Metamath Proof Explorer

Theorem leneg

Description: Negative of both sides of 'less than or equal to'. (Contributed by NM, 12-Sep-1999) (Proof shortened by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion leneg ⊢ A ∈ ℝ ∧ B ∈ ℝ → A ≤ B ↔ − B ≤ − A

Proof

Step Hyp Ref Expression
1 0re ⊢ 0 ∈ ℝ
2 lesub2 ⊢ A ∈ ℝ ∧ B ∈ ℝ ∧ 0 ∈ ℝ → A ≤ B ↔ 0 − B ≤ 0 − A
3 1 2 mp3an3 ⊢ A ∈ ℝ ∧ B ∈ ℝ → A ≤ B ↔ 0 − B ≤ 0 − A
4 df-neg ⊢ − B = 0 − B
5 df-neg ⊢ − A = 0 − A
6 4 5 breq12i ⊢ − B ≤ − A ↔ 0 − B ≤ 0 − A
7 3 6 bitr4di ⊢ A ∈ ℝ ∧ B ∈ ℝ → A ≤ B ↔ − B ≤ − A