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	nfint 3901	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 3902*	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 3903*	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 3904*	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 3905	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 3906	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 3907*	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 3908*	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 3909*	Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)
|- A C_ |^| { x e. B | A C_ x }
Theorem	ssmin 3910*	Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.)
|- A C_ |^| { x | ( A C_ x /\ ph ) }
Theorem	intmin 3911*	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 3912	Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)
|- ( A C_ B -> |^| B C_ |^| A )
Theorem	intssunim 3913*	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 3914*	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 3915*	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 3916*	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 3917*	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 3918*	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 3919*	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 3920*	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 3921	The intersection of a class containing the empty set is empty. (Contributed by NM, 24-Apr-2004.)
|- ( (/) e. A -> |^| A = (/) )
Theorem	intun 3922	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 3923	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 3924	The intersection of a pair is the intersection of its members. Closed form of intpr 3923. 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 3925	Intersection of a singleton. (Contributed by Stefan O'Rear, 22-Feb-2015.)
|- ( A e. V -> |^| { A } = A )
Theorem	intsn 3926	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
Theorem	uniintsnr 3927*	The union and intersection of a singleton are equal. See also eusn 3712. (Contributed by Jim Kingdon, 14-Aug-2018.)
|- ( E. x A = { x } -> U. A = |^| A )
Theorem	uniintabim 3928	The union and the intersection of a class abstraction are equal if there is a unique satisfying value of ph ( x ). (Contributed by Jim Kingdon, 14-Aug-2018.)
|- ( E! x ph -> U. { x | ph } = |^| { x | ph } )
Theorem	intunsn 3929	Theorem joining a singleton to an intersection. (Contributed by NM, 29-Sep-2002.)
|- B e. _V   =>    |- |^| ( A u. { B } ) = ( |^| A i^i B )
Theorem	rint0 3930	Relative intersection of an empty set. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- ( X = (/) -> ( A i^i |^| X ) = A )
Theorem	elrint 3931*	Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- ( X e. ( A i^i |^| B ) <-> ( X e. A /\ A. y e. B X e. y ) )
Theorem	elrint2 3932*	Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- ( X e. A -> ( X e. ( A i^i |^| B ) <-> A. y e. B X e. y ) )
2.1.20  Indexed union and intersection
Syntax	ciun 3933	Extend class notation to include indexed union. Note: Historically (prior to 21-Oct-2005), set.mm used the notation U. x e. A B, with the same union symbol as cuni 3856. While that syntax was unambiguous, it did not allow for LALR parsing of the syntax constructions in set.mm. The new syntax uses as distinguished symbol U_ instead of U. and does allow LALR parsing. Thanks to Peter Backes for suggesting this change.
class U_ x e. A B
Syntax	ciin 3934	Extend class notation to include indexed intersection. Note: Historically (prior to 21-Oct-2005), set.mm used the notation |^| x e. A B, with the same intersection symbol as cint 3891. Although that syntax was unambiguous, it did not allow for LALR parsing of the syntax constructions in set.mm. The new syntax uses a distinguished symbol |^|_ instead of |^| and does allow LALR parsing. Thanks to Peter Backes for suggesting this change.
class |^|_ x e. A B
Definition	df-iun 3935*	Define indexed union. Definition indexed union in [Stoll] p. 45. In most applications, A is independent of x (although this is not required by the definition), and B depends on x i.e. can be read informally as B ( x ). We call x the index, A the index set, and B the indexed set. In most books, x e. A is written as a subscript or underneath a union symbol U.. We use a special union symbol U_ to make it easier to distinguish from plain class union. In many theorems, you will see that x and A are in the same disjoint variable group (meaning A cannot depend on x) and that B and x do not share a disjoint variable group (meaning that can be thought of as B ( x ) i.e. can be substituted with a class expression containing x). An alternate definition tying indexed union to ordinary union is dfiun2 3967. Theorem uniiun 3987 provides a definition of ordinary union in terms of indexed union. (Contributed by NM, 27-Jun-1998.)
|- U_ x e. A B = { y | E. x e. A y e. B }
Definition	df-iin 3936*	Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 3935. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 3968. Theorem intiin 3988 provides a definition of ordinary intersection in terms of indexed intersection. (Contributed by NM, 27-Jun-1998.)
