P. 40 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 3901-4000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	opeq12 3901	Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)
|- ( ( A = C /\ B = D ) -> <. A , B >. = <. C , D >. )
Theorem	opeq1i 3902	Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
|- A = B   =>    |- <. A , C >. = <. B , C >.
Theorem	opeq2i 3903	Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
|- A = B   =>    |- <. C , A >. = <. C , B >.
Theorem	opeq12i 3904	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 3905	Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
|- ( ph -> A = B )   =>    |- ( ph -> <. A , C >. = <. B , C >. )
Theorem	opeq2d 3906	Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
|- ( ph -> A = B )   =>    |- ( ph -> <. C , A >. = <. C , B >. )
Theorem	opeq12d 3907	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 3908	Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
|- ( A = B -> <. A , C , D >. = <. B , C , D >. )
Theorem	oteq2 3909	Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
|- ( A = B -> <. C , A , D >. = <. C , B , D >. )
Theorem	oteq3 3910	Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
|- ( A = B -> <. C , D , A >. = <. C , D , B >. )
Theorem	oteq1d 3911	Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- ( ph -> A = B )   =>    |- ( ph -> <. A , C , D >. = <. B , C , D >. )
Theorem	oteq2d 3912	Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- ( ph -> A = B )   =>    |- ( ph -> <. C , A , D >. = <. C , B , D >. )
Theorem	oteq3d 3913	Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
|- ( ph -> A = B )   =>    |- ( ph -> <. C , D , A >. = <. C , D , B >. )
Theorem	oteq123d 3914	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 3915	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 3916	Deduction version of bound-variable hypothesis builder nfop 3915. This shows how the deduction version of a not-free theorem such as nfop 3915 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 3917	The ordered pair <. A , A >. in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)
|- A e. _V   =>    |- <. A , A >. = { { A } }
Theorem	ralunsn 3918*	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 3919*	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 3920	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 3921	Expansion of an ordered pair when the first member is a proper class. See also opprc 3920. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( -. A e. _V -> <. A , B >. = (/) )
Theorem	opprc2 3922	Expansion of an ordered pair when the second member is a proper class. See also opprc 3920. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( -. B e. _V -> <. A , B >. = (/) )
Theorem	oprcl 3923	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 3924	The power set of a singleton. (Contributed by Jim Kingdon, 12-Aug-2018.)
|- { (/) , { A } } C_ ~P { A }
Theorem	pwpw0ss 3925	Compute the power set of the power set of the empty set. (See pw0 3857 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 3926	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 3927	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 3928	Compute the power set of the power set of the power set of the empty set. (See also pw0 3857 and pwpw0ss 3925.) (Contributed by Jim Kingdon, 13-Aug-2018.)
|- ( { (/) , { (/) } } u. { { { (/) } } , { (/) , { (/) } } } ) C_ ~P { (/) , { (/) } }
Theorem	pwv 3929	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 3930	Extend class notation to include the union of a class. Read: "union (of) A".
class U. A
Definition	df-uni 3931*	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 3224. (Contributed by NM, 23-Aug-1993.)
|- U. A = { x | E. y ( x e. y /\ y e. A ) }
Theorem	dfuni2 3932*	Alternate definition of class union. (Contributed by NM, 28-Jun-1998.)
|- U. A = { x | E. y e. A x e. y }
Theorem	eluni 3933*	Membership in class union. (Contributed by NM, 22-May-1994.)
|- ( A e. U. B <-> E. x ( A e. x /\ x e. B ) )
Theorem	eluni2 3934*	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 3935	Membership in class union. (Contributed by NM, 24-Mar-1995.)
|- ( ( A e. B /\ B e. C ) -> A e. U. C )
Theorem	nfuni 3936	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 3937	Deduction version of nfuni 3936. (Contributed by NM, 18-Feb-2013.)
|- ( ph -> F/_ x A )   =>    |- ( ph -> F/_ x U. A )
Theorem	csbunig 3938	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 3939	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 3940	Inference of equality of two class unions. (Contributed by NM, 30-Aug-1993.)
|- A = B   =>    |- U. A = U. B
Theorem	unieqd 3941	Deduction of equality of two class unions. (Contributed by NM, 21-Apr-1995.)
|- ( ph -> A = B )   =>    |- ( ph -> U. A = U. B )
Theorem	eluniab 3942*	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 3943*	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 3944	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 3945	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 3946	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 3947	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 3948*	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 3949	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 3950	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 3951	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 3952	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 3953	Subclass relationship for subclass union. Inference form of uniss 3951. (Contributed by David Moews, 1-May-2017.)
|- A C_ B   =>    |- U. A C_ U. B
Theorem	unissd 3954	Subclass relationship for subclass union. Deduction form of uniss 3951. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A C_ B )   =>    |- ( ph -> U. A C_ U. B )
Theorem	uni0b 3955	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 3956*	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 3957	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 3958	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 3959	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 3960*	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 3961*	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 3962*	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 3963*	Relationship implying union. (Contributed by NM, 10-Nov-1999.)
|- ( ( A e. B /\ A. x e. B x C_ A ) -> A = U. B )
Theorem	unimax 3964*	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 3965	Extend class notation to include the intersection of a class. Read: "intersection (of) A".
class |^| A
Definition	df-int 3966*	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 3226. (Contributed by NM, 18-Aug-1993.)
|- |^| A = { x | A. y ( y e. A -> x e. y ) }
Theorem	dfint2 3967*	Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998.)
|- |^| A = { x | A. y e. A x e. y }
Theorem	inteq 3968	Equality law for intersection. (Contributed by NM, 13-Sep-1999.)
|- ( A = B -> |^| A = |^| B )
Theorem	inteqi 3969	Equality inference for class intersection. (Contributed by NM, 2-Sep-2003.)
|- A = B   =>    |- |^| A = |^| B
Theorem	inteqd 3970	Equality deduction for class intersection. (Contributed by NM, 2-Sep-2003.)
|- ( ph -> A = B )   =>    |- ( ph -> |^| A = |^| B )
Theorem	elint 3971*	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 3972*	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 3973*	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 3974	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 3975	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 3976*	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 3977*	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 3978*	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 3979	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 3980	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 3981*	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 3982*	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 3983*	Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)
|- A C_ |^| { x e. B | A C_ x }
Theorem	ssmin 3984*	Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.)
|- A C_ |^| { x | ( A C_ x /\ ph ) }
Theorem	intmin 3985*	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 3986	Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)
|- ( A C_ B -> |^| B C_ |^| A )
Theorem	intssunim 3987*	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 3988*	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 3989*	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 3990*	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 3991*	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 3992*	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 3993*	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 3994*	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 3995	The intersection of a class containing the empty set is empty. (Contributed by NM, 24-Apr-2004.)
|- ( (/) e. A -> |^| A = (/) )
Theorem	intun 3996	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 3997	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 3998	The intersection of a pair is the intersection of its members. Closed form of intpr 3997. 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 3999	Intersection of a singleton. (Contributed by Stefan O'Rear, 22-Feb-2015.)
|- ( A e. V -> |^| { A } = A )
Theorem	intsn 4000	The intersection of a singleton is its member. Theorem 70 of [Suppes] p. 41. (Contributed by NM, 29-Sep-2002.)
|- A e. _V   =>    |- |^| { A } = A