P. 47 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 4601-4700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	elpwg 4601	Membership in a power class. Theorem 86 of [Suppes] p. 47. See also elpw2g 5340. (Contributed by NM, 6-Aug-2000.) (Proof shortened by BJ, 31-Dec-2023.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
Theorem	elpw 4602	Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.) (Proof shortened by BJ, 31-Dec-2023.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)
Theorem	velpw 4603	Setvar variable membership in a power class. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
Theorem	elpwd 4604	Membership in a power class. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵)
Theorem	elpwi 4605	Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)
⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ⊆ 𝐵)
Theorem	elpwb 4606	Characterization of the elements of a power class. (Contributed by BJ, 29-Apr-2021.)
⊢ (𝐴 ∈ 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵))
Theorem	elpwid 4607	An element of a power class is a subclass. Deduction form of elpwi 4605. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
Theorem	elelpwi 4608	If 𝐴 belongs to a part of 𝐶, then 𝐴 belongs to 𝐶. (Contributed by FL, 3-Aug-2009.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝒫 𝐶) → 𝐴 ∈ 𝐶)
Theorem	sspw 4609	The powerclass preserves inclusion. See sspwb 5445 for the biconditional version. (Contributed by NM, 13-Oct-1996.) Extract forward implication of sspwb 5445 since it requires fewer axioms. (Revised by BJ, 13-Apr-2024.)
⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
Theorem	sspwi 4610	The powerclass preserves inclusion (inference form). (Contributed by BJ, 13-Apr-2024.)
⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ 𝒫 𝐴 ⊆ 𝒫 𝐵
Theorem	sspwd 4611	The powerclass preserves inclusion (deduction form). (Contributed by BJ, 13-Apr-2024.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
Theorem	pweq 4612	Equality theorem for power class. (Contributed by NM, 21-Jun-1993.) (Proof shortened by BJ, 13-Apr-2024.)
⊢ (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵)
Theorem	pweqALT 4613	Alternate proof of pweq 4612 directly from the definition. (Contributed by NM, 21-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵)
Theorem	pweqi 4614	Equality inference for power class. (Contributed by NM, 27-Nov-2013.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ 𝒫 𝐴 = 𝒫 𝐵
Theorem	pweqd 4615	Equality deduction for power class. (Contributed by NM, 27-Nov-2013.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → 𝒫 𝐴 = 𝒫 𝐵)
Theorem	pwunss 4616	The power class of the union of two classes includes the union of their power classes. Exercise 4.12(k) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.) Remove use of ax-sep 5293, ax-nul 5300, ax-pr 5423 and shorten proof. (Revised by BJ, 13-Apr-2024.)
⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵)
Theorem	nfpw 4617	Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥𝒫 𝐴
Theorem	pwidg 4618	A set is an element of its power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴)
Theorem	pwidb 4619	A class is an element of its powerclass if and only if it is a set. (Contributed by BJ, 31-Dec-2023.)
⊢ (𝐴 ∈ V ↔ 𝐴 ∈ 𝒫 𝐴)
Theorem	pwid 4620	A set is a member of its power class. Theorem 87 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 ∈ V    ⇒   ⊢ 𝐴 ∈ 𝒫 𝐴
Theorem	pwss 4621*	Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)
⊢ (𝒫 𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐵))
Theorem	pwundif 4622	Break up the power class of a union into a union of smaller classes. (Contributed by NM, 25-Mar-2007.) (Proof shortened by Thierry Arnoux, 20-Dec-2016.) Remove use of ax-sep 5293, ax-nul 5300, ax-pr 5423 and shorten proof. (Revised by BJ, 14-Apr-2024.)
⊢ 𝒫 (𝐴 ∪ 𝐵) = ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)
2.1.18  Unordered and ordered pairs
Theorem	snjust 4623*	Soundness justification theorem for df-sn 4625. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ {𝑥 ∣ 𝑥 = 𝐴} = {𝑦 ∣ 𝑦 = 𝐴}
Syntax	csn 4624	Extend class notation to include singleton.
class {𝐴}
Definition	df-sn 4625*	Define the singleton of a class. Definition 7.1 of [Quine] p. 48. For convenience, it is well-defined for proper classes, i.e., those that are not elements of V, see snprc 4717. For an alternate definition see dfsn2 4637. (Contributed by NM, 21-Jun-1993.)
⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴}
Syntax	cpr 4626	Extend class notation to include unordered pair.
class {𝐴, 𝐵}
Definition	df-pr 4627	Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. For example, 𝐴 ∈ {1, -1} → (𝐴↑2) = 1 (ex-pr 30214). They are unordered, so {𝐴, 𝐵} = {𝐵, 𝐴} as proven by prcom 4732. For a more traditional definition, but requiring a dummy variable, see dfpr2 4643. {𝐴, 𝐴} is also an unordered pair, but also a singleton because of {𝐴} = {𝐴, 𝐴} (see dfsn2 4637). Therefore, {𝐴, 𝐵} is called a proper (unordered) pair iff 𝐴 ≠ 𝐵 and 𝐴 and 𝐵 are sets. Note: ordered pairs are a completely different object defined below in df-op 4631. When the term "pair" is used without qualifier, it generally means "unordered pair", and the context makes it clear which version is meant. (Contributed by NM, 21-Jun-1993.)
⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
Syntax	ctp 4628	Extend class notation to include unordered triple (sometimes called "unordered triplet").
class {𝐴, 𝐵, 𝐶}
Definition	df-tp 4629	Define unordered triple of classes. Definition of [Enderton] p. 19. Note: ordered triples are a completely different object defined below in df-ot 4633. As with all tuples, when the term "triple" is used without qualifier, it means "ordered triple". (Contributed by NM, 9-Apr-1994.)
⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
Syntax	cop 4630	Extend class notation to include ordered pair.
class ⟨𝐴, 𝐵⟩
Definition	df-op 4631*	Definition of an ordered pair, equivalent to Kuratowski's definition {{𝐴}, {𝐴, 𝐵}} when the arguments are sets. Since the behavior of Kuratowski definition is not very useful for proper classes, we define it to be empty in this case (see opprc1 4893, opprc2 4894, and 0nelop 5492). For Kuratowski's actual definition when the arguments are sets, see dfop 4868. For the justifying theorem (for sets) see opth 5472. See dfopif 4866 for an equivalent formulation using the if operation. Definition 9.1 of [Quine] p. 58 defines an ordered pair unconditionally as ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}, which has different behavior from our df-op 4631 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 4631 was chosen because it often makes proofs shorter by eliminating unnecessary sethood hypotheses. There are other ways to define ordered pairs. The basic requirement is that two ordered pairs are equal iff their respective members are equal. In 1914 Norbert Wiener gave the first successful definition ⟨𝐴, 𝐵⟩2 = {{{𝐴}, ∅}, {{𝐵}}}, justified by opthwiener 5510. This was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition ⟨𝐴, 𝐵⟩3 = {𝐴, {𝐴, 𝐵}} is justified by opthreg 9627, but it requires the Axiom of Regularity for its justification and is not commonly used. A definition that also works for proper classes is ⟨𝐴, 𝐵⟩4 = ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})), justified by opthprc 5736. Nearly at the same time as Norbert Wiener, Felix Hausdorff proposed the following definition in "Grundzüge der Mengenlehre" ("Basics of Set Theory"), p. 32, in 1914: ⟨𝐴, 𝐵⟩5 = {{𝐴, 𝑂}, {𝐵, 𝑇}}. Hausdorff used 1 and 2 instead of 𝑂 and 𝑇, but actually any two different fixed sets will do (e.g., 𝑂 = ∅ and 𝑇 = {∅}, see 0nep0 5352). Furthermore, Hausdorff demanded that 𝑂 and 𝑇 are both different from 𝐴 as well as 𝐵, which is actually not necessary (at least not in full extent), see opthhausdorff0 5514 and opthhausdorff 5513. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 14247. An ordered pair of real numbers can also be represented by a complex number as shown by cru 12220. Kuratowski's ordered pair definition is standard for ZFC set theory, but it is very inconvenient to use in New Foundations theory because it is not type-level; a common alternate definition in New Foundations is the definition from [Rosser] p. 281. Since there are other ways to define ordered pairs, we discourage direct use of this definition so that most theorems won't depend on this particular construction; theorems will instead rely on dfopif 4866. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.)
