Filter Contains Greatest Element

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Theorem

Let $\struct{S, \preceq}$ be an ordered set with greatest element $\top$.

Let $\FF$ be a filter in $\struct{S, \preceq}$.

Then:

$\top \in \FF$

Proof

By Filter Axiom $\paren{1}$:

$\exists x \in \FF$

By definition of greatest element: $x \preceq \top$

By Filter Axiom $\paren{3}$:

$\top \in \FF$

$\blacksquare$