Let $A = \{a, b, c, d\}$, $B = \{e, f, g\}$. Suppose the number $X = abcd + efg$ is a prime. Then, we see that the numbers $2, 4, 6$ must belong to the same set $A$ or $B$, because, otherwise, both $abcd$ and $efg$ become even, and therefore $X$ must be even, and since $X \ne 2$, $X$ cannot be a prime. Also, the numbers $3$ and $6$ must belong to the same set $A$ or $B$, because, otherwise $X$ becomes a multiple of $3$ and since $X \ne 3$, $X$ cannot be a prime.
Consequently, we see that $A = \{2, 3, 4, 6\}$ and $B = \{1, 5, 7\}$, and we conclude that the only prime number of the form $abcd + efg$ is $2 \cdot 3 \cdot 4 \cdot 6 + 1 \cdot 5 \cdot 7 = 179$.