We call a set of formulas $\Sigma$ of a language $L$ consistent if there is no $\varphi$ in $L$ such that $\Sigma \vdash \varphi$ and $\Sigma \vdash \lnot \varphi$.
Apparently, an equivalent formulation is the following:
A set $\Sigma$ of formulas of $L$ is consistent iff $\Sigma \not\vdash \varphi$ for some sentence $\varphi$ of $L$.
The $\implies$ direction is clear: if we can prove all sentences then we can prove both $\varphi$ and $\lnot \varphi$ so that $\Sigma$ is inconsistent.
But I don't immediately see how to prove $\Longleftarrow$. Can someone explain this to me? Thanks!