Theorem usgrexmpl2trifr 47772

Description: 𝐺 is triangle-free. (Contributed by AV, 10-Aug-2025.)

Hypotheses

Ref	Expression
usgrexmpl2.v	⊢ 𝑉 = (0...5)
usgrexmpl2.e	⊢ 𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩
usgrexmpl2.g	⊢ 𝐺 = ⟨𝑉, 𝐸⟩

Assertion

Ref	Expression
usgrexmpl2trifr	⊢ ¬ ∃𝑡 𝑡 ∈ (GrTriangles‘𝐺)

Distinct variable group:   𝑡,𝐺

Allowed substitution hints:   𝐸(𝑡)   𝑉(𝑡)

Proof of Theorem usgrexmpl2trifr

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgrexmpl2.v	. . . . . . . . . 10 ⊢ 𝑉 = (0...5)
2		usgrexmpl2.e	. . . . . . . . . 10 ⊢ 𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩
3		usgrexmpl2.g	. . . . . . . . . 10 ⊢ 𝐺 = ⟨𝑉, 𝐸⟩
4	1, 2, 3	usgrexmpl2nb0 47766	. . . . . . . . 9 ⊢ (𝐺 NeighbVtx 0) = {1, 3, 5}
5	4	eleq2i 2830	. . . . . . . 8 ⊢ (𝑏 ∈ (𝐺 NeighbVtx 0) ↔ 𝑏 ∈ {1, 3, 5})
6		vex 3486	. . . . . . . . 9 ⊢ 𝑏 ∈ V
7	6	eltp 4712	. . . . . . . 8 ⊢ (𝑏 ∈ {1, 3, 5} ↔ (𝑏 = 1 ∨ 𝑏 = 3 ∨ 𝑏 = 5))
8	5, 7	bitri 275	. . . . . . 7 ⊢ (𝑏 ∈ (𝐺 NeighbVtx 0) ↔ (𝑏 = 1 ∨ 𝑏 = 3 ∨ 𝑏 = 5))
9	4	eleq2i 2830	. . . . . . . 8 ⊢ (𝑐 ∈ (𝐺 NeighbVtx 0) ↔ 𝑐 ∈ {1, 3, 5})
10		vex 3486	. . . . . . . . 9 ⊢ 𝑐 ∈ V
11	10	eltp 4712	. . . . . . . 8 ⊢ (𝑐 ∈ {1, 3, 5} ↔ (𝑐 = 1 ∨ 𝑐 = 3 ∨ 𝑐 = 5))
12	9, 11	bitri 275	. . . . . . 7 ⊢ (𝑐 ∈ (𝐺 NeighbVtx 0) ↔ (𝑐 = 1 ∨ 𝑐 = 3 ∨ 𝑐 = 5))
13		eqtr3 2760	. . . . . . . . . 10 ⊢ ((𝑏 = 1 ∧ 𝑐 = 1) → 𝑏 = 𝑐)
14	13	orcd 872	. . . . . . . . 9 ⊢ ((𝑏 = 1 ∧ 𝑐 = 1) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
15		ax-1ne0 11249	. . . . . . . . . . . . . . 15 ⊢ 1 ≠ 0
16		neeq1 3005	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 1 → (𝑏 ≠ 0 ↔ 1 ≠ 0))
17	15, 16	mpbiri 258	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 1 → 𝑏 ≠ 0)
18	17	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → 𝑏 ≠ 0)
19	18	neneqd 2947	. . . . . . . . . . . 12 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → ¬ 𝑏 = 0)
20	19	orcd 872	. . . . . . . . . . 11 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3))
21		3ne0 12395	. . . . . . . . . . . . . . 15 ⊢ 3 ≠ 0
22		neeq1 3005	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 3 → (𝑐 ≠ 0 ↔ 3 ≠ 0))
23	21, 22	mpbiri 258	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 3 → 𝑐 ≠ 0)
24	23	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → 𝑐 ≠ 0)
25	24	neneqd 2947	. . . . . . . . . . . 12 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → ¬ 𝑐 = 0)
26	25	olcd 873	. . . . . . . . . . 11 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0))
27	19	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1))
28	25	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0))
29	27, 28	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)))
30		2re 12363	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℝ
31		2lt3 12461	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 < 3
32	30, 31	gtneii 11398	. . . . . . . . . . . . . . . . . 18 ⊢ 3 ≠ 2
33		neeq1 3005	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = 3 → (𝑐 ≠ 2 ↔ 3 ≠ 2))
34	32, 33	mpbiri 258	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 3 → 𝑐 ≠ 2)
35	34	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → 𝑐 ≠ 2)
36	35	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → ¬ 𝑐 = 2)
37	36	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2))
38		1re 11286	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℝ
39		1lt3 12462	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 < 3
40	38, 39	gtneii 11398	. . . . . . . . . . . . . . . . . 18 ⊢ 3 ≠ 1
41		neeq1 3005	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = 3 → (𝑐 ≠ 1 ↔ 3 ≠ 1))
42	40, 41	mpbiri 258	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 3 → 𝑐 ≠ 1)
43	42	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → 𝑐 ≠ 1)
44	43	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → ¬ 𝑐 = 1)
45	44	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1))
46	37, 45	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)))
47		1ne2 12497	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ≠ 2
48		neeq1 3005	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 1 → (𝑏 ≠ 2 ↔ 1 ≠ 2))
49	47, 48	mpbiri 258	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 1 → 𝑏 ≠ 2)
50	49	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → 𝑏 ≠ 2)
51	50	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → ¬ 𝑏 = 2)
52	51	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3))
53	36	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))
54	52, 53	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
55	29, 46, 54	3jca 1128	. . . . . . . . . . . 12 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))))
56	38, 39	ltneii 11399	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ≠ 3
57		neeq1 3005	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 1 → (𝑏 ≠ 3 ↔ 1 ≠ 3))
58	56, 57	mpbiri 258	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 1 → 𝑏 ≠ 3)
59	58	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → 𝑏 ≠ 3)
60	59	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → ¬ 𝑏 = 3)
61	60	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4))
62		1lt4 12465	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 < 4
63	38, 62	ltneii 11399	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ≠ 4
64		neeq1 3005	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 1 → (𝑏 ≠ 4 ↔ 1 ≠ 4))
65	63, 64	mpbiri 258	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 1 → 𝑏 ≠ 4)
66	65	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → 𝑏 ≠ 4)
67	66	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → ¬ 𝑏 = 4)
68	67	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3))
69	61, 68	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → ((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)))
70	67	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5))
71		1lt5 12469	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 < 5
72	38, 71	ltneii 11399	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ≠ 5
73		neeq1 3005	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 1 → (𝑏 ≠ 5 ↔ 1 ≠ 5))
74	72, 73	mpbiri 258	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 1 → 𝑏 ≠ 5)
75	74	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → 𝑏 ≠ 5)
76	75	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → ¬ 𝑏 = 5)
77	76	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4))
78	70, 77	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)))
79	19	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5))
80	25	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))
81	79, 80	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
82	69, 78, 81	3jca 1128	. . . . . . . . . . . 12 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))
83	55, 82	jca 511	. . . . . . . . . . 11 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))
84	20, 26, 83	jca31 514	. . . . . . . . . 10 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))
85	84	olcd 873	. . . . . . . . 9 ⊢ ((𝑏 = 1 ∧ 𝑐 = 3) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
86	17	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → 𝑏 ≠ 0)
87	86	neneqd 2947	. . . . . . . . . . . 12 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → ¬ 𝑏 = 0)
88	87	orcd 872	. . . . . . . . . . 11 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3))
89	58	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → 𝑏 ≠ 3)
90	89	neneqd 2947	. . . . . . . . . . . 12 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → ¬ 𝑏 = 3)
91	90	orcd 872	. . . . . . . . . . 11 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0))
92	87	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1))
93		0re 11288	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ
94		5pos 12398	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 < 5
95	93, 94	gtneii 11398	. . . . . . . . . . . . . . . . . 18 ⊢ 5 ≠ 0
96		neeq1 3005	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = 5 → (𝑐 ≠ 0 ↔ 5 ≠ 0))
97	95, 96	mpbiri 258	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 5 → 𝑐 ≠ 0)
98	97	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → 𝑐 ≠ 0)
99	98	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → ¬ 𝑐 = 0)
100	99	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0))
101	92, 100	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)))
102		2lt5 12468	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 < 5
103	30, 102	gtneii 11398	. . . . . . . . . . . . . . . . . 18 ⊢ 5 ≠ 2
104		neeq1 3005	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = 5 → (𝑐 ≠ 2 ↔ 5 ≠ 2))
105	103, 104	mpbiri 258	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 5 → 𝑐 ≠ 2)
106	105	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → 𝑐 ≠ 2)
107	106	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → ¬ 𝑐 = 2)
108	107	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2))
109	49	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → 𝑏 ≠ 2)
110	109	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → ¬ 𝑏 = 2)
111	110	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1))
112	108, 111	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)))
113	110	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3))
114	90	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))
115	113, 114	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
116	101, 112, 115	3jca 1128	. . . . . . . . . . . 12 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))))
117	90	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4))
118	65	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → 𝑏 ≠ 4)
119	118	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → ¬ 𝑏 = 4)
120	119	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3))
121	117, 120	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → ((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)))
122	119	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5))
123	74	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → 𝑏 ≠ 5)
124	123	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → ¬ 𝑏 = 5)
125	124	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4))
126	122, 125	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)))
127	87	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5))
128	99	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))
129	127, 128	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
130	121, 126, 129	3jca 1128	. . . . . . . . . . . 12 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))
131	116, 130	jca 511	. . . . . . . . . . 11 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))
132	88, 91, 131	jca31 514	. . . . . . . . . 10 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))
133	132	olcd 873	. . . . . . . . 9 ⊢ ((𝑏 = 1 ∧ 𝑐 = 5) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
