$X$ and $Y$ are two max. size independent sets in a graph. Show that the subgraph $G[X \Delta Y]$ has a perfect matching.
What I've got: Take $A = X \setminus Y$ and $B = Y \setminus X$. Then $G[X \Delta Y]$ is a bipartite graph with color classes $A$ and $B$. Clearly, $|A| = |B|$. Furthermore, $\forall a \in A$, there must be at least one edge from $a$ to $B$ as otherwise $a$ could be added to $Y$, contradicting $X$'s maximum size. Similarly, $\forall b \in B$, there must be at least one edge from $b$ to $A$.
How do I proceed from here to prove the existence of a perfect matching between $A$ and $B$?