"Prove that the 'set of all sets' does not exist"
So, I'm going through "Introduction to set theory" by Karel Hrbacek, and Thomas Jech. The book proposes proving that the set of all sets doesn't exist. So here's what I came up with.
$V$ is a set $∧$ $B$ is a set $⇒$ $¬∃V,∀B [ B∈V ]$
Suppose $∃V,∀B [ B∈V ]$, then the Axiom Schema of Comprehension can be used
So $∀V,∃!T [ x∈T ⇔ x∈V ∧ x∉x]$ and $T$$=${$x∈V|x∉x$}. Now, if $∀B [ B∈V ]$ then $T∈V$; If $T∈T$ then $T∉T$, and if $T∉T$, $T∈T$.
So I have
$T∉T ⇔ T∈T$.
A contradiction.
$∴¬∃V,∀B [ B∈V ]$
So, Is the proof ok?