reserve W for Universe,
  H for ZF-formula,
  x,y,z,X for set,
  k for Variable,
  f for Function of VAR,W,
  u,v for Element of W;

theorem Th3:
  omega in W implies W |= the_axiom_of_infinity
proof
  assume omega in W;
  then reconsider u = omega as Element of W;
  now
    take u;
    thus u <> {};
    let v;
    assume
A1: v in u;
    then reconsider A = v as Ordinal;
    succ A in omega by A1,ORDINAL1:28;
    then succ A c= u by ORDINAL1:def 2;
    then reconsider w = succ A as Element of W by CLASSES1:def 1;
    take w;
    A in succ A by ORDINAL1:6;
    then v c= w & v <> w by ORDINAL1:def 2;
    hence v c< w & w in u by A1,ORDINAL1:28;
  end;
  hence thesis by ZFMODEL1:6;
end;