134	14, 85, 133	3jaodan 1431	. . . . . . . 8 ⊢ ((𝑏 = 1 ∧ (𝑐 = 1 ∨ 𝑐 = 3 ∨ 𝑐 = 5)) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
135		neeq1 3005	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 3 → (𝑏 ≠ 0 ↔ 3 ≠ 0))
136	21, 135	mpbiri 258	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 3 → 𝑏 ≠ 0)
137	136	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → 𝑏 ≠ 0)
138	137	neneqd 2947	. . . . . . . . . . . 12 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → ¬ 𝑏 = 0)
139	138	orcd 872	. . . . . . . . . . 11 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3))
140		neeq1 3005	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 1 → (𝑐 ≠ 0 ↔ 1 ≠ 0))
141	15, 140	mpbiri 258	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 1 → 𝑐 ≠ 0)
142	141	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → 𝑐 ≠ 0)
143	142	neneqd 2947	. . . . . . . . . . . 12 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → ¬ 𝑐 = 0)
144	143	olcd 873	. . . . . . . . . . 11 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0))
145	138	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1))
146	143	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0))
147	145, 146	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)))
148	58	necon2i 2977	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 3 → 𝑏 ≠ 1)
149	148	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → 𝑏 ≠ 1)
150	149	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → ¬ 𝑏 = 1)
151	150	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2))
152		neeq1 3005	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 3 → (𝑏 ≠ 2 ↔ 3 ≠ 2))
153	32, 152	mpbiri 258	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 3 → 𝑏 ≠ 2)
154	153	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → 𝑏 ≠ 2)
155	154	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → ¬ 𝑏 = 2)
156	155	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1))
157	151, 156	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)))
158	155	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3))
159		neeq1 3005	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = 1 → (𝑐 ≠ 2 ↔ 1 ≠ 2))
160	47, 159	mpbiri 258	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 1 → 𝑐 ≠ 2)
161	160	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → 𝑐 ≠ 2)
162	161	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → ¬ 𝑐 = 2)
163	162	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))
164	158, 163	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
165	147, 157, 164	3jca 1128	. . . . . . . . . . . 12 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))))
166		neeq1 3005	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = 1 → (𝑐 ≠ 4 ↔ 1 ≠ 4))
167	63, 166	mpbiri 258	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 1 → 𝑐 ≠ 4)
168	167	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → 𝑐 ≠ 4)
169	168	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → ¬ 𝑐 = 4)
170	169	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4))
171	42	necon2i 2977	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 1 → 𝑐 ≠ 3)
172	171	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → 𝑐 ≠ 3)
173	172	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → ¬ 𝑐 = 3)
174	173	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3))
175	170, 174	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → ((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)))
176		neeq1 3005	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = 1 → (𝑐 ≠ 5 ↔ 1 ≠ 5))
177	72, 176	mpbiri 258	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 1 → 𝑐 ≠ 5)
178	177	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → 𝑐 ≠ 5)
179	178	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → ¬ 𝑐 = 5)
180	179	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5))
181	169	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4))
182	180, 181	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)))
183	138	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5))
184	143	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))
185	183, 184	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
186	175, 182, 185	3jca 1128	. . . . . . . . . . . 12 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))
187	165, 186	jca 511	. . . . . . . . . . 11 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))
188	139, 144, 187	jca31 514	. . . . . . . . . 10 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))
189	188	olcd 873	. . . . . . . . 9 ⊢ ((𝑏 = 3 ∧ 𝑐 = 1) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
190		eqtr3 2760	. . . . . . . . . 10 ⊢ ((𝑏 = 3 ∧ 𝑐 = 3) → 𝑏 = 𝑐)
191	190	orcd 872	. . . . . . . . 9 ⊢ ((𝑏 = 3 ∧ 𝑐 = 3) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
192	136	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → 𝑏 ≠ 0)
193	192	neneqd 2947	. . . . . . . . . . . 12 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → ¬ 𝑏 = 0)
194	193	orcd 872	. . . . . . . . . . 11 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3))
195	97	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → 𝑐 ≠ 0)
196	195	neneqd 2947	. . . . . . . . . . . 12 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → ¬ 𝑐 = 0)
197	196	olcd 873	. . . . . . . . . . 11 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0))
198	193	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1))
199	196	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0))
200	198, 199	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)))
201	148	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → 𝑏 ≠ 1)
202	201	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → ¬ 𝑏 = 1)
203	202	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2))
204	177	necon2i 2977	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 5 → 𝑐 ≠ 1)
205	204	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → 𝑐 ≠ 1)
206	205	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → ¬ 𝑐 = 1)
207	206	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1))
208	203, 207	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)))
209	153	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → 𝑏 ≠ 2)
210	209	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → ¬ 𝑏 = 2)
211	210	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3))
212	105	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → 𝑐 ≠ 2)
213	212	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → ¬ 𝑐 = 2)
214	213	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))
215	211, 214	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
216	200, 208, 215	3jca 1128	. . . . . . . . . . . 12 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))))
217		4re 12373	. . . . . . . . . . . . . . . . . . 19 ⊢ 4 ∈ ℝ
218		4lt5 12466	. . . . . . . . . . . . . . . . . . 19 ⊢ 4 < 5
219	217, 218	gtneii 11398	. . . . . . . . . . . . . . . . . 18 ⊢ 5 ≠ 4
220		neeq1 3005	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = 5 → (𝑐 ≠ 4 ↔ 5 ≠ 4))
221	219, 220	mpbiri 258	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 5 → 𝑐 ≠ 4)
222	221	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → 𝑐 ≠ 4)
223	222	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → ¬ 𝑐 = 4)
224	223	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4))
225		3re 12369	. . . . . . . . . . . . . . . . . . 19 ⊢ 3 ∈ ℝ
226		3lt4 12463	. . . . . . . . . . . . . . . . . . 19 ⊢ 3 < 4
227	225, 226	ltneii 11399	. . . . . . . . . . . . . . . . . 18 ⊢ 3 ≠ 4
228		neeq1 3005	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 3 → (𝑏 ≠ 4 ↔ 3 ≠ 4))
229	227, 228	mpbiri 258	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 3 → 𝑏 ≠ 4)
230	229	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → 𝑏 ≠ 4)
231	230	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → ¬ 𝑏 = 4)
232	231	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3))
233	224, 232	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → ((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)))
234	231	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5))
235	223	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4))
236	234, 235	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)))
237	193	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5))
238	196	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))
239	237, 238	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
240	233, 236, 239	3jca 1128	. . . . . . . . . . . 12 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))
241	216, 240	jca 511	. . . . . . . . . . 11 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))
242	194, 197, 241	jca31 514	. . . . . . . . . 10 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))
243	242	olcd 873	. . . . . . . . 9 ⊢ ((𝑏 = 3 ∧ 𝑐 = 5) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
244	189, 191, 243	3jaodan 1431	. . . . . . . 8 ⊢ ((𝑏 = 3 ∧ (𝑐 = 1 ∨ 𝑐 = 3 ∨ 𝑐 = 5)) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
245	171	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → 𝑐 ≠ 3)
246	245	neneqd 2947	. . . . . . . . . . . 12 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → ¬ 𝑐 = 3)
247	246	olcd 873	. . . . . . . . . . 11 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3))
248	141	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → 𝑐 ≠ 0)
249	248	neneqd 2947	. . . . . . . . . . . 12 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → ¬ 𝑐 = 0)
250	249	olcd 873	. . . . . . . . . . 11 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0))
251		neeq1 3005	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 5 → (𝑏 ≠ 0 ↔ 5 ≠ 0))
252	95, 251	mpbiri 258	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 5 → 𝑏 ≠ 0)
253	252	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → 𝑏 ≠ 0)
254	253	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → ¬ 𝑏 = 0)
255	254	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1))
256	249	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0))
257	255, 256	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)))
258	74	necon2i 2977	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 5 → 𝑏 ≠ 1)
259	258	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → 𝑏 ≠ 1)
260	259	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → ¬ 𝑏 = 1)
261	260	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2))
262		neeq1 3005	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 5 → (𝑏 ≠ 2 ↔ 5 ≠ 2))
263	103, 262	mpbiri 258	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 5 → 𝑏 ≠ 2)
264	263	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → 𝑏 ≠ 2)
265	264	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → ¬ 𝑏 = 2)
266	265	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1))
267	261, 266	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)))
268	246	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3))
269	160	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → 𝑐 ≠ 2)
270	269	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → ¬ 𝑐 = 2)
271	270	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))
272	268, 271	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
273	257, 267, 272	3jca 1128	. . . . . . . . . . . 12 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))))
274		3lt5 12467	. . . . . . . . . . . . . . . . . . 19 ⊢ 3 < 5
275	225, 274	gtneii 11398	. . . . . . . . . . . . . . . . . 18 ⊢ 5 ≠ 3
276		neeq1 3005	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 5 → (𝑏 ≠ 3 ↔ 5 ≠ 3))
277	275, 276	mpbiri 258	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 5 → 𝑏 ≠ 3)
278	277	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → 𝑏 ≠ 3)
279	278	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → ¬ 𝑏 = 3)
280	279	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4))
281	246	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3))
282	280, 281	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → ((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)))
283	177	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → 𝑐 ≠ 5)
284	283	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → ¬ 𝑐 = 5)
285	284	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5))
286	167	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → 𝑐 ≠ 4)
287	286	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → ¬ 𝑐 = 4)
288	287	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4))
289	285, 288	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)))
290	254	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5))
291	249	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))
292	290, 291	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
293	282, 289, 292	3jca 1128	. . . . . . . . . . . 12 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))
294	273, 293	jca 511	. . . . . . . . . . 11 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))
295	247, 250, 294	jca31 514	. . . . . . . . . 10 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))
296	295	olcd 873	. . . . . . . . 9 ⊢ ((𝑏 = 5 ∧ 𝑐 = 1) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
297	252	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → 𝑏 ≠ 0)
298	297	neneqd 2947	. . . . . . . . . . . 12 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → ¬ 𝑏 = 0)
299	298	orcd 872	. . . . . . . . . . 11 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3))
300	23	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → 𝑐 ≠ 0)
301	300	neneqd 2947	. . . . . . . . . . . 12 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → ¬ 𝑐 = 0)
302	301	olcd 873	. . . . . . . . . . 11 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0))
303	298	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1))
304	301	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0))
305	303, 304	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)))
306	258	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → 𝑏 ≠ 1)
307	306	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → ¬ 𝑏 = 1)
308	307	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2))
309	42	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → 𝑐 ≠ 1)
310	309	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → ¬ 𝑐 = 1)
311	310	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1))
312	308, 311	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)))
313	263	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → 𝑏 ≠ 2)
314	313	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → ¬ 𝑏 = 2)
315	314	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3))
316	277	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → 𝑏 ≠ 3)
317	316	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → ¬ 𝑏 = 3)
318	317	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))
319	315, 318	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
320	305, 312, 319	3jca 1128	. . . . . . . . . . . 12 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))))
321	317	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4))
322		neeq1 3005	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 5 → (𝑏 ≠ 4 ↔ 5 ≠ 4))
323	219, 322	mpbiri 258	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 5 → 𝑏 ≠ 4)
324	323	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → 𝑏 ≠ 4)
325	324	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → ¬ 𝑏 = 4)
326	325	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3))
327	321, 326	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → ((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)))
328	325	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5))
329		neeq1 3005	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = 3 → (𝑐 ≠ 4 ↔ 3 ≠ 4))
330	227, 329	mpbiri 258	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 3 → 𝑐 ≠ 4)
331	330	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → 𝑐 ≠ 4)
332	331	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → ¬ 𝑐 = 4)
333	332	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4))
334	328, 333	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)))
335	298	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5))
336	301	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))
337	335, 336	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
338	327, 334, 337	3jca 1128	. . . . . . . . . . . 12 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))
339	320, 338	jca 511	. . . . . . . . . . 11 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))
340	299, 302, 339	jca31 514	. . . . . . . . . 10 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))
341	340	olcd 873	. . . . . . . . 9 ⊢ ((𝑏 = 5 ∧ 𝑐 = 3) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
342		eqtr3 2760	. . . . . . . . . 10 ⊢ ((𝑏 = 5 ∧ 𝑐 = 5) → 𝑏 = 𝑐)
343	342	orcd 872	. . . . . . . . 9 ⊢ ((𝑏 = 5 ∧ 𝑐 = 5) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
344	296, 341, 343	3jaodan 1431	. . . . . . . 8 ⊢ ((𝑏 = 5 ∧ (𝑐 = 1 ∨ 𝑐 = 3 ∨ 𝑐 = 5)) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
345	134, 244, 344	3jaoian 1430	. . . . . . 7 ⊢ (((𝑏 = 1 ∨ 𝑏 = 3 ∨ 𝑏 = 5) ∧ (𝑐 = 1 ∨ 𝑐 = 3 ∨ 𝑐 = 5)) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
346	8, 12, 345	syl2anb 597	. . . . . 6 ⊢ ((𝑏 ∈ (𝐺 NeighbVtx 0) ∧ 𝑐 ∈ (𝐺 NeighbVtx 0)) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
347	346	rgen2 3201	. . . . 5 ⊢ ∀𝑏 ∈ (𝐺 NeighbVtx 0)∀𝑐 ∈ (𝐺 NeighbVtx 0)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))
348	1, 2, 3	usgrexmpl2nb1 47767	. . . . . . . . 9 ⊢ (𝐺 NeighbVtx 1) = {0, 2}
349	348	eleq2i 2830	. . . . . . . 8 ⊢ (𝑏 ∈ (𝐺 NeighbVtx 1) ↔ 𝑏 ∈ {0, 2})
350	6	elpr 4672	. . . . . . . 8 ⊢ (𝑏 ∈ {0, 2} ↔ (𝑏 = 0 ∨ 𝑏 = 2))
351	349, 350	bitri 275	. . . . . . 7 ⊢ (𝑏 ∈ (𝐺 NeighbVtx 1) ↔ (𝑏 = 0 ∨ 𝑏 = 2))
352	348	eleq2i 2830	. . . . . . . 8 ⊢ (𝑐 ∈ (𝐺 NeighbVtx 1) ↔ 𝑐 ∈ {0, 2})
353	10	elpr 4672	. . . . . . . 8 ⊢ (𝑐 ∈ {0, 2} ↔ (𝑐 = 0 ∨ 𝑐 = 2))
354	352, 353	bitri 275	. . . . . . 7 ⊢ (𝑐 ∈ (𝐺 NeighbVtx 1) ↔ (𝑐 = 0 ∨ 𝑐 = 2))
355		eqtr3 2760	. . . . . . . . 9 ⊢ ((𝑏 = 0 ∧ 𝑐 = 0) → 𝑏 = 𝑐)
356	355	orcd 872	. . . . . . . 8 ⊢ ((𝑏 = 0 ∧ 𝑐 = 0) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
357		2ne0 12393	. . . . . . . . . . . . . 14 ⊢ 2 ≠ 0
358		neeq1 3005	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 2 → (𝑏 ≠ 0 ↔ 2 ≠ 0))
359	357, 358	mpbiri 258	. . . . . . . . . . . . 13 ⊢ (𝑏 = 2 → 𝑏 ≠ 0)
360	359	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → 𝑏 ≠ 0)
361	360	neneqd 2947	. . . . . . . . . . 11 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → ¬ 𝑏 = 0)
362	361	orcd 872	. . . . . . . . . 10 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3))
363	153	necon2i 2977	. . . . . . . . . . . . 13 ⊢ (𝑏 = 2 → 𝑏 ≠ 3)
364	363	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → 𝑏 ≠ 3)
365	364	neneqd 2947	. . . . . . . . . . 11 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → ¬ 𝑏 = 3)
366	365	orcd 872	. . . . . . . . . 10 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0))
367	361	orcd 872	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1))
368	49	necon2i 2977	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 2 → 𝑏 ≠ 1)
369	368	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → 𝑏 ≠ 1)
370	369	neneqd 2947	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → ¬ 𝑏 = 1)
371	370	orcd 872	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0))
372	367, 371	jca 511	. . . . . . . . . . . 12 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)))
373	370	orcd 872	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2))
374	141	necon2i 2977	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 0 → 𝑐 ≠ 1)
375	374	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → 𝑐 ≠ 1)
376	375	neneqd 2947	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → ¬ 𝑐 = 1)
377	376	olcd 873	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1))
378	373, 377	jca 511	. . . . . . . . . . . 12 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)))
379	23	necon2i 2977	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 0 → 𝑐 ≠ 3)
380	379	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → 𝑐 ≠ 3)
381	380	neneqd 2947	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → ¬ 𝑐 = 3)
382	381	olcd 873	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3))
383	365	orcd 872	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))
384	382, 383	jca 511	. . . . . . . . . . . 12 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
385	372, 378, 384	3jca 1128	. . . . . . . . . . 11 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))))
386	365	orcd 872	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4))
387	381	olcd 873	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3))
388	386, 387	jca 511	. . . . . . . . . . . 12 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → ((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)))
389	97	necon2i 2977	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 0 → 𝑐 ≠ 5)
390	389	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → 𝑐 ≠ 5)
391	390	neneqd 2947	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → ¬ 𝑐 = 5)
392	391	olcd 873	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5))
393		4pos 12396	. . . . . . . . . . . . . . . . . 18 ⊢ 0 < 4
394	93, 393	ltneii 11399	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≠ 4
395		neeq1 3005	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 0 → (𝑐 ≠ 4 ↔ 0 ≠ 4))
396	394, 395	mpbiri 258	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 0 → 𝑐 ≠ 4)
397	396	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → 𝑐 ≠ 4)
398	397	neneqd 2947	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → ¬ 𝑐 = 4)
399	398	olcd 873	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4))
