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Metamath Proof Explorer

Theorem nrmreg

Description: A normal T_1 space is regular Hausdorff. In other words, a T_4 space is T_3 . One can get away with slightly weaker assumptions; see nrmr0reg . (Contributed by Mario Carneiro, 25-Aug-2015)

Ref Expression
Assertion nrmreg 
|- ( ( J e. Nrm /\ J e. Fre ) -> J e. Reg )

Proof

Step Hyp Ref Expression
1 t1r0 
 |-  ( J e. Fre -> ( KQ ` J ) e. Fre )
2 nrmr0reg 
 |-  ( ( J e. Nrm /\ ( KQ ` J ) e. Fre ) -> J e. Reg )
3 1 2 sylan2 
 |-  ( ( J e. Nrm /\ J e. Fre ) -> J e. Reg )