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Theorem List for Intuitionistic Logic Explorer - 3801-3900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembrun 3801 The union of two binary relations. (Contributed by NM, 21-Dec-2008.)
 R  u.  S  R  S
 
Theorembrin 3802 The intersection of two relations. (Contributed by FL, 7-Oct-2008.)
 R  i^i  S  R  S
 
Theorembrdif 3803 The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.)
 R 
 \  S  R  S
 
Theoremsbcbrg 3804 Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 D  [.  ]. R C  [_  ]_ [_  ]_ R [_  ]_ C
 
Theoremsbcbr12g 3805* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
 D  [.  ]. R C  [_  ]_ R [_  ]_ C
 
Theoremsbcbr1g 3806* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
 D  [.  ]. R C  [_  ]_ R C
 
Theoremsbcbr2g 3807* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
 D  [.  ]. R C  R [_  ]_ C
 
2.1.23  Ordered-pair class abstractions (class builders)
 
Syntaxcopab 3808 Extend class notation to include ordered-pair class abstraction (class builder).
 { <. ,  >.  |  }
 
Syntaxcmpt 3809 Extend the definition of a class to include maps-to notation for defining a function via a rule.
 |->
 
Definitiondf-opab 3810* Define the class abstraction of a collection of ordered pairs. Definition 3.3 of [Monk1] p. 34. Usually and are distinct, although the definition doesn't strictly require it. The brace notation is called "class abstraction" by Quine; it is also (more commonly) called a "class builder" in the literature. (Contributed by NM, 4-Jul-1994.)

 { <. ,  >.  |  }  {  | 
 <. ,  >.  }
 
Definitiondf-mpt 3811* Define maps-to notation for defining a function via a rule. Read as "the function defined by the map from (in ) to ." The class expression is the value of the function at and normally contains the variable . Similar to the definition of mapping in [ChoquetDD] p. 2. (Contributed by NM, 17-Feb-2008.)
 |->  { <. ,  >.  |  }
 
Theoremopabss 3812* The collection of ordered pairs in a class is a subclass of it. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

 { <. ,  >.  |  R }  C_  R
 
Theoremopabbid 3813 Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

 F/   &     F/   &       =>     { <. ,  >.  |  }  { <. ,  >.  |  }
 
Theoremopabbidv 3814* Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (Contributed by NM, 15-May-1995.)
   =>     { <. ,  >.  |  }  { <. ,  >.  |  }
 
Theoremopabbii 3815 Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.)
   =>     { <. ,  >.  |  }  { <. ,  >.  |  }
 
Theoremnfopab 3816* Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999.) (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 11-Jul-2011.)

 F/   =>     F/_ { <. ,  >.  |  }
 
Theoremnfopab1 3817 The first abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
 F/_ { <. ,  >.  |  }
 
Theoremnfopab2 3818 The second abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
 F/_ { <. ,  >.  |  }
 
Theoremcbvopab 3819* Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.)

 F/   &     F/   &     F/   &     F/   &       =>    
 { <. ,  >.  |  }  {
 <. ,  >.  |  }
 
Theoremcbvopabv 3820* Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 15-Oct-1996.)
   =>    
 { <. ,  >.  |  }  {
 <. ,  >.  |  }
 
Theoremcbvopab1 3821* Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 6-Oct-2004.) (Revised by Mario Carneiro, 14-Oct-2016.)

 F/   &     F/   &       =>     { <. ,  >.  |  }  {
 <. ,  >.  |  }
 
Theoremcbvopab2 3822* Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.)

 F/   &     F/   &       =>     { <. ,  >.  |  }  {
 <. ,  >.  |  }
 
Theoremcbvopab1s 3823* Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 31-Jul-2003.)

 { <. ,  >.  |  }  {
 <. ,  >.  | 
 }
 
Theoremcbvopab1v 3824* Rule used to change the first bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
   =>    
 { <. ,  >.  |  }  {
 <. ,  >.  |  }
 
Theoremcbvopab2v 3825* Rule used to change the second bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 2-Sep-1999.)
   =>    
 { <. ,  >.  |  }  {
 <. ,  >.  |  }
 
Theoremcsbopabg 3826* Move substitution into a class abstraction. (Contributed by NM, 6-Aug-2007.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
 V  [_  ]_
 { <. ,  >.  |  }  {
 <. ,  >.  | 
 [.  ]. }
 
