Confused about proof that if graph G is 3-connected and |G| > 4, then it has an edge that, when contracted, produces another 3-connected graph

Question

I've been trying to understand the proof of the theorem in my textbook, however I'm stuck at one part of the proof

The first part of the proof assumes that such an edge doesn't exist. This means that for all edges in G, contracting the edge will result in a graph that is less than 3-connected. Therefore, G/xy will have a separator (A set of vertices that disconnects G) of size at most 2.

I don't understand the next part that says that the contracted edge is a member of the separator. Could someone please explain to me why this is the case?

Thank You