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Theorem dfac11 38173
Description: The right-hand side of this theorem (compare with ac4 9520), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 8674, 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 9165 . . 3 (CHOICE ↔ ∀𝑎𝑐𝑑𝑎 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑))
2 raleq 3291 . . . . . 6 (𝑎 = 𝑥 → (∀𝑑𝑎 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) ↔ ∀𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑)))
32exbidv 2005 . . . . 5 (𝑎 = 𝑥 → (∃𝑐𝑑𝑎 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) ↔ ∃𝑐𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑)))
43cbvalvw 2128 . . . 4 (∀𝑎𝑐𝑑𝑎 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) ↔ ∀𝑥𝑐𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑))
5 neeq1 3008 . . . . . . . . . 10 (𝑑 = 𝑧 → (𝑑 ≠ ∅ ↔ 𝑧 ≠ ∅))
6 fveq2 6348 . . . . . . . . . . 11 (𝑑 = 𝑧 → (𝑐𝑑) = (𝑐𝑧))
7 id 22 . . . . . . . . . . 11 (𝑑 = 𝑧𝑑 = 𝑧)
86, 7eleq12d 2847 . . . . . . . . . 10 (𝑑 = 𝑧 → ((𝑐𝑑) ∈ 𝑑 ↔ (𝑐𝑧) ∈ 𝑧))
95, 8imbi12d 334 . . . . . . . . 9 (𝑑 = 𝑧 → ((𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) ↔ (𝑧 ≠ ∅ → (𝑐𝑧) ∈ 𝑧)))
109cbvralv 3324 . . . . . . . 8 (∀𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑐𝑧) ∈ 𝑧))
11 fveq2 6348 . . . . . . . . . . . . . . 15 (𝑏 = 𝑧 → (𝑐𝑏) = (𝑐𝑧))
1211sneqd 4338 . . . . . . . . . . . . . 14 (𝑏 = 𝑧 → {(𝑐𝑏)} = {(𝑐𝑧)})
13 eqid 2774 . . . . . . . . . . . . . 14 (𝑏𝑥 ↦ {(𝑐𝑏)}) = (𝑏𝑥 ↦ {(𝑐𝑏)})
14 snex 5050 . . . . . . . . . . . . . 14 {(𝑐𝑧)} ∈ V
1512, 13, 14fvmpt 6441 . . . . . . . . . . . . 13 (𝑧𝑥 → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) = {(𝑐𝑧)})
16153ad2ant1 1154 . . . . . . . . . . . 12 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) = {(𝑐𝑧)})
17 simp3 1159 . . . . . . . . . . . . . . . 16 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → (𝑐𝑧) ∈ 𝑧)
1817snssd 4486 . . . . . . . . . . . . . . 15 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ⊆ 𝑧)
1914elpw 4313 . . . . . . . . . . . . . . 15 ({(𝑐𝑧)} ∈ 𝒫 𝑧 ↔ {(𝑐𝑧)} ⊆ 𝑧)
2018, 19sylibr 225 . . . . . . . . . . . . . 14 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ∈ 𝒫 𝑧)
21 snfi 8215 . . . . . . . . . . . . . . 15 {(𝑐𝑧)} ∈ Fin
2221a1i 11 . . . . . . . . . . . . . 14 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ∈ Fin)
2320, 22elind 3956 . . . . . . . . . . . . 13 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ∈ (𝒫 𝑧 ∩ Fin))
24 fvex 6359 . . . . . . . . . . . . . . 15 (𝑐𝑧) ∈ V
2524snnz 4455 . . . . . . . . . . . . . 14 {(𝑐𝑧)} ≠ ∅
2625a1i 11 . . . . . . . . . . . . 13 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ≠ ∅)
27 eldifsn 4464 . . . . . . . . . . . . 13 ({(𝑐𝑧)} ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}) ↔ ({(𝑐𝑧)} ∈ (𝒫 𝑧 ∩ Fin) ∧ {(𝑐𝑧)} ≠ ∅))
2823, 26, 27sylanbrc 573 . . . . . . . . . . . 12 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))
2916, 28eqeltrd 2853 . . . . . . . . . . 11 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))
30293exp 1139 . . . . . . . . . 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 3102 . . . . . . . 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 3358 . . . . . . . . 9 𝑥 ∈ V
3534mptex 6649 . . . . . . . 8 (𝑏𝑥 ↦ {(𝑐𝑏)}) ∈ V
36 fveq1 6347 . . . . . . . . . . 11 (𝑓 = (𝑏𝑥 ↦ {(𝑐𝑏)}) → (𝑓𝑧) = ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧))
