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Proof

(related to Corollary: Justification of Set Union)

• Let $A$ and $B$ be sets.

• By the axiom of pairing there is a set $Z$ containing $A$ and $B$ as elements:

• By the axiom of separation there is a subset $Z'\subseteq Z$ containing exactly $A$ and $B$ as elements:

• By the "axiom of union":https://www.bookofproofs.org/branches/axiom-of-union-ernst/ there is a set $Z^*$ containing the elements of $A$ or¹ the elements of $B.$

• By the axiom of separation there is a subset $Z^\dagger \subseteq Z^*$ containing exactly the elements of $A$ or¹ the elements of $B.$, i.e. $Z^\dagger =\{z\mid z\in A\vee z\in B\}.$

• This is the required set union $A\cup B:=Z^\dagger .$
• The axiom of extensionality ensures the uniqueness of such a set.
• Altogether, we have shown:

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References

Bibliography

1. Ebbinghaus, H.-D.: "Einführung in die Mengenlehre", BI Wisschenschaftsverlag, 1994, 3th Edition

Footnotes