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Theorem pwuncl 7753
Description: Power classes are closed under union. (Contributed by AV, 27-Feb-2024.)
Assertion
Ref Expression
pwuncl ((𝐴 ∈ 𝒫 𝑋𝐵 ∈ 𝒫 𝑋) → (𝐴𝐵) ∈ 𝒫 𝑋)

Proof of Theorem pwuncl
StepHypRef Expression
1 unexg 7732 . 2 ((𝐴 ∈ 𝒫 𝑋𝐵 ∈ 𝒫 𝑋) → (𝐴𝐵) ∈ V)
2 elpwi 4608 . . 3 (𝐴 ∈ 𝒫 𝑋𝐴𝑋)
3 elpwi 4608 . . 3 (𝐵 ∈ 𝒫 𝑋𝐵𝑋)
4 unss 4183 . . . 4 ((𝐴𝑋𝐵𝑋) ↔ (𝐴𝐵) ⊆ 𝑋)
54biimpi 215 . . 3 ((𝐴𝑋𝐵𝑋) → (𝐴𝐵) ⊆ 𝑋)
62, 3, 5syl2an 596 . 2 ((𝐴 ∈ 𝒫 𝑋𝐵 ∈ 𝒫 𝑋) → (𝐴𝐵) ⊆ 𝑋)
71, 6elpwd 4607 1 ((𝐴 ∈ 𝒫 𝑋𝐵 ∈ 𝒫 𝑋) → (𝐴𝐵) ∈ 𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  Vcvv 3474  cun 3945  wss 3947  𝒫 cpw 4601
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-pw 4603  df-sn 4628  df-pr 4630  df-uni 4908
This theorem is referenced by:  naddunif  8688  fiin  9413  fpwipodrs  18489  pwmnd  18814  cutlt  27408  clsk1indlem3  42779  isotone1  42784
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