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	ssiun2s 3701*	Subset relationship for an indexed union. (Contributed by NM, 26-Oct-2003.)
|- ( x = C -> B = D )   =>    |- ( C e. A -> D C_ U_ x e. A B )
Theorem	iunss2 3702*	A subclass condition on the members of two indexed classes C ( x ) and D ( y ) that implies a subclass relation on their indexed unions. Generalization of Proposition 8.6 of [TakeutiZaring] p. 59. Compare uniss2 3611. (Contributed by NM, 9-Dec-2004.)
|- ( A. x e. A E. y e. B C C_ D -> U_ x e. A C C_ U_ y e. B D )
Theorem	iunab 3703*	The indexed union of a class abstraction. (Contributed by NM, 27-Dec-2004.)
|- U_ x e. A { y | ph } = { y | E. x e. A ph }
Theorem	iunrab 3704*	The indexed union of a restricted class abstraction. (Contributed by NM, 3-Jan-2004.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
|- U_ x e. A { y e. B | ph } = { y e. B | E. x e. A ph }
Theorem	iunxdif2 3705*	Indexed union with a class difference as its index. (Contributed by NM, 10-Dec-2004.)
|- ( x = y -> C = D )   =>    |- ( A. x e. A E. y e. ( A \ B ) C C_ D -> U_ y e. ( A \ B ) D = U_ x e. A C )
Theorem	ssiinf 3706	Subset theorem for an indexed intersection. (Contributed by FL, 15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.)
|- F/_ x C   =>    |- ( C C_ |^|_ x e. A B <-> A. x e. A C C_ B )
Theorem	ssiin 3707*	Subset theorem for an indexed intersection. (Contributed by NM, 15-Oct-2003.)
|- ( C C_ |^|_ x e. A B <-> A. x e. A C C_ B )
Theorem	iinss 3708*	Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- ( E. x e. A B C_ C -> |^|_ x e. A B C_ C )
Theorem	iinss2 3709	An indexed intersection is included in any of its members. (Contributed by FL, 15-Oct-2012.)
|- ( x e. A -> |^|_ x e. A B C_ B )
Theorem	uniiun 3710*	Class union in terms of indexed union. Definition in [Stoll] p. 43. (Contributed by NM, 28-Jun-1998.)
|- U. A = U_ x e. A x
Theorem	intiin 3711*	Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)
|- |^| A = |^|_ x e. A x
Theorem	iunid 3712*	An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)
|- U_ x e. A { x } = A
Theorem	iun0 3713	An indexed union of the empty set is empty. (Contributed by NM, 26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- U_ x e. A (/) = (/)
Theorem	0iun 3714	An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- U_ x e. (/) A = (/)
Theorem	0iin 3715	An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)
|- |^|_ x e. (/) A = _V
Theorem	viin 3716*	Indexed intersection with a universal index class. (Contributed by NM, 11-Sep-2008.)
|- |^|_ x e. _V A = { y | A. x y e. A }
Theorem	iunn0m 3717*	There is an inhabited class in an indexed collection B ( x ) iff the indexed union of them is inhabited. (Contributed by Jim Kingdon, 16-Aug-2018.)
|- ( E. x e. A E. y y e. B <-> E. y y e. U_ x e. A B )
Theorem	iinab 3718*	Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.)
|- |^|_ x e. A { y | ph } = { y | A. x e. A ph }
Theorem	iinrabm 3719*	Indexed intersection of a restricted class builder. (Contributed by Jim Kingdon, 16-Aug-2018.)
|- ( E. x x e. A -> |^|_ x e. A { y e. B | ph } = { y e. B | A. x e. A ph } )
Theorem	iunin2 3720*	Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3710 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.)
|- U_ x e. A ( B i^i C ) = ( B i^i U_ x e. A C )
Theorem	iunin1 3721*	Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3710 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)
|- U_ x e. A ( C i^i B ) = ( U_ x e. A C i^i B )
Theorem	iundif2ss 3722*	Indexed union of class difference. Compare to theorem "De Morgan's laws" in [Enderton] p. 31. (Contributed by Jim Kingdon, 17-Aug-2018.)
|- U_ x e. A ( B \ C ) C_ ( B \ |^|_ x e. A C )
Theorem	2iunin 3723*	Rearrange indexed unions over intersection. (Contributed by NM, 18-Dec-2008.)
