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Theorem List for Metamath Proof Explorer - 4501-4600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremifeqda 4501 Separation of the values of the conditional operator. (Contributed by Alexander van der Vekens, 13-Apr-2018.)
((𝜑𝜓) → 𝐴 = 𝐶)    &   ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐶)       (𝜑 → if(𝜓, 𝐴, 𝐵) = 𝐶)
 
Theoremelimif 4502 Elimination of a conditional operator contained in a wff 𝜓. (Contributed by NM, 15-Feb-2005.) (Proof shortened by NM, 25-Apr-2019.)
(if(𝜑, 𝐴, 𝐵) = 𝐴 → (𝜓𝜒))    &   (if(𝜑, 𝐴, 𝐵) = 𝐵 → (𝜓𝜃))       (𝜓 ↔ ((𝜑𝜒) ∨ (¬ 𝜑𝜃)))
 
Theoremifbothda 4503 A wff 𝜃 containing a conditional operator is true when both of its cases are true. (Contributed by NM, 15-Feb-2015.)
(𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜃))    &   (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜃))    &   ((𝜂𝜑) → 𝜓)    &   ((𝜂 ∧ ¬ 𝜑) → 𝜒)       (𝜂𝜃)
 
Theoremifboth 4504 A wff 𝜃 containing a conditional operator is true when both of its cases are true. (Contributed by NM, 3-Sep-2006.) (Revised by Mario Carneiro, 15-Feb-2015.)
(𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜃))    &   (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜃))       ((𝜓𝜒) → 𝜃)
 
Theoremifid 4505 Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.)
if(𝜑, 𝐴, 𝐴) = 𝐴
 
Theoremeqif 4506 Expansion of an equality with a conditional operator. (Contributed by NM, 14-Feb-2005.)
(𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑𝐴 = 𝐵) ∨ (¬ 𝜑𝐴 = 𝐶)))
 
Theoremifval 4507 Another expression of the value of the if predicate, analogous to eqif 4506. See also the more specialized iftrue 4471 and iffalse 4474. (Contributed by BJ, 6-Apr-2019.)
(𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑𝐴 = 𝐵) ∧ (¬ 𝜑𝐴 = 𝐶)))
 
Theoremelif 4508 Membership in a conditional operator. (Contributed by NM, 14-Feb-2005.)
(𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ ((𝜑𝐴𝐵) ∨ (¬ 𝜑𝐴𝐶)))
 
Theoremifel 4509 Membership of a conditional operator. (Contributed by NM, 10-Sep-2005.)
(if(𝜑, 𝐴, 𝐵) ∈ 𝐶 ↔ ((𝜑𝐴𝐶) ∨ (¬ 𝜑𝐵𝐶)))
 
Theoremifcl 4510 Membership (closure) of a conditional operator. (Contributed by NM, 4-Apr-2005.)
((𝐴𝐶𝐵𝐶) → if(𝜑, 𝐴, 𝐵) ∈ 𝐶)
 
Theoremifcld 4511 Membership (closure) of a conditional operator, deduction form. (Contributed by SO, 16-Jul-2018.)
(𝜑𝐴𝐶)    &   (𝜑𝐵𝐶)       (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)
 
Theoremifcli 4512 Inference associated with ifcl 4510. Membership (closure) of a conditional operator. Also usable to keep a membership hypothesis for the weak deduction theorem dedth 4523 when the special case 𝐵𝐶 is provable. (Contributed by NM, 14-Aug-1999.) (Proof shortened by BJ, 1-Sep-2022.)
𝐴𝐶    &   𝐵𝐶       if(𝜑, 𝐴, 𝐵) ∈ 𝐶
 
Theoremifexd 4513 Existence of the conditional operator (deduction form). (Contributed by SN, 26-Jul-2024.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ V)
 
Theoremifexg 4514 Existence of the conditional operator (closed form). (Contributed by NM, 21-Mar-2011.) (Proof shortened by BJ, 1-Sep-2022.)
((𝐴𝑉𝐵𝑊) → if(𝜑, 𝐴, 𝐵) ∈ V)
 
Theoremifex 4515 Existence of the conditional operator (inference form). (Contributed by NM, 2-Sep-2004.)
𝐴 ∈ V    &   𝐵 ∈ V       if(𝜑, 𝐴, 𝐵) ∈ V
 
