# Set is Subset of Upper Closure

## Theorem

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

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

Then $X \subseteq X^\succeq$

where $X^\succeq$ denotes the upper closure of $X$.

## Proof

Let $x \in X$.

By definition of reflexivity:

$x \preceq x$

Thus by definition of upper closure:

$x \in X^\succeq$

$\blacksquare$