Theorem ltnsym

Description: 'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002)

	Ref	Expression
Assertion	ltnsym	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 𝐵 → ¬ 𝐵 < 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		axlttri	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 𝐵 ↔ ¬ ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ) )
2		pm2.46	⊢ ( ¬ ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) → ¬ 𝐵 < 𝐴 )
3	1 2	biimtrdi	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 𝐵 → ¬ 𝐵 < 𝐴 ) )