Theorem kmlem3 8974

Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. The right-hand side is part of the hypothesis of 4. (Contributed by NM, 25-Mar-2004.)

Assertion

Ref	Expression
kmlem3	⊢ ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ≠ ∅ ↔ ∃𝑣 ∈ 𝑧 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → ¬ 𝑣 ∈ (𝑧 ∩ 𝑤)))

Distinct variable group:   𝑥,𝑣,𝑤,𝑧

Proof of Theorem kmlem3

Step	Hyp	Ref	Expression
1		dfdif2 3583	. . . 4 ⊢ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) = {𝑣 ∈ 𝑧 ∣ ¬ 𝑣 ∈ ∪ (𝑥 ∖ {𝑧})}
2		dfnul3 3918	. . . . . 6 ⊢ ∅ = {𝑣 ∈ 𝑧 ∣ ¬ 𝑣 ∈ 𝑧}
3	2	uneq2i 3764	. . . . 5 ⊢ ({𝑣 ∈ 𝑧 ∣ ¬ 𝑣 ∈ ∪ (𝑥 ∖ {𝑧})} ∪ ∅) = ({𝑣 ∈ 𝑧 ∣ ¬ 𝑣 ∈ ∪ (𝑥 ∖ {𝑧})} ∪ {𝑣 ∈ 𝑧 ∣ ¬ 𝑣 ∈ 𝑧})
4		un0 3967	. . . . 5 ⊢ ({𝑣 ∈ 𝑧 ∣ ¬ 𝑣 ∈ ∪ (𝑥 ∖ {𝑧})} ∪ ∅) = {𝑣 ∈ 𝑧 ∣ ¬ 𝑣 ∈ ∪ (𝑥 ∖ {𝑧})}
5		unrab 3898	. . . . 5 ⊢ ({𝑣 ∈ 𝑧 ∣ ¬ 𝑣 ∈ ∪ (𝑥 ∖ {𝑧})} ∪ {𝑣 ∈ 𝑧 ∣ ¬ 𝑣 ∈ 𝑧}) = {𝑣 ∈ 𝑧 ∣ (¬ 𝑣 ∈ ∪ (𝑥 ∖ {𝑧}) ∨ ¬ 𝑣 ∈ 𝑧)}
6	3, 4, 5	3eqtr3i 2652	. . . 4 ⊢ {𝑣 ∈ 𝑧 ∣ ¬ 𝑣 ∈ ∪ (𝑥 ∖ {𝑧})} = {𝑣 ∈ 𝑧 ∣ (¬ 𝑣 ∈ ∪ (𝑥 ∖ {𝑧}) ∨ ¬ 𝑣 ∈ 𝑧)}
7		ianor 509	. . . . . . 7 ⊢ (¬ (𝑣 ∈ ∪ (𝑥 ∖ {𝑧}) ∧ 𝑣 ∈ 𝑧) ↔ (¬ 𝑣 ∈ ∪ (𝑥 ∖ {𝑧}) ∨ ¬ 𝑣 ∈ 𝑧))
8		eluni 4439	. . . . . . . . . 10 ⊢ (𝑣 ∈ ∪ (𝑥 ∖ {𝑧}) ↔ ∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑥 ∖ {𝑧})))
9	8	anbi1i 731	. . . . . . . . 9 ⊢ ((𝑣 ∈ ∪ (𝑥 ∖ {𝑧}) ∧ 𝑣 ∈ 𝑧) ↔ (∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑥 ∖ {𝑧})) ∧ 𝑣 ∈ 𝑧))
10		df-rex 2918	. . . . . . . . . 10 ⊢ (∃𝑤 ∈ 𝑥 ¬ (𝑧 ≠ 𝑤 → ¬ 𝑣 ∈ (𝑧 ∩ 𝑤)) ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ¬ (𝑧 ≠ 𝑤 → ¬ 𝑣 ∈ (𝑧 ∩ 𝑤))))
11		elin 3796	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ (𝑧 ∩ 𝑤) ↔ (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑤))
12	11	anbi2i 730	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ≠ 𝑤 ∧ 𝑣 ∈ (𝑧 ∩ 𝑤)) ↔ (𝑧 ≠ 𝑤 ∧ (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑤)))
13		df-an 386	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ≠ 𝑤 ∧ 𝑣 ∈ (𝑧 ∩ 𝑤)) ↔ ¬ (𝑧 ≠ 𝑤 → ¬ 𝑣 ∈ (𝑧 ∩ 𝑤)))
14	12, 13	bitr3i 266	. . . . . . . . . . . . 13 ⊢ ((𝑧 ≠ 𝑤 ∧ (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑤)) ↔ ¬ (𝑧 ≠ 𝑤 → ¬ 𝑣 ∈ (𝑧 ∩ 𝑤)))
15	14	anbi2i 730	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ 𝑥 ∧ (𝑧 ≠ 𝑤 ∧ (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑤))) ↔ (𝑤 ∈ 𝑥 ∧ ¬ (𝑧 ≠ 𝑤 → ¬ 𝑣 ∈ (𝑧 ∩ 𝑤))))
