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Theorem nosgnn0i 24384
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 24383 . . 3  |-  -.  (/)  e.  { 1o ,  2o }
2 nosgnn0i.1 . . . 4  |-  X  e. 
{ 1o ,  2o }
3 eleq1 2356 . . . 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 2461 . 2  |-  ( (/)  =/=  X  <->  -.  (/)  =  X )
75, 6mpbir 200 1  |-  (/)  =/=  X
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1632    e. wcel 1696    =/= wne 2459   (/)c0 3468   {cpr 3654   1oc1o 6488   2oc2o 6489
This theorem is referenced by:  sltres  24389  nobndlem2  24418  nobndlem4  24420  nobndlem5  24421  nobndlem6  24422  nobndlem8  24424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-nul 3469  df-sn 3659  df-pr 3660  df-suc 4414  df-1o 6495  df-2o 6496
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