Yes, but he's not getting in from work until gone midnight! If he was working 9-5 I would agree with you windy, but why should the whole house be up at 5.30am for no reason?
I think people lack understanding about late shifts/night work on here. If you work 9-5, you don't get home at 5.30pm and go to bed. You get home, have dinner, have a bath, do some housework, watch TV and have an "evening". Whereas if you work until midnight, you're expected to come home and go straight to bed - where's the chance to unwind?
So I'm guessing he probably doesn't get to sleep until 1am at the earliest, which means if he's expected to wake up with OP at 5.30am, he'll be running on 4.5 hours sleep a night, four days a week. Even when he gets up when OP goes to work, he's still running on less sleep than average. No wonder he's irritable in the mornings!
If they had several children or both had to be out of the door at 7am, then yes, he should help, but one 2yo who doesn't need to be out of the door until she starts nursery - no, I don't think it's necessary for both parents to wake up at stupid o'clock.