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Theorem dfacfin7 8831
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 3640 . 2  |-  ( ( _V  \  dom  card )  C_  Fin  <->  ( Fin  u.  ( _V  \  dom  card ) )  =  Fin )
2 dfac10 8569 . . . 4  |-  (CHOICE  <->  dom  card  =  _V )
3 finnum 8385 . . . . . . 7  |-  ( x  e.  Fin  ->  x  e.  dom  card )
43ssriv 3469 . . . . . 6  |-  Fin  C_  dom  card
5 ssequn2 3640 . . . . . 6  |-  ( Fin  C_  dom  card  <->  ( dom  card  u. 
Fin )  =  dom  card )
64, 5mpbi 212 . . . . 5  |-  ( dom 
card  u.  Fin )  =  dom  card
76eqeq1i 2430 . . . 4  |-  ( ( dom  card  u.  Fin )  =  _V  <->  dom  card  =  _V )
82, 7bitr4i 256 . . 3  |-  (CHOICE  <->  ( dom  card 
u.  Fin )  =  _V )
9 ssv 3485 . . . 4  |-  ( dom 
card  u.  Fin )  C_ 
_V
10 eqss 3480 . . . 4  |-  ( ( dom  card  u.  Fin )  =  _V  <->  ( ( dom  card  u.  Fin )  C_ 
_V  /\  _V  C_  ( dom  card  u.  Fin )
) )
119, 10mpbiran 927 . . 3  |-  ( ( dom  card  u.  Fin )  =  _V  <->  _V  C_  ( dom  card  u.  Fin )
)
12 ssundif 3880 . . 3  |-  ( _V  C_  ( dom  card  u.  Fin )  <->  ( _V  \  dom  card )  C_  Fin )
138, 11, 123bitri 275 . 2  |-  (CHOICE  <->  ( _V  \  dom  card )  C_  Fin )
14 dffin7-2 8830 . . 3  |- FinVII  =  ( Fin 
u.  ( _V  \  dom  card ) )
1514eqeq1i 2430 . 2  |-  (FinVII  =  Fin  <->  ( Fin  u.  ( _V  \  dom  card ) )  =  Fin )
161, 13, 153bitr4i 281 1  |-  (CHOICE  <-> FinVII  =  Fin )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    = wceq 1438   _Vcvv 3082    \ cdif 3434    u. cun 3435    C_ wss 3437   dom cdm 4851   Fincfn 7575   cardccrd 8372  CHOICEwac 8548  FinVIIcfin7 8716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-se 4811  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608  df-riota 6265  df-om 6705  df-wrecs 7034  df-recs 7096  df-er 7369  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-card 8376  df-ac 8549  df-fin7 8723
This theorem is referenced by:  fin71ac  8963
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