Theorem ltlestrd

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

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

Proof

Step	Hyp	Ref	Expression
1		ltstrd.1	|- ( ph -> A e. No )
2		ltstrd.2	|- ( ph -> B e. No )
3		ltstrd.3	|- ( ph -> C e. No )
4		ltlestrd.4	|- ( ph -> A
5		ltlestrd.5	|- ( ph -> B <_s C )
6		ltlestr	|- ( ( 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