Singleton is Chain

Theorem
Let $\struct {S, \preceq}$ be an ordered set.

Let $x \in S$.

Then $\set x$ is a chain of $\struct {S, \preceq}$.

Proof
It suffices to prove that
 * $\set x$ is connected

Let $y, z \in \set x$.

By definition of singleton:
 * $y = x$ and $z = x$

By definition of reflexivity;
 * $y \preceq z$

Thus
 * $y \preceq z$ or $z \preceq y$