Transitive Class/Examples/Singleton of Singleton of Empty Class is not Transitive

Example of Non-Transitive Class
Let $\O$ denote the empty set.

Consider the class $S$, defined as:


 * $S := \set {\set \O}$

$S$ is not transitive.

Proof
$S$ has $1$ element: $\set \O$.

$\set \O$ in turn has one element: $\O$.

But $\O$ is not itself an element of $S$.

Thus not all elements of elements of $S$ are themselves elements of $S$.

Hence by definition $S$ is not transitive.