This paper proves that ZFC and Peano arithmetic (PA) are inconsistent, the latter result being a corollary of the former. We introduce a metamathematical extension of ZFC that allows us to use statements in the proof that express logical consequence. We then show, by explicitly stating a contradiction, that the extended theory is inconsistent and that this immediately leads to the existence of a contradiction in ZFC. The contradiction is triggered by the conjunction of two properties of an infinite set, by means of which we express a strengthened form of the strong Goldbach conjecture. We use elementary number theory, with the constructive role of prime numbers within the natural numbers being an essential point.