Theorem nulssgt 27767

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		id 22	. 2 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ∈ 𝒫 No )
2		0ex 5282	. . 3 ⊢ ∅ ∈ V
3	2	a1i 11	. 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ∈ V)
4		elpwi 4587	. 2 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ⊆ No )
5		0ss 4380	. . 3 ⊢ ∅ ⊆ No
6	5	a1i 11	. 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ⊆ No )
7		noel 4318	. . . 4 ⊢ ¬ 𝑦 ∈ ∅
8	7	pm2.21i 119	. . 3 ⊢ (𝑦 ∈ ∅ → 𝑥 <s 𝑦)
9	8	3ad2ant3 1135	. 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ∅) → 𝑥 <s 𝑦)
10	1, 3, 4, 6, 9	ssltd 27760	1 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 <<s ∅)

Syntax hints:   → wi 4   ∈ wcel 2109  Vcvv 3464   ⊆ wss 3931  ∅c0 4313  𝒫 cpw 4580   class class class wbr 5124   No csur 27608   <s cslt 27609   <<s csslt 27749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-xp 5665  df-sslt 27750

This theorem is referenced by:  0sno  27795  1sno  27796  bday0s  27797  0slt1s  27798  bday0b  27799  bday1s  27800  cutneg  27802  lltropt  27841  made0  27842  elons2  28216  onscutlt  28222  onsiso  28226  bdayon  28230  onaddscl  28231  onmulscl  28232  n0scut  28283  n0sbday  28301  n0sfincut  28303  bdayn0p1  28315  zscut  28352  1p1e2s  28359  twocut  28366  addhalfcut  28391