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Theorem List for Intuitionistic Logic Explorer - 3401-3500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremundif1ss 3401 Absorption of difference by union. In classical logic, as Theorem 35 of [Suppes] p. 29, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
((𝐴𝐵) ∪ 𝐵) ⊆ (𝐴𝐵)

Theoremundif2ss 3402 Absorption of difference by union. In classical logic, as in Part of proof of Corollary 6K of [Enderton] p. 144, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
(𝐴 ∪ (𝐵𝐴)) ⊆ (𝐴𝐵)

Theoremundifabs 3403 Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)
(𝐴 ∪ (𝐴𝐵)) = 𝐴

Theoreminundifss 3404 The intersection and class difference of a class with another class are contained in the original class. In classical logic we'd be able to make a stronger statement: that everything in the original class is in the intersection or the difference (that is, this theorem would be equality rather than subset). (Contributed by Jim Kingdon, 4-Aug-2018.)
((𝐴𝐵) ∪ (𝐴𝐵)) ⊆ 𝐴

Theoremdisjdif2 3405 The difference of a class and a class disjoint from it is the original class. (Contributed by BJ, 21-Apr-2019.)
((𝐴𝐵) = ∅ → (𝐴𝐵) = 𝐴)

Theoremdifun2 3406 Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
((𝐴𝐵) ∖ 𝐵) = (𝐴𝐵)

Theoremundifss 3407 Union of complementary parts into whole. (Contributed by Jim Kingdon, 4-Aug-2018.)
(𝐴𝐵 ↔ (𝐴 ∪ (𝐵𝐴)) ⊆ 𝐵)

Theoremssdifin0 3408 A subset of a difference does not intersect the subtrahend. (Contributed by Jeff Hankins, 1-Sep-2013.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)
(𝐴 ⊆ (𝐵𝐶) → (𝐴𝐶) = ∅)

Theoremssdifeq0 3409 A class is a subclass of itself subtracted from another iff it is the empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.)
(𝐴 ⊆ (𝐵𝐴) ↔ 𝐴 = ∅)

Theoremssundifim 3410 A consequence of inclusion in the union of two classes. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.)
(𝐴 ⊆ (𝐵𝐶) → (𝐴𝐵) ⊆ 𝐶)

Theoremdifdifdirss 3411 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 3412 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 3413* Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1598). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) (Revised by Jim Kingdon, 7-Apr-2023.)
((∃𝑦 𝑦𝐴 ∧ ∀𝑥𝐴 𝜑) → ∃𝑥𝐴 𝜑)

Theoremr19.2mOLD 3414* Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1598). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) Obsolete version of r19.2m 3413 as of 7-Apr-2023. (Proof modification is discouraged.) (New usage is discouraged.)
((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝜑) → ∃𝑥𝐴 𝜑)

Theoremr19.3rm 3415* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 19-Dec-2018.)
𝑥𝜑       (∃𝑦 𝑦𝐴 → (𝜑 ↔ ∀𝑥𝐴 𝜑))

Theoremr19.28m 3416* 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 𝑥𝐴 → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝐴 𝜓)))

Theoremr19.3rmv 3417* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 6-Aug-2018.)
(∃𝑦 𝑦𝐴 → (𝜑 ↔ ∀𝑥𝐴 𝜑))

Theoremr19.9rmv 3418* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 5-Aug-2018.)
(∃𝑦 𝑦𝐴 → (𝜑 ↔ ∃𝑥𝐴 𝜑))

Theoremr19.28mv 3419* 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 𝑥𝐴 → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝐴 𝜓)))

Theoremr19.45mv 3420* Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
(∃𝑥 𝑥𝐴 → (∃𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝐴 𝜓)))

Theoremr19.44mv 3421* Restricted version of Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
(∃𝑦 𝑦𝐴 → (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓)))

Theoremr19.27m 3422* 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 𝑥𝐴 → (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑𝜓)))

Theoremr19.27mv 3423* 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 𝑥𝐴 → (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑𝜓)))

Theoremrzal 3424* Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴 = ∅ → ∀𝑥𝐴 𝜑)

Theoremrexn0 3425* Restricted existential quantification implies its restriction is nonempty (it is also inhabited as shown in rexm 3426). (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
(∃𝑥𝐴 𝜑𝐴 ≠ ∅)

Theoremrexm 3426* Restricted existential quantification implies its restriction is inhabited. (Contributed by Jim Kingdon, 16-Oct-2018.)
(∃𝑥𝐴 𝜑 → ∃𝑥 𝑥𝐴)

Theoremralidm 3427* Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.)
(∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑)

Theoremral0 3428 Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.)
𝑥 ∈ ∅ 𝜑

Theoremrgenm 3429* Generalization rule that eliminates an inhabited class requirement. (Contributed by Jim Kingdon, 5-Aug-2018.)
((∃𝑥 𝑥𝐴𝑥𝐴) → 𝜑)       𝑥𝐴 𝜑

