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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | pwidg 3601 | Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴) | ||
Theorem | pwid 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 | ||
Theorem | pwss 3603* | Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.) |
⊢ (𝒫 𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐵)) | ||
Syntax | csn 3604 | Extend class notation to include singleton. |
class {𝐴} | ||
Syntax | cpr 3605 | Extend class notation to include unordered pair. |
class {𝐴, 𝐵} | ||
Syntax | ctp 3606 | Extend class notation to include unordered triplet. |
class {𝐴, 𝐵, 𝐶} | ||
Syntax | cop 3607 | Extend class notation to include ordered pair. |
class ⟨𝐴, 𝐵⟩ | ||
Syntax | cotp 3608 | Extend class notation to include ordered triple. |
class ⟨𝐴, 𝐵, 𝐶⟩ | ||
Theorem | snjust 3609* | Soundness justification theorem for df-sn 3610. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ {𝑥 ∣ 𝑥 = 𝐴} = {𝑦 ∣ 𝑦 = 𝐴} | ||
Definition | df-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} = {𝑥 ∣ 𝑥 = 𝐴} | ||
Definition | df-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}) | ||
Definition | df-tp 3612 | 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 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, 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 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⟩ | ||
Theorem | sneq 3615 | Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | ||
Theorem | sneqi 3616 | Equality inference for singletons. (Contributed by NM, 22-Jan-2004.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ {𝐴} = {𝐵} | ||
Theorem | sneqd 3617 | Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → {𝐴} = {𝐵}) | ||
Theorem | dfsn2 3618 | Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) |
⊢ {𝐴} = {𝐴, 𝐴} | ||
Theorem | elsng 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.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)) | ||
Theorem | elsn 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) | ||
Theorem | velsn 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) | ||
Theorem | elsni 3622 | There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.) |
⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵) | ||
Theorem | dfpr2 3623* | 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 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.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))) | ||
Theorem | elpr 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 = 𝐵 ∨ 𝐴 = 𝐶)) | ||
Theorem | elpr2 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 = 𝐵 ∨ 𝐴 = 𝐶)) | ||
Theorem | elpri 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 = 𝐵 ∨ 𝐴 = 𝐶)) | ||
Theorem | nelpri 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.) |
⊢ 𝐴 ≠ 𝐵 & ⊢ 𝐴 ≠ 𝐶 ⇒ ⊢ ¬ 𝐴 ∈ {𝐵, 𝐶} | ||
Theorem | prneli 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.) |
⊢ 𝐴 ≠ 𝐵 & ⊢ 𝐴 ≠ 𝐶 ⇒ ⊢ 𝐴 ∉ {𝐵, 𝐶} | ||
Theorem | nelprd 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.) |
⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐴 ≠ 𝐶) ⇒ ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶}) | ||
Theorem | eldifpr 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}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷)) | ||
Theorem | rexdifpr 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 ↔ ∃𝑥 ∈ 𝐴 (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶 ∧ 𝜑)) | ||
Theorem | snidg 3633 | 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 3634 | 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 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} | ||
Theorem | vsnid 3636 | A setvar variable is a member of its singleton (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 𝑥 ∈ {𝑥} | ||
Theorem | elsn2g 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.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)) | ||
Theorem | elsn2 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) | ||
Theorem | nelsn 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.) |
⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 ∈ {𝐵}) | ||
Theorem | mosn 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} | ||
Theorem | ralsnsg 3641* | 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 3642* | 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 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) | ||
Theorem | ralsng 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 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)) | ||
Theorem | rexsng 3645* | Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)) | ||
Theorem | exsnrex 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} ↔ ∃𝑥 ∈ 𝑀 𝑀 = {𝑥}) | ||
Theorem | ralsn 3647* | 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 3648* | 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 3649 | 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 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}) ↔ (𝐴 ∈ 𝐵 ∧ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷 ∧ 𝐴 ≠ 𝐸))) | ||
Theorem | eltpi 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 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷)) | ||
Theorem | eltp 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 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷)) | ||
Theorem | dftp2 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 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)} | ||
Theorem | nfpr 3654 | Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥{𝐴, 𝐵} | ||
