Metamath Proof Explorer

Theorem grpn0

Description: A group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007) (Revised by Mario Carneiro, 2-Dec-2014)

Ref Expression
Assertion grpn0 
|- ( G e. Grp -> G =/= (/) )

Proof

Step Hyp Ref Expression
1 eqid 
 |-  ( Base ` G ) = ( Base ` G )
2 1 grpbn0 
 |-  ( G e. Grp -> ( Base ` G ) =/= (/) )
3 fveq2 
 |-  ( G = (/) -> ( Base ` G ) = ( Base ` (/) ) )
4 base0 
 |-  (/) = ( Base ` (/) )
5 3 4 eqtr4di 
 |-  ( G = (/) -> ( Base ` G ) = (/) )
6 5 necon3i 
 |-  ( ( Base ` G ) =/= (/) -> G =/= (/) )
7 2 6 syl 
 |-  ( G e. Grp -> G =/= (/) )