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Theorem nosgnn0i 27937
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 27936 . . 3  |-  -.  (/)  e.  { 1o ,  2o }
2 nosgnn0i.1 . . . 4  |-  X  e. 
{ 1o ,  2o }
3 eleq1 2523 . . . 4  |-  ( (/)  =  X  ->  ( (/)  e.  { 1o ,  2o } 
<->  X  e.  { 1o ,  2o } ) )
42, 3mpbiri 233 . . 3  |-  ( (/)  =  X  ->  (/)  e.  { 1o ,  2o } )
51, 4mto 176 . 2  |-  -.  (/)  =  X
6 df-ne 2646 . 2  |-  ( (/)  =/=  X  <->  -.  (/)  =  X )
75, 6mpbir 209 1  |-  (/)  =/=  X
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1370    e. wcel 1758    =/= wne 2644   (/)c0 3738   {cpr 3980   1oc1o 7016   2oc2o 7017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-nul 4522
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-v 3073  df-dif 3432  df-un 3434  df-nul 3739  df-sn 3979  df-pr 3981  df-suc 4826  df-1o 7023  df-2o 7024
This theorem is referenced by:  sltres  27942  nobndlem2  27971  nobndlem4  27973  nobndlem5  27974  nobndlem6  27975  nobndlem8  27977
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