⊢ ⟨𝐴, 𝐵⟩ = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
Syntax	cotp 4632	Extend class notation to include ordered triple.
class ⟨𝐴, 𝐵, 𝐶⟩
Definition	df-ot 4633	Define ordered triple of classes. Definition of ordered triple in [Stoll] p. 25. (Contributed by NM, 3-Apr-2015.)
⊢ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩
Theorem	sneq 4634	Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.)
⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})
Theorem	sneqi 4635	Equality inference for singletons. (Contributed by NM, 22-Jan-2004.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ {𝐴} = {𝐵}
Theorem	sneqd 4636	Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → {𝐴} = {𝐵})
Theorem	dfsn2 4637	Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
⊢ {𝐴} = {𝐴, 𝐴}
Theorem	elsng 4638	There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15 (generalized). (Contributed by NM, 13-Sep-1995.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
Theorem	elsn 4639	There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)
Theorem	velsn 4640	There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.)
⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
Theorem	elsni 4641	There is at most one element in a singleton. (Contributed by NM, 5-Jun-1994.)
⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
Theorem	absn 4642*	Condition for a class abstraction to be a singleton. Formerly part of proof of dfiota2 6495. (Contributed by Andrew Salmon, 30-Jun-2011.) (Revised by AV, 24-Aug-2022.)
⊢ ({𝑥 ∣ 𝜑} = {𝑌} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑌))
Theorem	dfpr2 4643*	Alternate definition of a pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
⊢ {𝐴, 𝐵} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)}
Theorem	dfsn2ALT 4644	Alternate definition of singleton, based on the (alternate) definition of pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by AV, 12-Jun-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ {𝐴} = {𝐴, 𝐴}
Theorem	elprg 4645	A member of a pair of classes is one or the other of them, and conversely as soon as it is a set. Exercise 1 of [TakeutiZaring] p. 15, generalized. (Contributed by NM, 13-Sep-1995.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
Theorem	elpri 4646	If a class is an element of a pair, then it is one of the two paired elements. (Contributed by Scott Fenton, 1-Apr-2011.)
⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
Theorem	elpr 4647	A member of a pair of classes is one or the other of them, and conversely as soon as it is a set. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
Theorem	elpr2g 4648	A member of a pair of sets is one or the other of them, and conversely. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 14-Oct-2005.) Generalize from sethood hypothesis to sethood antecedent. (Revised by BJ, 25-May-2024.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
Theorem	elpr2 4649	A member of a pair of sets is one or the other of them, and conversely. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 14-Oct-2005.) (Proof shortened by JJ, 23-Jul-2021.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
Theorem	elpr2OLD 4650	Obsolete version of elpr2 4649 as of 25-May-2024. (Contributed by NM, 14-Oct-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
Theorem	nelpr2 4651	If a class is not an element of an unordered pair, it is not the second listed element. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐶)
Theorem	nelpr1 4652	If a class is not an element of an unordered pair, it is not the first listed element. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐵)
Theorem	nelpri 4653	If an element doesn't match the items in an unordered pair, it is not in the unordered pair. (Contributed by David A. Wheeler, 10-May-2015.)
⊢ 𝐴 ≠ 𝐵    &   ⊢ 𝐴 ≠ 𝐶    ⇒   ⊢ ¬ 𝐴 ∈ {𝐵, 𝐶}
Theorem	prneli 4654	If an element doesn't match the items in an unordered pair, it is not in the unordered pair, using ∉. (Contributed by David A. Wheeler, 10-May-2015.)
⊢ 𝐴 ≠ 𝐵    &   ⊢ 𝐴 ≠ 𝐶    ⇒   ⊢ 𝐴 ∉ {𝐵, 𝐶}
Theorem	nelprd 4655	If an element doesn't match the items in an unordered pair, it is not in the unordered pair, deduction version. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐶)    ⇒   ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})
Theorem	eldifpr 4656	Membership in a set with two elements removed. Similar to eldifsn 4786 and eldiftp 4686. (Contributed by Mario Carneiro, 18-Jul-2017.)
