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Metamath Proof Explorer

Theorem mopnss

Description: An open set of a metric space is a subspace of its base set. (Contributed by NM, 3-Sep-2006)

		Ref	Expression
	Hypothesis	mopnval.1	$⊢ J = MetOpen (D)$
	Assertion	mopnss	$⊢ (D \in ∞Met (X) \land A \in J) \to A \subseteq X$

Step	Hyp	Ref	Expression
1		mopnval.1	$⊢ J = MetOpen (D)$
2	1	mopntopon	$⊢ D \in ∞Met (X) \to J \in TopOn (X)$
3		toponss	$⊢ (J \in TopOn (X) \land A \in J) \to A \subseteq X$
4	2 3	sylan	$⊢ (D \in ∞Met (X) \land A \in J) \to A \subseteq X$