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Metamath Proof Explorer

Theorem ltnegcon1i

Description: Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999)

Ref Expression
Hypotheses lt2.1 ⊢ A ∈ ℝ
lt2.2 ⊢ B ∈ ℝ
Assertion ltnegcon1i ⊢ − A < B ↔ − B < A

Proof

Step Hyp Ref Expression
1 lt2.1 ⊢ A ∈ ℝ
2 lt2.2 ⊢ B ∈ ℝ
3 ltnegcon1 ⊢ A ∈ ℝ ∧ B ∈ ℝ → − A < B ↔ − B < A
4 1 2 3 mp2an ⊢ − A < B ↔ − B < A