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Theorem List for Intuitionistic Logic Explorer - 3701-3800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremint0el 3701 The intersection of a class containing the empty set is empty. (Contributed by NM, 24-Apr-2004.)
 |-  ( (/)  e.  A  -> 
 |^| A  =  (/) )
 
Theoremintun 3702 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 3703 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 3704 The intersection of a pair is the intersection of its members. Closed form of intpr 3703. 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 3705 Intersection of a singleton. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( A  e.  V  -> 
 |^| { A }  =  A )
 
Theoremintsn 3706 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 3707* The union and intersection of a singleton are equal. See also eusn 3499. (Contributed by Jim Kingdon, 14-Aug-2018.)
 |-  ( E. x  A  =  { x }  ->  U. A  =  |^| A )
 
Theoremuniintabim 3708 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 3709 Theorem joining a singleton to an intersection. (Contributed by NM, 29-Sep-2002.)
 |-  B  e.  _V   =>    |-  |^| ( A  u.  { B } )  =  ( |^| A  i^i  B )
 
Theoremrint0 3710 Relative intersection of an empty set. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( X  =  (/)  ->  ( A  i^i  |^| X )  =  A )
 
Theoremelrint 3711* 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 3712* 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 3713 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 3636. 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 3714 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 3671. 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 3715* 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 distinct variable group (meaning  A cannot depend on  x) and that  B and  x do not share a distinct 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 3747. Theorem uniiun 3766 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 3716* Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 3715. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 3748. Theorem intiin 3767 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 3717* 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 3718* 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 3719* 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 3720 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 3721* 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 3722* 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 3723* 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 3724* 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 3725* Subclass theorem for indexed union. (Contributed by NM, 24-Jan-2012.)
 |-  ( A  C_  B  -> 
 |^|_ x  e.  B  C  C_  |^|_ x  e.  A  C )
 
Theoremiuneq1 3726* Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.)
 |-  ( A  =  B  -> 
 U_ x  e.  A  C  =  U_ x  e.  B  C )
 
Theoremiineq1 3727* Equality theorem for restricted existential quantifier. (Contributed by NM, 27-Jun-1998.)
 |-  ( A  =  B  -> 
 |^|_ x  e.  A  C  =  |^|_ x  e.  B  C )
 
Theoremss2iun 3728 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 3729 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 3730 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 3731 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 3732 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 3733 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 3734* 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 3735* 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 3736* 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 3737* 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 3738* 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 3739* 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 3740* 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 3741* 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 3742* 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 3743 Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.)
 |-  F/_ x U_ x  e.  A  B
 
Theoremnfii1 3744 Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.)
 |-  F/_ x |^|_ x  e.  A  B
 
Theoremdfiun2g 3745* 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 3746* 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 3747* 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 3748* 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 3749* 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 3750* 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 3751* 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 3752* 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 3753* 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 3754* 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 3755* 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 3756 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 3757* 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 3758* 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 3667. (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 3759* 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 3760* 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 3761* 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 3762 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 3763* 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 3764* 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 3765 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 3766* 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 3767* Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)
 |- 
 |^| A  =  |^|_ x  e.  A  x
 
Theoremiunid 3768* An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)
 |-  U_ x  e.  A  { x }  =  A
 
Theoremiun0 3769 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 3770 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 3771 An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)
 |-  |^|_ x  e.  (/)  A  =  _V
 
Theoremviin 3772* Indexed intersection with a universal index class. (Contributed by NM, 11-Sep-2008.)
 |-  |^|_ x  e.  _V  A  =  { y  |  A. x  y  e.  A }
 
Theoremiunn0m 3773* 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 3774* Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.)
 |-  |^|_ x  e.  A  {
 y  |  ph }  =  { y  |  A. x  e.  A  ph }
 
Theoremiinrabm 3775* 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 3776* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3766 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 3777* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3766 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 3778* 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 3779* 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 3780* 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 3781* 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 3782* 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 3783* 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 3784* Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  S )  =  A )
 
Theoremriinm 3785* 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 3786* 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 3787* 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 3788* 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 3789* 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
 
Theoremiunun 3790 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 3791 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 )
 
Theoremiunxiun 3792* 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 3793* 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 3794* 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 3795 Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
 |-  ( A  C_  ~P B  <->  U. A  C_  B )
 
Theorempwssb 3796* 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 )
 
Theoremelpwuni 3797 Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
 |-  ( B  e.  A  ->  ( A  C_  ~P B  <->  U. A  =  B ) )
 
Theoremiinpw 3798* 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
 
Theoremiunpwss 3799* 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
 
Theoremrintm 3800* 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 )
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