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	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \neg \left(A<B\wedge B<A\right)$$

Proof

Step	Hyp	Ref	Expression
1		ltso	$$⊢<\mathrm{Or}ℝ$$
2		so2nr	$$⊢\left(<\mathrm{Or}ℝ\wedge \left(A\in ℝ\wedge B\in ℝ\right)\right)\to \neg \left(A<B\wedge B<A\right)$$
3	1 2	mpan	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \neg \left(A<B\wedge B<A\right)$$