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Theorem List for Intuitionistic Logic Explorer - 3901-4000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelpw2g 3901 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.)
 V  ~P  C_
 
Theoremelpw2 3902 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 11-Oct-2007.)
 _V   =>     ~P  C_
 
Theorempwnss 3903 The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 V  ~P  C_
 
Theorempwne 3904 No set equals its power set. The sethood antecedent is necessary; compare pwv 3570. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
 V  ~P  =/=
 
Theoremrepizf2lem 3905 Lemma for repizf2 3906. If we have a function-like proposition which provides at most one value of for each in a set , we can change "at most one" to "exactly one" by restricting the values of to those values for which the proposition provides a value of . (Contributed by Jim Kingdon, 7-Sep-2018.)
 {  |  }
 
Theoremrepizf2 3906* Replacement. This version of replacement is stronger than repizf 3864 in the sense that does not need to map all values of in to a value of . The resulting set contains those elements for which there is a value of and in that sense, this theorem combines repizf 3864 with ax-sep 3866. Another variation would be  {  |  }  _V but we don't have a proof of that yet. (Contributed by Jim Kingdon, 7-Sep-2018.)

 F/   =>     {  |  }
 
2.2.5  Theorems requiring empty set existence
 
Theoremclass2seteq 3907* Equality theorem for classes and sets . (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.)
 V  {  |  _V }
 
Theorem0elpw 3908 Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)
 (/)  ~P
 
Theorem0nep0 3909 The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)
 (/)  =/=  { (/) }
 
Theorem0inp0 3910 Something cannot be equal to both the null set and the power set of the null set. (Contributed by NM, 30-Sep-2003.)
 (/)  { (/) }
 
Theoremunidif0 3911 The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)

 U.  \  { (/) } 
 U.
 
Theoremiin0imm 3912* An indexed intersection of the empty set, with an inhabited index set, is empty. (Contributed by Jim Kingdon, 29-Aug-2018.)
 |^|_  (/)  (/)
 
Theoremiin0r 3913* If an indexed intersection of the empty set is empty, the index set is non-empty. (Contributed by Jim Kingdon, 29-Aug-2018.)
 |^|_  (/)  (/)  =/=  (/)
 
Theoremintv 3914 The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.)

 |^| _V  (/)
 
Theoremaxpweq 3915* Two equivalent ways to express the Power Set Axiom. Note that ax-pow 3918 is not used by the proof. (Contributed by NM, 22-Jun-2009.)
 _V   =>     ~P  _V
 
2.2.6  Collection principle
 
Theorembnd 3916* A very strong generalization of the Axiom of Replacement (compare zfrep6 3865). Its strength lies in the rather profound fact that  , does not have to be a "function-like" wff, as it does in the standard Axiom of Replacement. This theorem is sometimes called the Boundedness Axiom. In the context of IZF, it is just a slight variation of ax-coll 3863. (Contributed by NM, 17-Oct-2004.)
 
Theorembnd2 3917* A variant of the Boundedness Axiom bnd 3916 that picks a subset out of a possibly proper class in which a property is true. (Contributed by NM, 4-Feb-2004.)
 _V   =>     C_
 
2.3  IZF Set Theory - add the Axioms of Power Sets and Pairing
 
2.3.1  Introduce the Axiom of Power Sets
 
Axiomax-pow 3918* Axiom of Power Sets. An axiom of Intuitionistic Zermelo-Fraenkel set theory. It states that a set exists that includes the power set of a given set i.e. contains every subset of . This is Axiom 8 of [Crosilla] p. "Axioms of CZF and IZF" except (a) unnecessary quantifiers are removed, and (b) Crosilla has a biconditional rather than an implication (but the two are equivalent by bm1.3ii 3869).

The variant axpow2 3920 uses explicit subset notation. A version using class notation is pwex 3923. (Contributed by NM, 5-Aug-1993.)

