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Theorem pwin 5579
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 4247 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ⊆ (𝐴𝐵))
2 velpw 4610 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
3 velpw 4610 . . . . 5 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
42, 3anbi12i 628 . . . 4 ((𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵) ↔ (𝑥𝐴𝑥𝐵))
5 velpw 4610 . . . 4 (𝑥 ∈ 𝒫 (𝐴𝐵) ↔ 𝑥 ⊆ (𝐴𝐵))
61, 4, 53bitr4i 303 . . 3 ((𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵) ↔ 𝑥 ∈ 𝒫 (𝐴𝐵))
76ineqri 4220 . 2 (𝒫 𝐴 ∩ 𝒫 𝐵) = 𝒫 (𝐴𝐵)
87eqcomi 2744 1 𝒫 (𝐴𝐵) = (𝒫 𝐴 ∩ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wcel 2106  cin 3962  wss 3963  𝒫 cpw 4605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-in 3970  df-ss 3980  df-pw 4607
This theorem is referenced by: (None)
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