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Theorem nosgnn0 28983
Description:  (/) is not a surreal sign. (Contributed by Scott Fenton, 16-Jun-2011.)
Assertion
Ref Expression
nosgnn0  |-  -.  (/)  e.  { 1o ,  2o }

Proof of Theorem nosgnn0
StepHypRef Expression
1 1n0 7137 . . . 4  |-  1o  =/=  (/)
2 necom 2731 . . . . 5  |-  ( 1o  =/=  (/)  <->  (/)  =/=  1o )
3 df-ne 2659 . . . . 5  |-  ( (/)  =/=  1o  <->  -.  (/)  =  1o )
42, 3bitri 249 . . . 4  |-  ( 1o  =/=  (/)  <->  -.  (/)  =  1o )
51, 4mpbi 208 . . 3  |-  -.  (/)  =  1o
6 nsuceq0 4953 . . . . 5  |-  suc  1o  =/=  (/)
7 necom 2731 . . . . . 6  |-  ( suc 
1o  =/=  (/)  <->  (/)  =/=  suc  1o )
8 df-2o 7123 . . . . . . 7  |-  2o  =  suc  1o
98neeq2i 2749 . . . . . 6  |-  ( (/)  =/=  2o  <->  (/)  =/=  suc  1o )
107, 9bitr4i 252 . . . . 5  |-  ( suc 
1o  =/=  (/)  <->  (/)  =/=  2o )
116, 10mpbi 208 . . . 4  |-  (/)  =/=  2o
12 df-ne 2659 . . . 4  |-  ( (/)  =/=  2o  <->  -.  (/)  =  2o )
1311, 12mpbi 208 . . 3  |-  -.  (/)  =  2o
145, 13pm3.2ni 851 . 2  |-  -.  ( (/)  =  1o  \/  (/)  =  2o )
15 0ex 4572 . . 3  |-  (/)  e.  _V
1615elpr 4040 . 2  |-  ( (/)  e.  { 1o ,  2o } 
<->  ( (/)  =  1o  \/  (/)  =  2o ) )
1714, 16mtbir 299 1  |-  -.  (/)  e.  { 1o ,  2o }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 368    = wceq 1374    e. wcel 1762    =/= wne 2657   (/)c0 3780   {cpr 4024   suc csuc 4875   1oc1o 7115   2oc2o 7116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-nul 4571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-v 3110  df-dif 3474  df-un 3476  df-nul 3781  df-sn 4023  df-pr 4025  df-suc 4879  df-1o 7122  df-2o 7123
This theorem is referenced by:  nosgnn0i  28984  sltres  28989  sltso  28994  nodenselem3  29008  nodenselem8  29013
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