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Theorem isfin3 8674
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 8666 . . 3  |- FinIII  =  { x  |  ~P x  e. FinIV }
21eleq2i 2519 . 2  |-  ( A  e. FinIII  <-> 
A  e.  { x  |  ~P x  e. FinIV } )
3 elex 3102 . . . 4  |-  ( ~P A  e. FinIV  ->  ~P A  e. 
_V )
4 pwexb 6592 . . . 4  |-  ( A  e.  _V  <->  ~P A  e.  _V )
53, 4sylibr 212 . . 3  |-  ( ~P A  e. FinIV  ->  A  e.  _V )
6 pweq 3996 . . . 4  |-  ( x  =  A  ->  ~P x  =  ~P A
)
76eleq1d 2510 . . 3  |-  ( x  =  A  ->  ( ~P x  e. FinIV  <->  ~P A  e. FinIV ) )
85, 7elab3 3237 . 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 1381    e. wcel 1802   {cab 2426   _Vcvv 3093   ~Pcpw 3993  FinIVcfin4 8658  FinIIIcfin3 8659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-rex 2797  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-pw 3995  df-sn 4011  df-pr 4013  df-uni 4231  df-fin3 8666
This theorem is referenced by:  fin23lem41  8730  isfin32i  8743  fin34  8768
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