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Metamath Proof Explorer

Theorem ntropn

Description: The interior of a subset of a topology's underlying set is open. (Contributed by NM, 11-Sep-2006) (Revised by Mario Carneiro, 11-Nov-2013)

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

Proof

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		inss1	$⊢ J \cap 𝒫 S \subseteq J$
4		uniopn	$⊢ (J \in Top \land J \cap 𝒫 S \subseteq J) \to ⋃ (J \cap 𝒫 S) \in J$
5	3 4	mpan2	$⊢ J \in Top \to ⋃ (J \cap 𝒫 S) \in J$
6	5	adantr	$⊢ (J \in Top \land S \subseteq X) \to ⋃ (J \cap 𝒫 S) \in J$
7	2 6	eqeltrd	$⊢ (J \in Top \land S \subseteq X) \to int (J) (S) \in J$