For each set $X$ there is a set containing all elements of the elements of $X$, formally

$$\forall X~\exists Y~\forall xz (x\in X\wedge z\in x\Rightarrow z\in Y).$$

The set, existence of which is ensured by the axiom of union, is denoted by $\bigcup X$ or, more detailed, $\bigcup_{u\in X} u.$ Both notations mean the following set:

$$z\in \bigcup X:\Leftrightarrow z\in u\text{ for an }u\in X.$$

| | | | | created: 2014-06-08 21:15:32 | modified: 2019-08-03 19:20:46 | by: *bookofproofs* | references: [983]

[983] **Ebbinghaus, H.-D.**: “Einführung in die Mengenlehre”, BI Wisschenschaftsverlag, 1994, 3