Theorem kur14lem7 35977

Description: Lemma for kur14 35981: main proof. The set 𝑇 here contains all the distinct combinations of 𝑘 and 𝑐 that can arise, and we prove here that applying 𝑘 or 𝑐 to any element of 𝑇 yields another element of 𝑇. In operator shorthand, we have 𝑇 = {𝐴, 𝑐𝐴, 𝑘𝐴 , 𝑐𝑘𝐴, 𝑘𝑐𝐴, 𝑐𝑘𝑐𝐴, 𝑘𝑐𝑘𝐴, 𝑐𝑘𝑐𝑘𝐴, 𝑘𝑐𝑘𝑐𝐴, 𝑐𝑘𝑐𝑘𝑐𝐴, 𝑘𝑐𝑘𝑐𝑘𝐴, 𝑐𝑘𝑐𝑘𝑐𝑘𝐴, 𝑘𝑐𝑘𝑐𝑘𝑐𝐴, 𝑐𝑘𝑐𝑘𝑐𝑘𝑐𝐴}. From the identities 𝑐𝑐𝐴 = 𝐴 and 𝑘𝑘𝐴 = 𝑘𝐴, we can reduce any operator combination containing two adjacent identical operators, which is why the list only contains alternating sequences. The reason the sequences don't keep going after a certain point is due to the identity 𝑘𝑐𝑘𝐴 = 𝑘𝑐𝑘𝑐𝑘𝑐𝑘𝐴, proved in kur14lem6 35976. (Contributed by Mario Carneiro, 11-Feb-2015.)

Hypotheses

Ref	Expression
kur14lem.j	⊢ 𝐽 ∈ Top
kur14lem.x	⊢ 𝑋 = ∪ 𝐽
kur14lem.k	⊢ 𝐾 = (cls‘𝐽)
kur14lem.i	⊢ 𝐼 = (int‘𝐽)
kur14lem.a	⊢ 𝐴 ⊆ 𝑋
kur14lem.b	⊢ 𝐵 = (𝑋 ∖ (𝐾‘𝐴))
kur14lem.c	⊢ 𝐶 = (𝐾‘(𝑋 ∖ 𝐴))
kur14lem.d	⊢ 𝐷 = (𝐼‘(𝐾‘𝐴))
kur14lem.t	⊢ 𝑇 = ((({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ∪ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))}))

Assertion

Ref	Expression
kur14lem7	⊢ (𝑁 ∈ 𝑇 → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))

Proof of Theorem kur14lem7

Step	Hyp	Ref	Expression
1		elun 4100	. . 3 ⊢ (𝑁 ∈ ((({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ∪ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))})) ↔ (𝑁 ∈ (({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ∨ 𝑁 ∈ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))})))
2		elun 4100	. . . . 5 ⊢ (𝑁 ∈ (({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ↔ (𝑁 ∈ ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∨ 𝑁 ∈ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}))
3		elun 4100	. . . . . . 7 ⊢ (𝑁 ∈ ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ↔ (𝑁 ∈ {𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∨ 𝑁 ∈ {𝐵, 𝐶, (𝐼‘𝐴)}))
4		eltpi 4649	. . . . . . . . 9 ⊢ (𝑁 ∈ {𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} → (𝑁 = 𝐴 ∨ 𝑁 = (𝑋 ∖ 𝐴) ∨ 𝑁 = (𝐾‘𝐴)))
5		kur14lem.a	. . . . . . . . . . 11 ⊢ 𝐴 ⊆ 𝑋
6		ssun1 4124	. . . . . . . . . . . . 13 ⊢ {𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ⊆ ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)})
7		ssun1 4124	. . . . . . . . . . . . . 14 ⊢ ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ⊆ (({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))})
8		ssun1 4124	. . . . . . . . . . . . . . 15 ⊢ (({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ⊆ ((({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ∪ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))}))
