KöMaL 2015 A637

Let $n$ be a positive integer. Let $\mathcal{F}$ be a family of sets that contains more than half of all subsets of an $n$ -element set $X$ . Prove that from $\mathcal{F}$ we can select $\lceil\log_2n\rceil+1$ sets that form a separating family on $X$ , i.e., for any two distinct elements of $X$ there is a selected set containing exactly one of the two elements.