Lemma 1.1.4 (Closure Properties of Grothendieck Universe).label Let $U$ be a Grothendieck universe, $u,v\in U$ and $t\suf u$ is a subset then $t,u\cup v\in U$.
Proof. By (GU2) $\cali{P}(u)\in U$ so by (GU1) $t\in U$. By (GU3) $\emp\in U$ so by (GU2) $\cali{P}(\emp),\cali{P}(\cali{P}(\emp))\in U$. Then $2=\curl{\bot,\top}\suf \cali{P}(\cali{P}(\emp))$ so $2\in U$. Define a function $f:2\to U$ by $f(\bot)=u$ and $f(\top)=v$. By (GU4) $u\cup v\in U$.$\square$
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