Theorem onsled

Description: Less-than or equal is the same as non-strict birthday comparison over surreal ordinals. Deduction version. (Contributed by Scott Fenton, 25-Feb-2026)

	Ref	Expression
Hypotheses	onsltd.1	|- ( ph -> A e. On_s )
	onsltd.2	|- ( ph -> B e. On_s )
Assertion	onsled	|- ( ph -> ( A <_s B <-> ( bday ` A ) C_ ( bday ` B ) ) )

Proof

Step	Hyp	Ref	Expression
1		onsltd.1	|- ( ph -> A e. On_s )
2		onsltd.2	|- ( ph -> B e. On_s )
3		onsle	|- ( ( A e. On_s /\ B e. On_s ) -> ( A <_s B <-> ( bday ` A ) C_ ( bday ` B ) ) )
4	1 2 3	syl2anc	|- ( ph -> ( A <_s B <-> ( bday ` A ) C_ ( bday ` B ) ) )