Theorem nulsltsd

Description: The empty set is less-than any set of surreals. Deduction version. (Contributed by Scott Fenton, 27-Feb-2026)

	Ref	Expression
Hypotheses	nulsltsd.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	nulsltsd.2	⊢ ( 𝜑 → 𝐴 ⊆ No )
Assertion	nulsltsd	⊢ ( 𝜑 → ∅ <<s 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		nulsltsd.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2		nulsltsd.2	⊢ ( 𝜑 → 𝐴 ⊆ No )
3	1 2	elpwd	⊢ ( 𝜑 → 𝐴 ∈ 𝒫 No )
4		nulslts	⊢ ( 𝐴 ∈ 𝒫 No → ∅ <<s 𝐴 )
5	3 4	syl	⊢ ( 𝜑 → ∅ <<s 𝐴 )