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Theorem fin45 8018
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 7941 . . 3  |-  ( A  e. FinIV  <-> 
A  ~<  ( A  +c  1o ) )
2 simpl 443 . . . . . . . . 9  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  A  =/=  (/) )
3 relen 6868 . . . . . . . . . . . 12  |-  Rel  ~~
43brrelexi 4729 . . . . . . . . . . 11  |-  ( A 
~~  ( A  +c  A )  ->  A  e.  _V )
54adantl 452 . . . . . . . . . 10  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  A  e.  _V )
6 0sdomg 6990 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
75, 6syl 15 . . . . . . . . 9  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  ( (/) 
~<  A  <->  A  =/=  (/) ) )
82, 7mpbird 223 . . . . . . . 8  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  (/)  ~<  A )
9 0sdom1dom 7060 . . . . . . . 8  |-  ( (/)  ~<  A 
<->  1o  ~<_  A )
108, 9sylib 188 . . . . . . 7  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  1o  ~<_  A )
11 cdadom2 7813 . . . . . . 7  |-  ( 1o  ~<_  A  ->  ( A  +c  1o )  ~<_  ( A  +c  A ) )
1210, 11syl 15 . . . . . 6  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  ( A  +c  1o )  ~<_  ( A  +c  A ) )
13 domen2 7004 . . . . . . 7  |-  ( A 
~~  ( A  +c  A )  ->  (
( A  +c  1o )  ~<_  A  <->  ( A  +c  1o )  ~<_  ( A  +c  A ) ) )
1413adantl 452 . . . . . 6  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  (
( A  +c  1o )  ~<_  A  <->  ( A  +c  1o )  ~<_  ( A  +c  A ) ) )
1512, 14mpbird 223 . . . . 5  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  ( A  +c  1o )  ~<_  A )
16 domnsym 6987 . . . . 5  |-  ( ( A  +c  1o )  ~<_  A  ->  -.  A  ~<  ( A  +c  1o ) )
1715, 16syl 15 . . . 4  |-  ( ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) )  ->  -.  A  ~<  ( A  +c  1o ) )
1817con2i 112 . . 3  |-  ( A 
~<  ( A  +c  1o )  ->  -.  ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) ) )
191, 18sylbi 187 . 2  |-  ( A  e. FinIV  ->  -.  ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) ) )
20 isfin5-2 8017 . 2  |-  ( A  e. FinIV  ->  ( A  e. FinV  <->  -.  ( A  =/=  (/)  /\  A  ~~  ( A  +c  A
) ) ) )
2119, 20mpbird 223 1  |-  ( A  e. FinIV  ->  A  e. FinV )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684    =/= wne 2446   _Vcvv 2788   (/)c0 3455   class class class wbr 4023  (class class class)co 5858   1oc1o 6472    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862    +c ccda 7793  FinIVcfin4 7906  FinVcfin5 7908
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-cda 7794  df-fin4 7913  df-fin5 7915
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