I'm trying to prove Mori-Nagata theorem.
Let $A$ be a noetherian domain, $K=\operatorname{Frac}(A)$ and $B$ be the integral closure of $A$ in $K$.
I want to show that for every nonzero $x\in B$, there are finitely many primes of $B$ of height $1$ containing $x$.
Replacing $A$ by $A[x]$, we may assume that $x\in A$.
It suffices to show that for every prime ideal $P$ of $B$ of height $1$ containing $x$, $P\cap A\in \operatorname{Ass}_A(A/xA)$.
(I know that the corresponding morphism $\operatorname{Spec}B\rightarrow \operatorname{Spec}A$ has finite fibres.)
This claim is equivalent to the following:
Let $P$ be a prime ideal of $B$ of height $1$, and put $p=P\cap A$.
Then $\operatorname{depth}A_p =1$.
How can I prove this? Any advice would be helpful.
Thank you.