Proving a language is not Semidecidable
Using contradiction suppose $L=\{\left&lt; M_1,M_2 \right&gt;|L(M_1)\subset L(M_2)\}$ is semi-decidable. So there exists Turing machine $T$ which for input $\left&lt;M_1,M_2\right&gt;$ 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 $\left&lt;w,M\right&gt;$ if $M$  doesn't accept $w$. To do so, you have to make another Turing machine $M'$ using $w$ that you are sure $w\notin L(M')$. Thus you can give $\left&lt; M',M\right&gt;$ as input to $T$ and look for its output. If it accept it means that $L(M')\subset L(M)$ which means $w\notin L(M)$.