUXGUXCUXHWPUVJUXBWQVHZV
        SUSUVJMDWEWEFJUMNUIZUGUHUXLLUIUJUHDUMJMUFWSWRWKUVHUVSWTVHAUVKFUVJNUIZUG
        UHZUXMLUIZUJUHZUVSVCUHZUVTAUXAUVKUXQVAUXKADFGHJLNMUVJQRSUEUBUFXAXBAUXPU
        VHUVSVCAUXNUVFUXOUVGUJAUXMOFUGAUXEOUXDUTUXMOVAUXIUXJUXBUXDONXCVHZXDAUXM
        OLUXRXEXHXFXGXIAUVRUVFUWIUJUHZCGCUMZOUTZUXTKUIZUOUPZUQZUVQUIZVCUHZUVHAU
        VRBGUVLOVAZUVLOUTZXJZUVNUOUPZUQZUVQUIZUYFAUVPUYKUVQABGUVOUYJAUVLGUTZVOZ
        UVMUYIUVNUOUYNUVMUYHUYGXJZUYIUYNUVLUSUTUWFUVMUYOYIAGUSUVLAUWGGUSUNSGXRW
        KXKAUWFUYMUWOXLUVLOXMVHUYHUYGXNXOXPXQXEAUYKHUTUYLUYFVAABFGHUYKUYDOUWIQR
        SAUYDHUTGFUYDVJUYDUOVKVLACGUYCFAUXTGUTZVOUYAUYBUOFAGFUXTKAGFKVJZKUOVKVL
        ZAKHUTUYQUYRVOUAAFGHKQRSVMVNZVPZXSAUOFUTZUYPAVUAFUOXTZAFUVGUWSYAAUWEVUA
        VUBYIRFYBWKYCZXLYDYEAGFOCKUSWEUOUYTSUOWEUTAYFYGAUYQUYRUYSWPYHAFGHUYDQRS
        VMYJZUWNAGFOKUYTUWNVQZAGUYCCUSOUOAUXTGOYKUTZVOUYAUYBUOVUFUYAYLAUXTGOYMY
        NYOSYPZBGUYJUYGUWIUVLUYDUIZUPZUYMUYJUYGUVNUYHUVNUOUPZUPZVUIUYGUYHUVNUOY
        QUYMUYGUVNVUHUPVUIVUKUYMUYGUVNUWIVUHUYGUVNUWIVAUYMUVLOKYTYNUUAUYMUYGVUH
        VUJUVNCUVLUYCVUJGUYDUXTUVLVAUYAUYHUYBUVNUOCBOUUBUXTUVLKYTUUCUYDUUDUYHUV
        NUOUVLKUUJYFUUKUUEUUFUUGUUHUUIUULWPXGAUXSUVFVCUHZUVHUYFAVULUVFUWIULZUJU
        HZUVHAUWCUWIUSUTZVUNVULVAUWPAUWEUWIFUTVUORVUEFUWIVGVHZUVFUWIUUMVHAVUMUV
        GUNZVUNUVHUNZAUVGUUNZUWJVUQAUWDVUSUWTUVGUUOWKAUWHUWJUWLUWMUUPUWIUVGUUQU
        URAVUMUSUTZUWDUWCVUQVURVDAVUOVUTVUPUWIUUSWKUWTUWPVUMUVGUVFUVAYRUVBUUTAU
        YEUVFUTZUYFVULUTZAFGOHUYDQRSVUDVUCUWOVUGUVCZAUYEUSUTZUWCUXSUSUTZVVAVVBY
        IAUWCVVAVVDUWPVVCUVFUYEVGVHUWPAUWCVUOVVEUWPVUPUVFUWIVRVHUYEUVFUXSUVDYRV
        NYSUVEYS $.

      $( Lemma for ~ cantnf .  This part of the proof is showing uniqueness of
         the Cantor normal form.  We already know that the relation ` T ` is a
         strict order, but we haven't shown it is a well-order yet.  But being
         a strict order is enough to show that two distinct ` F , G ` are ` T `
         -related as ` F < G ` or ` G < F ` , and WLOG assuming that
         ` F < G ` , we show that ` CNF ` respects this order and maps these
         two to different ordinals.  (Contributed by Mario Carneiro,
         28-May-2015.)  (Revised by AV, 2-Jul-2019.) $)
      cantnflem1 $p |- ( ph -> ( ( A CNF B ) ` F ) e. ( ( A CNF B ) ` G ) ) $=
        ( vu cv cfv wss c0 cif cmpt csuc wcel con0 cvv csupp cep mp1i 3syl wceq
        co wn wo syl cen wbr com simpld syl2anc bitrdi syl5ibcom simprd word wb
        eleqtrrd wf ffvelcdmd elssuni wa wi eqeltrrd sylibr eleq1 sseq2 anbi12d
        wf1o fveq2 sseq2d ifbid mpteq2dv fveq2d suceq eleq12d imbi12d f1ocnvfv2
        weq adantl ordelon sylancr a1i onsseleq syl2an2r simprl simprr jca comu
        eloni coe coa adantr oecl cfsupp cantnfs mpbid wne mpbird cdif 0ex ifid
        eqtrdi onelon ffvelcdmi sseldd ifeq1 eqeq12d ad2antrr isorel epeli fvex
        syl12anc 3bitr3g ontri1 bitrd expr eqidd crab nfcv eqtrd cuni ccnf ccnv
        cdm ovex oion uniexg wlim cantnflem1a n0i wwe ovexd cantnfcl oien breq1
        sucidg ensymb en0 bitri nnlim ioran sylanbrc unizlim mtbird orduniorsuc
        mtod oicl ord wiso oiiso isof1o f1ocnv f1of peano2b cantnflem1d eqeltrd
        mpd imbi2d ss0 sylan onsucb ordsssuc wtr ordtr trsuc cantnff ffvelcdmda
        wral oemapvali simp1d ne0d ifcld fmpttd wfun mptexd funmpt ssidd ifeq1d
        on0eln0 suppssr fsuppsssupp syl22anc mpbir2and elnn cantnfvalf suppssdm
        suppss2 fssdm oif omcl oaord syl3anc sselda adantlr onsssuc sucid sucex
        onss mpbii ordelsuc sseld sylbid cantnflem1c eleq1d mtbid con2d sylibrd
        vex impbid orbi1d bitr4d pm2.61ne mpteq2dva c1o wfn ifex ralrimivw eqid
        orcom fnmpt w3a suppvalfn nffvmpt1 nfv neeq1d cbvrabw fvmpt2d biantrurd
        nfne bitr4di rabbidva mpbiran necon3i iffalse necon1ai expimpd biimtrid
        dif1o syl5 rabssdv eqsstrd eqeq1 sseq1 orbi12d ifbieq1d cbvmptv ifeq1da
        3impia fvmpt ifeq2d ifor eqtr4di eqtr3d mpteq2ia eqtr4i cantnfp1 syldan
        cantnfsuc simp3d adantrr eleq2 rspcv syl3c oveq2d oveq1d eqtr4d embantd
        biimpd sylbird a1dd jaod com23 finds2 vtoclga mpcom mp2and eqeq2 orduni
        feqmptd con2bid ax-mp ordtri1 sylancl eleq2d ffvelcdm sylancom pm2.65da
        ontr2 eldifd imp ifbothda cantnfval 3eltr4d ) ABGBUHZNUUDZUUAZNUIZUJZVX
        NKUIZUKULZUMZFGUUBVCZUIZVXPUNZMUIZKVYBUILVYBUIZAVXPVXOUOZONUUCZUIZVXPUJ
        ZVYCVYEUOZAVXPVYDVXOAVXOUPUOZVXPUQUOVXPVYDUOLUKURVCZUQUOZVYLALUKURUUEVY
        MUSNUQUEUUFUTVXOUPUUGVXPUQUUPVAAVXOVXPVBZVDVXOVYDVBZAVYOVXOUKVBZVXOUUHZ
        VEZAVYQVDVYRVDZVYSVDAVYQVYMUKVBZAOVYMUOZWUAVDABCDEFGHIKLOPQRSTUAUBUCUDU
        UIZVYMOUUJVFAVXOVYMVGVHZVYQWUAAVYNVYMUSUUKZWUDALUKURUULZAWUEVXOVIUOZAFG
        HLNQRSUEUBUUMZVJZVYMUSNUQUEUUNVKVYQWUDUKVYMVGVHZWUAVXOUKVYMVGUUOWUJVYMU
        KVGVHWUAUKVYMUUQVYMUURUUSVLVMUVFAWUGVYTAWUEWUGWUHVNZVXOUUTVFVYQVYRUVAUV
        BVXOVOZVYOVYSVPAVYMUSNUEUVGZVXOUVCUTUVDAVYOVYPWULVYOVYPVEAWUMVXOUVEUTUV
        HUVQZVQZAVYIVXOUOZVYJAVYMVXOOVYHAVXOVYMNWHZVYMVXOVYHWHZVYMVXOVYHVRZAVXO
        VYMUSUSNUVIZWUQAVYNWUEWUTWUFWUIVYMUSNUQUEUVJVKZVXOVYMUSUSNUVKZVFZVXOVYM
        NUVLZVYMVXOVYHUVMZVAZWUCVSZVYIVXOVTVFZVXPVIUOZAVYGVYJWAZVYKWBZAVYDVIUOW
        VIAVXOVYDVIWUNWUKWCVXPUVNWDACUHZVXOUOZVYIWVLUJZWAZBGVXNWVLNUIZUJZVXSUKU
        LZUMZVYBUIZWVLUNZMUIZUOZWBZWBAWVKWBCVXPVIWVLVXPVBZWWDWVKAWWEWVOWVJWWCVY
        KWWEWVMVYGWVNVYJWVLVXPVXOWEWVLVXPVYIWFWGWWEWVTVYCWWBVYEWWEWVSVYAVYBWWEB
        GWVRVXTWWEWVQVXRVXSUKWWEWVPVXQVXNWVLVXPNWIWJWKWLWMWWEWWAVYDMWVLVXPWNWMW
        OWPUVRWWDUKVXOUOZVYIUKUJZWAZBGVXNUKNUIZUJZVXSUKULZUMZVYBUIZUKUNZMUIZUOZ
        WBUGUHZVXOUOZVYIWWQUJZWAZBGVXNWWQNUIZUJZVXSUKULZUMZVYBUIZWWQUNZMUIZUOZW
        BZWXFVXOUOZVYIWXFUJZWAZBGVXNWXFNUIZUJZVXSUKULZUMZVYBUIZWXFUNZMUIZUOZWBZ
        ACUGWVLUKVBZWVOWWHWWCWWPWYBWVMWWFWVNWWGWVLUKVXOWEWVLUKVYIWFWGWYBWVTWWMW
        WBWWOWYBWVSWWLVYBWYBBGWVRWWKWYBWVQWWJVXSUKWYBWVPWWIVXNWVLUKNWIWJWKWLWMW
        YBWWAWWNMWVLUKWNWMWOWPCUGWRZWVOWWTWWCWXHWYCWVMWWRWVNWWSWVLWWQVXOWEWVLWW
        QVYIWFWGWYCWVTWXEWWBWXGWYCWVSWXDVYBWYCBGWVRWXCWYCWVQWXBVXSUKWYCWVPWXAVX
        NWVLWWQNWIWJWKWLWMWYCWWAWXFMWVLWWQWNWMWOWPWVLWXFVBZWVOWXLWWCWXTWYDWVMWX
        JWVNWXKWVLWXFVXOWEWVLWXFVYIWFWGWYDWVTWXQWWBWXSWYDWVSWXPVYBWYDBGWVRWXOWY
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        XUOWWSXUBXUSAXUEXUNXULXUKXUGAWXJXULWWSXUMVVTVYIWWQUXOXDXPXUOVYIWXFUSVHZ
        WYEWXMUSVHZXUBYXTXUOWUTWUPWXJYYAYYBVPYUIAWUPXUNWVGXLXUQVXOVYMVYIWXFUSUS
        NYIYLVYIWXFYUJYJWYEWXMYUKYJYMXPWCXWPYXQYXRWBEWXMGXWKWXMVBZXWLYXQXWOYXRX
        WKWXMOVWAYYCXWMXVAXWNYXNXWKWXMKWIXWKWXMLWIYGWPVWBVWCVWDVWEVWFWOUYKVWHVW
        GYPVWIXUAXUCWXTWXIAXUCWXTWBWXJAWYLXUCWXTWYRXUCWYIWXQWYKWXSXUCWYHWXPVYBX
        UCBGWYGWXOXUCWYFWXNVXSUKXUCWYEWXMVXNVYIWXFNWIWJWKWLWMXUCWYJWXRMVYIWXFWN
        WMWOVMXLVWJVWKUYBVUPVWLXBVWMVWNVWOVWPAKVYAVYBAKBGVXSUMVYAABGFKXWBVWSABG
        VXSVXTVXRVXSVXSVBXYHVXSVXTVBXVRVXSUKVXSVXTVXSVWQUKVXTVXSVWQXVRVXRWAVXSY
        QXVRVXRVDZXYHXVRYYDVXQVXNUOZXYHXVRVXRYYEXVRXYOVXQUPUOZVXRYYEVDVPXYPAYYF
        XVQAXVLVXQGUOYYFSAVYMGVXQXXPAVYGVXQVYMUOWUOVXOVYMVXPNXXRYDVFYEGVXQYCVKZ
        XLVXNVXQYNVKVWTAXVQYYEXYHAXVQYYEWAZWAZVXSVXNLUIZUKYYIXVQXWQOVXNUOZVXSYY
        JVBZAXVQYYEXEZAXWQYYHYXSXLYYIOVXQUJZYYEYYKAYYNYYHAYYNVXQOUOZVDZAVXPVYIU
        OZYYOAVYJYYQVDZWVHAVYIVOZVXPVOZVYJYYRVPAXUEYYSXUGVYIXIVFWULYYTWUMVXOVWR
        VXAZVYIVXPVXBVXCXPAYYQVXQWYEUOZYYOAVXPVYIUSVHZVXQWYEUSVHZYYQUUUBAWUTVYG
        WUPUUUCUUUDVPWVAWUOWVGVXOVYMVXPVYIUSUSNYIYLVXPVYIOVYHYKYJVXQWYEVYINYKYJ
        YMAWYEOVXQWYQVXDYOUYEAOUPUOZYYFYYNYYPVPAXVLXWIUUUESXWSGOYCVKZYYGOVXQYNV
        KXRXLAXVQYYEXFZAUUUEYYHXYOYYNYYEWAYYKWBUUUFAXVQXYOYYEXYPVVTOVXQVXNVXHXD
        VWPXWPYYKYYLWBEVXNGEBWRZXWLYYKXWOYYLXWKVXNOVWAUUUHXWMVXSXWNYYJXWKVXNKWI
        XWKVXNLWIYGWPVWBVWCAYYHVXNGVYMXSUOYYJUKVBYYIVXNGVYMYYMYYIYVEVXPYURUOZYY
        IYVEWAZUUUIYYEYYIYYEYVEUUUGXLUUUJUUUIVXQYVCUOZYYEUUUJVXPYURUSVHZVXQYVCU
        SVHZUUUIUUUKUUUJWUTVYGYVNUUULUUUMVPAWUTYYHYVEWVAYHAVYGYYHYVEWUOYHYYIYVE
        WUSYVNAWUSYYHYVEWVFYHVYMVXOVXNVYHVXEVXFZVXOVYMVXPYURUSUSNYIYLVXPYURYVPY
        JVXQYVCYVQYJYMUUUJYVCVXNVXQYYIYVEWUQYVFAWUQYYHYVEWVCYHYVJVXFVXDYOXRUUUJ
        YURVXPUJZUUUIVDZUUUJYVNUUUOUUUNYURVXOVTVFUUUJYVSYYTUUUOUUUPVPUUUJYVTYVS
        UUUJWULYVNYVTWUMUUUNYWAXAYWCVFUUUAYURVXPVXBVXCXPVXGVXIAGFUQLUPVYMVXNUKX
        WHAVYMUWQSXXFUWTVVQYTYPVWIVXJVXKUYMYTWMAVYFVXOMUIVYEADFGHJLNMQRSUEUBUFV
        XLAVXOVYDMWUNWMYTVXM $.
    $}

