I am looking for a reference for the following fact.

Let $G$ be simple undirected connected planar graph with $\geq 2$ vertices. Then $G$ contains an edge $\{u,v\}$ such that $|N(u) \cap N(v)| \leq 2$, in other words the number of common neighbors of $\{u,v\}$ is at most $2$.

Here's a sketch of the proof. Any edge incident on a vertex of degree $\leq 3$ satisfies this requirement. If there is a vertex $v$ of degree $4$ for which none of its edges satisfy the requirement, then $N[v]$ must be a $K_5$ which is not possible; if there is a vertex of degree 5 for which none of its edges satisfy then $N[v]$ must contain a $K_{3,3}$. Since any planar graph contains a vertex of degree $\leq 5$ this is sufficient.

I would like to be able to skip the proof in my paper, using a reference to known work instead. Can anyone give me a reference for the above fact?