Theorem pwexr

Description: Converse of the Axiom of Power Sets. Note that it does not require ax-pow . (Contributed by NM, 11-Nov-2003)

	Ref	Expression
Assertion	pwexr	⊢ ( 𝒫 𝐴 ∈ 𝑉 → 𝐴 ∈ V )

Proof

Step	Hyp	Ref	Expression
1		unipw	⊢ ∪ 𝒫 𝐴 = 𝐴
2		uniexg	⊢ ( 𝒫 𝐴 ∈ 𝑉 → ∪ 𝒫 𝐴 ∈ V )
3	1 2	eqeltrrid	⊢ ( 𝒫 𝐴 ∈ 𝑉 → 𝐴 ∈ V )