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

Theoremtrel 4001 In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(Tr 𝐴 → ((𝐵𝐶𝐶𝐴) → 𝐵𝐴))

Theoremtrel3 4002 In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.)
(Tr 𝐴 → ((𝐵𝐶𝐶𝐷𝐷𝐴) → 𝐵𝐴))

Theoremtrss 4003 An element of a transitive class is a subset of the class. (Contributed by NM, 7-Aug-1994.)
(Tr 𝐴 → (𝐵𝐴𝐵𝐴))

Theoremtrin 4004 The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴𝐵))

Theoremtr0 4005 The empty set is transitive. (Contributed by NM, 16-Sep-1993.)
Tr ∅

Theoremtrv 4006 The universe is transitive. (Contributed by NM, 14-Sep-2003.)
Tr V

Theoremtriun 4007* The indexed union of a class of transitive sets is transitive. (Contributed by Mario Carneiro, 16-Nov-2014.)
(∀𝑥𝐴 Tr 𝐵 → Tr 𝑥𝐴 𝐵)

Theoremtruni 4008* 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.)
(∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)

Theoremtrint 4009* The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. (Contributed by Scott Fenton, 25-Feb-2011.)
(∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)

Theoremtrintssm 4010* 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.)
((Tr 𝐴 ∧ ∃𝑥 𝑥𝐴) → 𝐴𝐴)

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

2.2.1  Introduce the Axiom of Collection

Axiomax-coll 4011* Axiom of Collection. Axiom 7 of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). It is similar to bnd 4064 but uses a freeness hypothesis in place of one of the distinct variable constraints. (Contributed by Jim Kingdon, 23-Aug-2018.)
𝑏𝜑       (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑥𝑎𝑦𝑏 𝜑)

Theoremrepizf 4012* 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 4011. It is identical to zfrep6 4013 except for the choice of a freeness hypothesis rather than a distinct variable constraint between 𝑏 and 𝜑. (Contributed by Jim Kingdon, 23-Aug-2018.)
𝑏𝜑       (∀𝑥𝑎 ∃!𝑦𝜑 → ∃𝑏𝑥𝑎𝑦𝑏 𝜑)

Theoremzfrep6 4013* A version of the Axiom of Replacement. Normally 𝜑 would have free variables 𝑥 and 𝑦. Axiom 6 of [Kunen] p. 12. The Separation Scheme ax-sep 4014 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.)
(∀𝑥𝑧 ∃!𝑦𝜑 → ∃𝑤𝑥𝑧𝑦𝑤 𝜑)

2.2.2  Introduce the Axiom of Separation

Axiomax-sep 4014* 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 𝑦𝜑 condition replaced by a distinct variable constraint between 𝑦 and 𝜑).

The Separation Scheme is a weak form of Frege's Axiom of Comprehension, conditioning it (with 𝑥𝑧) so that it asserts the existence of a collection only if it is smaller than some other collection 𝑧 that already exists. This prevents Russell's paradox ru 2879. In some texts, this scheme is called "Aussonderung" or the Subset Axiom.

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

𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))

Theoremaxsep2 4015* A less restrictive version of the Separation Scheme ax-sep 4014, where variables 𝑥 and 𝑧 can both appear free in the wff 𝜑, which can therefore be thought of as 𝜑(𝑥, 𝑧). This version was derived from the more restrictive ax-sep 4014 with no additional set theory axioms. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))

Theoremzfauscl 4016* Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 4014, we invoke the Axiom of Extensionality (indirectly via vtocl 2712), which is needed for the justification of class variable notation. (Contributed by NM, 5-Aug-1993.)
𝐴 ∈ V       𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))

Theorembm1.3ii 4017* Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 4014. Similar to Theorem 1.3ii of [BellMachover] p. 463. (Contributed by NM, 5-Aug-1993.)
𝑥𝑦(𝜑𝑦𝑥)       𝑥𝑦(𝑦𝑥𝜑)

Theorema9evsep 4018* Derive a weakened version of ax-i9 1493, where 𝑥 and 𝑦 must be distinct, from Separation ax-sep 4014 and Extensionality ax-ext 2097. The theorem ¬ ∀𝑥¬ 𝑥 = 𝑦 also holds (ax9vsep 4019), but in intuitionistic logic 𝑥𝑥 = 𝑦 is stronger. (Contributed by Jim Kingdon, 25-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥 𝑥 = 𝑦

