P. 39 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 3801-3900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	neldifsn 3801	A is not in ( B \ { A } ). (Contributed by David Moews, 1-May-2017.)
|- -. A e. ( B \ { A } )
Theorem	neldifsnd 3802	A is not in ( B \ { A } ). Deduction form. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> -. A e. ( B \ { A } ) )
Theorem	rexdifsn 3803	Restricted existential quantification over a set with an element removed. (Contributed by NM, 4-Feb-2015.)
|- ( E. x e. ( A \ { B } ) ph <-> E. x e. A ( x =/= B /\ ph ) )
Theorem	snssb 3804	Characterization of the inclusion of a singleton in a class. (Contributed by BJ, 1-Jan-2025.)
|- ( { A } C_ B <-> ( A e. _V -> A e. B ) )
Theorem	snssg 3805	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.)
|- ( A e. V -> ( A e. B <-> { A } C_ B ) )
Theorem	snss 3806	The singleton of an element of a class is a subset of the class (inference form of snssg 3805). Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 21-Jun-1993.) (Proof shortened by BJ, 1-Jan-2025.)
|- A e. _V   =>    |- ( A e. B <-> { A } C_ B )
Theorem	snssgOLD 3807	Obsolete version of snssgOLD 3807 as of 1-Jan-2025. (Contributed by NM, 22-Jul-2001.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( A e. V -> ( A e. B <-> { A } C_ B ) )
Theorem	difsn 3808	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.)
|- ( -. A e. B -> ( B \ { A } ) = B )
Theorem	difprsnss 3809	Removal of a singleton from an unordered pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- ( { A , B } \ { A } ) C_ { B }
Theorem	difprsn1 3810	Removal of a singleton from an unordered pair. (Contributed by Thierry Arnoux, 4-Feb-2017.)
|- ( A =/= B -> ( { A , B } \ { A } ) = { B } )
Theorem	difprsn2 3811	Removal of a singleton from an unordered pair. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
|- ( A =/= B -> ( { A , B } \ { B } ) = { A } )
Theorem	diftpsn3 3812	Removal of a singleton from an unordered triple. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
|- ( ( A =/= C /\ B =/= C ) -> ( { A , B , C } \ { C } ) = { A , B } )
Theorem	difpr 3813	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.)
|- ( A \ { B , C } ) = ( ( A \ { B } ) \ { C } )
Theorem	difsnb 3814	( B \ { A } ) equals B if and only if A is not a member of B. Generalization of difsn 3808. (Contributed by David Moews, 1-May-2017.)
|- ( -. A e. B <-> ( B \ { A } ) = B )
Theorem	snssi 3815	The singleton of an element of a class is a subset of the class. (Contributed by NM, 6-Jun-1994.)
|- ( A e. B -> { A } C_ B )
Theorem	snssd 3816	The singleton of an element of a class is a subset of the class (deduction form). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- ( ph -> A e. B )   =>    |- ( ph -> { A } C_ B )
Theorem	difsnss 3817	If we remove a single element from a class then put it back in, we end up with a subset of the original class. If equality is decidable, we can replace subset with equality as seen in nndifsnid 6670. (Contributed by Jim Kingdon, 10-Aug-2018.)
|- ( B e. A -> ( ( A \ { B } ) u. { B } ) C_ A )
Theorem	pw0 3818	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.)
|- ~P (/) = { (/) }
Theorem	snsspr1 3819	A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 27-Aug-2004.)
|- { A } C_ { A , B }
Theorem	snsspr2 3820	A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 2-May-2009.)
|- { B } C_ { A , B }
Theorem	snsstp1 3821	A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
|- { A } C_ { A , B , C }
Theorem	snsstp2 3822	A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
|- { B } C_ { A , B , C }
Theorem	snsstp3 3823	A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
|- { C } C_ { A , B , C }
Theorem	prsstp12 3824	A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.)
|- { A , B } C_ { A , B , C }
Theorem	prsstp13 3825	A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.)
|- { A , C } C_ { A , B , C }
Theorem	prsstp23 3826	A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.)
|- { B , C } C_ { A , B , C }
Theorem	prss 3827	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.)
|- A e. _V   &    |- B e. _V   =>    |- ( ( A e. C /\ B e. C ) <-> { A , B } C_ C )
Theorem	prssg 3828	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.)
|- ( ( A e. V /\ B e. W ) -> ( ( A e. C /\ B e. C ) <-> { A , B } C_ C ) )
Theorem	prssi 3829	A pair of elements of a class is a subset of the class. (Contributed by NM, 16-Jan-2015.)
