It's claimed, e.g. in the formulation of Tarski–Grothendieck set theory, that the Tarski axiom implies that for every set, there is a Grothendieck universe containing it.
However, I can't see how the Tarski axiom implies that the set which is being constructed is e.g. transitive. If $x$ is the set generating $U$ via the Tarski axiom and $x=\{y\}$. How do I show that $y\in U$? None of the points in Tarskis definition put elements of $x$ directly into $U$, only sets containing these.
If we need some of the other axioms to do it, how?