    ${
      cantnf.c $e |- ( ph -> C e. ( A ^o B ) ) $.
      cantnf.s $e |- ( ph -> C C_ ran ( A CNF B ) ) $.
      cantnf.e $e |- ( ph -> (/) e. C ) $.
      $( Lemma for ~ cantnf .  (Contributed by Mario Carneiro, 28-May-2015.) $)
      cantnflem2 $p |- ( ph -> ( A e. ( On \ 2o ) /\ C e. ( On \ 1o ) ) ) $=
        ( con0 wcel c1o c2o cdif c0 wceq wo coe co oecl syl2anc onelon sylanbrc
        wss ondif1 eldifbd ssel syl5com mtod oe0m syl difss eqsstrdi syl5ibrcom
        oveq1 sseq1d oe1m eqimss 3syl jaod cpr elpri df2o3 eleq2s nsyl eldifd
        jca ) AFRUAUBSHRTUBSZAFRUALAFUCUDZFTUDZUEZFUASAVSFGUFUGZTULZAWAHTSZAHRT
        AHRSZUCHSVPAVTRSZHVTSZWCAFRSGRSZWDLMFGUHUIOVTHUJUIQHUMUKZUNAWEWAWBOVTTH
        UOUPUQAVQWAVRAWAVQUCGUFUGZTULAWHTGUBZTAWFWHWIUDMGURUSTGUTVAVQVTWHTFUCGU
        FVCVDVBAWAVRTGUFUGZTULZAWFWJTUDWKMGVEWJTVFVGVRVTWJTFTGUFVCVDVBVHUQVSFUC
        TVIUAFUCTVJVKVLVMVNWGVO $.

      cantnf.x $e |- X = U. |^| { c e. On | C e. ( A ^o c ) } $.
      cantnf.p $e |- P = ( iota d E. a e. On E. b e. ( A ^o X )
        ( d = <. a , b >. /\ ( ( ( A ^o X ) .o a ) +o b ) = C ) ) $.
      cantnf.y $e |- Y = ( 1st ` P ) $.
      cantnf.z $e |- Z = ( 2nd ` P ) $.
      ${
        cantnf.g $e |- ( ph -> G e. S ) $.
        cantnf.v $e |- ( ph -> ( ( A CNF B ) ` G ) = Z ) $.
        cantnf.f $e |- F = ( t e. B |-> if ( t = X , Y , ( G ` t ) ) ) $.
        $( Lemma for ~ cantnf .  Here we show existence of Cantor normal forms.
           Assuming (by transfinite induction) that every number less than
           ` C ` has a normal form, we can use ~ oeeu to factor ` C ` into the
           form ` ( ( A ^o X ) .o Y ) +o Z ` where ` 0 < Y < A ` and
           ` Z < ( A ^o X ) ` (and a fortiori ` X < B ` ).  Then since
           ` Z < ( A ^o X ) <_ ( A ^o X ) .o Y <_ C ` , ` Z ` has a normal
           form, and by appending the term ` ( A ^o X ) .o Y ` using ~ cantnfp1
           we get a normal form for ` C ` .  (Contributed by Mario Carneiro,
           28-May-2015.) $)
        cantnflem3 $p |- ( ph -> C e. ran ( A CNF B ) ) $=
          ( vk ccnf co cfv crn coe comu coa wcel wceq wss con0 c1o cdif w3a c2o
          c0 wa cantnflem2 eqid 3pm3.2i oeeui mpbiri simpld simp1d oecl syl2anc
          syl simp2d eldifad onelon wne dif1o simprbi wb on0eln0 mpbird omword1
          syl21anc omcl simp3d oaword1 simprd sseqtrd sstrd ontr2 oeord syl3anc
          wi mp2and csupp cv adantr suppssdm wf cfsupp wbr cantnfs mpbid sselda
          fssdm ffvelcdmd wfn cvv ffnd elexd elsuppfn simplbda cep coi cantnfle
          cmpo cseqom ex ssrdv cantnfp1 oveq2d 3eqtrd cantnff fnfvelrn eqeltrrd
          ) AMGHUQURZUSZIYQUTZAYRGOVAURZPVBURZNYQUSZVCURZUUAQVCURZIAMKVDZYRUUCV
          EZAFGHKMNOPUBUCUDUMAOHVDZYTGHVAURZVDZAYTIVFZIUUHVDZUUIAYTUUAIAYTVGVDZ
          PVGVDZVLPVDZYTUUAVFAGVGVDZOVGVDZUULUCAUUPPGVHVIVDZQYTVDZAUUPUUQUURVJZ
          UUDIVEZAGVGVKVIVDZIVGVHVIVDZVMZUUSUUTVMZABCDEGHIKLUBUCUDUEUFUGUHVNZUV
          CUVDOOVEZPPVEZQQVEZVJUVFUVGUVHOVOPVOQVOVPTRSUAGIOPJQOPQUIUJUKULVQVRWC
          ZVSZVTZGOWAWBZAUUOPGVDZUUMUCAPGVHAUUPUUQUURUVJWDZWEZGPWFWBZAUUNPVLWGZ
          AUUQUVQUVNUUQUVMUVQPGWHWIWCAUUMUUNUVQWJUVPPWKWCWLYTPWMWNAUUAUUDIAUUAV
          GVDZQVGVDZUUAUUDVFAUULUUMUVRUVLUVPYTPWOWBAUULUURUVSUVLAUUPUUQUURUVJWP
          ZYTQWFWBUUAQWQWBAUUSUUTUVIWRZWSWTUFAUULUUHVGVDZUUJUUKVMUUIXDUVLAUUOHV
          GVDZUWBUCUDGHWAWBYTIUUHXAWBXEAUUPUWCUVAUUGUUIWJUVKUDAUVAUVBUVEVSZOHGX
          BXCWLUVOABNVLXFURZOABXGZUWEVDZUWFOVDZAUWGVMZUWHGUWFVAURZYTVDZUWIUWJQV
          FZUURUWKUWIUWJUWJUWFNUSZVBURZQUWIUWJVGVDZUWMVGVDZVLUWMVDZUWJUWNVFUWIU
          UOUWFVGVDZUWOAUUOUWGUCXHZUWIUWCUWFHVDZUWRAUWCUWGUDXHZAUWEHUWFAHGUWENN
          VLXIAHGNXJZNVLXKXLZANKVDZUXBUXCVMUMAGHKNUBUCUDXMXNVSZXPXOZHUWFWFWBZGU
          WFWAWBZUWIUUOUWMGVDUWPUWSUWIHGUWFNAUXBUWGUXEXHUXFXQGUWMWFWBZUWIUWQUWM
          VLWGZAUWGUWTUXJANHXRUWCVLXSVDUWGUWTUXJVMWJAHGNUXEXTUDAVLIUHYAUWFNVGXS
          HVLYBXCYCUWIUWPUWQUXJWJUXIUWMWKWCWLUWJUWMWMWNUWIUWNUUBQUWIDGHUWFKUPNU
          WEYDYEZUPDXSXSGUPXGUXKUSZVAURUXLNUSVBURDXGVCURYGVLYHZUBUWSUXAUXKVOAUX
          DUWGUMXHUXMVOUXFYFAUUBQVEUWGUNXHWSWTAUURUWGUVTXHUWIUWOUULUWLUURVMUWKX
          DUXHAUULUWGUVLXHUWJQYTXAWBXEUWIUWRUUPUVAUWHUWKWJUXGAUUPUWGUVKXHAUVAUW
          GUWDXHUWFOGXBXCWLYIYJUOYKZWRAUUBQUUAVCUNYLUWAYMAYQKXRUUEYRYSVDAKUUHYQ
          AGHKUBUCUDYNXTAUUEUUFUXNVSKMYQYOWBYP $.
      $}

      $( Lemma for ~ cantnf .  Complete the induction step of ~ cantnflem3 .
         (Contributed by Mario Carneiro, 25-May-2015.) $)
      cantnflem4 $p |- ( ph -> C e. ran ( A CNF B ) ) $=
        ( vg vt cv ccnf co cfv wceq crn wcel wrex coe comu coa con0 wss c1o w3a
        cdif c2o wa cantnflem2 eqid 3pm3.2i oeeui mpbiri syl simpld simp1d oecl
        syl2anc simp2d eldifad onelon omcl simp3d oaword1 dif1o simprbi on0eln0
        c0 wne wb mpbird omword1 syl21anc sseldd simprd eleqtrd wfn cantnff ffn
        fvelrnb 3syl mpbid cif cmpt adantr simprl simprr cantnflem3 rexlimddv
        wf ) AUJULZFGUMUNZUONUPZHXMUQZURUJJANXOURZXNUJJUSZAHXONUDANFLUTUNZMVAUN
        ZNVBUNZHAXSXTNAXSVCURZNVCURZXSXTVDAXRVCURZMVCURZYAAFVCURZLVCURZYCTAYFMF
        VEVGURZNXRURZAYFYGYHVFZXTHUPZAFVCVHVGURHVCVEVGURVIZYIYJVIZABCDEFGHJKSTU
        AUBUCUDUEVJYKYLLLUPZMMUPZNNUPZVFYMYNYOLVKMVKNVKVLQOPRFHLMINLMNUFUGUHUIV
        MVNVOZVPZVQFLVRVSZAYEMFURZYDTAMFVEAYFYGYHYQVTZWAFMWBVSZXRMWCVSAYCYHYBYR
        AYFYGYHYQWDZXRNWBVSXSNWEVSAXRXSNAYCYDWIMURZXRXSVDYRUUAAUUCMWIWJZAYGUUDY
        TYGYSUUDMFWFWGVOAYDUUCUUDWKUUAMWHVOWLXRMWMWNUUBWOWOAYIYJYPWPWQWOAJFGUTU
        NZXMXKXMJWRXPXQWKAFGJSTUAWSJUUEXMWTUJJNXMXAXBXCAXLJURZXNVIZVIBCDEUKFGHI
        JKUKGUKULZLUPMUUHXLUOXDXEZXLLMNOPQRSAYEUUGTXFAGVCURUUGUAXFUBAHUUEURUUGU
        CXFAHXOVDUUGUDXFAWIHURUUGUEXFUFUGUHUIAUUFXNXGAUUFXNXHUUIVKXIXJ $.
    $}

