I'm working through an exercise which involves negation completeness.
Let $S = \{R\}$. Let x and y be distinct variables. Suppose we have the set $\phi = \{Rx \vee Ry\}$. Show "Not $\phi \vdash Rx$" and "Not $\phi \vdash \neg Rx$". In other words, I need to show that $\phi$ is not negation complete.
The only idea that I had was to do this by a case-by-case basis assuming each one is true.
For example:
Case 1: Suppose $\phi \vdash Rx$. This suggests that $\phi$ does not hold when $\neg Rx$. Consider $\neg Rx \wedge Ry$ where $\phi$ holds but $\neg Rx$, thus we have a contradiction.
Am I on the right track?