P. 48 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4701-4800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	reg3exmidlemwe 4701*	Lemma for reg3exmid 4702. Our counterexample A satisfies We. (Contributed by Jim Kingdon, 3-Oct-2021.)
|- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }   =>    |- _E We A
Theorem	reg3exmid 4702*	If any inhabited set satisfying df-wetr 4455 for _E has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Oct-2021.)
|- ( ( _E We z /\ E. w w e. z ) -> E. x e. z A. y e. z x C_ y )   =>    |- ( ph \/ -. ph )
Theorem	dcextest 4703*	If it is decidable whether { x | ph } is a set, then -. ph is decidable (where x does not occur in ph). From this fact, we can deduce (outside the formal system, since we cannot quantify over classes) that if it is decidable whether any class is a set, then "weak excluded middle" (that is, any negated proposition -. ph is decidable) holds. (Contributed by Jim Kingdon, 3-Jul-2022.)
|- DECID { x | ph } e. _V   =>    |- DECID -. ph
2.5.3  Transfinite induction
Theorem	tfi 4704*	The Principle of Transfinite Induction. Theorem 7.17 of [TakeutiZaring] p. 39. This principle states that if A is a class of ordinal numbers with the property that every ordinal number included in A also belongs to A, then every ordinal number is in A. (Contributed by NM, 18-Feb-2004.)
|- ( ( A C_ On /\ A. x e. On ( x C_ A -> x e. A ) ) -> A = On )
Theorem	tfis 4705*	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.)
|- ( x e. On -> ( A. y e. x [ y / x ] ph -> ph ) )   =>    |- ( x e. On -> ph )
Theorem	tfis2f 4706*	Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
|- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   &    |- ( x e. On -> ( A. y e. x ps -> ph ) )   =>    |- ( x e. On -> ph )
Theorem	tfis2 4707*	Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( x e. On -> ( A. y e. x ps -> ph ) )   =>    |- ( x e. On -> ph )
Theorem	tfis3 4708*	Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( x = A -> ( ph <-> ch ) )   &    |- ( x e. On -> ( A. y e. x ps -> ph ) )   =>    |- ( A e. On -> ch )
Theorem	tfisi 4709*	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.)
|- ( ph -> A e. V )   &    |- ( ph -> T e. On )   &    |- ( ( ph /\ ( R e. On /\ R C_ T ) /\ A. y ( S e. R -> ch ) ) -> ps )   &    |- ( x = y -> ( ps <-> ch ) )   &    |- ( x = A -> ( ps <-> th ) )   &    |- ( x = y -> R = S )   &    |- ( x = A -> R = T )   =>    |- ( ph -> th )
2.6  IZF Set Theory - add the Axiom of Infinity
2.6.1  Introduce the Axiom of Infinity
Axiom	ax-iinf 4710*	Axiom of Infinity. Axiom 5 of [Crosilla] p. "Axioms of CZF and IZF". (Contributed by Jim Kingdon, 16-Nov-2018.)
|- E. x ( (/) e. x /\ A. y ( y e. x -> suc y e. x ) )
Theorem	zfinf2 4711*	A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (Contributed by NM, 30-Aug-1993.)
|- E. x ( (/) e. x /\ A. y e. x suc y e. x )
2.6.2  The natural numbers
Syntax	com 4712	Extend class notation to include the class of natural numbers.
class om
Definition	df-iom 4713*	Define the class of natural numbers as the smallest inductive set, which is valid provided we assume the Axiom of Infinity. Definition 6.3 of [Eisenberg] p. 82. Note: the natural numbers om are a subset of the ordinal numbers df-on 4489. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers (df-inn 9238) with analogous properties and operations, but they will be different sets. We are unable to use the terms finite ordinal and natural number interchangeably, as shown at exmidonfin 7497. (Contributed by NM, 6-Aug-1994.) Use its alias dfom3 4714 instead for naming consistency with set.mm. (New usage is discouraged.)
