Preceding implies if Less Upper Bound then Greater Upper Bound

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

Let $x, y \in S$ such that
 * $x \preceq y$

Let $X \subseteq S$.

Then
 * $x$ is upper bound for $X \implies y$ is upper bound for $X$

and
 * $y$ is lower bound for $X \implies x$ is lower bound for $X$.

First Implication
Let $x$ be upper bound for $X$,

Let $z \in X$.

By definition of upper bound:
 * $z \preceq x$

Thus by definition of transitivity:
 * $z \preceq y$

Second Implication
This follows by mutatis mutandis.