TITLE: Why existence of proper class $\{x|x=x\}$ has a problem?
QUESTION [6 upvotes]: The Russel's Paradox, showing $X=\{x|x\notin x\}$ can't exist is not very hard.
If $X \in X$, then $X \notin X$ by definiition, in the other case, $X \notin X$, then $X \in X$ by definition. Both cases are impossible.
But how about whole things $X=\{x|x=x\}$? $X \in X$ probably cause the problem, but I don't know why violation of axiom of foundation in proper class is problem.

REPLY [8 votes]: Because in presence of the Axion of Separation (or Axiom of Specification), if the "universal set" $V = \{ x \mid x=x \}$ exists, we can have :

$R = \{ x \mid x \in V \land x \notin x \}$

and $R$ is the "illegal" Russell's set.

REPLY [6 votes]: Cantor's Theorem says that for any set $Y$, the power set of $Y$ is strictly bigger than $Y$. But $X$ contains all the elements of any set, and is therefore at least as big as any set, including its own power set, giving the contradiction 
$$
\mathcal P(X)\le X<\mathcal P(X)
$$