Minimal Element of an Ordinal

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

Proof
Let $$\left({S; \preceq}\right)$$ be an ordinal.

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

This exists by dint of an ordinal being a woset.

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

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

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