I'd be very thankful if you could help me with the following (I suppose basic) questions:
First of all, how is the union of a set defined: By ZF$4$, given a set $x$, there is a set consisting of all the elements of all the elements of $x$. Is this the union?
If yes, then if we take $X=\{\{1\}, \{1,2\}\}$, my notes claim that $\bigcup X=\{1,2\}$, but shouldn't it be $\{\{1\},\{2\},\{1,2\}\}$ since those are the elements of elements.
Another problem I have is the definition of brackets when using formulas - for example, if we take the definition of null set axiom we have the formula $∃x∀y(¬(y ∈ x))$ which I can't wrap my head around
Thanks!
Recall the definition:
$$\bigcup X = \{ y \mid \exists x \in X \colon y \in x \}$$
We have $1 \in \bigcup X$, since $y := 1 \in \{1\}$ and $x := \{1 \} \in X$.
Similarly $2 \in \bigcup X$, since $ 2 \in \{1,2 \}$ and $\{1,2\} \in X$.
It's now easy to verify that these are the only elements of $\bigcup X$ and hence that $\bigcup X = \{1,2\}$.