I'm having some difficulty understanding the axiom of union. As the axiom of union says:
For any set $x$ there is a set $y$ whose elements are precisely the elements of the elements of $x$.
So for example, for two sets $A = \{1,2\}, B = \{3,4\}$ I know by the axiom of pairs that a set $C=\{\{1,2\}, \{3,4\}\} $ exists, and by the axiom of unions that $\{1,2,3,4\}$ is also a set. But my problem comes when I apply the axiom of unions to a single element set (I can't find any examples online). For example:
Let define $0$ as $\emptyset$ and $1$ as $\{\emptyset\}$
Is $\bigcup \{1\}$ = $\{1\}$ or $\bigcup \{ 1\} = \{ \emptyset\} = 1$?
The axiom of union takes a layer of $\{\}$ off of elements. You have $1=\{\{\}\}=\{\emptyset\}$ and so applying the axiom of union to $\{1\}$ produces $1=\{\emptyset\}$. In general, applying it to $\{a\}$ produces $a$.