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Theorem pwundif 5246
 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 (𝐴𝐵) = ((𝒫 (𝐴𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)

Proof of Theorem pwundif
StepHypRef Expression
1 undif1 4265 . 2 ((𝒫 (𝐴𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴) = (𝒫 (𝐴𝐵) ∪ 𝒫 𝐴)
2 pwunss 5244 . . . . 5 (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵)
3 unss 4013 . . . . 5 ((𝒫 𝐴 ⊆ 𝒫 (𝐴𝐵) ∧ 𝒫 𝐵 ⊆ 𝒫 (𝐴𝐵)) ↔ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵))
42, 3mpbir 223 . . . 4 (𝒫 𝐴 ⊆ 𝒫 (𝐴𝐵) ∧ 𝒫 𝐵 ⊆ 𝒫 (𝐴𝐵))
54simpli 478 . . 3 𝒫 𝐴 ⊆ 𝒫 (𝐴𝐵)
6 ssequn2 4012 . . 3 (𝒫 𝐴 ⊆ 𝒫 (𝐴𝐵) ↔ (𝒫 (𝐴𝐵) ∪ 𝒫 𝐴) = 𝒫 (𝐴𝐵))
75, 6mpbi 222 . 2 (𝒫 (𝐴𝐵) ∪ 𝒫 𝐴) = 𝒫 (𝐴𝐵)
81, 7eqtr2i 2849 1 𝒫 (𝐴𝐵) = ((𝒫 (𝐴𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 386   = wceq 1658   ∖ cdif 3794   ∪ cun 3795   ⊆ wss 3797  𝒫 cpw 4377 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rab 3125  df-v 3415  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-pw 4379 This theorem is referenced by:  pwfilem  8528
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