Way Below Relation is Antisymmetric

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

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

Then
 * $x = y$

Proof
By Way Below implies Preceding:
 * $x \preceq y$ and $y \preceq x$

Thus by definition of antisymmetry:
 * $x = y$