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Metamath Proof Explorer

Theorem lenegi

Description: Negative of both sides of 'less than or equal to'. (Contributed by NM, 1-Aug-1999)

Ref Expression
Hypotheses lt2.1 ⊢ A ∈ ℝ
lt2.2 ⊢ B ∈ ℝ
Assertion lenegi ⊢ A ≤ B ↔ − B ≤ − A

Proof

Step Hyp Ref Expression
1 lt2.1 ⊢ A ∈ ℝ
2 lt2.2 ⊢ B ∈ ℝ
3 leneg ⊢ A ∈ ℝ ∧ B ∈ ℝ → A ≤ B ↔ − B ≤ − A
4 1 2 3 mp2an ⊢ A ≤ B ↔ − B ≤ − A