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Theorem pwin 5255
 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 4054 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ⊆ (𝐴𝐵))
2 selpw 4385 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
3 selpw 4385 . . . . 5 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
42, 3anbi12i 620 . . . 4 ((𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵) ↔ (𝑥𝐴𝑥𝐵))
5 selpw 4385 . . . 4 (𝑥 ∈ 𝒫 (𝐴𝐵) ↔ 𝑥 ⊆ (𝐴𝐵))
61, 4, 53bitr4i 295 . . 3 ((𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵) ↔ 𝑥 ∈ 𝒫 (𝐴𝐵))
76ineqri 4028 . 2 (𝒫 𝐴 ∩ 𝒫 𝐵) = 𝒫 (𝐴𝐵)
87eqcomi 2786 1 𝒫 (𝐴𝐵) = (𝒫 𝐴 ∩ 𝒫 𝐵)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 386   = wceq 1601   ∈ wcel 2106   ∩ cin 3790   ⊆ wss 3791  𝒫 cpw 4378 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-ext 2753 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-v 3399  df-in 3798  df-ss 3805  df-pw 4380 This theorem is referenced by: (None)
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