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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ral0 3301 | Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.) |
⊢ ∀x ∈ ∅ φ | ||
Theorem | rgenm 3302* | Generalization rule that eliminates an inhabited class requirement. (Contributed by Jim Kingdon, 5-Aug-2018.) |
⊢ ((∃x x ∈ A ∧ x ∈ A) → φ) ⇒ ⊢ ∀x ∈ A φ | ||
Theorem | ralf0 3303* | The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.) |
⊢ ¬ φ ⇒ ⊢ (∀x ∈ A φ ↔ A = ∅) | ||
Theorem | ralm 3304 | Inhabited classes and restricted quantification. (Contributed by Jim Kingdon, 6-Aug-2018.) |
⊢ ((∃x x ∈ A → ∀x ∈ A φ) ↔ ∀x ∈ A φ) | ||
Theorem | raaanlem 3305* | Special case of raaan 3306 where A is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.) |
⊢ Ⅎyφ & ⊢ Ⅎxψ ⇒ ⊢ (∃x x ∈ A → (∀x ∈ A ∀y ∈ A (φ ∧ ψ) ↔ (∀x ∈ A φ ∧ ∀y ∈ A ψ))) | ||
Theorem | raaan 3306* | Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.) |
⊢ Ⅎyφ & ⊢ Ⅎxψ ⇒ ⊢ (∀x ∈ A ∀y ∈ A (φ ∧ ψ) ↔ (∀x ∈ A φ ∧ ∀y ∈ A ψ)) | ||
Theorem | raaanv 3307* | Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.) |
⊢ (∀x ∈ A ∀y ∈ A (φ ∧ ψ) ↔ (∀x ∈ A φ ∧ ∀y ∈ A ψ)) | ||
Theorem | sbss 3308* | Set substitution into the first argument of a subset relation. (Contributed by Rodolfo Medina, 7-Jul-2010.) (Proof shortened by Mario Carneiro, 14-Nov-2016.) |
⊢ ([y / x]x ⊆ A ↔ y ⊆ A) | ||
Theorem | sbcssg 3309 | Distribute proper substitution through a subclass relation. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Alexander van der Vekens, 23-Jul-2017.) |
⊢ (A ∈ 𝑉 → ([A / x]B ⊆ 𝐶 ↔ ⦋A / x⦌B ⊆ ⦋A / x⦌𝐶)) | ||
Syntax | cif 3310 | Extend class notation to include the conditional operator. See df-if 3311 for a description. (In older databases this was denoted "ded".) |
class if(φ, A, B) | ||
Definition | df-if 3311* |
Define the conditional operator. Read if(φ, A,
B) as "if
φ then
A else B." See iftrue 3315 and iffalse 3317 for its
values. In mathematical literature, this operator is rarely defined
formally but is implicit in informal definitions such as "let
f(x)=0 if
x=0 and 1/x otherwise."
