- \(\bullet\) Show that Axiom 3.4 can be deduced from Axiom 3.1, Axiom 3.3 and Axiom 3.11.
- – \(A\) and \(B\) sets
-
\(\Vdash\) { Axiom 3.3 }
- \(C = \{ A, B \}\) exists
-
\(\vdash\) { Axiom 3.11 }
- There exists a set \(\bigcup C\) such that \(x\in\bigcup C\iff(x\in S\text{ for some }S\in C)\)
-
\(\vdash\) { Axiom 3.1, Axiom 3.3 \(S\) can be either \(A\) or \(B\) }
- There exists a set \(\bigcup C\) such that \(x\in\bigcup C\iff(x\in A \text{ or } x\in B \text{ or both})\)
- \(\vdash\) { let \(A \cup B := \bigcup C\) }
- \(\square\)