Ask yourself, if private member access disqualified the rest of the class from access, what would be the point of a private field? You declare a potentially complex object, and then literally can do nothing with it? You have a private region that can interact with itself, but which cannot have any effect on any public facing functionality? You use the private structures as middlemen to write to public fields that you then use? None of these scenarios makes that much sense. Access modifiers protect the class from external influences, but the OOP model assumes the programmer will take care of themselves within the class.
It is noteworthy that there is one condition in which private members can not be accessed: Inheritance. The base class private variables are there in a derived class, but cannot be referenced directly. To be clear,these are the base class' private variables. The child class has its own private scope that it can declare within and access normally.