How to show that ${\rm Ass}(M_{\mathfrak p})=\{\mathfrak p\}$?

Question

$R$ Noetherian, $M$ finitely generated module. If $P$ is minimal over $\operatorname{ann}M$, how is $M' = \ker (M \to M_P)$ a $P$-primary submodule?

While using this question and answer to solve my question, I found that I need to show that ${\rm Ass}(M_{\mathfrak p})=\{\mathfrak p\}$. Can someone help me to show that ${\rm Ass}(M_{\mathfrak p})=\{\mathfrak p\}$?