Theorem kmlem4 9909

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 2113	. . . . . . 7 ⊢ (𝑣 = 𝑤 → (𝑣 ∈ 𝑥 ↔ 𝑤 ∈ 𝑥))
2		neeq2 3007	. . . . . . 7 ⊢ (𝑣 = 𝑤 → (𝑧 ≠ 𝑣 ↔ 𝑧 ≠ 𝑤))
3	1, 2	anbi12d 631	. . . . . 6 ⊢ (𝑣 = 𝑤 → ((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) ↔ (𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤)))
4		elequ2 2121	. . . . . . 7 ⊢ (𝑣 = 𝑤 → (𝑦 ∈ 𝑣 ↔ 𝑦 ∈ 𝑤))
5	4	notbid 318	. . . . . 6 ⊢ (𝑣 = 𝑤 → (¬ 𝑦 ∈ 𝑣 ↔ ¬ 𝑦 ∈ 𝑤))
6	3, 5	imbi12d 345	. . . . 5 ⊢ (𝑣 = 𝑤 → (((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣) ↔ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ¬ 𝑦 ∈ 𝑤)))
7	6	spvv 2000	. . . 4 ⊢ (∀𝑣((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣) → ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ¬ 𝑦 ∈ 𝑤))
8		eldif 3897	. . . . 5 ⊢ (𝑦 ∈ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ↔ (𝑦 ∈ 𝑧 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∖ {𝑧})))
9		simpr 485	. . . . . 6 ⊢ ((𝑦 ∈ 𝑧 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∖ {𝑧})) → ¬ 𝑦 ∈ ∪ (𝑥 ∖ {𝑧}))
10		eluni 4842	. . . . . . . 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 1784	. . . . . . 7 ⊢ (∀𝑣 ¬ (𝑦 ∈ 𝑣 ∧ 𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ¬ ∃𝑣(𝑦 ∈ 𝑣 ∧ 𝑣 ∈ (𝑥 ∖ {𝑧})))
13		con2b 360	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑣 → ¬ 𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ (𝑣 ∈ (𝑥 ∖ {𝑧}) → ¬ 𝑦 ∈ 𝑣))
14		imnan 400	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑣 → ¬ 𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ¬ (𝑦 ∈ 𝑣 ∧ 𝑣 ∈ (𝑥 ∖ {𝑧})))
15		eldifsn 4720	. . . . . . . . . . 11 ⊢ (𝑣 ∈ (𝑥 ∖ {𝑧}) ↔ (𝑣 ∈ 𝑥 ∧ 𝑣 ≠ 𝑧))
16		necom 2997	. . . . . . . . . . . 12 ⊢ (𝑣 ≠ 𝑧 ↔ 𝑧 ≠ 𝑣)
17	16	anbi2i 623	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ 𝑥 ∧ 𝑣 ≠ 𝑧) ↔ (𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣))
18	15, 17	bitri 274	. . . . . . . . . 10 ⊢ (𝑣 ∈ (𝑥 ∖ {𝑧}) ↔ (𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣))
19	18	imbi1i 350	. . . . . . . . 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 1822	. . . . . . 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 3102	. 2 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ∀𝑦 ∈ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ¬ 𝑦 ∈ 𝑤)
27		disj 4381	. 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 396  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064   ∖ cdif 3884   ∩ cin 3886  ∅c0 4256  {csn 4561  ∪ cuni 4839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-v 3434  df-dif 3890  df-in 3894  df-nul 4257  df-sn 4562  df-uni 4840

This theorem is referenced by:  kmlem5  9910  kmlem11  9916