|- ( ( A e. C /\ B e. C ) -> { A , B } C_ C )
Theorem	prssd 3830	Deduction version of prssi 3829: A pair of elements of a class is a subset of the class. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> A e. C )   &    |- ( ph -> B e. C )   =>    |- ( ph -> { A , B } C_ C )
Theorem	prsspwg 3831	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.)
|- ( ( A e. V /\ B e. W ) -> ( { A , B } C_ ~P C <-> ( A C_ C /\ B C_ C ) ) )
Theorem	ssprss 3832	A pair as subset of a pair. (Contributed by AV, 26-Oct-2020.)
|- ( ( A e. V /\ B e. W ) -> ( { A , B } C_ { C , D } <-> ( ( A = C \/ A = D ) /\ ( B = C \/ B = D ) ) ) )
Theorem	ssprsseq 3833	A proper pair is a subset of a pair iff it is equal to the superset. (Contributed by AV, 26-Oct-2020.)
|- ( ( A e. V /\ B e. W /\ A =/= B ) -> ( { A , B } C_ { C , D } <-> { A , B } = { C , D } ) )
Theorem	sssnr 3834	Empty set and the singleton itself are subsets of a singleton. Concerning the converse, see exmidsssn 4290. (Contributed by Jim Kingdon, 10-Aug-2018.)
|- ( ( A = (/) \/ A = { B } ) -> A C_ { B } )
Theorem	sssnm 3835*	The inhabited subset of a singleton. (Contributed by Jim Kingdon, 10-Aug-2018.)
|- ( E. x x e. A -> ( A C_ { B } <-> A = { B } ) )
Theorem	eqsnm 3836*	Two ways to express that an inhabited set equals a singleton. (Contributed by Jim Kingdon, 11-Aug-2018.)
|- ( E. x x e. A -> ( A = { B } <-> A. x e. A x = B ) )
Theorem	ssprr 3837	The subsets of a pair. (Contributed by Jim Kingdon, 11-Aug-2018.)
|- ( ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) -> A C_ { B , C } )
Theorem	sstpr 3838	The subsets of a triple. (Contributed by Jim Kingdon, 11-Aug-2018.)
|- ( ( ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) \/ ( ( A = { D } \/ A = { B , D } ) \/ ( A = { C , D } \/ A = { B , C , D } ) ) ) -> A C_ { B , C , D } )
Theorem	tpss 3839	A triplet of elements of a class is a subset of the class. (Contributed by NM, 9-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   =>    |- ( ( A e. D /\ B e. D /\ C e. D ) <-> { A , B , C } C_ D )
Theorem	tpssi 3840	A triple of elements of a class is a subset of the class. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
|- ( ( A e. D /\ B e. D /\ C e. D ) -> { A , B , C } C_ D )
Theorem	sneqr 3841	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.)
|- A e. _V   =>    |- ( { A } = { B } -> A = B )
Theorem	snsssn 3842	If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.)
|- A e. _V   =>    |- ( { A } C_ { B } -> A = B )
Theorem	sneqrg 3843	Closed form of sneqr 3841. (Contributed by Scott Fenton, 1-Apr-2011.)
|- ( A e. V -> ( { A } = { B } -> A = B ) )
Theorem	sneqbg 3844	Two singletons of sets are equal iff their elements are equal. (Contributed by Scott Fenton, 16-Apr-2012.)
|- ( A e. V -> ( { A } = { B } <-> A = B ) )
Theorem	snsspw 3845	The singleton of a class is a subset of its power class. (Contributed by NM, 5-Aug-1993.)
|- { A } C_ ~P A
Theorem	prsspw 3846	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.)
|- A e. _V   &    |- B e. _V   =>    |- ( { A , B } C_ ~P C <-> ( A C_ C /\ B C_ C ) )
Theorem	preqr1g 3847	Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. Closed form of preqr1 3849. (Contributed by Jim Kingdon, 21-Sep-2018.)
|- ( ( A e. _V /\ B e. _V ) -> ( { A , C } = { B , C } -> A = B ) )
Theorem	preqr2g 3848	Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the second elements are equal. Closed form of preqr2 3850. (Contributed by Jim Kingdon, 21-Sep-2018.)
|- ( ( A e. _V /\ B e. _V ) -> ( { C , A } = { C , B } -> A = B ) )
Theorem	preqr1 3849	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.)
|- A e. _V   &    |- B e. _V   =>    |- ( { A , C } = { B , C } -> A = B )
Theorem	preqr2 3850	Reverse equality lemma for unordered pairs. If two unordered pairs have the same first element, the second elements are equal. (Contributed by NM, 5-Aug-1993.)
