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Given an infinite set of sets A - how can I prove in ZFC that the union of all the elements of A exists?

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See Wikipedia: Axiom of union. –  Martin Sleziak Aug 6 '12 at 9:57

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It is the axiom of union which assert this.

The axiom states that if $A$ is a set, then there exists a set $B$ such that $B=\bigcup A$, that is to say

For every $x$, $x\in B$ if and only if there exists $y$ such that $y\in A$ and $x\in y$.

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