|- om = |^| { x | ( (/) e. x /\ A. y e. x suc y e. x ) }
Theorem	dfom3 4714*	Alias for df-iom 4713. Use it instead of df-iom 4713 for naming consistency with set.mm. (Contributed by NM, 6-Aug-1994.)
|- om = |^| { x | ( (/) e. x /\ A. y e. x suc y e. x ) }
Theorem	omex 4715	The existence of omega (the class of natural numbers). Axiom 7 of [TakeutiZaring] p. 43. (Contributed by NM, 6-Aug-1994.)
|- om e. _V
2.6.3  Peano's postulates
Theorem	peano1 4716	Zero is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(1) of [TakeutiZaring] p. 42. (Contributed by NM, 15-May-1994.)
|- (/) e. om
Theorem	peano2 4717	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.)
|- ( A e. om -> suc A e. om )
Theorem	peano3 4718	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.)
|- ( A e. om -> suc A =/= (/) )
Theorem	peano4 4719	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.)
|- ( ( A e. om /\ B e. om ) -> ( suc A = suc B <-> A = B ) )
Theorem	peano5 4720*	The induction postulate: any class containing zero and closed under the successor operation contains all natural numbers. One of Peano's five postulates for arithmetic. Proposition 7.30(5) of [TakeutiZaring] p. 43. The more traditional statement of mathematical induction as a theorem schema, with a basis and an induction step, is derived from this theorem as Theorem findes 4725. (Contributed by NM, 18-Feb-2004.)
|- ( ( (/) e. A /\ A. x e. om ( x e. A -> suc x e. A ) ) -> om C_ A )
2.6.4  Finite induction (for finite ordinals)
Theorem	find 4721*	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.)
|- ( A C_ om /\ (/) e. A /\ A. x e. A suc x e. A )   =>    |- A = om
Theorem	finds 4722*	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.)
|- ( x = (/) -> ( ph <-> ps ) )   &    |- ( x = y -> ( ph <-> ch ) )   &    |- ( x = suc y -> ( ph <-> th ) )   &    |- ( x = A -> ( ph <-> ta ) )   &    |- ps   &    |- ( y e. om -> ( ch -> th ) )   =>    |- ( A e. om -> ta )
Theorem	finds2 4723*	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.)
|- ( x = (/) -> ( ph <-> ps ) )   &    |- ( x = y -> ( ph <-> ch ) )   &    |- ( x = suc y -> ( ph <-> th ) )   &    |- ( ta -> ps )   &    |- ( y e. om -> ( ta -> ( ch -> th ) ) )   =>    |- ( x e. om -> ( ta -> ph ) )
Theorem	finds1 4724*	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.)
|- ( x = (/) -> ( ph <-> ps ) )   &    |- ( x = y -> ( ph <-> ch ) )   &    |- ( x = suc y -> ( ph <-> th ) )   &    |- ps   &    |- ( y e. om -> ( ch -> th ) )   =>    |- ( x e. om -> ph )
Theorem	findes 4725	Finite induction with explicit substitution. The first hypothesis is the basis and the second is the induction step. Theorem Schema 22 of [Suppes] p. 136. This is an alternative for Metamath 100 proof #74. (Contributed by Raph Levien, 9-Jul-2003.)
|- [. (/) / x ]. ph   &    |- ( x e. om -> ( ph -> [. suc x / x ]. ph ) )   =>    |- ( x e. om -> ph )
2.6.5  The Natural Numbers (continued)
Theorem	nn0suc 4726*	A natural number is either 0 or a successor. Similar theorems for arbitrary sets or real numbers will not be provable (without the law of the excluded middle), but equality of natural numbers is decidable. (Contributed by NM, 27-May-1998.)
|- ( A e. om -> ( A = (/) \/ E. x e. om A = suc x ) )
Theorem	elomssom 4727	A natural number ordinal is, as a set, included in the set of natural number ordinals. (Contributed by NM, 21-Jun-1998.) Extract this result from the previous proof of elnn 4728. (Revised by BJ, 7-Aug-2024.)
|- ( A e. om -> A C_ om )
Theorem	elnn 4728	A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.)
