This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem ntrtop

Description: The interior of a topology's underlying set is the entire set. (Contributed by NM, 12-Sep-2006)

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

Step	Hyp	Ref	Expression
1		clscld.1	$⊢ X = ⋃ J$
2	1	topopn	$⊢ J \in Top \to X \in J$
3		ssid	$⊢ X \subseteq X$
4	1	isopn3	$⊢ (J \in Top \land X \subseteq X) \to (X \in J \leftrightarrow int (J) (X) = X)$
5	3 4	mpan2	$⊢ J \in Top \to (X \in J \leftrightarrow int (J) (X) = X)$
6	2 5	mpbid	$⊢ J \in Top \to int (J) (X) = X$