Probability Measure is Monotone/Proof 1

Proof
From Set Difference Union Second Set is Union:


 * $A \cup B = \paren {B \setminus A} \cup A$

From Set Difference Intersection with Second Set is Empty Set:


 * $\paren {B \setminus A} \cap A = \O$

From the Addition Law of Probability:
 * $\map \Pr {A \cup B} = \map \Pr {B \setminus A} + \map \Pr A$

From Union with Superset is Superset:


 * $A \subseteq B \implies A \cup B = B$

Thus:
 * $\map \Pr B = \map \Pr {B \setminus A} + \map \Pr A$

By definition of probability measure:
 * $\map \Pr {B \setminus A} \ge 0$

from which it follows that:
 * $\map \Pr B \ge \map \Pr A$