Hall's marriage theorem and a deck of cards

1. Induction on $|X|$. If $|X|=1$, the single vertex has a neighbour.

Let $|X|\ge2$.

*Case 1: $|N(A)|\ge|A|+1$ for every non-empty proper subset $A\subset X$.* Pick $x\in X$ and $y\in N(x)$; match them and delete both. For any non-empty $A\subseteq X\setminus\{x\}$, $|N'(A)|\ge|N(A)|-1\ge|A|$, so Hall's condition still holds and induction gives a matching of the rest.

*Case 2: some non-empty proper $A\subset X$ has $|N(A)|=|A|$.* By induction $A$ can be matched into $N(A)$. In the remaining graph, for $B\subseteq X\setminus A$,
$$|N(B)\setminus N(A)|=|N(A\cup B)|-|N(A)|\ge|B|,$$
so Hall's condition holds and induction matches $X\setminus A$. The two matchings together cover $X$. ∎

2. The cards. Let $X$ be the piles, $Y$ the ranks, and join a pile to a rank if the pile contains a card of that rank. Take any $k$ piles: they contain $4k$ cards, and each rank has exactly $4$ cards, so these piles contain cards of at least $k$ distinct ranks. By Hall there is a matching assigning to each pile a distinct rank present in it. ∎

Lesson. When you need a system of distinct representatives, check Hall's condition.