pgnbgreunbgrlem5 - Mathbox for Alexander van der Vekens

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Theorem pgnbgreunbgrlem5 48921

Description: Lemma 5 for pgnbgreunbgr 48923. Impossible cases. (Contributed by AV, 21-Nov-2025.)

**Hypotheses**
Ref	Expression
pgnbgreunbgr.g	⊢ 𝐺 = (5 gPetersenGr 2)
pgnbgreunbgr.v	⊢ 𝑉 = (Vtx‘𝐺)
pgnbgreunbgr.e	⊢ 𝐸 = (Edg‘𝐺)
pgnbgreunbgr.n	⊢ 𝑁 = (𝐺 NeighbVtx 𝑋)

**Assertion**
Ref	Expression
pgnbgreunbgrlem5	⊢ ((𝐿 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, (2^nd ‘𝑋)⟩ ∨ 𝐿 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩) → ((𝐾 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐾 = ⟨1, (2^nd ‘𝑋)⟩ ∨ 𝐾 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩) → ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))))

Distinct variable groups: 𝑦,𝑏 𝑦,𝐸 𝑦,𝐾 𝑦,𝐿 𝑦,𝑁 𝑦,𝑉 𝑦,𝑋 Allowed substitution hints: 𝐸(𝑏) 𝐺(𝑦, 𝑏) 𝐾(𝑏) 𝐿(𝑏) 𝑁(𝑏) 𝑉(𝑏) 𝑋(𝑏)

