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Theorem pwexg 3933
Description: Power set axiom expressed in class notation, with the sethood requirement as an antecedent. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Oct-2003.)
Assertion
Ref Expression
pwexg (𝐴𝑉 → 𝒫 𝐴 ∈ V)

Proof of Theorem pwexg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pweq 3362 . . 3 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
21eleq1d 2106 . 2 (𝑥 = 𝐴 → (𝒫 𝑥 ∈ V ↔ 𝒫 𝐴 ∈ V))
3 vex 2560 . . 3 𝑥 ∈ V
43pwex 3932 . 2 𝒫 𝑥 ∈ V
52, 4vtoclg 2613 1 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243  wcel 1393  Vcvv 2557  𝒫 cpw 3359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361
This theorem is referenced by:  abssexg  3934  snexgOLD  3935  snexg  3936  pwel  3954  uniexb  4205  xpexg  4452  fabexg  5077  fopwdom  6310
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