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Theorem nosgnn0 23723
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 6494 . . . 4  |-  1o  =/=  (/)
2 necom 2527 . . . . 5  |-  ( 1o  =/=  (/)  <->  (/)  =/=  1o )
3 df-ne 2448 . . . . 5  |-  ( (/)  =/=  1o  <->  -.  (/)  =  1o )
42, 3bitri 240 . . . 4  |-  ( 1o  =/=  (/)  <->  -.  (/)  =  1o )
51, 4mpbi 199 . . 3  |-  -.  (/)  =  1o
6 nsuceq0 4472 . . . . 5  |-  suc  1o  =/=  (/)
7 necom 2527 . . . . . 6  |-  ( suc 
1o  =/=  (/)  <->  (/)  =/=  suc  1o )
8 df-2o 6480 . . . . . . 7  |-  2o  =  suc  1o
98neeq2i 2457 . . . . . 6  |-  ( (/)  =/=  2o  <->  (/)  =/=  suc  1o )
107, 9bitr4i 243 . . . . 5  |-  ( suc 
1o  =/=  (/)  <->  (/)  =/=  2o )
116, 10mpbi 199 . . . 4  |-  (/)  =/=  2o
12 df-ne 2448 . . . 4  |-  ( (/)  =/=  2o  <->  -.  (/)  =  2o )
1311, 12mpbi 199 . . 3  |-  -.  (/)  =  2o
145, 13pm3.2ni 827 . 2  |-  -.  ( (/)  =  1o  \/  (/)  =  2o )
15 0ex 4150 . . 3  |-  (/)  e.  _V
1615elpr 3658 . 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 1623    e. wcel 1684    =/= wne 2446   (/)c0 3455   {cpr 3641   suc csuc 4394   1oc1o 6472   2oc2o 6473
This theorem is referenced by:  nosgnn0i  23724  sltres  23729  sltso  23734  nodenselem3  23748  nodenselem8  23753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-nul 3456  df-sn 3646  df-pr 3647  df-suc 4398  df-1o 6479  df-2o 6480
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