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Type | Label | Description |
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Statement | ||
Theorem | ifeqda 4501 | Separation of the values of the conditional operator. (Contributed by Alexander van der Vekens, 13-Apr-2018.) |
⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶) & ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = 𝐶) | ||
Theorem | elimif 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 → (𝜓 ↔ 𝜒)) & ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → (𝜓 ↔ 𝜃)) ⇒ ⊢ (𝜓 ↔ ((𝜑 ∧ 𝜒) ∨ (¬ 𝜑 ∧ 𝜃))) | ||
Theorem | ifbothda 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) → (𝜓 ↔ 𝜃)) & ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃)) & ⊢ ((𝜂 ∧ 𝜑) → 𝜓) & ⊢ ((𝜂 ∧ ¬ 𝜑) → 𝜒) ⇒ ⊢ (𝜂 → 𝜃) | ||
Theorem | ifboth 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) → (𝜓 ↔ 𝜃)) & ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃)) ⇒ ⊢ ((𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | ifid 4505 | Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.) |
⊢ if(𝜑, 𝐴, 𝐴) = 𝐴 | ||
Theorem | eqif 4506 | Expansion of an equality with a conditional operator. (Contributed by NM, 14-Feb-2005.) |
⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 = 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 = 𝐶))) | ||
Theorem | ifval 4507 | Another expression of the value of the if predicate, analogous to eqif 4506. See also the more specialized iftrue 4472 and iffalse 4475. (Contributed by BJ, 6-Apr-2019.) |
⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 → 𝐴 = 𝐵) ∧ (¬ 𝜑 → 𝐴 = 𝐶))) | ||
Theorem | elif 4508 | Membership in a conditional operator. (Contributed by NM, 14-Feb-2005.) |
⊢ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 ∈ 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 ∈ 𝐶))) | ||
Theorem | ifel 4509 | Membership of a conditional operator. (Contributed by NM, 10-Sep-2005.) |
⊢ (if(𝜑, 𝐴, 𝐵) ∈ 𝐶 ↔ ((𝜑 ∧ 𝐴 ∈ 𝐶) ∨ (¬ 𝜑 ∧ 𝐵 ∈ 𝐶))) | ||
Theorem | ifcl 4510 | Membership (closure) of a conditional operator. (Contributed by NM, 4-Apr-2005.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → if(𝜑, 𝐴, 𝐵) ∈ 𝐶) | ||
Theorem | ifcld 4511 | Membership (closure) of a conditional operator, deduction form. (Contributed by SO, 16-Jul-2018.) |
⊢ (𝜑 → 𝐴 ∈ 𝐶) & ⊢ (𝜑 → 𝐵 ∈ 𝐶) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶) | ||
Theorem | ifcli 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 4522 when the special case 𝐵 ∈ 𝐶 is provable. (Contributed by NM, 14-Aug-1999.) (Proof shortened by BJ, 1-Sep-2022.) |
⊢ 𝐴 ∈ 𝐶 & ⊢ 𝐵 ∈ 𝐶 ⇒ ⊢ if(𝜑, 𝐴, 𝐵) ∈ 𝐶 | ||
Theorem | ifexg 4513 | Conditional operator existence. (Contributed by NM, 21-Mar-2011.) (Proof shortened by BJ, 1-Sep-2022.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → if(𝜑, 𝐴, 𝐵) ∈ V) | ||
Theorem | ifex 4514 | Conditional operator existence. (Contributed by NM, 2-Sep-2004.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ if(𝜑, 𝐴, 𝐵) ∈ V | ||
Theorem | ifeqor 4515 | 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) | ||
Theorem | ifnot 4516 | 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) | ||
Theorem | ifan 4517 | Rewrite a conjunction in a conditional as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.) |
⊢ if((𝜑 ∧ 𝜓), 𝐴, 𝐵) = if(𝜑, if(𝜓, 𝐴, 𝐵), 𝐵) | ||
Theorem | ifor 4518 | Rewrite a disjunction in a conditional as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.) |
⊢ if((𝜑 ∨ 𝜓), 𝐴, 𝐵) = if(𝜑, 𝐴, if(𝜓, 𝐴, 𝐵)) | ||
Theorem | 2if2 4519 | Resolve two nested conditionals. (Contributed by Alexander van der Vekens, 27-Mar-2018.) |
⊢ ((𝜑 ∧ 𝜓) → 𝐷 = 𝐴) & ⊢ ((𝜑 ∧ ¬ 𝜓 ∧ 𝜃) → 𝐷 = 𝐵) & ⊢ ((𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜃) → 𝐷 = 𝐶) ⇒ ⊢ (𝜑 → 𝐷 = if(𝜓, 𝐴, if(𝜃, 𝐵, 𝐶))) | ||
Theorem | ifcomnan 4520 | Commute the conditions in two nested conditionals if both conditions are not simultaneously true. (Contributed by SO, 15-Jul-2018.) |
⊢ (¬ (𝜑 ∧ 𝜓) → if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = if(𝜓, 𝐵, if(𝜑, 𝐴, 𝐶))) | ||