    $d c ph $.
    $( The Cantor Normal Form theorem.  The function ` ( A CNF B ) ` , which
       maps a finitely supported function from ` B ` to ` A ` to the sum
       ` ( ( A ^o f ( a 1 ) ) o. a 1 ) +o `
       ` ( ( A ^o f ( a 2 ) ) o. a 2 ) +o ... ` over all indices ` a < B ` such
       that ` f ( a ) ` is nonzero, is an order isomorphism from the ordering
       ` T ` of finitely supported functions to the set ` ( A ^o B ) ` under
       the natural order.  Setting ` A = _om ` and letting ` B ` be arbitrarily
       large, the surjectivity of this function implies that every ordinal has
       a Cantor normal form (and injectivity, together with coherence
       ~ cantnfres , implies that such a representation is unique).
       (Contributed by Mario Carneiro, 28-May-2015.) $)
    cantnf $p |- ( ph -> ( A CNF B ) Isom T , _E ( S , ( A ^o B ) ) ) $=
      ( vt co cep cv cfv wcel c0 vf vg vd va vb vc vk vu vv wor coe wpo wfo wbr
      ccnf wi wral wiso oemapso word wwe con0 oecl syl2anc eloni syl ordwe weso
      sopo 4syl crn wceq cantnff frnd wss onss sseld weq eleq1w imbi12d r19.21v
      wf imbi2d wa wel wb ordelss sylan sselda pm5.5 ralbidva dfss3 bitr4di wne
      eleq1 cop crab cint cuni comu coa wrex c1st adantr simplrl simplrr simprl
      cio c2nd onelon on0eln0 biimpar eqid cantnflem4 csn cxp coi cdm cvv csupp
      cmpo cseqom fczsupp0 eqcomi oieq2 ax-mp cfsupp ne0i ad2antrl oveq1 neeq1d
      0ex a1i eqtri expr copab fveq2 ralbidv anbi12d fveq1 necon2d oe0m1 bitr3d
      syl5ibcom 3imtr4d impel fconstmpt fmptd fczfsuppd mpbir2and cantnfval cen
      cantnfs we0 oien mp2an en0 mpbi fveq2i seqom0g wfn ffnd fnfvelrn eqeltrrd
      eqtrdi pm2.61ne sylbid ex com23 a2i biimtrid tfis2 com3l mpdd ssrdv eqssd
      sylanbrc eleq12d imbi1d cbvrexvw eleq12 syl2an eqeqan12d rexbidv cbvopabv
      dffo2 bitrid simprll simprlr simprr cantnflem1 fvex epeli sylibr syl22anc
      ralrimivva soisoi ) AHIUJFGUKOZPULZHUWRFGUOOZUMZUAQZUBQZIUNZUXBUWTRZUXCUW
      TRZPUNZUPZUBHUQUAHUQHUWRIPUWTURABCDEFGHIJKLMUSAUWRUTZUWRPVAUWRPUJUWSAUWRV
      BSZUXIAFVBSZGVBSZUXJKLFGVCVDZUWRVEVFZUWRVGUWRPVHUWRPVIVJAHUWRUWTWBZUWTVKZ
      UWRVLUXAAFGHJKLVMZAUXPUWRAHUWRUWTUXQVNANUWRUXPANQZUWRSZUXRVBSZUXRUXPSZAUW
      RVBUXRAUXJUWRVBVOUXMUWRVPVFVQUXTAUXSUYAAUXSUYAUPZUPZACQZUWRSZUYDUXPSZUPZU
      PZNCNCVRZUYBUYGAUYIUXSUYEUYAUYFNCUWRVSNCUXPVSVTWCUYHCUXRUQAUYGCUXRUQZUPZU
      XTUYCAUYGCUXRWAUYKUYCUPUXTAUYJUYBAUXSUYJUYAAUXSUYJUYAUPAUXSWDZUYJUXRUXPVO
      ZUYAUYLUYJUYFCUXRUQUYMUYLUYGUYFCUXRUYLCNWEWDUYEUYGUYFWFUYLUXRUWRUYDAUXIUX
      SUXRUWRVOUXNUWRUXRWGWHWIUYEUYFWJVFWKCUXRUXPWLWMAUXSUYMUYAAUXSUYMWDZWDZUYA
      TUXPSUXRTUXRTUXPWOUYOUXRTWNZWDBCDEFGUXRUCQUDQZUEQZWPVLFUXRFUFQZUKOSUFVBWQ
      WRWSZUKOZUYQWTOUYRXAOUXRVLWDUEVUAXBUDVBXBUCXHZHIUYTVUBXCRZVUBXIRZUDUEUFUC
      JUYOUXKUYPAUXKUYNKXDZXDUYOUXLUYPAUXLUYNLXDZXDMAUXSUYMUYPXEAUXSUYMUYPXFUYO
      TUXRSZUYPUYOUXTVUGUYPWFUYOUXJUXSUXTAUXJUYNUXMXDAUXSUYMXGUWRUXRXJVDUXRXKVF
      XLUYTXMVUBXMVUCXMVUDXMXNUYOGTXOXPZUWTRZTUXPUYOVUITPXQZXRZUGDXSXSFUGQZVUJR
      ZUKOVUMVUHRWTODQZXAOYAZTYBZRZTUYODFGHUGVUHVUJVUPJVUEVUFTVUHTXTOZVLVUJVURP
      XQVLVURTGTYCYDTVURPYEYFUYOVUHHSZGFVUHWBZVUHTYGUNZUYOCGTFVUHUYOGTWNZTFSZUY
      DGSUYOTGUKOZTVLZFTWNZVVBVVCUYOFTVVDTUYOUWRTWNZFTVLZVVDTWNUXSVVGAUYMUWRUXR
      YHYIVVHUWRVVDTFTGUKYJYKUUDUUAUYOUXLVVBVVEWFVUFUXLTGSVVBVVEGXKGUUBUUCVFUYO
      UXKVVCVVFWFVUEFXKVFUUEGUYDYHUUFCGTUUGUUHAVVAUYNAGVBXSTLTXSSZAYLYMUUIXDAVU
      SVUTVVAWDWFUYNAFGHVUHJKLUUMXDUUJZVUPXMZUUKVUQTVUPRZTVUKTVUPVUKTUULUNZVUKT
      VLVVITPVAVVMYLPUUNTPVUJXSVUJXMUUOUUPVUKUUQUURUUSVVIVVLTVLYLVUOVUPTXSVVKUU
      TYFYNUVEUYOUWTHUVAVUSVUIUXPSUYOHUWRUWTAUXOUYNUXQXDUVBVVJHVUHUWTUVCVDUVDUV
      FYOUVGUVHUVIUVJYMUVKUVLUVMUVNUVOUVPHUWRUWTUWFUVQAUXHUAUBHHAUXBHSZUXCHSZWD
      ZUXDUXGAVVPUXDWDZWDZUXEUXFSUXGVVRUHUINEFGHIUGUXBUXCUGNXSXSFVULUXCTXTOPXQZ
      RZUKOVVTUXCRWTOUXRXAOYATYBZVVSUYSUXBRUYSUXCRSUFGWQWSZUFJAUXKVVQKXDAUXLVVQ
      LXDIVUNBQZRZVUNUYDRZSZDEWEZEQZVWCRZVWHUYDRZVLZUPZEGUQZWDZDGXBZBCYPUXRUHQZ
      RZUXRUIQZRZSZNEWEZVWHVWPRZVWHVWRRZVLZUPZEGUQZWDZNGXBZUHUIYPMVWOVXHBCUHUIV
      WOUXRVWCRZUXRUYDRZSZVXAVWKUPZEGUQZWDZNGXBBUHVRZCUIVRZWDZVXHVWNVXNDNGDNVRZ
      VWFVXKVWMVXMVXRVWDVXIVWEVXJVUNUXRVWCYQVUNUXRUYDYQUVRVXRVWLVXLEGVXRVWGVXAV
      WKDNVWHVSUVSYRYSUVTVXQVXNVXGNGVXQVXKVWTVXMVXFVXOVXIVWQVLVXJVWSVLVXKVWTWFV
      XPUXRVWCVWPYTUXRUYDVWRYTVXIVWQVXJVWSUWAUWBVXQVXLVXEEGVXQVWKVXDVXAVXOVXPVW
      IVXBVWJVXCVWHVWCVWPYTVWHUYDVWRYTUWCWCYRYSUWDUWGUWEYNAVVNVVOUXDUWHAVVNVVOU
      XDUWIAVVPUXDUWJVWBXMVVSXMVWAXMUWKUXEUXFUXCUWTUWLUWMUWNYOUWPUAUBHUWRIPUWTU
      WQUWO $.

    $( The lexicographic order on a function space of ordinals gives a
       well-ordering with order type equal to the ordinal exponential.  This
       provides an alternate definition of the ordinal exponential.
       (Contributed by Mario Carneiro, 28-May-2015.) $)
    oemapwe $p |- ( ph -> ( T We S /\ dom OrdIso ( T , S ) = ( A ^o B ) ) ) $=
      ( wwe wceq cep con0 wcel syl cvv coi cdm co word oecl syl2anc eloni ordwe
      coe 3syl ccnf wiso wb cantnf isowe mpbird ccnv wa isocnv wse ovex eqeltri
      dmex exse ax-mp eqid oieu sylancl mpbi2and simpld eqcomd jca ) AHINZHIUAZ
      UBZFGUIUCZOAVMVPPNZAVPQRZVPUDZVQAFQRGQRVRKLFGUEUFZVPUGZVPUHUJAHVPIPFGUKUC
      ZULZVMVQUMABCDEFGHIJKLMUNZHVPIPWBUOSUPZAVPVOAVPVOOZWBUQZVNOZAVSVPHPIWGULZ
      WFWHURZAVRVSVTWASAWCWIWDHVPIPWBUSSAVMHIUTZVSWIURWJUMWEHTRWKHWBUBTJWBFGUKV
      AVCVBHITVDVEHVPIVNWGVNVFVGVHVIVJVKVL $.

    $( An alternate definition of ~ df-cnf which relies on ~ cantnf .  (Note
       that although the use of ` S ` seems self-referential, one can use
       ~ cantnfdm to eliminate it.)  (Contributed by Mario Carneiro,
       28-May-2015.) $)
    cantnffval2 $p |- ( ph -> ( A CNF B ) = `' OrdIso ( T , S ) ) $=
      ( ccnf ccnv wceq cep con0 wcel cvv coi wrel coe wiso cantnf isof1o f1orel
      co wf1o 3syl dfrel2 sylib cdm word wa syl2anc eloni syl isocnv wwe wse wb
      oecl oemapwe simpld ovex dmex eqeltri exse ax-mp eqid oieu sylancl simprd
      mpbi2and cnveqd eqtr3d ) AFGNUHZOZOZVRHIUAZOAVRUBZVTVRPAHFGUCUHZIQVRUDZHW
      CVRUIWBABCDEFGHIJKLMUEZHWCIQVRUFHWCVRUGUJVRUKULAVSWAAWCWAUMZPZVSWAPZAWCUN
      ZWCHQIVSUDZWGWHUOZAWCRSZWIAFRSGRSWLKLFGVCUPWCUQURAWDWJWEHWCIQVRUSURAHIUTZ
      HIVAZWIWJUOWKVBAWMWFWCPABCDEFGHIJKLMVDVEHTSWNHVRUMTJVRFGNVFVGVHHITVIVJHWC
      IWAVSWAVKVLVMVOVNVPVQ $.
  $}

  ${
    $d w x y z A $.  $d w x y z B $.  $d x y z ph $.  $d x y z S $.
    cantnff1o.1 $e |- S = dom ( A CNF B ) $.
    cantnff1o.2 $e |- ( ph -> A e. On ) $.
    cantnff1o.3 $e |- ( ph -> B e. On ) $.
    $( Simplify the isomorphism of ~ cantnf to simple bijection.  (Contributed
       by Mario Carneiro, 30-May-2015.) $)
    cantnff1o $p |- ( ph -> ( A CNF B ) : S -1-1-onto-> ( A ^o B ) ) $=
      ( vz vx vy vw coe co cv cfv wcel wceq wi wral cep wa wrex copab ccnf wiso
      wf1o eqid cantnf isof1o syl ) ADBCLMZHNZINZOULJNZOPULKNZPUOUMOUOUNOQRKCSU
      AHCUBIJUCZTBCUDMZUEDUKUQUFAIJHKBCDUPEFGUPUGUHDUKUPTUQUIUJ $.
  $}

