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Theorem fin56 8821
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 386 . . . . 5  |-  ( A  =  (/)  ->  ( A  =  (/)  \/  A  ~~  1o ) )
2 sdom2en01 8730 . . . . 5  |-  ( A 
~<  2o  <->  ( A  =  (/)  \/  A  ~~  1o ) )
31, 2sylibr 215 . . . 4  |-  ( A  =  (/)  ->  A  ~<  2o )
43orcd 393 . . 3  |-  ( A  =  (/)  ->  ( A 
~<  2o  \/  A  ~<  ( A  X.  A ) ) )
5 onfin2 7770 . . . . . . . 8  |-  om  =  ( On  i^i  Fin )
6 inss2 3689 . . . . . . . 8  |-  ( On 
i^i  Fin )  C_  Fin
75, 6eqsstri 3500 . . . . . . 7  |-  om  C_  Fin
8 2onn 7349 . . . . . . 7  |-  2o  e.  om
97, 8sselii 3467 . . . . . 6  |-  2o  e.  Fin
10 relsdom 7584 . . . . . . 7  |-  Rel  ~<
1110brrelexi 4895 . . . . . 6  |-  ( A 
~<  ( A  +c  A
)  ->  A  e.  _V )
12 fidomtri 8426 . . . . . 6  |-  ( ( 2o  e.  Fin  /\  A  e.  _V )  ->  ( 2o  ~<_  A  <->  -.  A  ~<  2o ) )
139, 11, 12sylancr 667 . . . . 5  |-  ( A 
~<  ( A  +c  A
)  ->  ( 2o  ~<_  A 
<->  -.  A  ~<  2o ) )
14 xp2cda 8608 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( A  X.  2o )  =  ( A  +c  A
) )
1511, 14syl 17 . . . . . . . . 9  |-  ( A 
~<  ( A  +c  A
)  ->  ( A  X.  2o )  =  ( A  +c  A ) )
1615adantr 466 . . . . . . . 8  |-  ( ( A  ~<  ( A  +c  A )  /\  2o  ~<_  A )  ->  ( A  X.  2o )  =  ( A  +c  A
) )
17 xpdom2g 7674 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  2o 
~<_  A )  ->  ( A  X.  2o )  ~<_  ( A  X.  A ) )
1811, 17sylan 473 . . . . . . . 8  |-  ( ( A  ~<  ( A  +c  A )  /\  2o  ~<_  A )  ->  ( A  X.  2o )  ~<_  ( A  X.  A ) )
1916, 18eqbrtrrd 4448 . . . . . . 7  |-  ( ( A  ~<  ( A  +c  A )  /\  2o  ~<_  A )  ->  ( A  +c  A )  ~<_  ( A  X.  A ) )
20 sdomdomtr 7711 . . . . . . 7  |-  ( ( A  ~<  ( A  +c  A )  /\  ( A  +c  A )  ~<_  ( A  X.  A ) )  ->  A  ~<  ( A  X.  A ) )
2119, 20syldan 472 . . . . . 6  |-  ( ( A  ~<  ( A  +c  A )  /\  2o  ~<_  A )  ->  A  ~<  ( A  X.  A
) )
2221ex 435 . . . . 5  |-  ( A 
~<  ( A  +c  A
)  ->  ( 2o  ~<_  A  ->  A  ~<  ( A  X.  A ) ) )
2313, 22sylbird 238 . . . 4  |-  ( A 
~<  ( A  +c  A
)  ->  ( -.  A  ~<  2o  ->  A  ~<  ( A  X.  A
) ) )
2423orrd 379 . . 3  |-  ( A 
~<  ( A  +c  A
)  ->  ( A  ~<  2o  \/  A  ~<  ( A  X.  A ) ) )
254, 24jaoi 380 . 2  |-  ( ( A  =  (/)  \/  A  ~<  ( A  +c  A
) )  ->  ( A  ~<  2o  \/  A  ~<  ( A  X.  A
) ) )
26 isfin5 8727 . 2  |-  ( A  e. FinV  <-> 
( A  =  (/)  \/  A  ~<  ( A  +c  A ) ) )
27 isfin6 8728 . 2  |-  ( A  e. FinVI  <-> 
( A  ~<  2o  \/  A  ~<  ( A  X.  A ) ) )
2825, 26, 273imtr4i 269 1  |-  ( A  e. FinV  ->  A  e. FinVI )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1870   _Vcvv 3087    i^i cin 3441   (/)c0 3767   class class class wbr 4426    X. cxp 4852   Oncon0 5442  (class class class)co 6305   omcom 6706   1oc1o 7183   2oc2o 7184    ~~ cen 7574    ~<_ cdom 7575    ~< csdm 7576   Fincfn 7577    +c ccda 8595  FinVcfin5 8710  FinVIcfin6 8711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1o 7190  df-2o 7191  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-card 8372  df-cda 8596  df-fin5 8717  df-fin6 8718
This theorem is referenced by:  fin2so  31635
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