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Theorem isfin3 7922
Description: Definition of a III-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
isfin3  |-  ( A  e. FinIII  <->  ~P A  e. FinIV )

Proof of Theorem isfin3
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-fin3 7914 . . 3  |- FinIII  =  { x  |  ~P x  e. FinIV }
21eleq2i 2347 . 2  |-  ( A  e. FinIII  <-> 
A  e.  { x  |  ~P x  e. FinIV } )
3 elex 2796 . . . 4  |-  ( ~P A  e. FinIV  ->  ~P A  e. 
_V )
4 pwexb 4564 . . . 4  |-  ( A  e.  _V  <->  ~P A  e.  _V )
53, 4sylibr 203 . . 3  |-  ( ~P A  e. FinIV  ->  A  e.  _V )
6 pweq 3628 . . . 4  |-  ( x  =  A  ->  ~P x  =  ~P A
)
76eleq1d 2349 . . 3  |-  ( x  =  A  ->  ( ~P x  e. FinIV  <->  ~P A  e. FinIV ) )
85, 7elab3 2921 . 2  |-  ( A  e.  { x  |  ~P x  e. FinIV }  <->  ~P A  e. FinIV
)
92, 8bitri 240 1  |-  ( A  e. FinIII  <->  ~P A  e. FinIV )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684   {cab 2269   _Vcvv 2788   ~Pcpw 3625  FinIVcfin4 7906  FinIIIcfin3 7907
This theorem is referenced by:  fin23lem41  7978  isfin32i  7991  fin34  8016
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rex 2549  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-pw 3627  df-sn 3646  df-pr 3647  df-uni 3828  df-fin3 7914
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