Puzzle #216: Sandwiches

The local corner store has 12 sandwiches in the cooler. Five have corned beef, seven have cheddar cheese, and six have Swiss cheese. Each sandwich has at least one of the three ingredients, and there is at least one sandwich of each of the seven ingredient combinations. Exactly three sandwiches contain both types of cheese. Two sandwiches contain all three ingredients.

How sandwiches are there of each type?

Submit your answer to Gene Wirchenko <genew@telus.net>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, July 25, 2017 at noon Pacific Time. I will post the answer shortly after.