If you're going to count the emergent structure that forms from voluntary cooperation as "regulation", the word loses its meaning.

> Capitalism doesn't exist without private property law.

Sure it does.

Laws aren't self-enforcing: the the thing that makes laws "real" is violence (or the threat of violence) backing them up.

For those who are unable to avail themselves of remedies under the law - typically because they're engaging in illegal behavior - they can skip the law bit and move straight to violence.

Basically every criminal organization or individual operates in this manner.

I'm not saying it's nice to live this way - I very much like living in a place with a (mostly) functioning legal system. But Capitalism very much does exist without private property law.