MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pwundif Structured version   Visualization version   GIF version

Theorem pwundif 5258
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 4266 . 2 ((𝒫 (𝐴𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴) = (𝒫 (𝐴𝐵) ∪ 𝒫 𝐴)
2 pwunss 5256 . . . . 5 (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵)
3 unss 4009 . . . . 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 4008 . . 3 (𝒫 𝐴 ⊆ 𝒫 (𝐴𝐵) ↔ (𝒫 (𝐴𝐵) ∪ 𝒫 𝐴) = 𝒫 (𝐴𝐵))
75, 6mpbi 222 . 2 (𝒫 (𝐴𝐵) ∪ 𝒫 𝐴) = 𝒫 (𝐴𝐵)
81, 7eqtr2i 2802 1 𝒫 (𝐴𝐵) = ((𝒫 (𝐴𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 386   = wceq 1601  cdif 3788  cun 3789  wss 3791  𝒫 cpw 4378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-ext 2753
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-pw 4380
This theorem is referenced by:  pwfilem  8548
  Copyright terms: Public domain W3C validator