In ZFC, is there a set of objects that satisfy x=x

Question

Is there a set of objects that are equal to themselves? In other words, is this true: $$\exists y \forall x (x\in y \iff x=x)$$

Intuitively, It seems to be the set of all sets and hence false. How can I prove this?

Answer 1

It is not true.

If there did exist such a set $y$, then by the axiom of separation we'd have $$ \exists z \forall x [x \in z \Leftrightarrow (x \in y \wedge x \not\in x )] $$ That is, there would exist a set $z$ given by $z = \{ x \in y \mid x \not\in x \}$.

But then both $z \in z$ and $z \not\in z$ lead to a contradiction.

[This is Russell's paradox, for what it's worth.]