I have been told that Dilworth's Theorem implies Hall's Theorem. I don't seem to be able to show this implication, especially because I haven't done any graph theory in a while! Any help? If possible, I would like a graph theory proof (and not using systems of distinct representative).
I believe the main idea is to order the bipartite graph $G = X \cup Y$ by letting $x<y$ if and only if $x \in X$, $y\in Y$ and there is an edge between $x$ and $y$. Now if I could show that I can assume the longest anti-chain in $G$ contains the whole of $X$, I am done. But I am not sure this is true! I tried to apply Hall's Condition to a possibly "larger" anti-chain which does not contain some elements of X and show this is not possible, but I was unsuccessful.