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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 4191 | . . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ↔ 𝑥 ⊆ (𝐴 ∩ 𝐵)) | |
2 | velpw 4566 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
3 | velpw 4566 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵) | |
4 | 2, 3 | anbi12i 628 | . . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ 𝒫 𝐵) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵)) |
5 | velpw 4566 | . . . 4 ⊢ (𝑥 ∈ 𝒫 (𝐴 ∩ 𝐵) ↔ 𝑥 ⊆ (𝐴 ∩ 𝐵)) | |
6 | 1, 4, 5 | 3bitr4i 303 | . . 3 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ 𝒫 𝐵) ↔ 𝑥 ∈ 𝒫 (𝐴 ∩ 𝐵)) |
7 | 6 | ineqri 4165 | . 2 ⊢ (𝒫 𝐴 ∩ 𝒫 𝐵) = 𝒫 (𝐴 ∩ 𝐵) |
8 | 7 | eqcomi 2746 | 1 ⊢ 𝒫 (𝐴 ∩ 𝐵) = (𝒫 𝐴 ∩ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∩ cin 3910 ⊆ wss 3911 𝒫 cpw 4561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3448 df-in 3918 df-ss 3928 df-pw 4563 |
This theorem is referenced by: (None) |
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