Strictly Well-Founded Relation determines Strictly Minimal Elements

Theorem
Let $A$ be a class.

Let $\prec$ be a foundational relation on $A$.

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

Then $B$ has a $\prec$-minimal element.

Proof
Let $\prec' = \left({B \times B}\right) \cap {\prec}$.

by Restriction of Foundational Relation is Foundational, $\prec'$ is a foundational relation

By Well-Founded Relation Determines Minimal Elements/Lemma, $B$ has a $\prec'$-minimal element, $m$.

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

Also see
and weaker results that don't require the Axiom of Foundation.
 * Well-Founded Proper Relational Structure Determines Minimal Elements‎
 * Proper Well-Ordering Determines Smallest Elements