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Theorem pwundif 4539
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 5192, ax-nul 5199, ax-pr 5322 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 4086 . . . 4 𝐴 ⊆ (𝐴𝐵)
21sspwi 4527 . . 3 𝒫 𝐴 ⊆ 𝒫 (𝐴𝐵)
3 undif 4396 . . 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 4067 . 2 (𝒫 𝐴 ∪ (𝒫 (𝐴𝐵) ∖ 𝒫 𝐴)) = ((𝒫 (𝐴𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)
64, 5eqtr3i 2767 1 𝒫 (𝐴𝐵) = ((𝒫 (𝐴𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1543  cdif 3863  cun 3864  wss 3866  𝒫 cpw 4513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-pw 4515
This theorem is referenced by:  pwfilem  8855  pwfilemOLD  8970
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