Why can't a set have two elements of the same value? Suppose I have two sets, $A$ and $B$:
$$A = \{1, 2, 3, 4, 5\} \\ B = \{1, 1, 2, 3, 4\}$$
Set $A$ is valid, but set $B$ isn't because not all of its elements are unique. My question is, why can't sets contain duplicate elements?

To say $2$ sets are equal, show that each set is contained in the other. $\{1,1,2,3,4\} = \{1,2,3,4\}$ since any element on the left can be found on the right and vice-versa.
