You can be alone, and not feel lonely - and actually feel lonely when you're not alone.
Your SIL has reduced it to a simple equation and, as you probably understand yourself - it's a bit more complex.
As Ali23 above points out, there are different forms of loneliness.
I guess your SIL thought he was being helpful with his advice to 'get-out-there-and-meet-people'? But it's easier said than done.
The only time I have ever felt lonely was in a room - no, a house full of people at a charity coffee morning. I wasn't part of the 'circle' and didn't know any one of those who all appeared to know each other. I wandered around with a smile on my face, coffee in hand, clearly alone - but no-one spoke to me, and I didn't want to row myself into any of the little cliques of people milling around who seemed to be earnestly in conversation with each other, so I thanked the host for the invite and left.
If only it were as easy as your SIL appears to think it is!
I hope you enjoyed the day with your family nevertheless.