monotonicity of probability
Monotonicity says that a bigger event cannot be less likely than a smaller one contained inside it. If event A is a subset of event B — meaning every outcome that makes A happen also makes B happen — then P(A) <= P(B). It captures the plain idea that an easier-to-satisfy condition has at least as good a chance: 'rolling a 6' implies 'rolling an even number', so rolling a 6 cannot be more likely than rolling an even number.
It follows cleanly from the axioms. If A sits inside B, you can split B into two disjoint pieces: the part that is A, and the part of B left over outside A, namely B ∩ A^c. By additivity, P(B) = P(A) + P(B ∩ A^c). Since that leftover probability P(B ∩ A^c) is non-negative (the non-negativity axiom), adding it can only keep P(B) at or above P(A). The same one-line argument also shows the difference rule: P(B and not A) = P(B) - P(A) whenever A is contained in B.
A pair of immediate corollaries makes the whole probability scale finite and well behaved. Because every event A is a subset of the whole sample space Omega, monotonicity gives P(A) <= P(Omega) = 1; combined with non-negativity, this proves the bedrock fact that every probability lies between 0 and 1. Practically, monotonicity is also a sanity check: if a calculation ever hands you a sub-event with higher probability than the event containing it, you have made an error somewhere.
On a die, A = {6} is a subset of B = {2,4,6}. Monotonicity predicts P(A) <= P(B), and indeed 1/6 <= 3/6. The extra piece is B ∩ A^c = {2,4}, contributing exactly P(B) - P(A) = 3/6 - 1/6 = 2/6.
Containment forces an ordering of probabilities — and pins every probability into the range 0 to 1.
Monotonicity needs genuine containment (A inside B); two events that merely overlap have no guaranteed ordering of their probabilities.