Lemma 1 ("Principle of Noetherian induction")

Let E be an ordered set every non-empty subset of which admits a minimal element. Let F be a subset of E with the following property : if $a \in E$ is such that the relation $x < a$ implies $x \in F$, then $a \in F$. Then $F = E$.

Suppose $F \neq E$; then $CF$ would have a minimal element $b$. By definition, $x \in F$ for all $x < b$, which implies that $b \in F$, which is a contradiction.