reserve v,x,x1,x2,x3,x4,y,y1,y2,y3,y4,z,z1,z2 for object,
  X,X1,X2,X3,X4,Y,Y1,Y2,Y3,Y4,Y5,
  Z,Z1,Z2,Z3,Z4,Z5 for set;
reserve p for pair object;
reserve R for Relation;
reserve xx1 for Element of X1,
  xx2 for Element of X2,
  xx3 for Element of X3;

theorem Th34:
  X c= [:X,Y,Z:] or X c= [:Y,Z,X:] or X c= [:Z,X,Y:] implies X =  {}
proof
  assume that
A1: X c= [:X,Y,Z:] or X c= [:Y,Z,X:] or X c= [:Z,X,Y:] and
A2: X <> {};
  [:X,Y,Z:]<>{} or [:Y,Z,X:]<>{} or [:Z,X,Y:]<>{} by A1,A2;
  then reconsider X,Y,Z as non empty set by Th21;
  per cases by A1;
  suppose
A3: X c= [:X,Y,Z:];
    consider v such that
A4: v in X and
A5: for x,y,z st x in X or y in X holds v <> [x,y,z] by Th19;
    reconsider v as Element of [:X,Y,Z:] by A3,A4;
    v = [v`1_3,v`2_3,v`3_3];
    hence contradiction by A5;
  end;
  suppose
    X c= [:Y,Z,X:];
    then X c= [:[:Y,Z:],X:] by ZFMISC_1:def 3;
    hence thesis by ZFMISC_1:111;
  end;
  suppose
A6: X c= [:Z,X,Y:];
    consider v such that
A7: v in X and
A8: for z,x,y st z in X or x in X holds v <> [z,x,y] by Th19;
    reconsider v as Element of [:Z,X,Y:] by A6,A7;
    v = [v`1_3,v`2_3,v`3_3];
    hence thesis by A8;
  end;
end;
