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Theorem nosgnn0i 31796
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 31795 . . 3 ¬ ∅ ∈ {1𝑜, 2𝑜}
2 nosgnn0i.1 . . . 4 𝑋 ∈ {1𝑜, 2𝑜}
3 eleq1 2688 . . . 4 (∅ = 𝑋 → (∅ ∈ {1𝑜, 2𝑜} ↔ 𝑋 ∈ {1𝑜, 2𝑜}))
42, 3mpbiri 248 . . 3 (∅ = 𝑋 → ∅ ∈ {1𝑜, 2𝑜})
51, 4mto 188 . 2 ¬ ∅ = 𝑋
65neir 2796 1 ∅ ≠ 𝑋
Colors of variables: wff setvar class
Syntax hints:   = wceq 1482  wcel 1989  wne 2793  c0 3913  {cpr 4177  1𝑜c1o 7550  2𝑜c2o 7551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-nul 4787
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-v 3200  df-dif 3575  df-un 3577  df-nul 3914  df-sn 4176  df-pr 4178  df-suc 5727  df-1o 7557  df-2o 7558
This theorem is referenced by:  sltres  31799  noextenddif  31805  nolesgn2ores  31809  nosepnelem  31814  nosepdmlem  31817  nolt02o  31829  nosupbnd1lem3  31840  nosupbnd1lem5  31842  nosupbnd2lem1  31845
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