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Theorem dfac11 37139
Description: The right-hand side of this theorem (compare with ac4 9248), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 8448, 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 8895 . . 3 (CHOICE ↔ ∀𝑎𝑐𝑑𝑎 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑))
2 raleq 3130 . . . . . 6 (𝑎 = 𝑥 → (∀𝑑𝑎 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) ↔ ∀𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑)))
32exbidv 1847 . . . . 5 (𝑎 = 𝑥 → (∃𝑐𝑑𝑎 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) ↔ ∃𝑐𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑)))
43cbvalv 2272 . . . 4 (∀𝑎𝑐𝑑𝑎 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) ↔ ∀𝑥𝑐𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑))
5 neeq1 2852 . . . . . . . . . 10 (𝑑 = 𝑧 → (𝑑 ≠ ∅ ↔ 𝑧 ≠ ∅))
6 fveq2 6153 . . . . . . . . . . 11 (𝑑 = 𝑧 → (𝑐𝑑) = (𝑐𝑧))
7 id 22 . . . . . . . . . . 11 (𝑑 = 𝑧𝑑 = 𝑧)
86, 7eleq12d 2692 . . . . . . . . . 10 (𝑑 = 𝑧 → ((𝑐𝑑) ∈ 𝑑 ↔ (𝑐𝑧) ∈ 𝑧))
95, 8imbi12d 334 . . . . . . . . 9 (𝑑 = 𝑧 → ((𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) ↔ (𝑧 ≠ ∅ → (𝑐𝑧) ∈ 𝑧)))
109cbvralv 3162 . . . . . . . 8 (∀𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑐𝑧) ∈ 𝑧))
11 fveq2 6153 . . . . . . . . . . . . . . 15 (𝑏 = 𝑧 → (𝑐𝑏) = (𝑐𝑧))
1211sneqd 4165 . . . . . . . . . . . . . 14 (𝑏 = 𝑧 → {(𝑐𝑏)} = {(𝑐𝑧)})
13 eqid 2621 . . . . . . . . . . . . . 14 (𝑏𝑥 ↦ {(𝑐𝑏)}) = (𝑏𝑥 ↦ {(𝑐𝑏)})
14 snex 4874 . . . . . . . . . . . . . 14 {(𝑐𝑧)} ∈ V
1512, 13, 14fvmpt 6244 . . . . . . . . . . . . 13 (𝑧𝑥 → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) = {(𝑐𝑧)})
16153ad2ant1 1080 . . . . . . . . . . . 12 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) = {(𝑐𝑧)})
17 simp3 1061 . . . . . . . . . . . . . . . 16 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → (𝑐𝑧) ∈ 𝑧)
1817snssd 4314 . . . . . . . . . . . . . . 15 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ⊆ 𝑧)
1914elpw 4141 . . . . . . . . . . . . . . 15 ({(𝑐𝑧)} ∈ 𝒫 𝑧 ↔ {(𝑐𝑧)} ⊆ 𝑧)
2018, 19sylibr 224 . . . . . . . . . . . . . 14 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ∈ 𝒫 𝑧)
21 snfi 7989 . . . . . . . . . . . . . . 15 {(𝑐𝑧)} ∈ Fin
2221a1i 11 . . . . . . . . . . . . . 14 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ∈ Fin)
2320, 22elind 3781 . . . . . . . . . . . . 13 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ∈ (𝒫 𝑧 ∩ Fin))
24 fvex 6163 . . . . . . . . . . . . . . 15 (𝑐𝑧) ∈ V
2524snnz 4284 . . . . . . . . . . . . . 14 {(𝑐𝑧)} ≠ ∅
2625a1i 11 . . . . . . . . . . . . 13 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ≠ ∅)
27 eldifsn 4292 . . . . . . . . . . . . 13 ({(𝑐𝑧)} ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}) ↔ ({(𝑐𝑧)} ∈ (𝒫 𝑧 ∩ Fin) ∧ {(𝑐𝑧)} ≠ ∅))
2823, 26, 27sylanbrc 697 . . . . . . . . . . . 12 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))
2916, 28eqeltrd 2698 . . . . . . . . . . 11 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))
30293exp 1261 . . . . . . . . . 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 2945 . . . . . . . 8 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑐𝑧) ∈ 𝑧) → ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
3310, 32sylbi 207 . . . . . . 7 (∀𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) → ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
