P. 38 of Theorem List - New Foundations Explorer

Theorem List for New Foundations Explorer - 3701-3800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ifnot 3701	Negating the first argument swaps the last two arguments of a conditional operator. (Contributed by NM, 21-Jun-2007.)
⊢ if(¬ φ, A, B) = if(φ, B, A)
Theorem	ifan 3702	Rewrite a conjunction in an if statement as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ if((φ ∧ ψ), A, B) = if(φ, if(ψ, A, B), B)
Theorem	ifor 3703	Rewrite a disjunction in an if statement as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ if((φ ∨ ψ), A, B) = if(φ, A, if(ψ, A, B))
Theorem	dedth 3704	Weak deduction theorem that eliminates a hypothesis φ, making it become an antecedent. We assume that a proof exists for φ when the class variable A is replaced with a specific class B. The hypothesis χ should be assigned to the inference, and the inference's hypothesis eliminated with elimhyp 3711. If the inference has other hypotheses with class variable A, these can be kept by assigning keephyp 3717 to them. For more information, see the Deduction Theorem https://us.metamath.org/mpeuni/mmdeduction.html 3717. (Contributed by NM, 15-May-1999.)
⊢ (A = if(φ, A, B) → (ψ ↔ χ))    &   ⊢ χ    ⇒   ⊢ (φ → ψ)
Theorem	dedth2h 3705	Weak deduction theorem eliminating two hypotheses. This theorem is simpler to use than dedth2v 3708 but requires that each hypothesis has exactly one class variable. See also comments in dedth 3704. (Contributed by NM, 15-May-1999.)
⊢ (A = if(φ, A, C) → (χ ↔ θ))    &   ⊢ (B = if(ψ, B, D) → (θ ↔ τ))    &   ⊢ τ    ⇒   ⊢ ((φ ∧ ψ) → χ)
Theorem	dedth3h 3706	Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 3705. (Contributed by NM, 15-May-1999.)
⊢ (A = if(φ, A, D) → (θ ↔ τ))    &   ⊢ (B = if(ψ, B, R) → (τ ↔ η))    &   ⊢ (C = if(χ, C, S) → (η ↔ ζ))    &   ⊢ ζ    ⇒   ⊢ ((φ ∧ ψ ∧ χ) → θ)
Theorem	dedth4h 3707	Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 3705. (Contributed by NM, 16-May-1999.)
⊢ (A = if(φ, A, R) → (τ ↔ η))    &   ⊢ (B = if(ψ, B, S) → (η ↔ ζ))    &   ⊢ (C = if(χ, C, F) → (ζ ↔ σ))    &   ⊢ (D = if(θ, D, G) → (σ ↔ ρ))    &   ⊢ ρ    ⇒   ⊢ (((φ ∧ ψ) ∧ (χ ∧ θ)) → τ)
Theorem	dedth2v 3708	Weak deduction theorem for eliminating a hypothesis with 2 class variables. Note: if the hypothesis can be separated into two hypotheses, each with one class variable, then dedth2h 3705 is simpler to use. See also comments in dedth 3704. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
⊢ (A = if(φ, A, C) → (ψ ↔ χ))    &   ⊢ (B = if(φ, B, D) → (χ ↔ θ))    &   ⊢ θ    ⇒   ⊢ (φ → ψ)
Theorem	dedth3v 3709	Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 3708. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
⊢ (A = if(φ, A, D) → (ψ ↔ χ))    &   ⊢ (B = if(φ, B, R) → (χ ↔ θ))    &   ⊢ (C = if(φ, C, S) → (θ ↔ τ))    &   ⊢ τ    ⇒   ⊢ (φ → ψ)
Theorem	dedth4v 3710	Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 3708. (Contributed by NM, 21-Apr-2007.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
⊢ (A = if(φ, A, R) → (ψ ↔ χ))    &   ⊢ (B = if(φ, B, S) → (χ ↔ θ))    &   ⊢ (C = if(φ, C, T) → (θ ↔ τ))    &   ⊢ (D = if(φ, D, U) → (τ ↔ η))    &   ⊢ η    ⇒   ⊢ (φ → ψ)
Theorem	elimhyp 3711	Eliminate a hypothesis containing class variable A when it is known for a specific class B. For more information, see comments in dedth 3704. (Contributed by NM, 15-May-1999.)
