Ideal of the set-theoretic difference of two closed subsets of an affine scheme

Question

Consider the affine scheme $X=\operatorname{Spec} A$ and the closed subsets (not subschemes!) $Y,Z\subset X$.
I can prove the equality $j(Y\setminus Z)=j(Y): j(Z)$, where $j(Y)$ is the intersection of the prime ideals comprising $Y$ (as in the second page of $EGA_I$ !).
I would be grateful for a reference (not a proof) to this fact.