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Theorem fin56 8567
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 385 . . . . 5  |-  ( A  =  (/)  ->  ( A  =  (/)  \/  A  ~~  1o ) )
2 sdom2en01 8476 . . . . 5  |-  ( A 
~<  2o  <->  ( A  =  (/)  \/  A  ~~  1o ) )
31, 2sylibr 212 . . . 4  |-  ( A  =  (/)  ->  A  ~<  2o )
43orcd 392 . . 3  |-  ( A  =  (/)  ->  ( A 
~<  2o  \/  A  ~<  ( A  X.  A ) ) )
5 onfin2 7507 . . . . . . . 8  |-  om  =  ( On  i^i  Fin )
6 inss2 3576 . . . . . . . 8  |-  ( On 
i^i  Fin )  C_  Fin
75, 6eqsstri 3391 . . . . . . 7  |-  om  C_  Fin
8 2onn 7084 . . . . . . 7  |-  2o  e.  om
97, 8sselii 3358 . . . . . 6  |-  2o  e.  Fin
10 relsdom 7322 . . . . . . 7  |-  Rel  ~<
1110brrelexi 4884 . . . . . 6  |-  ( A 
~<  ( A  +c  A
)  ->  A  e.  _V )
12 fidomtri 8168 . . . . . 6  |-  ( ( 2o  e.  Fin  /\  A  e.  _V )  ->  ( 2o  ~<_  A  <->  -.  A  ~<  2o ) )
139, 11, 12sylancr 663 . . . . 5  |-  ( A 
~<  ( A  +c  A
)  ->  ( 2o  ~<_  A 
<->  -.  A  ~<  2o ) )
14 xp2cda 8354 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( A  X.  2o )  =  ( A  +c  A
) )
1511, 14syl 16 . . . . . . . . 9  |-  ( A 
~<  ( A  +c  A
)  ->  ( A  X.  2o )  =  ( A  +c  A ) )
1615adantr 465 . . . . . . . 8  |-  ( ( A  ~<  ( A  +c  A )  /\  2o  ~<_  A )  ->  ( A  X.  2o )  =  ( A  +c  A
) )
17 xpdom2g 7412 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  2o 
~<_  A )  ->  ( A  X.  2o )  ~<_  ( A  X.  A ) )
1811, 17sylan 471 . . . . . . . 8  |-  ( ( A  ~<  ( A  +c  A )  /\  2o  ~<_  A )  ->  ( A  X.  2o )  ~<_  ( A  X.  A ) )
1916, 18eqbrtrrd 4319 . . . . . . 7  |-  ( ( A  ~<  ( A  +c  A )  /\  2o  ~<_  A )  ->  ( A  +c  A )  ~<_  ( A  X.  A ) )
20 sdomdomtr 7449 . . . . . . 7  |-  ( ( A  ~<  ( A  +c  A )  /\  ( A  +c  A )  ~<_  ( A  X.  A ) )  ->  A  ~<  ( A  X.  A ) )
2119, 20syldan 470 . . . . . 6  |-  ( ( A  ~<  ( A  +c  A )  /\  2o  ~<_  A )  ->  A  ~<  ( A  X.  A
) )
2221ex 434 . . . . 5  |-  ( A 
~<  ( A  +c  A
)  ->  ( 2o  ~<_  A  ->  A  ~<  ( A  X.  A ) ) )
2313, 22sylbird 235 . . . 4  |-  ( A 
~<  ( A  +c  A
)  ->  ( -.  A  ~<  2o  ->  A  ~<  ( A  X.  A
) ) )
2423orrd 378 . . 3  |-  ( A 
~<  ( A  +c  A
)  ->  ( A  ~<  2o  \/  A  ~<  ( A  X.  A ) ) )
254, 24jaoi 379 . 2  |-  ( ( A  =  (/)  \/  A  ~<  ( A  +c  A
) )  ->  ( A  ~<  2o  \/  A  ~<  ( A  X.  A
) ) )
26 isfin5 8473 . 2  |-  ( A  e. FinV  <-> 
( A  =  (/)  \/  A  ~<  ( A  +c  A ) ) )
27 isfin6 8474 . 2  |-  ( A  e. FinVI  <-> 
( A  ~<  2o  \/  A  ~<  ( A  X.  A ) ) )
2825, 26, 273imtr4i 266 1  |-  ( A  e. FinV  ->  A  e. FinVI )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2977    i^i cin 3332   (/)c0 3642   class class class wbr 4297   Oncon0 4724    X. cxp 4843  (class class class)co 6096   omcom 6481   1oc1o 6918   2oc2o 6919    ~~ cen 7312    ~<_ cdom 7313    ~< csdm 7314   Fincfn 7315    +c ccda 8341  FinVcfin5 8456  FinVIcfin6 8457
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-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
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 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1o 6925  df-2o 6926  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114  df-cda 8342  df-fin5 8463  df-fin6 8464
This theorem is referenced by:  fin2so  28421
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