⊢ (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷))
Theorem	rexdifpr 4657	Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.)
⊢ (∃𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶})𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶 ∧ 𝜑))
Theorem	snidg 4658	A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 28-Oct-2003.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
Theorem	snidb 4659	A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.)
⊢ (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})
Theorem	snid 4660	A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 31-Dec-1993.)
⊢ 𝐴 ∈ V    ⇒   ⊢ 𝐴 ∈ {𝐴}
Theorem	vsnid 4661	A setvar variable is a member of its singleton. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 𝑥 ∈ {𝑥}
Theorem	elsn2g 4662	There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that 𝐵, rather than 𝐴, be a set. (Contributed by NM, 28-Oct-2003.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
Theorem	elsn2 4663	There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that 𝐵, rather than 𝐴, be a set. (Contributed by NM, 12-Jun-1994.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)
Theorem	nelsn 4664	If a class is not equal to the class in a singleton, then it is not in the singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020.) (Proof shortened by BJ, 4-May-2021.)
⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 ∈ {𝐵})
Theorem	rabeqsn 4665*	Conditions for a restricted class abstraction to be a singleton. (Contributed by AV, 18-Apr-2019.) (Proof shortened by AV, 26-Aug-2022.)
⊢ ({𝑥 ∈ 𝑉 ∣ 𝜑} = {𝑋} ↔ ∀𝑥((𝑥 ∈ 𝑉 ∧ 𝜑) ↔ 𝑥 = 𝑋))
Theorem	rabsssn 4666*	Conditions for a restricted class abstraction to be a subset of a singleton, i.e. to be a singleton or the empty set. (Contributed by AV, 18-Apr-2019.)
⊢ ({𝑥 ∈ 𝑉 ∣ 𝜑} ⊆ {𝑋} ↔ ∀𝑥 ∈ 𝑉 (𝜑 → 𝑥 = 𝑋))
Theorem	rabeqsnd 4667*	Conditions for a restricted class abstraction to be a singleton, in deduction form. (Contributed by Thierry Arnoux, 2-Dec-2021.)
⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝑥 = 𝐵)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝐵})
Theorem	ralsnsg 4668*	Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑))
Theorem	rexsns 4669*	Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by NM, 22-Aug-2018.)
⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑)
Theorem	rexsngf 4670*	Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.) (Revised by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))
Theorem	ralsngf 4671*	Restricted universal quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by AV, 3-Apr-2023.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))
Theorem	reusngf 4672*	Restricted existential uniqueness over a singleton. (Contributed by AV, 3-Apr-2023.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∃!𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))
Theorem	ralsng 4673*	Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.) Avoid ax-10 2130, ax-12 2164. (Revised by Gino Giotto, 30-Sep-2024.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))
Theorem	rexsng 4674*	Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.) Avoid ax-10 2130, ax-12 2164. (Revised by Gino Giotto, 30-Sep-2024.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))
Theorem	reusng 4675*	Restricted existential uniqueness over a singleton. (Contributed by AV, 3-Apr-2023.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∃!𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))
Theorem	2ralsng 4676*	Substitution expressed in terms of two quantifications over singletons. (Contributed by AV, 22-Dec-2019.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝜑 ↔ 𝜒))
Theorem	ralsngOLD 4677*	Obsolete version of ralsng 4673 as of 30-Sep-2024. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.) (Proof shortened by AV, 7-Apr-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))
Theorem	rexsngOLD 4678*	Obsolete version of rexsng 4674 as of 30-Sep-2024. (Contributed by NM, 29-Jan-2012.) (Proof shortened by AV, 7-Apr-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))
Theorem	rexreusng 4679*	Restricted existential uniqueness over a singleton is equivalent to a restricted existential quantification over a singleton. (Contributed by AV, 3-Apr-2023.)
⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ ∃!𝑥 ∈ {𝐴}𝜑))
Theorem	exsnrex 4680	There is a set being the element of a singleton if and only if there is an element of the singleton. (Contributed by Alexander van der Vekens, 1-Jan-2018.)
⊢ (∃𝑥 𝑀 = {𝑥} ↔ ∃𝑥 ∈ 𝑀 𝑀 = {𝑥})
Theorem	ralsn 4681*	Convert a universal quantification restricted to a singleton to a substitution. (Contributed by NM, 27-Apr-2009.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)
Theorem	rexsn 4682*	Convert an existential quantification restricted to a singleton to a substitution. (Contributed by Jeff Madsen, 5-Jan-2011.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)
Theorem	elpwunsn 4683	Membership in an extension of a power class. (Contributed by NM, 26-Mar-2007.)
⊢ (𝐴 ∈ (𝒫 (𝐵 ∪ {𝐶}) ∖ 𝒫 𝐵) → 𝐶 ∈ 𝐴)
Theorem	eqoreldif 4684	An element of a set is either equal to another element of the set or a member of the difference of the set and the singleton containing the other element. (Contributed by AV, 25-Aug-2020.) (Proof shortened by JJ, 23-Jul-2021.)
⊢ (𝐵 ∈ 𝐶 → (𝐴 ∈ 𝐶 ↔ (𝐴 = 𝐵 ∨ 𝐴 ∈ (𝐶 ∖ {𝐵}))))
Theorem	eltpg 4685	Members of an unordered triple of classes. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷)))
Theorem	eldiftp 4686	Membership in a set with three elements removed. Similar to eldifsn 4786 and eldifpr 4656. (Contributed by David A. Wheeler, 22-Jul-2017.)
⊢ (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷, 𝐸}) ↔ (𝐴 ∈ 𝐵 ∧ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷 ∧ 𝐴 ≠ 𝐸)))
Theorem	eltpi 4687	A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷))
Theorem	eltp 4688	A member of an unordered triple of classes is one of them. Special case of Exercise 1 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Apr-1994.) (Revised by Mario Carneiro, 11-Feb-2015.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷))
Theorem	dftp2 4689*	Alternate definition of unordered triple of classes. Special case of Definition 5.3 of [TakeutiZaring] p. 16. (Contributed by NM, 8-Apr-1994.)
⊢ {𝐴, 𝐵, 𝐶} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)}
Theorem	nfpr 4690	Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥{𝐴, 𝐵}
Theorem	ifpr 4691	Membership of a conditional operator in an unordered pair. (Contributed by NM, 17-Jun-2007.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → if(𝜑, 𝐴, 𝐵) ∈ {𝐴, 𝐵})
Theorem	ralprgf 4692*	Convert a restricted universal quantification over a pair to a conjunction, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 17-Sep-2011.) (Revised by AV, 8-Apr-2023.)
⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑥𝜒    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒)))
Theorem	rexprgf 4693*	Convert a restricted existential quantification over a pair to a disjunction, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 17-Sep-2011.) (Revised by AV, 2-Apr-2023.)
⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑥𝜒    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒)))
Theorem	ralprg 4694*	Convert a restricted universal quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) Avoid ax-10 2130, ax-12 2164. (Revised by Gino Giotto, 30-Sep-2024.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒)))
Theorem	ralprgOLD 4695*	Obsolete version of ralprg 4694 as of 30-Sep-2024. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) (Proof shortened by AV, 8-Apr-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒)))
Theorem	rexprg 4696*	Convert a restricted existential quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) Avoid ax-10 2130, ax-12 2164. (Revised by Gino Giotto, 30-Sep-2024.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒)))
Theorem	rexprgOLD 4697*	Obsolete version of rexprg 4696 as of 30-Sep-2024. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) (Proof shortened by AV, 8-Apr-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒)))
Theorem	raltpg 4698*	Convert a restricted universal quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜃))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)))
Theorem	rextpg 4699*	Convert a restricted existential quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜃))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∨ 𝜒 ∨ 𝜃)))
Theorem	ralpr 4700*	Convert a restricted universal quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    ⇒   ⊢ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒))