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Role of the set-theoretic universe in mathematics

It is famously known that the axiom of choice can be formalized in ZFC. More specifically, the axiom of choice can be formalized as the statement “every collection of non-empty sets can be well-ordered”. However, this statement is provably independent of ZFC.
Now, the question is, can it be proven in ZFC that the axiom of choice actually is true, and furthermore, that the model in which the axiom of choice is false is (using some sort of large cardinal notion) not a “mathematical universe”?
This question is a generalization of a question that I asked on MSE a while ago (which I’m not proud of):
What are the models of set theory in which the axiom of choice is false?

A:

The axiom of choice can be formalized in ZFC just by using the Axiom schema of replacement and the fact that the class of nonempty sets is equivalent to the class of ordinals:

Theorem: Suppose that $X$ is any set, and let $\{\varphi_n(x)\}_n$ be an enumeration of a sequence of first-order sentences with exactly one free variable each. Then the following are equivalent:

$\forall x\in X$, $\forall n\in\mathbb{N}$, $(\exists y\in X)(x=y\wedge\varphi_n(y))$;
For every $F\subseteq X\times\mathbb{N}$ with $\emptyset otin F$, there is an $f\colon F\to X$ such that the pair $(f\,[F]\,)(x)=y$ witnesses that $\exists y\in X(x=y\wedge\varphi_n(y))$ for all $n$.

Note that the idea here is to use the existence of an enumeration of the class of nonempty sets to create a formula which says “there is an enumeration of $X$”, and then to use replacement to get some $y\in X$ and a formula to

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