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Theorem dfacfin7 9809
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 4156 . 2 ((V ∖ dom card) ⊆ Fin ↔ (Fin ∪ (V ∖ dom card)) = Fin)
2 dfac10 9551 . . . 4 (CHOICE ↔ dom card = V)
3 finnum 9365 . . . . . . 7 (𝑥 ∈ Fin → 𝑥 ∈ dom card)
43ssriv 3968 . . . . . 6 Fin ⊆ dom card
5 ssequn2 4156 . . . . . 6 (Fin ⊆ dom card ↔ (dom card ∪ Fin) = dom card)
64, 5mpbi 231 . . . . 5 (dom card ∪ Fin) = dom card
76eqeq1i 2823 . . . 4 ((dom card ∪ Fin) = V ↔ dom card = V)
82, 7bitr4i 279 . . 3 (CHOICE ↔ (dom card ∪ Fin) = V)
9 ssv 3988 . . . 4 (dom card ∪ Fin) ⊆ V
10 eqss 3979 . . . 4 ((dom card ∪ Fin) = V ↔ ((dom card ∪ Fin) ⊆ V ∧ V ⊆ (dom card ∪ Fin)))
119, 10mpbiran 705 . . 3 ((dom card ∪ Fin) = V ↔ V ⊆ (dom card ∪ Fin))
12 ssundif 4429 . . 3 (V ⊆ (dom card ∪ Fin) ↔ (V ∖ dom card) ⊆ Fin)
138, 11, 123bitri 298 . 2 (CHOICE ↔ (V ∖ dom card) ⊆ Fin)
14 dffin7-2 9808 . . 3 FinVII = (Fin ∪ (V ∖ dom card))
1514eqeq1i 2823 . 2 (FinVII = Fin ↔ (Fin ∪ (V ∖ dom card)) = Fin)
161, 13, 153bitr4i 304 1 (CHOICE ↔ FinVII = Fin)
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528  Vcvv 3492  cdif 3930  cun 3931  wss 3933  dom cdm 5548  Fincfn 8497  cardccrd 9352  CHOICEwac 9529  FinVIIcfin7 9694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-om 7570  df-wrecs 7936  df-recs 7997  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-card 9356  df-ac 9530  df-fin7 9701
This theorem is referenced by:  fin71ac  9943
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