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Type | Label | Description |
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Statement | ||
Definition | df-pr 3601 | Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. They are unordered, so {𝐴, 𝐵} = {𝐵, 𝐴} as proven by prcom 3670. For a more traditional definition, but requiring a dummy variable, see dfpr2 3613. (Contributed by NM, 5-Aug-1993.) |
⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | ||
Definition | df-tp 3602 | 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 3603* |
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 3802 and opprc2 3803). For
Kuratowski's actual definition when the arguments are sets, see dfop 3779.
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 3603 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3603 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 3604 | 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 3605 | Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | ||
Theorem | sneqi 3606 | Equality inference for singletons. (Contributed by NM, 22-Jan-2004.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ {𝐴} = {𝐵} | ||
Theorem | sneqd 3607 | Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → {𝐴} = {𝐵}) | ||
Theorem | dfsn2 3608 | Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) |
⊢ {𝐴} = {𝐴, 𝐴} | ||
Theorem | elsng 3609 | 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 3610 | 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 3611 | 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 3612 | There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.) |
⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵) | ||
Theorem | dfpr2 3613* | 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 3614 | 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 3615 | 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 3616 | 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 3617 | 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 3618 | 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 3619 | 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 3620 | 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 3621 | Membership in a set with two elements removed. Similar to eldifsn 3721 and eldiftp 3640. (Contributed by Mario Carneiro, 18-Jul-2017.) |
⊢ (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷)) | ||
Theorem | rexdifpr 3622 | 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 3623 | 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 3624 | 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 3625 | 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 3626 | A setvar variable is a member of its singleton (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 𝑥 ∈ {𝑥} | ||
Theorem | elsn2g 3627 | 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 3628 | 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 3629 | 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 3630* | 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 3712 and snprc 3659. (Contributed by Jim Kingdon, 30-Aug-2018.) |
⊢ ∃*𝑥 𝑥 ∈ {𝐴} | ||
Theorem | ralsnsg 3631* | 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 3632* | 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 3633* | 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 3634* | 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 3635* | Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)) | ||
Theorem | exsnrex 3636 | 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 3637* | 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 3638* | 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 3639 | 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 3640 | Membership in a set with three elements removed. Similar to eldifsn 3721 and eldifpr 3621. (Contributed by David A. Wheeler, 22-Jul-2017.) |
⊢ (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷, 𝐸}) ↔ (𝐴 ∈ 𝐵 ∧ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷 ∧ 𝐴 ≠ 𝐸))) | ||
Theorem | eltpi 3641 | 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 3642 | 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 3643* | 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 3644 | Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥{𝐴, 𝐵} | ||
Theorem | ralprg 3645* | 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 3646* | 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 3647* | 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 3648* | 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 3649* | 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 3650* | 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 3651* | 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 3652* | 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 3653* | 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 3654 | Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥{𝐴} | ||
Theorem | csbsng 3655 | 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 3656 | 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 3657 | Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.) |
⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | ||
Theorem | disjpr2 3658 | 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 3659 | 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 3660* | Special case of r19.12 2583 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 3661* | Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.) |
⊢ (𝐵 ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ 𝑥 = 𝐵} = {𝐵}) | ||
Theorem | rabrsndc 3662* | 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 3663* | 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 3664 | 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 3665* | 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 3666 | Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) |
⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∣ 𝜑} = {𝐴}) → ∃!𝑥𝜑) | ||
Theorem | rabsneu 3667 | 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 3668* | Two ways to express "𝐴 is a singleton". (Contributed by NM, 30-Oct-2010.) |
⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 𝐴 = {𝑥}) | ||
Theorem | rabsnt 3669* | 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 3670 | Commutative law for unordered pairs. (Contributed by NM, 5-Aug-1993.) |
⊢ {𝐴, 𝐵} = {𝐵, 𝐴} | ||
Theorem | preq1 3671 | Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.) |
⊢ (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶}) | ||
Theorem | preq2 3672 | Equality theorem for unordered pairs. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵}) | ||
Theorem | preq12 3673 | Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷}) | ||
Theorem | preq1i 3674 | Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ {𝐴, 𝐶} = {𝐵, 𝐶} | ||
Theorem | preq2i 3675 | Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ {𝐶, 𝐴} = {𝐶, 𝐵} | ||
Theorem | preq12i 3676 | Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ {𝐴, 𝐶} = {𝐵, 𝐷} | ||
Theorem | preq1d 3677 | Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐶}) | ||
Theorem | preq2d 3678 | Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → {𝐶, 𝐴} = {𝐶, 𝐵}) | ||
Theorem | preq12d 3679 | Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐷}) | ||
Theorem | tpeq1 3680 | Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.) |
⊢ (𝐴 = 𝐵 → {𝐴, 𝐶, 𝐷} = {𝐵, 𝐶, 𝐷}) | ||
Theorem | tpeq2 3681 | Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.) |
⊢ (𝐴 = 𝐵 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷}) | ||
Theorem | tpeq3 3682 | Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.) |
⊢ (𝐴 = 𝐵 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵}) | ||
Theorem | tpeq1d 3683 | Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → {𝐴, 𝐶, 𝐷} = {𝐵, 𝐶, 𝐷}) | ||
Theorem | tpeq2d 3684 | Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷}) | ||
Theorem | tpeq3d 3685 | Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵}) | ||
Theorem | tpeq123d 3686 | 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 3687 | Rotation of the elements of an unordered triple. (Contributed by Alan Sare, 24-Oct-2011.) |
⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴} | ||
Theorem | tpcoma 3688 | Swap 1st and 2nd members of an undordered triple. (Contributed by NM, 22-May-2015.) |
⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶} | ||
Theorem | tpcomb 3689 | Swap 2nd and 3rd members of an undordered triple. (Contributed by NM, 22-May-2015.) |
⊢ {𝐴, 𝐵, 𝐶} = {𝐴, 𝐶, 𝐵} | ||
Theorem | tpass 3690 | Split off the first element of an unordered triple. (Contributed by Mario Carneiro, 5-Jan-2016.) |
⊢ {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶}) | ||
Theorem | qdass 3691 | Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.) |
⊢ ({𝐴, 𝐵} ∪ {𝐶, 𝐷}) = ({𝐴, 𝐵, 𝐶} ∪ {𝐷}) | ||
Theorem | qdassr 3692 | Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.) |
⊢ ({𝐴, 𝐵} ∪ {𝐶, 𝐷}) = ({𝐴} ∪ {𝐵, 𝐶, 𝐷}) | ||
Theorem | tpidm12 3693 | 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 3694 | 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 3695 | 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 3696 | 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 3697 | 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 3698 | 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 3699 | 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 3700 | 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} |
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