$ A \times (B \cup C) = (A \times B) \cup (A \times C) $

Let 

$p=(x,y) \in A \times (B \cup C)$. So $x \in A$ and $y \in (B \cup C)$. Three cases arise from here:

1. $y \in B$

2.$y \in C$

3.$y \in B \cap C$

Now proceeding from each case os trivial. But my question is that my textbook does not make third case which i do not understand why so. 

Thanks