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Proof

(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. $(A\cap B)\cap C=A\cap (B\cap C).$

Part 1: $(A\cap B)\cap C\subseteq A\cap (B\cap C).$

• Let $x\in (A\cap B)\cap C.$
• By the definition of set intersection, we have $x\in (A\cap B)\wedge x\in C.$
• 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 $(A\cap B)\cap C\subseteq A\cap (B\cap C).$

Part 2: $B\subseteq A$

• 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 $A\cap (B\cap C)\subseteq (A\cap B)\cap C.$

Conclusion

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

Bibliography

1. Kane, Jonathan: "Writing Proofs in Analysis", Springer, 2016