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	nelsn 3701	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.)
⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 ∈ {𝐵})
Theorem	mosn 3702*	A singleton has at most one element. This works whether 𝐴 is a proper class or not, and in that sense can be seen as encompassing both snmg 3785 and snprc 3731. (Contributed by Jim Kingdon, 30-Aug-2018.)
⊢ ∃*𝑥 𝑥 ∈ {𝐴}
Theorem	ralsnsg 3703*	Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑))
Theorem	ralsns 3704*	Substitution expressed in terms of quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)
⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑))
Theorem	rexsns 3705*	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)
Theorem	ralsng 3706*	Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))
Theorem	rexsng 3707*	Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))
Theorem	exsnrex 3708	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} ↔ ∃𝑥 ∈ 𝑀 𝑀 = {𝑥})
Theorem	ralsn 3709*	Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)
Theorem	rexsn 3710*	Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)
Theorem	eltpg 3711	Members of an unordered triple of classes. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷)))
Theorem	eldiftp 3712	Membership in a set with three elements removed. Similar to eldifsn 3795 and eldifpr 3693. (Contributed by David A. Wheeler, 22-Jul-2017.)
⊢ (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷, 𝐸}) ↔ (𝐴 ∈ 𝐵 ∧ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷 ∧ 𝐴 ≠ 𝐸)))
Theorem	eltpi 3713	A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷))
Theorem	eltp 3714	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 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷))
Theorem	dftp2 3715*	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 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)}
Theorem	nfpr 3716	Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥{𝐴, 𝐵}
Theorem	ralprg 3717*	Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒)))
Theorem	rexprg 3718*	Convert a quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒)))
Theorem	raltpg 3719*	Convert a quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜃))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)))
Theorem	rextpg 3720*	Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜃))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∨ 𝜒 ∨ 𝜃)))
Theorem	ralpr 3721*	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 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    ⇒   ⊢ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒))
Theorem	rexpr 3722*	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 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    ⇒   ⊢ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒))
Theorem	raltp 3723*	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 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜃))    ⇒   ⊢ (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃))
Theorem	rextp 3724*	Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜃))    ⇒   ⊢ (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∨ 𝜒 ∨ 𝜃))
Theorem	sbcsng 3725*	Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥 ∈ {𝐴}𝜑))
Theorem	nfsn 3726	Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥{𝐴}
Theorem	csbsng 3727	Distribute proper substitution through the singleton of a class. (Contributed by Alan Sare, 10-Nov-2012.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐵} = {⦋𝐴 / 𝑥⦌𝐵})
Theorem	disjsn 3728	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}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴)
Theorem	disjsn2 3729	Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.)
⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅)
Theorem	disjpr2 3730	The intersection of distinct unordered pairs is disjoint. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
⊢ (((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝐴 ≠ 𝐷 ∧ 𝐵 ≠ 𝐷)) → ({𝐴, 𝐵} ∩ {𝐶, 𝐷}) = ∅)
Theorem	snprc 3731	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, 5-Aug-1993.)
⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
Theorem	r19.12sn 3732*	Special case of r19.12 2637 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.) (Revised by BJ, 20-Dec-2021.)
⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ {𝐴}𝜑))
Theorem	rabsn 3733*	Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.)
⊢ (𝐵 ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ 𝑥 = 𝐵} = {𝐵})
Theorem	rabrsndc 3734*	A class abstraction over a decidable proposition restricted to a singleton is either the empty set or the singleton itself. (Contributed by Jim Kingdon, 8-Aug-2018.)
⊢ 𝐴 ∈ V    &   ⊢ DECID 𝜑    ⇒   ⊢ (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = ∅ ∨ 𝑀 = {𝐴}))
Theorem	euabsn2 3735*	Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
Theorem	euabsn 3736	Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by NM, 22-Feb-2004.)
⊢ (∃!𝑥𝜑 ↔ ∃𝑥{𝑥 ∣ 𝜑} = {𝑥})
Theorem	reusn 3737*	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.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦{𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦})
Theorem	absneu 3738	Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.)
⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∣ 𝜑} = {𝐴}) → ∃!𝑥𝜑)
Theorem	rabsneu 3739	Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) (Revised by Mario Carneiro, 23-Dec-2016.)
⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝐴}) → ∃!𝑥 ∈ 𝐵 𝜑)
Theorem	eusn 3740*	Two ways to express "𝐴 is a singleton". (Contributed by NM, 30-Oct-2010.)
⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 𝐴 = {𝑥})
Theorem	rabsnt 3741*	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 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → 𝜓)
Theorem	prcom 3742	Commutative law for unordered pairs. (Contributed by NM, 5-Aug-1993.)
⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
Theorem	preq1 3743	Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.)
⊢ (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})
Theorem	preq2 3744	Equality theorem for unordered pairs. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵})
Theorem	preq12 3745	Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.)
⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
Theorem	preq1i 3746	Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ {𝐴, 𝐶} = {𝐵, 𝐶}
Theorem	preq2i 3747	Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ {𝐶, 𝐴} = {𝐶, 𝐵}
Theorem	preq12i 3748	Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ {𝐴, 𝐶} = {𝐵, 𝐷}
Theorem	preq1d 3749	Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐶})
Theorem	preq2d 3750	Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → {𝐶, 𝐴} = {𝐶, 𝐵})
Theorem	preq12d 3751	Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐷})
Theorem	tpeq1 3752	Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
⊢ (𝐴 = 𝐵 → {𝐴, 𝐶, 𝐷} = {𝐵, 𝐶, 𝐷})
Theorem	tpeq2 3753	Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
⊢ (𝐴 = 𝐵 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷})
Theorem	tpeq3 3754	Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
⊢ (𝐴 = 𝐵 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵})
Theorem	tpeq1d 3755	Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → {𝐴, 𝐶, 𝐷} = {𝐵, 𝐶, 𝐷})
Theorem	tpeq2d 3756	Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷})
Theorem	tpeq3d 3757	Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵})
Theorem	tpeq123d 3758	Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    &   ⊢ (𝜑 → 𝐸 = 𝐹)    ⇒   ⊢ (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐹})
Theorem	tprot 3759	Rotation of the elements of an unordered triple. (Contributed by Alan Sare, 24-Oct-2011.)
⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
Theorem	tpcoma 3760	Swap 1st and 2nd members of an undordered triple. (Contributed by NM, 22-May-2015.)
⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶}
Theorem	tpcomb 3761	Swap 2nd and 3rd members of an undordered triple. (Contributed by NM, 22-May-2015.)
⊢ {𝐴, 𝐵, 𝐶} = {𝐴, 𝐶, 𝐵}
Theorem	tpass 3762	Split off the first element of an unordered triple. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶})
Theorem	qdass 3763	Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ ({𝐴, 𝐵} ∪ {𝐶, 𝐷}) = ({𝐴, 𝐵, 𝐶} ∪ {𝐷})
Theorem	qdassr 3764	Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)
⊢ ({𝐴, 𝐵} ∪ {𝐶, 𝐷}) = ({𝐴} ∪ {𝐵, 𝐶, 𝐷})
Theorem	tpidm12 3765	Unordered triple {𝐴, 𝐴, 𝐵} is just an overlong way to write {𝐴, 𝐵}. (Contributed by David A. Wheeler, 10-May-2015.)
⊢ {𝐴, 𝐴, 𝐵} = {𝐴, 𝐵}
Theorem	tpidm13 3766	Unordered triple {𝐴, 𝐵, 𝐴} is just an overlong way to write {𝐴, 𝐵}. (Contributed by David A. Wheeler, 10-May-2015.)
⊢ {𝐴, 𝐵, 𝐴} = {𝐴, 𝐵}
Theorem	tpidm23 3767	Unordered triple {𝐴, 𝐵, 𝐵} is just an overlong way to write {𝐴, 𝐵}. (Contributed by David A. Wheeler, 10-May-2015.)
⊢ {𝐴, 𝐵, 𝐵} = {𝐴, 𝐵}
Theorem	tpidm 3768	Unordered triple {𝐴, 𝐴, 𝐴} is just an overlong way to write {𝐴}. (Contributed by David A. Wheeler, 10-May-2015.)
