This question already has an answer here:
Given the following statement:
For a graph $G$, consider its minimum spanning tree $T$ and let $e = (a,b)$ be an edge that is not a light edge for a given cut $C$. Then $e$ never belongs to $T$.
Intuitively, I believe that the above statement must be true since in order to make an MST we always take (one of) the lightest edges available that cross the cut, but am I not sure how to approach a proof, or if my intuition is right.