Using Rice's theorem to prove undecidability of $E_{TM}$

Question

I saw this proof and I wondered if I could prove $E_{TM}$ with Rice's theorem similar to the one described in the answer. Can you do the same thing by letting $M$ to only accept empty strings? (the $M$ that is described in the answer of the proof)

So

1. If $x$ is empty, accept

2. If $x$ is any non-empty string, reject.

Answer 1

You have to be careful when you talk about accepting empty strings. Using the language from the linked question, what you are suggesting would yield $L(M_x)$ = $\{\epsilon\}$ instead of $L(M_x)$ = $\emptyset$.

You can use the same construction for machine $M_x$, but just change the the non-trivial property to "whether this machine accepts on no inputs". It's clearly a non-trivial property since it's not empty (there are machines with empty language), and it's not the set of all Turing Machines (since not all Turing Machines have empty language).