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Metamath Proof Explorer

Theorem ltnegi

Description: Negative of both sides of 'less than'. Theorem I.23 of Apostol p. 20. (Contributed by NM, 21-Jan-1997)

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

Proof

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