**Proof of Theorem pgnbgreunbgrlem5**
Step	Hyp	Ref	Expression
1		c0ex 11211	. . . . . 6 ⊢ 0 ∈ V
2		vex 3461	. . . . . 6 ⊢ 𝑦 ∈ V
3	1, 2	op2ndd 7999	. . . . 5 ⊢ (𝑋 = ⟨0, 𝑦⟩ → (2^nd ‘𝑋) = 𝑦)
4	3	adantr 486	. . . 4 ⊢ ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → (2^nd ‘𝑋) = 𝑦)
5		oveq1 7423	. . . . . . . . 9 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((2^nd ‘𝑋) + 1) = (𝑦 + 1))
6	5	oveq1d 7431	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (((2^nd ‘𝑋) + 1) mod 5) = ((𝑦 + 1) mod 5))
7	6	opeq2d 4847	. . . . . . 7 ⊢ ((2^nd ‘𝑋) = 𝑦 → ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ = ⟨0, ((𝑦 + 1) mod 5)⟩)
8	7	eqeq2d 2776	. . . . . 6 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐿 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ↔ 𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩))
9		opeq2 4841	. . . . . . 7 ⊢ ((2^nd ‘𝑋) = 𝑦 → ⟨1, (2^nd ‘𝑋)⟩ = ⟨1, 𝑦⟩)
10	9	eqeq2d 2776	. . . . . 6 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐿 = ⟨1, (2^nd ‘𝑋)⟩ ↔ 𝐿 = ⟨1, 𝑦⟩))
11		oveq1 7423	. . . . . . . . 9 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((2^nd ‘𝑋) − 1) = (𝑦 − 1))
12	11	oveq1d 7431	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (((2^nd ‘𝑋) − 1) mod 5) = ((𝑦 − 1) mod 5))
13	12	opeq2d 4847	. . . . . . 7 ⊢ ((2^nd ‘𝑋) = 𝑦 → ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩ = ⟨0, ((𝑦 − 1) mod 5)⟩)
14	13	eqeq2d 2776	. . . . . 6 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐿 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩ ↔ 𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩))
15	8, 10, 14	3orbi123d 1463	. . . . 5 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((𝐿 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, (2^nd ‘𝑋)⟩ ∨ 𝐿 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩) ↔ (𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, 𝑦⟩ ∨ 𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩)))
16	7	eqeq2d 2776	. . . . . 6 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐾 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ↔ 𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩))
17	9	eqeq2d 2776	. . . . . 6 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐾 = ⟨1, (2^nd ‘𝑋)⟩ ↔ 𝐾 = ⟨1, 𝑦⟩))
18	13	eqeq2d 2776	. . . . . 6 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐾 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩ ↔ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩))
19	16, 17, 18	3orbi123d 1463	. . . . 5 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((𝐾 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐾 = ⟨1, (2^nd ‘𝑋)⟩ ∨ 𝐾 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩) ↔ (𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∨ 𝐾 = ⟨1, 𝑦⟩ ∨ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩)))
20	15, 19	anbi12d 644	. . . 4 ⊢ ((2^nd ‘𝑋) = 𝑦 → (((𝐿 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, (2^nd ‘𝑋)⟩ ∨ 𝐿 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩) ∧ (𝐾 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐾 = ⟨1, (2^nd ‘𝑋)⟩ ∨ 𝐾 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩)) ↔ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, 𝑦⟩ ∨ 𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩) ∧ (𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∨ 𝐾 = ⟨1, 𝑦⟩ ∨ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩))))
21	4, 20	syl 18	. . 3 ⊢ ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → (((𝐿 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, (2^nd ‘𝑋)⟩ ∨ 𝐿 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩) ∧ (𝐾 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐾 = ⟨1, (2^nd ‘𝑋)⟩ ∨ 𝐾 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩)) ↔ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, 𝑦⟩ ∨ 𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩) ∧ (𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∨ 𝐾 = ⟨1, 𝑦⟩ ∨ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩))))
22		simpl 488	. . . . . . . . . . 11 ⊢ ((𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩) → 𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩)
23		simpr 490	. . . . . . . . . . 11 ⊢ ((𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩) → 𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩)
24	22, 23	neeq12d 3021	. . . . . . . . . 10 ⊢ ((𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩) → (𝐾 ≠ 𝐿 ↔ ⟨0, ((𝑦 + 1) mod 5)⟩ ≠ ⟨0, ((𝑦 + 1) mod 5)⟩))
25	24	ancoms 464	. . . . . . . . 9 ⊢ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩) → (𝐾 ≠ 𝐿 ↔ ⟨0, ((𝑦 + 1) mod 5)⟩ ≠ ⟨0, ((𝑦 + 1) mod 5)⟩))
