Theorem measle0

Description: If the measure of a given set is bounded by zero, it is zero. (Contributed by Thierry Arnoux, 20-Oct-2017)

	Ref	Expression
Assertion	measle0	⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ 𝑆 ∧ ( 𝑀 ‘ 𝐴 ) ≤ 0 ) → ( 𝑀 ‘ 𝐴 ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		simp3	⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ 𝑆 ∧ ( 𝑀 ‘ 𝐴 ) ≤ 0 ) → ( 𝑀 ‘ 𝐴 ) ≤ 0 )
2		measvxrge0	⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ 𝑆 ) → ( 𝑀 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) )
3		elxrge0	⊢ ( ( 𝑀 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) ↔ ( ( 𝑀 ‘ 𝐴 ) ∈ ℝ* ∧ 0 ≤ ( 𝑀 ‘ 𝐴 ) ) )
4	2 3	sylib	⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ 𝑆 ) → ( ( 𝑀 ‘ 𝐴 ) ∈ ℝ* ∧ 0 ≤ ( 𝑀 ‘ 𝐴 ) ) )
5	4	3adant3	⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ 𝑆 ∧ ( 𝑀 ‘ 𝐴 ) ≤ 0 ) → ( ( 𝑀 ‘ 𝐴 ) ∈ ℝ* ∧ 0 ≤ ( 𝑀 ‘ 𝐴 ) ) )
6	5	simprd	⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ 𝑆 ∧ ( 𝑀 ‘ 𝐴 ) ≤ 0 ) → 0 ≤ ( 𝑀 ‘ 𝐴 ) )
7	5	simpld	⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ 𝑆 ∧ ( 𝑀 ‘ 𝐴 ) ≤ 0 ) → ( 𝑀 ‘ 𝐴 ) ∈ ℝ* )
8		0xr	⊢ 0 ∈ ℝ*
9		xrletri3	⊢ ( ( ( 𝑀 ‘ 𝐴 ) ∈ ℝ* ∧ 0 ∈ ℝ* ) → ( ( 𝑀 ‘ 𝐴 ) = 0 ↔ ( ( 𝑀 ‘ 𝐴 ) ≤ 0 ∧ 0 ≤ ( 𝑀 ‘ 𝐴 ) ) ) )
10	7 8 9	sylancl	⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ 𝑆 ∧ ( 𝑀 ‘ 𝐴 ) ≤ 0 ) → ( ( 𝑀 ‘ 𝐴 ) = 0 ↔ ( ( 𝑀 ‘ 𝐴 ) ≤ 0 ∧ 0 ≤ ( 𝑀 ‘ 𝐴 ) ) ) )
11	1 6 10	mpbir2and	⊢ ( ( 𝑀 ∈ ( measures ‘ 𝑆 ) ∧ 𝐴 ∈ 𝑆 ∧ ( 𝑀 ‘ 𝐴 ) ≤ 0 ) → ( 𝑀 ‘ 𝐴 ) = 0 )