Theoremifeqor 4516 The possible values of a conditional operator. (Contributed by NM, 17-Jun-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(if(𝜑, 𝐴, 𝐵) = 𝐴 ∨ if(𝜑, 𝐴, 𝐵) = 𝐵)
 
Theoremifnot 4517 Negating the first argument swaps the last two arguments of a conditional operator. (Contributed by NM, 21-Jun-2007.)
if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴)
 
Theoremifan 4518 Rewrite a conjunction in a conditional as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)
if((𝜑𝜓), 𝐴, 𝐵) = if(𝜑, if(𝜓, 𝐴, 𝐵), 𝐵)
 
Theoremifor 4519 Rewrite a disjunction in a conditional as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)
if((𝜑𝜓), 𝐴, 𝐵) = if(𝜑, 𝐴, if(𝜓, 𝐴, 𝐵))
 
Theorem2if2 4520 Resolve two nested conditionals. (Contributed by Alexander van der Vekens, 27-Mar-2018.)
((𝜑𝜓) → 𝐷 = 𝐴)    &   ((𝜑 ∧ ¬ 𝜓𝜃) → 𝐷 = 𝐵)    &   ((𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜃) → 𝐷 = 𝐶)       (𝜑𝐷 = if(𝜓, 𝐴, if(𝜃, 𝐵, 𝐶)))
 
Theoremifcomnan 4521 Commute the conditions in two nested conditionals if both conditions are not simultaneously true. (Contributed by SO, 15-Jul-2018.)
(¬ (𝜑𝜓) → if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = if(𝜓, 𝐵, if(𝜑, 𝐴, 𝐶)))
 
Theoremcsbif 4522 Distribute proper substitution through the conditional operator. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 19-Aug-2018.)
𝐴 / 𝑥if(𝜑, 𝐵, 𝐶) = if([𝐴 / 𝑥]𝜑, 𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶)
 
2.1.16  The weak deduction theorem for set theory

This subsection contains a few results related to the weak deduction theorem in set theory. For the weak deduction theorem in propositional calculus, see the section beginning with elimh 1082. For more information on the weak deduction theorem, see the Weak Deduction Theorem page mmdeduction.html 1082.

In a Hilbert system of logic (which consists of a set of axioms, modus ponens, and the generalization rule), converting a deduction to a proof using the Deduction Theorem (taught in introductory logic books) involves an exponential increase of the number of steps as hypotheses are successively eliminated. Here is a trick that is not as general as the Deduction Theorem but requires only a linear increase in the number of steps.

The general problem: We want to convert a deduction P |- Q into a proof of the theorem |- P -> Q i.e., we want to eliminate the hypothesis P. Normally this is done using the Deduction (meta)Theorem, which looks at the microscopic steps of the deduction and usually doubles or triples the number of these microscopic steps for each hypothesis that is eliminated. We will look at a special case of this problem, without appealing to the Deduction Theorem.

We assume ZF with class notation. A and B are arbitrary (possibly proper) classes. P, Q, R, S and T are wffs.

We define the conditional operator, if(P, A, B), as follows: if(P, A, B) =def= { x | (x \in A & P) v (x \in B & -. P) } (where x does not occur in A, B, or P).

Lemma 1. A = if(P, A, B) -> (P <-> R), B = if(P, A, B) -> (S <-> R), S |- R Proof: Logic and Axiom of Extensionality.

Lemma 2. A = if(P, A, B) -> (Q <-> T), T |- P -> Q Proof: Logic and Axiom of Extensionality.

Here is a simple example that illustrates how it works. Suppose we have a deduction Ord A |- Tr A which means, "Assume A is an ordinal class. Then A is a transitive class." Note that A is a class variable that may be substituted with any class expression, so this is really a deduction scheme.

We want to convert this to a proof of the theorem (scheme) |- Ord A -> Tr A.

