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

Question

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

My Try:

$(x,y)\in A\times(B\cup C) $

$x\in A$ and $y\in(B\cup C)$

$(x\in A$ and $y\in B)$ or $(x\in A$ and $y\in C)$

$(x,y)\in A\times B$ or $(x,y)\in A\times C$

$(x,y)\in (A\times B)\cup(A\times C)$

$(x,y)\in (A\times B)\cup(A\times C)$

So, I proved $A\times(B\cap C)\subset(A\times B)\cup(A\times C)$

My question: Do I also need to prove $(A\times B)\cup(A\times C)\subset A\times(B\cap C)?$

Answer 1

Just write $\iff$ arrows between the statements.