Theorem haushmph

Description: Hausdorff-ness is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015)

	Ref	Expression
Assertion	haushmph	$$⊢J\simeq K\to \left(J\in \mathrm{Haus}\to K\in \mathrm{Haus}\right)$$

Proof

Step	Hyp	Ref	Expression
1		haustop	$$⊢J\in \mathrm{Haus}\to J\in \mathrm{Top}$$
2		cnhaus	$$⊢\left(J\in \mathrm{Haus}\wedge f:\bigcup K\underset{1-1}{⟶}\bigcup J\wedge f\in \left(K\mathrm{Cn}J\right)\right)\to K\in \mathrm{Haus}$$
3	1 2	haushmphlem	$$⊢J\simeq K\to \left(J\in \mathrm{Haus}\to K\in \mathrm{Haus}\right)$$