I was wondering if this proof was ok or if (as I suspect) I am wrong and if you could help point me in the right direction.

thanks for any help

Suppose that $\displaystyle \mathcal{F}$ is an inconsistent set of sentences. For each $\displaystyle G\epsilon\mathcal{F}$ let $\displaystyle \mathcal{F}_G$ be the set obtained from removing G from $\displaystyle \mathcal{F}$.

Prove that for any $\displaystyle G\epsilon\mathcal{F}$ $\displaystyle \mathcal{F}_G\vdash\lnot G$.

Proof

We can split into two cases.

1. $\displaystyle \mathcal{F}_G$ is still an inconsistent set and so it can prove anything, namely $\displaystyle \lnot G$

2. $\displaystyle \mathcal{F}$ is now consistent in which case G was false in the cases that $\displaystyle \mathcal{F}_G$ was satisfied and so $\displaystyle \lnot G$ is true when $\displaystyle \mathcal{F}_G$ is satisfied and so by definition $\displaystyle \mathcal{F}_G\models\lnot G$ and by completness $\displaystyle \mathcal{F}_G\vdashz\lnot G$