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Theorem dfacfin7 9181
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 3770 . 2 ((V ∖ dom card) ⊆ Fin ↔ (Fin ∪ (V ∖ dom card)) = Fin)
2 dfac10 8919 . . . 4 (CHOICE ↔ dom card = V)
3 finnum 8734 . . . . . . 7 (𝑥 ∈ Fin → 𝑥 ∈ dom card)
43ssriv 3592 . . . . . 6 Fin ⊆ dom card
5 ssequn2 3770 . . . . . 6 (Fin ⊆ dom card ↔ (dom card ∪ Fin) = dom card)
64, 5mpbi 220 . . . . 5 (dom card ∪ Fin) = dom card
76eqeq1i 2626 . . . 4 ((dom card ∪ Fin) = V ↔ dom card = V)
82, 7bitr4i 267 . . 3 (CHOICE ↔ (dom card ∪ Fin) = V)
9 ssv 3610 . . . 4 (dom card ∪ Fin) ⊆ V
10 eqss 3603 . . . 4 ((dom card ∪ Fin) = V ↔ ((dom card ∪ Fin) ⊆ V ∧ V ⊆ (dom card ∪ Fin)))
119, 10mpbiran 952 . . 3 ((dom card ∪ Fin) = V ↔ V ⊆ (dom card ∪ Fin))
12 ssundif 4030 . . 3 (V ⊆ (dom card ∪ Fin) ↔ (V ∖ dom card) ⊆ Fin)
138, 11, 123bitri 286 . 2 (CHOICE ↔ (V ∖ dom card) ⊆ Fin)
14 dffin7-2 9180 . . 3 FinVII = (Fin ∪ (V ∖ dom card))
1514eqeq1i 2626 . 2 (FinVII = Fin ↔ (Fin ∪ (V ∖ dom card)) = Fin)
161, 13, 153bitr4i 292 1 (CHOICE ↔ FinVII = Fin)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  Vcvv 3190  cdif 3557  cun 3558  wss 3560  dom cdm 5084  Fincfn 7915  cardccrd 8721  CHOICEwac 8898  FinVIIcfin7 9066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-om 7028  df-wrecs 7367  df-recs 7428  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725  df-ac 8899  df-fin7 9073
This theorem is referenced by:  fin71ac  9315
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