MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nulssgt Structured version   Visualization version   GIF version

Theorem nulssgt 27751
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 5311 . . 3 ∅ ∈ V
32a1i 11 . 2 (𝐴 ∈ 𝒫 No → ∅ ∈ V)
4 elpwi 4613 . 2 (𝐴 ∈ 𝒫 No 𝐴 No )
5 0ss 4400 . . 3 ∅ ⊆ No
65a1i 11 . 2 (𝐴 ∈ 𝒫 No → ∅ ⊆ No )
7 noel 4334 . . . 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 27744 1 (𝐴 ∈ 𝒫 No 𝐴 <<s ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  Vcvv 3473  wss 3949  c0 4326  𝒫 cpw 4606   class class class wbr 5152   No csur 27593   <s cslt 27594   <<s csslt 27733
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 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-sslt 27734
This theorem is referenced by:  0sno  27779  1sno  27780  bday0s  27781  0slt1s  27782  bday0b  27783  bday1s  27784  lltropt  27819  made0  27820  elons2  28171  n0scut  28223  n0sbday  28237
  Copyright terms: Public domain W3C validator