Proof

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

Let $x\in (A\cup B)\cup C.$
By the definition of set union, we have $x\in (A\cup B)\vee x\in C.$
Again, by the definition of set union, we have $(x\in A\vee x\in B)\vee x\in C.$
By the associativity of disjunction, we get $x\in A\vee (x\in B\vee x\in C).$
Again, by the definition of set union, $x\in A\vee x\in B\cup C.$
Again, by the definition of set union, we get finally $x\in A\cup (B\cup C).$
It follows that $(A\cup B)\cup C\subseteq A\cup (B\cup C).$

Let $x\in A\cup (B\cup C).$
By the definition of set union, we have $x\in A\vee x\in B\cup C.$
Again, by the definition of set union, we have $x\in A\vee (x\in B\vee x\in C).$
By the associativity of disjunction, we get $(x\in A\vee x\in B)\vee x\in C.$
Again, by the definition of set union, $x\in A\cup B\vee x\in C.$
Again, by the definition of set union, we get finally $x\in (A\cup B)\cup C.$
It follows that $A\cup (B\cup C)\subseteq (A\cup B)\cup C.$

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