⊢ (A = if(φ, A, B) → (φ ↔ ψ))    &   ⊢ (B = if(φ, A, B) → (χ ↔ ψ))    &   ⊢ χ    ⇒   ⊢ ψ
Theorem	elimhyp2v 3712	Eliminate a hypothesis containing 2 class variables. (Contributed by NM, 14-Aug-1999.)
⊢ (A = if(φ, A, C) → (φ ↔ χ))    &   ⊢ (B = if(φ, B, D) → (χ ↔ θ))    &   ⊢ (C = if(φ, A, C) → (τ ↔ η))    &   ⊢ (D = if(φ, B, D) → (η ↔ θ))    &   ⊢ τ    ⇒   ⊢ θ
Theorem	elimhyp3v 3713	Eliminate a hypothesis containing 3 class variables. (Contributed by NM, 14-Aug-1999.)
⊢ (A = if(φ, A, D) → (φ ↔ χ))    &   ⊢ (B = if(φ, B, R) → (χ ↔ θ))    &   ⊢ (C = if(φ, C, S) → (θ ↔ τ))    &   ⊢ (D = if(φ, A, D) → (η ↔ ζ))    &   ⊢ (R = if(φ, B, R) → (ζ ↔ σ))    &   ⊢ (S = if(φ, C, S) → (σ ↔ τ))    &   ⊢ η    ⇒   ⊢ τ
Theorem	elimhyp4v 3714	Eliminate a hypothesis containing 4 class variables (for use with the weak deduction theorem dedth 3704). (Contributed by NM, 16-Apr-2005.)
⊢ (A = if(φ, A, D) → (φ ↔ χ))    &   ⊢ (B = if(φ, B, R) → (χ ↔ θ))    &   ⊢ (C = if(φ, C, S) → (θ ↔ τ))    &   ⊢ (F = if(φ, F, G) → (τ ↔ ψ))    &   ⊢ (D = if(φ, A, D) → (η ↔ ζ))    &   ⊢ (R = if(φ, B, R) → (ζ ↔ σ))    &   ⊢ (S = if(φ, C, S) → (σ ↔ ρ))    &   ⊢ (G = if(φ, F, G) → (ρ ↔ ψ))    &   ⊢ η    ⇒   ⊢ ψ
Theorem	elimel 3715	Eliminate a membership hypothesis for weak deduction theorem, when special case B ∈ C is provable. (Contributed by NM, 15-May-1999.)
⊢ B ∈ C    ⇒   ⊢ if(A ∈ C, A, B) ∈ C
Theorem	elimdhyp 3716	Version of elimhyp 3711 where the hypothesis is deduced from the final antecedent. See ghomgrplem in set.mm for an example of its use. (Contributed by Paul Chapman, 25-Mar-2008.)
⊢ (φ → ψ)    &   ⊢ (A = if(φ, A, B) → (ψ ↔ χ))    &   ⊢ (B = if(φ, A, B) → (θ ↔ χ))    &   ⊢ θ    ⇒   ⊢ χ
Theorem	keephyp 3717	Transform a hypothesis ψ that we want to keep (but contains the same class variable A used in the eliminated hypothesis) for use with the weak deduction theorem. (Contributed by NM, 15-May-1999.)
⊢ (A = if(φ, A, B) → (ψ ↔ θ))    &   ⊢ (B = if(φ, A, B) → (χ ↔ θ))    &   ⊢ ψ    &   ⊢ χ    ⇒   ⊢ θ
Theorem	keephyp2v 3718	Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 3704). (Contributed by NM, 16-Apr-2005.)
