reserve X,Y,Z,X1,X2,X3,X4,X5,X6 for set, x,y for object;
reserve a,b,c for object, X,Y,Z,x,y,z for set;
reserve A,B,C,D for Ordinal;

theorem Th16:
  X c= A & X <> {} implies ex C st C in X & for B st B in X holds C c= B
proof
  set a = the Element of X;
  assume that
A1: X c= A and
A2: X <> {};
  a in X by A2;
  then consider Y such that
A3: Y in X and
A4: not ex a being object st a in X & a in Y by TARSKI:3;
  Y is Ordinal by A1,A3,Th9;
  then consider C such that
A5: C = Y;
  take C;
  thus C in X by A3,A5;
  let B;
  assume B in X;
  then not B in C by A4,A5;
  then B = C or C in B by Th10;
  hence thesis by Def2;
end;
