
theorem Th2:
  for X being non empty set st {{}} is Element of X holds not {{}} in Funcs X
  proof
    let X be non empty set;
    assume
A1: {{}} is Element of X;
    assume
A2: {{}} in Funcs X;
    set FAB = { (Funcs (A,B)) where A, B is Element of X : not contradiction };
    reconsider x = {{}} as Element of X by A1;
    consider y be set such that
A3: x in y in the set of all Funcs(A,B) where A,B is Element of X
      by A2,TARSKI:def 4;
    consider A,B be Element of X such that
A4: y = Funcs(A,B) by A3;
    consider f be Function such that
A5: x = f and
A6: dom f = A and
A7: rng f c= B by A3,A4,FUNCT_2:def 2;
    reconsider f as Function of A,B by A6,A7,FUNCT_2:2;
A8: {{}} = f by A5;
    {} in {{}} by TARSKI:def 1;
    then consider u,v be object such that
    u in A and v in B and
A9: {} = [u,v] by A8,ZFMISC_1:def 2;
    thus thesis by A9;
  end;
