 Principal prime in a Noetherian local ring lemma

If a principal prime ideal $(x)$ in a Noetherian local ring has positive height, then the ring is a domain. Localizing at $(x)$ shows that every annihilator of a power of $x$ lies in $(x)$; stabilization of the annihilator chain then proves that $x$ is a nonzerodivisor and that zero is prime.