Recall the definition of *cardinal definable*, where every set being cardinal definable is proved consistent relative to ZF + V=HOD. To re-iterate it:

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

Where: $\lambda_i <^\rho \kappa \iff \rho(\lambda_i) < \rho(\kappa)$, and $\rho$ is the rank function; and "*cardinal*" is defined after Scott's as an equivalence class under *bijection* of sets of the lowest possible rank.

Now, is the principle stating that

every set is cardinal definableconsistent with $\sf ZF + \neg AC$?

A related question replacing *cardinal* by *ordinal* in the above question would lead to $\sf V=HOD$, which is known to prove $\sf AC$ and so would be inconsistent with $\sf ZF + \neg AC$.