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.