Inclusion-exclusion principle

The inclusion-exclusion principle states that $|A\cup B|=$ $|A| + |B| - |A \cap B|$.

The inclusion-exclusion principle can be applied for three sets, by applying the same rule twice, $|A\cup (B\cup C)|$, giving us: ?

$$\begin{array}{rl}\boxed{|A\cup B\cup C|}& =|A\cup (B\cup C)|\\ & =|A|+|B\cup C|-|A\cap (B\cup C)|\\ & =|A|+|B|+|C|-|B\cap C|-|A\cap (B\cup C)|\\ & =|A|+|B|+|C|-|B\cap C|-|(A\cap B)\cup (A\cap C)|\\ & =|A|+|B|+|C|-|B\cap C|-(|A\cap B|+|A\cap C|-|(A\cap B)\cap (A\cap C)|)\\ & =\boxed{|A|+|B|+|C|-|B\cap C|-|A\cap C|-|A\cap C|+|A\cap B\cap C|}\end{array}$$