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Theorem elno 25606
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 2966 . 2  |-  ( A  e.  No  ->  A  e.  _V )
2 fex 5972 . . . 4  |-  ( ( A : x --> { 1o ,  2o }  /\  x  e.  On )  ->  A  e.  _V )
32ancoms 441 . . 3  |-  ( ( x  e.  On  /\  A : x --> { 1o ,  2o } )  ->  A  e.  _V )
43rexlimiva 2827 . 2  |-  ( E. x  e.  On  A : x --> { 1o ,  2o }  ->  A  e.  _V )
5 feq1 5579 . . . 4  |-  ( f  =  A  ->  (
f : x --> { 1o ,  2o }  <->  A :
x --> { 1o ,  2o } ) )
65rexbidv 2728 . . 3  |-  ( f  =  A  ->  ( E. x  e.  On  f : x --> { 1o ,  2o }  <->  E. x  e.  On  A : x --> { 1o ,  2o } ) )
7 df-no 25603 . . 3  |-  No  =  { f  |  E. x  e.  On  f : x --> { 1o ,  2o } }
86, 7elab2g 3086 . 2  |-  ( A  e.  _V  ->  ( A  e.  No  <->  E. x  e.  On  A : x --> { 1o ,  2o } ) )
91, 4, 8pm5.21nii 344 1  |-  ( A  e.  No  <->  E. x  e.  On  A : x --> { 1o ,  2o } )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653    e. wcel 1726   E.wrex 2708   _Vcvv 2958   {cpr 3817   Oncon0 4584   -->wf 5453   1oc1o 6720   2oc2o 6721   Nocsur 25600
This theorem is referenced by:  nofun  25609  nodmon  25610  norn  25611  elno2  25614  noreson  25620  noxpsgn  25625  nodenselem6  25646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-no 25603
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