True false question related to graph having a unique Minimum weight spanning tree

Question

• You have an undirected graph $G$
• $G$ has a cycle in it
• That cycle has an edge $e$
• e is a unique lightest weight edge in that cycle

Is it true that $e$ is part of every Minimum weight spanning tree of $G$?

I cant find any example to disprove this statement so I am assuming it is true. Do you think the same? or can you find a counter example?

Answer 1

ASCII art:

    A---1---B
    |\     /|
    | 1   1 |
    |  \ /  |
    1   C   1
    |  / \  |
    | 2   3 |
    |/     \|
    D---4---E

Consider the cycle $CDE$.