⊢ {𝐴, 𝐴, 𝐴} = {𝐴}
Theorem	tppreq3 3769	An unordered triple is an unordered pair if one of its elements is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
Theorem	prid1g 3770	An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵})
Theorem	prid2g 3771	An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)
⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐴, 𝐵})
Theorem	prid1 3772	An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 ∈ V    ⇒   ⊢ 𝐴 ∈ {𝐴, 𝐵}
Theorem	prid2 3773	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	ifpprsnssdc 3774	An unordered pair is a singleton or a subset of itself. This theorem is helpful to convert theorems about walks in arbitrary graphs into theorems about walks in pseudographs. (Contributed by AV, 27-Feb-2021.)
⊢ ((𝑃 = {𝐴, 𝐵} ∧ DECID 𝐴 = 𝐵) → if-(𝐴 = 𝐵, 𝑃 = {𝐴}, {𝐴, 𝐵} ⊆ 𝑃))
Theorem	prprc1 3775	A proper class vanishes in an unordered pair. (Contributed by NM, 5-Aug-1993.)
⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})
Theorem	prprc2 3776	A proper class vanishes in an unordered pair. (Contributed by NM, 22-Mar-2006.)
⊢ (¬ 𝐵 ∈ V → {𝐴, 𝐵} = {𝐴})
Theorem	prprc 3777	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 3778	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 3779	Closed theorem form of tpid1 3778. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐴, 𝐶, 𝐷})
Theorem	tpid2 3780	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 3781	Closed theorem form of tpid2 3780. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐴, 𝐷})
Theorem	tpid3g 3782	Closed theorem form of tpid3 3783. (Contributed by Alan Sare, 24-Oct-2011.)
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴})
Theorem	tpid3 3783	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 3784	The singleton of a set is not empty. It is also inhabited as shown at snmg 3785. (Contributed by NM, 14-Dec-2008.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅)
Theorem	snmg 3785*	The singleton of a set is inhabited. (Contributed by Jim Kingdon, 11-Aug-2018.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 ∈ {𝐴})
Theorem	snnz 3786	The singleton of a set is not empty. It is also inhabited as shown at snm 3787. (Contributed by NM, 10-Apr-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝐴} ≠ ∅
Theorem	snm 3787*	The singleton of a set is inhabited. (Contributed by Jim Kingdon, 11-Aug-2018.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∃𝑥 𝑥 ∈ {𝐴}
Theorem	snmb 3788*	A singleton is inhabited iff its argument is a set. (Contributed by Scott Fenton, 8-May-2018.) (Revised by Jim Kingdon, 29-Dec-2025.)
⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 ∈ {𝐴})
Theorem	prmg 3789*	A pair containing a set is inhabited. (Contributed by Jim Kingdon, 21-Sep-2018.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 ∈ {𝐴, 𝐵})
Theorem	prnz 3790	A pair containing a set is not empty. It is also inhabited (see prm 3791). (Contributed by NM, 9-Apr-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝐴, 𝐵} ≠ ∅
Theorem	prm 3791*	A pair containing a set is inhabited. (Contributed by Jim Kingdon, 21-Sep-2018.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∃𝑥 𝑥 ∈ {𝐴, 𝐵}
Theorem	prnzg 3792	A pair containing a set is not empty. It is also inhabited (see prmg 3789). (Contributed by FL, 19-Sep-2011.)
⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐵} ≠ ∅)
Theorem	tpnz 3793	A triplet containing a set is not empty. (Contributed by NM, 10-Apr-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝐴, 𝐵, 𝐶} ≠ ∅
Theorem	snssOLD 3794	Obsolete version of snss 3803 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 3795	Membership in a set with an element removed. (Contributed by NM, 10-Oct-2007.)
⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶))
Theorem	ssdifsn 3796	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 3797	Membership in a set with an element removed. (Contributed by NM, 10-Mar-2015.)
⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) → 𝐴 ≠ 𝐶)
Theorem	neldifsn 3798	𝐴 is not in (𝐵 ∖ {𝐴}). (Contributed by David Moews, 1-May-2017.)
⊢ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})
Theorem	neldifsnd 3799	𝐴 is not in (𝐵 ∖ {𝐴}). Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}))
Theorem	rexdifsn 3800	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))