pgnbgreunbgrlem6 - Mathbox for Alexander van der Vekens

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Theorem pgnbgreunbgrlem6 49495

Description: Lemma 6 for pgnbgreunbgr 49496. (Contributed by AV, 20-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
pgnbgreunbgrlem6	⊢ (((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))

Proof of Theorem pgnbgreunbgrlem6 Dummy variables 𝑦 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pgnbgreunbgr.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	nbgrcl 29973	. . . 4 ⊢ (𝐾 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋 ∈ 𝑉)
3		pgnbgreunbgr.n	. . . 4 ⊢ 𝑁 = (𝐺 NeighbVtx 𝑋)
4	2, 3	eleq2s 2882	. . 3 ⊢ (𝐾 ∈ 𝑁 → 𝑋 ∈ 𝑉)
5	4	3ad2ant1 1151	. 2 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) → 𝑋 ∈ 𝑉)
6		5eluz3 13050	. . . . . 6 ⊢ 5 ∈ (ℤ_≥‘3)
7		pglem 49462	. . . . . 6 ⊢ 2 ∈ (1..^(⌈‘(5 / 2)))
8		eqid 2763	. . . . . . 7 ⊢ (0..^5) = (0..^5)
9		eqid 2763	. . . . . . 7 ⊢ (1..^(⌈‘(5 / 2))) = (1..^(⌈‘(5 / 2)))
10		pgnbgreunbgr.g	. . . . . . 7 ⊢ 𝐺 = (5 gPetersenGr 2)
11	8, 9, 10, 1	gpgvtxel 49418	. . . . . 6 ⊢ ((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) → (𝑋 ∈ 𝑉 ↔ ∃𝑥 ∈ {0, 1}∃𝑦 ∈ (0..^5)𝑋 = ⟨𝑥, 𝑦⟩))
12	6, 7, 11	mp2an 705	. . . . 5 ⊢ (𝑋 ∈ 𝑉 ↔ ∃𝑥 ∈ {0, 1}∃𝑦 ∈ (0..^5)𝑋 = ⟨𝑥, 𝑦⟩)
13	12	biimpi 219	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ∃𝑥 ∈ {0, 1}∃𝑦 ∈ (0..^5)𝑋 = ⟨𝑥, 𝑦⟩)
14	13	adantl 487	. . 3 ⊢ ((((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) ∧ 𝑋 ∈ 𝑉) → ∃𝑥 ∈ {0, 1}∃𝑦 ∈ (0..^5)𝑋 = ⟨𝑥, 𝑦⟩)
15		vex 3458	. . . . . . . . 9 ⊢ 𝑥 ∈ V
16	15	elpr 4609	. . . . . . . 8 ⊢ (𝑥 ∈ {0, 1} ↔ (𝑥 = 0 ∨ 𝑥 = 1))
17	6, 7	pm3.2i 476	. . . . . . . . . . . . . . . . . . . 20 ⊢ (5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2))))
18		c0ex 11339	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ V
19		vex 3458	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑦 ∈ V
20	18, 19	op1std 8045	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑋 = ⟨0, 𝑦⟩ → (1^st ‘𝑋) = 0)
21	20	anim1ci 628	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0))
22	9, 10, 1, 3	gpgnbgrvtx0 49445	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0)) → 𝑁 = {⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩})
23	17, 21, 22	sylancr 599	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → 𝑁 = {⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩})
24		eleq2 2853	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 = {⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩} → (𝐾 ∈ 𝑁 ↔ 𝐾 ∈ {⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩}))
25		eleq2 2853	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 = {⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩} → (𝐿 ∈ 𝑁 ↔ 𝐿 ∈ {⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩}))
26	24, 25	anbi12d 644	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 = {⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩} → ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ↔ (𝐾 ∈ {⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩} ∧ 𝐿 ∈ {⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩})))
27	26	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) ∧ 𝑁 = {⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩}) → ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ↔ (𝐾 ∈ {⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩} ∧ 𝐿 ∈ {⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩})))
28		eltpi 4650	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐾 ∈ {⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨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)⟩))
29		eltpi 4650	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐿 ∈ {⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩} → (𝐿 = ⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, (2^nd ‘𝑋)⟩ ∨ 𝐿 = ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩))
30		pgnbgreunbgr.e	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝐸 = (Edg‘𝐺)
31	10, 1, 30, 3	pgnbgreunbgrlem5 49494	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((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⟩)))))
32	29, 31	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐿 ∈ {⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨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⟩)))))
33	28, 32	mpan9 516	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐾 ∈ {⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩} ∧ 𝐿 ∈ {⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨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⟩))))
