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Theorem kmlem2 8830
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 3766 . . . . . . . 8 (𝑦 = 𝑣 → (𝑧𝑦) = (𝑧𝑣))
21eleq2d 2669 . . . . . . 7 (𝑦 = 𝑣 → (𝑤 ∈ (𝑧𝑦) ↔ 𝑤 ∈ (𝑧𝑣)))
32eubidv 2474 . . . . . 6 (𝑦 = 𝑣 → (∃!𝑤 𝑤 ∈ (𝑧𝑦) ↔ ∃!𝑤 𝑤 ∈ (𝑧𝑣)))
43imbi2d 328 . . . . 5 (𝑦 = 𝑣 → ((𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦)) ↔ (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑣))))
54ralbidv 2965 . . . 4 (𝑦 = 𝑣 → (∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦)) ↔ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑣))))
65cbvexv 2258 . . 3 (∃𝑦𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦)) ↔ ∃𝑣𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑣)))
7 indi 3828 . . . . . . . . . . . 12 (𝑧 ∩ (𝑣 ∪ {𝑢})) = ((𝑧𝑣) ∪ (𝑧 ∩ {𝑢}))
8 elssuni 4394 . . . . . . . . . . . . . . . . 17 (𝑧𝑥𝑧 𝑥)
98ssneld 3566 . . . . . . . . . . . . . . . 16 (𝑧𝑥 → (¬ 𝑢 𝑥 → ¬ 𝑢𝑧))
10 disjsn 4188 . . . . . . . . . . . . . . . 16 ((𝑧 ∩ {𝑢}) = ∅ ↔ ¬ 𝑢𝑧)
119, 10syl6ibr 240 . . . . . . . . . . . . . . 15 (𝑧𝑥 → (¬ 𝑢 𝑥 → (𝑧 ∩ {𝑢}) = ∅))
1211impcom 444 . . . . . . . . . . . . . 14 ((¬ 𝑢 𝑥𝑧𝑥) → (𝑧 ∩ {𝑢}) = ∅)
1312uneq2d 3725 . . . . . . . . . . . . 13 ((¬ 𝑢 𝑥𝑧𝑥) → ((𝑧𝑣) ∪ (𝑧 ∩ {𝑢})) = ((𝑧𝑣) ∪ ∅))
14 un0 3915 . . . . . . . . . . . . 13 ((𝑧𝑣) ∪ ∅) = (𝑧𝑣)
1513, 14syl6eq 2656 . . . . . . . . . . . 12 ((¬ 𝑢 𝑥𝑧𝑥) → ((𝑧𝑣) ∪ (𝑧 ∩ {𝑢})) = (𝑧𝑣))
167, 15syl5req 2653 . . . . . . . . . . 11 ((¬ 𝑢 𝑥𝑧𝑥) → (𝑧𝑣) = (𝑧 ∩ (𝑣 ∪ {𝑢})))
1716eleq2d 2669 . . . . . . . . . 10 ((¬ 𝑢 𝑥𝑧𝑥) → (𝑤 ∈ (𝑧𝑣) ↔ 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))))
1817eubidv 2474 . . . . . . . . 9 ((¬ 𝑢 𝑥𝑧𝑥) → (∃!𝑤 𝑤 ∈ (𝑧𝑣) ↔ ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))))
1918imbi2d 328 . . . . . . . 8 ((¬ 𝑢 𝑥𝑧𝑥) → ((𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑣)) ↔ (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢})))))
2019ralbidva 2964 . . . . . . 7 𝑢 𝑥 → (∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑣)) ↔ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢})))))
21 vsnid 4152 . . . . . . . . . . . 12 𝑢 ∈ {𝑢}
2221olci 404 . . . . . . . . . . 11 (𝑢𝑣𝑢 ∈ {𝑢})
23 elun 3711 . . . . . . . . . . 11 (𝑢 ∈ (𝑣 ∪ {𝑢}) ↔ (𝑢𝑣𝑢 ∈ {𝑢}))
2422, 23mpbir 219 . . . . . . . . . 10 𝑢 ∈ (𝑣 ∪ {𝑢})
25 elssuni 4394 . . . . . . . . . . 11 ((𝑣 ∪ {𝑢}) ∈ 𝑥 → (𝑣 ∪ {𝑢}) ⊆ 𝑥)
2625sseld 3563 . . . . . . . . . 10 ((𝑣 ∪ {𝑢}) ∈ 𝑥 → (𝑢 ∈ (𝑣 ∪ {𝑢}) → 𝑢 𝑥))
2724, 26mpi 20 . . . . . . . . 9 ((𝑣 ∪ {𝑢}) ∈ 𝑥𝑢 𝑥)
2827con3i 148 . . . . . . . 8 𝑢 𝑥 → ¬ (𝑣 ∪ {𝑢}) ∈ 𝑥)
2928biantrurd 527 . . . . . . 7 𝑢 𝑥 → (∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))) ↔ (¬ (𝑣 ∪ {𝑢}) ∈ 𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))))))
3020, 29bitrd 266 . . . . . 6 𝑢 𝑥 → (∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑣)) ↔ (¬ (𝑣 ∪ {𝑢}) ∈ 𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))))))
31 vex 3172 . . . . . . . 8 𝑣 ∈ V
32 snex 4827 . . . . . . . 8 {𝑢} ∈ V
