aka union bound, sigma sub-additivity of Measures

It is either a theorem, or an axiom, depending on the formulation of Probability theory

Video

Let $A_1, A_2, ..., A_k$ be events (not necessarily independent); then,

$P(A_1 \cup A_2 \cup ... \cup A_k \leq P(A_1) + P(A_2) + ... + P(A_k)$

https://www.wikiwand.com/en/Boole's_inequality