Maximal Ideal WRT Filter Complement is Prime in Distributive Lattice/Lemma 3

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Lemma for Maximal Ideal WRT Filter Complement is Prime in Distributive Lattice

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 $F$.

Let $N = \set {x \in L: \exists m \in M: x \le m \vee a}$.

$M \subsetneq N$

Proof

Let $m \in M$.

Then:

$m \le \paren {m \vee a}$

so $m \in N$.

Thus $M \subseteq N$.

We have:

$a \le \paren {m \vee a}$

so:

$a \in N$

but:

$a \notin M$

Thus:

$M \subsetneq N$

$\blacksquare$