|- ( ( A e. B /\ B e. om ) -> A e. om )
Theorem	ordom 4729	Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.)
|- Ord om
Theorem	omelon2 4730	Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.)
|- ( om e. _V -> om e. On )
Theorem	omelon 4731	Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.)
|- om e. On
Theorem	nnon 4732	A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
|- ( A e. om -> A e. On )
Theorem	nnoni 4733	A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
|- A e. om   =>    |- A e. On
Theorem	nnord 4734	A natural number is ordinal. (Contributed by NM, 17-Oct-1995.)
|- ( A e. om -> Ord A )
Theorem	omsson 4735	Omega is a subset of On. (Contributed by NM, 13-Jun-1994.)
|- om C_ On
Theorem	limom 4736	Omega is a limit ordinal. Theorem 2.8 of [BellMachover] p. 473. (Contributed by NM, 26-Mar-1995.) (Proof rewritten by Jim Kingdon, 5-Jan-2019.)
|- Lim om
Theorem	peano2b 4737	A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.)
|- ( A e. om <-> suc A e. om )
Theorem	nnsuc 4738*	A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.)
|- ( ( A e. om /\ A =/= (/) ) -> E. x e. om A = suc x )
Theorem	nnsucpred 4739	The successor of the precedessor of a nonzero natural number. (Contributed by Jim Kingdon, 31-Jul-2022.)
|- ( ( A e. om /\ A =/= (/) ) -> suc U. A = A )
Theorem	nndceq0 4740	A natural number is either zero or nonzero. Decidable equality for natural numbers is a special case of the law of the excluded middle which holds in most constructive set theories including ours. (Contributed by Jim Kingdon, 5-Jan-2019.)
|- ( A e. om -> DECID A = (/) )
Theorem	0elnn 4741	A natural number is either the empty set or has the empty set as an element. (Contributed by Jim Kingdon, 23-Aug-2019.)
|- ( A e. om -> ( A = (/) \/ (/) e. A ) )
Theorem	nn0eln0 4742	A natural number is nonempty iff it contains the empty set. Although in constructive mathematics it is generally more natural to work with inhabited sets and ignore the whole concept of nonempty sets, in the specific case of natural numbers this theorem may be helpful in converting proofs which were written assuming excluded middle. (Contributed by Jim Kingdon, 28-Aug-2019.)
|- ( A e. om -> ( (/) e. A <-> A =/= (/) ) )
Theorem	nnregexmid 4743*	If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4657 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6732 or nntri3or 6726), sets of natural numbers are a different animal. (Contributed by Jim Kingdon, 6-Sep-2019.)
|- ( ( x C_ om /\ E. y y e. x ) -> E. y ( y e. x /\ A. z ( z e. y -> -. z e. x ) ) )   =>    |- ( ph \/ -. ph )
Theorem	omsinds 4744*	Strong (or "total") induction principle over om. (Contributed by Scott Fenton, 17-Jul-2015.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( x = A -> ( ph <-> ch ) )   &    |- ( x e. om -> ( A. y e. x ps -> ph ) )   =>    |- ( A e. om -> ch )
Theorem	nnpredcl 4745	The predecessor of a natural number is a natural number. This theorem is most interesting when the natural number is a successor (as seen in theorems like onsucuni2 4686) but also holds when it is (/) by uni0 3941. (Contributed by Jim Kingdon, 31-Jul-2022.)
|- ( A e. om -> U. A e. om )
Theorem	nnpredlt 4746	The predecessor (see nnpredcl 4745) of a nonzero natural number is less than (see df-iord 4487) that number. (Contributed by Jim Kingdon, 14-Sep-2024.)
|- ( ( A e. om /\ A =/= (/) ) -> U. A e. A )
2.6.6  Relations
Syntax	cxp 4747	Extend the definition of a class to include the cross product.
class ( A X. B )
Syntax	ccnv 4748	Extend the definition of a class to include the converse of a class.
class `' A
Syntax	cdm 4749	Extend the definition of a class to include the domain of a class.