Theorem | csbif 4521 | Distribute proper substitution through the conditional operator. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 19-Aug-2018.) |
⊢ ⦋𝐴 / 𝑥⦌if(𝜑, 𝐵, 𝐶) = if([𝐴 / 𝑥]𝜑, ⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶) | ||
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 1076. For more information on the weak deduction theorem, see the Weak Deduction Theorem page mmdeduction.html 1076. 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.) | ||
Theorem | dedth 4522 | 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 4529. If the inference has other hypotheses with class variable 𝐴, these can be kept by assigning keephyp 4535 to them. For more information, see the Weak Deduction Theorem page mmdeduction.html 4535. (Contributed by NM, 15-May-1999.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜒)) & ⊢ 𝜒 ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | dedth2h 4523 | Weak deduction theorem eliminating two hypotheses. This theorem is simpler to use than dedth2v 4526 but requires that each hypothesis have exactly one class variable. See also comments in dedth 4522. (Contributed by NM, 15-May-1999.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜒 ↔ 𝜃)) & ⊢ (𝐵 = if(𝜓, 𝐵, 𝐷) → (𝜃 ↔ 𝜏)) & ⊢ 𝜏 ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | ||
Theorem | dedth3h 4524 | Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 4523. (Contributed by NM, 15-May-1999.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜃 ↔ 𝜏)) & ⊢ (𝐵 = if(𝜓, 𝐵, 𝑅) → (𝜏 ↔ 𝜂)) & ⊢ (𝐶 = if(𝜒, 𝐶, 𝑆) → (𝜂 ↔ 𝜁)) & ⊢ 𝜁 ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | ||
Theorem | dedth4h 4525 | Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 4523. (Contributed by NM, 16-May-1999.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝑅) → (𝜏 ↔ 𝜂)) & ⊢ (𝐵 = if(𝜓, 𝐵, 𝑆) → (𝜂 ↔ 𝜁)) & ⊢ (𝐶 = if(𝜒, 𝐶, 𝐹) → (𝜁 ↔ 𝜎)) & ⊢ (𝐷 = if(𝜃, 𝐷, 𝐺) → (𝜎 ↔ 𝜌)) & ⊢ 𝜌 ⇒ ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → 𝜏) | ||
Theorem | dedth2v 4526 | 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 4523 is simpler to use. See also comments in dedth 4522. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜓 ↔ 𝜒)) & ⊢ (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒 ↔ 𝜃)) & ⊢ 𝜃 ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | dedth3v 4527 | Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 4526. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜓 ↔ 𝜒)) & ⊢ (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒 ↔ 𝜃)) & ⊢ (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃 ↔ 𝜏)) & ⊢ 𝜏 ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | dedth4v 4528 | Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 4526. (Contributed by NM, 21-Apr-2007.) (Proof shortened by Eric Schmidt, 28-Jul-2009.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝑅) → (𝜓 ↔ 𝜒)) & ⊢ (𝐵 = if(𝜑, 𝐵, 𝑆) → (𝜒 ↔ 𝜃)) & ⊢ (𝐶 = if(𝜑, 𝐶, 𝑇) → (𝜃 ↔ 𝜏)) & ⊢ (𝐷 = if(𝜑, 𝐷, 𝑈) → (𝜏 ↔ 𝜂)) & ⊢ 𝜂 ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | elimhyp 4529 | Eliminate a hypothesis containing class variable 𝐴 when it is known for a specific class 𝐵. For more information, see comments in dedth 4522. (Contributed by NM, 15-May-1999.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜑 ↔ 𝜓)) & ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜓)) & ⊢ 𝜒 ⇒ ⊢ 𝜓 | ||
Theorem | elimhyp2v 4530 | Eliminate a hypothesis containing 2 class variables. (Contributed by NM, 14-Aug-1999.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜑 ↔ 𝜒)) & ⊢ (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒 ↔ 𝜃)) & ⊢ (𝐶 = if(𝜑, 𝐴, 𝐶) → (𝜏 ↔ 𝜂)) & ⊢ (𝐷 = if(𝜑, 𝐵, 𝐷) → (𝜂 ↔ 𝜃)) & ⊢ 𝜏 ⇒ ⊢ 𝜃 | ||
Theorem | elimhyp3v 4531 | Eliminate a hypothesis containing 3 class variables. (Contributed by NM, 14-Aug-1999.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜑 ↔ 𝜒)) & ⊢ (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒 ↔ 𝜃)) & ⊢ (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃 ↔ 𝜏)) & ⊢ (𝐷 = if(𝜑, 𝐴, 𝐷) → (𝜂 ↔ 𝜁)) & ⊢ (𝑅 = if(𝜑, 𝐵, 𝑅) → (𝜁 ↔ 𝜎)) & ⊢ (𝑆 = if(𝜑, 𝐶, 𝑆) → (𝜎 ↔ 𝜏)) & ⊢ 𝜂 ⇒ ⊢ 𝜏 | ||