Theoremunopab 3827 Union of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)
 { <. ,  >.  |  }  u.  {
 <. ,  >.  |  }  { <. ,  >.  |  }
 
Theoremmpteq12f 3828 An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
 C  D  |->  C  |->  D
 
Theoremmpteq12dva 3829* An equality inference for the maps to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)
 C   &     D   =>     |->  C  |->  D
 
Theoremmpteq12dv 3830* An equality inference for the maps to notation. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 16-Dec-2013.)
 C   &     D   =>     |->  C  |->  D
 
Theoremmpteq12 3831* An equality theorem for the maps to notation. (Contributed by NM, 16-Dec-2013.)
 C  D  |->  C  |->  D
 
Theoremmpteq1 3832* An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
 |->  C  |->  C
 
Theoremmpteq1d 3833* An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 11-Jun-2016.)
   =>     |->  C  |->  C
 
Theoremmpteq2ia 3834 An equality inference for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
 C   =>     |->  |->  C
 
Theoremmpteq2i 3835 An equality inference for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
 C   =>     |->  |->  C
 
Theoremmpteq12i 3836 An equality inference for the maps to notation. (Contributed by Scott Fenton, 27-Oct-2010.) (Revised by Mario Carneiro, 16-Dec-2013.)
 C   &     D   =>     |->  C  |->  D
 
Theoremmpteq2da 3837 Slightly more general equality inference for the maps to notation. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.)

 F/   &     C   =>     |->  |->  C
 
Theoremmpteq2dva 3838* Slightly more general equality inference for the maps to notation. (Contributed by Scott Fenton, 25-Apr-2012.)
 C   =>     |->  |->  C
 
Theoremmpteq2dv 3839* An equality inference for the maps to notation. (Contributed by Mario Carneiro, 23-Aug-2014.)
 C   =>     |->  |->  C
 
Theoremnfmpt 3840* Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
 F/_   &     F/_   =>     F/_  |->
 
Theoremnfmpt1 3841 Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.)
 F/_  |->
 
Theoremcbvmpt 3842* Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 11-Sep-2011.)
 F/_   &     F/_ C   &     C   =>     |->  |->  C
 
Theoremcbvmptv 3843* Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by Mario Carneiro, 19-Feb-2013.)
 C   =>     |->  |->  C
 
Theoremmptv 3844* Function with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.)
 _V  |->  { <. ,  >.  |  }
 
2.1.24  Transitive classes
 
Syntaxwtr 3845 Extend wff notation to include transitive classes. Notation from [TakeutiZaring] p. 35.
 Tr
 
Definitiondf-tr 3846 Define the transitive class predicate. Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 3847 (which is suggestive of the word "transitive"), dftr3 3849, dftr4 3850, and dftr5 3848. The term "complete" is used instead of "transitive" in Definition 3 of [Suppes] p. 130. (Contributed by NM, 29-Aug-1993.)
 Tr  U.  C_
 
Theoremdftr2 3847* An alternate way of defining a transitive class. Exercise 7 of [TakeutiZaring] p. 40. (Contributed by NM, 24-Apr-1994.)
 Tr
 
Theoremdftr5 3848* An alternate way of defining a transitive class. (Contributed by NM, 20-Mar-2004.)
 Tr
 
Theoremdftr3 3849* An alternate way of defining a transitive class. Definition 7.1 of [TakeutiZaring] p. 35. (Contributed by NM, 29-Aug-1993.)
 Tr  C_
 
Theoremdftr4 3850 An alternate way of defining a transitive class. Definition of [Enderton] p. 71. (Contributed by NM, 29-Aug-1993.)
 Tr  C_  ~P
 
Theoremtreq 3851 Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.)
 Tr  Tr
 
Theoremtrel 3852 In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 Tr  C  C
 
Theoremtrel3 3853 In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.)
 Tr  C  C  D  D
 
Theoremtrss 3854 An element of a transitive class is a subset of the class. (Contributed by NM, 7-Aug-1994.)
 Tr  C_
 
Theoremtrin 3855 The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
 Tr 
 Tr  Tr  i^i
 
Theoremtr0 3856 The empty set is transitive. (Contributed by NM, 16-Sep-1993.)

 Tr  (/)
 
Theoremtrv 3857 The universe is transitive. (Contributed by NM, 14-Sep-2003.)

