Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfac11 Structured version   Visualization version   GIF version

Theorem dfac11 38309
Description: The right-hand side of this theorem (compare with ac4 9550), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 8704, despite not mentioning the cumulative hierarchy in any way as most consequences of regularity do.

This is definition (MC) of [Schechter] p. 141. EDITORIAL: the proof is not original with me of course but I lost my reference sometime after writing it.

A multiple choice function allows any total order to be extended to a choice function, which in turn defines a well-ordering. Since a well-ordering on a set defines a simple ordering of the power set, this allows the trivial well-ordering of the empty set to be transfinitely bootstrapped up the cumulative hierarchy to any desired level. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Stefan O'Rear, 1-Jun-2015.)

Assertion
Ref Expression
dfac11 (CHOICE ↔ ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
Distinct variable group:   𝑥,𝑧,𝑓

Proof of Theorem dfac11
Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfac3 9195 . . 3 (CHOICE ↔ ∀𝑎𝑐𝑑𝑎 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑))
2 raleq 3286 . . . . . 6 (𝑎 = 𝑥 → (∀𝑑𝑎 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) ↔ ∀𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑)))
32exbidv 2016 . . . . 5 (𝑎 = 𝑥 → (∃𝑐𝑑𝑎 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) ↔ ∃𝑐𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑)))
43cbvalvw 2136 . . . 4 (∀𝑎𝑐𝑑𝑎 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) ↔ ∀𝑥𝑐𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑))
5 neeq1 2999 . . . . . . . . . 10 (𝑑 = 𝑧 → (𝑑 ≠ ∅ ↔ 𝑧 ≠ ∅))
6 fveq2 6375 . . . . . . . . . . 11 (𝑑 = 𝑧 → (𝑐𝑑) = (𝑐𝑧))
7 id 22 . . . . . . . . . . 11 (𝑑 = 𝑧𝑑 = 𝑧)
86, 7eleq12d 2838 . . . . . . . . . 10 (𝑑 = 𝑧 → ((𝑐𝑑) ∈ 𝑑 ↔ (𝑐𝑧) ∈ 𝑧))
95, 8imbi12d 335 . . . . . . . . 9 (𝑑 = 𝑧 → ((𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) ↔ (𝑧 ≠ ∅ → (𝑐𝑧) ∈ 𝑧)))
109cbvralv 3319 . . . . . . . 8 (∀𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑐𝑧) ∈ 𝑧))
11 fveq2 6375 . . . . . . . . . . . . . . 15 (𝑏 = 𝑧 → (𝑐𝑏) = (𝑐𝑧))
1211sneqd 4346 . . . . . . . . . . . . . 14 (𝑏 = 𝑧 → {(𝑐𝑏)} = {(𝑐𝑧)})
13 eqid 2765 . . . . . . . . . . . . . 14 (𝑏𝑥 ↦ {(𝑐𝑏)}) = (𝑏𝑥 ↦ {(𝑐𝑏)})
14 snex 5064 . . . . . . . . . . . . . 14 {(𝑐𝑧)} ∈ V
1512, 13, 14fvmpt 6471 . . . . . . . . . . . . 13 (𝑧𝑥 → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) = {(𝑐𝑧)})
16153ad2ant1 1163 . . . . . . . . . . . 12 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) = {(𝑐𝑧)})
17 simp3 1168 . . . . . . . . . . . . . . . 16 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → (𝑐𝑧) ∈ 𝑧)
1817snssd 4494 . . . . . . . . . . . . . . 15 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ⊆ 𝑧)
1914elpw 4321 . . . . . . . . . . . . . . 15 ({(𝑐𝑧)} ∈ 𝒫 𝑧 ↔ {(𝑐𝑧)} ⊆ 𝑧)
2018, 19sylibr 225 . . . . . . . . . . . . . 14 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ∈ 𝒫 𝑧)
21 snfi 8245 . . . . . . . . . . . . . . 15 {(𝑐𝑧)} ∈ Fin
2221a1i 11 . . . . . . . . . . . . . 14 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ∈ Fin)
2320, 22elind 3960 . . . . . . . . . . . . 13 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ∈ (𝒫 𝑧 ∩ Fin))
24 fvex 6388 . . . . . . . . . . . . . . 15 (𝑐𝑧) ∈ V
2524snnz 4463 . . . . . . . . . . . . . 14 {(𝑐𝑧)} ≠ ∅
2625a1i 11 . . . . . . . . . . . . 13 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ≠ ∅)
27 eldifsn 4472 . . . . . . . . . . . . 13 ({(𝑐𝑧)} ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}) ↔ ({(𝑐𝑧)} ∈ (𝒫 𝑧 ∩ Fin) ∧ {(𝑐𝑧)} ≠ ∅))
2823, 26, 27sylanbrc 578 . . . . . . . . . . . 12 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))
2916, 28eqeltrd 2844 . . . . . . . . . . 11 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))
30293exp 1148 . . . . . . . . . 10 (𝑧𝑥 → (𝑧 ≠ ∅ → ((𝑐𝑧) ∈ 𝑧 → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
