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Theorem dfackm 10190
Description: Equivalence of the Axiom of Choice and Maes' AC ackm 10489. The proof consists of lemmas kmlem1 10174 through kmlem16 10189 and this final theorem. AC is not used for the proof. Note: bypassing the first step (i.e., replacing dfac5 10152 with biid 261) establishes the AC equivalence shown by Maes' writeup. The left-hand-side AC shown here was chosen because it is shorter to display. (Contributed by NM, 13-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.)
Assertion
Ref Expression
dfackm (CHOICE ↔ ∀𝑥𝑦𝑧𝑣𝑢((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣))))))
Distinct variable group:   𝑥,𝑦,𝑧,𝑣,𝑢

Proof of Theorem dfackm
Dummy variables 𝑤 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfac5 10152 . 2 (CHOICE ↔ ∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)))
2 eqid 2728 . . . . 5 {𝑡 ∣ ∃𝑥 𝑡 = ( (𝑥 ∖ {}))} = {𝑡 ∣ ∃𝑥 𝑡 = ( (𝑥 ∖ {}))}
32kmlem13 10186 . . . 4 (∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)) ↔ ∀𝑥(¬ ∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) → ∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))))
4 kmlem8 10181 . . . . 5 ((¬ ∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) → ∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))) ↔ (∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) ∨ ∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦))))
54albii 1814 . . . 4 (∀𝑥(¬ ∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) → ∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))) ↔ ∀𝑥(∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) ∨ ∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦))))
63, 5bitri 275 . . 3 (∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)) ↔ ∀𝑥(∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) ∨ ∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦))))
7 df-ne 2938 . . . . . . . . 9 (𝑦𝑣 ↔ ¬ 𝑦 = 𝑣)
87bicomi 223 . . . . . . . 8 𝑦 = 𝑣𝑦𝑣)
98anbi2i 622 . . . . . . 7 ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ↔ (𝑣𝑥𝑦𝑣))
109anbi1i 623 . . . . . 6 (((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣) ↔ ((𝑣𝑥𝑦𝑣) ∧ 𝑧𝑣))
1110imbi2i 336 . . . . 5 ((𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣)) ↔ (𝑧𝑦 → ((𝑣𝑥𝑦𝑣) ∧ 𝑧𝑣)))
12 biid 261 . . . . 5 ((𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣))) ↔ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣))))
13 biid 261 . . . . 5 (∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦) ↔ ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦))
1411, 12, 13kmlem16 10189 . . . 4 ((∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) ∨ ∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦))) ↔ ∃𝑦𝑧𝑣𝑢((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣))))))
1514albii 1814 . . 3 (∀𝑥(∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) ∨ ∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦))) ↔ ∀𝑥𝑦𝑧𝑣𝑢((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣))))))
166, 15bitri 275 . 2 (∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)) ↔ ∀𝑥𝑦𝑧𝑣𝑢((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣))))))
171, 16bitri 275 1 (CHOICE ↔ ∀𝑥𝑦𝑧𝑣𝑢((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wo 846  wal 1532   = wceq 1534  wex 1774  wcel 2099  ∃!weu 2558  {cab 2705  wne 2937  wral 3058  wrex 3067  cdif 3944  cin 3946  c0 4323  {csn 4629   cuni 4908  CHOICEwac 10139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-ac 10140
This theorem is referenced by:  axac3  10488  ackm  10489  axac2  10490
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