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Theorem kmlem2 9011
 Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)
Assertion
Ref Expression
kmlem2 (∃𝑦𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦)) ↔ ∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦))))
Distinct variable groups:   𝑥,𝑦,𝜑   𝑥,𝑤,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑧,𝑤)

Proof of Theorem kmlem2
Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ineq2 3841 . . . . . . . 8 (𝑦 = 𝑣 → (𝑧𝑦) = (𝑧𝑣))
21eleq2d 2716 . . . . . . 7 (𝑦 = 𝑣 → (𝑤 ∈ (𝑧𝑦) ↔ 𝑤 ∈ (𝑧𝑣)))
32eubidv 2518 . . . . . 6 (𝑦 = 𝑣 → (∃!𝑤 𝑤 ∈ (𝑧𝑦) ↔ ∃!𝑤 𝑤 ∈ (𝑧𝑣)))
43imbi2d 329 . . . . 5 (𝑦 = 𝑣 → ((𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦)) ↔ (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑣))))
54ralbidv 3015 . . . 4 (𝑦 = 𝑣 → (∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦)) ↔ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑣))))
65cbvexv 2311 . . 3 (∃𝑦𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦)) ↔ ∃𝑣𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑣)))
7 indi 3906 . . . . . . . . . . . 12 (𝑧 ∩ (𝑣 ∪ {𝑢})) = ((𝑧𝑣) ∪ (𝑧 ∩ {𝑢}))
8 elssuni 4499 . . . . . . . . . . . . . . . . 17 (𝑧𝑥𝑧 𝑥)
98ssneld 3638 . . . . . . . . . . . . . . . 16 (𝑧𝑥 → (¬ 𝑢 𝑥 → ¬ 𝑢𝑧))
10 disjsn 4278 . . . . . . . . . . . . . . . 16 ((𝑧 ∩ {𝑢}) = ∅ ↔ ¬ 𝑢𝑧)
119, 10syl6ibr 242 . . . . . . . . . . . . . . 15 (𝑧𝑥 → (¬ 𝑢 𝑥 → (𝑧 ∩ {𝑢}) = ∅))
1211impcom 445 . . . . . . . . . . . . . 14 ((¬ 𝑢 𝑥𝑧𝑥) → (𝑧 ∩ {𝑢}) = ∅)
1312uneq2d 3800 . . . . . . . . . . . . 13 ((¬ 𝑢 𝑥𝑧𝑥) → ((𝑧𝑣) ∪ (𝑧 ∩ {𝑢})) = ((𝑧𝑣) ∪ ∅))
14 un0 4000 . . . . . . . . . . . . 13 ((𝑧𝑣) ∪ ∅) = (𝑧𝑣)
1513, 14syl6eq 2701 . . . . . . . . . . . 12 ((¬ 𝑢 𝑥𝑧𝑥) → ((𝑧𝑣) ∪ (𝑧 ∩ {𝑢})) = (𝑧𝑣))
167, 15syl5req 2698 . . . . . . . . . . 11 ((¬ 𝑢 𝑥𝑧𝑥) → (𝑧𝑣) = (𝑧 ∩ (𝑣 ∪ {𝑢})))
1716eleq2d 2716 . . . . . . . . . 10 ((¬ 𝑢 𝑥𝑧𝑥) → (𝑤 ∈ (𝑧𝑣) ↔ 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))))
1817eubidv 2518 . . . . . . . . 9 ((¬ 𝑢 𝑥𝑧𝑥) → (∃!𝑤 𝑤 ∈ (𝑧𝑣) ↔ ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))))
1918imbi2d 329 . . . . . . . 8 ((¬ 𝑢 𝑥𝑧𝑥) → ((𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑣)) ↔ (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢})))))
2019ralbidva 3014 . . . . . . 7 𝑢 𝑥 → (∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑣)) ↔ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢})))))
21 vsnid 4242 . . . . . . . . . . . 12 𝑢 ∈ {𝑢}
2221olci 405 . . . . . . . . . . 11 (𝑢𝑣𝑢 ∈ {𝑢})
23 elun 3786 . . . . . . . . . . 11 (𝑢 ∈ (𝑣 ∪ {𝑢}) ↔ (𝑢𝑣𝑢 ∈ {𝑢}))
2422, 23mpbir 221 . . . . . . . . . 10 𝑢 ∈ (𝑣 ∪ {𝑢})
25 elssuni 4499 . . . . . . . . . . 11 ((𝑣 ∪ {𝑢}) ∈ 𝑥 → (𝑣 ∪ {𝑢}) ⊆ 𝑥)
2625sseld 3635 . . . . . . . . . 10 ((𝑣 ∪ {𝑢}) ∈ 𝑥 → (𝑢 ∈ (𝑣 ∪ {𝑢}) → 𝑢 𝑥))
2724, 26mpi 20 . . . . . . . . 9 ((𝑣 ∪ {𝑢}) ∈ 𝑥𝑢 𝑥)
2827con3i 150 . . . . . . . 8 𝑢 𝑥 → ¬ (𝑣 ∪ {𝑢}) ∈ 𝑥)
2928biantrurd 528 . . . . . . 7 𝑢 𝑥 → (∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))) ↔ (¬ (𝑣 ∪ {𝑢}) ∈ 𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))))))
