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Theorem kmlem2 9569
 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 4181 . . . . . . . 8 (𝑦 = 𝑣 → (𝑧𝑦) = (𝑧𝑣))
21eleq2d 2896 . . . . . . 7 (𝑦 = 𝑣 → (𝑤 ∈ (𝑧𝑦) ↔ 𝑤 ∈ (𝑧𝑣)))
32eubidv 2666 . . . . . 6 (𝑦 = 𝑣 → (∃!𝑤 𝑤 ∈ (𝑧𝑦) ↔ ∃!𝑤 𝑤 ∈ (𝑧𝑣)))
43imbi2d 343 . . . . 5 (𝑦 = 𝑣 → ((𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦)) ↔ (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑣))))
54ralbidv 3195 . . . 4 (𝑦 = 𝑣 → (∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦)) ↔ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑣))))
65cbvexvw 2038 . . 3 (∃𝑦𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦)) ↔ ∃𝑣𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑣)))
7 indi 4248 . . . . . . . . . . . 12 (𝑧 ∩ (𝑣 ∪ {𝑢})) = ((𝑧𝑣) ∪ (𝑧 ∩ {𝑢}))
8 elssuni 4859 . . . . . . . . . . . . . . . . 17 (𝑧𝑥𝑧 𝑥)
98ssneld 3967 . . . . . . . . . . . . . . . 16 (𝑧𝑥 → (¬ 𝑢 𝑥 → ¬ 𝑢𝑧))
10 disjsn 4639 . . . . . . . . . . . . . . . 16 ((𝑧 ∩ {𝑢}) = ∅ ↔ ¬ 𝑢𝑧)
119, 10syl6ibr 254 . . . . . . . . . . . . . . 15 (𝑧𝑥 → (¬ 𝑢 𝑥 → (𝑧 ∩ {𝑢}) = ∅))
1211impcom 410 . . . . . . . . . . . . . 14 ((¬ 𝑢 𝑥𝑧𝑥) → (𝑧 ∩ {𝑢}) = ∅)
1312uneq2d 4137 . . . . . . . . . . . . 13 ((¬ 𝑢 𝑥𝑧𝑥) → ((𝑧𝑣) ∪ (𝑧 ∩ {𝑢})) = ((𝑧𝑣) ∪ ∅))
14 un0 4342 . . . . . . . . . . . . 13 ((𝑧𝑣) ∪ ∅) = (𝑧𝑣)
1513, 14syl6eq 2870 . . . . . . . . . . . 12 ((¬ 𝑢 𝑥𝑧𝑥) → ((𝑧𝑣) ∪ (𝑧 ∩ {𝑢})) = (𝑧𝑣))
167, 15syl5req 2867 . . . . . . . . . . 11 ((¬ 𝑢 𝑥𝑧𝑥) → (𝑧𝑣) = (𝑧 ∩ (𝑣 ∪ {𝑢})))
1716eleq2d 2896 . . . . . . . . . 10 ((¬ 𝑢 𝑥𝑧𝑥) → (𝑤 ∈ (𝑧𝑣) ↔ 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))))
1817eubidv 2666 . . . . . . . . 9 ((¬ 𝑢 𝑥𝑧𝑥) → (∃!𝑤 𝑤 ∈ (𝑧𝑣) ↔ ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))))
1918imbi2d 343 . . . . . . . 8 ((¬ 𝑢 𝑥𝑧𝑥) → ((𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑣)) ↔ (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢})))))
2019ralbidva 3194 . . . . . . 7 𝑢 𝑥 → (∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑣)) ↔ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢})))))
21 vsnid 4594 . . . . . . . . . . . 12 𝑢 ∈ {𝑢}
2221olci 862 . . . . . . . . . . 11 (𝑢𝑣𝑢 ∈ {𝑢})
23 elun 4123 . . . . . . . . . . 11 (𝑢 ∈ (𝑣 ∪ {𝑢}) ↔ (𝑢𝑣𝑢 ∈ {𝑢}))
2422, 23mpbir 233 . . . . . . . . . 10 𝑢 ∈ (𝑣 ∪ {𝑢})
25 elssuni 4859 . . . . . . . . . . 11 ((𝑣 ∪ {𝑢}) ∈ 𝑥 → (𝑣 ∪ {𝑢}) ⊆ 𝑥)
2625sseld 3964 . . . . . . . . . 10 ((𝑣 ∪ {𝑢}) ∈ 𝑥 → (𝑢 ∈ (𝑣 ∪ {𝑢}) → 𝑢 𝑥))
2724, 26mpi 20 . . . . . . . . 9 ((𝑣 ∪ {𝑢}) ∈ 𝑥𝑢 𝑥)
2827con3i 157 . . . . . . . 8 𝑢 𝑥 → ¬ (𝑣 ∪ {𝑢}) ∈ 𝑥)
2928biantrurd 535 . . . . . . 7 𝑢 𝑥 → (∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))) ↔ (¬ (𝑣 ∪ {𝑢}) ∈ 𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))))))
3020, 29bitrd 281 . . . . . 6 𝑢 𝑥 → (∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑣)) ↔ (¬ (𝑣 ∪ {𝑢}) ∈ 𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))))))
31 vex 3496 . . . . . . . 8 𝑣 ∈ V
