Theorem pwexg 4292

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 3671	. . 3 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
2	1	eleq1d 2301	. 2 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ V ↔ 𝒫 𝐴 ∈ V))
3		vpwex 4291	. 2 ⊢ 𝒫 𝑥 ∈ V
4	2, 3	vtoclg 2874	1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2203  Vcvv 2812  𝒫 cpw 3668

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-in 3216  df-ss 3223  df-pw 3670

This theorem is referenced by:  pwexd  4293  abssexg  4294  pwex  4295  snexg  4296  pwel  4333  uniexb  4593  xpexg  4863  fabexg  5553  mapex  6887  pmvalg  6892  fopwdom  7088  ssenen  7104  2omapfi  7270  restid2  13453  toponsspwpwg  14879  tgdom  14929  distop  14942  epttop  14947  cldval  14956  ntrfval  14957  clsfval  14958  neifval  14997  neif  14998  neival  15000