This is related to Weibel Prop 4.4.5 of Homological Algebra.
A regular local ring $R$ is domain.
This is done by induction on dimension of ring $R$. Consider generators of maximal ideal $(x_1,\dots, x_n)$ with $n=dim(R)$. Then consider $R/x_1$. It follows that $R/x_1$ is regular local which indicates $R/x_1$ is domain. Hence $(x_1)$ is prime ideal. Suppose there is a $P\subset (x_1)$ where $P$ is prime.(i.e. We want to deduce $P=0$. Hence we have domain $R$.)
$\textbf{Q:}$ The book says $Q\subset (x_1)$. So $Q\subset\cap_i(x_1)^n$. Up to here, it is clear. However, the book says $\cap_i(x_1)^n=0$. Why $\cap_i(x_1)^n=0$? What is used here? I believe argument before that can't detect $\cap_i(x_1)^n=0$.(I am not sure whether the book assumed regular local is UFD here. If that is the case, then it holds but regular local being UFD is fairly non-trivial.)