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Theorem elpwpw 4990
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 4510 . 2 (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 ⊆ 𝒫 𝐵))
2 sspwuni 4988 . . 3 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
32anbi2i 625 . 2 ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝒫 𝐵) ↔ (𝐴 ∈ V ∧ 𝐴𝐵))
41, 3bitri 278 1 (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wcel 2112  Vcvv 3444  wss 3884  𝒫 cpw 4500   cuni 4803
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 2114  ax-9 2122  ax-11 2159  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ral 3114  df-v 3446  df-in 3891  df-ss 3901  df-pw 4502  df-uni 4804
This theorem is referenced by:  pwpwab  4991  elpwpwel  7473
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