Theorem | elimhyp4v 4532 | Eliminate a hypothesis containing 4 class variables (for use with the weak deduction theorem dedth 4522). (Contributed by NM, 16-Apr-2005.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜑 ↔ 𝜒)) & ⊢ (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒 ↔ 𝜃)) & ⊢ (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃 ↔ 𝜏)) & ⊢ (𝐹 = if(𝜑, 𝐹, 𝐺) → (𝜏 ↔ 𝜓)) & ⊢ (𝐷 = if(𝜑, 𝐴, 𝐷) → (𝜂 ↔ 𝜁)) & ⊢ (𝑅 = if(𝜑, 𝐵, 𝑅) → (𝜁 ↔ 𝜎)) & ⊢ (𝑆 = if(𝜑, 𝐶, 𝑆) → (𝜎 ↔ 𝜌)) & ⊢ (𝐺 = if(𝜑, 𝐹, 𝐺) → (𝜌 ↔ 𝜓)) & ⊢ 𝜂 ⇒ ⊢ 𝜓 | ||
Theorem | elimel 4533 | Eliminate a membership hypothesis for weak deduction theorem, when special case 𝐵 ∈ 𝐶 is provable. (Contributed by NM, 15-May-1999.) |
⊢ 𝐵 ∈ 𝐶 ⇒ ⊢ if(𝐴 ∈ 𝐶, 𝐴, 𝐵) ∈ 𝐶 | ||
Theorem | elimdhyp 4534 | Version of elimhyp 4529 where the hypothesis is deduced from the final antecedent. See divalg 15753 for an example of its use. (Contributed by Paul Chapman, 25-Mar-2008.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜒)) & ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜃 ↔ 𝜒)) & ⊢ 𝜃 ⇒ ⊢ 𝜒 | ||
Theorem | keephyp 4535 | 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) → (𝜓 ↔ 𝜃)) & ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃)) & ⊢ 𝜓 & ⊢ 𝜒 ⇒ ⊢ 𝜃 | ||
Theorem | keephyp2v 4536 | Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 4522). (Contributed by NM, 16-Apr-2005.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜓 ↔ 𝜒)) & ⊢ (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒 ↔ 𝜃)) & ⊢ (𝐶 = if(𝜑, 𝐴, 𝐶) → (𝜏 ↔ 𝜂)) & ⊢ (𝐷 = if(𝜑, 𝐵, 𝐷) → (𝜂 ↔ 𝜃)) & ⊢ 𝜓 & ⊢ 𝜏 ⇒ ⊢ 𝜃 | ||
Theorem | keephyp3v 4537 | Keep a hypothesis containing 3 class variables. (Contributed by NM, 27-Sep-1999.) |
⊢ (𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜌 ↔ 𝜒)) & ⊢ (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒 ↔ 𝜃)) & ⊢ (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃 ↔ 𝜏)) & ⊢ (𝐷 = if(𝜑, 𝐴, 𝐷) → (𝜂 ↔ 𝜁)) & ⊢ (𝑅 = if(𝜑, 𝐵, 𝑅) → (𝜁 ↔ 𝜎)) & ⊢ (𝑆 = if(𝜑, 𝐶, 𝑆) → (𝜎 ↔ 𝜏)) & ⊢ 𝜌 & ⊢ 𝜂 ⇒ ⊢ 𝜏 | ||
Syntax | cpw 4538 | 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 4539* | Soundness justification theorem for df-pw 4540. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ {𝑥 ∣ 𝑥 ⊆ 𝐴} = {𝑦 ∣ 𝑦 ⊆ 𝐴} | ||
Definition | df-pw 4540* | 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 28207). We will later introduce the Axiom of Power Sets ax-pow 5265, which can be expressed in class notation per pwexg 5278. Still later we will prove, in hashpw 13796, 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 = {𝑥 ∣ 𝑥 ⊆ 𝐴} | ||
Theorem | elpwg 4541 | Membership in a power class. Theorem 86 of [Suppes] p. 47. See also elpw2g 5246. (Contributed by NM, 6-Aug-2000.) (Proof shortened by BJ, 31-Dec-2023.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) | ||
Theorem | elpw 4542 | 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 𝐵 ↔ 𝐴 ⊆ 𝐵) | ||
Theorem | velpw 4543 | Setvar variable membership in a power class. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | ||
Theorem | elpwOLD 4544 | Obsolete proof of elpw 4542 as of 31-Dec-2023. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by NM, 31-Dec-1993.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵) | ||
Theorem | elpwgOLD 4545 | Obsolete proof of elpwg 4541 as of 31-Dec-2023. (Contributed by NM, 6-Aug-2000.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) | ||
Theorem | elpwd 4546 | Membership in a power class. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵) | ||
Theorem | elpwi 4547 | Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.) |
⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ⊆ 𝐵) | ||
Theorem | elpwb 4548 | Characterization of the elements of a power class. (Contributed by BJ, 29-Apr-2021.) |
⊢ (𝐴 ∈ 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵)) | ||
Theorem | elpwid 4549 | An element of a power class is a subclass. Deduction form of elpwi 4547. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | ||
