Theorem ltnegcon2i

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	ltnegcon2i	⊢ ( 𝐴 < - 𝐵 ↔ 𝐵 < - 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		lt2.1	⊢ 𝐴 ∈ ℝ
2		lt2.2	⊢ 𝐵 ∈ ℝ
3		ltnegcon2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < - 𝐵 ↔ 𝐵 < - 𝐴 ) )
4	1 2 3	mp2an	⊢ ( 𝐴 < - 𝐵 ↔ 𝐵 < - 𝐴 )