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Mirrors > Home > MPE Home > Th. List > pwundifOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of pwundif 4564 as of 26-Dec-2023. (Contributed by NM, 25-Mar-2007.) (Proof shortened by Thierry Arnoux, 20-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pwundifOLD | ⊢ 𝒫 (𝐴 ∪ 𝐵) = ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | undif1 4423 | . 2 ⊢ ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴) = (𝒫 (𝐴 ∪ 𝐵) ∪ 𝒫 𝐴) | |
2 | pwunss 4558 | . . . . 5 ⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵) | |
3 | unss 4159 | . . . . 5 ⊢ ((𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵) ∧ 𝒫 𝐵 ⊆ 𝒫 (𝐴 ∪ 𝐵)) ↔ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵)) | |
4 | 2, 3 | mpbir 233 | . . . 4 ⊢ (𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵) ∧ 𝒫 𝐵 ⊆ 𝒫 (𝐴 ∪ 𝐵)) |
5 | 4 | simpli 486 | . . 3 ⊢ 𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵) |
6 | ssequn2 4158 | . . 3 ⊢ (𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵) ↔ (𝒫 (𝐴 ∪ 𝐵) ∪ 𝒫 𝐴) = 𝒫 (𝐴 ∪ 𝐵)) | |
7 | 5, 6 | mpbi 232 | . 2 ⊢ (𝒫 (𝐴 ∪ 𝐵) ∪ 𝒫 𝐴) = 𝒫 (𝐴 ∪ 𝐵) |
8 | 1, 7 | eqtr2i 2845 | 1 ⊢ 𝒫 (𝐴 ∪ 𝐵) = ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ∖ cdif 3932 ∪ cun 3933 ⊆ wss 3935 𝒫 cpw 4538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-pw 4540 |
This theorem is referenced by: (None) |
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