Prove that no ordinal is an element of itself
If $\alpha\in\alpha$, then $\alpha$ is not strictly well-ordered by $\in$, since $x=\alpha$ is an element of $\alpha$ such that $x\in x$.
(Alternatively, if you include the axiom of regularity in your axioms, then no set $x$ can satisfy $x\in x$, since then $\{x\}$ would violate the axiom of regularity.)