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Theorem fin45 8822
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 8745 . . 3  |-  ( A  e. FinIV  <-> 
A  ~<  ( A  +c  1o ) )
2 simpl 459 . . . . . . . . 9  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  A  =/=  (/) )
3 relen 7574 . . . . . . . . . . . 12  |-  Rel  ~~
43brrelexi 4875 . . . . . . . . . . 11  |-  ( A 
~~  ( A  +c  A )  ->  A  e.  _V )
54adantl 468 . . . . . . . . . 10  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  A  e.  _V )
6 0sdomg 7701 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
75, 6syl 17 . . . . . . . . 9  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
82, 7mpbird 236 . . . . . . . 8  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  (/)  ~<  A )
9 0sdom1dom 7770 . . . . . . . 8  |-  ( (/)  ~<  A 
<->  1o  ~<_  A )
108, 9sylib 200 . . . . . . 7  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  1o  ~<_  A )
11 cdadom2 8617 . . . . . . 7  |-  ( 1o  ~<_  A  ->  ( A  +c  1o )  ~<_  ( A  +c  A ) )
1210, 11syl 17 . . . . . 6  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  ( A  +c  1o )  ~<_  ( A  +c  A ) )
13 domen2 7715 . . . . . . 7  |-  ( A 
~~  ( A  +c  A )  ->  (
( A  +c  1o )  ~<_  A  <->  ( A  +c  1o )  ~<_  ( A  +c  A ) ) )
1413adantl 468 . . . . . 6  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  (
( A  +c  1o )  ~<_  A  <->  ( A  +c  1o )  ~<_  ( A  +c  A ) ) )
1512, 14mpbird 236 . . . . 5  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  ( A  +c  1o )  ~<_  A )
16 domnsym 7698 . . . . 5  |-  ( ( A  +c  1o )  ~<_  A  ->  -.  A  ~<  ( A  +c  1o ) )
1715, 16syl 17 . . . 4  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  -.  A  ~<  ( A  +c  1o ) )
1817con2i 124 . . 3  |-  ( A 
~<  ( A  +c  1o )  ->  -.  ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) ) )
191, 18sylbi 199 . 2  |-  ( A  e. FinIV  ->  -.  ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) ) )
20 isfin5-2 8821 . 2  |-  ( A  e. FinIV  ->  ( A  e. FinV  <->  -.  ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) ) ) )
2119, 20mpbird 236 1  |-  ( A  e. FinIV  ->  A  e. FinV )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    e. wcel 1887    =/= wne 2622   _Vcvv 3045   (/)c0 3731   class class class wbr 4402  (class class class)co 6290   1oc1o 7175    ~~ cen 7566    ~<_ cdom 7567    ~< csdm 7568    +c ccda 8597  FinIVcfin4 8710  FinVcfin5 8712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-cda 8598  df-fin4 8717  df-fin5 8719
This theorem is referenced by:  fin2so  31932
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