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Theorem nosgnn0 24383
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 6510 . . . 4  |-  1o  =/=  (/)
2 necom 2540 . . . . 5  |-  ( 1o  =/=  (/)  <->  (/)  =/=  1o )
3 df-ne 2461 . . . . 5  |-  ( (/)  =/=  1o  <->  -.  (/)  =  1o )
42, 3bitri 240 . . . 4  |-  ( 1o  =/=  (/)  <->  -.  (/)  =  1o )
51, 4mpbi 199 . . 3  |-  -.  (/)  =  1o
6 nsuceq0 4488 . . . . 5  |-  suc  1o  =/=  (/)
7 necom 2540 . . . . . 6  |-  ( suc 
1o  =/=  (/)  <->  (/)  =/=  suc  1o )
8 df-2o 6496 . . . . . . 7  |-  2o  =  suc  1o
98neeq2i 2470 . . . . . 6  |-  ( (/)  =/=  2o  <->  (/)  =/=  suc  1o )
107, 9bitr4i 243 . . . . 5  |-  ( suc 
1o  =/=  (/)  <->  (/)  =/=  2o )
116, 10mpbi 199 . . . 4  |-  (/)  =/=  2o
12 df-ne 2461 . . . 4  |-  ( (/)  =/=  2o  <->  -.  (/)  =  2o )
1311, 12mpbi 199 . . 3  |-  -.  (/)  =  2o
145, 13pm3.2ni 827 . 2  |-  -.  ( (/)  =  1o  \/  (/)  =  2o )
15 0ex 4166 . . 3  |-  (/)  e.  _V
1615elpr 3671 . 2  |-  ( (/)  e.  { 1o ,  2o } 
<->  ( (/)  =  1o  \/  (/)  =  2o ) )
1714, 16mtbir 290 1  |-  -.  (/)  e.  { 1o ,  2o }
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 357    = wceq 1632    e. wcel 1696    =/= wne 2459   (/)c0 3468   {cpr 3654   suc csuc 4410   1oc1o 6488   2oc2o 6489
This theorem is referenced by:  nosgnn0i  24384  sltres  24389  sltso  24394  nodenselem3  24408  nodenselem8  24413
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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