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Theorem isfi 7529
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 7510 . . 3  |-  Fin  =  { y  |  E. x  e.  om  y  ~~  x }
21eleq2i 2538 . 2  |-  ( A  e.  Fin  <->  A  e.  { y  |  E. x  e.  om  y  ~~  x } )
3 relen 7511 . . . . 5  |-  Rel  ~~
43brrelexi 5032 . . . 4  |-  ( A 
~~  x  ->  A  e.  _V )
54rexlimivw 2945 . . 3  |-  ( E. x  e.  om  A  ~~  x  ->  A  e. 
_V )
6 breq1 4443 . . . 4  |-  ( y  =  A  ->  (
y  ~~  x  <->  A  ~~  x ) )
76rexbidv 2966 . . 3  |-  ( y  =  A  ->  ( E. x  e.  om  y  ~~  x  <->  E. x  e.  om  A  ~~  x
) )
85, 7elab3 3250 . 2  |-  ( A  e.  { y  |  E. x  e.  om  y  ~~  x }  <->  E. x  e.  om  A  ~~  x
)
92, 8bitri 249 1  |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1374    e. wcel 1762   {cab 2445   E.wrex 2808   _Vcvv 3106   class class class wbr 4440   omcom 6671    ~~ cen 7503   Fincfn 7506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-xp 4998  df-rel 4999  df-en 7507  df-fin 7510
This theorem is referenced by:  snfi  7586  php3  7693  onfin  7698  fisucdomOLD  7713  ominf  7722  isinf  7723  enfi  7726  ssnnfi  7729  ssfi  7730  dif1enOLD  7741  dif1en  7742  findcard  7748  findcard2  7749  findcard3  7752  nnsdomg  7768  isfiniteg  7769  unfi  7776  fiint  7786  pwfi  7804  finnum  8318  ficardom  8331  dif1card  8377  infpwfien  8432  ficard  8929  hashkf  12362  finminlem  29700
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