I believe it is true.
If $\displaystyle \binom{n}{r} = p$ for some prime $\displaystyle p$ and $\displaystyle 1 \lt r \lt n-1$, then,
If $\displaystyle n \ge p$, then $\displaystyle \binom{n}{r} \gt n \ge p$, as the binomial coefficients increase and then decrease as $\displaystyle r$ varies from $\displaystyle 0$ to $\displaystyle n$.
If $\displaystyle n \lt p$ then $\displaystyle \binom{n}{r}$ can never be divisible by $\displaystyle p$, as $\displaystyle n!$ is not divisible by $\displaystyle p$.
OR as hardmath succintly put it:
$p \mid \binom{n}{r} \Rightarrow p \le n \lt \binom{n}{r}$