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Suppose that there exists two unique minimum spanning trees, M and N. 
Let us say that M contains atleast one edge that is not in N, and has a
smaller weight than some edge in N. This must be true because the edges
are unique, so there must be a smallest different edge.\\

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So first choose the smallest edge that is in M but not in N. Lets call it E. Now add
that edge to the N mst. Adding any edge to a tree creates a unique cycle. So in
order to remove the cycle we have to remove one of the elements. Infact,
we can always remove an element that is smaller than the edge that was added.
This produces a smaller weight for tree N, contradicting the assumption that
it was a minimum spanning tree in the first place \\

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But how do we know that there must be a value larger than E in the cycle?
We know this because E is the smallest value in M, not in N. If E was the
largest value in the cycle, than it would mean than the two other values were also
in M. But then M would contain a cycle, and trees cannot contain cycles \\