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Theorem nosgnn0 27029
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 8438 . . . 4 1o ≠ ∅
21nesymi 2998 . . 3 ¬ ∅ = 1o
3 nsuceq0 6404 . . . . 5 suc 1o ≠ ∅
4 necom 2994 . . . . . 6 (suc 1o ≠ ∅ ↔ ∅ ≠ suc 1o)
5 df-2o 8417 . . . . . . 7 2o = suc 1o
65neeq2i 3006 . . . . . 6 (∅ ≠ 2o ↔ ∅ ≠ suc 1o)
74, 6bitr4i 278 . . . . 5 (suc 1o ≠ ∅ ↔ ∅ ≠ 2o)
83, 7mpbi 229 . . . 4 ∅ ≠ 2o
98neii 2942 . . 3 ¬ ∅ = 2o
102, 9pm3.2ni 880 . 2 ¬ (∅ = 1o ∨ ∅ = 2o)
11 0ex 5268 . . 3 ∅ ∈ V
1211elpr 4613 . 2 (∅ ∈ {1o, 2o} ↔ (∅ = 1o ∨ ∅ = 2o))
1310, 12mtbir 323 1 ¬ ∅ ∈ {1o, 2o}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 846   = wceq 1542  wcel 2107  wne 2940  c0 4286  {cpr 4592  suc csuc 6323  1oc1o 8409  2oc2o 8410
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5267
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-v 3449  df-dif 3917  df-un 3919  df-nul 4287  df-sn 4591  df-pr 4593  df-suc 6327  df-1o 8416  df-2o 8417
This theorem is referenced by:  nosgnn0i  27030  sltres  27033  noseponlem  27035  sltso  27047  nosepssdm  27057  nodenselem8  27062  nolt02olem  27065
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