Question 835126
Let X and Y be two sets. Let X be a subset of Y. So everything in X is also found in Y. 


If X is a subset then the only two things are possible: it is a proper subset or it is an improper subset. 


If it is a proper subset, then X will have less than Y (since X will be smaller). 


If X is an improper subset, then X will have the same number of items in set Y. This will force X to be equal to Y. 


If the two weren't equal for instance, then Y would have some element that is not in X, but that would make Y larger and hence X would be a proper subset. 


However, X is an improper subset, which would again force Y to not have that extra element. So this proves that X = Y.