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Type | Label | Description |
---|---|---|
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 4471 and iffalse 4474. (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 4523 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 | ifexd 4513 | Existence of the conditional operator (deduction form). (Contributed by SN, 26-Jul-2024.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ V) | ||
Theorem | ifexg 4514 | Existence of the conditional operator (closed form). (Contributed by NM, 21-Mar-2011.) (Proof shortened by BJ, 1-Sep-2022.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → if(𝜑, 𝐴, 𝐵) ∈ V) | ||
Theorem | ifex 4515 | Existence of the conditional operator (inference form). (Contributed by NM, 2-Sep-2004.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ if(𝜑, 𝐴, 𝐵) ∈ V | ||
Theorem | ifeqor 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) | ||
Theorem | ifnot 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) | ||
Theorem | ifan 4518 | 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 4519 | 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 4520 | 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 4521 | 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 4522 | 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 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.) | ||
Theorem | dedth 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) → (𝜓 ↔ 𝜒)) & ⊢ 𝜒 ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | dedth2h 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) → (𝜒 ↔ 𝜃)) & ⊢ (𝐵 = if(𝜓, 𝐵, 𝐷) → (𝜃 ↔ 𝜏)) & ⊢ 𝜏 ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | ||
Theorem | dedth3h 4525 | Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 4524. (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 4526 | Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 4524. (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 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) → (𝜓 ↔ 𝜒)) & ⊢ (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒 ↔ 𝜃)) & ⊢ 𝜃 ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | dedth3v 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) → (𝜓 ↔ 𝜒)) & ⊢ (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒 ↔ 𝜃)) & ⊢ (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃 ↔ 𝜏)) & ⊢ 𝜏 ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | dedth4v 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) → (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 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) → (𝜑 ↔ 𝜓)) & ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜓)) & ⊢ 𝜒 ⇒ ⊢ 𝜓 | ||
Theorem | elimhyp2v 4531 | 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 4532 | 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 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) → (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 4534 | 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 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.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜒)) & ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜃 ↔ 𝜒)) & ⊢ 𝜃 ⇒ ⊢ 𝜒 | ||
Theorem | keephyp 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) → (𝜓 ↔ 𝜃)) & ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃)) & ⊢ 𝜓 & ⊢ 𝜒 ⇒ ⊢ 𝜃 | ||
Theorem | keephyp2v 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) → (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 4538 | 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 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 | ||
Theorem | pwjust 4540* | Soundness justification theorem for df-pw 4541. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ {𝑥 ∣ 𝑥 ⊆ 𝐴} = {𝑦 ∣ 𝑦 ⊆ 𝐴} | ||
Definition | df-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 = {𝑥 ∣ 𝑥 ⊆ 𝐴} | ||
Theorem | elpwg 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.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) | ||
Theorem | elpw 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 𝐵 ↔ 𝐴 ⊆ 𝐵) | ||
Theorem | velpw 4544 | Setvar variable membership in a power class. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | ||
Theorem | elpwOLD 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 𝐵 ↔ 𝐴 ⊆ 𝐵) | ||
Theorem | elpwgOLD 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.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) | ||
Theorem | elpwd 4547 | Membership in a power class. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵) | ||
Theorem | elpwi 4548 | Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.) |
⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ⊆ 𝐵) | ||
Theorem | elpwb 4549 | Characterization of the elements of a power class. (Contributed by BJ, 29-Apr-2021.) |
⊢ (𝐴 ∈ 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵)) | ||
Theorem | elpwid 4550 | An element of a power class is a subclass. Deduction form of elpwi 4548. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | ||
Theorem | elelpwi 4551 | 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 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.) |
⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) | ||
Theorem | sspwi 4553 | The powerclass preserves inclusion (inference form). (Contributed by BJ, 13-Apr-2024.) |
⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ 𝒫 𝐴 ⊆ 𝒫 𝐵 | ||
Theorem | sspwd 4554 | The powerclass preserves inclusion (deduction form). (Contributed by BJ, 13-Apr-2024.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → 𝒫 𝐴 ⊆ 𝒫 𝐵) | ||
Theorem | pweq 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) | ||
