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Theorem pwexg 4141
Description: Power set axiom expressed in class notation, with the sethood requirement as an antecedent. (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 3546 . . 3 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
21eleq1d 2226 . 2 (𝑥 = 𝐴 → (𝒫 𝑥 ∈ V ↔ 𝒫 𝐴 ∈ V))
3 vpwex 4140 . 2 𝒫 𝑥 ∈ V
42, 3vtoclg 2772 1 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1335  wcel 2128  Vcvv 2712  𝒫 cpw 3543
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-ss 3115  df-pw 3545
This theorem is referenced by:  pwexd  4142  abssexg  4143  pwex  4144  snexg  4145  pwel  4178  uniexb  4433  xpexg  4700  fabexg  5357  mapex  6599  pmvalg  6604  fopwdom  6781  ssenen  6796  restid2  12371  toponsspwpwg  12431  tgdom  12483  distop  12496  epttop  12501  cldval  12510  ntrfval  12511  clsfval  12512  neifval  12551  neif  12552  neival  12554
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