Well-Founded Proper Relational Structure Determines Minimal Elements

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

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

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.

Proof
$B$ is not empty, so it has at least one element $x$.

By Singleton Subset, $\left\{ x \right\} \subseteq B$.

Since $\left\{ x \right\} \subseteq B$, it follows that $\left\{ x \right\}$ has a $\prec$-relational closure by Relational Closure Exists for Well-Founded Sets. The $\prec$-relational closure shall be denoted $y$.


 * $y \cap B \subseteq A$ by Intersection Subset


 * $y \cap B \ne \varnothing$ since $x \in y$ and $x \in B$

By the definition of foundational relation:


 * $\exists x \in \left({ y \cap B }\right): \forall w \in \left({ y \cap B }\right): w \not \prec x$ (1)

Suppose that $w \in B$ and $w \prec x$.

Since $x \in y$, it follows that $w \in y$, so $w \in \left({ B \cap y }\right)$ by the definition of intersection. This contradicts (1), so $w \not \prec x$ and $B$ has a minimal element.

Also See
Well-Founded Relation Determines Minimal Elements, a stronger version of this theorem that relies on the Axiom of Foundation.