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Theorem elpwb 3574
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 2741 . 2 (𝐴 ∈ 𝒫 𝐵𝐴 ∈ V)
2 elpwg 3572 . 2 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
31, 2biadan2 453 1 (𝐴 ∈ 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wcel 2141  Vcvv 2730  wss 3121  𝒫 cpw 3564
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-pw 3566
This theorem is referenced by:  elpwpw  3957  elpwpwel  4458
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