[026] The axiom of union 1 says that for each set \(A\) there is a set \(B\) that contains all the elements of the elements of \(A\); in symbols,
\[ ∀ A ∃ B, ∀ x, (x∈ B \iff (∃ y,y∈ A ∧ x∈ y))~ ~ . \]
This implies that this set is unique, by the axiom of extensionality [1Y8]; we indicate this set \(B\) with \({\underline⋃} A\) (so as not to confuse it with the symbol already introduced before).
For example if
\[ A = \{ \{ 1,3,\{ 5,2\} \} , \{ 7,19\} \} \]
then
\[ {\underline⋃}A = \{ 1,3,\{ 5,2\} , 7,19 \} \quad . \]
Given \(A_ 1,\ldots A_ k\) sets, let \(D=\{ A_ 1,\ldots A_ k\} \) 2 we define
\[ A_ 1∪ A_ 2\ldots ∪ A_ k {\stackrel{.}{=}}{\underline⋃}D \quad . \]