P. 37 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 3601-3700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	tpid1g 3601	Closed theorem form of tpid1 3600. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐴, 𝐶, 𝐷})
Theorem	tpid2 3602	One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ 𝐵 ∈ V    ⇒   ⊢ 𝐵 ∈ {𝐴, 𝐵, 𝐶}
Theorem	tpid2g 3603	Closed theorem form of tpid2 3602. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐴, 𝐷})
Theorem	tpid3g 3604	Closed theorem form of tpid3 3605. (Contributed by Alan Sare, 24-Oct-2011.)
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴})
Theorem	tpid3 3605	One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ 𝐶 ∈ V    ⇒   ⊢ 𝐶 ∈ {𝐴, 𝐵, 𝐶}
Theorem	snnzg 3606	The singleton of a set is not empty. (Contributed by NM, 14-Dec-2008.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅)
Theorem	snmg 3607*	The singleton of a set is inhabited. (Contributed by Jim Kingdon, 11-Aug-2018.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 ∈ {𝐴})
Theorem	snnz 3608	The singleton of a set is not empty. (Contributed by NM, 10-Apr-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝐴} ≠ ∅
Theorem	snm 3609*	The singleton of a set is inhabited. (Contributed by Jim Kingdon, 11-Aug-2018.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∃𝑥 𝑥 ∈ {𝐴}
Theorem	prmg 3610*	A pair containing a set is inhabited. (Contributed by Jim Kingdon, 21-Sep-2018.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 ∈ {𝐴, 𝐵})
Theorem	prnz 3611	A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝐴, 𝐵} ≠ ∅
Theorem	prm 3612*	A pair containing a set is inhabited. (Contributed by Jim Kingdon, 21-Sep-2018.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∃𝑥 𝑥 ∈ {𝐴, 𝐵}
Theorem	prnzg 3613	A pair containing a set is not empty. (Contributed by FL, 19-Sep-2011.)
⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐵} ≠ ∅)
Theorem	tpnz 3614	A triplet containing a set is not empty. (Contributed by NM, 10-Apr-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝐴, 𝐵, 𝐶} ≠ ∅
Theorem	snss 3615	The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)
Theorem	eldifsn 3616	Membership in a set with an element removed. (Contributed by NM, 10-Oct-2007.)
⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶))
Theorem	ssdifsn 3617	Subset of a set with an element removed. (Contributed by Emmett Weisz, 7-Jul-2021.) (Proof shortened by JJ, 31-May-2022.)
⊢ (𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴))
Theorem	eldifsni 3618	Membership in a set with an element removed. (Contributed by NM, 10-Mar-2015.)
⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) → 𝐴 ≠ 𝐶)
Theorem	neldifsn 3619	𝐴 is not in (𝐵 ∖ {𝐴}). (Contributed by David Moews, 1-May-2017.)
⊢ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})
Theorem	neldifsnd 3620	𝐴 is not in (𝐵 ∖ {𝐴}). Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}))
Theorem	rexdifsn 3621	Restricted existential quantification over a set with an element removed. (Contributed by NM, 4-Feb-2015.)
⊢ (∃𝑥 ∈ (𝐴 ∖ {𝐵})𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝑥 ≠ 𝐵 ∧ 𝜑))
Theorem	snssg 3622	The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵))
Theorem	difsn 3623	An element not in a set can be removed without affecting the set. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ (¬ 𝐴 ∈ 𝐵 → (𝐵 ∖ {𝐴}) = 𝐵)
Theorem	difprsnss 3624	Removal of a singleton from an unordered pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ ({𝐴, 𝐵} ∖ {𝐴}) ⊆ {𝐵}
Theorem	difprsn1 3625	Removal of a singleton from an unordered pair. (Contributed by Thierry Arnoux, 4-Feb-2017.)
⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
Theorem	difprsn2 3626	Removal of a singleton from an unordered pair. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})
Theorem	diftpsn3 3627	Removal of a singleton from an unordered triple. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
Theorem	difpr 3628	Removing two elements as pair of elements corresponds to removing each of the two elements as singletons. (Contributed by Alexander van der Vekens, 13-Jul-2018.)