  ${
    $d a b c d f w x y z A $.  $d a b c d f x y B $.  $d a b c d f w x y z F $.
    $d a b c d f x y G $.  $d a b c d f x y ph $.  $d c d w z R $.  $d c z S $.
    $d c d f x y U $.  $d f x Z $.
    wemapwe.t $e |- T = { <. x , y >. | E. z e. A ( ( x ` z ) S ( y ` z ) /\
        A. w e. A ( z R w -> ( x ` w ) = ( y ` w ) ) ) } $.
    wemapwe.u $e |- U = { x e. ( B ^m A ) | x finSupp Z } $.
    wemapwe.2 $e |- ( ph -> R We A ) $.
    wemapwe.3 $e |- ( ph -> S We B ) $.
    wemapwe.4 $e |- ( ph -> B =/= (/) ) $.
    wemapwe.5 $e |- F = OrdIso ( R , A ) $.
    wemapwe.6 $e |- G = OrdIso ( S , B ) $.
    wemapwe.7 $e |- Z = ( G ` (/) ) $.
    $( Construct lexicographic order on a function space based on a reverse
       well-ordering of the indices and a well-ordering of the values.
       (Contributed by Mario Carneiro, 29-May-2015.)  (Revised by AV,
       3-Jul-2019.) $)
    wemapwe $p |- ( ph -> T We U ) $=
      ( vf vc va vb vd cvv wcel wa wwe cxp cin ccnv ccom cmpt cfv wceq cdm wral
      cv wi wrex copab wbr ccnf co wf1o cfsupp cmap crab eqid cep simprr adantr
      wiso oiiso syl2anc isof1o syl simprl f1ocnv oiexg ad2antll dmexd ad2antrl
      3syl wne crn wfo f1ofo forn eqnetrd dm0rn0 necon3bii sylibr oif ffvelcdmi
      c0 con0 oion eqtrid mpbird weinxp fveq2 breq12d ralbidv anbi12d vex coexg
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      wb isorel syl12anc fco sylancl fvco3 sylancom fveq2d eqtrd bitr4d eqeq12d
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      ax-mp eqeltrid mapfien fveq2i f1ocnvfv1 breq2d rabbidv eqtr4d f1oeq3d coi
      cantnfdm coe oemapwe simpld f1owe sylc sylib wfn f1ofn breq1 imbi1d rexrn
      rexeqdv cnvexg eleq12 eqeqan12d imbi2d rexbidv brabga isocnv ssrab3 epeli
      fvex bitrdi simpr eleq12d raleqdv breq2 ralrn bitr3d bitr3id f1of1 f1fveq
      epel wf1 3bitrd anassrs ralbidva rexbidva 3bitr4rd 3bitr3d pm5.32rd df-xp
      opabbidv ineq12i ineq2i 3eqtr4g weeq1 we0 elmapex pm2.21d ralrimiv rabeq0
      con3i weeq2 mpbiri pm2.61d1 ) AGUHUIZFUHUIZUJZKJUKZAUXMUXNAUXMUJZKJKKULZU
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  $}

  ${
    $d x y A $.  $d x y B $.  $d x y C $.  $d x y D $.  $d x y ph $.
    $d x y F $.  $d x y G $.
    oef1o.f $e |- ( ph -> F : A -1-1-onto-> C ) $.
    oef1o.g $e |- ( ph -> G : B -1-1-onto-> D ) $.
    oef1o.a $e |- ( ph -> A e. ( On \ 1o ) ) $.
    oef1o.b $e |- ( ph -> B e. On ) $.
    oef1o.c $e |- ( ph -> C e. On ) $.
    oef1o.d $e |- ( ph -> D e. On ) $.
    oef1o.z $e |- ( ph -> ( F ` (/) ) = (/) ) $.
    oef1o.k $e |- K = ( y e. { x e. ( A ^m B ) | x finSupp (/) }
                           |-> ( F o. ( y o. `' G ) ) ) $.
    oef1o.h $e |- H = ( ( ( C CNF D ) o. K ) o. `' ( A CNF B ) ) $.
    $( A bijection of the base sets induces a bijection on ordinal
       exponentials.  (The assumption ` ( F `` (/) ) = (/) ` can be discharged
       using ~ fveqf1o .)  (Contributed by Mario Carneiro, 30-May-2015.)
       (Revised by AV, 3-Jul-2019.) $)
    oef1o $p |- ( ph -> H : ( A ^o B ) -1-1-onto-> ( C ^o D ) ) $=
      ( coe co ccnf ccom ccnv wf1o cdm eqid cantnff1o cv c0 cfsupp wbr cmap cfv
      crab cmpt con0 c1o cdif f1ocnv wcel ondif1 simprbi mapfien wceq wb f1oeq1
      ax-mp sylibr cantnfdm breq2d rabbidv eqtr4d f1oeq3d eldifad f1oeq2d f1oco
      syl mpbird syl2anc ) ADEUAUBZFGUAUBZFGUCUBZKUDZDEUCUBZUEZUDZUFZWBWCJUFZAW
      FUGZWCWEUFZWBWKWGUFZWIAWDUGZWCWDUFWKWNKUFZWLAFGWNWNUHPQUIAWOBUJZUKULUMZBD
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      KVQVTWKWNWCWDKVRWAAWKWBWFUFWMADEWKWKUHXMOUIWKWBWFVAVSWBWKWCWEWGVRWAJWHVFW
      JWIVGTWBWCJWHVHVIVJ $.
  $}

  ${
    $d k x z A $.  $d k u v w x y z I $.  $d x y M $.  $d u v w y ph $.
    $d f k u v w x y z F $.  $d u v w K $.  $d u v w y z T $.  $d u v x W $.
    $d f k u v w x y z G $.  $d f u v w x y H $.  $d k z S $.
    cnfcom.s $e |- S = dom ( _om CNF A ) $.
    cnfcom.a $e |- ( ph -> A e. On ) $.
    cnfcom.b $e |- ( ph -> B e. ( _om ^o A ) ) $.
    cnfcom.f $e |- F = ( `' ( _om CNF A ) ` B ) $.
    cnfcom.g $e |- G = OrdIso ( _E , ( F supp (/) ) ) $.
    cnfcom.h $e |- H = seqom ( ( k e. _V , z e. _V |-> ( M +o z ) ) , (/) ) $.
    cnfcom.t $e |- T = seqom ( ( k e. _V , f e. _V |-> K ) , (/) ) $.
    cnfcom.m $e |- M = ( ( _om ^o ( G ` k ) ) .o ( F ` ( G ` k ) ) ) $.
    cnfcom.k $e |- K = ( ( x e. M |-> ( dom f +o x ) ) u.
                         `' ( x e. dom f |-> ( M +o x ) ) ) $.
    ${
      cnfcom.1 $e |- ( ph -> I e. dom G ) $.
      ${
        cnfcom.2 $e |- ( ph -> O e. ( _om ^o ( G ` I ) ) ) $.
        cnfcom.3 $e |- ( ph -> ( T ` I ) : ( H ` I ) -1-1-onto-> O ) $.
        $( Lemma for ~ cnfcom .  (Contributed by Mario Carneiro, 30-May-2015.)
           (Revised by AV, 3-Jul-2019.) $)
        cnfcomlem $p |- ( ph -> ( T ` suc I ) : ( H ` suc I ) -1-1-onto->
          ( ( _om ^o ( G ` I ) ) .o ( F ` ( G ` I ) ) ) ) $=
          ( vy vu vv csuc cfv com coe co comu coa wf1o cv cmpt ccnv con0 omelon
          cun wcel c0 csupp suppssdm wf cfsupp wbr wa ccnf a1i cantnff1o f1ocnv
          f1of 3syl ffvelcdmd eqeltrid cantnfs mpbid simpld fssdm cdm ffvelcdmi
          cep oif syl sseldd onelon syl2anc oecl sylancr nnon omcl wwe cantnfcl
          simprd elnn cvv cantnfvalf eqid oacomf1o cmpo wceq seqomsuc weq oveq2
          nfcv cbvmptv simpl fveq2d oveq2d oveq12d eqtrid simpr dmeqd mpteq12dv
          oveq1d cnveqd uneq12d cbvmpo simprl f1odm sylan9eqr elexd fvexd mptex
          ovex fvex cnvex unex mpbird cseqom syl21anc wss wb adantr eqtrd mp2an
          ovmpod f1oeq1d oveq1i mpoeq3ia seqomeq12 eqtri cantnfsuc f1oeq2d cima
          sssucid sselid wral epelg biimpar wiso ovexd word oicl ordelss sselda
          oiiso isorel syl12anc epeli sylib ralrimiva wfun ffun ax-mp funimass4
          w3a simpll simplr peano1 simpr1 simpr2 cantnflt syl23anc wne wfn ffnd
          simpr3 0ex elsuppfn syl3anc biimtrdi on0eln0 omword1 oaabs2 f1oeq3d
          mpd ) AMULZLUMZMLUMZUNMKUMZUOUPZUWQJUMZUQUPZURUPZUWNGUMZUSZUWOUWTUXBU
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          HUWPPVUHYFWJYGZYAXTVWHVVAUXJVWHUIVUQVUTUWPUXIVWLVWHVUOUWTUXFURVWJYAXT
          YBYCAMUYLUFYHAMGYIUXLXBVFAUXHUXKUIUWTUXGUWRUWSUQYKYJUXJUIUWPUXIMLYLYJ
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          IJKLMQUXRVWMVDVOUXTUYEXMUAUXTUYEXRLICXBXBOVUGURUPZXFZVGYPZICXBXBVVTVU
          GURUPZXFZVGYPZUBVWOVWRXGVGVGXGVWPVWSXGICXBXBVWNVWQVWNVWQXGVVPXBVFVUGX
          BVFVMOVVTVUGURUDUUEVOUUFVGXDVWOVWRVGVGUUGUUBUUHZUUIYQUUJYOAUXAUWTUWOU
          XBAUWPUWRVFZUXNUWRUWTYRZUXAUWTXGAUXTUYEMUYLULZVFZUXSKMUUKUWQYRZVXARUY
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          WDUYBWHKXBUAUVCWMYTAMUYLUXFAUYLUUSUYMMUYLYRZUYBWHKUAUUTUFUYLMUVAWOZUV
          BAUYMVXIUFYTUYLUYBUXFMWHWHKUVDUVEWCVXFUWQMKYLUVFUVGUVHAKUVIZVXNVXEVXH
          YSUYLUYBKVJVXPUYOUYLUYBKUVJUVKVXOUIMUWQKUVLWOYOVWMVXDUXSVXEUVMZVMZCUN
          DUWQFIJKLMQUXRVXRVDVOUXTUYEVXQUVNUAUXTUYEVXQUVOVWTVGUNVFVXRUVPVOVWMVX
          DUXSVXEUVQVWMVXDUXSVXEUVRVWMVXDUXSVXEUWDUVSUVTVUAAUXPUXQVGUWSVFZVXBUY
          SUYTAVXSUWSVGUWAZAUYNVXTUYPAUYNUYAVXTVMZVXTAJDUWBUXTVGXBVFZUYNVYAYSAD
          UNJUYKUWCRVYBAUWEVOUWQJVCXBDVGUWFUWGUYAVXTXRUWHUWMAUXQVXSVXTYSUYTUWSU
          WIWJYOUWRUWSUWJYQUWPUWTUWQUWKYQUWLWC $.
      $}

