I have a quick question regarding to the proof that well-ordering theorem implies Maximal Principle. In the proof described here https://proofwiki.org/wiki/Well-Ordering_Theorem_implies_Hausdorff_Maximal_Principle The function is defined as
$$\rho(f:S_x \rightarrow \mathcal{P}(X))\begin{cases} f(S_x)\cup\ \{x\} &\mathrm{if}\ P(S_x,x) \\ f(S_x) & \mathrm{otherwise} \end{cases} $$
however, it doesn't seem to make sense to have $$f(S_x)\cup\{x\}$$ as for example if one has $X$ as $\{1, 2, 3, 4, 5, 6\}$ then $$f(1)=\{1\}$$ $$f(2)=\{\{1\},\{2\}\}$$ and so on. Thus $$f(S_3)=\{\{1\},\{\{1\},\{2\}\}\}$$ But I think the proof is intended to mean $$f(S_3)=\{1,2\}$$ So shouldn't the function be defined as the following?
$$\rho(f:S_x \rightarrow \mathcal{P}(X))\begin{cases} \cup_{z \in S_x}f(z)\cup\ \{x\} &\mathrm{if}\ P(S_x,x) \\ \cup_{z \in S_x}f(z) & \mathrm{otherwise} \end{cases} $$