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;

theorem
  z in [:X,{y}:] implies z`1 in X & z`2 = y
proof
  assume
A1: z in [:X,{y}:];
  then z`2 in {y} by Th4;
  hence thesis by A1,Th4,TARSKI:def 1;
end;
