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 Th2:
  X is closed_wrt_A1-A7 implies (o in X iff {o} in X) & (A in X
  implies union A in X)
proof
  assume
A1: X is closed_wrt_A1-A7;
  then
A2: X is closed_wrt_A2;
A3: now
    assume o in X;
    then {o,o} in X by A2;
    hence {o} in X by ENUMSET1:29;
  end;
A4: X is closed_wrt_A3 by A1;
  now
    assume {o} in X;
    then union {o} in X by A4;
    hence o in X by ZFMISC_1:25;
  end;
  hence o in X iff {o} in X by A3;
  assume A in X;
  hence thesis by A4;
end;
