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Theorem List for Metamath Proof Explorer - 4501-4600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdedth4v 4501 Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 4499. (Contributed by NM, 21-Apr-2007.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
(𝐴 = if(𝜑, 𝐴, 𝑅) → (𝜓𝜒))    &   (𝐵 = if(𝜑, 𝐵, 𝑆) → (𝜒𝜃))    &   (𝐶 = if(𝜑, 𝐶, 𝑇) → (𝜃𝜏))    &   (𝐷 = if(𝜑, 𝐷, 𝑈) → (𝜏𝜂))    &   𝜂       (𝜑𝜓)

Theoremelimhyp 4502 Eliminate a hypothesis containing class variable 𝐴 when it is known for a specific class 𝐵. For more information, see comments in dedth 4495. (Contributed by NM, 15-May-1999.)
(𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜑𝜓))    &   (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜓))    &   𝜒       𝜓

Theoremelimhyp2v 4503 Eliminate a hypothesis containing 2 class variables. (Contributed by NM, 14-Aug-1999.)
(𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜑𝜒))    &   (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒𝜃))    &   (𝐶 = if(𝜑, 𝐴, 𝐶) → (𝜏𝜂))    &   (𝐷 = if(𝜑, 𝐵, 𝐷) → (𝜂𝜃))    &   𝜏       𝜃

Theoremelimhyp3v 4504 Eliminate a hypothesis containing 3 class variables. (Contributed by NM, 14-Aug-1999.)
(𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜑𝜒))    &   (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒𝜃))    &   (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃𝜏))    &   (𝐷 = if(𝜑, 𝐴, 𝐷) → (𝜂𝜁))    &   (𝑅 = if(𝜑, 𝐵, 𝑅) → (𝜁𝜎))    &   (𝑆 = if(𝜑, 𝐶, 𝑆) → (𝜎𝜏))    &   𝜂       𝜏

Theoremelimhyp4v 4505 Eliminate a hypothesis containing 4 class variables (for use with the weak deduction theorem dedth 4495). (Contributed by NM, 16-Apr-2005.)
(𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜑𝜒))    &   (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒𝜃))    &   (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃𝜏))    &   (𝐹 = if(𝜑, 𝐹, 𝐺) → (𝜏𝜓))    &   (𝐷 = if(𝜑, 𝐴, 𝐷) → (𝜂𝜁))    &   (𝑅 = if(𝜑, 𝐵, 𝑅) → (𝜁𝜎))    &   (𝑆 = if(𝜑, 𝐶, 𝑆) → (𝜎𝜌))    &   (𝐺 = if(𝜑, 𝐹, 𝐺) → (𝜌𝜓))    &   𝜂       𝜓

Theoremelimel 4506 Eliminate a membership hypothesis for weak deduction theorem, when special case 𝐵𝐶 is provable. (Contributed by NM, 15-May-1999.)
𝐵𝐶       if(𝐴𝐶, 𝐴, 𝐵) ∈ 𝐶

Theoremelimdhyp 4507 Version of elimhyp 4502 where the hypothesis is deduced from the final antecedent. See divalg 15743 for an example of its use. (Contributed by Paul Chapman, 25-Mar-2008.)
(𝜑𝜓)    &   (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜒))    &   (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜃𝜒))    &   𝜃       𝜒

Theoremkeephyp 4508 Transform a hypothesis 𝜓 that we want to keep (but contains the same class variable 𝐴 used in the eliminated hypothesis) for use with the weak deduction theorem. (Contributed by NM, 15-May-1999.)
(𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜃))    &   (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜃))    &   𝜓    &   𝜒       𝜃

Theoremkeephyp2v 4509 Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 4495). (Contributed by NM, 16-Apr-2005.)
(𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜓𝜒))    &   (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒𝜃))    &   (𝐶 = if(𝜑, 𝐴, 𝐶) → (𝜏𝜂))    &   (𝐷 = if(𝜑, 𝐵, 𝐷) → (𝜂𝜃))    &   𝜓    &   𝜏       𝜃

