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Theorem elpwpwel 7706
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 7703 . . 3 (𝐴 ∈ V ↔ 𝐴 ∈ V)
21anbi1i 625 . 2 ((𝐴 ∈ V ∧ 𝐴𝐵) ↔ ( 𝐴 ∈ V ∧ 𝐴𝐵))
3 elpwpw 5067 . 2 (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴𝐵))
4 elpwb 4573 . 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 205  wa 397  wcel 2107  Vcvv 3448  wss 3915  𝒫 cpw 4565   cuni 4870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-pow 5325  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rab 3411  df-v 3450  df-in 3922  df-ss 3932  df-pw 4567  df-uni 4871
This theorem is referenced by:  elpwunicl  31515
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