16		eldifsn 4317	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ (𝑥 ∖ {𝑧}) ↔ (𝑤 ∈ 𝑥 ∧ 𝑤 ≠ 𝑧))
17		necom 2847	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ≠ 𝑧 ↔ 𝑧 ≠ 𝑤)
18	17	anbi2i 730	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑤 ≠ 𝑧) ↔ (𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤))
19	16, 18	bitri 264	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ (𝑥 ∖ {𝑧}) ↔ (𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤))
20	19	anbi2i 730	. . . . . . . . . . . . . 14 ⊢ (((𝑣 ∈ 𝑤 ∧ 𝑣 ∈ 𝑧) ∧ 𝑤 ∈ (𝑥 ∖ {𝑧})) ↔ ((𝑣 ∈ 𝑤 ∧ 𝑣 ∈ 𝑧) ∧ (𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤)))
21		ancom 466	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ∈ 𝑤 ∧ 𝑣 ∈ 𝑧) ↔ (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑤))
22	21	anbi2ci 732	. . . . . . . . . . . . . 14 ⊢ (((𝑣 ∈ 𝑤 ∧ 𝑣 ∈ 𝑧) ∧ (𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤)) ↔ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) ∧ (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑤)))
23		anass 681	. . . . . . . . . . . . . 14 ⊢ (((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) ∧ (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑤)) ↔ (𝑤 ∈ 𝑥 ∧ (𝑧 ≠ 𝑤 ∧ (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑤))))
24	20, 22, 23	3bitri 286	. . . . . . . . . . . . 13 ⊢ (((𝑣 ∈ 𝑤 ∧ 𝑣 ∈ 𝑧) ∧ 𝑤 ∈ (𝑥 ∖ {𝑧})) ↔ (𝑤 ∈ 𝑥 ∧ (𝑧 ≠ 𝑤 ∧ (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑤))))
25		an32 839	. . . . . . . . . . . . 13 ⊢ (((𝑣 ∈ 𝑤 ∧ 𝑣 ∈ 𝑧) ∧ 𝑤 ∈ (𝑥 ∖ {𝑧})) ↔ ((𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑥 ∖ {𝑧})) ∧ 𝑣 ∈ 𝑧))
26	24, 25	bitr3i 266	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ 𝑥 ∧ (𝑧 ≠ 𝑤 ∧ (𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑤))) ↔ ((𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑥 ∖ {𝑧})) ∧ 𝑣 ∈ 𝑧))
27	15, 26	bitr3i 266	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ 𝑥 ∧ ¬ (𝑧 ≠ 𝑤 → ¬ 𝑣 ∈ (𝑧 ∩ 𝑤))) ↔ ((𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑥 ∖ {𝑧})) ∧ 𝑣 ∈ 𝑧))
28	27	exbii 1774	. . . . . . . . . 10 ⊢ (∃𝑤(𝑤 ∈ 𝑥 ∧ ¬ (𝑧 ≠ 𝑤 → ¬ 𝑣 ∈ (𝑧 ∩ 𝑤))) ↔ ∃𝑤((𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑥 ∖ {𝑧})) ∧ 𝑣 ∈ 𝑧))
