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Theorem isfin6 7926
Description: Definition of a VI-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
isfin6  |-  ( A  e. FinVI  <-> 
( A  ~<  2o  \/  A  ~<  ( A  X.  A ) ) )

Proof of Theorem isfin6
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-fin6 7916 . . 3  |- FinVI  =  {
x  |  ( x 
~<  2o  \/  x  ~<  ( x  X.  x ) ) }
21eleq2i 2347 . 2  |-  ( A  e. FinVI  <-> 
A  e.  { x  |  ( x  ~<  2o  \/  x  ~<  (
x  X.  x ) ) } )
3 relsdom 6870 . . . . 5  |-  Rel  ~<
43brrelexi 4729 . . . 4  |-  ( A 
~<  2o  ->  A  e.  _V )
53brrelexi 4729 . . . 4  |-  ( A 
~<  ( A  X.  A
)  ->  A  e.  _V )
64, 5jaoi 368 . . 3  |-  ( ( A  ~<  2o  \/  A  ~<  ( A  X.  A ) )  ->  A  e.  _V )
7 breq1 4026 . . . 4  |-  ( x  =  A  ->  (
x  ~<  2o  <->  A  ~<  2o ) )
8 id 19 . . . . 5  |-  ( x  =  A  ->  x  =  A )
98, 8xpeq12d 4714 . . . . 5  |-  ( x  =  A  ->  (
x  X.  x )  =  ( A  X.  A ) )
108, 9breq12d 4036 . . . 4  |-  ( x  =  A  ->  (
x  ~<  ( x  X.  x )  <->  A  ~<  ( A  X.  A ) ) )
117, 10orbi12d 690 . . 3  |-  ( x  =  A  ->  (
( x  ~<  2o  \/  x  ~<  ( x  X.  x ) )  <->  ( A  ~<  2o  \/  A  ~<  ( A  X.  A ) ) ) )
126, 11elab3 2921 . 2  |-  ( A  e.  { x  |  ( x  ~<  2o  \/  x  ~<  ( x  X.  x ) ) }  <-> 
( A  ~<  2o  \/  A  ~<  ( A  X.  A ) ) )
132, 12bitri 240 1  |-  ( A  e. FinVI  <-> 
( A  ~<  2o  \/  A  ~<  ( A  X.  A ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    = wceq 1623    e. wcel 1684   {cab 2269   _Vcvv 2788   class class class wbr 4023    X. cxp 4687   2oc2o 6473    ~< csdm 6862  FinVIcfin6 7909
This theorem is referenced by:  fin56  8019  fin67  8021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-dom 6865  df-sdom 6866  df-fin6 7916
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