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Theorem nosgnn0i 27159
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 27158 . . 3 ¬ ∅ ∈ {1o, 2o}
2 nosgnn0i.1 . . . 4 𝑋 ∈ {1o, 2o}
3 eleq1 2821 . . . 4 (∅ = 𝑋 → (∅ ∈ {1o, 2o} ↔ 𝑋 ∈ {1o, 2o}))
42, 3mpbiri 257 . . 3 (∅ = 𝑋 → ∅ ∈ {1o, 2o})
51, 4mto 196 . 2 ¬ ∅ = 𝑋
65neir 2943 1 ∅ ≠ 𝑋
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wcel 2106  wne 2940  c0 4322  {cpr 4630  1oc1o 8458  2oc2o 8459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-v 3476  df-dif 3951  df-un 3953  df-nul 4323  df-sn 4629  df-pr 4631  df-suc 6370  df-1o 8465  df-2o 8466
This theorem is referenced by:  sltres  27162  noextenddif  27168  nolesgn2ores  27172  nosepnelem  27179  nosepdmlem  27183  nolt02o  27195  nosupbnd1lem3  27210  nosupbnd1lem5  27212  nosupbnd2lem1  27215
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