pgnbgreunbgrlem3 - Mathbox for Alexander van der Vekens

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Theorem pgnbgreunbgrlem3 48594

Description: Lemma 3 for pgnbgreunbgr 48601. (Contributed by AV, 18-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
pgnbgreunbgrlem3	⊢ (((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))

Proof of Theorem pgnbgreunbgrlem3 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 29404	. . . 4 ⊢ (𝐾 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋 ∈ 𝑉)
3		pgnbgreunbgr.n	. . . 4 ⊢ 𝑁 = (𝐺 NeighbVtx 𝑋)
4	2, 3	eleq2s 2854	. . 3 ⊢ (𝐾 ∈ 𝑁 → 𝑋 ∈ 𝑉)
5	4	3ad2ant1 1134	. 2 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) → 𝑋 ∈ 𝑉)
6		5eluz3 12833	. . . . . 6 ⊢ 5 ∈ (ℤ_≥‘3)
7		pglem 48567	. . . . . 6 ⊢ 2 ∈ (1..^(⌈‘(5 / 2)))
8		eqid 2736	. . . . . . 7 ⊢ (0..^5) = (0..^5)
9		eqid 2736	. . . . . . 7 ⊢ (1..^(⌈‘(5 / 2))) = (1..^(⌈‘(5 / 2)))
10		pgnbgreunbgr.g	. . . . . . 7 ⊢ 𝐺 = (5 gPetersenGr 2)
11	8, 9, 10, 1	gpgvtxel 48523	. . . . . 6 ⊢ ((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) → (𝑋 ∈ 𝑉 ↔ ∃𝑥 ∈ {0, 1}∃𝑦 ∈ (0..^5)𝑋 = ⟨𝑥, 𝑦⟩))
12	6, 7, 11	mp2an 693	. . . . 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 3433	. . . . . . . . 9 ⊢ 𝑥 ∈ V
16	15	elpr 4592	. . . . . . . 8 ⊢ (𝑥 ∈ {0, 1} ↔ (𝑥 = 0 ∨ 𝑥 = 1))
17		opeq1 4816	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → ⟨𝑥, 𝑦⟩ = ⟨0, 𝑦⟩)
18	17	eqeq2d 2747	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (𝑋 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨0, 𝑦⟩))
19	18	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑥 = 0 ∧ ((((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ (0..^5))) → (𝑋 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨0, 𝑦⟩))
20	6, 7	pm3.2i 470	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2))))
21	1	eleq2i 2828	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑋 ∈ 𝑉 ↔ 𝑋 ∈ (Vtx‘𝐺))
22	21	biimpi 216	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ (Vtx‘𝐺))
23		c0ex 11138	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 0 ∈ V
24		vex 3433	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑦 ∈ V
25	23, 24	op1std 7952	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑋 = ⟨0, 𝑦⟩ → (1^st ‘𝑋) = 0)
26	22, 25	anim12i 614	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑋 = ⟨0, 𝑦⟩) → (𝑋 ∈ (Vtx‘𝐺) ∧ (1^st ‘𝑋) = 0))
27		eqid 2736	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
28	9, 10, 27, 3	gpgnbgrvtx0 48550	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ (𝑋 ∈ (Vtx‘𝐺) ∧ (1^st ‘𝑋) = 0)) → 𝑁 = {⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩})
29	20, 26, 28	sylancr 588	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑋 = ⟨0, 𝑦⟩) → 𝑁 = {⟨0, (((2^nd ‘𝑋) + 1) mod 5)⟩, ⟨1, (2^nd ‘𝑋)⟩, ⟨0, (((2^nd ‘𝑋) − 1) mod 5)⟩})
30		eleq2 2825	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (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)⟩}))
31		eleq2 2825	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (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)⟩}))
32	30, 31	anbi12d 633	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (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)⟩})))
33	32	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑋 = ⟨0, 𝑦⟩) ∧ 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)⟩})))
34		eltpi 4632	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (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)⟩))
35		eltpi 4632	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (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)⟩))
36		pgnbgreunbgr.e	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 𝐸 = (Edg‘𝐺)
37	10, 1, 36, 3	pgnbgreunbgrlem1 48589	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((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 ∈ 𝑉 ∧ 𝑋 = ⟨0, 𝑦⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
38	35, 37	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (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 ∈ 𝑉 ∧ 𝑋 = ⟨0, 𝑦⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
39	34, 38	mpan9 506	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((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 ∈ 𝑉 ∧ 𝑋 = ⟨0, 𝑦⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
40	39	com12 32	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑋 = ⟨0, 𝑦⟩) → ((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, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
