Theorem kmlem2 9562

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	⊢ (∃𝑦∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑦)) ↔ ∃𝑦(¬ 𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑦))))

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.

Step	Hyp	Ref	Expression
1		ineq2 4133	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → (𝑧 ∩ 𝑦) = (𝑧 ∩ 𝑣))
2	1	eleq2d 2875	. . . . . . 7 ⊢ (𝑦 = 𝑣 → (𝑤 ∈ (𝑧 ∩ 𝑦) ↔ 𝑤 ∈ (𝑧 ∩ 𝑣)))
3	2	eubidv 2647	. . . . . 6 ⊢ (𝑦 = 𝑣 → (∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑦) ↔ ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑣)))
4	3	imbi2d 344	. . . . 5 ⊢ (𝑦 = 𝑣 → ((𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑦)) ↔ (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑣))))
5	4	ralbidv 3162	. . . 4 ⊢ (𝑦 = 𝑣 → (∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑦)) ↔ ∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑣))))
6	5	cbvexvw 2044	. . 3 ⊢ (∃𝑦∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑦)) ↔ ∃𝑣∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑣)))
7		indi 4200	. . . . . . . . . . . 12 ⊢ (𝑧 ∩ (𝑣 ∪ {𝑢})) = ((𝑧 ∩ 𝑣) ∪ (𝑧 ∩ {𝑢}))
8		elssuni 4830	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ 𝑥 → 𝑧 ⊆ ∪ 𝑥)
9	8	ssneld 3917	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑥 → (¬ 𝑢 ∈ ∪ 𝑥 → ¬ 𝑢 ∈ 𝑧))
10		disjsn 4607	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∩ {𝑢}) = ∅ ↔ ¬ 𝑢 ∈ 𝑧)
11	9, 10	syl6ibr 255	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑥 → (¬ 𝑢 ∈ ∪ 𝑥 → (𝑧 ∩ {𝑢}) = ∅))
12	11	impcom 411	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑢 ∈ ∪ 𝑥 ∧ 𝑧 ∈ 𝑥) → (𝑧 ∩ {𝑢}) = ∅)
13	12	uneq2d 4090	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑢 ∈ ∪ 𝑥 ∧ 𝑧 ∈ 𝑥) → ((𝑧 ∩ 𝑣) ∪ (𝑧 ∩ {𝑢})) = ((𝑧 ∩ 𝑣) ∪ ∅))
14		un0 4298	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∩ 𝑣) ∪ ∅) = (𝑧 ∩ 𝑣)
15	13, 14	eqtrdi 2849	. . . . . . . . . . . 12 ⊢ ((¬ 𝑢 ∈ ∪ 𝑥 ∧ 𝑧 ∈ 𝑥) → ((𝑧 ∩ 𝑣) ∪ (𝑧 ∩ {𝑢})) = (𝑧 ∩ 𝑣))
16	7, 15	syl5req 2846	. . . . . . . . . . 11 ⊢ ((¬ 𝑢 ∈ ∪ 𝑥 ∧ 𝑧 ∈ 𝑥) → (𝑧 ∩ 𝑣) = (𝑧 ∩ (𝑣 ∪ {𝑢})))
17	16	eleq2d 2875	. . . . . . . . . 10 ⊢ ((¬ 𝑢 ∈ ∪ 𝑥 ∧ 𝑧 ∈ 𝑥) → (𝑤 ∈ (𝑧 ∩ 𝑣) ↔ 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))))
18	17	eubidv 2647	. . . . . . . . 9 ⊢ ((¬ 𝑢 ∈ ∪ 𝑥 ∧ 𝑧 ∈ 𝑥) → (∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑣) ↔ ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))))
19	18	imbi2d 344	. . . . . . . 8 ⊢ ((¬ 𝑢 ∈ ∪ 𝑥 ∧ 𝑧 ∈ 𝑥) → ((𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑣)) ↔ (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢})))))
20	19	ralbidva 3161	. . . . . . 7 ⊢ (¬ 𝑢 ∈ ∪ 𝑥 → (∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑣)) ↔ ∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢})))))
21		vsnid 4562	. . . . . . . . . . . 12 ⊢ 𝑢 ∈ {𝑢}
22	21	olci 863	. . . . . . . . . . 11 ⊢ (𝑢 ∈ 𝑣 ∨ 𝑢 ∈ {𝑢})
23		elun 4076	. . . . . . . . . . 11 ⊢ (𝑢 ∈ (𝑣 ∪ {𝑢}) ↔ (𝑢 ∈ 𝑣 ∨ 𝑢 ∈ {𝑢}))
24	22, 23	mpbir 234	. . . . . . . . . 10 ⊢ 𝑢 ∈ (𝑣 ∪ {𝑢})
25		elssuni 4830	. . . . . . . . . . 11 ⊢ ((𝑣 ∪ {𝑢}) ∈ 𝑥 → (𝑣 ∪ {𝑢}) ⊆ ∪ 𝑥)
26	25	sseld 3914	. . . . . . . . . 10 ⊢ ((𝑣 ∪ {𝑢}) ∈ 𝑥 → (𝑢 ∈ (𝑣 ∪ {𝑢}) → 𝑢 ∈ ∪ 𝑥))
27	24, 26	mpi 20	. . . . . . . . 9 ⊢ ((𝑣 ∪ {𝑢}) ∈ 𝑥 → 𝑢 ∈ ∪ 𝑥)
28	27	con3i 157	. . . . . . . 8 ⊢ (¬ 𝑢 ∈ ∪ 𝑥 → ¬ (𝑣 ∪ {𝑢}) ∈ 𝑥)
29	28	biantrurd 536	. . . . . . 7 ⊢ (¬ 𝑢 ∈ ∪ 𝑥 → (∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))) ↔ (¬ (𝑣 ∪ {𝑢}) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))))))
