Strictly Well-Founded Relation determines Strictly Minimal Elements

Theorem
Let $A$ be a class.

Let $\RR$ 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 $\RR$.

Proof
First a lemma:

Lemma
Let $\RR' = \paren {B \times B} \cap \RR$.

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

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

By Minimal WRT Restriction, $m$ is a strictly minimal element under $\RR$ 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.