Let $A$ be the class of all ordinals within a model of ZFC.

Let $B$ be the class of all $t(\alpha)$ as defined in my theory where $\alpha \in Ord$ and $a,b,c,\dots \in t(\alpha) \implies a,b,c,\dots \in Ord$.

Let an element $\alpha$ of $A$ be considered "in" an element $\beta$ of $B$ if a well ordering of all elements $x,y,z,\dots$ of $\alpha$ appears somewhere within the sequence $(a,b,c,\dots) = \beta$.

Let $f$ be a function from $A$ onto $B$.

Define the element $\kappa$ of $B$ that is the sequence of all elements $\alpha \in A$ such that they are not elements of $f(\alpha)$.

Let $c$ be such that $c \in A$ and $f(c) = \kappa$.

$c \in \kappa \implies c \not\in f(c) = \kappa$

$c \not\in \kappa \implies c \in f(c) = \kappa$

There cannot exist a $c$ where $f(c) = \kappa$ implies the class B is strictly 'larger' than the class A.

That said, GÃ¶del and Cohen working in ZFC may not be comparable to what I'm doing here. Also, I'm not sure if anyone has ever proved the size of one proper class to be larger than the size of another, but perhaps I just did... I never know.