26		eqid 2765	. . . . . . . . . 10 ⊢ ⟨0, ((𝑦 + 1) mod 5)⟩ = ⟨0, ((𝑦 + 1) mod 5)⟩
27		eqneqall 2971	. . . . . . . . . 10 ⊢ (⟨0, ((𝑦 + 1) mod 5)⟩ = ⟨0, ((𝑦 + 1) mod 5)⟩ → (⟨0, ((𝑦 + 1) mod 5)⟩ ≠ ⟨0, ((𝑦 + 1) mod 5)⟩ → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
28	26, 27	ax-mp 5	. . . . . . . . 9 ⊢ (⟨0, ((𝑦 + 1) mod 5)⟩ ≠ ⟨0, ((𝑦 + 1) mod 5)⟩ → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))
29	25, 28	biimtrdi 256	. . . . . . . 8 ⊢ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩) → (𝐾 ≠ 𝐿 → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
30	29	impd 416	. . . . . . 7 ⊢ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))
31	30	ex 418	. . . . . 6 ⊢ (𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ → (𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
32		pgnbgreunbgr.g	. . . . . . . . . . . 12 ⊢ 𝐺 = (5 gPetersenGr 2)
33		pgnbgreunbgr.v	. . . . . . . . . . . 12 ⊢ 𝑉 = (Vtx‘𝐺)
34		pgnbgreunbgr.e	. . . . . . . . . . . 12 ⊢ 𝐸 = (Edg‘𝐺)
35		pgnbgreunbgr.n	. . . . . . . . . . . 12 ⊢ 𝑁 = (𝐺 NeighbVtx 𝑋)
36	32, 33, 34, 35	pgnbgreunbgrlem5lem1 48918	. . . . . . . . . . 11 ⊢ ((((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐾, ⟨1, 𝑏⟩} ∈ 𝐸) → ¬ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸)
37	36	pm2.21d 122	. . . . . . . . . 10 ⊢ ((((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐾, ⟨1, 𝑏⟩} ∈ 𝐸) → ({⟨1, 𝑏⟩, 𝐿} ∈ 𝐸 → 𝑋 = ⟨1, 𝑏⟩))
38	37	expimpd 459	. . . . . . . . 9 ⊢ (((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))
39	38	ex 418	. . . . . . . 8 ⊢ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))
40	39	adantld 496	. . . . . . 7 ⊢ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))
41	40	ex 418	. . . . . 6 ⊢ (𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ → (𝐾 = ⟨1, 𝑦⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
42	32, 33, 34, 35	pgnbgreunbgrlem5lem3 48920	. . . . . . . . . . 11 ⊢ ((((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐾, ⟨1, 𝑏⟩} ∈ 𝐸) → ¬ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸)
43	42	pm2.21d 122	. . . . . . . . . 10 ⊢ ((((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐾, ⟨1, 𝑏⟩} ∈ 𝐸) → ({⟨1, 𝑏⟩, 𝐿} ∈ 𝐸 → 𝑋 = ⟨1, 𝑏⟩))
44	43	expimpd 459	. . . . . . . . 9 ⊢ (((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))
45	44	ex 418	. . . . . . . 8 ⊢ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))
46	45	adantld 496	. . . . . . 7 ⊢ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))
47	46	ex 418	. . . . . 6 ⊢ (𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ → (𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
48	31, 41, 47	3jaod 1456	. . . . 5 ⊢ (𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ → ((𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∨ 𝐾 = ⟨1, 𝑦⟩ ∨ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
49		prcom 4700	. . . . . . . . . . . 12 ⊢ {𝐾, ⟨1, 𝑏⟩} = {⟨1, 𝑏⟩, 𝐾}
50	49	eleq1i 2856	. . . . . . . . . . 11 ⊢ ({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ↔ {⟨1, 𝑏⟩, 𝐾} ∈ 𝐸)
51		prcom 4700	. . . . . . . . . . . 12 ⊢ {⟨1, 𝑏⟩, 𝐿} = {𝐿, ⟨1, 𝑏⟩}
52	51	eleq1i 2856	. . . . . . . . . . 11 ⊢ ({⟨1, 𝑏⟩, 𝐿} ∈ 𝐸 ↔ {𝐿, ⟨1, 𝑏⟩} ∈ 𝐸)
53	50, 52	anbi12i 640	. . . . . . . . . 10 ⊢ (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) ↔ ({⟨1, 𝑏⟩, 𝐾} ∈ 𝐸 ∧ {𝐿, ⟨1, 𝑏⟩} ∈ 𝐸))
54	32, 33, 34, 35	pgnbgreunbgrlem5lem1 48918	. . . . . . . . . . . . 13 ⊢ ((((𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐿 = ⟨1, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐿, ⟨1, 𝑏⟩} ∈ 𝐸) → ¬ {⟨1, 𝑏⟩, 𝐾} ∈ 𝐸)
55	54	pm2.21d 122	. . . . . . . . . . . 12 ⊢ ((((𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐿 = ⟨1, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐿, ⟨1, 𝑏⟩} ∈ 𝐸) → ({⟨1, 𝑏⟩, 𝐾} ∈ 𝐸 → 𝑋 = ⟨1, 𝑏⟩))
56	55	ex 418	. . . . . . . . . . 11 ⊢ (((𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐿 = ⟨1, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → ({𝐿, ⟨1, 𝑏⟩} ∈ 𝐸 → ({⟨1, 𝑏⟩, 𝐾} ∈ 𝐸 → 𝑋 = ⟨1, 𝑏⟩)))
57	56	impcomd 417	. . . . . . . . . 10 ⊢ (((𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐿 = ⟨1, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({⟨1, 𝑏⟩, 𝐾} ∈ 𝐸 ∧ {𝐿, ⟨1, 𝑏⟩} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))
58	53, 57	biimtrid 245	. . . . . . . . 9 ⊢ (((𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐿 = ⟨1, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))
59	58	ex 418	. . . . . . . 8 ⊢ ((𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐿 = ⟨1, 𝑦⟩) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))
