Proving a language is not Semidecidable

Using contradiction suppose $L={left< M_1,M_2 right>|L(M_1)subset L(M_2)}$ is semi-decidable. So there exists Turing machine $T$ which for input $left<M_1,M_2right>$ if $L(M_1)subset L(M_2)$ will halt and accept.

We should use this Turing machine $T$ to make another Turing machine $T'$ which halt and accept on input $left<w,Mright>$ if $M$ doesn't accept $w$. To do so, you have to make another Turing machine $M'$ using $w$ that you are sure $wnotin L(M')$. Thus you can give $left< M',Mright>$ as input to $T$ and look for its output. If it accept it means that $L(M')subset L(M)$ which means $wnotin L(M)$.