P. 47 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4601-4700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	finds 4601*	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 4602*	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 4603*	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 4604	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 4605*	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 4606	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 4607. (Revised by BJ, 7-Aug-2024.)
|- ( A e. om -> A C_ om )
Theorem	elnn 4607	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 4608	Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.)
|- Ord om
Theorem	omelon2 4609	Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.)
|- ( om e. _V -> om e. On )
Theorem	omelon 4610	Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.)
|- om e. On
Theorem	nnon 4611	A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
|- ( A e. om -> A e. On )
Theorem	nnoni 4612	A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
|- A e. om   =>    |- A e. On
Theorem	nnord 4613	A natural number is ordinal. (Contributed by NM, 17-Oct-1995.)
|- ( A e. om -> Ord A )
Theorem	omsson 4614	Omega is a subset of On. (Contributed by NM, 13-Jun-1994.)
|- om C_ On
Theorem	limom 4615	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 4616	A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.)
|- ( A e. om <-> suc A e. om )
Theorem	nnsuc 4617*	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 4618	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 4619	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 4620	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 4621	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 4622*	If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4536 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6502 or nntri3or 6496), 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 4623*	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 4624	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 4565) but also holds when it is (/) by uni0 3838. (Contributed by Jim Kingdon, 31-Jul-2022.)
|- ( A e. om -> U. A e. om )
Theorem	nnpredlt 4625	The predecessor (see nnpredcl 4624) of a nonzero natural number is less than (see df-iord 4368) that number. (Contributed by Jim Kingdon, 14-Sep-2024.)
|- ( ( A e. om /\ A =/= (/) ) -> U. A e. A )
2.6.6  Relations
Syntax	cxp 4626	Extend the definition of a class to include the cross product.
class ( A X. B )
Syntax	ccnv 4627	Extend the definition of a class to include the converse of a class.
class `' A
Syntax	cdm 4628	Extend the definition of a class to include the domain of a class.
class dom A
Syntax	crn 4629	Extend the definition of a class to include the range of a class.
class ran A
Syntax	cres 4630	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 4631	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 4632	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 4633	Extend the definition of a wff to include the relation predicate. (Read: A is a relation.)
wff Rel A
Definition	df-xp 4634*	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 4635	Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 5081 and dfrel3 5088. (Contributed by NM, 1-Aug-1994.)
|- ( Rel A <-> A C_ ( _V X. _V ) )
Definition	df-cnv 4636*	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 4812 (see df-br 4006 and df-rel 4635 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 4637*	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 4638*	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 4639). For alternate definitions see dfdm2 5165, dfdm3 4816, and dfdm4 4821. 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 4639	Define the range of a class. For example, F = { <. 2 , 6 >., <. 3 , 9 >. } -> ran F = { 6 , 9 } . Contrast with domain (defined in df-dm 4638). For alternate definitions, see dfrn2 4817, dfrn3 4818, and dfrn4 5091. 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 4640	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 4641	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 4640) and range (df-rn 4639). For an alternate definition, see dfima2 4974. (Contributed by NM, 2-Aug-1994.)
|- ( A " B ) = ran ( A |` B )
Theorem	xpeq1 4642	Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.)
|- ( A = B -> ( A X. C ) = ( B X. C ) )
Theorem	xpeq2 4643	Equality theorem for cross product. (Contributed by NM, 5-Jul-1994.)
|- ( A = B -> ( C X. A ) = ( C X. B ) )
Theorem	elxpi 4644*	Membership in a cross product. Uses fewer axioms than elxp 4645. (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 4645*	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 4646*	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 4647	Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.)
|- ( ( A = B /\ C = D ) -> ( A X. C ) = ( B X. D ) )
Theorem	xpeq1i 4648	Equality inference for cross product. (Contributed by NM, 21-Dec-2008.)
|- A = B   =>    |- ( A X. C ) = ( B X. C )
Theorem	xpeq2i 4649	Equality inference for cross product. (Contributed by NM, 21-Dec-2008.)
|- A = B   =>    |- ( C X. A ) = ( C X. B )
Theorem	xpeq12i 4650	Equality inference for cross product. (Contributed by FL, 31-Aug-2009.)
|- A = B   &    |- C = D   =>    |- ( A X. C ) = ( B X. D )
Theorem	xpeq1d 4651	Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A X. C ) = ( B X. C ) )
Theorem	xpeq2d 4652	Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C X. A ) = ( C X. B ) )
Theorem	xpeq12d 4653	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 4654	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 4655	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 4656	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 4657	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 4658	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 4659	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 4660	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 4661	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 4662	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 4663	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 4664	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	rabxp 4665*	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 4666	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 4667	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 4668	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 4669	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 4670	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 4671	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 4672	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 4673	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 4674	A binary relation does not contain the empty set. (Contributed by AV, 15-Nov-2021.)
