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Metamath Proof Explorer

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	$⊢ ⋃ A \in V \to A \in V$

Step	Hyp	Ref	Expression
1		pwuni	$⊢ A \subseteq 𝒫 ⋃ A$
2		pwexg	$⊢ ⋃ A \in V \to 𝒫 ⋃ A \in V$
3		ssexg	$⊢ (A \subseteq 𝒫 ⋃ A \land 𝒫 ⋃ A \in V) \to A \in V$
4	1 2 3	sylancr	$⊢ ⋃ A \in V \to A \in V$