Step |
Hyp |
Ref |
Expression |
1 |
|
metf |
|- ( D e. ( Met ` X ) -> D : ( X X. X ) --> RR ) |
2 |
|
frel |
|- ( D : ( X X. X ) --> RR -> Rel D ) |
3 |
|
reldm0 |
|- ( Rel D -> ( D = (/) <-> dom D = (/) ) ) |
4 |
1 2 3
|
3syl |
|- ( D e. ( Met ` X ) -> ( D = (/) <-> dom D = (/) ) ) |
5 |
1
|
fdmd |
|- ( D e. ( Met ` X ) -> dom D = ( X X. X ) ) |
6 |
5
|
eqeq1d |
|- ( D e. ( Met ` X ) -> ( dom D = (/) <-> ( X X. X ) = (/) ) ) |
7 |
4 6
|
bitrd |
|- ( D e. ( Met ` X ) -> ( D = (/) <-> ( X X. X ) = (/) ) ) |
8 |
|
xpeq0 |
|- ( ( X X. X ) = (/) <-> ( X = (/) \/ X = (/) ) ) |
9 |
|
oridm |
|- ( ( X = (/) \/ X = (/) ) <-> X = (/) ) |
10 |
8 9
|
bitri |
|- ( ( X X. X ) = (/) <-> X = (/) ) |
11 |
7 10
|
bitrdi |
|- ( D e. ( Met ` X ) -> ( D = (/) <-> X = (/) ) ) |
12 |
11
|
necon3bid |
|- ( D e. ( Met ` X ) -> ( D =/= (/) <-> X =/= (/) ) ) |