Description: A singleton has 0 Lebesgue measure. (Contributed by Glauco Siliprandi, 11-Dec-2019)
Ref | Expression | ||
---|---|---|---|
Assertion | volsn | ⊢ ( 𝐴 ∈ ℝ → ( vol ‘ { 𝐴 } ) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snmbl | ⊢ ( 𝐴 ∈ ℝ → { 𝐴 } ∈ dom vol ) | |
2 | mblvol | ⊢ ( { 𝐴 } ∈ dom vol → ( vol ‘ { 𝐴 } ) = ( vol* ‘ { 𝐴 } ) ) | |
3 | 1 2 | syl | ⊢ ( 𝐴 ∈ ℝ → ( vol ‘ { 𝐴 } ) = ( vol* ‘ { 𝐴 } ) ) |
4 | ovolsn | ⊢ ( 𝐴 ∈ ℝ → ( vol* ‘ { 𝐴 } ) = 0 ) | |
5 | 3 4 | eqtrd | ⊢ ( 𝐴 ∈ ℝ → ( vol ‘ { 𝐴 } ) = 0 ) |