Theoremkeephyp3v 4510 Keep a hypothesis containing 3 class variables. (Contributed by NM, 27-Sep-1999.)
(𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜌𝜒))    &   (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒𝜃))    &   (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃𝜏))    &   (𝐷 = if(𝜑, 𝐴, 𝐷) → (𝜂𝜁))    &   (𝑅 = if(𝜑, 𝐵, 𝑅) → (𝜁𝜎))    &   (𝑆 = if(𝜑, 𝐶, 𝑆) → (𝜎𝜏))    &   𝜌    &   𝜂       𝜏

2.1.17  Power classes

Syntaxcpw 4511 Extend class notation to include power class. (The tilde in the Metamath token is meant to suggest the calligraphic font of the P.)
class 𝒫 𝐴

Theorempwjust 4512* Soundness justification theorem for df-pw 4513. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
{𝑥𝑥𝐴} = {𝑦𝑦𝐴}

Definitiondf-pw 4513* Define power class. Definition 5.10 of [TakeutiZaring] p. 17, but we also let it apply to proper classes, i.e. those that are not members of V. When applied to a set, this produces its power set. A power set of S is the set of all subsets of S, including the empty set and S itself. For example, if 𝐴 = {3, 5, 7}, then 𝒫 𝐴 = {∅, {3}, {5}, {7}, {3, 5}, {3, 7}, {5, 7}, {3, 5, 7}} (ex-pw 28212). We will later introduce the Axiom of Power Sets ax-pow 5243, which can be expressed in class notation per pwexg 5256. Still later we will prove, in hashpw 13793, that the size of the power set of a finite set is 2 raised to the power of the size of the set. (Contributed by NM, 24-Jun-1993.)
𝒫 𝐴 = {𝑥𝑥𝐴}

Theoremelpwg 4514 Membership in a power class. Theorem 86 of [Suppes] p. 47. See also elpw2g 5223. (Contributed by NM, 6-Aug-2000.) (Proof shortened by BJ, 31-Dec-2023.)
(𝐴𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))

Theoremelpw 4515 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 𝐵𝐴𝐵)

Theoremvelpw 4516 Setvar variable membership in a power class. (Contributed by David A. Wheeler, 8-Dec-2018.)
(𝑥 ∈ 𝒫 𝐴𝑥𝐴)

TheoremelpwOLD 4517 Obsolete proof of elpw 4515 as of 31-Dec-2023. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by NM, 31-Dec-1993.)
𝐴 ∈ V       (𝐴 ∈ 𝒫 𝐵𝐴𝐵)

TheoremelpwgOLD 4518 Obsolete proof of elpwg 4514 as of 31-Dec-2023. (Contributed by NM, 6-Aug-2000.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))

Theoremelpwd 4519 Membership in a power class. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐴𝐵)       (𝜑𝐴 ∈ 𝒫 𝐵)

Theoremelpwi 4520 Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)
(𝐴 ∈ 𝒫 𝐵𝐴𝐵)

Theoremelpwb 4521 Characterization of the elements of a power class. (Contributed by BJ, 29-Apr-2021.)
(𝐴 ∈ 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴𝐵))

Theoremelpwid 4522 An element of a power class is a subclass. Deduction form of elpwi 4520. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ 𝒫 𝐵)       (𝜑𝐴𝐵)

Theoremelelpwi 4523 If 𝐴 belongs to a part of 𝐶, then 𝐴 belongs to 𝐶. (Contributed by FL, 3-Aug-2009.)
((𝐴𝐵𝐵 ∈ 𝒫 𝐶) → 𝐴𝐶)

Theoremsspw 4524 The powerclass preserves inclusion. See sspwb 5319 for the biconditional version. (Contributed by NM, 13-Oct-1996.) Extract forward implication of sspwb 5319 since it requires fewer axioms. (Revised by BJ, 13-Apr-2024.)
(𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Theoremsspwi 4525 The powerclass preserves inclusion (inference form). (Contributed by BJ, 13-Apr-2024.)
𝐴𝐵       𝒫 𝐴 ⊆ 𝒫 𝐵