|- |^|_ x e. A B = { y | A. x e. A y e. B }
Theorem	eliun 3937*	Membership in indexed union. (Contributed by NM, 3-Sep-2003.)
|- ( A e. U_ x e. B C <-> E. x e. B A e. C )
Theorem	eliin 3938*	Membership in indexed intersection. (Contributed by NM, 3-Sep-2003.)
|- ( A e. V -> ( A e. |^|_ x e. B C <-> A. x e. B A e. C ) )
Theorem	iuncom 3939*	Commutation of indexed unions. (Contributed by NM, 18-Dec-2008.)
|- U_ x e. A U_ y e. B C = U_ y e. B U_ x e. A C
Theorem	iuncom4 3940	Commutation of union with indexed union. (Contributed by Mario Carneiro, 18-Jan-2014.)
|- U_ x e. A U. B = U. U_ x e. A B
Theorem	iunconstm 3941*	Indexed union of a constant class, i.e. where B does not depend on x. (Contributed by Jim Kingdon, 15-Aug-2018.)
|- ( E. x x e. A -> U_ x e. A B = B )
Theorem	iinconstm 3942*	Indexed intersection of a constant class, i.e. where B does not depend on x. (Contributed by Jim Kingdon, 19-Dec-2018.)
|- ( E. y y e. A -> |^|_ x e. A B = B )
Theorem	iuniin 3943*	Law combining indexed union with indexed intersection. Eq. 14 in [KuratowskiMostowski] p. 109. This theorem also appears as the last example at http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- U_ x e. A |^|_ y e. B C C_ |^|_ y e. B U_ x e. A C
Theorem	iunss1 3944*	Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- ( A C_ B -> U_ x e. A C C_ U_ x e. B C )
Theorem	iinss1 3945*	Subclass theorem for indexed union. (Contributed by NM, 24-Jan-2012.)
|- ( A C_ B -> |^|_ x e. B C C_ |^|_ x e. A C )
Theorem	iuneq1 3946*	Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.)
|- ( A = B -> U_ x e. A C = U_ x e. B C )
Theorem	iineq1 3947*	Equality theorem for restricted existential quantifier. (Contributed by NM, 27-Jun-1998.)
|- ( A = B -> |^|_ x e. A C = |^|_ x e. B C )
Theorem	ss2iun 3948	Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- ( A. x e. A B C_ C -> U_ x e. A B C_ U_ x e. A C )
Theorem	iuneq2 3949	Equality theorem for indexed union. (Contributed by NM, 22-Oct-2003.)
|- ( A. x e. A B = C -> U_ x e. A B = U_ x e. A C )
Theorem	iineq2 3950	Equality theorem for indexed intersection. (Contributed by NM, 22-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- ( A. x e. A B = C -> |^|_ x e. A B = |^|_ x e. A C )
Theorem	iuneq2i 3951	Equality inference for indexed union. (Contributed by NM, 22-Oct-2003.)
|- ( x e. A -> B = C )   =>    |- U_ x e. A B = U_ x e. A C
Theorem	iineq2i 3952	Equality inference for indexed intersection. (Contributed by NM, 22-Oct-2003.)
|- ( x e. A -> B = C )   =>    |- |^|_ x e. A B = |^|_ x e. A C
Theorem	iineq2d 3953	Equality deduction for indexed intersection. (Contributed by NM, 7-Dec-2011.)
|- F/ x ph   &    |- ( ( ph /\ x e. A ) -> B = C )   =>    |- ( ph -> |^|_ x e. A B = |^|_ x e. A C )
Theorem	iuneq2dv 3954*	Equality deduction for indexed union. (Contributed by NM, 3-Aug-2004.)
|- ( ( ph /\ x e. A ) -> B = C )   =>    |- ( ph -> U_ x e. A B = U_ x e. A C )
Theorem	iineq2dv 3955*	Equality deduction for indexed intersection. (Contributed by NM, 3-Aug-2004.)
