Theorem uniexr

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

	Ref	Expression
Assertion	uniexr	⊢ ( ∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ V )

Proof

Step	Hyp	Ref	Expression
1		pwuni	⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
2		pwexg	⊢ ( ∪ 𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ V )
3		ssexg	⊢ ( ( 𝐴 ⊆ 𝒫 ∪ 𝐴 ∧ 𝒫 ∪ 𝐴 ∈ V ) → 𝐴 ∈ V )
4	1 2 3	sylancr	⊢ ( ∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ V )