Maximal Ideal WRT Filter Complement is Prime in Distributive Lattice/Lemma 4

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

Then we have that:

$N \cap F = \O$

Proof

Aiming for a contradiction, suppose $x \in N \cap F$.

Then:

$x \in N$

so for some $m \in M$:

$x \le m \vee a$

Furthermore, $x \in F$.

So by the definition of a filter:

$m \vee a \in F$

But this contradicts our assumption that $\forall m \in M: m \vee a \notin F$.

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$\blacksquare$