Theoremax9vsep 4019* Derive a weakened version of ax-9 1494, where 𝑥 and 𝑦 must be distinct, from Separation ax-sep 4014 and Extensionality ax-ext 2097. In intuitionistic logic a9evsep 4018 is stronger and also holds. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ ∀𝑥 ¬ 𝑥 = 𝑦

2.2.3  Derive the Null Set Axiom

Theoremzfnuleu 4020* 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 4021). (Contributed by NM, 22-Dec-2007.)
𝑥𝑦 ¬ 𝑦𝑥       ∃!𝑥𝑦 ¬ 𝑦𝑥

Theoremaxnul 4021* 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 4014. 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 4020).

This theorem should not be referenced by any proof. Instead, use ax-nul 4022 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.)

𝑥𝑦 ¬ 𝑦𝑥

Axiomax-nul 4022* The Null Set Axiom of IZF set theory. It was derived as axnul 4021 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.)
𝑥𝑦 ¬ 𝑦𝑥

Theorem0ex 4023 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 4022. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
∅ ∈ V

Theoremcsbexga 4024 The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
((𝐴𝑉 ∧ ∀𝑥 𝐵𝑊) → 𝐴 / 𝑥𝐵 ∈ V)

Theoremcsbexa 4025 The existence of proper substitution into a class. (Contributed by NM, 7-Aug-2007.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
𝐴 ∈ V    &   𝐵 ∈ V       𝐴 / 𝑥𝐵 ∈ V

2.2.4  Theorems requiring subset and intersection existence

Theoremnalset 4026* No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.)
¬ ∃𝑥𝑦 𝑦𝑥

Theoremvnex 4027 The universal class does not exist as a set. (Contributed by NM, 4-Jul-2005.)
¬ ∃𝑥 𝑥 = V

Theoremvprc 4028 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 4029 The universal class does not belong to any class. (Contributed by FL, 31-Dec-2006.)
¬ V ∈ 𝐴

Theoreminex1 4030 Separation Scheme (Aussonderung) using class notation. Compare Exercise 4 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)
𝐴 ∈ V       (𝐴𝐵) ∈ V

Theoreminex2 4031 Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.)
𝐴 ∈ V       (𝐵𝐴) ∈ V

Theoreminex1g 4032 Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)
(𝐴𝑉 → (𝐴𝐵) ∈ V)

Theoremssex 4033 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 4014 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)
𝐵 ∈ V       (𝐴𝐵𝐴 ∈ V)

Theoremssexi 4034 The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.)
𝐵 ∈ V    &   𝐴𝐵       𝐴 ∈ V

Theoremssexg 4035 The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22 (generalized). (Contributed by NM, 14-Aug-1994.)
((𝐴𝐵𝐵𝐶) → 𝐴 ∈ V)

Theoremssexd 4036 A subclass of a set is a set. Deduction form of ssexg 4035. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐵𝐶)    &   (𝜑𝐴𝐵)       (𝜑𝐴 ∈ V)

Theoremdifexg 4037 Existence of a difference. (Contributed by NM, 26-May-1998.)
(𝐴𝑉 → (𝐴𝐵) ∈ V)

Theoremzfausab 4038* Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.)
𝐴 ∈ V       {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V

Theoremrabexg 4039* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.)
(𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)

Theoremrabex 4040* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.)
𝐴 ∈ V       {𝑥𝐴𝜑} ∈ V

Theoremelssabg 4041* Membership in a class abstraction involving a subset. Unlike elabg 2801, 𝐴 does not have to be a set. (Contributed by NM, 29-Aug-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐵𝑉 → (𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)} ↔ (𝐴𝐵𝜓)))

Theoreminteximm 4042* The intersection of an inhabited class exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
(∃𝑥 𝑥𝐴 𝐴 ∈ V)

Theoremintexr 4043 If the intersection of a class exists, the class is nonempty. (Contributed by Jim Kingdon, 27-Aug-2018.)
( 𝐴 ∈ V → 𝐴 ≠ ∅)

Theoremintnexr 4044 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.)
( 𝐴 = V → ¬ 𝐴 ∈ V)

