Theorem nulsslt

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

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

Proof

Step	Hyp	Ref	Expression
1		elex	⊢ ( 𝐴 ∈ 𝒫 No → 𝐴 ∈ V )
2		0ex	⊢ ∅ ∈ V
3	1 2	jctil	⊢ ( 𝐴 ∈ 𝒫 No → ( ∅ ∈ V ∧ 𝐴 ∈ V ) )
4		0ss	⊢ ∅ ⊆ No
5	4	a1i	⊢ ( 𝐴 ∈ 𝒫 No → ∅ ⊆ No )
6		elpwi	⊢ ( 𝐴 ∈ 𝒫 No → 𝐴 ⊆ No )
7		ral0	⊢ ∀ 𝑥 ∈ ∅ ∀ 𝑦 ∈ 𝐴 𝑥 <s 𝑦
8	7	a1i	⊢ ( 𝐴 ∈ 𝒫 No → ∀ 𝑥 ∈ ∅ ∀ 𝑦 ∈ 𝐴 𝑥 <s 𝑦 )
9	5 6 8	3jca	⊢ ( 𝐴 ∈ 𝒫 No → ( ∅ ⊆ No ∧ 𝐴 ⊆ No ∧ ∀ 𝑥 ∈ ∅ ∀ 𝑦 ∈ 𝐴 𝑥 <s 𝑦 ) )
10		brsslt	⊢ ( ∅ <<s 𝐴 ↔ ( ( ∅ ∈ V ∧ 𝐴 ∈ V ) ∧ ( ∅ ⊆ No ∧ 𝐴 ⊆ No ∧ ∀ 𝑥 ∈ ∅ ∀ 𝑦 ∈ 𝐴 𝑥 <s 𝑦 ) ) )
11	3 9 10	sylanbrc	⊢ ( 𝐴 ∈ 𝒫 No → ∅ <<s 𝐴 )