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Theorem nosgnn0 27703
Description: is not a surreal sign. (Contributed by Scott Fenton, 16-Jun-2011.)
Assertion
Ref Expression
nosgnn0 ¬ ∅ ∈ {1o, 2o}

Proof of Theorem nosgnn0
StepHypRef Expression
1 1n0 8526 . . . 4 1o ≠ ∅
21nesymi 2998 . . 3 ¬ ∅ = 1o
3 nsuceq0 6467 . . . . 5 suc 1o ≠ ∅
4 necom 2994 . . . . . 6 (suc 1o ≠ ∅ ↔ ∅ ≠ suc 1o)
5 df-2o 8507 . . . . . . 7 2o = suc 1o
65neeq2i 3006 . . . . . 6 (∅ ≠ 2o ↔ ∅ ≠ suc 1o)
74, 6bitr4i 278 . . . . 5 (suc 1o ≠ ∅ ↔ ∅ ≠ 2o)
83, 7mpbi 230 . . . 4 ∅ ≠ 2o
98neii 2942 . . 3 ¬ ∅ = 2o
102, 9pm3.2ni 881 . 2 ¬ (∅ = 1o ∨ ∅ = 2o)
11 0ex 5307 . . 3 ∅ ∈ V
1211elpr 4650 . 2 (∅ ∈ {1o, 2o} ↔ (∅ = 1o ∨ ∅ = 2o))
1310, 12mtbir 323 1 ¬ ∅ ∈ {1o, 2o}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 848   = wceq 1540  wcel 2108  wne 2940  c0 4333  {cpr 4628  suc csuc 6386  1oc1o 8499  2oc2o 8500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-un 3956  df-nul 4334  df-sn 4627  df-pr 4629  df-suc 6390  df-1o 8506  df-2o 8507
This theorem is referenced by:  nosgnn0i  27704  sltres  27707  noseponlem  27709  sltso  27721  nosepssdm  27731  nodenselem8  27736  nolt02olem  27739
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