Metamath Proof Explorer


Theorem metn0

Description: A metric space is nonempty iff its base set is nonempty. (Contributed by NM, 4-Oct-2007) (Revised by Mario Carneiro, 14-Aug-2015)

Ref Expression
Assertion metn0 D Met X D X

Proof

Step Hyp Ref Expression
1 metf D Met X D : X × X
2 frel D : X × X Rel D
3 reldm0 Rel D D = dom D =
4 1 2 3 3syl D Met X D = dom D =
5 1 fdmd D Met X dom D = X × X
6 5 eqeq1d D Met X dom D = X × X =
7 4 6 bitrd D Met X D = X × X =
8 xpeq0 X × X = X = X =
9 oridm X = X = X =
10 8 9 bitri X × X = X =
11 7 10 bitrdi D Met X D = X =
12 11 necon3bid D Met X D X