Translating "there is no number between two consecutive numbers"

Question

There is no number number strictly between two consecutive number.

Is my translation $$∀x ∈ Z | ∀y ∈ Z | ¬∃z ∈ Z \;(z<y ∧ z>x ∧ y=x+1)$$ correct?

Is $$∃x | ¬∃n ∈ Z \;(n<x<n+1) $$ also correct? I assume that they are just two different ways to say the same thing.

Answer 1

There is no number between two consecutive numbers.

Is my translation $$∀x ∈ Z | ∀y ∈ Z | ¬∃z ∈ Z \;(z<y ∧ z>x ∧ y=x+1)$$ correct?

Yes. Equivalent translations of "there is no integer strictly between two consecutive integers" include $$∀(x,y,z){∈}\mathbb Z^3 \;(x<z<y \implies y\ne x+1)$$ and $$∀(x,y,z){∈}\mathbb Z^3 \;(y=x+1\;\land\;z>x\implies z\not<y).$$

Is $$∃x ∈ Z | ¬∃n ∈ Z \;(n<x<n+1) $$ also correct? I assume that they are just two different ways to say the same thing.

Is there a typo? This suggestion says that there is some integer such that no consecutive pair of integers envelope it.