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Theorem List for Intuitionistic Logic Explorer - 3901-4000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrabex 3901* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.)

Theoremelssabg 3902* Membership in a class abstraction involving a subset. Unlike elabg 2688, does not have to be a set. (Contributed by NM, 29-Aug-2006.)

Theoreminteximm 3903* The intersection of an inhabited class exists. (Contributed by Jim Kingdon, 27-Aug-2018.)

Theoremintexr 3904 If the intersection of a class exists, the class is non-empty. (Contributed by Jim Kingdon, 27-Aug-2018.)

Theoremintnexr 3905 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.)

Theoremintexabim 3906 The intersection of an inhabited class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)

Theoremintexrabim 3907 The intersection of an inhabited restricted class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)

Theoremiinexgm 3908* The existence of an indexed union. is normally a free-variable parameter in , which should be read . (Contributed by Jim Kingdon, 28-Aug-2018.)

Theoreminuni 3909* The intersection of a union with a class is equal to the union of the intersections of each element of with . (Contributed by FL, 24-Mar-2007.)

Theoremelpw2g 3910 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.)

Theoremelpw2 3911 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 11-Oct-2007.)

Theorempwnss 3912 The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theorempwne 3913 No set equals its power set. The sethood antecedent is necessary; compare pwv 3579. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)

Theoremrepizf2lem 3914 Lemma for repizf2 3915. 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 3915* Replacement. This version of replacement is stronger than repizf 3873 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 3873 with ax-sep 3875. Another variation would be but we don't have a proof of that yet. (Contributed by Jim Kingdon, 7-Sep-2018.)

2.2.5  Theorems requiring empty set existence

Theoremclass2seteq 3916* Equality theorem for classes and sets . (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.)

Theorem0elpw 3917 Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)

Theorem0nep0 3918 The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)

Theorem0inp0 3919 Something cannot be equal to both the null set and the power set of the null set. (Contributed by NM, 30-Sep-2003.)

Theoremunidif0 3920 The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)

Theoremiin0imm 3921* An indexed intersection of the empty set, with an inhabited index set, is empty. (Contributed by Jim Kingdon, 29-Aug-2018.)

Theoremiin0r 3922* If an indexed intersection of the empty set is empty, the index set is non-empty. (Contributed by Jim Kingdon, 29-Aug-2018.)

Theoremintv 3923 The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.)

Theoremaxpweq 3924* Two equivalent ways to express the Power Set Axiom. Note that ax-pow 3927 is not used by the proof. (Contributed by NM, 22-Jun-2009.)

2.2.6  Collection principle

Theorembnd 3925* A very strong generalization of the Axiom of Replacement (compare zfrep6 3874). 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 3872. (Contributed by NM, 17-Oct-2004.)

Theorembnd2 3926* A variant of the Boundedness Axiom bnd 3925 that picks a subset out of a possibly proper class in which a property is true. (Contributed by NM, 4-Feb-2004.)

2.3  IZF Set Theory - add the Axioms of Power Sets and Pairing

2.3.1  Introduce the Axiom of Power Sets

Axiomax-pow 3927* 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 3878).

The variant axpow2 3929 uses explicit subset notation. A version using class notation is pwex 3932. (Contributed by NM, 5-Aug-1993.)

Theoremzfpow 3928* Axiom of Power Sets expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)

Theoremaxpow2 3929* A variant of the Axiom of Power Sets ax-pow 3927 using subset notation. Problem in {BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)

Theoremaxpow3 3930* A variant of the Axiom of Power Sets ax-pow 3927. 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.)

Theoremel 3931* Every set is an element of some other set. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theorempwex 3932 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.)

Theorempwexg 3933 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.)

Theoremabssexg 3934* Existence of a class of subsets. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

TheoremsnexgOLD 3935 A singleton whose element exists is a set. The 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 3936 and new proofs should use snexg 3936 instead. (Contributed by Jim Kingdon, 26-Jan-2019.) (New usage is discouraged.) (Proof modification is discouraged.) TODO: replace its uses by uses of snexg 3936 and then remove it.

Theoremsnexg 3936 A singleton whose element exists is a set. The 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.)

Theoremsnex 3937 A singleton whose element exists is a set. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.)

Theoremsnexprc 3938 A singleton whose element is a proper class is a set. The 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.)

Theoremp0ex 3939 The power set of the empty set (the ordinal 1) is a set. (Contributed by NM, 23-Dec-1993.)

Theorempp0ex 3940 (the ordinal 2) is a set. (Contributed by NM, 5-Aug-1993.)

Theoremord3ex 3941 The ordinal number 3 is a set, proved without the Axiom of Union. (Contributed by NM, 2-May-2009.)

Theoremdtruarb 3942* 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 4283 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 3943 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.)

2.3.2  Axiom of Pairing

Axiomax-pr 3944* 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 3878). (Contributed by NM, 14-Nov-2006.)

Theoremzfpair2 3945 Derive the abbreviated version of the Axiom of Pairing from ax-pr 3944. (Contributed by NM, 14-Nov-2006.)

TheoremprexgOLD 3946 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 3480, prprc1 3478, and prprc2 3479. This is a special case of prexg 3947 and new proofs should use prexg 3947 instead. (Contributed by Jim Kingdon, 25-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) TODO: replace its uses by uses of prexg 3947 and then remove it.

Theoremprexg 3947 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 3480, prprc1 3478, and prprc2 3479. (Contributed by Jim Kingdon, 16-Sep-2018.)

Theoremsnelpwi 3948 A singleton of a set belongs to the power class of a class containing the set. (Contributed by Alan Sare, 25-Aug-2011.)