Theorem | elelpwi 4550 | 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 | sspw 4551 | The powerclass preserves inclusion. See sspwb 5341 for the biconditional version. (Contributed by NM, 13-Oct-1996.) Extract forward implication of sspwb 5341 since it requires fewer axioms. (Revised by BJ, 13-Apr-2024.) |
⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) | ||
Theorem | sspwi 4552 | The powerclass preserves inclusion (inference form). (Contributed by BJ, 13-Apr-2024.) |
⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ 𝒫 𝐴 ⊆ 𝒫 𝐵 | ||
Theorem | sspwd 4553 | The powerclass preserves inclusion (deduction form). (Contributed by BJ, 13-Apr-2024.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → 𝒫 𝐴 ⊆ 𝒫 𝐵) | ||
Theorem | pweq 4554 | Equality theorem for power class. (Contributed by NM, 21-Jun-1993.) |
⊢ (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵) | ||
Theorem | pweqALT 4555 | Alternate proof of pweq 4554 directly from the definition. (Contributed by NM, 21-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵) | ||
Theorem | pweqi 4556 | Equality inference for power class. (Contributed by NM, 27-Nov-2013.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ 𝒫 𝐴 = 𝒫 𝐵 | ||
Theorem | pweqd 4557 | Equality deduction for power class. (Contributed by NM, 27-Nov-2013.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 𝒫 𝐴 = 𝒫 𝐵) | ||
Theorem | pwunss 4558 | 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 5202, ax-nul 5209, ax-pr 5329 and shorten proof. (Revised by BJ, 13-Apr-2024.) |
⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵) | ||
Theorem | nfpw 4559 | 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 4560 | A set is an element of its power set. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴) | ||
Theorem | pwidb 4561 | 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) | ||
Theorem | pwid 4562 | 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 4563* | Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.) |
⊢ (𝒫 𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐵)) | ||
Theorem | pwundif 4564 | 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 5202, ax-nul 5209, ax-pr 5329 and shorten proof. (Revised by BJ, 14-Apr-2024.) |
⊢ 𝒫 (𝐴 ∪ 𝐵) = ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴) | ||
Theorem | snjust 4565* | Soundness justification theorem for df-sn 4567. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ {𝑥 ∣ 𝑥 = 𝐴} = {𝑦 ∣ 𝑦 = 𝐴} | ||
Syntax | csn 4566 | Extend class notation to include singleton. |
class {𝐴} | ||
Definition | df-sn 4567* | 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 4652. For an alternate definition see dfsn2 4579. (Contributed by NM, 21-Jun-1993.) |
⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴} | ||
Syntax | cpr 4568 | Extend class notation to include unordered pair. |
class {𝐴, 𝐵} | ||
Definition | df-pr 4569 | Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. For example, 𝐴 ∈ {1, -1} → (𝐴↑2) = 1 (ex-pr 28208). They are unordered, so {𝐴, 𝐵} = {𝐵, 𝐴} as proven by prcom 4667. For a more traditional definition, but requiring a dummy variable, see dfpr2 4585. {𝐴, 𝐴} is also an unordered pair, but also a singleton because of {𝐴} = {𝐴, 𝐴} (see dfsn2 4579). Therefore, {𝐴, 𝐵} is called a proper (unordered) pair iff 𝐴 ≠ 𝐵 and 𝐴 and 𝐵 are sets. (Contributed by NM, 21-Jun-1993.) |
⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | ||
Syntax | ctp 4570 | Extend class notation to include unordered triplet. |
class {𝐴, 𝐵, 𝐶} | ||
Definition | df-tp 4571 | Define unordered triple of classes. Definition of [Enderton] p. 19. (Contributed by NM, 9-Apr-1994.) |
⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | ||
Syntax | cop 4572 | Extend class notation to include ordered pair. |
class 〈𝐴, 𝐵〉 | ||
Definition | df-op 4573* |
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 4826, opprc2 4827, and
0nelop 5385). For Kuratowski's actual definition when
the arguments are
sets, see dfop 4801. For the justifying theorem (for sets) see
opth 5367.
