Metamath Proof Explorer
Description: Contraposition of negative in 'less than'. (Contributed by Mario
Carneiro, 27-May-2016)
|
|
Ref |
Expression |
|
Hypotheses |
leidd.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
|
|
ltnegd.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
|
|
ltnegcon2d.3 |
⊢ ( 𝜑 → 𝐴 < - 𝐵 ) |
|
Assertion |
ltnegcon2d |
⊢ ( 𝜑 → 𝐵 < - 𝐴 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
leidd.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
ltnegd.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
ltnegcon2d.3 |
⊢ ( 𝜑 → 𝐴 < - 𝐵 ) |
4 |
|
ltnegcon2 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < - 𝐵 ↔ 𝐵 < - 𝐴 ) ) |
5 |
1 2 4
|
syl2anc |
⊢ ( 𝜑 → ( 𝐴 < - 𝐵 ↔ 𝐵 < - 𝐴 ) ) |
6 |
3 5
|
mpbid |
⊢ ( 𝜑 → 𝐵 < - 𝐴 ) |