Since each hand beats two other hands, you only need three possible outputs to beat any opposing hand.
(the missing condition is that every hand is beatable)
Each hand beats 2 distinct hands
With 3 distinct hands you can cover up to 6 distinct hands which is a superset of all possible hands
3 * 2 > 5
A --> D | E
B --> D | E
C --> D | E
D --> C | E
E --> D | C
A --> B | C
B --> C | D
C --> D | E
D --> E | A
E --> A | B