First to mention, I am just an undergraduate student with only very basic knowledge about logic and set theory. This is my general idea of how we define sets:
We first have first-order logic, which is like an "language"; it allows us to write down axioms to build up ZFC, then we have the notion of "set". Primitive notions are required in these steps. Similarly, we can define classes, which may not be sets.
Now, I want to build a new math object, which is not a set, but is not entirely independent from our ZFC set theory. (For example, given a set $A$, I want to define a math object $B$ such that $A\cup B=\varnothing.$ In this case, my new theory may need to extend the meaning of $\cup$ and $=$.) What are the steps that I have to go through, in order to build a new complete theory, and which probably doesn't contradict our original ZFC theory? What qualify my new theory to be a valid one?
P.s. Sorry if my question is unclear, I will like to clarify it if you spot any problem.