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Theorem pwuncl 7791
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 7764 . 2 ((𝐴 ∈ 𝒫 𝑋𝐵 ∈ 𝒫 𝑋) → (𝐴𝐵) ∈ V)
2 elpwi 4606 . . 3 (𝐴 ∈ 𝒫 𝑋𝐴𝑋)
3 elpwi 4606 . . 3 (𝐵 ∈ 𝒫 𝑋𝐵𝑋)
4 unss 4189 . . . 4 ((𝐴𝑋𝐵𝑋) ↔ (𝐴𝐵) ⊆ 𝑋)
54biimpi 216 . . 3 ((𝐴𝑋𝐵𝑋) → (𝐴𝐵) ⊆ 𝑋)
62, 3, 5syl2an 596 . 2 ((𝐴 ∈ 𝒫 𝑋𝐵 ∈ 𝒫 𝑋) → (𝐴𝐵) ⊆ 𝑋)
71, 6elpwd 4605 1 ((𝐴 ∈ 𝒫 𝑋𝐵 ∈ 𝒫 𝑋) → (𝐴𝐵) ∈ 𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2107  Vcvv 3479  cun 3948  wss 3950  𝒫 cpw 4599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-pw 4601  df-sn 4626  df-pr 4628  df-uni 4907
This theorem is referenced by:  naddunif  8732  fiin  9463  fpwipodrs  18586  pwmnd  18951  cutlt  27967  clsk1indlem3  44061  isotone1  44066  isgrtri  47915
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