proving $E_{TM}$ is undecidable using the halting language
Suppose that there is a Turing machine $T$ that decides $E_{TM}$.

Given a turning machine $M$ and an input $w$ you can construct a new Turing machine $M^*$ that decides whether $(M,w) \in H_{halt}$. $M^*$ operates as follows:


It first constructs a new Turing machine $M'$ that ignores its input, simulates $M$ on input $w$ and, once the simulation is complete, accepts.
It simulates $T$ with input $M'$ to decide whether $M' \in E_{TM}$.
If $M' \in E_{TM}$ then $M'$ does not accept any input, which implies that $M$ cannot halt on input $w$. In this case $M^*$ rejects.
If $M' \in E_{TM}$ then $M'$ accepts at least one input (and hence all inputs), meaning that $M$ must halt on input $w$. In this case $M^*$ accepts.