      $( Any ordinal ` B ` is equinumerous to the leading term of its Cantor
         normal form.  Here we show that bijection explicitly.  (Contributed by
         Mario Carneiro, 30-May-2015.)  (Revised by AV, 3-Jul-2019.) $)
      cnfcom $p |- ( ph -> ( T ` suc I ) : ( H ` suc I ) -1-1-onto->
        ( ( _om ^o ( G ` I ) ) .o ( F ` ( G ` I ) ) ) ) $=
        ( vw vy cdm wcel csuc cfv com coe co comu wf1o wi c0 csupp cep wwe con0
        omelon a1i ccnf ccnv wf cantnff1o f1of 3syl ffvelcdmd eqeltrid cantnfcl
        f1ocnv simprd elnn syl2anc wceq eleq1 suceq fveq2d fveq2 oveq2d oveq12d
        cv 2fveq3 f1oeq123d imbi12d imbi2d adantr simpr wss suppssdm cfsupp wbr
        wa cantnfs mpbid simpld fssdm onss syl sstrd oif ffvelcdmi ssel2 syl2an
        peano1 oen0 syl21anc cvv 0ex cmpo seqom0g ax-mp f1o0 wb coa f1oeq2 mp2b
        mpbir f1oeq1 mpbiri mp1i cnfcomlem ex word oicl ordtr trsuc mpan imim1i
        wtr ad2antrr simprl ad2antrl sseldd epeli syl31anc mpd eloni sucid wiso
        vex ovexd oiiso isorel syl12anc sucex fvex 3bitr3g mpbii ordsucss onsuc
        sylc oewordi nnon oecl omord2 oesuc eleqtrrd simprr exp32 expcom finds2
        a2d syl5 vtoclga mpcom ) AMKUHZUIZMUJZLUKZULMKUKZUMUNZUVNJUKZUOUNZUVLGU
        KZUPZUEMULUIZAUVKUVSUQZAUVKUVJULUIZUVTUEAJURUSUNZUTVAZUWBAULDFJKPULVBUI
        ZAVCVDZQTAJEULDVEUNZVFZUKFSAULDUMUNZFEUWHAFUWIUWGUPUWIFUWHUPUWIFUWHVGAU
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        WTUXLUYOUWNUXFGUWLURVTZWAUYOUWNUXFLUYPWAUYOUWQUXIUWRUXJUOUYOUWPUXHULUMU
        WLURKWBWCUWLURJKWFWDWGWHUWLUXNVRZUWMUXOUXAUYCUWLUXNUVJVSUYQUWOUXQUWSUYA
        UWTUYBUYQUWNUXPGUWLUXNVTZWAUYQUWNUXPLUYRWAUYQUWQUXSUWRUXTUOUYQUWPUXRULU
        MUWLUXNKWBWCUWLUXNJKWFWDWGWHUWLUXPVRZUWMUYEUXAUYMUWLUXPUVJVSUYSUWOUYGUW
        SUYKUWTUYLUYSUWNUYFGUWLUXPVTZWAUYSUWNUYFLUYTWAUYSUWQUYIUWRUYJUOUYSUWPUY
        HULUMUWLUXPKWBWCUWLUXPJKWFWDWGWHAUXEUXMAUXEWPZBCDEFGHIJKLURNOURPADVBUIZ
        UXEQWJAEUWIUIZUXERWJSTUAUBUCUDAUXEWKVUAUWEUXHVBUIZURULUIZURUXIUIUWEVUAV
        CVDAUWCVBWLUXHUWCUIVUDUXEAUWCDVBADULUWCJJURWMADULJVGZJURWNWOZAJFUIVUFVU
        GWPUWJAULDFJPUWFQWQWRWSZWTZAVUBDVBWLZQDXAXBZXCUVJUWCURKUWCUTKTXDZXEUWCV
        BUXHXFXGVUEVUAXHVDULUXHXIXJURGUKZURVRZURLUKZURVUMUPZVUAURXKUIZVUNXLIHXK
        XKNXMGURXKUBXNXOVUNVUPVUOURURUPZVURURURURUPZXPVUQVUOURVRVURVUSXQXLICXKX
        KOCWEXRUNXMLURXKUAXNVUOURURURXSXTYAVUOURVUMURYBYCYDYEYFAUXNULUIZUYDUYNU
        QUYDUYEUYCUQAVUTWPZUYNUYEUXOUYCUVJYMZUYEUXOUVJYGVVBUWCUTKTYHUVJYIXOUVJU
        XNYJYKZYLVVAUYEUYCUYMVVAUYEUYCUYMVVAUYEUYCWPZWPZBCDEFGHIJKLUXPNOUYAPAVU
        BVUTVVDQYNAVUCVUTVVDRYNSTUAUBUCUDVVAUYEUYCYOZVVEULUXRUJZUMUNZUYIUYAVVEV
        VGUYHWLZVVHUYIWLZVVEUYHYGZUXRUYHUIZVVIVVEUYHVBUIZVVKVVEDVBUYHAVUJVUTVVD
        VUKYNZVVEUWCDUYHAUWCDWLVUTVVDVUIYNZUYEUYHUWCUIVVAUYCUVJUWCUXPKVULXEYPYQ
        YQZUYHUUAXBVVEUXNUXPUIZVVLUXNUGUUDZUUBVVEUXNUXPUTWOZUXRUYHUTWOZVVQVVLVV
        EUVJUWCUTUTKUUCZUXOUYEVVSVVTXQAVWAVUTVVDAUWCXKUIUWDVWAAJURUSUUEAUWDUWBU
        WKWSUWCUTKXKTUUFVQYNUYEUXOVVAUYCVVCYPZVVFUVJUWCUXNUXPUTUTKUUGUUHUXNUXPU
        XNVVRUUIYRUXRUYHUXPKUUJYRUUKUULUXRUYHUUMUUOVVEVVGVBUIZVVMUWEVUEVVIVVJUQ
        VVEUXRVBUIZVWCVVEDVBUXRVVNVVEUWCDUXRVVOVVEUXOUXRUWCUIVWBUVJUWCUXNKVULXE
        XBYQZYQZUXRUUNXBVVPUWEVVEVCVDZVUEVVEXHVDZVVGUYHULUUPYSYTVVEUYAUXSULUOUN
        ZVVHVVEUXTULUIZUYAVWIUIZVVEDULUXRJAVUFVUTVVDVUHYNVWEVKZVVEUXTVBUIZUWEUX
        SVBUIZURUXSUIZVWJVWKXQVVEVWJVWMVWLUXTUUQXBVWGVVEUWEVWDVWNVWGVWFULUXRUUR
        VQVVEUWEVWDVUEVWOVWGVWFVWHULUXRXIXJUXTULUXSUUSYSWRVVEUWEVWDVVHVWIVRVWGV
        WFULUXRUUTVQUVAYQVVAUYEUYCUVBYEUVCUVFUVGUVDUVEUVHUVIYT $.
    $}

    $d k x z ph $.
    cnfcom.w $e |- W = ( G ` U. dom G ) $.
    ${
      cnfcom2.1 $e |- ( ph -> (/) e. B ) $.
      $( Lemma for ~ cnfcom2 .  (Contributed by Mario Carneiro, 30-May-2015.)
         (Revised by AV, 3-Jul-2019.) $)
      cnfcom2lem $p |- ( ph -> dom G = suc U. dom G ) $=
        ( cdm cuni wceq wn csuc c0 wlim wo wcel n0i syl wa com ccnf cfv csn cxp
        co cmpt cv wf cfsupp wbr ccnv coe wf1o con0 omelon a1i cantnff1o f1ocnv
        f1of 3syl ffvelcdmd eqeltrid cantnfs mpbid simpld adantr feqmptd eleq2i
        cdif dif0 cvv csupp wss cen simpr cep wwe ovexd cantnfcl syl2anc ensymd
        oien eqbrtrrd en0 sylib sylibr 0ex suppssr sylan2br mpteq2dva fconstmpt
        ss0b eqtrd eqtr4di fveq2d fveq2i f1ocnvfv2 eqtrid cantnf0 3eqtr3d mtand
        peano1 nnlim simpl2im ioran sylanbrc word wb oicl unizlim ax-mp sylnibr
        orduniorsuc mp1i ord mpd ) AKUGZYPUHZUIZUJYPYQUKUIZAYPULUIZYPUMZUNZYRAY
        TUJUUAUJZUUBUJAYTEULUIZAULEUOUUDUJUFEULUPUQAYTURZJUSDUTVDZVAZDULVBVCZUU
        FVAZEULUUEJUUHUUFUUEJBDULVEZUUHUUEJBDBVFZJVAZVEUUJUUEBDUSJADUSJVGZYTAUU
        MJULVHVIZAJFUOUUMUUNURAJEUUFVJZVAZFSAUSDVKVDZFEUUOAFUUQUUFVLZUUQFUUOVLU
        UQFUUOVGAUSDFPUSVMUOAVNVOZQVPZFUUQUUFVQUUQFUUOVRVSRVTWAZAUSDFJPUUSQWBWC
        WDWEZWFUUEBDUULULUUKDUOUUEUUKDULWHZUOUULULUIUVCDUUKDWIWGUUEDUSWJJVMULUU
        KULUVBUUEJULWKVDZULUIZUVDULWLUUEUVDULWMVIUVEUUEULUVDUUEYPULUVDWMAYTWNAY
        PUVDWMVIZYTAUVDWJUOUVDWOWPZUVFAJULWKWQAUVGYPUSUOZAUSDFJKPUUSQTUVAWRZWDU
        VDWOKWJTXAWSWEXBWTUVDXCXDUVDXKXEADVMUOYTQWEULWJUOUUEXFVOXGXHXIXLBDULXJX
        MXNAUUGEUIYTAUUGUUPUUFVAZEJUUPUUFSXOAUUREUUQUOUVJEUIUUTRFUUQEUUFXPWSXQW
        EAUUIULUIYTAUSDFPUUSQULUSUOAYAVOXRWEXSXTAUVGUVHUUCUVIYPYBYCYTUUAYDYEYPY
        FZYRUUBYGUVDWOKTYHZYPYIYJYKAYRYSUVKYRYSUNAUVLYPYLYMYNYO $.

      $( Any nonzero ordinal ` B ` is equinumerous to the leading term of its
         Cantor normal form.  (Contributed by Mario Carneiro, 30-May-2015.)
         (Revised by AV, 3-Jul-2019.) $)
      cnfcom2 $p |- ( ph -> ( T ` dom G ) : B -1-1-onto->
          ( ( _om ^o W ) .o ( F ` W ) ) ) $=
        ( cdm cuni csuc cfv com coe co comu wf1o con0 c0 cvv wcel ovex cep oion
        csupp ax-mp elexi uniex sucid cnfcom2lem eleqtrrid cnfcom oveq2i fveq2i
        wceq wb oveq12i f1oeq3 sylibr fveq2d f1oeq1d mpbird ccnf ccnv omelon wf
        a1i cantnff1o f1ocnv f1of 3syl ffvelcdmd eqeltrid cv coa cmpo cseqom wa
        mpoeq3ia eqid seqomeq12 mp2an eqtri cantnfval eqtr3di f1ocnvfv2 syl2anc
        oveq1i 3eqtr3d f1oeq2d mpbid ) AKUGZUHZUIZLUJZUKOULUMZOJUJZUNUMZXJGUJZU
        OZEXPXQUOAXRXMXPXLGUJZUOZAXMUKXKKUJZULUMZYAJUJZUNUMZXSUOZXTABCDEFGHIJKL
        XKMNPQRSTUAUBUCUDAXKXLXJXKXJXJUPJUQVCUMZURUSXJUPUSJUQVCUTYFVAKURTVBVDVE
        VFVGABCDEFGHIJKLMNOPQRSTUAUBUCUDUEUFVHZVIVJXPYDVMXTYEVNXNYBXOYCUNOYAUKU
        LUEVKOYAJUEVLVOXPYDXMXSVPVDVQAXMXPXQXSAXJXLGYGVRVSVTAXMEXPXQAXJLUJZEUKD
        WAUMZWBZUJZYIUJZXMEAJYIUJYHYLACUKDFIJKLPUKUPUSAWCWEZQTAJYKFSAUKDULUMZFE
        YJAFYNYIUOZYNFYJUOYNFYJWDAUKDFPYMQWFZFYNYIWGYNFYJWHWIRWJWKLICURURNCWLZW
        MUMZWNZUQWOZICURURUKIWLZKUJZULUMUUBJUJUNUMZYQWMUMZWNZUQWOZUAYSUUEVMUQUQ
        VMYTUUFVMICURURYRUUDYRUUDVMUUAURUSYQURUSWPNUUCYQWMUCXFWEWQUQWRYSUUEUQUQ
        WSWTXAXBJYKYISVLXCAXJXLLYGVRAYOEYNUSYLEVMYPRFYNEYIXDXEXGXHXI $.
    $}

    cnfcom3.1 $e |- ( ph -> _om C_ B ) $.
    $( Lemma for ~ cnfcom3 .  (Contributed by Mario Carneiro, 30-May-2015.)
       (Revised by AV, 4-Jul-2019.) $)
    cnfcom3lem $p |- ( ph -> W e. ( On \ 1o ) ) $=
      ( con0 wcel c0 wne c1o cdif cdm cuni cfv csupp co com suppssdm cfsupp wbr
      wf ccnf ccnv coe wf1o omelon a1i cantnff1o f1ocnv f1of ffvelcdmd eqeltrid
      3syl cantnfs mpbid simpld fssdm csuc cvv ovex cep ax-mp elexi uniex sucid
      wa peano1 sseldd cnfcom2lem eleqtrrid oif ffvelcdmi syl onelon syl2anc wn
      oion wss oecl sylancr ontri1 wceq fveq2i f1ocnvfv2 eqtrid adantr 1on wiso
      wb cv wwe ovexd cantnfcl oiiso ad2antrr ffvelcdm sylancom elssuni sylancl
      isof1o onuni isorel fvex epeli breq1i bitr3i 3bitr3g simplr eleq12d bitrd
      syl12anc mtbid wo onss sstrd sselda on0eqel ord ex mt3d el1o sylibr ssrdv
      cantnflt2 oe1 eleqtrdi eqeltrrd necon3bd mpd dif1o sylanbrc ) AOUGUHOUIUJ
      ZOUGUKULUHAOKUMZUNZKUOZUGUEADUGUHZUUPDUHUUPUGUHQAJUIUPUQZDUUPADURUURJJUIU
      SADURJVBZJUIUTVAZAJFUHZUUSUUTWGAJEURDVCUQZVDZUOZFSAURDVEUQZFEUVCAFUVEUVBV
      FZUVEFUVCVFUVEFUVCVBAURDFPURUGUHZAVGVHZQVIZFUVEUVBVJUVEFUVCVKVNRVLVMZAURD
      FJPUVHQVOVPVQVRZAUUOUUNUHZUUPUURUHAUUOUUOVSUUNUUOUUNUUNUGUURVTUHZUUNUGUHZ
      JUIUPWAUURWBKVTTWRWCZWDWEWFABCDEFGHIJKLMNOPQRSTUAUBUCUDUEAUREUIUFUIURUHZA
      WHVHWIWJWKZUUNUURUUOKUURWBKTWLWMWNWIDUUPWOWPVMAEURUHZWQZUUMAUREWSZUVSUFAU
      VGEUGUHZUVTUVSXJVGAUVEUGUHZEUVEUHZUWAAUVGUUQUWBVGQURDWTXARUVEEWOWPUREXBXA
      VPAUVROUIAOUIXCZUVRAUWDWGZJUVBUOZEURAUWFEXCUWDAUWFUVDUVBUOZEJUVDUVBSXDAUV
      FUWCUWGEXCUVIRFUVEEUVBXEWPXFXGUWEUWFURUKVEUQZURUWEURDUKFJPUVGUWEVGVHAUUQU
      WDQXGAUVAUWDUVJXGUVPUWEWHVHUKUGUHUWEXHVHUWEBUURUKUWEBXKZUURUHZUWIUKUHZUWE
      UWJWGZUWIUIXCZUWKUWLUWMUIUWIUHZUWLUUOUWIKVDZUOZUHZUWNUWLUWPUUOWSZUWQWQZUW
      LUWPUUNUHZUWRUWEUWJUURUUNUWOVBZUWTUWLUUNUURKVFZUURUUNUWOVFUXAUWLUUNUURWBW
      BKXIZUXBAUXCUWDUWJAUVMUURWBXLZUXCAJUIUPXMAUXDUUNURUHAURDFJKPUVHQTUVJXNVQU
      URWBKVTTXOWPXPZUUNUURWBWBKYAWNZUUNUURKVJUURUUNUWOVKVNUURUUNUWIUWOXQXRZUWP
      UUNXSWNUWLUWPUGUHZUUOUGUHZUWRUWSXJUWLUVNUWTUXHUVOUXGUUNUWPWOXAUVNUXIUVOUU
      NYBWCUWPUUOXBXTVPUWLUWQOUWPKUOZUHZUWNUWLUUOUWPWBVAZUUPUXJWBVAZUWQUXKUWLUX
      CUVLUWTUXLUXMXJUXEAUVLUWDUWJUVQXPUXGUUNUURUUOUWPWBWBKYCYLUUOUWPUWIUWOYDYE
      UXMOUXJWBVAUXKOUUPUXJWBUEYFOUXJUWPKYDYEYGYHUWLOUIUXJUWIAUWDUWJYIUWEUWJUXB
      UXJUWIXCUXFUUNUURUWIKXEXRYJYKYMUWLUWMUWNUWLUWIUGUHUWMUWNYNUWEUURUGUWIAUUR
      UGWSUWDAUURDUGUVKAUUQDUGWSQDYOWNYPXGYQUWIYRWNYSUUAUWIUUBUUCYTUUDUUEUVGUWH
      URXCVGURUUFWCUUGUUHYTUUIUUJOUGUUKUUL $.

