P. 38 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 3701-3800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	prid2 3701	An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐵 ∈ V    ⇒   ⊢ 𝐵 ∈ {𝐴, 𝐵}
Theorem	prprc1 3702	A proper class vanishes in an unordered pair. (Contributed by NM, 5-Aug-1993.)
⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})
Theorem	prprc2 3703	A proper class vanishes in an unordered pair. (Contributed by NM, 22-Mar-2006.)
⊢ (¬ 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴})
Theorem	prprc 3704	An unordered pair containing two proper classes is the empty set. (Contributed by NM, 22-Mar-2006.)
⊢ ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵} = ∅)
Theorem	tpid1 3705	One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ 𝐴 ∈ V    ⇒   ⊢ 𝐴 ∈ {𝐴, 𝐵, 𝐶}
Theorem	tpid1g 3706	Closed theorem form of tpid1 3705. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐴, 𝐶, 𝐷})
Theorem	tpid2 3707	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 3708	Closed theorem form of tpid2 3707. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐴, 𝐷})
Theorem	tpid3g 3709	Closed theorem form of tpid3 3710. (Contributed by Alan Sare, 24-Oct-2011.)
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴})
Theorem	tpid3 3710	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 3711	The singleton of a set is not empty. (Contributed by NM, 14-Dec-2008.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅)
Theorem	snmg 3712*	The singleton of a set is inhabited. (Contributed by Jim Kingdon, 11-Aug-2018.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 ∈ {𝐴})
Theorem	snnz 3713	The singleton of a set is not empty. (Contributed by NM, 10-Apr-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝐴} ≠ ∅
Theorem	snm 3714*	The singleton of a set is inhabited. (Contributed by Jim Kingdon, 11-Aug-2018.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∃𝑥 𝑥 ∈ {𝐴}
Theorem	prmg 3715*	A pair containing a set is inhabited. (Contributed by Jim Kingdon, 21-Sep-2018.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 ∈ {𝐴, 𝐵})
Theorem	prnz 3716	A pair containing a set is not empty. It is also inhabited (see prm 3717). (Contributed by NM, 9-Apr-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝐴, 𝐵} ≠ ∅
Theorem	prm 3717*	A pair containing a set is inhabited. (Contributed by Jim Kingdon, 21-Sep-2018.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∃𝑥 𝑥 ∈ {𝐴, 𝐵}
Theorem	prnzg 3718	A pair containing a set is not empty. It is also inhabited (see prmg 3715). (Contributed by FL, 19-Sep-2011.)
⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐵} ≠ ∅)
Theorem	tpnz 3719	A triplet containing a set is not empty. (Contributed by NM, 10-Apr-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝐴, 𝐵, 𝐶} ≠ ∅
Theorem	snssOLD 3720	Obsolete version of snss 3729 as of 1-Jan-2025. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)
Theorem	eldifsn 3721	Membership in a set with an element removed. (Contributed by NM, 10-Oct-2007.)
⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶))
Theorem	ssdifsn 3722	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 3723	Membership in a set with an element removed. (Contributed by NM, 10-Mar-2015.)
⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) → 𝐴 ≠ 𝐶)
Theorem	neldifsn 3724	𝐴 is not in (𝐵 ∖ {𝐴}). (Contributed by David Moews, 1-May-2017.)
⊢ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})
Theorem	neldifsnd 3725	𝐴 is not in (𝐵 ∖ {𝐴}). Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}))
Theorem	rexdifsn 3726	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	snssb 3727	Characterization of the inclusion of a singleton in a class. (Contributed by BJ, 1-Jan-2025.)
⊢ ({𝐴} ⊆ 𝐵 ↔ (𝐴 ∈ V → 𝐴 ∈ 𝐵))
Theorem	snssg 3728	The singleton formed on a set is included in a class if and only if the set is an element of that class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.) (Proof shortened by BJ, 1-Jan-2025.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵))
Theorem	snss 3729	The singleton of an element of a class is a subset of the class (inference form of snssg 3728). Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 21-Jun-1993.) (Proof shortened by BJ, 1-Jan-2025.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)
Theorem	snssgOLD 3730	Obsolete version of snssgOLD 3730 as of 1-Jan-2025. (Contributed by NM, 22-Jul-2001.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵))
Theorem	difsn 3731	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 3732	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 3733	Removal of a singleton from an unordered pair. (Contributed by Thierry Arnoux, 4-Feb-2017.)
⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
Theorem	difprsn2 3734	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 3735	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 3736	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 3737	(𝐵 ∖ {𝐴}) equals 𝐵 if and only if 𝐴 is not a member of 𝐵. Generalization of difsn 3731. (Contributed by David Moews, 1-May-2017.)
⊢ (¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)
Theorem	snssi 3738	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 3739	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 3740	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 6510. (Contributed by Jim Kingdon, 10-Aug-2018.)
⊢ (𝐵 ∈ 𝐴 → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ⊆ 𝐴)
Theorem	pw0 3741	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 3742	A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 27-Aug-2004.)
⊢ {𝐴} ⊆ {𝐴, 𝐵}
Theorem	snsspr2 3743	A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 2-May-2009.)
⊢ {𝐵} ⊆ {𝐴, 𝐵}
Theorem	snsstp1 3744	A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
⊢ {𝐴} ⊆ {𝐴, 𝐵, 𝐶}
Theorem	snsstp2 3745	A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
⊢ {𝐵} ⊆ {𝐴, 𝐵, 𝐶}
Theorem	snsstp3 3746	A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
⊢ {𝐶} ⊆ {𝐴, 𝐵, 𝐶}
Theorem	prsstp12 3747	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 3748	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 3749	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 3750	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 3751	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 3752	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	prssd 3753	Deduction version of prssi 3752: A pair of elements of a class is a subset of the class. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐶)    ⇒   ⊢ (𝜑 → {𝐴, 𝐵} ⊆ 𝐶)
Theorem	prsspwg 3754	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 3755	Empty set and the singleton itself are subsets of a singleton. Concerning the converse, see exmidsssn 4204. (Contributed by Jim Kingdon, 10-Aug-2018.)
⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵})
Theorem	sssnm 3756*	The inhabited subset of a singleton. (Contributed by Jim Kingdon, 10-Aug-2018.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐴 ⊆ {𝐵} ↔ 𝐴 = {𝐵}))
Theorem	eqsnm 3757*	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 3758	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 3759	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 3760	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 3761	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 3762	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 3763	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 3764	Closed form of sneqr 3762. (Contributed by Scott Fenton, 1-Apr-2011.)
⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
Theorem	sneqbg 3765	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 3766	The singleton of a class is a subset of its power class. (Contributed by NM, 5-Aug-1993.)
⊢ {𝐴} ⊆ 𝒫 𝐴
Theorem	prsspw 3767	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 3768	Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. Closed form of preqr1 3770. (Contributed by Jim Kingdon, 21-Sep-2018.)
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵))
Theorem	preqr2g 3769	Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the second elements are equal. Closed form of preqr2 3771. (Contributed by Jim Kingdon, 21-Sep-2018.)
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵))
Theorem	preqr1 3770	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 3771	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 3772	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 3773	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 3774	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 3775	Closed form of preq12b 3772. (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 3776	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 3777	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 3778	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 3779	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 3780	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 3781	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 3782	Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)
⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
Theorem	opeq1i 3783	Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩
Theorem	opeq2i 3784	Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩
Theorem	opeq12i 3785	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 3786	Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
Theorem	opeq2d 3787	Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)
Theorem	opeq12d 3788	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 3789	Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)
Theorem	oteq2 3790	Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)
Theorem	oteq3 3791	Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)
Theorem	oteq1d 3792	Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)
Theorem	oteq2d 3793	Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)
Theorem	oteq3d 3794	Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)
Theorem	oteq123d 3795	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 3796	Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥⟨𝐴, 𝐵⟩
Theorem	nfopd 3797	Deduction version of bound-variable hypothesis builder nfop 3796. This shows how the deduction version of a not-free theorem such as nfop 3796 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 3798	The ordered pair ⟨𝐴, 𝐴⟩ in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ⟨𝐴, 𝐴⟩ = {{𝐴}}
Theorem	ralunsn 3799*	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 3800*	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))))