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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  usgrexmpl2trifr Structured version   Visualization version   GIF version

Theorem usgrexmpl2trifr 47852
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.
StepHypRef 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
41, 2, 3usgrexmpl2nb0 47846 . . . . . . . . 9 (𝐺 NeighbVtx 0) = {1, 3, 5}
54eleq2i 2836 . . . . . . . 8 (𝑏 ∈ (𝐺 NeighbVtx 0) ↔ 𝑏 ∈ {1, 3, 5})
6 vex 3492 . . . . . . . . 9 𝑏 ∈ V
76eltp 4712 . . . . . . . 8 (𝑏 ∈ {1, 3, 5} ↔ (𝑏 = 1 ∨ 𝑏 = 3 ∨ 𝑏 = 5))
85, 7bitri 275 . . . . . . 7 (𝑏 ∈ (𝐺 NeighbVtx 0) ↔ (𝑏 = 1 ∨ 𝑏 = 3 ∨ 𝑏 = 5))
94eleq2i 2836 . . . . . . . 8 (𝑐 ∈ (𝐺 NeighbVtx 0) ↔ 𝑐 ∈ {1, 3, 5})
10 vex 3492 . . . . . . . . 9 𝑐 ∈ V
1110eltp 4712 . . . . . . . 8 (𝑐 ∈ {1, 3, 5} ↔ (𝑐 = 1 ∨ 𝑐 = 3 ∨ 𝑐 = 5))
129, 11bitri 275 . . . . . . 7 (𝑐 ∈ (𝐺 NeighbVtx 0) ↔ (𝑐 = 1 ∨ 𝑐 = 3 ∨ 𝑐 = 5))
13 eqtr3 2766 . . . . . . . . . 10 ((𝑏 = 1 ∧ 𝑐 = 1) → 𝑏 = 𝑐)
1413orcd 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 11253 . . . . . . . . . . . . . . 15 1 ≠ 0
16 neeq1 3009 . . . . . . . . . . . . . . 15 (𝑏 = 1 → (𝑏 ≠ 0 ↔ 1 ≠ 0))
1715, 16mpbiri 258 . . . . . . . . . . . . . 14 (𝑏 = 1 → 𝑏 ≠ 0)
1817adantr 480 . . . . . . . . . . . . 13 ((𝑏 = 1 ∧ 𝑐 = 3) → 𝑏 ≠ 0)
1918neneqd 2951 . . . . . . . . . . . 12 ((𝑏 = 1 ∧ 𝑐 = 3) → ¬ 𝑏 = 0)
2019orcd 872 . . . . . . . . . . 11 ((𝑏 = 1 ∧ 𝑐 = 3) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3))
21 3ne0 12399 . . . . . . . . . . . . . . 15 3 ≠ 0
22 neeq1 3009 . . . . . . . . . . . . . . 15 (𝑐 = 3 → (𝑐 ≠ 0 ↔ 3 ≠ 0))
2321, 22mpbiri 258 . . . . . . . . . . . . . 14 (𝑐 = 3 → 𝑐 ≠ 0)
2423adantl 481 . . . . . . . . . . . . 13 ((𝑏 = 1 ∧ 𝑐 = 3) → 𝑐 ≠ 0)
2524neneqd 2951 . . . . . . . . . . . 12 ((𝑏 = 1 ∧ 𝑐 = 3) → ¬ 𝑐 = 0)
2625olcd 873 . . . . . . . . . . 11 ((𝑏 = 1 ∧ 𝑐 = 3) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0))
2719orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 1 ∧ 𝑐 = 3) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1))
2825olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 1 ∧ 𝑐 = 3) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0))
2927, 28jca 511 . . . . . . . . . . . . 13 ((𝑏 = 1 ∧ 𝑐 = 3) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)))
30 2re 12367 . . . . . . . . . . . . . . . . . . 19 2 ∈ ℝ
31 2lt3 12465 . . . . . . . . . . . . . . . . . . 19 2 < 3
3230, 31gtneii 11402 . . . . . . . . . . . . . . . . . 18 3 ≠ 2
33 neeq1 3009 . . . . . . . . . . . . . . . . . 18 (𝑐 = 3 → (𝑐 ≠ 2 ↔ 3 ≠ 2))
3432, 33mpbiri 258 . . . . . . . . . . . . . . . . 17 (𝑐 = 3 → 𝑐 ≠ 2)
3534adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 1 ∧ 𝑐 = 3) → 𝑐 ≠ 2)
3635neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 1 ∧ 𝑐 = 3) → ¬ 𝑐 = 2)
3736olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 1 ∧ 𝑐 = 3) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2))
38 1re 11290 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℝ
39 1lt3 12466 . . . . . . . . . . . . . . . . . . 19 1 < 3
4038, 39gtneii 11402 . . . . . . . . . . . . . . . . . 18 3 ≠ 1
41 neeq1 3009 . . . . . . . . . . . . . . . . . 18 (𝑐 = 3 → (𝑐 ≠ 1 ↔ 3 ≠ 1))
4240, 41mpbiri 258 . . . . . . . . . . . . . . . . 17 (𝑐 = 3 → 𝑐 ≠ 1)
4342adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 1 ∧ 𝑐 = 3) → 𝑐 ≠ 1)
4443neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 1 ∧ 𝑐 = 3) → ¬ 𝑐 = 1)
4544olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 1 ∧ 𝑐 = 3) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1))
4637, 45jca 511 . . . . . . . . . . . . 13 ((𝑏 = 1 ∧ 𝑐 = 3) → ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)))
47 1ne2 12501 . . . . . . . . . . . . . . . . . 18 1 ≠ 2
48 neeq1 3009 . . . . . . . . . . . . . . . . . 18 (𝑏 = 1 → (𝑏 ≠ 2 ↔ 1 ≠ 2))
4947, 48mpbiri 258 . . . . . . . . . . . . . . . . 17 (𝑏 = 1 → 𝑏 ≠ 2)
5049adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 1 ∧ 𝑐 = 3) → 𝑏 ≠ 2)
5150neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 1 ∧ 𝑐 = 3) → ¬ 𝑏 = 2)
5251orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 1 ∧ 𝑐 = 3) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3))
5336olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 1 ∧ 𝑐 = 3) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))
5452, 53jca 511 . . . . . . . . . . . . 13 ((𝑏 = 1 ∧ 𝑐 = 3) → ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
5529, 46, 543jca 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))))
5638, 39ltneii 11403 . . . . . . . . . . . . . . . . . 18 1 ≠ 3
57 neeq1 3009 . . . . . . . . . . . . . . . . . 18 (𝑏 = 1 → (𝑏 ≠ 3 ↔ 1 ≠ 3))
5856, 57mpbiri 258 . . . . . . . . . . . . . . . . 17 (𝑏 = 1 → 𝑏 ≠ 3)
5958adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 1 ∧ 𝑐 = 3) → 𝑏 ≠ 3)
6059neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 1 ∧ 𝑐 = 3) → ¬ 𝑏 = 3)
6160orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 1 ∧ 𝑐 = 3) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4))
62 1lt4 12469 . . . . . . . . . . . . . . . . . . 19 1 < 4
6338, 62ltneii 11403 . . . . . . . . . . . . . . . . . 18 1 ≠ 4
64 neeq1 3009 . . . . . . . . . . . . . . . . . 18 (𝑏 = 1 → (𝑏 ≠ 4 ↔ 1 ≠ 4))
6563, 64mpbiri 258 . . . . . . . . . . . . . . . . 17 (𝑏 = 1 → 𝑏 ≠ 4)
6665adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 1 ∧ 𝑐 = 3) → 𝑏 ≠ 4)
6766neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 1 ∧ 𝑐 = 3) → ¬ 𝑏 = 4)
6867orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 1 ∧ 𝑐 = 3) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3))
6961, 68jca 511 . . . . . . . . . . . . 13 ((𝑏 = 1 ∧ 𝑐 = 3) → ((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)))
7067orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 1 ∧ 𝑐 = 3) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5))
71 1lt5 12473 . . . . . . . . . . . . . . . . . . 19 1 < 5
7238, 71ltneii 11403 . . . . . . . . . . . . . . . . . 18 1 ≠ 5
73 neeq1 3009 . . . . . . . . . . . . . . . . . 18 (𝑏 = 1 → (𝑏 ≠ 5 ↔ 1 ≠ 5))
7472, 73mpbiri 258 . . . . . . . . . . . . . . . . 17 (𝑏 = 1 → 𝑏 ≠ 5)
7574adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 1 ∧ 𝑐 = 3) → 𝑏 ≠ 5)
7675neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 1 ∧ 𝑐 = 3) → ¬ 𝑏 = 5)
7776orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 1 ∧ 𝑐 = 3) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4))
7870, 77jca 511 . . . . . . . . . . . . 13 ((𝑏 = 1 ∧ 𝑐 = 3) → ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)))
7919orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 1 ∧ 𝑐 = 3) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5))
8025olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 1 ∧ 𝑐 = 3) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))
8179, 80jca 511 . . . . . . . . . . . . 13 ((𝑏 = 1 ∧ 𝑐 = 3) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
8269, 78, 813jca 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))))
8355, 82jca 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)))))
8420, 26, 83jca31 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))))))
8584olcd 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)))))))
8617adantr 480 . . . . . . . . . . . . 13 ((𝑏 = 1 ∧ 𝑐 = 5) → 𝑏 ≠ 0)
8786neneqd 2951 . . . . . . . . . . . 12 ((𝑏 = 1 ∧ 𝑐 = 5) → ¬ 𝑏 = 0)
8887orcd 872 . . . . . . . . . . 11 ((𝑏 = 1 ∧ 𝑐 = 5) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3))
8958adantr 480 . . . . . . . . . . . . 13 ((𝑏 = 1 ∧ 𝑐 = 5) → 𝑏 ≠ 3)
9089neneqd 2951 . . . . . . . . . . . 12 ((𝑏 = 1 ∧ 𝑐 = 5) → ¬ 𝑏 = 3)
9190orcd 872 . . . . . . . . . . 11 ((𝑏 = 1 ∧ 𝑐 = 5) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0))
9287orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 1 ∧ 𝑐 = 5) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1))
93 0re 11292 . . . . . . . . . . . . . . . . . . 19 0 ∈ ℝ
94 5pos 12402 . . . . . . . . . . . . . . . . . . 19 0 < 5
9593, 94gtneii 11402 . . . . . . . . . . . . . . . . . 18 5 ≠ 0
96 neeq1 3009 . . . . . . . . . . . . . . . . . 18 (𝑐 = 5 → (𝑐 ≠ 0 ↔ 5 ≠ 0))
9795, 96mpbiri 258 . . . . . . . . . . . . . . . . 17 (𝑐 = 5 → 𝑐 ≠ 0)
9897adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 1 ∧ 𝑐 = 5) → 𝑐 ≠ 0)
9998neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 1 ∧ 𝑐 = 5) → ¬ 𝑐 = 0)
10099olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 1 ∧ 𝑐 = 5) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0))
10192, 100jca 511 . . . . . . . . . . . . 13 ((𝑏 = 1 ∧ 𝑐 = 5) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)))
102 2lt5 12472 . . . . . . . . . . . . . . . . . . 19 2 < 5
10330, 102gtneii 11402 . . . . . . . . . . . . . . . . . 18 5 ≠ 2
104 neeq1 3009 . . . . . . . . . . . . . . . . . 18 (𝑐 = 5 → (𝑐 ≠ 2 ↔ 5 ≠ 2))
105103, 104mpbiri 258 . . . . . . . . . . . . . . . . 17 (𝑐 = 5 → 𝑐 ≠ 2)
106105adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 1 ∧ 𝑐 = 5) → 𝑐 ≠ 2)
107106neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 1 ∧ 𝑐 = 5) → ¬ 𝑐 = 2)
108107olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 1 ∧ 𝑐 = 5) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2))
10949adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 1 ∧ 𝑐 = 5) → 𝑏 ≠ 2)
110109neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 1 ∧ 𝑐 = 5) → ¬ 𝑏 = 2)
111110orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 1 ∧ 𝑐 = 5) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1))
112108, 111jca 511 . . . . . . . . . . . . 13 ((𝑏 = 1 ∧ 𝑐 = 5) → ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)))
113110orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 1 ∧ 𝑐 = 5) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3))
11490orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 1 ∧ 𝑐 = 5) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))
115113, 114jca 511 . . . . . . . . . . . . 13 ((𝑏 = 1 ∧ 𝑐 = 5) → ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
