I am reading a set-theory's chapter in algebra course, which deals with classes, ordinal number.
I have a problems: the equivalence class of A under the equivalence relation of equipollence is a proper class (NBG)?
$V$ is the class of all sets
if $A,B$ is set, $A\sim B$ $:=\exists f\left( f\text{ is a map from }A\text{ to }B\wedge f\text{ is bijective}\right) $
$\sim$ $=:\left\{ \left( A,B\right) :A\sim B\wedge A\in V\wedge B\in V\right\} $
$\sim$ is equipollence and a equivalence relation on $V$
if $A\in V$, equivalence class $\left[ A\right] =:\left\{ B\in V:B\sim A\right\} $
Is $\left[ A\right]$ a proper class ? How to prove ?
Thank you