Theoremintexabim 4045 The intersection of an inhabited class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
(∃𝑥𝜑 {𝑥𝜑} ∈ V)

Theoremintexrabim 4046 The intersection of an inhabited restricted class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
(∃𝑥𝐴 𝜑 {𝑥𝐴𝜑} ∈ V)

Theoremiinexgm 4047* The existence of an indexed union. 𝑥 is normally a free-variable parameter in 𝐵, which should be read 𝐵(𝑥). (Contributed by Jim Kingdon, 28-Aug-2018.)
((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → 𝑥𝐴 𝐵 ∈ V)

Theoreminuni 4048* The intersection of a union 𝐴 with a class 𝐵 is equal to the union of the intersections of each element of 𝐴 with 𝐵. (Contributed by FL, 24-Mar-2007.)
( 𝐴𝐵) = {𝑥 ∣ ∃𝑦𝐴 𝑥 = (𝑦𝐵)}

Theoremelpw2g 4049 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.)
(𝐵𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))

Theoremelpw2 4050 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 11-Oct-2007.)
𝐵 ∈ V       (𝐴 ∈ 𝒫 𝐵𝐴𝐵)

Theorempwnss 4051 The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝐴𝑉 → ¬ 𝒫 𝐴𝐴)

Theorempwne 4052 No set equals its power set. The sethood antecedent is necessary; compare pwv 3703. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
(𝐴𝑉 → 𝒫 𝐴𝐴)

Theoremrepizf2lem 4053 Lemma for repizf2 4054. If we have a function-like proposition which provides at most one value of 𝑦 for each 𝑥 in a set 𝑤, we can change "at most one" to "exactly one" by restricting the values of 𝑥 to those values for which the proposition provides a value of 𝑦. (Contributed by Jim Kingdon, 7-Sep-2018.)
(∀𝑥𝑤 ∃*𝑦𝜑 ↔ ∀𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑)

Theoremrepizf2 4054* Replacement. This version of replacement is stronger than repizf 4012 in the sense that 𝜑 does not need to map all values of 𝑥 in 𝑤 to a value of 𝑦. The resulting set contains those elements for which there is a value of 𝑦 and in that sense, this theorem combines repizf 4012 with ax-sep 4014. Another variation would be 𝑥𝑤∃*𝑦𝜑 → {𝑦 ∣ ∃𝑥(𝑥𝑤𝜑)} ∈ V but we don't have a proof of that yet. (Contributed by Jim Kingdon, 7-Sep-2018.)
𝑧𝜑       (∀𝑥𝑤 ∃*𝑦𝜑 → ∃𝑧𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃𝑦𝑧 𝜑)

2.2.5  Theorems requiring empty set existence

Theoremclass2seteq 4055* Equality theorem for classes and sets . (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.)
(𝐴𝑉 → {𝑥𝐴𝐴 ∈ V} = 𝐴)

Theorem0elpw 4056 Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)
∅ ∈ 𝒫 𝐴

Theorem0nep0 4057 The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)
∅ ≠ {∅}

Theorem0inp0 4058 Something cannot be equal to both the null set and the power set of the null set. (Contributed by NM, 30-Sep-2003.)
(𝐴 = ∅ → ¬ 𝐴 = {∅})

Theoremunidif0 4059 The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)
(𝐴 ∖ {∅}) = 𝐴

Theoremiin0imm 4060* An indexed intersection of the empty set, with an inhabited index set, is empty. (Contributed by Jim Kingdon, 29-Aug-2018.)
(∃𝑦 𝑦𝐴 𝑥𝐴 ∅ = ∅)

Theoremiin0r 4061* If an indexed intersection of the empty set is empty, the index set is nonempty. (Contributed by Jim Kingdon, 29-Aug-2018.)
( 𝑥𝐴 ∅ = ∅ → 𝐴 ≠ ∅)

Theoremintv 4062 The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.)
V = ∅

Theoremaxpweq 4063* Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4066 is not used by the proof. (Contributed by NM, 22-Jun-2009.)
𝐴 ∈ V       (𝒫 𝐴 ∈ V ↔ ∃𝑥𝑦(∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))

