What you have quoted is a proof of negation, which is not the same as proof by contradiction. Your quoted part of the proof is constructive, ergo not by contrad

What you have quoted is a proof of negation, which is not the same as proof by contradiction. Your quoted part of the proof is constructive, ergo not by contradiction. Proof by contradiction has the following form: ~A -> false |- A. But the quoted proof has the form A -> false |- ~A.