The principle of non-contradiction states: "being is not non-being." Or, as Aristotle put it: "One cannot say of something that it is and that it is not in the same respect and at the same time."
This is a principle that rational beings instinctively know, and usually take for granted. It is also a fundamental law of logic and so, by definition, cannot be proven to be false.
Non-contradiction should not be confused with the principle of the excluded middle