Theorem nulsgts

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

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

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( 𝐴 ∈ 𝒫 No → 𝐴 ∈ 𝒫 No )
2		0ex	⊢ ∅ ∈ V
3	2	a1i	⊢ ( 𝐴 ∈ 𝒫 No → ∅ ∈ V )
4		elpwi	⊢ ( 𝐴 ∈ 𝒫 No → 𝐴 ⊆ No )
5		0ss	⊢ ∅ ⊆ No
6	5	a1i	⊢ ( 𝐴 ∈ 𝒫 No → ∅ ⊆ No )
7		noel	⊢ ¬ 𝑦 ∈ ∅
8	7	pm2.21i	⊢ ( 𝑦 ∈ ∅ → 𝑥 <s 𝑦 )
9	8	3ad2ant3	⊢ ( ( 𝐴 ∈ 𝒫 No ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ∅ ) → 𝑥 <s 𝑦 )
10	1 3 4 6 9	sltsd	⊢ ( 𝐴 ∈ 𝒫 No → 𝐴 <<s ∅ )