It seems to me that the original proof also uses "consecutive numbers are coprime" as the sole property. In particular, it looks at P = p_1 * ... * p_n and says

It seems to me that the original proof also uses "consecutive numbers are coprime" as the sole property. In particular, it looks at P = p_1 * ... * p_n and says that it is coprime to Q = P + 1; hence, there must exist another prime number.