Thm*  (P  P)

That is, no proposition is equivalent to its own negation.

Proof:

Suppose P  P.

Then P, since assuming P for the sake of argument would allow us to conclude P, contradicting the assumption P.

But from P, P itself now follows.

Sidenote:

Sometimes this theorem is proved considering both cases of P being true or false, and showing that each way leads to a contradiciton.
In contrast there is no case splitting in the argument shown here, just an argument that P would be false and therefore also true.

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