2.2.6  Collection principle

Theorembnd 4064* A very strong generalization of the Axiom of Replacement (compare zfrep6 4013). Its strength lies in the rather profound fact that 𝜑(𝑥, 𝑦) 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 4011. (Contributed by NM, 17-Oct-2004.)
(∀𝑥𝑧𝑦𝜑 → ∃𝑤𝑥𝑧𝑦𝑤 𝜑)

Theorembnd2 4065* A variant of the Boundedness Axiom bnd 4064 that picks a subset 𝑧 out of a possibly proper class 𝐵 in which a property is true. (Contributed by NM, 4-Feb-2004.)
𝐴 ∈ V       (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑧(𝑧𝐵 ∧ ∀𝑥𝐴𝑦𝑧 𝜑))

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

2.3.1  Introduce the Axiom of Power Sets

Axiomax-pow 4066* Axiom of Power Sets. An axiom of Intuitionistic Zermelo-Fraenkel set theory. It states that a set 𝑦 exists that includes the power set of a given set 𝑥 i.e. contains every subset of 𝑥. 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 4017).

The variant axpow2 4068 uses explicit subset notation. A version using class notation is pwex 4075. (Contributed by NM, 5-Aug-1993.)

𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)

Theoremzfpow 4067* Axiom of Power Sets expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
𝑥𝑦(∀𝑥(𝑥𝑦𝑥𝑧) → 𝑦𝑥)

Theoremaxpow2 4068* A variant of the Axiom of Power Sets ax-pow 4066 using subset notation. Problem in {BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
𝑦𝑧(𝑧𝑥𝑧𝑦)

Theoremaxpow3 4069* A variant of the Axiom of Power Sets ax-pow 4066. For any set 𝑥, there exists a set 𝑦 whose members are exactly the subsets of 𝑥 i.e. the power set of 𝑥. Axiom Pow of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
𝑦𝑧(𝑧𝑥𝑧𝑦)

Theoremel 4070* Every set is an element of some other set. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
𝑦 𝑥𝑦

Theoremvpwex 4071 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 4072 from vpwex 4071. (Revised by BJ, 10-Aug-2022.)
𝒫 𝑥 ∈ V

Theorempwexg 4072 Power set axiom expressed in class notation, with the sethood requirement as an antecedent. (Contributed by NM, 30-Oct-2003.)
(𝐴𝑉 → 𝒫 𝐴 ∈ V)

Theorempwexd 4073 Deduction version of the power set axiom. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)       (𝜑 → 𝒫 𝐴 ∈ V)

Theoremabssexg 4074* Existence of a class of subsets. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(𝐴𝑉 → {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)

Theorempwex 4075 Power set axiom expressed in class notation. (Contributed by NM, 21-Jun-1993.)
𝐴 ∈ V       𝒫 𝐴 ∈ V

Theoremsnexg 4076 A singleton whose element exists is a set. The 𝐴 ∈ 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.)
(𝐴𝑉 → {𝐴} ∈ V)

Theoremsnex 4077 A singleton whose element exists is a set. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.)
𝐴 ∈ V       {𝐴} ∈ V

Theoremsnexprc 4078 A singleton whose element is a proper class is a set. The ¬ 𝐴 ∈ 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.)
𝐴 ∈ V → {𝐴} ∈ V)

Theoremnotnotsnex 4079 A singleton is never a proper class. (Contributed by Mario Carneiro and Jim Kingdon, 3-Jul-2022.)
¬ ¬ {𝐴} ∈ V

Theoremp0ex 4080 The power set of the empty set (the ordinal 1) is a set. (Contributed by NM, 23-Dec-1993.)
{∅} ∈ V

Theorempp0ex 4081 {∅, {∅}} (the ordinal 2) is a set. (Contributed by NM, 5-Aug-1993.)
{∅, {∅}} ∈ V

Theoremord3ex 4082 The ordinal number 3 is a set, proved without the Axiom of Union. (Contributed by NM, 2-May-2009.)
{∅, {∅}, {∅, {∅}}} ∈ V

Theoremdtruarb 4083* 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 4442 in which we are given a set 𝑦 and go from there to a set 𝑥 which is not equal to it. (Contributed by Jim Kingdon, 2-Sep-2018.)
𝑥𝑦 ¬ 𝑥 = 𝑦

Theorempwuni 4084 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.)
𝐴 ⊆ 𝒫 𝐴

