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Theorem dfacfin7 10155

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 4117	. 2 ⊢ ((V ∖ dom card) ⊆ Fin ↔ (Fin ∪ (V ∖ dom card)) = Fin)
2		dfac10 9893	. . . 4 ⊢ (CHOICE ↔ dom card = V)
3		finnum 9706	. . . . . . 7 ⊢ (𝑥 ∈ Fin → 𝑥 ∈ dom card)
4	3	ssriv 3925	. . . . . 6 ⊢ Fin ⊆ dom card
5		ssequn2 4117	. . . . . 6 ⊢ (Fin ⊆ dom card ↔ (dom card ∪ Fin) = dom card)
6	4, 5	mpbi 229	. . . . 5 ⊢ (dom card ∪ Fin) = dom card
7	6	eqeq1i 2743	. . . 4 ⊢ ((dom card ∪ Fin) = V ↔ dom card = V)
8	2, 7	bitr4i 277	. . 3 ⊢ (CHOICE ↔ (dom card ∪ Fin) = V)
9		ssv 3945	. . . 4 ⊢ (dom card ∪ Fin) ⊆ V
10		eqss 3936	. . . 4 ⊢ ((dom card ∪ Fin) = V ↔ ((dom card ∪ Fin) ⊆ V ∧ V ⊆ (dom card ∪ Fin)))
11	9, 10	mpbiran 706	. . 3 ⊢ ((dom card ∪ Fin) = V ↔ V ⊆ (dom card ∪ Fin))
12		ssundif 4418	. . 3 ⊢ (V ⊆ (dom card ∪ Fin) ↔ (V ∖ dom card) ⊆ Fin)
13	8, 11, 12	3bitri 297	. 2 ⊢ (CHOICE ↔ (V ∖ dom card) ⊆ Fin)
14		dffin7-2 10154	. . 3 ⊢ Fin^VII = (Fin ∪ (V ∖ dom card))
15	14	eqeq1i 2743	. 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 205 = wceq 1539 Vcvv 3432 ∖ cdif 3884 ∪ cun 3885 ⊆ wss 3887 dom cdm 5589 Fincfn 8733 cardccrd 9693 CHOICEwac 9871 Fin^VIIcfin7 10040

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-card 9697 df-ac 9872 df-fin7 10047

This theorem is referenced by: fin71ac 10289

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