Every set is a union of singleton sets Is it possible to prove this statement? What axioms are necessary to conclude that every set on, for instance, $\mathbb{R}$ can be expressed as a union of singleton sets (sets containing exactly one element)?
Let $X=\{x : P\}$ where $P$ is a defining property of the elements of $X$, for instance $P$ can be $x>2$.
Then I can simply state that $$X = \bigcup_{P} X_i$$ where $\{X_i : i \in \mathbb{R_+}\}$ is a collection of all singleton sets of $\mathbb{R}$.
By the way, there are sets of bigger cardinality than of cardinality of all real numbers. In this case, using $\mathbb{R_+}$ as indexes for all singleton sets from such sets wouldn't be sufficient, right?

Maybe I'm losing some obvious, but I think that in your informal proof you are using at least:
Replacement: The set $Z = \{\{x\}:x\in X\}$ is the image of $X$ by the function  $x\longmapsto \{x\}$.
Union: $\bigcup Z$ is a set.
Extensionality: $X = \bigcup Z$ because both have the same elements.
