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

theorem
   X in U & X,A are_equipotent implies A in U
proof
  defpred P[Ordinal] means for X st X,$1 are_equipotent & X in U holds $1 in U;
A1:for A be Ordinal st
    for C be Ordinal st C in A holds P[C] holds P[A]
  proof
    let A be Ordinal such that
A2:   for C be Ordinal st C in A holds P[C];
    let S be set such that
A3:   S,A are_equipotent & S in U;
    consider f be Function such that
A4:   f is one-to-one & dom f = S & rng f = A by A3,WELLORD2:def 4;
    rng f c= U
    proof
      let y such that
A5:     y in rng f;
      reconsider B=y as Ordinal by A5,A4;
A6:     B|` f is one-to-one by A4,FUNCT_1:58;
A7:     rng (B|`f) = B by A4,RELAT_1:89,A5,ORDINAL1:def 2;
      dom(B|`f) in U by A3,A4,CLASSES1:def 1,RELAT_1:186;
      hence thesis by A7,A6,WELLORD2:def 4,A5,A4,A2;
    end;
    hence A in U by A4,A3,Th2;
  end;
  for O be Ordinal holds P[O] from ORDINAL1:sch 2(A1);
  hence thesis;
end;