Theoremsspwd 4526 The powerclass preserves inclusion (deduction form). (Contributed by BJ, 13-Apr-2024.)
(𝜑𝐴𝐵)       (𝜑 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Theorempweq 4527 Equality theorem for power class. (Contributed by NM, 21-Jun-1993.) (Proof shortened by BJ, 13-Apr-2024.)
(𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵)

TheorempweqALT 4528 Alternate proof of pweq 4527 directly from the definition. (Contributed by NM, 21-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵)

Theorempweqi 4529 Equality inference for power class. (Contributed by NM, 27-Nov-2013.)
𝐴 = 𝐵       𝒫 𝐴 = 𝒫 𝐵

Theorempweqd 4530 Equality deduction for power class. (Contributed by NM, 27-Nov-2013.)
(𝜑𝐴 = 𝐵)       (𝜑 → 𝒫 𝐴 = 𝒫 𝐵)

Theorempwunss 4531 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 5179, ax-nul 5186, ax-pr 5307 and shorten proof. (Revised by BJ, 13-Apr-2024.)
(𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵)

Theoremnfpw 4532 Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
𝑥𝐴       𝑥𝒫 𝐴

Theorempwidg 4533 A set is an element of its power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐴𝑉𝐴 ∈ 𝒫 𝐴)

Theorempwidb 4534 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)

Theorempwid 4535 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

Theorempwss 4536* Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)
(𝒫 𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))

Theorempwundif 4537 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 5179, ax-nul 5186, ax-pr 5307 and shorten proof. (Revised by BJ, 14-Apr-2024.)
𝒫 (𝐴𝐵) = ((𝒫 (𝐴𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)

2.1.18  Unordered and ordered pairs

Theoremsnjust 4538* Soundness justification theorem for df-sn 4540. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
{𝑥𝑥 = 𝐴} = {𝑦𝑦 = 𝐴}

Syntaxcsn 4539 Extend class notation to include singleton.
class {𝐴}

Definitiondf-sn 4540* 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 4627. For an alternate definition see dfsn2 4552. (Contributed by NM, 21-Jun-1993.)
{𝐴} = {𝑥𝑥 = 𝐴}

Syntaxcpr 4541 Extend class notation to include unordered pair.
class {𝐴, 𝐵}

Definitiondf-pr 4542 Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. For example, 𝐴 ∈ {1, -1} → (𝐴↑2) = 1 (ex-pr 28213). They are unordered, so {𝐴, 𝐵} = {𝐵, 𝐴} as proven by prcom 4642. For a more traditional definition, but requiring a dummy variable, see dfpr2 4558. {𝐴, 𝐴} is also an unordered pair, but also a singleton because of {𝐴} = {𝐴, 𝐴} (see dfsn2 4552). Therefore, {𝐴, 𝐵} is called a proper (unordered) pair iff 𝐴𝐵 and 𝐴 and 𝐵 are sets.

Note: ordered pairs are a completely different object defined below in df-op 4546. 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})

Syntaxctp 4543 Extend class notation to include unordered triple (sometimes called "unordered triplet").
class {𝐴, 𝐵, 𝐶}

Definitiondf-tp 4544 Define unordered triple of classes. Definition of [Enderton] p. 19.

