pgnbgreunbgrlem6 - Mathbox for Alexander van der Vekens

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Theorem pgnbgreunbgrlem6 48087

Description: Lemma 6 for pgnbgreunbgr 48088. (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 29238	. . . 4 ⊢ (𝐾 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋 ∈ 𝑉)
3		pgnbgreunbgr.n	. . . 4 ⊢ 𝑁 = (𝐺 NeighbVtx 𝑋)
4	2, 3	eleq2s 2846	. . 3 ⊢ (𝐾 ∈ 𝑁 → 𝑋 ∈ 𝑉)
5	4	3ad2ant1 1133	. 2 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) → 𝑋 ∈ 𝑉)
6		5eluz3 12818	. . . . . 6 ⊢ 5 ∈ (ℤ_≥‘3)
7		pglem 48055	. . . . . 6 ⊢ 2 ∈ (1..^(⌈‘(5 / 2)))
8		eqid 2729	. . . . . . 7 ⊢ (0..^5) = (0..^5)
9		eqid 2729	. . . . . . 7 ⊢ (1..^(⌈‘(5 / 2))) = (1..^(⌈‘(5 / 2)))
10		pgnbgreunbgr.g	. . . . . . 7 ⊢ 𝐺 = (5 gPetersenGr 2)
11	8, 9, 10, 1	gpgvtxel 48011	. . . . . 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 3448	. . . . . . . . 9 ⊢ 𝑥 ∈ V
16	15	elpr 4610	. . . . . . . 8 ⊢ (𝑥 ∈ {0, 1} ↔ (𝑥 = 0 ∨ 𝑥 = 1))
17	6, 7	pm3.2i 470	. . . . . . . . . . . . . . . . . . . 20 ⊢ (5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2))))
18		c0ex 11144	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ V
19		vex 3448	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑦 ∈ V
20	18, 19	op1std 7957	. . . . . . . . . . . . . . . . . . . . 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 48038	. . . . . . . . . . . . . . . . . . . 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 2817	. . . . . . . . . . . . . . . . . . . . . 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 2817	. . . . . . . . . . . . . . . . . . . . . 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 4648	. . . . . . . . . . . . . . . . . . . . . . 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 4648	. . . . . . . . . . . . . . . . . . . . . . . 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 48086	. . . . . . . . . . . . . . . . . . . . . . . 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 4833	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → ⟨𝑥, 𝑦⟩ = ⟨0, 𝑦⟩)
47	46	eqeq2d 2740	. . . . . . . . . . 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 4833	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → ⟨𝑥, 𝑦⟩ = ⟨1, 𝑦⟩)
51	50	eqeq2d 2740	. . . . . . . . . . . 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 2820	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑋 ∈ 𝑉 ↔ 𝑋 ∈ (Vtx‘𝐺))
54	53	biimpi 216	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ (Vtx‘𝐺))
55		1ex 11146	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 1 ∈ V
56	55, 19	op1std 7957	. . . . . . . . . . . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
59	9, 10, 58, 3	gpgnbgrvtx1 48039	. . . . . . . . . . . . . . . . . . . . . . 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 2817	. . . . . . . . . . . . . . . . . . . . . . . . 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 2817	. . . . . . . . . . . . . . . . . . . . . . . . 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 4648	. . . . . . . . . . . . . . . . . . . . . . . . . 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 4648	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 48082	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 3191	. . 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 1540 ∈ wcel 2109 ≠ wne 2925 ∃wrex 3053 {cpr 4587 {ctp 4589 ⟨cop 4591 ‘cfv 6499 (class class class)co 7369 1^st c1st 7945 2^nd c2nd 7946 0cc0 11044 1c1 11045 + caddc 11047 − cmin 11381 / cdiv 11811 2c2 12217 3c3 12218 5c5 12220 ℤ_≥cuz 12769 ..^cfzo 13591 ⌈cceil 13729 mod cmo 13807 Vtxcvtx 28899 Edgcedg 28950 NeighbVtx cnbgr 29235 gPetersenGr cgpg 48004

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9369 df-inf 9370 df-dju 9830 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-xnn0 12492 df-z 12506 df-dec 12626 df-uz 12770 df-rp 12928 df-ico 13288 df-fz 13445 df-fzo 13592 df-fl 13730 df-ceil 13731 df-mod 13808 df-seq 13943 df-exp 14003 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-dvds 16199 df-struct 17093 df-slot 17128 df-ndx 17140 df-base 17156 df-edgf 28892 df-vtx 28901 df-iedg 28902 df-edg 28951 df-upgr 28985 df-umgr 28986 df-usgr 29054 df-nbgr 29236 df-gpg 48005

This theorem is referenced by: pgnbgreunbgr 48088

W3C validator