9		kur14lem.t	. . . . . . . . . . . . . . 15 ⊢ 𝑇 = ((({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ∪ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))}))
10	8, 9	sseqtrri 3980	. . . . . . . . . . . . . 14 ⊢ (({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ⊆ 𝑇
11	7, 10	sstri 3940	. . . . . . . . . . . . 13 ⊢ ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ⊆ 𝑇
12	6, 11	sstri 3940	. . . . . . . . . . . 12 ⊢ {𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ⊆ 𝑇
13		kur14lem.j	. . . . . . . . . . . . . . . 16 ⊢ 𝐽 ∈ Top
14		kur14lem.x	. . . . . . . . . . . . . . . . 17 ⊢ 𝑋 = ∪ 𝐽
15	14	topopn 23224	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
16	13, 15	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 𝑋 ∈ 𝐽
17	16	elexi 3473	. . . . . . . . . . . . . 14 ⊢ 𝑋 ∈ V
18		difss 4083	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∖ 𝐴) ⊆ 𝑋
19	17, 18	ssexi 5284	. . . . . . . . . . . . 13 ⊢ (𝑋 ∖ 𝐴) ∈ V
20	19	tpid2 4731	. . . . . . . . . . . 12 ⊢ (𝑋 ∖ 𝐴) ∈ {𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)}
21	12, 20	sselii 3928	. . . . . . . . . . 11 ⊢ (𝑋 ∖ 𝐴) ∈ 𝑇
22		fvex 6898	. . . . . . . . . . . . 13 ⊢ (𝐾‘𝐴) ∈ V
23	22	tpid3 4734	. . . . . . . . . . . 12 ⊢ (𝐾‘𝐴) ∈ {𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)}
24	12, 23	sselii 3928	. . . . . . . . . . 11 ⊢ (𝐾‘𝐴) ∈ 𝑇
25	5, 21, 24	kur14lem1 35971	. . . . . . . . . 10 ⊢ (𝑁 = 𝐴 → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
26		kur14lem.k	. . . . . . . . . . . . 13 ⊢ 𝐾 = (cls‘𝐽)
27		kur14lem.i	. . . . . . . . . . . . 13 ⊢ 𝐼 = (int‘𝐽)
28	13, 14, 26, 27, 5	kur14lem4 35974	. . . . . . . . . . . 12 ⊢ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴
29	17, 5	ssexi 5284	. . . . . . . . . . . . . 14 ⊢ 𝐴 ∈ V
30	29	tpid1 4729	. . . . . . . . . . . . 13 ⊢ 𝐴 ∈ {𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)}
31	12, 30	sselii 3928	. . . . . . . . . . . 12 ⊢ 𝐴 ∈ 𝑇
32	28, 31	eqeltri 2857	. . . . . . . . . . 11 ⊢ (𝑋 ∖ (𝑋 ∖ 𝐴)) ∈ 𝑇
33		kur14lem.c	. . . . . . . . . . . 12 ⊢ 𝐶 = (𝐾‘(𝑋 ∖ 𝐴))
34		ssun2 4125	. . . . . . . . . . . . . 14 ⊢ {𝐵, 𝐶, (𝐼‘𝐴)} ⊆ ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)})
35	34, 11	sstri 3940	. . . . . . . . . . . . 13 ⊢ {𝐵, 𝐶, (𝐼‘𝐴)} ⊆ 𝑇
36	13, 14, 26, 27, 18	kur14lem3 35973	. . . . . . . . . . . . . . . 16 ⊢ (𝐾‘(𝑋 ∖ 𝐴)) ⊆ 𝑋
37	33, 36	eqsstri 3977	. . . . . . . . . . . . . . 15 ⊢ 𝐶 ⊆ 𝑋
38	17, 37	ssexi 5284	. . . . . . . . . . . . . 14 ⊢ 𝐶 ∈ V
39	38	tpid2 4731	. . . . . . . . . . . . 13 ⊢ 𝐶 ∈ {𝐵, 𝐶, (𝐼‘𝐴)}
40	35, 39	sselii 3928	. . . . . . . . . . . 12 ⊢ 𝐶 ∈ 𝑇