3736eleq1d 2838 . . . . . . . . . 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 3138 . . . . . . . 8 (𝑓 = (𝑏𝑥 ↦ {(𝑐𝑏)}) → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
4035, 39spcev 3456 . . . . . . 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 2013 . . . . 5 (∃𝑐𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) → ∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
4342alimi 1890 . . . 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 6359 . . . . . . 7 (𝑅1‘(rank‘𝑎)) ∈ V
4746pwex 4995 . . . . . 6 𝒫 (𝑅1‘(rank‘𝑎)) ∈ V
48 raleq 3291 . . . . . . 7 (𝑥 = 𝒫 (𝑅1‘(rank‘𝑎)) → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) ↔ ∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
4948exbidv 2005 . . . . . 6 (𝑥 = 𝒫 (𝑅1‘(rank‘𝑎)) → (∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) ↔ ∃𝑓𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
5047, 49spcv 3455 . . . . 5 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑓𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
51 rankon 8843 . . . . . . . 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 38172 . . . . . 6 (∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑏 𝑏 We (𝑅1‘(rank‘𝑎)))
5554exlimiv 2013 . . . . 5 (∃𝑓𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑏 𝑏 We (𝑅1‘(rank‘𝑎)))
56 vex 3358 . . . . . 6 𝑎 ∈ V
57 r1rankid 8907 . . . . . 6 (𝑎 ∈ V → 𝑎 ⊆ (𝑅1‘(rank‘𝑎)))
58 wess 5250 . . . . . . 7 (𝑎 ⊆ (𝑅1‘(rank‘𝑎)) → (𝑏 We (𝑅1‘(rank‘𝑎)) → 𝑏 We 𝑎))
5958eximdv 2001 . . . . . 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 2010 . . 3 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∀𝑎𝑏 𝑏 We 𝑎)
63 dfac8 9180 . . 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 1098  wal 1632   = wceq 1634  wex 1855  wcel 2148  wne 2946  wral 3064  Vcvv 3355  cdif 3726  cin 3728  wss 3729  c0 4073  𝒫 cpw 4307  {csn 4326  cmpt 4876   We wwe 5221  Oncon0 5877  cfv 6042  Fincfn 8130  𝑅1cr1 8810  rankcrnk 8811  CHOICEwac 9159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-rep 4917  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117  ax-reg 8674  ax-inf2 8723
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3or 1099  df-3an 1100  df-tru 1637  df-fal 1640  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-ral 3069  df-rex 3070  df-reu 3071  df-rmo 3072  df-rab 3073  df-v 3357  df-sbc 3594  df-csb 3689  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-pss 3745  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-tp 4331  df-op 4333  df-uni 4586  df-int 4623  df-iun 4667  df-br 4798  df-opab 4860  df-mpt 4877  df-tr 4900  df-id 5171  df-eprel 5176  df-po 5184  df-so 5185  df-fr 5222  df-se 5223  df-we 5224  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-pred 5834  df-ord 5880  df-on 5881  df-lim 5882  df-suc 5883  df-iota 6005  df-fun 6044  df-fn 6045  df-f 6046  df-f1 6047  df-fo 6048  df-f1o 6049  df-fv 6050  df-isom 6051  df-riota 6773  df-ov 6815  df-oprab 6816  df-mpt2 6817  df-om 7234  df-1st 7336  df-2nd 7337  df-wrecs 7580  df-recs 7642  df-rdg 7680  df-1o 7734  df-2o 7735  df-er 7917  df-map 8032  df-en 8131  df-fin 8134  df-sup 8525  df-r1 8812  df-rank 8813  df-card 8986  df-ac 9160
This theorem is referenced by: (None)
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