Theorem nulsgts 27927

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

Assertion

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

Proof of Theorem nulsgts

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

Step	Hyp	Ref	Expression
1		id 23	. 2 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ∈ 𝒫 No )
2		0ex 5262	. . 3 ⊢ ∅ ∈ V
3	2	a1i 11	. 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ∈ V)
4		elpwi 4565	. 2 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ⊆ No )
5		0ss 4357	. . 3 ⊢ ∅ ⊆ No
6	5	a1i 11	. 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ⊆ No )
7		noel 4293	. . . 4 ⊢ ¬ 𝑦 ∈ ∅
8	7	pm2.21i 120	. . 3 ⊢ (𝑦 ∈ ∅ → 𝑥 <s 𝑦)
9	8	3ad2ant3 1151	. 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ∅) → 𝑥 <s 𝑦)
10	1, 3, 4, 6, 9	sltsd 27919	1 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 <<s ∅)

Syntax hints:   → wi 4   ∈ wcel 2145  Vcvv 3457   ⊆ wss 3907  ∅c0 4288  𝒫 cpw 4558   class class class wbr 5105   No csur 27762   <s clts 27763   <<s cslts 27908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-slts 27909

This theorem is referenced by:  nulsgtsd  27929  0no  27960  1no  27961  bday0  27962  0lt1s  27963  bday0b  27964  bday1  27965  cutneg  27967  rightge0  27972  lltr  28013  made0  28014  elons2  28409  oncutlt  28415  oniso  28422  bdayons  28427  onaddscl  28428  onmulscl  28429  onsbnd  28432  n0cut  28485  n0bday  28503  n0fincut  28506  bdayn0p1  28520  zcuts  28558  twocut  28574  addhalfcut  28610