I did not once mention that property rights do not exist I simply gave a brief summery of what anarchy is and what it means to be an anarchist. I do not need to go into every detail of every corner of the ideology to explain what anarchy is.

You are right in the sense that you have the natural right to defend your property with the use of aggression but what you do not have the right to do is aggressively attack another's person or property. The government has a tendency to do this most often. Why should government have different rights to everybody else?

By saying that the act of aggression would be continuous under an anarchist society is pretty much like saying that everyone is a crazy lunatic socio path wanting to kill everyone and everything in it's path, which in hindsight couldn't be further from the truth.