Is it true $\emptyset \times A = \emptyset$?

Question

Let $\emptyset, A$ be an empty set and a set respectively.

Now consider Catesian product of the above.

$\emptyset \times A$

I think it should be $\emptyset $ but I can't prove it.

Answer 1

Yes it is true, since $x\in \emptyset\times A$ is there exists $a\in \emptyset $ and $a'\in A$ such that $(a,a')=x$.

But $a\in \emptyset $ is impossible, so there is no $x\in \emptyset\times A$.

So $\emptyset\times A=\emptyset$.