Conditions for Relation to be Well-Ordering

Theorem
Let $\struct {S, \RR}$ be a relational structure.

$\RR$ is a well-ordering :


 * $(1): \quad$ For all $x, y \in S$, either $x \mathrel \RR y$ or $y \mathrel \RR x$


 * $(2): \quad$ For every non-empty subset $T$ of $S$, there exists $z \in T$ such that:
 * $\forall x \in T: \paren {z \mathrel \RR x \iff x = z}$