    cnfcom.x $e |- X = ( u e. ( F ` W ) , v e. ( _om ^o W ) |->
      ( ( ( F ` W ) .o v ) +o u ) ) $.
    cnfcom.y $e |- Y = ( u e. ( F ` W ) , v e. ( _om ^o W ) |->
      ( ( ( _om ^o W ) .o u ) +o v ) ) $.
    cnfcom.n $e |- N = ( ( X o. `' Y ) o. ( T ` dom G ) ) $.
    $( Any infinite ordinal ` B ` is equinumerous to a power of ` _om ` .  (We
       are being careful here to show explicit bijections rather than simple
       equinumerosity because we want a uniform construction for ~ cnfcom3c .)
       (Contributed by Mario Carneiro, 28-May-2015.)  (Revised by AV,
       4-Jul-2019.) $)
    cnfcom3 $p |- ( ph -> N : B -1-1-onto-> ( _om ^o W ) ) $=
      ( com coe co ccnv ccom cdm cfv wf1o comu con0 omelon c0 csupp suppssdm wf
      wcel cfsupp wbr wa ccnf a1i cantnff1o f1ocnv f1of 3syl ffvelcdmd eqeltrid
      cantnfs mpbid simpld fssdm cuni csuc cvv ovex cep ax-mp elexi uniex sucid
      oion peano1 sseldd cnfcom2lem eleqtrrid oif ffvelcdmi onelon syl2anc oecl
      syl sylancr nnon omf1o wceq wne wfn wb ffnd 0ex elsuppfn syl3anc biimtrdi
      simpr mpd on0eln0 mpbird c1o cdif cnfcom3lem ondif1 simprbi omabs f1oeq3d
      syl22anc cnfcom2 f1oco f1oeq1 sylibr ) AGUORUPUQZSTURUSZMUTZIVAZUSZVBZGYN
      QVBZAYNRLVAZVCUQZYNYOVBZGUUBYQVBYSAUUBUUAYNVCUQZYOVBZUUCAYNVDVJZUUAVDVJZU
      UEAUOVDVJZRVDVJZUUFVEAFVDVJZRFVJZUUIUBALVFVGUQZFRAFUOUULLLVFVHAFUOLVIZLVF
      VKVLZALHVJUUMUUNVMALGUOFVNUQZURZVAHUDAUOFUPUQZHGUUPAHUUQUUOVBUUQHUUPVBUUQ
      HUUPVIAUOFHUAUUHAVEVOZUBVPHUUQUUOVQUUQHUUPVRVSUCVTWAAUOFHLUAUURUBWBWCWDZW
      EARYPWFZMVAZUULUJAUUTYPVJUVAUULVJAUUTUUTWGYPUUTYPYPVDUULWHVJYPVDVJLVFVGWI
      UULWJMWHUEWOWKWLWMWNABCFGHIJKLMNOPRUAUBUCUDUEUFUGUHUIUJAUOGVFUKVFUOVJAWPV
      OWQZWRWSYPUULUUTMUULWJMUEWTXAXEWAZWQZFRXBXCZUORXDXFAUUAUOVJZUUGAFUORLUUSU
      VDVTZUUAXGZXEEDYNUUATSUMULXHXCAUUDYNUUBYOAUVFVFUUAVJZUUIVFRVJZUUDYNXIUVGA
      UVIUUAVFXJZARUULVJZUVKUVCAUVLUUKUVKVMZUVKALFXKUUJVFWHVJZUVLUVMXLAFUOLUUSX
      MUBUVNAXNVORLVDWHFVFXOXPUUKUVKXRXQXSAUVFUUGUVIUVKXLUVGUVHUUAXTVSYAUVEARVD
      YBYCVJZUVJABCFGHIJKLMNOPRUAUBUCUDUEUFUGUHUIUJUKYDUVOUUIUVJRYEYFXEUUARYGYI
      YHWCABCFGHIJKLMNOPRUAUBUCUDUEUFUGUHUIUJUVBYJGUUBYNYOYQYKXCQYRXIYTYSXLUNGY
      NQYRYLWKYM $.
  $}

  ${
    $d b g k u v w x z A $.  $d u v K $.  $d g w L $.  $d x M $.  $d u v z T $.
    $d f k u v x z F $.  $d f k u v x z G $.  $d f u v x H $.  $d k z S $.
    $d u v w x W $.
    cnfcom3c.s $e |- S = dom ( _om CNF A ) $.
    cnfcom3c.f $e |- F = ( `' ( _om CNF A ) ` b ) $.
    cnfcom3c.g $e |- G = OrdIso ( _E , ( F supp (/) ) ) $.
    cnfcom3c.h $e |- H = seqom ( ( k e. _V , z e. _V |-> ( M +o z ) ) , (/) )
        $.
    cnfcom3c.t $e |- T = seqom ( ( k e. _V , f e. _V |-> K ) , (/) ) $.
    cnfcom3c.m $e |- M = ( ( _om ^o ( G ` k ) ) .o ( F ` ( G ` k ) ) ) $.
    cnfcom3c.k $e |- K = ( ( x e. M |-> ( dom f +o x ) ) u.
      `' ( x e. dom f |-> ( M +o x ) ) ) $.
    cnfcom3c.w $e |- W = ( G ` U. dom G ) $.
    cnfcom3c.x $e |- X = ( u e. ( F ` W ) , v e. ( _om ^o W ) |->
      ( ( ( F ` W ) .o v ) +o u ) ) $.
    cnfcom3c.y $e |- Y = ( u e. ( F ` W ) , v e. ( _om ^o W ) |->
      ( ( ( _om ^o W ) .o u ) +o v ) ) $.
    cnfcom3c.n $e |- N = ( ( X o. `' Y ) o. ( T ` dom G ) ) $.
    cnfcom3c.l $e |- L = ( b e. ( _om ^o A ) |-> N ) $.
    $( Lemma for ~ cnfcom3c .  (Contributed by Mario Carneiro, 30-May-2015.)
       (Revised by AV, 4-Jul-2019.) $)
    cnfcom3clem $p |- ( A e. On -> E. g A. b e. A ( _om C_ b ->
      E. w e. ( On \ 1o ) ( g ` b ) : b -1-1-onto-> ( _om ^o w ) ) ) $=
      ( con0 wcel com cv wss coe co cfv wf1o c1o cdif wrex wi wex w3a simp1 c2o
      wral omelon ondif2 mpbir2an oeworde sylancr simp2 sseldd simp3 cnfcom3lem
      1onn cnfcom3 cvv wceq f1of syl fexd fvmpt2 syl2anc f1oeq1d mpbird f1oeq3d
      oveq2 rspcev 3expia ralrimiva cmpt ovex mptex eqeltri nfmpt1 nfcxfr nfeq2
      wf fveq1 rexbidv imbi2d ralbid spcev ) FUOUPZUQUBURZUSZXLUQCURZUTVAZXLPVB
      ZVCZCUOVDVEZVFZVGZUBFVLZXMXLXOXLJURZVBZVCZCXRVFZVGZUBFVLZJVHXKXTUBFXKXLFU
      PZXMXSXKYHXMVIZSXRUPXLUQSUTVAZXPVCZXSYIABFXLGHIKLMNOQSUCXKYHXMVJZYIFUQFUT
      VAZXLYIUQUOVKVEUPZXKFYMUSYNUQUOUPVDUQUPVMWBUQVNVOYLUQFVPVQXKYHXMVRZVSZUDU
      EUFUGUHUIUJXKYHXMVTZWAYIYKXLYJRVCZYIABDEFXLGHIKLMNOQRSTUAUCYLYPUDUEUFUGUH
      UIUJYQUKULUMWCZYIXLYJXPRYIXLYMUPRWDUPXPRWEYPYIXLYJFRYIYRXLYJRXEYSXLYJRWFW
      GYOWHUBYMRWDPUNWIWJWKWLXQYKCSXRXNSWEXOYJXLXPXNSUQUTWNWMWOWJWPWQYGYAJPPUBY
      MRWRZWDUNUBYMRUQFUTWSWTXAYBPWEZYFXTUBFUBYBPUBPYTUNUBYMRXBXCXDUUAYEXSXMUUA
      YDXQCXRUUAXLXOYCXPXLYBPXFWKXGXHXIXJWG $.
  $}

  ${
    $d b f g k u v w x z A $.
    $( Wrap the construction of ~ cnfcom3 into an existential quantifier.  For
       any ` _om C_ b ` , there is a bijection from ` b ` to some power of
       ` _om ` .  Furthermore, this bijection is _canonical_ , which means that
       we can find a single function ` g ` which will give such bijections for
       every ` b ` less than some arbitrarily large bound ` A ` .  (Contributed
       by Mario Carneiro, 30-May-2015.) $)
    cnfcom3c $p |- ( A e. On -> E. g A. b e. A ( _om C_ b ->
      E. w e. ( On \ 1o ) ( g ` b ) : b -1-1-onto-> ( _om ^o w ) ) ) $=
      ( vx vz vv vu vk vf com co cdm cvv cv ccnv cfv coa cmpo eqid c0 csupp cep
      ccnf coi coe comu cmpt cun cseqom cuni ccom cnfcom3clem ) EFAGHBKBUDLZMZI
      JNNEKIODOUNPQZUAUBLUCUEZQZUFLURUPQUGLZJOMZEOZRLUHEUTUSVARLUHPUIZSUAUJZJCI
      UPUQIFNNUSFORLSUAUJZVBDKBUFLHGUQMZUKUQQZUPQZKVFUFLZVGGOZUGLHOZRLSZHGVGVHV
      HVJUGLVIRLSZPULVEVCQULZUHZUSVMVFVKVLDUOTUPTUQTVDTVCTUSTVBTVFTVKTVLTVMTVNT
      UM $.
  $}


$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
  Transitive closure of a relation
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)

  $( Declare new constant $)
  $c t++ $.

  $( Declare the syntax for the transitive closure of a class. $)
  cttrcl $a class t++ R $.

