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Theorem isfin5 8691
Description: Definition of a V-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
isfin5  |-  ( A  e. FinV  <-> 
( A  =  (/)  \/  A  ~<  ( A  +c  A ) ) )

Proof of Theorem isfin5
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-fin5 8681 . . 3  |- FinV  =  {
x  |  ( x  =  (/)  \/  x  ~<  ( x  +c  x
) ) }
21eleq2i 2545 . 2  |-  ( A  e. FinV  <-> 
A  e.  { x  |  ( x  =  (/)  \/  x  ~<  (
x  +c  x ) ) } )
3 id 22 . . . . 5  |-  ( A  =  (/)  ->  A  =  (/) )
4 0ex 4583 . . . . 5  |-  (/)  e.  _V
53, 4syl6eqel 2563 . . . 4  |-  ( A  =  (/)  ->  A  e. 
_V )
6 relsdom 7535 . . . . 5  |-  Rel  ~<
76brrelexi 5046 . . . 4  |-  ( A 
~<  ( A  +c  A
)  ->  A  e.  _V )
85, 7jaoi 379 . . 3  |-  ( ( A  =  (/)  \/  A  ~<  ( A  +c  A
) )  ->  A  e.  _V )
9 eqeq1 2471 . . . 4  |-  ( x  =  A  ->  (
x  =  (/)  <->  A  =  (/) ) )
10 id 22 . . . . 5  |-  ( x  =  A  ->  x  =  A )
1110, 10oveq12d 6313 . . . . 5  |-  ( x  =  A  ->  (
x  +c  x )  =  ( A  +c  A ) )
1210, 11breq12d 4466 . . . 4  |-  ( x  =  A  ->  (
x  ~<  ( x  +c  x )  <->  A  ~<  ( A  +c  A ) ) )
139, 12orbi12d 709 . . 3  |-  ( x  =  A  ->  (
( x  =  (/)  \/  x  ~<  ( x  +c  x ) )  <->  ( A  =  (/)  \/  A  ~<  ( A  +c  A ) ) ) )
148, 13elab3 3262 . 2  |-  ( A  e.  { x  |  ( x  =  (/)  \/  x  ~<  ( x  +c  x ) ) }  <-> 
( A  =  (/)  \/  A  ~<  ( A  +c  A ) ) )
152, 14bitri 249 1  |-  ( A  e. FinV  <-> 
( A  =  (/)  \/  A  ~<  ( A  +c  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 368    = wceq 1379    e. wcel 1767   {cab 2452   _Vcvv 3118   (/)c0 3790   class class class wbr 4453  (class class class)co 6295    ~< csdm 7527    +c ccda 8559  FinVcfin5 8674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-xp 5011  df-rel 5012  df-iota 5557  df-fv 5602  df-ov 6298  df-dom 7530  df-sdom 7531  df-fin5 8681
This theorem is referenced by:  isfin5-2  8783  fin56  8785
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