Is every set being cardinal definable consistent with ZF?

Question

$Define: X \text { is cardinal definable} \iff \\\exists \text { cardinal } \kappa \, \exists \text { cardinals } \lambda_1,.., \lambda_n <^\rho \kappa \ \exists \phi : \\ X=\{ y \in V_{\rho(\kappa)} \mid \phi^{V_{\rho (\kappa)}} (y,\lambda_1,..,\lambda_n)\}$

Where: $\lambda_i <^\rho \kappa \iff \rho(\lambda_i) < \rho(\kappa)$, and $\rho$ is the rank function; and "cardinal" is defined after Scott's.

Now, is the principle that every set is cardinal definable consistent with ZF?

Answer 1

Yes, assuming ZF is consistent it is, because it follows from ZFC + V=HOD. This is because the ordertype of the class of (ordinal-)cardinals is just that of the ordinals. That is, if $X$ is definable over $V_\alpha$ from ordinal parameter $\beta<\alpha$, then $X$ is definable over $V_{\aleph_\alpha}$ from ordinal parameter $\aleph_\beta$, since $\xi\mapsto\aleph_\xi$ restricted to $\alpha$ is also definable over $V_{\aleph_\alpha}$.