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Theorem elpwunicl 28547
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 4115 . . . . 5 (𝐴 ∈ 𝒫 𝒫 𝐵 → (𝐴 ∈ 𝒫 𝒫 𝐵𝐴 ⊆ 𝒫 𝐵))
31, 2syl 17 . . . 4 (𝜑 → (𝐴 ∈ 𝒫 𝒫 𝐵𝐴 ⊆ 𝒫 𝐵))
41, 3mpbid 220 . . 3 (𝜑𝐴 ⊆ 𝒫 𝐵)
5 pwuniss 28546 . . 3 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
64, 5syl 17 . 2 (𝜑 𝐴𝐵)
7 uniexg 6830 . . 3 (𝐴 ∈ 𝒫 𝒫 𝐵 𝐴 ∈ V)
8 elpwg 4115 . . 3 ( 𝐴 ∈ V → ( 𝐴 ∈ 𝒫 𝐵 𝐴𝐵))
91, 7, 83syl 18 . 2 (𝜑 → ( 𝐴 ∈ 𝒫 𝐵 𝐴𝐵))
106, 9mpbird 245 1 (𝜑 𝐴 ∈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wcel 1976  Vcvv 3172  wss 3539  𝒫 cpw 4107   cuni 4366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rex 2901  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-pw 4109  df-sn 4125  df-pr 4127  df-uni 4367
This theorem is referenced by:  ldgenpisyslem1  29346
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