Metamath Proof Explorer

Theorem 0ngrp

Description: The empty set is not a group. (Contributed by NM, 25-Apr-2007) (New usage is discouraged.)

Ref Expression
Assertion 0ngrp 
|- -. (/) e. GrpOp

Proof

Step Hyp Ref Expression
1 neirr 
 |-  -. (/) =/= (/)
2 rn0 
 |-  ran (/) = (/)
3 2 eqcomi 
 |-  (/) = ran (/)
4 3 grpon0 
 |-  ( (/) e. GrpOp -> (/) =/= (/) )
5 1 4 mto 
 |-  -. (/) e. GrpOp