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Theorem pwin 5441
 Description: The power class of the intersection of two classes is the intersection of their power classes. Exercise 4.12(j) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.)
Assertion
Ref Expression
pwin 𝒫 (𝐴𝐵) = (𝒫 𝐴 ∩ 𝒫 𝐵)

Proof of Theorem pwin
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ssin 4192 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ⊆ (𝐴𝐵))
2 velpw 4527 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
3 velpw 4527 . . . . 5 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
42, 3anbi12i 629 . . . 4 ((𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵) ↔ (𝑥𝐴𝑥𝐵))
5 velpw 4527 . . . 4 (𝑥 ∈ 𝒫 (𝐴𝐵) ↔ 𝑥 ⊆ (𝐴𝐵))
61, 4, 53bitr4i 306 . . 3 ((𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵) ↔ 𝑥 ∈ 𝒫 (𝐴𝐵))
76ineqri 4165 . 2 (𝒫 𝐴 ∩ 𝒫 𝐵) = 𝒫 (𝐴𝐵)
87eqcomi 2833 1 𝒫 (𝐴𝐵) = (𝒫 𝐴 ∩ 𝒫 𝐵)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1538   ∈ wcel 2115   ∩ cin 3918   ⊆ wss 3919  𝒫 cpw 4522 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-in 3926  df-ss 3936  df-pw 4524 This theorem is referenced by: (None)
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