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

Lemma for Maximal Ideal WRT Filter Complement is Prime in Distributive Lattice
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:
 * $M \subsetneq N$