41	33, 40	eqeltrri 2858	. . . . . . . . . . 11 ⊢ (𝐾‘(𝑋 ∖ 𝐴)) ∈ 𝑇
42	18, 32, 41	kur14lem1 35971	. . . . . . . . . 10 ⊢ (𝑁 = (𝑋 ∖ 𝐴) → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
43	13, 14, 26, 27, 5	kur14lem3 35973	. . . . . . . . . . 11 ⊢ (𝐾‘𝐴) ⊆ 𝑋
44		kur14lem.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (𝑋 ∖ (𝐾‘𝐴))
45		difss 4083	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∖ (𝐾‘𝐴)) ⊆ 𝑋
46	44, 45	eqsstri 3977	. . . . . . . . . . . . . . 15 ⊢ 𝐵 ⊆ 𝑋
47	17, 46	ssexi 5284	. . . . . . . . . . . . . 14 ⊢ 𝐵 ∈ V
48	47	tpid1 4729	. . . . . . . . . . . . 13 ⊢ 𝐵 ∈ {𝐵, 𝐶, (𝐼‘𝐴)}
49	35, 48	sselii 3928	. . . . . . . . . . . 12 ⊢ 𝐵 ∈ 𝑇
50	44, 49	eqeltrri 2858	. . . . . . . . . . 11 ⊢ (𝑋 ∖ (𝐾‘𝐴)) ∈ 𝑇
51	13, 14, 26, 27, 5	kur14lem5 35975	. . . . . . . . . . . 12 ⊢ (𝐾‘(𝐾‘𝐴)) = (𝐾‘𝐴)
52	51, 24	eqeltri 2857	. . . . . . . . . . 11 ⊢ (𝐾‘(𝐾‘𝐴)) ∈ 𝑇
53	43, 50, 52	kur14lem1 35971	. . . . . . . . . 10 ⊢ (𝑁 = (𝐾‘𝐴) → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
54	25, 42, 53	3jaoi 1454	. . . . . . . . 9 ⊢ ((𝑁 = 𝐴 ∨ 𝑁 = (𝑋 ∖ 𝐴) ∨ 𝑁 = (𝐾‘𝐴)) → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
55	4, 54	syl 18	. . . . . . . 8 ⊢ (𝑁 ∈ {𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
56		eltpi 4649	. . . . . . . . 9 ⊢ (𝑁 ∈ {𝐵, 𝐶, (𝐼‘𝐴)} → (𝑁 = 𝐵 ∨ 𝑁 = 𝐶 ∨ 𝑁 = (𝐼‘𝐴)))
57	44	difeq2i 4071	. . . . . . . . . . . . 13 ⊢ (𝑋 ∖ 𝐵) = (𝑋 ∖ (𝑋 ∖ (𝐾‘𝐴)))
58	13, 14, 26, 27, 43	kur14lem4 35974	. . . . . . . . . . . . 13 ⊢ (𝑋 ∖ (𝑋 ∖ (𝐾‘𝐴))) = (𝐾‘𝐴)
59	57, 58	eqtri 2784	. . . . . . . . . . . 12 ⊢ (𝑋 ∖ 𝐵) = (𝐾‘𝐴)
60	59, 24	eqeltri 2857	. . . . . . . . . . 11 ⊢ (𝑋 ∖ 𝐵) ∈ 𝑇
61		ssun2 4125	. . . . . . . . . . . . 13 ⊢ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))} ⊆ (({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))})
62	61, 10	sstri 3940	. . . . . . . . . . . 12 ⊢ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))} ⊆ 𝑇
63		fvex 6898	. . . . . . . . . . . . 13 ⊢ (𝐾‘𝐵) ∈ V
64	63	tpid1 4729	. . . . . . . . . . . 12 ⊢ (𝐾‘𝐵) ∈ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}
65	62, 64	sselii 3928	. . . . . . . . . . 11 ⊢ (𝐾‘𝐵) ∈ 𝑇
66	46, 60, 65	kur14lem1 35971	. . . . . . . . . 10 ⊢ (𝑁 = 𝐵 → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
67	33	difeq2i 4071	. . . . . . . . . . . . 13 ⊢ (𝑋 ∖ 𝐶) = (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝐴)))
68	13, 14, 26, 27, 5	kur14lem2 35972	. . . . . . . . . . . . 13 ⊢ (𝐼‘𝐴) = (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝐴)))
69	67, 68	eqtr4i 2787	. . . . . . . . . . . 12 ⊢ (𝑋 ∖ 𝐶) = (𝐼‘𝐴)
70		fvex 6898	. . . . . . . . . . . . . 14 ⊢ (𝐼‘𝐴) ∈ V
71	70	tpid3 4734	. . . . . . . . . . . . 13 ⊢ (𝐼‘𝐴) ∈ {𝐵, 𝐶, (𝐼‘𝐴)}
