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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 )