Theorem pwundifOLD 5454

Description: Obsolete proof of pwundif 4562 as of 26-Dec-2023. (Contributed by NM, 25-Mar-2007.) (Proof shortened by Thierry Arnoux, 20-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
pwundifOLD	⊢ 𝒫 (𝐴 ∪ 𝐵) = ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)

Proof of Theorem pwundifOLD

Step	Hyp	Ref	Expression
1		undif1 4421	. 2 ⊢ ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴) = (𝒫 (𝐴 ∪ 𝐵) ∪ 𝒫 𝐴)
2		pwunss 4556	. . . . 5 ⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵)
3		unss 4157	. . . . 5 ⊢ ((𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵) ∧ 𝒫 𝐵 ⊆ 𝒫 (𝐴 ∪ 𝐵)) ↔ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵))
4	2, 3	mpbir 233	. . . 4 ⊢ (𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵) ∧ 𝒫 𝐵 ⊆ 𝒫 (𝐴 ∪ 𝐵))
5	4	simpli 486	. . 3 ⊢ 𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵)
6		ssequn2 4156	. . 3 ⊢ (𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵) ↔ (𝒫 (𝐴 ∪ 𝐵) ∪ 𝒫 𝐴) = 𝒫 (𝐴 ∪ 𝐵))
7	5, 6	mpbi 232	. 2 ⊢ (𝒫 (𝐴 ∪ 𝐵) ∪ 𝒫 𝐴) = 𝒫 (𝐴 ∪ 𝐵)
8	1, 7	eqtr2i 2844	1 ⊢ 𝒫 (𝐴 ∪ 𝐵) = ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)

Syntax hints:   ∧ wa 398   = wceq 1536   ∖ cdif 3930   ∪ cun 3931   ⊆ wss 3933  𝒫 cpw 4536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-rab 3146  df-v 3495  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-pw 4538

This theorem is referenced by: (None)