Math
posted by B.B. .
Use sets to verify that 8>6. Here is how I answered this question: I will use sets A and B with eight and six elements. A={8,7,6,5,4,3,2,1} B={6,5,4,3,2,1}. Using the symbol 8 to represent n(A) and the symbol 6 to represent n(B), I will write 8>6 (read eight is greater than six). The set that runs out of elements is said to have fewer elements than the other set, and its whole number is less than that of the other set. For B and A , I write 6<8 (read six is less than eight).

Without using the formal definition of the integers and of the numbers 6 and 8, you can't prove that 8 > 6. Any demonstration that doesn't use the definitions will end up using what it is supposed to verify.
So, in your case, if you don't use the fact that 8 > 6, you can't construct the sets A and B in the first place. E.g., you could just as well have defined:
A = {8,7,6,5,4,3,2,1}
B= {8,5,4,3,2,1}
Then, analogous to your proof, I can use the symbol 6 to represent n(A) and the symbol 8 to represent n(B).
Then B is a subset of A, so you could say that 8 < 6. 
Thanks for the help.