34	33	com12 33	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐾 ∈ {⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩} ∧ 𝐿 ∈ {⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨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⟩))))
35	34	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) ∧ 𝑁 = {⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩}) → ((𝐾 ∈ {⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩} ∧ 𝐿 ∈ {⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨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⟩))))
36	27, 35	sylbid 243	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) ∧ 𝑁 = {⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩}) → ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
37	23, 36	mpdan 700	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
38	37	com12 33	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
39	38	expd 421	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑋 = ⟨0, 𝑦⟩ → (𝑋 ∈ 𝑉 → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))))
40	39	com24 96	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (𝑋 ∈ 𝑉 → (𝑋 = ⟨0, 𝑦⟩ → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))))
41	40	expd 421	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝐾 ≠ 𝐿 → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (𝑋 ∈ 𝑉 → (𝑋 = ⟨0, 𝑦⟩ → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))))
42	41	3impia 1135	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (𝑋 ∈ 𝑉 → (𝑋 = ⟨0, 𝑦⟩ → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))))
43	42	expdimp 458	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) → (𝑦 ∈ (0..^5) → (𝑋 ∈ 𝑉 → (𝑋 = ⟨0, 𝑦⟩ → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))))
44	43	com23 87	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) → (𝑋 ∈ 𝑉 → (𝑦 ∈ (0..^5) → (𝑋 = ⟨0, 𝑦⟩ → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))))
45	44	imp31 423	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ (0..^5)) → (𝑋 = ⟨0, 𝑦⟩ → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))
46		opeq1 4845	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → ⟨𝑥, 𝑦⟩ = ⟨0, 𝑦⟩)
47	46	eqeq2d 2774	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (𝑋 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨0, 𝑦⟩))
48	47	imbi1d 344	. . . . . . . . . 10 ⊢ (𝑥 = 0 → ((𝑋 = ⟨𝑥, 𝑦⟩ → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)) ↔ (𝑋 = ⟨0, 𝑦⟩ → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
49	45, 48	imbitrrid 249	. . . . . . . . 9 ⊢ (𝑥 = 0 → (((((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ (0..^5)) → (𝑋 = ⟨𝑥, 𝑦⟩ → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
50		opeq1 4845	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → ⟨𝑥, 𝑦⟩ = ⟨1, 𝑦⟩)
51	50	eqeq2d 2774	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (𝑋 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨1, 𝑦⟩))
52	51	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑥 = 1 ∧ ((((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ (0..^5))) → (𝑋 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨1, 𝑦⟩))
53	1	eleq2i 2856	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑋 ∈ 𝑉 ↔ 𝑋 ∈ (Vtx‘𝐺))
54	53	biimpi 219	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ (Vtx‘𝐺))
55		1ex 11342	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 1 ∈ V
56	55, 19	op1std 8045	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑋 = ⟨1, 𝑦⟩ → (1^st ‘𝑋) = 1)
57	54, 56	anim12i 625	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑋 = ⟨1, 𝑦⟩) → (𝑋 ∈ (Vtx‘𝐺) ∧ (1^st ‘𝑋) = 1))
58		eqid 2763	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
59	9, 10, 58, 3	gpgnbgrvtx1 49446	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ (𝑋 ∈ (Vtx‘𝐺) ∧ (1^st ‘𝑋) = 1)) → 𝑁 = {⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩, ⟨0, (2^nd ‘𝑋)⟩, ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩})
60	17, 57, 59	sylancr 599	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑋 = ⟨1, 𝑦⟩) → 𝑁 = {⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩, ⟨0, (2^nd ‘𝑋)⟩, ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩})
61		eleq2 2853	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 = {⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩, ⟨0, (2^nd ‘𝑋)⟩, ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩} → (𝐾 ∈ 𝑁 ↔ 𝐾 ∈ {⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩, ⟨0, (2^nd ‘𝑋)⟩, ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩}))
62		eleq2 2853	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 = {⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩, ⟨0, (2^nd ‘𝑋)⟩, ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩} → (𝐿 ∈ 𝑁 ↔ 𝐿 ∈ {⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩, ⟨0, (2^nd ‘𝑋)⟩, ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩}))
