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 (𝐴 ∪ 𝐵) = ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)

Proof of Theorem pwundif

Step	Hyp	Ref	Expression
1		ssun1 4086	. . . 4 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
2	1	sspwi 4527	. . 3 ⊢ 𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵)
3		undif 4396	. . 3 ⊢ (𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵) ↔ (𝒫 𝐴 ∪ (𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴)) = 𝒫 (𝐴 ∪ 𝐵))
4	2, 3	mpbi 233	. 2 ⊢ (𝒫 𝐴 ∪ (𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴)) = 𝒫 (𝐴 ∪ 𝐵)
5		uncom 4067	. 2 ⊢ (𝒫 𝐴 ∪ (𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴)) = ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)
6	4, 5	eqtr3i 2767	1 ⊢ 𝒫 (𝐴 ∪ 𝐵) = ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)

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