Pizza Positioning Puzzle

Seating Arrangements

Me ( I'm Johnny) and my brothers Vinny, Mickey, Smitty, Gino, and Pauly always take the same seat when we have our weekly pizza party at our favorite Pizza place - and we always sit at a round table.

The following diagram indicates our seating.

Seat Positions

The following is known:

1. Smitty's seat is separated from Johnny's seat by exactly one of our other brothers.

2. Mickey's seat number differs from Vinny's and Smitty's, positively or negatively, by 2 and 5 in one order or the other.

3. Vinny's number is 1 larger than Gino's number.

4. Johnny's number is either 1 larger or 1 smaller than Pauly's number.

Which brother sits in which seat?

Mickey's seat number differs from either Vinny or Smitty by 5, so that means Mickey must be in seat 1 or 6.
Vinny differs from Mickey by 2 or 5, so he must be in seats 1, 3, 4, or 6, but he can't be in seat 1 because his seat number is larger than Gino's.
Vinny also can't be in seat 6. If he were in seat 6, Smitty would be in seat 3 (2 apart from Mickey) and Gino would be in seat 5 (one less than Vinny). But this leaves nowhere for Johnny and Pauly to sit one apart from each other.
Vinny also can't be in seat 4. If he were, Gino would be in seat 3 and with Mickey in 6 and Smitty in 1, and again there is no way for Johnny and Pauly to sit one apart from each other.
So Vinny must be in seat 3. Then Gino is in seat 2. Mickey is in 1, and Smitty is in 6. This leaves Johnny and Pauly in 4 and 5. Johnny must be in 4 based on clue 1.

So:
1 Mickey
2 Gino
3 Vinny
4 Johnny
5 Pauly
6 Smitty