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Theorem dfacfin7 8284
Description: Axiom of Choice equivalent: the VII-finite sets are the same as I-finite sets. (Contributed by Mario Carneiro, 18-May-2015.)
Assertion
Ref Expression
dfacfin7  |-  (CHOICE  <-> FinVII  =  Fin )

Proof of Theorem dfacfin7
StepHypRef Expression
1 ssequn2 3522 . 2  |-  ( ( _V  \  dom  card )  C_  Fin  <->  ( Fin  u.  ( _V  \  dom  card ) )  =  Fin )
2 dfac10 8022 . . . 4  |-  (CHOICE  <->  dom  card  =  _V )
3 finnum 7840 . . . . . . 7  |-  ( x  e.  Fin  ->  x  e.  dom  card )
43ssriv 3354 . . . . . 6  |-  Fin  C_  dom  card
5 ssequn2 3522 . . . . . 6  |-  ( Fin  C_  dom  card  <->  ( dom  card  u. 
Fin )  =  dom  card )
64, 5mpbi 201 . . . . 5  |-  ( dom 
card  u.  Fin )  =  dom  card
76eqeq1i 2445 . . . 4  |-  ( ( dom  card  u.  Fin )  =  _V  <->  dom  card  =  _V )
82, 7bitr4i 245 . . 3  |-  (CHOICE  <->  ( dom  card 
u.  Fin )  =  _V )
9 ssv 3370 . . . 4  |-  ( dom 
card  u.  Fin )  C_ 
_V
10 eqss 3365 . . . 4  |-  ( ( dom  card  u.  Fin )  =  _V  <->  ( ( dom  card  u.  Fin )  C_ 
_V  /\  _V  C_  ( dom  card  u.  Fin )
) )
119, 10mpbiran 886 . . 3  |-  ( ( dom  card  u.  Fin )  =  _V  <->  _V  C_  ( dom  card  u.  Fin )
)
12 ssundif 3713 . . 3  |-  ( _V  C_  ( dom  card  u.  Fin )  <->  ( _V  \  dom  card )  C_  Fin )
138, 11, 123bitri 264 . 2  |-  (CHOICE  <->  ( _V  \  dom  card )  C_  Fin )
14 dffin7-2 8283 . . 3  |- FinVII  =  ( Fin 
u.  ( _V  \  dom  card ) )
1514eqeq1i 2445 . 2  |-  (FinVII  =  Fin  <->  ( Fin  u.  ( _V  \  dom  card ) )  =  Fin )
161, 13, 153bitr4i 270 1  |-  (CHOICE  <-> FinVII  =  Fin )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653   _Vcvv 2958    \ cdif 3319    u. cun 3320    C_ wss 3322   dom cdm 4881   Fincfn 7112   cardccrd 7827  CHOICEwac 8001  FinVIIcfin7 8169
This theorem is referenced by:  fin71ac  8416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-riota 6552  df-recs 6636  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-card 7831  df-ac 8002  df-fin7 8176
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