Or another way to prove it, is that the maximum xor you can achieve by only using numbers in the range $[0; 50]$ is $63$, and the minimum one is $0$, so you can have at most $64$ different numbers. If you have $7$ numbers such that for every subset its xor is unique, then you will have $2^7 = 128$ different numbers; that is a contradiction. * If for subsets $A, B$ their xors are equal, then you can find $C = A \, \Delta \, B$ (symmetric difference) whose xor is $0$.
Or another way to prove it, is that the maximum xor you can achieve by only using numbers in the range $[0; 50]$ is $63$, and the minimum one is $0$, so you can have at most $64$ different numbers. If you have $7$ numbers such that for every subset its xor is unique, then you will have $2^7 = 128$ different numbers; that is a contradiction. * If for subsets $A, B$ their xors are equal, then you can find $C = A \, \Delta \, B$ (symmetric difference) whose xor is $0$.