Proof
(related to Corollary: There is no set of all sets)
 We have seen in the above explanation how the axiom of separation avoids the Russell's Paradox.
 In particular, we have seen that for every set $X$ the set $\{z\in X\mid z\not\in z\}$ is welldefined (exists).
 Therefore, if a "set of all sets" existed, it would have to contain $\{z\in X\mid z\not\in z\}$ as an element.
 But for all $X$ we have $\{z\in X\mid z\not\in z\}\not\in X.$
 Therefore, a "set of all sets" does not exist.
∎
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References
Bibliography
 Hoffmann, Dirk W.: "Grenzen der Mathematik  Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011
 Ebbinghaus, H.D.: "Einführung in die Mengenlehre", BI Wisschenschaftsverlag, 1994, 3th Edition