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Theorem fin56 8019
Description: Every V-finite set is VI-finite because multiplication dominates addition for cardinals. (Contributed by Stefan O'Rear, 29-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)
Assertion
Ref Expression
fin56  |-  ( A  e. FinV  ->  A  e. FinVI )

Proof of Theorem fin56
StepHypRef Expression
1 orc 374 . . . . 5  |-  ( A  =  (/)  ->  ( A  =  (/)  \/  A  ~~  1o ) )
2 sdom2en01 7928 . . . . 5  |-  ( A 
~<  2o  <->  ( A  =  (/)  \/  A  ~~  1o ) )
31, 2sylibr 203 . . . 4  |-  ( A  =  (/)  ->  A  ~<  2o )
43orcd 381 . . 3  |-  ( A  =  (/)  ->  ( A 
~<  2o  \/  A  ~<  ( A  X.  A ) ) )
5 onfin2 7052 . . . . . . . 8  |-  om  =  ( On  i^i  Fin )
6 inss2 3390 . . . . . . . 8  |-  ( On 
i^i  Fin )  C_  Fin
75, 6eqsstri 3208 . . . . . . 7  |-  om  C_  Fin
8 2onn 6638 . . . . . . 7  |-  2o  e.  om
97, 8sselii 3177 . . . . . 6  |-  2o  e.  Fin
10 relsdom 6870 . . . . . . 7  |-  Rel  ~<
1110brrelexi 4729 . . . . . 6  |-  ( A 
~<  ( A  +c  A
)  ->  A  e.  _V )
12 fidomtri 7626 . . . . . 6  |-  ( ( 2o  e.  Fin  /\  A  e.  _V )  ->  ( 2o  ~<_  A  <->  -.  A  ~<  2o ) )
139, 11, 12sylancr 644 . . . . 5  |-  ( A 
~<  ( A  +c  A
)  ->  ( 2o  ~<_  A 
<->  -.  A  ~<  2o ) )
14 xp2cda 7806 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( A  X.  2o )  =  ( A  +c  A
) )
1511, 14syl 15 . . . . . . . . 9  |-  ( A 
~<  ( A  +c  A
)  ->  ( A  X.  2o )  =  ( A  +c  A ) )
1615adantr 451 . . . . . . . 8  |-  ( ( A  ~<  ( A  +c  A )  /\  2o  ~<_  A )  ->  ( A  X.  2o )  =  ( A  +c  A
) )
17 xpdom2g 6958 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  2o 
~<_  A )  ->  ( A  X.  2o )  ~<_  ( A  X.  A ) )
1811, 17sylan 457 . . . . . . . 8  |-  ( ( A  ~<  ( A  +c  A )  /\  2o  ~<_  A )  ->  ( A  X.  2o )  ~<_  ( A  X.  A ) )
1916, 18eqbrtrrd 4045 . . . . . . 7  |-  ( ( A  ~<  ( A  +c  A )  /\  2o  ~<_  A )  ->  ( A  +c  A )  ~<_  ( A  X.  A ) )
20 sdomdomtr 6994 . . . . . . 7  |-  ( ( A  ~<  ( A  +c  A )  /\  ( A  +c  A )  ~<_  ( A  X.  A ) )  ->  A  ~<  ( A  X.  A ) )
2119, 20syldan 456 . . . . . 6  |-  ( ( A  ~<  ( A  +c  A )  /\  2o  ~<_  A )  ->  A  ~<  ( A  X.  A
) )
2221ex 423 . . . . 5  |-  ( A 
~<  ( A  +c  A
)  ->  ( 2o  ~<_  A  ->  A  ~<  ( A  X.  A ) ) )
2313, 22sylbird 226 . . . 4  |-  ( A 
~<  ( A  +c  A
)  ->  ( -.  A  ~<  2o  ->  A  ~<  ( A  X.  A
) ) )
2423orrd 367 . . 3  |-  ( A 
~<  ( A  +c  A
)  ->  ( A  ~<  2o  \/  A  ~<  ( A  X.  A ) ) )
254, 24jaoi 368 . 2  |-  ( ( A  =  (/)  \/  A  ~<  ( A  +c  A
) )  ->  ( A  ~<  2o  \/  A  ~<  ( A  X.  A
) ) )
26 isfin5 7925 . 2  |-  ( A  e. FinV  <-> 
( A  =  (/)  \/  A  ~<  ( A  +c  A ) ) )
27 isfin6 7926 . 2  |-  ( A  e. FinVI  <-> 
( A  ~<  2o  \/  A  ~<  ( A  X.  A ) ) )
2825, 26, 273imtr4i 257 1  |-  ( A  e. FinV  ->  A  e. FinVI )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151   (/)c0 3455   class class class wbr 4023   Oncon0 4392   omcom 4656    X. cxp 4687  (class class class)co 5858   1oc1o 6472   2oc2o 6473    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862   Fincfn 6863    +c ccda 7793  FinVcfin5 7908  FinVIcfin6 7909
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-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-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-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-1o 6479  df-2o 6480  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-cda 7794  df-fin5 7915  df-fin6 7916
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