A market satisfies the Law of One Price if every two self-financing strategies that replicate the same claim have the same initial value.
An inconsistent pricing strategy is a self-financing strategy $\phi$ with $V_T(\phi)\equiv 0$ and $V_0(\phi) < 0$.
Prove the Law of One Price holds if and only if there does not exist an inconsistent pricing strategy.
Attempted proof - Suppose we have two self financing strategies $\phi$ and $\psi$ that replicates some claim $X$ such that $V_0(\phi) = V_0(\psi)$. Hence we cannot satisfy the condition of $V_0(\phi) < 0$ nor $V_0(\psi) < 0$ so there is no inconsistent pricing strategy.
I am not sure how to show the converse and whether this is rigorous enough. Any suggestions are greatly appreciated.