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Theorem elpwb 4608
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 3501 . 2 (𝐴 ∈ 𝒫 𝐵𝐴 ∈ V)
2 elpwg 4603 . 2 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
31, 2biadanii 822 1 (𝐴 ∈ 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wcel 2108  Vcvv 3480  wss 3951  𝒫 cpw 4600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-ss 3968  df-pw 4602
This theorem is referenced by:  elpwpw  5102  elpwpwel  7787  onsupcl2  43237  onsupuni2  43242  onsupintrab2  43244  onuniintrab2  43247
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