Number of prime divisors of a proper closed subset of a scheme

Question

Let $X$ be a noetherian integral separated scheme which is regular in codimension one. Let $Z$ be a proper closed subset of $X$. There is a claim I could not figure out: $Z$ can contain at most finitely many prime divisors of $X$ (as usual, a prime divisor of $X$ is a closed integral subscheme of $X$ of codimension one). Could anyone explain why this is true?

Answer 1

Here is my argument: since $Z$ is a proper closed subset $\mathrm{codim}(Z)\geq1$. If $\mathrm{codim}(Z)\geq2$, there is no prime divisor of $X$ condtained in $Z$. Let $\mathrm{codim}(Z)=1$. In this case, we see that any prime divisor $Y$ of $X$ contained in $Z$ is an irreducible component of $Z$. Since $X$ is noetherian, so is $Z$. This implies that there are only finitely many irreducible components of $Z$, i.e. $Z$ can contain only finitely many prime divisors of $Z$.

Does this make sense?