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Theorem List for Metamath Proof Explorer - 4301-4400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremeqsn 4301* Two ways to express that a nonempty set equals a singleton. (Contributed by NM, 15-Dec-2007.) (Proof shortened by JJ, 23-Jul-2021.)
(𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵))

TheoremeqsnOLD 4302* Obsolete proof of eqsn 4301 as of 23-Jul-2021. (Contributed by NM, 15-Dec-2007.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵))

Theoremissn 4303* A sufficient condition for a (nonempty) set to be a singleton. (Contributed by AV, 20-Sep-2020.)
(∃𝑥𝐴𝑦𝐴 𝑥 = 𝑦 → ∃𝑧 𝐴 = {𝑧})

Theoremn0snor2el 4304* A nonempty set is either a singleton or contains at least two different elements. (Contributed by AV, 20-Sep-2020.)
(𝐴 ≠ ∅ → (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ∨ ∃𝑧 𝐴 = {𝑧}))

Theoremssunpr 4305 Possible values for a set sandwiched between another set and it plus a singleton. (Contributed by Mario Carneiro, 2-Jul-2016.)
((𝐵𝐴𝐴 ⊆ (𝐵 ∪ {𝐶, 𝐷})) ↔ ((𝐴 = 𝐵𝐴 = (𝐵 ∪ {𝐶})) ∨ (𝐴 = (𝐵 ∪ {𝐷}) ∨ 𝐴 = (𝐵 ∪ {𝐶, 𝐷}))))

Theoremsspr 4306 The subsets of a pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Mario Carneiro, 2-Jul-2016.)
(𝐴 ⊆ {𝐵, 𝐶} ↔ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})))

Theoremsstp 4307 The subsets of a triple. (Contributed by Mario Carneiro, 2-Jul-2016.)
(𝐴 ⊆ {𝐵, 𝐶, 𝐷} ↔ (((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) ∨ ((𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}) ∨ (𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷}))))

Theoremtpss 4308 A triplet of elements of a class is a subset of the class. (Contributed by NM, 9-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       ((𝐴𝐷𝐵𝐷𝐶𝐷) ↔ {𝐴, 𝐵, 𝐶} ⊆ 𝐷)

Theoremtpssi 4309 A triple of elements of a class is a subset of the class. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
((𝐴𝐷𝐵𝐷𝐶𝐷) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)

Theoremsneqrg 4310 Closed form of sneqr 4311. (Contributed by Scott Fenton, 1-Apr-2011.) (Proof shortened by JJ, 23-Jul-2021.)
(𝐴𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))

Theoremsneqr 4311 If the singletons of two sets are equal, the two sets are equal. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 27-Aug-1993.)
𝐴 ∈ V       ({𝐴} = {𝐵} → 𝐴 = 𝐵)

Theoremsnsssn 4312 If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.)
𝐴 ∈ V       ({𝐴} ⊆ {𝐵} → 𝐴 = 𝐵)

TheoremsneqrgOLD 4313 Obsolete proof of sneqrg 4310 as of 23-Jul-2021. (Contributed by Scott Fenton, 1-Apr-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))

Theoremsneqbg 4314 Two singletons of sets are equal iff their elements are equal. (Contributed by Scott Fenton, 16-Apr-2012.)
(𝐴𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))

Theoremsnsspw 4315 The singleton of a class is a subset of its power class. (Contributed by NM, 21-Jun-1993.)
{𝐴} ⊆ 𝒫 𝐴

Theoremprsspw 4316 An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by OpenAI, 25-Mar-2020.)
𝐴 ∈ V    &   𝐵 ∈ V       ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴𝐶𝐵𝐶))

Theorempreq1b 4317 Biconditional equality lemma for unordered pairs, deduction form. Two unordered pairs have the same second element iff the first elements are equal. (Contributed by AV, 18-Dec-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵))

Theorempreq2b 4318 Biconditional equality lemma for unordered pairs, deduction form. Two unordered pairs have the same first element iff the second elements are equal. (Contributed by AV, 18-Dec-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ 𝐴 = 𝐵))

Theorempreqr1 4319 Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. (Contributed by NM, 18-Oct-1995.)
𝐴 ∈ V    &   𝐵 ∈ V       ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵)

Theorempreqr1OLD 4320 Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. (Contributed by NM, 18-Oct-1995.) Obsolete version of preqr1 4319 as of 18-Dec-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ V    &   𝐵 ∈ V       ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵)

Theorempreqr2 4321 Reverse equality lemma for unordered pairs. If two unordered pairs have the same first element, the second elements are equal. (Contributed by NM, 15-Jul-1993.)
𝐴 ∈ V    &   𝐵 ∈ V       ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵)

