reserve m for Cardinal,
  A,B,C for Ordinal,
  x,y,z,X,Y,Z,W for set,
  f for Function;

theorem Th2:
  W is Tarski & x in W & y in W implies {x} in W & {x,y} in W
proof
  defpred C[object] means $1 = {};
  assume that
A1: W is Tarski and
A2: x in W and
A3: y in W;
A4: {x} c= bool x by ZFMISC_1:68;
  bool x in W by A1,A2;
  hence {x} in W by A1,A4,CLASSES1:def 1;
  then
A5: bool {x} in W by A1;
  deffunc g(object) = x;
  deffunc f(object) = y;
  consider f such that
A6: dom f = {{},{x}} &
for z being object st z in {{},{x}} holds (C[z] implies f.z =
  f(z)) & (not C[z] implies f.z = g(z)) from PARTFUN1:sch 1;
  {x,y} c= rng f
  proof
    let z be object;
A7: {} in {{},{x}} by TARSKI:def 2;
A8: {x} in {{},{x}} by TARSKI:def 2;
    assume z in {x,y};
    then z = x or z = y by TARSKI:def 2;
    then f.{} = z or f.{x} = z by A6,A7,A8;
    hence thesis by A6,A7,A8,FUNCT_1:def 3;
  end;
  then
A9: card {x,y} c= card {{},{x}} by A6,CARD_1:12;
A10: {x,y} c= W
  by A2,A3,TARSKI:def 2;
  bool {x} = {{},{x}} by ZFMISC_1:24;
  then card {{},{x}} in card W by A1,A5,Th1;
  then card {x,y} in card W by A9,ORDINAL1:12;
  hence thesis by A1,A10,CLASSES1:1;
end;
