ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pwexg GIF version

Theorem pwexg 4099
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 3508 . . 3 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
21eleq1d 2206 . 2 (𝑥 = 𝐴 → (𝒫 𝑥 ∈ V ↔ 𝒫 𝐴 ∈ V))
3 vpwex 4098 . 2 𝒫 𝑥 ∈ V
42, 3vtoclg 2741 1 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331  wcel 1480  Vcvv 2681  𝒫 cpw 3505
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072  df-ss 3079  df-pw 3507
This theorem is referenced by:  pwexd  4100  abssexg  4101  pwex  4102  snexg  4103  pwel  4135  uniexb  4389  xpexg  4648  fabexg  5305  mapex  6541  pmvalg  6546  fopwdom  6723  ssenen  6738  restid2  12118  toponsspwpwg  12178  tgdom  12230  distop  12243  epttop  12248  cldval  12257  ntrfval  12258  clsfval  12259  neifval  12298  neif  12299  neival  12301
  Copyright terms: Public domain W3C validator