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Also, no way Tyrion is a partial Targaryn. If tywin even remotely thought it possible he would have killed tyrion.
Wrong! He said he couldnt prove he wasnt a Lannister, technically it was his wife's child...
Unless you are familiar with English lit or antiquity, the insult of being a cuckold is unfamiliar. The idea that another man seduced the Lion's wife and seeded a child under the Lion's nose would be a huge insult and a point of much ridicule "limp d-ck old man couldnt please his wife". For Tywin who cares more about "legacy" than his children, to be known that he was cuckolded would be enormous. So he can NEVER disown Tyrion, even if he had the proof.