Description: The distance function of a metric space is symmetric. Definition 14-1.1(c) of Gleason p. 223. (Contributed by NM, 27-Aug-2006) (Revised by Mario Carneiro, 20-Aug-2015)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | metsym | |- ( ( D e. ( Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) = ( B D A ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | metxmet | |- ( D e. ( Met ` X ) -> D e. ( *Met ` X ) ) |
|
| 2 | xmetsym | |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) = ( B D A ) ) |
|
| 3 | 1 2 | syl3an1 | |- ( ( D e. ( Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) = ( B D A ) ) |