Note: ordered triples are a completely different object defined below in df-ot 4548. 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})

Syntaxcop 4545 Extend class notation to include ordered pair.
class 𝐴, 𝐵

Definitiondf-op 4546* 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 4802, opprc2 4803, and 0nelop 5363). For Kuratowski's actual definition when the arguments are sets, see dfop 4775. For the justifying theorem (for sets) see opth 5345. See dfopif 4773 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 4546 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 4546 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, 𝐵2 = {{{𝐴}, ∅}, {{𝐵}}}, justified by opthwiener 5381. This was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition 𝐴, 𝐵3 = {𝐴, {𝐴, 𝐵}} is justified by opthreg 9069, 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, 𝐵4 = ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})), justified by opthprc 5593. 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, 𝐵5 = {{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 5235). 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 5385 and opthhausdorff 5384. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 13626. An ordered pair of real numbers can also be represented by a complex number as shown by cru 11617. 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 4773. (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}})}

Syntaxcotp 4547 Extend class notation to include ordered triple.
class 𝐴, 𝐵, 𝐶

Definitiondf-ot 4548 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

Theoremsneq 4549 Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.)
(𝐴 = 𝐵 → {𝐴} = {𝐵})

Theoremsneqi 4550 Equality inference for singletons. (Contributed by NM, 22-Jan-2004.)
𝐴 = 𝐵       {𝐴} = {𝐵}

Theoremsneqd 4551 Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝐴} = {𝐵})

Theoremdfsn2 4552 Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
{𝐴} = {𝐴, 𝐴}

Theoremelsng 4553 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.)
(𝐴𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))

Theoremelsn 4554 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)

Theoremvelsn 4555 There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.)
(𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)

Theoremelsni 4556 There is at most one element in a singleton. (Contributed by NM, 5-Jun-1994.)
(𝐴 ∈ {𝐵} → 𝐴 = 𝐵)

Theoremabsn 4557* Condition for a class abstraction to be a singleton. Formerly part of proof of dfiota2 6294. (Contributed by Andrew Salmon, 30-Jun-2011.) (Revised by AV, 24-Aug-2022.)
({𝑥𝜑} = {𝑌} ↔ ∀𝑥(𝜑𝑥 = 𝑌))

Theoremdfpr2 4558* Alternate definition of a pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
{𝐴, 𝐵} = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵)}

Theoremdfsn2ALT 4559 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}

Theoremelprg 4560 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.)
(𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))

Theoremelpri 4561 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 = 𝐵𝐴 = 𝐶))

Theoremelpr 4562 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 = 𝐵𝐴 = 𝐶))

Theoremelpr2g 4563 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.)
((𝐵𝑉𝐶𝑊) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))

Theoremelpr2 4564 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 = 𝐵𝐴 = 𝐶))

Theoremelpr2OLD 4565 Obsolete version of elpr2 4564 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 = 𝐵𝐴 = 𝐶))

Theoremnelpr2 4566 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.)
(𝜑𝐴𝑉)    &   (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})       (𝜑𝐴𝐶)

Theoremnelpr1 4567 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.)
(𝜑𝐴𝑉)    &   (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})       (𝜑𝐴𝐵)

Theoremnelpri 4568 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.)
𝐴𝐵    &   𝐴𝐶        ¬ 𝐴 ∈ {𝐵, 𝐶}

Theoremprneli 4569 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.)
𝐴𝐵    &   𝐴𝐶       𝐴 ∉ {𝐵, 𝐶}

Theoremnelprd 4570 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.)
(𝜑𝐴𝐵)    &   (𝜑𝐴𝐶)       (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})

Theoremeldifpr 4571 Membership in a set with two elements removed. Similar to eldifsn 4693 and eldiftp 4598. (Contributed by Mario Carneiro, 18-Jul-2017.)
(𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷}) ↔ (𝐴𝐵𝐴𝐶𝐴𝐷))

Theoremrexdifpr 4572 Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.)
(∃𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶})𝜑 ↔ ∃𝑥𝐴 (𝑥𝐵𝑥𝐶𝜑))

Theoremsnidg 4573 A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 28-Oct-2003.)
(𝐴𝑉𝐴 ∈ {𝐴})

Theoremsnidb 4574 A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.)
(𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})

Theoremsnid 4575 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}

Theoremvsnid 4576 A setvar variable is a member of its singleton. (Contributed by David A. Wheeler, 8-Dec-2018.)
𝑥 ∈ {𝑥}

