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Theorem dfacfin7 10248

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 4129	. 2 ⊢ ((V ∖ dom card) ⊆ Fin ↔ (Fin ∪ (V ∖ dom card)) = Fin)
2		dfac10 9986	. . . 4 ⊢ (CHOICE ↔ dom card = V)
3		finnum 9797	. . . . . . 7 ⊢ (𝑥 ∈ Fin → 𝑥 ∈ dom card)
4	3	ssriv 3935	. . . . . 6 ⊢ Fin ⊆ dom card
5		ssequn2 4129	. . . . . 6 ⊢ (Fin ⊆ dom card ↔ (dom card ∪ Fin) = dom card)
6	4, 5	mpbi 229	. . . . 5 ⊢ (dom card ∪ Fin) = dom card
7	6	eqeq1i 2741	. . . 4 ⊢ ((dom card ∪ Fin) = V ↔ dom card = V)
8	2, 7	bitr4i 277	. . 3 ⊢ (CHOICE ↔ (dom card ∪ Fin) = V)
9		ssv 3955	. . . 4 ⊢ (dom card ∪ Fin) ⊆ V
10		eqss 3946	. . . 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 4431	. . 3 ⊢ (V ⊆ (dom card ∪ Fin) ↔ (V ∖ dom card) ⊆ Fin)
13	8, 11, 12	3bitri 296	. 2 ⊢ (CHOICE ↔ (V ∖ dom card) ⊆ Fin)
14		dffin7-2 10247	. . 3 ⊢ Fin^VII = (Fin ∪ (V ∖ dom card))
15	14	eqeq1i 2741	. 2 ⊢ (Fin^VII = Fin ↔ (Fin ∪ (V ∖ dom card)) = Fin)
16	1, 13, 15	3bitr4i 302	1 ⊢ (CHOICE ↔ Fin^VII = Fin)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 = wceq 1540 Vcvv 3441 ∖ cdif 3894 ∪ cun 3895 ⊆ wss 3897 dom cdm 5614 Fincfn 8796 cardccrd 9784 CHOICEwac 9964 Fin^VIIcfin7 10133

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4852 df-int 4894 df-iun 4940 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-se 5570 df-we 5571 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-isom 6482 df-riota 7286 df-ov 7332 df-om 7773 df-2nd 7892 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-1o 8359 df-er 8561 df-en 8797 df-dom 8798 df-sdom 8799 df-fin 8800 df-card 9788 df-ac 9965 df-fin7 10140

This theorem is referenced by: fin71ac 10382

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