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Theorem nosgnn0 31936
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 7620 . . . 4 1𝑜 ≠ ∅
21nesymi 2880 . . 3 ¬ ∅ = 1𝑜
3 nsuceq0 5843 . . . . 5 suc 1𝑜 ≠ ∅
4 necom 2876 . . . . . 6 (suc 1𝑜 ≠ ∅ ↔ ∅ ≠ suc 1𝑜)
5 df-2o 7606 . . . . . . 7 2𝑜 = suc 1𝑜
65neeq2i 2888 . . . . . 6 (∅ ≠ 2𝑜 ↔ ∅ ≠ suc 1𝑜)
74, 6bitr4i 267 . . . . 5 (suc 1𝑜 ≠ ∅ ↔ ∅ ≠ 2𝑜)
83, 7mpbi 220 . . . 4 ∅ ≠ 2𝑜
98neii 2825 . . 3 ¬ ∅ = 2𝑜
102, 9pm3.2ni 917 . 2 ¬ (∅ = 1𝑜 ∨ ∅ = 2𝑜)
11 0ex 4823 . . 3 ∅ ∈ V
1211elpr 4231 . 2 (∅ ∈ {1𝑜, 2𝑜} ↔ (∅ = 1𝑜 ∨ ∅ = 2𝑜))
1310, 12mtbir 312 1 ¬ ∅ ∈ {1𝑜, 2𝑜}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 382   = wceq 1523  wcel 2030  wne 2823  c0 3948  {cpr 4212  suc csuc 5763  1𝑜c1o 7598  2𝑜c2o 7599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-v 3233  df-dif 3610  df-un 3612  df-nul 3949  df-sn 4211  df-pr 4213  df-suc 5767  df-1o 7605  df-2o 7606
This theorem is referenced by:  nosgnn0i  31937  sltres  31940  noseponlem  31942  sltso  31952  nosepssdm  31961  nodenselem8  31966  nolt02olem  31969
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