Maximal Ideal WRT Filter Complement is Prime in Distributive Lattice/Assertions

Maximal Ideal WRT Filter Complement is Prime in Distributive Lattice Assertions
Let $\struct {S, \vee, \wedge, \preceq}$ be a distributive lattice.

Let $F$ be a filter in $L$.

Let $M$ be an ideal in $L$ which is disjoint from $F$ such that:
 * no ideal in $L$ larger than $M$ is disjoint from $F$.