reserve N for Cardinal;
reserve M for Aleph;
reserve X for non empty set;
reserve Y,Z,Z1,Z2,Y1,Y2,Y3,Y4 for Subset of X;
reserve S for Subset-Family of X;
reserve x for set;

theorem Th2:
  { X } is non empty Subset-Family of X & not {} in { X } & for Y1,
Y2 holds (Y1 in { X } & Y2 in { X } implies Y1 /\ Y2 in { X }) & ( Y1 in { X }
  & Y1 c= Y2 implies Y2 in { X })
proof
A1: X c= X;
  { X } c= bool X
  proof
    let x be object;
    assume x in { X };
    then x=X by TARSKI:def 1;
    hence thesis by A1;
  end;
  hence { X } is non empty Subset-Family of X;
  thus not {} in { X } by TARSKI:def 1;
  let Y1,Y2;
  thus Y1 in { X } & Y2 in { X } implies Y1 /\ Y2 in { X }
  proof
    assume Y1 in { X } & Y2 in { X };
    then Y1 =X & Y2 = X by TARSKI:def 1;
    hence thesis by TARSKI:def 1;
  end;
  thus Y1 in { X } & Y1 c= Y2 implies Y2 in { X }
  proof
    assume Y1 in { X } & Y1 c= Y2;
    then X c= Y2 by TARSKI:def 1;
    then Y2 = X;
    hence thesis by TARSKI:def 1;
  end;
end;
