The theorem 0.3 in Folland's Real analysis book, proves that Zorn's Lemma implies Well-Ordering.
In the theorem, we construct on any set $X$ a collection $\mathcal{W}$ of all well-orderings of subsets of $X$. In the collection $\mathcal{W}, (E_{1},\leq_{1})\leq (E_{2},\leq_{2})$ if
- $E_{1}\subset E_{2}$ and $\leq_{1}$ agrees with $\leq_{2}$ on $E_{1}$.
- If $x\in E_{2}\setminus E_{1}$, then $y\leq_{2}x$ for all $y\in E_{1}$.
I was able to show that any collection $\{(E_{\alpha},\leq_{\alpha}\}_{\alpha\in A}$ which is linearly ordered, then the ordering $(E,\leq_{\mathcal{P}})$ where $E = \bigcup_{\alpha\in A}E_{\alpha}$ and ordering between any two elements is done using the fact that any two elements lie in some $E_{\alpha}$ and I was able to prove that this way of ordering is well defined.
But I am stuck at proving that $(E,\leq_{\mathcal{P}})$ is well-ordered.