116101, 112, 1153jca 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))))
11790orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 1 ∧ 𝑐 = 5) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4))
11865adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 1 ∧ 𝑐 = 5) → 𝑏 ≠ 4)
119118neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 1 ∧ 𝑐 = 5) → ¬ 𝑏 = 4)
120119orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 1 ∧ 𝑐 = 5) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3))
121117, 120jca 511 . . . . . . . . . . . . 13 ((𝑏 = 1 ∧ 𝑐 = 5) → ((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)))
122119orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 1 ∧ 𝑐 = 5) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5))
12374adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 1 ∧ 𝑐 = 5) → 𝑏 ≠ 5)
124123neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 1 ∧ 𝑐 = 5) → ¬ 𝑏 = 5)
125124orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 1 ∧ 𝑐 = 5) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4))
126122, 125jca 511 . . . . . . . . . . . . 13 ((𝑏 = 1 ∧ 𝑐 = 5) → ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)))
12787orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 1 ∧ 𝑐 = 5) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5))
12899olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 1 ∧ 𝑐 = 5) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))
129127, 128jca 511 . . . . . . . . . . . . 13 ((𝑏 = 1 ∧ 𝑐 = 5) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
130121, 126, 1293jca 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))))
131116, 130jca 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)))))
13288, 91, 131jca31 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))))))
133132olcd 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)))))))
13414, 85, 1333jaodan 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 3009 . . . . . . . . . . . . . . 15 (𝑏 = 3 → (𝑏 ≠ 0 ↔ 3 ≠ 0))
13621, 135mpbiri 258 . . . . . . . . . . . . . 14 (𝑏 = 3 → 𝑏 ≠ 0)
137136adantr 480 . . . . . . . . . . . . 13 ((𝑏 = 3 ∧ 𝑐 = 1) → 𝑏 ≠ 0)
138137neneqd 2951 . . . . . . . . . . . 12 ((𝑏 = 3 ∧ 𝑐 = 1) → ¬ 𝑏 = 0)
139138orcd 872 . . . . . . . . . . 11 ((𝑏 = 3 ∧ 𝑐 = 1) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3))
140 neeq1 3009 . . . . . . . . . . . . . . 15 (𝑐 = 1 → (𝑐 ≠ 0 ↔ 1 ≠ 0))
14115, 140mpbiri 258 . . . . . . . . . . . . . 14 (𝑐 = 1 → 𝑐 ≠ 0)
142141adantl 481 . . . . . . . . . . . . 13 ((𝑏 = 3 ∧ 𝑐 = 1) → 𝑐 ≠ 0)
143142neneqd 2951 . . . . . . . . . . . 12 ((𝑏 = 3 ∧ 𝑐 = 1) → ¬ 𝑐 = 0)
144143olcd 873 . . . . . . . . . . 11 ((𝑏 = 3 ∧ 𝑐 = 1) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0))
145138orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 3 ∧ 𝑐 = 1) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1))
146143olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 3 ∧ 𝑐 = 1) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0))
147145, 146jca 511 . . . . . . . . . . . . 13 ((𝑏 = 3 ∧ 𝑐 = 1) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)))
14858necon2i 2981 . . . . . . . . . . . . . . . . 17 (𝑏 = 3 → 𝑏 ≠ 1)
149148adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 3 ∧ 𝑐 = 1) → 𝑏 ≠ 1)
150149neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 3 ∧ 𝑐 = 1) → ¬ 𝑏 = 1)
151150orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 3 ∧ 𝑐 = 1) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2))
152 neeq1 3009 . . . . . . . . . . . . . . . . . 18 (𝑏 = 3 → (𝑏 ≠ 2 ↔ 3 ≠ 2))
15332, 152mpbiri 258 . . . . . . . . . . . . . . . . 17 (𝑏 = 3 → 𝑏 ≠ 2)
154153adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 3 ∧ 𝑐 = 1) → 𝑏 ≠ 2)
155154neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 3 ∧ 𝑐 = 1) → ¬ 𝑏 = 2)
156155orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 3 ∧ 𝑐 = 1) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1))
157151, 156jca 511 . . . . . . . . . . . . 13 ((𝑏 = 3 ∧ 𝑐 = 1) → ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)))
158155orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 3 ∧ 𝑐 = 1) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3))
159 neeq1 3009 . . . . . . . . . . . . . . . . . 18 (𝑐 = 1 → (𝑐 ≠ 2 ↔ 1 ≠ 2))
16047, 159mpbiri 258 . . . . . . . . . . . . . . . . 17 (𝑐 = 1 → 𝑐 ≠ 2)
161160adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 3 ∧ 𝑐 = 1) → 𝑐 ≠ 2)
162161neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 3 ∧ 𝑐 = 1) → ¬ 𝑐 = 2)
163162olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 3 ∧ 𝑐 = 1) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))
164158, 163jca 511 . . . . . . . . . . . . 13 ((𝑏 = 3 ∧ 𝑐 = 1) → ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
165147, 157, 1643jca 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 3009 . . . . . . . . . . . . . . . . . 18 (𝑐 = 1 → (𝑐 ≠ 4 ↔ 1 ≠ 4))
16763, 166mpbiri 258 . . . . . . . . . . . . . . . . 17 (𝑐 = 1 → 𝑐 ≠ 4)
168167adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 3 ∧ 𝑐 = 1) → 𝑐 ≠ 4)
169168neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 3 ∧ 𝑐 = 1) → ¬ 𝑐 = 4)
170169olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 3 ∧ 𝑐 = 1) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4))
17142necon2i 2981 . . . . . . . . . . . . . . . . 17 (𝑐 = 1 → 𝑐 ≠ 3)
172171adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 3 ∧ 𝑐 = 1) → 𝑐 ≠ 3)
173172neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 3 ∧ 𝑐 = 1) → ¬ 𝑐 = 3)
174173olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 3 ∧ 𝑐 = 1) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3))
175170, 174jca 511 . . . . . . . . . . . . 13 ((𝑏 = 3 ∧ 𝑐 = 1) → ((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)))
176 neeq1 3009 . . . . . . . . . . . . . . . . . 18 (𝑐 = 1 → (𝑐 ≠ 5 ↔ 1 ≠ 5))
17772, 176mpbiri 258 . . . . . . . . . . . . . . . . 17 (𝑐 = 1 → 𝑐 ≠ 5)
178177adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 3 ∧ 𝑐 = 1) → 𝑐 ≠ 5)
179178neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 3 ∧ 𝑐 = 1) → ¬ 𝑐 = 5)
180179olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 3 ∧ 𝑐 = 1) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5))
181169olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 3 ∧ 𝑐 = 1) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4))
182180, 181jca 511 . . . . . . . . . . . . 13 ((𝑏 = 3 ∧ 𝑐 = 1) → ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)))
183138orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 3 ∧ 𝑐 = 1) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5))
184143olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 3 ∧ 𝑐 = 1) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))
185183, 184jca 511 . . . . . . . . . . . . 13 ((𝑏 = 3 ∧ 𝑐 = 1) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
186175, 182, 1853jca 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))))
187165, 186jca 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)))))
188139, 144, 187jca31 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))))))
189188olcd 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 2766 . . . . . . . . . 10 ((𝑏 = 3 ∧ 𝑐 = 3) → 𝑏 = 𝑐)
191190orcd 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)))))))
192136adantr 480 . . . . . . . . . . . . 13 ((𝑏 = 3 ∧ 𝑐 = 5) → 𝑏 ≠ 0)
193192neneqd 2951 . . . . . . . . . . . 12 ((𝑏 = 3 ∧ 𝑐 = 5) → ¬ 𝑏 = 0)
194193orcd 872 . . . . . . . . . . 11 ((𝑏 = 3 ∧ 𝑐 = 5) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3))
19597adantl 481 . . . . . . . . . . . . 13 ((𝑏 = 3 ∧ 𝑐 = 5) → 𝑐 ≠ 0)
196195neneqd 2951 . . . . . . . . . . . 12 ((𝑏 = 3 ∧ 𝑐 = 5) → ¬ 𝑐 = 0)
197196olcd 873 . . . . . . . . . . 11 ((𝑏 = 3 ∧ 𝑐 = 5) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0))
198193orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 3 ∧ 𝑐 = 5) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1))
199196olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 3 ∧ 𝑐 = 5) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0))
200198, 199jca 511 . . . . . . . . . . . . 13 ((𝑏 = 3 ∧ 𝑐 = 5) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)))
201148adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 3 ∧ 𝑐 = 5) → 𝑏 ≠ 1)
202201neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 3 ∧ 𝑐 = 5) → ¬ 𝑏 = 1)
203202orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 3 ∧ 𝑐 = 5) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2))
204177necon2i 2981 . . . . . . . . . . . . . . . . 17 (𝑐 = 5 → 𝑐 ≠ 1)
205204adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 3 ∧ 𝑐 = 5) → 𝑐 ≠ 1)
206205neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 3 ∧ 𝑐 = 5) → ¬ 𝑐 = 1)
207206olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 3 ∧ 𝑐 = 5) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1))
208203, 207jca 511 . . . . . . . . . . . . 13 ((𝑏 = 3 ∧ 𝑐 = 5) → ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)))
209153adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 3 ∧ 𝑐 = 5) → 𝑏 ≠ 2)
210209neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 3 ∧ 𝑐 = 5) → ¬ 𝑏 = 2)
211210orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 3 ∧ 𝑐 = 5) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3))
212105adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 3 ∧ 𝑐 = 5) → 𝑐 ≠ 2)
213212neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 3 ∧ 𝑐 = 5) → ¬ 𝑐 = 2)
214213olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 3 ∧ 𝑐 = 5) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))
215211, 214jca 511 . . . . . . . . . . . . 13 ((𝑏 = 3 ∧ 𝑐 = 5) → ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
216200, 208, 2153jca 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 12377 . . . . . . . . . . . . . . . . . . 19 4 ∈ ℝ
218 4lt5 12470 . . . . . . . . . . . . . . . . . . 19 4 < 5
219217, 218gtneii 11402 . . . . . . . . . . . . . . . . . 18 5 ≠ 4
