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Theorem fin45 8576
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 8499 . . 3  |-  ( A  e. FinIV  <-> 
A  ~<  ( A  +c  1o ) )
2 simpl 457 . . . . . . . . 9  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  A  =/=  (/) )
3 relen 7330 . . . . . . . . . . . 12  |-  Rel  ~~
43brrelexi 4894 . . . . . . . . . . 11  |-  ( A 
~~  ( A  +c  A )  ->  A  e.  _V )
54adantl 466 . . . . . . . . . 10  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  A  e.  _V )
6 0sdomg 7455 . . . . . . . . . 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 7525 . . . . . . . 8  |-  ( (/)  ~<  A 
<->  1o  ~<_  A )
108, 9sylib 196 . . . . . . 7  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  1o  ~<_  A )
11 cdadom2 8371 . . . . . . 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 7469 . . . . . . 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 7452 . . . . 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 8575 . 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 1756    =/= wne 2620   _Vcvv 2987   (/)c0 3652   class class class wbr 4307  (class class class)co 6106   1oc1o 6928    ~~ cen 7322    ~<_ cdom 7323    ~< csdm 7324    +c ccda 8351  FinIVcfin4 8464  FinVcfin5 8466
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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-reu 2737  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-1st 6592  df-2nd 6593  df-recs 6847  df-rdg 6881  df-1o 6935  df-er 7116  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-cda 8352  df-fin4 8471  df-fin5 8473
This theorem is referenced by:  fin2so  28435
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