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Theorem List for Metamath Proof Explorer - 4301-4400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremralnralall 4301* A contradiction concerning restricted generalization for a nonempty set implies anything. (Contributed by Alexander van der Vekens, 4-Sep-2018.)
(𝐴 ≠ ∅ → ((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 ¬ 𝜑) → 𝜓))

Theoremfalseral0 4302* A false statement can only be true for elements of an empty set. (Contributed by AV, 30-Oct-2020.)
((∀𝑥 ¬ 𝜑 ∧ ∀𝑥𝐴 𝜑) → 𝐴 = ∅)

Theoremraaan 4303* Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.)
𝑦𝜑    &   𝑥𝜓       (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))

Theoremraaanv 4304* Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)
(∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))

Theoremsbss 4305* Set substitution into the first argument of a subset relation. (Contributed by Rodolfo Medina, 7-Jul-2010.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)

Theoremsbcssg 4306 Distribute proper substitution through a subclass relation. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Alexander van der Vekens, 23-Jul-2017.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))

2.1.15  The conditional operator for classes

This subsection introduces the conditional operator for classes, denoted by if(𝜑, 𝐴, 𝐵) (see df-if 4308). It is the analogue for classes of the conditional operator for propositions, denoted by if-(𝜑, 𝜓, 𝜒) (see df-ifp 1047).

Syntaxcif 4307 Extend class notation to include the conditional operator for classes.
class if(𝜑, 𝐴, 𝐵)

Definitiondf-if 4308* Definition of the conditional operator for classes. The expression if(𝜑, 𝐴, 𝐵) is read "if 𝜑 then 𝐴 else 𝐵". See iftrue 4313 and iffalse 4316 for its values. In the mathematical literature, this operator is rarely defined formally but is implicit in informal definitions such as "let f(x)=0 if x=0 and 1/x otherwise".

An important use for us is in conjunction with the weak deduction theorem, which is described in the next section, beginning at dedth 4363. (Contributed by NM, 15-May-1999.)

if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}

Theoremdfif2 4309* An alternate definition of the conditional operator df-if 4308 with one fewer connectives (but probably less intuitive to understand). (Contributed by NM, 30-Jan-2006.)
if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐵𝜑) → (𝑥𝐴𝜑))}

Theoremdfif6 4310* An alternate definition of the conditional operator df-if 4308 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
if(𝜑, 𝐴, 𝐵) = ({𝑥𝐴𝜑} ∪ {𝑥𝐵 ∣ ¬ 𝜑})

Theoremifeq1 4311 Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
(𝐴 = 𝐵 → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐶))

Theoremifeq2 4312 Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
(𝐴 = 𝐵 → if(𝜑, 𝐶, 𝐴) = if(𝜑, 𝐶, 𝐵))

Theoremiftrue 4313 Value of the conditional operator when its first argument is true. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)

Theoremiftruei 4314 Inference associated with iftrue 4313. (Contributed by BJ, 7-Oct-2018.)
𝜑       if(𝜑, 𝐴, 𝐵) = 𝐴

Theoremiftrued 4315 Value of the conditional operator when its first argument is true. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝜒)       (𝜑 → if(𝜒, 𝐴, 𝐵) = 𝐴)

Theoremiffalse 4316 Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.)
𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)

Theoremiffalsei 4317 Inference associated with iffalse 4316. (Contributed by BJ, 7-Oct-2018.)
¬ 𝜑       if(𝜑, 𝐴, 𝐵) = 𝐵

Theoremiffalsed 4318 Value of the conditional operator when its first argument is false. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → ¬ 𝜒)       (𝜑 → if(𝜒, 𝐴, 𝐵) = 𝐵)

Theoremifnefalse 4319 When values are unequal, but an "if" condition checks if they are equal, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs versus applying iffalse 4316 directly in this case. It happens, e.g., in oevn0 7881. (Contributed by David A. Wheeler, 15-May-2015.)
(𝐴𝐵 → if(𝐴 = 𝐵, 𝐶, 𝐷) = 𝐷)

Theoremifsb 4320 Distribute a function over an if-clause. (Contributed by Mario Carneiro, 14-Aug-2013.)
(if(𝜑, 𝐴, 𝐵) = 𝐴𝐶 = 𝐷)    &   (if(𝜑, 𝐴, 𝐵) = 𝐵𝐶 = 𝐸)       𝐶 = if(𝜑, 𝐷, 𝐸)

Theoremdfif3 4321* Alternate definition of the conditional operator df-if 4308. Note that 𝜑 is independent of 𝑥 i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.) (Revised by Mario Carneiro, 8-Sep-2013.)
𝐶 = {𝑥𝜑}       if(𝜑, 𝐴, 𝐵) = ((𝐴𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶)))

Theoremdfif4 4322* Alternate definition of the conditional operator df-if 4308. Note that 𝜑 is independent of 𝑥 i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.)
𝐶 = {𝑥𝜑}       if(𝜑, 𝐴, 𝐵) = ((𝐴𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵𝐶)))