29		19.41v 1914	. . . . . . . . . 10 ⊢ (∃𝑤((𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑥 ∖ {𝑧})) ∧ 𝑣 ∈ 𝑧) ↔ (∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑥 ∖ {𝑧})) ∧ 𝑣 ∈ 𝑧))
30	10, 28, 29	3bitri 286	. . . . . . . . 9 ⊢ (∃𝑤 ∈ 𝑥 ¬ (𝑧 ≠ 𝑤 → ¬ 𝑣 ∈ (𝑧 ∩ 𝑤)) ↔ (∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑥 ∖ {𝑧})) ∧ 𝑣 ∈ 𝑧))
31		rexnal 2995	. . . . . . . . 9 ⊢ (∃𝑤 ∈ 𝑥 ¬ (𝑧 ≠ 𝑤 → ¬ 𝑣 ∈ (𝑧 ∩ 𝑤)) ↔ ¬ ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → ¬ 𝑣 ∈ (𝑧 ∩ 𝑤)))
32	9, 30, 31	3bitr2ri 289	. . . . . . . 8 ⊢ (¬ ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → ¬ 𝑣 ∈ (𝑧 ∩ 𝑤)) ↔ (𝑣 ∈ ∪ (𝑥 ∖ {𝑧}) ∧ 𝑣 ∈ 𝑧))
33	32	con1bii 346	. . . . . . 7 ⊢ (¬ (𝑣 ∈ ∪ (𝑥 ∖ {𝑧}) ∧ 𝑣 ∈ 𝑧) ↔ ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → ¬ 𝑣 ∈ (𝑧 ∩ 𝑤)))
34	7, 33	bitr3i 266	. . . . . 6 ⊢ ((¬ 𝑣 ∈ ∪ (𝑥 ∖ {𝑧}) ∨ ¬ 𝑣 ∈ 𝑧) ↔ ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → ¬ 𝑣 ∈ (𝑧 ∩ 𝑤)))
35	34	a1i 11	. . . . 5 ⊢ (𝑣 ∈ 𝑧 → ((¬ 𝑣 ∈ ∪ (𝑥 ∖ {𝑧}) ∨ ¬ 𝑣 ∈ 𝑧) ↔ ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → ¬ 𝑣 ∈ (𝑧 ∩ 𝑤))))
36	35	rabbiia 3185	. . . 4 ⊢ {𝑣 ∈ 𝑧 ∣ (¬ 𝑣 ∈ ∪ (𝑥 ∖ {𝑧}) ∨ ¬ 𝑣 ∈ 𝑧)} = {𝑣 ∈ 𝑧 ∣ ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → ¬ 𝑣 ∈ (𝑧 ∩ 𝑤))}
37	1, 6, 36	3eqtri 2648	. . 3 ⊢ (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) = {𝑣 ∈ 𝑧 ∣ ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → ¬ 𝑣 ∈ (𝑧 ∩ 𝑤))}
38	37	neeq1i 2858	. 2 ⊢ ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ≠ ∅ ↔ {𝑣 ∈ 𝑧 ∣ ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → ¬ 𝑣 ∈ (𝑧 ∩ 𝑤))} ≠ ∅)
39		rabn0 3958	. 2 ⊢ ({𝑣 ∈ 𝑧 ∣ ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → ¬ 𝑣 ∈ (𝑧 ∩ 𝑤))} ≠ ∅ ↔ ∃𝑣 ∈ 𝑧 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → ¬ 𝑣 ∈ (𝑧 ∩ 𝑤)))
40	38, 39	bitri 264	1 ⊢ ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ≠ ∅ ↔ ∃𝑣 ∈ 𝑧 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → ¬ 𝑣 ∈ (𝑧 ∩ 𝑤)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573  ∅c0 3915  {csn 4177  ∪ cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-nul 3916  df-sn 4178  df-uni 4437

This theorem is referenced by:  kmlem13  8984