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Theorem pwundifOLD 5434
 Description: Obsolete proof of pwundif 4537 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 (𝐴𝐵) = ((𝒫 (𝐴𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)

Proof of Theorem pwundifOLD
StepHypRef Expression
1 undif1 4396 . 2 ((𝒫 (𝐴𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴) = (𝒫 (𝐴𝐵) ∪ 𝒫 𝐴)
2 pwunss 4531 . . . . 5 (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵)
3 unss 4135 . . . . 5 ((𝒫 𝐴 ⊆ 𝒫 (𝐴𝐵) ∧ 𝒫 𝐵 ⊆ 𝒫 (𝐴𝐵)) ↔ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵))
42, 3mpbir 234 . . . 4 (𝒫 𝐴 ⊆ 𝒫 (𝐴𝐵) ∧ 𝒫 𝐵 ⊆ 𝒫 (𝐴𝐵))
54simpli 487 . . 3 𝒫 𝐴 ⊆ 𝒫 (𝐴𝐵)
6 ssequn2 4134 . . 3 (𝒫 𝐴 ⊆ 𝒫 (𝐴𝐵) ↔ (𝒫 (𝐴𝐵) ∪ 𝒫 𝐴) = 𝒫 (𝐴𝐵))
75, 6mpbi 233 . 2 (𝒫 (𝐴𝐵) ∪ 𝒫 𝐴) = 𝒫 (𝐴𝐵)
81, 7eqtr2i 2846 1 𝒫 (𝐴𝐵) = ((𝒫 (𝐴𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1538   ∖ cdif 3905   ∪ cun 3906   ⊆ wss 3908  𝒫 cpw 4511 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-11 2161  ax-12 2178  ax-ext 2794 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-rab 3139  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-pw 4513 This theorem is referenced by: (None)
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