MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfacfin7 Structured version   Visualization version   GIF version

Theorem dfacfin7 10436
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
StepHypRef Expression
1 ssequn2 4198 . 2 ((V ∖ dom card) ⊆ Fin ↔ (Fin ∪ (V ∖ dom card)) = Fin)
2 dfac10 10175 . . . 4 (CHOICE ↔ dom card = V)
3 finnum 9985 . . . . . . 7 (𝑥 ∈ Fin → 𝑥 ∈ dom card)
43ssriv 3998 . . . . . 6 Fin ⊆ dom card
5 ssequn2 4198 . . . . . 6 (Fin ⊆ dom card ↔ (dom card ∪ Fin) = dom card)
64, 5mpbi 230 . . . . 5 (dom card ∪ Fin) = dom card
76eqeq1i 2739 . . . 4 ((dom card ∪ Fin) = V ↔ dom card = V)
82, 7bitr4i 278 . . 3 (CHOICE ↔ (dom card ∪ Fin) = V)
9 ssv 4019 . . . 4 (dom card ∪ Fin) ⊆ V
10 eqss 4010 . . . 4 ((dom card ∪ Fin) = V ↔ ((dom card ∪ Fin) ⊆ V ∧ V ⊆ (dom card ∪ Fin)))
119, 10mpbiran 709 . . 3 ((dom card ∪ Fin) = V ↔ V ⊆ (dom card ∪ Fin))
12 ssundif 4493 . . 3 (V ⊆ (dom card ∪ Fin) ↔ (V ∖ dom card) ⊆ Fin)
138, 11, 123bitri 297 . 2 (CHOICE ↔ (V ∖ dom card) ⊆ Fin)
14 dffin7-2 10435 . . 3 FinVII = (Fin ∪ (V ∖ dom card))
1514eqeq1i 2739 . 2 (FinVII = Fin ↔ (Fin ∪ (V ∖ dom card)) = Fin)
161, 13, 153bitr4i 303 1 (CHOICE ↔ FinVII = Fin)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1536  Vcvv 3477  cdif 3959  cun 3960  wss 3962  dom cdm 5688  Fincfn 8983  cardccrd 9972  CHOICEwac 10152  FinVIIcfin7 10321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-1o 8504  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-card 9976  df-ac 10153  df-fin7 10328
This theorem is referenced by:  fin71ac  10570
  Copyright terms: Public domain W3C validator