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Theorem List for Intuitionistic Logic Explorer - 3501-3600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoreminundifss 3501 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 3502 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 3503 Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
((𝐴𝐵) ∖ 𝐵) = (𝐴𝐵)
 
Theoremundifss 3504 Union of complementary parts into whole. (Contributed by Jim Kingdon, 4-Aug-2018.)
(𝐴𝐵 ↔ (𝐴 ∪ (𝐵𝐴)) ⊆ 𝐵)
 
Theoremssdifin0 3505 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 3506 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 3507 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 3508 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 3509 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 3510* Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1638). 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 3511* Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1638). 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 3510 as of 7-Apr-2023. (Proof modification is discouraged.) (New usage is discouraged.)
((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝜑) → ∃𝑥𝐴 𝜑)
 
Theoremr19.3rm 3512* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 19-Dec-2018.)
𝑥𝜑       (∃𝑦 𝑦𝐴 → (𝜑 ↔ ∀𝑥𝐴 𝜑))
 
Theoremr19.28m 3513* 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 3514* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 6-Aug-2018.)
(∃𝑦 𝑦𝐴 → (𝜑 ↔ ∀𝑥𝐴 𝜑))
 
Theoremr19.9rmv 3515* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 5-Aug-2018.)
(∃𝑦 𝑦𝐴 → (𝜑 ↔ ∃𝑥𝐴 𝜑))
 
Theoremr19.28mv 3516* 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 3517* 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 3518* Restricted version of Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
(∃𝑦 𝑦𝐴 → (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓)))
 
Theoremr19.27m 3519* 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 3520* 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 3521* Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴 = ∅ → ∀𝑥𝐴 𝜑)
 
Theoremrexn0 3522* Restricted existential quantification implies its restriction is nonempty (it is also inhabited as shown in rexm 3523). (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
(∃𝑥𝐴 𝜑𝐴 ≠ ∅)
 
Theoremrexm 3523* Restricted existential quantification implies its restriction is inhabited. (Contributed by Jim Kingdon, 16-Oct-2018.)
(∃𝑥𝐴 𝜑 → ∃𝑥 𝑥𝐴)
 
Theoremralidm 3524* Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.)
(∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑)
 
Theoremral0 3525 Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.)
𝑥 ∈ ∅ 𝜑
 
Theoremrgenm 3526* Generalization rule that eliminates an inhabited class requirement. (Contributed by Jim Kingdon, 5-Aug-2018.)
((∃𝑥 𝑥𝐴𝑥𝐴) → 𝜑)       𝑥𝐴 𝜑
 
Theoremralf0 3527* The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.)
¬ 𝜑       (∀𝑥𝐴 𝜑𝐴 = ∅)
 
Theoremralm 3528 Inhabited classes and restricted quantification. (Contributed by Jim Kingdon, 6-Aug-2018.)
((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) ↔ ∀𝑥𝐴 𝜑)
 
Theoremraaanlem 3529* Special case of raaan 3530 where 𝐴 is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.)
𝑦𝜑    &   𝑥𝜓       (∃𝑥 𝑥𝐴 → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
 
Theoremraaan 3530* Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.)
𝑦𝜑    &   𝑥𝜓       (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))
 
Theoremraaanv 3531* Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)
(∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))
 
Theoremsbss 3532* 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 3533 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 3534 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 3535 Extend class notation to include the conditional operator. See df-if 3536 for a description. (In older databases this was denoted "ded".)
class if(𝜑, 𝐴, 𝐵)
 
Definitiondf-if 3536* Define the conditional operator. Read if(𝜑, 𝐴, 𝐵) as "if 𝜑 then 𝐴 else 𝐵". See iftrue 3540 and iffalse 3543 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 835). (Contributed by NM, 15-May-1999.)

if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
 
Theoremdfif6 3537* An alternate definition of the conditional operator df-if 3536 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
if(𝜑, 𝐴, 𝐵) = ({𝑥𝐴𝜑} ∪ {𝑥𝐵 ∣ ¬ 𝜑})
 
Theoremifeq1 3538 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 3539 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 3540 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 3541 Inference associated with iftrue 3540. (Contributed by BJ, 7-Oct-2018.)
𝜑       if(𝜑, 𝐴, 𝐵) = 𝐴
 
Theoremiftrued 3542 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 3543 Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.)
𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
 
Theoremiffalsei 3544 Inference associated with iffalse 3543. (Contributed by BJ, 7-Oct-2018.)
¬ 𝜑       if(𝜑, 𝐴, 𝐵) = 𝐵
 
Theoremiffalsed 3545 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 3546 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 3543 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)
(𝐴𝐵 → if(𝐴 = 𝐵, 𝐶, 𝐷) = 𝐷)
 
Theoremifsbdc 3547 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 3548* Alternate definition of the conditional operator df-if 3536. 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)))
 
Theoremifssun 3549 A conditional class is included in the union of its two alternatives. (Contributed by BJ, 15-Aug-2024.)
if(𝜑, 𝐴, 𝐵) ⊆ (𝐴𝐵)
 
Theoremifidss 3550 A conditional class whose two alternatives are equal is included in that alternative. With excluded middle, we can prove it is equal to it. (Contributed by BJ, 15-Aug-2024.)
if(𝜑, 𝐴, 𝐴) ⊆ 𝐴
 
