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Theorem isfin6 8581
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 8571 . . 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 7428 . . . . 5  |-  Rel  ~<
43brrelexi 4988 . . . 4  |-  ( A 
~<  2o  ->  A  e.  _V )
53brrelexi 4988 . . . 4  |-  ( A 
~<  ( A  X.  A
)  ->  A  e.  _V )
64, 5jaoi 379 . . 3  |-  ( ( A  ~<  2o  \/  A  ~<  ( A  X.  A ) )  ->  A  e.  _V )
7 breq1 4404 . . . 4  |-  ( x  =  A  ->  (
x  ~<  2o  <->  A  ~<  2o ) )
8 id 22 . . . . 5  |-  ( x  =  A  ->  x  =  A )
98, 8xpeq12d 4974 . . . . 5  |-  ( x  =  A  ->  (
x  X.  x )  =  ( A  X.  A ) )
108, 9breq12d 4414 . . . 4  |-  ( x  =  A  ->  (
x  ~<  ( x  X.  x )  <->  A  ~<  ( A  X.  A ) ) )
117, 10orbi12d 709 . . 3  |-  ( x  =  A  ->  (
( x  ~<  2o  \/  x  ~<  ( x  X.  x ) )  <->  ( A  ~<  2o  \/  A  ~<  ( A  X.  A ) ) ) )
126, 11elab3 3220 . 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 368    = wceq 1370    e. wcel 1758   {cab 2439   _Vcvv 3078   class class class wbr 4401    X. cxp 4947   2oc2o 7025    ~< csdm 7420  FinVIcfin6 8564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-br 4402  df-opab 4460  df-xp 4955  df-rel 4956  df-dom 7423  df-sdom 7424  df-fin6 8571
This theorem is referenced by:  fin56  8674  fin67  8676
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