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Theorem nosgnn0 33861
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 8318 . . . 4 1o ≠ ∅
21nesymi 3001 . . 3 ¬ ∅ = 1o
3 nsuceq0 6346 . . . . 5 suc 1o ≠ ∅
4 necom 2997 . . . . . 6 (suc 1o ≠ ∅ ↔ ∅ ≠ suc 1o)
5 df-2o 8298 . . . . . . 7 2o = suc 1o
65neeq2i 3009 . . . . . 6 (∅ ≠ 2o ↔ ∅ ≠ suc 1o)
74, 6bitr4i 277 . . . . 5 (suc 1o ≠ ∅ ↔ ∅ ≠ 2o)
83, 7mpbi 229 . . . 4 ∅ ≠ 2o
98neii 2945 . . 3 ¬ ∅ = 2o
102, 9pm3.2ni 878 . 2 ¬ (∅ = 1o ∨ ∅ = 2o)
11 0ex 5231 . . 3 ∅ ∈ V
1211elpr 4584 . 2 (∅ ∈ {1o, 2o} ↔ (∅ = 1o ∨ ∅ = 2o))
1310, 12mtbir 323 1 ¬ ∅ ∈ {1o, 2o}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 844   = wceq 1539  wcel 2106  wne 2943  c0 4256  {cpr 4563  suc csuc 6268  1oc1o 8290  2oc2o 8291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-nul 5230
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-sn 4562  df-pr 4564  df-suc 6272  df-1o 8297  df-2o 8298
This theorem is referenced by:  nosgnn0i  33862  sltres  33865  noseponlem  33867  sltso  33879  nosepssdm  33889  nodenselem8  33894  nolt02olem  33897
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