⊢ (𝐴 ∖ {𝐵, 𝐶}) = ((𝐴 ∖ {𝐵}) ∖ {𝐶})
Theorem	difsnb 3629	(𝐵 ∖ {𝐴}) equals 𝐵 if and only if 𝐴 is not a member of 𝐵. Generalization of difsn 3623. (Contributed by David Moews, 1-May-2017.)
⊢ (¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)
Theorem	snssi 3630	The singleton of an element of a class is a subset of the class. (Contributed by NM, 6-Jun-1994.)
⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵)
Theorem	snssd 3631	The singleton of an element of a class is a subset of the class (deduction form). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    ⇒   ⊢ (𝜑 → {𝐴} ⊆ 𝐵)
Theorem	difsnss 3632	If we remove a single element from a class then put it back in, we end up with a subset of the original class. If equality is decidable, we can replace subset with equality as seen in nndifsnid 6357. (Contributed by Jim Kingdon, 10-Aug-2018.)
⊢ (𝐵 ∈ 𝐴 → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ⊆ 𝐴)
Theorem	pw0 3633	Compute the power set of the empty set. Theorem 89 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ 𝒫 ∅ = {∅}
Theorem	snsspr1 3634	A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 27-Aug-2004.)
⊢ {𝐴} ⊆ {𝐴, 𝐵}
Theorem	snsspr2 3635	A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 2-May-2009.)
⊢ {𝐵} ⊆ {𝐴, 𝐵}
Theorem	snsstp1 3636	A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
⊢ {𝐴} ⊆ {𝐴, 𝐵, 𝐶}
Theorem	snsstp2 3637	A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
⊢ {𝐵} ⊆ {𝐴, 𝐵, 𝐶}
Theorem	snsstp3 3638	A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
⊢ {𝐶} ⊆ {𝐴, 𝐵, 𝐶}
Theorem	prsstp12 3639	A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.)
⊢ {𝐴, 𝐵} ⊆ {𝐴, 𝐵, 𝐶}
Theorem	prsstp13 3640	A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.)
⊢ {𝐴, 𝐶} ⊆ {𝐴, 𝐵, 𝐶}
Theorem	prsstp23 3641	A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.)
⊢ {𝐵, 𝐶} ⊆ {𝐴, 𝐵, 𝐶}
Theorem	prss 3642	A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶)
Theorem	prssg 3643	A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 22-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶))
Theorem	prssi 3644	A pair of elements of a class is a subset of the class. (Contributed by NM, 16-Jan-2015.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → {𝐴, 𝐵} ⊆ 𝐶)
Theorem	prsspwg 3645	An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by Thierry Arnoux, 3-Oct-2016.) (Revised by NM, 18-Jan-2018.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶)))
Theorem	sssnr 3646	Empty set and the singleton itself are subsets of a singleton. Concerning the converse, see exmidsssn 4085. (Contributed by Jim Kingdon, 10-Aug-2018.)
⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵})
Theorem	sssnm 3647*	The inhabited subset of a singleton. (Contributed by Jim Kingdon, 10-Aug-2018.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐴 ⊆ {𝐵} ↔ 𝐴 = {𝐵}))
Theorem	eqsnm 3648*	Two ways to express that an inhabited set equals a singleton. (Contributed by Jim Kingdon, 11-Aug-2018.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐴 = {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵))
Theorem	ssprr 3649	The subsets of a pair. (Contributed by Jim Kingdon, 11-Aug-2018.)
⊢ (((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) → 𝐴 ⊆ {𝐵, 𝐶})
Theorem	sstpr 3650	The subsets of a triple. (Contributed by Jim Kingdon, 11-Aug-2018.)