220 neeq1 3009 . . . . . . . . . . . . . . . . . 18 (𝑐 = 5 → (𝑐 ≠ 4 ↔ 5 ≠ 4))
221219, 220mpbiri 258 . . . . . . . . . . . . . . . . 17 (𝑐 = 5 → 𝑐 ≠ 4)
222221adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 3 ∧ 𝑐 = 5) → 𝑐 ≠ 4)
223222neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 3 ∧ 𝑐 = 5) → ¬ 𝑐 = 4)
224223olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 3 ∧ 𝑐 = 5) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4))
225 3re 12373 . . . . . . . . . . . . . . . . . . 19 3 ∈ ℝ
226 3lt4 12467 . . . . . . . . . . . . . . . . . . 19 3 < 4
227225, 226ltneii 11403 . . . . . . . . . . . . . . . . . 18 3 ≠ 4
228 neeq1 3009 . . . . . . . . . . . . . . . . . 18 (𝑏 = 3 → (𝑏 ≠ 4 ↔ 3 ≠ 4))
229227, 228mpbiri 258 . . . . . . . . . . . . . . . . 17 (𝑏 = 3 → 𝑏 ≠ 4)
230229adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 3 ∧ 𝑐 = 5) → 𝑏 ≠ 4)
231230neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 3 ∧ 𝑐 = 5) → ¬ 𝑏 = 4)
232231orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 3 ∧ 𝑐 = 5) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3))
233224, 232jca 511 . . . . . . . . . . . . 13 ((𝑏 = 3 ∧ 𝑐 = 5) → ((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)))
234231orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 3 ∧ 𝑐 = 5) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5))
235223olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 3 ∧ 𝑐 = 5) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4))
236234, 235jca 511 . . . . . . . . . . . . 13 ((𝑏 = 3 ∧ 𝑐 = 5) → ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)))
237193orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 3 ∧ 𝑐 = 5) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5))
238196olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 3 ∧ 𝑐 = 5) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))
239237, 238jca 511 . . . . . . . . . . . . 13 ((𝑏 = 3 ∧ 𝑐 = 5) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
240233, 236, 2393jca 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))))
241216, 240jca 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)))))
242194, 197, 241jca31 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))))))
243242olcd 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)))))))
244189, 191, 2433jaodan 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)))))))
245171adantl 481 . . . . . . . . . . . . 13 ((𝑏 = 5 ∧ 𝑐 = 1) → 𝑐 ≠ 3)
246245neneqd 2951 . . . . . . . . . . . 12 ((𝑏 = 5 ∧ 𝑐 = 1) → ¬ 𝑐 = 3)
247246olcd 873 . . . . . . . . . . 11 ((𝑏 = 5 ∧ 𝑐 = 1) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3))
248141adantl 481 . . . . . . . . . . . . 13 ((𝑏 = 5 ∧ 𝑐 = 1) → 𝑐 ≠ 0)
249248neneqd 2951 . . . . . . . . . . . 12 ((𝑏 = 5 ∧ 𝑐 = 1) → ¬ 𝑐 = 0)
250249olcd 873 . . . . . . . . . . 11 ((𝑏 = 5 ∧ 𝑐 = 1) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0))
251 neeq1 3009 . . . . . . . . . . . . . . . . . 18 (𝑏 = 5 → (𝑏 ≠ 0 ↔ 5 ≠ 0))
25295, 251mpbiri 258 . . . . . . . . . . . . . . . . 17 (𝑏 = 5 → 𝑏 ≠ 0)
253252adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 5 ∧ 𝑐 = 1) → 𝑏 ≠ 0)
254253neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 5 ∧ 𝑐 = 1) → ¬ 𝑏 = 0)
255254orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 5 ∧ 𝑐 = 1) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1))
256249olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 5 ∧ 𝑐 = 1) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0))
257255, 256jca 511 . . . . . . . . . . . . 13 ((𝑏 = 5 ∧ 𝑐 = 1) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)))
25874necon2i 2981 . . . . . . . . . . . . . . . . 17 (𝑏 = 5 → 𝑏 ≠ 1)
259258adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 5 ∧ 𝑐 = 1) → 𝑏 ≠ 1)
260259neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 5 ∧ 𝑐 = 1) → ¬ 𝑏 = 1)
261260orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 5 ∧ 𝑐 = 1) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2))
262 neeq1 3009 . . . . . . . . . . . . . . . . . 18 (𝑏 = 5 → (𝑏 ≠ 2 ↔ 5 ≠ 2))
263103, 262mpbiri 258 . . . . . . . . . . . . . . . . 17 (𝑏 = 5 → 𝑏 ≠ 2)
264263adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 5 ∧ 𝑐 = 1) → 𝑏 ≠ 2)
265264neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 5 ∧ 𝑐 = 1) → ¬ 𝑏 = 2)
266265orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 5 ∧ 𝑐 = 1) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1))
267261, 266jca 511 . . . . . . . . . . . . 13 ((𝑏 = 5 ∧ 𝑐 = 1) → ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)))
268246olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 5 ∧ 𝑐 = 1) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3))
269160adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 5 ∧ 𝑐 = 1) → 𝑐 ≠ 2)
270269neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 5 ∧ 𝑐 = 1) → ¬ 𝑐 = 2)
271270olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 5 ∧ 𝑐 = 1) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))
272268, 271jca 511 . . . . . . . . . . . . 13 ((𝑏 = 5 ∧ 𝑐 = 1) → ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
273257, 267, 2723jca 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 12471 . . . . . . . . . . . . . . . . . . 19 3 < 5
275225, 274gtneii 11402 . . . . . . . . . . . . . . . . . 18 5 ≠ 3
276 neeq1 3009 . . . . . . . . . . . . . . . . . 18 (𝑏 = 5 → (𝑏 ≠ 3 ↔ 5 ≠ 3))
277275, 276mpbiri 258 . . . . . . . . . . . . . . . . 17 (𝑏 = 5 → 𝑏 ≠ 3)
278277adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 5 ∧ 𝑐 = 1) → 𝑏 ≠ 3)
279278neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 5 ∧ 𝑐 = 1) → ¬ 𝑏 = 3)
280279orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 5 ∧ 𝑐 = 1) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4))
281246olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 5 ∧ 𝑐 = 1) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3))
282280, 281jca 511 . . . . . . . . . . . . 13 ((𝑏 = 5 ∧ 𝑐 = 1) → ((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)))
283177adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 5 ∧ 𝑐 = 1) → 𝑐 ≠ 5)
284283neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 5 ∧ 𝑐 = 1) → ¬ 𝑐 = 5)
285284olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 5 ∧ 𝑐 = 1) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5))
286167adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 5 ∧ 𝑐 = 1) → 𝑐 ≠ 4)
287286neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 5 ∧ 𝑐 = 1) → ¬ 𝑐 = 4)
288287olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 5 ∧ 𝑐 = 1) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4))
289285, 288jca 511 . . . . . . . . . . . . 13 ((𝑏 = 5 ∧ 𝑐 = 1) → ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)))
290254orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 5 ∧ 𝑐 = 1) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5))
291249olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 5 ∧ 𝑐 = 1) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))
292290, 291jca 511 . . . . . . . . . . . . 13 ((𝑏 = 5 ∧ 𝑐 = 1) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
293282, 289, 2923jca 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))))
294273, 293jca 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)))))
295247, 250, 294jca31 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))))))
296295olcd 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)))))))
297252adantr 480 . . . . . . . . . . . . 13 ((𝑏 = 5 ∧ 𝑐 = 3) → 𝑏 ≠ 0)
298297neneqd 2951 . . . . . . . . . . . 12 ((𝑏 = 5 ∧ 𝑐 = 3) → ¬ 𝑏 = 0)
299298orcd 872 . . . . . . . . . . 11 ((𝑏 = 5 ∧ 𝑐 = 3) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3))
30023adantl 481 . . . . . . . . . . . . 13 ((𝑏 = 5 ∧ 𝑐 = 3) → 𝑐 ≠ 0)
301300neneqd 2951 . . . . . . . . . . . 12 ((𝑏 = 5 ∧ 𝑐 = 3) → ¬ 𝑐 = 0)
302301olcd 873 . . . . . . . . . . 11 ((𝑏 = 5 ∧ 𝑐 = 3) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0))
303298orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 5 ∧ 𝑐 = 3) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1))
304301olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 5 ∧ 𝑐 = 3) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0))
305303, 304jca 511 . . . . . . . . . . . . 13 ((𝑏 = 5 ∧ 𝑐 = 3) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)))
306258adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 5 ∧ 𝑐 = 3) → 𝑏 ≠ 1)
307306neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 5 ∧ 𝑐 = 3) → ¬ 𝑏 = 1)
308307orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 5 ∧ 𝑐 = 3) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2))
30942adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 5 ∧ 𝑐 = 3) → 𝑐 ≠ 1)
310309neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 5 ∧ 𝑐 = 3) → ¬ 𝑐 = 1)
311310olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 5 ∧ 𝑐 = 3) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1))
312308, 311jca 511 . . . . . . . . . . . . 13 ((𝑏 = 5 ∧ 𝑐 = 3) → ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)))
313263adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 5 ∧ 𝑐 = 3) → 𝑏 ≠ 2)
314313neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 5 ∧ 𝑐 = 3) → ¬ 𝑏 = 2)
315314orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 5 ∧ 𝑐 = 3) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3))
316277adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 5 ∧ 𝑐 = 3) → 𝑏 ≠ 3)
317316neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 5 ∧ 𝑐 = 3) → ¬ 𝑏 = 3)
318317orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 5 ∧ 𝑐 = 3) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))
319315, 318jca 511 . . . . . . . . . . . . 13 ((𝑏 = 5 ∧ 𝑐 = 3) → ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
320305, 312, 3193jca 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))))
321317orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 5 ∧ 𝑐 = 3) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4))
