This is an endeavor to salvage "Nested Selection" principle presented in a prior posting.
Define
$ \begin{align} Y \text { is } \Phi \text{-image of } X \iff &\forall a \in X \exists b \in Y: \Phi(b,a), \\& \forall b \in Y \exists a \in X: \Phi(b,a)\end{align}$
Nested Selection: If $\Phi$ is a transitive asymmetric binary relation, then if $G$ is an infinite set of pairwise disjoint infinite sets, that is strictly well ordered by relation ${\small \langle\! \langle}\Phi\text{-image} \small \rangle \!\rangle$, with limit elements of $G$ having each element $l$ in them, bearing the $\Phi$ relation to each $k$ that is an element of a prior element of $G$, that is we have $\Phi(l,k)$; then there exists a choice set $C$ on $G$ such that for any two distinct elements $a,b \in C$ we have: $\Phi(a,b) \lor \Phi(b,a)$.
By $C$ being a choice set on $G$ it means that $C \subseteq \bigcup G$ and $C$ shares exactly one element with each element of $G$.
The above is a theorem of $\sf ZFGC$, since for any $G$ meeting the above hypothesis, we'd have an enumeration $(g_\alpha)_{\alpha \in \Omega}$ of its elements after the well-ordering relation ${\small \langle\! \langle}\Phi\text{-image} \small \rangle \!\rangle$, where each successor is a $\Phi \text{-image}$ of its predecessor, and so the following map $f$ would exist:
$\begin{align} f: G \to \bigcup G, \ & g_\lambda \mapsto c` g_\lambda \text { if } \forall \kappa: \lambda \neq \kappa+1, \\ & g_{\alpha+1} \mapsto c`\{x \in g_{\alpha+1} \mid \Phi(x, f`g_\alpha) \} \end{align}$
Where $c$ is the global choice function.
Is Nested Selection equivalent to $\sf AC$ over $\sf ZF$?
