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Theorem nosgnn0 32345
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 7847 . . . 4 1o ≠ ∅
21nesymi 3056 . . 3 ¬ ∅ = 1o
3 nsuceq0 6047 . . . . 5 suc 1o ≠ ∅
4 necom 3052 . . . . . 6 (suc 1o ≠ ∅ ↔ ∅ ≠ suc 1o)
5 df-2o 7832 . . . . . . 7 2o = suc 1o
65neeq2i 3064 . . . . . 6 (∅ ≠ 2o ↔ ∅ ≠ suc 1o)
74, 6bitr4i 270 . . . . 5 (suc 1o ≠ ∅ ↔ ∅ ≠ 2o)
83, 7mpbi 222 . . . 4 ∅ ≠ 2o
98neii 3001 . . 3 ¬ ∅ = 2o
102, 9pm3.2ni 909 . 2 ¬ (∅ = 1o ∨ ∅ = 2o)
11 0ex 5016 . . 3 ∅ ∈ V
1211elpr 4422 . 2 (∅ ∈ {1o, 2o} ↔ (∅ = 1o ∨ ∅ = 2o))
1310, 12mtbir 315 1 ¬ ∅ ∈ {1o, 2o}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 878   = wceq 1656  wcel 2164  wne 2999  c0 4146  {cpr 4401  suc csuc 5969  1oc1o 7824  2oc2o 7825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-nul 5015
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-v 3416  df-dif 3801  df-un 3803  df-nul 4147  df-sn 4400  df-pr 4402  df-suc 5973  df-1o 7831  df-2o 7832
This theorem is referenced by:  nosgnn0i  32346  sltres  32349  noseponlem  32351  sltso  32361  nosepssdm  32370  nodenselem8  32375  nolt02olem  32378
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