# Shortest path tree from each vertex implies a unique MST?

Given a connected, undirected graph G, edge-weighted (positive), prove that

If G has a spanning tree T which, for each vertex r in G, is a shortest path tree from r, then G has a unique MST.

I know how to show that given a T like that, it is also a MST, but how can I show that it's unique? I tried to assume by contradiction that it isn't, but I can't see how to get a contradiction from that

Pick any edge $e$ not in $T$. If you add $e$ to $T$, you will get a cycle. Can you show that $e$ is the unique heaviest edge in that cycle?