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Metamath Proof Explorer

Theorem eltop2

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

		Ref	Expression
	Assertion	eltop2	$⊢ J \in Top \to (A \in J \leftrightarrow \forall x \in A \exists y \in J (x \in y \land y \subseteq A))$

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		eltg2b	$⊢ J \in Top \to (A \in topGen (J) \leftrightarrow \forall x \in A \exists y \in J (x \in y \land y \subseteq A))$
4	2 3	bitr3d	$⊢ J \in Top \to (A \in J \leftrightarrow \forall x \in A \exists y \in J (x \in y \land y \subseteq A))$