class dom A
Syntax	crn 4750	Extend the definition of a class to include the range of a class.
class ran A
Syntax	cres 4751	Extend the definition of a class to include the restriction of a class. (Read: The restriction of A to B.)
class ( A |` B )
Syntax	cima 4752	Extend the definition of a class to include the image of a class. (Read: The image of B under A.)
class ( A " B )
Syntax	ccom 4753	Extend the definition of a class to include the composition of two classes. (Read: The composition of A and B.)
class ( A o. B )
Syntax	wrel 4754	Extend the definition of a wff to include the relation predicate. (Read: A is a relation.)
wff Rel A
Definition	df-xp 4755*	Define the Cartesian product of two classes. This is also sometimes called the "cross product" but that term also has other meanings; we intentionally choose a less ambiguous term. Definition 9.11 of [Quine] p. 64. For example, ( { 1 , 5 } X. { 2 , 7 } ) = ( { <. 1 , 2 >. , <. 1 , 7 >. } u. { <. 5 , 2 >. , <. 5 , 7 >. } ). Another example is that the set of rational numbers is defined using the Cartesian product as ( ZZ X. NN ); the left- and right-hand sides of the Cartesian product represent the top (integer) and bottom (natural) numbers of a fraction. (Contributed by NM, 4-Jul-1994.)
|- ( A X. B ) = { <. x , y >. | ( x e. A /\ y e. B ) }
Definition	df-rel 4756	Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 5213 and dfrel3 5220. (Contributed by NM, 1-Aug-1994.)
|- ( Rel A <-> A C_ ( _V X. _V ) )
Definition	df-cnv 4757*	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 4938 (see df-br 4110 and df-rel 4756 for more on relations). For example, `' { <. 2 , 6 >. , <. 3 , 9 >. } = { <. 6 , 2 >. , <. 9 , 3 >. }. We use Quine's breve accent (smile) notation. Like Quine, we use it as a prefix, which eliminates the need for parentheses. "Converse" is Quine's terminology. Some authors use a "minus one" exponent and call it "inverse", especially when the argument is a function, although this is not in general a genuine inverse. (Contributed by NM, 4-Jul-1994.)
|- `' A = { <. x , y >. | y A x }
Definition	df-co 4758*	Define the composition of two classes. Definition 6.6(3) of [TakeutiZaring] p. 24. Note that Definition 7 of [Suppes] p. 63 reverses A and B, uses a slash instead of o., and calls the operation "relative product". (Contributed by NM, 4-Jul-1994.)
|- ( A o. B ) = { <. x , y >. | E. z ( x B z /\ z A y ) }
Definition	df-dm 4759*	Define the domain of a class. Definition 3 of [Suppes] p. 59. For example, F = { <. 2 , 6 >., <. 3 , 9 >. } -> dom F = { 2 , 3 } . Contrast with range (defined in df-rn 4760). For alternate definitions see dfdm2 5297, dfdm3 4942, and dfdm4 4948. The notation " dom " is used by Enderton; other authors sometimes use script D. (Contributed by NM, 1-Aug-1994.)
|- dom A = { x | E. y x A y }
Definition	df-rn 4760	Define the range of a class. For example, F = { <. 2 , 6 >., <. 3 , 9 >. } -> ran F = { 6 , 9 } . Contrast with domain (defined in df-dm 4759). For alternate definitions, see dfrn2 4943, dfrn3 4944, and dfrn4 5223. The notation " ran " is used by Enderton; other authors sometimes use script R or script W. (Contributed by NM, 1-Aug-1994.)
|- ran A = dom `' A
Definition	df-res 4761	Define the restriction of a class. Definition 6.6(1) of [TakeutiZaring] p. 24. For example, ( F = { <. 2 , 6 >. , <. 3 , 9 >. } /\ B = { 1 , 2 } ) -> ( F |` B ) = { <. 2 , 6 >. }. We do not introduce a special syntax for the corestriction of a class: it will be expressed either as the intersection ( A i^i ( _V X. B ) ) or as the converse of the restricted converse. (Contributed by NM, 2-Aug-1994.)
