Theorem pwexg 4319

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	⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)

Proof of Theorem pwexg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pweq 3691	. . 3 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
2	1	eleq1d 2307	. 2 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ V ↔ 𝒫 𝐴 ∈ V))
3		vpwex 4318	. 2 ⊢ 𝒫 𝑥 ∈ V
4	2, 3	vtoclg 2883	1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)

This proof depends on syntax axioms:   → wi 4   = wceq 1402   ∈ wcel 2209  Vcvv 2821  𝒫 cpw 3688

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-pw 3690

This theorem is used by:  pwexd  4320  abssexg  4321  pwex  4322  snexg  4323  pwel  4360  uniexb  4621  xpexg  4891  fabexg  5582  mapex  6932  pmvalg  6937  fopwdom  7141  ssenen  7157  2omapfi  7327  indv  9305  restid2  13662  toponsspwpwg  15260  tgdom  15310  distop  15323  epttop  15328  cldval  15337  ntrfval  15338  clsfval  15339  neifval  15378  neif  15379  neival  15381