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Theorem isfin6 7942
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 7932 . . 3  |- FinVI  =  {
x  |  ( x 
~<  2o  \/  x  ~<  ( x  X.  x ) ) }
21eleq2i 2360 . 2  |-  ( A  e. FinVI  <-> 
A  e.  { x  |  ( x  ~<  2o  \/  x  ~<  (
x  X.  x ) ) } )
3 relsdom 6886 . . . . 5  |-  Rel  ~<
43brrelexi 4745 . . . 4  |-  ( A 
~<  2o  ->  A  e.  _V )
53brrelexi 4745 . . . 4  |-  ( A 
~<  ( A  X.  A
)  ->  A  e.  _V )
64, 5jaoi 368 . . 3  |-  ( ( A  ~<  2o  \/  A  ~<  ( A  X.  A ) )  ->  A  e.  _V )
7 breq1 4042 . . . 4  |-  ( x  =  A  ->  (
x  ~<  2o  <->  A  ~<  2o ) )
8 id 19 . . . . 5  |-  ( x  =  A  ->  x  =  A )
98, 8xpeq12d 4730 . . . . 5  |-  ( x  =  A  ->  (
x  X.  x )  =  ( A  X.  A ) )
108, 9breq12d 4052 . . . 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 2934 . 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 1632    e. wcel 1696   {cab 2282   _Vcvv 2801   class class class wbr 4039    X. cxp 4703   2oc2o 6489    ~< csdm 6878  FinVIcfin6 7925
This theorem is referenced by:  fin56  8035  fin67  8037
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  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-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-dom 6881  df-sdom 6882  df-fin6 7932
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