Equivalence of Definitions of Lattice Filter

Theorem
Let $\struct {S, \vee, \wedge, \preceq}$ be a lattice.

Let $F \subseteq S$ be a non-empty subset of $S$.

Definition 1 implies Definition 2
Let $F$ satisify the lattice filter axioms.

To show that $F$ is a meet semilattice filter it is sufficient to show:

Let $x \in F, y \in S : x \preceq y$.

By the lattice filter axioms, $F$ is a sublattice of $\struct {S, \vee, \wedge, \preceq}$, so:
 * $x \vee y \in F$

From Preceding iff Join equals Larger Operand:
 * $y = x \vee y$

Hence:
 * $y \in F$

The result follows.

Definition 2 implies Definition 1
Let $F$ be a meet semilattice filter of $\struct {S, \wedge, \preceq}$.

To show that $F$ is a lattice filter of $\struct {S, \vee, \wedge, \preceq}$ it is sufficient to show:

Let $x \in F, a \in S$.

By definition of join:
 * $x \preceq x \vee a$

By definition of meet semilattice filter, $F$ is an upper section, so:
 * $x \vee a \in F$

The result follows.