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Theorem List for Metamath Proof Explorer - 4201-4300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremelpwi 4201 Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)
(𝐴 ∈ 𝒫 𝐵𝐴𝐵)

Theoremelpwb 4202 Characterization of the elements of a power class. (Contributed by BJ, 29-Apr-2021.)
(𝐴 ∈ 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴𝐵))

Theoremelpwid 4203 An element of a power class is a subclass. Deduction form of elpwi 4201. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ 𝒫 𝐵)       (𝜑𝐴𝐵)

Theoremelelpwi 4204 If 𝐴 belongs to a part of 𝐶 then 𝐴 belongs to 𝐶. (Contributed by FL, 3-Aug-2009.)
((𝐴𝐵𝐵 ∈ 𝒫 𝐶) → 𝐴𝐶)

Theoremnfpw 4205 Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
𝑥𝐴       𝑥𝒫 𝐴

Theorempwidg 4206 Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐴𝑉𝐴 ∈ 𝒫 𝐴)

Theorempwid 4207 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 4208* Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)
(𝒫 𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))

2.1.17  Unordered and ordered pairs

Theoremsnjust 4209* Soundness justification theorem for df-sn 4211. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
{𝑥𝑥 = 𝐴} = {𝑦𝑦 = 𝐴}

Syntaxcsn 4210 Extend class notation to include singleton.
class {𝐴}

Definitiondf-sn 4211* 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, although it is not very meaningful in this case. For an alternate definition see dfsn2 4223. (Contributed by NM, 21-Jun-1993.)
{𝐴} = {𝑥𝑥 = 𝐴}

Syntaxcpr 4212 Extend class notation to include unordered pair.
class {𝐴, 𝐵}

Definitiondf-pr 4213 Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. For example, 𝐴 ∈ {1, -1} → (𝐴↑2) = 1 (ex-pr 27417). They are unordered, so {𝐴, 𝐵} = {𝐵, 𝐴} as proven by prcom 4299. For a more traditional definition, but requiring a dummy variable, see dfpr2 4228. {𝐴, 𝐴} is also an unordered pair, but also a singleton because of {𝐴} = {𝐴, 𝐴} (see dfsn2 4223). Therefore, {𝐴, 𝐵} is called a proper (unordered) pair iff 𝐴𝐵 and 𝐴 and 𝐵 are sets. (Contributed by NM, 21-Jun-1993.)
{𝐴, 𝐵} = ({𝐴} ∪ {𝐵})

Syntaxctp 4214 Extend class notation to include unordered triplet.
class {𝐴, 𝐵, 𝐶}

Definitiondf-tp 4215 Define unordered triple of classes. Definition of [Enderton] p. 19. (Contributed by NM, 9-Apr-1994.)
{𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})

Syntaxcop 4216 Extend class notation to include ordered pair.
class 𝐴, 𝐵

Definitiondf-op 4217* 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 4457, opprc2 4458, and 0nelop 4989). For Kuratowski's actual definition when the arguments are sets, see dfop 4432. For the justifying theorem (for sets) see opth 4974. See dfopif 4430 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 4217 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 4217 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 5005. 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 8553, 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 5201. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 13097. An ordered pair of real numbers can also be represented by a complex number as shown by cru 11050. 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 4430. (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 4218 Extend class notation to include ordered triple.
class 𝐴, 𝐵, 𝐶

Definitiondf-ot 4219 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 4220 Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.)
(𝐴 = 𝐵 → {𝐴} = {𝐵})

Theoremsneqi 4221 Equality inference for singletons. (Contributed by NM, 22-Jan-2004.)
𝐴 = 𝐵       {𝐴} = {𝐵}

Theoremsneqd 4222 Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝐴} = {𝐵})

Theoremdfsn2 4223 Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
{𝐴} = {𝐴, 𝐴}

Theoremelsng 4224 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 4225 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 4226 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 4227 There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.)
(𝐴 ∈ {𝐵} → 𝐴 = 𝐵)

Theoremdfpr2 4228* Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
{𝐴, 𝐵} = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵)}

Theoremelprg 4229 A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15, generalized. (Contributed by NM, 13-Sep-1995.)
(𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))

Theoremelpri 4230 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 4231 A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)
𝐴 ∈ V       (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))

Theoremelpr2 4232 A member of an unordered pair of classes is one or the other of them. 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 4233 Obsolete proof of elpr2 4232 as of 23-Jul-2021. (Contributed by NM, 14-Oct-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐵 ∈ V    &   𝐶 ∈ V       (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))

Theoremnelpri 4234 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 4235 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 4236 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 4237 Membership in a set with two elements removed. Similar to eldifsn 4350 and eldiftp 4260. (Contributed by Mario Carneiro, 18-Jul-2017.)
(𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷}) ↔ (𝐴𝐵𝐴𝐶𝐴𝐷))

