pgnbgreunbgrlem4 - Mathbox for Alexander van der Vekens

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Theorem pgnbgreunbgrlem4 48479

Description: Lemma 4 for pgnbgreunbgr 48485. (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
pgnbgreunbgrlem4	⊢ ((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⟩)))))

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 pgnbgreunbgrlem4**
Step	Hyp	Ref	Expression
1		1ex 11140	. . . . . 6 ⊢ 1 ∈ V
2		vex 3446	. . . . . 6 ⊢ 𝑦 ∈ V
3	1, 2	op2ndd 7954	. . . . 5 ⊢ (𝑋 = ⟨1, 𝑦⟩ → (2^nd ‘𝑋) = 𝑦)
4		oveq1 7375	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((2^nd ‘𝑋) + 2) = (𝑦 + 2))
5	4	oveq1d 7383	. . . . . . . . . 10 ⊢ ((2^nd ‘𝑋) = 𝑦 → (((2^nd ‘𝑋) + 2) mod 5) = ((𝑦 + 2) mod 5))
6	5	opeq2d 4838	. . . . . . . . 9 ⊢ ((2^nd ‘𝑋) = 𝑦 → ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ = ⟨1, ((𝑦 + 2) mod 5)⟩)
7	6	eqeq2d 2748	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐿 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ↔ 𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩))
8		opeq2 4832	. . . . . . . . 9 ⊢ ((2^nd ‘𝑋) = 𝑦 → ⟨0, (2^nd ‘𝑋)⟩ = ⟨0, 𝑦⟩)
9	8	eqeq2d 2748	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐿 = ⟨0, (2^nd ‘𝑋)⟩ ↔ 𝐿 = ⟨0, 𝑦⟩))
10		oveq1 7375	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((2^nd ‘𝑋) − 2) = (𝑦 − 2))
11	10	oveq1d 7383	. . . . . . . . . 10 ⊢ ((2^nd ‘𝑋) = 𝑦 → (((2^nd ‘𝑋) − 2) mod 5) = ((𝑦 − 2) mod 5))
12	11	opeq2d 4838	. . . . . . . . 9 ⊢ ((2^nd ‘𝑋) = 𝑦 → ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ = ⟨1, ((𝑦 − 2) mod 5)⟩)
13	12	eqeq2d 2748	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐿 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ ↔ 𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩))
14	7, 9, 13	3orbi123d 1438	. . . . . . 7 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((𝐿 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ∨ 𝐿 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝐿 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩) ↔ (𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∨ 𝐿 = ⟨0, 𝑦⟩ ∨ 𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩)))
15	6	eqeq2d 2748	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐾 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ↔ 𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩))
16	8	eqeq2d 2748	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐾 = ⟨0, (2^nd ‘𝑋)⟩ ↔ 𝐾 = ⟨0, 𝑦⟩))
17	12	eqeq2d 2748	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ ↔ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩))
18	15, 16, 17	3orbi123d 1438	. . . . . . 7 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((𝐾 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ∨ 𝐾 = ⟨0, (2^nd ‘𝑋)⟩ ∨ 𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩) ↔ (𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∨ 𝐾 = ⟨0, 𝑦⟩ ∨ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩)))
19	14, 18	anbi12d 633	. . . . . 6 ⊢ ((2^nd ‘𝑋) = 𝑦 → (((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)⟩)) ↔ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∨ 𝐿 = ⟨0, 𝑦⟩ ∨ 𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩) ∧ (𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∨ 𝐾 = ⟨0, 𝑦⟩ ∨ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩))))
