If we have mice in our garden rats will stay away is this correct?

Last year we had a rat problem - one got in the house from the back garden. Now that we seemed to have got rid of the rat problem I am catching rather a lot of mice in the same garden approximately 1 per day in the past week. I mentioned it to some friends who happen to live in the country and they told me that if we have mice we will not have rats as they do not mix - how true is this ?? I think the main reason for them in our garden is the pond as we do not put out food scraps for wildlife nor do we have pets or have a lot of waste / hiding places for them but the neighbours do. I would rather live with the mice than exterminate them and have the rats come back - has anyone heard this saying before or have any idea if it might be true ? ?