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Metamath Proof Explorer

Theorem mopnuni

Description: The union of all open sets in a metric space is its underlying set. (Contributed by NM, 4-Sep-2006) (Revised by Mario Carneiro, 12-Nov-2013)

		Ref	Expression
	Hypothesis	mopnval.1	$⊢ J = MetOpen (D)$
	Assertion	mopnuni	$⊢ D \in ∞Met (X) \to X = ⋃ J$

Proof

Step	Hyp	Ref	Expression
1		mopnval.1	$⊢ J = MetOpen (D)$
2	1	mopntopon	$⊢ D \in ∞Met (X) \to J \in TopOn (X)$
3		toponuni	$⊢ J \in TopOn (X) \to X = ⋃ J$
4	2 3	syl	$⊢ D \in ∞Met (X) \to X = ⋃ J$