Theorem nulssgt 32034

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

Assertion

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

Proof of Theorem nulssgt

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3243	. . 3 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ∈ V)
2		0ex 4823	. . 3 ⊢ ∅ ∈ V
3	1, 2	jctir 560	. 2 ⊢ (𝐴 ∈ 𝒫 No → (𝐴 ∈ V ∧ ∅ ∈ V))
4		elpwi 4201	. . 3 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ⊆ No )
5		0ss 4005	. . . 4 ⊢ ∅ ⊆ No
6	5	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝒫 No → ∅ ⊆ No )
7		ral0 4109	. . . . 5 ⊢ ∀𝑦 ∈ ∅ 𝑥 <s 𝑦
8	7	rgenw 2953	. . . 4 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ∅ 𝑥 <s 𝑦
9	8	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝒫 No → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ∅ 𝑥 <s 𝑦)
10	4, 6, 9	3jca 1261	. 2 ⊢ (𝐴 ∈ 𝒫 No → (𝐴 ⊆ No ∧ ∅ ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ∅ 𝑥 <s 𝑦))
11		brsslt 32025	. 2 ⊢ (𝐴 <<s ∅ ↔ ((𝐴 ∈ V ∧ ∅ ∈ V) ∧ (𝐴 ⊆ No ∧ ∅ ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ∅ 𝑥 <s 𝑦)))
12	3, 10, 11	sylanbrc 699	1 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 <<s ∅)

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054   ∈ wcel 2030  ∀wral 2941  Vcvv 3231   ⊆ wss 3607  ∅c0 3948  𝒫 cpw 4191   class class class wbr 4685   No csur 31918   <s cslt 31919   <<s csslt 32021

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-sslt 32022

This theorem is referenced by: (None)