# In an indirect proof, is it possible to reject the assumption based on contradiction of a premise?

In an indirect proof, is it possible to reject the assumption based on contradiction of a premise?
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Irene Simon
This is a proof that $x\in A\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}\mathrm{¬}P$. It can be simplified, since everything between "Assume $P$" and "Derive $x\notin A$" is just a proof that $P\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}x\notin A$, from which by contrapositive we get $x\in A\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}\mathrm{¬}P$, which was to be demonstrated.
You can't in general conclude from this that $P$ is always false, of course.

brasocas6
Yes. Deriving a contradiction from the assumption of $P$ is a proof for $\mathrm{¬}P$.
Although apparently similar, this is not actually Reduction ad Absurdum.
This is the Rule of Negation Introduction. Unlike RAA this is an intuitionistically valid rule of inference.
$\frac{\frac{\mathrm{\Sigma },\mathrm{¬}P⊢\mathrm{\perp }}{\mathrm{\Sigma }⊢\mathrm{¬}\mathrm{¬}P}\mathrm{¬}\mathsf{I}}{\mathrm{\Sigma }⊢P}\mathrm{¬}\mathrm{¬}\mathsf{E}$