 Prime lattice ideal

A proper lattice ideal $I$ is prime if $x\wedge y\in I$ implies $x\in I$ or $y\in I$. Its complement is a prime filter of a distributive lattice when the lattice is distributive: downward closure of the ideal makes the complement upward closed, primeness makes it meet-closed, and join closure makes that filter prime. Equivalently, its complement indicator is a homomorphism of lattices to the two-element lattice. Given $b\nleqq a$ in a distributive lattice, maximize an ideal containing $a$ and avoiding $b$ using Zorn lemma. If $x,y$ lie outside it, the two enlarged ideals contain $b$, so $b\le i\vee x$ and $b\le j\vee y$ for some $i,j\in I$. Distributivity shows that $x\wedge y\in I$ would force $b\in I$, a contradiction. Thus this maximal avoiding ideal is prime and separates the two specified elements.