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	snsssn 4601	If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ({𝐴} ⊆ {𝐵} → 𝐴 = 𝐵)
Theorem	mosneq 4602*	There exists at most one set whose singleton is equal to a given class. See also moeq 3588. (Contributed by BJ, 24-Sep-2022.)
⊢ ∃*𝑥{𝑥} = 𝐴
Theorem	sneqbg 4603	Two singletons of sets are equal iff their elements are equal. (Contributed by Scott Fenton, 16-Apr-2012.)
⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))
Theorem	snsspw 4604	The singleton of a class is a subset of its power class. (Contributed by NM, 21-Jun-1993.)
⊢ {𝐴} ⊆ 𝒫 𝐴
Theorem	prsspw 4605	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 ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶))
Theorem	preq1b 4606	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.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵))
Theorem	preq2b 4607	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.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ 𝐴 = 𝐵))
Theorem	preqr1 4608	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)
Theorem	preqr2 4609	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)
Theorem	preq12b 4610	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 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))
Theorem	prel12OLD 4611	Obsolete as of 16-Jun-2022. Use prel12g 4627 instead. (Contributed by NM, 17-Oct-1996.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    &   ⊢ 𝐷 ∈ V    ⇒   ⊢ (¬ 𝐴 = 𝐵 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷})))
Theorem	opthpr 4612	An unordered pair has the ordered pair property (compare opth 5176) under certain conditions. (Contributed by NM, 27-Mar-2007.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    &   ⊢ 𝐷 ∈ V    ⇒   ⊢ (𝐴 ≠ 𝐷 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
Theorem	preqr1g 4613	Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. Closed form of preqr1 4608. (Contributed by AV, 29-Jan-2021.) (Revised by AV, 18-Sep-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵))
Theorem	preq12bg 4614	Closed form of preq12b 4610. (Contributed by Scott Fenton, 28-Mar-2014.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
Theorem	prel12gOLD 4615	Obsolete version of prel12g 4627 as of 12-Jun-2022. (Contributed by AV, 9-Dec-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → (¬ 𝐴 = 𝐵 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷}))))
Theorem	prneimg 4616	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.)
⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → (((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∨ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)) → {𝐴, 𝐵} ≠ {𝐶, 𝐷}))
Theorem	prnebg 4617	A (proper) pair is not equal to another (maybe improper) 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.)
⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → (((𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∨ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷)) ↔ {𝐴, 𝐵} ≠ {𝐶, 𝐷}))
Theorem	pr1eqbg 4618	A (proper) pair is equal to another (maybe improper) pair containing one element of the first pair if and only if the other element of the first pair is contained in the second pair. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) ∧ 𝐴 ≠ 𝐵) → (𝐴 = 𝐶 ↔ {𝐴, 𝐵} = {𝐵, 𝐶}))
Theorem	pr1nebg 4619	A (proper) pair is not equal to another (maybe improper) pair containing one element of the first pair if and only if the other element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) ∧ 𝐴 ≠ 𝐵) → (𝐴 ≠ 𝐶 ↔ {𝐴, 𝐵} ≠ {𝐵, 𝐶}))
Theorem	preqsnd 4620	Equivalence for a pair equal to a singleton, deduction form. (Contributed by Thierry Arnoux, 27-Dec-2016.) (Revised by AV, 13-Jun-2022.)
⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → 𝐵 ∈ V)    ⇒   ⊢ (𝜑 → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)))
Theorem	preqsndOLD 4621	Obsolete version of preqsnd 4620 as of 12-Jun-2022: Hypothesis preqsndOLD.3 is not needed. (Contributed by Thierry Arnoux, 27-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → 𝐵 ∈ V)    &   ⊢ (𝜑 → 𝐶 ∈ V)    ⇒   ⊢ (𝜑 → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)))
Theorem	prnesn 4622	A proper unordered pair is not a (proper or improper) singleton. (Contributed by AV, 13-Jun-2022.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≠ {𝐶})
Theorem	prneprprc 4623	A proper unordered pair is not an improper unordered pair. (Contributed by AV, 13-Jun-2022.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ ¬ 𝐶 ∈ V) → {𝐴, 𝐵} ≠ {𝐶, 𝐷})
Theorem	preqsn 4624	Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.) (Proof revised by AV, 12-Jun-2022.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶))
Theorem	preqsnOLD 4625	Obsolete proof of preqsn 4624 as of 12-Jun-2022. (Contributed by NM, 3-Jun-2008.) (Proof shortened by JJ, 23-Jul-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶))
Theorem	preq12nebg 4626	Equality relationship for two proper unordered pairs. (Contributed by AV, 12-Jun-2022.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
Theorem	prel12g 4627	Equality of two unordered pairs. (Contributed by NM, 17-Oct-1996.) (Revised by AV, 9-Dec-2018.) (Revised by AV, 12-Jun-2022.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷})))
Theorem	opthprneg 4628	An unordered pair has the ordered pair property (compare opth 5176) under certain conditions. Variant of opthpr 4612 in closed form. (Contributed by AV, 13-Jun-2022.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐷)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
Theorem	elpreqprlem 4629*	Lemma for elpreqpr 4630. (Contributed by Scott Fenton, 7-Dec-2020.) (Revised by AV, 9-Dec-2020.)
