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Theorem isfi 6901
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 6883 . . 3  |-  Fin  =  { y  |  E. x  e.  om  y  ~~  x }
21eleq2i 2360 . 2  |-  ( A  e.  Fin  <->  A  e.  { y  |  E. x  e.  om  y  ~~  x } )
3 relen 6884 . . . . 5  |-  Rel  ~~
43brrelexi 4745 . . . 4  |-  ( A 
~~  x  ->  A  e.  _V )
54rexlimivw 2676 . . 3  |-  ( E. x  e.  om  A  ~~  x  ->  A  e. 
_V )
6 breq1 4042 . . . 4  |-  ( y  =  A  ->  (
y  ~~  x  <->  A  ~~  x ) )
76rexbidv 2577 . . 3  |-  ( y  =  A  ->  ( E. x  e.  om  y  ~~  x  <->  E. x  e.  om  A  ~~  x
) )
85, 7elab3 2934 . 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 1632    e. wcel 1696   {cab 2282   E.wrex 2557   _Vcvv 2801   class class class wbr 4039   omcom 4672    ~~ cen 6876   Fincfn 6879
This theorem is referenced by:  snfi  6957  php3  7063  onfin  7067  fisucdomOLD  7082  ominf  7091  isinf  7092  enfi  7095  ssnnfi  7098  ssfi  7099  dif1enOLD  7106  dif1en  7107  findcard  7113  findcard2  7114  findcard3  7116  nnsdomg  7132  isfiniteg  7133  unfi  7140  fiint  7149  pwfi  7167  finnum  7597  ficardom  7610  dif1card  7654  infpwfien  7705  ficard  8203  hashkf  11355  finminlem  26334
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-en 6880  df-fin 6883
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