30	20, 29	bitrd 282	. . . . . 6 ⊢ (¬ 𝑢 ∈ ∪ 𝑥 → (∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑣)) ↔ (¬ (𝑣 ∪ {𝑢}) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))))))
31		vex 3444	. . . . . . . 8 ⊢ 𝑣 ∈ V
32		snex 5297	. . . . . . . 8 ⊢ {𝑢} ∈ V
33	31, 32	unex 7449	. . . . . . 7 ⊢ (𝑣 ∪ {𝑢}) ∈ V
34		eleq1 2877	. . . . . . . . 9 ⊢ (𝑦 = (𝑣 ∪ {𝑢}) → (𝑦 ∈ 𝑥 ↔ (𝑣 ∪ {𝑢}) ∈ 𝑥))
35	34	notbid 321	. . . . . . . 8 ⊢ (𝑦 = (𝑣 ∪ {𝑢}) → (¬ 𝑦 ∈ 𝑥 ↔ ¬ (𝑣 ∪ {𝑢}) ∈ 𝑥))
36		ineq2 4133	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑣 ∪ {𝑢}) → (𝑧 ∩ 𝑦) = (𝑧 ∩ (𝑣 ∪ {𝑢})))
37	36	eleq2d 2875	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑣 ∪ {𝑢}) → (𝑤 ∈ (𝑧 ∩ 𝑦) ↔ 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))))
38	37	eubidv 2647	. . . . . . . . . 10 ⊢ (𝑦 = (𝑣 ∪ {𝑢}) → (∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑦) ↔ ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))))
39	38	imbi2d 344	. . . . . . . . 9 ⊢ (𝑦 = (𝑣 ∪ {𝑢}) → ((𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑦)) ↔ (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢})))))
40	39	ralbidv 3162	. . . . . . . 8 ⊢ (𝑦 = (𝑣 ∪ {𝑢}) → (∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑦)) ↔ ∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢})))))
41	35, 40	anbi12d 633	. . . . . . 7 ⊢ (𝑦 = (𝑣 ∪ {𝑢}) → ((¬ 𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑦))) ↔ (¬ (𝑣 ∪ {𝑢}) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢}))))))
42	33, 41	spcev 3555	. . . . . 6 ⊢ ((¬ (𝑣 ∪ {𝑢}) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ (𝑣 ∪ {𝑢})))) → ∃𝑦(¬ 𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑦))))
43	30, 42	syl6bi 256	. . . . 5 ⊢ (¬ 𝑢 ∈ ∪ 𝑥 → (∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑣)) → ∃𝑦(¬ 𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑦)))))
44		vuniex 7445	. . . . . 6 ⊢ ∪ 𝑥 ∈ V
45		eleq2 2878	. . . . . . . 8 ⊢ (𝑦 = ∪ 𝑥 → (𝑢 ∈ 𝑦 ↔ 𝑢 ∈ ∪ 𝑥))
46	45	notbid 321	. . . . . . 7 ⊢ (𝑦 = ∪ 𝑥 → (¬ 𝑢 ∈ 𝑦 ↔ ¬ 𝑢 ∈ ∪ 𝑥))
47	46	exbidv 1922	. . . . . 6 ⊢ (𝑦 = ∪ 𝑥 → (∃𝑢 ¬ 𝑢 ∈ 𝑦 ↔ ∃𝑢 ¬ 𝑢 ∈ ∪ 𝑥))
48		nalset 5181	. . . . . . . 8 ⊢ ¬ ∃𝑦∀𝑢 𝑢 ∈ 𝑦
49		alexn 1846	. . . . . . . 8 ⊢ (∀𝑦∃𝑢 ¬ 𝑢 ∈ 𝑦 ↔ ¬ ∃𝑦∀𝑢 𝑢 ∈ 𝑦)
50	48, 49	mpbir 234	. . . . . . 7 ⊢ ∀𝑦∃𝑢 ¬ 𝑢 ∈ 𝑦
51	50	spi 2181	. . . . . 6 ⊢ ∃𝑢 ¬ 𝑢 ∈ 𝑦
52	44, 47, 51	vtocl 3507	. . . . 5 ⊢ ∃𝑢 ¬ 𝑢 ∈ ∪ 𝑥
53	43, 52	exlimiiv 1932	. . . 4 ⊢ (∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑣)) → ∃𝑦(¬ 𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑦))))
54	53	exlimiv 1931	. . 3 ⊢ (∃𝑣∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑣)) → ∃𝑦(¬ 𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑦))))
55	6, 54	sylbi 220	. 2 ⊢ (∃𝑦∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑦)) → ∃𝑦(¬ 𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑦))))
56		exsimpr 1870	. 2 ⊢ (∃𝑦(¬ 𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑦))) → ∃𝑦∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑦)))
57	55, 56	impbii 212	1 ⊢ (∃𝑦∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑦)) ↔ ∃𝑦(¬ 𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 (𝜑 → ∃!𝑤 𝑤 ∈ (𝑧 ∩ 𝑦))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃!weu 2628  ∀wral 3106   ∪ cun 3879   ∩ cin 3880  ∅c0 4243  {csn 4525  ∪ cuni 4800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-uni 4801

This theorem is referenced by:  kmlem8  9568