Theorem nulsgts 28162

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 5261	. . 3 ⊢ ∅ ∈ V
3	2	a1i 11	. 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ∈ V)
4		elpwi 4564	. 2 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 ⊆ No)
5		0ss 4350	. . 3 ⊢ ∅ ⊆ No
6	5	a1i 11	. 2 ⊢ (𝐴 ∈ 𝒫 No → ∅ ⊆ No)
7		noel 4284	. . . 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 28154	1 ⊢ (𝐴 ∈ 𝒫 No → 𝐴 <<s ∅)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Vcvv 3451   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557   class class class wbr 5103  Nocsur 27997   <s clts 27998   <<s cslts 28143

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 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-slts 28144

This theorem is used by:  nulsgtsd  28164  0no  28195  1no  28196  bday0  28197  0lt1s  28198  bday0b  28199  bday1  28200  cutneg  28202  rightge0  28207  lltr  28248  made0  28249  elons2  28644  oncutlt  28650  oniso  28657  bdayons  28662  onaddscl  28663  onmulscl  28664  onsbnd  28667  n0cut  28720  n0bday  28738  n0fincut  28741  bdayn0p1  28755  zcuts  28793  twocut  28809  addhalfcut  28845