Something that doesn't divide a factorial

I claim that $n$ is a prime number. So , suppose $n$ is not a prime number. This means we can express $n=p\times q$ for some positive integers $p,q$ such that $1<p,q<n$ . Let $S$ be a set such that $S=\{1,2,3,4\dots, p , \dots , q ,\dots ,(n-1)\}$ . Note that $(n-1)!$ is the product of all elements of $S$ .Thus, $p|(n-1)!$ , $q|(n-1)!$ which means $pq|(n-1)! \Rightarrow n|(n-1)!$ . So $n$ must be prime so as to not divide $(n-1)!$ , thus the second largest two digit prime number is $\boxed{89}$ .

As pointed by Svatejas S , the answer is trivial by Wilson's Theorem.

This is a beautiful way of asking for the penultimate prime number between 1 and 100