63	61, 62	anbi12d 644	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 = {⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩, ⟨0, (2^nd ‘𝑋)⟩, ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩} → ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ↔ (𝐾 ∈ {⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩, ⟨0, (2^nd ‘𝑋)⟩, ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩} ∧ 𝐿 ∈ {⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩, ⟨0, (2^nd ‘𝑋)⟩, ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩})))
64	63	adantl 487	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑋 = ⟨1, 𝑦⟩) ∧ 𝑁 = {⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩, ⟨0, (2^nd ‘𝑋)⟩, ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩}) → ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ↔ (𝐾 ∈ {⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩, ⟨0, (2^nd ‘𝑋)⟩, ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩} ∧ 𝐿 ∈ {⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩, ⟨0, (2^nd ‘𝑋)⟩, ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩})))
65		eltpi 4650	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐾 ∈ {⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩, ⟨0, (2^nd ‘𝑋)⟩, ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩} → (𝐾 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ∨ 𝐾 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩))
66		eltpi 4650	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐿 ∈ {⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩, ⟨0, (2^nd ‘𝑋)⟩, ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩} → (𝐿 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ∨ 𝐿 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝐿 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩))
67	10, 1, 30, 3	pgnbgreunbgrlem4 49490	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐿 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ∨ 𝐿 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝐿 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩) → ((𝐾 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ∨ 𝐾 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩) → ((𝑋 ∈ 𝑉 ∧ 𝑋 = ⟨1, 𝑦⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))))
68	66, 67	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐿 ∈ {⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩, ⟨0, (2^nd ‘𝑋)⟩, ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩} → ((𝐾 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ∨ 𝐾 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩) → ((𝑋 ∈ 𝑉 ∧ 𝑋 = ⟨1, 𝑦⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))))
69	65, 68	mpan9 516	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐾 ∈ {⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩, ⟨0, (2^nd ‘𝑋)⟩, ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩} ∧ 𝐿 ∈ {⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩, ⟨0, (2^nd ‘𝑋)⟩, ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩}) → ((𝑋 ∈ 𝑉 ∧ 𝑋 = ⟨1, 𝑦⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
70	69	com12 33	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑋 = ⟨1, 𝑦⟩) → ((𝐾 ∈ {⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩, ⟨0, (2^nd ‘𝑋)⟩, ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩} ∧ 𝐿 ∈ {⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩, ⟨0, (2^nd ‘𝑋)⟩, ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩}) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
71	70	adantr 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑋 = ⟨1, 𝑦⟩) ∧ 𝑁 = {⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩, ⟨0, (2^nd ‘𝑋)⟩, ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩}) → ((𝐾 ∈ {⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩, ⟨0, (2^nd ‘𝑋)⟩, ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩} ∧ 𝐿 ∈ {⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩, ⟨0, (2^nd ‘𝑋)⟩, ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩}) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
72	64, 71	sylbid 243	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑋 = ⟨1, 𝑦⟩) ∧ 𝑁 = {⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩, ⟨0, (2^nd ‘𝑋)⟩, ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩}) → ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
73	60, 72	mpdan 700	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑋 = ⟨1, 𝑦⟩) → ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
