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Theorem dfacfin7 8239
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 3484 . 2  |-  ( ( _V  \  dom  card )  C_  Fin  <->  ( Fin  u.  ( _V  \  dom  card ) )  =  Fin )
2 dfac10 7977 . . . 4  |-  (CHOICE  <->  dom  card  =  _V )
3 finnum 7795 . . . . . . 7  |-  ( x  e.  Fin  ->  x  e.  dom  card )
43ssriv 3316 . . . . . 6  |-  Fin  C_  dom  card
5 ssequn2 3484 . . . . . 6  |-  ( Fin  C_  dom  card  <->  ( dom  card  u. 
Fin )  =  dom  card )
64, 5mpbi 200 . . . . 5  |-  ( dom 
card  u.  Fin )  =  dom  card
76eqeq1i 2415 . . . 4  |-  ( ( dom  card  u.  Fin )  =  _V  <->  dom  card  =  _V )
82, 7bitr4i 244 . . 3  |-  (CHOICE  <->  ( dom  card 
u.  Fin )  =  _V )
9 ssv 3332 . . . 4  |-  ( dom 
card  u.  Fin )  C_ 
_V
10 eqss 3327 . . . 4  |-  ( ( dom  card  u.  Fin )  =  _V  <->  ( ( dom  card  u.  Fin )  C_ 
_V  /\  _V  C_  ( dom  card  u.  Fin )
) )
119, 10mpbiran 885 . . 3  |-  ( ( dom  card  u.  Fin )  =  _V  <->  _V  C_  ( dom  card  u.  Fin )
)
12 ssundif 3675 . . 3  |-  ( _V  C_  ( dom  card  u.  Fin )  <->  ( _V  \  dom  card )  C_  Fin )
138, 11, 123bitri 263 . 2  |-  (CHOICE  <->  ( _V  \  dom  card )  C_  Fin )
14 dffin7-2 8238 . . 3  |- FinVII  =  ( Fin 
u.  ( _V  \  dom  card ) )
1514eqeq1i 2415 . 2  |-  (FinVII  =  Fin  <->  ( Fin  u.  ( _V  \  dom  card ) )  =  Fin )
161, 13, 153bitr4i 269 1  |-  (CHOICE  <-> FinVII  =  Fin )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649   _Vcvv 2920    \ cdif 3281    u. cun 3282    C_ wss 3284   dom cdm 4841   Fincfn 7072   cardccrd 7782  CHOICEwac 7956  FinVIIcfin7 8124
This theorem is referenced by:  fin71ac  8371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6512  df-recs 6596  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-card 7786  df-ac 7957  df-fin7 8131
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