I would like to show that if {$A_{i}$: i$\in$I} $\subseteq$ $A$ $\in$ $\mathbb{U}$, then $\bigcup_{i \in I}$$A_{i}$ $\in$ $\mathbb{U}$, where $\mathbb{U}$ is a universe and the capital $A's$ are all sets (p. 162-163, Lectures on the Hyperreals, William Goldberg).
A universe is a strongly transitive set $\mathbb{U}$ such that:
1) if $a,b\in\mathbb{U}$, then ${a,b}\in\mathbb{U}$
2) if there are sets $A,B\in\mathbb{U}$, then $A\cup{B}\in\mathbb{U}$.
3) if there is a set $A\in\mathbb{U}$, then $\mathcal{P}(A)\in\mathbb{U}$
Strong Transitivity: for any set $A\in\mathbb{U}$, there exists a transitive set $B\in\mathbb{U}$ with $A\subseteq{B}\subseteq{\mathbb{U}}$. Transitivity of $B$ means that $c\in{C}\in{B}\implies{c\in{B}}$