3130a2d 29 . . . . . . . . 9 (𝑧𝑥 → ((𝑧 ≠ ∅ → (𝑐𝑧) ∈ 𝑧) → (𝑧 ≠ ∅ → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
3231ralimia 3097 . . . . . . . 8 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑐𝑧) ∈ 𝑧) → ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
3310, 32sylbi 208 . . . . . . 7 (∀𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) → ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
34 vex 3353 . . . . . . . . 9 𝑥 ∈ V
3534mptex 6679 . . . . . . . 8 (𝑏𝑥 ↦ {(𝑐𝑏)}) ∈ V
36 fveq1 6374 . . . . . . . . . . 11 (𝑓 = (𝑏𝑥 ↦ {(𝑐𝑏)}) → (𝑓𝑧) = ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧))
3736eleq1d 2829 . . . . . . . . . 10 (𝑓 = (𝑏𝑥 ↦ {(𝑐𝑏)}) → ((𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}) ↔ ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
3837imbi2d 331 . . . . . . . . 9 (𝑓 = (𝑏𝑥 ↦ {(𝑐𝑏)}) → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) ↔ (𝑧 ≠ ∅ → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
3938ralbidv 3133 . . . . . . . 8 (𝑓 = (𝑏𝑥 ↦ {(𝑐𝑏)}) → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
4035, 39spcev 3452 . . . . . . 7 (∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
4133, 40syl 17 . . . . . 6 (∀𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) → ∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
4241exlimiv 2025 . . . . 5 (∃𝑐𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) → ∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
4342alimi 1906 . . . 4 (∀𝑥𝑐𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) → ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
444, 43sylbi 208 . . 3 (∀𝑎𝑐𝑑𝑎 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) → ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
451, 44sylbi 208 . 2 (CHOICE → ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
46 fvex 6388 . . . . . . 7 (𝑅1‘(rank‘𝑎)) ∈ V
4746pwex 5016 . . . . . 6 𝒫 (𝑅1‘(rank‘𝑎)) ∈ V
48 raleq 3286 . . . . . . 7 (𝑥 = 𝒫 (𝑅1‘(rank‘𝑎)) → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) ↔ ∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
4948exbidv 2016 . . . . . 6 (𝑥 = 𝒫 (𝑅1‘(rank‘𝑎)) → (∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) ↔ ∃𝑓𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
5047, 49spcv 3451 . . . . 5 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑓𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
51 rankon 8873 . . . . . . . 8 (rank‘𝑎) ∈ On
5251a1i 11 . . . . . . 7 (∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → (rank‘𝑎) ∈ On)
53 id 22 . . . . . . 7 (∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
5452, 53aomclem8 38308 . . . . . 6 (∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑏 𝑏 We (𝑅1‘(rank‘𝑎)))
5554exlimiv 2025 . . . . 5 (∃𝑓𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑏 𝑏 We (𝑅1‘(rank‘𝑎)))
56 vex 3353 . . . . . 6 𝑎 ∈ V
57 r1rankid 8937 . . . . . 6 (𝑎 ∈ V → 𝑎 ⊆ (𝑅1‘(rank‘𝑎)))
58 wess 5264 . . . . . . 7 (𝑎 ⊆ (𝑅1‘(rank‘𝑎)) → (𝑏 We (𝑅1‘(rank‘𝑎)) → 𝑏 We 𝑎))
5958eximdv 2012 . . . . . 6 (𝑎 ⊆ (𝑅1‘(rank‘𝑎)) → (∃𝑏 𝑏 We (𝑅1‘(rank‘𝑎)) → ∃𝑏 𝑏 We 𝑎))
6056, 57, 59mp2b 10 . . . . 5 (∃𝑏 𝑏 We (𝑅1‘(rank‘𝑎)) → ∃𝑏 𝑏 We 𝑎)
6150, 55, 603syl 18 . . . 4 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑏 𝑏 We 𝑎)
6261alrimiv 2022 . . 3 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∀𝑎𝑏 𝑏 We 𝑎)
63 dfac8 9210 . . 3 (CHOICE ↔ ∀𝑎𝑏 𝑏 We 𝑎)
6462, 63sylibr 225 . 2 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → CHOICE)
6545, 64impbii 200 1 (CHOICE ↔ ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  w3a 1107  wal 1650   = wceq 1652  wex 1874  wcel 2155  wne 2937  wral 3055  Vcvv 3350  cdif 3729  cin 3731  wss 3732  c0 4079  𝒫 cpw 4315  {csn 4334  cmpt 4888   We wwe 5235  Oncon0 5908  cfv 6068  Fincfn 8160  𝑅1cr1 8840  rankcrnk 8841  CHOICEwac 9189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-reg 8704  ax-inf2 8753
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-er 7947  df-map 8062  df-en 8161  df-fin 8164  df-sup 8555  df-r1 8842  df-rank 8843  df-card 9016  df-ac 9190
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator