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, $\exists x \in B: B \cap x = \varnothing$

Proof
Because $\Epsilon$ creates a well-ordering on any ordinal and the initial segments of $x$ are sets, we may conclude that $B$ has an $\Epsilon$-minimal element from Well-Founded Relation Determines Minimal Elements/Special Case