Let $K$ be a number field and $\mathcal O$ its ring of algebraic integers.
Let $\mathfrak p_1,\ldots , \mathfrak p_n$ prime ideals of $\mathcal O$ with $\mathfrak p_i\neq\mathfrak p_j$ if $i\neq j$.
Let $ \pi_j\in \mathfrak p_j\setminus \displaystyle\prod_{l\neq j} \mathfrak p_l$
My question is: $\pi_j\notin \mathfrak p_i\setminus \displaystyle\prod_{l\neq i} \mathfrak p_l$ if $i\neq j$?
Thank you all.