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Theorem pwundif 4526
 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 5171, ax-nul 5178, ax-pr 5299 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 4102 . . . 4 𝐴 ⊆ (𝐴𝐵)
21sspwi 4514 . . 3 𝒫 𝐴 ⊆ 𝒫 (𝐴𝐵)
3 undif 4391 . . 3 (𝒫 𝐴 ⊆ 𝒫 (𝐴𝐵) ↔ (𝒫 𝐴 ∪ (𝒫 (𝐴𝐵) ∖ 𝒫 𝐴)) = 𝒫 (𝐴𝐵))
42, 3mpbi 233 . 2 (𝒫 𝐴 ∪ (𝒫 (𝐴𝐵) ∖ 𝒫 𝐴)) = 𝒫 (𝐴𝐵)
5 uncom 4083 . 2 (𝒫 𝐴 ∪ (𝒫 (𝐴𝐵) ∖ 𝒫 𝐴)) = ((𝒫 (𝐴𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)
64, 5eqtr3i 2823 1 𝒫 (𝐴𝐵) = ((𝒫 (𝐴𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∖ cdif 3880   ∪ cun 3881   ⊆ wss 3883  𝒫 cpw 4500 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-rab 3115  df-v 3444  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-pw 4502 This theorem is referenced by:  pwfilem  8820
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