Theorem sst1

Description: A topology finer than a T_1 topology is T_1. (Contributed by Mario Carneiro, 25-Aug-2015)

	Ref	Expression
Hypothesis	t1sep.1	$$⊢X=\bigcup J$$
Assertion	sst1	$$⊢\left(J\in \mathrm{Fre}\wedge K\in \mathrm{TopOn}\left(X\right)\wedge J\subseteq K\right)\to K\in \mathrm{Fre}$$

Proof

Step	Hyp	Ref	Expression
1		t1sep.1	$$⊢X=\bigcup J$$
2		t1top	$$⊢J\in \mathrm{Fre}\to J\in \mathrm{Top}$$
3		cnt1	$$⊢\left(J\in \mathrm{Fre}\wedge \left({\mathrm{I}\upharpoonright }_X\right):X\underset{1-1}{⟶}X\wedge {\mathrm{I}\upharpoonright }_X\in \left(K\mathrm{Cn}J\right)\right)\to K\in \mathrm{Fre}$$
4	1 2 3	sshauslem	$$⊢\left(J\in \mathrm{Fre}\wedge K\in \mathrm{TopOn}\left(X\right)\wedge J\subseteq K\right)\to K\in \mathrm{Fre}$$