(related to Proposition: Set Intersection is Associative)

- Suppose that $A,B,C$ are sets.
- We will show that the set intersection is associative, i.e. $(A\cap B)\cap C=A\cap (B\cap C).$

- Let $x\in (A\cap B)\cap C.$
- By the definition of set intersection, we have $x\in (A\cap B)\wedge x\in C.$
- Again, by the definition of set intersection, we have $(x\in A\wedge x\in B)\wedge x\in C.$
- By the associativity of conjunction, we get $x\in A\wedge (x\in B\wedge x\in C).$
- Again, by the definition of set intersection, $x\in A\wedge x\in B\cap C.$
- Again, by the definition of set intersection, we get finally $x\in A\cap (B\cap C).$
- It follows that $(A\cap B)\cap C\subseteq A\cap (B\cap C).$

- Let $x\in A\cap (B\cap C).$
- By the definition of set intersection, we have $x\in A\wedge x\in B\cap C.$
- Again, by the definition of set intersection, we have $x\in A\wedge (x\in B\wedge x\in C).$
- By the associativity of conjunction, we get $(x\in A\wedge x\in B)\wedge x\in C.$
- Again, by the definition of set intersection, $x\in A\cap B\wedge x\in C.$
- Again, by the definition of set intersection, we get finally $x\in (A\cap B)\cap C.$
- It follows that $A\cap (B\cap C)\subseteq (A\cap B)\cap C.$

- It follows from the equality of sets that $A\cap (B\cap C)=(A\cap B)\cap C.$∎

**Kane, Jonathan**: "Writing Proofs in Analysis", Springer, 2016