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Theorem List for Metamath Proof Explorer - 3901-4000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremiineq2dv 3901* 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 3902* 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 3903* 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 3904* 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 3905 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 3906 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 3907 Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.)
 |-  F/_ x U_ x  e.  A  B
 
Theoremnfii1 3908 Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.)
 |-  F/_ x |^|_ x  e.  A  B
 
Theoremdfiun2g 3909* 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 3910* 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 3911* 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 3912* 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 }
 
Theoremcbviun 3913* 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 3914* 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 3915* 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 3916* 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 3917* 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 3918* 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 3919 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 3920* 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 3921* 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 3832. (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 3922* 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 3923* 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 3924* 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 3925 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 3926* 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 3927* 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 3928 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 3929* 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 3930* Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)
 |- 
 |^| A  =  |^|_ x  e.  A  x
 
Theoremiunid 3931* An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)
 |-  U_ x  e.  A  { x }  =  A
 
Theoremiun0 3932 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 3933 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 3934 An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)
 |-  |^|_ x  e.  (/)  A  =  _V
 
Theoremviin 3935* Indexed intersection with a universal index class. When  A doesn't depend on  x, this evaluates to  A by 19.3 1760 and abid2 2375. When  A  =  x, this evaluates to  (/) by intiin 3930 and intv 4158. (Contributed by NM, 11-Sep-2008.)
 |-  |^|_ x  e.  _V  A  =  { y  |  A. x  y  e.  A }
 
Theoremiunn0 3936* 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 3937* Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.)
 |-  |^|_ x  e.  A  {
 y  |  ph }  =  { y  |  A. x  e.  A  ph }
 
Theoremiinrab 3938* 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 3939* 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 3940* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3929 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 3941* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3929 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 3942* Indexed intersection of union. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 3930 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)
 |-  |^|_ x  e.  A  ( B  u.  C )  =  ( B  u.  |^|_
 x  e.  A  C )
 
Theoremiundif2 3943* Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 3930 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)
 |-  U_ x  e.  A  ( B  \  C )  =  ( B  \  |^|_
 x  e.  A  C )
 
Theorem2iunin 3944* 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 3945* Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 3929 to recover Enderton's theorem. (Contributed by NM, 5-Oct-2006.)
 |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  ( B  \  C )  =  ( B  \  U_ x  e.  A  C ) )
 
Theoremiinin2 3946* Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 3930 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 3947* Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 3930 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 ) )
 
Theoremelriin 3948* 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 3949* Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  S )  =  A )
 
Theoremriinn0 3950* 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 3951* 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 3952* 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 3953* 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 3954* 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 3955* 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 3956 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 3957 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 3958* 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 3959* 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 3960* 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 3961 Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
 |-  ( A  C_  ~P B  <->  U. A  C_  B )
 
Theorempwssb 3962* 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 3963 Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
 |-  ( B  e.  A  ->  ( A  C_  ~P B  <->  U. A  =  B ) )
 
Theoremiinpw 3964* 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 3965* 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 3966 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 3967 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 3968* 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 3969* 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 3970 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 3971 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 3972* 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 3973* 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 3974* 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 3975* 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 3976* 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 3977* 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 3978* 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 3979 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 3980 Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- 
 F/ xDisj  x  e.  A B
 
Theoremdisjor 3981* 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 3982* 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.)
 |-  ( 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 )  =  (/) ) )
 
Theoremdisjors 3983* Two ways to say that a collection 
B ( i ) for  i  e.  A is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  (Disj  x  e.  A B 
 <-> 
 A. i  e.  A  A. j  e.  A  ( i  =  j  \/  ( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/) ) )
 
Theoremdisji2 3984* Property of a disjoint collection: if  B ( X )  =  C and  B ( Y )  =  D, and  X  =/=  Y, then  C and  D are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  ( x  =  X  ->  B  =  C )   &    |-  ( x  =  Y  ->  B  =  D )   =>    |-  ( (Disj  x  e.  A B  /\  ( X  e.  A  /\  Y  e.  A )  /\  X  =/=  Y )  ->  ( C  i^i  D )  =  (/) )
 
Theoremdisji 3985* Property of a disjoint collection: if  B ( X )  =  C and  B ( Y )  =  D have a common element  Z, then  X  =  Y. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  ( x  =  X  ->  B  =  C )   &    |-  ( x  =  Y  ->  B  =  D )   =>    |-  ( (Disj  x  e.  A B  /\  ( X  e.  A  /\  Y  e.  A )  /\  ( Z  e.  C  /\  Z  e.  D ) )  ->  X  =  Y )
 