Theoremelsn2g 4577 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.)
(𝐵𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))

Theoremelsn2 4578 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)

Theoremnelsn 4579 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.)
(𝐴𝐵 → ¬ 𝐴 ∈ {𝐵})

Theoremrabeqsn 4580* Conditions for a restricted class abstraction to be a singleton. (Contributed by AV, 18-Apr-2019.) (Proof shortened by AV, 26-Aug-2022.)
({𝑥𝑉𝜑} = {𝑋} ↔ ∀𝑥((𝑥𝑉𝜑) ↔ 𝑥 = 𝑋))

Theoremrabsssn 4581* 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.)
({𝑥𝑉𝜑} ⊆ {𝑋} ↔ ∀𝑥𝑉 (𝜑𝑥 = 𝑋))

Theoremralsnsg 4582* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
(𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))

Theoremrexsns 4583* 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)

Theoremrexsngf 4584* Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.) (Revised by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑𝜓))

Theoremralsngf 4585* Restricted universal quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by AV, 3-Apr-2023.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑𝜓))

Theoremreusngf 4586* Restricted existential uniqueness over a singleton. (Contributed by AV, 3-Apr-2023.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∃!𝑥 ∈ {𝐴}𝜑𝜓))

Theoremralsng 4587* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.) (Proof shortened by AV, 7-Apr-2023.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑𝜓))

Theoremrexsng 4588* Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.) (Proof shortened by AV, 7-Apr-2023.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑𝜓))

Theoremreusng 4589* Restricted existential uniqueness over a singleton. (Contributed by AV, 3-Apr-2023.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∃!𝑥 ∈ {𝐴}𝜑𝜓))

Theorem2ralsng 4590* Substitution expressed in terms of two quantifications over singletons. (Contributed by AV, 22-Dec-2019.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))       ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝜑𝜒))

Theoremrexreusng 4591* Restricted existential uniqueness over a singleton is equivalent to a restricted existential quantification over a singleton. (Contributed by AV, 3-Apr-2023.)
(𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ ∃!𝑥 ∈ {𝐴}𝜑))

Theoremexsnrex 4592 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} ↔ ∃𝑥𝑀 𝑀 = {𝑥})

Theoremralsn 4593* Convert a universal quantification restricted to a singleton to a substitution. (Contributed by NM, 27-Apr-2009.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥 ∈ {𝐴}𝜑𝜓)

Theoremrexsn 4594* Convert an existential quantification restricted to a singleton to a substitution. (Contributed by Jeff Madsen, 5-Jan-2011.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∃𝑥 ∈ {𝐴}𝜑𝜓)

Theoremelpwunsn 4595 Membership in an extension of a power class. (Contributed by NM, 26-Mar-2007.)
(𝐴 ∈ (𝒫 (𝐵 ∪ {𝐶}) ∖ 𝒫 𝐵) → 𝐶𝐴)

Theoremeqoreldif 4596 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.)
(𝐵𝐶 → (𝐴𝐶 ↔ (𝐴 = 𝐵𝐴 ∈ (𝐶 ∖ {𝐵}))))

Theoremeltpg 4597 Members of an unordered triple of classes. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.)
(𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵𝐴 = 𝐶𝐴 = 𝐷)))

Theoremeldiftp 4598 Membership in a set with three elements removed. Similar to eldifsn 4693 and eldifpr 4571. (Contributed by David A. Wheeler, 22-Jul-2017.)
(𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷, 𝐸}) ↔ (𝐴𝐵 ∧ (𝐴𝐶𝐴𝐷𝐴𝐸)))

Theoremeltpi 4599 A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015.)
(𝐴 ∈ {𝐵, 𝐶, 𝐷} → (𝐴 = 𝐵𝐴 = 𝐶𝐴 = 𝐷))

Theoremeltp 4600 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 = 𝐵𝐴 = 𝐶𝐴 = 𝐷))

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