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Theorem fin56 8064
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 7973 . . . . 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 7095 . . . . . . . 8  |-  om  =  ( On  i^i  Fin )
6 inss2 3424 . . . . . . . 8  |-  ( On 
i^i  Fin )  C_  Fin
75, 6eqsstri 3242 . . . . . . 7  |-  om  C_  Fin
8 2onn 6680 . . . . . . 7  |-  2o  e.  om
97, 8sselii 3211 . . . . . 6  |-  2o  e.  Fin
10 relsdom 6913 . . . . . . 7  |-  Rel  ~<
1110brrelexi 4766 . . . . . 6  |-  ( A 
~<  ( A  +c  A
)  ->  A  e.  _V )
12 fidomtri 7671 . . . . . 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 7851 . . . . . . . . . 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 7001 . . . . . . . . 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 4082 . . . . . . 7  |-  ( ( A  ~<  ( A  +c  A )  /\  2o  ~<_  A )  ->  ( A  +c  A )  ~<_  ( A  X.  A ) )
20 sdomdomtr 7037 . . . . . . 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 7970 . 2  |-  ( A  e. FinV  <-> 
( A  =  (/)  \/  A  ~<  ( A  +c  A ) ) )
27 isfin6 7971 . 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 1633    e. wcel 1701   _Vcvv 2822    i^i cin 3185   (/)c0 3489   class class class wbr 4060   Oncon0 4429   omcom 4693    X. cxp 4724  (class class class)co 5900   1oc1o 6514   2oc2o 6515    ~~ cen 6903    ~<_ cdom 6904    ~< csdm 6905   Fincfn 6906    +c ccda 7838  FinVcfin5 7953  FinVIcfin6 7954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1o 6521  df-2o 6522  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-card 7617  df-cda 7839  df-fin5 7960  df-fin6 7961
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