If we replace the axiom of Regularity in $\sf ZF$ by the scheme of Acyclicity, which is:

$$\begin{align} n=2,3,\dots;\ & \neg \exists x_1,\dots , \exists x_n: \\ &x_1 \in x_2 \land \dots \land x_{n-1} \in x_n \ \land \\ &x_1=x_n \end{align}$$

Call this Acyclic ZF.

Does Corets' principle of every set is equinumerouse to some well-founded set, hold in Acyclic ZF.