I am working on a translation from French about, ultimately, the axiom of constructibility. In the opening paragraphs, the author describes how we are going to assume the ZF axioms are consistent to create a stronger theory with this axiom. Then, he goes on to say that it is harmless to also assume that there is a transitive ZF model (a set model). Then, he continues by making a precise statement about this which I translated as "More precisely, the theory of Zermelo-Fraenkel, including the single binary predicate $\in$ admits an expansion of the theory with $\in$ to include a unary predicate $\tau$ with an axiom saying a there is a unique set which is transitive and a ZF model by checking with $\tau$." I'm not sure if I got it right, so my question is:
How do you expand a model of ZF + the axiom of consistency to include that the model is transitive by adding a unary predicate?