|- A e. _V   &    |- B e. _V   =>    |- ( { C , A } = { C , B } -> A = B )
Theorem	preq12b 3851	Equality relationship for two unordered pairs. (Contributed by NM, 17-Oct-1996.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- D e. _V   =>    |- ( { A , B } = { C , D } <-> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) )
Theorem	prel12 3852	Equality of two unordered pairs. (Contributed by NM, 17-Oct-1996.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- D e. _V   =>    |- ( -. A = B -> ( { A , B } = { C , D } <-> ( A e. { C , D } /\ B e. { C , D } ) ) )
Theorem	opthpr 3853	A way to represent ordered pairs using unordered pairs with distinct members. (Contributed by NM, 27-Mar-2007.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- D e. _V   =>    |- ( A =/= D -> ( { A , B } = { C , D } <-> ( A = C /\ B = D ) ) )
Theorem	preq12bg 3854	Closed form of preq12b 3851. (Contributed by Scott Fenton, 28-Mar-2014.)
|- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) ) -> ( { A , B } = { C , D } <-> ( ( A = C /\ B = D ) \/ ( A = D /\ B = C ) ) ) )
Theorem	prneimg 3855	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.)
|- ( ( ( A e. U /\ B e. V ) /\ ( C e. X /\ D e. Y ) ) -> ( ( ( A =/= C /\ A =/= D ) \/ ( B =/= C /\ B =/= D ) ) -> { A , B } =/= { C , D } ) )
Theorem	preqsn 3856	Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   =>    |- ( { A , B } = { C } <-> ( A = B /\ B = C ) )
Theorem	elpr2elpr 3857*	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.)
|- ( ( X e. V /\ Y e. V /\ A e. { X , Y } ) -> E. b e. V { X , Y } = { A , b } )
Theorem	dfopg 3858	Value of the ordered pair when the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)
|- ( ( A e. V /\ B e. W ) -> <. A , B >. = { { A } , { A , B } } )
Theorem	dfop 3859	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.)
|- A e. _V   &    |- B e. _V   =>    |- <. A , B >. = { { A } , { A , B } }
Theorem	opeq1 3860	Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( A = B -> <. A , C >. = <. B , C >. )
Theorem	opeq2 3861	Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( A = B -> <. C , A >. = <. C , B >. )
Theorem	opeq12 3862	Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)
|- ( ( A = C /\ B = D ) -> <. A , B >. = <. C , D >. )
Theorem	opeq1i 3863	Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
|- A = B   =>    |- <. A , C >. = <. B , C >.
Theorem	opeq2i 3864	Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
|- A = B   =>    |- <. C , A >. = <. C , B >.
Theorem	opeq12i 3865	Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
|- A = B   &    |- C = D   =>    |- <. A , C >. = <. B , D >.
Theorem	opeq1d 3866	Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
|- ( ph -> A = B )   =>    |- ( ph -> <. A , C >. = <. B , C >. )
Theorem	opeq2d 3867	Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
|- ( ph -> A = B )   =>    |- ( ph -> <. C , A >. = <. C , B >. )
Theorem	opeq12d 3868	Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> <. A , C >. = <. B , D >. )
Theorem	oteq1 3869	Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
|- ( A = B -> <. A , C , D >. = <. B , C , D >. )
Theorem	oteq2 3870	Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
|- ( A = B -> <. C , A , D >. = <. C , B , D >. )
Theorem	oteq3 3871	Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
|- ( A = B -> <. C , D , A >. = <. C , D , B >. )
Theorem	oteq1d 3872	Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- ( ph -> A = B )   =>    |- ( ph -> <. A , C , D >. = <. B , C , D >. )
Theorem	oteq2d 3873	Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- ( ph -> A = B )   =>    |- ( ph -> <. C , A , D >. = <. C , B , D >. )
Theorem	oteq3d 3874	Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- ( ph -> A = B )   =>    |- ( ph -> <. C , D , A >. = <. C , D , B >. )
Theorem	oteq123d 3875	Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   &    |- ( ph -> E = F )   =>    |- ( ph -> <. A , C , E >. = <. B , D , F >. )
Theorem	nfop 3876	Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/_ x <. A , B >.
