Form of the smallest vertex cover in a bipartite graph

I'm trying to write a proof of Konig's theorem using Menger's theorem. However, I got stuck along the way. In order to move forward I'd (apparently) need to show the following fact.

Let G be a bipartite graph with partite sets X and Y. The smallest vertex cover of G is of the form $(X-A)\cup N(A)$ for some $A\subseteq X$.

I tried doing a proof by contradiction but to no avail. Maybe the thesis I'm trying to show is false in the first place? Any help would be greatly appreciated.

I'm trying to write a proof of Konig's theorem using Menger's theorem. However, I got stuck along the way. In order to move forward I'd (apparently) need to show the following fact.

Let G be a bipartite graph with partite sets X and Y. The smallest vertex cover of G is of the form $(X-A)\cup N(A)$ for some $A\subseteq X$.

I tried doing a proof by contradiction but to no avail. Maybe the thesis I'm trying to show is false in the first place? Any help would be greatly appreciated.