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Theorem pwin 4942
 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 3797 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ⊆ (𝐴𝐵))
2 selpw 4115 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
3 selpw 4115 . . . . 5 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
42, 3anbi12i 729 . . . 4 ((𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵) ↔ (𝑥𝐴𝑥𝐵))
5 selpw 4115 . . . 4 (𝑥 ∈ 𝒫 (𝐴𝐵) ↔ 𝑥 ⊆ (𝐴𝐵))
61, 4, 53bitr4i 291 . . 3 ((𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵) ↔ 𝑥 ∈ 𝒫 (𝐴𝐵))
76ineqri 3768 . 2 (𝒫 𝐴 ∩ 𝒫 𝐵) = 𝒫 (𝐴𝐵)
87eqcomi 2619 1 𝒫 (𝐴𝐵) = (𝒫 𝐴 ∩ 𝒫 𝐵)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110 This theorem is referenced by: (None)
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