Theorem son2lpi

Description: A strict order relation has no 2-cycle loops. (Contributed by NM, 10-Feb-1996) (Revised by Mario Carneiro, 10-May-2013)

	Ref	Expression
Hypotheses	soi.1	$$⊢R\mathrm{Or}S$$
	soi.2	$$⊢R\subseteq S\times S$$
Assertion	son2lpi	$$⊢\neg \left(ARB\wedge BRA\right)$$

Proof

Step	Hyp	Ref	Expression
1		soi.1	$$⊢R\mathrm{Or}S$$
2		soi.2	$$⊢R\subseteq S\times S$$
3	1 2	soirri	$$⊢\neg ARA$$
4	1 2	sotri	$$⊢\left(ARB\wedge BRA\right)\to ARA$$
5	3 4	mto	$$⊢\neg \left(ARB\wedge BRA\right)$$