Theorem sdomnsym

Description: Strict dominance is asymmetric. Theorem 21(ii) of Suppes p. 97. (Contributed by NM, 8-Jun-1998)

	Ref	Expression
Assertion	sdomnsym	$$⊢A\prec B\to \neg B\prec A$$

Proof

Step	Hyp	Ref	Expression
1		sdomnen	$$⊢A\prec B\to \neg A\approx B$$
2		sdomdom	$$⊢A\prec B\to A\preccurlyeq B$$
3		sdomdom	$$⊢B\prec A\to B\preccurlyeq A$$
4		sbth	$$⊢\left(A\preccurlyeq B\wedge B\preccurlyeq A\right)\to A\approx B$$
5	2 3 4	syl2an	$$⊢\left(A\prec B\wedge B\prec A\right)\to A\approx B$$
6	1 5	mtand	$$⊢A\prec B\to \neg B\prec A$$