That depends on how the income is spent. If people spend their extra income on leisure activities, luxury goods, etc then it'll absolutely mean that a high household income will be a good measure of richness. That's obvious.
However, if dual income households decide to spend their income on investing in a better house, what will happen is that it'll push house prices up - a finite supply of housing stock means there'll be competition for the best homes, so people will have to pay more for them. That demand side pressure will filter down the housing ladder and pull all the prices up. After thirty or forty years this will have had the effect on house prices that single income families just can't afford to buy. Further, it'll mean that dual income families have to retain two incomes, so parents wanting to stay at home when they have children simply won't be able to afford to.
This has already happened in the UK. Lots of pundits are predicting a crash because the first-time buyers can't afford to buy, meaning no one is buying the lower price houses, and that's going to pull the whole thing down.
Mind you, the pundits have been saying as much for about twenty years.