Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > pwin | Structured version Visualization version 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 3797 | . . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ↔ 𝑥 ⊆ (𝐴 ∩ 𝐵)) | |
2 | selpw 4115 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
3 | selpw 4115 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵) | |
4 | 2, 3 | anbi12i 729 | . . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ 𝒫 𝐵) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵)) |
5 | selpw 4115 | . . . 4 ⊢ (𝑥 ∈ 𝒫 (𝐴 ∩ 𝐵) ↔ 𝑥 ⊆ (𝐴 ∩ 𝐵)) | |
6 | 1, 4, 5 | 3bitr4i 291 | . . 3 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ 𝒫 𝐵) ↔ 𝑥 ∈ 𝒫 (𝐴 ∩ 𝐵)) |
7 | 6 | ineqri 3768 | . 2 ⊢ (𝒫 𝐴 ∩ 𝒫 𝐵) = 𝒫 (𝐴 ∩ 𝐵) |
8 | 7 | eqcomi 2619 | 1 ⊢ 𝒫 (𝐴 ∩ 𝐵) = (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) |
Copyright terms: Public domain | W3C validator |