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Theorem fin71num 8585
Description: A well-orderable set is VII-finite iff it is I-finite. Thus, even without choice, on the class of well-orderable sets all eight definitions of finite set coincide. (Contributed by Mario Carneiro, 18-May-2015.)
Assertion
Ref Expression
fin71num  |-  ( A  e.  dom  card  ->  ( A  e. FinVII 
<->  A  e.  Fin )
)

Proof of Theorem fin71num
StepHypRef Expression
1 isfin7-2 8584 . 2  |-  ( A  e.  dom  card  ->  ( A  e. FinVII 
<->  ( A  e.  dom  card 
->  A  e.  Fin ) ) )
2 biimt 335 . 2  |-  ( A  e.  dom  card  ->  ( A  e.  Fin  <->  ( A  e.  dom  card  ->  A  e. 
Fin ) ) )
31, 2bitr4d 256 1  |-  ( A  e.  dom  card  ->  ( A  e. FinVII 
<->  A  e.  Fin )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    e. wcel 1756   dom cdm 4859   Fincfn 7329   cardccrd 8124  FinVIIcfin7 8472
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 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
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 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-sbc 3206  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-int 4148  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-om 6496  df-er 7120  df-en 7330  df-dom 7331  df-sdom 7332  df-fin 7333  df-card 8128  df-fin7 8479
This theorem is referenced by:  fin2so  28439
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