Theoremrexdifpr 4238 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 4239 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 4240 A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.)
(𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})

Theoremsnid 4241 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 4242 A setvar variable is a member of its singleton. (Contributed by David A. Wheeler, 8-Dec-2018.)
𝑥 ∈ {𝑥}

Theoremelsn2g 4243 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 4244 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 4245 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 4246* Conditions for a restricted class abstraction to be a singleton. (Contributed by AV, 18-Apr-2019.)
({𝑥𝑉𝜑} = {𝑋} ↔ ∀𝑥((𝑥𝑉𝜑) ↔ 𝑥 = 𝑋))

Theoremrabsssn 4247* 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 4248* 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 4249* 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)

Theoremralsng 4250* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑𝜓))

Theoremrexsng 4251* Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑𝜓))

Theorem2ralsng 4252* 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))

Theoremexsnrex 4253 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 4254* Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥 ∈ {𝐴}𝜑𝜓)

Theoremrexsn 4255* Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∃𝑥 ∈ {𝐴}𝜑𝜓)

Theoremelpwunsn 4256 Membership in an extension of a power class. (Contributed by NM, 26-Mar-2007.)
(𝐴 ∈ (𝒫 (𝐵 ∪ {𝐶}) ∖ 𝒫 𝐵) → 𝐶𝐴)

Theoremeqoreldif 4257 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}))))

TheoremeqoreldifOLD 4258 Obsolete proof of eqoreldif 4257 as of 23-Jul-2021. (Contributed by AV, 25-Aug-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐵𝐶 → (𝐴𝐶 ↔ (𝐴 = 𝐵𝐴 ∈ (𝐶 ∖ {𝐵}))))

Theoremeltpg 4259 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 4260 Membership in a set with three elements removed. Similar to eldifsn 4350 and eldifpr 4237. (Contributed by David A. Wheeler, 22-Jul-2017.)
(𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷, 𝐸}) ↔ (𝐴𝐵 ∧ (𝐴𝐶𝐴𝐷𝐴𝐸)))

Theoremeltpi 4261 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 4262 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))

Theoremdftp2 4263* 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 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)}

Theoremnfpr 4264 Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)
𝑥𝐴    &   𝑥𝐵       𝑥{𝐴, 𝐵}

Theoremifpr 4265 Membership of a conditional operator in an unordered pair. (Contributed by NM, 17-Jun-2007.)
((𝐴𝐶𝐵𝐷) → if(𝜑, 𝐴, 𝐵) ∈ {𝐴, 𝐵})

Theoremralprg 4266* Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))       ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))

Theoremrexprg 4267* Convert a quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))       ((𝐴𝑉𝐵𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))

Theoremraltpg 4268* Convert a quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))    &   (𝑥 = 𝐶 → (𝜑𝜃))       ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃)))

Theoremrextpg 4269* Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))    &   (𝑥 = 𝐶 → (𝜑𝜃))       ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃)))

Theoremralpr 4270* Convert a 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 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))       (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒))

Theoremrexpr 4271* Convert an existential quantification over a pair to a disjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))       (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒))

Theoremraltp 4272* Convert a quantification over a triple to a conjunction. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))    &   (𝑥 = 𝐶 → (𝜑𝜃))       (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃))

Theoremrextp 4273* Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))    &   (𝑥 = 𝐶 → (𝜑𝜃))       (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃))

Theoremnfsn 4274 Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.)
𝑥𝐴       𝑥{𝐴}

Theoremcsbsng 4275 Distribute proper substitution through the singleton of a class. csbsng 4275 is derived from the virtual deduction proof csbsngVD 39443. (Contributed by Alan Sare, 10-Nov-2012.)
(𝐴𝑉𝐴 / 𝑥{𝐵} = {𝐴 / 𝑥𝐵})

Theoremcsbprg 4276 Distribute proper substitution through a pair of classes. (Contributed by Alexander van der Vekens, 4-Sep-2018.)
(𝐶𝑉𝐶 / 𝑥{𝐴, 𝐵} = {𝐶 / 𝑥𝐴, 𝐶 / 𝑥𝐵})

Theoremelinsn 4277 If the intersection of two classes is a (proper) singleton, then the singleton element is a member of both classes. (Contributed by AV, 30-Dec-2021.)
((𝐴𝑉 ∧ (𝐵𝐶) = {𝐴}) → (𝐴𝐵𝐴𝐶))

Theoremdisjsn 4278 Intersection with the singleton of a non-member is disjoint. (Contributed by NM, 22-May-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵𝐴)