⊢ (A = if(φ, A, C) → (ψ ↔ χ))    &   ⊢ (B = if(φ, B, D) → (χ ↔ θ))    &   ⊢ (C = if(φ, A, C) → (τ ↔ η))    &   ⊢ (D = if(φ, B, D) → (η ↔ θ))    &   ⊢ ψ    &   ⊢ τ    ⇒   ⊢ θ
Theorem	keephyp3v 3719	Keep a hypothesis containing 3 class variables. (Contributed by NM, 27-Sep-1999.)
⊢ (A = if(φ, A, D) → (ρ ↔ χ))    &   ⊢ (B = if(φ, B, R) → (χ ↔ θ))    &   ⊢ (C = if(φ, C, S) → (θ ↔ τ))    &   ⊢ (D = if(φ, A, D) → (η ↔ ζ))    &   ⊢ (R = if(φ, B, R) → (ζ ↔ σ))    &   ⊢ (S = if(φ, C, S) → (σ ↔ τ))    &   ⊢ ρ    &   ⊢ η    ⇒   ⊢ τ
Theorem	keepel 3720	Keep a membership hypothesis for weak deduction theorem, when special case B ∈ C is provable. (Contributed by NM, 14-Aug-1999.)
⊢ A ∈ C    &   ⊢ B ∈ C    ⇒   ⊢ if(φ, A, B) ∈ C
Theorem	ifex 3721	Conditional operator existence. (Contributed by NM, 2-Sep-2004.)
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ if(φ, A, B) ∈ V
Theorem	ifexg 3722	Conditional operator existence. (Contributed by NM, 21-Mar-2011.)
⊢ ((A ∈ V ∧ B ∈ W) → if(φ, A, B) ∈ V)
2.1.15  Power classes
Syntax	cpw 3723	Extend class notation to include power class. (The tilde in the Metamath token is meant to suggest the calligraphic font of the P.)
class ℘A
Theorem	pwjust 3724*	Soundness justification theorem for df-pw 3725. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ {x ∣ x ⊆ A} = {y ∣ y ⊆ A}
Definition	df-pw 3725*	Define power class. Definition 5.10 of [TakeutiZaring] p. 17, but we also let it apply to proper classes, i.e. those that are not members of V. When applied to a set, this produces its power set. A power set of S is the set of all subsets of S, including the empty set and S itself. For example, if A = { 3 , 5 , 7 }, then ℘A = {∅, { 3 }, { 5 }, { 7 }, { 3 , 5 }, { 3 , 7 }, { 5 , 7 }, { 3 , 5 , 7 }} (ex-pw in set.mm). We will later introduce the Axiom of Power Sets ax-pow in set.mm, which can be expressed in class notation per pwexg 4329. Still later we will prove, in hashpw in set.mm, that the size of the power set of a finite set is 2 raised to the power of the size of the set. (Contributed by NM, 5-Aug-1993.)
⊢ ℘A = {x ∣ x ⊆ A}
Theorem	pweq 3726	Equality theorem for power class. (Contributed by NM, 5-Aug-1993.)
⊢ (A = B → ℘A = ℘B)
Theorem	pweqi 3727	Equality inference for power class. (Contributed by NM, 27-Nov-2013.)
⊢ A = B    ⇒   ⊢ ℘A = ℘B
Theorem	pweqd 3728	Equality deduction for power class. (Contributed by NM, 27-Nov-2013.)
⊢ (φ → A = B)    ⇒   ⊢ (φ → ℘A = ℘B)
Theorem	elpw 3729	Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.)
⊢ A ∈ V    ⇒   ⊢ (A ∈ ℘B ↔ A ⊆ B)
Theorem	elpwg 3730	Membership in a power class. Theorem 86 of [Suppes] p. 47. See also elpw2g in set.mm. (Contributed by NM, 6-Aug-2000.)
⊢ (A ∈ V → (A ∈ ℘B ↔ A ⊆ B))
Theorem	elpwi 3731	Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)
⊢ (A ∈ ℘B → A ⊆ B)
Theorem	elpwid 3732	An element of a power class is a subclass. Deduction form of elpwi 3731. (Contributed by David Moews, 1-May-2017.)