 Tr  _V
 
Theoremtriun 3858* The indexed union of a class of transitive sets is transitive. (Contributed by Mario Carneiro, 16-Nov-2014.)
 Tr  Tr  U_
 
Theoremtruni 3859* The union of a class of transitive sets is transitive. Exercise 5(a) of [Enderton] p. 73. (Contributed by Scott Fenton, 21-Feb-2011.) (Proof shortened by Mario Carneiro, 26-Apr-2014.)
 Tr  Tr  U.
 
Theoremtrint 3860* The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. (Contributed by Scott Fenton, 25-Feb-2011.)
 Tr  Tr  |^|
 
Theoremtrintssm 3861* If is transitive and inhabited, then  |^| is a subset of . (Contributed by Jim Kingdon, 22-Aug-2018.)
 Tr  |^|  C_
 
Theoremtrint0m 3862* Any inhabited transitive class includes its intersection. Similar to Exercise 2 in [TakeutiZaring] p. 44. (Contributed by Jim Kingdon, 22-Aug-2018.)
 Tr  |^|  C_
 
2.2  IZF Set Theory - add the Axioms of Collection and Separation
 
2.2.1  Introduce the Axiom of Collection
 
Axiomax-coll 3863* Axiom of Collection. Axiom 7 of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). It is similar to bnd 3916 but uses a freeness hypothesis in place of one of the distinct variable constraints. (Contributed by Jim Kingdon, 23-Aug-2018.)

 F/ b   =>     a  b  a  b
 
Theoremrepizf 3864* Axiom of Replacement. Axiom 7' of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). In our context this is not an axiom, but a theorem proved from ax-coll 3863. It is identical to zfrep6 3865 except for the choice of a freeness hypothesis rather than a distinct variable constraint between  b and . (Contributed by Jim Kingdon, 23-Aug-2018.)

 F/ b   =>     a  b  a  b
 
Theoremzfrep6 3865* A version of the Axiom of Replacement. Normally would have free variables and . Axiom 6 of [Kunen] p. 12. The Separation Scheme ax-sep 3866 cannot be derived from this version and must be stated as a separate axiom in an axiom system (such as Kunen's) that uses this version. (Contributed by NM, 10-Oct-2003.)
 
2.2.2  Introduce the Axiom of Separation
 
Axiomax-sep 3866* The Axiom of Separation of IZF set theory. Axiom 6 of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed, and with a  F/ condition replaced by a distinct variable constraint between and ).

The Separation Scheme is a weak form of Frege's Axiom of Comprehension, conditioning it (with ) so that it asserts the existence of a collection only if it is smaller than some other collection that already exists. This prevents Russell's paradox ru 2757. In some texts, this scheme is called "Aussonderung" or the Subset Axiom.

(Contributed by NM, 11-Sep-2006.)

 
Theoremaxsep2 3867* A less restrictive version of the Separation Scheme ax-sep 3866, where variables and can both appear free in the wff , which can therefore be thought of as  , . This version was derived from the more restrictive ax-sep 3866 with no additional set theory axioms. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
 
Theoremzfauscl 3868* Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 3866, we invoke the Axiom of Extensionality (indirectly via vtocl 2602), which is needed for the justification of class variable notation. (Contributed by NM, 5-Aug-1993.)
 _V   =>   
 
Theorembm1.3ii 3869* Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 3866. Similar to Theorem 1.3ii of [BellMachover] p. 463. (Contributed by NM, 5-Aug-1993.)
   =>   
 
Theorema9evsep 3870* Derive a weakened version of ax-i9 1420, where and must be distinct, from Separation ax-sep 3866 and Extensionality ax-ext 2019. The theorem also holds (ax9vsep 3871), but in intuitionistic logic is stronger. (Contributed by Jim Kingdon, 25-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
 
Theoremax9vsep 3871* Derive a weakened version of ax-9 1421, where and must be distinct, from Separation ax-sep 3866 and Extensionality ax-ext 2019. In intuitionistic logic a9evsep 3870 is stronger and also holds. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 
2.2.3  Derive the Null Set Axiom
 
Theoremzfnuleu 3872* Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2022 to strengthen the hypothesis in the form of axnul 3873). (Contributed by NM, 22-Dec-2007.)
   =>   
 
Theoremaxnul 3873* The Null Set Axiom of ZF set theory: there exists a set with no elements. Axiom of Empty Set of [Enderton] p. 18. In some textbooks, this is presented as a separate axiom; here we show it can be derived from Separation ax-sep 3866. This version of the Null Set Axiom tells us that at least one empty set exists, but does not tell us that it is unique - we need the Axiom of Extensionality to do that (see zfnuleu 3872).

