I want to make a statement like this:
$\forall x\forall y\in S, (x>y) \lor (x=y) \lor (x<y)$
Where S is the set of all cardinalities. The problem is that such a set doesn't exist and to describe all cardinalities one needs to use a proper class.
I want to know how I can frame the above statement, which is essentially that the cardinal numbers obey the law of trichotomy, without having to refer to bijections in place of cardinal numbers. I want to treat the cardinal numbers as their own objects.
Well, remember that bounded quantifies don't really exist. It's a shorthand, $\forall x\in S, \varphi$ is really $\forall x(x\in S\rightarrow \varphi)$.
Now combine this with the fact that since classes are not objects, writing $x\in S$, when $S$ is a proper class, is just a shorthand for $\psi(x)$, where $\psi$ is the formula defining $S$.
And from here you are all done.