72	35, 71	sselii 3928	. . . . . . . . . . . 12 ⊢ (𝐼‘𝐴) ∈ 𝑇
73	69, 72	eqeltri 2857	. . . . . . . . . . 11 ⊢ (𝑋 ∖ 𝐶) ∈ 𝑇
74	13, 14, 26, 27, 18	kur14lem5 35975	. . . . . . . . . . . . 13 ⊢ (𝐾‘(𝐾‘(𝑋 ∖ 𝐴))) = (𝐾‘(𝑋 ∖ 𝐴))
75	33	fveq2i 6888	. . . . . . . . . . . . 13 ⊢ (𝐾‘𝐶) = (𝐾‘(𝐾‘(𝑋 ∖ 𝐴)))
76	74, 75, 33	3eqtr4i 2794	. . . . . . . . . . . 12 ⊢ (𝐾‘𝐶) = 𝐶
77	76, 40	eqeltri 2857	. . . . . . . . . . 11 ⊢ (𝐾‘𝐶) ∈ 𝑇
78	37, 73, 77	kur14lem1 35971	. . . . . . . . . 10 ⊢ (𝑁 = 𝐶 → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
79		difss 4083	. . . . . . . . . . . 12 ⊢ (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝐴))) ⊆ 𝑋
80	68, 79	eqsstri 3977	. . . . . . . . . . 11 ⊢ (𝐼‘𝐴) ⊆ 𝑋
81	69	difeq2i 4071	. . . . . . . . . . . . 13 ⊢ (𝑋 ∖ (𝑋 ∖ 𝐶)) = (𝑋 ∖ (𝐼‘𝐴))
82	13, 14, 26, 27, 37	kur14lem4 35974	. . . . . . . . . . . . 13 ⊢ (𝑋 ∖ (𝑋 ∖ 𝐶)) = 𝐶
83	81, 82	eqtr3i 2786	. . . . . . . . . . . 12 ⊢ (𝑋 ∖ (𝐼‘𝐴)) = 𝐶
84	83, 40	eqeltri 2857	. . . . . . . . . . 11 ⊢ (𝑋 ∖ (𝐼‘𝐴)) ∈ 𝑇
85		fvex 6898	. . . . . . . . . . . . 13 ⊢ (𝐾‘(𝐼‘𝐴)) ∈ V
86	85	tpid3 4734	. . . . . . . . . . . 12 ⊢ (𝐾‘(𝐼‘𝐴)) ∈ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}
87	62, 86	sselii 3928	. . . . . . . . . . 11 ⊢ (𝐾‘(𝐼‘𝐴)) ∈ 𝑇
88	80, 84, 87	kur14lem1 35971	. . . . . . . . . 10 ⊢ (𝑁 = (𝐼‘𝐴) → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
89	66, 78, 88	3jaoi 1454	. . . . . . . . 9 ⊢ ((𝑁 = 𝐵 ∨ 𝑁 = 𝐶 ∨ 𝑁 = (𝐼‘𝐴)) → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
90	56, 89	syl 18	. . . . . . . 8 ⊢ (𝑁 ∈ {𝐵, 𝐶, (𝐼‘𝐴)} → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
91	55, 90	jaoi 871	. . . . . . 7 ⊢ ((𝑁 ∈ {𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∨ 𝑁 ∈ {𝐵, 𝐶, (𝐼‘𝐴)}) → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
92	3, 91	sylbi 220	. . . . . 6 ⊢ (𝑁 ∈ ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
93		eltpi 4649	. . . . . . 7 ⊢ (𝑁 ∈ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))} → (𝑁 = (𝐾‘𝐵) ∨ 𝑁 = 𝐷 ∨ 𝑁 = (𝐾‘(𝐼‘𝐴))))
94	13, 14, 26, 27, 46	kur14lem3 35973	. . . . . . . . 9 ⊢ (𝐾‘𝐵) ⊆ 𝑋
95	13, 14, 26, 27, 43	kur14lem2 35972	. . . . . . . . . . 11 ⊢ (𝐼‘(𝐾‘𝐴)) = (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘𝐴))))
96		kur14lem.d	. . . . . . . . . . 11 ⊢ 𝐷 = (𝐼‘(𝐾‘𝐴))
97	44	fveq2i 6888	. . . . . . . . . . . 12 ⊢ (𝐾‘𝐵) = (𝐾‘(𝑋 ∖ (𝐾‘𝐴)))
98	97	difeq2i 4071	. . . . . . . . . . 11 ⊢ (𝑋 ∖ (𝐾‘𝐵)) = (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘𝐴))))
99	95, 96, 98	3eqtr4i 2794	. . . . . . . . . 10 ⊢ 𝐷 = (𝑋 ∖ (𝐾‘𝐵))
100	96, 95	eqtri 2784	. . . . . . . . . . . . . 14 ⊢ 𝐷 = (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘𝐴))))
