Rank Function Property of Well-Founded Relation

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

Let $\struct {T, \prec}$ be a strictly well-ordered set.

Let there exist a rank function $\operatorname {rk}: S \to T$, that is:
 * $\forall x, y \in S: \paren {x \ne y \text { and } \tuple {x, y} \in \RR} \implies \map {\operatorname {rk} } x \prec \map {\operatorname {rk} } y$

Then $\RR$ is a well-founded relation.