Given a connected, undirected, weighted graph $G$, the condition
The maximum-weight edge in any cycle of $G$ is unique.
is not necessary for $G$ to have a unique minimum spanning tree (MST).
However, is this condition sufficient for $G$ to have a unique MST?
Answer My Own Question by attempting to prove that the condition above is sufficient for a graph $G$ to have a unique MST. I am asking for reviews. Thanks in advance.
First, for each cycle, the unique heaviest edge in it must be not in any MST. Consider the remaining graph $G'$, obtained by keeping removing the unique heaviest edges in cycles from $G$. All MSTs are part of $G'$. If we can show that $G'$ is a spanning tree, then $G'$ is the unique MST of $G$.
$G'$ is connected because each edge removed belongs to a cycle when it was removed.
$G'$ is acyclic because we are keeping breaking cycles.
Thus, $G'$ is a spanning tree.