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Theorem nosgnn0 27718
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 8525 . . . 4 1o ≠ ∅
21nesymi 2996 . . 3 ¬ ∅ = 1o
3 nsuceq0 6469 . . . . 5 suc 1o ≠ ∅
4 necom 2992 . . . . . 6 (suc 1o ≠ ∅ ↔ ∅ ≠ suc 1o)
5 df-2o 8506 . . . . . . 7 2o = suc 1o
65neeq2i 3004 . . . . . 6 (∅ ≠ 2o ↔ ∅ ≠ suc 1o)
74, 6bitr4i 278 . . . . 5 (suc 1o ≠ ∅ ↔ ∅ ≠ 2o)
83, 7mpbi 230 . . . 4 ∅ ≠ 2o
98neii 2940 . . 3 ¬ ∅ = 2o
102, 9pm3.2ni 880 . 2 ¬ (∅ = 1o ∨ ∅ = 2o)
11 0ex 5313 . . 3 ∅ ∈ V
1211elpr 4655 . 2 (∅ ∈ {1o, 2o} ↔ (∅ = 1o ∨ ∅ = 2o))
1310, 12mtbir 323 1 ¬ ∅ ∈ {1o, 2o}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 847   = wceq 1537  wcel 2106  wne 2938  c0 4339  {cpr 4633  suc csuc 6388  1oc1o 8498  2oc2o 8499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-v 3480  df-dif 3966  df-un 3968  df-nul 4340  df-sn 4632  df-pr 4634  df-suc 6392  df-1o 8505  df-2o 8506
This theorem is referenced by:  nosgnn0i  27719  sltres  27722  noseponlem  27724  sltso  27736  nosepssdm  27746  nodenselem8  27751  nolt02olem  27754
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