I am studying set theory. In most of the references on set theory such as the Jech's book, they start with the axioms of ZFC and the first-order language of set theory and eventually they start to talk about certain 'objects' called sets. I see a certain logic jump. I want to understand how is formally defined the sentence '$x$ is a set' from the first-order language. I understand that this sentence means that we can find some well-formed formula $\varphi(u)$ with a free variable $u$ where the sentence $(\exists x)\varphi(x)$ is true in ZFC. Am I right?
When you work in standard set theory, everything is a set.
In other words, formulas in the language of set theory do cannot even be applied to things that are not sets.
So if you want a formula $\varphi(x)$ which is true when $x$ is a set and false if $x$ is not a set, this means that you want a $\varphi(x)$ that is always true, period. You can use any propositional tautology you want for that, such as $x\in x\to x\in x$, but that won't make you any wiser about sets.