Russell, with his paradox, proves that the set $\{x:x\notin x\}$ of all sets that are not members of themselves doesn't exist. So, he demonstrates that the set $\{x:p(x)\}$ doesn't exist necessary (it will not exist if $p(x)$ leads to a contradiction).
But Zermelo answers (axiom of subsets):
If the set $A$ exists, and if $p$ is a predicate, then the set $\{x \in A : p(x)\}$ of all elements in $A$ satisfying $p$ also exists.
According to axiom of subsets, the set $W = \{x \in \mathbb{R} : x \notin x\}$ exists. But I don't see what this set is. For instance, do we have $\pi \in W$ (i.e $\pi \notin \pi$)?