P. 43 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4201-4300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	difexg 4201	Existence of a difference. (Contributed by NM, 26-May-1998.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ V)
Theorem	zfausab 4202*	Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V
Theorem	rabexg 4203*	Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
Theorem	rabex 4204*	Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V
Theorem	rabexd 4205*	Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 4206. (Contributed by AV, 16-Jul-2019.)
⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓}    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐵 ∈ V)
Theorem	rabex2 4206*	Separation Scheme in terms of a restricted class abstraction. (Contributed by AV, 16-Jul-2019.) (Revised by AV, 26-Mar-2021.)
⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓}    &   ⊢ 𝐴 ∈ V    ⇒   ⊢ 𝐵 ∈ V
Theorem	rab2ex 4207*	A class abstraction based on a class abstraction based on a set is a set. (Contributed by AV, 16-Jul-2019.) (Revised by AV, 26-Mar-2021.)
⊢ 𝐵 = {𝑦 ∈ 𝐴 ∣ 𝜓}    &   ⊢ 𝐴 ∈ V    ⇒   ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} ∈ V
Theorem	elssabg 4208*	Membership in a class abstraction involving a subset. Unlike elabg 2926, 𝐴 does not have to be a set. (Contributed by NM, 29-Aug-2006.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ (𝑥 ⊆ 𝐵 ∧ 𝜑)} ↔ (𝐴 ⊆ 𝐵 ∧ 𝜓)))
Theorem	inteximm 4209*	The intersection of an inhabited class exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝐴 ∈ V)
Theorem	intexr 4210	If the intersection of a class exists, the class is nonempty. (Contributed by Jim Kingdon, 27-Aug-2018.)
⊢ (∩ 𝐴 ∈ V → 𝐴 ≠ ∅)
Theorem	intnexr 4211	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)
Theorem	intexabim 4212	The intersection of an inhabited class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
⊢ (∃𝑥𝜑 → ∩ {𝑥 ∣ 𝜑} ∈ V)
Theorem	intexrabim 4213	The intersection of an inhabited restricted class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∩ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
Theorem	iinexgm 4214*	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 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V)
Theorem	inuni 4215*	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.)
⊢ (∪ 𝐴 ∩ 𝐵) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵)}
Theorem	elpw2g 4216	Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
Theorem	elpw2 4217	Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 11-Oct-2007.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)
Theorem	elpwi2 4218	Membership in a power class. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Proof shortened by Wolf Lammen, 26-May-2024.)
⊢ 𝐵 ∈ 𝑉    &   ⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ 𝐴 ∈ 𝒫 𝐵
Theorem	pwnss 4219	The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.)
⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝐴 ⊆ 𝐴)
Theorem	pwne 4220	No set equals its power set. The sethood antecedent is necessary; compare pwv 3863. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≠ 𝐴)
Theorem	repizf2lem 4221	Lemma for repizf2 4222. 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.)
⊢ (∀𝑥 ∈ 𝑤 ∃*𝑦𝜑 ↔ ∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑)
Theorem	repizf2 4222*	Replacement. This version of replacement is stronger than repizf 4176 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 4176 with ax-sep 4178. Another variation would be ∀𝑥 ∈ 𝑤∃*𝑦𝜑 → {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)} ∈ V but we don't have a proof of that yet. (Contributed by Jim Kingdon, 7-Sep-2018.)
⊢ Ⅎ𝑧𝜑    ⇒   ⊢ (∀𝑥 ∈ 𝑤 ∃*𝑦𝜑 → ∃𝑧∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃𝑦 ∈ 𝑧 𝜑)
2.2.5  Theorems requiring empty set existence
Theorem	class2seteq 4223*	Equality theorem for classes and sets . (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = 𝐴)
Theorem	0elpw 4224	Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)
⊢ ∅ ∈ 𝒫 𝐴
Theorem	0nep0 4225	The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)
⊢ ∅ ≠ {∅}
Theorem	0inp0 4226	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 = {∅})
Theorem	unidif0 4227	The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)
⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴
Theorem	iin0imm 4228*	An indexed intersection of the empty set, with an inhabited index set, is empty. (Contributed by Jim Kingdon, 29-Aug-2018.)
⊢ (∃𝑦 𝑦 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 ∅ = ∅)
Theorem	iin0r 4229*	If an indexed intersection of the empty set is empty, the index set is nonempty. (Contributed by Jim Kingdon, 29-Aug-2018.)
