Strictly Well-Founded Relation is Antireflexive/Corollary

Theorem

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

Suppose that $S$ is non-empty.

Then $\preceq$ is not a strictly well-founded 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 Strictly Well-Founded Relation is Antireflexive, $\preceq$ is not a strictly well-founded relation.

$\blacksquare$