In the absence of excluded middle, this will tend to be useful where φ is decidable (in the sense of df-dc 734). (Contributed by NM, 15-May-1999.) |
⊢ if(φ, A, B) = {x ∣ ((x ∈ A ∧ φ) ∨ (x ∈ B ∧ ¬ φ))} | ||
Theorem | dfif6 3312* | An alternate definition of the conditional operator df-if 3311 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.) |
⊢ if(φ, A, B) = ({x ∈ A ∣ φ} ∪ {x ∈ B ∣ ¬ φ}) | ||
Theorem | ifeq1 3313 | Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ (A = B → if(φ, A, 𝐶) = if(φ, B, 𝐶)) | ||
Theorem | ifeq2 3314 | Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ (A = B → if(φ, 𝐶, A) = if(φ, 𝐶, B)) | ||
Theorem | iftrue 3315 | Value of the conditional operator when its first argument is true. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
⊢ (φ → if(φ, A, B) = A) | ||
Theorem | iftruei 3316 | Inference associated with iftrue 3315. (Contributed by BJ, 7-Oct-2018.) |
⊢ φ ⇒ ⊢ if(φ, A, B) = A | ||
Theorem | iffalse 3317 | Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.) |
⊢ (¬ φ → if(φ, A, B) = B) | ||
Theorem | iffalsei 3318 | Inference associated with iffalse 3317. (Contributed by BJ, 7-Oct-2018.) |
⊢ ¬ φ ⇒ ⊢ if(φ, A, B) = B | ||
Theorem | ifnefalse 3319 | When values are unequal, but an "if" condition checks if they are equal, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs vs. applying iffalse 3317 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.) |
⊢ (A ≠ B → if(A = B, 𝐶, 𝐷) = 𝐷) | ||
Theorem | dfif3 3320* | Alternate definition of the conditional operator df-if 3311. Note that φ is independent of x i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ 𝐶 = {x ∣ φ} ⇒ ⊢ if(φ, A, B) = ((A ∩ 𝐶) ∪ (B ∩ (V ∖ 𝐶))) | ||
Theorem | ifeq12 3321 | Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.) |
⊢ ((A = B ∧ 𝐶 = 𝐷) → if(φ, A, 𝐶) = if(φ, B, 𝐷)) | ||
Theorem | ifeq1d 3322 | Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.) |
⊢ (φ → A = B) ⇒ ⊢ (φ → if(ψ, A, 𝐶) = if(ψ, B, 𝐶)) | ||
Theorem | ifeq2d 3323 | Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.) |
⊢ (φ → A = B) ⇒ ⊢ (φ → if(ψ, 𝐶, A) = if(ψ, 𝐶, B)) | ||
Theorem | ifeq12d 3324 | Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.) |
⊢ (φ → A = B) & ⊢ (φ → 𝐶 = 𝐷) ⇒ ⊢ (φ → if(ψ, A, 𝐶) = if(ψ, B, 𝐷)) | ||
Theorem | ifbi 3325 | Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.) |
⊢ ((φ ↔ ψ) → if(φ, A, B) = if(ψ, A, B)) | ||
Theorem | ifbid 3326 | Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.) |
⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → if(ψ, A, B) = if(χ, A, B)) | ||
Theorem | ifbieq1d 3327 | Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018.) |
⊢ (φ → (ψ ↔ χ)) & ⊢ (φ → A = B) ⇒ ⊢ (φ → if(ψ, A, 𝐶) = if(χ, B, 𝐶)) | ||
Theorem | ifbieq2i 3328 | Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ (φ ↔ ψ) & ⊢ A = B ⇒ ⊢ if(φ, 𝐶, A) = if(ψ, 𝐶, B) | ||
Theorem | ifbieq2d 3329 | 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 3330 | 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 3331 | 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 3332 | Deduction version of nfif 3333. (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 3333 | 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) | ||
Syntax | cpw 3334 | 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 3335* | Soundness justification theorem for df-pw 3336. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ {x ∣ x ⊆ A} = {y ∣ y ⊆ A} | ||
Definition | df-pw 3336* | 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 3337 | Equality theorem for power class. (Contributed by NM, 5-Aug-1993.) |
⊢ (A = B → 𝒫 A = 𝒫 B) | ||
Theorem | pweqi 3338 | Equality inference for power class. (Contributed by NM, 27-Nov-2013.) |
⊢ A = B ⇒ ⊢ 𝒫 A = 𝒫 B | ||
Theorem | pweqd 3339 | Equality deduction for power class. (Contributed by NM, 27-Nov-2013.) |
⊢ (φ → A = B) ⇒ ⊢ (φ → 𝒫 A = 𝒫 B) | ||
Theorem | elpw 3340 | 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 3341* | Setvar variable membership in a power class (common case). See elpw 3340. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (x ∈ 𝒫 A ↔ x ⊆ A) | ||
Theorem | elpwg 3342 | 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 3343 | Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.) |
⊢ (A ∈ 𝒫 B → A ⊆ B) | ||
Theorem | elpwid 3344 | An element of a power class is a subclass. Deduction form of elpwi 3343. (Contributed by David Moews, 1-May-2017.) |
⊢ (φ → A ∈ 𝒫 B) ⇒ ⊢ (φ → A ⊆ B) | ||
Theorem | elelpwi 3345 | 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 3346 | 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 3347 | Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ (A ∈ 𝑉 → A ∈ 𝒫 A) | ||
Theorem | pwid 3348 | 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 3349* | Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.) |
⊢ (𝒫 A ⊆ B ↔ ∀x(x ⊆ A → x ∈ B)) | ||
Syntax | csn 3350 | Extend class notation to include singleton. |
