Theorem sletri3d

Description: Trichotomy law for surreal less-than or equal. Deduction version. (Contributed by Scott Fenton, 25-Feb-2026)

	Ref	Expression
Hypotheses	sled.1	|- ( ph -> A e. No )
	sled.2	|- ( ph -> B e. No )
Assertion	sletri3d	|- ( ph -> ( A = B <-> ( A <_s B /\ B <_s A ) ) )

Proof

Step	Hyp	Ref	Expression
1		sled.1	|- ( ph -> A e. No )
2		sled.2	|- ( ph -> B e. No )
3		sletri3	|- ( ( A e. No /\ B e. No ) -> ( A = B <-> ( A <_s B /\ B <_s A ) ) )
4	1 2 3	syl2anc	|- ( ph -> ( A = B <-> ( A <_s B /\ B <_s A ) ) )