Theorem 1 (Zermelo)

Every set E can be well-ordered.

Let $\mathfrak{S} = \mathfrak{P}(\mathrm{E}) - \{\mathrm{E}\}$ be the set of all subsets of E other than E itself. For each $\mathrm{X} \in \mathfrak{S}$ let $p(\mathrm{X}) = \tau_x \ (x \in \mathrm{E} - \mathrm{X})$; since the relation $\mathrm{X} \in \mathfrak{S}$ implies $(\exists x)(x \in \mathrm{E} - \mathrm{X})$, we have $p(\mathrm{X}) \in \mathrm{E} - \mathrm{X}$ (Chapter I, § 4, no. 1) and therefore $p(\mathrm{X}) \notin \mathrm{X}$. We may therefore apply Lemma 3, and consequently there exists a well-ordering on a subset M of E such that $\mathrm{M} \notin \mathfrak{S}$; but the only subset of E which does not belong to $\mathfrak{S}$ is E itself, and the theorem is proved.