This is an exercise problem (Ex.3) from the excellent lecture note by Jeff Erickson Lecture 20: Minimum Spanning Trees [Fa’13] .
Prove that an edge-weighted graph $G$ has a unique minimum spanning tree if and only if the following conditions hold
For any partition of the vertices of $G$ into two subsets, the minimum-weight edge with one endpoint in each subset is unique.
The maximum-weight edge in any cycle of $G$ is unique.
Consider the "$\Rightarrow$" direction and the following graph $G$.
$G$ has a unique MST. However, for the partition $\{A\}$ and $\{B,C\}$, the minimum-weight crossing edge is not unique.
Did I misunderstand some points? Or if there are flaws in the theorem, how can we fix it?