# Help with a logical derivation of set theoretical statement

1. ### julypraise

110
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 $$\alpha\in\mathbb{R}$$ ($$\alpha$$ is a cut) is fixed, $$\beta$$ is the set of all $$p$$ with the following property:

There exists $$r>0$$ such that $$-p-r\notin\alpha$$.​

From the given above, I need to derive that

if $$q\in\alpha$$, then $$q\notin\beta$$.​

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 $$\beta$$ such that

$$\beta=\left\{p|\exists r\in\mathbb{Q} (r>0 \wedge -p-r \notin \alpha)\right\}$$,​

I derived

$$p \notin \beta \leftrightarrow \forall r \in \mathbb{Q} (r>0 \to -p-r\in\alpha)$$.​

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.

Last edited: Feb 12, 2011