I work at a four-year teaching school, where we pride ourselves on teaching pure math, proof, and a rather obsessive carefulness of work. Recently I have been criticized for saying that "Let $x \in A$" is often a good way to begin a proof of a statement about all elements of $A$. The criticism is based on the objection that $A$ could be empty, in which case there is no $x$ to be in $A$. The issue affects quite a lot of mathematical content, because more than half of our proofs are proofs of universal statements, and most of them begin this way.
I have objected that if $A$ is empty, then any universal statement $\forall x \in A ...$ is vacuously true, but people are telling me that this needs to be dealt with as a special case, or else the proof is technically incorrect, etc.
I have appealed to normal mathematical conventions, without success. Our department prides itself on being more careful than normal working mathematicians. Convention can do what it will, but we intend to be right!
I have appealed to serious logic, by talking about the underlying meaning of "Let $x \in A$." In my reading it plays a dual role of symbol introduction ("Use $x$ to represent a single thing") and assumption ("Assume $x \in A$"). But these arguments have no traction -- "our students can't be expected to understand clever subtleties of metalogic."
I fear the only option remaining is to appeal to authority -- some specific authority who says this act of "Let $x \in A$" has some sort of seal of approval. Maybe such an authority is here?
From this question you might thing I work with fools, but they're really very wonderful and intelligent people, and the sense of family here is unusually strong. Like family, they drive me out of my head sometimes. Probably it's mutual.
So I suppose I have two questions: 1. Am I right? 2. Is there any hope for me to persuade my colleagues that I'm right?
(3. But social advice would be welcome too.)