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Theorem List for Intuitionistic Logic Explorer - 3701-3800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremuniiun 3701* Class union in terms of indexed union. Definition in [Stoll] p. 43. (Contributed by NM, 28-Jun-1998.)

 U.  U_
 
Theoremintiin 3702* Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)

 |^|  |^|_
 
Theoremiunid 3703* An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)
 U_  { }
 
Theoremiun0 3704 An indexed union of the empty set is empty. (Contributed by NM, 26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 U_  (/)  (/)
 
Theorem0iun 3705 An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 U_  (/)  (/)
 
Theorem0iin 3706 An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)
 |^|_  (/)  _V
 
Theoremviin 3707* Indexed intersection with a universal index class. (Contributed by NM, 11-Sep-2008.)
 |^|_  _V  {  |  }
 
Theoremiunn0m 3708* There is an inhabited class in an indexed collection iff the indexed union of them is inhabited. (Contributed by Jim Kingdon, 16-Aug-2018.)
 U_
 
Theoremiinab 3709* Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.)
 |^|_  {  |  }  {  |  }
 
Theoremiinrabm 3710* Indexed intersection of a restricted class builder. (Contributed by Jim Kingdon, 16-Aug-2018.)
 |^|_  {  |  }  {  |  }
 
Theoremiunin2 3711* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3701 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.)
 U_  i^i  C  i^i  U_  C
 
Theoremiunin1 3712* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3701 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)
 U_  C  i^i  U_  C  i^i
 
Theoremiundif2ss 3713* Indexed union of class difference. Compare to theorem "De Morgan's laws" in [Enderton] p. 31. (Contributed by Jim Kingdon, 17-Aug-2018.)
 U_  \  C  C_  \  |^|_  C
 
Theorem2iunin 3714* Rearrange indexed unions over intersection. (Contributed by NM, 18-Dec-2008.)
 U_  U_  C  i^i  D  U_  C  i^i  U_  D
 
Theoremiindif2m 3715* Indexed intersection of class difference. Compare to Theorem "De Morgan's laws" in [Enderton] p. 31. (Contributed by Jim Kingdon, 17-Aug-2018.)
 |^|_  \  C  \ 
 U_  C
 
Theoremiinin2m 3716* Indexed intersection of intersection. Compare to Theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Jim Kingdon, 17-Aug-2018.)
 |^|_  i^i  C  i^i  |^|_  C
 
Theoremiinin1m 3717* Indexed intersection of intersection. Compare to Theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Jim Kingdon, 17-Aug-2018.)
 |^|_  C  i^i  |^|_  C  i^i
 
Theoremelriin 3718* Elementhood in a relative intersection. (Contributed by Mario Carneiro, 30-Dec-2016.)
 i^i  |^|_  X  S  X  S
 
Theoremriin0 3719* Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 X  (/)  i^i  |^|_  X  S
 
Theoremriinm 3720* Relative intersection of an inhabited family. (Contributed by Jim Kingdon, 19-Aug-2018.)
 X  S  C_  X 
 i^i  |^|_  X  S  |^|_  X  S
 
Theoremiinxsng 3721* 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.)
 C   =>     V  |^|_  { }  C
 
Theoremiinxprg 3722* Indexed intersection with an unordered pair index. (Contributed by NM, 25-Jan-2012.)
 C  D   &     C  E   =>     V  W  |^|_  { ,  } C  D  i^i  E
 
Theoremiunxsng 3723* A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.)
 C   =>     V  U_  { }  C
 
Theoremiunxsn 3724* 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.)
 _V   &     C   =>     U_  { }  C
 
Theoremiunun 3725 Separate a union in an indexed union. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
 U_  u.  C  U_  u.  U_  C
 
Theoremiunxun 3726 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_  u.  C  U_  C  u.  U_  C
 
Theoremiunxiun 3727* Separate an indexed union in the index of an indexed union. (Contributed by Mario Carneiro, 5-Dec-2016.)
 U_  U_  C  U_  U_  C
 
Theoremiinuniss 3728* 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.)
 u.  |^|  C_  |^|_  u.
 
Theoremiununir 3729* 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.)
 u.  U.  U_  u.  (/)  (/)
 
Theoremsspwuni 3730 Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
 C_  ~P  U.  C_
 
Theorempwssb 3731* Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006.)
 C_  ~P  C_
 
Theoremelpwuni 3732 Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
 C_  ~P  U.
 
Theoremiinpw 3733* 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 |^|  |^|_  ~P
 
Theoremiunpwss 3734* 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_  ~P  C_  ~P U.
 
Theoremrintm 3735* Relative intersection of an inhabited class. (Contributed by Jim Kingdon, 19-Aug-2018.)
 X  C_  ~P  X  i^i  |^| X  |^| X
 
2.1.21  Disjointness
 
Syntaxwdisj 3736 Extend wff notation to include the statement that a family of classes , for , is a disjoint family.
Disj
 
Definitiondf-disj 3737* A collection of classes is disjoint when for each element , it is in for at most one . (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by NM, 16-Jun-2017.)
Disj
 
Theoremdfdisj2 3738* Alternate definition for disjoint classes. (Contributed by NM, 17-Jun-2017.)
Disj
 
Theoremdisjss2 3739 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.)
 C_  C Disj  C Disj
 
Theoremdisjeq2 3740 Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
 C Disj Disj  C
 
Theoremdisjeq2dv 3741* Equality deduction for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
 C   =>    Disj Disj  C
 
