Theorem pwexg 3907

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 3337	. . 3 ⊢ (x = A → 𝒫 x = 𝒫 A)
2	1	eleq1d 2088	. 2 ⊢ (x = A → (𝒫 x ∈ V ↔ 𝒫 A ∈ V))
3		vex 2538	. . 3 ⊢ x ∈ V
4	3	pwex 3906	. 2 ⊢ 𝒫 x ∈ V
5	2, 4	vtoclg 2590	1 ⊢ (A ∈ 𝑉 → 𝒫 A ∈ V)

Syntax hints:   → wi 4   = wceq 1228   ∈ wcel 1374  Vcvv 2535  𝒫 cpw 3334

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-in 2901  df-ss 2908  df-pw 3336

This theorem is referenced by:  abssexg  3908  snexgOLD  3909  snexg  3910  pwel  3928  uniexb  4155  xpexg  4379  fabexg  5002