That's a fair statement to make, and one that I have heard before but the definition of what constitutes the working class is arguable thus most people predefine it per scope of conversation, and not in a global sense.
For example, in the UK, commonly doctors and lawyers have not been considered part of the working class, regardless of if their investments outside of their profession.
Because a doctor can work independently, and hire other workers like nurses, and attendants, and own all of their tools of trade outright, they are part of the petit bourgeois.
Now you say that petit bourgeois is part of the working class, but many would disagree as Karl Marx and Friedrich Engels popularized a more fine grained definition of social classes as follows:
1. the working class - factory workers, peasants, and people who earn only by their labour
2. lumpenproletariat (commonly considered to include vagabonds, criminals or the 'unthinking poor')
3. professional middle-classes (engineers or tradesmen who do not typically hire employees)
3. petit-bourgeoisie - Professionals and small scale managers who hire workers but work alongside them
4. haute bourgeoisie - landed aristocracy & other capitalists who live from investment alone
Now if you are not marxists, you may drop some of the classes or simply call them something else. The lumpenproletariat are sometimes called the "underclass", and in advanced capitalist economies the line between professional middle-classes and petit-bourgeois blurs due to capital infusions from the top. For example, startup owners should rightly be professional middle class, but are upgraded due to VC infusion.
There are other ways to define the classes as well that don't depend on monetary wealth----for example the ladder from: working class to bourgeoisies to gentry to elite/aristocrats, so the terms themselves are fairly overloaded.
Analysing class warfare is a bit of my hobby. I blame my sassy liberal arts professors. :)
