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Theorem isfin2 7936
Description: Definition of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
isfin2  |-  ( A  e.  V  ->  ( A  e. FinII 
<-> 
A. y  e.  ~P  ~P A ( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y ) ) )
Distinct variable group:    y, A
Allowed substitution hint:    V( y)

Proof of Theorem isfin2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pweq 3641 . . . 4  |-  ( x  =  A  ->  ~P x  =  ~P A
)
21pweqd 3643 . . 3  |-  ( x  =  A  ->  ~P ~P x  =  ~P ~P A )
32raleqdv 2755 . 2  |-  ( x  =  A  ->  ( A. y  e.  ~P  ~P x ( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y )  <->  A. y  e.  ~P  ~P A ( ( y  =/=  (/)  /\ [ C.]  Or  y )  ->  U. y  e.  y ) ) )
4 df-fin2 7928 . 2  |- FinII  =  {
x  |  A. y  e.  ~P  ~P x ( ( y  =/=  (/)  /\ [ C.]  Or  y )  ->  U. y  e.  y ) }
53, 4elab2g 2929 1  |-  ( A  e.  V  ->  ( A  e. FinII 
<-> 
A. y  e.  ~P  ~P A ( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   (/)c0 3468   ~Pcpw 3638   U.cuni 3843    Or wor 4329   [ C.] crpss 6292  FinIIcfin2 7921
This theorem is referenced by:  fin2i  7937  isfin2-2  7961  ssfin2  7962  enfin2i  7963  fin12  8055  fin1a2s  8056
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803  df-in 3172  df-ss 3179  df-pw 3640  df-fin2 7928
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