Theoremralf0 3430* The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.)
¬ 𝜑       (∀𝑥𝐴 𝜑𝐴 = ∅)

Theoremralm 3431 Inhabited classes and restricted quantification. (Contributed by Jim Kingdon, 6-Aug-2018.)
((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) ↔ ∀𝑥𝐴 𝜑)

Theoremraaanlem 3432* Special case of raaan 3433 where 𝐴 is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.)
𝑦𝜑    &   𝑥𝜓       (∃𝑥 𝑥𝐴 → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))

Theoremraaan 3433* Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.)
𝑦𝜑    &   𝑥𝜓       (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))

Theoremraaanv 3434* Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)
(∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))

Theoremsbss 3435* 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 3436 Distribute proper substitution through a subclass relation. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Alexander van der Vekens, 23-Jul-2017.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))

Theoremdcun 3437 The union of two decidable classes is decidable. (Contributed by Jim Kingdon, 5-Oct-2022.)
(𝜑DECID 𝑘𝐴)    &   (𝜑DECID 𝑘𝐵)       (𝜑DECID 𝑘 ∈ (𝐴𝐵))

2.1.15  Conditional operator

Syntaxcif 3438 Extend class notation to include the conditional operator. See df-if 3439 for a description. (In older databases this was denoted "ded".)
class if(𝜑, 𝐴, 𝐵)

Definitiondf-if 3439* Define the conditional operator. Read if(𝜑, 𝐴, 𝐵) as "if 𝜑 then 𝐴 else 𝐵." See iftrue 3443 and iffalse 3446 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 803). (Contributed by NM, 15-May-1999.)

if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}

Theoremdfif6 3440* An alternate definition of the conditional operator df-if 3439 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
if(𝜑, 𝐴, 𝐵) = ({𝑥𝐴𝜑} ∪ {𝑥𝐵 ∣ ¬ 𝜑})

Theoremifeq1 3441 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 3442 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 3443 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 3444 Inference associated with iftrue 3443. (Contributed by BJ, 7-Oct-2018.)
𝜑       if(𝜑, 𝐴, 𝐵) = 𝐴

Theoremiftrued 3445 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 3446 Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.)
𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)

Theoremiffalsei 3447 Inference associated with iffalse 3446. (Contributed by BJ, 7-Oct-2018.)
¬ 𝜑       if(𝜑, 𝐴, 𝐵) = 𝐵

Theoremiffalsed 3448 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 3449 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 versus applying iffalse 3446 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)
(𝐴𝐵 → if(𝐴 = 𝐵, 𝐶, 𝐷) = 𝐷)

Theoremifsbdc 3450 Distribute a function over an if-clause. (Contributed by Jim Kingdon, 1-Jan-2022.)
(if(𝜑, 𝐴, 𝐵) = 𝐴𝐶 = 𝐷)    &   (if(𝜑, 𝐴, 𝐵) = 𝐵𝐶 = 𝐸)       (DECID 𝜑𝐶 = if(𝜑, 𝐷, 𝐸))

Theoremdfif3 3451* Alternate definition of the conditional operator df-if 3439. 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 3452 Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.)
((𝐴 = 𝐵𝐶 = 𝐷) → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐷))

Theoremifeq1d 3453 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
(𝜑𝐴 = 𝐵)       (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))

Theoremifeq2d 3454 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
(𝜑𝐴 = 𝐵)       (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))

Theoremifeq12d 3455 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 3456 Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
((𝜑𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))

Theoremifbid 3457 Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.)
(𝜑 → (𝜓𝜒))       (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐴, 𝐵))

Theoremifbieq1d 3458 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 3459 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 3460 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 3461 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 3462 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 3463 Deduction version of nfif 3464. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑𝑥if(𝜓, 𝐴, 𝐵))

Theoremnfif 3464 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)

Theoremifcldadc 3465 Conditional closure. (Contributed by Jim Kingdon, 11-Jan-2022.)
((𝜑𝜓) → 𝐴𝐶)    &   ((𝜑 ∧ ¬ 𝜓) → 𝐵𝐶)    &   (𝜑DECID 𝜓)       (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)

Theoremifeq1dadc 3466 Conditional equality. (Contributed by Jim Kingdon, 1-Jan-2022.)
((𝜑𝜓) → 𝐴 = 𝐵)    &   (𝜑DECID 𝜓)       (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))

Theoremifbothdadc 3467 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) → (𝜓𝜃))    &   (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜃))    &   ((𝜂𝜑) → 𝜓)    &   ((𝜂 ∧ ¬ 𝜑) → 𝜒)    &   (𝜂DECID 𝜑)       (𝜂𝜃)

Theoremifbothdc 3468 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) → (𝜓𝜃))    &   (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜃))       ((𝜓𝜒DECID 𝜑) → 𝜃)

