Theorem nulssgt

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

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

Proof

Step	Hyp	Ref	Expression
1		elex	⊢ ( 𝐴 ∈ 𝒫 No → 𝐴 ∈ V )
2		0ex	⊢ ∅ ∈ V
3	1 2	jctir	⊢ ( 𝐴 ∈ 𝒫 No → ( 𝐴 ∈ V ∧ ∅ ∈ V ) )
4		elpwi	⊢ ( 𝐴 ∈ 𝒫 No → 𝐴 ⊆ No )
5		0ss	⊢ ∅ ⊆ No
6	5	a1i	⊢ ( 𝐴 ∈ 𝒫 No → ∅ ⊆ No )
7		ral0	⊢ ∀ 𝑦 ∈ ∅ 𝑥 <s 𝑦
8	7	rgenw	⊢ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ ∅ 𝑥 <s 𝑦
9	8	a1i	⊢ ( 𝐴 ∈ 𝒫 No → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ ∅ 𝑥 <s 𝑦 )
10	4 6 9	3jca	⊢ ( 𝐴 ∈ 𝒫 No → ( 𝐴 ⊆ No ∧ ∅ ⊆ No ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ ∅ 𝑥 <s 𝑦 ) )
11		brsslt	⊢ ( 𝐴 <<s ∅ ↔ ( ( 𝐴 ∈ V ∧ ∅ ∈ V ) ∧ ( 𝐴 ⊆ No ∧ ∅ ⊆ No ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ ∅ 𝑥 <s 𝑦 ) ) )
12	3 10 11	sylanbrc	⊢ ( 𝐴 ∈ 𝒫 No → 𝐴 <<s ∅ )