graphs – Can an edge that is never a clear edge be part of the minimal spanning tree?

Let $ G = (V, E) $ to be an undirected graph with arbitrary weighting. Let us define a Cut $ C = (S, V setminus S) $ of $ G $ to be a partition of $ G $ in two sets, $ S $ and $ V setminus S $.

For a cut $ C $, say one edge $ e $ crosses $ C $ if and only if an end point of $ e $ is a summit in $ S $ and the other is a summit $ V setminus $. Finally call an edge $ e $ be a clear edge running through the cup $ C $ if and only if $ e $ crosses the cup $ C $ and has a minimum weight compared to all the other edges that cross the cut $ C $.

My question is: can an edge $ e $ it's never a light edge for any cut $ C $ be part of a tree covering minimum of $ G $?

I think this is true, simply because we can consider $ G $ to be a tree then select an edge and make its weight bigger than all the others. Then, as there is only a minimal covering tree, this cutting edge will surely be part of it, and for any cut of $ G $ it will never be a light edge. Is my reasoning correct?