20		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩) → 𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩)
21		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩) → 𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩)
22	20, 21	neeq12d 2994	. . . . . . . . . . . 12 ⊢ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩) → (𝐾 ≠ 𝐿 ↔ ⟨1, ((𝑦 + 2) mod 5)⟩ ≠ ⟨1, ((𝑦 + 2) mod 5)⟩))
23		eqid 2737	. . . . . . . . . . . . 13 ⊢ ⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨1, ((𝑦 + 2) mod 5)⟩
24		eqneqall 2944	. . . . . . . . . . . . 13 ⊢ (⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨1, ((𝑦 + 2) mod 5)⟩ → (⟨1, ((𝑦 + 2) mod 5)⟩ ≠ ⟨1, ((𝑦 + 2) mod 5)⟩ → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))))
25	23, 24	ax-mp 5	. . . . . . . . . . . 12 ⊢ (⟨1, ((𝑦 + 2) mod 5)⟩ ≠ ⟨1, ((𝑦 + 2) mod 5)⟩ → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩)))
26	22, 25	biimtrdi 253	. . . . . . . . . . 11 ⊢ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩) → (𝐾 ≠ 𝐿 → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))))
27	26	impd 410	. . . . . . . . . 10 ⊢ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩)))
28	27	ex 412	. . . . . . . . 9 ⊢ (𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ → (𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))))
29		prcom 4691	. . . . . . . . . . . . . . . 16 ⊢ {⟨1, 𝑏⟩, ⟨0, 𝑦⟩} = {⟨0, 𝑦⟩, ⟨1, 𝑏⟩}
30	29	eleq1i 2828	. . . . . . . . . . . . . . 15 ⊢ ({⟨1, 𝑏⟩, ⟨0, 𝑦⟩} ∈ 𝐸 ↔ {⟨0, 𝑦⟩, ⟨1, 𝑏⟩} ∈ 𝐸)
31		5eluz3 12808	. . . . . . . . . . . . . . . . 17 ⊢ 5 ∈ (ℤ_≥‘3)
32		pglem 48451	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ (1..^(⌈‘(5 / 2)))
33		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (1..^(⌈‘(5 / 2))) = (1..^(⌈‘(5 / 2)))
34		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (0..^5) = (0..^5)
35		pgnbgreunbgr.g	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐺 = (5 gPetersenGr 2)
36		pgnbgreunbgr.e	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐸 = (Edg‘𝐺)
37	33, 34, 35, 36	gpgedgiov 48425	. . . . . . . . . . . . . . . . 17 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ (𝑦 ∈ (0..^5) ∧ 𝑏 ∈ (0..^5))) → ({⟨0, 𝑦⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ↔ 𝑦 = 𝑏))
38	31, 32, 37	mpanl12 703	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ (0..^5) ∧ 𝑏 ∈ (0..^5)) → ({⟨0, 𝑦⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ↔ 𝑦 = 𝑏))
39	38	ancoms 458	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → ({⟨0, 𝑦⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ↔ 𝑦 = 𝑏))
40	30, 39	bitrid 283	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → ({⟨1, 𝑏⟩, ⟨0, 𝑦⟩} ∈ 𝐸 ↔ 𝑦 = 𝑏))
41		opeq2 4832	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑏 → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩)
42	40, 41	biimtrdi 253	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → ({⟨1, 𝑏⟩, ⟨0, 𝑦⟩} ∈ 𝐸 → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))
43	42	adantld 490	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({⟨1, ((𝑦 + 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, ⟨0, 𝑦⟩} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))
44		preq1 4692	. . . . . . . . . . . . . . 15 ⊢ (𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩ → {𝐾, ⟨1, 𝑏⟩} = {⟨1, ((𝑦 + 2) mod 5)⟩, ⟨1, 𝑏⟩})
