Use of "Let" in proofs I'm confused about the use of the word let in proofs. For instance, if we want to prove that $A\subset B$ we can start by saying: 'let $x\in A$'. Are we assuming then that $A$ is non-empty?
Another example: We can prove that if $A\times B\subset C\times D $ then $A \subset C$ and $B \subset D$ starting with let $a\in A$ and $b\in B$ But that is false if $B$ is empty.
What is wrong with that proof?

There are two confusing and different uses of “Let.”
In this case, when proving $A\subset B,$ you are saying “Let $x$ run through all elements of $A.$” Then you prove that $x\in B,$ too, proving that:
$$\forall x:x\in A\implies x\in B\tag1$$
In this case, you don’t really need to consider $A$ empty.
This is a special case of a general type of proof. If $P(x)$ and $Q(x)$ are statements, and we want to prove:
$$\forall x:P(x)\implies Q(x)\tag2$$ we need not know whether $P(x)$ is ever true or not, we just need to show if $P(x)$ is true, we can conclude $Q(x).$
There are theorems like “If $n$ is an odd perfect number, then <something else about $n$>.” The proofs might start, “Let $n$ be an odd perfect number….” We still don’t know if odd perfect numbers exist, but we can still prove things about them.
The other meaning would be where you only needed one element of $A.$ Then saying, “Let $x\in A\dots$” requires you to prove $A$ is non-empty first.
It does take some time to get used to these two usages, since nothing in the language, only the context, hints at the meaning.
It might be better in the first example to say, “If $x\in A,$ then… “ rather than “Let $x\in A…$”