400	392, 399	jca 511	. . . . . . . . . . . 12 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)))
401	361	orcd 872	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5))
402	263	necon2i 2977	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 2 → 𝑏 ≠ 5)
403	402	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → 𝑏 ≠ 5)
404	403	neneqd 2947	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → ¬ 𝑏 = 5)
405	404	orcd 872	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))
406	401, 405	jca 511	. . . . . . . . . . . 12 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
407	388, 400, 406	3jca 1128	. . . . . . . . . . 11 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))
408	385, 407	jca 511	. . . . . . . . . 10 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))
409	362, 366, 408	jca31 514	. . . . . . . . 9 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))
410	409	olcd 873	. . . . . . . 8 ⊢ ((𝑏 = 2 ∧ 𝑐 = 0) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
411	34	necon2i 2977	. . . . . . . . . . . . 13 ⊢ (𝑐 = 2 → 𝑐 ≠ 3)
412	411	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → 𝑐 ≠ 3)
413	412	neneqd 2947	. . . . . . . . . . 11 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → ¬ 𝑐 = 3)
414	413	olcd 873	. . . . . . . . . 10 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3))
415		neeq1 3005	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 2 → (𝑐 ≠ 0 ↔ 2 ≠ 0))
416	357, 415	mpbiri 258	. . . . . . . . . . . . 13 ⊢ (𝑐 = 2 → 𝑐 ≠ 0)
417	416	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → 𝑐 ≠ 0)
418	417	neneqd 2947	. . . . . . . . . . 11 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → ¬ 𝑐 = 0)
419	418	olcd 873	. . . . . . . . . 10 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0))
420	160	necon2i 2977	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 2 → 𝑐 ≠ 1)
421	420	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → 𝑐 ≠ 1)
422	421	neneqd 2947	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → ¬ 𝑐 = 1)
423	422	olcd 873	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1))
424	418	olcd 873	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0))
425	423, 424	jca 511	. . . . . . . . . . . 12 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)))
426	17	necon2i 2977	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 0 → 𝑏 ≠ 1)
427	426	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → 𝑏 ≠ 1)
428	427	neneqd 2947	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → ¬ 𝑏 = 1)
429	428	orcd 872	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2))
430	359	necon2i 2977	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 0 → 𝑏 ≠ 2)
431	430	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → 𝑏 ≠ 2)
432	431	neneqd 2947	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → ¬ 𝑏 = 2)
433	432	orcd 872	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1))
434	429, 433	jca 511	. . . . . . . . . . . 12 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)))
435	413	olcd 873	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3))
436	136	necon2i 2977	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 0 → 𝑏 ≠ 3)
437	436	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → 𝑏 ≠ 3)
438	437	neneqd 2947	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → ¬ 𝑏 = 3)
439	438	orcd 872	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))
440	435, 439	jca 511	. . . . . . . . . . . 12 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
441	425, 434, 440	3jca 1128	. . . . . . . . . . 11 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))))
442	438	orcd 872	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4))
443	413	olcd 873	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3))
444	442, 443	jca 511	. . . . . . . . . . . 12 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → ((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)))
445		neeq1 3005	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 0 → (𝑏 ≠ 4 ↔ 0 ≠ 4))
446	394, 445	mpbiri 258	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 0 → 𝑏 ≠ 4)
447	446	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → 𝑏 ≠ 4)
448	447	neneqd 2947	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → ¬ 𝑏 = 4)
449	448	orcd 872	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5))
450	252	necon2i 2977	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 0 → 𝑏 ≠ 5)
451	450	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → 𝑏 ≠ 5)
452	451	neneqd 2947	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → ¬ 𝑏 = 5)
453	452	orcd 872	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4))
454	449, 453	jca 511	. . . . . . . . . . . 12 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)))
455	105	necon2i 2977	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 2 → 𝑐 ≠ 5)
456	455	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → 𝑐 ≠ 5)
457	456	neneqd 2947	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → ¬ 𝑐 = 5)
458	457	olcd 873	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5))
459	418	olcd 873	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))
460	458, 459	jca 511	. . . . . . . . . . . 12 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
461	444, 454, 460	3jca 1128	. . . . . . . . . . 11 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))
462	441, 461	jca 511	. . . . . . . . . 10 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))
463	414, 419, 462	jca31 514	. . . . . . . . 9 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))
464	463	olcd 873	. . . . . . . 8 ⊢ ((𝑏 = 0 ∧ 𝑐 = 2) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
465	359	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → 𝑏 ≠ 0)
466	465	neneqd 2947	. . . . . . . . . . 11 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → ¬ 𝑏 = 0)
467	466	orcd 872	. . . . . . . . . 10 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3))
468	416	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → 𝑐 ≠ 0)
469	468	neneqd 2947	. . . . . . . . . . 11 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → ¬ 𝑐 = 0)
470	469	olcd 873	. . . . . . . . . 10 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0))
471	466	orcd 872	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1))
472	469	olcd 873	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0))
473	471, 472	jca 511	. . . . . . . . . . . 12 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)))
474	368	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → 𝑏 ≠ 1)
475	474	neneqd 2947	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → ¬ 𝑏 = 1)
476	475	orcd 872	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2))
477	420	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → 𝑐 ≠ 1)
478	477	neneqd 2947	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → ¬ 𝑐 = 1)
479	478	olcd 873	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1))
480	476, 479	jca 511	. . . . . . . . . . . 12 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)))
481	411	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → 𝑐 ≠ 3)
482	481	neneqd 2947	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → ¬ 𝑐 = 3)
483	482	olcd 873	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3))
484	363	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → 𝑏 ≠ 3)
485	484	neneqd 2947	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → ¬ 𝑏 = 3)
486	485	orcd 872	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))
487	483, 486	jca 511	. . . . . . . . . . . 12 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
488	473, 480, 487	3jca 1128	. . . . . . . . . . 11 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))))
489	485	orcd 872	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4))
490		2lt4 12464	. . . . . . . . . . . . . . . . . 18 ⊢ 2 < 4
491	30, 490	ltneii 11399	. . . . . . . . . . . . . . . . 17 ⊢ 2 ≠ 4
492		neeq1 3005	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 2 → (𝑏 ≠ 4 ↔ 2 ≠ 4))
493	491, 492	mpbiri 258	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 2 → 𝑏 ≠ 4)
494	493	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → 𝑏 ≠ 4)
495	494	neneqd 2947	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → ¬ 𝑏 = 4)
496	495	orcd 872	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3))
497	489, 496	jca 511	. . . . . . . . . . . 12 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → ((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)))
498	495	orcd 872	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5))
499	402	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → 𝑏 ≠ 5)
500	499	neneqd 2947	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → ¬ 𝑏 = 5)
501	500	orcd 872	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4))
502	498, 501	jca 511	. . . . . . . . . . . 12 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)))
503	466	orcd 872	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5))
504	469	olcd 873	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))
505	503, 504	jca 511	. . . . . . . . . . . 12 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
506	497, 502, 505	3jca 1128	. . . . . . . . . . 11 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))
507	488, 506	jca 511	. . . . . . . . . 10 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))
508	467, 470, 507	jca31 514	. . . . . . . . 9 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))
509	508	olcd 873	. . . . . . . 8 ⊢ ((𝑏 = 2 ∧ 𝑐 = 2) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
510	356, 410, 464, 509	ccase 1038	. . . . . . 7 ⊢ (((𝑏 = 0 ∨ 𝑏 = 2) ∧ (𝑐 = 0 ∨ 𝑐 = 2)) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
511	351, 354, 510	syl2anb 597	. . . . . 6 ⊢ ((𝑏 ∈ (𝐺 NeighbVtx 1) ∧ 𝑐 ∈ (𝐺 NeighbVtx 1)) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
512	511	rgen2 3201	. . . . 5 ⊢ ∀𝑏 ∈ (𝐺 NeighbVtx 1)∀𝑐 ∈ (𝐺 NeighbVtx 1)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))
513	1, 2, 3	usgrexmpl2nb2 47768	. . . . . . . . 9 ⊢ (𝐺 NeighbVtx 2) = {1, 3}
514	513	eleq2i 2830	. . . . . . . 8 ⊢ (𝑏 ∈ (𝐺 NeighbVtx 2) ↔ 𝑏 ∈ {1, 3})
515	6	elpr 4672	. . . . . . . 8 ⊢ (𝑏 ∈ {1, 3} ↔ (𝑏 = 1 ∨ 𝑏 = 3))
516	514, 515	bitri 275	. . . . . . 7 ⊢ (𝑏 ∈ (𝐺 NeighbVtx 2) ↔ (𝑏 = 1 ∨ 𝑏 = 3))
517	513	eleq2i 2830	. . . . . . . 8 ⊢ (𝑐 ∈ (𝐺 NeighbVtx 2) ↔ 𝑐 ∈ {1, 3})
518	10	elpr 4672	. . . . . . . 8 ⊢ (𝑐 ∈ {1, 3} ↔ (𝑐 = 1 ∨ 𝑐 = 3))
519	517, 518	bitri 275	. . . . . . 7 ⊢ (𝑐 ∈ (𝐺 NeighbVtx 2) ↔ (𝑐 = 1 ∨ 𝑐 = 3))
520	14, 189, 85, 191	ccase 1038	. . . . . . 7 ⊢ (((𝑏 = 1 ∨ 𝑏 = 3) ∧ (𝑐 = 1 ∨ 𝑐 = 3)) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
521	516, 519, 520	syl2anb 597	. . . . . 6 ⊢ ((𝑏 ∈ (𝐺 NeighbVtx 2) ∧ 𝑐 ∈ (𝐺 NeighbVtx 2)) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
522	521	rgen2 3201	. . . . 5 ⊢ ∀𝑏 ∈ (𝐺 NeighbVtx 2)∀𝑐 ∈ (𝐺 NeighbVtx 2)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))