|- U_ x e. A U_ y e. B ( C i^i D ) = ( U_ x e. A C i^i U_ y e. B D )
Theorem	iindif2m 3724*	Indexed intersection of class difference. Compare to Theorem "De Morgan's laws" in [Enderton] p. 31. (Contributed by Jim Kingdon, 17-Aug-2018.)
|- ( E. x x e. A -> |^|_ x e. A ( B \ C ) = ( B \ U_ x e. A C ) )
Theorem	iinin2m 3725*	Indexed intersection of intersection. Compare to Theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Jim Kingdon, 17-Aug-2018.)
|- ( E. x x e. A -> |^|_ x e. A ( B i^i C ) = ( B i^i |^|_ x e. A C ) )
Theorem	iinin1m 3726*	Indexed intersection of intersection. Compare to Theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Jim Kingdon, 17-Aug-2018.)
|- ( E. x x e. A -> |^|_ x e. A ( C i^i B ) = ( |^|_ x e. A C i^i B ) )
Theorem	elriin 3727*	Elementhood in a relative intersection. (Contributed by Mario Carneiro, 30-Dec-2016.)
|- ( B e. ( A i^i |^|_ x e. X S ) <-> ( B e. A /\ A. x e. X B e. S ) )
Theorem	riin0 3728*	Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- ( X = (/) -> ( A i^i |^|_ x e. X S ) = A )
Theorem	riinm 3729*	Relative intersection of an inhabited family. (Contributed by Jim Kingdon, 19-Aug-2018.)
|- ( ( A. x e. X S C_ A /\ E. x x e. X ) -> ( A i^i |^|_ x e. X S ) = |^|_ x e. X S )
Theorem	iinxsng 3730*	A singleton index picks out an instance of an indexed intersection's argument. (Contributed by NM, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
|- ( x = A -> B = C )   =>    |- ( A e. V -> |^|_ x e. { A } B = C )
Theorem	iinxprg 3731*	Indexed intersection with an unordered pair index. (Contributed by NM, 25-Jan-2012.)
|- ( x = A -> C = D )   &    |- ( x = B -> C = E )   =>    |- ( ( A e. V /\ B e. W ) -> |^|_ x e. { A , B } C = ( D i^i E ) )
Theorem	iunxsng 3732*	A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.)
|- ( x = A -> B = C )   =>    |- ( A e. V -> U_ x e. { A } B = C )
Theorem	iunxsn 3733*	A singleton index picks out an instance of an indexed union's argument. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 25-Jun-2016.)
|- A e. _V   &    |- ( x = A -> B = C )   =>    |- U_ x e. { A } B = C
Theorem	iunun 3734	Separate a union in an indexed union. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
|- U_ x e. A ( B u. C ) = ( U_ x e. A B u. U_ x e. A C )
Theorem	iunxun 3735	Separate a union in the index of an indexed union. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
|- U_ x e. ( A u. B ) C = ( U_ x e. A C u. U_ x e. B C )
Theorem	iunxiun 3736*	Separate an indexed union in the index of an indexed union. (Contributed by Mario Carneiro, 5-Dec-2016.)
|- U_ x e. U_ y e. A B C = U_ y e. A U_ x e. B C
Theorem	iinuniss 3737*	A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33 but with equality changed to subset. (Contributed by Jim Kingdon, 19-Aug-2018.)
|- ( A u. |^| B ) C_ |^|_ x e. B ( A u. x )
Theorem	iununir 3738*	A relationship involving union and indexed union. Exercise 25 of [Enderton] p. 33 but with biconditional changed to implication. (Contributed by Jim Kingdon, 19-Aug-2018.)
|- ( ( A u. U. B ) = U_ x e. B ( A u. x ) -> ( B = (/) -> A = (/) ) )
Theorem	sspwuni 3739	Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
|- ( A C_ ~P B <-> U. A C_ B )
Theorem	pwssb 3740*	Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006.)
|- ( A C_ ~P B <-> A. x e. A x C_ B )
Theorem	elpwuni 3741	Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
|- ( B e. A -> ( A C_ ~P B <-> U. A = B ) )
Theorem	iinpw 3742*	The power class of an intersection in terms of indexed intersection. Exercise 24(a) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.)
|- ~P |^| A = |^|_ x e. A ~P x
Theorem	iunpwss 3743*	Inclusion of an indexed union of a power class in the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.)