Theorempreq12b 4322 Equality relationship for two unordered pairs. (Contributed by NM, 17-Oct-1996.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V       ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))

Theoremprel12 4323 Equality of two unordered pairs. (Contributed by NM, 17-Oct-1996.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V       𝐴 = 𝐵 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷})))

Theoremopthpr 4324 An unordered pair has the ordered pair property (compare opth 4871) under certain conditions. (Contributed by NM, 27-Mar-2007.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V       (𝐴𝐷 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Theorempreqr1g 4325 Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. Closed form of preqr1 4319. (Contributed by AV, 29-Jan-2021.) (Revised by AV, 18-Sep-2021.)
((𝐴𝑉𝐵𝑊) → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵))

Theorempreq12bg 4326 Closed form of preq12b 4322. (Contributed by Scott Fenton, 28-Mar-2014.)
(((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))

Theoremprel12g 4327 Closed form of prel12 4323. (Contributed by AV, 9-Dec-2018.)
(((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) → (¬ 𝐴 = 𝐵 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷}))))

Theoremprneimg 4328 Two pairs are not equal if at least one element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
(((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌)) → (((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)) → {𝐴, 𝐵} ≠ {𝐶, 𝐷}))

Theoremprnebg 4329 A (proper) pair is not equal to another (maybe inproper) pair if and only if an element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 16-Jan-2018.)
(((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌) ∧ 𝐴𝐵) → (((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)) ↔ {𝐴, 𝐵} ≠ {𝐶, 𝐷}))

Theorempreqsnd 4330 Equivalence for a pair equal to a singleton, deduction form. (Contributed by Thierry Arnoux, 27-Dec-2016.)
(𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)       (𝜑 → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶)))

Theorempreqsn 4331 Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.) (Proof shortened by JJ, 23-Jul-2021.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵𝐵 = 𝐶))

TheorempreqsnOLD 4332 Obsolete proof of preqsn 4331 as of 23-Jul-2021. (Contributed by NM, 3-Jun-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵𝐵 = 𝐶))

Theoremelpreqprlem 4333* Lemma for elpreqpr 4334. (Contributed by Scott Fenton, 7-Dec-2020.) (Revised by AV, 9-Dec-2020.)
(𝐵𝑉 → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥})

Theoremelpreqpr 4334* Equality and membership rule for pairs. (Contributed by Scott Fenton, 7-Dec-2020.)
(𝐴 ∈ {𝐵, 𝐶} → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})

Theoremelpreqprb 4335* A set is an element of an unordered pair iff there is another (maybe the same) set which is an element of the unordered pair. (Proposed by BJ, 8-Dec-2020.) (Contributed by AV, 9-Dec-2020.)
(𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥}))

Theoremelpr2elpr 4336* For an element of an unordered pair which is a subset of a given set, there is another (maybe the same) element of the given set being an element of the unordered pair. (Contributed by AV, 5-Dec-2020.)
((𝑋𝑉𝑌𝑉𝐴 ∈ {𝑋, 𝑌}) → ∃𝑏𝑉 {𝑋, 𝑌} = {𝐴, 𝑏})

Theoremdfopif 4337 Rewrite df-op 4132 using if. When both arguments are sets, it reduces to the standard Kuratowski definition; otherwise, it is defined to be the empty set. Avoid directly depending on this detail so that theorems will not depend on the Kuratowski construction. (Contributed by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.)
𝐴, 𝐵⟩ = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅)

Theoremdfopg 4338 Value of the ordered pair when the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.)
((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})

Theoremdfop 4339 Value of an ordered pair when the arguments are sets, with the conclusion corresponding to Kuratowski's original definition. (Contributed by NM, 25-Jun-1998.) (Avoid depending on this detail.)
𝐴 ∈ V    &   𝐵 ∈ V       𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}

Theoremopeq1 4340 Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
(𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)

Theoremopeq2 4341 Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
(𝐴 = 𝐵 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)

Theoremopeq12 4342 Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)
((𝐴 = 𝐶𝐵 = 𝐷) → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)

Theoremopeq1i 4343 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
𝐴 = 𝐵       𝐴, 𝐶⟩ = ⟨𝐵, 𝐶

Theoremopeq2i 4344 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
𝐴 = 𝐵       𝐶, 𝐴⟩ = ⟨𝐶, 𝐵

Theoremopeq12i 4345 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
𝐴 = 𝐵    &   𝐶 = 𝐷       𝐴, 𝐶⟩ = ⟨𝐵, 𝐷

Theoremopeq1d 4346 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)

