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Theorem List for Intuitionistic Logic Explorer - 4001-4100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtrint 4001* The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. (Contributed by Scott Fenton, 25-Feb-2011.)

Theoremtrintssm 4002* Any inhabited transitive class includes its intersection. Similar to Exercise 3 in [TakeutiZaring] p. 44 (which mistakenly does not include the inhabitedness hypothesis). (Contributed by Jim Kingdon, 22-Aug-2018.)

2.2  IZF Set Theory - add the Axioms of Collection and Separation

2.2.1  Introduce the Axiom of Collection

Axiomax-coll 4003* Axiom of Collection. Axiom 7 of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). It is similar to bnd 4056 but uses a freeness hypothesis in place of one of the distinct variable constraints. (Contributed by Jim Kingdon, 23-Aug-2018.)

Theoremrepizf 4004* 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 4003. It is identical to zfrep6 4005 except for the choice of a freeness hypothesis rather than a distinct variable constraint between and . (Contributed by Jim Kingdon, 23-Aug-2018.)

Theoremzfrep6 4005* A version of the Axiom of Replacement. Normally would have free variables and . Axiom 6 of [Kunen] p. 12. The Separation Scheme ax-sep 4006 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 4006* 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 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 2877. In some texts, this scheme is called "Aussonderung" or the Subset Axiom.

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

Theoremaxsep2 4007* A less restrictive version of the Separation Scheme ax-sep 4006, 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 4006 with no additional set theory axioms. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Theoremzfauscl 4008* Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 4006, we invoke the Axiom of Extensionality (indirectly via vtocl 2711), which is needed for the justification of class variable notation. (Contributed by NM, 5-Aug-1993.)

Theorembm1.3ii 4009* Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 4006. Similar to Theorem 1.3ii of [BellMachover] p. 463. (Contributed by NM, 5-Aug-1993.)

Theorema9evsep 4010* Derive a weakened version of ax-i9 1493, where and must be distinct, from Separation ax-sep 4006 and Extensionality ax-ext 2097. The theorem also holds (ax9vsep 4011), but in intuitionistic logic is stronger. (Contributed by Jim Kingdon, 25-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax9vsep 4011* Derive a weakened version of ax-9 1494, where and must be distinct, from Separation ax-sep 4006 and Extensionality ax-ext 2097. In intuitionistic logic a9evsep 4010 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 4012* Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2100 to strengthen the hypothesis in the form of axnul 4013). (Contributed by NM, 22-Dec-2007.)

Theoremaxnul 4013* 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 4006. 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 4012).

This theorem should not be referenced by any proof. Instead, use ax-nul 4014 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 4014* The Null Set Axiom of IZF set theory. It was derived as axnul 4013 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 4015 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 4014. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremcsbexga 4016 The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.)

Theoremcsbexa 4017 The existence of proper substitution into a class. (Contributed by NM, 7-Aug-2007.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

2.2.4  Theorems requiring subset and intersection existence

Theoremnalset 4018* No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.)

Theoremvnex 4019 The universal class does not exist as a set. (Contributed by NM, 4-Jul-2005.)

Theoremvprc 4020 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.)

Theoremnvel 4021 The universal class does not belong to any class. (Contributed by FL, 31-Dec-2006.)

Theoreminex1 4022 Separation Scheme (Aussonderung) using class notation. Compare Exercise 4 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)

Theoreminex2 4023 Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.)

Theoreminex1g 4024 Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)

Theoremssex 4025 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 4006 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)

Theoremssexi 4026 The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.)

Theoremssexg 4027 The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22 (generalized). (Contributed by NM, 14-Aug-1994.)

Theoremssexd 4028 A subclass of a set is a set. Deduction form of ssexg 4027. (Contributed by David Moews, 1-May-2017.)

Theoremdifexg 4029 Existence of a difference. (Contributed by NM, 26-May-1998.)

Theoremzfausab 4030* Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.)

Theoremrabexg 4031* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.)

Theoremrabex 4032* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.)

Theoremelssabg 4033* Membership in a class abstraction involving a subset. Unlike elabg 2799, does not have to be a set. (Contributed by NM, 29-Aug-2006.)

Theoreminteximm 4034* The intersection of an inhabited class exists. (Contributed by Jim Kingdon, 27-Aug-2018.)

Theoremintexr 4035 If the intersection of a class exists, the class is nonempty. (Contributed by Jim Kingdon, 27-Aug-2018.)