74	73	com12 33	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → ((𝑋 ∈ 𝑉 ∧ 𝑋 = ⟨1, 𝑦⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
75	74	expd 421	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑋 ∈ 𝑉 → (𝑋 = ⟨1, 𝑦⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))))
76	75	com23 87	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑋 = ⟨1, 𝑦⟩ → (𝑋 ∈ 𝑉 → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))))
77	76	com24 96	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (𝑋 ∈ 𝑉 → (𝑋 = ⟨1, 𝑦⟩ → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))))
78	77	expd 421	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝐾 ≠ 𝐿 → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (𝑋 ∈ 𝑉 → (𝑋 = ⟨1, 𝑦⟩ → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))))
79	78	3impia 1135	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (𝑋 ∈ 𝑉 → (𝑋 = ⟨1, 𝑦⟩ → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))))
80	79	expdimp 458	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) → (𝑦 ∈ (0..^5) → (𝑋 ∈ 𝑉 → (𝑋 = ⟨1, 𝑦⟩ → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))))
81	80	com23 87	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) → (𝑋 ∈ 𝑉 → (𝑦 ∈ (0..^5) → (𝑋 = ⟨1, 𝑦⟩ → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))))
82	81	imp31 423	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ (0..^5)) → (𝑋 = ⟨1, 𝑦⟩ → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))
83	82	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑥 = 1 ∧ ((((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ (0..^5))) → (𝑋 = ⟨1, 𝑦⟩ → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))
84	52, 83	sylbid 243	. . . . . . . . . 10 ⊢ ((𝑥 = 1 ∧ ((((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ (0..^5))) → (𝑋 = ⟨𝑥, 𝑦⟩ → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))
85	84	ex 418	. . . . . . . . 9 ⊢ (𝑥 = 1 → (((((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ (0..^5)) → (𝑋 = ⟨𝑥, 𝑦⟩ → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
86	49, 85	jaoi 871	. . . . . . . 8 ⊢ ((𝑥 = 0 ∨ 𝑥 = 1) → (((((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ (0..^5)) → (𝑋 = ⟨𝑥, 𝑦⟩ → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
87	16, 86	sylbi 220	. . . . . . 7 ⊢ (𝑥 ∈ {0, 1} → (((((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ (0..^5)) → (𝑋 = ⟨𝑥, 𝑦⟩ → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
88	87	expd 421	. . . . . 6 ⊢ (𝑥 ∈ {0, 1} → ((((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) ∧ 𝑋 ∈ 𝑉) → (𝑦 ∈ (0..^5) → (𝑋 = ⟨𝑥, 𝑦⟩ → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))))
89	88	com12 33	. . . . 5 ⊢ ((((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) ∧ 𝑋 ∈ 𝑉) → (𝑥 ∈ {0, 1} → (𝑦 ∈ (0..^5) → (𝑋 = ⟨𝑥, 𝑦⟩ → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))))
90	89	impd 416	. . . 4 ⊢ ((((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) ∧ 𝑋 ∈ 𝑉) → ((𝑥 ∈ {0, 1} ∧ 𝑦 ∈ (0..^5)) → (𝑋 = ⟨𝑥, 𝑦⟩ → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
91	90	rexlimdvv 3222	. . 3 ⊢ ((((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) ∧ 𝑋 ∈ 𝑉) → (∃𝑥 ∈ {0, 1}∃𝑦 ∈ (0..^5)𝑋 = ⟨𝑥, 𝑦⟩ → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)))
92	14, 91	mpd 16	. 2 ⊢ ((((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) ∧ 𝑋 ∈ 𝑉) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))
93	5, 92	mpidan 702	1 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (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 ∨ wo 861 ∨ w3o 1102 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∃wrex 3090 {cpr 4590 {ctp 4592 ⟨cop 4840 ‘cfv 6566 (class class class)co 7451 1^st c1st 8033 2^nd c2nd 8034 0cc0 11239 1c1 11240 + caddc 11242 − cmin 11580 / cdiv 12013 2c2 12437 3c3 12438 5c5 12440 ℤ_≥cuz 13005 ..^cfzo 13828 ⌈cceil 13971 mod cmo 14049 Vtxcvtx 29631 Edgcedg 29682 NeighbVtx cnbgr 29970 gPetersenGr cgpg 49411

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-oadd 8518 df-er 8755 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-sup 9471 df-inf 9472 df-dju 10019 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-xnn0 12720 df-z 12734 df-dec 12855 df-uz 13006 df-rp 13161 df-ico 13522 df-fz 13680 df-fzo 13829 df-fl 13972 df-ceil 13973 df-mod 14050 df-seq 14185 df-exp 14245 df-hash 14516 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-dvds 16467 df-struct 17369 df-slot 17404 df-ndx 17416 df-base 17432 df-edgf 29624 df-vtx 29633 df-iedg 29634 df-edg 29683 df-upgr 29717 df-umgr 29718 df-usgr 29789 df-nbgr 29971 df-gpg 49412

This theorem is used by: pgnbgreunbgr 49496

W3C validator