P. 38 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 3701-3800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	opeq2 3701	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 3702	Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)
|- ( ( A = C /\ B = D ) -> <. A , B >. = <. C , D >. )
Theorem	opeq1i 3703	Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
|- A = B   =>    |- <. A , C >. = <. B , C >.
Theorem	opeq2i 3704	Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
|- A = B   =>    |- <. C , A >. = <. C , B >.
Theorem	opeq12i 3705	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 3706	Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
|- ( ph -> A = B )   =>    |- ( ph -> <. A , C >. = <. B , C >. )
Theorem	opeq2d 3707	Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
|- ( ph -> A = B )   =>    |- ( ph -> <. C , A >. = <. C , B >. )
Theorem	opeq12d 3708	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 3709	Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
|- ( A = B -> <. A , C , D >. = <. B , C , D >. )
Theorem	oteq2 3710	Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
|- ( A = B -> <. C , A , D >. = <. C , B , D >. )
Theorem	oteq3 3711	Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
|- ( A = B -> <. C , D , A >. = <. C , D , B >. )
Theorem	oteq1d 3712	Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- ( ph -> A = B )   =>    |- ( ph -> <. A , C , D >. = <. B , C , D >. )
Theorem	oteq2d 3713	Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- ( ph -> A = B )   =>    |- ( ph -> <. C , A , D >. = <. C , B , D >. )
Theorem	oteq3d 3714	Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- ( ph -> A = B )   =>    |- ( ph -> <. C , D , A >. = <. C , D , B >. )
Theorem	oteq123d 3715	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 3716	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 3717	Deduction version of bound-variable hypothesis builder nfop 3716. This shows how the deduction version of a not-free theorem such as nfop 3716 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 3718	The ordered pair <. A , A >. in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)
|- A e. _V   =>    |- <. A , A >. = { { A } }
Theorem	ralunsn 3719*	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 3720*	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 3721	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 3722	Expansion of an ordered pair when the first member is a proper class. See also opprc 3721. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( -. A e. _V -> <. A , B >. = (/) )
Theorem	opprc2 3723	Expansion of an ordered pair when the second member is a proper class. See also opprc 3721. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( -. B e. _V -> <. A , B >. = (/) )
Theorem	oprcl 3724	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 3725	The power set of a singleton. (Contributed by Jim Kingdon, 12-Aug-2018.)
|- { (/) , { A } } C_ ~P { A }
Theorem	pwpw0ss 3726	Compute the power set of the power set of the empty set. (See pw0 3662 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 3727	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 3728	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 3729	Compute the power set of the power set of the power set of the empty set. (See also pw0 3662 and pwpw0ss 3726.) (Contributed by Jim Kingdon, 13-Aug-2018.)
|- ( { (/) , { (/) } } u. { { { (/) } } , { (/) , { (/) } } } ) C_ ~P { (/) , { (/) } }
Theorem	pwv 3730	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 3731	Extend class notation to include the union of a class (read: 'union A')
class U. A
Definition	df-uni 3732*	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, { { 1 , 3 } , { 1 , 8 } } = { 1 , 3 , 8 } . This is similar to the union of two classes df-un 3070. (Contributed by NM, 23-Aug-1993.)
|- U. A = { x | E. y ( x e. y /\ y e. A ) }
Theorem	dfuni2 3733*	Alternate definition of class union. (Contributed by NM, 28-Jun-1998.)
|- U. A = { x | E. y e. A x e. y }
Theorem	eluni 3734*	Membership in class union. (Contributed by NM, 22-May-1994.)
|- ( A e. U. B <-> E. x ( A e. x /\ x e. B ) )
Theorem	eluni2 3735*	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 3736	Membership in class union. (Contributed by NM, 24-Mar-1995.)
|- ( ( A e. B /\ B e. C ) -> A e. U. C )
Theorem	nfuni 3737	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 3738	Deduction version of nfuni 3737. (Contributed by NM, 18-Feb-2013.)
|- ( ph -> F/_ x A )   =>    |- ( ph -> F/_ x U. A )
Theorem	csbunig 3739	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 3740	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 )
Theorem	unieqi 3741	Inference of equality of two class unions. (Contributed by NM, 30-Aug-1993.)
|- A = B   =>    |- U. A = U. B
Theorem	unieqd 3742	Deduction of equality of two class unions. (Contributed by NM, 21-Apr-1995.)
|- ( ph -> A = B )   =>    |- ( ph -> U. A = U. B )
Theorem	eluniab 3743*	Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.)
|- ( A e. U. { x | ph } <-> E. x ( A e. x /\ ph ) )
Theorem	elunirab 3744*	Membership in union of a class abstraction. (Contributed by NM, 4-Oct-2006.)
|- ( A e. U. { x e. B | ph } <-> E. x e. B ( A e. x /\ ph ) )
Theorem	unipr 3745	The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 23-Aug-1993.)
|- A e. _V   &    |- B e. _V   =>    |- U. { A , B } = ( A u. B )
Theorem	uniprg 3746	The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 25-Aug-2006.)
