
theorem Th15:
  for X being non empty set
  st for a being set st a in X holds a is cardinal number
  ex A being Cardinal st A in X & A = meet X
proof
  let X be non empty set;
  assume A1: for a being set st a in X holds a is cardinal number;
  defpred P[Ordinal] means $1 in X & $1 is cardinal number;
  A2: ex A being Ordinal st P[A]
  proof
    consider A being object such that
      A3: A in X by XBOOLE_0:def 1;
    reconsider A as Ordinal by A1, A3;
    take A;
    thus thesis by A1, A3;
  end;
  consider A being Ordinal such that
    A4: P[A] & for B being Ordinal st P[B] holds A c= B
    from ORDINAL1:sch 1(A2);
  reconsider A as Cardinal by A4;
  take A;
  thus A in X by A4;
  A5: meet X c= A by A4, SETFAM_1:3;
  now
    let x be object;
    assume A6: x in A;
    now
      let Y be set;
      assume A7: Y in X;
      then reconsider B = Y as Ordinal by A1;
      B in X & B is cardinal number by A1, A7;
      then A c= B by A4;
      hence x in Y by A6;
    end;
    hence x in meet X by SETFAM_1:def 1;
  end;
  then A c= meet X by TARSKI:def 3;
  hence A = meet X by A5, XBOOLE_0:def 10;
end;