Theorem | pweqALT 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) | ||
Theorem | pweqi 4557 | Equality inference for power class. (Contributed by NM, 27-Nov-2013.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ 𝒫 𝐴 = 𝒫 𝐵 | ||
Theorem | pweqd 4558 | Equality deduction for power class. (Contributed by NM, 27-Nov-2013.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 𝒫 𝐴 = 𝒫 𝐵) | ||
Theorem | pwunss 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 (𝐴 ∪ 𝐵) | ||
Theorem | nfpw 4560 | 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 4561 | A set is an element of its power set. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴) | ||
Theorem | pwidb 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) | ||
Theorem | pwid 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 | ||
Theorem | pwss 4564* | Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.) |
⊢ (𝒫 𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐵)) | ||
Theorem | pwundif 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 (𝐴 ∪ 𝐵) = ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴) | ||
Theorem | snjust 4566* | Soundness justification theorem for df-sn 4568. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ {𝑥 ∣ 𝑥 = 𝐴} = {𝑦 ∣ 𝑦 = 𝐴} | ||
Syntax | csn 4567 | Extend class notation to include singleton. |
class {𝐴} | ||
Definition | df-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} = {𝑥 ∣ 𝑥 = 𝐴} | ||
Syntax | cpr 4569 | Extend class notation to include unordered pair. |
class {𝐴, 𝐵} | ||
Definition | df-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 𝐴 ≠ 𝐵 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}) | ||
Syntax | ctp 4571 | Extend class notation to include unordered triple (sometimes called "unordered triplet"). |
class {𝐴, 𝐵, 𝐶} | ||
Definition | df-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}) | ||
Syntax | cop 4573 | Extend class notation to include ordered pair. |
class 〈𝐴, 𝐵〉 | ||
Definition | df-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, 𝐵〉2 = {{{𝐴}, ∅}, {{𝐵}}}, justified by opthwiener 5431. 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 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, 𝐵〉4 = ((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, 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 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}})} | ||
Syntax | cotp 4575 | Extend class notation to include ordered triple. |
class 〈𝐴, 𝐵, 𝐶〉 | ||
Definition | df-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〉 | ||
Theorem | sneq 4577 | Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.) |
⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | ||
Theorem | sneqi 4578 | Equality inference for singletons. (Contributed by NM, 22-Jan-2004.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ {𝐴} = {𝐵} | ||
Theorem | sneqd 4579 | Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → {𝐴} = {𝐵}) | ||
Theorem | dfsn2 4580 | Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) |
⊢ {𝐴} = {𝐴, 𝐴} | ||
Theorem | elsng 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.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)) | ||
Theorem | elsn 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) | ||
Theorem | velsn 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) | ||
Theorem | elsni 4584 | There is at most one element in a singleton. (Contributed by NM, 5-Jun-1994.) |
⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵) | ||
Theorem | absn 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.) |
⊢ ({𝑥 ∣ 𝜑} = {𝑌} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑌)) | ||
Theorem | dfpr2 4586* | Alternate definition of a pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.) |
⊢ {𝐴, 𝐵} = {𝑥 ∣ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)} | ||
Theorem | dfsn2ALT 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} | ||
Theorem | elprg 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.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))) | ||
Theorem | elpri 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 = 𝐵 ∨ 𝐴 = 𝐶)) | ||
Theorem | elpr 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 = 𝐵 ∨ 𝐴 = 𝐶)) | ||
Theorem | elpr2g 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.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))) | ||
Theorem | elpr2 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 = 𝐵 ∨ 𝐴 = 𝐶)) | ||
Theorem | elpr2OLD 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 = 𝐵 ∨ 𝐴 = 𝐶)) | ||
Theorem | nelpr2 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.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶}) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐶) | ||
Theorem | nelpr1 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.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶}) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐵) | ||
Theorem | nelpri 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.) |
⊢ 𝐴 ≠ 𝐵 & ⊢ 𝐴 ≠ 𝐶 ⇒ ⊢ ¬ 𝐴 ∈ {𝐵, 𝐶} | ||
Theorem | prneli 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.) |
⊢ 𝐴 ≠ 𝐵 & ⊢ 𝐴 ≠ 𝐶 ⇒ ⊢ 𝐴 ∉ {𝐵, 𝐶} | ||
Theorem | nelprd 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.) |
⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐴 ≠ 𝐶) ⇒ ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶}) | ||
Theorem | eldifpr 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}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷)) | ||
Theorem | rexdifpr 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 ↔ ∃𝑥 ∈ 𝐴 (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶 ∧ 𝜑)) |
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