In the set theory NFU (described by M. Randall Holmes in "Elementary Set Theory with a Universal Set"), it is possible to define the set of all sets, and the set of all one-element sets. An object is a set if and only if $\emptyset$ is a subset of it, so we can define $V^*$, the set of all sets, as $\{x\ |\ \emptyset \subseteq x\}$. We can also define the set of all one-element sets, $V^1$, as $\{x\ |\ x \text{ has exactly one element}\}$.
Is there a bijection between $V^*$ and $V^1$? Clearly, since $V^1$ is a subset of $V^*$, we can define an injection $f : V^1 \to V^*, f(x) = x$. It's not obvious how one could define an injection $V^* \to V^1$. The "most obvious" candidate, $g(x) = \{x\}$, does not exist; its definition is not stratified.
For finite sets, we can define an injection easily enough:
$\begin{align} g(\emptyset) &= \{(0, \textit{anything})\}\\ g(\{x\}) &= \{(1, x)\}\\ g(\{x,y\}) &= \{(2, x, y)\}\\ g(\{x,y,z\}) &= \{(3, x, y, z)\} \end{align}$
And so on. But this definition cannot be extended to infinite sets.