 reserve x,y,X1,X2,X3,X4,X5,X6,Y,Y1,Y2,Y3,Y4,Y5,Z,Z1,Z2,Z3,Z4,Z5 for set;
 reserve X for non empty set;

theorem
  not ( X1 in X2 & X2 in X3 & X3 in X1)
proof
A1: X2 in { X1,X2,X3 } & X3 in { X1,X2,X3 } by ENUMSET1:def 1;
A2: X1 in { X1,X2,X3 } by ENUMSET1:def 1;
  then consider T being set such that
A3: T in { X1,X2,X3 } and
A4: { X1,X2,X3 } misses T by Th1;
  T = X1 or T = X2 or T = X3 by A3,ENUMSET1:def 1;
  hence thesis by A2,A1,A4,XBOOLE_0:3;
end;