45	44	eleq1d 2822	. . . . . . . . . . . . . 14 ⊢ (𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩ → ({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ↔ {⟨1, ((𝑦 + 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸))
46		preq2 4693	. . . . . . . . . . . . . . 15 ⊢ (𝐿 = ⟨0, 𝑦⟩ → {⟨1, 𝑏⟩, 𝐿} = {⟨1, 𝑏⟩, ⟨0, 𝑦⟩})
47	46	eleq1d 2822	. . . . . . . . . . . . . 14 ⊢ (𝐿 = ⟨0, 𝑦⟩ → ({⟨1, 𝑏⟩, 𝐿} ∈ 𝐸 ↔ {⟨1, 𝑏⟩, ⟨0, 𝑦⟩} ∈ 𝐸))
48	45, 47	bi2anan9r 640	. . . . . . . . . . . . 13 ⊢ ((𝐿 = ⟨0, 𝑦⟩ ∧ 𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) ↔ ({⟨1, ((𝑦 + 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, ⟨0, 𝑦⟩} ∈ 𝐸)))
49	48	imbi1d 341	. . . . . . . . . . . 12 ⊢ ((𝐿 = ⟨0, 𝑦⟩ ∧ 𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩) → ((({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩) ↔ (({⟨1, ((𝑦 + 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, ⟨0, 𝑦⟩} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩)))
50	43, 49	imbitrrid 246	. . . . . . . . . . 11 ⊢ ((𝐿 = ⟨0, 𝑦⟩ ∧ 𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩)))
51	50	adantld 490	. . . . . . . . . 10 ⊢ ((𝐿 = ⟨0, 𝑦⟩ ∧ 𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩)))
52	51	ex 412	. . . . . . . . 9 ⊢ (𝐿 = ⟨0, 𝑦⟩ → (𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))))
53		prcom 4691	. . . . . . . . . . . . . . . 16 ⊢ {⟨1, ((𝑦 + 2) mod 5)⟩, ⟨1, 𝑏⟩} = {⟨1, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩}
54	53	eleq1i 2828	. . . . . . . . . . . . . . 15 ⊢ ({⟨1, ((𝑦 + 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ↔ {⟨1, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)
55		prcom 4691	. . . . . . . . . . . . . . . 16 ⊢ {⟨1, 𝑏⟩, ⟨1, ((𝑦 − 2) mod 5)⟩} = {⟨1, ((𝑦 − 2) mod 5)⟩, ⟨1, 𝑏⟩}
56	55	eleq1i 2828	. . . . . . . . . . . . . . 15 ⊢ ({⟨1, 𝑏⟩, ⟨1, ((𝑦 − 2) mod 5)⟩} ∈ 𝐸 ↔ {⟨1, ((𝑦 − 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸)
57	54, 56	anbi12ci 630	. . . . . . . . . . . . . 14 ⊢ (({⟨1, ((𝑦 + 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, ⟨1, ((𝑦 − 2) mod 5)⟩} ∈ 𝐸) ↔ ({⟨1, ((𝑦 − 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
58		5nn 12243	. . . . . . . . . . . . . . . . 17 ⊢ 5 ∈ ℕ
59	58	nnzi 12527	. . . . . . . . . . . . . . . 16 ⊢ 5 ∈ ℤ
60		uzid 12778	. . . . . . . . . . . . . . . 16 ⊢ (5 ∈ ℤ → 5 ∈ (ℤ_≥‘5))
61	59, 60	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 5 ∈ (ℤ_≥‘5)
62		4t2e8 12320	. . . . . . . . . . . . . . . . . . 19 ⊢ (4 · 2) = 8
63	62	oveq1i 7378	. . . . . . . . . . . . . . . . . 18 ⊢ ((4 · 2) mod 5) = (8 mod 5)
64		8mod5e3 47720	. . . . . . . . . . . . . . . . . 18 ⊢ (8 mod 5) = 3
65	63, 64	eqtri 2760	. . . . . . . . . . . . . . . . 17 ⊢ ((4 · 2) mod 5) = 3
66		3ne0 12263	. . . . . . . . . . . . . . . . 17 ⊢ 3 ≠ 0
67	65, 66	eqnetri 3003	. . . . . . . . . . . . . . . 16 ⊢ ((4 · 2) mod 5) ≠ 0
68	32, 67	pm3.2i 470	. . . . . . . . . . . . . . 15 ⊢ (2 ∈ (1..^(⌈‘(5 / 2))) ∧ ((4 · 2) mod 5) ≠ 0)
69	33, 34, 35, 36	gpgedg2iv 48427	. . . . . . . . . . . . . . 15 ⊢ ((5 ∈ (ℤ_≥‘5) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) ∧ (2 ∈ (1..^(⌈‘(5 / 2))) ∧ ((4 · 2) mod 5) ≠ 0)) → (({⟨1, ((𝑦 − 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸) ↔ 𝑏 = 𝑦))
70	61, 68, 69	mp3an13 1455	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({⟨1, ((𝑦 − 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸) ↔ 𝑏 = 𝑦))