Theorem | ralprg 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 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒))) | ||
Theorem | rexprg 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 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒))) | ||
Theorem | raltpg 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 → (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 3658* | 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 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 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) ⇒ ⊢ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒)) | ||
Theorem | rexpr 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 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) ⇒ ⊢ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒)) | ||
Theorem | raltp 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 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜃)) ⇒ ⊢ (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)) | ||
Theorem | rextp 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 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜃)) ⇒ ⊢ (∃𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∨ 𝜒 ∨ 𝜃)) | ||
Theorem | sbcsng 3663* | 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 3664 | Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥{𝐴} | ||
Theorem | csbsng 3665 | 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 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}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴) | ||
Theorem | disjsn2 3667 | Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.) |
⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | ||
Theorem | disjpr2 3668 | 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 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} = ∅) | ||
Theorem | r19.12sn 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.) |
⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ {𝐴}𝜑)) | ||
Theorem | rabsn 3671* | Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.) |
⊢ (𝐵 ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ 𝑥 = 𝐵} = {𝐵}) | ||
Theorem | rabrsndc 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})) | ||
Theorem | euabsn2 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{𝑥 ∣ 𝜑} = {𝑦}) | ||
Theorem | euabsn 3674 | 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 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.) |
⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦{𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦}) | ||
Theorem | absneu 3676 | Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) |
⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∣ 𝜑} = {𝐴}) → ∃!𝑥𝜑) | ||
Theorem | rabsneu 3677 | 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 3678* | Two ways to express "𝐴 is a singleton". (Contributed by NM, 30-Oct-2010.) |
⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 𝐴 = {𝑥}) | ||
Theorem | rabsnt 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 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → 𝜓) | ||
Theorem | prcom 3680 | Commutative law for unordered pairs. (Contributed by NM, 5-Aug-1993.) |
⊢ {𝐴, 𝐵} = {𝐵, 𝐴} | ||
Theorem | preq1 3681 | Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.) |
⊢ (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶}) | ||
Theorem | preq2 3682 | Equality theorem for unordered pairs. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵}) | ||
Theorem | preq12 3683 | Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷}) | ||
Theorem | preq1i 3684 | Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ {𝐴, 𝐶} = {𝐵, 𝐶} | ||
Theorem | preq2i 3685 | Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ {𝐶, 𝐴} = {𝐶, 𝐵} | ||
Theorem | preq12i 3686 | Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ {𝐴, 𝐶} = {𝐵, 𝐷} | ||
Theorem | preq1d 3687 | Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐶}) | ||
Theorem | preq2d 3688 | Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → {𝐶, 𝐴} = {𝐶, 𝐵}) | ||
Theorem | preq12d 3689 | Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐷}) | ||
Theorem | tpeq1 3690 | Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.) |
⊢ (𝐴 = 𝐵 → {𝐴, 𝐶, 𝐷} = {𝐵, 𝐶, 𝐷}) | ||
Theorem | tpeq2 3691 | Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.) |
⊢ (𝐴 = 𝐵 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷}) | ||
Theorem | tpeq3 3692 | Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.) |
⊢ (𝐴 = 𝐵 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵}) | ||
Theorem | tpeq1d 3693 | Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → {𝐴, 𝐶, 𝐷} = {𝐵, 𝐶, 𝐷}) | ||
Theorem | tpeq2d 3694 | Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷}) | ||
Theorem | tpeq3d 3695 | Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵}) | ||
Theorem | tpeq123d 3696 | 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 3697 | Rotation of the elements of an unordered triple. (Contributed by Alan Sare, 24-Oct-2011.) |
⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴} | ||
Theorem | tpcoma 3698 | Swap 1st and 2nd members of an undordered triple. (Contributed by NM, 22-May-2015.) |
⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶} | ||
Theorem | tpcomb 3699 | Swap 2nd and 3rd members of an undordered triple. (Contributed by NM, 22-May-2015.) |
⊢ {𝐴, 𝐵, 𝐶} = {𝐴, 𝐶, 𝐵} | ||
Theorem | tpass 3700 | Split off the first element of an unordered triple. (Contributed by Mario Carneiro, 5-Jan-2016.) |
⊢ {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶}) |
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