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Theorem isfi 6662
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 6644 . . 3  |-  Fin  =  { y  |  E. x  e.  om  y  ~~  x }
21eleq2i 2207 . 2  |-  ( A  e.  Fin  <->  A  e.  { y  |  E. x  e.  om  y  ~~  x } )
3 relen 6645 . . . . 5  |-  Rel  ~~
43brrelex1i 4589 . . . 4  |-  ( A 
~~  x  ->  A  e.  _V )
54rexlimivw 2548 . . 3  |-  ( E. x  e.  om  A  ~~  x  ->  A  e. 
_V )
6 breq1 3939 . . . 4  |-  ( y  =  A  ->  (
y  ~~  x  <->  A  ~~  x ) )
76rexbidv 2439 . . 3  |-  ( y  =  A  ->  ( E. x  e.  om  y  ~~  x  <->  E. x  e.  om  A  ~~  x
) )
85, 7elab3 2839 . 2  |-  ( A  e.  { y  |  E. x  e.  om  y  ~~  x }  <->  E. x  e.  om  A  ~~  x
)
92, 8bitri 183 1  |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x
)
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1332    e. wcel 1481   {cab 2126   E.wrex 2418   _Vcvv 2689   class class class wbr 3936   omcom 4511    ~~ cen 6639   Fincfn 6641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-xp 4552  df-rel 4553  df-en 6642  df-fin 6644
This theorem is referenced by:  snfig  6715  fict  6769  fidceq  6770  nnfi  6773  enfi  6774  ssfilem  6776  dif1enen  6781  php5fin  6783  fisbth  6784  fin0  6786  fin0or  6787  diffitest  6788  findcard  6789  findcard2  6790  findcard2s  6791  diffisn  6794  infnfi  6796  fientri3  6810  unsnfi  6814  unsnfidcex  6815  unsnfidcel  6816  fiintim  6824  fidcenumlemim  6847  finnum  7055  hashcl  10558  hashen  10561  fihashdom  10580  hashun  10582  zfz1iso  10615
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