Theorem ltnsymi

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

	Ref	Expression
Hypotheses	lt.1	$$⊢A\in ℝ$$
	lt.2	$$⊢B\in ℝ$$
Assertion	ltnsymi	$$⊢A<B\to \neg B<A$$

Proof

Step	Hyp	Ref	Expression
1		lt.1	$$⊢A\in ℝ$$
2		lt.2	$$⊢B\in ℝ$$
3		ltnsym	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A<B\to \neg B<A\right)$$
4	1 2 3	mp2an	$$⊢A<B\to \neg B<A$$