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Theorem elno 25518
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 2928 . 2  |-  ( A  e.  No  ->  A  e.  _V )
2 fex 5932 . . . 4  |-  ( ( A : x --> { 1o ,  2o }  /\  x  e.  On )  ->  A  e.  _V )
32ancoms 440 . . 3  |-  ( ( x  e.  On  /\  A : x --> { 1o ,  2o } )  ->  A  e.  _V )
43rexlimiva 2789 . 2  |-  ( E. x  e.  On  A : x --> { 1o ,  2o }  ->  A  e.  _V )
5 feq1 5539 . . . 4  |-  ( f  =  A  ->  (
f : x --> { 1o ,  2o }  <->  A :
x --> { 1o ,  2o } ) )
65rexbidv 2691 . . 3  |-  ( f  =  A  ->  ( E. x  e.  On  f : x --> { 1o ,  2o }  <->  E. x  e.  On  A : x --> { 1o ,  2o } ) )
7 df-no 25515 . . 3  |-  No  =  { f  |  E. x  e.  On  f : x --> { 1o ,  2o } }
86, 7elab2g 3048 . 2  |-  ( A  e.  _V  ->  ( A  e.  No  <->  E. x  e.  On  A : x --> { 1o ,  2o } ) )
91, 4, 8pm5.21nii 343 1  |-  ( A  e.  No  <->  E. x  e.  On  A : x --> { 1o ,  2o } )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649    e. wcel 1721   E.wrex 2671   _Vcvv 2920   {cpr 3779   Oncon0 4545   -->wf 5413   1oc1o 6680   2oc2o 6681   Nocsur 25512
This theorem is referenced by:  nofun  25521  nodmon  25522  norn  25523  elno2  25526  noreson  25532  noxpsgn  25537  nodenselem6  25558
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-no 25515
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