The catch is that we must be able to prove "Ord A" for at least one object A (and this is what makes it weaker than the ordinary Deduction Theorem). However, it is easy to prove |- Ord 0 (the empty set is ordinal). (For a typical textbook "theorem", i.e., deduction, there is usually at least one object satisfying each hypothesis, otherwise the theorem would not be very useful. We can always go back to the standard Deduction Theorem for those hypotheses where this is not the case.) Continuing with the example:

Equality axioms (and Extensionality) yield |- A = if(Ord A, A, 0) -> (Ord A <-> Ord if(Ord A, A, 0)) (1) |- 0 = if(Ord A, A, 0) -> (Ord 0 <-> Ord if(Ord A, A, 0)) (2) From (1), (2) and |- Ord 0, Lemma 1 yields |- Ord if(Ord A, A, 0) (3) From (3) and substituting if(Ord A, A, 0) for A in the original deduction, |- Tr if(Ord A, A, 0) (4) Equality axioms (and Extensionality) yield |- A = if(Ord A, A, 0) -> (Tr A <-> Tr if(Ord A, A, 0)) (5) From (4) and (5), Lemma 2 yields |- Ord A -> Tr A (Q.E.D.)

 
Theoremdedth 4523 Weak deduction theorem that eliminates a hypothesis 𝜑, making it become an antecedent. We assume that a proof exists for 𝜑 when the class variable 𝐴 is replaced with a specific class 𝐵. The hypothesis 𝜒 should be assigned to the inference, and the inference hypothesis eliminated with elimhyp 4530. If the inference has other hypotheses with class variable 𝐴, these can be kept by assigning keephyp 4536 to them. For more information, see the Weak Deduction Theorem page mmdeduction.html 4536. (Contributed by NM, 15-May-1999.)
(𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜒))    &   𝜒       (𝜑𝜓)
 
Theoremdedth2h 4524 Weak deduction theorem eliminating two hypotheses. This theorem is simpler to use than dedth2v 4527 but requires that each hypothesis have exactly one class variable. See also comments in dedth 4523. (Contributed by NM, 15-May-1999.)
(𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜒𝜃))    &   (𝐵 = if(𝜓, 𝐵, 𝐷) → (𝜃𝜏))    &   𝜏       ((𝜑𝜓) → 𝜒)
 
Theoremdedth3h 4525 Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 4524. (Contributed by NM, 15-May-1999.)
(𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜃𝜏))    &   (𝐵 = if(𝜓, 𝐵, 𝑅) → (𝜏𝜂))    &   (𝐶 = if(𝜒, 𝐶, 𝑆) → (𝜂𝜁))    &   𝜁       ((𝜑𝜓𝜒) → 𝜃)
 
Theoremdedth4h 4526 Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 4524. (Contributed by NM, 16-May-1999.)
(𝐴 = if(𝜑, 𝐴, 𝑅) → (𝜏𝜂))    &   (𝐵 = if(𝜓, 𝐵, 𝑆) → (𝜂𝜁))    &   (𝐶 = if(𝜒, 𝐶, 𝐹) → (𝜁𝜎))    &   (𝐷 = if(𝜃, 𝐷, 𝐺) → (𝜎𝜌))    &   𝜌       (((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)
 
Theoremdedth2v 4527 Weak deduction theorem for eliminating a hypothesis with 2 class variables. Note: if the hypothesis can be separated into two hypotheses, each with one class variable, then dedth2h 4524 is simpler to use. See also comments in dedth 4523. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
(𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜓𝜒))    &   (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒𝜃))    &   𝜃       (𝜑𝜓)
 
Theoremdedth3v 4528 Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 4527. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
(𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜓𝜒))    &   (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒𝜃))    &   (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃𝜏))    &   𝜏       (𝜑𝜓)
 
Theoremdedth4v 4529 Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 4527. (Contributed by NM, 21-Apr-2007.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
(𝐴 = if(𝜑, 𝐴, 𝑅) → (𝜓𝜒))    &   (𝐵 = if(𝜑, 𝐵, 𝑆) → (𝜒𝜃))    &   (𝐶 = if(𝜑, 𝐶, 𝑇) → (𝜃𝜏))    &   (𝐷 = if(𝜑, 𝐷, 𝑈) → (𝜏𝜂))    &   𝜂       (𝜑𝜓)
 
Theoremelimhyp 4530 Eliminate a hypothesis containing class variable 𝐴 when it is known for a specific class 𝐵. For more information, see comments in dedth 4523. (Contributed by NM, 15-May-1999.)
(𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜑𝜓))    &   (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜓))    &   𝜒       𝜓
 
