Comparing covers

Question

Considering the Zariski topology, let $$V = \bigcup_{i \in I} U_i$$ be a maximal open cover of $V$ by basic open sets. Similarly, let $$V' = \bigcup_{j \in J} W_j$$ by the maximal open cover of $V'$ by basic open sets. If $V \subset V'$ then $$\bigcup_{i \in I} U_i \subset \bigcup_{j \in J} W_j.$$ Suppose the cover is not maximal, therefore $$V = \bigcup_{i \in I} U_i' \ \text{and} \ V' = \bigcup_{j \in J} W_j'.$$ How could we prove that if $V \subset V'$ then $$\bigcup_{i \in I} U_i' \subset \bigcup_{j \in J} W_j'?$$