class {A} | ||
Syntax | cpr 3351 | Extend class notation to include unordered pair. |
class {A, B} | ||
Syntax | ctp 3352 | Extend class notation to include unordered triplet. |
class {A, B, 𝐶} | ||
Syntax | cop 3353 | Extend class notation to include ordered pair. |
class ⟨A, B⟩ | ||
Syntax | cotp 3354 | Extend class notation to include ordered triple. |
class ⟨A, B, 𝐶⟩ | ||
Theorem | snjust 3355* | Soundness justification theorem for df-sn 3356. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ {x ∣ x = A} = {y ∣ y = A} | ||
Definition | df-sn 3356* | 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 3364. (Contributed by NM, 5-Aug-1993.) |
⊢ {A} = {x ∣ x = A} | ||
Definition | df-pr 3357 | Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. They are unordered, so {A, B} = {B, A} as proven by prcom 3420. For a more traditional definition, but requiring a dummy variable, see dfpr2 3366. (Contributed by NM, 5-Aug-1993.) |
⊢ {A, B} = ({A} ∪ {B}) | ||
Definition | df-tp 3358 | 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 3359* |
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 3545 and opprc2 3546). For
Kuratowski's actual definition when the arguments are sets, see dfop 3522.
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 3359 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3359 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 3360 | 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 3361 | Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) |
⊢ (A = B → {A} = {B}) | ||
Theorem | sneqi 3362 | Equality inference for singletons. (Contributed by NM, 22-Jan-2004.) |
⊢ A = B ⇒ ⊢ {A} = {B} | ||
Theorem | sneqd 3363 | Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.) |
⊢ (φ → A = B) ⇒ ⊢ (φ → {A} = {B}) | ||
Theorem | dfsn2 3364 | Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) |
⊢ {A} = {A, A} | ||
Theorem | elsn 3365* | There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) |
⊢ (x ∈ {A} ↔ x = A) | ||
Theorem | dfpr2 3366* | 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 3367 | 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 3368 | 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 3369 | 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 3370 | 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 3371 | 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 | elsncg 3372 | There is only 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 | elsnc 3373 | There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.) |
⊢ A ∈ V ⇒ ⊢ (A ∈ {B} ↔ A = B) | ||
Theorem | elsni 3374 | There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.) |
⊢ (A ∈ {B} → A = B) | ||
Theorem | snidg 3375 | 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 3376 | 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 3377 | 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 | ssnid 3378 | A setvar variable is a member of its singleton (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ x ∈ {x} | ||
Theorem | elsnc2g 3379 | 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 | elsnc2 3380 | 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 3381* | 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 3460 and snprc 3409. (Contributed by Jim Kingdon, 30-Aug-2018.) |
⊢ ∃*x x ∈ {A} | ||
Theorem | ralsns 3382* | 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 3383* | 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 | rexsnsOLD 3384* | Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) Obsolete as of 22-Aug-2018. Use rexsns 3383 instead. (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (A ∈ 𝑉 → (∃x ∈ {A}φ ↔ [A / x]φ)) | ||
Theorem | ralsng 3385* | 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 3386* | Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.) |
⊢ (x = A → (φ ↔ ψ)) ⇒ ⊢ (A ∈ 𝑉 → (∃x ∈ {A}φ ↔ ψ)) | ||
Theorem | exsnrex 3387 | 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 3388* | 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 3389* | 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 3390 | 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 3391 | 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 3392 | 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 3393* | 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 3394 | Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.) |
⊢ ℲxA & ⊢ ℲxB ⇒ ⊢ Ⅎx{A, B} | ||
Theorem | ralprg 3395* | 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 3396* | 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 3397* | 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 3398* | 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 3399* | 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 3400* | 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)) |
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