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Theorem nosgnn0i 24313
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  e. 
{ 1o ,  2o }
Assertion
Ref Expression
nosgnn0i  |-  (/)  =/=  X

Proof of Theorem nosgnn0i
StepHypRef Expression
1 nosgnn0 24312 . . 3  |-  -.  (/)  e.  { 1o ,  2o }
2 nosgnn0i.1 . . . 4  |-  X  e. 
{ 1o ,  2o }
3 eleq1 2343 . . . 4  |-  ( (/)  =  X  ->  ( (/)  e.  { 1o ,  2o } 
<->  X  e.  { 1o ,  2o } ) )
42, 3mpbiri 224 . . 3  |-  ( (/)  =  X  ->  (/)  e.  { 1o ,  2o } )
51, 4mto 167 . 2  |-  -.  (/)  =  X
6 df-ne 2448 . 2  |-  ( (/)  =/=  X  <->  -.  (/)  =  X )
75, 6mpbir 200 1  |-  (/)  =/=  X
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684    =/= wne 2446   (/)c0 3455   {cpr 3641   1oc1o 6472   2oc2o 6473
This theorem is referenced by:  sltres  24318  nobndlem2  24347  nobndlem4  24349  nobndlem5  24350  nobndlem6  24351  nobndlem8  24353
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-nul 3456  df-sn 3646  df-pr 3647  df-suc 4398  df-1o 6479  df-2o 6480
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