Theorem lenegcon1i

Description: Contraposition of negative in 'less than or equal to'. (Contributed by NM, 6-Apr-2005)

	Ref	Expression
Hypotheses	lt2.1	⊢ 𝐴 ∈ ℝ
	lt2.2	⊢ 𝐵 ∈ ℝ
Assertion	lenegcon1i	⊢ ( - 𝐴 ≤ 𝐵 ↔ - 𝐵 ≤ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		lt2.1	⊢ 𝐴 ∈ ℝ
2		lt2.2	⊢ 𝐵 ∈ ℝ
3		lenegcon1	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( - 𝐴 ≤ 𝐵 ↔ - 𝐵 ≤ 𝐴 ) )
4	1 2 3	mp2an	⊢ ( - 𝐴 ≤ 𝐵 ↔ - 𝐵 ≤ 𝐴 )