Prove that if an integer n > 2, then there exist a prime number p such that n < p < n!

My proof.

Let p be a prime divisor of n!-1, then $\displaystyle p \leq n!-1 \leq n!$.

If p = n, then p|n!, but that is impossible since p|n!-1.

Now, how do I prove that p is not < n?