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Theorem pwundif 4588
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.)
Assertion
Ref Expression
pwundif 𝒫 (𝐴𝐵) = ((𝒫 (𝐴𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)

Proof of Theorem pwundif
StepHypRef Expression
1 ssun1 4136 . . . 4 𝐴 ⊆ (𝐴𝐵)
21sspwi 4576 . . 3 𝒫 𝐴 ⊆ 𝒫 (𝐴𝐵)
3 undif 4445 . . 3 (𝒫 𝐴 ⊆ 𝒫 (𝐴𝐵) ↔ (𝒫 𝐴 ∪ (𝒫 (𝐴𝐵) ∖ 𝒫 𝐴)) = 𝒫 (𝐴𝐵))
42, 3mpbi 229 . 2 (𝒫 𝐴 ∪ (𝒫 (𝐴𝐵) ∖ 𝒫 𝐴)) = 𝒫 (𝐴𝐵)
5 uncom 4117 . 2 (𝒫 𝐴 ∪ (𝒫 (𝐴𝐵) ∖ 𝒫 𝐴)) = ((𝒫 (𝐴𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)
64, 5eqtr3i 2763 1 𝒫 (𝐴𝐵) = ((𝒫 (𝐴𝐵) ∖ 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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