This is an exercise problem (Ex.3) from the excellent lecture note by Jeff Erickson Lecture 20: Minimum Spanning Trees [Fa13]. Prove that an edge-weighted graph $G$ has a unique minimum spanning tree if and only if the  following conditions holdFor 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?
Necessary and sufficient condition for unique minimum spanning tree
graph theory;spanning trees
Answer my own question by simply copying the comment made by @JeffE, the author of the lecture noteOops! Yes, this is a bug. (Note to self: Change every instance of Prove to Prove or disprove.)  JeffE