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Theorem isfin3 8577
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 8569 . . 3  |- FinIII  =  { x  |  ~P x  e. FinIV }
21eleq2i 2532 . 2  |-  ( A  e. FinIII  <-> 
A  e.  { x  |  ~P x  e. FinIV } )
3 elex 3087 . . . 4  |-  ( ~P A  e. FinIV  ->  ~P A  e. 
_V )
4 pwexb 6498 . . . 4  |-  ( A  e.  _V  <->  ~P A  e.  _V )
53, 4sylibr 212 . . 3  |-  ( ~P A  e. FinIV  ->  A  e.  _V )
6 pweq 3972 . . . 4  |-  ( x  =  A  ->  ~P x  =  ~P A
)
76eleq1d 2523 . . 3  |-  ( x  =  A  ->  ( ~P x  e. FinIV  <->  ~P A  e. FinIV ) )
85, 7elab3 3220 . 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 1370    e. wcel 1758   {cab 2439   _Vcvv 3078   ~Pcpw 3969  FinIVcfin4 8561  FinIIIcfin3 8562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-rex 2805  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-pw 3971  df-sn 3987  df-pr 3989  df-uni 4201  df-fin3 8569
This theorem is referenced by:  fin23lem41  8633  isfin32i  8646  fin34  8671
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