Theorem leltstrd

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 )
	leltstrd.4	|- ( ph -> A <_s B )
	leltstrd.5	|- ( ph -> B
Assertion	leltstrd	|- ( 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		leltstrd.4	|- ( ph -> A <_s B )
5		leltstrd.5	|- ( ph -> B
6		leltstr	|- ( ( A e. No /\ B e. No /\ C e. No ) -> ( ( A <_s B /\ B A
7	1 2 3 6	syl3anc	|- ( ph -> ( ( A <_s B /\ B A
8	4 5 7	mp2and	|- ( ph -> A