Well-Founded Proper Relational Structure Determines Minimal Elements

Theorem
Let $A$ and $B$ be classes.

Let $\prec$ be a foundational relation.

Furthermore, let every $\prec$-initial segment of $A$ be a small class.

Suppose $B \subset A$ and $B \ne \varnothing$.

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