Strictly Well-Founded Relation is Antireflexive/Corollary

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

Suppose that $S$ is non-empty.

Then $\preceq$ is not a foundational relation.

Proof
Since $S$ is non-empty, it has an element $x$.

By the definition of ordering, $\preceq$ is a reflexive relation.

Thus $x \preceq x$.

By Foundational Relation is Antireflexive, $\preceq$ is not a foundational relation.