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Theorem List for Metamath Proof Explorer - 4101-4200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremiuncom 4101* 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 4102 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
 
Theoremiunconst 4103* Indexed union of a constant class, i.e. where  B does not depend on  x. (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( A  =/=  (/)  ->  U_ x  e.  A  B  =  B )
 
Theoremiinconst 4104* Indexed intersection of a constant class, i.e. where  B does not depend on  x. (Contributed by Mario Carneiro, 6-Feb-2015.)
 |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  B  =  B )
 
Theoremiuniin 4105* 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 4106* 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 4107* Subclass theorem for indexed union. (Contributed by NM, 24-Jan-2012.)
 |-  ( A  C_  B  -> 
 |^|_ x  e.  B  C  C_  |^|_ x  e.  A  C )
 
Theoremiuneq1 4108* Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.)
 |-  ( A  =  B  -> 
 U_ x  e.  A  C  =  U_ x  e.  B  C )
 
Theoremiineq1 4109* Equality theorem for restricted existential quantifier. (Contributed by NM, 27-Jun-1998.)
 |-  ( A  =  B  -> 
 |^|_ x  e.  A  C  =  |^|_ x  e.  B  C )
 
Theoremss2iun 4110 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 4111 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 4112 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 4113 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 4114 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 4115 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 4116* 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 4117* 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 4118* 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 4119* 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 4120* 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 )
 
Theoremnfiun 4121 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
 
Theoremnfiin 4122 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 4123 Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.)
 |-  F/_ x U_ x  e.  A  B
 
Theoremnfii1 4124 Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.)
 |-  F/_ x |^|_ x  e.  A  B
 
Theoremdfiun2g 4125* 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 4126* 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 4127* 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 4128* 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 4129* 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 4130* 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 4131* 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 4132* 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 4133* 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 4134* 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 4135* 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 4136 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 4137* 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 4138* 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 4048. (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 4139* 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 4140* 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 4141* 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 4142 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 4143* 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 4144* 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 4145 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 4146* 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 4147* Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)
 |- 
 |^| A  =  |^|_ x  e.  A  x
 
Theoremiunid 4148* An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)
 |-  U_ x  e.  A  { x }  =  A
 
Theoremiun0 4149 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 4150 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 4151 An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)
 |-  |^|_ x  e.  (/)  A  =  _V
 
Theoremviin 4152* Indexed intersection with a universal index class. When  A doesn't depend on  x, this evaluates to  A by 19.3 1792 and abid2 2555. When  A  =  x, this evaluates to  (/) by intiin 4147 and intv 4378. (Contributed by NM, 11-Sep-2008.)
 |-  |^|_ x  e.  _V  A  =  { y  |  A. x  y  e.  A }
 
Theoremiunn0 4153* There is a non-empty class in an indexed collection  B ( x ) iff the indexed union of them is non-empty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( E. x  e.  A  B  =/=  (/)  <->  U_ x  e.  A  B  =/=  (/) )
 
Theoremiinab 4154* Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.)
 |-  |^|_ x  e.  A  {
 y  |  ph }  =  { y  |  A. x  e.  A  ph }
 
Theoremiinrab 4155* Indexed intersection of a restricted class builder. (Contributed by NM, 6-Dec-2011.)
 |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  { y  e.  B  |  ph }  =  { y  e.  B  |  A. x  e.  A  ph
 } )
 
Theoremiinrab2 4156* Indexed intersection of a restricted class builder. (Contributed by NM, 6-Dec-2011.)
 |-  ( |^|_ x  e.  A  { y  e.  B  |  ph }  i^i  B )  =  { y  e.  B  |  A. x  e.  A  ph }
 
Theoremiunin2 4157* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4146 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 4158* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4146 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 )
 
Theoremiinun2 4159* Indexed intersection of union. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4147 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)
 |-  |^|_ x  e.  A  ( B  u.  C )  =  ( B  u.  |^|_
 x  e.  A  C )
 
Theoremiundif2 4160* Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 4147 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)
 |-  U_ x  e.  A  ( B  \  C )  =  ( B  \  |^|_
 x  e.  A  C )
 
Theorem2iunin 4161* 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 )
 
Theoremiindif2 4162* Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 4146 to recover Enderton's theorem. (Contributed by NM, 5-Oct-2006.)
 |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  ( B  \  C )  =  ( B  \  U_ x  e.  A  C ) )
 
