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Theorem pwin 4046
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 3186 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ⊆ (𝐴𝐵))
2 vex 2577 . . . . . 6 𝑥 ∈ V
32elpw 3392 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
42elpw 3392 . . . . 5 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
53, 4anbi12i 441 . . . 4 ((𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵) ↔ (𝑥𝐴𝑥𝐵))
62elpw 3392 . . . 4 (𝑥 ∈ 𝒫 (𝐴𝐵) ↔ 𝑥 ⊆ (𝐴𝐵))
71, 5, 63bitr4i 205 . . 3 ((𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵) ↔ 𝑥 ∈ 𝒫 (𝐴𝐵))
87ineqri 3157 . 2 (𝒫 𝐴 ∩ 𝒫 𝐵) = 𝒫 (𝐴𝐵)
98eqcomi 2060 1 𝒫 (𝐴𝐵) = (𝒫 𝐴 ∩ 𝒫 𝐵)
Colors of variables: wff set class
Syntax hints:  wa 101   = wceq 1259  wcel 1409  cin 2943  wss 2944  𝒫 cpw 3386
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2951  df-ss 2958  df-pw 3388
This theorem is referenced by: (None)
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