  ${
    $d R f n m x y $.
    $( Define the transitive closure of a class.  This is the smallest relation
       containing ` R ` (or more precisely, the relation ` ( R |`` _V ) `
       induced by ` R ` ) and having the transitive property.  Definition from
       [Levy] p. 59, who denotes it as ` R * ` and calls it the "ancestral" of
       ` R ` .  (Contributed by Scott Fenton, 17-Oct-2024.) $)
    df-ttrcl $a |- t++ R = { <. x , y >. | E. n e. ( _om \ 1o )
     E. f ( f Fn suc n /\ ( ( f ` (/) ) = x /\ ( f ` n ) = y ) /\
     A. m e. n ( f ` m ) R ( f ` suc m ) ) } $.
  $}

  ${
    $d R f m n x y $.  $d S f m n x y $.
    $( Equality theorem for transitive closure.  (Contributed by Scott Fenton,
       17-Oct-2024.) $)
    ttrcleq $p |- ( R = S -> t++ R = t++ S ) $=
      ( vf vn vx vy vm wceq csuc wfn cfv wbr wral w3a wex copab cttrcl df-ttrcl
      cv wrex com c1o cdif breq ralbidv 3anbi3d exbidv rexbidv opabbidv 3eqtr4g
      c0 wa ) ABHZCSZDSZIJZUKUNKESHUOUNKFSHULZGSZUNKZURIUNKZALZGUOMZNZCOZDUAUBU
      CZTZEFPUPUQUSUTBLZGUOMZNZCOZDVETZEFPAQBQUMVFVKEFUMVDVJDVEUMVCVICUMVBVHUPU
      QUMVAVGGUOUSUTABUDUEUFUGUHUIEFACGDREFBCGDRUJ $.
  $}

  ${
    $d R y z n f a $.  $d ph y z n f a $.  $d x y z n f a $.
    nfttrcld.1 $e |- ( ph -> F/_ x R ) $.
    $( Bound variable hypothesis builder for transitive closure.  Deduction
       form.  (Contributed by Scott Fenton, 26-Oct-2024.) $)
    nfttrcld $p |- ( ph -> F/_ x t++ R ) $=
      ( vf vn vy vz va cttrcl cv csuc wfn c0 cfv wceq wa nfv nfcvd nfvd wbr w3a
      wral wex com c1o cdif copab df-ttrcl nfbrd nfraldw nf3and nfrexdw nfopabd
      wrex nfexd nfcxfrd ) ABCJEKZFKZLMZNUROGKPUSUROHKPQZIKZUROZVBLUROZCUAZIUSU
      CZUBZEUDZFUEUFUGZUOZGHUHGHCEIFUIAVJGHBAGRAHRAVHBFVIAFRABVISAVGBEAERAUTVAV
      FBAUTBTAVABTAVEBIUSAIRABUSSABVCVDCABVCSDABVDSUJUKULUPUMUNUQ $.
  $}

  ${
    nfttrcl.1 $e |- F/_ x R $.
    $( Bound variable hypothesis builder for transitive closure.  (Contributed
       by Scott Fenton, 17-Oct-2024.) $)
    nfttrcl $p |- F/_ x t++ R $=
      ( cttrcl wnfc wtru a1i nfttrcld mptru ) ABDEFABABEFCGHI $.
  $}

  ${
    $d R f n m x y $.
    $( The transitive closure of a class is a relation.  (Contributed by Scott
       Fenton, 17-Oct-2024.) $)
    relttrcl $p |- Rel t++ R $=
      ( vf vn vx vy vm cv csuc wfn c0 cfv wceq wa wbr wral w3a wex com c1o cdif
      wrex cttrcl df-ttrcl relopabi ) BGZCGZHIJUEKDGLUFUEKEGLMFGZUEKUGHUEKANFUF
      OPBQCRSTUADEAUBDEABFCUCUD $.
  $}

  ${
    $d A x y n f a $.  $d B x y n f a $.  $d R x y n f a $.
    $( Characterization of elements of the transitive closure of a relation.
       (Contributed by Scott Fenton, 18-Aug-2024.) $)
    brttrcl $p |- ( A t++ R B <-> E. n e. ( _om \ 1o ) E. f
     ( f Fn suc n /\ ( ( f ` (/) ) = A /\ ( f ` n ) = B ) /\
     A. a e. n ( f ` a ) R ( f ` suc a ) ) ) $=
      ( vx vy wbr cvv wcel wa cv csuc c0 cfv wceq w3a wex wrex wfn wral com c1o
      cttrcl cdif relttrcl brrelex12i fvex eleq1 mpbii anim12i 3ad2ant2 exlimiv
      rexlimivw eqeq2 anbi1d 3anbi2d exbidv rexbidv anbi2d df-ttrcl pm5.21nii
      brabg ) ABCUEZIAJKZBJKZLZDMZEMZNUAZOVIPZAQZVJVIPZBQZLZFMZVIPVQNVIPCIFVJUB
      ZRZDSZEUCUDUFZTZABVECUGUHVTVHEWAVSVHDVPVKVHVRVMVFVOVGVMVLJKVFOVIUIVLAJUJU
      KVOVNJKVGVJVIUIVNBJUJUKULUMUNUOVKVLGMZQZVNHMZQZLZVRRZDSZEWATVKVMWFLZVRRZD
      SZEWATWBGHABJJVEWCAQZWIWLEWAWMWHWKDWMWGWJVKVRWMWDVMWFWCAVLUPUQURUSUTWEBQZ
      WLVTEWAWNWKVSDWNWJVPVKVRWNWFVOVMWEBVNUPVAURUSUTGHCDFEVBVDVC $.
  $}

  ${
    $d A m n f a $.  $d B m n f a $.  $d R m n f a $.
    $( Characterization of elements of the transitive closure of a relation.
       (Contributed by Scott Fenton, 24-Aug-2024.) $)
    brttrcl2 $p |- ( A t++ R B <-> E. n e. _om E. f
     ( f Fn suc suc n /\ ( ( f ` (/) ) = A /\ ( f ` suc n ) = B ) /\
     A. a e. suc n ( f ` a ) R ( f ` suc a ) ) ) $=
      ( vm wbr cv csuc wfn c0 cfv wceq wa wex com cdif wrex wcel cttrcl w3a c1o
      wral brttrcl df-1o difeq2i eleq2i wb peano1 eldifsucnn dif0 rexeqi 3bitri
      ax-mp anbi1i r19.41v bitr4i exbii df-rex rexcom4 3bitr4i vex sucex fneq2d
      suceq fveqeq2 anbi2d raleq 3anbi123d exbidv ceqsexv rexbii ) ABCUAHDIZGIZ
      JZKZLVNMANZVOVNMBNZOZFIZVNMWAJVNMCHZFVOUDZUBZDPZGQUCRZSZVOEIZJZNZWEOZGPZE
      QSZVNWIJZKZVRWIVNMBNZOZWBFWIUDZUBZDPZEQSABCDGFUEVOWFTZWEOZGPWKEQSZGPWGWMX
      BXCGXBWJEQSZWEOXCXAXDWEXAVOQLJZRZTZWJEQLRZSZXDWFXFVOUCXEQUFUGUHLQTXGXIUIU
      JELVOUKUOWJEXHQQULUMUNUPWJWEEQUQURUSWEGWFUTWKEGQVAVBWLWTEQWEWTGWIWHEVCVDW
      JWDWSDWJVQWOVTWQWCWRWJVPWNVNVOWIVFVEWJVSWPVRVOWIBVNVGVHWBFVOWIVIVJVKVLVMU
      N $.
  $}

  ${
    $d R x y f n m $.
    $( If ` R ` is a relation, then it is a subclass of its transitive closure.
       (Contributed by Scott Fenton, 17-Oct-2024.) $)
    ssttrcl $p |- ( Rel R -> R C_ t++ R ) $=
      ( vx vy vf vn vm cv wcel wbr csuc wfn c0 cfv wceq wral w3a com c1o eqid
      wa wrel cttrcl wss cop wi wal wex cdif wrex wn 1onn onirri eldif mpbir2an
      1on cif cmpt weq vex ifex fnmpti pm3.2i 1oex sucex mptex fneq1 fveq1 word
      onordi 0elsuc ax-mp iftrue fvmpt eqtrdi eqeq1d sucid eqeq1 ifbid cvv neii
      1n0 iffalsei eqeltri eqtri anbi12d breq12d 3anbi123d spcev mp3an12 fneq2d
      suceq fveqeq2 anbi2d raleq csn df1o2 raleqi 0ex fveq2 df-1o eqtr4di ralsn
      fveq2d bitri bitrdi exbidv rspcev sylancr df-br brttrcl bitr3i gen2 ssrel
      3imtr3i mpbiri ) AUAAAUBZUCBGZCGZUDZAHZXSXPHZUEZCUFBUFYBBCXQXRAIZDGZEGZJZ
      KZLYDMZXQNZYEYDMXRNZTZFGZYDMZYLJZYDMZAIZFYEOZPZDUGZEQRUHZUIZXTYAYCRYTHZYD
      RJZKZYIRYDMZXRNZTZYHUUEAIZPZDUGZUUAUUBRQHRRHUJUKRUOULRQRUMUNFUUCYLLNZXQXR
      UPZUQZUUCKZBBURZCCURZTZYCUUJFUUCUULUUMUUKXQXRBUSZCUSZUTUUMSZVAUUOUUPXQSXR
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      UUHYPFYERWNUVKYPFLWOZOUUHYPFRUVLWPWQYPUUHFLWRUUKYMYHYOUUEAYLLYDWSUUKYNRYD
      UUKYNLJRYLLWKWTXAXCWFXBXDXEWGXFXGXHXQXRAXIUUAXQXRXPIYAXQXRADEFXJXQXRXPXIX
      KXNXLBCAXPXMXO $.
  $}

  ${
    $d R a b c f g h n m p q x y z $.
    $( The transitive closure of a class is transitive.  (Contributed by Scott
       Fenton, 17-Oct-2024.) $)
    ttrcltr $p |- ( t++ R o. t++ R ) C_ t++ R $=
      ( vn vp vc vq cv csuc c0 cfv wceq wa wbr com c1o wrex wcel coa adantr syl
      wb vx vy vf vz va vg vm vb vh wfn wral w3a wex co eldifi nnacl syl2an wss
      cdif wn eldif word nnord ordtri1 sylancr biimpar sylbi nnaword1 mpbid cif
      crio 0elsuc eleq1 fveq2 eqeq2 riotabidv ifbieq12d fvex ifex fvmpt iftrued
      fveq2d eqtrd sylan9eq sucid df-1o eleq2i peano1 eldifsucnn rexeqi anbi12i
      ax-mp 3bitri reeanv peano2 sylan syl2anc nnasuc notbid rexlimivv iffalsed
      suceq adantl weq simpr syl3anc breq12d elelsuc ordsucelsuc biimpa 3brtr4d
      wel elnn ancoms syl2an2r oveq2 eqeq1d mpbird wreu wrmo nnasmo reu5 riota2
      sylanbrc rspcdva cvv vex ordelsuc nfcv nfov nfeq1 rspcev eqeq2d syl5ibcom
      riota2f ex 1onn fveq1 3anbi123d brttrcl cttrcl relco cop 1on onordi sstrd
      ccom eldifd cmpt eqid simpl2l ovex mp1i difeq2i dif0 ordsucsssuc sseqtrrd
      bitr4i oveq12 eleq12d syl5ibrcom nnacan riota5 3eqtrd simpr2r rspcv mpan9
      simprl3 adantlr wo w3o ordtri3or 3orel3 syl5com simprr3 ordelss nnawordex
      cbvrexvw sylib simprr simpllr eqeltrd simprl ad4antlr nnaord eqcom bitrdi
      reximssdv rexbidva risset sylibr bitrd riotacl nfriota1 syl2an2 rexlimdva
      eqidd rexbidv mpd nfsuc eqtr4di eqsstrrid 0ex simpl2r simpr2l eqtr4d con0
      wi nnon oa1suc iftruei ifbieq1d eqtr3id jaod syld imp pm2.61dan ralrimiva
      breqtrrd fnmpti sucex mptex fneq1 anbi12d ralbidv mp3an1 syl21anc fveqeq2
      spcev fneq2d anbi2d raleq exbidv exlimdvv exlimiv opelco 2rexbii 3bitr4ri
      eeanv exbii bitri df-br bitr3i 3imtr4i relssi ) UAUBAUUAZVUFUUGZVUFVUFVUF
      UUBUCFZBFZGZUJZHVUHIZUAFZJZVUIVUHIZUDFZJZKZUEFZVUHIZVUSGZVUHIZALZUEVUIUKZ
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      XCBUGVWDVWDVUIVWDPZVVGVWDPZKZVWBVXCUCUFVXIVWBVXCVXIVUIVVGQUNZVWDPVWBVWGVX
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      VUHVRWXSVVFVRVSVTSWXOVWQVXKPZWUJWYGJVYOWXNXYIVXIWXNXYITZVWBVXIVYMXYJVYNVW
      OVXJXISRXJCVWQWUCWYGVXKWUDVWHVWQJZVYPWYBVYQWUBWYCWYFVWHVWQVUJVMVWHVWQVUHV
      NXYKWUAWYEVVFXYKVYTWYDEMVWHVWQVYSVOVPWBVQWUTWYBWYCWYFVWQVUHVRWYEVVFVRVSVT
      SXKUXRWUDVXKUJZWUFWUHKZWULVXRCVXKWUCWUDVYPVYQWUBVWHVUHVRWUAVVFVRVSWUTUXTV
      XQXYLXYMWULULUIWUDCVXKWUCVXJWVIUYAUYBVWGWUDJZVXLXYLVXOXYMVXPWULVXKVWGWUDU
      YCXYNVWLWUFVXNWUHXYNVWKWUEVUMHVWGWUDYRXQXYNVXMWUGVVLVXJVWGWUDYRXQUYDXYNVW
      SWUKDVXJXYNVWPWUIVWRWUJAVWOVWGWUDYRVWQVWGWUDYRXGUYEYSUYIUYFUYGVXBVXRCVXJV
      WDWVJVXAVXQUIWVJVWJVXLVWNVXOVWTVXPWVJVWIVXKVWGVWHVXJXBUYJWVJVWMVXNVWLVWHV
      XJVVLVWGUYHUYKVWSDVWHVXJUYLYSUYMYLXOYPUYNWTUYOVXEVUMVUPVUFLZVUPVVLVUFLZKZ
      UDUMVWFUDVUMVVLVUFVUFUAYGUBYGUYPXYQVWEUDVVEUCUMZVWAUFUMZKZUGVWDOBVWDOXYRB
      VWDOZXYSUGVWDOZKVWEXYQXYRXYSBUGVWDVWDWNVWCXYTBUGVWDVWDVVEVWAUCUFUYSUYQXYO
      YUAXYPYUBVUMVUPAUCBUEYTVUPVVLAUFUGUHYTWKUYRUYTVUAVXFVUMVVLVUFLVXCVUMVVLVU
      FVUBVUMVVLAUICDYTVUCVUDVUE $.
  $}

  ${
    $d R f n a x y $.
    $( The transitive closure of ` R ` restricted to ` _V ` is the same as the
       transitive closure of ` R ` itself.  (Contributed by Scott Fenton,
       20-Oct-2024.) $)
    ttrclresv $p |- t++ ( R |` _V ) = t++ R $=
      ( vf vn vx vy va cv csuc cfv wceq cvv wbr wral w3a wrex copab cttrcl fvex
      wex df-ttrcl wfn c0 wa cres com c1o cdif wcel brresi mpbiran ralbii exbii
      3anbi3i rexbii opabbii 3eqtr4i ) BGZCGZHUAZUBUQIDGJURUQIEGJUCZFGZUQIZVAHZ
      UQIZAKUDZLZFURMZNZBSZCUEUFUGZOZDEPUSUTVBVDALZFURMZNZBSZCVJOZDEPVEQAQVKVPD
      EVIVOCVJVHVNBVGVMUSUTVFVLFURVFVBKUHVLVAUQRKVBVDAVCUQRUIUJUKUMULUNUODEVEBF
      CTDEABFCTUP $.
  $}

  $( Composition law for the transitive closure of a relation.  (Contributed by
     Scott Fenton, 20-Oct-2024.) $)
  ttrclco $p |- ( t++ R o. R ) C_ t++ R $=
    ( cvv cres cttrcl ccom wrel wss ssttrcl coss2 mp2b ttrcltr sstri wceq relco
    relres dfrel3 mpbi resco ttrclresv coeq1i 3eqtr3i 3sstr3i ) ABCZDZUCEZUDADZ
    AEZUFUEUDUDEZUDUCFUCUDGUEUHGABOUCHUCUDUDIJUCKLUDAEZBCZUIUEUGUIFUJUIMUDANUIP
    QUDABRUDUFAASZTUAUKUB $.

