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Metamath Proof Explorer

Theorem ntrss2

Description: A subset includes its interior. (Contributed by NM, 3-Oct-2007) (Revised by Mario Carneiro, 11-Nov-2013)

		Ref	Expression
	Hypothesis	clscld.1	$⊢ X = ⋃ J$
	Assertion	ntrss2	$⊢ (J \in Top \land S \subseteq X) \to int (J) (S) \subseteq S$

Step	Hyp	Ref	Expression
1		clscld.1	$⊢ X = ⋃ J$
2	1	ntrval	$⊢ (J \in Top \land S \subseteq X) \to int (J) (S) = ⋃ (J \cap 𝒫 S)$
3		inss2	$⊢ J \cap 𝒫 S \subseteq 𝒫 S$
4	3	unissi	$⊢ ⋃ (J \cap 𝒫 S) \subseteq ⋃ 𝒫 S$
5		unipw	$⊢ ⋃ 𝒫 S = S$
6	4 5	sseqtri	$⊢ ⋃ (J \cap 𝒫 S) \subseteq S$
7	2 6	eqsstrdi	$⊢ (J \in Top \land S \subseteq X) \to int (J) (S) \subseteq S$