Distinct weights imply a unique minimum spanning tree

For a finite connected weighted graph with distinct edge weights, Kruskal's algorithm produces the unique minimum spanning tree. If another optimal tree existed, take an edge belonging only to the algorithm's tree and slightly increase its weight without changing the strict edge order. The algorithm then returns exactly the same tree, which must still be optimal, but the other tree's weight is unchanged and is now strictly smaller, a contradiction. Distinct weights are sufficient, not necessary: a triangle with weights $1,1,2$ has a unique optimum formed from the two weight-one edges.