In the set theory of NBG, the class existence theorem says that for all predicative well formed formulas $\varphi$ (wff's in which the variables quantify only over sets) there is a (unique up to extensionality) class, $A$, such that if $x_1, x_2, ..., x_n$ are the free variables of $\varphi$ then the universal closure of $(x_1, x_2, ..., x_n) \in A \iff \varphi(x_1, x_2, ..., x_n)$ is true.
So in some sense we "can" define a function that takes as a parameter a predicative wff, let's call it $\varphi$ and outputs the unique class such that the universal closure of $(x_1, x_2, ..., x_n) \in A \iff \varphi(x_1, x_2, ..., x_n)$ is true.
But we can sort of define an inverse function, call it $g$. For each class $A$ define $g(A)= "x \in A"$. And now one can see that $g$ is "injective". So in some sense, the cardinality of the whole classes is less that that of the predicative wff's, which the latter is clearly at most countable. And this result looks like a contradiction, like wouldn't it imply that the Von Neumann universe is at most countable?
I feel like there is a huge flaw in my reasoning such that I am making my function $g$ that operates on objects of the language($A$, the classes) and outputs objects in the metalanguage(predicative wff's).
Can someone explain the flaw in this reasoning? I possess a limited understanding of formal logic and the axiomatic set theory of NBG, without reaching an advanced level of expertise.
The formula $x\in A$ uses the class $A$ as a parameter. There are only countably many formulas, but if we allow arbitrary classes to appear as parameters in these formulas, then there are at least as many formulas with parameters as there are classes!