Theoremintnexr 4036 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 4037 The intersection of an inhabited class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)

Theoremintexrabim 4038 The intersection of an inhabited restricted class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)

Theoremiinexgm 4039* 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 4040* 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 4041 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.)

Theoremelpw2 4042 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 11-Oct-2007.)

Theorempwnss 4043 The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theorempwne 4044 No set equals its power set. The sethood antecedent is necessary; compare pwv 3701. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)

Theoremrepizf2lem 4045 Lemma for repizf2 4046. 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 4046* Replacement. This version of replacement is stronger than repizf 4004 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 4004 with ax-sep 4006. 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 4047* Equality theorem for classes and sets . (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.)

Theorem0elpw 4048 Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)

Theorem0nep0 4049 The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)

Theorem0inp0 4050 Something cannot be equal to both the null set and the power set of the null set. (Contributed by NM, 30-Sep-2003.)

Theoremunidif0 4051 The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)

Theoremiin0imm 4052* An indexed intersection of the empty set, with an inhabited index set, is empty. (Contributed by Jim Kingdon, 29-Aug-2018.)

Theoremiin0r 4053* If an indexed intersection of the empty set is empty, the index set is nonempty. (Contributed by Jim Kingdon, 29-Aug-2018.)

Theoremintv 4054 The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.)

Theoremaxpweq 4055* Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4058 is not used by the proof. (Contributed by NM, 22-Jun-2009.)

2.2.6  Collection principle

Theorembnd 4056* A very strong generalization of the Axiom of Replacement (compare zfrep6 4005). 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 4003. (Contributed by NM, 17-Oct-2004.)

Theorembnd2 4057* A variant of the Boundedness Axiom bnd 4056 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 4058* 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 4009).

The variant axpow2 4060 uses explicit subset notation. A version using class notation is pwex 4067. (Contributed by NM, 5-Aug-1993.)

Theoremzfpow 4059* Axiom of Power Sets expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)

Theoremaxpow2 4060* A variant of the Axiom of Power Sets ax-pow 4058 using subset notation. Problem in {BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)

Theoremaxpow3 4061* A variant of the Axiom of Power Sets ax-pow 4058. 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 4062* Every set is an element of some other set. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremvpwex 4063 Power set axiom: the powerclass of a set is a set. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Revised to prove pwexg 4064 from vpwex 4063. (Revised by BJ, 10-Aug-2022.)

Theorempwexg 4064 Power set axiom expressed in class notation, with the sethood requirement as an antecedent. (Contributed by NM, 30-Oct-2003.)

Theorempwexd 4065 Deduction version of the power set axiom. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Theoremabssexg 4066* Existence of a class of subsets. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theorempwex 4067 Power set axiom expressed in class notation. (Contributed by NM, 21-Jun-1993.)

Theoremsnexg 4068 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 4069 A singleton whose element exists is a set. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.)

Theoremsnexprc 4070 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.)

Theoremnotnotsnex 4071 A singleton is never a proper class. (Contributed by Mario Carneiro and Jim Kingdon, 3-Jul-2022.)

Theoremp0ex 4072 The power set of the empty set (the ordinal 1) is a set. (Contributed by NM, 23-Dec-1993.)

Theorempp0ex 4073 (the ordinal 2) is a set. (Contributed by NM, 5-Aug-1993.)

Theoremord3ex 4074 The ordinal number 3 is a set, proved without the Axiom of Union. (Contributed by NM, 2-May-2009.)

Theoremdtruarb 4075* 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 4434 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 4076 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.)

Theoremundifexmid 4077* Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3409 and undifdcss 6764 are provable, the full statement implies excluded middle as shown here. (Contributed by Jim Kingdon, 16-Jun-2022.)

2.3.2  A notation for excluded middle

Syntaxwem 4078 Formula for an abbreviation of excluded middle.
EXMID

Definitiondf-exmid 4079 The expression EXMID will be used as a readable shorthand for any form of the law of the excluded middle; this is a useful shorthand largely because it hides statements of the form "for any proposition" in a system which can only quantify over sets, not propositions.

To see how this compares with other ways of expressing excluded middle, compare undifexmid 4077 with exmidundif 4089. The former may be more recognizable as excluded middle because it is in terms of propositions, and the proof may be easier to follow for much the same reason (it just has to show and in the the relevant parts of the proof). The latter, however, has the key advantage of being able to prove both directions of the biconditional. To state that excluded middle implies a proposition is hard to do gracefully without EXMID, because there is no way to write a hypothesis for an arbitrary proposition; instead the hypothesis would need to be the particular instance of excluded middle which that proof needs. Or to say it another way, EXMID implies DECID by exmidexmid 4080 but there is no good way to express the converse.

