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Theorem nosgnn0 33494
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 8143 . . . 4 1o ≠ ∅
21nesymi 2991 . . 3 ¬ ∅ = 1o
3 nsuceq0 6246 . . . . 5 suc 1o ≠ ∅
4 necom 2987 . . . . . 6 (suc 1o ≠ ∅ ↔ ∅ ≠ suc 1o)
5 df-2o 8125 . . . . . . 7 2o = suc 1o
65neeq2i 2999 . . . . . 6 (∅ ≠ 2o ↔ ∅ ≠ suc 1o)
74, 6bitr4i 281 . . . . 5 (suc 1o ≠ ∅ ↔ ∅ ≠ 2o)
83, 7mpbi 233 . . . 4 ∅ ≠ 2o
98neii 2936 . . 3 ¬ ∅ = 2o
102, 9pm3.2ni 880 . 2 ¬ (∅ = 1o ∨ ∅ = 2o)
11 0ex 5172 . . 3 ∅ ∈ V
1211elpr 4536 . 2 (∅ ∈ {1o, 2o} ↔ (∅ = 1o ∨ ∅ = 2o))
1310, 12mtbir 326 1 ¬ ∅ ∈ {1o, 2o}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 846   = wceq 1542  wcel 2113  wne 2934  c0 4209  {cpr 4515  suc csuc 6168  1oc1o 8117  2oc2o 8118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-ext 2710  ax-nul 5171
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-ne 2935  df-v 3399  df-dif 3844  df-un 3846  df-nul 4210  df-sn 4514  df-pr 4516  df-suc 6172  df-1o 8124  df-2o 8125
This theorem is referenced by:  nosgnn0i  33495  sltres  33498  noseponlem  33500  sltso  33512  nosepssdm  33522  nodenselem8  33527  nolt02olem  33530
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