is it possible to using std::pair as key of std::set without any duplicates each element?

I'm trying to store struct in std::set using two integers in struct as key. I know that is possible to using std::pair as key of std::set.

struct TYPE{
  pair<int, int> nums;
  ... // some other things

  TYPE(){}
  TYPE(int first, int second) { nums = make_pair(first, second); }
  bool operator<(const TYPE &rhs) const{
    return nums < rhs.nums;
  }
};
set<TYPE> nums_set;
nums_set.insert(TYPE(1, 2));
nums_set.insert(TYPE(1, 4));
nums_set.insert(TYPE(5, 2));
// size of set : 3, (1,2)(1,4)(5,2)
auto it = nums_set.find(TYPE(1, 2)); // find return (1,2)

However, I want searchable and no duplicates std::set, regardless of the order of the elements in std::pair. like this :

nums_set.insert(TYPE(1, 2));
nums_set.insert(TYPE(1, 4)); // failed. duplicate 1
nums_set.insert(TYPE(4, 2)); // failed. duplicate 2
nums_set.insert(TYPE(4, 1)); // failed. duplicate 1
nums_set.insert(TYPE(3, 4));
// size of set : 2, (1,2)(3,4)
auto it = nums_set.find(TYPE(2, 7));  // find return (1,2).

easiest solution so far is use std::vector instead and do duplicate check before inserting into the vector like this:

auto fn = [](const TYPE& e, const TYPE&& t){
    return e.nums.first == t.nums.first ||
            e.nums.first == t.nums.second ||
            e.nums.second == t.nums.first ||
            e.nums.second == t.nums.second;
};
vector<TYPE> nums_vec;
nums_vec.push_back(TYPE(1, 2));
if(nums_vec.end() == find_if(nums_vec.begin(), nums_vec.end(), 
  bind(fn, placeholders::_1, TYPE(1,4))))
{  nums_vec.push_back(TYPE(1,4)); }

However, I feel it's not a good idea to do this for every insert to avoid duplication. So, my question is, is it possible to implement these features with std::set too?