I am little confused about the proof given here http://euclid.colorado.edu/~monkd/m6730/gradsets05.pdf
On the second page, when defining $P$, the author says that $B\subset A$ and $(B,<)$ is a well-ordering structure. Isn't this exactly what we want to prove? How do we know that $B$ can be well-ordered? What happens if $P$ is empty?