reserve V for Universe,
  a,b,x,y,z,x9,y9 for Element of V,
  X for Subclass of V,
  o,p,q,r,s,t,u,a1,a2,a3,A,B,C,D for set,
  K,L,M for Ordinal,
  n for Element of omega,
  fs for finite Subset of omega,
  e,g,h for Function,
  E for non empty set,
  f for Function of VAR,E,
  k,k1 for Element of NAT,
  v1,v2,v3 for Element of VAR,
  H,H9 for ZF-formula;

theorem Th6:
  X is closed_wrt_A1-A7 & o in X & p in X implies {o,p} in X & [o,p ] in X
proof
  assume that
A1: X is closed_wrt_A1-A7 and
A2: o in X and
A3: p in X;
  reconsider a=o,b=p as Element of V by A2,A3;
A4: {o} in X by A1,A2,Th2;
A5: X is closed_wrt_A2 by A1;
  hence {o,p} in X by A2,A3;
  {a,b} in X by A2,A3,A5;
  hence thesis by A5,A4;
end;
