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Theorem List for Intuitionistic Logic Explorer - 3301-3400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdifdifdirss 3301 Distributive law for class difference. In classical logic, as in Exercise 4.8 of [Stoll] p. 16, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
((AB) ∖ 𝐶) ⊆ ((A𝐶) ∖ (B𝐶))

Theoremuneqdifeqim 3302 Two ways that A and B can "partition" 𝐶 (when A and B don't overlap and A is a part of 𝐶). In classical logic, the second implication would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.)
((A𝐶 (AB) = ∅) → ((AB) = 𝐶 → (𝐶A) = B))

Theoremr19.2m 3303* Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1526). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
((x x A x A φ) → x A φ)

Theoremr19.3rm 3304* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 19-Dec-2018.)
xφ       (y y A → (φx A φ))

Theoremr19.28m 3305* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
xφ       (x x A → (x A (φ ψ) ↔ (φ x A ψ)))

Theoremr19.3rmv 3306* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 6-Aug-2018.)
(y y A → (φx A φ))

Theoremr19.9rmv 3307* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 5-Aug-2018.)
(y y A → (φx A φ))

Theoremr19.28mv 3308* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.)
(x x A → (x A (φ ψ) ↔ (φ x A ψ)))

Theoremr19.45mv 3309* Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
(x x A → (x A (φ ψ) ↔ (φ x A ψ)))

Theoremr19.27m 3310* Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
xψ       (x x A → (x A (φ ψ) ↔ (x A φ ψ)))

Theoremr19.27mv 3311* Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
(x x A → (x A (φ ψ) ↔ (x A φ ψ)))

Theoremrzal 3312* Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(A = ∅ → x A φ)

Theoremrexn0 3313* Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
(x A φA ≠ ∅)

Theoremrexm 3314* Restricted existential quantification implies its restriction is inhabited. (Contributed by Jim Kingdon, 16-Oct-2018.)
(x A φx x A)

Theoremralidm 3315* Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.)
(x A x A φx A φ)

Theoremral0 3316 Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.)
x φ

Theoremrgenm 3317* Generalization rule that eliminates an inhabited class requirement. (Contributed by Jim Kingdon, 5-Aug-2018.)
((x x A x A) → φ)       x A φ

Theoremralf0 3318* The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.)
¬ φ       (x A φA = ∅)

Theoremralm 3319 Inhabited classes and restricted quantification. (Contributed by Jim Kingdon, 6-Aug-2018.)
((x x Ax A φ) ↔ x A φ)

Theoremraaanlem 3320* Special case of raaan 3321 where A is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.)
yφ    &   xψ       (x x A → (x A y A (φ ψ) ↔ (x A φ y A ψ)))

Theoremraaan 3321* Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.)
yφ    &   xψ       (x A y A (φ ψ) ↔ (x A φ y A ψ))

Theoremraaanv 3322* Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)
(x A y A (φ ψ) ↔ (x A φ y A ψ))

Theoremsbss 3323* 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]xAyA)

Theoremsbcssg 3324 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]B𝐶A / xBA / x𝐶))

2.1.15  Conditional operator

Syntaxcif 3325 Extend class notation to include the conditional operator. See df-if 3326 for a description. (In older databases this was denoted "ded".)
class if(φ, A, B)

Definitiondf-if 3326* Define the conditional operator. Read if(φ, A, B) as "if φ then A else B." See iftrue 3330 and iffalse 3333 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 742). (Contributed by NM, 15-May-1999.)

if(φ, A, B) = {x ∣ ((x A φ) (x B ¬ φ))}

Theoremdfif6 3327* An alternate definition of the conditional operator df-if 3326 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
if(φ, A, B) = ({x Aφ} ∪ {x B ∣ ¬ φ})

Theoremifeq1 3328 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))

Theoremifeq2 3329 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))

Theoremiftrue 3330 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)

