I want to check if there is anything wrong with my proof below:
Let S be the set whose elements are prime ideals of A. Impose the following relation on S. $P_1 \leq P_2$ iff $P_2 \subset P_1$. Let $(J)_{\alpha \in U}$ be a chain then Consider $J_{min} = \bigcap_{\alpha \in U} J_{\alpha}$, then we can prove that $J_{min}$ is an ideal. Moreover it is prime. So every chain has an upper bound so by zorn's lemma maximal element exist and by construction this maximal element is the minimal element with respect to inclusion.