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Let G be an undirected graph with distinct edge weights.
Let T be the MST in G.
Let (u, v) be any edge in T. Show that there is a cut (S; V-S) such that (u; v) is the minimum weight edge in this cut.
I’ll give it a shoot, let’s consider a cut such that e = (u, v) is the only of its edges belonging to T. Suppose there’s another edge e’ in the cut with w(e’) < w(e), then we could form another ST including e’ and dropping e, which would have smaller weight, absurd.