Let $R$ be a unital commutative ring, $P$ $\subseteq$ $R$ a prime ideal, $X\subseteq P$ a subset. Show there exists a minimal (inclusion minimal) prime ideal contained in $P$ which contains $X$.
My approach: Let $\langle X\rangle$ be the ideal generated by $X$.
$P^c$ is a multiplicative set, as $P$ is prime. $\langle X\rangle\subseteq P\subseteq R$ and $P^c\cap\langle X\rangle=\emptyset $ so by Zorn's lemma there exist prime ideal $I_1$ satisfying $\langle X\rangle\subseteq I_1\subseteq P$. As $I_1$ is prime, it's complement is a multiplicative subset which does not intersect $\langle X\rangle$, hence I can find prime ideal $I_2$ satisfying $\langle X\rangle\subseteq I_2 \subseteq I_1$ and so on... I've got an infinite descending chain of prime ideals and I can't be sure that the process will stop.
Hints will be appreciated, thank you.
Edit: Can I conclude that every prime ideal contains a minimal prime ideal?