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Metamath Proof Explorer

Theorem le0neg2

Description: Comparison of a number and its negative to zero. (Contributed by NM, 24-Aug-1999)

Ref Expression
Assertion le0neg2 ⊢ A ∈ ℝ → 0 ≤ A ↔ − A ≤ 0

Proof

Step Hyp Ref Expression
1 0re ⊢ 0 ∈ ℝ
2 leneg ⊢ 0 ∈ ℝ ∧ A ∈ ℝ → 0 ≤ A ↔ − A ≤ − 0
3 1 2 mpan ⊢ A ∈ ℝ → 0 ≤ A ↔ − A ≤ − 0
4 neg0 ⊢ − 0 = 0
5 4 breq2i ⊢ − A ≤ − 0 ↔ − A ≤ 0
6 3 5 bitrdi ⊢ A ∈ ℝ → 0 ≤ A ↔ − A ≤ 0