Equivalence of Definitions of Inconsistent (Logic)

Theorem
Let $\LL$ be a logical language.

Let $\mathscr P$ be a proof system for $\LL$.

Definition $(1)$ implies Definition $(2)$
Let $\FF$ be an inconsistent set of logical formulas by definition $1$.

Let $\phi$ be an arbitrary logical formula in $\FF$.

Then :
 * $\phi \land \lnot \phi$

is a logical formula in $\FF$.

Thus $\FF$ is an inconsistent set of logical formulas by definition $2$.

Definition $(2)$ implies Definition $(1)$
Let $\FF$ be an inconsistent set of logical formulas by definition $2$.

Then by definition:


 * $\exists \phi \in \F: \FF \vdash_{\mathscr P} \paren {\phi \land \neg \phi}$

That is, $\phi \land \neg \phi$ is a logical formula in $\FF$.

Let $\psi$ be an arbitrary logical formula.

By the Rule of Explosion:


 * $\forall \psi: \paren {\phi \land \neg \phi} \implies \psi$

That is:
 * $\forall \psi: \FF \vdash_{\mathscr P} \psi$

That is:
 * $\psi \in \FF$

As $\psi$ is arbitrary, it follows that every logical formula is a provable consequence of $\FF$.

Thus $\FF$ is an inconsistent set of logical formulas by definition $1$.