60	59	adantld 496	. . . . . . 7 ⊢ ((𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐿 = ⟨1, 𝑦⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))
61	60	expcom 419	. . . . . 6 ⊢ (𝐿 = ⟨1, 𝑦⟩ → (𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
62		simpr 490	. . . . . . . . . 10 ⊢ ((𝐿 = ⟨1, 𝑦⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → 𝐾 = ⟨1, 𝑦⟩)
63		simpl 488	. . . . . . . . . 10 ⊢ ((𝐿 = ⟨1, 𝑦⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → 𝐿 = ⟨1, 𝑦⟩)
64	62, 63	neeq12d 3021	. . . . . . . . 9 ⊢ ((𝐿 = ⟨1, 𝑦⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → (𝐾 ≠ 𝐿 ↔ ⟨1, 𝑦⟩ ≠ ⟨1, 𝑦⟩))
65		eqid 2765	. . . . . . . . . 10 ⊢ ⟨1, 𝑦⟩ = ⟨1, 𝑦⟩
66		eqneqall 2971	. . . . . . . . . 10 ⊢ (⟨1, 𝑦⟩ = ⟨1, 𝑦⟩ → (⟨1, 𝑦⟩ ≠ ⟨1, 𝑦⟩ → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
67	65, 66	ax-mp 5	. . . . . . . . 9 ⊢ (⟨1, 𝑦⟩ ≠ ⟨1, 𝑦⟩ → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))
68	64, 67	biimtrdi 256	. . . . . . . 8 ⊢ ((𝐿 = ⟨1, 𝑦⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → (𝐾 ≠ 𝐿 → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
69	68	impd 416	. . . . . . 7 ⊢ ((𝐿 = ⟨1, 𝑦⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))
70	69	ex 418	. . . . . 6 ⊢ (𝐿 = ⟨1, 𝑦⟩ → (𝐾 = ⟨1, 𝑦⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
71	32, 33, 34, 35	pgnbgreunbgrlem5lem2 48919	. . . . . . . . . . . . 13 ⊢ ((((𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐿 = ⟨1, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐿, ⟨1, 𝑏⟩} ∈ 𝐸) → ¬ {⟨1, 𝑏⟩, 𝐾} ∈ 𝐸)
72	71	pm2.21d 122	. . . . . . . . . . . 12 ⊢ ((((𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐿 = ⟨1, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐿, ⟨1, 𝑏⟩} ∈ 𝐸) → ({⟨1, 𝑏⟩, 𝐾} ∈ 𝐸 → 𝑋 = ⟨1, 𝑏⟩))
73	72	ex 418	. . . . . . . . . . 11 ⊢ (((𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐿 = ⟨1, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → ({𝐿, ⟨1, 𝑏⟩} ∈ 𝐸 → ({⟨1, 𝑏⟩, 𝐾} ∈ 𝐸 → 𝑋 = ⟨1, 𝑏⟩)))
74	73	impcomd 417	. . . . . . . . . 10 ⊢ (((𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐿 = ⟨1, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({⟨1, 𝑏⟩, 𝐾} ∈ 𝐸 ∧ {𝐿, ⟨1, 𝑏⟩} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))
75	53, 74	biimtrid 245	. . . . . . . . 9 ⊢ (((𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐿 = ⟨1, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))
76	75	ex 418	. . . . . . . 8 ⊢ ((𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐿 = ⟨1, 𝑦⟩) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))
77	76	adantld 496	. . . . . . 7 ⊢ ((𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐿 = ⟨1, 𝑦⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))
78	77	expcom 419	. . . . . 6 ⊢ (𝐿 = ⟨1, 𝑦⟩ → (𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
79	61, 70, 78	3jaod 1456	. . . . 5 ⊢ (𝐿 = ⟨1, 𝑦⟩ → ((𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∨ 𝐾 = ⟨1, 𝑦⟩ ∨ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
80	32, 33, 34, 35	pgnbgreunbgrlem5lem3 48920	. . . . . . . . . . . . 13 ⊢ ((((𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐿, ⟨1, 𝑏⟩} ∈ 𝐸) → ¬ {⟨1, 𝑏⟩, 𝐾} ∈ 𝐸)
81	80	pm2.21d 122	. . . . . . . . . . . 12 ⊢ ((((𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐿, ⟨1, 𝑏⟩} ∈ 𝐸) → ({⟨1, 𝑏⟩, 𝐾} ∈ 𝐸 → 𝑋 = ⟨1, 𝑏⟩))
82	81	ex 418	. . . . . . . . . . 11 ⊢ (((𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → ({𝐿, ⟨1, 𝑏⟩} ∈ 𝐸 → ({⟨1, 𝑏⟩, 𝐾} ∈ 𝐸 → 𝑋 = ⟨1, 𝑏⟩)))
83	82	impcomd 417	. . . . . . . . . 10 ⊢ (((𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({⟨1, 𝑏⟩, 𝐾} ∈ 𝐸 ∧ {𝐿, ⟨1, 𝑏⟩} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))
84	53, 83	biimtrid 245	. . . . . . . . 9 ⊢ (((𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))
85	84	ex 418	. . . . . . . 8 ⊢ ((𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))
86	85	adantld 496	. . . . . . 7 ⊢ ((𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∧ 𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))
87	86	expcom 419	. . . . . 6 ⊢ (𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ → (𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
88	32, 33, 34, 35	pgnbgreunbgrlem5lem2 48919	. . . . . . . . . . 11 ⊢ ((((𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐾, ⟨1, 𝑏⟩} ∈ 𝐸) → ¬ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸)
89	88	pm2.21d 122	. . . . . . . . . 10 ⊢ ((((𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐾, ⟨1, 𝑏⟩} ∈ 𝐸) → ({⟨1, 𝑏⟩, 𝐿} ∈ 𝐸 → 𝑋 = ⟨1, 𝑏⟩))
90	89	expimpd 459	. . . . . . . . 9 ⊢ (((𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))
