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Theorem pwunss 4542
Description: The power class of the union of two classes includes the union of their power classes. Exercise 4.12(k) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.) Remove use of ax-sep 5189, ax-nul 5196, ax-pr 5317 and shorten proof. (Revised by BJ, 13-Apr-2024.)
Assertion
Ref Expression
pwunss (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵)

Proof of Theorem pwunss
StepHypRef Expression
1 ssun1 4134 . . 3 𝐴 ⊆ (𝐴𝐵)
21sspwi 4536 . 2 𝒫 𝐴 ⊆ 𝒫 (𝐴𝐵)
3 ssun2 4135 . . 3 𝐵 ⊆ (𝐴𝐵)
43sspwi 4536 . 2 𝒫 𝐵 ⊆ 𝒫 (𝐴𝐵)
52, 4unssi 4147 1 (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  cun 3917  wss 3919  𝒫 cpw 4522
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-pw 4524
This theorem is referenced by:  pwundifOLD  5444  pwun  5445
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