Theorem pwexg 4315

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 4314	. 2 ⊢ 𝒫 𝑥 ∈ V
4	2, 3	vtoclg 2883	1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)

Syntax hints:   → wi 4   = wceq 1402   ∈ wcel 2209  Vcvv 2821  𝒫 cpw 3688

This theorem was proved from 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 4247  ax-pow 4309

This theorem 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 referenced by:  pwexd  4316  abssexg  4317  pwex  4318  snexg  4319  pwel  4356  uniexb  4617  xpexg  4887  fabexg  5577  mapex  6922  pmvalg  6927  fopwdom  7130  ssenen  7146  2omapfi  7314  restid2  13585  toponsspwpwg  15106  tgdom  15156  distop  15169  epttop  15174  cldval  15183  ntrfval  15184  clsfval  15185  neifval  15224  neif  15225  neival  15227