Theorem pwundif 4945

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.)

Assertion

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

Proof of Theorem pwundif

Step	Hyp	Ref	Expression
1		undif1 3995	. 2 ⊢ ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴) = (𝒫 (𝐴 ∪ 𝐵) ∪ 𝒫 𝐴)
2		pwunss 4943	. . . . 5 ⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵)
3		unss 3749	. . . . 5 ⊢ ((𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵) ∧ 𝒫 𝐵 ⊆ 𝒫 (𝐴 ∪ 𝐵)) ↔ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵))
4	2, 3	mpbir 220	. . . 4 ⊢ (𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵) ∧ 𝒫 𝐵 ⊆ 𝒫 (𝐴 ∪ 𝐵))
5	4	simpli 473	. . 3 ⊢ 𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵)
6		ssequn2 3748	. . 3 ⊢ (𝒫 𝐴 ⊆ 𝒫 (𝐴 ∪ 𝐵) ↔ (𝒫 (𝐴 ∪ 𝐵) ∪ 𝒫 𝐴) = 𝒫 (𝐴 ∪ 𝐵))
7	5, 6	mpbi 219	. 2 ⊢ (𝒫 (𝐴 ∪ 𝐵) ∪ 𝒫 𝐴) = 𝒫 (𝐴 ∪ 𝐵)
8	1, 7	eqtr2i 2633	1 ⊢ 𝒫 (𝐴 ∪ 𝐵) = ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)

Syntax hints:   ∧ wa 383   = wceq 1475   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  𝒫 cpw 4108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110

This theorem is referenced by:  pwfilem  8143