I'm not sure that I understand your point. I'm not arguing that increasing minimum wage is the only cause of inflation. I was saying that like inflation, increasing minimum wage affects everyone in the economy eventually.

Don't you agree that if you arbitrarily increase wages for a significant segment of the population, that segment will now want more of the goods and services they could not previously afford? Increased demand will drive up prices for those goods and services unless supply increases at the same time. If prices do go up, those with more bargaining power will seek higher returns for their labor. The net effect will be that almost everyone will see their income rise to some extent, but the supply of goods and services they are chasing is going to remain essentially the same. So almost all of us will be paying more for the same stuff. The only way that won't happen is if supply could magically increase at no extra cost – but that can't happen because we've raised the wages of all the people that supply those goods and services.

To reiterate: the scenario you lay out (taken to the conclusion that eventually all earners experience the same relative change) is not possible, because increasing minimum wage does not increase monetary supply. This is the key way it differs from true inflation. Ultimately, those extra wages – both higher minimum wages and, as you point out, higher wages of those with more bargaining power – have to come from somewhere. Ideally, that somewhere is the extremely high earners who have disproportionately gained income and wealth since the 1970s, thus reducing the relative gap in purchasing power between low and high-income earners.

I don't disagree that the effects you describe happen; my point is that, because of controls on monetary supply, they cannot happen equally to everyone. Nor can a minimum wage hike cause inflation on its own. (Of course, one could coordinate a change in monetary policy to effect inflation which negates the change in minimum wage in exactly the way you describe. The assumption is that the government is not so stupid as to enact two self-defeating polices.)

Of course, whether a minimum wage hike is an appropriate way to achieve the goal of reducing income and wealth inequality is debatable, but it does have a (non-transient) economic effect, because the value of money is anchored by other, more influential, aspects of monetary policy than the minimum wage (namely, the federal reserve rate).