101		difss 4083	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘𝐴)))) ⊆ 𝑋
102	100, 101	eqsstri 3977	. . . . . . . . . . . . 13 ⊢ 𝐷 ⊆ 𝑋
103	17, 102	ssexi 5284	. . . . . . . . . . . 12 ⊢ 𝐷 ∈ V
104	103	tpid2 4731	. . . . . . . . . . 11 ⊢ 𝐷 ∈ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}
105	62, 104	sselii 3928	. . . . . . . . . 10 ⊢ 𝐷 ∈ 𝑇
106	99, 105	eqeltrri 2858	. . . . . . . . 9 ⊢ (𝑋 ∖ (𝐾‘𝐵)) ∈ 𝑇
107	13, 14, 26, 27, 46	kur14lem5 35975	. . . . . . . . . 10 ⊢ (𝐾‘(𝐾‘𝐵)) = (𝐾‘𝐵)
108	107, 65	eqeltri 2857	. . . . . . . . 9 ⊢ (𝐾‘(𝐾‘𝐵)) ∈ 𝑇
109	94, 106, 108	kur14lem1 35971	. . . . . . . 8 ⊢ (𝑁 = (𝐾‘𝐵) → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
110	99	difeq2i 4071	. . . . . . . . . . 11 ⊢ (𝑋 ∖ 𝐷) = (𝑋 ∖ (𝑋 ∖ (𝐾‘𝐵)))
111	13, 14, 26, 27, 94	kur14lem4 35974	. . . . . . . . . . 11 ⊢ (𝑋 ∖ (𝑋 ∖ (𝐾‘𝐵))) = (𝐾‘𝐵)
112	110, 111	eqtri 2784	. . . . . . . . . 10 ⊢ (𝑋 ∖ 𝐷) = (𝐾‘𝐵)
113	112, 65	eqeltri 2857	. . . . . . . . 9 ⊢ (𝑋 ∖ 𝐷) ∈ 𝑇
114		ssun1 4124	. . . . . . . . . . 11 ⊢ {(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ⊆ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))})
115		ssun2 4125	. . . . . . . . . . . 12 ⊢ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))}) ⊆ ((({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ∪ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))}))
116	115, 9	sseqtrri 3980	. . . . . . . . . . 11 ⊢ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))}) ⊆ 𝑇
117	114, 116	sstri 3940	. . . . . . . . . 10 ⊢ {(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ⊆ 𝑇
118		fvex 6898	. . . . . . . . . . 11 ⊢ (𝐾‘𝐷) ∈ V
119	118	tpid2 4731	. . . . . . . . . 10 ⊢ (𝐾‘𝐷) ∈ {(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))}
120	117, 119	sselii 3928	. . . . . . . . 9 ⊢ (𝐾‘𝐷) ∈ 𝑇
121	102, 113, 120	kur14lem1 35971	. . . . . . . 8 ⊢ (𝑁 = 𝐷 → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
122	13, 14, 26, 27, 80	kur14lem3 35973	. . . . . . . . 9 ⊢ (𝐾‘(𝐼‘𝐴)) ⊆ 𝑋
123	13, 14, 26, 27, 37	kur14lem2 35972	. . . . . . . . . . 11 ⊢ (𝐼‘𝐶) = (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝐶)))
124	69	fveq2i 6888	. . . . . . . . . . . 12 ⊢ (𝐾‘(𝑋 ∖ 𝐶)) = (𝐾‘(𝐼‘𝐴))
125	124	difeq2i 4071	. . . . . . . . . . 11 ⊢ (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝐶))) = (𝑋 ∖ (𝐾‘(𝐼‘𝐴)))
126	123, 125	eqtri 2784	. . . . . . . . . 10 ⊢ (𝐼‘𝐶) = (𝑋 ∖ (𝐾‘(𝐼‘𝐴)))
127		fvex 6898	. . . . . . . . . . . 12 ⊢ (𝐼‘𝐶) ∈ V
128	127	tpid1 4729	. . . . . . . . . . 11 ⊢ (𝐼‘𝐶) ∈ {(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))}
129	117, 128	sselii 3928	. . . . . . . . . 10 ⊢ (𝐼‘𝐶) ∈ 𝑇
