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Theorem isfin3 8665
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 8657 . . 3  |- FinIII  =  { x  |  ~P x  e. FinIV }
21eleq2i 2538 . 2  |-  ( A  e. FinIII  <-> 
A  e.  { x  |  ~P x  e. FinIV } )
3 elex 3115 . . . 4  |-  ( ~P A  e. FinIV  ->  ~P A  e. 
_V )
4 pwexb 6582 . . . 4  |-  ( A  e.  _V  <->  ~P A  e.  _V )
53, 4sylibr 212 . . 3  |-  ( ~P A  e. FinIV  ->  A  e.  _V )
6 pweq 4006 . . . 4  |-  ( x  =  A  ->  ~P x  =  ~P A
)
76eleq1d 2529 . . 3  |-  ( x  =  A  ->  ( ~P x  e. FinIV  <->  ~P A  e. FinIV ) )
85, 7elab3 3250 . 2  |-  ( A  e.  { x  |  ~P x  e. FinIV }  <->  ~P A  e. FinIV
)
92, 8bitri 249 1  |-  ( A  e. FinIII  <->  ~P A  e. FinIV )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1374    e. wcel 1762   {cab 2445   _Vcvv 3106   ~Pcpw 4003  FinIVcfin4 8649  FinIIIcfin3 8650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-rex 2813  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-pw 4005  df-sn 4021  df-pr 4023  df-uni 4239  df-fin3 8657
This theorem is referenced by:  fin23lem41  8721  isfin32i  8734  fin34  8759
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