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Theorem dfacfin7 10307

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 4139	. 2 ⊢ ((V ∖ dom card) ⊆ Fin ↔ (Fin ∪ (V ∖ dom card)) = Fin)
2		dfac10 10046	. . . 4 ⊢ (CHOICE ↔ dom card = V)
3		finnum 9858	. . . . . . 7 ⊢ (𝑥 ∈ Fin → 𝑥 ∈ dom card)
4	3	ssriv 3935	. . . . . 6 ⊢ Fin ⊆ dom card
5		ssequn2 4139	. . . . . 6 ⊢ (Fin ⊆ dom card ↔ (dom card ∪ Fin) = dom card)
6	4, 5	mpbi 230	. . . . 5 ⊢ (dom card ∪ Fin) = dom card
7	6	eqeq1i 2739	. . . 4 ⊢ ((dom card ∪ Fin) = V ↔ dom card = V)
8	2, 7	bitr4i 278	. . 3 ⊢ (CHOICE ↔ (dom card ∪ Fin) = V)
9		ssv 3956	. . . 4 ⊢ (dom card ∪ Fin) ⊆ V
10		eqss 3947	. . . 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 4438	. . 3 ⊢ (V ⊆ (dom card ∪ Fin) ↔ (V ∖ dom card) ⊆ Fin)
13	8, 11, 12	3bitri 297	. 2 ⊢ (CHOICE ↔ (V ∖ dom card) ⊆ Fin)
14		dffin7-2 10306	. . 3 ⊢ Fin^VII = (Fin ∪ (V ∖ dom card))
15	14	eqeq1i 2739	. 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 1541 Vcvv 3438 ∖ cdif 3896 ∪ cun 3897 ⊆ wss 3899 dom cdm 5622 Fincfn 8881 cardccrd 9845 CHOICEwac 10023 Fin^VIIcfin7 10192

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7313 df-ov 7359 df-om 7807 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-1o 8395 df-er 8633 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-card 9849 df-ac 10024 df-fin7 10199

This theorem is referenced by: fin71ac 10441

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