The cycle property on graphs states, that
For any cycle $C$ in a graph, if the weight of an edge $e$ of $C$ is larger than the individual weights of all other edges of $C$, then this edge cannot belong to a minimum spanning tree.
Now I'm interested in generalizing this statement to matroids, but I cannot find a way of proving or disproving it.
Does the following hold?
For any cycle $C$ in a matroid, if the weight of an element $e\in C$ is larger than the individual weights of all other elements in $C$, then $e$ cannot belong to a minimal basis of the matroid.