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Theorem pwunss 4612
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 5289, ax-nul 5296, ax-pr 5417 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 4164 . . 3 𝐴 ⊆ (𝐴𝐵)
21sspwi 4606 . 2 𝒫 𝐴 ⊆ 𝒫 (𝐴𝐵)
3 ssun2 4165 . . 3 𝐵 ⊆ (𝐴𝐵)
43sspwi 4606 . 2 𝒫 𝐵 ⊆ 𝒫 (𝐴𝐵)
52, 4unssi 4177 1 (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  cun 3938  wss 3940  𝒫 cpw 4594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-un 3945  df-in 3947  df-ss 3957  df-pw 4596
This theorem is referenced by:  pwun  5562
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