This definition and how we use it is easiest to understand (and most appropriate to assign the name "excluded middle" to) if we assume ax-sep 4006, in which case EXMID means that all propositions are decidable (see exmidexmid 4080 and notice that it relies on ax-sep 4006). If we instead work with ax-bdsep 12774, EXMID as defined here means that all bounded propositions are decidable.

(Contributed by Mario Carneiro and Jim Kingdon, 18-Jun-2022.)

EXMID DECID

Theoremexmidexmid 4080 EXMID implies that an arbitrary proposition is decidable. That is, EXMID captures the usual meaning of excluded middle when stated in terms of propositions.

To get other propositional statements which are equivalent to excluded middle, combine this with notnotrdc 811, peircedc 882, or condc 821.

(Contributed by Jim Kingdon, 18-Jun-2022.)

EXMID DECID

Theoremexmid01 4081 Excluded middle is equivalent to saying any subset of is either or . (Contributed by BJ and Jim Kingdon, 18-Jun-2022.)
EXMID

Theorempwntru 4082 A slight strengthening of pwtrufal 12884. (Contributed by Mario Carneiro and Jim Kingdon, 12-Sep-2023.)

Theoremexmid1dc 4083* A convenience theorem for proving that something implies EXMID. Think of this as an alternative to using a proposition, as in proofs like undifexmid 4077 or ordtriexmid 4397. In this context can be thought of as "x is true". (Contributed by Jim Kingdon, 21-Nov-2023.)
DECID        EXMID

Theoremexmidn0m 4084* Excluded middle is equivalent to any set being empty or inhabited. (Contributed by Jim Kingdon, 5-Mar-2023.)
EXMID

Theoremexmidsssn 4085* Excluded middle is equivalent to the biconditionalized version of sssnr 3646 for sets. (Contributed by Jim Kingdon, 5-Mar-2023.)
EXMID

Theoremexmidsssnc 4086* Excluded middle in terms of subsets of a singleton. This is similar to exmid01 4081 but lets you choose any set as the element of the singleton rather than just . It is similar to exmidsssn 4085 but for a particular set rather than all sets. (Contributed by Jim Kingdon, 29-Jul-2023.)
EXMID

Theoremexmid0el 4087 Excluded middle is equivalent to decidability of being an element of an arbitrary set. (Contributed by Jim Kingdon, 18-Jun-2022.)
EXMID DECID

Theoremexmidel 4088* Excluded middle is equivalent to decidability of membership for two arbitrary sets. (Contributed by Jim Kingdon, 18-Jun-2022.)
EXMID DECID

Theoremexmidundif 4089* Excluded middle is equivalent to every subset having a complement. That is, the union of a subset and its relative complement being the whole set. Although special cases such as undifss 3409 and undifdcss 6764 are provable, the full statement is equivalent to excluded middle as shown here. (Contributed by Jim Kingdon, 18-Jun-2022.)
EXMID

Theoremexmidundifim 4090* Excluded middle is equivalent to every subset having a complement. Variation of exmidundif 4089 with an implication rather than a biconditional. (Contributed by Jim Kingdon, 16-Feb-2023.)
EXMID

2.3.3  Axiom of Pairing

Axiomax-pr 4091* 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 4009). (Contributed by NM, 14-Nov-2006.)

Theoremzfpair2 4092 Derive the abbreviated version of the Axiom of Pairing from ax-pr 4091. (Contributed by NM, 14-Nov-2006.)

Theoremprexg 4093 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 3599, prprc1 3597, and prprc2 3598. (Contributed by Jim Kingdon, 16-Sep-2018.)

Theoremsnelpwi 4094 A singleton of a set belongs to the power class of a class containing the set. (Contributed by Alan Sare, 25-Aug-2011.)

Theoremsnelpw 4095 A singleton of a set belongs to the power class of a class containing the set. (Contributed by NM, 1-Apr-1998.)

Theoremprelpwi 4096 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 4097* A theorem similar to extensionality, requiring the existence of a singleton. Exercise 8 of [TakeutiZaring] p. 16. (Contributed by NM, 10-Aug-1993.)

Theoremsspwb 4098 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 4099 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 4100 Membership of a power class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.)

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