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	iundif2ss 3801*	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 3802*	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 3803*	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 3804*	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 3805*	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 3806*	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 3807*	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 3808*	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 3809*	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 3810*	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 3811*	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 3812*	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	iunxsngf 3813*	A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.) (Revised by Thierry Arnoux, 2-May-2020.)
|- F/_ x C   &    |- ( x = A -> B = C )   =>    |- ( A e. V -> U_ x e. { A } B = C )
Theorem	iunun 3814	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 3815	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 3816*	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 3817*	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 3818*	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 3819	Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
|- ( A C_ ~P B <-> U. A C_ B )
Theorem	pwssb 3820*	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	elpwpw 3821	Characterization of the elements of a double power class: they are exactly the sets whose union is included in that class. (Contributed by BJ, 29-Apr-2021.)
|- ( A e. ~P ~P B <-> ( A e. _V /\ U. A C_ B ) )
Theorem	pwpwab 3822*	The double power class written as a class abstraction: the class of sets whose union is included in the given class. (Contributed by BJ, 29-Apr-2021.)
|- ~P ~P A = { x | U. x C_ A }
Theorem	pwpwssunieq 3823*	The class of sets whose union is equal to a given class is included in the double power class of that class. (Contributed by BJ, 29-Apr-2021.)
|- { x | U. x = A } C_ ~P ~P A
Theorem	elpwuni 3824	Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
|- ( B e. A -> ( A C_ ~P B <-> U. A = B ) )
Theorem	iinpw 3825*	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 3826*	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 3827*	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 3828	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 3829*	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 3830*	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 3831	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 3832	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 3833*	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 3834*	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 3835*	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 3836*	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 3837*	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 3838*	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 3839*	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 3840*	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 3841	Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- F/ xDisj x e. A B
Theorem	disjnim 3842*	If a collection B ( i ) for i e. A is disjoint, then pairs are disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Jim Kingdon, 6-Oct-2022.)
|- ( i = j -> B = C )   =>    |- (Disj i e. A B -> A. i e. A A. j e. A ( i =/= j -> ( B i^i C ) = (/) ) )
Theorem	disjnims 3843*	If a collection B ( i ) for i e. A is disjoint, then pairs are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by Jim Kingdon, 7-Oct-2022.)
|- (Disj x e. A B -> A. i e. A A. j e. A ( i =/= j -> ( [_ i / x ]_ B i^i [_ j / x ]_ B ) = (/) ) )
Theorem	disji2 3844*	Property of a disjoint collection: if B ( X ) = C and B ( Y ) = D, and X =/= Y, then C and D are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- ( x = X -> B = C )   &    |- ( x = Y -> B = D )   =>    |- ( (Disj x e. A B /\ ( X e. A /\ Y e. A ) /\ X =/= Y ) -> ( C i^i D ) = (/) )
Theorem	invdisj 3845*	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	disjiun 3846*	A disjoint collection yields disjoint indexed unions for disjoint index sets. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 14-Nov-2016.)
|- ( (Disj x e. A B /\ ( C C_ A /\ D C_ A /\ ( C i^i D ) = (/) ) ) -> ( U_ x e. C B i^i U_ x e. D B ) = (/) )
Theorem	sndisj 3847	Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- Disj x e. A { x }
Theorem	0disj 3848	Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- Disj x e. A (/)
Theorem	disjxsn 3849*	A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- Disj x e. { A } B
Theorem	disjx0 3850	An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- Disj x e. (/) B
2.1.22  Binary relations
Syntax	wbr 3851	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 3852	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 4714). (Contributed by NM, 31-Dec-1993.)
|- ( A R B <-> <. A , B >. e. R )
Theorem	breq 3853	Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)
|- ( R = S -> ( A R B <-> A S B ) )
Theorem	breq1 3854	Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
|- ( A = B -> ( A R C <-> B R C ) )
Theorem	breq2 3855	Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
|- ( A = B -> ( C R A <-> C R B ) )
Theorem	breq12 3856	Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)
|- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )
Theorem	breqi 3857	Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.)
|- R = S   =>    |- ( A R B <-> A S B )
Theorem	breq1i 3858	Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
|- A = B   =>    |- ( A R C <-> B R C )
Theorem	breq2i 3859	Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
|- A = B   =>    |- ( C R A <-> C R B )
Theorem	breq12i 3860	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 3861	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A R C <-> B R C ) )
Theorem	breqd 3862	Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C A D <-> C B D ) )
Theorem	breq2d 3863	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C R A <-> C R B ) )
Theorem	breq12d 3864	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 3865	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 3866	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 3867	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 3868	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 3869	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 3870	Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
|- A = B   &    |- B R C   =>    |- A R C
Theorem	eqbrtrd 3871	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 3872	Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
|- A = B   &    |- A R C   =>    |- B R C
Theorem	eqbrtrrd 3873	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 3874	Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
|- A R B   &    |- B = C   =>    |- A R C
Theorem	breqtrd 3875	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 3876	Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
|- A R B   &    |- C = B   =>    |- A R C
Theorem	breqtrrd 3877	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 3878	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 3879	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 3880	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 3881	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 3882	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 3883	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 3884	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 3885	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 3886	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 3887	B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
|- A R B   &    |- ( ph -> C = B )   =>    |- ( ph -> A R C )
Theorem	syl6eqbr 3888	A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)
|- ( ph -> A = B )   &    |- B R C   =>    |- ( ph -> A R C )
Theorem	syl6eqbrr 3889	A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)
|- ( ph -> B = A )   &    |- B R C   =>    |- ( ph -> A R C )
Theorem	syl6breq 3890	A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
|- ( ph -> A R B )   &    |- B = C   =>    |- ( ph -> A R C )
Theorem	syl6breqr 3891	A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
|- ( ph -> A R B )   &    |- C = B   =>    |- ( ph -> A R C )
Theorem	ssbrd 3892	Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.)
|- ( ph -> A C_ B )   =>    |- ( ph -> ( C A D -> C B D ) )
Theorem	ssbri 3893	Inference from a subclass relationship of binary relations. (Contributed by NM, 28-Mar-2007.) (Revised by Mario Carneiro, 8-Feb-2015.)
|- A C_ B   =>    |- ( C A D -> C B D )
Theorem	nfbrd 3894	Deduction version of bound-variable hypothesis builder nfbr 3895. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 14-Oct-2016.)
|- ( ph -> F/_ x A )   &    |- ( ph -> F/_ x R )   &    |- ( ph -> F/_ x B )   =>    |- ( ph -> F/ x A R B )
Theorem	nfbr 3895	Bound-variable hypothesis builder for binary relation. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
|- F/_ x A   &    |- F/_ x R   &    |- F/_ x B   =>    |- F/ x A R B
Theorem	brab1 3896*	Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.)
|- ( x R A <-> x e. { z | z R A } )
Theorem	brun 3897	The union of two binary relations. (Contributed by NM, 21-Dec-2008.)
|- ( A ( R u. S ) B <-> ( A R B \/ A S B ) )
Theorem	brin 3898	The intersection of two relations. (Contributed by FL, 7-Oct-2008.)
|- ( A ( R i^i S ) B <-> ( A R B /\ A S B ) )
Theorem	brdif 3899	The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.)
|- ( A ( R \ S ) B <-> ( A R B /\ -. A S B ) )
Theorem	sbcbrg 3900	Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- ( A e. D -> ( [. A / x ]. B R C <-> [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C ) )