34 vex 3192 . . . . . . . . 9 𝑥 ∈ V
3534mptex 6446 . . . . . . . 8 (𝑏𝑥 ↦ {(𝑐𝑏)}) ∈ V
36 fveq1 6152 . . . . . . . . . . 11 (𝑓 = (𝑏𝑥 ↦ {(𝑐𝑏)}) → (𝑓𝑧) = ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧))
3736eleq1d 2683 . . . . . . . . . 10 (𝑓 = (𝑏𝑥 ↦ {(𝑐𝑏)}) → ((𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}) ↔ ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
3837imbi2d 330 . . . . . . . . 9 (𝑓 = (𝑏𝑥 ↦ {(𝑐𝑏)}) → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) ↔ (𝑧 ≠ ∅ → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
3938ralbidv 2981 . . . . . . . 8 (𝑓 = (𝑏𝑥 ↦ {(𝑐𝑏)}) → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
4035, 39spcev 3289 . . . . . . 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 1855 . . . . 5 (∃𝑐𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) → ∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
4342alimi 1736 . . . 4 (∀𝑥𝑐𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) → ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
444, 43sylbi 207 . . 3 (∀𝑎𝑐𝑑𝑎 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) → ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
451, 44sylbi 207 . 2 (CHOICE → ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
46 fvex 6163 . . . . . . 7 (𝑅1‘(rank‘𝑎)) ∈ V
4746pwex 4813 . . . . . 6 𝒫 (𝑅1‘(rank‘𝑎)) ∈ V
48 raleq 3130 . . . . . . 7 (𝑥 = 𝒫 (𝑅1‘(rank‘𝑎)) → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) ↔ ∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
4948exbidv 1847 . . . . . 6 (𝑥 = 𝒫 (𝑅1‘(rank‘𝑎)) → (∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) ↔ ∃𝑓𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
5047, 49spcv 3288 . . . . 5 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑓𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
51 rankon 8609 . . . . . . . 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 37138 . . . . . 6 (∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑏 𝑏 We (𝑅1‘(rank‘𝑎)))
5554exlimiv 1855 . . . . 5 (∃𝑓𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑏 𝑏 We (𝑅1‘(rank‘𝑎)))
56 vex 3192 . . . . . 6 𝑎 ∈ V
57 r1rankid 8673 . . . . . 6 (𝑎 ∈ V → 𝑎 ⊆ (𝑅1‘(rank‘𝑎)))
58 wess 5066 . . . . . . 7 (𝑎 ⊆ (𝑅1‘(rank‘𝑎)) → (𝑏 We (𝑅1‘(rank‘𝑎)) → 𝑏 We 𝑎))
5958eximdv 1843 . . . . . 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 1852 . . 3 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∀𝑎𝑏 𝑏 We 𝑎)
63 dfac8 8908 . . 3 (CHOICE ↔ ∀𝑎𝑏 𝑏 We 𝑎)
6462, 63sylibr 224 . 2 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → CHOICE)
6545, 64impbii 199 1 (CHOICE ↔ ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1036  wal 1478   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2907  Vcvv 3189  cdif 3556  cin 3558  wss 3559  c0 3896  𝒫 cpw 4135  {csn 4153  cmpt 4678   We wwe 5037  Oncon0 5687  cfv 5852  Fincfn 7906  𝑅1cr1 8576  rankcrnk 8577  CHOICEwac 8889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-reg 8448  ax-inf2 8489
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-er 7694  df-map 7811  df-en 7907  df-fin 7910  df-sup 8299  df-r1 8578  df-rank 8579  df-card 8716  df-ac 8890
This theorem is referenced by: (None)
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