An argument with premises Γ and conclusion φ is shown to be valid. What follows if we now combine Γ with the negation of φ into a single set?

C. The resulting set cannot be satisfied by any assignment

Testing validity by reductio works because establishing validity guarantees no assignment can simultaneously satisfy the premises and the denial of the conclusion. Inconsistency does not mean the original premises were flawed, nor does it selectively entail only original claims.