Following what is written at the 10th chapter of the "Teoria de Cojuntos, una introduccion" by Fernando Herandez (For the sake of completeness I should say that this is also present at the 11th chapter of "The Axiom of Choice" by Thomas Jech).
For every infinite cardinal number $\kappa$, let $\aleph(\kappa)$ be the Hartogs number of $\kappa$, i.e., the least ordinal which cannot be embedded by a one-to-one mapping in a set of cardinality $\kappa$. For every $\kappa$, $\aleph(\kappa)$ is an aleph, viz. the least aleph $\aleph$ such that $\aleph\not\le\kappa$.
If $\kappa$ is an infinite cardinal and $\aleph$ is ana aleph, and if
then either $\kappa\ge\aleph$ or $\kappa\le\aleph$. In particular, if
then $\kappa$ is an aleph.
The Axiom of Choice is equivalet to say that for two cardinal $\kappa$ and $\lambda$ it's result $\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\kappa+\lambda=\kappa*\lambda$.
Proof. We will show that under the hypothesis of the theorem any infinite cardinal is an aleph. Let $\kappa$ be an infinite cardinal, by Lemma 10.5 it follows that $\kappa\le\aleph(\kappa)$, which says that $\kappa$ is an aleph.
So I don't understand the proof of theorem 10.6: could someone explain to me why what is here written prove the theorem?