Theoreminvdisj 3986* If there is a function  C ( y ) such that  C ( y )  =  x for all  y  e.  B
( x ), then the sets  B ( x ) for distinct  x  e.  A are disjoint. (Contributed by Mario Carneiro, 10-Dec-2016.)
 |-  ( A. x  e.  A  A. y  e.  B  C  =  x 
 -> Disj 
 x  e.  A B )
 
Theoremdisjiun 3987* A disjoint collection yields disjoint indexed unions for disjoint index sets. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 14-Nov-2016.)
 |-  ( (Disj  x  e.  A B  /\  ( C 
 C_  A  /\  D  C_  A  /\  ( C  i^i  D )  =  (/) ) )  ->  ( U_ x  e.  C  B  i^i  U_ x  e.  D  B )  =  (/) )
 
TheoremdisjiunOLD 3988* A disjoint collection yields disjoint indexed unions for disjoint index sets. (Contributed by Mario Carneiro, 26-Mar-2015.) (New usage is discouraged.)
 |-  ( ( A. y E* x ( x  e.  A  /\  y  e.  B )  /\  ( C  C_  A  /\  D  C_  A  /\  ( C  i^i  D )  =  (/) ) )  ->  ( U_ x  e.  C  B  i^i  U_ x  e.  D  B )  =  (/) )
 
Theoremsndisj 3989 Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- Disj  x  e.  A { x }
 
Theorem0disj 3990 Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- Disj  x  e.  A (/)
 
Theoremdisjxsn 3991* A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- Disj  x  e.  { A } B
 
Theoremdisjx0 3992 An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- Disj  x  e.  (/) B
 
Theoremdisjprg 3993* A pair collection is disjoint iff the two sets in the family have empty intersection. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  ( x  =  A  ->  C  =  D )   &    |-  ( x  =  B  ->  C  =  E )   =>    |-  ( ( A  e.  V  /\  B  e.  V  /\  A  =/=  B ) 
 ->  (Disj  x  e.  { A ,  B } C  <->  ( D  i^i  E )  =  (/) ) )
 
Theoremdisjxiun 3994* An indexed union of a disjoint collection of disjoint collections is disjoint if each component is disjoint, and the disjoint unions in the collection are also disjoint. Note that  B ( y ) and  C
( x ) may have the displayed free variables. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  (Disj  y  e.  A B  ->  (Disj  x  e.  U_ y  e.  A  B C  <->  ( A. y  e.  A Disj  x  e.  B C  /\ Disj  y  e.  A U_ x  e.  B  C ) ) )
 
Theoremdisjxun 3995* The union of two disjoint collections. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |-  ( x  =  y 
 ->  C  =  D )   =>    |-  ( ( A  i^i  B )  =  (/)  ->  (Disj  x  e.  ( A  u.  B ) C  <->  (Disj  x  e.  A C  /\ Disj  x  e.  B C  /\  A. x  e.  A  A. y  e.  B  ( C  i^i  D )  =  (/) ) ) )
 
Theoremdisjss3 3996* Expand a disjoint collection with any number of empty sets. (Contributed by Mario Carneiro, 15-Nov-2016.)
 |-  ( ( A  C_  B  /\  A. x  e.  ( B  \  A ) C  =  (/) )  ->  (Disj  x  e.  A C  <-> Disj  x  e.  B C ) )
 
2.1.22  Binary relations
 
Syntaxwbr 3997 Extend wff notation to include the general binary relation predicate. Note that the syntax is simply three class symbols in a row. Since binary relations are the only possible wff expressions consisting of three class expressions in a row, the syntax is unambiguous. (For an example of how syntax could become ambiguous if we are not careful, see the comment in cneg 9006.)
 wff  A R B
 
Definitiondf-br 3998 Define a general binary relation. Note that the syntax is simply three class symbols in a row. Definition 6.18 of [TakeutiZaring] p. 29 generalized to arbitrary classes. Class  R often denotes a relation such as " < " that compares two classes  A and 
B, which might be numbers such as  1 and  2 (see df-ltxr 8840 for the specific definition of  <). As a wff, relations are true or false. For example,  ( R  =  { <. 2 ,  6
>. ,  <. 3 ,  9 >. }  ->  3 R 9 ) (ex-br 20762). Often class  R meets the  Rel criteria to be defined in df-rel 4676, and in particular  R may be a function (see df-fun 4683). This definition of relations is well-defined, although not very meaningful, when classes  A and/or  B are proper classes (i.e. are not sets). On the other hand, we often find uses for this definition when  R is a proper class. (Contributed by NM, 31-Dec-1993.)
 |-  ( A R B  <->  <. A ,  B >.  e.  R )
 
Theorembreq 3999 Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)
 |-  ( R  =  S  ->  ( A R B  <->  A S B ) )
 
Theorembreq1 4000 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
 |-  ( A  =  B  ->  ( A R C  <->  B R C ) )
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