Proof of greatest element of a non-empty set

Here is a write-up which I think only uses concepts up to that point in the book:

Let $z$ be the smallest element of $B$. Assume $\forall a \in A \; (a \not = z)$.

This means $a<z$.

Now notice that $z \not = 1$ since $A$ is not empty.

This means $z = S(w)$ for some $w$. (Theorem 1.1 in that chapter of the book).

But this says

$$a < S(w) = z$$

meaning

$$a \leq w < z$$

so $w \in B$ and $z$ is not minimal.