Theoremiftruei 3331 Inference associated with iftrue 3330. (Contributed by BJ, 7-Oct-2018.)
φ       if(φ, A, B) = A

Theoremiftrued 3332 Value of the conditional operator when its first argument is true. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(φχ)       (φ → if(χ, A, B) = A)

Theoremiffalse 3333 Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.)
φ → if(φ, A, B) = B)

Theoremiffalsei 3334 Inference associated with iffalse 3333. (Contributed by BJ, 7-Oct-2018.)
¬ φ       if(φ, A, B) = B

Theoremiffalsed 3335 Value of the conditional operator when its first argument is false. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(φ → ¬ χ)       (φ → if(χ, A, B) = B)

Theoremifnefalse 3336 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 3333 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)
(AB → if(A = B, 𝐶, 𝐷) = 𝐷)

Theoremdfif3 3337* Alternate definition of the conditional operator df-if 3326. 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 = {xφ}       if(φ, A, B) = ((A𝐶) ∪ (B ∩ (V ∖ 𝐶)))

Theoremifeq12 3338 Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.)
((A = B 𝐶 = 𝐷) → if(φ, A, 𝐶) = if(φ, B, 𝐷))

Theoremifeq1d 3339 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
(φA = B)       (φ → if(ψ, A, 𝐶) = if(ψ, B, 𝐶))

Theoremifeq2d 3340 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
(φA = B)       (φ → if(ψ, 𝐶, A) = if(ψ, 𝐶, B))

Theoremifeq12d 3341 Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.)
(φA = B)    &   (φ𝐶 = 𝐷)       (φ → if(ψ, A, 𝐶) = if(ψ, B, 𝐷))

Theoremifbi 3342 Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
((φψ) → if(φ, A, B) = if(ψ, A, B))

Theoremifbid 3343 Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.)
(φ → (ψχ))       (φ → if(ψ, A, B) = if(χ, A, B))

Theoremifbieq1d 3344 Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018.)
(φ → (ψχ))    &   (φA = B)       (φ → if(ψ, A, 𝐶) = if(χ, B, 𝐶))

Theoremifbieq2i 3345 Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
(φψ)    &   A = B       if(φ, 𝐶, A) = if(ψ, 𝐶, B)

Theoremifbieq2d 3346 Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
(φ → (ψχ))    &   (φA = B)       (φ → if(ψ, 𝐶, A) = if(χ, 𝐶, B))

Theoremifbieq12i 3347 Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.)
(φψ)    &   A = 𝐶    &   B = 𝐷       if(φ, A, B) = if(ψ, 𝐶, 𝐷)

Theoremifbieq12d 3348 Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.)
(φ → (ψχ))    &   (φA = 𝐶)    &   (φB = 𝐷)       (φ → if(ψ, A, B) = if(χ, 𝐶, 𝐷))

Theoremnfifd 3349 Deduction version of nfif 3350. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.)
(φ → Ⅎxψ)    &   (φxA)    &   (φxB)       (φxif(ψ, A, B))

Theoremnfif 3350 Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
xφ    &   xA    &   xB       xif(φ, A, B)

2.1.16  Power classes

Syntaxcpw 3351 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

Theorempwjust 3352* Soundness justification theorem for df-pw 3353. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
{xxA} = {yyA}

Definitiondf-pw 3353* 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 = {xxA}

Theorempweq 3354 Equality theorem for power class. (Contributed by NM, 5-Aug-1993.)
(A = B → 𝒫 A = 𝒫 B)

Theorempweqi 3355 Equality inference for power class. (Contributed by NM, 27-Nov-2013.)
A = B       𝒫 A = 𝒫 B

Theorempweqd 3356 Equality deduction for power class. (Contributed by NM, 27-Nov-2013.)
(φA = B)       (φ → 𝒫 A = 𝒫 B)

Theoremelpw 3357 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.)
A V       (A 𝒫 BAB)

