Theorem ltnsym2

Description: 'Less than' is antisymmetric and irreflexive. (Contributed by NM, 13-Aug-2005) (Proof shortened by Andrew Salmon, 19-Nov-2011)

	Ref	Expression
Assertion	ltnsym2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ¬ ( 𝐴 < 𝐵 ∧ 𝐵 < 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		ltso	⊢ < Or ℝ
2		so2nr	⊢ ( ( < Or ℝ ∧ ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ) → ¬ ( 𝐴 < 𝐵 ∧ 𝐵 < 𝐴 ) )
3	1 2	mpan	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ¬ ( 𝐴 < 𝐵 ∧ 𝐵 < 𝐴 ) )