|- ( A |` B ) = ( A i^i ( B X. _V ) )
Definition	df-ima 4762	Define the image of a class (as restricted by another class). Definition 6.6(2) of [TakeutiZaring] p. 24. For example, ( F = { <. 2 , 6 >., <. 3 , 9 >. } /\ B = { 1 , 2 } ) -> ( F " B ) = { 6 } . Contrast with restriction (df-res 4761) and range (df-rn 4760). For an alternate definition, see dfima2 5103. (Contributed by NM, 2-Aug-1994.)
|- ( A " B ) = ran ( A |` B )
Theorem	xpeq1 4763	Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.)
|- ( A = B -> ( A X. C ) = ( B X. C ) )
Theorem	xpeq2 4764	Equality theorem for cross product. (Contributed by NM, 5-Jul-1994.)
|- ( A = B -> ( C X. A ) = ( C X. B ) )
Theorem	elxpi 4765*	Membership in a cross product. Uses fewer axioms than elxp 4766. (Contributed by NM, 4-Jul-1994.)
|- ( A e. ( B X. C ) -> E. x E. y ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) )
Theorem	elxp 4766*	Membership in a cross product. (Contributed by NM, 4-Jul-1994.)
|- ( A e. ( B X. C ) <-> E. x E. y ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) )
Theorem	elxp2 4767*	Membership in a cross product. (Contributed by NM, 23-Feb-2004.)
|- ( A e. ( B X. C ) <-> E. x e. B E. y e. C A = <. x , y >. )
Theorem	xpeq12 4768	Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.)
|- ( ( A = B /\ C = D ) -> ( A X. C ) = ( B X. D ) )
Theorem	xpeq1i 4769	Equality inference for cross product. (Contributed by NM, 21-Dec-2008.)
|- A = B   =>    |- ( A X. C ) = ( B X. C )
Theorem	xpeq2i 4770	Equality inference for cross product. (Contributed by NM, 21-Dec-2008.)
|- A = B   =>    |- ( C X. A ) = ( C X. B )
Theorem	xpeq12i 4771	Equality inference for cross product. (Contributed by FL, 31-Aug-2009.)
|- A = B   &    |- C = D   =>    |- ( A X. C ) = ( B X. D )
Theorem	xpeq1d 4772	Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A X. C ) = ( B X. C ) )
Theorem	xpeq2d 4773	Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C X. A ) = ( C X. B ) )
Theorem	xpeq12d 4774	Equality deduction for Cartesian product. (Contributed by NM, 8-Dec-2013.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A X. C ) = ( B X. D ) )
Theorem	sqxpeqd 4775	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.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A X. A ) = ( B X. B ) )
Theorem	nfxp 4776	Bound-variable hypothesis builder for cross product. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/_ x ( A X. B )
Theorem	0nelxp 4777	The empty set is not a member of a cross product. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- -. (/) e. ( A X. B )
Theorem	0nelelxp 4778	A member of a cross product (ordered pair) doesn't contain the empty set. (Contributed by NM, 15-Dec-2008.)
|- ( C e. ( A X. B ) -> -. (/) e. C )
Theorem	opelxp 4779	Ordered pair membership in a cross product. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( <. A , B >. e. ( C X. D ) <-> ( A e. C /\ B e. D ) )
Theorem	brxp 4780	Binary relation on a cross product. (Contributed by NM, 22-Apr-2004.)
|- ( A ( C X. D ) B <-> ( A e. C /\ B e. D ) )
Theorem	opelxpi 4781	Ordered pair membership in a cross product (implication). (Contributed by NM, 28-May-1995.)
|- ( ( A e. C /\ B e. D ) -> <. A , B >. e. ( C X. D ) )
Theorem	opelxpd 4782	Ordered pair membership in a Cartesian product, deduction form. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
|- ( ph -> A e. C )   &    |- ( ph -> B e. D )   =>    |- ( ph -> <. A , B >. e. ( C X. D ) )
Theorem	opelxp1 4783	The first member of an ordered pair of classes in a cross product belongs to first cross product argument. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( <. A , B >. e. ( C X. D ) -> A e. C )
Theorem	opelxp2 4784	The second member of an ordered pair of classes in a cross product belongs to second cross product argument. (Contributed by Mario Carneiro, 26-Apr-2015.)
