Theorem nulssgt 27825

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 5304	. . 3 ⊢ ∅ ∈ V
3	2	a1i 11	. 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ∈ V)
4		elpwi 4604	. 2 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ⊆ No )
5		0ss 4394	. . 3 ⊢ ∅ ⊆ No
6	5	a1i 11	. 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ⊆ No )
7		noel 4330	. . . 4 ⊢ ¬ 𝑦 ∈ ∅
8	7	pm2.21i 119	. . 3 ⊢ (𝑦 ∈ ∅ → 𝑥 <s 𝑦)
9	8	3ad2ant3 1132	. 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ∅) → 𝑥 <s 𝑦)
10	1, 3, 4, 6, 9	ssltd 27818	1 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 <<s ∅)

Syntax hints:   → wi 4   ∈ wcel 2099  Vcvv 3462   ⊆ wss 3946  ∅c0 4322  𝒫 cpw 4597   class class class wbr 5145   No csur 27666   <s cslt 27667   <<s csslt 27807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-br 5146  df-opab 5208  df-xp 5680  df-sslt 27808

This theorem is referenced by:  0sno  27853  1sno  27854  bday0s  27855  0slt1s  27856  bday0b  27857  bday1s  27858  lltropt  27893  made0  27894  elons2  28249  n0scut  28303  n0sbday  28317