Theoremdisjsn2 4279 Two distinct singletons are disjoint. (Contributed by NM, 25-May-1998.)
(𝐴𝐵 → ({𝐴} ∩ {𝐵}) = ∅)

Theoremdisjpr2 4280 Two completely distinct unordered pairs are disjoint. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Proof shortened by JJ, 23-Jul-2021.)
(((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → ({𝐴, 𝐵} ∩ {𝐶, 𝐷}) = ∅)

Theoremdisjpr2OLD 4281 Obsolete proof of disjpr2 4280 as of 23-Jul-2021. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
(((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → ({𝐴, 𝐵} ∩ {𝐶, 𝐷}) = ∅)

Theoremdisjprsn 4282 The disjoint intersection of an unordered pair and a singleton. (Contributed by AV, 23-Jan-2021.)
((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} ∩ {𝐶}) = ∅)

Theoremdisjtpsn 4283 The disjoint intersection of an unordered triple and a singleton. (Contributed by AV, 14-Nov-2021.)
((𝐴𝐷𝐵𝐷𝐶𝐷) → ({𝐴, 𝐵, 𝐶} ∩ {𝐷}) = ∅)

Theoremdisjtp2 4284 Two completely distinct unordered triples are disjoint. (Contributed by AV, 14-Nov-2021.)
(((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ (𝐴𝐸𝐵𝐸𝐶𝐸) ∧ (𝐴𝐹𝐵𝐹𝐶𝐹)) → ({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸, 𝐹}) = ∅)

Theoremsnprc 4285 The singleton of a proper class (one that doesn't exist) is the empty set. Theorem 7.2 of [Quine] p. 48. (Contributed by NM, 21-Jun-1993.)
𝐴 ∈ V ↔ {𝐴} = ∅)

Theoremsnnzb 4286 A singleton is nonempty iff its argument is a set. (Contributed by Scott Fenton, 8-May-2018.)
(𝐴 ∈ V ↔ {𝐴} ≠ ∅)

Theoremr19.12sn 4287* Special case of r19.12 3092 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.) (Revised by BJ, 18-Mar-2020.)
(𝐴𝑉 → (∃𝑥 ∈ {𝐴}∀𝑦𝐵 𝜑 ↔ ∀𝑦𝐵𝑥 ∈ {𝐴}𝜑))

Theoremrabsn 4288* Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.)
(𝐵𝐴 → {𝑥𝐴𝑥 = 𝐵} = {𝐵})

Theoremrabsnifsb 4289* A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by AV, 21-Jul-2019.)
{𝑥 ∈ {𝐴} ∣ 𝜑} = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅)

Theoremrabsnif 4290* A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 21-Jul-2019.)
(𝑥 = 𝐴 → (𝜑𝜓))       {𝑥 ∈ {𝐴} ∣ 𝜑} = if(𝜓, {𝐴}, ∅)

Theoremrabrsn 4291* A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by Alexander van der Vekens, 22-Dec-2017.) (Proof shortened by AV, 21-Jul-2019.)
(𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = ∅ ∨ 𝑀 = {𝐴}))

Theoremeuabsn2 4292* Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by Mario Carneiro, 14-Nov-2016.)
(∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})

Theoremeuabsn 4293 Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by NM, 22-Feb-2004.)
(∃!𝑥𝜑 ↔ ∃𝑥{𝑥𝜑} = {𝑥})

Theoremreusn 4294* A way to express restricted existential uniqueness of a wff: its restricted class abstraction is a singleton. (Contributed by NM, 30-May-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
(∃!𝑥𝐴 𝜑 ↔ ∃𝑦{𝑥𝐴𝜑} = {𝑦})

Theoremabsneu 4295 Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.)
((𝐴𝑉 ∧ {𝑥𝜑} = {𝐴}) → ∃!𝑥𝜑)

Theoremrabsneu 4296 Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) (Revised by Mario Carneiro, 23-Dec-2016.)
((𝐴𝑉 ∧ {𝑥𝐵𝜑} = {𝐴}) → ∃!𝑥𝐵 𝜑)

Theoremeusn 4297* Two ways to express "𝐴 is a singleton." (Contributed by NM, 30-Oct-2010.)
(∃!𝑥 𝑥𝐴 ↔ ∃𝑥 𝐴 = {𝑥})

Theoremrabsnt 4298* Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by NM, 29-May-2006.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
𝐵 ∈ V    &   (𝑥 = 𝐵 → (𝜑𝜓))       ({𝑥𝐴𝜑} = {𝐵} → 𝜓)

Theoremprcom 4299 Commutative law for unordered pairs. (Contributed by NM, 15-Jul-1993.)
{𝐴, 𝐵} = {𝐵, 𝐴}

Theorempreq1 4300 Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.)
(𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})

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