Metamath Proof Explorer
Description: Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999) (Proof shortened by Andrew Salmon, 19-Nov-2011)
|
|
Ref |
Expression |
|
Hypotheses |
lt2.1 |
⊢ 𝐴 ∈ ℝ |
|
|
lt2.2 |
⊢ 𝐵 ∈ ℝ |
|
Assertion |
lesub0i |
⊢ ( ( 0 ≤ 𝐴 ∧ 𝐵 ≤ ( 𝐵 − 𝐴 ) ) ↔ 𝐴 = 0 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
lt2.1 |
⊢ 𝐴 ∈ ℝ |
2 |
|
lt2.2 |
⊢ 𝐵 ∈ ℝ |
3 |
|
lesub0 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 0 ≤ 𝐴 ∧ 𝐵 ≤ ( 𝐵 − 𝐴 ) ) ↔ 𝐴 = 0 ) ) |
4 |
1 2 3
|
mp2an |
⊢ ( ( 0 ≤ 𝐴 ∧ 𝐵 ≤ ( 𝐵 − 𝐴 ) ) ↔ 𝐴 = 0 ) |