130	126, 129	eqeltrri 2858	. . . . . . . . 9 ⊢ (𝑋 ∖ (𝐾‘(𝐼‘𝐴))) ∈ 𝑇
131	13, 14, 26, 27, 80	kur14lem5 35975	. . . . . . . . . 10 ⊢ (𝐾‘(𝐾‘(𝐼‘𝐴))) = (𝐾‘(𝐼‘𝐴))
132	131, 87	eqeltri 2857	. . . . . . . . 9 ⊢ (𝐾‘(𝐾‘(𝐼‘𝐴))) ∈ 𝑇
133	122, 130, 132	kur14lem1 35971	. . . . . . . 8 ⊢ (𝑁 = (𝐾‘(𝐼‘𝐴)) → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
134	109, 121, 133	3jaoi 1454	. . . . . . 7 ⊢ ((𝑁 = (𝐾‘𝐵) ∨ 𝑁 = 𝐷 ∨ 𝑁 = (𝐾‘(𝐼‘𝐴))) → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
135	93, 134	syl 18	. . . . . 6 ⊢ (𝑁 ∈ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))} → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
136	92, 135	jaoi 871	. . . . 5 ⊢ ((𝑁 ∈ ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∨ 𝑁 ∈ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
137	2, 136	sylbi 220	. . . 4 ⊢ (𝑁 ∈ (({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
138		elun 4100	. . . . 5 ⊢ (𝑁 ∈ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))}) ↔ (𝑁 ∈ {(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∨ 𝑁 ∈ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))}))
139		eltpi 4649	. . . . . . 7 ⊢ (𝑁 ∈ {(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} → (𝑁 = (𝐼‘𝐶) ∨ 𝑁 = (𝐾‘𝐷) ∨ 𝑁 = (𝐼‘(𝐾‘𝐵))))
140		difss 4083	. . . . . . . . . 10 ⊢ (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝐶))) ⊆ 𝑋
141	123, 140	eqsstri 3977	. . . . . . . . 9 ⊢ (𝐼‘𝐶) ⊆ 𝑋
142	126	difeq2i 4071	. . . . . . . . . . 11 ⊢ (𝑋 ∖ (𝐼‘𝐶)) = (𝑋 ∖ (𝑋 ∖ (𝐾‘(𝐼‘𝐴))))
143	13, 14, 26, 27, 122	kur14lem4 35974	. . . . . . . . . . 11 ⊢ (𝑋 ∖ (𝑋 ∖ (𝐾‘(𝐼‘𝐴)))) = (𝐾‘(𝐼‘𝐴))
144	142, 143	eqtri 2784	. . . . . . . . . 10 ⊢ (𝑋 ∖ (𝐼‘𝐶)) = (𝐾‘(𝐼‘𝐴))
145	144, 87	eqeltri 2857	. . . . . . . . 9 ⊢ (𝑋 ∖ (𝐼‘𝐶)) ∈ 𝑇
146		ssun2 4125	. . . . . . . . . . 11 ⊢ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))} ⊆ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))})
147	146, 116	sstri 3940	. . . . . . . . . 10 ⊢ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))} ⊆ 𝑇
148		fvex 6898	. . . . . . . . . . 11 ⊢ (𝐾‘(𝐼‘𝐶)) ∈ V
149	148	prid1 4723	. . . . . . . . . 10 ⊢ (𝐾‘(𝐼‘𝐶)) ∈ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))}
150	147, 149	sselii 3928	. . . . . . . . 9 ⊢ (𝐾‘(𝐼‘𝐶)) ∈ 𝑇
151	141, 145, 150	kur14lem1 35971	. . . . . . . 8 ⊢ (𝑁 = (𝐼‘𝐶) → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
152	13, 14, 26, 27, 102	kur14lem3 35973	. . . . . . . . 9 ⊢ (𝐾‘𝐷) ⊆ 𝑋
153	99	fveq2i 6888	. . . . . . . . . . . 12 ⊢ (𝐾‘𝐷) = (𝐾‘(𝑋 ∖ (𝐾‘𝐵)))
154	153	difeq2i 4071	. . . . . . . . . . 11 ⊢ (𝑋 ∖ (𝐾‘𝐷)) = (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘𝐵))))