41	40	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑋 = ⟨0, 𝑦⟩) ∧ 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, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
42	33, 41	sylbid 240	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑋 = ⟨0, 𝑦⟩) ∧ 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, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
43	29, 42	mpdan 688	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑋 = ⟨0, 𝑦⟩) → ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
44	43	com12 32	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → ((𝑋 ∈ 𝑉 ∧ 𝑋 = ⟨0, 𝑦⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
45	44	expd 415	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑋 ∈ 𝑉 → (𝑋 = ⟨0, 𝑦⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
46	45	com23 86	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑋 = ⟨0, 𝑦⟩ → (𝑋 ∈ 𝑉 → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
47	46	com24 95	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (𝑋 ∈ 𝑉 → (𝑋 = ⟨0, 𝑦⟩ → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
48	47	expd 415	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝐾 ≠ 𝐿 → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (𝑋 ∈ 𝑉 → (𝑋 = ⟨0, 𝑦⟩ → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))))
49	48	3impia 1118	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (𝑋 ∈ 𝑉 → (𝑋 = ⟨0, 𝑦⟩ → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
50	49	expdimp 452	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) → (𝑦 ∈ (0..^5) → (𝑋 ∈ 𝑉 → (𝑋 = ⟨0, 𝑦⟩ → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
51	50	com23 86	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) → (𝑋 ∈ 𝑉 → (𝑦 ∈ (0..^5) → (𝑋 = ⟨0, 𝑦⟩ → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
52	51	imp31 417	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ (0..^5)) → (𝑋 = ⟨0, 𝑦⟩ → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))
53	52	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑥 = 0 ∧ ((((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ (0..^5))) → (𝑋 = ⟨0, 𝑦⟩ → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))
54	19, 53	sylbid 240	. . . . . . . . . 10 ⊢ ((𝑥 = 0 ∧ ((((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ (0..^5))) → (𝑋 = ⟨𝑥, 𝑦⟩ → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))
55	54	ex 412	. . . . . . . . 9 ⊢ (𝑥 = 0 → (((((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ (0..^5)) → (𝑋 = ⟨𝑥, 𝑦⟩ → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
56		1ex 11140	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ∈ V
57	56, 24	op1std 7952	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑋 = ⟨1, 𝑦⟩ → (1^st ‘𝑋) = 1)
58	57	anim1ci 617	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 1))
59	9, 10, 1, 3	gpgnbgrvtx1 48551	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ (𝑋 ∈ 𝑉 ∧ (1^st ‘𝑋) = 1)) → 𝑁 = {⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩, ⟨0, (2^nd ‘𝑋)⟩, ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩})
60	20, 58, 59	sylancr 588	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → 𝑁 = {⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩, ⟨0, (2^nd ‘𝑋)⟩, ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩})
61		eleq2 2825	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (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 2825	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (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 633	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (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	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) ∧ 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 4632	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (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 4632	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (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, 36, 3	pgnbgreunbgrlem2 48593	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((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 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
68	66, 67	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (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 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
69	65, 68	mpan9 506	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((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 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
70	69	com12 32	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((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, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
