Metamath Proof Explorer

Theorem pwexb

Description: The Axiom of Power Sets and its converse. A class is a set iff its power class is a set. (Contributed by NM, 11-Nov-2003)

		Ref	Expression
	Assertion	pwexb	$⊢ A \in V \leftrightarrow 𝒫 A \in V$

Step	Hyp	Ref	Expression
1		pwexg	$⊢ A \in V \to 𝒫 A \in V$
2		pwexr	$⊢ 𝒫 A \in V \to A \in V$
3	1 2	impbii	$⊢ A \in V \leftrightarrow 𝒫 A \in V$