reserve X,Y,Z for set,
        x,y,z for object,
        A,B,C for Ordinal;
reserve U for Grothendieck;

theorem
  A in U implies Rank A in U
proof
  defpred P[Ordinal] means $1 in U implies Rank $1 in U;
A1:for A st for C st C in A holds P[C] holds P[A]
  proof
    let A such that
A2: for C st C in A holds P[C];
    assume
A3: A in U;
    deffunc BR(Ordinal)=bool Rank $1;
    consider br be Sequence such that
A4:   dom br = A & for O be Ordinal st O in A holds br.O =BR(O)
      from ORDINAL2:sch 2;
A5: rng br c= U
    proof
      let y be object;
      assume y in rng br;
      then consider B be object such that
A6:     B in dom br & br.B = y by FUNCT_1:def 3;
      reconsider B as Ordinal by A6;
      B in U by A6,A4,A3,Th13;
      then bool Rank B in U & br.B = bool Rank B by A2,A4,A6,Def1;
      hence thesis by A6;
    end;
A7: union rng br c= Rank A
    proof
      let z;
      assume z in union rng br;
      then consider y be set such that
A8:   z in y & y in rng br by TARSKI:def 4;
      consider B be object such that
A9:   B in dom br & br.B = y by A8,FUNCT_1:def 3;
      reconsider B as Ordinal by A9;
      br.B = bool Rank B by A4,A9;
      hence thesis by A8,A9,A4,Th3;
    end;
    Rank A c= union rng br
    proof
      let x such that
A10:  x in Rank A;
      reconsider X=x as set by TARSKI:1;
      consider B such that
A11:  B in A & X in bool Rank B by A10,Th3;
A12:  br.B in rng br by A11,A4,FUNCT_1:def 3;
      br.B = bool Rank B by A11,A4;
      hence thesis by A11,A12,TARSKI:def 4;
    end;
    then Rank A = union rng br by A7;
    hence thesis by A5,Def3,A4,A3;
  end;
  for O be Ordinal holds P[O] from ORDINAL1:sch 2(A1);
  hence thesis;
end;
