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Theorem nosgnn0 33167
 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 8121 . . . 4 1o ≠ ∅
21nesymi 3075 . . 3 ¬ ∅ = 1o
3 nsuceq0 6273 . . . . 5 suc 1o ≠ ∅
4 necom 3071 . . . . . 6 (suc 1o ≠ ∅ ↔ ∅ ≠ suc 1o)
5 df-2o 8105 . . . . . . 7 2o = suc 1o
65neeq2i 3083 . . . . . 6 (∅ ≠ 2o ↔ ∅ ≠ suc 1o)
74, 6bitr4i 280 . . . . 5 (suc 1o ≠ ∅ ↔ ∅ ≠ 2o)
83, 7mpbi 232 . . . 4 ∅ ≠ 2o
98neii 3020 . . 3 ¬ ∅ = 2o
102, 9pm3.2ni 877 . 2 ¬ (∅ = 1o ∨ ∅ = 2o)
11 0ex 5213 . . 3 ∅ ∈ V
1211elpr 4592 . 2 (∅ ∈ {1o, 2o} ↔ (∅ = 1o ∨ ∅ = 2o))
1310, 12mtbir 325 1 ¬ ∅ ∈ {1o, 2o}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 843   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  ∅c0 4293  {cpr 4571  suc csuc 6195  1oc1o 8097  2oc2o 8098 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-nul 5212 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-v 3498  df-dif 3941  df-un 3943  df-nul 4294  df-sn 4570  df-pr 4572  df-suc 6199  df-1o 8104  df-2o 8105 This theorem is referenced by:  nosgnn0i  33168  sltres  33171  noseponlem  33173  sltso  33183  nosepssdm  33192  nodenselem8  33197  nolt02olem  33200
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