Subset of Ordinals has Minimal Element

Theorem
Let $A$ be an ordinal (we shall allow $A$ to be a proper class).

Let $B$ be a nonempty subset of $A$.

Then $B$ has an $\Epsilon$-minimal element.

That is:
 * $\exists x \in B: B \cap x = \O$

Proof
We have that $\Epsilon$ creates a well-ordering on any ordinal.

Also, the initial segments of $x$ are sets.

Therefore from Proper Well-Ordering Determines Smallest Elements:
 * $B$ has an $\Epsilon$-minimal element.