Set is Subset of Finite Suprema Set

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

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

Then $X \subseteq \map {\mathrm {finsups} } X$

where $\map {\mathrm {finsups} } X$ denotes finite suprema set of $X$.

Proof
Let $x \in X$.

By Supremum of Singleton:
 * $\set x$ admits a supremum and $\sup \set x = x$

By definitions of subset and singleton:
 * $\set x \subseteq X$

By Singleton is Finite:
 * $\set x$ is a finite set.

Thus by definition of finite suprema set:
 * $x \in \map {\mathrm {finsups} } X$