If we have a first order set theory $T$, i.e. its language is first order logic with the membership symbol added as the sole extra-logical primitive. Supposed that $T$ proves the existence of a set $\mathcal M$ and a set $\mathcal E$ of ordered pairs in $\mathcal M$: such that if we replace each atomic formula $x \in y$ by $[\exists p \in \mathcal E (\text{ordered-pair}(p,x,y))]$ and bound all quantifiers to $\mathcal M$, in all axioms of $T$, then $T$ proves that each replacing sentence is true, i.e. if $\alpha$ was an axiom of $T$ then $T$ proves that the replacement sentence $\alpha'$ of $\alpha$ to be true.
Of course the ternary relation "ordered-pair" is definable in $T$ and provable in it to satisfy the basic property of ordered pairs, also it satisfies the existence of Cartesian products between any two sets in terms of them, and domains, co-domains, and ranges of sets of them.
Suppose that $T$ meets conditions for Godels incompleteness arguments, through interpreting Robinson's arithmetic $\sf Q$ for example, and of course having a recursively enumerable set of theorems. Then does that always mean that $T$ proves the existence of a model of it, and thus it proves its own consistency and thus $T$ is inconsistent?