Prove that a cut edge is in every spanning tree of a graph 
Given a simple and connected graph $G = (V,E)$, and an edge $e \in E$. Prove:
$e$ is a cut edge if and only if $e$ is in every spanning tree of $G$.

I have been thinking about this question for a long time and have made no progress.

Hint ("only if"): Imagine you have a spanning tree in the graph which doesn't contain the cut-edge. What happens to the graph if you remove this cut edge? What happens to the spanning tree?
Hint ("if"): What happens if you remove this "indispensable" edge (the one which is in every spanning tree)? Can the resulting graph have any spanning tree? What kinds of graphs don't have any spanning tree?