Theoremdfif5 4323* Alternate definition of the conditional operator df-if 4308. Note that 𝜑 is independent of 𝑥 i.e. a constant true or false (see also ab0orv 4184). (Contributed by Gérard Lang, 18-Aug-2013.)
𝐶 = {𝑥𝜑}       if(𝜑, 𝐴, 𝐵) = ((𝐴𝐵) ∪ (((𝐴𝐵) ∩ 𝐶) ∪ ((𝐵𝐴) ∩ (V ∖ 𝐶))))

Theoremifeq12 4324 Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.)
((𝐴 = 𝐵𝐶 = 𝐷) → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐷))

Theoremifeq1d 4325 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
(𝜑𝐴 = 𝐵)       (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))

Theoremifeq2d 4326 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
(𝜑𝐴 = 𝐵)       (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))

Theoremifeq12d 4327 Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐷))

Theoremifbi 4328 Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
((𝜑𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))

Theoremifbid 4329 Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.)
(𝜑 → (𝜓𝜒))       (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐴, 𝐵))

Theoremifbieq1d 4330 Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018.)
(𝜑 → (𝜓𝜒))    &   (𝜑𝐴 = 𝐵)       (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜒, 𝐵, 𝐶))

Theoremifbieq2i 4331 Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝜓)    &   𝐴 = 𝐵       if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)

Theoremifbieq2d 4332 Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑 → (𝜓𝜒))    &   (𝜑𝐴 = 𝐵)       (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐵))

Theoremifbieq12i 4333 Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.)
(𝜑𝜓)    &   𝐴 = 𝐶    &   𝐵 = 𝐷       if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷)

Theoremifbieq12d 4334 Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝜑 → (𝜓𝜒))    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷))

Theoremnfifd 4335 Deduction form of nfif 4336. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑𝑥if(𝜓, 𝐴, 𝐵))

Theoremnfif 4336 Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐵       𝑥if(𝜑, 𝐴, 𝐵)

Theoremifeq1da 4337 Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝜑𝜓) → 𝐴 = 𝐵)       (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))

Theoremifeq2da 4338 Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵)       (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))

Theoremifeq12da 4339 Equivalence deduction for conditional operators. (Contributed by Wolf Lammen, 24-Jun-2021.)
((𝜑𝜓) → 𝐴 = 𝐶)    &   ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷)       (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))

Theoremifbieq12d2 4340 Equivalence deduction for conditional operators. (Contributed by Thierry Arnoux, 14-Feb-2017.) (Proof shortened by Wolf Lammen, 24-Jun-2021.)
(𝜑 → (𝜓𝜒))    &   ((𝜑𝜓) → 𝐴 = 𝐶)    &   ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷)       (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷))

Theoremifclda 4341 Conditional closure. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝜑𝜓) → 𝐴𝐶)    &   ((𝜑 ∧ ¬ 𝜓) → 𝐵𝐶)       (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)

Theoremifeqda 4342 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 4343 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 4344 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 4345 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 4346 Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.)
if(𝜑, 𝐴, 𝐴) = 𝐴

Theoremeqif 4347 Expansion of an equality with a conditional operator. (Contributed by NM, 14-Feb-2005.)
(𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑𝐴 = 𝐵) ∨ (¬ 𝜑𝐴 = 𝐶)))

Theoremifval 4348 Another expression of the value of the if predicate, analogous to eqif 4347. See also the more specialized iftrue 4313 and iffalse 4316. (Contributed by BJ, 6-Apr-2019.)
(𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑𝐴 = 𝐵) ∧ (¬ 𝜑𝐴 = 𝐶)))

Theoremelif 4349 Membership in a conditional operator. (Contributed by NM, 14-Feb-2005.)
(𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ ((𝜑𝐴𝐵) ∨ (¬ 𝜑𝐴𝐶)))

Theoremifel 4350 Membership of a conditional operator. (Contributed by NM, 10-Sep-2005.)
(if(𝜑, 𝐴, 𝐵) ∈ 𝐶 ↔ ((𝜑𝐴𝐶) ∨ (¬ 𝜑𝐵𝐶)))

Theoremifcl 4351 Membership (closure) of a conditional operator. (Contributed by NM, 4-Apr-2005.)
((𝐴𝐶𝐵𝐶) → if(𝜑, 𝐴, 𝐵) ∈ 𝐶)

Theoremifcld 4352 Membership (closure) of a conditional operator, deduction form. (Contributed by SO, 16-Jul-2018.)
(𝜑𝐴𝐶)    &   (𝜑𝐵𝐶)       (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)

Theoremifcli 4353 Inference associated with ifcl 4351. Membership (closure) of a conditional operator. Also usable to keep a membership hypothesis for the weak deduction theorem dedth 4363 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

Theoremifexg 4354 Conditional operator existence. (Contributed by NM, 21-Mar-2011.) (Proof shortened by BJ, 1-Sep-2022.)
((𝐴𝑉𝐵𝑊) → if(𝜑, 𝐴, 𝐵) ∈ V)

Theoremifex 4355 Conditional operator existence. (Contributed by NM, 2-Sep-2004.)
𝐴 ∈ V    &   𝐵 ∈ V       if(𝜑, 𝐴, 𝐵) ∈ V

Theoremifeqor 4356 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 4357 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 4358 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 4359 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 4360 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 4361 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 4362 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 1064. For more information on the weak deduction theorem, see the Weak Deduction Theorem page mmdeduction.html.

