P. 46 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 4501-4600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dfif3 4501*	Alternate definition of the conditional operator df-if 4487. 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 = {𝑥 ∣ 𝜑}    ⇒   ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶)))
Theorem	dfif4 4502*	Alternate definition of the conditional operator df-if 4487. Note that 𝜑 is independent of 𝑥 i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.)
⊢ 𝐶 = {𝑥 ∣ 𝜑}    ⇒   ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∪ 𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵 ∪ 𝐶)))
Theorem	dfif5 4503*	Alternate definition of the conditional operator df-if 4487. Note that 𝜑 is independent of 𝑥 i.e. a constant true or false (see also ab0orv 4338). (Contributed by Gérard Lang, 18-Aug-2013.)
⊢ 𝐶 = {𝑥 ∣ 𝜑}    ⇒   ⊢ if(𝜑, 𝐴, 𝐵) = ((𝐴 ∩ 𝐵) ∪ (((𝐴 ∖ 𝐵) ∩ 𝐶) ∪ ((𝐵 ∖ 𝐴) ∩ (V ∖ 𝐶))))
Theorem	ifssun 4504	A conditional class is included in the union of its two alternatives. (Contributed by BJ, 15-Aug-2024.)
⊢ if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵)
Theorem	ifeq12 4505	Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐷))
Theorem	ifeq1d 4506	Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
Theorem	ifeq2d 4507	Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
Theorem	ifeq12d 4508	Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐷))
Theorem	ifbi 4509	Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
Theorem	ifbid 4510	Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐴, 𝐵))
Theorem	ifbieq1d 4511	Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜒, 𝐵, 𝐶))
Theorem	ifbieq2i 4512	Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ 𝐴 = 𝐵    ⇒   ⊢ if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)
Theorem	ifbieq2d 4513	Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐵))
Theorem	ifbieq12i 4514	Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷)
Theorem	ifbieq12d 4515	Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷))
Theorem	nfifd 4516	Deduction form of nfif 4517. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥if(𝜓, 𝐴, 𝐵))
Theorem	nfif 4517	Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥if(𝜑, 𝐴, 𝐵)
Theorem	ifeq1da 4518	Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
Theorem	ifeq2da 4519	Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
Theorem	ifeq12da 4520	Equivalence deduction for conditional operators. (Contributed by Wolf Lammen, 24-Jun-2021.)
⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶)    &   ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
Theorem	ifbieq12d2 4521	Equivalence deduction for conditional operators. (Contributed by Thierry Arnoux, 14-Feb-2017.) (Proof shortened by Wolf Lammen, 24-Jun-2021.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶)    &   ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷))
Theorem	ifclda 4522	Conditional closure. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 ∈ 𝐶)    ⇒   ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)
Theorem	ifeqda 4523	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 4524	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 4525	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 4526	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 4527	Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.)
⊢ if(𝜑, 𝐴, 𝐴) = 𝐴
Theorem	eqif 4528	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 4529	Another expression of the value of the if predicate, analogous to eqif 4528. See also the more specialized iftrue 4492 and iffalse 4495. (Contributed by BJ, 6-Apr-2019.)
⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 → 𝐴 = 𝐵) ∧ (¬ 𝜑 → 𝐴 = 𝐶)))
Theorem	elif 4530	Membership in a conditional operator. (Contributed by NM, 14-Feb-2005.)
⊢ (𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ ((𝜑 ∧ 𝐴 ∈ 𝐵) ∨ (¬ 𝜑 ∧ 𝐴 ∈ 𝐶)))
Theorem	ifel 4531	Membership of a conditional operator. (Contributed by NM, 10-Sep-2005.)
⊢ (if(𝜑, 𝐴, 𝐵) ∈ 𝐶 ↔ ((𝜑 ∧ 𝐴 ∈ 𝐶) ∨ (¬ 𝜑 ∧ 𝐵 ∈ 𝐶)))
Theorem	ifcl 4532	Membership (closure) of a conditional operator. (Contributed by NM, 4-Apr-2005.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → if(𝜑, 𝐴, 𝐵) ∈ 𝐶)
Theorem	ifcld 4533	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 4534	Inference associated with ifcl 4532. Membership (closure) of a conditional operator. Also usable to keep a membership hypothesis for the weak deduction theorem dedth 4545 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 4535	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 4536	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 4537	Existence of the conditional operator (inference form). (Contributed by NM, 2-Sep-2004.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ if(𝜑, 𝐴, 𝐵) ∈ V
Theorem	ifeqor 4538	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 4539	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 4540	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 4541	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 4542	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 4543	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 4544	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)
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 1097. For more information on the weak deduction theorem, see the Weak Deduction Theorem page mmdeduction.html 1097. 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 4545	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 4552. If the inference has other hypotheses with class variable 𝐴, these can be kept by assigning keephyp 4558 to them. For more information, see the Weak Deduction Theorem page mmdeduction.html 4558. (Contributed by NM, 15-May-1999.)
⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜒))    &   ⊢ 𝜒    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	dedth2h 4546	Weak deduction theorem eliminating two hypotheses. This theorem is simpler to use than dedth2v 4549 but requires that each hypothesis have exactly one class variable. See also comments in dedth 4545. (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 4547	Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 4546. (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 4548	Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 4546. (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 4549	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 4546 is simpler to use. See also comments in dedth 4545. (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 4550	Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 4549. (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 4551	Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 4549. (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 4552	Eliminate a hypothesis containing class variable 𝐴 when it is known for a specific class 𝐵. For more information, see comments in dedth 4545. (Contributed by NM, 15-May-1999.)
⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜑 ↔ 𝜓))    &   ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜓))    &   ⊢ 𝜒    ⇒   ⊢ 𝜓
Theorem	elimhyp2v 4553	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 4554	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 4555	Eliminate a hypothesis containing 4 class variables (for use with the weak deduction theorem dedth 4545). (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 4556	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 4557	Version of elimhyp 4552 where the hypothesis is deduced from the final antecedent. See divalg 16460 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 4558	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 4559	Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 4545). (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 4560	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
2.1.17  Power classes
Syntax	cpw 4561	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 4562*	Soundness justification theorem for df-pw 4563. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ {𝑥 ∣ 𝑥 ⊆ 𝐴} = {𝑦 ∣ 𝑦 ⊆ 𝐴}
Definition	df-pw 4563*	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 30746). We will later introduce the Axiom of Power Sets ax-pow 5336, which can be expressed in class notation per pwexg 5349. Still later we will prove, in hashpw 14473, 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 4564	Membership in a power class. Theorem 86 of [Suppes] p. 47. See also elpw2g 5303. (Contributed by NM, 6-Aug-2000.) (Proof shortened by BJ, 31-Dec-2023.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
Theorem	elpw 4565	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 4566	Setvar variable membership in a power class. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
Theorem	elpwd 4567	Membership in a power class. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵)
Theorem	elpwi 4568	Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)
⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ⊆ 𝐵)
Theorem	elpwb 4569	Characterization of the elements of a power class. (Contributed by BJ, 29-Apr-2021.)
⊢ (𝐴 ∈ 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵))
Theorem	elpwid 4570	An element of a power class is a subclass. Deduction form of elpwi 4568. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
Theorem	elelpwi 4571	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 4572	The powerclass preserves inclusion. See sspwb 5430 for the biconditional version. (Contributed by NM, 13-Oct-1996.) Extract forward implication of sspwb 5430 since it requires fewer axioms. (Revised by BJ, 13-Apr-2024.)
⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
Theorem	sspwi 4573	The powerclass preserves inclusion (inference form). (Contributed by BJ, 13-Apr-2024.)
⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ 𝒫 𝐴 ⊆ 𝒫 𝐵
Theorem	sspwd 4574	The powerclass preserves inclusion (deduction form). (Contributed by BJ, 13-Apr-2024.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
Theorem	pweq 4575	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 4576	Alternate proof of pweq 4575 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 4577	Equality inference for power class. (Contributed by NM, 27-Nov-2013.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ 𝒫 𝐴 = 𝒫 𝐵
Theorem	pweqd 4578	Equality deduction for power class. (Contributed by NM, 27-Nov-2013.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → 𝒫 𝐴 = 𝒫 𝐵)
Theorem	pwunss 4579	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 5256, ax-nul 5268, ax-pr 5404 and shorten proof. (Revised by BJ, 13-Apr-2024.)
⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵)
Theorem	nfpw 4580	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 4581	A set is an element of its power set. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴)
Theorem	pwidgOLD 4582	Obsolete version of pwidg 4581 as of 10-Jun-2026. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴)
Theorem	pwidb 4583	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 4584	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 4585*	Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)
⊢ (𝒫 𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐵))
Theorem	pwundif 4586	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 5256, ax-nul 5268, ax-pr 5404 and shorten proof. (Revised by BJ, 14-Apr-2024.)
⊢ 𝒫 (𝐴 ∪ 𝐵) = ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)
2.1.18  Unordered and ordered pairs
Theorem	snjust 4587*	Soundness justification theorem for df-sn 4589. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ {𝑥 ∣ 𝑥 = 𝐴} = {𝑦 ∣ 𝑦 = 𝐴}
Syntax	csn 4588	Extend class notation to include singleton.
class {𝐴}
Definition	df-sn 4589*	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 4682. For an alternate definition see dfsn2 4601. (Contributed by NM, 21-Jun-1993.)
⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴}
Syntax	cpr 4590	Extend class notation to include unordered pair.
class {𝐴, 𝐵}
Definition	df-pr 4591	Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. For example, 𝐴 ∈ {1, -1} → (𝐴↑2) = 1 (ex-pr 30747). They are unordered, so {𝐴, 𝐵} = {𝐵, 𝐴} as proven by prcom 4697. For a more traditional definition, but requiring a dummy variable, see dfpr2 4609. {𝐴, 𝐴} is also an unordered pair, but also a singleton because of {𝐴} = {𝐴, 𝐴} (see dfsn2 4601). 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 4595. 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 4592	Extend class notation to include unordered triple (sometimes called "unordered triplet").
class {𝐴, 𝐵, 𝐶}
Definition	df-tp 4593	Define unordered triple of classes. Definition of [Enderton] p. 19. Note: ordered triples are a completely different object defined below in df-ot 4597. 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 4594	Extend class notation to include ordered pair.
class ⟨𝐴, 𝐵⟩
Definition	df-op 4595*	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 4861, opprc2 4862, and 0nelop 5479). For Kuratowski's actual definition when the arguments are sets, see dfop 4836. For the justifying theorem (for sets) see opth 5458. See dfopif 4834 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 4595 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 4595 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 5497. 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 9586, 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 5725. 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 5328). 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 5501 and opthhausdorff 5500. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 14306. An ordered pair of real numbers can also be represented by a complex number as shown by cru 12209. 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 4834. (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 4596	Extend class notation to include ordered triple.
class ⟨𝐴, 𝐵, 𝐶⟩
Definition	df-ot 4597	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 4598	Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.)
⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})
Theorem	sneqi 4599	Equality inference for singletons. (Contributed by NM, 22-Jan-2004.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ {𝐴} = {𝐵}
Theorem	sneqd 4600	Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → {𝐴} = {𝐵})