⊢ (φ → A ∈ ℘B)    ⇒   ⊢ (φ → A ⊆ B)
Theorem	elelpwi 3733	If A belongs to a part of C then A belongs to C. (Contributed by FL, 3-Aug-2009.)
⊢ ((A ∈ B ∧ B ∈ ℘C) → A ∈ C)
Theorem	nfpw 3734	Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
⊢ ℲxA    ⇒   ⊢ Ⅎx℘A
Theorem	pwidg 3735	Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ (A ∈ V → A ∈ ℘A)
Theorem	pwid 3736	A set is a member of its power class. Theorem 87 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.)
⊢ A ∈ V    ⇒   ⊢ A ∈ ℘A
Theorem	pwss 3737*	Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)
⊢ (℘A ⊆ B ↔ ∀x(x ⊆ A → x ∈ B))
2.1.16  Unordered and ordered pairs
Syntax	csn 3738	Extend class notation to include singleton.
class {A}
Syntax	cpr 3739	Extend class notation to include unordered pair.
class {A, B}
Syntax	ctp 3740	Extend class notation to include unordered triplet.
class {A, B, C}
Theorem	snjust 3741*	Soundness justification theorem for df-sn 3742. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ {x ∣ x = A} = {y ∣ y = A}
Definition	df-sn 3742*	Define the singleton of a class. Definition 7.1 of [Quine] p. 48. For convenience, it is well-defined for proper classes, i.e., those that are not elements of V, although it is not very meaningful in this case. For an alternate definition see dfsn2 3748. (Contributed by NM, 5-Aug-1993.)
⊢ {A} = {x ∣ x = A}
Definition	df-pr 3743	Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. For example, A ∈ { 1 , -u 1 } → (A ^ 2 ) = 1 (ex-pr in set.mm). They are unordered, so {A, B} = {B, A} as proven by prcom 3799. For a more traditional definition, but requiring a dummy variable, see dfpr2 3750. (Contributed by NM, 5-Aug-1993.)
⊢ {A, B} = ({A} ∪ {B})
Definition	df-tp 3744	Define unordered triple of classes. Definition of [Enderton] p. 19. (Contributed by NM, 9-Apr-1994.)
⊢ {A, B, C} = ({A, B} ∪ {C})
Theorem	sneq 3745	Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.)
⊢ (A = B → {A} = {B})
Theorem	sneqi 3746	Equality inference for singletons. (Contributed by NM, 22-Jan-2004.)
⊢ A = B    ⇒   ⊢ {A} = {B}
Theorem	sneqd 3747	Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.)
⊢ (φ → A = B)    ⇒   ⊢ (φ → {A} = {B})
Theorem	dfsn2 3748	Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
⊢ {A} = {A, A}
Theorem	elsn 3749*	There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.)
⊢ (x ∈ {A} ↔ x = A)
Theorem	dfpr2 3750*	Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
⊢ {A, B} = {x ∣ (x = A ∨ x = B)}
Theorem	elprg 3751	A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15, generalized. (Contributed by NM, 13-Sep-1995.)
⊢ (A ∈ V → (A ∈ {B, C} ↔ (A = B ∨ A = C)))
Theorem	elpr 3752	A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)
⊢ A ∈ V    ⇒   ⊢ (A ∈ {B, C} ↔ (A = B ∨ A = C))
Theorem	elpr2 3753	A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 14-Oct-2005.)
⊢ B ∈ V    &   ⊢ C ∈ V    ⇒   ⊢ (A ∈ {B, C} ↔ (A = B ∨ A = C))
Theorem	elpri 3754	If a class is an element of a pair, then it is one of the two paired elements. (Contributed by Scott Fenton, 1-Apr-2011.)
⊢ (A ∈ {B, C} → (A = B ∨ A = C))
Theorem	nelpri 3755	If an element doesn't match the items in an unordered pair, it is not in the unordered pair. (Contributed by David A. Wheeler, 10-May-2015.)
