ZFC: Sets are classes - notational question I am following some lecture notes I found online to study axiomatic set theory, more precisely ZFC. The lectures have also been recorded and can be found here.
The first lecture is mostly an introduction and mentions that we use first-order logic to derive sentences, which as far as I understand are mathematical statements formed according to some specific rules (syntax) that are either true or false. We also give the sentences some meaning, e.g. we think of variables as sets and $\in$ is thought of as membership. To tell whether a sentence is true we reduce it to simpler sentences, but at some point we simply stop and assume that certain sentences are true and these are our axioms. Again to give this some meaning, we also pretend there is a universe of sets and these axioms can be interpreted as statements about this universe of sets that we intuitivelky believe to be true.
Even though I have not studied logic this seems to make sense to me. In the second lecture classes are informally defined as collections of sets. I understand why classes might be useful and I also understand that with this informal definition they might not be sets. This means we cannot directly talk about them in our axiomatic system. However, we would still like to be able to do so. The trick is to define classes in terms of a property of sets, i.e. a formula with one free variable. So we can define a class as
$\{x: \varphi(x)\}$ where $\varphi(x)$ is a formula with the free variable $x$.
We can then deal with classes by using the formulas defining them, e.g. if $c$ and $D$ are defined by the formulas $\varphi(x)$ and $\psi(x)$, then we formally state $C=D$ by saying $\forall x (\varphi(x) \iff \psi(x))$. We also allow formulas with multiple free variables and call the additional variables parameters, which means they are fixed sets.
So far so good. Next, the lecture notes mention the following:

Again, I think it's clear that we can form the class defined by the formula $y \in x$ and intuitively a collection of all members of the set $x$ should be the set $x$. However, the notation $x = \{y : y \in x\}$ confuses me.
Is this an abuse of notation? We basically just introduce a shorthand notation for the class $\{y : y \in x\}$ and we call this $x$ as well, right? This is because we think of this class as the set $x$ even though as mentioned earlier classes do not make any sense in our axiomatic system. So to state it differently this is an informal definition that says that we will call this class $x$ similar to what we do in "everyday" mathematics.
I would appreciate if anyone could clarify this.
Thanks a lot!
 A: If you read that passage carefully you will see that it says given a set $x$....
Once the set $x$ is given, it makes sense to write a mathematical sentence about the set $x$, expressed in the language of set theory, perhaps using set builder notation (such as $\{y \mid y \in x\}$). And it also makes sense to discuss the truth or falsity of that sentence.
Thus, the sentence $x = \{y \mid y \in x\}$ makes perfect sense. In more common language it says the set $x$ is equal to the set of all $y$ such that $y \in x$.
One thing that ought to become clear pretty soon in your lecture notes (or in any book/course on set theory) is that axioms of ZFC are carefully formulated in such a way that lets you use set builder notation to express those sets whose existence is being postulated by the axioms. In particular this is true of the axiom scheme of specification, which let's you use the given set $x$ to express a new set $\{y \mid y \in x\}$, although as it turns out this "new" set is equal to $x$ itself.
