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Theorem isfi 6885
Description: Express " A is finite." Definition 10.29 of [TakeutiZaring] p. 91 (whose " Fin " is a predicate instead of a class). (Contributed by NM, 22-Aug-2008.)
Assertion
Ref Expression
isfi  |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x
)
Distinct variable group:    x, A

Proof of Theorem isfi
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-fin 6867 . . 3  |-  Fin  =  { y  |  E. x  e.  om  y  ~~  x }
21eleq2i 2347 . 2  |-  ( A  e.  Fin  <->  A  e.  { y  |  E. x  e.  om  y  ~~  x } )
3 relen 6868 . . . . 5  |-  Rel  ~~
43brrelexi 4729 . . . 4  |-  ( A 
~~  x  ->  A  e.  _V )
54rexlimivw 2663 . . 3  |-  ( E. x  e.  om  A  ~~  x  ->  A  e. 
_V )
6 breq1 4026 . . . 4  |-  ( y  =  A  ->  (
y  ~~  x  <->  A  ~~  x ) )
76rexbidv 2564 . . 3  |-  ( y  =  A  ->  ( E. x  e.  om  y  ~~  x  <->  E. x  e.  om  A  ~~  x
) )
85, 7elab3 2921 . 2  |-  ( A  e.  { y  |  E. x  e.  om  y  ~~  x }  <->  E. x  e.  om  A  ~~  x
)
92, 8bitri 240 1  |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   _Vcvv 2788   class class class wbr 4023   omcom 4656    ~~ cen 6860   Fincfn 6863
This theorem is referenced by:  snfi  6941  php3  7047  onfin  7051  fisucdomOLD  7066  ominf  7075  isinf  7076  enfi  7079  ssnnfi  7082  ssfi  7083  dif1enOLD  7090  dif1en  7091  findcard  7097  findcard2  7098  findcard3  7100  nnsdomg  7116  isfiniteg  7117  unfi  7124  fiint  7133  pwfi  7151  finnum  7581  ficardom  7594  dif1card  7638  infpwfien  7689  ficard  8187  hashkf  11339  finminlem  26231
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-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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  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-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-en 6864  df-fin 6867
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