This theorem should not be referenced by any proof. Instead, use ax-nul 3874 below so that the uses of the Null Set Axiom can be more easily identified. (Contributed by Jeff Hoffman, 3-Feb-2008.) (Revised by NM, 4-Feb-2008.) (New usage is discouraged.) (Proof modification is discouraged.)

 
Axiomax-nul 3874* The Null Set Axiom of IZF set theory. It was derived as axnul 3873 above and is therefore redundant, but we state it as a separate axiom here so that its uses can be identified more easily. Axiom 4 of [Crosilla] p. "Axioms of CZF and IZF". (Contributed by NM, 7-Aug-2003.)
 
Theorem0ex 3875 The Null Set Axiom of ZF set theory: the empty set exists. Corollary 5.16 of [TakeutiZaring] p. 20. For the unabbreviated version, see ax-nul 3874. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 (/)  _V
 
Theoremcsbexga 3876 The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
 V  W  [_  ]_  _V
 
Theoremcsbexa 3877 The existence of proper substitution into a class. (Contributed by NM, 7-Aug-2007.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 _V   &     _V   =>     [_  ]_  _V
 
2.2.4  Theorems requiring subset and intersection existence
 
Theoremnalset 3878* No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.)
 
Theoremvprc 3879 The universal class is not a member of itself (and thus is not a set). Proposition 5.21 of [TakeutiZaring] p. 21; our proof, however, does not depend on the Axiom of Regularity. (Contributed by NM, 23-Aug-1993.)
 _V  _V
 
Theoremnvel 3880 The universal class doesn't belong to any class. (Contributed by FL, 31-Dec-2006.)
 _V
 
Theoremvnex 3881 The universal class does not exist. (Contributed by NM, 4-Jul-2005.)
 _V
 
Theoreminex1 3882 Separation Scheme (Aussonderung) using class notation. Compare Exercise 4 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)
 _V   =>     i^i  _V
 
Theoreminex2 3883 Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.)
 _V   =>     i^i  _V
 
Theoreminex1g 3884 Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)
 V  i^i  _V
 
Theoremssex 3885 The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22. This is one way to express the Axiom of Separation ax-sep 3866 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)
 _V   =>     C_  _V
 
Theoremssexi 3886 The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.)
 _V   &     C_    =>     _V
 
Theoremssexg 3887 The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22 (generalized). (Contributed by NM, 14-Aug-1994.)
 C_  C  _V
 
Theoremssexd 3888 A subclass of a set is a set. Deduction form of ssexg 3887. (Contributed by David Moews, 1-May-2017.)
 C   &     C_    =>     _V
 
Theoremdifexg 3889 Existence of a difference. (Contributed by NM, 26-May-1998.)
 V  \  _V
 
Theoremzfausab 3890* Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.)
 _V   =>    
 {  |  }  _V
 
Theoremrabexg 3891* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.)
 V  {  |  }  _V
 
Theoremrabex 3892* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.)
 _V   =>    
 {  |  }  _V
 
Theoremelssabg 3893* Membership in a class abstraction involving a subset. Unlike elabg 2682, does not have to be a set. (Contributed by NM, 29-Aug-2006.)
   =>     V  {  |  C_  }  C_
 
Theoreminteximm 3894* The intersection of an inhabited class exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
 |^|  _V
 
Theoremintexr 3895 If the intersection of a class exists, the class is non-empty. (Contributed by Jim Kingdon, 27-Aug-2018.)
 |^|  _V  =/=  (/)
 
Theoremintnexr 3896 If a class intersection is the universe, it is not a set. In classical logic this would be an equivalence. (Contributed by Jim Kingdon, 27-Aug-2018.)
 |^|  _V  |^|  _V
 
Theoremintexabim 3897 The intersection of an inhabited class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
 |^| {  | 
 }  _V
 
Theoremintexrabim 3898 The intersection of an inhabited restricted class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
 |^| {  |  }  _V
 
Theoremiinexgm 3899* The existence of an indexed union. is normally a free-variable parameter in , which should be read . (Contributed by Jim Kingdon, 28-Aug-2018.)
 C  |^|_  _V
 
Theoreminuni 3900* The intersection of a union  U. with a class is equal to the union of the intersections of each element of with . (Contributed by FL, 24-Mar-2007.)
 U.  i^i  U. {  |  i^i  }
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