Metamath Proof Explorer

Theorem haushmph

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

Ref Expression
Assertion haushmph 
|- ( J ~= K -> ( J e. Haus -> K e. Haus ) )

Proof

Step Hyp Ref Expression
1 haustop 
 |-  ( J e. Haus -> J e. Top )
2 cnhaus 
 |-  ( ( J e. Haus /\ f : U. K -1-1-> U. J /\ f e. ( K Cn J ) ) -> K e. Haus )
3 1 2 haushmphlem 
 |-  ( J ~= K -> ( J e. Haus -> K e. Haus ) )