The transitive closure of $A$, $TC(A)$, is a transitive set containing $A$ with the property that for every set $K$, if $K$ is transitive and $A \subseteq K$, then $TC(A) \subseteq K$.
$HC_{\beth_\omega} = \{A:\forall x \in TC(\{A\}),|x|<\beth_\omega\}$ is a model of ZFC - Union. $\{\beth_0,\beth_1,\beth_2,\beth_3,...\}$ belongs to $HC_{\beth_\omega}$ but doesn't have a transitive closure in $HC_{\beth_\omega}$, and so the existence of transitive closures for every set can't be proven in ZFC - Union.
Can something similar be proven about ZFC - Replacement?
