Theorem sshaus

Description: A topology finer than a Hausdorff topology is Hausdorff. (Contributed by Mario Carneiro, 2-Mar-2015)

	Ref	Expression
Hypothesis	t1sep.1	$$⊢X=\bigcup J$$
Assertion	sshaus	$$⊢\left(J\in \mathrm{Haus}\wedge K\in \mathrm{TopOn}\left(X\right)\wedge J\subseteq K\right)\to K\in \mathrm{Haus}$$

Proof

Step	Hyp	Ref	Expression
1		t1sep.1	$$⊢X=\bigcup J$$
2		haustop	$$⊢J\in \mathrm{Haus}\to J\in \mathrm{Top}$$
3		cnhaus	$$⊢\left(J\in \mathrm{Haus}\wedge \left({\mathrm{I}\upharpoonright }_X\right):X\underset{1-1}{⟶}X\wedge {\mathrm{I}\upharpoonright }_X\in \left(K\mathrm{Cn}J\right)\right)\to K\in \mathrm{Haus}$$
4	1 2 3	sshauslem	$$⊢\left(J\in \mathrm{Haus}\wedge K\in \mathrm{TopOn}\left(X\right)\wedge J\subseteq K\right)\to K\in \mathrm{Haus}$$