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Theorem isfin3 8140
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 8132 . . 3  |- FinIII  =  { x  |  ~P x  e. FinIV }
21eleq2i 2476 . 2  |-  ( A  e. FinIII  <-> 
A  e.  { x  |  ~P x  e. FinIV } )
3 elex 2932 . . . 4  |-  ( ~P A  e. FinIV  ->  ~P A  e. 
_V )
4 pwexb 4720 . . . 4  |-  ( A  e.  _V  <->  ~P A  e.  _V )
53, 4sylibr 204 . . 3  |-  ( ~P A  e. FinIV  ->  A  e.  _V )
6 pweq 3770 . . . 4  |-  ( x  =  A  ->  ~P x  =  ~P A
)
76eleq1d 2478 . . 3  |-  ( x  =  A  ->  ( ~P x  e. FinIV  <->  ~P A  e. FinIV ) )
85, 7elab3 3057 . 2  |-  ( A  e.  { x  |  ~P x  e. FinIV }  <->  ~P A  e. FinIV
)
92, 8bitri 241 1  |-  ( A  e. FinIII  <->  ~P A  e. FinIV )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649    e. wcel 1721   {cab 2398   _Vcvv 2924   ~Pcpw 3767  FinIVcfin4 8124  FinIIIcfin3 8125
This theorem is referenced by:  fin23lem41  8196  isfin32i  8209  fin34  8234
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-rex 2680  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-pw 3769  df-sn 3788  df-pr 3789  df-uni 3984  df-fin3 8132
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