Theorem ltnsymi

Description: 'Less than' is not symmetric. (Contributed by NM, 6-May-1999)

	Ref	Expression
Hypotheses	lt.1	⊢ 𝐴 ∈ ℝ
	lt.2	⊢ 𝐵 ∈ ℝ
Assertion	ltnsymi	⊢ ( 𝐴 < 𝐵 → ¬ 𝐵 < 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		lt.1	⊢ 𝐴 ∈ ℝ
2		lt.2	⊢ 𝐵 ∈ ℝ
3		ltnsym	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 𝐵 → ¬ 𝐵 < 𝐴 ) )
4	1 2 3	mp2an	⊢ ( 𝐴 < 𝐵 → ¬ 𝐵 < 𝐴 )