Private inheritance from vector, exposing equality operator

I need a vector which stores elements in sorted order (I need constant-time random access too, so not std::set). I am trying to implement that by privately inheriting std::vector.

template <typename T, typename Tcomp = less<T> >
class SortedVector : private vector<T> {

    Tcomp _comparator;


public:
    using vector<T>::operator[];
    using vector<T>::operator==;
    using vector<T>::size;

I have a couple of questions:

1. Is it a bad idea to inherit std::vector privately?
2. I get the following error: SortedVector.h:26:30: error: no members matching ‘std::vector<std::basic_string<char> >::operator==’ in ‘class std::vector<std::basic_string<char> >’

Any advice?