I have a question about FA CUP I want forum answer

[wbvs]

suppose stoke lose out in the Quarter Finals of the FA Cup , and at
the end of the season all the teams except stoke finishes 1 to 7
in the EPL standings, remember the FA Cup give the winner qualification
to the europa league and if the team that wins already qualify
it goes to the next team . In this scenario , would stoke qualify
for the europa league ?