Theorem kmlem4 10154

Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 26-Mar-2004.)

Assertion

Ref	Expression
kmlem4	⊢ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑤) = ∅)

Distinct variable group:   𝑥,𝑤,𝑧

Proof of Theorem kmlem4

Dummy variables 𝑣 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elequ1 2112	. . . . . . 7 ⊢ (𝑣 = 𝑤 → (𝑣 ∈ 𝑥 ↔ 𝑤 ∈ 𝑥))
2		neeq2 3003	. . . . . . 7 ⊢ (𝑣 = 𝑤 → (𝑧 ≠ 𝑣 ↔ 𝑧 ≠ 𝑤))
3	1, 2	anbi12d 630	. . . . . 6 ⊢ (𝑣 = 𝑤 → ((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) ↔ (𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤)))
4		elequ2 2120	. . . . . . 7 ⊢ (𝑣 = 𝑤 → (𝑦 ∈ 𝑣 ↔ 𝑦 ∈ 𝑤))
5	4	notbid 318	. . . . . 6 ⊢ (𝑣 = 𝑤 → (¬ 𝑦 ∈ 𝑣 ↔ ¬ 𝑦 ∈ 𝑤))
6	3, 5	imbi12d 344	. . . . 5 ⊢ (𝑣 = 𝑤 → (((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣) ↔ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ¬ 𝑦 ∈ 𝑤)))
7	6	spvv 1999	. . . 4 ⊢ (∀𝑣((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣) → ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ¬ 𝑦 ∈ 𝑤))
8		eldif 3958	. . . . 5 ⊢ (𝑦 ∈ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ↔ (𝑦 ∈ 𝑧 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∖ {𝑧})))
9		simpr 484	. . . . . 6 ⊢ ((𝑦 ∈ 𝑧 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∖ {𝑧})) → ¬ 𝑦 ∈ ∪ (𝑥 ∖ {𝑧}))
10		eluni 4911	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ (𝑥 ∖ {𝑧}) ↔ ∃𝑣(𝑦 ∈ 𝑣 ∧ 𝑣 ∈ (𝑥 ∖ {𝑧})))
11	10	notbii 320	. . . . . . 7 ⊢ (¬ 𝑦 ∈ ∪ (𝑥 ∖ {𝑧}) ↔ ¬ ∃𝑣(𝑦 ∈ 𝑣 ∧ 𝑣 ∈ (𝑥 ∖ {𝑧})))
12		alnex 1782	. . . . . . 7 ⊢ (∀𝑣 ¬ (𝑦 ∈ 𝑣 ∧ 𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ¬ ∃𝑣(𝑦 ∈ 𝑣 ∧ 𝑣 ∈ (𝑥 ∖ {𝑧})))
13		con2b 359	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑣 → ¬ 𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ (𝑣 ∈ (𝑥 ∖ {𝑧}) → ¬ 𝑦 ∈ 𝑣))
14		imnan 399	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑣 → ¬ 𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ¬ (𝑦 ∈ 𝑣 ∧ 𝑣 ∈ (𝑥 ∖ {𝑧})))
15		eldifsn 4790	. . . . . . . . . . 11 ⊢ (𝑣 ∈ (𝑥 ∖ {𝑧}) ↔ (𝑣 ∈ 𝑥 ∧ 𝑣 ≠ 𝑧))
16		necom 2993	. . . . . . . . . . . 12 ⊢ (𝑣 ≠ 𝑧 ↔ 𝑧 ≠ 𝑣)
17	16	anbi2i 622	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ 𝑥 ∧ 𝑣 ≠ 𝑧) ↔ (𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣))
18	15, 17	bitri 275	. . . . . . . . . 10 ⊢ (𝑣 ∈ (𝑥 ∖ {𝑧}) ↔ (𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣))
19	18	imbi1i 349	. . . . . . . . 9 ⊢ ((𝑣 ∈ (𝑥 ∖ {𝑧}) → ¬ 𝑦 ∈ 𝑣) ↔ ((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣))
20	13, 14, 19	3bitr3i 301	. . . . . . . 8 ⊢ (¬ (𝑦 ∈ 𝑣 ∧ 𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣))
21	20	albii 1820	. . . . . . 7 ⊢ (∀𝑣 ¬ (𝑦 ∈ 𝑣 ∧ 𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ∀𝑣((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣))
22	11, 12, 21	3bitr2i 299	. . . . . 6 ⊢ (¬ 𝑦 ∈ ∪ (𝑥 ∖ {𝑧}) ↔ ∀𝑣((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣))
23	9, 22	sylib 217	. . . . 5 ⊢ ((𝑦 ∈ 𝑧 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∖ {𝑧})) → ∀𝑣((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣))
24	8, 23	sylbi 216	. . . 4 ⊢ (𝑦 ∈ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) → ∀𝑣((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣))
25	7, 24	syl11 33	. . 3 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → (𝑦 ∈ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) → ¬ 𝑦 ∈ 𝑤))
26	25	ralrimiv 3144	. 2 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ∀𝑦 ∈ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ¬ 𝑦 ∈ 𝑤)
27		disj 4447	. 2 ⊢ (((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑤) = ∅ ↔ ∀𝑦 ∈ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ¬ 𝑦 ∈ 𝑤)
28	26, 27	sylibr 233	1 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑤) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1780   ∈ wcel 2105   ≠ wne 2939  ∀wral 3060   ∖ cdif 3945   ∩ cin 3947  ∅c0 4322  {csn 4628  ∪ cuni 4908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-v 3475  df-dif 3951  df-in 3955  df-nul 4323  df-sn 4629  df-uni 4909

This theorem is referenced by:  kmlem5  10155  kmlem11  10161