322 neeq1 3009 . . . . . . . . . . . . . . . . . 18 (𝑏 = 5 → (𝑏 ≠ 4 ↔ 5 ≠ 4))
323219, 322mpbiri 258 . . . . . . . . . . . . . . . . 17 (𝑏 = 5 → 𝑏 ≠ 4)
324323adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 5 ∧ 𝑐 = 3) → 𝑏 ≠ 4)
325324neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 5 ∧ 𝑐 = 3) → ¬ 𝑏 = 4)
326325orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 5 ∧ 𝑐 = 3) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3))
327321, 326jca 511 . . . . . . . . . . . . 13 ((𝑏 = 5 ∧ 𝑐 = 3) → ((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)))
328325orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 5 ∧ 𝑐 = 3) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5))
329 neeq1 3009 . . . . . . . . . . . . . . . . . 18 (𝑐 = 3 → (𝑐 ≠ 4 ↔ 3 ≠ 4))
330227, 329mpbiri 258 . . . . . . . . . . . . . . . . 17 (𝑐 = 3 → 𝑐 ≠ 4)
331330adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 5 ∧ 𝑐 = 3) → 𝑐 ≠ 4)
332331neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 5 ∧ 𝑐 = 3) → ¬ 𝑐 = 4)
333332olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 5 ∧ 𝑐 = 3) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4))
334328, 333jca 511 . . . . . . . . . . . . 13 ((𝑏 = 5 ∧ 𝑐 = 3) → ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)))
335298orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 5 ∧ 𝑐 = 3) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5))
336301olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 5 ∧ 𝑐 = 3) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))
337335, 336jca 511 . . . . . . . . . . . . 13 ((𝑏 = 5 ∧ 𝑐 = 3) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
338327, 334, 3373jca 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))))
339320, 338jca 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)))))
340299, 302, 339jca31 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))))))
341340olcd 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 2766 . . . . . . . . . 10 ((𝑏 = 5 ∧ 𝑐 = 5) → 𝑏 = 𝑐)
343342orcd 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)))))))
344296, 341, 3433jaodan 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)))))))
345134, 244, 3443jaoian 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)))))))
3468, 12, 345syl2anb 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)))))))
347346rgen2 3205 . . . . 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))))))
3481, 2, 3usgrexmpl2nb1 47847 . . . . . . . . 9 (𝐺 NeighbVtx 1) = {0, 2}
349348eleq2i 2836 . . . . . . . 8 (𝑏 ∈ (𝐺 NeighbVtx 1) ↔ 𝑏 ∈ {0, 2})
3506elpr 4672 . . . . . . . 8 (𝑏 ∈ {0, 2} ↔ (𝑏 = 0 ∨ 𝑏 = 2))
351349, 350bitri 275 . . . . . . 7 (𝑏 ∈ (𝐺 NeighbVtx 1) ↔ (𝑏 = 0 ∨ 𝑏 = 2))
352348eleq2i 2836 . . . . . . . 8 (𝑐 ∈ (𝐺 NeighbVtx 1) ↔ 𝑐 ∈ {0, 2})
35310elpr 4672 . . . . . . . 8 (𝑐 ∈ {0, 2} ↔ (𝑐 = 0 ∨ 𝑐 = 2))
354352, 353bitri 275 . . . . . . 7 (𝑐 ∈ (𝐺 NeighbVtx 1) ↔ (𝑐 = 0 ∨ 𝑐 = 2))
355 eqtr3 2766 . . . . . . . . 9 ((𝑏 = 0 ∧ 𝑐 = 0) → 𝑏 = 𝑐)
356355orcd 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 12397 . . . . . . . . . . . . . 14 2 ≠ 0
358 neeq1 3009 . . . . . . . . . . . . . 14 (𝑏 = 2 → (𝑏 ≠ 0 ↔ 2 ≠ 0))
359357, 358mpbiri 258 . . . . . . . . . . . . 13 (𝑏 = 2 → 𝑏 ≠ 0)
360359adantr 480 . . . . . . . . . . . 12 ((𝑏 = 2 ∧ 𝑐 = 0) → 𝑏 ≠ 0)
361360neneqd 2951 . . . . . . . . . . 11 ((𝑏 = 2 ∧ 𝑐 = 0) → ¬ 𝑏 = 0)
362361orcd 872 . . . . . . . . . 10 ((𝑏 = 2 ∧ 𝑐 = 0) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3))
363153necon2i 2981 . . . . . . . . . . . . 13 (𝑏 = 2 → 𝑏 ≠ 3)
364363adantr 480 . . . . . . . . . . . 12 ((𝑏 = 2 ∧ 𝑐 = 0) → 𝑏 ≠ 3)
365364neneqd 2951 . . . . . . . . . . 11 ((𝑏 = 2 ∧ 𝑐 = 0) → ¬ 𝑏 = 3)
366365orcd 872 . . . . . . . . . 10 ((𝑏 = 2 ∧ 𝑐 = 0) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0))
367361orcd 872 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 0) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1))
36849necon2i 2981 . . . . . . . . . . . . . . . 16 (𝑏 = 2 → 𝑏 ≠ 1)
369368adantr 480 . . . . . . . . . . . . . . 15 ((𝑏 = 2 ∧ 𝑐 = 0) → 𝑏 ≠ 1)
370369neneqd 2951 . . . . . . . . . . . . . 14 ((𝑏 = 2 ∧ 𝑐 = 0) → ¬ 𝑏 = 1)
371370orcd 872 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 0) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0))
372367, 371jca 511 . . . . . . . . . . . 12 ((𝑏 = 2 ∧ 𝑐 = 0) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)))
373370orcd 872 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 0) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2))
374141necon2i 2981 . . . . . . . . . . . . . . . 16 (𝑐 = 0 → 𝑐 ≠ 1)
375374adantl 481 . . . . . . . . . . . . . . 15 ((𝑏 = 2 ∧ 𝑐 = 0) → 𝑐 ≠ 1)
376375neneqd 2951 . . . . . . . . . . . . . 14 ((𝑏 = 2 ∧ 𝑐 = 0) → ¬ 𝑐 = 1)
377376olcd 873 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 0) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1))
378373, 377jca 511 . . . . . . . . . . . 12 ((𝑏 = 2 ∧ 𝑐 = 0) → ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)))
37923necon2i 2981 . . . . . . . . . . . . . . . 16 (𝑐 = 0 → 𝑐 ≠ 3)
380379adantl 481 . . . . . . . . . . . . . . 15 ((𝑏 = 2 ∧ 𝑐 = 0) → 𝑐 ≠ 3)
381380neneqd 2951 . . . . . . . . . . . . . 14 ((𝑏 = 2 ∧ 𝑐 = 0) → ¬ 𝑐 = 3)
382381olcd 873 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 0) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3))
383365orcd 872 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 0) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))
384382, 383jca 511 . . . . . . . . . . . 12 ((𝑏 = 2 ∧ 𝑐 = 0) → ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
385372, 378, 3843jca 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))))
386365orcd 872 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 0) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4))
387381olcd 873 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 0) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3))
388386, 387jca 511 . . . . . . . . . . . 12 ((𝑏 = 2 ∧ 𝑐 = 0) → ((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)))
38997necon2i 2981 . . . . . . . . . . . . . . . 16 (𝑐 = 0 → 𝑐 ≠ 5)
390389adantl 481 . . . . . . . . . . . . . . 15 ((𝑏 = 2 ∧ 𝑐 = 0) → 𝑐 ≠ 5)
391390neneqd 2951 . . . . . . . . . . . . . 14 ((𝑏 = 2 ∧ 𝑐 = 0) → ¬ 𝑐 = 5)
392391olcd 873 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 0) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5))
393 4pos 12400 . . . . . . . . . . . . . . . . . 18 0 < 4
39493, 393ltneii 11403 . . . . . . . . . . . . . . . . 17 0 ≠ 4
395 neeq1 3009 . . . . . . . . . . . . . . . . 17 (𝑐 = 0 → (𝑐 ≠ 4 ↔ 0 ≠ 4))
396394, 395mpbiri 258 . . . . . . . . . . . . . . . 16 (𝑐 = 0 → 𝑐 ≠ 4)
397396adantl 481 . . . . . . . . . . . . . . 15 ((𝑏 = 2 ∧ 𝑐 = 0) → 𝑐 ≠ 4)
398397neneqd 2951 . . . . . . . . . . . . . 14 ((𝑏 = 2 ∧ 𝑐 = 0) → ¬ 𝑐 = 4)
399398olcd 873 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 0) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4))
400392, 399jca 511 . . . . . . . . . . . 12 ((𝑏 = 2 ∧ 𝑐 = 0) → ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)))
401361orcd 872 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 0) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5))
402263necon2i 2981 . . . . . . . . . . . . . . . 16 (𝑏 = 2 → 𝑏 ≠ 5)
403402adantr 480 . . . . . . . . . . . . . . 15 ((𝑏 = 2 ∧ 𝑐 = 0) → 𝑏 ≠ 5)
404403neneqd 2951 . . . . . . . . . . . . . 14 ((𝑏 = 2 ∧ 𝑐 = 0) → ¬ 𝑏 = 5)
405404orcd 872 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 0) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))
406401, 405jca 511 . . . . . . . . . . . 12 ((𝑏 = 2 ∧ 𝑐 = 0) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
407388, 400, 4063jca 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))))
408385, 407jca 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)))))
409362, 366, 408jca31 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))))))
410409olcd 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)))))))
41134necon2i 2981 . . . . . . . . . . . . 13 (𝑐 = 2 → 𝑐 ≠ 3)
412411adantl 481 . . . . . . . . . . . 12 ((𝑏 = 0 ∧ 𝑐 = 2) → 𝑐 ≠ 3)
413412neneqd 2951 . . . . . . . . . . 11 ((𝑏 = 0 ∧ 𝑐 = 2) → ¬ 𝑐 = 3)
414413olcd 873 . . . . . . . . . 10 ((𝑏 = 0 ∧ 𝑐 = 2) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3))
415 neeq1 3009 . . . . . . . . . . . . . 14 (𝑐 = 2 → (𝑐 ≠ 0 ↔ 2 ≠ 0))
416357, 415mpbiri 258 . . . . . . . . . . . . 13 (𝑐 = 2 → 𝑐 ≠ 0)
417416adantl 481 . . . . . . . . . . . 12 ((𝑏 = 0 ∧ 𝑐 = 2) → 𝑐 ≠ 0)
418417neneqd 2951 . . . . . . . . . . 11 ((𝑏 = 0 ∧ 𝑐 = 2) → ¬ 𝑐 = 0)
419418olcd 873 . . . . . . . . . 10 ((𝑏 = 0 ∧ 𝑐 = 2) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0))
420160necon2i 2981 . . . . . . . . . . . . . . . 16 (𝑐 = 2 → 𝑐 ≠ 1)
421420adantl 481 . . . . . . . . . . . . . . 15 ((𝑏 = 0 ∧ 𝑐 = 2) → 𝑐 ≠ 1)
422421neneqd 2951 . . . . . . . . . . . . . 14 ((𝑏 = 0 ∧ 𝑐 = 2) → ¬ 𝑐 = 1)
423422olcd 873 . . . . . . . . . . . . 13 ((𝑏 = 0 ∧ 𝑐 = 2) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1))
424418olcd 873 . . . . . . . . . . . . 13 ((𝑏 = 0 ∧ 𝑐 = 2) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0))
425423, 424jca 511 . . . . . . . . . . . 12 ((𝑏 = 0 ∧ 𝑐 = 2) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)))
42617necon2i 2981 . . . . . . . . . . . . . . . 16 (𝑏 = 0 → 𝑏 ≠ 1)
427426adantr 480 . . . . . . . . . . . . . . 15 ((𝑏 = 0 ∧ 𝑐 = 2) → 𝑏 ≠ 1)
428427neneqd 2951 . . . . . . . . . . . . . 14 ((𝑏 = 0 ∧ 𝑐 = 2) → ¬ 𝑏 = 1)
429428orcd 872 . . . . . . . . . . . . 13 ((𝑏 = 0 ∧ 𝑐 = 2) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2))
430359necon2i 2981 . . . . . . . . . . . . . . . 16 (𝑏 = 0 → 𝑏 ≠ 2)
431430adantr 480 . . . . . . . . . . . . . . 15 ((𝑏 = 0 ∧ 𝑐 = 2) → 𝑏 ≠ 2)
432431neneqd 2951 . . . . . . . . . . . . . 14 ((𝑏 = 0 ∧ 𝑐 = 2) → ¬ 𝑏 = 2)
433432orcd 872 . . . . . . . . . . . . 13 ((𝑏 = 0 ∧ 𝑐 = 2) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1))
