Two vertices $x$ and $y$ form an arc that is part of a minimum capacity $s-t$ cut in a directed graph $G = (V, A)$. Prove that another minimum capacity $s-t$ cut cannot exist in $G$ if it only contains the $y$ vertex as part of its arcs and not the $x$ vertex.
I am not entirely sure how to approach this proof. I realize that I probably need to cut $G$ into two sections $S^*$ and $S^/$. I was then thinking about using the cut lemma to show that the cardinality of any flow has to be equal to the flow out of a vertex subtracted by the flow into a vertex. But I am not entirely sure how I would apply that to this problem specifically. I was thinking that I may have to treat $G$ as a bipartite graph but that did not lead me to a good conclusion either.
I would be extremely appreciative of any assistance!