Lemma 2

Every prime ideal of a ring $A$ contains a minimal prime ideal of $A$.

It is sufficient, by Zorn's Lemma, to show that the set P of prime ideals, ordered by the relation $\supseteq$, is inductive. Now, if $G$ is a non-empty totally ordered subset of P, the intersection $p_0$ of the ideals $p \in G$ is also a prime ideal: for if $x \notin p_0$ and $y \notin p_0$, there exists an ideal $p \in G$ such that $x \notin p$ and $y \notin p$, whence $xy \notin p$ and a fortiori $xy \notin p_0$.