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Theorem fin56 8790
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 8699 . . . . 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 7728 . . . . . . . 8  |-  om  =  ( On  i^i  Fin )
6 inss2 3715 . . . . . . . 8  |-  ( On 
i^i  Fin )  C_  Fin
75, 6eqsstri 3529 . . . . . . 7  |-  om  C_  Fin
8 2onn 7307 . . . . . . 7  |-  2o  e.  om
97, 8sselii 3496 . . . . . 6  |-  2o  e.  Fin
10 relsdom 7542 . . . . . . 7  |-  Rel  ~<
1110brrelexi 5049 . . . . . 6  |-  ( A 
~<  ( A  +c  A
)  ->  A  e.  _V )
12 fidomtri 8391 . . . . . 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 8577 . . . . . . . . . 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 7632 . . . . . . . . 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 4478 . . . . . . 7  |-  ( ( A  ~<  ( A  +c  A )  /\  2o  ~<_  A )  ->  ( A  +c  A )  ~<_  ( A  X.  A ) )
20 sdomdomtr 7669 . . . . . . 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 8696 . 2  |-  ( A  e. FinV  <-> 
( A  =  (/)  \/  A  ~<  ( A  +c  A ) ) )
27 isfin6 8697 . 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 1395    e. wcel 1819   _Vcvv 3109    i^i cin 3470   (/)c0 3793   class class class wbr 4456   Oncon0 4887    X. cxp 5006  (class class class)co 6296   omcom 6699   1oc1o 7141   2oc2o 7142    ~~ cen 7532    ~<_ cdom 7533    ~< csdm 7534   Fincfn 7535    +c ccda 8564  FinVcfin5 8679  FinVIcfin6 8680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1o 7148  df-2o 7149  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-fin5 8686  df-fin6 8687
This theorem is referenced by:  fin2so  30245
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