Working in bi-sorted $L_{\omega_1,\omega} (=, \in)$, if we write $\sf ZFC + Classes$ as it is; i.e., in bi-sorted $L_{\omega, \omega} (=,\in)$, and add the following definability rule written in bi-sorted $L_{\omega_1,\omega} (=, \in)$ :

**Definability rule:** if $\phi_1,\phi_2,\phi_3,...$ are all formulas in bi-sorted $L_{\omega,\omega}(=,\in)$, in which only symbol "$y$" occurs free, and it never occurs bound, then:
$$\forall X: \bigvee_{i \in \mathbb N} X=\{y \mid \phi_i\}$$
This would ensure that all classes are pointwise definable in bi-sorted $L_{\omega,\omega}(=,\in)$

Would "$\sf ZFC + Classes + Definability \ rule$", manage to prove that all classes are countable?