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Theorem List for Intuitionistic Logic Explorer - 3801-3900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremint0el 3801 The intersection of a class containing the empty set is empty. (Contributed by NM, 24-Apr-2004.)
 |-  ( (/)  e.  A  -> 
 |^| A  =  (/) )
 
Theoremintun 3802 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 )
 
Theoremintpr 3803 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 )
 
Theoremintprg 3804 The intersection of a pair is the intersection of its members. Closed form of intpr 3803. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| { A ,  B }  =  ( A  i^i  B ) )
 
Theoremintsng 3805 Intersection of a singleton. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( A  e.  V  -> 
 |^| { A }  =  A )
 
Theoremintsn 3806 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
 
Theoremuniintsnr 3807* The union and intersection of a singleton are equal. See also eusn 3597. (Contributed by Jim Kingdon, 14-Aug-2018.)
 |-  ( E. x  A  =  { x }  ->  U. A  =  |^| A )
 
Theoremuniintabim 3808 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 } )
 
Theoremintunsn 3809 Theorem joining a singleton to an intersection. (Contributed by NM, 29-Sep-2002.)
 |-  B  e.  _V   =>    |-  |^| ( A  u.  { B } )  =  ( |^| A  i^i  B )
 
Theoremrint0 3810 Relative intersection of an empty set. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( X  =  (/)  ->  ( A  i^i  |^| X )  =  A )
 
Theoremelrint 3811* 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
 ) )
 
Theoremelrint2 3812* 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
 
Syntaxciun 3813 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 3736. 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
 
Syntaxciin 3814 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 3771. 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
 
Definitiondf-iun 3815* 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 3847. Theorem uniiun 3866 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 }
 
Definitiondf-iin 3816* Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 3815. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 3848. Theorem intiin 3867 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 }
 
Theoremeliun 3817* Membership in indexed union. (Contributed by NM, 3-Sep-2003.)
 |-  ( A  e.  U_ x  e.  B  C  <->  E. x  e.  B  A  e.  C )
 
Theoremeliin 3818* 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 )
 )
 
Theoremiuncom 3819* 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
 
Theoremiuncom4 3820 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
 
Theoremiunconstm 3821* 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 )
 
Theoremiinconstm 3822* 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 )
 
Theoremiuniin 3823* 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
 
Theoremiunss1 3824* 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 )
 
Theoremiinss1 3825* Subclass theorem for indexed union. (Contributed by NM, 24-Jan-2012.)
 |-  ( A  C_  B  -> 
 |^|_ x  e.  B  C  C_  |^|_ x  e.  A  C )
 
Theoremiuneq1 3826* Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.)
 |-  ( A  =  B  -> 
 U_ x  e.  A  C  =  U_ x  e.  B  C )
 
Theoremiineq1 3827* Equality theorem for restricted existential quantifier. (Contributed by NM, 27-Jun-1998.)
 |-  ( A  =  B  -> 
 |^|_ x  e.  A  C  =  |^|_ x  e.  B  C )
 
Theoremss2iun 3828 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 )
 
Theoremiuneq2 3829 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 )
 
Theoremiineq2 3830 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 )
 
Theoremiuneq2i 3831 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
 
Theoremiineq2i 3832 Equality inference for indexed intersection. (Contributed by NM, 22-Oct-2003.)
 |-  ( x  e.  A  ->  B  =  C )   =>    |-  |^|_
 x  e.  A  B  =  |^|_ x  e.  A  C
 
Theoremiineq2d 3833 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 )
 
Theoremiuneq2dv 3834* 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 )
 
Theoremiineq2dv 3835* 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 )
 
Theoremiuneq1d 3836* 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 )
 
Theoremiuneq12d 3837* 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 )
 
Theoremiuneq2d 3838* 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 )
 
Theoremnfiunxy 3839* 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
 
Theoremnfiinxy 3840* 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
 
Theoremnfiunya 3841* 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
 
Theoremnfiinya 3842* 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
 
Theoremnfiu1 3843 Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.)
 |-  F/_ x U_ x  e.  A  B
 
Theoremnfii1 3844 Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.)
 |-  F/_ x |^|_ x  e.  A  B
 
Theoremdfiun2g 3845* 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 }
 )
 
Theoremdfiin2g 3846* 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 }
 )
 
Theoremdfiun2 3847* 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 }
 
Theoremdfiin2 3848* 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 }
 
Theoremdfiunv2 3849* 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 }
 
Theoremcbviun 3850* 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
 
Theoremcbviin 3851* 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
 
Theoremcbviunv 3852* 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
 
Theoremcbviinv 3853* 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
 
Theoremiunss 3854* 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 )
 
Theoremssiun 3855* 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 )
 
Theoremssiun2 3856 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 )
 
Theoremssiun2s 3857* 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 )
 
Theoremiunss2 3858* 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 3767. (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 )
 
Theoremiunab 3859* 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 }
 
Theoremiunrab 3860* 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
 }
 
Theoremiunxdif2 3861* 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 )
 
Theoremssiinf 3862 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 )
 
Theoremssiin 3863* 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 )
 
Theoremiinss 3864* 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 )
 
Theoremiinss2 3865 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 )
 
Theoremuniiun 3866* 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
 
Theoremintiin 3867* Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)
 |- 
 |^| A  =  |^|_ x  e.  A  x
 
Theoremiunid 3868* An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)
 |-  U_ x  e.  A  { x }  =  A
 
Theoremiun0 3869 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  (/) 
 =  (/)
 
Theorem0iun 3870 An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  U_ x  e.  (/)  A  =  (/)
 
Theorem0iin 3871 An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)
 |-  |^|_ x  e.  (/)  A  =  _V
 
Theoremviin 3872* Indexed intersection with a universal index class. (Contributed by NM, 11-Sep-2008.)
 |-  |^|_ x  e.  _V  A  =  { y  |  A. x  y  e.  A }
 
Theoremiunn0m 3873* 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 )
 
Theoremiinab 3874* Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.)
 |-  |^|_ x  e.  A  {
 y  |  ph }  =  { y  |  A. x  e.  A  ph }
 
Theoremiinrabm 3875* 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
 } )
 
Theoremiunin2 3876* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3866 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 )
 
Theoremiunin1 3877* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3866 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 )
 
Theoremiundif2ss 3878* 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 )
 
Theorem2iunin 3879* 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 )
 
Theoremiindif2m 3880* 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 ) )
 
Theoremiinin2m 3881* 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 ) )
 
Theoremiinin1m 3882* 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 ) )
 
Theoremelriin 3883* 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 ) )
 
Theoremriin0 3884* Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  S )  =  A )
 
Theoremriinm 3885* 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 )
 
Theoremiinxsng 3886* 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 )
 
Theoremiinxprg 3887* 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 ) )
 
Theoremiunxsng 3888* 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 )
 
Theoremiunxsn 3889* 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
 
Theoremiunxsngf 3890* 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 )
 
Theoremiunun 3891 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 )
 
Theoremiunxun 3892 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 )
 
Theoremiunxprg 3893* 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 ) )
 
Theoremiunxiun 3894* 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
 
Theoremiinuniss 3895* 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 )
 
Theoremiununir 3896* 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  =  (/) ) )
 
Theoremsspwuni 3897 Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
 |-  ( A  C_  ~P B  <->  U. A  C_  B )
 
Theorempwssb 3898* 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 )
 
Theoremelpwpw 3899 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 ) )
 
Theorempwpwab 3900* 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 }
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