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

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.

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).