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	ifeq2d 3601	Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
Theorem	ifeq12d 3602	Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐷))
Theorem	ifbi 3603	Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
Theorem	ifbid 3604	Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐴, 𝐵))
Theorem	ifbieq1d 3605	Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜒, 𝐵, 𝐶))
Theorem	ifbieq2i 3606	Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ 𝐴 = 𝐵    ⇒   ⊢ if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)
Theorem	ifbieq2d 3607	Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐵))
Theorem	ifbieq12i 3608	Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷)
Theorem	ifbieq12d 3609	Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷))
Theorem	nfifd 3610	Deduction version of nfif 3611. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥if(𝜓, 𝐴, 𝐵))
Theorem	nfif 3611	Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥if(𝜑, 𝐴, 𝐵)
Theorem	ifcldadc 3612	Conditional closure. (Contributed by Jim Kingdon, 11-Jan-2022.)
⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 ∈ 𝐶)    &   ⊢ (𝜑 → DECID 𝜓)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)
Theorem	ifeq1dadc 3613	Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐵)    &   ⊢ (𝜑 → DECID 𝜓)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
Theorem	ifeq2dadc 3614	Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵)    &   ⊢ (𝜑 → DECID 𝜓)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
Theorem	ifeqdadc 3615	Separation of the values of the conditional operator. (Contributed by Alexander van der Vekens, 13-Apr-2018.)
⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶)    &   ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐶)    &   ⊢ (𝜑 → DECID 𝜓)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = 𝐶)
Theorem	ifbothdadc 3616	A formula 𝜃 containing a decidable conditional operator is true when both of its cases are true. (Contributed by Jim Kingdon, 3-Jun-2022.)
⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜃))    &   ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃))    &   ⊢ ((𝜂 ∧ 𝜑) → 𝜓)    &   ⊢ ((𝜂 ∧ ¬ 𝜑) → 𝜒)    &   ⊢ (𝜂 → DECID 𝜑)    ⇒   ⊢ (𝜂 → 𝜃)
Theorem	ifbothdc 3617	A wff 𝜃 containing a conditional operator is true when both of its cases are true. (Contributed by Jim Kingdon, 8-Aug-2021.)
⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜃))    &   ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃))    ⇒   ⊢ ((𝜓 ∧ 𝜒 ∧ DECID 𝜑) → 𝜃)
Theorem	ifiddc 3618	Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.)
⊢ (DECID 𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴)
Theorem	eqifdc 3619	Expansion of an equality with a conditional operator. (Contributed by Jim Kingdon, 28-Jul-2022.)
⊢ (DECID 𝜑 → (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 = 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 = 𝐶))))
Theorem	ifcldcd 3620	Membership (closure) of a conditional operator, deduction form. (Contributed by Jim Kingdon, 8-Aug-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐶)    &   ⊢ (𝜑 → DECID 𝜓)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)
Theorem	ifnotdc 3621	Negating the first argument swaps the last two arguments of a conditional operator. (Contributed by NM, 21-Jun-2007.)
⊢ (DECID 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴))
Theorem	2if2dc 3622	Resolve two nested conditionals. (Contributed by Alexander van der Vekens, 27-Mar-2018.)
⊢ ((𝜑 ∧ 𝜓) → 𝐷 = 𝐴)    &   ⊢ ((𝜑 ∧ ¬ 𝜓 ∧ 𝜃) → 𝐷 = 𝐵)    &   ⊢ ((𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜃) → 𝐷 = 𝐶)    &   ⊢ (𝜑 → DECID 𝜓)    &   ⊢ ((𝜑 ∧ ¬ 𝜓) → DECID 𝜃)    ⇒   ⊢ (𝜑 → 𝐷 = if(𝜓, 𝐴, if(𝜃, 𝐵, 𝐶)))
Theorem	ifandc 3623	Rewrite a conjunction in a conditional as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ (DECID 𝜑 → if((𝜑 ∧ 𝜓), 𝐴, 𝐵) = if(𝜑, if(𝜓, 𝐴, 𝐵), 𝐵))
Theorem	ifordc 3624	Rewrite a disjunction in a conditional as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ (DECID 𝜑 → if((𝜑 ∨ 𝜓), 𝐴, 𝐵) = if(𝜑, 𝐴, if(𝜓, 𝐴, 𝐵)))
Theorem	ifmdc 3625	If a conditional class is inhabited, then the condition is decidable. This shows that conditionals are not very useful unless one can prove the condition decidable. (Contributed by BJ, 24-Sep-2022.)
