Minimal Element of an Ordinal

Theorem
The minimal element of any ordinal is the empty set.

Proof
Let $$S$$ be an ordinal.

Let the minimal element of $$S$$ be $$s_0$$.

This exists by dint of an ordinal being a woset.

From Ordering on an Ordinal is Subset Relation, $$S$$ is well-ordered by $$\subseteq$$.

So, by definition of an ordinal, $$s_0 = \left\{{s \in S: s \subset s_0}\right\}$$.

But as $$s_0$$ is minimal, there are no elements of $$S$$ which are a subset of it.

So $$\left\{{s \in S: s \subset s_0}\right\} = \varnothing$$, hence the result.