It's well known that the nilradical of a commutative ring with identity $A$ is the intersection of all the prime ideals of $A$.

Every proof I found (e.g. in the classical "Commutative Algebra" by Atiyah and Macdonald) uses Zorn's lemma to prove that $x \notin Nil(A) \Rightarrow x \notin \cap_{\mathfrak{p}\in Spec(A)} \mathfrak{p}$ (the other way is immediate). Does anybody know a proof that doesn't involve it?