Singleton of Bottom is Ideal

Theorem
Let $\left({S, \preceq}\right)$ be a bounded below ordered set.

Then
 * $\left\{ {\bot}\right\}$ is an ideal in $\left({S, \preceq}\right)$

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

Non-empty
By definition of singleton:
 * $\bot \in \left\{ {\bot}\right\}$

By definition:
 * $\left\{ {\bot}\right\}$ is a non-empty set.

Directed
Thus by Singleton is Directed and Filtered Subset:
 * $\left\{ {\bot}\right\}$ is directed.

Lower
Let $x \in \left\{ {\bot}\right\}, y \in S$ 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:
 * $y \in \left\{ {\bot}\right\}$

Thus by definition:
 * $\left\{ {\bot}\right\}$ is a lower set.

Thus by definition:
 * $\left\{ {\bot}\right\}$ is an ideal in $\left({S, \preceq}\right)$