Theorem	nfopd 3877	Deduction version of bound-variable hypothesis builder nfop 3876. This shows how the deduction version of a not-free theorem such as nfop 3876 can be created from the corresponding not-free inference theorem. (Contributed by NM, 4-Feb-2008.)
|- ( ph -> F/_ x A )   &    |- ( ph -> F/_ x B )   =>    |- ( ph -> F/_ x <. A , B >. )
Theorem	opid 3878	The ordered pair <. A , A >. in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)
|- A e. _V   =>    |- <. A , A >. = { { A } }
Theorem	ralunsn 3879*	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.)
|- ( x = B -> ( ph <-> ps ) )   =>    |- ( B e. C -> ( A. x e. ( A u. { B } ) ph <-> ( A. x e. A ph /\ ps ) ) )
Theorem	2ralunsn 3880*	Double restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.)
|- ( x = B -> ( ph <-> ch ) )   &    |- ( y = B -> ( ph <-> ps ) )   &    |- ( x = B -> ( ps <-> th ) )   =>    |- ( 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 ) ) ) )
Theorem	opprc 3881	Expansion of an ordered pair when either member is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)
|- ( -. ( A e. _V /\ B e. _V ) -> <. A , B >. = (/) )
Theorem	opprc1 3882	Expansion of an ordered pair when the first member is a proper class. See also opprc 3881. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( -. A e. _V -> <. A , B >. = (/) )
Theorem	opprc2 3883	Expansion of an ordered pair when the second member is a proper class. See also opprc 3881. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( -. B e. _V -> <. A , B >. = (/) )
Theorem	oprcl 3884	If an ordered pair has an element, then its arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)
|- ( C e. <. A , B >. -> ( A e. _V /\ B e. _V ) )
Theorem	pwsnss 3885	The power set of a singleton. (Contributed by Jim Kingdon, 12-Aug-2018.)
|- { (/) , { A } } C_ ~P { A }
Theorem	pwpw0ss 3886	Compute the power set of the power set of the empty set. (See pw0 3818 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48 (but with subset in place of equality). (Contributed by Jim Kingdon, 12-Aug-2018.)
|- { (/) , { (/) } } C_ ~P { (/) }
Theorem	pwprss 3887	The power set of an unordered pair. (Contributed by Jim Kingdon, 13-Aug-2018.)
|- ( { (/) , { A } } u. { { B } , { A , B } } ) C_ ~P { A , B }
Theorem	pwtpss 3888	The power set of an unordered triple. (Contributed by Jim Kingdon, 13-Aug-2018.)
|- ( ( { (/) , { A } } u. { { B } , { A , B } } ) u. ( { { C } , { A , C } } u. { { B , C } , { A , B , C } } ) ) C_ ~P { A , B , C }
Theorem	pwpwpw0ss 3889	Compute the power set of the power set of the power set of the empty set. (See also pw0 3818 and pwpw0ss 3886.) (Contributed by Jim Kingdon, 13-Aug-2018.)
|- ( { (/) , { (/) } } u. { { { (/) } } , { (/) , { (/) } } } ) C_ ~P { (/) , { (/) } }
Theorem	pwv 3890	The power class of the universe is the universe. Exercise 4.12(d) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)
|- ~P _V = _V
2.1.18  The union of a class
Syntax	cuni 3891	Extend class notation to include the union of a class. Read: "union (of) A".
class U. A
Definition	df-uni 3892*	Define the union of a class i.e. the collection of all members of the members of the class. Definition 5.5 of [TakeutiZaring] p. 16. For example, U. { { 1 , 3 } , { 1 , 8 } } = { 1 , 3 , 8 }. This is similar to the union of two classes df-un 3202. (Contributed by NM, 23-Aug-1993.)
|- U. A = { x | E. y ( x e. y /\ y e. A ) }
Theorem	dfuni2 3893*	Alternate definition of class union. (Contributed by NM, 28-Jun-1998.)
|- U. A = { x | E. y e. A x e. y }
Theorem	eluni 3894*	Membership in class union. (Contributed by NM, 22-May-1994.)
|- ( A e. U. B <-> E. x ( A e. x /\ x e. B ) )
Theorem	eluni2 3895*	Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999.)
|- ( A e. U. B <-> E. x e. B A e. x )
Theorem	elunii 3896	Membership in class union. (Contributed by NM, 24-Mar-1995.)
|- ( ( A e. B /\ B e. C ) -> A e. U. C )
Theorem	nfuni 3897	Bound-variable hypothesis builder for union. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- F/_ x A   =>    |- F/_ x U. A
Theorem	nfunid 3898	Deduction version of nfuni 3897. (Contributed by NM, 18-Feb-2013.)
|- ( ph -> F/_ x A )   =>    |- ( ph -> F/_ x U. A )
Theorem	csbunig 3899	Distribute proper substitution through the union of a class. (Contributed by Alan Sare, 10-Nov-2012.)
|- ( A e. V -> [_ A / x ]_ U. B = U. [_ A / x ]_ B )
Theorem	unieq 3900	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.)
|- ( A = B -> U. A = U. B )