Theorem volge0

Description: The volume of a set is always nonnegative. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Assertion	volge0	⊢ ( 𝐴 ∈ dom vol → 0 ≤ ( vol ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		mblss	⊢ ( 𝐴 ∈ dom vol → 𝐴 ⊆ ℝ )
2		ovolge0	⊢ ( 𝐴 ⊆ ℝ → 0 ≤ ( vol* ‘ 𝐴 ) )
3	1 2	syl	⊢ ( 𝐴 ∈ dom vol → 0 ≤ ( vol* ‘ 𝐴 ) )
4		mblvol	⊢ ( 𝐴 ∈ dom vol → ( vol ‘ 𝐴 ) = ( vol* ‘ 𝐴 ) )
5	3 4	breqtrrd	⊢ ( 𝐴 ∈ dom vol → 0 ≤ ( vol ‘ 𝐴 ) )