155	13, 14, 26, 27, 94	kur14lem2 35972	. . . . . . . . . . 11 ⊢ (𝐼‘(𝐾‘𝐵)) = (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘𝐵))))
156	154, 155	eqtr4i 2787	. . . . . . . . . 10 ⊢ (𝑋 ∖ (𝐾‘𝐷)) = (𝐼‘(𝐾‘𝐵))
157		fvex 6898	. . . . . . . . . . . 12 ⊢ (𝐼‘(𝐾‘𝐵)) ∈ V
158	157	tpid3 4734	. . . . . . . . . . 11 ⊢ (𝐼‘(𝐾‘𝐵)) ∈ {(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))}
159	117, 158	sselii 3928	. . . . . . . . . 10 ⊢ (𝐼‘(𝐾‘𝐵)) ∈ 𝑇
160	156, 159	eqeltri 2857	. . . . . . . . 9 ⊢ (𝑋 ∖ (𝐾‘𝐷)) ∈ 𝑇
161	13, 14, 26, 27, 102	kur14lem5 35975	. . . . . . . . . 10 ⊢ (𝐾‘(𝐾‘𝐷)) = (𝐾‘𝐷)
162	161, 120	eqeltri 2857	. . . . . . . . 9 ⊢ (𝐾‘(𝐾‘𝐷)) ∈ 𝑇
163	152, 160, 162	kur14lem1 35971	. . . . . . . 8 ⊢ (𝑁 = (𝐾‘𝐷) → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
164		difss 4083	. . . . . . . . . 10 ⊢ (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘𝐵)))) ⊆ 𝑋
165	155, 164	eqsstri 3977	. . . . . . . . 9 ⊢ (𝐼‘(𝐾‘𝐵)) ⊆ 𝑋
166	156	difeq2i 4071	. . . . . . . . . . 11 ⊢ (𝑋 ∖ (𝑋 ∖ (𝐾‘𝐷))) = (𝑋 ∖ (𝐼‘(𝐾‘𝐵)))
167	13, 14, 26, 27, 152	kur14lem4 35974	. . . . . . . . . . 11 ⊢ (𝑋 ∖ (𝑋 ∖ (𝐾‘𝐷))) = (𝐾‘𝐷)
168	166, 167	eqtr3i 2786	. . . . . . . . . 10 ⊢ (𝑋 ∖ (𝐼‘(𝐾‘𝐵))) = (𝐾‘𝐷)
169	168, 120	eqeltri 2857	. . . . . . . . 9 ⊢ (𝑋 ∖ (𝐼‘(𝐾‘𝐵))) ∈ 𝑇
170	13, 14, 26, 27, 5, 44	kur14lem6 35976	. . . . . . . . . 10 ⊢ (𝐾‘(𝐼‘(𝐾‘𝐵))) = (𝐾‘𝐵)
171	170, 65	eqeltri 2857	. . . . . . . . 9 ⊢ (𝐾‘(𝐼‘(𝐾‘𝐵))) ∈ 𝑇
172	165, 169, 171	kur14lem1 35971	. . . . . . . 8 ⊢ (𝑁 = (𝐼‘(𝐾‘𝐵)) → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
173	151, 163, 172	3jaoi 1454	. . . . . . 7 ⊢ ((𝑁 = (𝐼‘𝐶) ∨ 𝑁 = (𝐾‘𝐷) ∨ 𝑁 = (𝐼‘(𝐾‘𝐵))) → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
174	139, 173	syl 18	. . . . . 6 ⊢ (𝑁 ∈ {(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
175		elpri 4608	. . . . . . 7 ⊢ (𝑁 ∈ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))} → (𝑁 = (𝐾‘(𝐼‘𝐶)) ∨ 𝑁 = (𝐼‘(𝐾‘(𝐼‘𝐴)))))
176	13, 14, 26, 27, 141	kur14lem3 35973	. . . . . . . . 9 ⊢ (𝐾‘(𝐼‘𝐶)) ⊆ 𝑋
177	126	fveq2i 6888	. . . . . . . . . . . 12 ⊢ (𝐾‘(𝐼‘𝐶)) = (𝐾‘(𝑋 ∖ (𝐾‘(𝐼‘𝐴))))
178	177	difeq2i 4071	. . . . . . . . . . 11 ⊢ (𝑋 ∖ (𝐾‘(𝐼‘𝐶))) = (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘(𝐼‘𝐴)))))
179	13, 14, 26, 27, 122	kur14lem2 35972	. . . . . . . . . . 11 ⊢ (𝐼‘(𝐾‘(𝐼‘𝐴))) = (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘(𝐼‘𝐴)))))
180	178, 179	eqtr4i 2787	. . . . . . . . . 10 ⊢ (𝑋 ∖ (𝐾‘(𝐼‘𝐶))) = (𝐼‘(𝐾‘(𝐼‘𝐴)))
181		fvex 6898	. . . . . . . . . . . 12 ⊢ (𝐼‘(𝐾‘(𝐼‘𝐴))) ∈ V
182	181	prid2 4724	. . . . . . . . . . 11 ⊢ (𝐼‘(𝐾‘(𝐼‘𝐴))) ∈ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))}
183	147, 182	sselii 3928	. . . . . . . . . 10 ⊢ (𝐼‘(𝐾‘(𝐼‘𝐴))) ∈ 𝑇
