Question about "earned income" when it comes to having both a Roth/Traditional 401k and a Roth/Traditional IRA...

I am just starting this whole investing journey so I am trying to educate myself on these rules.

A hypothetical: Let's say I only make 10,000 a year... and I put it ALL into my job's ROTH 401k as I earn the money (100% of each paycheck)...

However, I also want to open a Roth Ira and put the max 7000 allowed into it (I would do this from my personal savings account since I already put 100% of my paychecks this year into my Roth 401k with my employer, so I don't have any actual "earned income" leftover after the 401k to put into the IRA)...

IS THIS EVEN ALLOWED??

From what I understand I can only put "earned income" per year into a Roth Ira, but if I already put all of this year's earned income into my 401k then does that mean I have no earned income left that I am LEGALLY allowed to put into a Roth Ira since that would exceed the amount of money I earned this year (10,000 + 7,000 would be 17,000 total which exceeds my yearly earned income of 10,000)....?

Thanks in advance for anyone's advice/help!