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Theorem pwexg 4166
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 3569 . . 3 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
21eleq1d 2239 . 2 (𝑥 = 𝐴 → (𝒫 𝑥 ∈ V ↔ 𝒫 𝐴 ∈ V))
3 vpwex 4165 . 2 𝒫 𝑥 ∈ V
42, 3vtoclg 2790 1 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1348  wcel 2141  Vcvv 2730  𝒫 cpw 3566
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-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160
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 3568
This theorem is referenced by:  pwexd  4167  abssexg  4168  pwex  4169  snexg  4170  pwel  4203  uniexb  4458  xpexg  4725  fabexg  5385  mapex  6632  pmvalg  6637  fopwdom  6814  ssenen  6829  restid2  12588  toponsspwpwg  12814  tgdom  12866  distop  12879  epttop  12884  cldval  12893  ntrfval  12894  clsfval  12895  neifval  12934  neif  12935  neival  12937
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