|- U_ x e. A ~P x C_ ~P U. A
Theorem	rintm 3744*	Relative intersection of an inhabited class. (Contributed by Jim Kingdon, 19-Aug-2018.)
|- ( ( X C_ ~P A /\ E. x x e. X ) -> ( A i^i |^| X ) = |^| X )
2.1.21  Disjointness
Syntax	wdisj 3745	Extend wff notation to include the statement that a family of classes B ( x ), for x e. A, is a disjoint family.
wff Disj x e. A B
Definition	df-disj 3746*	A collection of classes B ( x ) is disjoint when for each element y, it is in B ( x ) for at most one x. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by NM, 16-Jun-2017.)
|- (Disj x e. A B <-> A. y E* x e. A y e. B )
Theorem	dfdisj2 3747*	Alternate definition for disjoint classes. (Contributed by NM, 17-Jun-2017.)
|- (Disj x e. A B <-> A. y E* x ( x e. A /\ y e. B ) )
Theorem	disjss2 3748	If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- ( A. x e. A B C_ C -> (Disj x e. A C -> Disj x e. A B ) )
Theorem	disjeq2 3749	Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- ( A. x e. A B = C -> (Disj x e. A B <-> Disj x e. A C ) )
Theorem	disjeq2dv 3750*	Equality deduction for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- ( ( ph /\ x e. A ) -> B = C )   =>    |- ( ph -> (Disj x e. A B <-> Disj x e. A C ) )
Theorem	disjss1 3751*	A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- ( A C_ B -> (Disj x e. B C -> Disj x e. A C ) )
Theorem	disjeq1 3752*	Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- ( A = B -> (Disj x e. A C <-> Disj x e. B C ) )
Theorem	disjeq1d 3753*	Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- ( ph -> A = B )   =>    |- ( ph -> (Disj x e. A C <-> Disj x e. B C ) )
Theorem	disjeq12d 3754*	Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> (Disj x e. A C <-> Disj x e. B D ) )
Theorem	cbvdisj 3755*	Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- F/_ y B   &    |- F/_ x C   &    |- ( x = y -> B = C )   =>    |- (Disj x e. A B <-> Disj y e. A C )
Theorem	cbvdisjv 3756*	Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 11-Dec-2016.)
|- ( x = y -> B = C )   =>    |- (Disj x e. A B <-> Disj y e. A C )
Theorem	nfdisjv 3757*	Bound-variable hypothesis builder for disjoint collection. (Contributed by Jim Kingdon, 19-Aug-2018.)
|- F/_ y A   &    |- F/_ y B   =>    |- F/ yDisj x e. A B
Theorem	nfdisj1 3758	Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- F/ xDisj x e. A B
Theorem	invdisj 3759*	If there is a function C ( y ) such that C ( y ) = x for all y e. B ( x ), then the sets B ( x ) for distinct x e. A are disjoint. (Contributed by Mario Carneiro, 10-Dec-2016.)
|- ( A. x e. A A. y e. B C = x -> Disj x e. A B )
Theorem	sndisj 3760	Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- Disj x e. A { x }
Theorem	0disj 3761	Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- Disj x e. A (/)
Theorem	disjxsn 3762*	A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- Disj x e. { A } B
Theorem	disjx0 3763	An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- Disj x e. (/) B
2.1.22  Binary relations
Syntax	wbr 3764	Extend wff notation to include the general binary relation predicate. Note that the syntax is simply three class symbols in a row. Since binary relations are the only possible wff expressions consisting of three class expressions in a row, the syntax is unambiguous.
wff A R B
Definition	df-br 3765	Define a general binary relation. Note that the syntax is simply three class symbols in a row. Definition 6.18 of [TakeutiZaring] p. 29 generalized to arbitrary classes. This definition of relations is well-defined, although not very meaningful, when classes A and/or B are proper classes (i.e. are not sets). On the other hand, we often find uses for this definition when R is a proper class (see for example iprc 4600). (Contributed by NM, 31-Dec-1993.)
|- ( A R B <-> <. A , B >. e. R )
Theorem	breq 3766	Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)
|- ( R = S -> ( A R B <-> A S B ) )
Theorem	breq1 3767	Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
|- ( A = B -> ( A R C <-> B R C ) )
Theorem	breq2 3768	Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
|- ( A = B -> ( C R A <-> C R B ) )
Theorem	breq12 3769	Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)
|- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )
Theorem	breqi 3770	Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.)
