$small (x) \equiv_{df} \\\exists y \subset WF \cup N ([y \in V \lor |y|<|WF \cup N|] \land |x| \leq |y|)$
Where $|x| \leq |y| \equiv_{df} \exists f (f:x \rightarrowtail y)$, and $|x| < |y| \equiv_{df} |x| \leq |y| \ \land |y| \not\leq |x|$
Where $\rightarrowtail$ denotes "injections", $V$ is the set of all sets (i.e. elements of classes), $WF$ is the class of all well founded sets, and $N$ is the set of all Frege's naturals. Where a Frege natural is a set that is an equivalence class of finite sets under equivalence relation "set-bijection"
Now "ML + every hereditarily small class is a set", does prove all rules of ZF+Global choice over WF, and ZFC - foundation over the hereditarily small sets realm.
Question 1: What's the consistency strength of this theory?
Question 2: Can this theory prove global choice over the realm of hereditarily small sets?