3331, 32unex 6828 . . . . . . 7 (𝑣 ∪ {𝑢}) ∈ V
34 eleq1 2672 . . . . . . . . 9 (𝑦 = (𝑣 ∪ {𝑢}) → (𝑦𝑥 ↔ (𝑣 ∪ {𝑢}) ∈ 𝑥))
3534notbid 306 . . . . . . . 8 (𝑦 = (𝑣 ∪ {𝑢}) → (¬ 𝑦𝑥 ↔ ¬ (𝑣 ∪ {𝑢}) ∈ 𝑥))
36 ineq2 3766 . . . . . . . . . . . 12 (𝑦 = (𝑣 ∪ {𝑢}) → (𝑧𝑦) = (𝑧 ∩ (𝑣 ∪ {𝑢})))
3736eleq2d 2669 . . . . . . . . . . 11 (𝑦 = (𝑣 ∪ {𝑢}) → (𝑤 ∈ (𝑧𝑦) ↔ 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))))
3837eubidv 2474 . . . . . . . . . 10 (𝑦 = (𝑣 ∪ {𝑢}) → (∃!𝑤 𝑤 ∈ (𝑧𝑦) ↔ ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))))
3938imbi2d 328 . . . . . . . . 9 (𝑦 = (𝑣 ∪ {𝑢}) → ((𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦)) ↔ (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢})))))
4039ralbidv 2965 . . . . . . . 8 (𝑦 = (𝑣 ∪ {𝑢}) → (∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦)) ↔ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢})))))
4135, 40anbi12d 742 . . . . . . 7 (𝑦 = (𝑣 ∪ {𝑢}) → ((¬ 𝑦𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦))) ↔ (¬ (𝑣 ∪ {𝑢}) ∈ 𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))))))
4233, 41spcev 3269 . . . . . 6 ((¬ (𝑣 ∪ {𝑢}) ∈ 𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢})))) → ∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦))))
4330, 42syl6bi 241 . . . . 5 𝑢 𝑥 → (∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑣)) → ∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦)))))
44 vuniex 6826 . . . . . 6 𝑥 ∈ V
45 eleq2 2673 . . . . . . . 8 (𝑦 = 𝑥 → (𝑢𝑦𝑢 𝑥))
4645notbid 306 . . . . . . 7 (𝑦 = 𝑥 → (¬ 𝑢𝑦 ↔ ¬ 𝑢 𝑥))
4746exbidv 1836 . . . . . 6 (𝑦 = 𝑥 → (∃𝑢 ¬ 𝑢𝑦 ↔ ∃𝑢 ¬ 𝑢 𝑥))
48 nalset 4715 . . . . . . . 8 ¬ ∃𝑦𝑢 𝑢𝑦
49 alexn 1759 . . . . . . . 8 (∀𝑦𝑢 ¬ 𝑢𝑦 ↔ ¬ ∃𝑦𝑢 𝑢𝑦)
5048, 49mpbir 219 . . . . . . 7 𝑦𝑢 ¬ 𝑢𝑦
5150spi 2040 . . . . . 6 𝑢 ¬ 𝑢𝑦
5244, 47, 51vtocl 3228 . . . . 5 𝑢 ¬ 𝑢 𝑥
5343, 52exlimiiv 1845 . . . 4 (∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑣)) → ∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦))))
5453exlimiv 1844 . . 3 (∃𝑣𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑣)) → ∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦))))
556, 54sylbi 205 . 2 (∃𝑦𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦)) → ∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦))))
56 exsimpr 1783 . 2 (∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦))) → ∃𝑦𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦)))
5755, 56impbii 197 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 194  wo 381  wa 382  wal 1472   = wceq 1474  wex 1694  wcel 1976  ∃!weu 2454  wral 2892  cun 3534  cin 3535  c0 3870  {csn 4121   cuni 4363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pr 4825  ax-un 6821
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ral 2897  df-rex 2898  df-v 3171  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-sn 4122  df-pr 4124  df-uni 4364
This theorem is referenced by:  kmlem8  8836
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