523		c0ex 11280	. . . . . 6 ⊢ 0 ∈ V
524		1ex 11282	. . . . . 6 ⊢ 1 ∈ V
525		2ex 12366	. . . . . 6 ⊢ 2 ∈ V
526		oveq2 7453	. . . . . . 7 ⊢ (𝑎 = 0 → (𝐺 NeighbVtx 𝑎) = (𝐺 NeighbVtx 0))
527	526	raleqdv 3329	. . . . . . 7 ⊢ (𝑎 = 0 → (∀𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))) ↔ ∀𝑐 ∈ (𝐺 NeighbVtx 0)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))))
528	526, 527	raleqbidv 3349	. . . . . 6 ⊢ (𝑎 = 0 → (∀𝑏 ∈ (𝐺 NeighbVtx 𝑎)∀𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))) ↔ ∀𝑏 ∈ (𝐺 NeighbVtx 0)∀𝑐 ∈ (𝐺 NeighbVtx 0)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))))
529		oveq2 7453	. . . . . . 7 ⊢ (𝑎 = 1 → (𝐺 NeighbVtx 𝑎) = (𝐺 NeighbVtx 1))
530	529	raleqdv 3329	. . . . . . 7 ⊢ (𝑎 = 1 → (∀𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))) ↔ ∀𝑐 ∈ (𝐺 NeighbVtx 1)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))))
531	529, 530	raleqbidv 3349	. . . . . 6 ⊢ (𝑎 = 1 → (∀𝑏 ∈ (𝐺 NeighbVtx 𝑎)∀𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))) ↔ ∀𝑏 ∈ (𝐺 NeighbVtx 1)∀𝑐 ∈ (𝐺 NeighbVtx 1)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))))
532		oveq2 7453	. . . . . . 7 ⊢ (𝑎 = 2 → (𝐺 NeighbVtx 𝑎) = (𝐺 NeighbVtx 2))
533	532	raleqdv 3329	. . . . . . 7 ⊢ (𝑎 = 2 → (∀𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))) ↔ ∀𝑐 ∈ (𝐺 NeighbVtx 2)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))))
534	532, 533	raleqbidv 3349	. . . . . 6 ⊢ (𝑎 = 2 → (∀𝑏 ∈ (𝐺 NeighbVtx 𝑎)∀𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))) ↔ ∀𝑏 ∈ (𝐺 NeighbVtx 2)∀𝑐 ∈ (𝐺 NeighbVtx 2)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))))
535	523, 524, 525, 528, 531, 534	raltp 4730	. . . . 5 ⊢ (∀𝑎 ∈ {0, 1, 2}∀𝑏 ∈ (𝐺 NeighbVtx 𝑎)∀𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))) ↔ (∀𝑏 ∈ (𝐺 NeighbVtx 0)∀𝑐 ∈ (𝐺 NeighbVtx 0)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))) ∧ ∀𝑏 ∈ (𝐺 NeighbVtx 1)∀𝑐 ∈ (𝐺 NeighbVtx 1)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))) ∧ ∀𝑏 ∈ (𝐺 NeighbVtx 2)∀𝑐 ∈ (𝐺 NeighbVtx 2)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))))
536	347, 512, 522, 535	mpbir3an 1341	. . . 4 ⊢ ∀𝑎 ∈ {0, 1, 2}∀𝑏 ∈ (𝐺 NeighbVtx 𝑎)∀𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))
537	1, 2, 3	usgrexmpl2nb3 47769	. . . . . . . . 9 ⊢ (𝐺 NeighbVtx 3) = {0, 2, 4}
538	537	eleq2i 2830	. . . . . . . 8 ⊢ (𝑏 ∈ (𝐺 NeighbVtx 3) ↔ 𝑏 ∈ {0, 2, 4})
539	6	eltp 4712	. . . . . . . 8 ⊢ (𝑏 ∈ {0, 2, 4} ↔ (𝑏 = 0 ∨ 𝑏 = 2 ∨ 𝑏 = 4))
540	538, 539	bitri 275	. . . . . . 7 ⊢ (𝑏 ∈ (𝐺 NeighbVtx 3) ↔ (𝑏 = 0 ∨ 𝑏 = 2 ∨ 𝑏 = 4))
541	537	eleq2i 2830	. . . . . . . 8 ⊢ (𝑐 ∈ (𝐺 NeighbVtx 3) ↔ 𝑐 ∈ {0, 2, 4})
542	10	eltp 4712	. . . . . . . 8 ⊢ (𝑐 ∈ {0, 2, 4} ↔ (𝑐 = 0 ∨ 𝑐 = 2 ∨ 𝑐 = 4))
543	541, 542	bitri 275	. . . . . . 7 ⊢ (𝑐 ∈ (𝐺 NeighbVtx 3) ↔ (𝑐 = 0 ∨ 𝑐 = 2 ∨ 𝑐 = 4))
544	330	necon2i 2977	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 4 → 𝑐 ≠ 3)
545	544	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → 𝑐 ≠ 3)
546	545	neneqd 2947	. . . . . . . . . . . 12 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → ¬ 𝑐 = 3)
547	546	olcd 873	. . . . . . . . . . 11 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3))
548	436	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → 𝑏 ≠ 3)
549	548	neneqd 2947	. . . . . . . . . . . 12 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → ¬ 𝑏 = 3)
550	549	orcd 872	. . . . . . . . . . 11 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0))
551	167	necon2i 2977	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 4 → 𝑐 ≠ 1)
552	551	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → 𝑐 ≠ 1)
553	552	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → ¬ 𝑐 = 1)
554	553	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1))
555	426	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → 𝑏 ≠ 1)
556	555	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → ¬ 𝑏 = 1)
557	556	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0))
558	554, 557	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)))
559	556	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2))
560	430	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → 𝑏 ≠ 2)
561	560	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → ¬ 𝑏 = 2)
562	561	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1))
563	559, 562	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)))
564	546	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3))
565	549	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))
566	564, 565	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
567	558, 563, 566	3jca 1128	. . . . . . . . . . . 12 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))))
568	549	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4))
569	546	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3))
570	568, 569	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → ((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)))
571	446	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → 𝑏 ≠ 4)
572	571	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → ¬ 𝑏 = 4)
573	572	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5))
574	450	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → 𝑏 ≠ 5)
575	574	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → ¬ 𝑏 = 5)
576	575	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4))
577	573, 576	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)))
578	221	necon2i 2977	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 4 → 𝑐 ≠ 5)
579	578	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → 𝑐 ≠ 5)
580	579	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → ¬ 𝑐 = 5)
581	580	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5))
582	396	necon2i 2977	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 4 → 𝑐 ≠ 0)
583	582	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → 𝑐 ≠ 0)
584	583	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → ¬ 𝑐 = 0)
585	584	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))
586	581, 585	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
587	570, 577, 586	3jca 1128	. . . . . . . . . . . 12 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))
588	567, 587	jca 511	. . . . . . . . . . 11 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))
589	547, 550, 588	jca31 514	. . . . . . . . . 10 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))
590	589	olcd 873	. . . . . . . . 9 ⊢ ((𝑏 = 0 ∧ 𝑐 = 4) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
591	356, 464, 590	3jaodan 1431	. . . . . . . 8 ⊢ ((𝑏 = 0 ∧ (𝑐 = 0 ∨ 𝑐 = 2 ∨ 𝑐 = 4)) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
592	359	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → 𝑏 ≠ 0)
593	592	neneqd 2947	. . . . . . . . . . . 12 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → ¬ 𝑏 = 0)
594	593	orcd 872	. . . . . . . . . . 11 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3))
595	582	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → 𝑐 ≠ 0)
596	595	neneqd 2947	. . . . . . . . . . . 12 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → ¬ 𝑐 = 0)
597	596	olcd 873	. . . . . . . . . . 11 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0))
598	593	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1))
599	596	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0))
600	598, 599	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)))
601	368	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → 𝑏 ≠ 1)
602	601	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → ¬ 𝑏 = 1)
603	602	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2))
604	551	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → 𝑐 ≠ 1)
605	604	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → ¬ 𝑐 = 1)
606	605	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1))
607	603, 606	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)))
608	544	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → 𝑐 ≠ 3)
609	608	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → ¬ 𝑐 = 3)
610	609	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3))
611	363	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → 𝑏 ≠ 3)
612	611	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → ¬ 𝑏 = 3)
613	612	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))
614	610, 613	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
615	600, 607, 614	3jca 1128	. . . . . . . . . . . 12 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))))
616	612	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4))
617	609	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3))
618	616, 617	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → ((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)))
619	493	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → 𝑏 ≠ 4)
620	619	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → ¬ 𝑏 = 4)
621	620	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5))
622	402	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → 𝑏 ≠ 5)
623	622	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → ¬ 𝑏 = 5)
624	623	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4))
625	621, 624	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)))
626	593	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5))
627	596	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))
628	626, 627	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
629	618, 625, 628	3jca 1128	. . . . . . . . . . . 12 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))
630	615, 629	jca 511	. . . . . . . . . . 11 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))
631	594, 597, 630	jca31 514	. . . . . . . . . 10 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))
632	631	olcd 873	. . . . . . . . 9 ⊢ ((𝑏 = 2 ∧ 𝑐 = 4) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
633	410, 509, 632	3jaodan 1431	. . . . . . . 8 ⊢ ((𝑏 = 2 ∧ (𝑐 = 0 ∨ 𝑐 = 2 ∨ 𝑐 = 4)) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
634	446	necon2i 2977	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 4 → 𝑏 ≠ 0)
635	634	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → 𝑏 ≠ 0)
636	635	neneqd 2947	. . . . . . . . . . . 12 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → ¬ 𝑏 = 0)
637	636	orcd 872	. . . . . . . . . . 11 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3))
638	229	necon2i 2977	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 4 → 𝑏 ≠ 3)
639	638	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → 𝑏 ≠ 3)
640	639	neneqd 2947	. . . . . . . . . . . 12 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → ¬ 𝑏 = 3)
641	640	orcd 872	. . . . . . . . . . 11 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0))
642	636	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1))
