Theorem pwexg

Description: Power set axiom expressed in class notation, with the sethood requirement as an antecedent. (Contributed by NM, 30-Oct-2003)

	Ref	Expression
Assertion	pwexg	$$⊢A\in V\to 𝒫A\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		pweq	$$⊢x=A\to 𝒫x=𝒫A$$
2	1	eleq1d	$$⊢x=A\to \left(𝒫x\in \mathrm{V}\leftrightarrow 𝒫A\in \mathrm{V}\right)$$
3		vpwex	$$⊢𝒫x\in \mathrm{V}$$
4	2 3	vtoclg	$$⊢A\in V\to 𝒫A\in \mathrm{V}$$