Proof that a Language is not context-free using the Pumping Lemma

$$
L={a^ib^jc^k ;| ;i, j, k in N ; and ; i <k<j}
$$

I need to show that this language is not context-free with the help of the Pumping Lemma.
My first intuition is, that there exist 5 different cases, i.e. the middle part, let’s call it vwx, consists of

  1. only a’s
  2. only b’s
  3. only c’s
  4. a’s and b’s
  5. b’s and c’s

and I need to find a pumping constant, which excludes the new word from the above defined language. However, I am having a hard time how show that formally and precisely. Any hints are highly appreciated!