 
Theoremzfpow 3919* Axiom of Power Sets expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
 
Theoremaxpow2 3920* A variant of the Axiom of Power Sets ax-pow 3918 using subset notation. Problem in {BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
 C_
 
Theoremaxpow3 3921* A variant of the Axiom of Power Sets ax-pow 3918. For any set , there exists a set whose members are exactly the subsets of i.e. the power set of . Axiom Pow of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
 C_
 
Theoremel 3922* Every set is an element of some other set. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 
Theorempwex 3923 Power set axiom expressed in class notation. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 _V   =>    
 ~P  _V
 
Theorempwexg 3924 Power set axiom expressed in class notation, with the sethood requirement as an antecedent. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Oct-2003.)
 V  ~P  _V
 
Theoremabssexg 3925* Existence of a class of subsets. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 V  {  |  C_  }  _V
 
TheoremsnexgOLD 3926 A singleton whose element exists is a set. The  _V case of Theorem 7.12 of [Quine] p. 51, proved using only Extensionality, Power Set, and Separation. Replacement is not needed. This is a special case of snexg 3927 and new proofs should use snexg 3927 instead. (Contributed by Jim Kingdon, 26-Jan-2019.) (New usage is discouraged.) (Proof modification is discouraged.) TODO: replace its uses by uses of snexg 3927 and then remove it.
 _V  { }  _V
 
Theoremsnexg 3927 A singleton whose element exists is a set. The  _V case of Theorem 7.12 of [Quine] p. 51, proved using only Extensionality, Power Set, and Separation. Replacement is not needed. (Contributed by Jim Kingdon, 1-Sep-2018.)
 V  { }  _V
 
Theoremsnex 3928 A singleton whose element exists is a set. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.)
 _V   =>    
 { }  _V
 
Theoremsnexprc 3929 A singleton whose element is a proper class is a set. The  _V case of Theorem 7.12 of [Quine] p. 51, proved using only Extensionality, Power Set, and Separation. Replacement is not needed. (Contributed by Jim Kingdon, 1-Sep-2018.)
 _V  { }  _V
 
Theoremp0ex 3930 The power set of the empty set (the ordinal 1) is a set. (Contributed by NM, 23-Dec-1993.)

 { (/) }  _V
 
Theorempp0ex 3931  { (/) ,  { (/)
} } (the ordinal 2) is a set. (Contributed by NM, 5-Aug-1993.)

 { (/) ,  { (/) } }  _V
 
Theoremord3ex 3932 The ordinal number 3 is a set, proved without the Axiom of Union. (Contributed by NM, 2-May-2009.)

 { (/) ,  { (/) } ,  { (/) ,  { (/) } } }  _V
 
Theoremdtruarb 3933* At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). This theorem asserts the existence of two sets which do not equal each other; compare with dtruex 4237 in which we are given a set and go from there to a set which is not equal to it. (Contributed by Jim Kingdon, 2-Sep-2018.)
 
Theorempwuni 3934 A class is a subclass of the power class of its union. Exercise 6(b) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.)
 C_  ~P U.
 
2.3.2  Axiom of Pairing
 
Axiomax-pr 3935* The Axiom of Pairing of IZF set theory. Axiom 2 of [Crosilla] p. "Axioms of CZF and IZF", except (a) unnecessary quantifiers are removed, and (b) Crosilla has a biconditional rather than an implication (but the two are equivalent by bm1.3ii 3869). (Contributed by NM, 14-Nov-2006.)
 
Theoremzfpair2 3936 Derive the abbreviated version of the Axiom of Pairing from ax-pr 3935. (Contributed by NM, 14-Nov-2006.)