Theoremelimhyp2v 4531 Eliminate a hypothesis containing 2 class variables. (Contributed by NM, 14-Aug-1999.)
(𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜑𝜒))    &   (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒𝜃))    &   (𝐶 = if(𝜑, 𝐴, 𝐶) → (𝜏𝜂))    &   (𝐷 = if(𝜑, 𝐵, 𝐷) → (𝜂𝜃))    &   𝜏       𝜃
 
Theoremelimhyp3v 4532 Eliminate a hypothesis containing 3 class variables. (Contributed by NM, 14-Aug-1999.)
(𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜑𝜒))    &   (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒𝜃))    &   (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃𝜏))    &   (𝐷 = if(𝜑, 𝐴, 𝐷) → (𝜂𝜁))    &   (𝑅 = if(𝜑, 𝐵, 𝑅) → (𝜁𝜎))    &   (𝑆 = if(𝜑, 𝐶, 𝑆) → (𝜎𝜏))    &   𝜂       𝜏
 
Theoremelimhyp4v 4533 Eliminate a hypothesis containing 4 class variables (for use with the weak deduction theorem dedth 4523). (Contributed by NM, 16-Apr-2005.)
(𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜑𝜒))    &   (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒𝜃))    &   (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃𝜏))    &   (𝐹 = if(𝜑, 𝐹, 𝐺) → (𝜏𝜓))    &   (𝐷 = if(𝜑, 𝐴, 𝐷) → (𝜂𝜁))    &   (𝑅 = if(𝜑, 𝐵, 𝑅) → (𝜁𝜎))    &   (𝑆 = if(𝜑, 𝐶, 𝑆) → (𝜎𝜌))    &   (𝐺 = if(𝜑, 𝐹, 𝐺) → (𝜌𝜓))    &   𝜂       𝜓
 
Theoremelimel 4534 Eliminate a membership hypothesis for weak deduction theorem, when special case 𝐵𝐶 is provable. (Contributed by NM, 15-May-1999.)
𝐵𝐶       if(𝐴𝐶, 𝐴, 𝐵) ∈ 𝐶
 
Theoremelimdhyp 4535 Version of elimhyp 4530 where the hypothesis is deduced from the final antecedent. See divalg 16108 for an example of its use. (Contributed by Paul Chapman, 25-Mar-2008.)
(𝜑𝜓)    &   (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜒))    &   (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜃𝜒))    &   𝜃       𝜒
 
Theoremkeephyp 4536 Transform a hypothesis 𝜓 that we want to keep (but contains the same class variable 𝐴 used in the eliminated hypothesis) for use with the weak deduction theorem. (Contributed by NM, 15-May-1999.)
(𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜃))    &   (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜃))    &   𝜓    &   𝜒       𝜃
 
Theoremkeephyp2v 4537 Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 4523). (Contributed by NM, 16-Apr-2005.)
(𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜓𝜒))    &   (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒𝜃))    &   (𝐶 = if(𝜑, 𝐴, 𝐶) → (𝜏𝜂))    &   (𝐷 = if(𝜑, 𝐵, 𝐷) → (𝜂𝜃))    &   𝜓    &   𝜏       𝜃
 
Theoremkeephyp3v 4538 Keep a hypothesis containing 3 class variables. (Contributed by NM, 27-Sep-1999.)
(𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜌𝜒))    &   (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒𝜃))    &   (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃𝜏))    &   (𝐷 = if(𝜑, 𝐴, 𝐷) → (𝜂𝜁))    &   (𝑅 = if(𝜑, 𝐵, 𝑅) → (𝜁𝜎))    &   (𝑆 = if(𝜑, 𝐶, 𝑆) → (𝜎𝜏))    &   𝜌    &   𝜂       𝜏
 
2.1.17  Power classes
 
Syntaxcpw 4539 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 4540* Soundness justification theorem for df-pw 4541. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
{𝑥𝑥𝐴} = {𝑦𝑦𝐴}
 
Definitiondf-pw 4541* 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 = {3, 5, 7}, then 𝒫 𝐴 = {∅, {3}, {5}, {7}, {3, 5}, {3, 7}, {5, 7}, {3, 5, 7}} (ex-pw 28787). We will later introduce the Axiom of Power Sets ax-pow 5292, which can be expressed in class notation per pwexg 5305. Still later we will prove, in hashpw 14147, 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, 24-Jun-1993.)
𝒫 𝐴 = {𝑥𝑥𝐴}
 
