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Theorem dfacfin7 9810
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 4134 . 2 ((V ∖ dom card) ⊆ Fin ↔ (Fin ∪ (V ∖ dom card)) = Fin)
2 dfac10 9552 . . . 4 (CHOICE ↔ dom card = V)
3 finnum 9365 . . . . . . 7 (𝑥 ∈ Fin → 𝑥 ∈ dom card)
43ssriv 3946 . . . . . 6 Fin ⊆ dom card
5 ssequn2 4134 . . . . . 6 (Fin ⊆ dom card ↔ (dom card ∪ Fin) = dom card)
64, 5mpbi 233 . . . . 5 (dom card ∪ Fin) = dom card
76eqeq1i 2827 . . . 4 ((dom card ∪ Fin) = V ↔ dom card = V)
82, 7bitr4i 281 . . 3 (CHOICE ↔ (dom card ∪ Fin) = V)
9 ssv 3966 . . . 4 (dom card ∪ Fin) ⊆ V
10 eqss 3957 . . . 4 ((dom card ∪ Fin) = V ↔ ((dom card ∪ Fin) ⊆ V ∧ V ⊆ (dom card ∪ Fin)))
119, 10mpbiran 708 . . 3 ((dom card ∪ Fin) = V ↔ V ⊆ (dom card ∪ Fin))
12 ssundif 4405 . . 3 (V ⊆ (dom card ∪ Fin) ↔ (V ∖ dom card) ⊆ Fin)
138, 11, 123bitri 300 . 2 (CHOICE ↔ (V ∖ dom card) ⊆ Fin)
14 dffin7-2 9809 . . 3 FinVII = (Fin ∪ (V ∖ dom card))
1514eqeq1i 2827 . 2 (FinVII = Fin ↔ (Fin ∪ (V ∖ dom card)) = Fin)
161, 13, 153bitr4i 306 1 (CHOICE ↔ FinVII = Fin)
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  Vcvv 3469  cdif 3905  cun 3906  wss 3908  dom cdm 5532  Fincfn 8496  cardccrd 9352  CHOICEwac 9530  FinVIIcfin7 9695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-om 7566  df-wrecs 7934  df-recs 7995  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-card 9356  df-ac 9531  df-fin7 9702
This theorem is referenced by:  fin71ac  9944
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