So the question asks, given that we have a undirected graph with unique edge weights, prove that the graph has a unique minimum spanning tree. My Proof: If the graph has unique edge weights, we can list the edges as $E_1,E_2,\ldots, E_N, \text{ s.t. } E_1 < E_2 < E_3 < \cdots < E_N.$
Then applying Kruskal's Algorithm, we can say that the edges that will be in our minimum spanning tree will be all edges in the above list such that as we go through the list, the next element added does not create a cycle. Since the only time we are faced with multiple decision points is when there are two edges of the same weight to choose from, and since the edge weights are distinct, we are never confronted with a decision to choose between two edges, and hence our minimum spanning tree is unique.
But I feel as though the part about decision points is flimsy. Any advice?