Theorem nulsgts 28006

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 5275	. . 3 ⊢ ∅ ∈ V
3	2	a1i 11	. 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ∈ V)
4		elpwi 4574	. 2 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ⊆ No )
5		0ss 4360	. . 3 ⊢ ∅ ⊆ No
6	5	a1i 11	. 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ⊆ No )
7		noel 4294	. . . 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 27998	1 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 <<s ∅)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  Vcvv 3458   ⊆ wss 3908  ∅c0 4289  𝒫 cpw 4567   class class class wbr 5114   No csur 27841   <s clts 27842   <<s cslts 27987

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 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

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 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5672  df-slts 27988

This theorem is used by:  nulsgtsd  28008  0no  28039  1no  28040  bday0  28041  0lt1s  28042  bday0b  28043  bday1  28044  cutneg  28046  rightge0  28051  lltr  28092  made0  28093  elons2  28488  oncutlt  28494  oniso  28501  bdayons  28506  onaddscl  28507  onmulscl  28508  onsbnd  28511  n0cut  28564  n0bday  28582  n0fincut  28585  bdayn0p1  28599  zcuts  28637  twocut  28653  addhalfcut  28689