Singleton is Chain

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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 $z \preceq y$

$\blacksquare$

Sources