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Theorem elpwb 3445
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 2633 . 2 (𝐴 ∈ 𝒫 𝐵𝐴 ∈ V)
2 elpwg 3443 . 2 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
31, 2biadan2 445 1 (𝐴 ∈ 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wcel 1439  Vcvv 2622  wss 3002  𝒫 cpw 3435
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2624  df-in 3008  df-ss 3015  df-pw 3437
This theorem is referenced by:  elpwpw  3823  elpwpwel  4312
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