Axiom of Foundation (Strong Form)/Proof 1

Theorem
Let $B$ be a class.

Suppose $B$ is not empty.

Then $B$ has an $\Epsilon$-minimal element, where $\Epsilon$ is the epsilon relation.

Proof
By Epsilon is Foundational, $\Epsilon$, the epsilon relation, is a foundational relation on $B$.

By Epsilon Relation is Proper, $\left({\mathbb U, \Epsilon}\right)$ is a proper relational structure, where $\mathbb U$ is the universal class.

By Well-Founded Proper Relational Structure Determines Minimal Elements, $B$ has an $\Epsilon$-minimal element.