# Singleton is Chain

## Theorem

Let $\left({S, \preceq}\right)$ be an ordered set.

Let $x \in S$.

Then $\left\{ {x}\right\}$ is a chain of $\left({S, \preceq}\right)$.

## Proof

It suffices to prove that

$\left\{ {x}\right\}$ is connected

Let $y, z \in \left\{ {x}\right\}$

By definition of singleton:

$y = x$ and $z = x$

By definition of reflexivity;

$y \preceq z$

Thus

$y \preceq z$ or $z \preceq y$

$\blacksquare$