Theorem kmlem4 9579

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 2121	. . . . . . 7 ⊢ (𝑣 = 𝑤 → (𝑣 ∈ 𝑥 ↔ 𝑤 ∈ 𝑥))
2		neeq2 3079	. . . . . . 7 ⊢ (𝑣 = 𝑤 → (𝑧 ≠ 𝑣 ↔ 𝑧 ≠ 𝑤))
3	1, 2	anbi12d 632	. . . . . 6 ⊢ (𝑣 = 𝑤 → ((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) ↔ (𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤)))
4		elequ2 2129	. . . . . . 7 ⊢ (𝑣 = 𝑤 → (𝑦 ∈ 𝑣 ↔ 𝑦 ∈ 𝑤))
5	4	notbid 320	. . . . . 6 ⊢ (𝑣 = 𝑤 → (¬ 𝑦 ∈ 𝑣 ↔ ¬ 𝑦 ∈ 𝑤))
6	3, 5	imbi12d 347	. . . . 5 ⊢ (𝑣 = 𝑤 → (((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣) ↔ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ¬ 𝑦 ∈ 𝑤)))
7	6	spvv 2003	. . . 4 ⊢ (∀𝑣((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣) → ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ¬ 𝑦 ∈ 𝑤))
8		eldif 3946	. . . . 5 ⊢ (𝑦 ∈ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ↔ (𝑦 ∈ 𝑧 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∖ {𝑧})))
9		simpr 487	. . . . . 6 ⊢ ((𝑦 ∈ 𝑧 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∖ {𝑧})) → ¬ 𝑦 ∈ ∪ (𝑥 ∖ {𝑧}))
10		eluni 4841	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ (𝑥 ∖ {𝑧}) ↔ ∃𝑣(𝑦 ∈ 𝑣 ∧ 𝑣 ∈ (𝑥 ∖ {𝑧})))
11	10	notbii 322	. . . . . . 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 362	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑣 → ¬ 𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ (𝑣 ∈ (𝑥 ∖ {𝑧}) → ¬ 𝑦 ∈ 𝑣))
14		imnan 402	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑣 → ¬ 𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ¬ (𝑦 ∈ 𝑣 ∧ 𝑣 ∈ (𝑥 ∖ {𝑧})))
15		eldifsn 4719	. . . . . . . . . . 11 ⊢ (𝑣 ∈ (𝑥 ∖ {𝑧}) ↔ (𝑣 ∈ 𝑥 ∧ 𝑣 ≠ 𝑧))
16		necom 3069	. . . . . . . . . . . 12 ⊢ (𝑣 ≠ 𝑧 ↔ 𝑧 ≠ 𝑣)
17	16	anbi2i 624	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ 𝑥 ∧ 𝑣 ≠ 𝑧) ↔ (𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣))
18	15, 17	bitri 277	. . . . . . . . . 10 ⊢ (𝑣 ∈ (𝑥 ∖ {𝑧}) ↔ (𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣))
19	18	imbi1i 352	. . . . . . . . 9 ⊢ ((𝑣 ∈ (𝑥 ∖ {𝑧}) → ¬ 𝑦 ∈ 𝑣) ↔ ((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣))
20	13, 14, 19	3bitr3i 303	. . . . . . . 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 301	. . . . . 6 ⊢ (¬ 𝑦 ∈ ∪ (𝑥 ∖ {𝑧}) ↔ ∀𝑣((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣))
23	9, 22	sylib 220	. . . . 5 ⊢ ((𝑦 ∈ 𝑧 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∖ {𝑧})) → ∀𝑣((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣))
24	8, 23	sylbi 219	. . . 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 3181	. 2 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ∀𝑦 ∈ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ¬ 𝑦 ∈ 𝑤)
27		disj 4399	. 2 ⊢ (((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑤) = ∅ ↔ ∀𝑦 ∈ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ¬ 𝑦 ∈ 𝑤)
28	26, 27	sylibr 236	1 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑤) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398  ∀wal 1535   = wceq 1537  ∃wex 1780   ∈ wcel 2114   ≠ wne 3016  ∀wral 3138   ∖ cdif 3933   ∩ cin 3935  ∅c0 4291  {csn 4567  ∪ cuni 4838

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-v 3496  df-dif 3939  df-in 3943  df-nul 4292  df-sn 4568  df-uni 4839

This theorem is referenced by:  kmlem5  9580  kmlem11  9586