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;

theorem Th20:
  X <> {} implies ex v st v in X & not ex x1,x2,x3,x4 st (x1 in X
  or x2 in X) & v = [x1,x2,x3,x4]
proof
  assume X <> {};
  then consider Y such that
A1: Y in X and
A2: for Y1,Y2,Y3,Y4,Y5 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5
  in Y holds Y1 misses X by XREGULAR:6;
  take v = Y;
  thus v in X by A1;
  given x1,x2,x3,x4 such that
A3: x1 in X or x2 in X and
A4: v = [x1,x2,x3,x4];
  set Y1 = { x1,x2 }, Y2 = { Y1,{x1} }, Y3 = { Y2,x3 }, Y4 = { Y3, {Y2}}, Y5 =
  { Y4,x4 };
A5: Y3 in Y4 & Y4 in Y5 by TARSKI:def 2;
A6: Y5 in Y by A4,TARSKI:def 2;
A7: x1 in Y1 & x2 in Y1 by TARSKI:def 2;
  Y1 in Y2 & Y2 in Y3 by TARSKI:def 2;
  hence contradiction by A2,A7,A5,A6,A3,XBOOLE_0:3;
end;
