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Theorem dfacfin7 10439

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 ↔ Fin^VII = Fin)

**Proof of Theorem dfacfin7**
Step	Hyp	Ref	Expression
1		ssequn2 4189	. 2 ⊢ ((V ∖ dom card) ⊆ Fin ↔ (Fin ∪ (V ∖ dom card)) = Fin)
2		dfac10 10178	. . . 4 ⊢ (CHOICE ↔ dom card = V)
3		finnum 9988	. . . . . . 7 ⊢ (𝑥 ∈ Fin → 𝑥 ∈ dom card)
4	3	ssriv 3987	. . . . . 6 ⊢ Fin ⊆ dom card
5		ssequn2 4189	. . . . . 6 ⊢ (Fin ⊆ dom card ↔ (dom card ∪ Fin) = dom card)
6	4, 5	mpbi 230	. . . . 5 ⊢ (dom card ∪ Fin) = dom card
7	6	eqeq1i 2742	. . . 4 ⊢ ((dom card ∪ Fin) = V ↔ dom card = V)
8	2, 7	bitr4i 278	. . 3 ⊢ (CHOICE ↔ (dom card ∪ Fin) = V)
9		ssv 4008	. . . 4 ⊢ (dom card ∪ Fin) ⊆ V
10		eqss 3999	. . . 4 ⊢ ((dom card ∪ Fin) = V ↔ ((dom card ∪ Fin) ⊆ V ∧ V ⊆ (dom card ∪ Fin)))
11	9, 10	mpbiran 709	. . 3 ⊢ ((dom card ∪ Fin) = V ↔ V ⊆ (dom card ∪ Fin))
12		ssundif 4488	. . 3 ⊢ (V ⊆ (dom card ∪ Fin) ↔ (V ∖ dom card) ⊆ Fin)
13	8, 11, 12	3bitri 297	. 2 ⊢ (CHOICE ↔ (V ∖ dom card) ⊆ Fin)
14		dffin7-2 10438	. . 3 ⊢ Fin^VII = (Fin ∪ (V ∖ dom card))
15	14	eqeq1i 2742	. 2 ⊢ (Fin^VII = Fin ↔ (Fin ∪ (V ∖ dom card)) = Fin)
16	1, 13, 15	3bitr4i 303	1 ⊢ (CHOICE ↔ Fin^VII = Fin)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 = wceq 1540 Vcvv 3480 ∖ cdif 3948 ∪ cun 3949 ⊆ wss 3951 dom cdm 5685 Fincfn 8985 cardccrd 9975 CHOICEwac 10155 Fin^VIIcfin7 10324

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-ac 10156 df-fin7 10331

This theorem is referenced by: fin71ac 10573

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