Basic Proof by Contradiction of the Zero Product Property

Prove the Zero Producty Property in real numbers that:
If ab=0 then a=0 or b=0

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...call that ~(X or Y) is equivalent to (~X AND ~Y). So ~Q is (~a=0 and ~b=0). I'm going to rewrite ~Q as (a~=0 and b~=0) where ~= means "is not equal to".

So we assume ab=0 and a~=0 and b~-0 (that's P and ~Q). Now the second and third parts of the assumption mean that a is some nonzero number, and b is some nonzero number. Then we're ...