First, assume A⊆B. Now, suppose x∈A∪C. Then x∈A or x∈C
case 1: If x∈A, then x∈B because A⊆B. Thus, x∈B or x∈C.
case 2: If x∈C, then x∈B or x∈C.
So, either way, x∈B∪C. Therefore, A∪C⊆B∪C. ☐
Remember this definition:
A⊆B ⇔ ∀x(x∈A⇒x∈B)
Do you understand how the definition was used to prove the above theorem?