MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isfin2 Structured version   Unicode version

Theorem isfin2 8166
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 3794 . . . 4  |-  ( x  =  A  ->  ~P x  =  ~P A
)
21pweqd 3796 . . 3  |-  ( x  =  A  ->  ~P ~P x  =  ~P ~P A )
32raleqdv 2902 . 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 8158 . 2  |- FinII  =  {
x  |  A. y  e.  ~P  ~P x ( ( y  =/=  (/)  /\ [ C.]  Or  y )  ->  U. y  e.  y ) }
53, 4elab2g 3076 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 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   (/)c0 3620   ~Pcpw 3791   U.cuni 4007    Or wor 4494   [ C.] crpss 6513  FinIIcfin2 8151
This theorem is referenced by:  fin2i  8167  isfin2-2  8191  ssfin2  8192  enfin2i  8193  fin12  8285  fin1a2s  8286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-v 2950  df-in 3319  df-ss 3326  df-pw 3793  df-fin2 8158
  Copyright terms: Public domain W3C validator