⊢ (𝐵 ∈ 𝑉 → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥})
Theorem	elpreqpr 4630*	Equality and membership rule for pairs. (Contributed by Scott Fenton, 7-Dec-2020.)
⊢ (𝐴 ∈ {𝐵, 𝐶} → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})
Theorem	elpreqprb 4631*	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.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥}))
Theorem	elpr2elpr 4632*	For an element 𝐴 of an unordered pair which is a subset of a given set 𝑉, there is another (maybe the same) element 𝑏 of the given set 𝑉 being an element of the unordered pair. (Contributed by AV, 5-Dec-2020.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝐴 ∈ {𝑋, 𝑌}) → ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝐴, 𝑏})
Theorem	dfopif 4633	Rewrite df-op 4405 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}}, ∅)
Theorem	dfopg 4634	Value of the ordered pair when the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
Theorem	dfop 4635	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}}
Theorem	opeq1 4636	Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
Theorem	opeq2 4637	Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)
Theorem	opeq12 4638	Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)
⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
Theorem	opeq1i 4639	Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩
Theorem	opeq2i 4640	Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩
Theorem	opeq12i 4641	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⟩
Theorem	opeq1d 4642	Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
Theorem	opeq2d 4643	Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)
Theorem	opeq12d 4644	Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩)
Theorem	oteq1 4645	Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)
Theorem	oteq2 4646	Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)
Theorem	oteq3 4647	Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)
Theorem	oteq1d 4648	Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)
Theorem	oteq2d 4649	Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)
Theorem	oteq3d 4650	Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)
Theorem	oteq123d 4651	Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    &   ⊢ (𝜑 → 𝐸 = 𝐹)    ⇒   ⊢ (𝜑 → ⟨𝐴, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐹⟩)
Theorem	nfop 4652	Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥⟨𝐴, 𝐵⟩
Theorem	nfopd 4653	Deduction version of bound-variable hypothesis builder nfop 4652. This shows how the deduction version of a not-free theorem such as nfop 4652 can be created from the corresponding not-free inference theorem. (Contributed by NM, 4-Feb-2008.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥⟨𝐴, 𝐵⟩)
Theorem	csbopg 4654	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.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⟨𝐶, 𝐷⟩ = ⟨⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟩)
Theorem	opidg 4655	The ordered pair ⟨𝐴, 𝐴⟩ in Kuratowski's representation. Closed form of opid 4656. (Contributed by Peter Mazsa, 22-Jul-2019.) (Avoid depending on this detail.)
⊢ (𝐴 ∈ 𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}})
Theorem	opid 4656	The ordered pair ⟨𝐴, 𝐴⟩ in Kuratowski's representation. Inference form of opidg 4655. (Contributed by FL, 28-Dec-2011.) (Proof shortened by AV, 16-Feb-2022.) (Avoid depending on this detail.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ⟨𝐴, 𝐴⟩ = {{𝐴}}
Theorem	ralunsn 4657*	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 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)))
Theorem	2ralunsn 4658*	Double restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.)
⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})∀𝑦 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ∧ (∀𝑦 ∈ 𝐴 𝜒 ∧ 𝜃))))
Theorem	opprc 4659	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⟩ = ∅)
Theorem	opprc1 4660	Expansion of an ordered pair when the first member is a proper class. See also opprc 4659. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ (¬ 𝐴 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
Theorem	opprc2 4661	Expansion of an ordered pair when the second member is a proper class. See also opprc 4659. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ (¬ 𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
Theorem	oprcl 4662	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))
Theorem	pwsn 4663	The power set of a singleton. (Contributed by NM, 5-Jun-2006.)
⊢ 𝒫 {𝐴} = {∅, {𝐴}}
Theorem	pwsnALT 4664	Alternate proof of pwsn 4663, more direct. (Contributed by NM, 5-Jun-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝒫 {𝐴} = {∅, {𝐴}}
Theorem	pwpr 4665	The power set of an unordered pair. (Contributed by NM, 1-May-2009.)
⊢ 𝒫 {𝐴, 𝐵} = ({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}})
Theorem	pwtp 4666	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}}))
Theorem	pwpwpw0 4667	Compute the power set of the power set of the power set of the empty set. (See also pw0 4574 and pwpw0 4575.) (Contributed by NM, 2-May-2009.)
⊢ 𝒫 {∅, {∅}} = ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})
Theorem	pwv 4668	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
Theorem	prproe 4669*	For an element of a proper unordered pair of elements of a class 𝑉, there is another (different) element of the class 𝑉 which is an element of the proper pair. (Contributed by AV, 18-Dec-2021.)
