Different definitions of the Grothendieck universe In Wikipedia the Grothendieck universe is defined as a set $U$ with the following properties:

*

*if $X\in Y\in U$, then $X\in U$,

*if $X,Y\in U$, then $\{X,Y\}\in U$,

*if $X\in U$, then $2^X\in U$,

*if $I\in U$ and $\{X_i;\ i\in I\}\subseteq U$, then $\bigcup_{i\in I}X_i\in U$.

Recently I found another definition at this website:
a set $U$ is a Grothendieck universe if it has the following properties:

*

*if $X\in Y\in U$, then $X\in U$,

*if $X,Y\in U$, then $\{X,Y\}\in U$,

*if $X\in U$, then $2^X\in U$,

*if $X\in U$, then $\cup X\in U$
(the difference is in the last property).
I suppose these two definitions are equivalent, but I don't understand why. Can anybody explain this to me?
 A: The second definition is wrong.  For instance, the set $V_{\omega+\omega}$ satisfies the second definition, but not the first.  It fails to satisfy the first definition since it contains $\omega$ and every element of the family $(V_{\omega+n})_{n\in\omega}$ but it does not contain their union (namely, $V_{\omega+\omega}$ itself).  However, this union cannot be written as the union of any element of $V_{\omega+\omega}$ (and indeed the union of any element of $V_{\omega+\omega}$ is still an element of $V_{\omega+\omega}$) so this does not violate the second definition.
A: I misread the second definition! They are in fact not equivalent, and the second definition is wrong. To see this, note that $V_{\omega+\omega}$ satisfies Definition $2$ but not Definition $1$.
I think it's much better to rephrase the last clause as follows:

If $D\in U$, $C\subseteq U$, and there is a surjection $D\rightarrow C$, then $C\in U$.

This amounts to saying that $U$ satisfies replacement as calculated in $V$ (this is stronger than merely $U$ satisfying the replacement axioms internally). Together with the other rules, this implies that if $U$ is a Grothendieck universe containing $\omega$ then $U=V_\alpha$ for $\alpha$ a regular strong limit cardinal, or (strongly) inaccessible cardinal. So Grothendieck universes are really just (small) large cardinals in disguise.
