Please note that neither the set resulting in the axiom of pairing, nor the respective set resulting in the "axiom of union": are required to contain exactly the initial two sets, respectively their elements. This is indicated schematically in the above diagrams by some additional elements in comparison to the initial sets. But if we apply both axioms, we are now able to justify the union of sets.# Corollary: Justification of Set Union

(related to Axiom: Axiom of Union)

The set union $A\cup B$ of two arbitrary sets $A$ and $B$ is well-defined.

Proofs: 1

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  1. Ebbinghaus, H.-D.: "Einf├╝hrung in die Mengenlehre", BI Wisschenschaftsverlag, 1994, 3th Edition