Theoremifeq12 3551 Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.)
((𝐴 = 𝐵𝐶 = 𝐷) → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐷))
 
Theoremifeq1d 3552 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
(𝜑𝐴 = 𝐵)       (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
 
Theoremifeq2d 3553 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
(𝜑𝐴 = 𝐵)       (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
 
Theoremifeq12d 3554 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 3555 Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
((𝜑𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
 
Theoremifbid 3556 Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.)
(𝜑 → (𝜓𝜒))       (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐴, 𝐵))
 
Theoremifbieq1d 3557 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 3558 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 3559 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 3560 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 3561 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 3562 Deduction version of nfif 3563. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑𝑥if(𝜓, 𝐴, 𝐵))
 
Theoremnfif 3563 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 3564 Conditional closure. (Contributed by Jim Kingdon, 11-Jan-2022.)
((𝜑𝜓) → 𝐴𝐶)    &   ((𝜑 ∧ ¬ 𝜓) → 𝐵𝐶)    &   (𝜑DECID 𝜓)       (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)
 
Theoremifeq1dadc 3565 Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝜑𝜓) → 𝐴 = 𝐵)    &   (𝜑DECID 𝜓)       (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
 
Theoremifeq2dadc 3566 Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵)    &   (𝜑DECID 𝜓)       (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
 
Theoremifbothdadc 3567 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 3568 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 3569 Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.)
(DECID 𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴)
 
Theoremeqifdc 3570 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 3571 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)
 
Theoremifnotdc 3572 Negating the first argument swaps the last two arguments of a conditional operator. (Contributed by NM, 21-Jun-2007.)
(DECID 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴))
 
Theoremifandc 3573 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))
 
Theoremifordc 3574 Rewrite a disjunction in a conditional as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)
(DECID 𝜑 → if((𝜑𝜓), 𝐴, 𝐵) = if(𝜑, 𝐴, if(𝜓, 𝐴, 𝐵)))
 
Theoremifmdc 3575 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 3576 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 3577* Soundness justification theorem for df-pw 3578. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
{𝑥𝑥𝐴} = {𝑦𝑦𝐴}
 
Definitiondf-pw 3578* 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 3579 Equality theorem for power class. (Contributed by NM, 5-Aug-1993.)
(𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵)
 
Theorempweqi 3580 Equality inference for power class. (Contributed by NM, 27-Nov-2013.)
𝐴 = 𝐵       𝒫 𝐴 = 𝒫 𝐵
 
Theorempweqd 3581 Equality deduction for power class. (Contributed by NM, 27-Nov-2013.)
(𝜑𝐴 = 𝐵)       (𝜑 → 𝒫 𝐴 = 𝒫 𝐵)
 
Theoremelpw 3582 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.)
𝐴 ∈ V       (𝐴 ∈ 𝒫 𝐵𝐴𝐵)
 
Theoremvelpw 3583* Setvar variable membership in a power class (common case). See elpw 3582. (Contributed by David A. Wheeler, 8-Dec-2018.)
(𝑥 ∈ 𝒫 𝐴𝑥𝐴)
 
Theoremelpwg 3584 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 6-Aug-2000.)
(𝐴𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
 
Theoremelpwi 3585 Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)
(𝐴 ∈ 𝒫 𝐵𝐴𝐵)
 
Theoremelpwb 3586 Characterization of the elements of a power class. (Contributed by BJ, 29-Apr-2021.)
(𝐴 ∈ 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴𝐵))
 
Theoremelpwid 3587 An element of a power class is a subclass. Deduction form of elpwi 3585. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ 𝒫 𝐵)       (𝜑𝐴𝐵)
 
Theoremelelpwi 3588 If 𝐴 belongs to a part of 𝐶 then 𝐴 belongs to 𝐶. (Contributed by FL, 3-Aug-2009.)
((𝐴𝐵𝐵 ∈ 𝒫 𝐶) → 𝐴𝐶)
 
Theoremnfpw 3589 Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
𝑥𝐴       𝑥𝒫 𝐴
 
Theorempwidg 3590 Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐴𝑉𝐴 ∈ 𝒫 𝐴)
 
Theorempwid 3591 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 3592* Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)
(𝒫 𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 
2.1.17  Unordered and ordered pairs
 
Syntaxcsn 3593 Extend class notation to include singleton.
class {𝐴}
 
Syntaxcpr 3594 Extend class notation to include unordered pair.
class {𝐴, 𝐵}
 
Syntaxctp 3595 Extend class notation to include unordered triplet.
class {𝐴, 𝐵, 𝐶}
 
Syntaxcop 3596 Extend class notation to include ordered pair.
class 𝐴, 𝐵
 
Syntaxcotp 3597 Extend class notation to include ordered triple.
class 𝐴, 𝐵, 𝐶
 
Theoremsnjust 3598* Soundness justification theorem for df-sn 3599. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
{𝑥𝑥 = 𝐴} = {𝑦𝑦 = 𝐴}
 
Definitiondf-sn 3599* 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 3607. (Contributed by NM, 5-Aug-1993.)
{𝐴} = {𝑥𝑥 = 𝐴}
 
Definitiondf-pr 3600 Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. They are unordered, so {𝐴, 𝐵} = {𝐵, 𝐴} as proven by prcom 3669. For a more traditional definition, but requiring a dummy variable, see dfpr2 3612. (Contributed by NM, 5-Aug-1993.)
{𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
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