⊢ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) → DECID 𝜑)
Theorem	ifnetruedc 3626	Deduce truth from a conditional operator value. (Contributed by Thierry Arnoux, 20-Feb-2025.)
⊢ ((DECID 𝜑 ∧ 𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) → 𝜑)
Theorem	ifnefals 3627	Deduce falsehood from a conditional operator value. (Contributed by Thierry Arnoux, 20-Feb-2025.)
⊢ ((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) → ¬ 𝜑)
Theorem	ifnebibdc 3628	The converse of ifbi 3603 holds if the two values are not equal. (Contributed by Thierry Arnoux, 20-Feb-2025.)
⊢ ((DECID 𝜑 ∧ DECID 𝜓 ∧ 𝐴 ≠ 𝐵) → (if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵) ↔ (𝜑 ↔ 𝜓)))
2.1.16  Power classes
Syntax	cpw 3629	Extend class notation to include power class. (The tilde in the Metamath token is meant to suggest the calligraphic font of the P.)
class 𝒫 𝐴
Theorem	pwjust 3630*	Soundness justification theorem for df-pw 3631. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ {𝑥 ∣ 𝑥 ⊆ 𝐴} = {𝑦 ∣ 𝑦 ⊆ 𝐴}
Definition	df-pw 3631*	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 is { 3 , 5 , 7 }, then 𝒫 𝐴 is { (/) , { 3 } , { 5 } , { 7 } , { 3 , 5 } , { 3 , 7 } , { 5 , 7 } , { 3 , 5 , 7 } }. We will later introduce the Axiom of Power Sets. Still later we will prove 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.)
⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
Theorem	pweq 3632	Equality theorem for power class. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵)
Theorem	pweqi 3633	Equality inference for power class. (Contributed by NM, 27-Nov-2013.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ 𝒫 𝐴 = 𝒫 𝐵
Theorem	pweqd 3634	Equality deduction for power class. (Contributed by NM, 27-Nov-2013.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → 𝒫 𝐴 = 𝒫 𝐵)
Theorem	elpw 3635	Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)
Theorem	velpw 3636*	Setvar variable membership in a power class (common case). See elpw 3635. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
Theorem	elpwg 3637	Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 6-Aug-2000.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
Theorem	elpwi 3638	Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)
⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ⊆ 𝐵)
Theorem	elpwb 3639	Characterization of the elements of a power class. (Contributed by BJ, 29-Apr-2021.)
⊢ (𝐴 ∈ 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵))
Theorem	elpwid 3640	An element of a power class is a subclass. Deduction form of elpwi 3638. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
Theorem	elelpwi 3641	If 𝐴 belongs to a part of 𝐶 then 𝐴 belongs to 𝐶. (Contributed by FL, 3-Aug-2009.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝒫 𝐶) → 𝐴 ∈ 𝐶)
Theorem	nfpw 3642	Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥𝒫 𝐴
Theorem	pwidg 3643	Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴)
Theorem	pwid 3644	A set is a member of its power class. Theorem 87 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 ∈ V    ⇒   ⊢ 𝐴 ∈ 𝒫 𝐴
Theorem	pwss 3645*	Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)
⊢ (𝒫 𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐵))
2.1.17  Unordered and ordered pairs
Syntax	csn 3646	Extend class notation to include singleton.
class {𝐴}
Syntax	cpr 3647	Extend class notation to include unordered pair.
class {𝐴, 𝐵}
Syntax	ctp 3648	Extend class notation to include unordered triplet.
class {𝐴, 𝐵, 𝐶}
Syntax	cop 3649	Extend class notation to include ordered pair.
class ⟨𝐴, 𝐵⟩
Syntax	cotp 3650	Extend class notation to include ordered triple.
class ⟨𝐴, 𝐵, 𝐶⟩
Theorem	snjust 3651*	Soundness justification theorem for df-sn 3652. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ {𝑥 ∣ 𝑥 = 𝐴} = {𝑦 ∣ 𝑦 = 𝐴}
Definition	df-sn 3652*	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 3660. (Contributed by NM, 5-Aug-1993.)
⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴}
Definition	df-pr 3653	Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. They are unordered, so {𝐴, 𝐵} = {𝐵, 𝐴} as proven by prcom 3722. For a more traditional definition, but requiring a dummy variable, see dfpr2 3665. (Contributed by NM, 5-Aug-1993.)
⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
Definition	df-tp 3654	Define unordered triple of classes. Definition of [Enderton] p. 19. (Contributed by NM, 9-Apr-1994.)
⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
Definition	df-op 3655*	Definition of an ordered pair, equivalent to Kuratowski's definition {{𝐴}, {𝐴, 𝐵}} when the arguments are sets. Since the behavior of Kuratowski definition is not very useful for proper classes, we define it to be empty in this case (see opprc1 3858 and opprc2 3859). For Kuratowski's actual definition when the arguments are sets, see dfop 3835. Definition 9.1 of [Quine] p. 58 defines an ordered pair unconditionally as ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}, which has different behavior from our df-op 3655 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3655 was chosen because it often makes proofs shorter by eliminating unnecessary sethood hypotheses. There are other ways to define ordered pairs. The basic requirement is that two ordered pairs are equal iff their respective members are equal. In 1914 Norbert Wiener gave the first successful definition ⟨𝐴, 𝐵⟩2 = {{{𝐴}, ∅}, {{𝐵}}}. This was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition is ⟨𝐴, 𝐵⟩3 = {𝐴, {𝐴, 𝐵}}, but it requires the Axiom of Regularity for its justification and is not commonly used. Finally, an ordered pair of real numbers can be represented by a complex number. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ ⟨𝐴, 𝐵⟩ = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
Definition	df-ot 3656	Define ordered triple of classes. Definition of ordered triple in [Stoll] p. 25. (Contributed by NM, 3-Apr-2015.)
⊢ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩
Theorem	sneq 3657	Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})
Theorem	sneqi 3658	Equality inference for singletons. (Contributed by NM, 22-Jan-2004.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ {𝐴} = {𝐵}
Theorem	sneqd 3659	Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → {𝐴} = {𝐵})
Theorem	dfsn2 3660	Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
⊢ {𝐴} = {𝐴, 𝐴}
Theorem	elsng 3661	There is exactly 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	elsn 3662	There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)
Theorem	velsn 3663	There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.)
⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
Theorem	elsni 3664	There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.)
⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
Theorem	dfpr2 3665*	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 3666	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 3667	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 3668	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 3669	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 3670	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	prneli 3671	If an element doesn't match the items in an unordered pair, it is not in the unordered pair, using ∉. (Contributed by David A. Wheeler, 10-May-2015.)
⊢ 𝐴 ≠ 𝐵    &   ⊢ 𝐴 ≠ 𝐶    ⇒   ⊢ 𝐴 ∉ {𝐵, 𝐶}
Theorem	nelprd 3672	If an element doesn't match the items in an unordered pair, it is not in the unordered pair, deduction version. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐶)    ⇒   ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})
Theorem	eldifpr 3673	Membership in a set with two elements removed. Similar to eldifsn 3774 and eldiftp 3692. (Contributed by Mario Carneiro, 18-Jul-2017.)
⊢ (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷))
Theorem	rexdifpr 3674	Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.)
⊢ (∃𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶})𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶 ∧ 𝜑))
Theorem	snidg 3675	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 3676	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 3677	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	vsnid 3678	A setvar variable is a member of its singleton (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ 𝑥 ∈ {𝑥}
Theorem	elsn2g 3679	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	elsn2 3680	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	nelsn 3681	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 3682*	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 3764 and snprc 3711. (Contributed by Jim Kingdon, 30-Aug-2018.)
⊢ ∃*𝑥 𝑥 ∈ {𝐴}
Theorem	ralsnsg 3683*	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 3684*	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 3685*	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 3686*	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 3687*	Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))
Theorem	exsnrex 3688	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 3689*	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 3690*	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 3691	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 3692	Membership in a set with three elements removed. Similar to eldifsn 3774 and eldifpr 3673. (Contributed by David A. Wheeler, 22-Jul-2017.)
⊢ (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷, 𝐸}) ↔ (𝐴 ∈ 𝐵 ∧ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷 ∧ 𝐴 ≠ 𝐸)))
Theorem	eltpi 3693	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 3694	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 3695*	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 3696	Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥{𝐴, 𝐵}
Theorem	ralprg 3697*	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 3698*	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 3699*	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 3700*	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)))