MathNet 2013 Question 71

Let E be a set of size n, and let $\mathscr F$ be a family of subsets of E with $|\mathscr F|=2^{n-1}$ such that A∩B∩C is nonempty for all A,B,C in $\mathscr F$. Prove that the intersection of all members of $\mathscr F$ is nonempty.