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Theorem elno 24300
Description: Membership in the surreals. (Shortened proof on 2012-Apr-14, SF). (Contributed by Scott Fenton, 11-Jun-2011.)
Assertion
Ref Expression
elno  |-  ( A  e.  No  <->  E. x  e.  On  A : x --> { 1o ,  2o } )
Distinct variable group:    x, A

Proof of Theorem elno
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 elex 2796 . 2  |-  ( A  e.  No  ->  A  e.  _V )
2 fex 5749 . . . 4  |-  ( ( A : x --> { 1o ,  2o }  /\  x  e.  On )  ->  A  e.  _V )
32ancoms 439 . . 3  |-  ( ( x  e.  On  /\  A : x --> { 1o ,  2o } )  ->  A  e.  _V )
43rexlimiva 2662 . 2  |-  ( E. x  e.  On  A : x --> { 1o ,  2o }  ->  A  e.  _V )
5 feq1 5375 . . . 4  |-  ( f  =  A  ->  (
f : x --> { 1o ,  2o }  <->  A :
x --> { 1o ,  2o } ) )
65rexbidv 2564 . . 3  |-  ( f  =  A  ->  ( E. x  e.  On  f : x --> { 1o ,  2o }  <->  E. x  e.  On  A : x --> { 1o ,  2o } ) )
7 df-no 24297 . . 3  |-  No  =  { f  |  E. x  e.  On  f : x --> { 1o ,  2o } }
86, 7elab2g 2916 . 2  |-  ( A  e.  _V  ->  ( A  e.  No  <->  E. x  e.  On  A : x --> { 1o ,  2o } ) )
91, 4, 8pm5.21nii 342 1  |-  ( A  e.  No  <->  E. x  e.  On  A : x --> { 1o ,  2o } )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788   {cpr 3641   Oncon0 4392   -->wf 5251   1oc1o 6472   2oc2o 6473   Nocsur 24294
This theorem is referenced by:  nofun  24303  nodmon  24304  norn  24305  elno2  24308  noreson  24314  noxpsgn  24319  nodenselem6  24340
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-no 24297
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