MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nosgnn0i Structured version   Visualization version   GIF version

Theorem nosgnn0i 26959
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 26958 . . 3 ¬ ∅ ∈ {1o, 2o}
2 nosgnn0i.1 . . . 4 𝑋 ∈ {1o, 2o}
3 eleq1 2825 . . . 4 (∅ = 𝑋 → (∅ ∈ {1o, 2o} ↔ 𝑋 ∈ {1o, 2o}))
42, 3mpbiri 257 . . 3 (∅ = 𝑋 → ∅ ∈ {1o, 2o})
51, 4mto 196 . 2 ¬ ∅ = 𝑋
65neir 2944 1 ∅ ≠ 𝑋
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wcel 2106  wne 2941  c0 4280  {cpr 4586  1oc1o 8397  2oc2o 8398
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 2708  ax-nul 5261
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 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-v 3445  df-dif 3911  df-un 3913  df-nul 4281  df-sn 4585  df-pr 4587  df-suc 6321  df-1o 8404  df-2o 8405
This theorem is referenced by:  sltres  26962  noextenddif  26968  nolesgn2ores  26972  nosepnelem  26979  nosepdmlem  26983  nolt02o  26995  nosupbnd1lem3  27010  nosupbnd1lem5  27012  nosupbnd2lem1  27015
  Copyright terms: Public domain W3C validator