Theorem haustop

Description: A Hausdorff space is a topology. (Contributed by NM, 5-Mar-2007)

	Ref	Expression
Assertion	haustop	$$⊢J\in \mathrm{Haus}\to J\in \mathrm{Top}$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢\bigcup J=\bigcup J$$
2	1	ishaus	$$⊢J\in \mathrm{Haus}\leftrightarrow \left(J\in \mathrm{Top}\wedge \forall x\in \bigcup J\forall y\in \bigcup J\left(x\ne y\to \exists n\in J\exists m\in J\left(x\in n\wedge y\in m\wedge n\cap m=\varnothing \right)\right)\right)$$
3	2	simplbi	$$⊢J\in \mathrm{Haus}\to J\in \mathrm{Top}$$