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Theorem elpwunicl 32390
Description: Closure of a set union with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 21-Jun-2020.) (Proof shortened by BJ, 6-Apr-2024.)
Hypothesis
Ref Expression
elpwunicl.1 (𝜑𝐴 ∈ 𝒫 𝒫 𝐵)
Assertion
Ref Expression
elpwunicl (𝜑 𝐴 ∈ 𝒫 𝐵)

Proof of Theorem elpwunicl
StepHypRef Expression
1 elpwunicl.1 . 2 (𝜑𝐴 ∈ 𝒫 𝒫 𝐵)
2 elpwpwel 7767 . 2 (𝐴 ∈ 𝒫 𝒫 𝐵 𝐴 ∈ 𝒫 𝐵)
31, 2sylib 217 1 (𝜑 𝐴 ∈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  𝒫 cpw 4598   cuni 4903
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 2696  ax-sep 5294  ax-pow 5359  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rab 3420  df-v 3465  df-in 3946  df-ss 3956  df-pw 4600  df-uni 4904
This theorem is referenced by:  ldgenpisyslem1  33839
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