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Type | Label | Description |
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Statement | ||
Theorem | pwjust 3401* | Soundness justification theorem for df-pw 3402. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ {𝑥 ∣ 𝑥 ⊆ 𝐴} = {𝑦 ∣ 𝑦 ⊆ 𝐴} | ||
Definition | df-pw 3402* | 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 3403 | Equality theorem for power class. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵) | ||
Theorem | pweqi 3404 | Equality inference for power class. (Contributed by NM, 27-Nov-2013.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ 𝒫 𝐴 = 𝒫 𝐵 | ||
Theorem | pweqd 3405 | Equality deduction for power class. (Contributed by NM, 27-Nov-2013.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 𝒫 𝐴 = 𝒫 𝐵) | ||
Theorem | elpw 3406 | 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 | selpw 3407* | Setvar variable membership in a power class (common case). See elpw 3406. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | ||
Theorem | elpwg 3408 | 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 3409 | Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.) |
⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ⊆ 𝐵) | ||
Theorem | elpwid 3410 | An element of a power class is a subclass. Deduction form of elpwi 3409. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | ||
Theorem | elelpwi 3411 | 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 3412 | 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 3413 | Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴) | ||
Theorem | pwid 3414 | 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 3415* | Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.) |
⊢ (𝒫 𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐵)) | ||
Syntax | csn 3416 | Extend class notation to include singleton. |
class {𝐴} | ||
Syntax | cpr 3417 | Extend class notation to include unordered pair. |
class {𝐴, 𝐵} | ||
Syntax | ctp 3418 | Extend class notation to include unordered triplet. |
class {𝐴, 𝐵, 𝐶} | ||
Syntax | cop 3419 | Extend class notation to include ordered pair. |
class 〈𝐴, 𝐵〉 | ||
Syntax | cotp 3420 | Extend class notation to include ordered triple. |
class 〈𝐴, 𝐵, 𝐶〉 | ||
Theorem | snjust 3421* | Soundness justification theorem for df-sn 3422. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ {𝑥 ∣ 𝑥 = 𝐴} = {𝑦 ∣ 𝑦 = 𝐴} | ||
Definition | df-sn 3422* | 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 3430. (Contributed by NM, 5-Aug-1993.) |
⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴} | ||
Definition | df-pr 3423 | Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. They are unordered, so {𝐴, 𝐵} = {𝐵, 𝐴} as proven by prcom 3486. For a more traditional definition, but requiring a dummy variable, see dfpr2 3435. (Contributed by NM, 5-Aug-1993.) |
⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | ||
Definition | df-tp 3424 | 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 3425* |
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 3612 and opprc2 3613). For
Kuratowski's actual definition when the arguments are sets, see dfop 3589.
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 3425 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3425 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 3426 | 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 3427 | Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | ||
Theorem | sneqi 3428 | Equality inference for singletons. (Contributed by NM, 22-Jan-2004.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ {𝐴} = {𝐵} | ||
Theorem | sneqd 3429 | Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → {𝐴} = {𝐵}) | ||
Theorem | dfsn2 3430 | Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) |
⊢ {𝐴} = {𝐴, 𝐴} | ||
Theorem | elsng 3431 | 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 3432 | 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 3433 | 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 3434 | There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.) |
⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵) | ||
Theorem | dfpr2 3435* | 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 3436 | 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 3437 | 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 3438 | 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 3439 | 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 3440 | 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 | snidg 3441 | 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 3442 | 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 3443 | 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 3444 | A setvar variable is a member of its singleton (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 𝑥 ∈ {𝑥} | ||
Theorem | elsn2g 3445 | 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 3446 | 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 | mosn 3447* | 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 3526 and snprc 3475. (Contributed by Jim Kingdon, 30-Aug-2018.) |
⊢ ∃*𝑥 𝑥 ∈ {𝐴} | ||
Theorem | ralsnsg 3448* | 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 3449* | 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 3450* | 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 3451* | 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 3452* | Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)) | ||
Theorem | exsnrex 3453 | 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 3454* | 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 3455* | 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 3456 | 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 3457 | 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 3458 | 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 3459* | 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 3460 | Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥{𝐴, 𝐵} | ||
Theorem | ralprg 3461* | 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 3462* | 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 3463* | 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 3464* | 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 3465* | 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 3466* | 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 3467* | 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 3468* | 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 3469* | 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 3470 | Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥{𝐴} | ||
Theorem | csbsng 3471 | 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 3472 | 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 3473 | Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.) |
⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | ||
Theorem | disjpr2 3474 | 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 3475 | 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 3476* | Special case of r19.12 2471 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 3477* | Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.) |
⊢ (𝐵 ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ 𝑥 = 𝐵} = {𝐵}) | ||
Theorem | rabrsndc 3478* | 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 3479* | 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 3480 | 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 3481* | 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 3482 | Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) |
⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∣ 𝜑} = {𝐴}) → ∃!𝑥𝜑) | ||
Theorem | rabsneu 3483 | 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 3484* | Two ways to express "𝐴 is a singleton." (Contributed by NM, 30-Oct-2010.) |
⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 𝐴 = {𝑥}) | ||
Theorem | rabsnt 3485* | 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 3486 | Commutative law for unordered pairs. (Contributed by NM, 5-Aug-1993.) |
⊢ {𝐴, 𝐵} = {𝐵, 𝐴} | ||
Theorem | preq1 3487 | Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.) |
⊢ (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶}) | ||
Theorem | preq2 3488 | Equality theorem for unordered pairs. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵}) | ||
Theorem | preq12 3489 | Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷}) | ||
Theorem | preq1i 3490 | Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ {𝐴, 𝐶} = {𝐵, 𝐶} | ||
Theorem | preq2i 3491 | Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ {𝐶, 𝐴} = {𝐶, 𝐵} | ||
Theorem | preq12i 3492 | Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ {𝐴, 𝐶} = {𝐵, 𝐷} | ||
Theorem | preq1d 3493 | Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐶}) | ||
Theorem | preq2d 3494 | Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → {𝐶, 𝐴} = {𝐶, 𝐵}) | ||
Theorem | preq12d 3495 | Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐷}) | ||
Theorem | tpeq1 3496 | Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.) |
⊢ (𝐴 = 𝐵 → {𝐴, 𝐶, 𝐷} = {𝐵, 𝐶, 𝐷}) | ||
Theorem | tpeq2 3497 | Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.) |
⊢ (𝐴 = 𝐵 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷}) | ||
Theorem | tpeq3 3498 | Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.) |
⊢ (𝐴 = 𝐵 → {𝐶, 𝐷, 𝐴} = {𝐶, 𝐷, 𝐵}) | ||
Theorem | tpeq1d 3499 | Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → {𝐴, 𝐶, 𝐷} = {𝐵, 𝐶, 𝐷}) | ||
Theorem | tpeq2d 3500 | Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → {𝐶, 𝐴, 𝐷} = {𝐶, 𝐵, 𝐷}) |
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