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Theorem elpwpwel 4396
 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 4394 . . 3 (𝐴 ∈ V ↔ 𝐴 ∈ V)
21anbi1i 453 . 2 ((𝐴 ∈ V ∧ 𝐴𝐵) ↔ ( 𝐴 ∈ V ∧ 𝐴𝐵))
3 elpwpw 3899 . 2 (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴𝐵))
4 elpwb 3520 . 2 ( 𝐴 ∈ 𝒫 𝐵 ↔ ( 𝐴 ∈ V ∧ 𝐴𝐵))
52, 3, 43bitr4i 211 1 (𝐴 ∈ 𝒫 𝒫 𝐵 𝐴 ∈ 𝒫 𝐵)
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104   ∈ wcel 1480  Vcvv 2686   ⊆ wss 3071  𝒫 cpw 3510  ∪ cuni 3736 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-un 4355 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512  df-uni 3737 This theorem is referenced by: (None)
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