Contrary to popular belief, increasing the minimum wage actually causes more wealth inequality for the following reasons.
1. Rich corporations can handle a minimum wage increase while small businesses cannot. Small business owners who cannot afford increased labor costs go out of business and become poor.
2. When small businesses leave the competitive market, major corporations do not have to worry about competition. Less competition means more profitability for the market leaders because prices aren't being driven down as they naturally would.
3. Upward mobility of the poor is decreased because increased labor costs are a barrier to entry for new businesses.
You can read justification for each reason at the following address:
http://www.brainofjt.com/2013/12/minimum-wage-poverty.html