To $\sf ZF - Regularity$ add the following axiom:
Hereditary size: $\forall x \ \exists H_x \ \exists f (f: x \rightarrowtail H_x)$
Where: $H_x= \{y: \forall z \in TC(\{y\})\exists f (f: z \rightarrowtail x)\}$
Where $TC$ is the transitive closure operator on sets defined in the usual manner.
Is this axiom equivalent to $\sf AC$ over the rest of axioms of $\sf ZF - Regularity$?
The reason for asking this question, is that if the above proves to be weaker than $\sf AC$ then we have a sort of a definition for cardinality in absence of regularity and choice but with the above axiom holding. That definition would be:
The cardinality of a set x is the set of all sets equi-numeorus (i.e. bijective) to x that are hereditarily subnumerous (injective) to x.
