HomeHome Intuitionistic Logic Explorer
Theorem List (p. 37 of 152)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 3601-3700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempwidg 3601 Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐴𝑉𝐴 ∈ 𝒫 𝐴)
 
Theorempwid 3602 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
 
Theorempwss 3603* Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)
(𝒫 𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 
2.1.17  Unordered and ordered pairs
 
Syntaxcsn 3604 Extend class notation to include singleton.
class {𝐴}
 
Syntaxcpr 3605 Extend class notation to include unordered pair.
class {𝐴, 𝐵}
 
Syntaxctp 3606 Extend class notation to include unordered triplet.
class {𝐴, 𝐵, 𝐶}
 
Syntaxcop 3607 Extend class notation to include ordered pair.
class 𝐴, 𝐵
 
Syntaxcotp 3608 Extend class notation to include ordered triple.
class 𝐴, 𝐵, 𝐶
 
Theoremsnjust 3609* Soundness justification theorem for df-sn 3610. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
{𝑥𝑥 = 𝐴} = {𝑦𝑦 = 𝐴}
 
Definitiondf-sn 3610* 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 3618. (Contributed by NM, 5-Aug-1993.)
{𝐴} = {𝑥𝑥 = 𝐴}
 
Definitiondf-pr 3611 Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. They are unordered, so {𝐴, 𝐵} = {𝐵, 𝐴} as proven by prcom 3680. For a more traditional definition, but requiring a dummy variable, see dfpr2 3623. (Contributed by NM, 5-Aug-1993.)
{𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
 
Definitiondf-tp 3612 Define unordered triple of classes. Definition of [Enderton] p. 19. (Contributed by NM, 9-Apr-1994.)
{𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
 
Definitiondf-op 3613* 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 3812 and opprc2 3813). For Kuratowski's actual definition when the arguments are sets, see dfop 3789.

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 3613 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3613 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, 𝐵2 = {{{𝐴}, ∅}, {{𝐵}}}. This was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition is 𝐴, 𝐵3 = {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}})}
 
Definitiondf-ot 3614 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
 
Theoremsneq 3615 Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.)
(𝐴 = 𝐵 → {𝐴} = {𝐵})
 
Theoremsneqi 3616 Equality inference for singletons. (Contributed by NM, 22-Jan-2004.)
𝐴 = 𝐵       {𝐴} = {𝐵}
 
Theoremsneqd 3617 Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝐴} = {𝐵})
 
Theoremdfsn2 3618 Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
{𝐴} = {𝐴, 𝐴}
 
Theoremelsng 3619 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.)
(𝐴𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
 
Theoremelsn 3620 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)
 
Theoremvelsn 3621 There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.)
(𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
 
Theoremelsni 3622 There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.)
(𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
 
Theoremdfpr2 3623* Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
{𝐴, 𝐵} = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵)}
 
Theoremelprg 3624 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.)
(𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
 
Theoremelpr 3625 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 = 𝐵𝐴 = 𝐶))
 
Theoremelpr2 3626 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 = 𝐵𝐴 = 𝐶))
 
Theoremelpri 3627 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 = 𝐵𝐴 = 𝐶))
 
Theoremnelpri 3628 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.)
𝐴𝐵    &   𝐴𝐶        ¬ 𝐴 ∈ {𝐵, 𝐶}
 
Theoremprneli 3629 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.)
𝐴𝐵    &   𝐴𝐶       𝐴 ∉ {𝐵, 𝐶}
 
Theoremnelprd 3630 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.)
(𝜑𝐴𝐵)    &   (𝜑𝐴𝐶)       (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})
 
Theoremeldifpr 3631 Membership in a set with two elements removed. Similar to eldifsn 3731 and eldiftp 3650. (Contributed by Mario Carneiro, 18-Jul-2017.)
(𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷}) ↔ (𝐴𝐵𝐴𝐶𝐴𝐷))
 
Theoremrexdifpr 3632 Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.)
(∃𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶})𝜑 ↔ ∃𝑥𝐴 (𝑥𝐵𝑥𝐶𝜑))
 
Theoremsnidg 3633 A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 28-Oct-2003.)
(𝐴𝑉𝐴 ∈ {𝐴})
 
