Let $S=\{1,2,3,\ldots, 2n\}.$ If $P\subseteq S$ such that $x,y\in P \rightarrow x\nmid y$ and $y \nmid x$ then what is the maximum cardinality of $P$

Question

Let $S=\{1,2,3,\ldots, 2n\}.$ If $P\subseteq S$ such that if $x,y\in P \rightarrow x\nmid y$ and $y \nmid x$ Then what is the maximum cardinality of $P$

I know that $\{n+1, n+2,\ldots, 2n\}\subseteq P$ so $|P|\geq n$

How can I proceed from here. I somehow want to use Pigeonhole principle.