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Metamath Proof Explorer

Theorem lenegcon1i

Description: Contraposition of negative in 'less than or equal to'. (Contributed by NM, 6-Apr-2005)

Ref Expression
Hypotheses lt2.1 ⊢ A ∈ ℝ
lt2.2 ⊢ B ∈ ℝ
Assertion lenegcon1i ⊢ − A ≤ B ↔ − B ≤ A

Proof

Step Hyp Ref Expression
1 lt2.1 ⊢ A ∈ ℝ
2 lt2.2 ⊢ B ∈ ℝ
3 lenegcon1 ⊢ A ∈ ℝ ∧ B ∈ ℝ → − A ≤ B ↔ − B ≤ A
4 1 2 3 mp2an ⊢ − A ≤ B ↔ − B ≤ A