I've been having a bit of trouble trying to prove this. I don't want an answer to this question (it's an assignment problem), just some assistance on how to proceed.
I know that in order to prove that a graph is non-planar, we have to show that it contains a subgraph that is a subdivision of $K_5$ or $K_{3,3}$.
In order to show this, we must then show that there exists a set of 5 vertices such that there are 4 disjoint paths between each one and the other four (where each internal vertex in these paths has degree 2), or that there exist six vertices such that there exist disjoint paths from three of them to the other three (each internal vertex has degree 2).
I'm just not sure how I would go about showing the above.