434429, 433jca 511 . . . . . . . . . . . 12 ((𝑏 = 0 ∧ 𝑐 = 2) → ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)))
435413olcd 873 . . . . . . . . . . . . 13 ((𝑏 = 0 ∧ 𝑐 = 2) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3))
436136necon2i 2981 . . . . . . . . . . . . . . . 16 (𝑏 = 0 → 𝑏 ≠ 3)
437436adantr 480 . . . . . . . . . . . . . . 15 ((𝑏 = 0 ∧ 𝑐 = 2) → 𝑏 ≠ 3)
438437neneqd 2951 . . . . . . . . . . . . . 14 ((𝑏 = 0 ∧ 𝑐 = 2) → ¬ 𝑏 = 3)
439438orcd 872 . . . . . . . . . . . . 13 ((𝑏 = 0 ∧ 𝑐 = 2) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))
440435, 439jca 511 . . . . . . . . . . . 12 ((𝑏 = 0 ∧ 𝑐 = 2) → ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
441425, 434, 4403jca 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))))
442438orcd 872 . . . . . . . . . . . . 13 ((𝑏 = 0 ∧ 𝑐 = 2) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4))
443413olcd 873 . . . . . . . . . . . . 13 ((𝑏 = 0 ∧ 𝑐 = 2) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3))
444442, 443jca 511 . . . . . . . . . . . 12 ((𝑏 = 0 ∧ 𝑐 = 2) → ((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)))
445 neeq1 3009 . . . . . . . . . . . . . . . . 17 (𝑏 = 0 → (𝑏 ≠ 4 ↔ 0 ≠ 4))
446394, 445mpbiri 258 . . . . . . . . . . . . . . . 16 (𝑏 = 0 → 𝑏 ≠ 4)
447446adantr 480 . . . . . . . . . . . . . . 15 ((𝑏 = 0 ∧ 𝑐 = 2) → 𝑏 ≠ 4)
448447neneqd 2951 . . . . . . . . . . . . . 14 ((𝑏 = 0 ∧ 𝑐 = 2) → ¬ 𝑏 = 4)
449448orcd 872 . . . . . . . . . . . . 13 ((𝑏 = 0 ∧ 𝑐 = 2) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5))
450252necon2i 2981 . . . . . . . . . . . . . . . 16 (𝑏 = 0 → 𝑏 ≠ 5)
451450adantr 480 . . . . . . . . . . . . . . 15 ((𝑏 = 0 ∧ 𝑐 = 2) → 𝑏 ≠ 5)
452451neneqd 2951 . . . . . . . . . . . . . 14 ((𝑏 = 0 ∧ 𝑐 = 2) → ¬ 𝑏 = 5)
453452orcd 872 . . . . . . . . . . . . 13 ((𝑏 = 0 ∧ 𝑐 = 2) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4))
454449, 453jca 511 . . . . . . . . . . . 12 ((𝑏 = 0 ∧ 𝑐 = 2) → ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)))
455105necon2i 2981 . . . . . . . . . . . . . . . 16 (𝑐 = 2 → 𝑐 ≠ 5)
456455adantl 481 . . . . . . . . . . . . . . 15 ((𝑏 = 0 ∧ 𝑐 = 2) → 𝑐 ≠ 5)
457456neneqd 2951 . . . . . . . . . . . . . 14 ((𝑏 = 0 ∧ 𝑐 = 2) → ¬ 𝑐 = 5)
458457olcd 873 . . . . . . . . . . . . 13 ((𝑏 = 0 ∧ 𝑐 = 2) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5))
459418olcd 873 . . . . . . . . . . . . 13 ((𝑏 = 0 ∧ 𝑐 = 2) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))
460458, 459jca 511 . . . . . . . . . . . 12 ((𝑏 = 0 ∧ 𝑐 = 2) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
461444, 454, 4603jca 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))))
462441, 461jca 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)))))
463414, 419, 462jca31 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))))))
464463olcd 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)))))))
465359adantr 480 . . . . . . . . . . . 12 ((𝑏 = 2 ∧ 𝑐 = 2) → 𝑏 ≠ 0)
466465neneqd 2951 . . . . . . . . . . 11 ((𝑏 = 2 ∧ 𝑐 = 2) → ¬ 𝑏 = 0)
467466orcd 872 . . . . . . . . . 10 ((𝑏 = 2 ∧ 𝑐 = 2) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3))
468416adantl 481 . . . . . . . . . . . 12 ((𝑏 = 2 ∧ 𝑐 = 2) → 𝑐 ≠ 0)
469468neneqd 2951 . . . . . . . . . . 11 ((𝑏 = 2 ∧ 𝑐 = 2) → ¬ 𝑐 = 0)
470469olcd 873 . . . . . . . . . 10 ((𝑏 = 2 ∧ 𝑐 = 2) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0))
471466orcd 872 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 2) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1))
472469olcd 873 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 2) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0))
473471, 472jca 511 . . . . . . . . . . . 12 ((𝑏 = 2 ∧ 𝑐 = 2) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)))
474368adantr 480 . . . . . . . . . . . . . . 15 ((𝑏 = 2 ∧ 𝑐 = 2) → 𝑏 ≠ 1)
475474neneqd 2951 . . . . . . . . . . . . . 14 ((𝑏 = 2 ∧ 𝑐 = 2) → ¬ 𝑏 = 1)
476475orcd 872 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 2) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2))
477420adantl 481 . . . . . . . . . . . . . . 15 ((𝑏 = 2 ∧ 𝑐 = 2) → 𝑐 ≠ 1)
478477neneqd 2951 . . . . . . . . . . . . . 14 ((𝑏 = 2 ∧ 𝑐 = 2) → ¬ 𝑐 = 1)
479478olcd 873 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 2) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1))
480476, 479jca 511 . . . . . . . . . . . 12 ((𝑏 = 2 ∧ 𝑐 = 2) → ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)))
481411adantl 481 . . . . . . . . . . . . . . 15 ((𝑏 = 2 ∧ 𝑐 = 2) → 𝑐 ≠ 3)
482481neneqd 2951 . . . . . . . . . . . . . 14 ((𝑏 = 2 ∧ 𝑐 = 2) → ¬ 𝑐 = 3)
483482olcd 873 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 2) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3))
484363adantr 480 . . . . . . . . . . . . . . 15 ((𝑏 = 2 ∧ 𝑐 = 2) → 𝑏 ≠ 3)
485484neneqd 2951 . . . . . . . . . . . . . 14 ((𝑏 = 2 ∧ 𝑐 = 2) → ¬ 𝑏 = 3)
486485orcd 872 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 2) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))
487483, 486jca 511 . . . . . . . . . . . 12 ((𝑏 = 2 ∧ 𝑐 = 2) → ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
488473, 480, 4873jca 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))))
489485orcd 872 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 2) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4))
490 2lt4 12468 . . . . . . . . . . . . . . . . . 18 2 < 4
49130, 490ltneii 11403 . . . . . . . . . . . . . . . . 17 2 ≠ 4
492 neeq1 3009 . . . . . . . . . . . . . . . . 17 (𝑏 = 2 → (𝑏 ≠ 4 ↔ 2 ≠ 4))
493491, 492mpbiri 258 . . . . . . . . . . . . . . . 16 (𝑏 = 2 → 𝑏 ≠ 4)
494493adantr 480 . . . . . . . . . . . . . . 15 ((𝑏 = 2 ∧ 𝑐 = 2) → 𝑏 ≠ 4)
495494neneqd 2951 . . . . . . . . . . . . . 14 ((𝑏 = 2 ∧ 𝑐 = 2) → ¬ 𝑏 = 4)
496495orcd 872 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 2) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3))
497489, 496jca 511 . . . . . . . . . . . 12 ((𝑏 = 2 ∧ 𝑐 = 2) → ((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)))
498495orcd 872 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 2) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5))
499402adantr 480 . . . . . . . . . . . . . . 15 ((𝑏 = 2 ∧ 𝑐 = 2) → 𝑏 ≠ 5)
500499neneqd 2951 . . . . . . . . . . . . . 14 ((𝑏 = 2 ∧ 𝑐 = 2) → ¬ 𝑏 = 5)
501500orcd 872 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 2) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4))
502498, 501jca 511 . . . . . . . . . . . 12 ((𝑏 = 2 ∧ 𝑐 = 2) → ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)))
503466orcd 872 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 2) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5))
504469olcd 873 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 2) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))
505503, 504jca 511 . . . . . . . . . . . 12 ((𝑏 = 2 ∧ 𝑐 = 2) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
506497, 502, 5053jca 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))))
507488, 506jca 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)))))
508467, 470, 507jca31 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))))))
509508olcd 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)))))))
510356, 410, 464, 509ccase 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)))))))
511351, 354, 510syl2anb 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)))))))
512511rgen2 3205 . . . . 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))))))
5131, 2, 3usgrexmpl2nb2 47848 . . . . . . . . 9 (𝐺 NeighbVtx 2) = {1, 3}
514513eleq2i 2836 . . . . . . . 8 (𝑏 ∈ (𝐺 NeighbVtx 2) ↔ 𝑏 ∈ {1, 3})
5156elpr 4672 . . . . . . . 8 (𝑏 ∈ {1, 3} ↔ (𝑏 = 1 ∨ 𝑏 = 3))
516514, 515bitri 275 . . . . . . 7 (𝑏 ∈ (𝐺 NeighbVtx 2) ↔ (𝑏 = 1 ∨ 𝑏 = 3))
517513eleq2i 2836 . . . . . . . 8 (𝑐 ∈ (𝐺 NeighbVtx 2) ↔ 𝑐 ∈ {1, 3})
51810elpr 4672 . . . . . . . 8 (𝑐 ∈ {1, 3} ↔ (𝑐 = 1 ∨ 𝑐 = 3))
519517, 518bitri 275 . . . . . . 7 (𝑐 ∈ (𝐺 NeighbVtx 2) ↔ (𝑐 = 1 ∨ 𝑐 = 3))
52014, 189, 85, 191ccase 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)))))))
521516, 519, 520syl2anb 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)))))))
522521rgen2 3205 . . . . 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 11284 . . . . . 6 0 ∈ V
524 1ex 11286 . . . . . 6 1 ∈ V
525 2ex 12370 . . . . . 6 2 ∈ V
526 oveq2 7456 . . . . . . 7 (𝑎 = 0 → (𝐺 NeighbVtx 𝑎) = (𝐺 NeighbVtx 0))
527526raleqdv 3334 . . . . . . 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))))))))
528526, 527raleqbidv 3354 . . . . . 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 7456 . . . . . . 7 (𝑎 = 1 → (𝐺 NeighbVtx 𝑎) = (𝐺 NeighbVtx 1))
530529raleqdv 3334 . . . . . . 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))))))))
531529, 530raleqbidv 3354 . . . . . 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 7456 . . . . . . 7 (𝑎 = 2 → (𝐺 NeighbVtx 𝑎) = (𝐺 NeighbVtx 2))
533532raleqdv 3334 . . . . . . 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))))))))
534532, 533raleqbidv 3354 . . . . . 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))))))))
535523, 524, 525, 528, 531, 534raltp 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))))))))
536347, 512, 522, 535mpbir3an 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))))))
5371, 2, 3usgrexmpl2nb3 47849 . . . . . . . . 9 (𝐺 NeighbVtx 3) = {0, 2, 4}
538537eleq2i 2836 . . . . . . . 8 (𝑏 ∈ (𝐺 NeighbVtx 3) ↔ 𝑏 ∈ {0, 2, 4})
5396eltp 4712 . . . . . . . 8 (𝑏 ∈ {0, 2, 4} ↔ (𝑏 = 0 ∨ 𝑏 = 2 ∨ 𝑏 = 4))
540538, 539bitri 275 . . . . . . 7 (𝑏 ∈ (𝐺 NeighbVtx 3) ↔ (𝑏 = 0 ∨ 𝑏 = 2 ∨ 𝑏 = 4))
541537eleq2i 2836 . . . . . . . 8 (𝑐 ∈ (𝐺 NeighbVtx 3) ↔ 𝑐 ∈ {0, 2, 4})
54210eltp 4712 . . . . . . . 8 (𝑐 ∈ {0, 2, 4} ↔ (𝑐 = 0 ∨ 𝑐 = 2 ∨ 𝑐 = 4))
543541, 542bitri 275 . . . . . . 7 (𝑐 ∈ (𝐺 NeighbVtx 3) ↔ (𝑐 = 0 ∨ 𝑐 = 2 ∨ 𝑐 = 4))
544330necon2i 2981 . . . . . . . . . . . . . 14 (𝑐 = 4 → 𝑐 ≠ 3)
545544adantl 481 . . . . . . . . . . . . 13 ((𝑏 = 0 ∧ 𝑐 = 4) → 𝑐 ≠ 3)
