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Theorem elpwpwel 7786
Description: A class belongs to a double power class if and only if its union belongs to the power class. (Contributed by BJ, 22-Jan-2023.)
Assertion
Ref Expression
elpwpwel (𝐴 ∈ 𝒫 𝒫 𝐵 𝐴 ∈ 𝒫 𝐵)

Proof of Theorem elpwpwel
StepHypRef Expression
1 uniexb 7783 . . 3 (𝐴 ∈ V ↔ 𝐴 ∈ V)
21anbi1i 624 . 2 ((𝐴 ∈ V ∧ 𝐴𝐵) ↔ ( 𝐴 ∈ V ∧ 𝐴𝐵))
3 elpwpw 5107 . 2 (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴𝐵))
4 elpwb 4613 . 2 ( 𝐴 ∈ 𝒫 𝐵 ↔ ( 𝐴 ∈ V ∧ 𝐴𝐵))
52, 3, 43bitr4i 303 1 (𝐴 ∈ 𝒫 𝒫 𝐵 𝐴 ∈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wcel 2106  Vcvv 3478  wss 3963  𝒫 cpw 4605   cuni 4912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-pow 5371  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-in 3970  df-ss 3980  df-pw 4607  df-uni 4913
This theorem is referenced by:  elpwunicl  32575
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