Theorem nulsgts 28049

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 5268	. . 3 ⊢ ∅ ∈ V
3	2	a1i 11	. 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ∈ V)
4		elpwi 4567	. 2 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ⊆ No )
5		0ss 4353	. . 3 ⊢ ∅ ⊆ No
6	5	a1i 11	. 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ⊆ No )
7		noel 4287	. . . 4 ⊢ ¬ 𝑦 ∈ ∅
8	7	pm2.21i 120	. . 3 ⊢ (𝑦 ∈ ∅ → 𝑥 <s 𝑦)
9	8	3ad2ant3 1153	. 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ∅) → 𝑥 <s 𝑦)
10	1, 3, 4, 6, 9	sltsd 28041	1 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 <<s ∅)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Vcvv 3453   ⊆ wss 3902  ∅c0 4282  𝒫 cpw 4560   class class class wbr 5107   No csur 27884   <s clts 27885   <<s cslts 28030

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-slts 28031

This theorem is used by:  nulsgtsd  28051  0no  28082  1no  28083  bday0  28084  0lt1s  28085  bday0b  28086  bday1  28087  cutneg  28089  rightge0  28094  lltr  28135  made0  28136  elons2  28531  oncutlt  28537  oniso  28544  bdayons  28549  onaddscl  28550  onmulscl  28551  onsbnd  28554  n0cut  28607  n0bday  28625  n0fincut  28628  bdayn0p1  28642  zcuts  28680  twocut  28696  addhalfcut  28732