HomeHome Intuitionistic Logic Explorer
Theorem List (p. 40 of 142)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 3901-4000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnfiunya 3901* 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 3902* 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 3903 Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.)
 |-  F/_ x U_ x  e.  A  B
 
Theoremnfii1 3904 Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.)
 |-  F/_ x |^|_ x  e.  A  B
 
Theoremdfiun2g 3905* 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 3906* 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 3907* 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 3908* 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 3909* 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 3910* 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 3911* 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 3912* 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 3913* 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 3914* 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 3915* 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 3916 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 3917* 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 3918* 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 3827. (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 3919* 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 3920* 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 3921* 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 3922 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 3923* 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 3924* 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 3925 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 3926* 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 3927* Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)
 |- 
 |^| A  =  |^|_ x  e.  A  x
 
Theoremiunid 3928* An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)
 |-  U_ x  e.  A  { x }  =  A
 
Theoremiun0 3929 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 3930 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 3931 An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)
 |-  |^|_ x  e.  (/)  A  =  _V
 
Theoremviin 3932* Indexed intersection with a universal index class. (Contributed by NM, 11-Sep-2008.)
 |-  |^|_ x  e.  _V  A  =  { y  |  A. x  y  e.  A }
 
Theoremiunn0m 3933* 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 3934* Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.)
 |-  |^|_ x  e.  A  {
 y  |  ph }  =  { y  |  A. x  e.  A  ph }
 
Theoremiinrabm 3935* 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 3936* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3926 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 3937* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3926 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 3938* 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 3939* 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 3940* 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 3941* 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 3942* 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 3943* 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 3944* Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  S )  =  A )
 
Theoremriinm 3945* 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 3946* 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 3947* 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 3948* 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 3949* 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 3950* 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 3951 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 3952 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 3953* 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 3954* 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 3955* 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 3956* 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 3957 Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
 |-  ( A  C_  ~P B  <->  U. A  C_  B )
 
Theorempwssb 3958* 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 3959 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 3960* 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 }
 
Theorempwpwssunieq 3961* 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
 
Theoremelpwuni 3962 Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
 |-  ( B  e.  A  ->  ( A  C_  ~P B  <->  U. A  =  B ) )
 
Theoremiinpw 3963* 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 3964* 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 3965* 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 )
 
2.1.21  Disjointness
 
Syntaxwdisj 3966 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 3967* 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 3968* 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 3969 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 3970 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 3971* 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 3972* 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 3973* 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 3974* 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 3975* 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 3976* 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 3977* 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 )
 
Theoremnfdisjv 3978* Bound-variable hypothesis builder for disjoint collection. (Contributed by Jim Kingdon, 19-Aug-2018.)
 |-  F/_ y A   &    |-  F/_ y B   =>    |-  F/ yDisj  x  e.  A  B
 
Theoremnfdisj1 3979 Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- 
 F/ xDisj  x  e.  A  B
 
Theoremdisjnim 3980* If a collection  B ( i ) for  i  e.  A is disjoint, then pairs are disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Jim Kingdon, 6-Oct-2022.)
 |-  ( i  =  j 
 ->  B  =  C )   =>    |-  (Disj  i  e.  A  B  ->  A. i  e.  A  A. j  e.  A  ( i  =/=  j  ->  ( B  i^i  C )  =  (/) ) )
 
Theoremdisjnims 3981* If a collection  B ( i ) for  i  e.  A is disjoint, then pairs are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by Jim Kingdon, 7-Oct-2022.)
 |-  (Disj  x  e.  A  B  ->  A. i  e.  A  A. j  e.  A  ( i  =/=  j  ->  ( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/) ) )
 
Theoremdisji2 3982* 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 )  =  (/) )
 
Theoreminvdisj 3983* 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 3984* 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 )  =  (/) )
 
Theoremsndisj 3985 Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- Disj  x  e.  A  { x }
 
Theorem0disj 3986 Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- Disj  x  e.  A  (/)
 
Theoremdisjxsn 3987* A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- Disj  x  e.  { A } B
 
Theoremdisjx0 3988 An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 |- Disj  x  e.  (/)  B
 
2.1.22  Binary relations
 
Syntaxwbr 3989 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.
 wff  A R B
 
Definitiondf-br 3990 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. 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 (see for example iprc 4879). (Contributed by NM, 31-Dec-1993.)
 |-  ( A R B  <->  <. A ,  B >.  e.  R )
 
Theorembreq 3991 Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)
 |-  ( R  =  S  ->  ( A R B  <->  A S B ) )
 
Theorembreq1 3992 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
 |-  ( A  =  B  ->  ( A R C  <->  B R C ) )
 
Theorembreq2 3993 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
 |-  ( A  =  B  ->  ( C R A  <->  C R B ) )
 
Theorembreq12 3994 Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A R C 
 <->  B R D ) )
 
Theorembreqi 3995 Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.)
 |-  R  =  S   =>    |-  ( A R B 
 <->  A S B )
 
Theorembreq1i 3996 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
 |-  A  =  B   =>    |-  ( A R C 
 <->  B R C )
 
Theorembreq2i 3997 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
 |-  A  =  B   =>    |-  ( C R A 
 <->  C R B )
 
Theorembreq12i 3998 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A R C  <->  B R D )
 
Theorembreq1d 3999 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A R C  <->  B R C ) )
 
Theorembreqd 4000 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C A D  <->  C B D ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14113
  Copyright terms: Public domain < Previous  Next >