|- ( <. A , B >. e. ( C X. D ) -> B e. D )
Theorem	otelxp1 4785	The first member of an ordered triple of classes in a cross product belongs to first cross product argument. (Contributed by NM, 28-May-2008.)
|- ( <. <. A , B >. , C >. e. ( ( R X. S ) X. T ) -> A e. R )
Theorem	opabssxpd 4786*	An ordered-pair class abstraction is a subset of a Cartesian product. Formerly part of proof for opabex2 6388. (Contributed by AV, 26-Nov-2021.)
|- ( ( ph /\ ps ) -> x e. A )   &    |- ( ( ph /\ ps ) -> y e. B )   =>    |- ( ph -> { <. x , y >. | ps } C_ ( A X. B ) )
Theorem	rabxp 4787*	Membership in a class builder restricted to a cross product. (Contributed by NM, 20-Feb-2014.)
|- ( x = <. y , z >. -> ( ph <-> ps ) )   =>    |- { x e. ( A X. B ) | ph } = { <. y , z >. | ( y e. A /\ z e. B /\ ps ) }
Theorem	brrelex12 4788	A true binary relation on a relation implies the arguments are sets. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)
|- ( ( Rel R /\ A R B ) -> ( A e. _V /\ B e. _V ) )
Theorem	brrelex1 4789	A true binary relation on a relation implies the first argument is a set. (This is a property of our ordered pair definition.) (Contributed by NM, 18-May-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( ( Rel R /\ A R B ) -> A e. _V )
Theorem	brrelex 4790	A true binary relation on a relation implies the first argument is a set. (This is a property of our ordered pair definition.) (Contributed by NM, 18-May-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( ( Rel R /\ A R B ) -> A e. _V )
Theorem	brrelex2 4791	A true binary relation on a relation implies the second argument is a set. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)
|- ( ( Rel R /\ A R B ) -> B e. _V )
Theorem	brrelex12i 4792	Two classes that are related by a binary relation are sets. (An artifact of our ordered pair definition.) (Contributed by BJ, 3-Oct-2022.)
|- Rel R   =>    |- ( A R B -> ( A e. _V /\ B e. _V ) )
Theorem	brrelex1i 4793	The first argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by NM, 4-Jun-1998.)
|- Rel R   =>    |- ( A R B -> A e. _V )
Theorem	brrelex2i 4794	The second argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)
|- Rel R   =>    |- ( A R B -> B e. _V )
Theorem	nprrel 4795	No proper class is related to anything via any relation. (Contributed by Roy F. Longton, 30-Jul-2005.)
|- Rel R   &    |- -. A e. _V   =>    |- -. A R B
Theorem	0nelrel 4796	A binary relation does not contain the empty set. (Contributed by AV, 15-Nov-2021.)
|- ( Rel R -> (/) e/ R )
Theorem	fconstmpt 4797*	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.)
|- ( A X. { B } ) = ( x e. A |-> B )
Theorem	vtoclr 4798*	Variable to class conversion of transitive relation. (Contributed by NM, 9-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- Rel R   &    |- ( ( x R y /\ y R z ) -> x R z )   =>    |- ( ( A R B /\ B R C ) -> A R C )
Theorem	opelvvg 4799	Ordered pair membership in the universal class of ordered pairs. (Contributed by Mario Carneiro, 3-May-2015.)
|- ( ( A e. V /\ B e. W ) -> <. A , B >. e. ( _V X. _V ) )
Theorem	opelvv 4800	Ordered pair membership in the universal class of ordered pairs. (Contributed by NM, 22-Aug-2013.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- A e. _V   &    |- B e. _V   =>    |- <. A , B >. e. ( _V X. _V )