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Theorem dfacfin7 8115
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 3424 . 2  |-  ( ( _V  \  dom  card )  C_  Fin  <->  ( Fin  u.  ( _V  \  dom  card ) )  =  Fin )
2 dfac10 7853 . . . 4  |-  (CHOICE  <->  dom  card  =  _V )
3 finnum 7671 . . . . . . 7  |-  ( x  e.  Fin  ->  x  e.  dom  card )
43ssriv 3260 . . . . . 6  |-  Fin  C_  dom  card
5 ssequn2 3424 . . . . . 6  |-  ( Fin  C_  dom  card  <->  ( dom  card  u. 
Fin )  =  dom  card )
64, 5mpbi 199 . . . . 5  |-  ( dom 
card  u.  Fin )  =  dom  card
76eqeq1i 2365 . . . 4  |-  ( ( dom  card  u.  Fin )  =  _V  <->  dom  card  =  _V )
82, 7bitr4i 243 . . 3  |-  (CHOICE  <->  ( dom  card 
u.  Fin )  =  _V )
9 ssv 3274 . . . 4  |-  ( dom 
card  u.  Fin )  C_ 
_V
10 eqss 3270 . . . 4  |-  ( ( dom  card  u.  Fin )  =  _V  <->  ( ( dom  card  u.  Fin )  C_ 
_V  /\  _V  C_  ( dom  card  u.  Fin )
) )
119, 10mpbiran 884 . . 3  |-  ( ( dom  card  u.  Fin )  =  _V  <->  _V  C_  ( dom  card  u.  Fin )
)
12 ssundif 3613 . . 3  |-  ( _V  C_  ( dom  card  u.  Fin )  <->  ( _V  \  dom  card )  C_  Fin )
138, 11, 123bitri 262 . 2  |-  (CHOICE  <->  ( _V  \  dom  card )  C_  Fin )
14 dffin7-2 8114 . . 3  |- FinVII  =  ( Fin 
u.  ( _V  \  dom  card ) )
1514eqeq1i 2365 . 2  |-  (FinVII  =  Fin  <->  ( Fin  u.  ( _V  \  dom  card ) )  =  Fin )
161, 13, 153bitr4i 268 1  |-  (CHOICE  <-> FinVII  =  Fin )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1642   _Vcvv 2864    \ cdif 3225    u. cun 3226    C_ wss 3228   dom cdm 4771   Fincfn 6951   cardccrd 7658  CHOICEwac 7832  FinVIIcfin7 8000
This theorem is referenced by:  fin71ac  8248
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-riota 6391  df-recs 6475  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-card 7662  df-ac 7833  df-fin7 8007
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