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Theorem isfi 6771
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
StepHypRef Expression
1 df-fin 6753 . . 3  |-  Fin  =  { y  |  E. x  e.  om  y  ~~  x }
21eleq2i 2317 . 2  |-  ( A  e.  Fin  <->  A  e.  { y  |  E. x  e.  om  y  ~~  x } )
3 relen 6754 . . . . 5  |-  Rel  ~~
43brrelexi 4636 . . . 4  |-  ( A 
~~  x  ->  A  e.  _V )
54rexlimivw 2625 . . 3  |-  ( E. x  e.  om  A  ~~  x  ->  A  e. 
_V )
6 breq1 3923 . . . 4  |-  ( y  =  A  ->  (
y  ~~  x  <->  A  ~~  x ) )
76rexbidv 2528 . . 3  |-  ( y  =  A  ->  ( E. x  e.  om  y  ~~  x  <->  E. x  e.  om  A  ~~  x
) )
85, 7elab3 2858 . 2  |-  ( A  e.  { y  |  E. x  e.  om  y  ~~  x }  <->  E. x  e.  om  A  ~~  x
)
92, 8bitri 242 1  |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x
)
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1619    e. wcel 1621   {cab 2239   E.wrex 2510   _Vcvv 2727   class class class wbr 3920   omcom 4547    ~~ cen 6746   Fincfn 6749
This theorem is referenced by:  snfi  6826  php3  6932  onfin  6936  fisucdomOLD  6951  ominf  6960  isinf  6961  enfi  6964  ssnnfi  6967  ssfi  6968  dif1enOLD  6975  dif1en  6976  findcard  6982  findcard2  6983  findcard3  6985  nnsdomg  7001  isfiniteg  7002  unfi  7009  fiint  7018  pwfi  7035  finnum  7465  ficardom  7478  dif1card  7522  infpwfien  7573  ficard  8069  hashkf  11217  finminlem  25397
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-xp 4594  df-rel 4595  df-en 6750  df-fin 6753
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