Consistency of a particular theory Take a first order language with Parameter:
Constant Symbol: $\emptyset$
Equality Relation and Belonging to Relation as Predicate Symbol
Now take theory in this first order language with axioms:


*

*$(\forall x)(\forall y)\Bigg((x=y) \longleftrightarrow \Big((\forall z)(z \in x \longleftrightarrow z \in y)\Big)\Bigg)$

*$(\forall x)(\forall y)\Bigg((x\neq y) \longleftrightarrow \Big((\exists z)\big(z \in x \land z \notin y)\lor (z \notin x \land z \in y)\big)\Big)\Bigg)$

*$(\forall x)(\forall y)(\forall z)\Big(\big(x=y) \land (x \in z)\big) \longrightarrow (y \in z)\Big)$

*$(\forall x)(x \notin \emptyset)$
Question: Is this theory consistent?
Explanation of terms: Consistency means prove there doesn't exist $\phi$ such that this theory proves $\phi$ as well as $\neg\phi$.
Consistency I am not taking whether there exist model of this theory in set theory.

Yes - for example, it is true in the structure with a single element $E$ (for emptyset), and where the binary relation is empty. It also has less stupid examples: given any transitive set $X$, the pair $(X, \in)$ is a model of this theory.
(Note that having a model easily implies consistency - this is the soundness theorem. More surprisingly, the converse is also true - this is the completeness theorem.)

Based on this, I'm not really sure you wrote the axioms you intend. Also, unless I'm misreading, (2) is the same as (1), and (3) follows from the semantics of equality in first-order logic. Did you mean to write other axioms?