71	57, 70	bitrid 283	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({⟨1, ((𝑦 + 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, ⟨1, ((𝑦 − 2) mod 5)⟩} ∈ 𝐸) ↔ 𝑏 = 𝑦))
72	41	eqcoms 2745	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑦 → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩)
73	71, 72	biimtrdi 253	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({⟨1, ((𝑦 + 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, ⟨1, ((𝑦 − 2) mod 5)⟩} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))
74		preq2 4693	. . . . . . . . . . . . . . 15 ⊢ (𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ → {⟨1, 𝑏⟩, 𝐿} = {⟨1, 𝑏⟩, ⟨1, ((𝑦 − 2) mod 5)⟩})
75	74	eleq1d 2822	. . . . . . . . . . . . . 14 ⊢ (𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ → ({⟨1, 𝑏⟩, 𝐿} ∈ 𝐸 ↔ {⟨1, 𝑏⟩, ⟨1, ((𝑦 − 2) mod 5)⟩} ∈ 𝐸))
76	45, 75	bi2anan9r 640	. . . . . . . . . . . . 13 ⊢ ((𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) ↔ ({⟨1, ((𝑦 + 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, ⟨1, ((𝑦 − 2) mod 5)⟩} ∈ 𝐸)))
77	76	imbi1d 341	. . . . . . . . . . . 12 ⊢ ((𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩) → ((({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩) ↔ (({⟨1, ((𝑦 + 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, ⟨1, ((𝑦 − 2) mod 5)⟩} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩)))
78	73, 77	imbitrrid 246	. . . . . . . . . . 11 ⊢ ((𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩)))
79	78	adantld 490	. . . . . . . . . 10 ⊢ ((𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩)))
80	79	ex 412	. . . . . . . . 9 ⊢ (𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ → (𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))))
81	28, 52, 80	3jaoi 1431	. . . . . . . 8 ⊢ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∨ 𝐿 = ⟨0, 𝑦⟩ ∨ 𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩) → (𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))))
82	39, 41	biimtrdi 253	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → ({⟨0, 𝑦⟩, ⟨1, 𝑏⟩} ∈ 𝐸 → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))
83	82	adantrd 491	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({⟨0, 𝑦⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))
84		preq1 4692	. . . . . . . . . . . . . . 15 ⊢ (𝐾 = ⟨0, 𝑦⟩ → {𝐾, ⟨1, 𝑏⟩} = {⟨0, 𝑦⟩, ⟨1, 𝑏⟩})
85	84	eleq1d 2822	. . . . . . . . . . . . . 14 ⊢ (𝐾 = ⟨0, 𝑦⟩ → ({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ↔ {⟨0, 𝑦⟩, ⟨1, 𝑏⟩} ∈ 𝐸))
86		preq2 4693	. . . . . . . . . . . . . . 15 ⊢ (𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ → {⟨1, 𝑏⟩, 𝐿} = {⟨1, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩})
87	86	eleq1d 2822	. . . . . . . . . . . . . 14 ⊢ (𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ → ({⟨1, 𝑏⟩, 𝐿} ∈ 𝐸 ↔ {⟨1, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
88	85, 87	bi2anan9r 640	. . . . . . . . . . . . 13 ⊢ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) ↔ ({⟨0, 𝑦⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
89	88	imbi1d 341	. . . . . . . . . . . 12 ⊢ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) → ((({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩) ↔ (({⟨0, 𝑦⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩)))
90	83, 89	imbitrrid 246	. . . . . . . . . . 11 ⊢ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩)))
