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

Metamath Proof Explorer

Theorem eltop3

Description: Membership in a topology. (Contributed by NM, 19-Jul-2006)

		Ref	Expression
	Assertion	eltop3	$⊢ J \in Top \to (A \in J \leftrightarrow \exists x (x \subseteq J \land A = ⋃ x))$

Step	Hyp	Ref	Expression
1		tgtop	$⊢ J \in Top \to topGen (J) = J$
2	1	eleq2d	$⊢ J \in Top \to (A \in topGen (J) \leftrightarrow A \in J)$
3		eltg3	$⊢ J \in Top \to (A \in topGen (J) \leftrightarrow \exists x (x \subseteq J \land A = ⋃ x))$
4	2 3	bitr3d	$⊢ J \in Top \to (A \in J \leftrightarrow \exists x (x \subseteq J \land A = ⋃ x))$