# How to Prove: Let m, n $∈ \mathbb{Z}$. If m $\le$ n $\le$ m then m = n.

Need help proving this. Any help would be greatly appreciated. Thank you!

Let m, n $\in$ $\mathbb{Z}$. If m $\le$ n $\le$ m then m = n.

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What have you tried so far for this problem? –  NasuSama Feb 18 at 3:23
What is your definition of $\le$? Do you know it is antisymmetric? Presumably not, because this is proving it. We have to know what you have available to prove this. –  Ross Millikan Feb 18 at 3:25
@NasuSama I know that by definition this means either m<n and n<m or m=n, but I'm having trouble proving m=n. –  user127835 Feb 18 at 3:26
@RossMillikan If m>n it means m-n$\in \mathbb{N}$. If m $\ge$ n it means m>n or m=n. –  user127835 Feb 18 at 3:27
Now, the statement seems reasonable. –  NasuSama Feb 18 at 3:28

You are given $(m \lt n) \vee (m=n)$, so assume $m \lt n$. Can you show $n \not \lt m$?

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That makes it easy to prove $n \not \lt m$, doesn't it? –  Ross Millikan Feb 18 at 3:41

$m\leq n\leq m$ implies:

$1.$ $m$ is less than or equal to $n$

$2.$ $m$ is greater than or equal to $n$

Taking the intersection of the above two conditions, we get $m=n$.

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By assuming $n\neq m$, you have either $n<m$ or $m<n$. Then assume one of it, say $n<m$, together with $n\geq m$. Can you get a contradiction?
Let's just start at the most complex side and do what comes naturally: \begin{align} & m \le n \;\land\; n \le m \\ \equiv & \qquad \text{"basic property of $\;\le\;$, twice -- to introduce $\;=\;$ as in our goal"} \\ & (m < n \lor m = n) \;\land\; (n < m \lor n = m) \\ \equiv & \qquad \text{"logic: factor out $\;m = n\;$ using distribution"} \\ & (m < n \;\land\; n < m) \lor m = n \\ \equiv & \qquad \text{"... -- to get rid of the left hand part"} \\ & \ldots & \end{align} Can you finish this calculation, given the rules you know about $\;<\;$?