There is a page in Metamath (updated link) that claims the following:
Later Grothendieck invented the concept of Grothendieck's universes and showed they were equal to transitive Tarski's classes.
Transitive Tarski classes are mentioned, e.g. in this post. A Tarski class $U$ satisfies the following conditions (given here):
- If $A\in U$ and $B \subseteq A$, then $B \in U$;
- If $A\in U$, then $P(A)\in U$;
- If $A \subseteq U$, and $|A|\neq|U|$, then $A \in U$.
A transitive Tarski class additionally satisfies:
- If $A\in U$ and $a\in A$, then $a\in U$.
Is anyone aware of the exact reference for this claim? Or, in other words: Where can I find Grothendieck's proof?