3020, 29bitrd 268 . . . . . 6 𝑢 𝑥 → (∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑣)) ↔ (¬ (𝑣 ∪ {𝑢}) ∈ 𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))))))
31 vex 3234 . . . . . . . 8 𝑣 ∈ V
32 snex 4938 . . . . . . . 8 {𝑢} ∈ V
3331, 32unex 6998 . . . . . . 7 (𝑣 ∪ {𝑢}) ∈ V
34 eleq1 2718 . . . . . . . . 9 (𝑦 = (𝑣 ∪ {𝑢}) → (𝑦𝑥 ↔ (𝑣 ∪ {𝑢}) ∈ 𝑥))
3534notbid 307 . . . . . . . 8 (𝑦 = (𝑣 ∪ {𝑢}) → (¬ 𝑦𝑥 ↔ ¬ (𝑣 ∪ {𝑢}) ∈ 𝑥))
36 ineq2 3841 . . . . . . . . . . . 12 (𝑦 = (𝑣 ∪ {𝑢}) → (𝑧𝑦) = (𝑧 ∩ (𝑣 ∪ {𝑢})))
3736eleq2d 2716 . . . . . . . . . . 11 (𝑦 = (𝑣 ∪ {𝑢}) → (𝑤 ∈ (𝑧𝑦) ↔ 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))))
3837eubidv 2518 . . . . . . . . . 10 (𝑦 = (𝑣 ∪ {𝑢}) → (∃!𝑤 𝑤 ∈ (𝑧𝑦) ↔ ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))))
3938imbi2d 329 . . . . . . . . 9 (𝑦 = (𝑣 ∪ {𝑢}) → ((𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦)) ↔ (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢})))))
4039ralbidv 3015 . . . . . . . 8 (𝑦 = (𝑣 ∪ {𝑢}) → (∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦)) ↔ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢})))))
4135, 40anbi12d 747 . . . . . . 7 (𝑦 = (𝑣 ∪ {𝑢}) → ((¬ 𝑦𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦))) ↔ (¬ (𝑣 ∪ {𝑢}) ∈ 𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))))))
4233, 41spcev 3331 . . . . . 6 ((¬ (𝑣 ∪ {𝑢}) ∈ 𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢})))) → ∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦))))
4330, 42syl6bi 243 . . . . 5 𝑢 𝑥 → (∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑣)) → ∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦)))))
44 vuniex 6996 . . . . . 6 𝑥 ∈ V
45 eleq2 2719 . . . . . . . 8 (𝑦 = 𝑥 → (𝑢𝑦𝑢 𝑥))
4645notbid 307 . . . . . . 7 (𝑦 = 𝑥 → (¬ 𝑢𝑦 ↔ ¬ 𝑢 𝑥))
4746exbidv 1890 . . . . . 6 (𝑦 = 𝑥 → (∃𝑢 ¬ 𝑢𝑦 ↔ ∃𝑢 ¬ 𝑢 𝑥))
48 nalset 4828 . . . . . . . 8 ¬ ∃𝑦𝑢 𝑢𝑦
49 alexn 1811 . . . . . . . 8 (∀𝑦𝑢 ¬ 𝑢𝑦 ↔ ¬ ∃𝑦𝑢 𝑢𝑦)
5048, 49mpbir 221 . . . . . . 7 𝑦𝑢 ¬ 𝑢𝑦
5150spi 2092 . . . . . 6 𝑢 ¬ 𝑢𝑦
5244, 47, 51vtocl 3290 . . . . 5 𝑢 ¬ 𝑢 𝑥
5343, 52exlimiiv 1899 . . . 4 (∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑣)) → ∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦))))
5453exlimiv 1898 . . 3 (∃𝑣𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑣)) → ∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦))))
556, 54sylbi 207 . 2 (∃𝑦𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦)) → ∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦))))
56 exsimpr 1836 . 2 (∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦))) → ∃𝑦𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦)))
5755, 56impbii 199 1 (∃𝑦𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦)) ↔ ∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383  ∀wal 1521   = wceq 1523  ∃wex 1744   ∈ wcel 2030  ∃!weu 2498  ∀wral 2941   ∪ cun 3605   ∩ cin 3606  ∅c0 3948  {csn 4210  ∪ cuni 4468 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-sn 4211  df-pr 4213  df-uni 4469 This theorem is referenced by:  kmlem8  9017
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