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Theorem isfin6 8671
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 8661 . . 3  |- FinVI  =  {
x  |  ( x 
~<  2o  \/  x  ~<  ( x  X.  x ) ) }
21eleq2i 2532 . 2  |-  ( A  e. FinVI  <-> 
A  e.  { x  |  ( x  ~<  2o  \/  x  ~<  (
x  X.  x ) ) } )
3 relsdom 7516 . . . . 5  |-  Rel  ~<
43brrelexi 5029 . . . 4  |-  ( A 
~<  2o  ->  A  e.  _V )
53brrelexi 5029 . . . 4  |-  ( A 
~<  ( A  X.  A
)  ->  A  e.  _V )
64, 5jaoi 377 . . 3  |-  ( ( A  ~<  2o  \/  A  ~<  ( A  X.  A ) )  ->  A  e.  _V )
7 breq1 4442 . . . 4  |-  ( x  =  A  ->  (
x  ~<  2o  <->  A  ~<  2o ) )
8 id 22 . . . . 5  |-  ( x  =  A  ->  x  =  A )
98sqxpeqd 5014 . . . . 5  |-  ( x  =  A  ->  (
x  X.  x )  =  ( A  X.  A ) )
108, 9breq12d 4452 . . . 4  |-  ( x  =  A  ->  (
x  ~<  ( x  X.  x )  <->  A  ~<  ( A  X.  A ) ) )
117, 10orbi12d 707 . . 3  |-  ( x  =  A  ->  (
( x  ~<  2o  \/  x  ~<  ( x  X.  x ) )  <->  ( A  ~<  2o  \/  A  ~<  ( A  X.  A ) ) ) )
126, 11elab3 3250 . 2  |-  ( A  e.  { x  |  ( x  ~<  2o  \/  x  ~<  ( x  X.  x ) ) }  <-> 
( A  ~<  2o  \/  A  ~<  ( A  X.  A ) ) )
132, 12bitri 249 1  |-  ( A  e. FinVI  <-> 
( A  ~<  2o  \/  A  ~<  ( A  X.  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 366    = wceq 1398    e. wcel 1823   {cab 2439   _Vcvv 3106   class class class wbr 4439    X. cxp 4986   2oc2o 7116    ~< csdm 7508  FinVIcfin6 8654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-dom 7511  df-sdom 7512  df-fin6 8661
This theorem is referenced by:  fin56  8764  fin67  8766
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