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Theorem dfacfin7 10468

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 4212	. 2 ⊢ ((V ∖ dom card) ⊆ Fin ↔ (Fin ∪ (V ∖ dom card)) = Fin)
2		dfac10 10207	. . . 4 ⊢ (CHOICE ↔ dom card = V)
3		finnum 10017	. . . . . . 7 ⊢ (𝑥 ∈ Fin → 𝑥 ∈ dom card)
4	3	ssriv 4012	. . . . . 6 ⊢ Fin ⊆ dom card
5		ssequn2 4212	. . . . . 6 ⊢ (Fin ⊆ dom card ↔ (dom card ∪ Fin) = dom card)
6	4, 5	mpbi 230	. . . . 5 ⊢ (dom card ∪ Fin) = dom card
7	6	eqeq1i 2745	. . . 4 ⊢ ((dom card ∪ Fin) = V ↔ dom card = V)
8	2, 7	bitr4i 278	. . 3 ⊢ (CHOICE ↔ (dom card ∪ Fin) = V)
9		ssv 4033	. . . 4 ⊢ (dom card ∪ Fin) ⊆ V
10		eqss 4024	. . . 4 ⊢ ((dom card ∪ Fin) = V ↔ ((dom card ∪ Fin) ⊆ V ∧ V ⊆ (dom card ∪ Fin)))
11	9, 10	mpbiran 708	. . 3 ⊢ ((dom card ∪ Fin) = V ↔ V ⊆ (dom card ∪ Fin))
12		ssundif 4511	. . 3 ⊢ (V ⊆ (dom card ∪ Fin) ↔ (V ∖ dom card) ⊆ Fin)
13	8, 11, 12	3bitri 297	. 2 ⊢ (CHOICE ↔ (V ∖ dom card) ⊆ Fin)
14		dffin7-2 10467	. . 3 ⊢ Fin^VII = (Fin ∪ (V ∖ dom card))
15	14	eqeq1i 2745	. 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 1537 Vcvv 3488 ∖ cdif 3973 ∪ cun 3974 ⊆ wss 3976 dom cdm 5700 Fincfn 9003 cardccrd 10004 CHOICEwac 10184 Fin^VIIcfin7 10353

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-om 7904 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-1o 8522 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-card 10008 df-ac 10185 df-fin7 10360

This theorem is referenced by: fin71ac 10602

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