71	70	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) ∧ 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, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
72	64, 71	sylbid 240	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) ∧ 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, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
73	60, 72	mpdan 688	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
74	73	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → ((𝑋 = ⟨1, 𝑦⟩ ∧ 𝑋 ∈ 𝑉) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
75	74	expd 415	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑋 = ⟨1, 𝑦⟩ → (𝑋 ∈ 𝑉 → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
76	75	com24 95	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (𝑋 ∈ 𝑉 → (𝑋 = ⟨1, 𝑦⟩ → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
77	76	expd 415	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝐾 ≠ 𝐿 → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (𝑋 ∈ 𝑉 → (𝑋 = ⟨1, 𝑦⟩ → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))))
78	77	3impia 1118	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (𝑋 ∈ 𝑉 → (𝑋 = ⟨1, 𝑦⟩ → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
79	78	expdimp 452	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) → (𝑦 ∈ (0..^5) → (𝑋 ∈ 𝑉 → (𝑋 = ⟨1, 𝑦⟩ → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
80	79	com23 86	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) → (𝑋 ∈ 𝑉 → (𝑦 ∈ (0..^5) → (𝑋 = ⟨1, 𝑦⟩ → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
81	80	imp31 417	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ (0..^5)) → (𝑋 = ⟨1, 𝑦⟩ → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))
82		opeq1 4816	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → ⟨𝑥, 𝑦⟩ = ⟨1, 𝑦⟩)
83	82	eqeq2d 2747	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝑋 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨1, 𝑦⟩))
84	83	imbi1d 341	. . . . . . . . . 10 ⊢ (𝑥 = 1 → ((𝑋 = ⟨𝑥, 𝑦⟩ → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)) ↔ (𝑋 = ⟨1, 𝑦⟩ → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
85	81, 84	imbitrrid 246	. . . . . . . . 9 ⊢ (𝑥 = 1 → (((((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ (0..^5)) → (𝑋 = ⟨𝑥, 𝑦⟩ → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
86	55, 85	jaoi 858	. . . . . . . 8 ⊢ ((𝑥 = 0 ∨ 𝑥 = 1) → (((((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ (0..^5)) → (𝑋 = ⟨𝑥, 𝑦⟩ → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 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, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 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, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 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, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 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, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))))
91	90	rexlimdvv 3193	. . 3 ⊢ ((((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) ∧ 𝑋 ∈ 𝑉) → (∃𝑥 ∈ {0, 1}∃𝑦 ∈ (0..^5)𝑋 = ⟨𝑥, 𝑦⟩ → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))
92	14, 91	mpd 15	. 2 ⊢ ((((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) ∧ 𝑋 ∈ 𝑉) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))
93	5, 92	mpidan 690	1 ⊢ (((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝐾 ≠ 𝐿) ∧ 𝑏 ∈ (0..^5)) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∨ w3o 1086 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ∃wrex 3061 {cpr 4569 {ctp 4571 ⟨cop 4573 ‘cfv 6498 (class class class)co 7367 1^st c1st 7940 2^nd c2nd 7941 0cc0 11038 1c1 11039 + caddc 11041 − cmin 11377 / cdiv 11807 2c2 12236 3c3 12237 5c5 12239 ℤ_≥cuz 12788 ..^cfzo 13608 ⌈cceil 13750 mod cmo 13828 Vtxcvtx 29065 Edgcedg 29116 NeighbVtx cnbgr 29401 gPetersenGr cgpg 48516

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 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 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-inf 9356 df-dju 9825 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-xnn0 12511 df-z 12525 df-dec 12645 df-uz 12789 df-rp 12943 df-ico 13304 df-fz 13462 df-fzo 13609 df-fl 13751 df-ceil 13752 df-mod 13829 df-hash 14293 df-dvds 16222 df-struct 17117 df-slot 17152 df-ndx 17164 df-base 17180 df-edgf 29058 df-vtx 29067 df-iedg 29068 df-edg 29117 df-upgr 29151 df-umgr 29152 df-usgr 29220 df-nbgr 29402 df-gpg 48517

This theorem is referenced by: pgnbgreunbgr 48601

W3C validator