Theoremifiddc 3469 Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.)
(DECID 𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴)

Theoremeqifdc 3470 Expansion of an equality with a conditional operator. (Contributed by Jim Kingdon, 28-Jul-2022.)
(DECID 𝜑 → (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑𝐴 = 𝐵) ∨ (¬ 𝜑𝐴 = 𝐶))))

Theoremifcldcd 3471 Membership (closure) of a conditional operator, deduction form. (Contributed by Jim Kingdon, 8-Aug-2021.)
(𝜑𝐴𝐶)    &   (𝜑𝐵𝐶)    &   (𝜑DECID 𝜓)       (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)

Theoremifandc 3472 Rewrite a conjunction in a conditional as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)
(DECID 𝜑 → if((𝜑𝜓), 𝐴, 𝐵) = if(𝜑, if(𝜓, 𝐴, 𝐵), 𝐵))

Theoremifmdc 3473 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)

2.1.16  Power classes

Syntaxcpw 3474 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 3475* Soundness justification theorem for df-pw 3476. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
{𝑥𝑥𝐴} = {𝑦𝑦𝐴}

Definitiondf-pw 3476* 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 3477 Equality theorem for power class. (Contributed by NM, 5-Aug-1993.)
(𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵)

Theorempweqi 3478 Equality inference for power class. (Contributed by NM, 27-Nov-2013.)
𝐴 = 𝐵       𝒫 𝐴 = 𝒫 𝐵

Theorempweqd 3479 Equality deduction for power class. (Contributed by NM, 27-Nov-2013.)
(𝜑𝐴 = 𝐵)       (𝜑 → 𝒫 𝐴 = 𝒫 𝐵)

Theoremelpw 3480 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.)
𝐴 ∈ V       (𝐴 ∈ 𝒫 𝐵𝐴𝐵)

Theoremselpw 3481* Setvar variable membership in a power class (common case). See elpw 3480. (Contributed by David A. Wheeler, 8-Dec-2018.)
(𝑥 ∈ 𝒫 𝐴𝑥𝐴)

Theoremelpwg 3482 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 6-Aug-2000.)
(𝐴𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))

Theoremelpwi 3483 Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)
(𝐴 ∈ 𝒫 𝐵𝐴𝐵)

Theoremelpwb 3484 Characterization of the elements of a power class. (Contributed by BJ, 29-Apr-2021.)
(𝐴 ∈ 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴𝐵))

Theoremelpwid 3485 An element of a power class is a subclass. Deduction form of elpwi 3483. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ 𝒫 𝐵)       (𝜑𝐴𝐵)

Theoremelelpwi 3486 If 𝐴 belongs to a part of 𝐶 then 𝐴 belongs to 𝐶. (Contributed by FL, 3-Aug-2009.)
((𝐴𝐵𝐵 ∈ 𝒫 𝐶) → 𝐴𝐶)

Theoremnfpw 3487 Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
𝑥𝐴       𝑥𝒫 𝐴

Theorempwidg 3488 Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐴𝑉𝐴 ∈ 𝒫 𝐴)

Theorempwid 3489 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 3490* Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)
(𝒫 𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))

2.1.17  Unordered and ordered pairs

Syntaxcsn 3491 Extend class notation to include singleton.
class {𝐴}

Syntaxcpr 3492 Extend class notation to include unordered pair.
class {𝐴, 𝐵}

Syntaxctp 3493 Extend class notation to include unordered triplet.
class {𝐴, 𝐵, 𝐶}

Syntaxcop 3494 Extend class notation to include ordered pair.
class 𝐴, 𝐵

Syntaxcotp 3495 Extend class notation to include ordered triple.
class 𝐴, 𝐵, 𝐶

Theoremsnjust 3496* Soundness justification theorem for df-sn 3497. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
{𝑥𝑥 = 𝐴} = {𝑦𝑦 = 𝐴}

Definitiondf-sn 3497* 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 3505. (Contributed by NM, 5-Aug-1993.)
{𝐴} = {𝑥𝑥 = 𝐴}

Definitiondf-pr 3498 Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. They are unordered, so {𝐴, 𝐵} = {𝐵, 𝐴} as proven by prcom 3563. For a more traditional definition, but requiring a dummy variable, see dfpr2 3510. (Contributed by NM, 5-Aug-1993.)
{𝐴, 𝐵} = ({𝐴} ∪ {𝐵})

Definitiondf-tp 3499 Define unordered triple of classes. Definition of [Enderton] p. 19. (Contributed by NM, 9-Apr-1994.)
{𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})

Definitiondf-op 3500* 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 3691 and opprc2 3692). For Kuratowski's actual definition when the arguments are sets, see dfop 3668.

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 3500 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3500 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, 𝐵2 = {{{𝐴}, ∅}, {{𝐵}}}. This was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition is 𝐴, 𝐵3 = {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}})}

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