P. 41 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4001-4100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	uniintsnr 4001*	The union and intersection of a singleton are equal. See also eusn 3781. (Contributed by Jim Kingdon, 14-Aug-2018.)
|- ( E. x A = { x } -> U. A = |^| A )
Theorem	uniintabim 4002	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 4003	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 4004	Relative intersection of an empty set. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- ( X = (/) -> ( A i^i |^| X ) = A )
Theorem	elrint 4005*	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 4006*	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 4007	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 3930. 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 4008	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 3965. 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 4009*	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 4041. Theorem uniiun 4061 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 4010*	Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 4009. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 4042. Theorem intiin 4062 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 4011*	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 4012*	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 4013*	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 4014	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 4015*	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 4016*	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 4017*	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 4018*	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 4019*	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 4020*	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 4021*	Equality theorem for restricted existential quantifier. (Contributed by NM, 27-Jun-1998.)
|- ( A = B -> |^|_ x e. A C = |^|_ x e. B C )
Theorem	ss2iun 4022	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 4023	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 4024	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 4025	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 4026	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 4027	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 4028*	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 4029*	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 4030*	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 4031*	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 4032*	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 4033*	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 4034*	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 4035*	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 4036*	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 4037	Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.)
|- F/_ x U_ x e. A B
Theorem	nfii1 4038	Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.)
|- F/_ x |^|_ x e. A B
Theorem	dfiun2g 4039*	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 4040*	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 4041*	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 4042*	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 4043*	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 4044*	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 4045*	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 4046*	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 4047*	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 4048*	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 4049*	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 4050	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 4051*	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 4052*	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 3961. (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 4053*	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 4054*	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 4055*	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 4056*	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 4057	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 4058*	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 4059*	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 4060	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 4061*	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 4062*	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 4063*	An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)
|- U_ x e. A { x } = A
Theorem	iun0 4064	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 4065	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 4066	An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)
|- |^|_ x e. (/) A = _V
Theorem	viin 4067*	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 4068*	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 4069*	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 4070*	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 4071*	Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4061 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 4072*	Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4061 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 4073*	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 4074*	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 4075*	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 4076*	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 4077*	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 4078*	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 4079*	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 4080*	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 4081*	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 4082*	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 4083*	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 4084*	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 4085*	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 4086	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 4087	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	iunxprg 4088*	A pair index picks out two instances of an indexed union's argument. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
|- ( x = A -> C = D )   &    |- ( x = B -> C = E )   =>    |- ( ( A e. V /\ B e. W ) -> U_ x e. { A , B } C = ( D u. E ) )
Theorem	iunxiun 4089*	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 4090*	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 4091*	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 4092	Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
|- ( A C_ ~P B <-> U. A C_ B )
Theorem	pwssb 4093*	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 4094	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 4095*	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 4096*	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 4097	Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
|- ( B e. A -> ( A C_ ~P B <-> U. A = B ) )
Theorem	iinpw 4098*	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 4099*	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 4100*	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 )