Solution
= Solution
Fix any edge $xy$. After deleting $x$ and $y$, every nonempty subset $A$ of $X\setminus\{x\}$ satisfies $|\Gamma(A)\setminus\{y\}|\geq|A|$, by the strict inequality in the question. The <Hall marriage theorem> gives a <matching in a graph> for the remainder; adding $xy$ gives a matching covering $X$ and containing the prescribed edge. The case $|X|=1$ is immediate.