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Theorem isfin3 8012
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 8004 . . 3  |- FinIII  =  { x  |  ~P x  e. FinIV }
21eleq2i 2422 . 2  |-  ( A  e. FinIII  <-> 
A  e.  { x  |  ~P x  e. FinIV } )
3 elex 2872 . . . 4  |-  ( ~P A  e. FinIV  ->  ~P A  e. 
_V )
4 pwexb 4646 . . . 4  |-  ( A  e.  _V  <->  ~P A  e.  _V )
53, 4sylibr 203 . . 3  |-  ( ~P A  e. FinIV  ->  A  e.  _V )
6 pweq 3704 . . . 4  |-  ( x  =  A  ->  ~P x  =  ~P A
)
76eleq1d 2424 . . 3  |-  ( x  =  A  ->  ( ~P x  e. FinIV  <->  ~P A  e. FinIV ) )
85, 7elab3 2997 . 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 1642    e. wcel 1710   {cab 2344   _Vcvv 2864   ~Pcpw 3701  FinIVcfin4 7996  FinIIIcfin3 7997
This theorem is referenced by:  fin23lem41  8068  isfin32i  8081  fin34  8106
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-rex 2625  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-pw 3703  df-sn 3722  df-pr 3723  df-uni 3909  df-fin3 8004
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