⊢ (∩ 𝑥 ∈ 𝐴 ∅ = ∅ → 𝐴 ≠ ∅)
Theorem	intv 4230	The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.)
⊢ ∩ V = ∅
Theorem	axpweq 4231*	Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4234 is not used by the proof. (Contributed by NM, 22-Jun-2009.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝒫 𝐴 ∈ V ↔ ∃𝑥∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴) → 𝑦 ∈ 𝑥))
2.2.6  Collection principle
Theorem	bnd 4232*	A very strong generalization of the Axiom of Replacement (compare zfrep6 4177). 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 4175. (Contributed by NM, 17-Oct-2004.)
⊢ (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)
Theorem	bnd2 4233*	A variant of the Boundedness Axiom bnd 4232 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    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑧(𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜑))
2.3  IZF Set Theory - add the Axioms of Power Sets and Pairing
2.3.1  Introduce the Axiom of Power Sets
Axiom	ax-pow 4234*	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 4181). The variant axpow2 4236 uses explicit subset notation. A version using class notation is pwex 4243. (Contributed by NM, 5-Aug-1993.)
⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
Theorem	zfpow 4235*	Axiom of Power Sets expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
⊢ ∃𝑥∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)
Theorem	axpow2 4236*	A variant of the Axiom of Power Sets ax-pow 4234 using subset notation. Problem in {BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦)
Theorem	axpow3 4237*	A variant of the Axiom of Power Sets ax-pow 4234. 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.)
⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦)
Theorem	el 4238*	Every set is an element of some other set. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ ∃𝑦 𝑥 ∈ 𝑦
Theorem	vpwex 4239	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 4240 from vpwex 4239. (Revised by BJ, 10-Aug-2022.)
⊢ 𝒫 𝑥 ∈ V
Theorem	pwexg 4240	Power set axiom expressed in class notation, with the sethood requirement as an antecedent. (Contributed by NM, 30-Oct-2003.)
⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
Theorem	pwexd 4241	Deduction version of the power set axiom. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝒫 𝐴 ∈ V)
Theorem	abssexg 4242*	Existence of a class of subsets. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V)
Theorem	pwex 4243	Power set axiom expressed in class notation. (Contributed by NM, 21-Jun-1993.)
⊢ 𝐴 ∈ V    ⇒   ⊢ 𝒫 𝐴 ∈ V
Theorem	snexg 4244	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.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)
Theorem	snex 4245	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
Theorem	snexprc 4246	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)
Theorem	notnotsnex 4247	A singleton is never a proper class. (Contributed by Mario Carneiro and Jim Kingdon, 3-Jul-2022.)
⊢ ¬ ¬ {𝐴} ∈ V
Theorem	p0ex 4248	The power set of the empty set (the ordinal 1) is a set. (Contributed by NM, 23-Dec-1993.)
⊢ {∅} ∈ V
Theorem	pp0ex 4249	{∅, {∅}} (the ordinal 2) is a set. (Contributed by NM, 5-Aug-1993.)
⊢ {∅, {∅}} ∈ V
Theorem	ord3ex 4250	The ordinal number 3 is a set, proved without the Axiom of Union. (Contributed by NM, 2-May-2009.)
⊢ {∅, {∅}, {∅, {∅}}} ∈ V
Theorem	dtruarb 4251*	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 4625 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.)
⊢ ∃𝑥∃𝑦 ¬ 𝑥 = 𝑦
Theorem	pwuni 4252	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
Theorem	undifexmid 4253*	Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3549 and undifdcss 7046 are provable, the full statement implies excluded middle as shown here. (Contributed by Jim Kingdon, 16-Jun-2022.)
⊢ (𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦)    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
2.3.2  A notation for excluded middle
Syntax	wem 4254	Formula for an abbreviation of excluded middle.
wff EXMID
Definition	df-exmid 4255	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 4253 with exmidundif 4266. 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 4256 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 4178, in which case EXMID means that all propositions are decidable (see exmidexmid 4256 and notice that it relies on ax-sep 4178). If we instead work with ax-bdsep 16019, 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))
Theorem	exmidexmid 4256	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 845, peircedc 916, or condc 855. (Contributed by Jim Kingdon, 18-Jun-2022.)
⊢ (EXMID → DECID 𝜑)
Theorem	ss1o0el1 4257	A subclass of {∅} contains the empty set if and only if it equals {∅}. (Contributed by BJ and Jim Kingdon, 9-Aug-2024.)
