Luca Trevisan wrote, " The oracle $C$ tells us that we cannot have a relativizing proof that derives the $𝑁𝑃 ⊈ 𝑃/𝑝𝑜𝑙𝑦$ conclusion from the $𝑃 ≠𝑁𝑃$ assumption, so a theorem such as Karp-Lipton, which derives (via relativizing arguments) the $𝑁𝑃 ⊈ 𝑃/𝑝𝑜𝑙𝑦$ conclusion from a stronger assumption, is about as much as we can hope to prove using relativizing arguments. – " This is replied to the question that "The fact there exists $C$ s.t. $P^C ≠ NP^C$, but $NP^C \subset P/poly^C$ ... what does this give us?".
He wrote that Karp-Lipton theorem is proved by relativizing arguments. How do we know that this theorem is derived from relativizing arguments? I remember the proof and it has nothing about relativizing (I don't want to sketch the proof here since it takes some time and you can find it in Arora and Barak's textbook).