  $( Composition law for the transitive closure of a relation.  (Contributed by
     Scott Fenton, 20-Oct-2024.) $)
  cottrcl $p |- ( R o. t++ R ) C_ t++ R $=
    ( cvv cres cttrcl ccom wss wrel relres ssttrcl ax-mp coss1 ttrcltr crn wceq
    sstri ssv cores ttrclresv coeq2i eqtri 3sstr3i ) ABCZUBDZEZUCAADZEZUEUDUCUC
    EZUCUBUCFZUDUGFUBGUHABHUBIJUBUCUCKJUBLOUDAUCEZUFUCMZBFUDUINUJPAUCBQJUCUEAAR
    ZSTUKUA $.

  ${
    $d R x y z f g n m a b i $.  $d S x y z f g n m a b i $.
    $( If ` R ` is a subclass of ` S ` and ` S ` is transitive, then the
       transitive closure of ` R ` is a subclass of ` S ` .  (Contributed by
       Scott Fenton, 20-Oct-2024.) $)
    ttrclss $p |- ( ( R C_ S /\ ( S o. S ) C_ S ) -> t++ R C_ S ) $=
      ( vy vf vn va vg wa cv wcel wi wal csuc c0 cfv wceq wbr wral w3a suceq vx
      vm vi vz vb wss ccom cop cttrcl wfn wex com wrex c1o fneq2d df-1o eqtr4di
      syl anbi2d csn df1o2 eqtrdi raleqdv 0ex fveq2 fveq2d breq12d ralsn bitrdi
      fveqeq2d 3anbi123d exbidv imbi1d albidv imbi2d weq cbvralvw fneq1 anbi12d
      fveq1 eqeq1d ralbidv cbvexvw eqeq2 3anbi2d imbi12d cbvalvw breq12 3adant1
      breq2 biimpa ssbr adantr syl5 exlimdv alrimiv fvex simpr1 sssucid fnssres
      spcv cres sylancl peano2 ad2antrr nnord 0elsuc fvresd simpr2l eqtrd sucex
      word vex sucid mp1i simplr3 elelsuc adantl rspcdva wb ordsucelsuc 3brtr4d
      fvres ralrimiva resex spcev syl121anc simplrl simpr3 ralimdv sylc simpr2r
      rspcv mpsyl breqtrd breq1 sylibr albii sylib df-br simplrr mpan2d embantd
      brco ssbrd ex com23 3impia a2d finds com12 ralrimiv ralcom4 r19.23v bitri
      3exp brttrcl2 bitr3i imbi12i wrel relttrcl ssrel ax-mp ) ABUFZBBUGZBUFZHZ
      UAIZCIZUHZAUIZJZUVJBJZKZCLZUALZUVKBUFZUVGUVOUAUVGDIZEIZMZMZUJZNUVROZUVHPZ
      UVTUVROUVIPZHZFIZUVROZUWGMZUVROZAQZFUVTRZSZDUKZEULUMZUVHUVIBQZKZCLZUVOUVG
      UWNUWPKZCLZEULRZUWRUVGUWTEULUVSULJUVGUWTUVGUVRUBIZMZMZUJZUWDUXCUVROUVIPZH
      ZUWKFUXCRZSZDUKZUWPKZCLZKUVGUVRNMZMZUJZUWDUNUVROZUVIPZHZUWCUXPAQZSZDUKZUW
      PKZCLZKUVGGIZUCIZMZMZUJZNUYDOZUVHPZUYFUYDOZUDIZPZHZUEIZUYDOZUYOMZUYDOZAQZ
      UEUYFRZSZGUKZUVHUYLBQZKZUDLZKUVGUVRUYGMZUJZUWDUYGUVROZUVIPZHZUWKFUYGRZSZD
      UKZUWPKZCLZKUVGUWTKUBUCUVSUXBNPZUXLUYCUVGVUPUXKUYBCVUPUXJUYAUWPVUPUXIUXTD
      VUPUXEUXOUXGUXRUXHUXSVUPUXDUXNUVRVUPUXCUXMPUXDUXNPUXBNTZUXCUXMTURUOVUPUXF
      UXQUWDVUPUXCUNUVIUVRVUPUXCUXMUNVUQUPUQZVJUSVUPUXHUWKFNUTZRUXSVUPUWKFUXCVU
      SVUPUXCUNVUSVURVAVBVCUWKUXSFNVDUWGNPZUWHUWCUWJUXPAUWGNUVRVEVUTUWIUNUVRVUT
      UWIUXMUNUWGNTUPUQVFVGVHVIVKVLVMVNVOUBUCVPZUXLVUEUVGVVAUXLUYHUYJUYKUVIPZHZ
      UYTSZGUKZUWPKZCLVUEVVAUXKVVFCVVAUXJVVEUWPVVAUXJUVRUYGUJZUWDUYFUVROZUVIPZH
      ZUYOUVROZUYQUVROZAQZUEUYFRZSZDUKVVEVVAUXIVVODVVAUXEVVGUXGVVJUXHVVNVVAUXDU
      YGUVRVVAUXCUYFPUXDUYGPUXBUYETZUXCUYFTURUOVVAUXFVVIUWDVVAUXCUYFUVIUVRVVPVJ
      USVVAUXHUWKFUYFRVVNVVAUWKFUXCUYFVVPVCUWKVVMFUEUYFFUEVPZUWHVVKUWJVVLAUWGUY
      OUVRVEVVQUWIUYQUVRUWGUYOTVFVGZVQVIVKVLVVOVVDDGDGVPZVVGUYHVVJVVCVVNUYTUYGU
      VRUYDVRVVSUWDUYJVVIVVBVVSUWCUYIUVHNUVRUYDVTWAVVSVVHUYKUVIUYFUVRUYDVTWAVSV
      VSVVMUYSUEUYFVVSVVKUYPVVLUYRAUYOUVRUYDVTUYQUVRUYDVTVGWBVKWCVIVMVNVVFVUDCU
      DCUDVPZVVEVUBUWPVUCVVTVVDVUAGVVTVVCUYNUYHUYTVVTVVBUYMUYJUVIUYLUYKWDUSWEVL
      UVIUYLUVHBWJWFWGVIVOUXBUYFPZUXLVUOUVGVWAUXKVUNCVWAUXJVUMUWPVWAUXIVULDVWAU
      XEVUGUXGVUJUXHVUKVWAUXDVUFUVRVWAUXCUYGPUXDVUFPUXBUYFTZUXCUYGTURUOVWAUXFVU
      IUWDVWAUXCUYGUVIUVRVWBVJUSVWAUWKFUXCUYGVWBVCVKVLVMVNVOUBEVPZUXLUWTUVGVWCU
      XKUWSCVWCUXJUWNUWPVWCUXIUWMDVWCUXEUWBUXGUWFUXHUWLVWCUXDUWAUVRVWCUXCUVTPUX
      DUWAPUXBUVSTZUXCUVTTURUOVWCUXFUWEUWDVWCUXCUVTUVIUVRVWDVJUSVWCUWKFUXCUVTVW
      DVCVKVLVMVNVOUVGUYBCUVGUXTUWPDUXTUVHUVIAQZUVGUWPUXRUXSVWEUXOUXRUXSVWEUWCU
      VHUXPUVIAWHWKWIUVDVWEUWPKUVFABUVHUVIWLWMWNWOWPUYEULJZUVGVUEVUOVWFUVGVUEVU
      OVWFUVGVUESZVUNCVWGVULUWPDVWFUVGVUEVULUWPKZVUEUYHUYJUYKVVHPZHZUYTSZGUKZUV
      HVVHBQZKZVWFUVGHZVWHVUDVWNUDVVHUYFUVRWQZUYLVVHPZVUBVWLVUCVWMVWQVUAVWKGVWQ
      UYNVWJUYHUYTVWQUYMVWIUYJUYLVVHUYKWDUSWEVLUYLVVHUVHBWJZWFXAVWOVULVWNUWPVWO
      VULVWNUWPKVWOVULHZVWLVWMUWPVWSUVRUYGXBZUYGUJZNVWTOZUVHPZUYFVWTOZVVHPZUYOV
      WTOZUYQVWTOZAQZUEUYFRZVWLVWSVUGUYGVUFUFVXAVWOVUGVUJVUKWRUYGWSVUFUYGUVRWTX
      CVWSVXBUWCUVHVWSNUYGUVRVWSUYFXLZNUYGJVWSUYFULJZVXJVWFVXKUVGVULUYEXDXEUYFX
      FURZUYFXGURXHUWDVUIVUGVUKVWOXIXJUYFUYGJZVXEVWSUYFUYEUCXMXKXNZUYFUYGUVRYCX
      OVWSVXHUEUYFVWSUYOUYFJZHZVVKVVLVXFVXGAVXPUWKVVMFUYGUYOVVRVUGVUJVUKVWOVXOX
      PVXOUYOUYGJVWSUYOUYFXQXRZXSVXPUYOUYGUVRVXQXHVXPUYQUYGUVRVWSVXOUYQUYGJZVWS
      VXJVXOVXRXTVXLUYOUYFYAURWKXHYBYDVWKVXAVXCVXEHZVXISGVWTUVRUYGDXMYEUYDVWTPZ
      UYHVXAVWJVXSUYTVXIUYGUYDVWTVRVXTUYJVXCVWIVXEVXTUYIVXBUVHNUYDVWTVTWAVXTUYK
      VXDVVHUYFUYDVWTVTWAVSVXTUYSVXHUEUYFVXTUYPVXFUYRVXGAUYOUYDVWTVTUYQUYDVWTVT
      VGWBVKYFYGVWSVWMVVHUVIBQZUWPVWSVVHVUHUVIBVXMVWSUWHUWJBQZFUYGRZVVHVUHBQZVX
      NVWSUVDVUKVYCVWFUVDUVFVULYHVWOVUGVUJVUKYIUVDUWKVYBFUYGABUWHUWJWLYJYKVYBVY
      DFUYFUYGUWGUYFPZUWHVVHUWJVUHBUWGUYFUVRVEVYEUWIUYGUVRUWGUYFTVFVGYMYNUWDVUI
      VUGVUKVWOYLYOVWMVYAHZUVHUVIUVEQZVWSUWPVYFVUCUYLUVIBQZHZUDUKVYGVYIVYFUDVVH
      VWPVWQVUCVWMVYHVYAVWRUYLVVHUVIBYPVSYFUDUVHUVIBBUAXMCXMUUDYQVWSUVEBUVHUVIV
      WFUVDUVFVULUUAUUEWNUUBUUCUUFUUGWNUUHWOWPUUPUUIUUJUUKUULUXAUWSEULRZCLUWRUW
      SECULUUMVYJUWQCUWNUWPEULUUNYRUUOYSUWQUVNCUWOUVLUWPUVMUWOUVHUVIUVKQUVLUVHU
      VIADEFUUQUVHUVIUVKYTUURUVHUVIBYTUUSYRYSWPUVKUUTUVQUVPXTAUVAUACUVKBUVBUVCY
      Q $.
  $}

  ${
    $d R x y f n a $.
    $( The domain of a transitive closure is the same as the domain of the
       original class.  (Contributed by Scott Fenton, 26-Oct-2024.) $)
    dmttrcl $p |- dom t++ R = dom R $=
      ( vf vn vx vy va cttrcl cdm cv csuc c0 cfv wceq wbr com c1o wcel wss cvv
      wa wfn wral w3a wex cdif wrex cab copab df-ttrcl dmeq