Theoremelpwg 4542 Membership in a power class. Theorem 86 of [Suppes] p. 47. See also elpw2g 5272. (Contributed by NM, 6-Aug-2000.) (Proof shortened by BJ, 31-Dec-2023.)
(𝐴𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
 
Theoremelpw 4543 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.) (Proof shortened by BJ, 31-Dec-2023.)
𝐴 ∈ V       (𝐴 ∈ 𝒫 𝐵𝐴𝐵)
 
Theoremvelpw 4544 Setvar variable membership in a power class. (Contributed by David A. Wheeler, 8-Dec-2018.)
(𝑥 ∈ 𝒫 𝐴𝑥𝐴)
 
TheoremelpwOLD 4545 Obsolete proof of elpw 4543 as of 31-Dec-2023. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by NM, 31-Dec-1993.)
𝐴 ∈ V       (𝐴 ∈ 𝒫 𝐵𝐴𝐵)
 
TheoremelpwgOLD 4546 Obsolete proof of elpwg 4542 as of 31-Dec-2023. (Contributed by NM, 6-Aug-2000.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
 
Theoremelpwd 4547 Membership in a power class. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐴𝐵)       (𝜑𝐴 ∈ 𝒫 𝐵)
 
Theoremelpwi 4548 Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)
(𝐴 ∈ 𝒫 𝐵𝐴𝐵)
 
Theoremelpwb 4549 Characterization of the elements of a power class. (Contributed by BJ, 29-Apr-2021.)
(𝐴 ∈ 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴𝐵))
 
Theoremelpwid 4550 An element of a power class is a subclass. Deduction form of elpwi 4548. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ 𝒫 𝐵)       (𝜑𝐴𝐵)
 
Theoremelelpwi 4551 If 𝐴 belongs to a part of 𝐶, then 𝐴 belongs to 𝐶. (Contributed by FL, 3-Aug-2009.)
((𝐴𝐵𝐵 ∈ 𝒫 𝐶) → 𝐴𝐶)
 
Theoremsspw 4552 The powerclass preserves inclusion. See sspwb 5368 for the biconditional version. (Contributed by NM, 13-Oct-1996.) Extract forward implication of sspwb 5368 since it requires fewer axioms. (Revised by BJ, 13-Apr-2024.)
(𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
 
Theoremsspwi 4553 The powerclass preserves inclusion (inference form). (Contributed by BJ, 13-Apr-2024.)
𝐴𝐵       𝒫 𝐴 ⊆ 𝒫 𝐵
 
Theoremsspwd 4554 The powerclass preserves inclusion (deduction form). (Contributed by BJ, 13-Apr-2024.)
(𝜑𝐴𝐵)       (𝜑 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
 
Theorempweq 4555 Equality theorem for power class. (Contributed by NM, 21-Jun-1993.) (Proof shortened by BJ, 13-Apr-2024.)
(𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵)
 
TheorempweqALT 4556 Alternate proof of pweq 4555 directly from the definition. (Contributed by NM, 21-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵)
 
Theorempweqi 4557 Equality inference for power class. (Contributed by NM, 27-Nov-2013.)
𝐴 = 𝐵       𝒫 𝐴 = 𝒫 𝐵
 
Theorempweqd 4558 Equality deduction for power class. (Contributed by NM, 27-Nov-2013.)
(𝜑𝐴 = 𝐵)       (𝜑 → 𝒫 𝐴 = 𝒫 𝐵)
 
Theorempwunss 4559 The power class of the union of two classes includes the union of their power classes. Exercise 4.12(k) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.) Remove use of ax-sep 5227, ax-nul 5234, ax-pr 5356 and shorten proof. (Revised by BJ, 13-Apr-2024.)
(𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵)
 
Theoremnfpw 4560 Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
𝑥𝐴       𝑥𝒫 𝐴
 
Theorempwidg 4561 A set is an element of its power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐴𝑉𝐴 ∈ 𝒫 𝐴)
 
