In your example it is Mon June 7th. Sun June 13th would be "next Sunday" because it is still 6 days away. However, if today is Fri June 11th, and you wanted to refer to Sun June 13th, we would say "this Sunday" because it is so close!
Also remember that the day of the week comes at the beginning of most date constructions. After that, then the month and the date (depends where you are which way round you put these), then the year. But the day always comes at the beginning.
Monday, June 7th, 2010
Monday, 7th June, 2010