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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 4604 . . 3 (𝐴 ∈ 𝒫 𝑋𝐴𝑋)
3 elpwi 4604 . . 3 (𝐵 ∈ 𝒫 𝑋𝐵𝑋)
4 unss 4179 . . . 4 ((𝐴𝑋𝐵𝑋) ↔ (𝐴𝐵) ⊆ 𝑋)
54biimpi 215 . . 3 ((𝐴𝑋𝐵𝑋) → (𝐴𝐵) ⊆ 𝑋)
62, 3, 5syl2an 595 . 2 ((𝐴 ∈ 𝒫 𝑋𝐵 ∈ 𝒫 𝑋) → (𝐴𝐵) ⊆ 𝑋)
71, 6elpwd 4603 1 ((𝐴 ∈ 𝒫 𝑋𝐵 ∈ 𝒫 𝑋) → (𝐴𝐵) ∈ 𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2098  Vcvv 3468  cun 3941  wss 3943  𝒫 cpw 4597
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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-pw 4599  df-sn 4624  df-pr 4626  df-uni 4903
This theorem is referenced by:  naddunif  8691  fiin  9416  fpwipodrs  18503  pwmnd  18860  cutlt  27803  clsk1indlem3  43351  isotone1  43356
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