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Theorem isfin5 8480
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 8470 . . 3  |- FinV  =  {
x  |  ( x  =  (/)  \/  x  ~<  ( x  +c  x
) ) }
21eleq2i 2507 . 2  |-  ( A  e. FinV  <-> 
A  e.  { x  |  ( x  =  (/)  \/  x  ~<  (
x  +c  x ) ) } )
3 id 22 . . . . 5  |-  ( A  =  (/)  ->  A  =  (/) )
4 0ex 4434 . . . . 5  |-  (/)  e.  _V
53, 4syl6eqel 2531 . . . 4  |-  ( A  =  (/)  ->  A  e. 
_V )
6 relsdom 7329 . . . . 5  |-  Rel  ~<
76brrelexi 4891 . . . 4  |-  ( A 
~<  ( A  +c  A
)  ->  A  e.  _V )
85, 7jaoi 379 . . 3  |-  ( ( A  =  (/)  \/  A  ~<  ( A  +c  A
) )  ->  A  e.  _V )
9 eqeq1 2449 . . . 4  |-  ( x  =  A  ->  (
x  =  (/)  <->  A  =  (/) ) )
10 id 22 . . . . 5  |-  ( x  =  A  ->  x  =  A )
1110, 10oveq12d 6121 . . . . 5  |-  ( x  =  A  ->  (
x  +c  x )  =  ( A  +c  A ) )
1210, 11breq12d 4317 . . . 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 3125 . 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 1369    e. wcel 1756   {cab 2429   _Vcvv 2984   (/)c0 3649   class class class wbr 4304  (class class class)co 6103    ~< csdm 7321    +c ccda 8348  FinVcfin5 8463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-xp 4858  df-rel 4859  df-iota 5393  df-fv 5438  df-ov 6106  df-dom 7324  df-sdom 7325  df-fin5 8470
This theorem is referenced by:  isfin5-2  8572  fin56  8574
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