|- ( ( A e. V /\ B e. W ) -> U. { A , B } = ( A u. B ) )
Theorem	unisn 3747	A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 30-Aug-1993.)
|- A e. _V   =>    |- U. { A } = A
Theorem	unisng 3748	A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 13-Aug-2002.)
|- ( A e. V -> U. { A } = A )
Theorem	dfnfc2 3749*	An alternate statement of the effective freeness of a class A, when it is a set. (Contributed by Mario Carneiro, 14-Oct-2016.)
|- ( A. x A e. V -> ( F/_ x A <-> A. y F/ x y = A ) )
Theorem	uniun 3750	The class union of the union of two classes. Theorem 8.3 of [Quine] p. 53. (Contributed by NM, 20-Aug-1993.)
|- U. ( A u. B ) = ( U. A u. U. B )
Theorem	uniin 3751	The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. (Contributed by NM, 4-Dec-2003.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- U. ( A i^i B ) C_ ( U. A i^i U. B )
Theorem	uniss 3752	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.)
|- ( A C_ B -> U. A C_ U. B )
Theorem	ssuni 3753	Subclass relationship for class union. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- ( ( A C_ B /\ B e. C ) -> A C_ U. C )
Theorem	unissi 3754	Subclass relationship for subclass union. Inference form of uniss 3752. (Contributed by David Moews, 1-May-2017.)
|- A C_ B   =>    |- U. A C_ U. B
Theorem	unissd 3755	Subclass relationship for subclass union. Deduction form of uniss 3752. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A C_ B )   =>    |- ( ph -> U. A C_ U. B )
Theorem	uni0b 3756	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.)
|- ( U. A = (/) <-> A C_ { (/) } )
Theorem	uni0c 3757*	The union of a set is empty iff all of its members are empty. (Contributed by NM, 16-Aug-2006.)
|- ( U. A = (/) <-> A. x e. A x = (/) )
Theorem	uni0 3758	The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Reproved without relying on ax-nul by Eric Schmidt.) (Contributed by NM, 16-Sep-1993.) (Revised by Eric Schmidt, 4-Apr-2007.)
|- U. (/) = (/)
Theorem	elssuni 3759	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.)
|- ( A e. B -> A C_ U. B )
Theorem	unissel 3760	Condition turning a subclass relationship for union into an equality. (Contributed by NM, 18-Jul-2006.)
|- ( ( U. A C_ B /\ B e. A ) -> U. A = B )
Theorem	unissb 3761*	Relationship involving membership, subset, and union. Exercise 5 of [Enderton] p. 26 and its converse. (Contributed by NM, 20-Sep-2003.)
|- ( U. A C_ B <-> A. x e. A x C_ B )
Theorem	uniss2 3762*	A subclass condition on the members of two classes that implies a subclass relation on their unions. Proposition 8.6 of [TakeutiZaring] p. 59. (Contributed by NM, 22-Mar-2004.)
|- ( A. x e. A E. y e. B x C_ y -> U. A C_ U. B )
Theorem	unidif 3763*	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.)
|- ( A. x e. A E. y e. ( A \ B ) x C_ y -> U. ( A \ B ) = U. A )
Theorem	ssunieq 3764*	Relationship implying union. (Contributed by NM, 10-Nov-1999.)
|- ( ( A e. B /\ A. x e. B x C_ A ) -> A = U. B )
Theorem	unimax 3765*	Any member of a class is the largest of those members that it includes. (Contributed by NM, 13-Aug-2002.)
|- ( A e. B -> U. { x e. B | x C_ A } = A )
2.1.19  The intersection of a class
Syntax	cint 3766	Extend class notation to include the intersection of a class (read: 'intersect A').
class |^| A
Definition	df-int 3767*	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 3072. (Contributed by NM, 18-Aug-1993.)
|- |^| A = { x | A. y ( y e. A -> x e. y ) }
Theorem	dfint2 3768*	Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998.)
|- |^| A = { x | A. y e. A x e. y }
Theorem	inteq 3769	Equality law for intersection. (Contributed by NM, 13-Sep-1999.)
|- ( A = B -> |^| A = |^| B )
Theorem	inteqi 3770	Equality inference for class intersection. (Contributed by NM, 2-Sep-2003.)
|- A = B   =>    |- |^| A = |^| B
Theorem	inteqd 3771	Equality deduction for class intersection. (Contributed by NM, 2-Sep-2003.)
|- ( ph -> A = B )   =>    |- ( ph -> |^| A = |^| B )
Theorem	elint 3772*	Membership in class intersection. (Contributed by NM, 21-May-1994.)
|- A e. _V   =>    |- ( A e. |^| B <-> A. x ( x e. B -> A e. x ) )
Theorem	elint2 3773*	Membership in class intersection. (Contributed by NM, 14-Oct-1999.)