643	65	necon2i 2977	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 4 → 𝑏 ≠ 1)
644	643	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → 𝑏 ≠ 1)
645	644	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → ¬ 𝑏 = 1)
646	645	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0))
647	642, 646	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)))
648	416	necon2i 2977	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 0 → 𝑐 ≠ 2)
649	648	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → 𝑐 ≠ 2)
650	649	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → ¬ 𝑐 = 2)
651	650	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2))
652	374	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → 𝑐 ≠ 1)
653	652	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → ¬ 𝑐 = 1)
654	653	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1))
655	651, 654	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)))
656	379	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → 𝑐 ≠ 3)
657	656	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → ¬ 𝑐 = 3)
658	657	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3))
659	640	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))
660	658, 659	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
661	647, 655, 660	3jca 1128	. . . . . . . . . . . 12 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))))
662	640	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4))
663	657	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3))
664	662, 663	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → ((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)))
665	389	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → 𝑐 ≠ 5)
666	665	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → ¬ 𝑐 = 5)
667	666	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5))
668	396	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → 𝑐 ≠ 4)
669	668	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → ¬ 𝑐 = 4)
670	669	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4))
671	667, 670	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)))
672	636	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5))
673	323	necon2i 2977	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 4 → 𝑏 ≠ 5)
674	673	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → 𝑏 ≠ 5)
675	674	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → ¬ 𝑏 = 5)
676	675	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))
677	672, 676	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
678	664, 671, 677	3jca 1128	. . . . . . . . . . . 12 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))
679	661, 678	jca 511	. . . . . . . . . . 11 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))
680	637, 641, 679	jca31 514	. . . . . . . . . 10 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))
681	680	olcd 873	. . . . . . . . 9 ⊢ ((𝑏 = 4 ∧ 𝑐 = 0) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
682	634	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → 𝑏 ≠ 0)
683	682	neneqd 2947	. . . . . . . . . . . 12 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → ¬ 𝑏 = 0)
684	683	orcd 872	. . . . . . . . . . 11 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3))
685	416	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → 𝑐 ≠ 0)
686	685	neneqd 2947	. . . . . . . . . . . 12 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → ¬ 𝑐 = 0)
687	686	olcd 873	. . . . . . . . . . 11 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0))
688	683	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1))
689	686	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0))
690	688, 689	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)))
691	643	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → 𝑏 ≠ 1)
692	691	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → ¬ 𝑏 = 1)
693	692	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2))
694	420	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → 𝑐 ≠ 1)
695	694	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → ¬ 𝑐 = 1)
696	695	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1))
697	693, 696	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)))
698	493	necon2i 2977	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 4 → 𝑏 ≠ 2)
699	698	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → 𝑏 ≠ 2)
700	699	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → ¬ 𝑏 = 2)
701	700	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3))
702	638	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → 𝑏 ≠ 3)
703	702	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → ¬ 𝑏 = 3)
704	703	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))
705	701, 704	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
706	690, 697, 705	3jca 1128	. . . . . . . . . . . 12 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))))
707	703	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4))
708	411	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → 𝑐 ≠ 3)
709	708	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → ¬ 𝑐 = 3)
710	709	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3))
711	707, 710	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → ((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)))
712	455	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → 𝑐 ≠ 5)
713	712	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → ¬ 𝑐 = 5)
714	713	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5))
715		neeq1 3005	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = 2 → (𝑐 ≠ 4 ↔ 2 ≠ 4))
716	491, 715	mpbiri 258	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 2 → 𝑐 ≠ 4)
717	716	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → 𝑐 ≠ 4)
718	717	neneqd 2947	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → ¬ 𝑐 = 4)
719	718	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4))
720	714, 719	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)))
721	683	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5))
722	686	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))
723	721, 722	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
724	711, 720, 723	3jca 1128	. . . . . . . . . . . 12 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))
725	706, 724	jca 511	. . . . . . . . . . 11 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))
726	684, 687, 725	jca31 514	. . . . . . . . . 10 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))
727	726	olcd 873	. . . . . . . . 9 ⊢ ((𝑏 = 4 ∧ 𝑐 = 2) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
728		eqtr3 2760	. . . . . . . . . 10 ⊢ ((𝑏 = 4 ∧ 𝑐 = 4) → 𝑏 = 𝑐)
729	728	orcd 872	. . . . . . . . 9 ⊢ ((𝑏 = 4 ∧ 𝑐 = 4) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
730	681, 727, 729	3jaodan 1431	. . . . . . . 8 ⊢ ((𝑏 = 4 ∧ (𝑐 = 0 ∨ 𝑐 = 2 ∨ 𝑐 = 4)) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
731	591, 633, 730	3jaoian 1430	. . . . . . 7 ⊢ (((𝑏 = 0 ∨ 𝑏 = 2 ∨ 𝑏 = 4) ∧ (𝑐 = 0 ∨ 𝑐 = 2 ∨ 𝑐 = 4)) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
732	540, 543, 731	syl2anb 597	. . . . . 6 ⊢ ((𝑏 ∈ (𝐺 NeighbVtx 3) ∧ 𝑐 ∈ (𝐺 NeighbVtx 3)) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
733	732	rgen2 3201	. . . . 5 ⊢ ∀𝑏 ∈ (𝐺 NeighbVtx 3)∀𝑐 ∈ (𝐺 NeighbVtx 3)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))
734	1, 2, 3	usgrexmpl2nb4 47770	. . . . . . . . 9 ⊢ (𝐺 NeighbVtx 4) = {3, 5}
735	734	eleq2i 2830	. . . . . . . 8 ⊢ (𝑏 ∈ (𝐺 NeighbVtx 4) ↔ 𝑏 ∈ {3, 5})
736	6	elpr 4672	. . . . . . . 8 ⊢ (𝑏 ∈ {3, 5} ↔ (𝑏 = 3 ∨ 𝑏 = 5))
737	735, 736	bitri 275	. . . . . . 7 ⊢ (𝑏 ∈ (𝐺 NeighbVtx 4) ↔ (𝑏 = 3 ∨ 𝑏 = 5))
738	734	eleq2i 2830	. . . . . . . 8 ⊢ (𝑐 ∈ (𝐺 NeighbVtx 4) ↔ 𝑐 ∈ {3, 5})
739	10	elpr 4672	. . . . . . . 8 ⊢ (𝑐 ∈ {3, 5} ↔ (𝑐 = 3 ∨ 𝑐 = 5))
740	738, 739	bitri 275	. . . . . . 7 ⊢ (𝑐 ∈ (𝐺 NeighbVtx 4) ↔ (𝑐 = 3 ∨ 𝑐 = 5))
741	191, 341, 243, 343	ccase 1038	. . . . . . 7 ⊢ (((𝑏 = 3 ∨ 𝑏 = 5) ∧ (𝑐 = 3 ∨ 𝑐 = 5)) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
742	737, 740, 741	syl2anb 597	. . . . . 6 ⊢ ((𝑏 ∈ (𝐺 NeighbVtx 4) ∧ 𝑐 ∈ (𝐺 NeighbVtx 4)) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
743	742	rgen2 3201	. . . . 5 ⊢ ∀𝑏 ∈ (𝐺 NeighbVtx 4)∀𝑐 ∈ (𝐺 NeighbVtx 4)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))
744	1, 2, 3	usgrexmpl2nb5 47771	. . . . . . . . 9 ⊢ (𝐺 NeighbVtx 5) = {0, 4}
745	744	eleq2i 2830	. . . . . . . 8 ⊢ (𝑏 ∈ (𝐺 NeighbVtx 5) ↔ 𝑏 ∈ {0, 4})
746	6	elpr 4672	. . . . . . . 8 ⊢ (𝑏 ∈ {0, 4} ↔ (𝑏 = 0 ∨ 𝑏 = 4))
747	745, 746	bitri 275	. . . . . . 7 ⊢ (𝑏 ∈ (𝐺 NeighbVtx 5) ↔ (𝑏 = 0 ∨ 𝑏 = 4))
748	744	eleq2i 2830	. . . . . . . 8 ⊢ (𝑐 ∈ (𝐺 NeighbVtx 5) ↔ 𝑐 ∈ {0, 4})
749	10	elpr 4672	. . . . . . . 8 ⊢ (𝑐 ∈ {0, 4} ↔ (𝑐 = 0 ∨ 𝑐 = 4))
750	748, 749	bitri 275	. . . . . . 7 ⊢ (𝑐 ∈ (𝐺 NeighbVtx 5) ↔ (𝑐 = 0 ∨ 𝑐 = 4))
751	356, 681, 590, 729	ccase 1038	. . . . . . 7 ⊢ (((𝑏 = 0 ∨ 𝑏 = 4) ∧ (𝑐 = 0 ∨ 𝑐 = 4)) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
752	747, 750, 751	syl2anb 597	. . . . . 6 ⊢ ((𝑏 ∈ (𝐺 NeighbVtx 5) ∧ 𝑐 ∈ (𝐺 NeighbVtx 5)) → (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
753	752	rgen2 3201	. . . . 5 ⊢ ∀𝑏 ∈ (𝐺 NeighbVtx 5)∀𝑐 ∈ (𝐺 NeighbVtx 5)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))
754		3ex 12371	. . . . . 6 ⊢ 3 ∈ V
755		4nn0 12568	. . . . . . 7 ⊢ 4 ∈ ℕ0
756	755	elexi 3506	. . . . . 6 ⊢ 4 ∈ V
757		5nn0 12569	. . . . . . 7 ⊢ 5 ∈ ℕ0
758	757	elexi 3506	. . . . . 6 ⊢ 5 ∈ V
759		oveq2 7453	. . . . . . 7 ⊢ (𝑎 = 3 → (𝐺 NeighbVtx 𝑎) = (𝐺 NeighbVtx 3))
760	759	raleqdv 3329	. . . . . . 7 ⊢ (𝑎 = 3 → (∀𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))) ↔ ∀𝑐 ∈ (𝐺 NeighbVtx 3)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))))
761	759, 760	raleqbidv 3349	. . . . . 6 ⊢ (𝑎 = 3 → (∀𝑏 ∈ (𝐺 NeighbVtx 𝑎)∀𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))) ↔ ∀𝑏 ∈ (𝐺 NeighbVtx 3)∀𝑐 ∈ (𝐺 NeighbVtx 3)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))))
762		oveq2 7453	. . . . . . 7 ⊢ (𝑎 = 4 → (𝐺 NeighbVtx 𝑎) = (𝐺 NeighbVtx 4))
763	762	raleqdv 3329	. . . . . . 7 ⊢ (𝑎 = 4 → (∀𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))) ↔ ∀𝑐 ∈ (𝐺 NeighbVtx 4)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))))
764	762, 763	raleqbidv 3349	. . . . . 6 ⊢ (𝑎 = 4 → (∀𝑏 ∈ (𝐺 NeighbVtx 𝑎)∀𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))) ↔ ∀𝑏 ∈ (𝐺 NeighbVtx 4)∀𝑐 ∈ (𝐺 NeighbVtx 4)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))))
765		oveq2 7453	. . . . . . 7 ⊢ (𝑎 = 5 → (𝐺 NeighbVtx 𝑎) = (𝐺 NeighbVtx 5))
766	765	raleqdv 3329	. . . . . . 7 ⊢ (𝑎 = 5 → (∀𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))) ↔ ∀𝑐 ∈ (𝐺 NeighbVtx 5)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))))
767	765, 766	raleqbidv 3349	. . . . . 6 ⊢ (𝑎 = 5 → (∀𝑏 ∈ (𝐺 NeighbVtx 𝑎)∀𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))) ↔ ∀𝑏 ∈ (𝐺 NeighbVtx 5)∀𝑐 ∈ (𝐺 NeighbVtx 5)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))))
