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Theorem isfi 6271
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 6254 . . 3  |-  Fin  =  { y  |  E. x  e.  om  y  ~~  x }
21eleq2i 2120 . 2  |-  ( A  e.  Fin  <->  A  e.  { y  |  E. x  e.  om  y  ~~  x } )
3 relen 6255 . . . . 5  |-  Rel  ~~
43brrelexi 4411 . . . 4  |-  ( A 
~~  x  ->  A  e.  _V )
54rexlimivw 2446 . . 3  |-  ( E. x  e.  om  A  ~~  x  ->  A  e. 
_V )
6 breq1 3794 . . . 4  |-  ( y  =  A  ->  (
y  ~~  x  <->  A  ~~  x ) )
76rexbidv 2344 . . 3  |-  ( y  =  A  ->  ( E. x  e.  om  y  ~~  x  <->  E. x  e.  om  A  ~~  x
) )
85, 7elab3 2716 . 2  |-  ( A  e.  { y  |  E. x  e.  om  y  ~~  x }  <->  E. x  e.  om  A  ~~  x
)
92, 8bitri 177 1  |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x
)
Colors of variables: wff set class
Syntax hints:    <-> wb 102    = wceq 1259    e. wcel 1409   {cab 2042   E.wrex 2324   _Vcvv 2574   class class class wbr 3791   omcom 4340    ~~ cen 6249   Fincfn 6251
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-xp 4378  df-rel 4379  df-en 6252  df-fin 6254
This theorem is referenced by:  snfig  6321  fidceq  6360  nnfi  6363  enfi  6364  ssfiexmid  6366  php5fin  6369  fisbth  6370  fin0  6372  fin0or  6373  diffitest  6374  findcard  6375  findcard2  6376  findcard2s  6377  diffisn  6380  fientri3  6383  finnum  6420
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