Axiom of separation The Axiom of separation states that, if A is a set then $\{a \in A ;\Phi(a)\}$ is a set.
Given a set $B \subseteq A$, Suppose I define $B=\{ a \in A ; a\notin B \}$.
This, of course leads to a contradiction. Because we define $B$ by elements not from $B$. My queation is: what part of the axioms sais that this kind of definition is not possible?
Thank you!

You were given a set and named it $B$.
You defined another set, and named it $B$.
Just because you've given them the same names doesn't mean they are actually the same set. Your contradiction only appears because you've confused yourself and thought the two sets were the same since you gave them the same name.
