Q: $p_n$ is never a divisor of $n!$ I just wanted to know if this will suffice as proof, as the title suggests, I want to prove that $p_n$ is never a divisor of $n!$ unless $n=2$, and I feel like stating QED after making the following statement regarding the p-adic valuation of $n!$ for the $n^{th}$ prime, but yes this is very brief and so I need criticism of course:
$$\sum _{j=1}^{ \bigl\lfloor {\frac {\ln  \left( n \right) }{\ln 
 \left( p_{{n}} \right) }} \bigr\rfloor +1} \Bigl\lfloor {\frac {n}{{p
_{{n}}}^{j}}} \Bigr\rfloor =0 \,\,\,\,\forall n: n \gt 1 \land n \in \mathbb N
$$
