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	tpnz 3701	A triplet containing a set is not empty. (Contributed by NM, 10-Apr-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {𝐴, 𝐵, 𝐶} ≠ ∅
Theorem	snss 3702	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 3703	Membership in a set with an element removed. (Contributed by NM, 10-Oct-2007.)
⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶))
Theorem	ssdifsn 3704	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 3705	Membership in a set with an element removed. (Contributed by NM, 10-Mar-2015.)
⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) → 𝐴 ≠ 𝐶)
Theorem	neldifsn 3706	𝐴 is not in (𝐵 ∖ {𝐴}). (Contributed by David Moews, 1-May-2017.)
⊢ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})
Theorem	neldifsnd 3707	𝐴 is not in (𝐵 ∖ {𝐴}). Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}))
Theorem	rexdifsn 3708	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 3709	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 3710	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 3711	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 3712	Removal of a singleton from an unordered pair. (Contributed by Thierry Arnoux, 4-Feb-2017.)
⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
Theorem	difprsn2 3713	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 3714	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 3715	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 3716	(𝐵 ∖ {𝐴}) equals 𝐵 if and only if 𝐴 is not a member of 𝐵. Generalization of difsn 3710. (Contributed by David Moews, 1-May-2017.)
⊢ (¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)
Theorem	snssi 3717	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 3718	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 3719	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 6475. (Contributed by Jim Kingdon, 10-Aug-2018.)
⊢ (𝐵 ∈ 𝐴 → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ⊆ 𝐴)
Theorem	pw0 3720	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 3721	A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 27-Aug-2004.)
⊢ {𝐴} ⊆ {𝐴, 𝐵}
Theorem	snsspr2 3722	A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 2-May-2009.)
⊢ {𝐵} ⊆ {𝐴, 𝐵}
Theorem	snsstp1 3723	A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
⊢ {𝐴} ⊆ {𝐴, 𝐵, 𝐶}
Theorem	snsstp2 3724	A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
⊢ {𝐵} ⊆ {𝐴, 𝐵, 𝐶}
Theorem	snsstp3 3725	A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
⊢ {𝐶} ⊆ {𝐴, 𝐵, 𝐶}
Theorem	prsstp12 3726	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 3727	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 3728	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 3729	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 3730	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 3731	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 3732	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 3733	Empty set and the singleton itself are subsets of a singleton. Concerning the converse, see exmidsssn 4181. (Contributed by Jim Kingdon, 10-Aug-2018.)
⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵})
Theorem	sssnm 3734*	The inhabited subset of a singleton. (Contributed by Jim Kingdon, 10-Aug-2018.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐴 ⊆ {𝐵} ↔ 𝐴 = {𝐵}))
Theorem	eqsnm 3735*	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 3736	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 3737	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 3738	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 3739	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 3740	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 3741	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 3742	Closed form of sneqr 3740. (Contributed by Scott Fenton, 1-Apr-2011.)
⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
Theorem	sneqbg 3743	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 3744	The singleton of a class is a subset of its power class. (Contributed by NM, 5-Aug-1993.)
⊢ {𝐴} ⊆ 𝒫 𝐴
Theorem	prsspw 3745	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 3746	Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. Closed form of preqr1 3748. (Contributed by Jim Kingdon, 21-Sep-2018.)
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵))
Theorem	preqr2g 3747	Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the second elements are equal. Closed form of preqr2 3749. (Contributed by Jim Kingdon, 21-Sep-2018.)
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵))
Theorem	preqr1 3748	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 3749	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 3750	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 3751	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 3752	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 3753	Closed form of preq12b 3750. (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 3754	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 3755	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 3756	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 3757	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 3758	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 3759	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 3760	Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)
⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
Theorem	opeq1i 3761	Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩
Theorem	opeq2i 3762	Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩
Theorem	opeq12i 3763	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 3764	Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
Theorem	opeq2d 3765	Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)
Theorem	opeq12d 3766	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 3767	Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)
Theorem	oteq2 3768	Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)
Theorem	oteq3 3769	Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)
Theorem	oteq1d 3770	Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)
Theorem	oteq2d 3771	Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)
Theorem	oteq3d 3772	Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)
Theorem	oteq123d 3773	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 3774	Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥⟨𝐴, 𝐵⟩
Theorem	nfopd 3775	Deduction version of bound-variable hypothesis builder nfop 3774. This shows how the deduction version of a not-free theorem such as nfop 3774 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 3776	The ordered pair ⟨𝐴, 𝐴⟩ in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ⟨𝐴, 𝐴⟩ = {{𝐴}}
Theorem	ralunsn 3777*	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 3778*	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 3779	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 3780	Expansion of an ordered pair when the first member is a proper class. See also opprc 3779. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ (¬ 𝐴 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
Theorem	opprc2 3781	Expansion of an ordered pair when the second member is a proper class. See also opprc 3779. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ (¬ 𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
Theorem	oprcl 3782	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 3783	The power set of a singleton. (Contributed by Jim Kingdon, 12-Aug-2018.)
⊢ {∅, {𝐴}} ⊆ 𝒫 {𝐴}
Theorem	pwpw0ss 3784	Compute the power set of the power set of the empty set. (See pw0 3720 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 3785	The power set of an unordered pair. (Contributed by Jim Kingdon, 13-Aug-2018.)
⊢ ({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ⊆ 𝒫 {𝐴, 𝐵}
Theorem	pwtpss 3786	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 3787	Compute the power set of the power set of the power set of the empty set. (See also pw0 3720 and pwpw0ss 3784.) (Contributed by Jim Kingdon, 13-Aug-2018.)
⊢ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) ⊆ 𝒫 {∅, {∅}}
Theorem	pwv 3788	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
Syntax	cuni 3789	Extend class notation to include the union of a class. Read: "union (of) 𝐴".
class ∪ 𝐴
Definition	df-uni 3790*	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}. This is similar to the union of two classes df-un 3120. (Contributed by NM, 23-Aug-1993.)
⊢ ∪ 𝐴 = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)}
Theorem	dfuni2 3791*	Alternate definition of class union. (Contributed by NM, 28-Jun-1998.)
⊢ ∪ 𝐴 = {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦}
Theorem	eluni 3792*	Membership in class union. (Contributed by NM, 22-May-1994.)
⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))
Theorem	eluni2 3793*	Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999.)
⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥)
Theorem	elunii 3794	Membership in class union. (Contributed by NM, 24-Mar-1995.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ ∪ 𝐶)
Theorem	nfuni 3795	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 3796	Deduction version of nfuni 3795. (Contributed by NM, 18-Feb-2013.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴)
Theorem	csbunig 3797	Distribute proper substitution through the union of a class. (Contributed by Alan Sare, 10-Nov-2012.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌∪ 𝐵 = ∪ ⦋𝐴 / 𝑥⦌𝐵)
Theorem	unieq 3798	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 3799	Inference of equality of two class unions. (Contributed by NM, 30-Aug-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ∪ 𝐴 = ∪ 𝐵
Theorem	unieqd 3800	Deduction of equality of two class unions. (Contributed by NM, 21-Apr-1995.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ∪ 𝐴 = ∪ 𝐵)