91	90	adantld 490	. . . . . . . . . 10 ⊢ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩)))
92	91	ex 412	. . . . . . . . 9 ⊢ (𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ → (𝐾 = ⟨0, 𝑦⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))))
93		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝐿 = ⟨0, 𝑦⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) → 𝐾 = ⟨0, 𝑦⟩)
94		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝐿 = ⟨0, 𝑦⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) → 𝐿 = ⟨0, 𝑦⟩)
95	93, 94	neeq12d 2994	. . . . . . . . . . . 12 ⊢ ((𝐿 = ⟨0, 𝑦⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) → (𝐾 ≠ 𝐿 ↔ ⟨0, 𝑦⟩ ≠ ⟨0, 𝑦⟩))
96		eqid 2737	. . . . . . . . . . . . 13 ⊢ ⟨0, 𝑦⟩ = ⟨0, 𝑦⟩
97		eqneqall 2944	. . . . . . . . . . . . 13 ⊢ (⟨0, 𝑦⟩ = ⟨0, 𝑦⟩ → (⟨0, 𝑦⟩ ≠ ⟨0, 𝑦⟩ → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))))
98	96, 97	ax-mp 5	. . . . . . . . . . . 12 ⊢ (⟨0, 𝑦⟩ ≠ ⟨0, 𝑦⟩ → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩)))
99	95, 98	biimtrdi 253	. . . . . . . . . . 11 ⊢ ((𝐿 = ⟨0, 𝑦⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) → (𝐾 ≠ 𝐿 → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))))
100	99	impd 410	. . . . . . . . . 10 ⊢ ((𝐿 = ⟨0, 𝑦⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩)))
101	100	ex 412	. . . . . . . . 9 ⊢ (𝐿 = ⟨0, 𝑦⟩ → (𝐾 = ⟨0, 𝑦⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))))
102	82	adantrd 491	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({⟨0, 𝑦⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, ⟨1, ((𝑦 − 2) mod 5)⟩} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))
103	84	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) → {𝐾, ⟨1, 𝑏⟩} = {⟨0, 𝑦⟩, ⟨1, 𝑏⟩})
104	103	eleq1d 2822	. . . . . . . . . . . . . 14 ⊢ ((𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) → ({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ↔ {⟨0, 𝑦⟩, ⟨1, 𝑏⟩} ∈ 𝐸))
105	74	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) → {⟨1, 𝑏⟩, 𝐿} = {⟨1, 𝑏⟩, ⟨1, ((𝑦 − 2) mod 5)⟩})
106	105	eleq1d 2822	. . . . . . . . . . . . . 14 ⊢ ((𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) → ({⟨1, 𝑏⟩, 𝐿} ∈ 𝐸 ↔ {⟨1, 𝑏⟩, ⟨1, ((𝑦 − 2) mod 5)⟩} ∈ 𝐸))
107	104, 106	anbi12d 633	. . . . . . . . . . . . 13 ⊢ ((𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) ↔ ({⟨0, 𝑦⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, ⟨1, ((𝑦 − 2) mod 5)⟩} ∈ 𝐸)))
108	107	imbi1d 341	. . . . . . . . . . . 12 ⊢ ((𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) → ((({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩) ↔ (({⟨0, 𝑦⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, ⟨1, ((𝑦 − 2) mod 5)⟩} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩)))
109	102, 108	imbitrrid 246	. . . . . . . . . . 11 ⊢ ((𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩)))
110	109	adantld 490	. . . . . . . . . 10 ⊢ ((𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩)))
111	110	ex 412	. . . . . . . . 9 ⊢ (𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ → (𝐾 = ⟨0, 𝑦⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))))
112	92, 101, 111	3jaoi 1431	. . . . . . . 8 ⊢ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∨ 𝐿 = ⟨0, 𝑦⟩ ∨ 𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩) → (𝐾 = ⟨0, 𝑦⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))))