⊢ ((((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) ∨ ((𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}) ∨ (𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷}))) → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
Theorem	tpss 3651	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    ⇒   ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ↔ {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
Theorem	tpssi 3652	A triple of elements of a class is a subset of the class. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
Theorem	sneqr 3653	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)
Theorem	snsssn 3654	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	sneqrg 3655	Closed form of sneqr 3653. (Contributed by Scott Fenton, 1-Apr-2011.)
⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
Theorem	sneqbg 3656	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 3657	The singleton of a class is a subset of its power class. (Contributed by NM, 5-Aug-1993.)
⊢ {𝐴} ⊆ 𝒫 𝐴
Theorem	prsspw 3658	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.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶))
Theorem	preqr1g 3659	Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. Closed form of preqr1 3661. (Contributed by Jim Kingdon, 21-Sep-2018.)
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵))
Theorem	preqr2g 3660	Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the second elements are equal. Closed form of preqr2 3662. (Contributed by Jim Kingdon, 21-Sep-2018.)
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵))
Theorem	preqr1 3661	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 3662	Reverse equality lemma for unordered pairs. If two unordered pairs have the same first element, the second elements are equal. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵)
Theorem	preq12b 3663	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	prel12 3664	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})))
Theorem	opthpr 3665	A way to represent ordered pairs using unordered pairs with distinct members. (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	preq12bg 3666	Closed form of preq12b 3663. (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	prneimg 3667	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	preqsn 3668	Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶))
Theorem	dfopg 3669	Value of the ordered pair when the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
Theorem	dfop 3670	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.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
Theorem	opeq1 3671	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 3672	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 3673	Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)
⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
Theorem	opeq1i 3674	Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩
Theorem	opeq2i 3675	Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩
Theorem	opeq12i 3676	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 3677	Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
Theorem	opeq2d 3678	Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)
Theorem	opeq12d 3679	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 3680	Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)
Theorem	oteq2 3681	Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)
Theorem	oteq3 3682	Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)
Theorem	oteq1d 3683	Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)
Theorem	oteq2d 3684	Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)
Theorem	oteq3d 3685	Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)
Theorem	oteq123d 3686	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 3687	Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥⟨𝐴, 𝐵⟩
Theorem	nfopd 3688	Deduction version of bound-variable hypothesis builder nfop 3687. This shows how the deduction version of a not-free theorem such as nfop 3687 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	opid 3689	The ordered pair ⟨𝐴, 𝐴⟩ in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ⟨𝐴, 𝐴⟩ = {{𝐴}}
Theorem	ralunsn 3690*	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 3691*	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 3692	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 3693	Expansion of an ordered pair when the first member is a proper class. See also opprc 3692. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ (¬ 𝐴 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
Theorem	opprc2 3694	Expansion of an ordered pair when the second member is a proper class. See also opprc 3692. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ (¬ 𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
Theorem	oprcl 3695	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	pwsnss 3696	The power set of a singleton. (Contributed by Jim Kingdon, 12-Aug-2018.)
⊢ {∅, {𝐴}} ⊆ 𝒫 {𝐴}
Theorem	pwpw0ss 3697	Compute the power set of the power set of the empty set. (See pw0 3633 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48 (but with subset in place of equality). (Contributed by Jim Kingdon, 12-Aug-2018.)
⊢ {∅, {∅}} ⊆ 𝒫 {∅}
Theorem	pwprss 3698	The power set of an unordered pair. (Contributed by Jim Kingdon, 13-Aug-2018.)
⊢ ({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ⊆ 𝒫 {𝐴, 𝐵}
Theorem	pwtpss 3699	The power set of an unordered triple. (Contributed by Jim Kingdon, 13-Aug-2018.)
⊢ (({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ∪ ({{𝐶}, {𝐴, 𝐶}} ∪ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}})) ⊆ 𝒫 {𝐴, 𝐵, 𝐶}
Theorem	pwpwpw0ss 3700	Compute the power set of the power set of the power set of the empty set. (See also pw0 3633 and pwpw0ss 3697.) (Contributed by Jim Kingdon, 13-Aug-2018.)
⊢ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) ⊆ 𝒫 {∅, {∅}}