Theorem sdomtr

Description: Strict dominance is transitive. Theorem 21(iii) of Suppes p. 97. (Contributed by NM, 9-Jun-1998)

	Ref	Expression
Assertion	sdomtr	$$⊢\left(A\prec B\wedge B\prec C\right)\to A\prec C$$

Proof

Step	Hyp	Ref	Expression
1		sdomdom	$$⊢A\prec B\to A\preccurlyeq B$$
2		domsdomtr	$$⊢\left(A\preccurlyeq B\wedge B\prec C\right)\to A\prec C$$
3	1 2	sylan	$$⊢\left(A\prec B\wedge B\prec C\right)\to A\prec C$$