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Theorem dfacfin7 8829
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 3607 . 2  |-  ( ( _V  \  dom  card )  C_  Fin  <->  ( Fin  u.  ( _V  \  dom  card ) )  =  Fin )
2 dfac10 8567 . . . 4  |-  (CHOICE  <->  dom  card  =  _V )
3 finnum 8382 . . . . . . 7  |-  ( x  e.  Fin  ->  x  e.  dom  card )
43ssriv 3436 . . . . . 6  |-  Fin  C_  dom  card
5 ssequn2 3607 . . . . . 6  |-  ( Fin  C_  dom  card  <->  ( dom  card  u. 
Fin )  =  dom  card )
64, 5mpbi 212 . . . . 5  |-  ( dom 
card  u.  Fin )  =  dom  card
76eqeq1i 2456 . . . 4  |-  ( ( dom  card  u.  Fin )  =  _V  <->  dom  card  =  _V )
82, 7bitr4i 256 . . 3  |-  (CHOICE  <->  ( dom  card 
u.  Fin )  =  _V )
9 ssv 3452 . . . 4  |-  ( dom 
card  u.  Fin )  C_ 
_V
10 eqss 3447 . . . 4  |-  ( ( dom  card  u.  Fin )  =  _V  <->  ( ( dom  card  u.  Fin )  C_ 
_V  /\  _V  C_  ( dom  card  u.  Fin )
) )
119, 10mpbiran 929 . . 3  |-  ( ( dom  card  u.  Fin )  =  _V  <->  _V  C_  ( dom  card  u.  Fin )
)
12 ssundif 3851 . . 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 8828 . . 3  |- FinVII  =  ( Fin 
u.  ( _V  \  dom  card ) )
1514eqeq1i 2456 . 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 1444   _Vcvv 3045    \ cdif 3401    u. cun 3402    C_ wss 3404   dom cdm 4834   Fincfn 7569   cardccrd 8369  CHOICEwac 8546  FinVIIcfin7 8714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-om 6693  df-wrecs 7028  df-recs 7090  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-ac 8547  df-fin7 8721
This theorem is referenced by:  fin71ac  8961
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