reserve A, B, X, Y for set;

theorem Th7:
  <:pr2(X,Y),pr1(X,Y):> is one-to-one
proof
  set f = <:pr2(X,Y),pr1(X,Y):>;
  let x, y be object such that
A1: x in dom f and
A2: y in dom f and
A3: f.x = f.y;
A4: dom f = [:X,Y:] by Th4;
  then consider x1, x2 being object such that
A5: x1 in X & x2 in Y and
A6: x = [x1,x2] by A1,ZFMISC_1:def 2;
  consider y1, y2 being object such that
A7: y1 in X & y2 in Y and
A8: y = [y1,y2] by A4,A2,ZFMISC_1:def 2;
A9: f.(y1,y2) = [y2,y1] by A7,Lm1;
A10: f.(x1,x2) = [x2,x1] by A5,Lm1;
  then x1 = y1 by A3,A6,A8,A9,XTUPLE_0:1;
  hence thesis by A3,A6,A8,A10,A9,XTUPLE_0:1;
end;
