Set is Subset of Finite Suprema Set

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

Let $X$ be a subset of $S$.

Then $X \subseteq \operatorname{finsups}\left({X}\right)$

where $\operatorname{finsups}\left({X}\right)$ denotes finite suprema set of $X$.

Proof
Let $x \in X$.

By Supremum of Singleton:
 * $\left\{ {x}\right\}$ admits a supremum and $\sup \left\{ {x}\right\} = x$

By definitions of subset and singleton:
 * $\left\{ {x}\right\} \subseteq X$

By Singleton is Finite:
 * $\left\{ {x}\right\}$ is a finite set.

Thus by definition of finite suprema set:
 * $x \in \operatorname{finsups}\left({X}\right)$