Theoremundifexmid 4085* Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3411 and undifdcss 6777 are provable, the full statement implies excluded middle as shown here. (Contributed by Jim Kingdon, 16-Jun-2022.)
(𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) = 𝑦)       (𝜑 ∨ ¬ 𝜑)

2.3.2  A notation for excluded middle

Syntaxwem 4086 Formula for an abbreviation of excluded middle.
wff EXMID

Definitiondf-exmid 4087 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 4085 with exmidundif 4097. 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 𝜑 and ¬ 𝜑 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 𝜑 ∨ ¬ 𝜑 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 𝜑 by exmidexmid 4088 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 4014, in which case EXMID means that all propositions are decidable (see exmidexmid 4088 and notice that it relies on ax-sep 4014). If we instead work with ax-bdsep 12893, EXMID as defined here means that all bounded propositions are decidable.

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

(EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))

Theoremexmidexmid 4088 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.)

(EXMIDDECID 𝜑)

Theoremexmid01 4089 Excluded middle is equivalent to saying any subset of {∅} is either or {∅}. (Contributed by BJ and Jim Kingdon, 18-Jun-2022.)
(EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))

Theorempwntru 4090 A slight strengthening of pwtrufal 13003. (Contributed by Mario Carneiro and Jim Kingdon, 12-Sep-2023.)
((𝐴 ⊆ {∅} ∧ 𝐴 ≠ {∅}) → 𝐴 = ∅)

Theoremexmid1dc 4091* A convenience theorem for proving that something implies EXMID. Think of this as an alternative to using a proposition, as in proofs like undifexmid 4085 or ordtriexmid 4405. In this context 𝑥 = {∅} can be thought of as "x is true". (Contributed by Jim Kingdon, 21-Nov-2023.)
((𝜑𝑥 ⊆ {∅}) → DECID 𝑥 = {∅})       (𝜑EXMID)

Theoremexmidn0m 4092* Excluded middle is equivalent to any set being empty or inhabited. (Contributed by Jim Kingdon, 5-Mar-2023.)
(EXMID ↔ ∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥))

Theoremexmidsssn 4093* Excluded middle is equivalent to the biconditionalized version of sssnr 3648 for sets. (Contributed by Jim Kingdon, 5-Mar-2023.)
(EXMID ↔ ∀𝑥𝑦(𝑥 ⊆ {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})))

Theoremexmidsssnc 4094* Excluded middle in terms of subsets of a singleton. This is similar to exmid01 4089 but lets you choose any set as the element of the singleton rather than just . It is similar to exmidsssn 4093 but for a particular set 𝐵 rather than all sets. (Contributed by Jim Kingdon, 29-Jul-2023.)
(𝐵𝑉 → (EXMID ↔ ∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵}))))

Theoremexmid0el 4095 Excluded middle is equivalent to decidability of being an element of an arbitrary set. (Contributed by Jim Kingdon, 18-Jun-2022.)
(EXMID ↔ ∀𝑥DECID ∅ ∈ 𝑥)

Theoremexmidel 4096* Excluded middle is equivalent to decidability of membership for two arbitrary sets. (Contributed by Jim Kingdon, 18-Jun-2022.)
(EXMID ↔ ∀𝑥𝑦DECID 𝑥𝑦)

Theoremexmidundif 4097* 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 3411 and undifdcss 6777 are provable, the full statement is equivalent to excluded middle as shown here. (Contributed by Jim Kingdon, 18-Jun-2022.)
(EXMID ↔ ∀𝑥𝑦(𝑥𝑦 ↔ (𝑥 ∪ (𝑦𝑥)) = 𝑦))

Theoremexmidundifim 4098* Excluded middle is equivalent to every subset having a complement. Variation of exmidundif 4097 with an implication rather than a biconditional. (Contributed by Jim Kingdon, 16-Feb-2023.)
(EXMID ↔ ∀𝑥𝑦(𝑥𝑦 → (𝑥 ∪ (𝑦𝑥)) = 𝑦))

2.3.3  Axiom of Pairing

Axiomax-pr 4099* 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 4017). (Contributed by NM, 14-Nov-2006.)
𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)

Theoremzfpair2 4100 Derive the abbreviated version of the Axiom of Pairing from ax-pr 4099. (Contributed by NM, 14-Nov-2006.)
{𝑥, 𝑦} ∈ V

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