Theorem dfacfin7 10321

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

Step	Hyp	Ref	Expression
1		ssequn2 4129	. 2 ⊢ ((V ∖ dom card) ⊆ Fin ↔ (Fin ∪ (V ∖ dom card)) = Fin)
2		dfac10 10060	. . . 4 ⊢ (CHOICE ↔ dom card = V)
3		finnum 9872	. . . . . . 7 ⊢ (𝑥 ∈ Fin → 𝑥 ∈ dom card)
4	3	ssriv 3925	. . . . . 6 ⊢ Fin ⊆ dom card
5		ssequn2 4129	. . . . . 6 ⊢ (Fin ⊆ dom card ↔ (dom card ∪ Fin) = dom card)
6	4, 5	mpbi 230	. . . . 5 ⊢ (dom card ∪ Fin) = dom card
7	6	eqeq1i 2741	. . . 4 ⊢ ((dom card ∪ Fin) = V ↔ dom card = V)
8	2, 7	bitr4i 278	. . 3 ⊢ (CHOICE ↔ (dom card ∪ Fin) = V)
9		ssv 3946	. . . 4 ⊢ (dom card ∪ Fin) ⊆ V
10		eqss 3937	. . . 4 ⊢ ((dom card ∪ Fin) = V ↔ ((dom card ∪ Fin) ⊆ V ∧ V ⊆ (dom card ∪ Fin)))
11	9, 10	mpbiran 710	. . 3 ⊢ ((dom card ∪ Fin) = V ↔ V ⊆ (dom card ∪ Fin))
12		ssundif 4427	. . 3 ⊢ (V ⊆ (dom card ∪ Fin) ↔ (V ∖ dom card) ⊆ Fin)
13	8, 11, 12	3bitri 297	. 2 ⊢ (CHOICE ↔ (V ∖ dom card) ⊆ Fin)
14		dffin7-2 10320	. . 3 ⊢ FinVII = (Fin ∪ (V ∖ dom card))
15	14	eqeq1i 2741	. 2 ⊢ (FinVII = Fin ↔ (Fin ∪ (V ∖ dom card)) = Fin)
16	1, 13, 15	3bitr4i 303	1 ⊢ (CHOICE ↔ FinVII = Fin)

Syntax hints:   ↔ wb 206   = wceq 1542  Vcvv 3429   ∖ cdif 3886   ∪ cun 3887   ⊆ wss 3889  dom cdm 5631  Fincfn 8893  cardccrd 9859  CHOICEwac 10037  Fin^VIIcfin7 10206

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-1o 8405  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-card 9863  df-ac 10038  df-fin7 10213

This theorem is referenced by:  fin71ac  10455