⊢ ((𝐶 ∈ {𝐴, 𝐵} ∧ 𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵})
Theorem	3elpr2eq 4670	If there are three elements in a proper unordered pair, and two of them are different from the third one, the two must be equal. (Contributed by AV, 19-Dec-2021.)
⊢ (((𝑋 ∈ {𝐴, 𝐵} ∧ 𝑌 ∈ {𝐴, 𝐵} ∧ 𝑍 ∈ {𝐴, 𝐵}) ∧ (𝑌 ≠ 𝑋 ∧ 𝑍 ≠ 𝑋)) → 𝑌 = 𝑍)
2.1.19  The union of a class
Syntax	cuni 4671	Extend class notation to include the union of a class. Read: "union (of) 𝐴".
class ∪ 𝐴
Definition	df-uni 4672*	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 27858). This is similar to the union of two classes df-un 3797. (Contributed by NM, 23-Aug-1993.)
⊢ ∪ 𝐴 = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)}
Theorem	dfuni2 4673*	Alternate definition of class union. (Contributed by NM, 28-Jun-1998.)
⊢ ∪ 𝐴 = {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦}
Theorem	eluni 4674*	Membership in class union. (Contributed by NM, 22-May-1994.)
⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))
Theorem	eluni2 4675*	Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999.)
⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥)
Theorem	elunii 4676	Membership in class union. (Contributed by NM, 24-Mar-1995.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ ∪ 𝐶)
Theorem	nfuni 4677	Bound-variable hypothesis builder for union. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥∪ 𝐴
Theorem	nfunid 4678	Deduction version of nfuni 4677. (Contributed by NM, 18-Feb-2013.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴)
Theorem	unieq 4679	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)
Theorem	unieqi 4680	Inference of equality of two class unions. (Contributed by NM, 30-Aug-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ∪ 𝐴 = ∪ 𝐵
Theorem	unieqd 4681	Deduction of equality of two class unions. (Contributed by NM, 21-Apr-1995.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ∪ 𝐴 = ∪ 𝐵)
Theorem	eluniab 4682*	Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.)
⊢ (𝐴 ∈ ∪ {𝑥 ∣ 𝜑} ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝜑))
Theorem	elunirab 4683*	Membership in union of a class abstraction. (Contributed by NM, 4-Oct-2006.)
⊢ (𝐴 ∈ ∪ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝜑))
Theorem	unipr 4684	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} = (𝐴 ∪ 𝐵)
Theorem	uniprg 4685	The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 25-Aug-2006.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
Theorem	unisng 4686	A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 13-Aug-2002.)
⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
Theorem	unisn 4687	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
Theorem	unisnOLD 4688	Obsolete proof of unisn 4687 as of 16-Sep-2022. (Contributed by NM, 30-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∪ {𝐴} = 𝐴
Theorem	unisngOLD 4689	Obsolete proof of unisng 4686 as of 16-Sep-2022. (Contributed by NM, 13-Aug-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
Theorem	unisn3 4690*	Union of a singleton in the form of a restricted class abstraction. (Contributed by NM, 3-Jul-2008.)
⊢ (𝐴 ∈ 𝐵 → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 = 𝐴} = 𝐴)
Theorem	dfnfc2 4691*	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 𝐴 ∈ 𝑉 → (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴))
Theorem	uniun 4692	The class union of the union of two classes. Theorem 8.3 of [Quine] p. 53. (Contributed by NM, 20-Aug-1993.)
⊢ ∪ (𝐴 ∪ 𝐵) = (∪ 𝐴 ∪ ∪ 𝐵)
Theorem	uniin 4693	The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uniinqs 8110 for a condition where equality holds. (Contributed by NM, 4-Dec-2003.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ ∪ (𝐴 ∩ 𝐵) ⊆ (∪ 𝐴 ∩ ∪ 𝐵)
Theorem	uniss 4694	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.)
⊢ (𝐴 ⊆ 𝐵 → ∪ 𝐴 ⊆ ∪ 𝐵)
Theorem	ssuni 4695	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.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ⊆ ∪ 𝐶)
Theorem	unissi 4696	Subclass relationship for subclass union. Inference form of uniss 4694. (Contributed by David Moews, 1-May-2017.)
⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ ∪ 𝐴 ⊆ ∪ 𝐵
Theorem	unissd 4697	Subclass relationship for subclass union. Deduction form of uniss 4694. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → ∪ 𝐴 ⊆ ∪ 𝐵)
Theorem	uni0b 4698	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 ⊆ {∅})
Theorem	uni0c 4699*	The union of a set is empty iff all of its members are empty. (Contributed by NM, 16-Aug-2006.)
⊢ (∪ 𝐴 = ∅ ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅)
Theorem	uni0 4700	The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Reproved without relying on ax-nul 5025 by Eric Schmidt.) (Contributed by NM, 16-Sep-1993.) (Revised by Eric Schmidt, 4-Apr-2007.)
⊢ ∪ ∅ = ∅