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Theorem elpwpwel 7474
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 7471 . . 3 (𝐴 ∈ V ↔ 𝐴 ∈ V)
21anbi1i 626 . 2 ((𝐴 ∈ V ∧ 𝐴𝐵) ↔ ( 𝐴 ∈ V ∧ 𝐴𝐵))
3 elpwpw 4999 . 2 (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴𝐵))
4 elpwb 4521 . 2 ( 𝐴 ∈ 𝒫 𝐵 ↔ ( 𝐴 ∈ V ∧ 𝐴𝐵))
52, 3, 43bitr4i 306 1 (𝐴 ∈ 𝒫 𝒫 𝐵 𝐴 ∈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wcel 2114  Vcvv 3469  wss 3908  𝒫 cpw 4511   cuni 4813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-pow 5243  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-ral 3135  df-rab 3139  df-v 3471  df-in 3915  df-ss 3925  df-pw 4513  df-uni 4814
This theorem is referenced by:  elpwunicl  30313
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