Proof of the negation of an existential Quantifier This was an exercise in my book and I was wondering if I got it correct.
Proof ¬(∃x)A(x)⟺(∀x)¬A(x)

* Let, U be any arbitrarily chosen Universe and A be any arbitrarily chosen open sentence. 

* (∃x)A(x), implies that there exists at least one object in U that satisfies A(x).

* ¬(∃x)A(x), implies that (∃x)A(x) is false in U

* (∃x)A(x), is false in U when the truth set of A(x) is empty in U.

* Since, the truth set is empty it means that that no object exists in U that satisfies A(x).

* Since, no object in U satisfies A(x) it would imply that all objects would satisfy ¬A(x) in U.

* (∀x)¬A(x), implies that all objects in U do not satisfy A(x).

* Therefore, ¬(∃x)A(x)⟺(∀x)¬A(x)

 A: Your phrasing of the proof is very terse, to the point where it can really only be understood by someone who already knows the reasoning you're trying to express. You should strive to write your proofs with at least some complete prose sentences that describe what you're doing, rather than just leave it to the reader to find a possible train of thoughts that connect the semi-symbolic sentence fragments on the paper. Remember that proofs are a matter of communicating ideas, not just writing down symbols that look profound.
(Also remember to use correct spelling and typography for those prose sentences. The first word in a sentence should be capitalized, for example).
That being said, it looks like what you're actually proving is that
$$ U\vDash \neg (\exists x) A(x) \quad\iff\quad U\vDash (\forall x)\neg A(x) $$
for every $U$. In order to get from there to
$$ \vDash \neg (\exists x) A(x) \leftrightarrow (\forall x)\neg A(x) $$
I would expect some explicit reasoning that involves the meaning of the $\leftrightarrow$ symbol, as well as what it means for a formula to be logically valid.