⊢ (𝐴 ⊆ {∅} → (∅ ∈ 𝐴 ↔ 𝐴 = {∅}))
Theorem	exmid01 4258	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 = {∅})))
Theorem	pwntru 4259	A slight strengthening of pwtrufal 16136. (Contributed by Mario Carneiro and Jim Kingdon, 12-Sep-2023.)
⊢ ((𝐴 ⊆ {∅} ∧ 𝐴 ≠ {∅}) → 𝐴 = ∅)
Theorem	exmid1dc 4260*	A convenience theorem for proving that something implies EXMID. Think of this as an alternative to using a proposition, as in proofs like undifexmid 4253 or ordtriexmid 4587. In this context 𝑥 = {∅} can be thought of as "x is true". (Contributed by Jim Kingdon, 21-Nov-2023.)
⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → DECID 𝑥 = {∅})    ⇒   ⊢ (𝜑 → EXMID)
Theorem	exmidn0m 4261*	Excluded middle is equivalent to any set being empty or inhabited. (Contributed by Jim Kingdon, 5-Mar-2023.)
⊢ (EXMID ↔ ∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦 ∈ 𝑥))
Theorem	exmidsssn 4262*	Excluded middle is equivalent to the biconditionalized version of sssnr 3807 for sets. (Contributed by Jim Kingdon, 5-Mar-2023.)
⊢ (EXMID ↔ ∀𝑥∀𝑦(𝑥 ⊆ {𝑦} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑦})))
Theorem	exmidsssnc 4263*	Excluded middle in terms of subsets of a singleton. This is similar to exmid01 4258 but lets you choose any set as the element of the singleton rather than just ∅. It is similar to exmidsssn 4262 but for a particular set 𝐵 rather than all sets. (Contributed by Jim Kingdon, 29-Jul-2023.)
⊢ (𝐵 ∈ 𝑉 → (EXMID ↔ ∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵}))))
Theorem	exmid0el 4264	Excluded middle is equivalent to decidability of ∅ being an element of an arbitrary set. (Contributed by Jim Kingdon, 18-Jun-2022.)
⊢ (EXMID ↔ ∀𝑥DECID ∅ ∈ 𝑥)
Theorem	exmidel 4265*	Excluded middle is equivalent to decidability of membership for two arbitrary sets. (Contributed by Jim Kingdon, 18-Jun-2022.)
⊢ (EXMID ↔ ∀𝑥∀𝑦DECID 𝑥 ∈ 𝑦)
Theorem	exmidundif 4266*	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 3549 and undifdcss 7046 are provable, the full statement is equivalent to excluded middle as shown here. (Contributed by Jim Kingdon, 18-Jun-2022.)
⊢ (EXMID ↔ ∀𝑥∀𝑦(𝑥 ⊆ 𝑦 ↔ (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦))
Theorem	exmidundifim 4267*	Excluded middle is equivalent to every subset having a complement. Variation of exmidundif 4266 with an implication rather than a biconditional. (Contributed by Jim Kingdon, 16-Feb-2023.)
⊢ (EXMID ↔ ∀𝑥∀𝑦(𝑥 ⊆ 𝑦 → (𝑥 ∪ (𝑦 ∖ 𝑥)) = 𝑦))
Theorem	exmid1stab 4268*	If every proposition is stable, excluded middle follows. We are thinking of 𝑥 as a proposition and 𝑥 = {∅} as "𝑥 is true". (Contributed by Jim Kingdon, 28-Nov-2023.)
⊢ ((𝜑 ∧ 𝑥 ⊆ {∅}) → STAB 𝑥 = {∅})    ⇒   ⊢ (𝜑 → EXMID)
2.3.3  Axiom of Pairing
Axiom	ax-pr 4269*	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 4181). (Contributed by NM, 14-Nov-2006.)
⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)
Theorem	zfpair2 4270	Derive the abbreviated version of the Axiom of Pairing from ax-pr 4269. (Contributed by NM, 14-Nov-2006.)
⊢ {𝑥, 𝑦} ∈ V
Theorem	prexg 4271	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 3753, prprc1 3751, and prprc2 3752. (Contributed by Jim Kingdon, 16-Sep-2018.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V)
Theorem	snelpwg 4272	A singleton of a set is a member of the powerclass of a class if and only if that set is a member of that class. (Contributed by NM, 1-Apr-1998.) Put in closed form and avoid ax-nul 4186. (Revised by BJ, 17-Jan-2025.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵))
Theorem	snelpwi 4273	A singleton of a set belongs to the power class of a class containing the set. (Contributed by Alan Sare, 25-Aug-2011.)
⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ 𝒫 𝐵)
Theorem	snelpw 4274	A singleton of a set belongs to the power class of a class containing the set. (Contributed by NM, 1-Apr-1998.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵)
Theorem	prelpw 4275	An unordered pair of two sets is a member of the powerclass of a class if and only if the two sets are members of that class. (Contributed by AV, 8-Jan-2020.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ∈ 𝒫 𝐶))
Theorem	prelpwi 4276	A pair of two sets belongs to the power class of a class containing those two sets. (Contributed by Thierry Arnoux, 10-Mar-2017.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → {𝐴, 𝐵} ∈ 𝒫 𝐶)
Theorem	rext 4277*	A theorem similar to extensionality, requiring the existence of a singleton. Exercise 8 of [TakeutiZaring] p. 16. (Contributed by NM, 10-Aug-1993.)
⊢ (∀𝑧(𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → 𝑥 = 𝑦)
Theorem	sspwb 4278	Classes are subclasses if and only if their power classes are subclasses. Exercise 18 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)
⊢ (𝐴 ⊆ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
Theorem	unipw 4279	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.)
⊢ ∪ 𝒫 𝐴 = 𝐴
Theorem	pwel 4280	Membership of a power class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.)
⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 ∪ 𝐵)
Theorem	pwtr 4281	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 𝐴 ↔ Tr 𝒫 𝐴)
Theorem	ssextss 4282*	An extensionality-like principle defining subclass in terms of subsets. (Contributed by NM, 30-Jun-2004.)
⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵))
Theorem	ssext 4283*	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.)
⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵))
Theorem	nssssr 4284*	Negation of subclass relationship. Compare nssr 3261. (Contributed by Jim Kingdon, 17-Sep-2018.)
⊢ (∃𝑥(𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵) → ¬ 𝐴 ⊆ 𝐵)
Theorem	pweqb 4285	Classes are equal if and only if their power classes are equal. Exercise 19 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)
⊢ (𝐴 = 𝐵 ↔ 𝒫 𝐴 = 𝒫 𝐵)
Theorem	intid 4286*	The intersection of all sets to which a set belongs is the singleton of that set. (Contributed by NM, 5-Jun-2009.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} = {𝐴}
Theorem	euabex 4287	The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.)
⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V)
Theorem	mss 4288*	An inhabited class (even if proper) has an inhabited subset. (Contributed by Jim Kingdon, 17-Sep-2018.)
⊢ (∃𝑦 𝑦 ∈ 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ ∃𝑧 𝑧 ∈ 𝑥))
Theorem	exss 4289*	Restricted existence in a class (even if proper) implies restricted existence in a subset. (Contributed by NM, 23-Aug-2003.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑))
Theorem	opexg 4290	An ordered pair of sets is a set. (Contributed by Jim Kingdon, 11-Jan-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ ∈ V)
Theorem	opex 4291	An ordered pair of sets is a set. (Contributed by Jim Kingdon, 24-Sep-2018.) (Revised by Mario Carneiro, 24-May-2019.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ⟨𝐴, 𝐵⟩ ∈ V
Theorem	otexg 4292	An ordered triple of sets is a set. (Contributed by Jim Kingdon, 19-Sep-2018.)
⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ⟨𝐴, 𝐵, 𝐶⟩ ∈ V)
Theorem	elop 4293	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.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐴 ∈ ⟨𝐵, 𝐶⟩ ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶}))
Theorem	opi1 4294	One of the two elements in an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ {𝐴} ∈ ⟨𝐴, 𝐵⟩
Theorem	opi2 4295	One of the two elements of an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵⟩
2.3.4  Ordered pair theorem
Theorem	opm 4296*	An ordered pair is inhabited iff the arguments are sets. (Contributed by Jim Kingdon, 21-Sep-2018.)
⊢ (∃𝑥 𝑥 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	opnzi 4297	An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 4296). (Contributed by Mario Carneiro, 26-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ⟨𝐴, 𝐵⟩ ≠ ∅
Theorem	opth1 4298	Equality of the first members of equal ordered pairs. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐴 = 𝐶)
Theorem	opth 4299	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.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	opthg 4300	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.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))