Theorem probinc

Description: A probability law is increasing with regard to event set inclusion. (Contributed by Thierry Arnoux, 10-Feb-2017)

	Ref	Expression
Assertion	probinc	⊢ ( ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃 ) ∧ 𝐴 ⊆ 𝐵 ) → ( 𝑃 ‘ 𝐴 ) ≤ ( 𝑃 ‘ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		simpl1	⊢ ( ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃 ) ∧ 𝐴 ⊆ 𝐵 ) → 𝑃 ∈ Prob )
2		domprobmeas	⊢ ( 𝑃 ∈ Prob → 𝑃 ∈ ( measures ‘ dom 𝑃 ) )
3	1 2	syl	⊢ ( ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃 ) ∧ 𝐴 ⊆ 𝐵 ) → 𝑃 ∈ ( measures ‘ dom 𝑃 ) )
4		simpl2	⊢ ( ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃 ) ∧ 𝐴 ⊆ 𝐵 ) → 𝐴 ∈ dom 𝑃 )
5		simpl3	⊢ ( ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃 ) ∧ 𝐴 ⊆ 𝐵 ) → 𝐵 ∈ dom 𝑃 )
6		simpr	⊢ ( ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃 ) ∧ 𝐴 ⊆ 𝐵 ) → 𝐴 ⊆ 𝐵 )
7	3 4 5 6	measssd	⊢ ( ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃 ) ∧ 𝐴 ⊆ 𝐵 ) → ( 𝑃 ‘ 𝐴 ) ≤ ( 𝑃 ‘ 𝐵 ) )