|- ( ( ph /\ x e. A ) -> B = C )   =>    |- ( ph -> |^|_ x e. A B = |^|_ x e. A C )
Theorem	iuneq1d 3956*	Equality theorem for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)
|- ( ph -> A = B )   =>    |- ( ph -> U_ x e. A C = U_ x e. B C )
Theorem	iuneq12d 3957*	Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> U_ x e. A C = U_ x e. B D )
Theorem	iuneq2d 3958*	Equality deduction for indexed union. (Contributed by Drahflow, 22-Oct-2015.)
|- ( ph -> B = C )   =>    |- ( ph -> U_ x e. A B = U_ x e. A C )
Theorem	nfiunxy 3959*	Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)
|- F/_ y A   &    |- F/_ y B   =>    |- F/_ y U_ x e. A B
Theorem	nfiinxy 3960*	Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.)
|- F/_ y A   &    |- F/_ y B   =>    |- F/_ y |^|_ x e. A B
Theorem	nfiunya 3961*	Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)
|- F/_ y A   &    |- F/_ y B   =>    |- F/_ y U_ x e. A B
Theorem	nfiinya 3962*	Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.)
|- F/_ y A   &    |- F/_ y B   =>    |- F/_ y |^|_ x e. A B
Theorem	nfiu1 3963	Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.)
|- F/_ x U_ x e. A B
Theorem	nfii1 3964	Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.)
|- F/_ x |^|_ x e. A B
Theorem	dfiun2g 3965*	Alternate definition of indexed union when B is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- ( A. x e. A B e. C -> U_ x e. A B = U. { y | E. x e. A y = B } )
Theorem	dfiin2g 3966*	Alternate definition of indexed intersection when B is a set. (Contributed by Jeff Hankins, 27-Aug-2009.)
|- ( A. x e. A B e. C -> |^|_ x e. A B = |^| { y | E. x e. A y = B } )
Theorem	dfiun2 3967*	Alternate definition of indexed union when B is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 27-Jun-1998.) (Revised by David Abernethy, 19-Jun-2012.)
|- B e. _V   =>    |- U_ x e. A B = U. { y | E. x e. A y = B }
Theorem	dfiin2 3968*	Alternate definition of indexed intersection when B is a set. Definition 15(b) of [Suppes] p. 44. (Contributed by NM, 28-Jun-1998.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- B e. _V   =>    |- |^|_ x e. A B = |^| { y | E. x e. A y = B }
Theorem	dfiunv2 3969*	Define double indexed union. (Contributed by FL, 6-Nov-2013.)
|- U_ x e. A U_ y e. B C = { z | E. x e. A E. y e. B z e. C }
Theorem	cbviun 3970*	Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.)
|- F/_ y B   &    |- F/_ x C   &    |- ( x = y -> B = C )   =>    |- U_ x e. A B = U_ y e. A C
Theorem	cbviin 3971*	Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.)
|- F/_ y B   &    |- F/_ x C   &    |- ( x = y -> B = C )   =>    |- |^|_ x e. A B = |^|_ y e. A C
Theorem	cbviunv 3972*	Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 15-Sep-2003.)
|- ( x = y -> B = C )   =>    |- U_ x e. A B = U_ y e. A C
Theorem	cbviinv 3973*	Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.)
|- ( x = y -> B = C )   =>    |- |^|_ x e. A B = |^|_ y e. A C
Theorem	iunss 3974*	Subset theorem for an indexed union. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- ( U_ x e. A B C_ C <-> A. x e. A B C_ C )
Theorem	ssiun 3975*	Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- ( E. x e. A C C_ B -> C C_ U_ x e. A B )
Theorem	ssiun2 3976	Identity law for subset of an indexed union. (Contributed by NM, 12-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|- ( x e. A -> B C_ U_ x e. A B )
Theorem	ssiun2s 3977*	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 3978*	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 3887. (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	iunssd 3979*	Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
|- ( ( ph /\ x e. A ) -> B C_ C )   =>    |- ( ph -> U_ x e. A B C_ C )
Theorem	iunab 3980*	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 3981*	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 3982*	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 3983	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 3984*	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 3985*	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 3986	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 3987*	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 3988*	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 3989*	An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)
|- U_ x e. A { x } = A
Theorem	iun0 3990	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 3991	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 3992	An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)
|- |^|_ x e. (/) A = _V
Theorem	viin 3993*	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 3994*	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 3995*	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 3996*	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 3997*	Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3987 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 3998*	Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3987 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 3999*	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 4000*	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 )