546545neneqd 2951 . . . . . . . . . . . 12 ((𝑏 = 0 ∧ 𝑐 = 4) → ¬ 𝑐 = 3)
547546olcd 873 . . . . . . . . . . 11 ((𝑏 = 0 ∧ 𝑐 = 4) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3))
548436adantr 480 . . . . . . . . . . . . 13 ((𝑏 = 0 ∧ 𝑐 = 4) → 𝑏 ≠ 3)
549548neneqd 2951 . . . . . . . . . . . 12 ((𝑏 = 0 ∧ 𝑐 = 4) → ¬ 𝑏 = 3)
550549orcd 872 . . . . . . . . . . 11 ((𝑏 = 0 ∧ 𝑐 = 4) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0))
551167necon2i 2981 . . . . . . . . . . . . . . . . 17 (𝑐 = 4 → 𝑐 ≠ 1)
552551adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 0 ∧ 𝑐 = 4) → 𝑐 ≠ 1)
553552neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 0 ∧ 𝑐 = 4) → ¬ 𝑐 = 1)
554553olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 0 ∧ 𝑐 = 4) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1))
555426adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 0 ∧ 𝑐 = 4) → 𝑏 ≠ 1)
556555neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 0 ∧ 𝑐 = 4) → ¬ 𝑏 = 1)
557556orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 0 ∧ 𝑐 = 4) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0))
558554, 557jca 511 . . . . . . . . . . . . 13 ((𝑏 = 0 ∧ 𝑐 = 4) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)))
559556orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 0 ∧ 𝑐 = 4) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2))
560430adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 0 ∧ 𝑐 = 4) → 𝑏 ≠ 2)
561560neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 0 ∧ 𝑐 = 4) → ¬ 𝑏 = 2)
562561orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 0 ∧ 𝑐 = 4) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1))
563559, 562jca 511 . . . . . . . . . . . . 13 ((𝑏 = 0 ∧ 𝑐 = 4) → ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)))
564546olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 0 ∧ 𝑐 = 4) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3))
565549orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 0 ∧ 𝑐 = 4) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))
566564, 565jca 511 . . . . . . . . . . . . 13 ((𝑏 = 0 ∧ 𝑐 = 4) → ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
567558, 563, 5663jca 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))))
568549orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 0 ∧ 𝑐 = 4) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4))
569546olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 0 ∧ 𝑐 = 4) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3))
570568, 569jca 511 . . . . . . . . . . . . 13 ((𝑏 = 0 ∧ 𝑐 = 4) → ((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)))
571446adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 0 ∧ 𝑐 = 4) → 𝑏 ≠ 4)
572571neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 0 ∧ 𝑐 = 4) → ¬ 𝑏 = 4)
573572orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 0 ∧ 𝑐 = 4) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5))
574450adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 0 ∧ 𝑐 = 4) → 𝑏 ≠ 5)
575574neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 0 ∧ 𝑐 = 4) → ¬ 𝑏 = 5)
576575orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 0 ∧ 𝑐 = 4) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4))
577573, 576jca 511 . . . . . . . . . . . . 13 ((𝑏 = 0 ∧ 𝑐 = 4) → ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)))
578221necon2i 2981 . . . . . . . . . . . . . . . . 17 (𝑐 = 4 → 𝑐 ≠ 5)
579578adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 0 ∧ 𝑐 = 4) → 𝑐 ≠ 5)
580579neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 0 ∧ 𝑐 = 4) → ¬ 𝑐 = 5)
581580olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 0 ∧ 𝑐 = 4) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5))
582396necon2i 2981 . . . . . . . . . . . . . . . . 17 (𝑐 = 4 → 𝑐 ≠ 0)
583582adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 0 ∧ 𝑐 = 4) → 𝑐 ≠ 0)
584583neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 0 ∧ 𝑐 = 4) → ¬ 𝑐 = 0)
585584olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 0 ∧ 𝑐 = 4) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))
586581, 585jca 511 . . . . . . . . . . . . 13 ((𝑏 = 0 ∧ 𝑐 = 4) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
587570, 577, 5863jca 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))))
588567, 587jca 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)))))
589547, 550, 588jca31 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))))))
590589olcd 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)))))))
591356, 464, 5903jaodan 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)))))))
592359adantr 480 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 4) → 𝑏 ≠ 0)
593592neneqd 2951 . . . . . . . . . . . 12 ((𝑏 = 2 ∧ 𝑐 = 4) → ¬ 𝑏 = 0)
594593orcd 872 . . . . . . . . . . 11 ((𝑏 = 2 ∧ 𝑐 = 4) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3))
595582adantl 481 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 4) → 𝑐 ≠ 0)
596595neneqd 2951 . . . . . . . . . . . 12 ((𝑏 = 2 ∧ 𝑐 = 4) → ¬ 𝑐 = 0)
597596olcd 873 . . . . . . . . . . 11 ((𝑏 = 2 ∧ 𝑐 = 4) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0))
598593orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 2 ∧ 𝑐 = 4) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1))
599596olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 2 ∧ 𝑐 = 4) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0))
600598, 599jca 511 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 4) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)))
601368adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 2 ∧ 𝑐 = 4) → 𝑏 ≠ 1)
602601neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 2 ∧ 𝑐 = 4) → ¬ 𝑏 = 1)
603602orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 2 ∧ 𝑐 = 4) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2))
604551adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 2 ∧ 𝑐 = 4) → 𝑐 ≠ 1)
605604neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 2 ∧ 𝑐 = 4) → ¬ 𝑐 = 1)
606605olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 2 ∧ 𝑐 = 4) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1))
607603, 606jca 511 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 4) → ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)))
608544adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 2 ∧ 𝑐 = 4) → 𝑐 ≠ 3)
609608neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 2 ∧ 𝑐 = 4) → ¬ 𝑐 = 3)
610609olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 2 ∧ 𝑐 = 4) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3))
611363adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 2 ∧ 𝑐 = 4) → 𝑏 ≠ 3)
612611neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 2 ∧ 𝑐 = 4) → ¬ 𝑏 = 3)
613612orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 2 ∧ 𝑐 = 4) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))
614610, 613jca 511 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 4) → ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
615600, 607, 6143jca 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))))
616612orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 2 ∧ 𝑐 = 4) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4))
617609olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 2 ∧ 𝑐 = 4) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3))
618616, 617jca 511 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 4) → ((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)))
619493adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 2 ∧ 𝑐 = 4) → 𝑏 ≠ 4)
620619neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 2 ∧ 𝑐 = 4) → ¬ 𝑏 = 4)
621620orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 2 ∧ 𝑐 = 4) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5))
622402adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 2 ∧ 𝑐 = 4) → 𝑏 ≠ 5)
623622neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 2 ∧ 𝑐 = 4) → ¬ 𝑏 = 5)
624623orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 2 ∧ 𝑐 = 4) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4))
625621, 624jca 511 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 4) → ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)))
626593orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 2 ∧ 𝑐 = 4) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5))
627596olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 2 ∧ 𝑐 = 4) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))
628626, 627jca 511 . . . . . . . . . . . . 13 ((𝑏 = 2 ∧ 𝑐 = 4) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
629618, 625, 6283jca 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))))
630615, 629jca 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)))))
631594, 597, 630jca31 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))))))
632631olcd 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)))))))
633410, 509, 6323jaodan 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)))))))
634446necon2i 2981 . . . . . . . . . . . . . 14 (𝑏 = 4 → 𝑏 ≠ 0)
635634adantr 480 . . . . . . . . . . . . 13 ((𝑏 = 4 ∧ 𝑐 = 0) → 𝑏 ≠ 0)
636635neneqd 2951 . . . . . . . . . . . 12 ((𝑏 = 4 ∧ 𝑐 = 0) → ¬ 𝑏 = 0)
637636orcd 872 . . . . . . . . . . 11 ((𝑏 = 4 ∧ 𝑐 = 0) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3))
638229necon2i 2981 . . . . . . . . . . . . . 14 (𝑏 = 4 → 𝑏 ≠ 3)
639638adantr 480 . . . . . . . . . . . . 13 ((𝑏 = 4 ∧ 𝑐 = 0) → 𝑏 ≠ 3)
640639neneqd 2951 . . . . . . . . . . . 12 ((𝑏 = 4 ∧ 𝑐 = 0) → ¬ 𝑏 = 3)
641640orcd 872 . . . . . . . . . . 11 ((𝑏 = 4 ∧ 𝑐 = 0) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0))
642636orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 4 ∧ 𝑐 = 0) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1))
64365necon2i 2981 . . . . . . . . . . . . . . . . 17 (𝑏 = 4 → 𝑏 ≠ 1)
644643adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 4 ∧ 𝑐 = 0) → 𝑏 ≠ 1)
