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Theorem pwunss 4623
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 5302, ax-nul 5312, ax-pr 5438 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 4188 . . 3 𝐴 ⊆ (𝐴𝐵)
21sspwi 4617 . 2 𝒫 𝐴 ⊆ 𝒫 (𝐴𝐵)
3 ssun2 4189 . . 3 𝐵 ⊆ (𝐴𝐵)
43sspwi 4617 . 2 𝒫 𝐵 ⊆ 𝒫 (𝐴𝐵)
52, 4unssi 4201 1 (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  cun 3961  wss 3963  𝒫 cpw 4605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-ss 3980  df-pw 4607
This theorem is referenced by:  pwun  5581
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