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 Th19:
  X <> {} implies ex v st v in X & not ex x,y,z st (x in X or y in
  X) & v = [x,y,z]
proof
  assume X <> {};
  then consider Y such that
A1: Y in X and
A2: not ex Y1,Y2,Y3 st Y1 in Y2 & Y2 in Y3 & Y3 in Y & not Y1 misses X
  by XREGULAR:4;
  take v = Y;
  thus v in X by A1;
  given x,y,z such that
A3: x in X or y in X and
A4: v = [x,y,z];
  set Y1 = { x,y }, Y2 = { Y1,{x} }, Y3 = { Y2,z };
A5: x in Y1 & y in Y1 by TARSKI:def 2;
A6: Y3 in Y by A4,TARSKI:def 2;
  Y1 in Y2 & Y2 in Y3 by TARSKI:def 2;
  hence contradiction by A2,A5,A6,A3,XBOOLE_0:3;