⊢ A ≠ B    &   ⊢ A ≠ C    ⇒   ⊢ ¬ A ∈ {B, C}
Theorem	elsncg 3756	There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15 (generalized). (Contributed by NM, 13-Sep-1995.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ (A ∈ V → (A ∈ {B} ↔ A = B))
Theorem	elsnc 3757	There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)
⊢ A ∈ V    ⇒   ⊢ (A ∈ {B} ↔ A = B)
Theorem	elsni 3758	There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.)
⊢ (A ∈ {B} → A = B)
Theorem	snidg 3759	A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 28-Oct-2003.)
⊢ (A ∈ V → A ∈ {A})
Theorem	snidb 3760	A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.)
⊢ (A ∈ V ↔ A ∈ {A})
Theorem	snid 3761	A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 31-Dec-1993.)
⊢ A ∈ V    ⇒   ⊢ A ∈ {A}
Theorem	elsnc2g 3762	There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that B, rather than A, be a set. (Contributed by NM, 28-Oct-2003.)
⊢ (B ∈ V → (A ∈ {B} ↔ A = B))
Theorem	elsnc2 3763	There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that B, rather than A, be a set. (Contributed by NM, 12-Jun-1994.)
⊢ B ∈ V    ⇒   ⊢ (A ∈ {B} ↔ A = B)
Theorem	ralsns 3764*	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 3765*	Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)
⊢ (A ∈ V → (∃x ∈ {A}φ ↔ [̣A / x]̣φ))
Theorem	ralsng 3766*	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 3767*	Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.)
⊢ (x = A → (φ ↔ ψ))    ⇒   ⊢ (A ∈ V → (∃x ∈ {A}φ ↔ ψ))
Theorem	ralsn 3768*	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 3769*	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 3770	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	eltpi 3771	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 3772	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 3773*	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 3774	Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)
⊢ ℲxA    &   ⊢ ℲxB    ⇒   ⊢ Ⅎx{A, B}
Theorem	ifpr 3775	Membership of a conditional operator in an unordered pair. (Contributed by NM, 17-Jun-2007.)
⊢ ((A ∈ C ∧ B ∈ D) → if(φ, A, B) ∈ {A, B})
Theorem	ralprg 3776*	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 3777*	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 3778*	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 3779*	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 3780*	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 3781*	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 3782*	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 3783*	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 3784*	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 3785	Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.)
⊢ ℲxA    ⇒   ⊢ Ⅎx{A}
Theorem	csbsng 3786	Distribute proper substitution through the singleton of a class. csbsng 3786 is derived from the virtual deduction proof csbsngVD in set.mm. (Contributed by Alan Sare, 10-Nov-2012.)
⊢ (A ∈ V → [A / x]{B} = {[A / x]B})
Theorem	disjsn 3787	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 3788	Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.)
⊢ (A ≠ B → ({A} ∩ {B}) = ∅)
Theorem	snprc 3789	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 3790*	Special case of r19.12 2728 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.)
⊢ A ∈ V    ⇒   ⊢ (∃x ∈ {A}∀y ∈ B φ ↔ ∀y ∈ B ∃x ∈ {A}φ)
Theorem	rabsn 3791*	Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.)
⊢ (B ∈ A → {x ∈ A ∣ x = B} = {B})
Theorem	euabsn2 3792*	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 3793	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 3794*	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 3795	Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.)
⊢ ((A ∈ V ∧ {x ∣ φ} = {A}) → ∃!xφ)
Theorem	rabsneu 3796	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 3797*	Two ways to express "A is a singleton." (Contributed by NM, 30-Oct-2010.)
⊢ (∃!x x ∈ A ↔ ∃x A = {x})
Theorem	rabsnt 3798*	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 3799	Commutative law for unordered pairs. (Contributed by NM, 5-Aug-1993.)
⊢ {A, B} = {B, A}
Theorem	preq1 3800	Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.)
⊢ (A = B → {A, C} = {B, C})