Theorempwidb 4562 A class is an element of its powerclass if and only if it is a set. (Contributed by BJ, 31-Dec-2023.)
(𝐴 ∈ V ↔ 𝐴 ∈ 𝒫 𝐴)
 
Theorempwid 4563 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 4564* Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)
(𝒫 𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 
Theorempwundif 4565 Break up the power class of a union into a union of smaller classes. (Contributed by NM, 25-Mar-2007.) (Proof shortened by Thierry Arnoux, 20-Dec-2016.) Remove use of ax-sep 5227, ax-nul 5234, ax-pr 5356 and shorten proof. (Revised by BJ, 14-Apr-2024.)
𝒫 (𝐴𝐵) = ((𝒫 (𝐴𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)
 
2.1.18  Unordered and ordered pairs
 
Theoremsnjust 4566* Soundness justification theorem for df-sn 4568. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
{𝑥𝑥 = 𝐴} = {𝑦𝑦 = 𝐴}
 
Syntaxcsn 4567 Extend class notation to include singleton.
class {𝐴}
 
Definitiondf-sn 4568* 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, see snprc 4659. For an alternate definition see dfsn2 4580. (Contributed by NM, 21-Jun-1993.)
{𝐴} = {𝑥𝑥 = 𝐴}
 
Syntaxcpr 4569 Extend class notation to include unordered pair.
class {𝐴, 𝐵}
 
Definitiondf-pr 4570 Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. For example, 𝐴 ∈ {1, -1} → (𝐴↑2) = 1 (ex-pr 28788). They are unordered, so {𝐴, 𝐵} = {𝐵, 𝐴} as proven by prcom 4674. For a more traditional definition, but requiring a dummy variable, see dfpr2 4586. {𝐴, 𝐴} is also an unordered pair, but also a singleton because of {𝐴} = {𝐴, 𝐴} (see dfsn2 4580). Therefore, {𝐴, 𝐵} is called a proper (unordered) pair iff 𝐴𝐵 and 𝐴 and 𝐵 are sets.

Note: ordered pairs are a completely different object defined below in df-op 4574. When the term "pair" is used without qualifier, it generally means "unordered pair", and the context makes it clear which version is meant. (Contributed by NM, 21-Jun-1993.)

{𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
 
Syntaxctp 4571 Extend class notation to include unordered triple (sometimes called "unordered triplet").
class {𝐴, 𝐵, 𝐶}
 
Definitiondf-tp 4572 Define unordered triple of classes. Definition of [Enderton] p. 19.

Note: ordered triples are a completely different object defined below in df-ot 4576. As with all tuples, when the term "triple" is used without qualifier, it means "ordered triple". (Contributed by NM, 9-Apr-1994.)

{𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
 
Syntaxcop 4573 Extend class notation to include ordered pair.
class 𝐴, 𝐵
 
Definitiondf-op 4574* 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 4834, opprc2 4835, and 0nelop 5413). For Kuratowski's actual definition when the arguments are sets, see dfop 4809. For the justifying theorem (for sets) see opth 5394. See dfopif 4806 for an equivalent formulation using the if operation.

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 4574 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 4574 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}}}, justified by opthwiener 5431. This was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition 𝐴, 𝐵3 = {𝐴, {𝐴, 𝐵}} is justified by opthreg 9352, but it requires the Axiom of Regularity for its justification and is not commonly used. A definition that also works for proper classes is 𝐴, 𝐵4 = ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})), justified by opthprc 5651. Nearly at the same time as Norbert Wiener, Felix Hausdorff proposed the following definition in "Grundzüge der Mengenlehre" ("Basics of Set Theory"), p. 32, in 1914: 𝐴, 𝐵5 = {{𝐴, 𝑂}, {𝐵, 𝑇}}. Hausdorff used 1 and 2 instead of 𝑂 and 𝑇, but actually any two different fixed sets will do (e.g., 𝑂 = ∅ and 𝑇 = {∅}, see 0nep0 5284). Furthermore, Hausdorff demanded that 𝑂 and 𝑇 are both different from 𝐴 as well as 𝐵, which is actually not necessary (at least not in full extent), see opthhausdorff0 5435 and opthhausdorff 5434. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 13980. An ordered pair of real numbers can also be represented by a complex number as shown by cru 11963. Kuratowski's ordered pair definition is standard for ZFC set theory, but it is very inconvenient to use in New Foundations theory because it is not type-level; a common alternate definition in New Foundations is the definition from [Rosser] p. 281.

