I'm starting to study about the Ternary A-semirings and studying the paper of Daddi and Pawar entitle: Ideal Theory in Commutative Ternary A-semirings, which was published in Int. Math. Forum vol. 7 (2012) no. 42, pp. 2085-2091. In Theorem 3.3, they show that if $P$ is a maximal ideal with respect to $T$, then $P$ is prime. I understand that they will be prove by contrapositive proof of the definition of prime, which is likely to assume that $A\nsubseteq P, B\nsubseteq P$ and $C\nsubseteq P$. I do not understand why you assume that $P\subset A, P\subset B$ and $P\subset C$.
Please explain the reason for me to understand. Thank you very much.