What is the relationship of the following to other axioms of $\sf ZFC$?

$\sf WB$: Every set $A$ is in bijection with a set well-founded by $\in$.

Obviously, $\sf ZF$ implies $\sf WB$ (because every set is well-founded) and $\sf ZFC-Reg$ implies $\sf WB$ (because every set is equinumerous to an ordinal, which is well-founded). Is it known if $\sf ZF-Reg$ implies $\sf WB$? Another equivalent statement is:

$\sf WB'$: Every set injects into ${\rm WF}:=\bigcup_\alpha V_\alpha$.

(Since every set well-founded by $\in$ is in $\rm WF$, $\sf WB\to WB'$, and for the converse note that the range $B\subseteq {\rm WF}$ of any injection from set $A$ must also be a set so it is contained in $V_\delta$ where $\delta$ is the supremum of the ranks of elements of $B$.) When written this way it seems hard to believe that it could be false even without Regularity.



The whole point of urelements is that you can replace them with Quine atoms. Namely sets of the form $x=\{x\}$. Now when you do a permutations model construction, the well founded sets will continue to satisfy choice, if it was true before. Take any non-well orderable set, it cannot inject into a subclass where every set can be well ordered.


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