Way Below implies Preceding

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

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

where $\ll$ denotes element is way below second element.

Then
 * $x \preceq y$

Proof
By Singleton is Directed and Filtered Subset:
 * $\left\{ {y}\right\}$ is directed.

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

By definition of reflexivity:
 * $y \preceq \sup \left\{ {y}\right\}$

By definition of way below:
 * $\exists d \in \left\{ {y}\right\}: x \preceq d$

Thus by definition of singleton:
 * $x \preceq y$