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Theorem nosgnn0 25613
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 6739 . . . 4  |-  1o  =/=  (/)
2 necom 2685 . . . . 5  |-  ( 1o  =/=  (/)  <->  (/)  =/=  1o )
3 df-ne 2601 . . . . 5  |-  ( (/)  =/=  1o  <->  -.  (/)  =  1o )
42, 3bitri 241 . . . 4  |-  ( 1o  =/=  (/)  <->  -.  (/)  =  1o )
51, 4mpbi 200 . . 3  |-  -.  (/)  =  1o
6 nsuceq0 4661 . . . . 5  |-  suc  1o  =/=  (/)
7 necom 2685 . . . . . 6  |-  ( suc 
1o  =/=  (/)  <->  (/)  =/=  suc  1o )
8 df-2o 6725 . . . . . . 7  |-  2o  =  suc  1o
98neeq2i 2612 . . . . . 6  |-  ( (/)  =/=  2o  <->  (/)  =/=  suc  1o )
107, 9bitr4i 244 . . . . 5  |-  ( suc 
1o  =/=  (/)  <->  (/)  =/=  2o )
116, 10mpbi 200 . . . 4  |-  (/)  =/=  2o
12 df-ne 2601 . . . 4  |-  ( (/)  =/=  2o  <->  -.  (/)  =  2o )
1311, 12mpbi 200 . . 3  |-  -.  (/)  =  2o
145, 13pm3.2ni 828 . 2  |-  -.  ( (/)  =  1o  \/  (/)  =  2o )
15 0ex 4339 . . 3  |-  (/)  e.  _V
1615elpr 3832 . 2  |-  ( (/)  e.  { 1o ,  2o } 
<->  ( (/)  =  1o  \/  (/)  =  2o ) )
1714, 16mtbir 291 1  |-  -.  (/)  e.  { 1o ,  2o }
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 358    = wceq 1652    e. wcel 1725    =/= wne 2599   (/)c0 3628   {cpr 3815   suc csuc 4583   1oc1o 6717   2oc2o 6718
This theorem is referenced by:  nosgnn0i  25614  sltres  25619  sltso  25624  nodenselem3  25638  nodenselem8  25643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4338
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-v 2958  df-dif 3323  df-un 3325  df-nul 3629  df-sn 3820  df-pr 3821  df-suc 4587  df-1o 6724  df-2o 6725
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