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Theorem dfackm 9581
Description: Equivalence of the Axiom of Choice and Maes' AC ackm 9876. The proof consists of lemmas kmlem1 9565 through kmlem16 9580 and this final theorem. AC is not used for the proof. Note: bypassing the first step (i.e. replacing dfac5 9543 with biid 262) 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 9543 . 2 (CHOICE ↔ ∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)))
2 eqid 2826 . . . . 5 {𝑡 ∣ ∃𝑥 𝑡 = ( (𝑥 ∖ {}))} = {𝑡 ∣ ∃𝑥 𝑡 = ( (𝑥 ∖ {}))}
32kmlem13 9577 . . . 4 (∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)) ↔ ∀𝑥(¬ ∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) → ∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))))
4 kmlem8 9572 . . . . 5 ((¬ ∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) → ∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))) ↔ (∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) ∨ ∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦))))
54albii 1813 . . . 4 (∀𝑥(¬ ∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) → ∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧𝑦))) ↔ ∀𝑥(∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) ∨ ∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦))))
63, 5bitri 276 . . 3 (∀𝑥((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)) ↔ ∀𝑥(∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) ∨ ∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦))))
7 df-ne 3022 . . . . . . . . 9 (𝑦𝑣 ↔ ¬ 𝑦 = 𝑣)
87bicomi 225 . . . . . . . 8 𝑦 = 𝑣𝑦𝑣)
98anbi2i 622 . . . . . . 7 ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ↔ (𝑣𝑥𝑦𝑣))
109anbi1i 623 . . . . . 6 (((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣) ↔ ((𝑣𝑥𝑦𝑣) ∧ 𝑧𝑣))
1110imbi2i 337 . . . . 5 ((𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣)) ↔ (𝑧𝑦 → ((𝑣𝑥𝑦𝑣) ∧ 𝑧𝑣)))
12 biid 262 . . . . 5 ((𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣))) ↔ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣))))
13 biid 262 . . . . 5 (∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦) ↔ ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦))
1411, 12, 13kmlem16 9580 . . . 4 ((∃𝑧𝑥𝑣𝑧𝑤𝑥 (𝑧𝑤𝑣 ∈ (𝑧𝑤)) ∨ ∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦))) ↔ ∃𝑦𝑧𝑣𝑢((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑣𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧𝑣))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑣𝑧𝑣𝑦) ∧ ((𝑢𝑧𝑢𝑦) → 𝑢 = 𝑣))))))
1514albii 1813 . . 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 276 . 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 276 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 207  wa 396  wo 843  wal 1528   = wceq 1530  wex 1773  wcel 2107  ∃!weu 2651  {cab 2804  wne 3021  wral 3143  wrex 3144  cdif 3937  cin 3939  c0 4295  {csn 4564   cuni 4837  CHOICEwac 9530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ac 9531
This theorem is referenced by:  axac3  9875  ackm  9876  axac2  9877
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