A question was recently asked about Theorem 1 of Suppes's axiomatic set theory asking why the empty set was used. I have a related question. Why, in the formula:
$(\exists{C})(\forall{x})(x \in C \iff x \in \emptyset \& (x \ne x))$
does Suppes use the empty set as the set of which $A$ is a subset? That is, wouldn't the theorem be equally valid if he had made this first line:
$(\exists{C})(\forall{x})(x \in C \iff x \in B \& (x \ne x))$
In other words, isn't the superset irrelevant (it can be $B$, $\emptyset$ or any other set) if ultimately what we are proving is that no $x$ can be a member of $A$ because x≠x is absurd?
At this stage in the development of the theory, no sets have been defined except the empty set, (whose existence is posited as a primitive constant of the theory). It is the only primitive formula involving set that can be used in the axiom schema of separation.