Strictly Well-Founded Relation determines Strictly Minimal Elements

Theorem
Let $A$ be a class.

Let $\prec$ be a strictly well-founded relation on $A$.

Let $B$ be a nonempty class such that $B \subseteq A$.

Then $B$ has a strictly minimal element under $\prec$.

Proof
Let $\prec' = \paren {B \times B} \cap {\prec}$.

By Restriction of Strictly Well-Founded Relation is Strictly Well-Founded, $\prec'$ is a strictly well-founded relation

By the lemma:
 * $B$ has a strictly minimal element $m$ under $\prec'$.

By Minimal WRT Restriction, $m$ is $\prec$-minimal in $B$.

Also see

 * Well-Founded Proper Relational Structure Determines Minimal Elements‎
 * Proper Well-Ordering Determines Smallest Elements

weaker results that do not require the Axiom of Foundation.