Power Set Width

Probability Level 2

Let $X$ be a set of subsets of $\{1,2,3,4\}$ such that no element of $X$ is completely contained in any other element of $X$ : that is, for any two distinct subsets $A,B \in X$ , $A \nsubseteq B$ and $B \nsubseteq A$ .

What is the maximum possible value of $|X|$ ?

(Bonus: generalize to $\{1,2,\ldots,n\}$ .)