 { ,  }  _V
 
TheoremprexgOLD 3937 The Axiom of Pairing using class variables. Theorem 7.13 of [Quine] p. 51, but restricted to classes which exist. For proper classes, see prprc 3471, prprc1 3469, and prprc2 3470. This is a special case of prexg 3938 and new proofs should use prexg 3938 instead. (Contributed by Jim Kingdon, 25-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) TODO: replace its uses by uses of prexg 3938 and then remove it.
 _V  _V  { ,  }  _V
 
Theoremprexg 3938 The Axiom of Pairing using class variables. Theorem 7.13 of [Quine] p. 51, but restricted to classes which exist. For proper classes, see prprc 3471, prprc1 3469, and prprc2 3470. (Contributed by Jim Kingdon, 16-Sep-2018.)
 V  W  { ,  }  _V
 
Theoremsnelpwi 3939 A singleton of a set belongs to the power class of a class containing the set. (Contributed by Alan Sare, 25-Aug-2011.)
 { }  ~P
 
Theoremsnelpw 3940 A singleton of a set belongs to the power class of a class containing the set. (Contributed by NM, 1-Apr-1998.)
 _V   =>     { }  ~P
 
Theoremprelpwi 3941 A pair of two sets belongs to the power class of a class containing those two sets. (Contributed by Thierry Arnoux, 10-Mar-2017.)
 C  C  { ,  }  ~P C
 
Theoremrext 3942* A theorem similar to extensionality, requiring the existence of a singleton. Exercise 8 of [TakeutiZaring] p. 16. (Contributed by NM, 10-Aug-1993.)
 
Theoremsspwb 3943 Classes are subclasses if and only if their power classes are subclasses. Exercise 18 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)
 C_  ~P  C_  ~P
 
Theoremunipw 3944 A class equals the union of its power class. Exercise 6(a) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.) (Proof shortened by Alan Sare, 28-Dec-2008.)

 U. ~P
 
Theorempwel 3945 Membership of a power class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.)
 ~P  ~P ~P U.
 
Theorempwtr 3946 A class is transitive iff its power class is transitive. (Contributed by Alan Sare, 25-Aug-2011.) (Revised by Mario Carneiro, 15-Jun-2014.)
 Tr  Tr  ~P
 
Theoremssextss 3947* An extensionality-like principle defining subclass in terms of subsets. (Contributed by NM, 30-Jun-2004.)
 C_  C_  C_
 
Theoremssext 3948* An extensionality-like principle that uses the subset instead of the membership relation: two classes are equal iff they have the same subsets. (Contributed by NM, 30-Jun-2004.)
 C_  C_
 
Theoremnssssr 3949* Negation of subclass relationship. Compare nssr 2997. (Contributed by Jim Kingdon, 17-Sep-2018.)
 C_  C_  C_
 
Theorempweqb 3950 Classes are equal if and only if their power classes are equal. Exercise 19 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)
 ~P  ~P
 
Theoremintid 3951* The intersection of all sets to which a set belongs is the singleton of that set. (Contributed by NM, 5-Jun-2009.)
 _V   =>    
 |^| {  |  }  { }
 
Theoremeuabex 3952 The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.)
 {  |  }  _V
 
Theoremmss 3953* An inhabited class (even if proper) has an inhabited subset. (Contributed by Jim Kingdon, 17-Sep-2018.)
 C_
 
Theoremexss 3954* Restricted existence in a class (even if proper) implies restricted existence in a subset. (Contributed by NM, 23-Aug-2003.)
 C_
 
Theoremopexg 3955 An ordered pair of sets is a set. (Contributed by Jim Kingdon, 11-Jan-2019.)
 V  W  <. ,  >.  _V
 
TheoremopexgOLD 3956 An ordered pair of sets is a set. This is a special case of opexg 3955 and new proofs should use opexg 3955 instead. (Contributed by Jim Kingdon, 19-Sep-2018.) (New usage is discouraged.) (Proof modification is discouraged.) TODO: replace its uses by uses of opexg 3955 and then remove it.
 _V  _V  <. ,  >.  _V
 
Theoremopex 3957 An ordered pair of sets is a set. (Contributed by Jim Kingdon, 24-Sep-2018.) (Revised by Mario Carneiro, 24-May-2019.)
 _V   &     _V   =>     <. ,  >.  _V
 
Theoremotexg 3958 An ordered triple of sets is a set. (Contributed by Jim Kingdon, 19-Sep-2018.)
 U  V  C  W  <. ,  ,  C >.  _V
 
Theoremelop 3959 An ordered pair has two elements. Exercise 3 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
 _V   &     _V   &     C  _V   =>     <. ,  C >.  { }  { ,  C }
 
Theoremopi1 3960 One of the two elements in an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
 _V   &     _V   =>     { }  <. ,  >.
 
