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Theorem nosgnn0 27578
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 8502 . . . 4 1o ≠ ∅
21nesymi 2993 . . 3 ¬ ∅ = 1o
3 nsuceq0 6446 . . . . 5 suc 1o ≠ ∅
4 necom 2989 . . . . . 6 (suc 1o ≠ ∅ ↔ ∅ ≠ suc 1o)
5 df-2o 8481 . . . . . . 7 2o = suc 1o
65neeq2i 3001 . . . . . 6 (∅ ≠ 2o ↔ ∅ ≠ suc 1o)
74, 6bitr4i 278 . . . . 5 (suc 1o ≠ ∅ ↔ ∅ ≠ 2o)
83, 7mpbi 229 . . . 4 ∅ ≠ 2o
98neii 2937 . . 3 ¬ ∅ = 2o
102, 9pm3.2ni 879 . 2 ¬ (∅ = 1o ∨ ∅ = 2o)
11 0ex 5301 . . 3 ∅ ∈ V
1211elpr 4647 . 2 (∅ ∈ {1o, 2o} ↔ (∅ = 1o ∨ ∅ = 2o))
1310, 12mtbir 323 1 ¬ ∅ ∈ {1o, 2o}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 846   = wceq 1534  wcel 2099  wne 2935  c0 4318  {cpr 4626  suc csuc 6365  1oc1o 8473  2oc2o 8474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-nul 5300
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-v 3471  df-dif 3947  df-un 3949  df-nul 4319  df-sn 4625  df-pr 4627  df-suc 6369  df-1o 8480  df-2o 8481
This theorem is referenced by:  nosgnn0i  27579  sltres  27582  noseponlem  27584  sltso  27596  nosepssdm  27606  nodenselem8  27611  nolt02olem  27614
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