If A and B are any two sets, prove that
P(A) = P(B) implies A = B.

Explanation

Given, P(A) = P(B) `" … (i)"`
To prove A = B
Let `x in A` implies there exists a subset X of A such that `x in X`.
Now, `XsubeAimpliesX inP(A)`
`implies XsubeB`
`implies x inB" "[because x in X]`
Thus, `x in A implies x in B`
`therefore A sube B" ... (ii)"`
Let `y in B` implies there exists a subset Y of B such that `y in Y`.
Now, `YsubeBimpliesYinP(B)`
`implies YinP(A)" "[becauseP(B)=P(A)]`
`implies YsubeA`
`implies yinA" "[becausey in Y]`
Thus, `y inB implies y inA`
`therefore B subeA" ... (iii)"`
From Eqs. (ii) and (iii), we have A = B