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Theorem elpwb 4548
Description: Characterization of the elements of a power class. (Contributed by BJ, 29-Apr-2021.)
Assertion
Ref Expression
elpwb (𝐴 ∈ 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴𝐵))

Proof of Theorem elpwb
StepHypRef Expression
1 elex 3510 . 2 (𝐴 ∈ 𝒫 𝐵𝐴 ∈ V)
2 elpwg 4541 . 2 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
31, 2biadanii 818 1 (𝐴 ∈ 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wcel 2105  Vcvv 3492  wss 3933  𝒫 cpw 4535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-in 3940  df-ss 3949  df-pw 4537
This theorem is referenced by:  elpwpw  5015  elpwpwel  7478
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