Theoremsnelpw 3949 A singleton of a set belongs to the power class of a class containing the set. (Contributed by NM, 1-Apr-1998.)

Theoremprelpwi 3950 A pair of two sets belongs to the power class of a class containing those two sets. (Contributed by Thierry Arnoux, 10-Mar-2017.)

Theoremrext 3951* A theorem similar to extensionality, requiring the existence of a singleton. Exercise 8 of [TakeutiZaring] p. 16. (Contributed by NM, 10-Aug-1993.)

Theoremsspwb 3952 Classes are subclasses if and only if their power classes are subclasses. Exercise 18 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)

Theoremunipw 3953 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.)

Theorempwel 3954 Membership of a power class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.)

Theorempwtr 3955 A class is transitive iff its power class is transitive. (Contributed by Alan Sare, 25-Aug-2011.) (Revised by Mario Carneiro, 15-Jun-2014.)

Theoremssextss 3956* An extensionality-like principle defining subclass in terms of subsets. (Contributed by NM, 30-Jun-2004.)

Theoremssext 3957* 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.)

Theoremnssssr 3958* Negation of subclass relationship. Compare nssr 3003. (Contributed by Jim Kingdon, 17-Sep-2018.)

Theorempweqb 3959 Classes are equal if and only if their power classes are equal. Exercise 19 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)

Theoremintid 3960* The intersection of all sets to which a set belongs is the singleton of that set. (Contributed by NM, 5-Jun-2009.)

Theoremeuabex 3961 The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.)

Theoremmss 3962* An inhabited class (even if proper) has an inhabited subset. (Contributed by Jim Kingdon, 17-Sep-2018.)

Theoremexss 3963* Restricted existence in a class (even if proper) implies restricted existence in a subset. (Contributed by NM, 23-Aug-2003.)

Theoremopexg 3964 An ordered pair of sets is a set. (Contributed by Jim Kingdon, 11-Jan-2019.)

TheoremopexgOLD 3965 An ordered pair of sets is a set. This is a special case of opexg 3964 and new proofs should use opexg 3964 instead. (Contributed by Jim Kingdon, 19-Sep-2018.) (New usage is discouraged.) (Proof modification is discouraged.) TODO: replace its uses by uses of opexg 3964 and then remove it.

Theoremopex 3966 An ordered pair of sets is a set. (Contributed by Jim Kingdon, 24-Sep-2018.) (Revised by Mario Carneiro, 24-May-2019.)

Theoremotexg 3967 An ordered triple of sets is a set. (Contributed by Jim Kingdon, 19-Sep-2018.)

Theoremelop 3968 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.)

Theoremopi1 3969 One of the two elements in an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopi2 3970 One of the two elements of an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)

2.3.3  Ordered pair theorem

Theoremopm 3971* An ordered pair is inhabited iff the arguments are sets. (Contributed by Jim Kingdon, 21-Sep-2018.)

Theoremopnzi 3972 An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 3971). (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremopth1 3973 Equality of the first members of equal ordered pairs. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopth 3974 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 and are not required to be sets due our specific ordered pair definition. (Contributed by NM, 28-May-1995.)

Theoremopthg 3975 Ordered pair theorem. and 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.)

Theoremopthg2 3976 Ordered pair theorem. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopth2 3977 Ordered pair theorem. (Contributed by NM, 21-Sep-2014.)

Theoremotth2 3978 Ordered triple theorem, with triple express with ordered pairs. (Contributed by NM, 1-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremotth 3979 Ordered triple theorem. (Contributed by NM, 25-Sep-2014.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremeqvinop 3980* A variable introduction law for ordered pairs. Analog of Lemma 15 of [Monk2] p. 109. (Contributed by NM, 28-May-1995.)

Theoremcopsexg 3981* Substitution of class for ordered pair . (Contributed by NM, 27-Dec-1996.) (Revised by Andrew Salmon, 11-Jul-2011.)

Theoremcopsex2t 3982* Closed theorem form of copsex2g 3983. (Contributed by NM, 17-Feb-2013.)

Theoremcopsex2g 3983* Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.)

Theoremcopsex4g 3984* An implicit substitution inference for 2 ordered pairs. (Contributed by NM, 5-Aug-1995.)

Theorem0nelop 3985 A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremopeqex 3986 Equivalence of existence implied by equality of ordered pairs. (Contributed by NM, 28-May-2008.)

Theoremopcom 3987 An ordered pair commutes iff its members are equal. (Contributed by NM, 28-May-2009.)

Theoremmoop2 3988* "At most one" property of an ordered pair. (Contributed by NM, 11-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopeqsn 3989 Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)

Theoremopeqpr 3990 Equivalence for an ordered pair equal to an unordered pair. (Contributed by NM, 3-Jun-2008.)

Theoremeuotd 3991* Prove existential uniqueness for an ordered triple. (Contributed by Mario Carneiro, 20-May-2015.)

Theoremuniop 3992 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.)

Theoremuniopel 3993 Ordered pair membership is inherited by class union. (Contributed by NM, 13-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)

2.3.4  Ordered-pair class abstractions (cont.)

Theoremopabid 3994 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 3995* Membership in a class abstraction of pairs. (Contributed by NM, 24-Mar-1998.)

TheoremopelopabsbALT 3996* The law of concretion in terms of substitutions. Less general than opelopabsb 3997, 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 3997* The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Revised by Mario Carneiro, 18-Nov-2016.)

Theorembrabsb 3998* The law of concretion in terms of substitutions. (Contributed by NM, 17-Mar-2008.)

Theoremopelopabt 3999* Closed theorem form of opelopab 4008. (Contributed by NM, 19-Feb-2013.)

Theoremopelopabga 4000* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.)

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