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Theorem elpwpw 3907
 Description: Characterization of the elements of a double power class: they are exactly the sets whose union is included in that class. (Contributed by BJ, 29-Apr-2021.)
Assertion
Ref Expression
elpwpw (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴𝐵))

Proof of Theorem elpwpw
StepHypRef Expression
1 elpwb 3525 . 2 (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 ⊆ 𝒫 𝐵))
2 sspwuni 3905 . . 3 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
32anbi2i 453 . 2 ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝒫 𝐵) ↔ (𝐴 ∈ V ∧ 𝐴𝐵))
41, 3bitri 183 1 (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴𝐵))
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104   ∈ wcel 1481  Vcvv 2689   ⊆ wss 3076  𝒫 cpw 3515  ∪ cuni 3744 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-in 3082  df-ss 3089  df-pw 3517  df-uni 3745 This theorem is referenced by:  pwpwab  3908  elpwpwel  4404
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