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Theorem fin45 8228
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 8151 . . 3  |-  ( A  e. FinIV  <-> 
A  ~<  ( A  +c  1o ) )
2 simpl 444 . . . . . . . . 9  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  A  =/=  (/) )
3 relen 7073 . . . . . . . . . . . 12  |-  Rel  ~~
43brrelexi 4877 . . . . . . . . . . 11  |-  ( A 
~~  ( A  +c  A )  ->  A  e.  _V )
54adantl 453 . . . . . . . . . 10  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  A  e.  _V )
6 0sdomg 7195 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
75, 6syl 16 . . . . . . . . 9  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
82, 7mpbird 224 . . . . . . . 8  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  (/)  ~<  A )
9 0sdom1dom 7265 . . . . . . . 8  |-  ( (/)  ~<  A 
<->  1o  ~<_  A )
108, 9sylib 189 . . . . . . 7  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  1o  ~<_  A )
11 cdadom2 8023 . . . . . . 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 7209 . . . . . . 7  |-  ( A 
~~  ( A  +c  A )  ->  (
( A  +c  1o )  ~<_  A  <->  ( A  +c  1o )  ~<_  ( A  +c  A ) ) )
1413adantl 453 . . . . . 6  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  (
( A  +c  1o )  ~<_  A  <->  ( A  +c  1o )  ~<_  ( A  +c  A ) ) )
1512, 14mpbird 224 . . . . 5  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  ( A  +c  1o )  ~<_  A )
16 domnsym 7192 . . . . 5  |-  ( ( A  +c  1o )  ~<_  A  ->  -.  A  ~<  ( A  +c  1o ) )
1715, 16syl 16 . . . 4  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  -.  A  ~<  ( A  +c  1o ) )
1817con2i 114 . . 3  |-  ( A 
~<  ( A  +c  1o )  ->  -.  ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) ) )
191, 18sylbi 188 . 2  |-  ( A  e. FinIV  ->  -.  ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) ) )
20 isfin5-2 8227 . 2  |-  ( A  e. FinIV  ->  ( A  e. FinV  <->  -.  ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) ) ) )
2119, 20mpbird 224 1  |-  ( A  e. FinIV  ->  A  e. FinV )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1721    =/= wne 2567   _Vcvv 2916   (/)c0 3588   class class class wbr 4172  (class class class)co 6040   1oc1o 6676    ~~ cen 7065    ~<_ cdom 7066    ~< csdm 7067    +c ccda 8003  FinIVcfin4 8116  FinVcfin5 8118
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-recs 6592  df-rdg 6627  df-1o 6683  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-cda 8004  df-fin4 8123  df-fin5 8125
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