Theorem dfacfin7 9810

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 4110	. 2 ⊢ ((V ∖ dom card) ⊆ Fin ↔ (Fin ∪ (V ∖ dom card)) = Fin)
2		dfac10 9548	. . . 4 ⊢ (CHOICE ↔ dom card = V)
3		finnum 9361	. . . . . . 7 ⊢ (𝑥 ∈ Fin → 𝑥 ∈ dom card)
4	3	ssriv 3919	. . . . . 6 ⊢ Fin ⊆ dom card
5		ssequn2 4110	. . . . . 6 ⊢ (Fin ⊆ dom card ↔ (dom card ∪ Fin) = dom card)
6	4, 5	mpbi 233	. . . . 5 ⊢ (dom card ∪ Fin) = dom card
7	6	eqeq1i 2803	. . . 4 ⊢ ((dom card ∪ Fin) = V ↔ dom card = V)
8	2, 7	bitr4i 281	. . 3 ⊢ (CHOICE ↔ (dom card ∪ Fin) = V)
9		ssv 3939	. . . 4 ⊢ (dom card ∪ Fin) ⊆ V
10		eqss 3930	. . . 4 ⊢ ((dom card ∪ Fin) = V ↔ ((dom card ∪ Fin) ⊆ V ∧ V ⊆ (dom card ∪ Fin)))
11	9, 10	mpbiran 708	. . 3 ⊢ ((dom card ∪ Fin) = V ↔ V ⊆ (dom card ∪ Fin))
12		ssundif 4391	. . 3 ⊢ (V ⊆ (dom card ∪ Fin) ↔ (V ∖ dom card) ⊆ Fin)
13	8, 11, 12	3bitri 300	. 2 ⊢ (CHOICE ↔ (V ∖ dom card) ⊆ Fin)
14		dffin7-2 9809	. . 3 ⊢ FinVII = (Fin ∪ (V ∖ dom card))
15	14	eqeq1i 2803	. 2 ⊢ (FinVII = Fin ↔ (Fin ∪ (V ∖ dom card)) = Fin)
16	1, 13, 15	3bitr4i 306	1 ⊢ (CHOICE ↔ FinVII = Fin)

Syntax hints:   ↔ wb 209   = wceq 1538  Vcvv 3441   ∖ cdif 3878   ∪ cun 3879   ⊆ wss 3881  dom cdm 5519  Fincfn 8492  cardccrd 9348  CHOICEwac 9526  Fin^VIIcfin7 9695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-om 7561  df-wrecs 7930  df-recs 7991  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-card 9352  df-ac 9527  df-fin7 9702

This theorem is referenced by:  fin71ac  9944