|- R = S   =>    |- ( A R B <-> A S B )
Theorem	breq1i 3771	Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
|- A = B   =>    |- ( A R C <-> B R C )
Theorem	breq2i 3772	Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
|- A = B   =>    |- ( C R A <-> C R B )
Theorem	breq12i 3773	Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
|- A = B   &    |- C = D   =>    |- ( A R C <-> B R D )
Theorem	breq1d 3774	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A R C <-> B R C ) )
Theorem	breqd 3775	Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C A D <-> C B D ) )
Theorem	breq2d 3776	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C R A <-> C R B ) )
Theorem	breq12d 3777	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A R C <-> B R D ) )
Theorem	breq123d 3778	Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
|- ( ph -> A = B )   &    |- ( ph -> R = S )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A R C <-> B S D ) )
Theorem	breqan12d 3779	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
|- ( ph -> A = B )   &    |- ( ps -> C = D )   =>    |- ( ( ph /\ ps ) -> ( A R C <-> B R D ) )
Theorem	breqan12rd 3780	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
|- ( ph -> A = B )   &    |- ( ps -> C = D )   =>    |- ( ( ps /\ ph ) -> ( A R C <-> B R D ) )
Theorem	nbrne1 3781	Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
|- ( ( A R B /\ -. A R C ) -> B =/= C )
Theorem	nbrne2 3782	Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
|- ( ( A R C /\ -. B R C ) -> A =/= B )
Theorem	eqbrtri 3783	Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
|- A = B   &    |- B R C   =>    |- A R C
Theorem	eqbrtrd 3784	Substitution of equal classes into a binary relation. (Contributed by NM, 8-Oct-1999.)
|- ( ph -> A = B )   &    |- ( ph -> B R C )   =>    |- ( ph -> A R C )
Theorem	eqbrtrri 3785	Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
|- A = B   &    |- A R C   =>    |- B R C
Theorem	eqbrtrrd 3786	Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
|- ( ph -> A = B )   &    |- ( ph -> A R C )   =>    |- ( ph -> B R C )
Theorem	breqtri 3787	Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
|- A R B   &    |- B = C   =>    |- A R C
Theorem	breqtrd 3788	Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
|- ( ph -> A R B )   &    |- ( ph -> B = C )   =>    |- ( ph -> A R C )
Theorem	breqtrri 3789	Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
|- A R B   &    |- C = B   =>    |- A R C
Theorem	breqtrrd 3790	Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
|- ( ph -> A R B )   &    |- ( ph -> C = B )   =>    |- ( ph -> A R C )
Theorem	3brtr3i 3791	Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
|- A R B   &    |- A = C   &    |- B = D   =>    |- C R D
Theorem	3brtr4i 3792	Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
|- A R B   &    |- C = A   &    |- D = B   =>    |- C R D
Theorem	3brtr3d 3793	Substitution of equality into both sides of a binary relation. (Contributed by NM, 18-Oct-1999.)
|- ( ph -> A R B )   &    |- ( ph -> A = C )   &    |- ( ph -> B = D )   =>    |- ( ph -> C R D )
Theorem	3brtr4d 3794	Substitution of equality into both sides of a binary relation. (Contributed by NM, 21-Feb-2005.)
|- ( ph -> A R B )   &    |- ( ph -> C = A )   &    |- ( ph -> D = B )   =>    |- ( ph -> C R D )
Theorem	3brtr3g 3795	Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
|- ( ph -> A R B )   &    |- A = C   &    |- B = D   =>    |- ( ph -> C R D )
Theorem	3brtr4g 3796	Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
|- ( ph -> A R B )   &    |- C = A   &    |- D = B   =>    |- ( ph -> C R D )
Theorem	syl5eqbr 3797	B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
|- A = B   &    |- ( ph -> B R C )   =>    |- ( ph -> A R C )
Theorem	syl5eqbrr 3798	B chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.)
|- B = A   &    |- ( ph -> B R C )   =>    |- ( ph -> A R C )
Theorem	syl5breq 3799	B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
|- A R B   &    |- ( ph -> B = C )   =>    |- ( ph -> A R C )
Theorem	syl5breqr 3800	B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
|- A R B   &    |- ( ph -> C = B )   =>    |- ( ph -> A R C )