Gödel's proof of his incompleteness theorems makes use of Gödel numbering, which is a device that allows a theory of arithmetic $S$ (e.g. PRA) to express and reason about metamathematical statements about a theory $T$ ($T$ must satisfy some conditions that I am omitting for the sake of brevity).
Gödel numbering seems however unnecessary in the case one decides to carry metamathematics in an adequately powerful set theory, say ZFC just to fix one, as opposed to a theory of arithmetic. As done in proof theory, one defines formal languages, axioms, theorems, deductions, rules of inference, theories, etc. as mathematical objects (which are ultimately sets). Having taken care of this, we can view ZFC as a mathematical object in the universe of ZFC, or to be more precise there is an object in the universe of ZFC which is a carbon-copy of ZFC. To avoid confusion, I will refer to this carbon-copy object as $\overline{\text{ZFC}}$ and statements in the language of $\overline{\text{ZFC}}$ will be overlined. Now the sentence "$\overline{(\emptyset = \emptyset) \wedge \neg (\emptyset = \emptyset})$ is not a theorem of $\overline{\text{ZFC}}$" is a sentence in ZFC which asserts the consistency of $\overline{\text{ZFC}}$.
Do the incompleteness theorems prevent this sentence from being provable (assuming ZFC is consistent), even though Gödel numbering was not used to implement it?