Theoremopeq2d 4347 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)

Theoremopeq12d 4348 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩)

Theoremoteq1 4349 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
(𝐴 = 𝐵 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)

Theoremoteq2 4350 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
(𝐴 = 𝐵 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)

Theoremoteq3 4351 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
(𝐴 = 𝐵 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)

Theoremoteq1d 4352 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)

Theoremoteq2d 4353 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)

Theoremoteq3d 4354 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)

Theoremoteq123d 4355 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)    &   (𝜑𝐸 = 𝐹)       (𝜑 → ⟨𝐴, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐹⟩)

Theoremnfop 4356 Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.)
𝑥𝐴    &   𝑥𝐵       𝑥𝐴, 𝐵

Theoremnfopd 4357 Deduction version of bound-variable hypothesis builder nfop 4356. This shows how the deduction version of a not-free theorem such as nfop 4356 can be created from the corresponding not-free inference theorem. (Contributed by NM, 4-Feb-2008.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑𝑥𝐴, 𝐵⟩)

Theoremcsbopg 4358 Distribution of class substitution over ordered pairs. (Contributed by Drahflow, 25-Sep-2015.) (Revised by Mario Carneiro, 29-Oct-2015.) (Revised by ML, 25-Oct-2020.)
(𝐴𝑉𝐴 / 𝑥𝐶, 𝐷⟩ = ⟨𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟩)

Theoremopid 4359 The ordered pair 𝐴, 𝐴 in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.) (Avoid depending on this detail.)
𝐴 ∈ V       𝐴, 𝐴⟩ = {{𝐴}}

Theoremralunsn 4360* Restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 23-Apr-2015.)
(𝑥 = 𝐵 → (𝜑𝜓))       (𝐵𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑥𝐴 𝜑𝜓)))

Theorem2ralunsn 4361* Double restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.)
(𝑥 = 𝐵 → (𝜑𝜒))    &   (𝑦 = 𝐵 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜓𝜃))       (𝐵𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})∀𝑦 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ ((∀𝑥𝐴𝑦𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓) ∧ (∀𝑦𝐴 𝜒𝜃))))

Theoremopprc 4362 Expansion of an ordered pair when either member is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)
(¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)

Theoremopprc1 4363 Expansion of an ordered pair when the first member is a proper class. See also opprc 4362. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐴 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)

Theoremopprc2 4364 Expansion of an ordered pair when the second member is a proper class. See also opprc 4362. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)

Theoremoprcl 4365 If an ordered pair has an element, then its arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)
(𝐶 ∈ ⟨𝐴, 𝐵⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Theorempwsn 4366 The power set of a singleton. (Contributed by NM, 5-Jun-2006.)
𝒫 {𝐴} = {∅, {𝐴}}

TheorempwsnALT 4367 Alternate proof of pwsn 4366, more direct. (Contributed by NM, 5-Jun-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
𝒫 {𝐴} = {∅, {𝐴}}

Theorempwpr 4368 The power set of an unordered pair. (Contributed by NM, 1-May-2009.)
𝒫 {𝐴, 𝐵} = ({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}})

Theorempwtp 4369 The power set of an unordered triple. (Contributed by Mario Carneiro, 2-Jul-2016.)
𝒫 {𝐴, 𝐵, 𝐶} = (({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ∪ ({{𝐶}, {𝐴, 𝐶}} ∪ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}}))

Theorempwpwpw0 4370 Compute the power set of the power set of the power set of the empty set. (See also pw0 4283 and pwpw0 4284.) (Contributed by NM, 2-May-2009.)
𝒫 {∅, {∅}} = ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})

Theorempwv 4371 The power class of the universe is the universe. Exercise 4.12(d) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)
𝒫 V = V

2.1.18  The union of a class

Syntaxcuni 4372 Extend class notation to include the union of a class (read: 'union 𝐴')
class 𝐴

Definitiondf-uni 4373* Define the union of a class i.e. the collection of all members of the members of the class. Definition 5.5 of [TakeutiZaring] p. 16. For example, {{1, 3}, {1, 8}} = {1, 3, 8} (ex-uni 26675). This is similar to the union of two classes df-un 3545. (Contributed by NM, 23-Aug-1993.)
𝐴 = {𝑥 ∣ ∃𝑦(𝑥𝑦𝑦𝐴)}

Theoremdfuni2 4374* Alternate definition of class union. (Contributed by NM, 28-Jun-1998.)
𝐴 = {𝑥 ∣ ∃𝑦𝐴 𝑥𝑦}

Theoremeluni 4375* Membership in class union. (Contributed by NM, 22-May-1994.)
(𝐴 𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵))