|- A e. _V   =>    |- ( A e. |^| B <-> A. x e. B A e. x )
Theorem	elintg 3774*	Membership in class intersection, with the sethood requirement expressed as an antecedent. (Contributed by NM, 20-Nov-2003.)
|- ( A e. V -> ( A e. |^| B <-> A. x e. B A e. x ) )
Theorem	elinti 3775	Membership in class intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- ( A e. |^| B -> ( C e. B -> A e. C ) )
Theorem	nfint 3776	Bound-variable hypothesis builder for intersection. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
|- F/_ x A   =>    |- F/_ x |^| A
Theorem	elintab 3777*	Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993.)
|- A e. _V   =>    |- ( A e. |^| { x | ph } <-> A. x ( ph -> A e. x ) )
Theorem	elintrab 3778*	Membership in the intersection of a class abstraction. (Contributed by NM, 17-Oct-1999.)
|- A e. _V   =>    |- ( A e. |^| { x e. B | ph } <-> A. x e. B ( ph -> A e. x ) )
Theorem	elintrabg 3779*	Membership in the intersection of a class abstraction. (Contributed by NM, 17-Feb-2007.)
|- ( A e. V -> ( A e. |^| { x e. B | ph } <-> A. x e. B ( ph -> A e. x ) ) )
Theorem	int0 3780	The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.)
|- |^| (/) = _V
Theorem	intss1 3781	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.)
|- ( A e. B -> |^| B C_ A )
Theorem	ssint 3782*	Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse. (Contributed by NM, 14-Oct-1999.)
|- ( A C_ |^| B <-> A. x e. B A C_ x )
Theorem	ssintab 3783*	Subclass of the intersection of a class abstraction. (Contributed by NM, 31-Jul-2006.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- ( A C_ |^| { x | ph } <-> A. x ( ph -> A C_ x ) )
Theorem	ssintub 3784*	Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)
|- A C_ |^| { x e. B | A C_ x }
Theorem	ssmin 3785*	Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.)
|- A C_ |^| { x | ( A C_ x /\ ph ) }
Theorem	intmin 3786*	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.)
|- ( A e. B -> |^| { x e. B | A C_ x } = A )
Theorem	intss 3787	Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)
|- ( A C_ B -> |^| B C_ |^| A )
Theorem	intssunim 3788*	The intersection of an inhabited set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)
|- ( E. x x e. A -> |^| A C_ U. A )
Theorem	ssintrab 3789*	Subclass of the intersection of a restricted class builder. (Contributed by NM, 30-Jan-2015.)
|- ( A C_ |^| { x e. B | ph } <-> A. x e. B ( ph -> A C_ x ) )
Theorem	intssuni2m 3790*	Subclass relationship for intersection and union. (Contributed by Jim Kingdon, 14-Aug-2018.)
|- ( ( A C_ B /\ E. x x e. A ) -> |^| A C_ U. B )
Theorem	intminss 3791*	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.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( A e. B /\ ps ) -> |^| { x e. B | ph } C_ A )
Theorem	intmin2 3792*	Any set is the smallest of all sets that include it. (Contributed by NM, 20-Sep-2003.)
|- A e. _V   =>    |- |^| { x | A C_ x } = A
Theorem	intmin3 3793*	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.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ps   =>    |- ( A e. V -> |^| { x | ph } C_ A )
Theorem	intmin4 3794*	Elimination of a conjunct in a class intersection. (Contributed by NM, 31-Jul-2006.)
|- ( A C_ |^| { x | ph } -> |^| { x | ( A C_ x /\ ph ) } = |^| { x | ph } )
Theorem	intab 3795*	The intersection of a special case of a class abstraction. y may be free in ph and A, which can be thought of a ph ( y ) and A ( y ). (Contributed by NM, 28-Jul-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
|- A e. _V   &    |- { x | E. y ( ph /\ x = A ) } e. _V   =>    |- |^| { x | A. y ( ph -> A e. x ) } = { x | E. y ( ph /\ x = A ) }
Theorem	int0el 3796	The intersection of a class containing the empty set is empty. (Contributed by NM, 24-Apr-2004.)
|- ( (/) e. A -> |^| A = (/) )
Theorem	intun 3797	The class intersection of the union of two classes. Theorem 78 of [Suppes] p. 42. (Contributed by NM, 22-Sep-2002.)
|- |^| ( A u. B ) = ( |^| A i^i |^| B )
Theorem	intpr 3798	The intersection of a pair is the intersection of its members. Theorem 71 of [Suppes] p. 42. (Contributed by NM, 14-Oct-1999.)
|- A e. _V   &    |- B e. _V   =>    |- |^| { A , B } = ( A i^i B )
Theorem	intprg 3799	The intersection of a pair is the intersection of its members. Closed form of intpr 3798. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.)
|- ( ( A e. V /\ B e. W ) -> |^| { A , B } = ( A i^i B ) )
Theorem	intsng 3800	Intersection of a singleton. (Contributed by Stefan O'Rear, 22-Feb-2015.)
|- ( A e. V -> |^| { A } = A )