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Theorem nulssgt 27681
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.
StepHypRef Expression
1 id 22 . 2 (𝐴 ∈ 𝒫 No 𝐴 ∈ 𝒫 No )
2 0ex 5300 . . 3 ∅ ∈ V
32a1i 11 . 2 (𝐴 ∈ 𝒫 No → ∅ ∈ V)
4 elpwi 4604 . 2 (𝐴 ∈ 𝒫 No 𝐴 No )
5 0ss 4391 . . 3 ∅ ⊆ No
65a1i 11 . 2 (𝐴 ∈ 𝒫 No → ∅ ⊆ No )
7 noel 4325 . . . 4 ¬ 𝑦 ∈ ∅
87pm2.21i 119 . . 3 (𝑦 ∈ ∅ → 𝑥 <s 𝑦)
983ad2ant3 1132 . 2 ((𝐴 ∈ 𝒫 No 𝑥𝐴𝑦 ∈ ∅) → 𝑥 <s 𝑦)
101, 3, 4, 6, 9ssltd 27674 1 (𝐴 ∈ 𝒫 No 𝐴 <<s ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  Vcvv 3468  wss 3943  c0 4317  𝒫 cpw 4597   class class class wbr 5141   No csur 27523   <s cslt 27524   <<s csslt 27663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-sslt 27664
This theorem is referenced by:  0sno  27709  1sno  27710  bday0s  27711  0slt1s  27712  bday0b  27713  bday1s  27714  lltropt  27749  made0  27750  elons2  28101  n0scut  28153  n0sbday  28167
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