Theorem dfacfin7 9537

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 4014	. 2 ⊢ ((V ∖ dom card) ⊆ Fin ↔ (Fin ∪ (V ∖ dom card)) = Fin)
2		dfac10 9275	. . . 4 ⊢ (CHOICE ↔ dom card = V)
3		finnum 9088	. . . . . . 7 ⊢ (𝑥 ∈ Fin → 𝑥 ∈ dom card)
4	3	ssriv 3832	. . . . . 6 ⊢ Fin ⊆ dom card
5		ssequn2 4014	. . . . . 6 ⊢ (Fin ⊆ dom card ↔ (dom card ∪ Fin) = dom card)
6	4, 5	mpbi 222	. . . . 5 ⊢ (dom card ∪ Fin) = dom card
7	6	eqeq1i 2831	. . . 4 ⊢ ((dom card ∪ Fin) = V ↔ dom card = V)
8	2, 7	bitr4i 270	. . 3 ⊢ (CHOICE ↔ (dom card ∪ Fin) = V)
9		ssv 3851	. . . 4 ⊢ (dom card ∪ Fin) ⊆ V
10		eqss 3843	. . . 4 ⊢ ((dom card ∪ Fin) = V ↔ ((dom card ∪ Fin) ⊆ V ∧ V ⊆ (dom card ∪ Fin)))
11	9, 10	mpbiran 702	. . 3 ⊢ ((dom card ∪ Fin) = V ↔ V ⊆ (dom card ∪ Fin))
12		ssundif 4276	. . 3 ⊢ (V ⊆ (dom card ∪ Fin) ↔ (V ∖ dom card) ⊆ Fin)
13	8, 11, 12	3bitri 289	. 2 ⊢ (CHOICE ↔ (V ∖ dom card) ⊆ Fin)
14		dffin7-2 9536	. . 3 ⊢ FinVII = (Fin ∪ (V ∖ dom card))
15	14	eqeq1i 2831	. 2 ⊢ (FinVII = Fin ↔ (Fin ∪ (V ∖ dom card)) = Fin)
16	1, 13, 15	3bitr4i 295	1 ⊢ (CHOICE ↔ FinVII = Fin)

Syntax hints:   ↔ wb 198   = wceq 1658  Vcvv 3415   ∖ cdif 3796   ∪ cun 3797   ⊆ wss 3799  dom cdm 5343  Fincfn 8223  cardccrd 9075  CHOICEwac 9252  Fin^VIIcfin7 9422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rmo 3126  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-int 4699  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-se 5303  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-isom 6133  df-riota 6867  df-om 7328  df-wrecs 7673  df-recs 7735  df-er 8010  df-en 8224  df-dom 8225  df-sdom 8226  df-fin 8227  df-card 9079  df-ac 9253  df-fin7 9429

This theorem is referenced by:  fin71ac  9671