Theorem dfacfin7 9821

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 ↔ FinVII = Fin)

Proof of Theorem dfacfin7

Step	Hyp	Ref	Expression
1		ssequn2 4159	. 2 ⊢ ((V ∖ dom card) ⊆ Fin ↔ (Fin ∪ (V ∖ dom card)) = Fin)
2		dfac10 9563	. . . 4 ⊢ (CHOICE ↔ dom card = V)
3		finnum 9377	. . . . . . 7 ⊢ (𝑥 ∈ Fin → 𝑥 ∈ dom card)
4	3	ssriv 3971	. . . . . 6 ⊢ Fin ⊆ dom card
5		ssequn2 4159	. . . . . 6 ⊢ (Fin ⊆ dom card ↔ (dom card ∪ Fin) = dom card)
6	4, 5	mpbi 232	. . . . 5 ⊢ (dom card ∪ Fin) = dom card
7	6	eqeq1i 2826	. . . 4 ⊢ ((dom card ∪ Fin) = V ↔ dom card = V)
8	2, 7	bitr4i 280	. . 3 ⊢ (CHOICE ↔ (dom card ∪ Fin) = V)
9		ssv 3991	. . . 4 ⊢ (dom card ∪ Fin) ⊆ V
10		eqss 3982	. . . 4 ⊢ ((dom card ∪ Fin) = V ↔ ((dom card ∪ Fin) ⊆ V ∧ V ⊆ (dom card ∪ Fin)))
11	9, 10	mpbiran 707	. . 3 ⊢ ((dom card ∪ Fin) = V ↔ V ⊆ (dom card ∪ Fin))
12		ssundif 4433	. . 3 ⊢ (V ⊆ (dom card ∪ Fin) ↔ (V ∖ dom card) ⊆ Fin)
13	8, 11, 12	3bitri 299	. 2 ⊢ (CHOICE ↔ (V ∖ dom card) ⊆ Fin)
14		dffin7-2 9820	. . 3 ⊢ FinVII = (Fin ∪ (V ∖ dom card))
15	14	eqeq1i 2826	. 2 ⊢ (FinVII = Fin ↔ (Fin ∪ (V ∖ dom card)) = Fin)
16	1, 13, 15	3bitr4i 305	1 ⊢ (CHOICE ↔ FinVII = Fin)

Syntax hints:   ↔ wb 208   = wceq 1537  Vcvv 3494   ∖ cdif 3933   ∪ cun 3934   ⊆ wss 3936  dom cdm 5555  Fincfn 8509  cardccrd 9364  CHOICEwac 9541  Fin^VIIcfin7 9706

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-om 7581  df-wrecs 7947  df-recs 8008  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-card 9368  df-ac 9542  df-fin7 9713

This theorem is referenced by:  fin71ac  9955