The maximum value of the size of a subset that satisfies a specific condition

Question

Let $N_n= \{1,2,...,n\}$ and $P_n$ be power set of $N_n$. $X\subset P_n$ satisfy below condition.

For all $x,y \in X$, $x\cap y \not= \emptyset$. For all $x,y \in X$, $x \subset y$ implies $x=y$.

Then, How can we find maximum value of $|X|$?

I guess maximum value of $|X|$= $n \choose {[\frac{n}{2}]+1}$. But I'm not sure...