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Mirrors > Home > ILE Home > Th. List > pwin | GIF version |
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.) |
Ref | Expression |
---|---|
pwin | ⊢ 𝒫 (𝐴 ∩ 𝐵) = (𝒫 𝐴 ∩ 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssin 3349 | . . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ↔ 𝑥 ⊆ (𝐴 ∩ 𝐵)) | |
2 | vex 2733 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | 2 | elpw 3570 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) |
4 | 2 | elpw 3570 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵) |
5 | 3, 4 | anbi12i 457 | . . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ 𝒫 𝐵) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵)) |
6 | 2 | elpw 3570 | . . . 4 ⊢ (𝑥 ∈ 𝒫 (𝐴 ∩ 𝐵) ↔ 𝑥 ⊆ (𝐴 ∩ 𝐵)) |
7 | 1, 5, 6 | 3bitr4i 211 | . . 3 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ 𝒫 𝐵) ↔ 𝑥 ∈ 𝒫 (𝐴 ∩ 𝐵)) |
8 | 7 | ineqri 3320 | . 2 ⊢ (𝒫 𝐴 ∩ 𝒫 𝐵) = 𝒫 (𝐴 ∩ 𝐵) |
9 | 8 | eqcomi 2174 | 1 ⊢ 𝒫 (𝐴 ∩ 𝐵) = (𝒫 𝐴 ∩ 𝒫 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1348 ∈ wcel 2141 ∩ cin 3120 ⊆ wss 3121 𝒫 cpw 3564 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-in 3127 df-ss 3134 df-pw 3566 |
This theorem is referenced by: (None) |
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