Theoremopi2 3961 One of the two elements of an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
 _V   &     _V   =>     { ,  }  <. ,  >.
 
2.3.3  Ordered pair theorem
 
Theoremopm 3962* An ordered pair is inhabited iff the arguments are sets. (Contributed by Jim Kingdon, 21-Sep-2018.)
 <. ,  >.  _V  _V
 
Theoremopnzi 3963 An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 3962). (Contributed by Mario Carneiro, 26-Apr-2015.)
 _V   &     _V   =>     <. ,  >.  =/=  (/)
 
Theoremopth1 3964 Equality of the first members of equal ordered pairs. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
 _V   &     _V   =>     <. ,  >.  <. C ,  D >.  C
 
Theoremopth 3965 The ordered pair theorem. If two ordered pairs are equal, their first elements are equal and their second elements are equal. Exercise 6 of [TakeutiZaring] p. 16. Note that  C and  D are not required to be sets due our specific ordered pair definition. (Contributed by NM, 28-May-1995.)
 _V   &     _V   =>     <. ,  >.  <. C ,  D >.  C  D
 
Theoremopthg 3966 Ordered pair theorem.  C and  D are not required to be sets under our specific ordered pair definition. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)
 V  W  <. ,  >.  <. C ,  D >.  C  D
 
Theoremopthg2 3967 Ordered pair theorem. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)
 C  V  D  W  <. ,  >.  <. C ,  D >.  C  D
 
Theoremopth2 3968 Ordered pair theorem. (Contributed by NM, 21-Sep-2014.)
 C  _V   &     D  _V   =>     <. ,  >.  <. C ,  D >.  C  D
 
Theoremotth2 3969 Ordered triple theorem, with triple express with ordered pairs. (Contributed by NM, 1-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)
 _V   &     _V   &     R  _V   =>     <. <. ,  >. ,  R >.  <. <. C ,  D >. ,  S >.  C  D  R  S
 
Theoremotth 3970 Ordered triple theorem. (Contributed by NM, 25-Sep-2014.) (Revised by Mario Carneiro, 26-Apr-2015.)
 _V   &     _V   &     R  _V   =>     <. ,  ,  R >.  <. C ,  D ,  S >.  C  D  R  S
 
Theoremeqvinop 3971* A variable introduction law for ordered pairs. Analog of Lemma 15 of [Monk2] p. 109. (Contributed by NM, 28-May-1995.)
 _V   &     C  _V   =>     <. ,  C >.  <. ,  >. 
 <. ,  >. 
 <. ,  C >.
 
Theoremcopsexg 3972* Substitution of class for ordered pair 
<. ,  >.. (Contributed by NM, 27-Dec-1996.) (Revised by Andrew Salmon, 11-Jul-2011.)
 <. ,  >.  <. ,  >.
 
Theoremcopsex2t 3973* Closed theorem form of copsex2g 3974. (Contributed by NM, 17-Feb-2013.)
 V  W  <. ,  >.  <. ,  >.
 
Theoremcopsex2g 3974* Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.)
   =>     V  W  <. ,  >.  <. ,  >.
 
Theoremcopsex4g 3975* An implicit substitution inference for 2 ordered pairs. (Contributed by NM, 5-Aug-1995.)
 C  D    =>     R  S  C  R  D  S  <. ,  >.  <. ,  >.  <. C ,  D >.  <. ,  >.
 
Theorem0nelop 3976 A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
 (/)  <. ,  >.
 
Theoremopeqex 3977 Equivalence of existence implied by equality of ordered pairs. (Contributed by NM, 28-May-2008.)
 <. ,  >.  <. C ,  D >.  _V  _V  C  _V  D  _V
 
Theoremopcom 3978 An ordered pair commutes iff its members are equal. (Contributed by NM, 28-May-2009.)
 _V   &     _V   =>     <. ,  >.  <. ,  >.
 
