Jesus Christ people. Raising minimum wage will do NOTHING but raise rent prices and every other price. Think about this. If I owned an apartment building and had a bunch of poor people living there and suddenly minimum wage gets a 200% increase?! Guess what MY ASS is doing right away? Increasing rent costs every fucking chance I get, because I know those low life losers were making it somehow before, therefore I also know that they now have extra money, and why should they get to keep it? Fuck that and fuck them. That shit is mine. Why? Because I know they have it to give, this is capitalism, and I am a business man. Therefore, I will take their money because that's just what you do here.
I raise my rent a little, and my neighbors will raise their rent prices to match, and their neighbors will do the same. In 2 or 3 years rent will go up by 20% and it will keep climbing and climbing until those minimum wagers are right back where they started; poor and struggling just like before.
The minimum wage has relatively little effect on inflation because not that many people make it and its effects are generally concentrated in a few economic sectors.
As for your idea that they would be no better off, no. Your scenario only makes sense if every building owner in the city did the same thing as you suggest, which is 1.) unlikely due to this being capitalism and all and 2.) it's probably illegal collusion.
