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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 5203, ax-nul 5210, ax-pr 5330 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 4148 | . . . 4 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) | |
2 | 1 | sspwi 4553 | . . 3 ⊢ 𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵) |
3 | undif 4430 | . . 3 ⊢ (𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵) ↔ (𝒫 𝐴 ∪ (𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴)) = 𝒫 (𝐴 ∪ 𝐵)) | |
4 | 2, 3 | mpbi 232 | . 2 ⊢ (𝒫 𝐴 ∪ (𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴)) = 𝒫 (𝐴 ∪ 𝐵) |
5 | uncom 4129 | . 2 ⊢ (𝒫 𝐴 ∪ (𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴)) = ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴) | |
6 | 4, 5 | eqtr3i 2846 | 1 ⊢ 𝒫 (𝐴 ∪ 𝐵) = ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∖ cdif 3933 ∪ cun 3934 ⊆ wss 3936 𝒫 cpw 4539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-pw 4541 |
This theorem is referenced by: pwfilem 8818 |
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