Theoremdisjss1 3742* A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
 C_ Disj  C Disj  C
 
Theoremdisjeq1 3743* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj  C Disj  C
 
Theoremdisjeq1d 3744* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
   =>    Disj  C Disj  C
 
Theoremdisjeq12d 3745* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
   &     C  D   =>    Disj  C Disj  D
 
Theoremcbvdisj 3746* Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
 F/_   &     F/_ C   &     C   =>    Disj Disj  C
 
Theoremcbvdisjv 3747* Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 11-Dec-2016.)
 C   =>    Disj Disj  C
 
Theoremnfdisjv 3748* Bound-variable hypothesis builder for disjoint collection. (Contributed by Jim Kingdon, 19-Aug-2018.)
 F/_   &     F/_   =>     F/Disj
 
Theoremnfdisj1 3749 Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)

 F/Disj
 
Theoreminvdisj 3750* If there is a function  C such that  C for all , then the sets for distinct are disjoint. (Contributed by Mario Carneiro, 10-Dec-2016.)
 C Disj
 
Theoremsndisj 3751 Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj  { }
 
Theorem0disj 3752 Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj  (/)
 
Theoremdisjxsn 3753* A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj  { }
 
Theoremdisjx0 3754 An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj  (/)
 
2.1.22  Binary relations
 
Syntaxwbr 3755 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.
 R
 
Definitiondf-br 3756 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 and/or 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 4543). (Contributed by NM, 31-Dec-1993.)
 R  <. ,  >.  R
 
Theorembreq 3757 Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)
 R  S  R  S
 
Theorembreq1 3758 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
 R C  R C
 
Theorembreq2 3759 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
 C R  C R
 
Theorembreq12 3760 Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)
 C  D  R C  R D
 
Theorembreqi 3761 Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.)
 R  S   =>     R  S
 
Theorembreq1i 3762 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
   =>     R C  R C
 
Theorembreq2i 3763 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
   =>     C R  C R
 
Theorembreq12i 3764 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
   &     C  D   =>     R C  R D
 
Theorembreq1d 3765 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
   =>     R C  R C
 
Theorembreqd 3766 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
   =>     C D  C D
 
Theorembreq2d 3767 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
   =>     C R  C R
 
Theorembreq12d 3768 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
   &     C  D   =>     R C  R D
 
Theorembreq123d 3769 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
   &     R  S   &     C  D   =>     R C  S D
 
Theorembreqan12d 3770 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
   &     C  D   =>     R C  R D
 
Theorembreqan12rd 3771 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
   &     C  D   =>     R C  R D
 
Theoremnbrne1 3772 Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
 R  R C  =/=  C
 
Theoremnbrne2 3773 Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
 R C  R C  =/=
 
Theoremeqbrtri 3774 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
   &     R C   =>     R C
 
Theoremeqbrtrd 3775 Substitution of equal classes into a binary relation. (Contributed by NM, 8-Oct-1999.)
   &     R C   =>     R C
 
Theoremeqbrtrri 3776 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
   &     R C   =>     R C
 
Theoremeqbrtrrd 3777 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
   &     R C   =>     R C
 
Theorembreqtri 3778 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
 R   &     C   =>     R C
 
Theorembreqtrd 3779 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
 R   &     C   =>     R C
 
Theorembreqtrri 3780 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
 R   &     C    =>     R C
 
Theorembreqtrrd 3781 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
 R   &     C    =>     R C
 
Theorem3brtr3i 3782 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
 R   &     C   &     D   =>     C R D
 
Theorem3brtr4i 3783 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
 R   &     C    &     D    =>     C R D
 
Theorem3brtr3d 3784 Substitution of equality into both sides of a binary relation. (Contributed by NM, 18-Oct-1999.)
 R   &     C   &     D   =>     C R D
 
Theorem3brtr4d 3785 Substitution of equality into both sides of a binary relation. (Contributed by NM, 21-Feb-2005.)
 R   &     C    &     D    =>     C R D
 
Theorem3brtr3g 3786 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
 R   &     C   &     D   =>     C R D
 
Theorem3brtr4g 3787 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
 R   &     C    &     D    =>     C R D
 
Theoremsyl5eqbr 3788 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
   &     R C   =>     R C
 
Theoremsyl5eqbrr 3789 B chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.)
   &     R C   =>     R C
 
Theoremsyl5breq 3790 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
 R   &     C   =>     R C
 
Theoremsyl5breqr 3791 B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
 R   &     C    =>     R C
 
Theoremsyl6eqbr 3792 A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)
   &     R C   =>     R C
 
Theoremsyl6eqbrr 3793 A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)
   &     R C   =>     R C
 
Theoremsyl6breq 3794 A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
 R   &     C   =>     R C
 
Theoremsyl6breqr 3795 A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
 R   &     C    =>     R C
 
Theoremssbrd 3796 Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.)
 C_    =>     C D  C D
 
Theoremssbri 3797 Inference from a subclass relationship of binary relations. (Contributed by NM, 28-Mar-2007.) (Revised by Mario Carneiro, 8-Feb-2015.)
 C_    =>     C D  C D
 
Theoremnfbrd 3798 Deduction version of bound-variable hypothesis builder nfbr 3799. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 14-Oct-2016.)
 F/_   &     F/_ R   &     F/_   =>     F/  R
 
Theoremnfbr 3799 Bound-variable hypothesis builder for binary relation. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
 F/_   &     F/_ R   &     F/_   =>    
 F/  R
 
Theorembrab1 3800* Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.)
 R  {  |  R }
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