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	⊢ ( 𝜑 → 𝐴 ∈ Ons )
	onsltd.2	⊢ ( 𝜑 → 𝐵 ∈ Ons )
Assertion	onsled	⊢ ( 𝜑 → ( 𝐴 ≤s 𝐵 ↔ ( bday ‘ 𝐴 ) ⊆ ( bday ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		onsltd.1	⊢ ( 𝜑 → 𝐴 ∈ Ons )
2		onsltd.2	⊢ ( 𝜑 → 𝐵 ∈ Ons )
3		onsle	⊢ ( ( 𝐴 ∈ Ons ∧ 𝐵 ∈ Ons ) → ( 𝐴 ≤s 𝐵 ↔ ( bday ‘ 𝐴 ) ⊆ ( bday ‘ 𝐵 ) ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 ≤s 𝐵 ↔ ( bday ‘ 𝐴 ) ⊆ ( bday ‘ 𝐵 ) ) )