645644neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 4 ∧ 𝑐 = 0) → ¬ 𝑏 = 1)
646645orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 4 ∧ 𝑐 = 0) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0))
647642, 646jca 511 . . . . . . . . . . . . 13 ((𝑏 = 4 ∧ 𝑐 = 0) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)))
648416necon2i 2981 . . . . . . . . . . . . . . . . 17 (𝑐 = 0 → 𝑐 ≠ 2)
649648adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 4 ∧ 𝑐 = 0) → 𝑐 ≠ 2)
650649neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 4 ∧ 𝑐 = 0) → ¬ 𝑐 = 2)
651650olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 4 ∧ 𝑐 = 0) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2))
652374adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 4 ∧ 𝑐 = 0) → 𝑐 ≠ 1)
653652neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 4 ∧ 𝑐 = 0) → ¬ 𝑐 = 1)
654653olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 4 ∧ 𝑐 = 0) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1))
655651, 654jca 511 . . . . . . . . . . . . 13 ((𝑏 = 4 ∧ 𝑐 = 0) → ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)))
656379adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 4 ∧ 𝑐 = 0) → 𝑐 ≠ 3)
657656neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 4 ∧ 𝑐 = 0) → ¬ 𝑐 = 3)
658657olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 4 ∧ 𝑐 = 0) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3))
659640orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 4 ∧ 𝑐 = 0) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))
660658, 659jca 511 . . . . . . . . . . . . 13 ((𝑏 = 4 ∧ 𝑐 = 0) → ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
661647, 655, 6603jca 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))))
662640orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 4 ∧ 𝑐 = 0) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4))
663657olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 4 ∧ 𝑐 = 0) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3))
664662, 663jca 511 . . . . . . . . . . . . 13 ((𝑏 = 4 ∧ 𝑐 = 0) → ((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)))
665389adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 4 ∧ 𝑐 = 0) → 𝑐 ≠ 5)
666665neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 4 ∧ 𝑐 = 0) → ¬ 𝑐 = 5)
667666olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 4 ∧ 𝑐 = 0) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5))
668396adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 4 ∧ 𝑐 = 0) → 𝑐 ≠ 4)
669668neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 4 ∧ 𝑐 = 0) → ¬ 𝑐 = 4)
670669olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 4 ∧ 𝑐 = 0) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4))
671667, 670jca 511 . . . . . . . . . . . . 13 ((𝑏 = 4 ∧ 𝑐 = 0) → ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)))
672636orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 4 ∧ 𝑐 = 0) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5))
673323necon2i 2981 . . . . . . . . . . . . . . . . 17 (𝑏 = 4 → 𝑏 ≠ 5)
674673adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 4 ∧ 𝑐 = 0) → 𝑏 ≠ 5)
675674neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 4 ∧ 𝑐 = 0) → ¬ 𝑏 = 5)
676675orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 4 ∧ 𝑐 = 0) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))
677672, 676jca 511 . . . . . . . . . . . . 13 ((𝑏 = 4 ∧ 𝑐 = 0) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
678664, 671, 6773jca 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))))
679661, 678jca 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)))))
680637, 641, 679jca31 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))))))
681680olcd 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)))))))
682634adantr 480 . . . . . . . . . . . . 13 ((𝑏 = 4 ∧ 𝑐 = 2) → 𝑏 ≠ 0)
683682neneqd 2951 . . . . . . . . . . . 12 ((𝑏 = 4 ∧ 𝑐 = 2) → ¬ 𝑏 = 0)
684683orcd 872 . . . . . . . . . . 11 ((𝑏 = 4 ∧ 𝑐 = 2) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3))
685416adantl 481 . . . . . . . . . . . . 13 ((𝑏 = 4 ∧ 𝑐 = 2) → 𝑐 ≠ 0)
686685neneqd 2951 . . . . . . . . . . . 12 ((𝑏 = 4 ∧ 𝑐 = 2) → ¬ 𝑐 = 0)
687686olcd 873 . . . . . . . . . . 11 ((𝑏 = 4 ∧ 𝑐 = 2) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0))
688683orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 4 ∧ 𝑐 = 2) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1))
689686olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 4 ∧ 𝑐 = 2) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0))
690688, 689jca 511 . . . . . . . . . . . . 13 ((𝑏 = 4 ∧ 𝑐 = 2) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)))
691643adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 4 ∧ 𝑐 = 2) → 𝑏 ≠ 1)
692691neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 4 ∧ 𝑐 = 2) → ¬ 𝑏 = 1)
693692orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 4 ∧ 𝑐 = 2) → (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2))
694420adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 4 ∧ 𝑐 = 2) → 𝑐 ≠ 1)
695694neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 4 ∧ 𝑐 = 2) → ¬ 𝑐 = 1)
696695olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 4 ∧ 𝑐 = 2) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1))
697693, 696jca 511 . . . . . . . . . . . . 13 ((𝑏 = 4 ∧ 𝑐 = 2) → ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)))
698493necon2i 2981 . . . . . . . . . . . . . . . . 17 (𝑏 = 4 → 𝑏 ≠ 2)
699698adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 4 ∧ 𝑐 = 2) → 𝑏 ≠ 2)
700699neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 4 ∧ 𝑐 = 2) → ¬ 𝑏 = 2)
701700orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 4 ∧ 𝑐 = 2) → (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3))
702638adantr 480 . . . . . . . . . . . . . . . 16 ((𝑏 = 4 ∧ 𝑐 = 2) → 𝑏 ≠ 3)
703702neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 4 ∧ 𝑐 = 2) → ¬ 𝑏 = 3)
704703orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 4 ∧ 𝑐 = 2) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2))
705701, 704jca 511 . . . . . . . . . . . . 13 ((𝑏 = 4 ∧ 𝑐 = 2) → ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
706690, 697, 7053jca 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))))
707703orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 4 ∧ 𝑐 = 2) → (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4))
708411adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 4 ∧ 𝑐 = 2) → 𝑐 ≠ 3)
709708neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 4 ∧ 𝑐 = 2) → ¬ 𝑐 = 3)
710709olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 4 ∧ 𝑐 = 2) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3))
711707, 710jca 511 . . . . . . . . . . . . 13 ((𝑏 = 4 ∧ 𝑐 = 2) → ((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)))
712455adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 4 ∧ 𝑐 = 2) → 𝑐 ≠ 5)
713712neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 4 ∧ 𝑐 = 2) → ¬ 𝑐 = 5)
714713olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 4 ∧ 𝑐 = 2) → (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5))
715 neeq1 3009 . . . . . . . . . . . . . . . . . 18 (𝑐 = 2 → (𝑐 ≠ 4 ↔ 2 ≠ 4))
716491, 715mpbiri 258 . . . . . . . . . . . . . . . . 17 (𝑐 = 2 → 𝑐 ≠ 4)
717716adantl 481 . . . . . . . . . . . . . . . 16 ((𝑏 = 4 ∧ 𝑐 = 2) → 𝑐 ≠ 4)
718717neneqd 2951 . . . . . . . . . . . . . . 15 ((𝑏 = 4 ∧ 𝑐 = 2) → ¬ 𝑐 = 4)
719718olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 4 ∧ 𝑐 = 2) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4))
720714, 719jca 511 . . . . . . . . . . . . 13 ((𝑏 = 4 ∧ 𝑐 = 2) → ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)))
721683orcd 872 . . . . . . . . . . . . . 14 ((𝑏 = 4 ∧ 𝑐 = 2) → (¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5))
722686olcd 873 . . . . . . . . . . . . . 14 ((𝑏 = 4 ∧ 𝑐 = 2) → (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0))
723721, 722jca 511 . . . . . . . . . . . . 13 ((𝑏 = 4 ∧ 𝑐 = 2) → ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
724711, 720, 7233jca 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))))
725706, 724jca 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)))))
726684, 687, 725jca31 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))))))
727726olcd 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 2766 . . . . . . . . . 10 ((𝑏 = 4 ∧ 𝑐 = 4) → 𝑏 = 𝑐)
729728orcd 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)))))))
730681, 727, 7293jaodan 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)))))))
731591, 633, 7303jaoian 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)))))))
732540, 543, 731syl2anb 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)))))))
733732rgen2 3205 . . . . 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))))))
7341, 2, 3usgrexmpl2nb4 47850 . . . . . . . . 9 (𝐺 NeighbVtx 4) = {3, 5}
735734eleq2i 2836 . . . . . . . 8 (𝑏 ∈ (𝐺 NeighbVtx 4) ↔ 𝑏 ∈ {3, 5})
7366elpr 4672 . . . . . . . 8 (𝑏 ∈ {3, 5} ↔ (𝑏 = 3 ∨ 𝑏 = 5))
737735, 736bitri 275 . . . . . . 7 (𝑏 ∈ (𝐺 NeighbVtx 4) ↔ (𝑏 = 3 ∨ 𝑏 = 5))
738734eleq2i 2836 . . . . . . . 8 (𝑐 ∈ (𝐺 NeighbVtx 4) ↔ 𝑐 ∈ {3, 5})
73910elpr 4672 . . . . . . . 8 (𝑐 ∈ {3, 5} ↔ (𝑐 = 3 ∨ 𝑐 = 5))
740738, 739bitri 275 . . . . . . 7 (𝑐 ∈ (𝐺 NeighbVtx 4) ↔ (𝑐 = 3 ∨ 𝑐 = 5))
741191, 341, 243, 343ccase 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)))))))
742737, 740, 741syl2anb 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)))))))
743742rgen2 3205 . . . . 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))))))
7441, 2, 3usgrexmpl2nb5 47851 . . . . . . . . 9 (𝐺 NeighbVtx 5) = {0, 4}
745744eleq2i 2836 . . . . . . . 8 (𝑏 ∈ (𝐺 NeighbVtx 5) ↔ 𝑏 ∈ {0, 4})
7466elpr 4672 . . . . . . . 8 (𝑏 ∈ {0, 4} ↔ (𝑏 = 0 ∨ 𝑏 = 4))
747745, 746bitri 275 . . . . . . 7 (𝑏 ∈ (𝐺 NeighbVtx 5) ↔ (𝑏 = 0 ∨ 𝑏 = 4))
748744eleq2i 2836 . . . . . . . 8 (𝑐 ∈ (𝐺 NeighbVtx 5) ↔ 𝑐 ∈ {0, 4})