32 snex 5322 . . . . . . . 8 {𝑢} ∈ V
3331, 32unex 7461 . . . . . . 7 (𝑣 ∪ {𝑢}) ∈ V
34 eleq1 2898 . . . . . . . . 9 (𝑦 = (𝑣 ∪ {𝑢}) → (𝑦𝑥 ↔ (𝑣 ∪ {𝑢}) ∈ 𝑥))
3534notbid 320 . . . . . . . 8 (𝑦 = (𝑣 ∪ {𝑢}) → (¬ 𝑦𝑥 ↔ ¬ (𝑣 ∪ {𝑢}) ∈ 𝑥))
36 ineq2 4181 . . . . . . . . . . . 12 (𝑦 = (𝑣 ∪ {𝑢}) → (𝑧𝑦) = (𝑧 ∩ (𝑣 ∪ {𝑢})))
3736eleq2d 2896 . . . . . . . . . . 11 (𝑦 = (𝑣 ∪ {𝑢}) → (𝑤 ∈ (𝑧𝑦) ↔ 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))))
3837eubidv 2666 . . . . . . . . . 10 (𝑦 = (𝑣 ∪ {𝑢}) → (∃!𝑤 𝑤 ∈ (𝑧𝑦) ↔ ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))))
3938imbi2d 343 . . . . . . . . 9 (𝑦 = (𝑣 ∪ {𝑢}) → ((𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦)) ↔ (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢})))))
4039ralbidv 3195 . . . . . . . 8 (𝑦 = (𝑣 ∪ {𝑢}) → (∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦)) ↔ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢})))))
4135, 40anbi12d 632 . . . . . . 7 (𝑦 = (𝑣 ∪ {𝑢}) → ((¬ 𝑦𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦))) ↔ (¬ (𝑣 ∪ {𝑢}) ∈ 𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))))))
4233, 41spcev 3605 . . . . . 6 ((¬ (𝑣 ∪ {𝑢}) ∈ 𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢})))) → ∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦))))
4330, 42syl6bi 255 . . . . 5 𝑢 𝑥 → (∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑣)) → ∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦)))))
44 vuniex 7457 . . . . . 6 𝑥 ∈ V
45 eleq2 2899 . . . . . . . 8 (𝑦 = 𝑥 → (𝑢𝑦𝑢 𝑥))
4645notbid 320 . . . . . . 7 (𝑦 = 𝑥 → (¬ 𝑢𝑦 ↔ ¬ 𝑢 𝑥))
4746exbidv 1916 . . . . . 6 (𝑦 = 𝑥 → (∃𝑢 ¬ 𝑢𝑦 ↔ ∃𝑢 ¬ 𝑢 𝑥))
48 nalset 5208 . . . . . . . 8 ¬ ∃𝑦𝑢 𝑢𝑦
49 alexn 1839 . . . . . . . 8 (∀𝑦𝑢 ¬ 𝑢𝑦 ↔ ¬ ∃𝑦𝑢 𝑢𝑦)
5048, 49mpbir 233 . . . . . . 7 𝑦𝑢 ¬ 𝑢𝑦
5150spi 2176 . . . . . 6 𝑢 ¬ 𝑢𝑦
5244, 47, 51vtocl 3558 . . . . 5 𝑢 ¬ 𝑢 𝑥
5343, 52exlimiiv 1926 . . . 4 (∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑣)) → ∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦))))
5453exlimiv 1925 . . 3 (∃𝑣𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑣)) → ∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦))))
556, 54sylbi 219 . 2 (∃𝑦𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦)) → ∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦))))
56 exsimpr 1864 . 2 (∃𝑦𝑦𝑥 ∧ ∀𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦))) → ∃𝑦𝑧𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧𝑦)))
5755, 56impbii 211 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 208   ∧ wa 398   ∨ wo 843  ∀wal 1529   = wceq 1531  ∃wex 1774   ∈ wcel 2108  ∃!weu 2647  ∀wral 3136   ∪ cun 3932   ∩ cin 3933  ∅c0 4289  {csn 4559  ∪ cuni 4830 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 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-sn 4560  df-pr 4562  df-uni 4831 This theorem is referenced by:  kmlem8  9575
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