# Power Set Width

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 $$\{1,2,\ldots,n\}$$.)

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