113	64, 66	eqnetri 3003	. . . . . . . . . . . . . . . 16 ⊢ (8 mod 5) ≠ 0
114	63, 113	eqnetri 3003	. . . . . . . . . . . . . . 15 ⊢ ((4 · 2) mod 5) ≠ 0
115		simpll 767	. . . . . . . . . . . . . . . 16 ⊢ (((5 ∈ (ℤ_≥‘5) ∧ ((4 · 2) mod 5) ≠ 0) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → 5 ∈ (ℤ_≥‘5))
116		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((5 ∈ (ℤ_≥‘5) ∧ ((4 · 2) mod 5) ≠ 0) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)))
117	32	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (5 ∈ (ℤ_≥‘5) → 2 ∈ (1..^(⌈‘(5 / 2))))
118	117	anim1i 616	. . . . . . . . . . . . . . . . 17 ⊢ ((5 ∈ (ℤ_≥‘5) ∧ ((4 · 2) mod 5) ≠ 0) → (2 ∈ (1..^(⌈‘(5 / 2))) ∧ ((4 · 2) mod 5) ≠ 0))
119	118	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((5 ∈ (ℤ_≥‘5) ∧ ((4 · 2) mod 5) ≠ 0) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (2 ∈ (1..^(⌈‘(5 / 2))) ∧ ((4 · 2) mod 5) ≠ 0))
120	115, 116, 119, 69	syl3anc 1374	. . . . . . . . . . . . . . 15 ⊢ (((5 ∈ (ℤ_≥‘5) ∧ ((4 · 2) mod 5) ≠ 0) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({⟨1, ((𝑦 − 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸) ↔ 𝑏 = 𝑦))
121	61, 114, 120	mpanl12 703	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({⟨1, ((𝑦 − 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸) ↔ 𝑏 = 𝑦))
122	121, 72	biimtrdi 253	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({⟨1, ((𝑦 − 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))
123	122	adantl 481	. . . . . . . . . . . 12 ⊢ ((⟨1, ((𝑦 − 2) mod 5)⟩ ≠ ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({⟨1, ((𝑦 − 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))
124	123	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩) → ((⟨1, ((𝑦 − 2) mod 5)⟩ ≠ ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({⟨1, ((𝑦 − 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩)))
125		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩) → 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩)
126		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩) → 𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩)
127	125, 126	neeq12d 2994	. . . . . . . . . . . . 13 ⊢ ((𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩) → (𝐾 ≠ 𝐿 ↔ ⟨1, ((𝑦 − 2) mod 5)⟩ ≠ ⟨1, ((𝑦 + 2) mod 5)⟩))
128	127	ancoms 458	. . . . . . . . . . . 12 ⊢ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩) → (𝐾 ≠ 𝐿 ↔ ⟨1, ((𝑦 − 2) mod 5)⟩ ≠ ⟨1, ((𝑦 + 2) mod 5)⟩))
129	128	anbi1d 632	. . . . . . . . . . 11 ⊢ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ↔ (⟨1, ((𝑦 − 2) mod 5)⟩ ≠ ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)))))
130		preq1 4692	. . . . . . . . . . . . . 14 ⊢ (𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩ → {𝐾, ⟨1, 𝑏⟩} = {⟨1, ((𝑦 − 2) mod 5)⟩, ⟨1, 𝑏⟩})
131	130	eleq1d 2822	. . . . . . . . . . . . 13 ⊢ (𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩ → ({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ↔ {⟨1, ((𝑦 − 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸))
132	131, 87	bi2anan9r 640	. . . . . . . . . . . 12 ⊢ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) ↔ ({⟨1, ((𝑦 − 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
133	132	imbi1d 341	. . . . . . . . . . 11 ⊢ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩) → ((({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩) ↔ (({⟨1, ((𝑦 − 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩)))
