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Theorem elpwunicl 29699
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 4310 . . . . 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 29698 . . 3 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
64, 5syl 17 . 2 (𝜑 𝐴𝐵)
7 uniexg 7121 . . 3 (𝐴 ∈ 𝒫 𝒫 𝐵 𝐴 ∈ V)
8 elpwg 4310 . . 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 2139  Vcvv 3340  wss 3715  𝒫 cpw 4302   cuni 4588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rex 3056  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-pw 4304  df-sn 4322  df-pr 4324  df-uni 4589
This theorem is referenced by:  ldgenpisyslem1  30556
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