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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ifbieq2d 3501 | 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 3502 | 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 3503 | 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 3504 | Deduction version of nfif 3505. (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 3505 | 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 3506 | 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 3507 | Conditional equality. (Contributed by Jim Kingdon, 1-Jan-2022.) |
⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) & ⊢ (𝜑 → DECID 𝜓) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) | ||
Theorem | ifbothdadc 3508 | 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 3509 | 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 3510 | Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.) |
⊢ (DECID 𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴) | ||
Theorem | eqifdc 3511 | 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 3512 | 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 | ifandc 3513 | 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 | ifmdc 3514 | 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) | ||
Syntax | cpw 3515 | 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 3516* | Soundness justification theorem for df-pw 3517. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ {𝑥 ∣ 𝑥 ⊆ 𝐴} = {𝑦 ∣ 𝑦 ⊆ 𝐴} | ||
Definition | df-pw 3517* | 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 3518 | Equality theorem for power class. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵) | ||
Theorem | pweqi 3519 | Equality inference for power class. (Contributed by NM, 27-Nov-2013.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ 𝒫 𝐴 = 𝒫 𝐵 | ||
Theorem | pweqd 3520 | Equality deduction for power class. (Contributed by NM, 27-Nov-2013.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 𝒫 𝐴 = 𝒫 𝐵) | ||
Theorem | elpw 3521 | 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 3522* | Setvar variable membership in a power class (common case). See elpw 3521. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | ||
Theorem | elpwg 3523 | 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 3524 | Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.) |
⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ⊆ 𝐵) | ||
Theorem | elpwb 3525 | Characterization of the elements of a power class. (Contributed by BJ, 29-Apr-2021.) |
⊢ (𝐴 ∈ 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵)) | ||
Theorem | elpwid 3526 | An element of a power class is a subclass. Deduction form of elpwi 3524. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | ||
Theorem | elelpwi 3527 | 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 3528 | 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 3529 | Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴) | ||
Theorem | pwid 3530 | 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 3531* | Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.) |
⊢ (𝒫 𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐵)) | ||
Syntax | csn 3532 | Extend class notation to include singleton. |
class {𝐴} | ||
Syntax | cpr 3533 | Extend class notation to include unordered pair. |
class {𝐴, 𝐵} | ||
Syntax | ctp 3534 | Extend class notation to include unordered triplet. |
class {𝐴, 𝐵, 𝐶} | ||
Syntax | cop 3535 | Extend class notation to include ordered pair. |
class 〈𝐴, 𝐵〉 | ||
Syntax | cotp 3536 | Extend class notation to include ordered triple. |
class 〈𝐴, 𝐵, 𝐶〉 | ||
Theorem | snjust 3537* | Soundness justification theorem for df-sn 3538. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ {𝑥 ∣ 𝑥 = 𝐴} = {𝑦 ∣ 𝑦 = 𝐴} | ||
Definition | df-sn 3538* | 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 3546. (Contributed by NM, 5-Aug-1993.) |
⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴} | ||
Definition | df-pr 3539 | Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. They are unordered, so {𝐴, 𝐵} = {𝐵, 𝐴} as proven by prcom 3607. For a more traditional definition, but requiring a dummy variable, see dfpr2 3551. (Contributed by NM, 5-Aug-1993.) |
⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | ||
Definition | df-tp 3540 | 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 3541* |
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 3735 and opprc2 3736). For
Kuratowski's actual definition when the arguments are sets, see dfop 3712.
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 3541 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3541 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 3542 | 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 3543 | Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | ||
Theorem | sneqi 3544 | Equality inference for singletons. (Contributed by NM, 22-Jan-2004.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ {𝐴} = {𝐵} | ||
Theorem | sneqd 3545 | Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → {𝐴} = {𝐵}) | ||
Theorem | dfsn2 3546 | Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) |
⊢ {𝐴} = {𝐴, 𝐴} | ||
Theorem | elsng 3547 | 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 3548 | 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 3549 | 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 3550 | There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.) |
⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵) | ||
Theorem | dfpr2 3551* | 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 3552 | 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 3553 | 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 3554 | 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 3555 | 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 3556 | 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 3557 | 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 3558 | 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 3559 | Membership in a set with two elements removed. Similar to eldifsn 3658 and eldiftp 3577. (Contributed by Mario Carneiro, 18-Jul-2017.) |
⊢ (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷)) | ||
Theorem | rexdifpr 3560 | 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 3561 | 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 3562 | 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 3563 | 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 3564 | A setvar variable is a member of its singleton (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 𝑥 ∈ {𝑥} | ||
Theorem | elsn2g 3565 | 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 3566 | 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 3567* | 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 3649 and snprc 3596. (Contributed by Jim Kingdon, 30-Aug-2018.) |
⊢ ∃*𝑥 𝑥 ∈ {𝐴} | ||
Theorem | ralsnsg 3568* | 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 3569* | 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 3570* | 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 3571* | 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 3572* | Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)) | ||
Theorem | exsnrex 3573 | 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 3574* | 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 3575* | 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 3576 | 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 3577 | Membership in a set with three elements removed. Similar to eldifsn 3658 and eldifpr 3559. (Contributed by David A. Wheeler, 22-Jul-2017.) |
⊢ (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷, 𝐸}) ↔ (𝐴 ∈ 𝐵 ∧ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷 ∧ 𝐴 ≠ 𝐸))) | ||
Theorem | eltpi 3578 | 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 3579 | 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 3580* | 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 3581 | Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥{𝐴, 𝐵} | ||
Theorem | ralprg 3582* | 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 3583* | 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 3584* | 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 3585* | 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 3586* | 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 3587* | 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 3588* | 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 3589* | 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 3590* | 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 3591 | Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥{𝐴} | ||
Theorem | csbsng 3592 | 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 3593 | 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 3594 | Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.) |
⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | ||
Theorem | disjpr2 3595 | 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 3596 | 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 3597* | Special case of r19.12 2541 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 3598* | Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.) |
⊢ (𝐵 ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ 𝑥 = 𝐵} = {𝐵}) | ||
Theorem | rabrsndc 3599* | 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 3600* | 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}) |
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