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Metamath Proof Explorer

Theorem mopnin

Description: The intersection of two open sets of a metric space is open. (Contributed by NM, 4-Sep-2006) (Revised by Mario Carneiro, 23-Dec-2013)

		Ref	Expression
	Hypothesis	mopni.1	$⊢ J = MetOpen (D)$
	Assertion	mopnin	$⊢ (D \in ∞Met (X) \land A \in J \land B \in J) \to A \cap B \in J$

Proof

Step	Hyp	Ref	Expression
1		mopni.1	$⊢ J = MetOpen (D)$
2	1	mopntop	$⊢ D \in ∞Met (X) \to J \in Top$
3		inopn	$⊢ (J \in Top \land A \in J \land B \in J) \to A \cap B \in J$
4	2 3	syl3an1	$⊢ (D \in ∞Met (X) \land A \in J \land B \in J) \to A \cap B \in J$