Recall the definition of *[cardinal definable][1]*, where every set being cardinal definable is [proved][2] 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 definable* consistent with $\sf ZF + \neg AC$?


  [1]: https://mathoverflow.net/questions/412534/is-every-set-being-cardinal-definable-consistent-with-zf
  [2]: https://mathoverflow.net/questions/412534/is-every-set-being-cardinal-definable-consistent-with-zf/412536#412536