intersection of the empty set and vacuous truth

Solution 1:

There’s nothing wrong with the ‘vacuous truth’ part of the argument. It’s perfectly correct that if $X$ is any set, then $\left\{x\in X:x\in\bigcap\varnothing\right\}=X$. To see this, note that if $x\in X$, then $x\notin\bigcap\varnothing$ if and only if there is an $A\in\varnothing$ such that $x\notin A$, and since there is no $A\in\varnothing$ at all, this is not the case.

The problem with the argument is that nothing in $\mathsf{ZF}$ permits the formation of $\left\{x:x\in\bigcap\varnothing\right\}$: this an example of unrestricted comprehension, which is not permitted in $\mathsf{ZF}$. $\mathsf{ZF}$ permits only restricted comprehension, using a formula to pick elements from an already existing set, not from the universe at large.

Solution 2:

Your confusion about how the intersection over a set can result in a proper class is justified.

In some places the definition of the intersection is bounded, so the result is always a set, i.e. $$\bigcap\mathcal A=\left\{x\in\bigcup\mathcal A\mid\forall A\in\mathcal A.x\in A\right\}$$

The philosophical justification is that the intersection over a set should result in a set, so we take only elements from $\bigcup\cal A$, which by the axiom of union is a set. The result is only different for the empty set, that is if $\cal A\neq\varnothing$ then we can easily forget about this bound, but when $\cal A=\varnothing$ we need to decide whether or not we do that.

This is the set theoretical equivalent of $0^0$ being indeterminate in analysis.

And as a side remark, vacuous argument reside in the logic, not in the axiomatic systems.