Help with a logical derivation of set theoretical statement

julypraise

1. The problem statement, all variables and given/known data
This is actually from the proof of Dedekind's cut in Rudin's Principles of Mathematical analysis on the page 19. It says when [tex]\alpha\in\mathbb{R}[/tex] ([tex]\alpha[/tex] is a cut) is fixed, [tex]\beta[/tex] is the set of all [tex]p[/tex] with the following property:

There exists [tex]r>0[/tex] such that [tex]-p-r\notin\alpha[/tex].

From the given above, I need to derive that

if [tex]q\in\alpha[/tex], then [tex]q\notin\beta[/tex].

But I cannot reach this statement as my explanation for this is in the below.

2. Relevant equations

3. The attempt at a solution
The draft I have done so far is that, as defining [tex]\beta[/tex] such that

[tex]\beta=\left\{p|\exists r\in\mathbb{Q} (r>0 \wedge -p-r \notin \alpha)\right\}[/tex],

I derived

[tex]p \notin \beta \leftrightarrow
\forall r \in \mathbb{Q} (r>0 \to -p-r\in\alpha)[/tex].

And I'm stucked here. From the last statement, I cannot derive the conclusion I was meant to derive. If anyone gives me help, I will give thanks.

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