Reference request for the independence of $ \text{Con}(\mathsf{ZFC}) $. Gödel’s Second Incompleteness Theorem says that if $ \mathsf{ZFC} $ is consistent, then $ \mathsf{ZFC} \nvdash \text{Con}(\mathsf{ZFC}) $, i.e., $ \text{Con}(\mathsf{ZFC}) $ is not provable in $ \mathsf{ZFC} $.
Does anyone know of an authoritative reference that contains the claim that $ \neg \text{Con}(\mathsf{ZFC}) $ is also not provable in $ \mathsf{ZFC} $ if $ \mathsf{ZFC} $ is consistent?
Thanks!

This is just not true.
Suppose that $\sf ZFC+\operatorname{Con}(ZFC)$ is just inconsistent, and the additional assumption is the one causing the inconsistency. What does that mean? It means that in every model of $\sf ZFC$ it holds that $\lnot\operatorname{Con}\sf (ZFC)$, and therefore it is provable from $\sf ZFC$ that $\lnot\operatorname{Con}\sf (ZFC)$.
Of course if you believe that inaccessible cardinals, for example, are not inconsistent with $\sf ZFC$, then you have all the reason to believe that $\sf ZFC$ does not prove that $\lnot\operatorname{Con}\sf (ZFC)$. Why? Because $\sf ZFC+\exists\kappa\text{ inaccessible}\vdash\operatorname{Con}(ZFC)$.