Theoremselpw 3358* Setvar variable membership in a power class (common case). See elpw 3357. (Contributed by David A. Wheeler, 8-Dec-2018.)
(x 𝒫 AxA)

Theoremelpwg 3359 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 6-Aug-2000.)
(A 𝑉 → (A 𝒫 BAB))

Theoremelpwi 3360 Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)
(A 𝒫 BAB)

Theoremelpwid 3361 An element of a power class is a subclass. Deduction form of elpwi 3360. (Contributed by David Moews, 1-May-2017.)
(φA 𝒫 B)       (φAB)

Theoremelelpwi 3362 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)

Theoremnfpw 3363 Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
xA       x𝒫 A

Theorempwidg 3364 Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(A 𝑉A 𝒫 A)

Theorempwid 3365 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

Theorempwss 3366* Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)
(𝒫 ABx(xAx B))

2.1.17  Unordered and ordered pairs

Syntaxcsn 3367 Extend class notation to include singleton.
class {A}

Syntaxcpr 3368 Extend class notation to include unordered pair.
class {A, B}

Syntaxctp 3369 Extend class notation to include unordered triplet.
class {A, B, 𝐶}

Syntaxcop 3370 Extend class notation to include ordered pair.
class A, B

Syntaxcotp 3371 Extend class notation to include ordered triple.
class A, B, 𝐶

Theoremsnjust 3372* Soundness justification theorem for df-sn 3373. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
{xx = A} = {yy = A}

Definitiondf-sn 3373* 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 3381. (Contributed by NM, 5-Aug-1993.)
{A} = {xx = A}

Definitiondf-pr 3374 Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. They are unordered, so {A, B} = {B, A} as proven by prcom 3437. For a more traditional definition, but requiring a dummy variable, see dfpr2 3383. (Contributed by NM, 5-Aug-1993.)
{A, B} = ({A} ∪ {B})

Definitiondf-tp 3375 Define unordered triple of classes. Definition of [Enderton] p. 19. (Contributed by NM, 9-Apr-1994.)
{A, B, 𝐶} = ({A, B} ∪ {𝐶})

Definitiondf-op 3376* 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 3562 and opprc2 3563). For Kuratowski's actual definition when the arguments are sets, see dfop 3539.

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 3376 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3376 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}})}

Definitiondf-ot 3377 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

Theoremsneq 3378 Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.)
(A = B → {A} = {B})

Theoremsneqi 3379 Equality inference for singletons. (Contributed by NM, 22-Jan-2004.)
A = B       {A} = {B}

Theoremsneqd 3380 Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.)
(φA = B)       (φ → {A} = {B})

Theoremdfsn2 3381 Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
{A} = {A, A}

Theoremelsn 3382* There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.)
(x {A} ↔ x = A)

Theoremdfpr2 3383* 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)}

Theoremelprg 3384 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)))

Theoremelpr 3385 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))

Theoremelpr2 3386 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))

Theoremelpri 3387 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))

Theoremnelpri 3388 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.)
AB    &   A𝐶        ¬ A {B, 𝐶}

Theoremelsncg 3389 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))

Theoremelsnc 3390 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)

Theoremelsni 3391 There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.)
(A {B} → A = B)

Theoremsnidg 3392 A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 28-Oct-2003.)
(A 𝑉A {A})

Theoremsnidb 3393 A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.)
(A V ↔ A {A})

Theoremsnid 3394 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}

Theoremssnid 3395 A setvar variable is a member of its singleton (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
x {x}

Theoremelsnc2g 3396 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))

Theoremelsnc2 3397 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)

Theoremmosn 3398* 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 3477 and snprc 3426. (Contributed by Jim Kingdon, 30-Aug-2018.)
∃*x x {A}

Theoremralsns 3399* Substitution expressed in terms of quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)
(A 𝑉 → (x {A}φ[A / x]φ))

Theoremrexsns 3400* 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)

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