768	754, 756, 758, 761, 764, 767	raltp 4730	. . . . 5 ⊢ (∀𝑎 ∈ {3, 4, 5}∀𝑏 ∈ (𝐺 NeighbVtx 𝑎)∀𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))) ↔ (∀𝑏 ∈ (𝐺 NeighbVtx 3)∀𝑐 ∈ (𝐺 NeighbVtx 3)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))) ∧ ∀𝑏 ∈ (𝐺 NeighbVtx 4)∀𝑐 ∈ (𝐺 NeighbVtx 4)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))) ∧ ∀𝑏 ∈ (𝐺 NeighbVtx 5)∀𝑐 ∈ (𝐺 NeighbVtx 5)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))))
769	733, 743, 753, 768	mpbir3an 1341	. . . 4 ⊢ ∀𝑎 ∈ {3, 4, 5}∀𝑏 ∈ (𝐺 NeighbVtx 𝑎)∀𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))
770		ralunb 4214	. . . 4 ⊢ (∀𝑎 ∈ ({0, 1, 2} ∪ {3, 4, 5})∀𝑏 ∈ (𝐺 NeighbVtx 𝑎)∀𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))) ↔ (∀𝑎 ∈ {0, 1, 2}∀𝑏 ∈ (𝐺 NeighbVtx 𝑎)∀𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))) ∧ ∀𝑎 ∈ {3, 4, 5}∀𝑏 ∈ (𝐺 NeighbVtx 𝑎)∀𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))))
771	536, 769, 770	mpbir2an 710	. . 3 ⊢ ∀𝑎 ∈ ({0, 1, 2} ∪ {3, 4, 5})∀𝑏 ∈ (𝐺 NeighbVtx 𝑎)∀𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))
772		ianor 982	. . . . . 6 ⊢ (¬ (𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))) ↔ (¬ 𝑏 ≠ 𝑐 ∨ ¬ {𝑏, 𝑐} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))))
773		nne 2946	. . . . . . 7 ⊢ (¬ 𝑏 ≠ 𝑐 ↔ 𝑏 = 𝑐)
774		ioran 984	. . . . . . . . . 10 ⊢ (¬ (((𝑏 = 0 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 0)) ∨ ((((𝑏 = 0 ∧ 𝑐 = 1) ∨ (𝑏 = 1 ∧ 𝑐 = 0)) ∨ ((𝑏 = 1 ∧ 𝑐 = 2) ∨ (𝑏 = 2 ∧ 𝑐 = 1)) ∨ ((𝑏 = 2 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 2))) ∨ (((𝑏 = 3 ∧ 𝑐 = 4) ∨ (𝑏 = 4 ∧ 𝑐 = 3)) ∨ ((𝑏 = 4 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 4)) ∨ ((𝑏 = 0 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 0))))) ↔ (¬ ((𝑏 = 0 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 0)) ∧ ¬ ((((𝑏 = 0 ∧ 𝑐 = 1) ∨ (𝑏 = 1 ∧ 𝑐 = 0)) ∨ ((𝑏 = 1 ∧ 𝑐 = 2) ∨ (𝑏 = 2 ∧ 𝑐 = 1)) ∨ ((𝑏 = 2 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 2))) ∨ (((𝑏 = 3 ∧ 𝑐 = 4) ∨ (𝑏 = 4 ∧ 𝑐 = 3)) ∨ ((𝑏 = 4 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 4)) ∨ ((𝑏 = 0 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 0))))))
775		ioran 984	. . . . . . . . . . . 12 ⊢ (¬ ((𝑏 = 0 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 0)) ↔ (¬ (𝑏 = 0 ∧ 𝑐 = 3) ∧ ¬ (𝑏 = 3 ∧ 𝑐 = 0)))
776		ianor 982	. . . . . . . . . . . . 13 ⊢ (¬ (𝑏 = 0 ∧ 𝑐 = 3) ↔ (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3))
777		ianor 982	. . . . . . . . . . . . 13 ⊢ (¬ (𝑏 = 3 ∧ 𝑐 = 0) ↔ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0))
778	776, 777	anbi12i 627	. . . . . . . . . . . 12 ⊢ ((¬ (𝑏 = 0 ∧ 𝑐 = 3) ∧ ¬ (𝑏 = 3 ∧ 𝑐 = 0)) ↔ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)))
779	775, 778	bitri 275	. . . . . . . . . . 11 ⊢ (¬ ((𝑏 = 0 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 0)) ↔ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)))
780		ioran 984	. . . . . . . . . . . 12 ⊢ (¬ ((((𝑏 = 0 ∧ 𝑐 = 1) ∨ (𝑏 = 1 ∧ 𝑐 = 0)) ∨ ((𝑏 = 1 ∧ 𝑐 = 2) ∨ (𝑏 = 2 ∧ 𝑐 = 1)) ∨ ((𝑏 = 2 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 2))) ∨ (((𝑏 = 3 ∧ 𝑐 = 4) ∨ (𝑏 = 4 ∧ 𝑐 = 3)) ∨ ((𝑏 = 4 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 4)) ∨ ((𝑏 = 0 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 0)))) ↔ (¬ (((𝑏 = 0 ∧ 𝑐 = 1) ∨ (𝑏 = 1 ∧ 𝑐 = 0)) ∨ ((𝑏 = 1 ∧ 𝑐 = 2) ∨ (𝑏 = 2 ∧ 𝑐 = 1)) ∨ ((𝑏 = 2 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 2))) ∧ ¬ (((𝑏 = 3 ∧ 𝑐 = 4) ∨ (𝑏 = 4 ∧ 𝑐 = 3)) ∨ ((𝑏 = 4 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 4)) ∨ ((𝑏 = 0 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 0)))))
781		3ioran 1106	. . . . . . . . . . . . . 14 ⊢ (¬ (((𝑏 = 0 ∧ 𝑐 = 1) ∨ (𝑏 = 1 ∧ 𝑐 = 0)) ∨ ((𝑏 = 1 ∧ 𝑐 = 2) ∨ (𝑏 = 2 ∧ 𝑐 = 1)) ∨ ((𝑏 = 2 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 2))) ↔ (¬ ((𝑏 = 0 ∧ 𝑐 = 1) ∨ (𝑏 = 1 ∧ 𝑐 = 0)) ∧ ¬ ((𝑏 = 1 ∧ 𝑐 = 2) ∨ (𝑏 = 2 ∧ 𝑐 = 1)) ∧ ¬ ((𝑏 = 2 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 2))))
782		ioran 984	. . . . . . . . . . . . . . . 16 ⊢ (¬ ((𝑏 = 0 ∧ 𝑐 = 1) ∨ (𝑏 = 1 ∧ 𝑐 = 0)) ↔ (¬ (𝑏 = 0 ∧ 𝑐 = 1) ∧ ¬ (𝑏 = 1 ∧ 𝑐 = 0)))
783		ianor 982	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝑏 = 0 ∧ 𝑐 = 1) ↔ (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1))
784		ianor 982	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝑏 = 1 ∧ 𝑐 = 0) ↔ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0))
785	783, 784	anbi12i 627	. . . . . . . . . . . . . . . 16 ⊢ ((¬ (𝑏 = 0 ∧ 𝑐 = 1) ∧ ¬ (𝑏 = 1 ∧ 𝑐 = 0)) ↔ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)))
786	782, 785	bitri 275	. . . . . . . . . . . . . . 15 ⊢ (¬ ((𝑏 = 0 ∧ 𝑐 = 1) ∨ (𝑏 = 1 ∧ 𝑐 = 0)) ↔ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)))
787		ioran 984	. . . . . . . . . . . . . . . 16 ⊢ (¬ ((𝑏 = 1 ∧ 𝑐 = 2) ∨ (𝑏 = 2 ∧ 𝑐 = 1)) ↔ (¬ (𝑏 = 1 ∧ 𝑐 = 2) ∧ ¬ (𝑏 = 2 ∧ 𝑐 = 1)))
788		ianor 982	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝑏 = 1 ∧ 𝑐 = 2) ↔ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2))
789		ianor 982	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝑏 = 2 ∧ 𝑐 = 1) ↔ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1))
790	788, 789	anbi12i 627	. . . . . . . . . . . . . . . 16 ⊢ ((¬ (𝑏 = 1 ∧ 𝑐 = 2) ∧ ¬ (𝑏 = 2 ∧ 𝑐 = 1)) ↔ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)))
791	787, 790	bitri 275	. . . . . . . . . . . . . . 15 ⊢ (¬ ((𝑏 = 1 ∧ 𝑐 = 2) ∨ (𝑏 = 2 ∧ 𝑐 = 1)) ↔ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)))
792		ioran 984	. . . . . . . . . . . . . . . 16 ⊢ (¬ ((𝑏 = 2 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 2)) ↔ (¬ (𝑏 = 2 ∧ 𝑐 = 3) ∧ ¬ (𝑏 = 3 ∧ 𝑐 = 2)))
793		ianor 982	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝑏 = 2 ∧ 𝑐 = 3) ↔ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3))
794		ianor 982	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝑏 = 3 ∧ 𝑐 = 2) ↔ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))
795	793, 794	anbi12i 627	. . . . . . . . . . . . . . . 16 ⊢ ((¬ (𝑏 = 2 ∧ 𝑐 = 3) ∧ ¬ (𝑏 = 3 ∧ 𝑐 = 2)) ↔ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
796	792, 795	bitri 275	. . . . . . . . . . . . . . 15 ⊢ (¬ ((𝑏 = 2 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 2)) ↔ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
797	786, 791, 796	3anbi123i 1155	. . . . . . . . . . . . . 14 ⊢ ((¬ ((𝑏 = 0 ∧ 𝑐 = 1) ∨ (𝑏 = 1 ∧ 𝑐 = 0)) ∧ ¬ ((𝑏 = 1 ∧ 𝑐 = 2) ∨ (𝑏 = 2 ∧ 𝑐 = 1)) ∧ ¬ ((𝑏 = 2 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 2))) ↔ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))))
798	781, 797	bitri 275	. . . . . . . . . . . . 13 ⊢ (¬ (((𝑏 = 0 ∧ 𝑐 = 1) ∨ (𝑏 = 1 ∧ 𝑐 = 0)) ∨ ((𝑏 = 1 ∧ 𝑐 = 2) ∨ (𝑏 = 2 ∧ 𝑐 = 1)) ∨ ((𝑏 = 2 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 2))) ↔ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))))
799		3ioran 1106	. . . . . . . . . . . . . 14 ⊢ (¬ (((𝑏 = 3 ∧ 𝑐 = 4) ∨ (𝑏 = 4 ∧ 𝑐 = 3)) ∨ ((𝑏 = 4 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 4)) ∨ ((𝑏 = 0 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 0))) ↔ (¬ ((𝑏 = 3 ∧ 𝑐 = 4) ∨ (𝑏 = 4 ∧ 𝑐 = 3)) ∧ ¬ ((𝑏 = 4 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 4)) ∧ ¬ ((𝑏 = 0 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 0))))
800		ioran 984	. . . . . . . . . . . . . . . 16 ⊢ (¬ ((𝑏 = 3 ∧ 𝑐 = 4) ∨ (𝑏 = 4 ∧ 𝑐 = 3)) ↔ (¬ (𝑏 = 3 ∧ 𝑐 = 4) ∧ ¬ (𝑏 = 4 ∧ 𝑐 = 3)))
801		ianor 982	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝑏 = 3 ∧ 𝑐 = 4) ↔ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4))
802		ianor 982	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝑏 = 4 ∧ 𝑐 = 3) ↔ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3))
803	801, 802	anbi12i 627	. . . . . . . . . . . . . . . 16 ⊢ ((¬ (𝑏 = 3 ∧ 𝑐 = 4) ∧ ¬ (𝑏 = 4 ∧ 𝑐 = 3)) ↔ ((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)))
804	800, 803	bitri 275	. . . . . . . . . . . . . . 15 ⊢ (¬ ((𝑏 = 3 ∧ 𝑐 = 4) ∨ (𝑏 = 4 ∧ 𝑐 = 3)) ↔ ((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)))
805		ioran 984	. . . . . . . . . . . . . . . 16 ⊢ (¬ ((𝑏 = 4 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 4)) ↔ (¬ (𝑏 = 4 ∧ 𝑐 = 5) ∧ ¬ (𝑏 = 5 ∧ 𝑐 = 4)))
806		ianor 982	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝑏 = 4 ∧ 𝑐 = 5) ↔ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5))
807		ianor 982	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝑏 = 5 ∧ 𝑐 = 4) ↔ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4))
808	806, 807	anbi12i 627	. . . . . . . . . . . . . . . 16 ⊢ ((¬ (𝑏 = 4 ∧ 𝑐 = 5) ∧ ¬ (𝑏 = 5 ∧ 𝑐 = 4)) ↔ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)))
809	805, 808	bitri 275	. . . . . . . . . . . . . . 15 ⊢ (¬ ((𝑏 = 4 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 4)) ↔ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)))
810		ioran 984	. . . . . . . . . . . . . . . 16 ⊢ (¬ ((𝑏 = 0 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 0)) ↔ (¬ (𝑏 = 0 ∧ 𝑐 = 5) ∧ ¬ (𝑏 = 5 ∧ 𝑐 = 0)))
811		ianor 982	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝑏 = 0 ∧ 𝑐 = 5) ↔ (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5))
812		ianor 982	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (𝑏 = 5 ∧ 𝑐 = 0) ↔ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))
813	811, 812	anbi12i 627	. . . . . . . . . . . . . . . 16 ⊢ ((¬ (𝑏 = 0 ∧ 𝑐 = 5) ∧ ¬ (𝑏 = 5 ∧ 𝑐 = 0)) ↔ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
814	810, 813	bitri 275	. . . . . . . . . . . . . . 15 ⊢ (¬ ((𝑏 = 0 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 0)) ↔ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
815	804, 809, 814	3anbi123i 1155	. . . . . . . . . . . . . 14 ⊢ ((¬ ((𝑏 = 3 ∧ 𝑐 = 4) ∨ (𝑏 = 4 ∧ 𝑐 = 3)) ∧ ¬ ((𝑏 = 4 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 4)) ∧ ¬ ((𝑏 = 0 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 0))) ↔ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))
816	799, 815	bitri 275	. . . . . . . . . . . . 13 ⊢ (¬ (((𝑏 = 3 ∧ 𝑐 = 4) ∨ (𝑏 = 4 ∧ 𝑐 = 3)) ∨ ((𝑏 = 4 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 4)) ∨ ((𝑏 = 0 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 0))) ↔ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))
817	798, 816	anbi12i 627	. . . . . . . . . . . 12 ⊢ ((¬ (((𝑏 = 0 ∧ 𝑐 = 1) ∨ (𝑏 = 1 ∧ 𝑐 = 0)) ∨ ((𝑏 = 1 ∧ 𝑐 = 2) ∨ (𝑏 = 2 ∧ 𝑐 = 1)) ∨ ((𝑏 = 2 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 2))) ∧ ¬ (((𝑏 = 3 ∧ 𝑐 = 4) ∨ (𝑏 = 4 ∧ 𝑐 = 3)) ∨ ((𝑏 = 4 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 4)) ∨ ((𝑏 = 0 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 0)))) ↔ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))
818	780, 817	bitri 275	. . . . . . . . . . 11 ⊢ (¬ ((((𝑏 = 0 ∧ 𝑐 = 1) ∨ (𝑏 = 1 ∧ 𝑐 = 0)) ∨ ((𝑏 = 1 ∧ 𝑐 = 2) ∨ (𝑏 = 2 ∧ 𝑐 = 1)) ∨ ((𝑏 = 2 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 2))) ∨ (((𝑏 = 3 ∧ 𝑐 = 4) ∨ (𝑏 = 4 ∧ 𝑐 = 3)) ∨ ((𝑏 = 4 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 4)) ∨ ((𝑏 = 0 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 0)))) ↔ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))
