reserve A,B,C for Ordinal,
  X,X1,Y,Y1,Z for set,a,b,b1,b2,x,y,z for object,
  R for Relation,
  f,g,h for Function,
  k,m,n for Nat;
reserve M,N for Cardinal;

theorem Th13:
  card X in card bool X
proof
  deffunc f(object) = {$1};
  consider f such that
A1: dom f = X &
for x being object st x in X holds f.x = f(x) from FUNCT_1:sch 3;
A2: rng f c= bool X
  proof
    let x be object;
    assume x in rng f;
    then consider y being object such that
A3: y in dom f and
A4: x = f.y by FUNCT_1:def 3;
A5: { y } c= X
    by A1,A3,TARSKI:def 1;
    f.y = { y } by A1,A3;
    hence thesis by A4,A5;
  end;
A6: card X <> card bool X by Th4,Th12;
  f is one-to-one
  proof
    let x,y be object;
    assume that
A7: x in dom f & y in dom f and
A8: f.x = f.y;
    f.x = { x } & f.y = { y } by A1,A7;
    hence thesis by A8,ZFMISC_1:3;
  end;
  then card X c= card bool X by A1,A2,Th9;
  hence thesis by A6,ORDINAL1:11,XBOOLE_0:def 8;
end;
