What would a likely new minimum wage be? Over $6.00 certainly, perhaps $7.00. Why would we raise the minimum wage? To 'help the poor', of course.
But...if raising the minimum wage from $5 to $7 'helps the poor', wouldn't it help them even more to raise it to $10, or $20, or $40? Why stop at $7? Who wouldn't benefit from getting $20 an hour instead of $5?
Let's suppose a $20 minimum wage goes into effect tomorrow. What would happen? The most obvious effect is that companies with minimum wage employees will suddenly discover their payroll has quadrupled...but, unfortunately, their gross income has not magically quadrupled to go along with it. Thus, a $20 minimum wage would lead to a huge amount of layoffs, as companies are forced to find a equilibrium between productivity and expense. This seems inarguable...
Here's the point: how can anyone argue that raising the minimum wage to $7 will have no impact on unemployment, if it's blinding obvious that raising it to $20 would? Obviously, the layoff effect itself will be exponentially greater at $20 than at $7, but if it's clear that (higher minimum wage == layoffs) in large cases, wouldn't it be proportionally just as true in small cases as well?
There's undoubtedly still a policy argument in favor of a higher minimum wage even accepting a result of higher unemployment, but using an 'extreme case' thought experiment should make it clear that if large changes have certain consequences, then small changes will also have (proportionally smaller, yet undeniably present) consequences as well...