Theoremiinin2 4163* Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4147 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  ( B  i^i  C )  =  ( B  i^i  |^|_ x  e.  A  C ) )
 
Theoremiinin1 4164* Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4147 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  ( C  i^i  B )  =  ( |^|_ x  e.  A  C  i^i  B ) )
 
Theoremiinvdif 4165* The indexed intersection of a complement. (Contributed by Gérard Lang, 5-Aug-2018.)
 |-  |^|_ x  e.  A  ( _V  \  B )  =  ( _V  \  U_ x  e.  A  B )
 
Theoremelriin 4166* 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 4167* Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  S )  =  A )
 
Theoremriinn0 4168* Relative intersection of a nonempty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( ( A. x  e.  X  S  C_  A  /\  X  =/=  (/) )  ->  ( A  i^i  |^|_ x  e.  X  S )  = 
 |^|_ x  e.  X  S )
 
Theoremriinrab 4169* Relative intersection of a relative abstraction. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( A  i^i  |^|_ x  e.  X  { y  e.  A  |  ph } )  =  { y  e.  A  |  A. x  e.  X  ph
 }
 
Theoremiinxsng 4170* 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 4171* 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 4172* 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 4173* 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 4174 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 4175 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 4176* 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
 
Theoremiinuni 4177* A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
 |-  ( A  u.  |^| B )  =  |^|_ x  e.  B  ( A  u.  x )
 
Theoremiununi 4178* A relationship involving union and indexed union. Exercise 25 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
 |-  ( ( B  =  (/) 
 ->  A  =  (/) )  <->  ( A  u.  U. B )  =  U_ x  e.  B  ( A  u.  x ) )
 
Theoremsspwuni 4179 Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
 |-  ( A  C_  ~P B  <->  U. A  C_  B )
 
Theorempwssb 4180* 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 4181 Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
 |-  ( B  e.  A  ->  ( A  C_  ~P B  <->  U. A  =  B ) )
 
Theoremiinpw 4182* 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 4183* 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
 
Theoremrintn0 4184 Relative intersection of a nonempty set. (Contributed by Stefan O'Rear, 3-Apr-2015.) (Revised by Mario Carneiro, 5-Jun-2015.)
 |-  ( ( X  C_  ~P A  /\  X  =/=  (/) )  ->  ( A  i^i  |^| X )  = 
 |^| X )
 
2.1.21  Disjointness
 
Syntaxwdisj 4185 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
 
Definitiondf-disj 4186* 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 )
 
Theoremdfdisj2 4187* 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 ) )
 
Theoremdisjss2 4188 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 ) )
 
Theoremdisjeq2 4189 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 ) )
 
Theoremdisjeq2dv 4190* 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 ) )
 
Theoremdisjss1 4191* 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 ) )
 
Theoremdisjeq1 4192* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  ( A  =  B  ->  (Disj  x  e.  A C 
 <-> Disj  x  e.  B C ) )
 
Theoremdisjeq1d 4193* 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 ) )
 
Theoremdisjeq12d 4194* 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 ) )
 
Theoremcbvdisj 4195* 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 )
 
Theoremcbvdisjv 4196* 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 )
 
Theoremnfdisj 4197 Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  F/_ y A   &    |-  F/_ y B   =>    |-  F/ yDisj  x  e.  A B
 
Theoremnfdisj1 4198 Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- 
 F/ xDisj  x  e.  A B
 
Theoremdisjor 4199* Two ways to say that a collection 
B ( i ) for  i  e.  A is disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 14-Nov-2016.)
 |-  ( i  =  j 
 ->  B  =  C )   =>    |-  (Disj  i  e.  A B  <->  A. i  e.  A  A. j  e.  A  (
 i  =  j  \/  ( B  i^i  C )  =  (/) ) )
 
TheoremdisjmoOLD 4200* Two ways to say that a collection 
B ( i ) for  i  e.  A is disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( i  =  j 
 ->  B  =  C )   =>    |-  ( A. x E* i
 ( i  e.  A  /\  x  e.  B ) 
 <-> 
 A. i  e.  A  A. j  e.  A  ( i  =  j  \/  ( B  i^i  C )  =  (/) ) )
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