A Song of Ice and Fire: Why didn't Tywin Lannister disinherit Tyrion if he hates him so much?

Why didn't Tywin disinherit Tyrion?

But...

Tywin did disinherit Tyrion. And to ensure that he stayed disinherited he planned to send him to the Wall. Though that plan went down the toilet. Literally.

And the reason why he did so is below.

Artwork by Magali Villeneuve

Under Westerosi law, Tyrion is the rightful heir to Casterly Rock. Jaime has forfeited his right to it by joining the King's Guard.

"What do I want, you ask? I'll tell you what I want. I want what is mine by rights. I want Casterly Rock."

Hiw father's mouth grew hard. "Your brother's birthright?"

"The knights of the Kingsguard are forbidden to marry, to father children, and to hold land....The day Jaime put on that white cloak, he gave up his claim to Casterly Rock, but you never once have you acknowledged it....I want you to stand up before the realm and proclaim that I am your son and your lawful heir.

Lord Tywin's eyes were a pale green flecked with gold, as luminous as they were merciless. ""Casterly Rock," he declared in a flat cold dead tone. And then, "Never."...

"Why?"

"You ask that? You, who killed your mother to come into the world? You are an ill-made, devious, disobedient, spiteful little creature full of envy, lust and low cunning. Men's laws give you the right to bear my name and display my colors since I cannot prove you are not mine. To teach me humility, the gods have condemned me to watch you waddle about wearing that proud lion that was my father's sigil and his father's before him. But either gods nor me shall ever compel me to let you turn Casterly Rock into your whorehouse."...

..."Go back to your bed, Tyrion, and speak to me no more of your rights to Casterly Rock. You shall have your reward, but it shall be one I deem appropriate to your servic and station.

Tywin does what he wants.