Theorem volss

Description: The Lebesgue measure is monotone with respect to set inclusion. (Contributed by Thierry Arnoux, 17-Oct-2017)

	Ref	Expression
Assertion	volss	⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ 𝐴 ⊆ 𝐵 ) → ( vol ‘ 𝐴 ) ≤ ( vol ‘ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		simp3	⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ 𝐴 ⊆ 𝐵 ) → 𝐴 ⊆ 𝐵 )
2		mblss	⊢ ( 𝐵 ∈ dom vol → 𝐵 ⊆ ℝ )
3	2	3ad2ant2	⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ 𝐴 ⊆ 𝐵 ) → 𝐵 ⊆ ℝ )
4		ovolss	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ ) → ( vol* ‘ 𝐴 ) ≤ ( vol* ‘ 𝐵 ) )
5	1 3 4	syl2anc	⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ 𝐴 ⊆ 𝐵 ) → ( vol* ‘ 𝐴 ) ≤ ( vol* ‘ 𝐵 ) )
6		mblvol	⊢ ( 𝐴 ∈ dom vol → ( vol ‘ 𝐴 ) = ( vol* ‘ 𝐴 ) )
7	6	3ad2ant1	⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ 𝐴 ⊆ 𝐵 ) → ( vol ‘ 𝐴 ) = ( vol* ‘ 𝐴 ) )
8		mblvol	⊢ ( 𝐵 ∈ dom vol → ( vol ‘ 𝐵 ) = ( vol* ‘ 𝐵 ) )
9	8	3ad2ant2	⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ 𝐴 ⊆ 𝐵 ) → ( vol ‘ 𝐵 ) = ( vol* ‘ 𝐵 ) )
10	5 7 9	3brtr4d	⊢ ( ( 𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol ∧ 𝐴 ⊆ 𝐵 ) → ( vol ‘ 𝐴 ) ≤ ( vol ‘ 𝐵 ) )