pgnbgreunbgrlem6 - Mathbox for Alexander van der Vekens

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Theorem pgnbgreunbgrlem6 48312

Description: Lemma 6 for pgnbgreunbgr 48313. (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 29357	. . . 4 ⊢ (𝐾 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋 ∈ 𝑉)
3		pgnbgreunbgr.n	. . . 4 ⊢ 𝑁 = (𝐺 NeighbVtx 𝑋)
4	2, 3	eleq2s 2852	. . 3 ⊢ (𝐾 ∈ 𝑁 → 𝑋 ∈ 𝑉)
5	4	3ad2ant1 1133	. 2 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) → 𝑋 ∈ 𝑉)
6		5eluz3 12794	. . . . . 6 ⊢ 5 ∈ (ℤ_≥‘3)
7		pglem 48279	. . . . . 6 ⊢ 2 ∈ (1..^(⌈‘(5 / 2)))
8		eqid 2734	. . . . . . 7 ⊢ (0..^5) = (0..^5)
9		eqid 2734	. . . . . . 7 ⊢ (1..^(⌈‘(5 / 2))) = (1..^(⌈‘(5 / 2)))
10		pgnbgreunbgr.g	. . . . . . 7 ⊢ 𝐺 = (5 gPetersenGr 2)
11	8, 9, 10, 1	gpgvtxel 48235	. . . . . 6 ⊢ ((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) → (𝑋 ∈ 𝑉 ↔ ∃𝑥 ∈ {0, 1}∃𝑦 ∈ (0..^5)𝑋 = ⟨𝑥, 𝑦⟩))
12	6, 7, 11	mp2an 692	. . . . 5 ⊢ (𝑋 ∈ 𝑉 ↔ ∃𝑥 ∈ {0, 1}∃𝑦 ∈ (0..^5)𝑋 = ⟨𝑥, 𝑦⟩)
13	12	biimpi 216	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ∃𝑥 ∈ {0, 1}∃𝑦 ∈ (0..^5)𝑋 = ⟨𝑥, 𝑦⟩)
14	13	adantl 481	. . 3 ⊢ ((((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) ∧ 𝑋 ∈ 𝑉) → ∃𝑥 ∈ {0, 1}∃𝑦 ∈ (0..^5)𝑋 = ⟨𝑥, 𝑦⟩)
15		vex 3442	. . . . . . . . 9 ⊢ 𝑥 ∈ V
16	15	elpr 4603	. . . . . . . 8 ⊢ (𝑥 ∈ {0, 1} ↔ (𝑥 = 0 ∨ 𝑥 = 1))
17	6, 7	pm3.2i 470	. . . . . . . . . . . . . . . . . . . 20 ⊢ (5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2))))
18		c0ex 11124	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ V
19		vex 3442	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑦 ∈ V
20	18, 19	op1std 7941	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑋 = ⟨0, 𝑦⟩ → (1^st ‘𝑋) = 0)
21	20	anim1ci 616	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑋 = ⟨0, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 0))
22	9, 10, 1, 3	gpgnbgrvtx0 48262	. . . . . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . . . . . . . . . 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 632	. . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . 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 4643	. . . . . . . . . . . . . . . . . . . . . . 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 4643	. . . . . . . . . . . . . . . . . . . . . . . 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 48311	. . . . . . . . . . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . . . . . . . . . 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 506	. . . . . . . . . . . . . . . . . . . . . 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 32	. . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . 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 240	. . . . . . . . . . . . . . . . . . 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 687	. . . . . . . . . . . . . . . . . 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 32	. . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . . 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 95	. . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . 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 1117	. . . . . . . . . . . . 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 452	. . . . . . . . . . . 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 86	. . . . . . . . . . 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 417	. . . . . . . . . 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 4827	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → ⟨𝑥, 𝑦⟩ = ⟨0, 𝑦⟩)
47	46	eqeq2d 2745	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (𝑋 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨0, 𝑦⟩))
48	47	imbi1d 341	. . . . . . . . . 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 246	. . . . . . . . 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 4827	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → ⟨𝑥, 𝑦⟩ = ⟨1, 𝑦⟩)
51	50	eqeq2d 2745	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (𝑋 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨1, 𝑦⟩))
52	51	adantr 480	. . . . . . . . . . 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 2826	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑋 ∈ 𝑉 ↔ 𝑋 ∈ (Vtx‘𝐺))
54	53	biimpi 216	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ (Vtx‘𝐺))
55		1ex 11126	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 1 ∈ V
56	55, 19	op1std 7941	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑋 = ⟨1, 𝑦⟩ → (1^st ‘𝑋) = 1)
57	54, 56	anim12i 613	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑋 = ⟨1, 𝑦⟩) → (𝑋 ∈ (Vtx‘𝐺) ∧ (1^st ‘𝑋) = 1))
58		eqid 2734	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
59	9, 10, 58, 3	gpgnbgrvtx1 48263	. . . . . . . . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . . . . . . . . . . . . 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 632	. . . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . 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 4643	. . . . . . . . . . . . . . . . . . . . . . . . . 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 4643	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 48307	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . . . . . . . . . . . . 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 506	. . . . . . . . . . . . . . . . . . . . . . . . 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 32	. . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . 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 240	. . . . . . . . . . . . . . . . . . . . . 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 687	. . . . . . . . . . . . . . . . . . . . 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 32	. . . . . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . . . . . 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 86	. . . . . . . . . . . . . . . . . 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 95	. . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . . 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 1117	. . . . . . . . . . . . . . 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 452	. . . . . . . . . . . . . 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 86	. . . . . . . . . . . . 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 417	. . . . . . . . . . . 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 481	. . . . . . . . . . 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 240	. . . . . . . . . 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 412	. . . . . . . . 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 857	. . . . . . . 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 217	. . . . . . 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 415	. . . . . 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 32	. . . . 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 410	. . . 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 3190	. . 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 15	. 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 689	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

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∨ w3o 1085 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2930 ∃wrex 3058 {cpr 4580 {ctp 4582 ⟨cop 4584 ‘cfv 6490 (class class class)co 7356 1^st c1st 7929 2^nd c2nd 7930 0cc0 11024 1c1 11025 + caddc 11027 − cmin 11362 / cdiv 11792 2c2 12198 3c3 12199 5c5 12201 ℤ_≥cuz 12749 ..^cfzo 13568 ⌈cceil 13709 mod cmo 13787 Vtxcvtx 29018 Edgcedg 29069 NeighbVtx cnbgr 29354 gPetersenGr cgpg 48228

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-er 8633 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-sup 9343 df-inf 9344 df-dju 9811 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-xnn0 12473 df-z 12487 df-dec 12606 df-uz 12750 df-rp 12904 df-ico 13265 df-fz 13422 df-fzo 13569 df-fl 13710 df-ceil 13711 df-mod 13788 df-seq 13923 df-exp 13983 df-hash 14252 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-dvds 16178 df-struct 17072 df-slot 17107 df-ndx 17119 df-base 17135 df-edgf 29011 df-vtx 29020 df-iedg 29021 df-edg 29070 df-upgr 29104 df-umgr 29105 df-usgr 29173 df-nbgr 29355 df-gpg 48229

This theorem is referenced by: pgnbgreunbgr 48313

W3C validator