Theoremeluni2 4376* Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999.)
(𝐴 𝐵 ↔ ∃𝑥𝐵 𝐴𝑥)

Theoremelunii 4377 Membership in class union. (Contributed by NM, 24-Mar-1995.)
((𝐴𝐵𝐵𝐶) → 𝐴 𝐶)

Theoremnfuni 4378 Bound-variable hypothesis builder for union. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
𝑥𝐴       𝑥 𝐴

Theoremnfunid 4379 Deduction version of nfuni 4378. (Contributed by NM, 18-Feb-2013.)
(𝜑𝑥𝐴)       (𝜑𝑥 𝐴)

Theoremunieq 4380 Equality theorem for class union. Exercise 15 of [TakeutiZaring] p. 18. (Contributed by NM, 10-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(𝐴 = 𝐵 𝐴 = 𝐵)

Theoremunieqi 4381 Inference of equality of two class unions. (Contributed by NM, 30-Aug-1993.)
𝐴 = 𝐵        𝐴 = 𝐵

Theoremunieqd 4382 Deduction of equality of two class unions. (Contributed by NM, 21-Apr-1995.)
(𝜑𝐴 = 𝐵)       (𝜑 𝐴 = 𝐵)

Theoremeluniab 4383* Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.)
(𝐴 {𝑥𝜑} ↔ ∃𝑥(𝐴𝑥𝜑))

Theoremelunirab 4384* Membership in union of a class abstraction. (Contributed by NM, 4-Oct-2006.)
(𝐴 {𝑥𝐵𝜑} ↔ ∃𝑥𝐵 (𝐴𝑥𝜑))

Theoremunipr 4385 The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 23-Aug-1993.)
𝐴 ∈ V    &   𝐵 ∈ V        {𝐴, 𝐵} = (𝐴𝐵)

Theoremuniprg 4386 The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 25-Aug-2006.)
((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} = (𝐴𝐵))

Theoremunisn 4387 A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 30-Aug-1993.)
𝐴 ∈ V        {𝐴} = 𝐴

Theoremunisng 4388 A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 13-Aug-2002.)
(𝐴𝑉 {𝐴} = 𝐴)

Theoremunisn3 4389* Union of a singleton in the form of a restricted class abstraction. (Contributed by NM, 3-Jul-2008.)
(𝐴𝐵 {𝑥𝐵𝑥 = 𝐴} = 𝐴)

Theoremdfnfc2 4390* An alternative statement of the effective freeness of a class 𝐴, when it is a set. (Contributed by Mario Carneiro, 14-Oct-2016.) (Proof shortened by JJ, 26-Jul-2021.)
(∀𝑥 𝐴𝑉 → (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦 = 𝐴))

Theoremdfnfc2OLD 4391* Obsolete proof of dfnfc2 4390 as of 26-Jul-2021. (Contributed by Mario Carneiro, 14-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝐴𝑉 → (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦 = 𝐴))

Theoremuniun 4392 The class union of the union of two classes. Theorem 8.3 of [Quine] p. 53. (Contributed by NM, 20-Aug-1993.)
(𝐴𝐵) = ( 𝐴 𝐵)

Theoremuniin 4393 The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uniinqs 7714 for a condition where equality holds. (Contributed by NM, 4-Dec-2003.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(𝐴𝐵) ⊆ ( 𝐴 𝐵)

Theoremuniss 4394 Subclass relationship for class union. Theorem 61 of [Suppes] p. 39. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(𝐴𝐵 𝐴 𝐵)

Theoremssuni 4395 Subclass relationship for class union. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by JJ, 26-Jul-2021.)
((𝐴𝐵𝐵𝐶) → 𝐴 𝐶)

TheoremssuniOLD 4396 Obsolete proof of ssuni 4395 as of 26-Jul-2021. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵𝐵𝐶) → 𝐴 𝐶)

Theoremunissi 4397 Subclass relationship for subclass union. Inference form of uniss 4394. (Contributed by David Moews, 1-May-2017.)
𝐴𝐵        𝐴 𝐵

Theoremunissd 4398 Subclass relationship for subclass union. Deduction form of uniss 4394. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)       (𝜑 𝐴 𝐵)

Theoremuni0b 4399 The union of a set is empty iff the set is included in the singleton of the empty set. (Contributed by NM, 12-Sep-2004.)
( 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})

Theoremuni0c 4400* The union of a set is empty iff all of its members are empty. (Contributed by NM, 16-Aug-2006.)
( 𝐴 = ∅ ↔ ∀𝑥𝐴 𝑥 = ∅)

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