"I do not have a cat."
Translation:Je n'ai pas de chat.
the direct object of a verb in negative does not have an indefinite or partitive article:
- je n'ai pas de chat
- je n'ai plus de chat
- je n'ai jamais (eu) de chat
But why do you use 'de' here? So far, I've only really seen that used with food.
"de" is a common preposition and very versatile. You will see it a lot, and in various usages.
The point here is that the direct object of a verb in the negative form loses its article and "de" is used instead. This is valid for any noun, be it countable or not, singular or plural.