184	180, 183	eqeltri 2857	. . . . . . . . 9 ⊢ (𝑋 ∖ (𝐾‘(𝐼‘𝐶))) ∈ 𝑇
185	13, 14, 26, 27, 141	kur14lem5 35975	. . . . . . . . . 10 ⊢ (𝐾‘(𝐾‘(𝐼‘𝐶))) = (𝐾‘(𝐼‘𝐶))
186	185, 150	eqeltri 2857	. . . . . . . . 9 ⊢ (𝐾‘(𝐾‘(𝐼‘𝐶))) ∈ 𝑇
187	176, 184, 186	kur14lem1 35971	. . . . . . . 8 ⊢ (𝑁 = (𝐾‘(𝐼‘𝐶)) → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
188		difss 4083	. . . . . . . . . 10 ⊢ (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘(𝐼‘𝐴))))) ⊆ 𝑋
189	179, 188	eqsstri 3977	. . . . . . . . 9 ⊢ (𝐼‘(𝐾‘(𝐼‘𝐴))) ⊆ 𝑋
190	180	difeq2i 4071	. . . . . . . . . . 11 ⊢ (𝑋 ∖ (𝑋 ∖ (𝐾‘(𝐼‘𝐶)))) = (𝑋 ∖ (𝐼‘(𝐾‘(𝐼‘𝐴))))
191	13, 14, 26, 27, 176	kur14lem4 35974	. . . . . . . . . . 11 ⊢ (𝑋 ∖ (𝑋 ∖ (𝐾‘(𝐼‘𝐶)))) = (𝐾‘(𝐼‘𝐶))
192	190, 191	eqtr3i 2786	. . . . . . . . . 10 ⊢ (𝑋 ∖ (𝐼‘(𝐾‘(𝐼‘𝐴)))) = (𝐾‘(𝐼‘𝐶))
193	192, 150	eqeltri 2857	. . . . . . . . 9 ⊢ (𝑋 ∖ (𝐼‘(𝐾‘(𝐼‘𝐴)))) ∈ 𝑇
194	13, 14, 26, 27, 18, 68	kur14lem6 35976	. . . . . . . . . 10 ⊢ (𝐾‘(𝐼‘(𝐾‘(𝐼‘𝐴)))) = (𝐾‘(𝐼‘𝐴))
195	194, 87	eqeltri 2857	. . . . . . . . 9 ⊢ (𝐾‘(𝐼‘(𝐾‘(𝐼‘𝐴)))) ∈ 𝑇
196	189, 193, 195	kur14lem1 35971	. . . . . . . 8 ⊢ (𝑁 = (𝐼‘(𝐾‘(𝐼‘𝐴))) → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
197	187, 196	jaoi 871	. . . . . . 7 ⊢ ((𝑁 = (𝐾‘(𝐼‘𝐶)) ∨ 𝑁 = (𝐼‘(𝐾‘(𝐼‘𝐴)))) → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
198	175, 197	syl 18	. . . . . 6 ⊢ (𝑁 ∈ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))} → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
199	174, 198	jaoi 871	. . . . 5 ⊢ ((𝑁 ∈ {(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∨ 𝑁 ∈ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))}) → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
200	138, 199	sylbi 220	. . . 4 ⊢ (𝑁 ∈ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))}) → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
201	137, 200	jaoi 871	. . 3 ⊢ ((𝑁 ∈ (({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ∨ 𝑁 ∈ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))})) → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
202	1, 201	sylbi 220	. 2 ⊢ (𝑁 ∈ ((({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ∪ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))})) → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))
203	202, 9	eleq2s 2879	1 ⊢ (𝑁 ∈ 𝑇 → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   = wceq 1570   ∈ wcel 2145   ∖ cdif 3896   ∪ cun 3897   ⊆ wss 3899  {cpr 4586  {ctp 4588  ∪ cuni 4867  ‘cfv 6538  Topctop 23211  intcnt 23335  clsccl 23336

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-top 23212  df-cld 23337  df-ntr 23338  df-cls 23339

This theorem is used by:  kur14lem9  35979