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Theorem fin45 8784
Description: Every IV-finite set is V-finite: if we can pack two copies of the set into itself, we can certainly leave space. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Proof shortened by Mario Carneiro, 18-May-2015.)
Assertion
Ref Expression
fin45  |-  ( A  e. FinIV  ->  A  e. FinV )

Proof of Theorem fin45
StepHypRef Expression
1 isfin4-3 8707 . . 3  |-  ( A  e. FinIV  <-> 
A  ~<  ( A  +c  1o ) )
2 simpl 457 . . . . . . . . 9  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  A  =/=  (/) )
3 relen 7533 . . . . . . . . . . . 12  |-  Rel  ~~
43brrelexi 5046 . . . . . . . . . . 11  |-  ( A 
~~  ( A  +c  A )  ->  A  e.  _V )
54adantl 466 . . . . . . . . . 10  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  A  e.  _V )
6 0sdomg 7658 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
75, 6syl 16 . . . . . . . . 9  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
82, 7mpbird 232 . . . . . . . 8  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  (/)  ~<  A )
9 0sdom1dom 7729 . . . . . . . 8  |-  ( (/)  ~<  A 
<->  1o  ~<_  A )
108, 9sylib 196 . . . . . . 7  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  1o  ~<_  A )
11 cdadom2 8579 . . . . . . 7  |-  ( 1o  ~<_  A  ->  ( A  +c  1o )  ~<_  ( A  +c  A ) )
1210, 11syl 16 . . . . . 6  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  ( A  +c  1o )  ~<_  ( A  +c  A ) )
13 domen2 7672 . . . . . . 7  |-  ( A 
~~  ( A  +c  A )  ->  (
( A  +c  1o )  ~<_  A  <->  ( A  +c  1o )  ~<_  ( A  +c  A ) ) )
1413adantl 466 . . . . . 6  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  (
( A  +c  1o )  ~<_  A  <->  ( A  +c  1o )  ~<_  ( A  +c  A ) ) )
1512, 14mpbird 232 . . . . 5  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  ( A  +c  1o )  ~<_  A )
16 domnsym 7655 . . . . 5  |-  ( ( A  +c  1o )  ~<_  A  ->  -.  A  ~<  ( A  +c  1o ) )
1715, 16syl 16 . . . 4  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  -.  A  ~<  ( A  +c  1o ) )
1817con2i 120 . . 3  |-  ( A 
~<  ( A  +c  1o )  ->  -.  ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) ) )
191, 18sylbi 195 . 2  |-  ( A  e. FinIV  ->  -.  ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) ) )
20 isfin5-2 8783 . 2  |-  ( A  e. FinIV  ->  ( A  e. FinV  <->  -.  ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) ) ) )
2119, 20mpbird 232 1  |-  ( A  e. FinIV  ->  A  e. FinV )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1767    =/= wne 2662   _Vcvv 3118   (/)c0 3790   class class class wbr 4453  (class class class)co 6295   1oc1o 7135    ~~ cen 7525    ~<_ cdom 7526    ~< csdm 7527    +c ccda 8559  FinIVcfin4 8672  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-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-cda 8560  df-fin4 8679  df-fin5 8681
This theorem is referenced by:  fin2so  29974
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