Theoremmoop2 3979* "At most one" property of an ordered pair. (Contributed by NM, 11-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
 _V   =>     <. ,  >.
 
Theoremopeqsn 3980 Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)
 _V   &     _V   &     C  _V   =>     <. ,  >.  { C }  C  { }
 
Theoremopeqpr 3981 Equivalence for an ordered pair equal to an unordered pair. (Contributed by NM, 3-Jun-2008.)
 _V   &     _V   &     C  _V   &     D  _V   =>     <. ,  >.  { C ,  D }  C  { }  D  { ,  }  C  { ,  }  D  { }
 
Theoremeuotd 3982* Prove existential uniqueness for an ordered triple. (Contributed by Mario Carneiro, 20-May-2015.)
 _V   &     _V   &     C  _V   &     a  b  c  C   =>     a b c  <.
 a ,  b ,  c >.
 
Theoremuniop 3983 The union of an ordered pair. Theorem 65 of [Suppes] p. 39. (Contributed by NM, 17-Aug-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
 _V   &     _V   =>     U. <. ,  >.  { ,  }
 
Theoremuniopel 3984 Ordered pair membership is inherited by class union. (Contributed by NM, 13-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
 _V   &     _V   =>     <. ,  >.  C  U. <. ,  >.  U. C
 
2.3.4  Ordered-pair class abstractions (cont.)
 
Theoremopabid 3985 The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 <. ,  >.  { <. ,  >.  |  }
 
Theoremelopab 3986* Membership in a class abstraction of pairs. (Contributed by NM, 24-Mar-1998.)
 { <. ,  >.  |  }  <. ,  >.
 
TheoremopelopabsbALT 3987* The law of concretion in terms of substitutions. Less general than opelopabsb 3988, but having a much shorter proof. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
 <. ,  >.  { <. ,  >.  |  }
 
Theoremopelopabsb 3988* The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Revised by Mario Carneiro, 18-Nov-2016.)
 <. ,  >.  { <. ,  >.  |  }  [.  ]. [.  ].
 
Theorembrabsb 3989* The law of concretion in terms of substitutions. (Contributed by NM, 17-Mar-2008.)
 R  { <. ,  >.  |  }   =>     R  [.  ]. [.  ].
 
Theoremopelopabt 3990* Closed theorem form of opelopab 3999. (Contributed by NM, 19-Feb-2013.)
 V  W  <. ,  >.  { <. ,  >.  |  }
 
Theoremopelopabga 3991* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.)
   =>     V  W  <. ,  >.  { <. ,  >.  |  }
 
Theorembrabga 3992* The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.)
   &     R  { <. ,  >.  |  }   =>     V  W  R
 
Theoremopelopab2a 3993* Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 19-Dec-2013.)
   =>     C  D  <. ,  >.  { <. ,  >.  |  C  D  }
 
Theoremopelopaba 3994* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.)
 _V   &     _V   &       =>     <. ,  >.  { <. ,  >.  |  }
 
Theorembraba 3995* The law of concretion for a binary relation. (Contributed by NM, 19-Dec-2013.)
 _V   &     _V   &       &     R  { <. ,  >.  |  }   =>     R
 
Theoremopelopabg 3996* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 19-Dec-2013.)
   &       =>     V  W  <. ,  >.  { <. ,  >.  |  }
 
Theorembrabg 3997* The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 19-Dec-2013.)
   &       &     R  { <. ,  >.  |  }   =>     C  D  R
 
Theoremopelopab2 3998* Ordered pair membership in an ordered pair class abstraction. (Contributed by NM, 14-Oct-2007.) (Revised by Mario Carneiro, 19-Dec-2013.)
   &       =>     C  D  <. ,  >.  { <. ,  >.  |  C  D  }
 
Theoremopelopab 3999* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 16-May-1995.)
 _V   &     _V   &       &       =>     <. ,  >.  { <. ,  >.  |  }
 
Theorembrab 4000* The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.)
 _V   &     _V   &       &       &     R  { <. ,  >.  |  }   =>     R
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