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Theorem elpwpwel 4471
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 4469 . . 3 (𝐴 ∈ V ↔ 𝐴 ∈ V)
21anbi1i 458 . 2 ((𝐴 ∈ V ∧ 𝐴𝐵) ↔ ( 𝐴 ∈ V ∧ 𝐴𝐵))
3 elpwpw 3970 . 2 (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴𝐵))
4 elpwb 3584 . 2 ( 𝐴 ∈ 𝒫 𝐵 ↔ ( 𝐴 ∈ V ∧ 𝐴𝐵))
52, 3, 43bitr4i 212 1 (𝐴 ∈ 𝒫 𝒫 𝐵 𝐴 ∈ 𝒫 𝐵)
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  wcel 2148  Vcvv 2737  wss 3129  𝒫 cpw 3574   cuni 3807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-un 4429
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-in 3135  df-ss 3142  df-pw 3576  df-uni 3808
This theorem is referenced by: (None)
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