Step |
Hyp |
Ref |
Expression |
1 |
|
clwwlkn |
|- ( 0 ClWWalksN G ) = { w e. ( ClWWalks ` G ) | ( # ` w ) = 0 } |
2 |
|
rabeq0 |
|- ( { w e. ( ClWWalks ` G ) | ( # ` w ) = 0 } = (/) <-> A. w e. ( ClWWalks ` G ) -. ( # ` w ) = 0 ) |
3 |
|
0re |
|- 0 e. RR |
4 |
3
|
ltnri |
|- -. 0 < 0 |
5 |
|
breq2 |
|- ( ( # ` w ) = 0 -> ( 0 < ( # ` w ) <-> 0 < 0 ) ) |
6 |
4 5
|
mtbiri |
|- ( ( # ` w ) = 0 -> -. 0 < ( # ` w ) ) |
7 |
|
clwwlkgt0 |
|- ( w e. ( ClWWalks ` G ) -> 0 < ( # ` w ) ) |
8 |
6 7
|
nsyl3 |
|- ( w e. ( ClWWalks ` G ) -> -. ( # ` w ) = 0 ) |
9 |
2 8
|
mprgbir |
|- { w e. ( ClWWalks ` G ) | ( # ` w ) = 0 } = (/) |
10 |
1 9
|
eqtri |
|- ( 0 ClWWalksN G ) = (/) |