If $\Sigma_1$ and $\Sigma_2$ are consistent sets and if $\Sigma_1 \vdash \alpha$ for every $\alpha \in \Sigma_2$, is $\Sigma_1 \cup \Sigma_2$ consistent? Intuitively I think it is consistent, but I am not sure how to prove it.
I would also like to know if $\Sigma_1 \vdash \alpha$ for every $\alpha$ such that $\Sigma_2 \vdash \alpha$ is $\Sigma_1 \cup \Sigma_2$ consistent?
Finally, are any difference(s) between the first and second question?