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Theorem isfi 7541
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 7522 . . 3  |-  Fin  =  { y  |  E. x  e.  om  y  ~~  x }
21eleq2i 2521 . 2  |-  ( A  e.  Fin  <->  A  e.  { y  |  E. x  e.  om  y  ~~  x } )
3 relen 7523 . . . . 5  |-  Rel  ~~
43brrelexi 5030 . . . 4  |-  ( A 
~~  x  ->  A  e.  _V )
54rexlimivw 2932 . . 3  |-  ( E. x  e.  om  A  ~~  x  ->  A  e. 
_V )
6 breq1 4440 . . . 4  |-  ( y  =  A  ->  (
y  ~~  x  <->  A  ~~  x ) )
76rexbidv 2954 . . 3  |-  ( y  =  A  ->  ( E. x  e.  om  y  ~~  x  <->  E. x  e.  om  A  ~~  x
) )
85, 7elab3 3239 . 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 1383    e. wcel 1804   {cab 2428   E.wrex 2794   _Vcvv 3095   class class class wbr 4437   omcom 6685    ~~ cen 7515   Fincfn 7518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-xp 4995  df-rel 4996  df-en 7519  df-fin 7522
This theorem is referenced by:  snfi  7598  php3  7705  onfin  7710  fisucdomOLD  7725  ominf  7734  isinf  7735  enfi  7738  ssnnfi  7741  ssfi  7742  dif1enOLD  7754  dif1en  7755  findcard  7761  findcard2  7762  findcard3  7765  nnsdomg  7781  isfiniteg  7782  unfi  7789  fiint  7799  pwfi  7817  finnum  8332  ficardom  8345  dif1card  8391  infpwfien  8446  ficard  8943  hashkf  12389  finminlem  30112
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