See dfopif 4799 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 4573 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 4573 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}}}, justified by opthwiener 5403. This was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition 〈𝐴, 𝐵〉3 = {𝐴, {𝐴, 𝐵}} is justified by opthreg 9080, 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, 𝐵〉4 = ((𝐴 × {∅}) ∪ (𝐵 × {{∅}})), justified by opthprc 5615. 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, 𝐵〉5 = {{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 5257). 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 5407 and opthhausdorff 5406. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 13629. An ordered pair of real numbers can also be represented by a complex number as shown by cru 11629. 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 4799. (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}})} | ||
Syntax | cotp 4574 | Extend class notation to include ordered triple. |
class 〈𝐴, 𝐵, 𝐶〉 | ||
Definition | df-ot 4575 | 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 4576 | Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.) |
⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | ||
Theorem | sneqi 4577 | Equality inference for singletons. (Contributed by NM, 22-Jan-2004.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ {𝐴} = {𝐵} | ||
Theorem | sneqd 4578 | Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → {𝐴} = {𝐵}) | ||
Theorem | dfsn2 4579 | Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) |
⊢ {𝐴} = {𝐴, 𝐴} | ||
Theorem | elsng 4580 | 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.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)) | ||
Theorem | elsn 4581 | 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) | ||
Theorem | velsn 4582 | There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.) |
⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | ||
Theorem | elsni 4583 | There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.) |
⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵) | ||
Theorem | absn 4584* | Condition for a class abstraction to be a singleton. Formerly part of proof of dfiota2 6314. (Contributed by Andrew Salmon, 30-Jun-2011.) (Revised by AV, 24-Aug-2022.) |
⊢ ({𝑥 ∣ 𝜑} = {𝑌} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑌)) | ||
Theorem | dfpr2 4585* | 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 | dfsn2ALT 4586 | Alternate definition of singleton, based on the (alternate) definition of unordered 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} | ||
Theorem | elprg 4587 | 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 | elpri 4588 | 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 | elpr 4589 | 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 4590 | 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.) (Proof shortened by JJ, 23-Jul-2021.) |
⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) | ||
Theorem | nelpr2 4591 | 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.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶}) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐶) | ||
Theorem | nelpr1 4592 | 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.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶}) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐵) | ||
Theorem | nelpri 4593 | 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 | prneli 4594 | 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.) |
⊢ 𝐴 ≠ 𝐵 & ⊢ 𝐴 ≠ 𝐶 ⇒ ⊢ 𝐴 ∉ {𝐵, 𝐶} | ||
Theorem | nelprd 4595 | 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.) |
⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐴 ≠ 𝐶) ⇒ ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶}) | ||
Theorem | eldifpr 4596 | Membership in a set with two elements removed. Similar to eldifsn 4718 and eldiftp 4623. (Contributed by Mario Carneiro, 18-Jul-2017.) |
⊢ (𝐴 ∈ (𝐵 ∖ {𝐶, 𝐷}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷)) | ||
Theorem | rexdifpr 4597 | Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.) |
⊢ (∃𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶})𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶 ∧ 𝜑)) | ||
Theorem | snidg 4598 | 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 4599 | 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 4600 | 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} |
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