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 4363 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 4370. If the inference has other hypotheses with class variable 𝐴, these can be kept by assigning keephyp 4376 to them. For more information, see the Weak Deduction Theorem page mmdeduction.html. (Contributed by NM, 15-May-1999.)
(𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜒))    &   𝜒       (𝜑𝜓)

Theoremdedth2h 4364 Weak deduction theorem eliminating two hypotheses. This theorem is simpler to use than dedth2v 4367 but requires that each hypothesis have exactly one class variable. See also comments in dedth 4363. (Contributed by NM, 15-May-1999.)
(𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜒𝜃))    &   (𝐵 = if(𝜓, 𝐵, 𝐷) → (𝜃𝜏))    &   𝜏       ((𝜑𝜓) → 𝜒)

Theoremdedth3h 4365 Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 4364. (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 4366 Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 4364. (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 4367 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 4364 is simpler to use. See also comments in dedth 4363. (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 4368 Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 4367. (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 4369 Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 4367. (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 4370 Eliminate a hypothesis containing class variable 𝐴 when it is known for a specific class 𝐵. For more information, see comments in dedth 4363. (Contributed by NM, 15-May-1999.)
(𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜑𝜓))    &   (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜓))    &   𝜒       𝜓

Theoremelimhyp2v 4371 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 4372 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 4373 Eliminate a hypothesis containing 4 class variables (for use with the weak deduction theorem dedth 4363). (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 4374 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 4375 Version of elimhyp 4370 where the hypothesis is deduced from the final antecedent. See divalg 15537 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 4376 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 4377 Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 4363). (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 4378 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 4379 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 4380* Soundness justification theorem for df-pw 4381. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
{𝑥𝑥𝐴} = {𝑦𝑦𝐴}

Definitiondf-pw 4381* 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 27865). We will later introduce the Axiom of Power Sets ax-pow 5079, which can be expressed in class notation per pwexg 5092. Still later we will prove, in hashpw 13541, 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}

Theorempweq 4382 Equality theorem for power class. (Contributed by NM, 21-Jun-1993.)
(𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵)

Theorempweqi 4383 Equality inference for power class. (Contributed by NM, 27-Nov-2013.)
𝐴 = 𝐵       𝒫 𝐴 = 𝒫 𝐵

Theorempweqd 4384 Equality deduction for power class. (Contributed by NM, 27-Nov-2013.)
(𝜑𝐴 = 𝐵)       (𝜑 → 𝒫 𝐴 = 𝒫 𝐵)

Theoremelpw 4385 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.)
𝐴 ∈ V       (𝐴 ∈ 𝒫 𝐵𝐴𝐵)

Theoremselpw 4386* Setvar variable membership in a power class. See elpw 4385. (Contributed by David A. Wheeler, 8-Dec-2018.)
(𝑥 ∈ 𝒫 𝐴𝑥𝐴)

Theoremelpwg 4387 Membership in a power class. Theorem 86 of [Suppes] p. 47. See also elpw2g 5063. (Contributed by NM, 6-Aug-2000.)
(𝐴𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))

Theoremelpwd 4388 Membership in a power class. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐴𝐵)       (𝜑𝐴 ∈ 𝒫 𝐵)

Theoremelpwi 4389 Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)
(𝐴 ∈ 𝒫 𝐵𝐴𝐵)

Theoremelpwb 4390 Characterization of the elements of a power class. (Contributed by BJ, 29-Apr-2021.)
(𝐴 ∈ 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴𝐵))

Theoremelpwid 4391 An element of a power class is a subclass. Deduction form of elpwi 4389. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ 𝒫 𝐵)       (𝜑𝐴𝐵)

Theoremelelpwi 4392 If 𝐴 belongs to a part of 𝐶 then 𝐴 belongs to 𝐶. (Contributed by FL, 3-Aug-2009.)
((𝐴𝐵𝐵 ∈ 𝒫 𝐶) → 𝐴𝐶)

Theoremnfpw 4393 Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
𝑥𝐴       𝑥𝒫 𝐴

Theorempwidg 4394 Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐴𝑉𝐴 ∈ 𝒫 𝐴)

Theorempwid 4395 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 4396* Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)
(𝒫 𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))

2.1.18  Unordered and ordered pairs

Theoremsnjust 4397* Soundness justification theorem for df-sn 4399. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
{𝑥𝑥 = 𝐴} = {𝑦𝑦 = 𝐴}

Syntaxcsn 4398 Extend class notation to include singleton.
class {𝐴}

Definitiondf-sn 4399* 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 4484. For an alternate definition see dfsn2 4411. (Contributed by NM, 21-Jun-1993.)
{𝐴} = {𝑥𝑥 = 𝐴}

Syntaxcpr 4400 Extend class notation to include unordered pair.
class {𝐴, 𝐵}

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