91	90	ex 418	. . . . . . . 8 ⊢ ((𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))
92	91	adantld 496	. . . . . . 7 ⊢ ((𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐾 = ⟨1, 𝑦⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))
93	92	ex 418	. . . . . 6 ⊢ (𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ → (𝐾 = ⟨1, 𝑦⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
94		simpr 490	. . . . . . . . . 10 ⊢ ((𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) → 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩)
95		simpl 488	. . . . . . . . . 10 ⊢ ((𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) → 𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩)
96	94, 95	neeq12d 3021	. . . . . . . . 9 ⊢ ((𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) → (𝐾 ≠ 𝐿 ↔ ⟨0, ((𝑦 − 1) mod 5)⟩ ≠ ⟨0, ((𝑦 − 1) mod 5)⟩))
97		eqid 2765	. . . . . . . . . 10 ⊢ ⟨0, ((𝑦 − 1) mod 5)⟩ = ⟨0, ((𝑦 − 1) mod 5)⟩
98		eqneqall 2971	. . . . . . . . . 10 ⊢ (⟨0, ((𝑦 − 1) mod 5)⟩ = ⟨0, ((𝑦 − 1) mod 5)⟩ → (⟨0, ((𝑦 − 1) mod 5)⟩ ≠ ⟨0, ((𝑦 − 1) mod 5)⟩ → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
99	97, 98	ax-mp 5	. . . . . . . . 9 ⊢ (⟨0, ((𝑦 − 1) mod 5)⟩ ≠ ⟨0, ((𝑦 − 1) mod 5)⟩ → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))
100	96, 99	biimtrdi 256	. . . . . . . 8 ⊢ ((𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) → (𝐾 ≠ 𝐿 → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
101	100	impd 416	. . . . . . 7 ⊢ ((𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ ∧ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))
102	101	ex 418	. . . . . 6 ⊢ (𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ → (𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
103	87, 93, 102	3jaod 1456	. . . . 5 ⊢ (𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩ → ((𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∨ 𝐾 = ⟨1, 𝑦⟩ ∨ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
104	48, 79, 103	3jaoi 1454	. . . 4 ⊢ ((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, 𝑦⟩ ∨ 𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩) → ((𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∨ 𝐾 = ⟨1, 𝑦⟩ ∨ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
105	104	imp 412	. . 3 ⊢ (((𝐿 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, 𝑦⟩ ∨ 𝐿 = ⟨0, ((𝑦 − 1) mod 5)⟩) ∧ (𝐾 = ⟨0, ((𝑦 + 1) mod 5)⟩ ∨ 𝐾 = ⟨1, 𝑦⟩ ∨ 𝐾 = ⟨0, ((𝑦 − 1) mod 5)⟩)) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))
106	21, 105	biimtrdi 256	. 2 ⊢ ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → (((𝐿 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, (2^nd ‘𝑋)⟩ ∨ 𝐿 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩) ∧ (𝐾 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐾 = ⟨1, (2^nd ‘𝑋)⟩ ∨ 𝐾 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩)) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
107	106	expdcom 420	1 ⊢ ((𝐿 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, (2^nd ‘𝑋)⟩ ∨ 𝐿 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩) → ((𝐾 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐾 = ⟨1, (2^nd ‘𝑋)⟩ ∨ 𝐾 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩) → ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ w3o 1102 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 {cpr 4593 ⟨cop 4597 ‘cfv 6540 (class class class)co 7416 2^nd c2nd 7987 0cc0 11111 1c1 11112 + caddc 11114 − cmin 11452 2c2 12306 5c5 12309 ..^cfzo 13695 mod cmo 13916 Vtxcvtx 29377 Edgcedg 29428 NeighbVtx cnbgr 29716 gPetersenGr cgpg 48838

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188 ax-pre-sup 11189

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7865 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-oadd 8459 df-er 8696 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9405 df-inf 9406 df-dju 9899 df-card 9937 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-div 11883 df-nn 12245 df-2 12314 df-3 12315 df-4 12316 df-5 12317 df-6 12318 df-7 12319 df-8 12320 df-9 12321 df-n0 12516 df-xnn0 12589 df-z 12603 df-dec 12724 df-uz 12875 df-rp 13029 df-ico 13390 df-fz 13548 df-fzo 13696 df-fl 13839 df-ceil 13840 df-mod 13917 df-seq 14052 df-exp 14112 df-hash 14381 df-cj 15170 df-re 15171 df-im 15172 df-sqrt 15306 df-abs 15307 df-dvds 16329 df-struct 17225 df-slot 17260 df-ndx 17272 df-base 17288 df-edgf 29370 df-vtx 29379 df-iedg 29380 df-edg 29429 df-umgr 29464 df-usgr 29535 df-gpg 48839

This theorem is used by: pgnbgreunbgrlem6 48922

W3C validator