Singleton of Bottom is Ideal

Theorem
Let $\struct {S, \preceq}$ be a bounded below ordered set.

Then
 * $\set \bot$ is an ideal in $\struct {S, \preceq}$

where $\bot$ denotes the smallest element in $S$.

Non-empty
By definition of singleton:
 * $\bot \in \set \bot$

By definition:
 * $\set \bot$ is a non-empty set.

Directed
Thus by Singleton is Directed and Filtered Subset:
 * $\set \bot$ is directed.

Lower
Let $x \in \set \bot, y \in S$ such that
 * $y \preceq x$

By definition of singleton:
 * $x = \bot$

By definition of smallest element:
 * $\bot \preceq y$

By definition of antisymmetry:
 * $y = \bot$

Thus by definition of singleton:
 * $y \in \set \bot$

Thus by definition:
 * $\set \bot$ is a lower set.

Thus by definition:
 * $\set \bot$ is an ideal in $\struct {S, \preceq}$