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Theorem nosgnn0i 33294
 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 33293 . . 3 ¬ ∅ ∈ {1o, 2o}
2 nosgnn0i.1 . . . 4 𝑋 ∈ {1o, 2o}
3 eleq1 2877 . . . 4 (∅ = 𝑋 → (∅ ∈ {1o, 2o} ↔ 𝑋 ∈ {1o, 2o}))
42, 3mpbiri 261 . . 3 (∅ = 𝑋 → ∅ ∈ {1o, 2o})
51, 4mto 200 . 2 ¬ ∅ = 𝑋
65neir 2990 1 ∅ ≠ 𝑋
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∅c0 4243  {cpr 4527  1oc1o 8081  2oc2o 8082 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770  ax-nul 5175 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-v 3443  df-dif 3884  df-un 3886  df-nul 4244  df-sn 4526  df-pr 4528  df-suc 6166  df-1o 8088  df-2o 8089 This theorem is referenced by:  sltres  33297  noextenddif  33303  nolesgn2ores  33307  nosepnelem  33312  nosepdmlem  33315  nolt02o  33327  nosupbnd1lem3  33338  nosupbnd1lem5  33340  nosupbnd2lem1  33343
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