Theoremsnidb 3634 A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.)
(𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})
 
Theoremsnid 3635 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}
 
Theoremvsnid 3636 A setvar variable is a member of its singleton (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
𝑥 ∈ {𝑥}
 
Theoremelsn2g 3637 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.)
(𝐵𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
 
Theoremelsn2 3638 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)
 
Theoremnelsn 3639 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.)
(𝐴𝐵 → ¬ 𝐴 ∈ {𝐵})
 
Theoremmosn 3640* 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 3722 and snprc 3669. (Contributed by Jim Kingdon, 30-Aug-2018.)
∃*𝑥 𝑥 ∈ {𝐴}
 
Theoremralsnsg 3641* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
(𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
 
Theoremralsns 3642* Substitution expressed in terms of quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)
(𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
 
Theoremrexsns 3643* 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)
 
Theoremralsng 3644* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑𝜓))
 
Theoremrexsng 3645* Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑𝜓))
 
Theoremexsnrex 3646 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} ↔ ∃𝑥𝑀 𝑀 = {𝑥})
 
Theoremralsn 3647* Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥 ∈ {𝐴}𝜑𝜓)
 
Theoremrexsn 3648* Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∃𝑥 ∈ {𝐴}𝜑𝜓)
 
Theoremeltpg 3649 Members of an unordered triple of classes. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.)
(𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵𝐴 = 𝐶𝐴 = 𝐷)))
 
Theoremeldiftp 3650 Membership in a set with three elements removed. Similar to eldifsn 3731 and eldifpr 3631. (Contributed by David A. Wheeler, 22-Jul-2017.)
(𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷, 𝐸}) ↔ (𝐴𝐵 ∧ (𝐴𝐶𝐴𝐷𝐴𝐸)))
 
Theoremeltpi 3651 A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015.)
(𝐴 ∈ {𝐵, 𝐶, 𝐷} → (𝐴 = 𝐵𝐴 = 𝐶𝐴 = 𝐷))
 
Theoremeltp 3652 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 = 𝐵𝐴 = 𝐶𝐴 = 𝐷))
 
Theoremdftp2 3653* 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 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)}
 
Theoremnfpr 3654 Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)
𝑥𝐴    &   𝑥𝐵       𝑥{𝐴, 𝐵}
 
Theoremralprg 3655* Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))       ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
 
Theoremrexprg 3656* Convert a quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))       ((𝐴𝑉𝐵𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
 
Theoremraltpg 3657* Convert a quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))    &   (𝑥 = 𝐶 → (𝜑𝜃))       ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃)))
 
Theoremrextpg 3658* Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))    &   (𝑥 = 𝐶 → (𝜑𝜃))       ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃)))
 
Theoremralpr 3659* 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 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))       (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒))
 
Theoremrexpr 3660* 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 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))       (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒))
 
Theoremraltp 3661* 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 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))    &   (𝑥 = 𝐶 → (𝜑𝜃))       (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃))
 
Theoremrextp 3662* Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))    &   (𝑥 = 𝐶 → (𝜑𝜃))       (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃))
 
Theoremsbcsng 3663* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥 ∈ {𝐴}𝜑))
 
Theoremnfsn 3664 Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.)
𝑥𝐴       𝑥{𝐴}
 
Theoremcsbsng 3665 Distribute proper substitution through the singleton of a class. (Contributed by Alan Sare, 10-Nov-2012.)
(𝐴𝑉𝐴 / 𝑥{𝐵} = {𝐴 / 𝑥𝐵})
 
Theoremdisjsn 3666 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}) = ∅ ↔ ¬ 𝐵𝐴)
 
Theoremdisjsn2 3667 Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.)
(𝐴𝐵 → ({𝐴} ∩ {𝐵}) = ∅)
 
Theoremdisjpr2 3668 The intersection of distinct unordered pairs is disjoint. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
(((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐷𝐵𝐷)) → ({𝐴, 𝐵} ∩ {𝐶, 𝐷}) = ∅)
 
Theoremsnprc 3669 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} = ∅)
 
