KöMaL 2006 A396

The elements of families $\mathcal{A}$ and $\mathcal{B}$ are finite sets, each having at most $n$ elements. Suppose that for an arbitrary finite set $F$ there exist $X\in\mathcal{A}$ and $Y\in\mathcal{B}$ such that $X \cap Y \cap F$=Ø. Prove that there exist $X\in\mathcal{A}$ and $Y\in\mathcal{B}$ such that $X \cap Y$=Ø.