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Theorem pwundif 5171
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 4187 . 2 ((𝒫 (𝐴𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴) = (𝒫 (𝐴𝐵) ∪ 𝒫 𝐴)
2 pwunss 5169 . . . . 5 (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵)
3 unss 3930 . . . . 5 ((𝒫 𝐴 ⊆ 𝒫 (𝐴𝐵) ∧ 𝒫 𝐵 ⊆ 𝒫 (𝐴𝐵)) ↔ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵))
42, 3mpbir 221 . . . 4 (𝒫 𝐴 ⊆ 𝒫 (𝐴𝐵) ∧ 𝒫 𝐵 ⊆ 𝒫 (𝐴𝐵))
54simpli 476 . . 3 𝒫 𝐴 ⊆ 𝒫 (𝐴𝐵)
6 ssequn2 3929 . . 3 (𝒫 𝐴 ⊆ 𝒫 (𝐴𝐵) ↔ (𝒫 (𝐴𝐵) ∪ 𝒫 𝐴) = 𝒫 (𝐴𝐵))
75, 6mpbi 220 . 2 (𝒫 (𝐴𝐵) ∪ 𝒫 𝐴) = 𝒫 (𝐴𝐵)
81, 7eqtr2i 2783 1 𝒫 (𝐴𝐵) = ((𝒫 (𝐴𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1632  cdif 3712  cun 3713  wss 3715  𝒫 cpw 4302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-pw 4304
This theorem is referenced by:  pwfilem  8427
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