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Theorem pwundif 4629
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 5302, ax-nul 5312, ax-pr 5438 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 4188 . . . 4 𝐴 ⊆ (𝐴𝐵)
21sspwi 4617 . . 3 𝒫 𝐴 ⊆ 𝒫 (𝐴𝐵)
3 undif 4488 . . 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 4168 . 2 (𝒫 𝐴 ∪ (𝒫 (𝐴𝐵) ∖ 𝒫 𝐴)) = ((𝒫 (𝐴𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)
64, 5eqtr3i 2765 1 𝒫 (𝐴𝐵) = ((𝒫 (𝐴𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  cdif 3960  cun 3961  wss 3963  𝒫 cpw 4605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-pw 4607
This theorem is referenced by:  pwfilem  9354
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