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 )