theorem
  id X c= id Y iff X c= Y
proof
  thus id X c= id Y implies X c= Y
  proof
    assume
A1: id X c= id Y;
    let x be object;
    assume x in X;
    then [x,x] in id X by RELAT_1:def 10;
    hence thesis by A1,RELAT_1:def 10;
  end;
  assume
A2: X c= Y;
  let z be object;
  assume
A3: z in id X;
  then consider x,x9 being object such that
A4: x in X and
  x9 in X and
A5: z = [x,x9] by ZFMISC_1:84;
  x = x9 by A3,A5,RELAT_1:def 10;
  hence thesis by A2,A4,A5,RELAT_1:def 10;
end;
