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Mirrors > Home > MPE Home > Th. List > pwundif | Structured version Visualization version GIF version |
Description: Break up the power class of a union into a union of smaller classes. (Contributed by NM, 25-Mar-2007.) (Proof shortened by Thierry Arnoux, 20-Dec-2016.) Remove use of ax-sep 5260, ax-nul 5267, ax-pr 5388 and shorten proof. (Revised by BJ, 14-Apr-2024.) |
Ref | Expression |
---|---|
pwundif | ⊢ 𝒫 (𝐴 ∪ 𝐵) = ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun1 4136 | . . . 4 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
2 | 1 | sspwi 4576 | . . 3 ⊢ 𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵) |
3 | undif 4445 | . . 3 ⊢ (𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵) ↔ (𝒫 𝐴 ∪ (𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴)) = 𝒫 (𝐴 ∪ 𝐵)) | |
4 | 2, 3 | mpbi 229 | . 2 ⊢ (𝒫 𝐴 ∪ (𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴)) = 𝒫 (𝐴 ∪ 𝐵) |
5 | uncom 4117 | . 2 ⊢ (𝒫 𝐴 ∪ (𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴)) = ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴) | |
6 | 4, 5 | eqtr3i 2763 | 1 ⊢ 𝒫 (𝐴 ∪ 𝐵) = ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∖ cdif 3911 ∪ cun 3912 ⊆ wss 3914 𝒫 cpw 4564 |
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 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-pw 4566 |
This theorem is referenced by: pwfilem 9127 pwfilemOLD 9296 |
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