What's wrong with the following line of reasoning, if so?
The comprehension axiom of Zermelo's set theory would be provable by the other axioms, if the following was provable:
($*$) All subclasses of a set are sets.
Then, especially all definable subclasses of a set were sets, which is essentially what the comprehension axiom says.
Since I assume that the comprehension axiom is not provable by the other axioms, ($*$) must not be provable. What does this mean? Are there models of set theory with subclasses of sets that are not sets?