Since there are other ways to define ordered pairs, we discourage direct use of this definition so that most theorems won't depend on this particular construction; theorems will instead rely on dfopif 4806. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.)

𝐴, 𝐵⟩ = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
 
Syntaxcotp 4575 Extend class notation to include ordered triple.
class 𝐴, 𝐵, 𝐶
 
Definitiondf-ot 4576 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 4577 Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.)
(𝐴 = 𝐵 → {𝐴} = {𝐵})
 
Theoremsneqi 4578 Equality inference for singletons. (Contributed by NM, 22-Jan-2004.)
𝐴 = 𝐵       {𝐴} = {𝐵}
 
Theoremsneqd 4579 Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝐴} = {𝐵})
 
Theoremdfsn2 4580 Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
{𝐴} = {𝐴, 𝐴}
 
Theoremelsng 4581 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.)
(𝐴𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
 
Theoremelsn 4582 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)
 
Theoremvelsn 4583 There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.)
(𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
 
Theoremelsni 4584 There is at most one element in a singleton. (Contributed by NM, 5-Jun-1994.)
(𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
 
Theoremabsn 4585* Condition for a class abstraction to be a singleton. Formerly part of proof of dfiota2 6390. (Contributed by Andrew Salmon, 30-Jun-2011.) (Revised by AV, 24-Aug-2022.)
({𝑥𝜑} = {𝑌} ↔ ∀𝑥(𝜑𝑥 = 𝑌))
 
Theoremdfpr2 4586* Alternate definition of a pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
{𝐴, 𝐵} = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵)}
 
Theoremdfsn2ALT 4587 Alternate definition of singleton, based on the (alternate) definition of pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by AV, 12-Jun-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
{𝐴} = {𝐴, 𝐴}
 
Theoremelprg 4588 A member of a pair of classes is one or the other of them, and conversely as soon as it is a set. Exercise 1 of [TakeutiZaring] p. 15, generalized. (Contributed by NM, 13-Sep-1995.)
(𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
 
Theoremelpri 4589 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 = 𝐵𝐴 = 𝐶))
 
Theoremelpr 4590 A member of a pair of classes is one or the other of them, and conversely as soon as it is a set. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)
𝐴 ∈ V       (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))
 
Theoremelpr2g 4591 A member of a pair of sets is one or the other of them, and conversely. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 14-Oct-2005.) Generalize from sethood hypothesis to sethood antecedent. (Revised by BJ, 25-May-2024.)
((𝐵𝑉𝐶𝑊) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
 
Theoremelpr2 4592 A member of a pair of sets is one or the other of them, and conversely. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 14-Oct-2005.) (Proof shortened by JJ, 23-Jul-2021.)
𝐵 ∈ V    &   𝐶 ∈ V       (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))
 
Theoremelpr2OLD 4593 Obsolete version of elpr2 4592 as of 25-May-2024. (Contributed by NM, 14-Oct-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐵 ∈ V    &   𝐶 ∈ V       (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))
 
Theoremnelpr2 4594 If a class is not an element of an unordered pair, it is not the second listed element. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})       (𝜑𝐴𝐶)
 
Theoremnelpr1 4595 If a class is not an element of an unordered pair, it is not the first listed element. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})       (𝜑𝐴𝐵)
 
Theoremnelpri 4596 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}
 
Theoremprneli 4597 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.)
𝐴𝐵    &   𝐴𝐶       𝐴 ∉ {𝐵, 𝐶}
 
Theoremnelprd 4598 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.)
(𝜑𝐴𝐵)    &   (𝜑𝐴𝐶)       (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})
 
Theoremeldifpr 4599 Membership in a set with two elements removed. Similar to eldifsn 4726 and eldiftp 4628. (Contributed by Mario Carneiro, 18-Jul-2017.)
(𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷}) ↔ (𝐴𝐵𝐴𝐶𝐴𝐷))
 
Theoremrexdifpr 4600 Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.)
(∃𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶})𝜑 ↔ ∃𝑥𝐴 (𝑥𝐵𝑥𝐶𝜑))
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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46480
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