Axiom of Regularity - Transitive set

Question

I just managed to confuse myself completely while studying for Set Theory. We have the Axiom of regularity: $$\forall S (S\not= \emptyset \rightarrow (\exists x\in S)(S\cap x=\emptyset))$$ Now a set is transitive, if $x\in T$ implies $x\subset T$.
I don't understand anymore how that can be while we have the Axiom of regularity.
Doesn't follow from the definition of a transitive set, that $x\cap T = x$?

Maybe if someone knows a good source for a more detailed explanation of the Axiom of regularity or has the time to explain me some more?

Thanks and best,

Luca

Answer 1

You are right. If $x$ is transitive, and $y\in x$ then $x\cap y=y$.

Hint: Show that a transitive set $x$ must either be empty, or $\varnothing\in x$.