Theorem nulslts

Description: The empty set is less-than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021)

	Ref	Expression
Assertion	nulslts	⊢ ( 𝐴 ∈ 𝒫 No → ∅ <<s 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		0ex	⊢ ∅ ∈ V
2	1	a1i	⊢ ( 𝐴 ∈ 𝒫 No → ∅ ∈ V )
3		elex	⊢ ( 𝐴 ∈ 𝒫 No → 𝐴 ∈ V )
4		0ss	⊢ ∅ ⊆ No
5	4	a1i	⊢ ( 𝐴 ∈ 𝒫 No → ∅ ⊆ No )
6		elpwi	⊢ ( 𝐴 ∈ 𝒫 No → 𝐴 ⊆ No )
7		noel	⊢ ¬ 𝑥 ∈ ∅
8	7	pm2.21i	⊢ ( 𝑥 ∈ ∅ → 𝑥 <s 𝑦 )
9	8	3ad2ant2	⊢ ( ( 𝐴 ∈ 𝒫 No ∧ 𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴 ) → 𝑥 <s 𝑦 )
10	2 3 5 6 9	sltsd	⊢ ( 𝐴 ∈ 𝒫 No → ∅ <<s 𝐴 )