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Theorem nosgnn0i 27704
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 27703 . . 3 ¬ ∅ ∈ {1o, 2o}
2 nosgnn0i.1 . . . 4 𝑋 ∈ {1o, 2o}
3 eleq1 2829 . . . 4 (∅ = 𝑋 → (∅ ∈ {1o, 2o} ↔ 𝑋 ∈ {1o, 2o}))
42, 3mpbiri 258 . . 3 (∅ = 𝑋 → ∅ ∈ {1o, 2o})
51, 4mto 197 . 2 ¬ ∅ = 𝑋
65neir 2943 1 ∅ ≠ 𝑋
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108  wne 2940  c0 4333  {cpr 4628  1oc1o 8499  2oc2o 8500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-un 3956  df-nul 4334  df-sn 4627  df-pr 4629  df-suc 6390  df-1o 8506  df-2o 8507
This theorem is referenced by:  sltres  27707  noextenddif  27713  nolesgn2ores  27717  nosepnelem  27724  nosepdmlem  27728  nolt02o  27740  nosupbnd1lem3  27755  nosupbnd1lem5  27757  nosupbnd2lem1  27760
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