Well if you are good through {a} = {c, d} = {a, d} you are golden. the set {a} has 1 element, the other two have 2 elements, for finite sets to be equal, the cardinality must be the same. but that means c,d and a,d are really they same thing and they must all actually be a to have the sets be equal. So that gives us a=c=d, but recall which case we are in, for a=b. so we get b=a=c=d, which means that in particular a=c and b=d, they are just all the same too. It is kind of like the trivial case I guess. Does that explain it?