74910elpr 4672 . . . . . . . 8 (𝑐 ∈ {0, 4} ↔ (𝑐 = 0 ∨ 𝑐 = 4))
750748, 749bitri 275 . . . . . . 7 (𝑐 ∈ (𝐺 NeighbVtx 5) ↔ (𝑐 = 0 ∨ 𝑐 = 4))
751356, 681, 590, 729ccase 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)))))))
752747, 750, 751syl2anb 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)))))))
753752rgen2 3205 . . . . 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 12375 . . . . . 6 3 ∈ V
755 4nn0 12572 . . . . . . 7 4 ∈ ℕ0
756755elexi 3511 . . . . . 6 4 ∈ V
757 5nn0 12573 . . . . . . 7 5 ∈ ℕ0
758757elexi 3511 . . . . . 6 5 ∈ V
759 oveq2 7456 . . . . . . 7 (𝑎 = 3 → (𝐺 NeighbVtx 𝑎) = (𝐺 NeighbVtx 3))
760759raleqdv 3334 . . . . . . 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))))))))
761759, 760raleqbidv 3354 . . . . . 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 7456 . . . . . . 7 (𝑎 = 4 → (𝐺 NeighbVtx 𝑎) = (𝐺 NeighbVtx 4))
763762raleqdv 3334 . . . . . . 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))))))))
764762, 763raleqbidv 3354 . . . . . 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 7456 . . . . . . 7 (𝑎 = 5 → (𝐺 NeighbVtx 𝑎) = (𝐺 NeighbVtx 5))
766765raleqdv 3334 . . . . . . 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))))))))
767765, 766raleqbidv 3354 . . . . . 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))))))))
768754, 756, 758, 761, 764, 767raltp 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))))))))
769733, 743, 753, 768mpbir3an 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 4220 . . . 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))))))))
771536, 769, 770mpbir2an 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 (¬ (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))) ↔ (¬ 𝑏𝑐 ∨ ¬ {𝑏, 𝑐} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))))
773 nne 2950 . . . . . . 7 𝑏𝑐𝑏 = 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))
778776, 777anbi12i 627 . . . . . . . . . . . 12 ((¬ (𝑏 = 0 ∧ 𝑐 = 3) ∧ ¬ (𝑏 = 3 ∧ 𝑐 = 0)) ↔ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 0)))
779775, 778bitri 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))
785783, 784anbi12i 627 . . . . . . . . . . . . . . . 16 ((¬ (𝑏 = 0 ∧ 𝑐 = 1) ∧ ¬ (𝑏 = 1 ∧ 𝑐 = 0)) ↔ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 1) ∧ (¬ 𝑏 = 1 ∨ ¬ 𝑐 = 0)))
786782, 785bitri 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))
790788, 789anbi12i 627 . . . . . . . . . . . . . . . 16 ((¬ (𝑏 = 1 ∧ 𝑐 = 2) ∧ ¬ (𝑏 = 2 ∧ 𝑐 = 1)) ↔ ((¬ 𝑏 = 1 ∨ ¬ 𝑐 = 2) ∧ (¬ 𝑏 = 2 ∨ ¬ 𝑐 = 1)))
791787, 790bitri 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))
795793, 794anbi12i 627 . . . . . . . . . . . . . . . 16 ((¬ (𝑏 = 2 ∧ 𝑐 = 3) ∧ ¬ (𝑏 = 3 ∧ 𝑐 = 2)) ↔ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
796792, 795bitri 275 . . . . . . . . . . . . . . 15 (¬ ((𝑏 = 2 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 2)) ↔ ((¬ 𝑏 = 2 ∨ ¬ 𝑐 = 3) ∧ (¬ 𝑏 = 3 ∨ ¬ 𝑐 = 2)))
797786, 791, 7963anbi123i 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))))
798781, 797bitri 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))
803801, 802anbi12i 627 . . . . . . . . . . . . . . . 16 ((¬ (𝑏 = 3 ∧ 𝑐 = 4) ∧ ¬ (𝑏 = 4 ∧ 𝑐 = 3)) ↔ ((¬ 𝑏 = 3 ∨ ¬ 𝑐 = 4) ∧ (¬ 𝑏 = 4 ∨ ¬ 𝑐 = 3)))
804800, 803bitri 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))
808806, 807anbi12i 627 . . . . . . . . . . . . . . . 16 ((¬ (𝑏 = 4 ∧ 𝑐 = 5) ∧ ¬ (𝑏 = 5 ∧ 𝑐 = 4)) ↔ ((¬ 𝑏 = 4 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 4)))
809805, 808bitri 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))
813811, 812anbi12i 627 . . . . . . . . . . . . . . . 16 ((¬ (𝑏 = 0 ∧ 𝑐 = 5) ∧ ¬ (𝑏 = 5 ∧ 𝑐 = 0)) ↔ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
814810, 813bitri 275 . . . . . . . . . . . . . . 15 (¬ ((𝑏 = 0 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 0)) ↔ ((¬ 𝑏 = 0 ∨ ¬ 𝑐 = 5) ∧ (¬ 𝑏 = 5 ∨ ¬ 𝑐 = 0)))
815804, 809, 8143anbi123i 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))))
816799, 815bitri 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))))
817798, 816anbi12i 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)))))
818780, 817bitri 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)))))
819779, 818anbi12i 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))))))
820774, 819bitri 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))))))
8216, 10, 523, 524preq12b 4875 . . . . . . . . . . . 12 ({𝑏, 𝑐} = {0, 1} ↔ ((𝑏 = 0 ∧ 𝑐 = 1) ∨ (𝑏 = 1 ∧ 𝑐 = 0)))
8226, 10, 524, 525preq12b 4875 . . . . . . . . . . . 12 ({𝑏, 𝑐} = {1, 2} ↔ ((𝑏 = 1 ∧ 𝑐 = 2) ∨ (𝑏 = 2 ∧ 𝑐 = 1)))
8236, 10, 525, 754preq12b 4875 . . . . . . . . . . . 12 ({𝑏, 𝑐} = {2, 3} ↔ ((𝑏 = 2 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 2)))
824821, 822, 8233orbi123i 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))))
8256, 10, 754, 756preq12b 4875 . . . . . . . . . . . 12 ({𝑏, 𝑐} = {3, 4} ↔ ((𝑏 = 3 ∧ 𝑐 = 4) ∨ (𝑏 = 4 ∧ 𝑐 = 3)))
8266, 10, 756, 758preq12b 4875 . . . . . . . . . . . 12 ({𝑏, 𝑐} = {4, 5} ↔ ((𝑏 = 4 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 4)))
8276, 10, 523, 758preq12b 4875 . . . . . . . . . . . 12 ({𝑏, 𝑐} = {0, 5} ↔ ((𝑏 = 0 ∧ 𝑐 = 5) ∨ (𝑏 = 5 ∧ 𝑐 = 0)))
828825, 826, 8273orbi123i 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))))
829824, 828orbi12i 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)))))
830829orbi2i 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))))))
831820, 830xchnxbir 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 4176 . . . . . . . . 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 5452 . . . . . . . . . . . 12 {𝑏, 𝑐} ∈ V
834833elsn 4663 . . . . . . . . . . 11 ({𝑏, 𝑐} ∈ {{0, 3}} ↔ {𝑏, 𝑐} = {0, 3})
8356, 10, 523, 754preq12b 4875 . . . . . . . . . . 11 ({𝑏, 𝑐} = {0, 3} ↔ ((𝑏 = 0 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 0)))
836834, 835bitri 275 . . . . . . . . . 10 ({𝑏, 𝑐} ∈ {{0, 3}} ↔ ((𝑏 = 0 ∧ 𝑐 = 3) ∨ (𝑏 = 3 ∧ 𝑐 = 0)))
837 elun 4176 . . . . . . . . . . 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}}))
838833eltp 4712 . . . . . . . . . . . 12 ({𝑏, 𝑐} ∈ {{0, 1}, {1, 2}, {2, 3}} ↔ ({𝑏, 𝑐} = {0, 1} ∨ {𝑏, 𝑐} = {1, 2} ∨ {𝑏, 𝑐} = {2, 3}))
839833eltp 4712 . . . . . . . . . . . 12 ({𝑏, 𝑐} ∈ {{3, 4}, {4, 5}, {0, 5}} ↔ ({𝑏, 𝑐} = {3, 4} ∨ {𝑏, 𝑐} = {4, 5} ∨ {𝑏, 𝑐} = {0, 5}))
840838, 839orbi12i 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})))
841837, 840bitri 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})))
842836, 841orbi12i 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}))))
843832, 842bitri 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}))))
844831, 843xchnxbir 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))))))
845773, 844orbi12i 913 . . . . . 6 ((¬ 𝑏𝑐 ∨ ¬ {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)))))))
846772, 845bitr2i 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)))))) ↔ ¬ (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))))
8478463ralbii 3136 . . . 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) ¬ (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))))
848 ralnex3 3140 . . . 4 (∀𝑎 ∈ ({0, 1, 2} ∪ {3, 4, 5})∀𝑏 ∈ (𝐺 NeighbVtx 𝑎)∀𝑐 ∈ (𝐺 NeighbVtx 𝑎) ¬ (𝑏𝑐 ∧ {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)(𝑏𝑐 ∧ {𝑏, 𝑐} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))))
849847, 848bitri 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)(𝑏𝑐 ∧ {𝑏, 𝑐} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))))
850771, 849mpbi 230 . 2 ¬ ∃𝑎 ∈ ({0, 1, 2} ∪ {3, 4, 5})∃𝑏 ∈ (𝐺 NeighbVtx 𝑎)∃𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏𝑐 ∧ {𝑏, 𝑐} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}})))
8511, 2, 3usgrexmpl2 47842 . . 3 𝐺 ∈ USGraph
8521, 2, 3usgrexmpl2vtx 47843 . . . . 5 (Vtx‘𝐺) = ({0, 1, 2} ∪ {3, 4, 5})
853852eqcomi 2749 . . . 4 ({0, 1, 2} ∪ {3, 4, 5}) = (Vtx‘𝐺)
8541, 2, 3usgrexmpl2edg 47844 . . . . 5 (Edg‘𝐺) = ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))
855854eqcomi 2749 . . . 4 ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}})) = (Edg‘𝐺)
856 eqid 2740 . . . 4 (𝐺 NeighbVtx 𝑎) = (𝐺 NeighbVtx 𝑎)
857853, 855, 856usgrgrtrirex 47799 . . 3 (𝐺 ∈ USGraph → (∃𝑡 𝑡 ∈ (GrTriangles‘𝐺) ↔ ∃𝑎 ∈ ({0, 1, 2} ∪ {3, 4, 5})∃𝑏 ∈ (𝐺 NeighbVtx 𝑎)∃𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏𝑐 ∧ {𝑏, 𝑐} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}})))))
858851, 857ax-mp 5 . 2 (∃𝑡 𝑡 ∈ (GrTriangles‘𝐺) ↔ ∃𝑎 ∈ ({0, 1, 2} ∪ {3, 4, 5})∃𝑏 ∈ (𝐺 NeighbVtx 𝑎)∃𝑐 ∈ (𝐺 NeighbVtx 𝑎)(𝑏𝑐 ∧ {𝑏, 𝑐} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))))
859850, 858mtbir 323 1 ¬ ∃𝑡 𝑡 ∈ (GrTriangles‘𝐺)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395  wo 846  w3o 1086  w3a 1087   = wceq 1537  wex 1777  wcel 2108  wne 2946  wral 3067  wrex 3076  cun 3974  {csn 4648  {cpr 4650  {ctp 4652  cop 4654  cfv 6573  (class class class)co 7448  0cc0 11184  1c1 11185  2c2 12348  3c3 12349  4c4 12350  5c5 12351  0cn0 12553  ...cfz 13567  ⟨“cs7 14895  Vtxcvtx 29031  Edgcedg 29082  USGraphcusgr 29184   NeighbVtx cnbgr 29367  GrTrianglescgrtri 47788
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 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261
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 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-3o 8524  df-oadd 8526  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-dju 9970  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-n0 12554  df-xnn0 12626  df-z 12640  df-uz 12904  df-fz 13568  df-fzo 13712  df-hash 14380  df-word 14563  df-concat 14619  df-s1 14644  df-s2 14897  df-s3 14898  df-s4 14899  df-s5 14900  df-s6 14901  df-s7 14902  df-vtx 29033  df-iedg 29034  df-edg 29083  df-uhgr 29093  df-upgr 29117  df-umgr 29118  df-uspgr 29185  df-usgr 29186  df-nbgr 29368  df-grtri 47789
This theorem is referenced by:  usgrexmpl12ngric  47853  usgrexmpl12ngrlic  47854
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