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 Th17:
  X is closed_wrt_A1-A7 & a in X implies {{[0-element_of(V),x], [
  1-element_of(V),x]} : x in a} in X
proof
  assume that
A1: X is closed_wrt_A1-A7 and
A2: a in X;
  set f={[0-element_of(V),1-element_of(V)],[1-element_of(V),1-element_of(V)]};
A3: 1-element_of(V) in X by A1,Lm13;
  then
A4: [1-element_of(V),1-element_of(V)] in X by A1,Th6;
  set F={{[1-element_of(V),x]}:x in a};
A5: X is closed_wrt_A4 by A1;
A6: F in X
  proof
    reconsider s={1-element_of(V)} as Element of V;
A7: F={{[y,x]} where y,x: y in s & x in a}
    proof
      thus F c= {{[y,x]} where y,x: y in s & x in a}
      proof
        reconsider y=1-element_of(V) as Element of V;
        let p be object;
        assume p in F;
        then
A8:     ex x st p={[1-element_of(V),x]} & x in a;
        y in s by TARSKI:def 1;
        hence thesis by A8;
      end;
      let p be object;
      assume p in {{[y,x]} where y,x: y in s & x in a};
      then consider y,x such that
A9:   p={[y,x]} & y in s and
A10:  x in a;
      p={[1-element_of(V),x]} by A9,TARSKI:def 1;
      hence thesis by A10;
    end;
    1-element_of(V) in X by A1,Lm13;
    then {1-element_of(V)} in X by A1,Th2;
    hence thesis by A2,A5,A7;
  end;
  then reconsider F9=F as Element of V;
  set f9={f};
  set Z={{[0-element_of(V),x],[1-element_of(V),x]}: x in a};
A11: Z={x(#)y: x in f9 & y in F9}
  proof
    thus Z c= {x(#)y: x in f9 & y in F9}
    proof
      reconsider x9=f as Element of V;
      let p be object;
A12:  x9 in f9 by TARSKI:def 1;
      assume p in Z;
      then consider x such that
A13:  p={[0-element_of(V),x],[1-element_of(V),x]} & x in a;
      reconsider y={[1-element_of(V),x]} as Element of V;
      y in F9 & p=x9(#)y by A13,Lm14;
      hence thesis by A12;
    end;
    let p be object;
    assume p in {x(#)y: x in f9 & y in F9};
    then consider x,y such that
A14: p=x(#)y and
A15: x in f9 and
A16: y in F9;
    consider x9 such that
A17: y={[1-element_of(V),x9]} and
A18: x9 in a by A16;
    x={[0-element_of(V),1-element_of(V)],[1-element_of(V), 1-element_of(V
    )]} by A15,TARSKI:def 1;
    then p={[0-element_of(V),x9],[1-element_of(V),x9]} by A14,A17,Lm14;
    hence thesis by A18;
  end;
  0-element_of(V) in X by A1,Lm13;
  then [0-element_of(V),1-element_of(V)] in X by A1,A3,Th6;
  then f in X by A1,A4,Th6;
  then
A19: {f} in X by A1,Th2;
  X is closed_wrt_A7 by A1;
  hence thesis by A19,A6,A11;
end;