|- ( Rel R -> (/) e/ R )
Theorem	fconstmpt 4675*	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 4676*	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 4677	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 4678	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 )
Theorem	opthprc 4679	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.)
|- ( ( ( A X. { (/) } ) u. ( B X. { { (/) } } ) ) = ( ( C X. { (/) } ) u. ( D X. { { (/) } } ) ) <-> ( A = C /\ B = D ) )
Theorem	brel 4680	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.)
|- R C_ ( C X. D )   =>    |- ( A R B -> ( A e. C /\ B e. D ) )
Theorem	brab2a 4681*	Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 9-Nov-2015.)
|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )   &    |- R = { <. x , y >. | ( ( x e. C /\ y e. D ) /\ ph ) }   =>    |- ( A R B <-> ( ( A e. C /\ B e. D ) /\ ps ) )
Theorem	elxp3 4682*	Membership in a cross product. (Contributed by NM, 5-Mar-1995.)
|- ( A e. ( B X. C ) <-> E. x E. y ( <. x , y >. = A /\ <. x , y >. e. ( B X. C ) ) )
Theorem	opeliunxp 4683	Membership in a union of cross products. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 1-Jan-2017.)
|- ( <. x , C >. e. U_ x e. A ( { x } X. B ) <-> ( x e. A /\ C e. B ) )
Theorem	xpundi 4684	Distributive law for cross product over union. Theorem 103 of [Suppes] p. 52. (Contributed by NM, 12-Aug-2004.)
|- ( A X. ( B u. C ) ) = ( ( A X. B ) u. ( A X. C ) )
Theorem	xpundir 4685	Distributive law for cross product over union. Similar to Theorem 103 of [Suppes] p. 52. (Contributed by NM, 30-Sep-2002.)
|- ( ( A u. B ) X. C ) = ( ( A X. C ) u. ( B X. C ) )
Theorem	xpiundi 4686*	Distributive law for cross product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)
|- ( C X. U_ x e. A B ) = U_ x e. A ( C X. B )
Theorem	xpiundir 4687*	Distributive law for cross product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)
|- ( U_ x e. A B X. C ) = U_ x e. A ( B X. C )
Theorem	iunxpconst 4688*	Membership in a union of cross products when the second factor is constant. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- U_ x e. A ( { x } X. B ) = ( A X. B )
Theorem	xpun 4689	The cross product of two unions. (Contributed by NM, 12-Aug-2004.)
|- ( ( A u. B ) X. ( C u. D ) ) = ( ( ( A X. C ) u. ( A X. D ) ) u. ( ( B X. C ) u. ( B X. D ) ) )
Theorem	elvv 4690*	Membership in universal class of ordered pairs. (Contributed by NM, 4-Jul-1994.)
|- ( A e. ( _V X. _V ) <-> E. x E. y A = <. x , y >. )
Theorem	elvvv 4691*	Membership in universal class of ordered triples. (Contributed by NM, 17-Dec-2008.)
|- ( A e. ( ( _V X. _V ) X. _V ) <-> E. x E. y E. z A = <. <. x , y >. , z >. )
Theorem	elvvuni 4692	An ordered pair contains its union. (Contributed by NM, 16-Sep-2006.)
|- ( A e. ( _V X. _V ) -> U. A e. A )
Theorem	mosubopt 4693*	"At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-Aug-2007.)
|- ( A. y A. z E* x ph -> E* x E. y E. z ( A = <. y , z >. /\ ph ) )
Theorem	mosubop 4694*	"At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-May-1995.)
|- E* x ph   =>    |- E* x E. y E. z ( A = <. y , z >. /\ ph )
Theorem	brinxp2 4695	Intersection of binary relation with Cartesian product. (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( A ( R i^i ( C X. D ) ) B <-> ( A e. C /\ B e. D /\ A R B ) )
Theorem	brinxp 4696	Intersection of binary relation with Cartesian product. (Contributed by NM, 9-Mar-1997.)
|- ( ( A e. C /\ B e. D ) -> ( A R B <-> A ( R i^i ( C X. D ) ) B ) )
Theorem	poinxp 4697	Intersection of partial order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
|- ( R Po A <-> ( R i^i ( A X. A ) ) Po A )
Theorem	soinxp 4698	Intersection of linear order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
|- ( R Or A <-> ( R i^i ( A X. A ) ) Or A )
Theorem	seinxp 4699	Intersection of set-like relation with cross product of its field. (Contributed by Mario Carneiro, 22-Jun-2015.)
|- ( R Se A <-> ( R i^i ( A X. A ) ) Se A )
Theorem	posng 4700	Partial ordering of a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.)
|- ( ( Rel R /\ A e. _V ) -> ( R Po { A } <-> -. A R A ) )