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Theorem elpwunicl 28754
 Description: Closure of a set union with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 21-Jun-2020.)
Hypotheses
Ref Expression
elpwunicl.1 (𝜑𝐵𝑉)
elpwunicl.2 (𝜑𝐴 ∈ 𝒫 𝒫 𝐵)
Assertion
Ref Expression
elpwunicl (𝜑 𝐴 ∈ 𝒫 𝐵)

Proof of Theorem elpwunicl
StepHypRef Expression
1 elpwunicl.2 . . . 4 (𝜑𝐴 ∈ 𝒫 𝒫 𝐵)
2 elpwg 4116 . . . . 5 (𝐴 ∈ 𝒫 𝒫 𝐵 → (𝐴 ∈ 𝒫 𝒫 𝐵𝐴 ⊆ 𝒫 𝐵))
31, 2syl 17 . . . 4 (𝜑 → (𝐴 ∈ 𝒫 𝒫 𝐵𝐴 ⊆ 𝒫 𝐵))
41, 3mpbid 221 . . 3 (𝜑𝐴 ⊆ 𝒫 𝐵)
5 pwuniss 28753 . . 3 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
64, 5syl 17 . 2 (𝜑 𝐴𝐵)
7 uniexg 6853 . . 3 (𝐴 ∈ 𝒫 𝒫 𝐵 𝐴 ∈ V)
8 elpwg 4116 . . 3 ( 𝐴 ∈ V → ( 𝐴 ∈ 𝒫 𝐵 𝐴𝐵))
91, 7, 83syl 18 . 2 (𝜑 → ( 𝐴 ∈ 𝒫 𝐵 𝐴𝐵))
106, 9mpbird 246 1 (𝜑 𝐴 ∈ 𝒫 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110  df-sn 4126  df-pr 4128  df-uni 4373 This theorem is referenced by:  ldgenpisyslem1  29553
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