Axiom of Foundation (Strong Form)

Theorem
Let $B$ be a class.

Suppose $B$ is not empty.

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

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

The union of $x$ is its $\in$-initial segment by the definition of union.

Therefore, every $\in$-initial segment is a small class by the Axiom of Unions.

By the fact that Nonempty Subsets of Well-Founded Relations have Minimal Elements, $B$ has an $\in$-minimal element.