Theorem slttrd

Description: Surreal less than is transitive. (Contributed by Scott Fenton, 8-Dec-2021)

	Ref	Expression
Hypotheses	slttrd.1	|- ( ph -> A e. No )
	slttrd.2	|- ( ph -> B e. No )
	slttrd.3	|- ( ph -> C e. No )
	slttrd.4	|- ( ph -> A
	slttrd.5	|- ( ph -> B
Assertion	slttrd	|- ( ph -> A

Proof

Step	Hyp	Ref	Expression
1		slttrd.1	|- ( ph -> A e. No )
2		slttrd.2	|- ( ph -> B e. No )
3		slttrd.3	|- ( ph -> C e. No )
4		slttrd.4	|- ( ph -> A
5		slttrd.5	|- ( ph -> B
6		slttr	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( A A
7	1 2 3 6	syl3anc	|- ( ph -> ( ( A A
8	4 5 7	mp2and	|- ( ph -> A