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 Th15:
  (for x st x in X holds x is Ordinal & x c= X) implies
   X is epsilon-transitive epsilon-connected
proof
  assume
A1: for x st x in X holds x is Ordinal & x c= X;
  thus X is epsilon-transitive
  by A1;
  let x,y;
  assume x in X & y in X;
  then x is Ordinal & y is Ordinal by A1;
  hence thesis by Th10;
end;
