thus ( X = Y implies for w being rational number holds
( w in X iff w in Y ) ) ; :: thesis: ( ( for w being rational number holds
( w in X iff w in Y ) ) implies X = Y )
assume for w being rational number holds
( w in X iff w in Y ) ; :: thesis: X = Y
then ( X c= Y & Y c= X ) by Def10;
hence X = Y by XBOOLE_0:def 10; :: thesis: verum