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Theorem pwpwuni 38250
 Description: Relationship between power class and union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Assertion
Ref Expression
pwpwuni (𝐴𝑉 → (𝐴 ∈ 𝒫 𝒫 𝐵 𝐴 ∈ 𝒫 𝐵))

Proof of Theorem pwpwuni
StepHypRef Expression
1 elpwg 4116 . 2 (𝐴𝑉 → (𝐴 ∈ 𝒫 𝒫 𝐵𝐴 ⊆ 𝒫 𝐵))
2 sspwuni 4547 . . 3 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
32a1i 11 . 2 (𝐴𝑉 → (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵))
4 uniexg 6853 . . . 4 (𝐴𝑉 𝐴 ∈ V)
5 elpwg 4116 . . . 4 ( 𝐴 ∈ V → ( 𝐴 ∈ 𝒫 𝐵 𝐴𝐵))
64, 5syl 17 . . 3 (𝐴𝑉 → ( 𝐴 ∈ 𝒫 𝐵 𝐴𝐵))
76bicomd 212 . 2 (𝐴𝑉 → ( 𝐴𝐵 𝐴 ∈ 𝒫 𝐵))
81, 3, 73bitrd 293 1 (𝐴𝑉 → (𝐴 ∈ 𝒫 𝒫 𝐵 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-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-ral 2901  df-rex 2902  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110  df-uni 4373 This theorem is referenced by:  psmeasurelem  39363
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