Not Preceding implies There Exists Meet Irreducible Element Not Preceding

Theorem
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a bounded below continuous lattice.

Let $x, y \in S$ such that
 * $y \npreceq x$

Then
 * $\exists p \in S: p$ is meet irreducible and $x \preceq p$ and $y \npreceq p$

Proof
By definition of continuous:
 * $L$ satisfies axiom of approximation

and
 * $\forall x \in S: x^\ll$ is directed.

By Axiom of Approximation in Up-Complete Semilattice
 * $\exists u \in S: u \ll y \land u \npreceq x$

By Way Below implies There Exists Way Below Open Filter Subset of Way Above Closure:
 * there exists way below open filter $F$ in $L$: $y \in F \subseteq u^\gg$

By Way Above Closure is Subset of Upper Closure of Element:
 * $u^\gg \subseteq u^\succeq$

By definition of upper set:
 * $x \notin F$

By definition of relative complement:
 * $x \in \relcomp S F$

By Upper Way Below Open Subset Complement is Non Empty implies There Exists Maximal Element of Complement:
 * $\exists m \in \relcomp S F: x \preceq m \land m = \max \relcomp S F$

Thus by Maximal Element of Complement of Filter is Meet Irreducible:
 * $m$ is meet irreducible.

Thus
 * $x \preceq m$


 * $y \preceq m$

By definition of upper set:
 * $m \in F$

This contradicts $m \in \relcomp S F$ by definition of greatest element.