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Theorem elpwunicl 29676
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 4308 . . . . 5 (𝐴 ∈ 𝒫 𝒫 𝐵 → (𝐴 ∈ 𝒫 𝒫 𝐵𝐴 ⊆ 𝒫 𝐵))
31, 2syl 17 . . . 4 (𝜑 → (𝐴 ∈ 𝒫 𝒫 𝐵𝐴 ⊆ 𝒫 𝐵))
41, 3mpbid 222 . . 3 (𝜑𝐴 ⊆ 𝒫 𝐵)
5 pwuniss 29675 . . 3 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
64, 5syl 17 . 2 (𝜑 𝐴𝐵)
7 uniexg 7118 . . 3 (𝐴 ∈ 𝒫 𝒫 𝐵 𝐴 ∈ V)
8 elpwg 4308 . . 3 ( 𝐴 ∈ V → ( 𝐴 ∈ 𝒫 𝐵 𝐴𝐵))
91, 7, 83syl 18 . 2 (𝜑 → ( 𝐴 ∈ 𝒫 𝐵 𝐴𝐵))
106, 9mpbird 247 1 (𝜑 𝐴 ∈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wcel 2137  Vcvv 3338  wss 3713  𝒫 cpw 4300   cuni 4586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-rex 3054  df-v 3340  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-pw 4302  df-sn 4320  df-pr 4322  df-uni 4587
This theorem is referenced by:  ldgenpisyslem1  30533
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