Theorem kmlem4 10107

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 2148	. . . . . . 7 ⊢ (𝑣 = 𝑤 → (𝑣 ∈ 𝑥 ↔ 𝑤 ∈ 𝑥))
2		neeq2 3019	. . . . . . 7 ⊢ (𝑣 = 𝑤 → (𝑧 ≠ 𝑣 ↔ 𝑧 ≠ 𝑤))
3	1, 2	anbi12d 641	. . . . . 6 ⊢ (𝑣 = 𝑤 → ((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) ↔ (𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤)))
4		elequ2 2156	. . . . . . 7 ⊢ (𝑣 = 𝑤 → (𝑦 ∈ 𝑣 ↔ 𝑦 ∈ 𝑤))
5	4	notbid 320	. . . . . 6 ⊢ (𝑣 = 𝑤 → (¬ 𝑦 ∈ 𝑣 ↔ ¬ 𝑦 ∈ 𝑤))
6	3, 5	imbi12d 346	. . . . 5 ⊢ (𝑣 = 𝑤 → (((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣) ↔ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ¬ 𝑦 ∈ 𝑤)))
7	6	spvv 2007	. . . 4 ⊢ (∀𝑣((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣) → ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ¬ 𝑦 ∈ 𝑤))
8		eldif 3914	. . . . 5 ⊢ (𝑦 ∈ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ↔ (𝑦 ∈ 𝑧 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∖ {𝑧})))
9		eluni 4867	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ (𝑥 ∖ {𝑧}) ↔ ∃𝑣(𝑦 ∈ 𝑣 ∧ 𝑣 ∈ (𝑥 ∖ {𝑧})))
10	9	notbii 322	. . . . . . 7 ⊢ (¬ 𝑦 ∈ ∪ (𝑥 ∖ {𝑧}) ↔ ¬ ∃𝑣(𝑦 ∈ 𝑣 ∧ 𝑣 ∈ (𝑥 ∖ {𝑧})))
11		alnex 1800	. . . . . . 7 ⊢ (∀𝑣 ¬ (𝑦 ∈ 𝑣 ∧ 𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ¬ ∃𝑣(𝑦 ∈ 𝑣 ∧ 𝑣 ∈ (𝑥 ∖ {𝑧})))
12		con2b 361	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑣 → ¬ 𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ (𝑣 ∈ (𝑥 ∖ {𝑧}) → ¬ 𝑦 ∈ 𝑣))
13		imnan 403	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑣 → ¬ 𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ¬ (𝑦 ∈ 𝑣 ∧ 𝑣 ∈ (𝑥 ∖ {𝑧})))
14		eldifsn 4745	. . . . . . . . . . 11 ⊢ (𝑣 ∈ (𝑥 ∖ {𝑧}) ↔ (𝑣 ∈ 𝑥 ∧ 𝑣 ≠ 𝑧))
15		necom 3009	. . . . . . . . . . . 12 ⊢ (𝑣 ≠ 𝑧 ↔ 𝑧 ≠ 𝑣)
16	15	anbi2i 632	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ 𝑥 ∧ 𝑣 ≠ 𝑧) ↔ (𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣))
17	14, 16	bitri 277	. . . . . . . . . 10 ⊢ (𝑣 ∈ (𝑥 ∖ {𝑧}) ↔ (𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣))
18	17	imbi1i 351	. . . . . . . . 9 ⊢ ((𝑣 ∈ (𝑥 ∖ {𝑧}) → ¬ 𝑦 ∈ 𝑣) ↔ ((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣))
19	12, 13, 18	3bitr3i 303	. . . . . . . 8 ⊢ (¬ (𝑦 ∈ 𝑣 ∧ 𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣))
20	19	albii 1838	. . . . . . 7 ⊢ (∀𝑣 ¬ (𝑦 ∈ 𝑣 ∧ 𝑣 ∈ (𝑥 ∖ {𝑧})) ↔ ∀𝑣((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣))
21	10, 11, 20	3bitr2i 301	. . . . . 6 ⊢ (¬ 𝑦 ∈ ∪ (𝑥 ∖ {𝑧}) ↔ ∀𝑣((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣))
22	21	bilani 508	. . . . 5 ⊢ ((𝑦 ∈ 𝑧 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∖ {𝑧})) → ∀𝑣((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣))
23	8, 22	sylbi 219	. . . 4 ⊢ (𝑦 ∈ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) → ∀𝑣((𝑣 ∈ 𝑥 ∧ 𝑧 ≠ 𝑣) → ¬ 𝑦 ∈ 𝑣))
24	7, 23	syl11 33	. . 3 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → (𝑦 ∈ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) → ¬ 𝑦 ∈ 𝑤))
25	24	ralrimiv 3152	. 2 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ∀𝑦 ∈ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ¬ 𝑦 ∈ 𝑤)
26		disj 4403	. 2 ⊢ (((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑤) = ∅ ↔ ∀𝑦 ∈ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ¬ 𝑦 ∈ 𝑤)
27	25, 26	sylibr 236	1 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑤) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1557   = wceq 1559  ∃wex 1798   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075   ∖ cdif 3901   ∩ cin 3903  ∅c0 4285  {csn 4581  ∪ cuni 4864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-v 3455  df-dif 3907  df-in 3911  df-nul 4286  df-sn 4582  df-uni 4865

This theorem is referenced by:  kmlem5  10108  kmlem11  10114