Theorem pwexg 3924

Description: Power set axiom expressed in class notation, with the sethood requirement as an antecedent. Axiom 4 of [TakeutiZaring] p. 17. (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 3354	. . 3 ⊢ (x = A → 𝒫 x = 𝒫 A)
2	1	eleq1d 2103	. 2 ⊢ (x = A → (𝒫 x ∈ V ↔ 𝒫 A ∈ V))
3		vex 2554	. . 3 ⊢ x ∈ V
4	3	pwex 3923	. 2 ⊢ 𝒫 x ∈ V
5	2, 4	vtoclg 2607	1 ⊢ (A ∈ 𝑉 → 𝒫 A ∈ V)

Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  Vcvv 2551  𝒫 cpw 3351

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353

This theorem is referenced by:  abssexg  3925  snexgOLD  3926  snexg  3927  pwel  3945  uniexb  4171  xpexg  4395  fabexg  5020  fopwdom  6246