Theoremr19.12sn 3670* Special case of r19.12 2593 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.) (Revised by BJ, 20-Dec-2021.)
(𝐴𝑉 → (∃𝑥 ∈ {𝐴}∀𝑦𝐵 𝜑 ↔ ∀𝑦𝐵𝑥 ∈ {𝐴}𝜑))
 
Theoremrabsn 3671* Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.)
(𝐵𝐴 → {𝑥𝐴𝑥 = 𝐵} = {𝐵})
 
Theoremrabrsndc 3672* 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}))
 
Theoremeuabsn2 3673* Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by Mario Carneiro, 14-Nov-2016.)
(∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
 
Theoremeuabsn 3674 Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by NM, 22-Feb-2004.)
(∃!𝑥𝜑 ↔ ∃𝑥{𝑥𝜑} = {𝑥})
 
Theoremreusn 3675* 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.)
(∃!𝑥𝐴 𝜑 ↔ ∃𝑦{𝑥𝐴𝜑} = {𝑦})
 
Theoremabsneu 3676 Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.)
((𝐴𝑉 ∧ {𝑥𝜑} = {𝐴}) → ∃!𝑥𝜑)
 
Theoremrabsneu 3677 Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) (Revised by Mario Carneiro, 23-Dec-2016.)
((𝐴𝑉 ∧ {𝑥𝐵𝜑} = {𝐴}) → ∃!𝑥𝐵 𝜑)
 
Theoremeusn 3678* Two ways to express "𝐴 is a singleton". (Contributed by NM, 30-Oct-2010.)
(∃!𝑥 𝑥𝐴 ↔ ∃𝑥 𝐴 = {𝑥})
 
Theoremrabsnt 3679* 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 → (𝜑𝜓))       ({𝑥𝐴𝜑} = {𝐵} → 𝜓)
 
Theoremprcom 3680 Commutative law for unordered pairs. (Contributed by NM, 5-Aug-1993.)
{𝐴, 𝐵} = {𝐵, 𝐴}
 
Theorempreq1 3681 Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.)
(𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})
 
Theorempreq2 3682 Equality theorem for unordered pairs. (Contributed by NM, 5-Aug-1993.)
(𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵})
 
Theorempreq12 3683 Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.)
((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
 
Theorempreq1i 3684 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
𝐴 = 𝐵       {𝐴, 𝐶} = {𝐵, 𝐶}
 
Theorempreq2i 3685 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
𝐴 = 𝐵       {𝐶, 𝐴} = {𝐶, 𝐵}
 
Theorempreq12i 3686 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
𝐴 = 𝐵    &   𝐶 = 𝐷       {𝐴, 𝐶} = {𝐵, 𝐷}
 
Theorempreq1d 3687 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐶})
 
Theorempreq2d 3688 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝐶, 𝐴} = {𝐶, 𝐵})
 
Theorempreq12d 3689 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐷})
 
Theoremtpeq1 3690 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
(𝐴 = 𝐵 → {𝐴, 𝐶, 𝐷} = {𝐵, 𝐶, 𝐷})
 
Theoremtpeq2 3691 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
(𝐴 = 𝐵 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷})
 
Theoremtpeq3 3692 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
(𝐴 = 𝐵 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵})
 
Theoremtpeq1d 3693 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝐴, 𝐶, 𝐷} = {𝐵, 𝐶, 𝐷})
 
Theoremtpeq2d 3694 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷})
 
Theoremtpeq3d 3695 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵})
 
Theoremtpeq123d 3696 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)    &   (𝜑𝐸 = 𝐹)       (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐹})
 
Theoremtprot 3697 Rotation of the elements of an unordered triple. (Contributed by Alan Sare, 24-Oct-2011.)
{𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
 
Theoremtpcoma 3698 Swap 1st and 2nd members of an undordered triple. (Contributed by NM, 22-May-2015.)
{𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶}
 
Theoremtpcomb 3699 Swap 2nd and 3rd members of an undordered triple. (Contributed by NM, 22-May-2015.)
{𝐴, 𝐵, 𝐶} = {𝐴, 𝐶, 𝐵}
 
Theoremtpass 3700 Split off the first element of an unordered triple. (Contributed by Mario Carneiro, 5-Jan-2016.)
{𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶})
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15104
  Copyright terms: Public domain < Previous  Next >