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Theorem isfin3 8181
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 8173 . . 3  |- FinIII  =  { x  |  ~P x  e. FinIV }
21eleq2i 2502 . 2  |-  ( A  e. FinIII  <-> 
A  e.  { x  |  ~P x  e. FinIV } )
3 elex 2966 . . . 4  |-  ( ~P A  e. FinIV  ->  ~P A  e. 
_V )
4 pwexb 4756 . . . 4  |-  ( A  e.  _V  <->  ~P A  e.  _V )
53, 4sylibr 205 . . 3  |-  ( ~P A  e. FinIV  ->  A  e.  _V )
6 pweq 3804 . . . 4  |-  ( x  =  A  ->  ~P x  =  ~P A
)
76eleq1d 2504 . . 3  |-  ( x  =  A  ->  ( ~P x  e. FinIV  <->  ~P A  e. FinIV ) )
85, 7elab3 3091 . 2  |-  ( A  e.  { x  |  ~P x  e. FinIV }  <->  ~P A  e. FinIV
)
92, 8bitri 242 1  |-  ( A  e. FinIII  <->  ~P A  e. FinIV )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653    e. wcel 1726   {cab 2424   _Vcvv 2958   ~Pcpw 3801  FinIVcfin4 8165  FinIIIcfin3 8166
This theorem is referenced by:  fin23lem41  8237  isfin32i  8250  fin34  8275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rex 2713  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-pw 3803  df-sn 3822  df-pr 3823  df-uni 4018  df-fin3 8173
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