In my text book the proof of well ordering principle goes like this:
Let A be an arbitrarily given nonempty set which is to be well-ordered. Consider the family A* of all well-ordered sets ($A_0$, $\le _0$), where $A_0 \subseteq A$. We partially order A* by writing ($A_0, \subseteq _0) \subseteq ^*(A_1, \subseteq _1)$ if and only if
(i) $A_0 \subseteq A_1$
(ii) x, y $\in A_0$ and x $\le _0y$ imply x $\subseteq _1 y$
(iii) x $\in A_1 - A_0$ implies y $\le _1 x$ for all y $\in A_0$
The proof goes like this but they say nothing about whether A* can be empty or not. Shouldn't they first proof that A* is nonempty(well-defined?)? If it's okay not to show that, why is it so?