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Theorem nosgnn0i 31056
Description: If 𝑋 is a surreal sign, then it is not null. (Contributed by Scott Fenton, 3-Aug-2011.)
Hypothesis
Ref Expression
nosgnn0i.1 𝑋 ∈ {1𝑜, 2𝑜}
Assertion
Ref Expression
nosgnn0i ∅ ≠ 𝑋

Proof of Theorem nosgnn0i
StepHypRef Expression
1 nosgnn0 31055 . . 3 ¬ ∅ ∈ {1𝑜, 2𝑜}
2 nosgnn0i.1 . . . 4 𝑋 ∈ {1𝑜, 2𝑜}
3 eleq1 2676 . . . 4 (∅ = 𝑋 → (∅ ∈ {1𝑜, 2𝑜} ↔ 𝑋 ∈ {1𝑜, 2𝑜}))
42, 3mpbiri 247 . . 3 (∅ = 𝑋 → ∅ ∈ {1𝑜, 2𝑜})
51, 4mto 187 . 2 ¬ ∅ = 𝑋
65neir 2785 1 ∅ ≠ 𝑋
Colors of variables: wff setvar class
Syntax hints:   = wceq 1475  wcel 1977  wne 2780  c0 3874  {cpr 4127  1𝑜c1o 7440  2𝑜c2o 7441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-pr 4128  df-suc 5646  df-1o 7447  df-2o 7448
This theorem is referenced by:  sltres  31061  nobndlem2  31092  nobndlem4  31094  nobndlem5  31095  nobndlem6  31096  nobndlem8  31098
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