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Theorem pwuncl 7772
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 7751 . 2 ((𝐴 ∈ 𝒫 𝑋𝐵 ∈ 𝒫 𝑋) → (𝐴𝐵) ∈ V)
2 elpwi 4610 . . 3 (𝐴 ∈ 𝒫 𝑋𝐴𝑋)
3 elpwi 4610 . . 3 (𝐵 ∈ 𝒫 𝑋𝐵𝑋)
4 unss 4184 . . . 4 ((𝐴𝑋𝐵𝑋) ↔ (𝐴𝐵) ⊆ 𝑋)
54biimpi 215 . . 3 ((𝐴𝑋𝐵𝑋) → (𝐴𝐵) ⊆ 𝑋)
62, 3, 5syl2an 595 . 2 ((𝐴 ∈ 𝒫 𝑋𝐵 ∈ 𝒫 𝑋) → (𝐴𝐵) ⊆ 𝑋)
71, 6elpwd 4609 1 ((𝐴 ∈ 𝒫 𝑋𝐵 ∈ 𝒫 𝑋) → (𝐴𝐵) ∈ 𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2099  Vcvv 3471  cun 3945  wss 3947  𝒫 cpw 4603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-pw 4605  df-sn 4630  df-pr 4632  df-uni 4909
This theorem is referenced by:  naddunif  8714  fiin  9446  fpwipodrs  18532  pwmnd  18889  cutlt  27865  clsk1indlem3  43473  isotone1  43478
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