Theorem onsltd

Description: Less-than is the same as 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	onsltd	|- ( ph -> ( A ( bday ` A ) e. ( bday ` B ) ) )

Proof

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