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Theorem pwundif 4623
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 5295, ax-nul 5305, ax-pr 5431 and shorten proof. (Revised by BJ, 14-Apr-2024.)
Assertion
Ref Expression
pwundif 𝒫 (𝐴𝐵) = ((𝒫 (𝐴𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)

Proof of Theorem pwundif
StepHypRef Expression
1 ssun1 4177 . . . 4 𝐴 ⊆ (𝐴𝐵)
21sspwi 4611 . . 3 𝒫 𝐴 ⊆ 𝒫 (𝐴𝐵)
3 undif 4481 . . 3 (𝒫 𝐴 ⊆ 𝒫 (𝐴𝐵) ↔ (𝒫 𝐴 ∪ (𝒫 (𝐴𝐵) ∖ 𝒫 𝐴)) = 𝒫 (𝐴𝐵))
42, 3mpbi 230 . 2 (𝒫 𝐴 ∪ (𝒫 (𝐴𝐵) ∖ 𝒫 𝐴)) = 𝒫 (𝐴𝐵)
5 uncom 4157 . 2 (𝒫 𝐴 ∪ (𝒫 (𝐴𝐵) ∖ 𝒫 𝐴)) = ((𝒫 (𝐴𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)
64, 5eqtr3i 2766 1 𝒫 (𝐴𝐵) = ((𝒫 (𝐴𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  cdif 3947  cun 3948  wss 3950  𝒫 cpw 4599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-pw 4601
This theorem is referenced by:  pwfilem  9357
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