819	779, 818	anbi12i 627	. . . . . . . . . 10 ⊢ ((¬ ((𝑏 = 0 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 0)) ∧ ¬ ((((𝑏 = 0 ∧ 𝑐 = 1) ∨ (𝑏 = 1 ∧ 𝑐 = 0)) ∨ ((𝑏 = 1 ∧ 𝑐 = 2) ∨ (𝑏 = 2 ∧ 𝑐 = 1)) ∨ ((𝑏 = 2 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 2))) ∨ (((𝑏 = 3 ∧ 𝑐 = 4) ∨ (𝑏 = 4 ∧ 𝑐 = 3)) ∨ ((𝑏 = 4 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 4)) ∨ ((𝑏 = 0 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 0))))) ↔ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))
820	774, 819	bitri 275	. . . . . . . . 9 ⊢ (¬ (((𝑏 = 0 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 0)) ∨ ((((𝑏 = 0 ∧ 𝑐 = 1) ∨ (𝑏 = 1 ∧ 𝑐 = 0)) ∨ ((𝑏 = 1 ∧ 𝑐 = 2) ∨ (𝑏 = 2 ∧ 𝑐 = 1)) ∨ ((𝑏 = 2 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 2))) ∨ (((𝑏 = 3 ∧ 𝑐 = 4) ∨ (𝑏 = 4 ∧ 𝑐 = 3)) ∨ ((𝑏 = 4 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 4)) ∨ ((𝑏 = 0 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 0))))) ↔ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))
821	6, 10, 523, 524	preq12b 4875	. . . . . . . . . . . 12 ⊢ ({𝑏, 𝑐} = {0, 1} ↔ ((𝑏 = 0 ∧ 𝑐 = 1) ∨ (𝑏 = 1 ∧ 𝑐 = 0)))
822	6, 10, 524, 525	preq12b 4875	. . . . . . . . . . . 12 ⊢ ({𝑏, 𝑐} = {1, 2} ↔ ((𝑏 = 1 ∧ 𝑐 = 2) ∨ (𝑏 = 2 ∧ 𝑐 = 1)))
823	6, 10, 525, 754	preq12b 4875	. . . . . . . . . . . 12 ⊢ ({𝑏, 𝑐} = {2, 3} ↔ ((𝑏 = 2 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 2)))
824	821, 822, 823	3orbi123i 1156	. . . . . . . . . . 11 ⊢ (({𝑏, 𝑐} = {0, 1} ∨ {𝑏, 𝑐} = {1, 2} ∨ {𝑏, 𝑐} = {2, 3}) ↔ (((𝑏 = 0 ∧ 𝑐 = 1) ∨ (𝑏 = 1 ∧ 𝑐 = 0)) ∨ ((𝑏 = 1 ∧ 𝑐 = 2) ∨ (𝑏 = 2 ∧ 𝑐 = 1)) ∨ ((𝑏 = 2 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 2))))
825	6, 10, 754, 756	preq12b 4875	. . . . . . . . . . . 12 ⊢ ({𝑏, 𝑐} = {3, 4} ↔ ((𝑏 = 3 ∧ 𝑐 = 4) ∨ (𝑏 = 4 ∧ 𝑐 = 3)))
826	6, 10, 756, 758	preq12b 4875	. . . . . . . . . . . 12 ⊢ ({𝑏, 𝑐} = {4, 5} ↔ ((𝑏 = 4 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 4)))
827	6, 10, 523, 758	preq12b 4875	. . . . . . . . . . . 12 ⊢ ({𝑏, 𝑐} = {0, 5} ↔ ((𝑏 = 0 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 0)))
828	825, 826, 827	3orbi123i 1156	. . . . . . . . . . 11 ⊢ (({𝑏, 𝑐} = {3, 4} ∨ {𝑏, 𝑐} = {4, 5} ∨ {𝑏, 𝑐} = {0, 5}) ↔ (((𝑏 = 3 ∧ 𝑐 = 4) ∨ (𝑏 = 4 ∧ 𝑐 = 3)) ∨ ((𝑏 = 4 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 4)) ∨ ((𝑏 = 0 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 0))))
829	824, 828	orbi12i 913	. . . . . . . . . 10 ⊢ ((({𝑏, 𝑐} = {0, 1} ∨ {𝑏, 𝑐} = {1, 2} ∨ {𝑏, 𝑐} = {2, 3}) ∨ ({𝑏, 𝑐} = {3, 4} ∨ {𝑏, 𝑐} = {4, 5} ∨ {𝑏, 𝑐} = {0, 5})) ↔ ((((𝑏 = 0 ∧ 𝑐 = 1) ∨ (𝑏 = 1 ∧ 𝑐 = 0)) ∨ ((𝑏 = 1 ∧ 𝑐 = 2) ∨ (𝑏 = 2 ∧ 𝑐 = 1)) ∨ ((𝑏 = 2 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 2))) ∨ (((𝑏 = 3 ∧ 𝑐 = 4) ∨ (𝑏 = 4 ∧ 𝑐 = 3)) ∨ ((𝑏 = 4 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 4)) ∨ ((𝑏 = 0 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 0)))))
830	829	orbi2i 911	. . . . . . . . 9 ⊢ ((((𝑏 = 0 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 0)) ∨ (({𝑏, 𝑐} = {0, 1} ∨ {𝑏, 𝑐} = {1, 2} ∨ {𝑏, 𝑐} = {2, 3}) ∨ ({𝑏, 𝑐} = {3, 4} ∨ {𝑏, 𝑐} = {4, 5} ∨ {𝑏, 𝑐} = {0, 5}))) ↔ (((𝑏 = 0 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 0)) ∨ ((((𝑏 = 0 ∧ 𝑐 = 1) ∨ (𝑏 = 1 ∧ 𝑐 = 0)) ∨ ((𝑏 = 1 ∧ 𝑐 = 2) ∨ (𝑏 = 2 ∧ 𝑐 = 1)) ∨ ((𝑏 = 2 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 2))) ∨ (((𝑏 = 3 ∧ 𝑐 = 4) ∨ (𝑏 = 4 ∧ 𝑐 = 3)) ∨ ((𝑏 = 4 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 4)) ∨ ((𝑏 = 0 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 0))))))
831	820, 830	xchnxbir 333	. . . . . . . 8 ⊢ (¬ (((𝑏 = 0 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 0)) ∨ (({𝑏, 𝑐} = {0, 1} ∨ {𝑏, 𝑐} = {1, 2} ∨ {𝑏, 𝑐} = {2, 3}) ∨ ({𝑏, 𝑐} = {3, 4} ∨ {𝑏, 𝑐} = {4, 5} ∨ {𝑏, 𝑐} = {0, 5}))) ↔ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))
832		elun 4170	. . . . . . . . 9 ⊢ ({𝑏, 𝑐} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}})) ↔ ({𝑏, 𝑐} ∈ {{0, 3}} ∨ {𝑏, 𝑐} ∈ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}})))
833		prex 5455	. . . . . . . . . . . 12 ⊢ {𝑏, 𝑐} ∈ V
834	833	elsn 4663	. . . . . . . . . . 11 ⊢ ({𝑏, 𝑐} ∈ {{0, 3}} ↔ {𝑏, 𝑐} = {0, 3})
835	6, 10, 523, 754	preq12b 4875	. . . . . . . . . . 11 ⊢ ({𝑏, 𝑐} = {0, 3} ↔ ((𝑏 = 0 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 0)))
836	834, 835	bitri 275	. . . . . . . . . 10 ⊢ ({𝑏, 𝑐} ∈ {{0, 3}} ↔ ((𝑏 = 0 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 0)))
837		elun 4170	. . . . . . . . . . 11 ⊢ ({𝑏, 𝑐} ∈ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}) ↔ ({𝑏, 𝑐} ∈ {{0, 1}, {1, 2}, {2, 3}} ∨ {𝑏, 𝑐} ∈ {{3, 4}, {4, 5}, {0, 5}}))
838	833	eltp 4712	. . . . . . . . . . . 12 ⊢ ({𝑏, 𝑐} ∈ {{0, 1}, {1, 2}, {2, 3}} ↔ ({𝑏, 𝑐} = {0, 1} ∨ {𝑏, 𝑐} = {1, 2} ∨ {𝑏, 𝑐} = {2, 3}))
839	833	eltp 4712	. . . . . . . . . . . 12 ⊢ ({𝑏, 𝑐} ∈ {{3, 4}, {4, 5}, {0, 5}} ↔ ({𝑏, 𝑐} = {3, 4} ∨ {𝑏, 𝑐} = {4, 5} ∨ {𝑏, 𝑐} = {0, 5}))
840	838, 839	orbi12i 913	. . . . . . . . . . 11 ⊢ (({𝑏, 𝑐} ∈ {{0, 1}, {1, 2}, {2, 3}} ∨ {𝑏, 𝑐} ∈ {{3, 4}, {4, 5}, {0, 5}}) ↔ (({𝑏, 𝑐} = {0, 1} ∨ {𝑏, 𝑐} = {1, 2} ∨ {𝑏, 𝑐} = {2, 3}) ∨ ({𝑏, 𝑐} = {3, 4} ∨ {𝑏, 𝑐} = {4, 5} ∨ {𝑏, 𝑐} = {0, 5})))
841	837, 840	bitri 275	. . . . . . . . . 10 ⊢ ({𝑏, 𝑐} ∈ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}) ↔ (({𝑏, 𝑐} = {0, 1} ∨ {𝑏, 𝑐} = {1, 2} ∨ {𝑏, 𝑐} = {2, 3}) ∨ ({𝑏, 𝑐} = {3, 4} ∨ {𝑏, 𝑐} = {4, 5} ∨ {𝑏, 𝑐} = {0, 5})))
842	836, 841	orbi12i 913	. . . . . . . . 9 ⊢ (({𝑏, 𝑐} ∈ {{0, 3}} ∨ {𝑏, 𝑐} ∈ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}})) ↔ (((𝑏 = 0 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 0)) ∨ (({𝑏, 𝑐} = {0, 1} ∨ {𝑏, 𝑐} = {1, 2} ∨ {𝑏, 𝑐} = {2, 3}) ∨ ({𝑏, 𝑐} = {3, 4} ∨ {𝑏, 𝑐} = {4, 5} ∨ {𝑏, 𝑐} = {0, 5}))))
843	832, 842	bitri 275	. . . . . . . 8 ⊢ ({𝑏, 𝑐} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}})) ↔ (((𝑏 = 0 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 0)) ∨ (({𝑏, 𝑐} = {0, 1} ∨ {𝑏, 𝑐} = {1, 2} ∨ {𝑏, 𝑐} = {2, 3}) ∨ ({𝑏, 𝑐} = {3, 4} ∨ {𝑏, 𝑐} = {4, 5} ∨ {𝑏, 𝑐} = {0, 5}))))
844	831, 843	xchnxbir 333	. . . . . . 7 ⊢ (¬ {𝑏, 𝑐} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}})) ↔ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))))))
845	773, 844	orbi12i 913	. . . . . 6 ⊢ ((¬ 𝑏 ≠ 𝑐 ∨ ¬ {𝑏, 𝑐} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))) ↔ (𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))))
846	772, 845	bitr2i 276	. . . . 5 ⊢ ((𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))) ↔ ¬ (𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))))
847	846	3ralbii 3132	. . . 4 ⊢ (∀𝑎 ∈ ({0, 1, 2} ∪ {3, 4, 5})∀𝑏 ∈ (𝐺 NeighbVtx 𝑎)∀𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))) ↔ ∀𝑎 ∈ ({0, 1, 2} ∪ {3, 4, 5})∀𝑏 ∈ (𝐺 NeighbVtx 𝑎)∀𝑐 ∈ (𝐺 NeighbVtx 𝑎) ¬ (𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))))
848		ralnex3 3136	. . . 4 ⊢ (∀𝑎 ∈ ({0, 1, 2} ∪ {3, 4, 5})∀𝑏 ∈ (𝐺 NeighbVtx 𝑎)∀𝑐 ∈ (𝐺 NeighbVtx 𝑎) ¬ (𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))) ↔ ¬ ∃𝑎 ∈ ({0, 1, 2} ∪ {3, 4, 5})∃𝑏 ∈ (𝐺 NeighbVtx 𝑎)∃𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))))
849	847, 848	bitri 275	. . 3 ⊢ (∀𝑎 ∈ ({0, 1, 2} ∪ {3, 4, 5})∀𝑏 ∈ (𝐺 NeighbVtx 𝑎)∀𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 = 𝑐 ∨ (((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)) ∧ ((((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)) ∧ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)) ∧ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))) ∧ (((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)) ∧ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)) ∧ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))))) ↔ ¬ ∃𝑎 ∈ ({0, 1, 2} ∪ {3, 4, 5})∃𝑏 ∈ (𝐺 NeighbVtx 𝑎)∃𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))))
850	771, 849	mpbi 230	. 2 ⊢ ¬ ∃𝑎 ∈ ({0, 1, 2} ∪ {3, 4, 5})∃𝑏 ∈ (𝐺 NeighbVtx 𝑎)∃𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}})))
851	1, 2, 3	usgrexmpl2 47762	. . 3 ⊢ 𝐺 ∈ USGraph
852	1, 2, 3	usgrexmpl2vtx 47763	. . . . 5 ⊢ (Vtx‘𝐺) = ({0, 1, 2} ∪ {3, 4, 5})
853	852	eqcomi 2743	. . . 4 ⊢ ({0, 1, 2} ∪ {3, 4, 5}) = (Vtx‘𝐺)
854	1, 2, 3	usgrexmpl2edg 47764	. . . . 5 ⊢ (Edg‘𝐺) = ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))
855	854	eqcomi 2743	. . . 4 ⊢ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}})) = (Edg‘𝐺)
856		eqid 2734	. . . 4 ⊢ (𝐺 NeighbVtx 𝑎) = (𝐺 NeighbVtx 𝑎)
857	853, 855, 856	usgrgrtrirex 47728	. . 3 ⊢ (𝐺 ∈ USGraph → (∃𝑡 𝑡 ∈ (GrTriangles‘𝐺) ↔ ∃𝑎 ∈ ({0, 1, 2} ∪ {3, 4, 5})∃𝑏 ∈ (𝐺 NeighbVtx 𝑎)∃𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}})))))
858	851, 857	ax-mp 5	. 2 ⊢ (∃𝑡 𝑡 ∈ (GrTriangles‘𝐺) ↔ ∃𝑎 ∈ ({0, 1, 2} ∪ {3, 4, 5})∃𝑏 ∈ (𝐺 NeighbVtx 𝑎)∃𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))))
859	850, 858	mtbir 323	1 ⊢ ¬ ∃𝑡 𝑡 ∈ (GrTriangles‘𝐺)

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∨ wo 846   ∨ w3o 1086   ∧ w3a 1087   = wceq 1537  ∃wex 1777   ∈ wcel 2103   ≠ wne 2942  ∀wral 3063  ∃wrex 3072   ∪ cun 3968  {csn 4648  {cpr 4650  {ctp 4652  ⟨cop 4654  ‘cfv 6572  (class class class)co 7445  0cc0 11180  1c1 11181  2c2 12344  3c3 12345  4c4 12346  5c5 12347  ℕ₀cn0 12549  ...cfz 13563  ⟨“cs7 14891  Vtxcvtx 29022  Edgcedg 29073  USGraphcusgr 29175   NeighbVtx cnbgr 29358  GrTrianglescgrtri 47718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766  ax-cnex 11236  ax-resscn 11237  ax-1cn 11238  ax-icn 11239  ax-addcl 11240  ax-addrcl 11241  ax-mulcl 11242  ax-mulrcl 11243  ax-mulcom 11244  ax-addass 11245  ax-mulass 11246  ax-distr 11247  ax-i2m1 11248  ax-1ne0 11249  ax-1rid 11250  ax-rnegex 11251  ax-rrecex 11252  ax-cnre 11253  ax-pre-lttri 11254  ax-pre-lttrn 11255  ax-pre-ltadd 11256  ax-pre-mulgt0 11257

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4973  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-pred 6331  df-ord 6397  df-on 6398  df-lim 6399  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-riota 7401  df-ov 7448  df-oprab 7449  df-mpo 7450  df-om 7900  df-1st 8026  df-2nd 8027  df-frecs 8318  df-wrecs 8349  df-recs 8423  df-rdg 8462  df-1o 8518  df-2o 8519  df-3o 8520  df-oadd 8522  df-er 8759  df-en 9000  df-dom 9001  df-sdom 9002  df-fin 9003  df-dju 9966  df-card 10004  df-pnf 11322  df-mnf 11323  df-xr 11324  df-ltxr 11325  df-le 11326  df-sub 11518  df-neg 11519  df-nn 12290  df-2 12352  df-3 12353  df-4 12354  df-5 12355  df-6 12356  df-7 12357  df-n0 12550  df-xnn0 12622  df-z 12636  df-uz 12900  df-fz 13564  df-fzo 13708  df-hash 14376  df-word 14559  df-concat 14615  df-s1 14640  df-s2 14893  df-s3 14894  df-s4 14895  df-s5 14896  df-s6 14897  df-s7 14898  df-vtx 29024  df-iedg 29025  df-edg 29074  df-uhgr 29084  df-upgr 29108  df-umgr 29109  df-uspgr 29176  df-usgr 29177  df-nbgr 29359  df-grtri 47719

This theorem is referenced by:  usgrexmpl12ngric  47773