134	124, 129, 133	3imtr4d 294	. . . . . . . . . 10 ⊢ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩)))
135	134	ex 412	. . . . . . . . 9 ⊢ (𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ → (𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))))
136	37	ancom2s 651	. . . . . . . . . . . . . . . . 17 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → ({⟨0, 𝑦⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ↔ 𝑦 = 𝑏))
137		equcom 2020	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑏 ↔ 𝑏 = 𝑦)
138	136, 137	bitrdi 287	. . . . . . . . . . . . . . . 16 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → ({⟨0, 𝑦⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ↔ 𝑏 = 𝑦))
139	30, 138	bitrid 283	. . . . . . . . . . . . . . 15 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → ({⟨1, 𝑏⟩, ⟨0, 𝑦⟩} ∈ 𝐸 ↔ 𝑏 = 𝑦))
140	31, 32, 139	mpanl12 703	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → ({⟨1, 𝑏⟩, ⟨0, 𝑦⟩} ∈ 𝐸 ↔ 𝑏 = 𝑦))
141	140, 72	biimtrdi 253	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → ({⟨1, 𝑏⟩, ⟨0, 𝑦⟩} ∈ 𝐸 → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))
142	141	adantld 490	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({⟨1, ((𝑦 − 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, ⟨0, 𝑦⟩} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))
143	142	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({⟨1, ((𝑦 − 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, ⟨0, 𝑦⟩} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))
144	130	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝐿 = ⟨0, 𝑦⟩ ∧ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩) → {𝐾, ⟨1, 𝑏⟩} = {⟨1, ((𝑦 − 2) mod 5)⟩, ⟨1, 𝑏⟩})
145	144	eleq1d 2822	. . . . . . . . . . . . 13 ⊢ ((𝐿 = ⟨0, 𝑦⟩ ∧ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩) → ({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ↔ {⟨1, ((𝑦 − 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸))
146	47	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐿 = ⟨0, 𝑦⟩ ∧ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩) → ({⟨1, 𝑏⟩, 𝐿} ∈ 𝐸 ↔ {⟨1, 𝑏⟩, ⟨0, 𝑦⟩} ∈ 𝐸))
147	145, 146	anbi12d 633	. . . . . . . . . . . 12 ⊢ ((𝐿 = ⟨0, 𝑦⟩ ∧ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) ↔ ({⟨1, ((𝑦 − 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, ⟨0, 𝑦⟩} ∈ 𝐸)))
148	147	imbi1d 341	. . . . . . . . . . 11 ⊢ ((𝐿 = ⟨0, 𝑦⟩ ∧ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩) → ((({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩) ↔ (({⟨1, ((𝑦 − 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, ⟨0, 𝑦⟩} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩)))
149	143, 148	imbitrrid 246	. . . . . . . . . 10 ⊢ ((𝐿 = ⟨0, 𝑦⟩ ∧ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩)))
150	149	ex 412	. . . . . . . . 9 ⊢ (𝐿 = ⟨0, 𝑦⟩ → (𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))))
151		eqeq2 2749	. . . . . . . . . . 11 ⊢ (⟨1, ((𝑦 − 2) mod 5)⟩ = 𝐿 → (𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩ ↔ 𝐾 = 𝐿))
152	151	eqcoms 2745	. . . . . . . . . 10 ⊢ (𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ → (𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩ ↔ 𝐾 = 𝐿))
153		eqneqall 2944	. . . . . . . . . . 11 ⊢ (𝐾 = 𝐿 → (𝐾 ≠ 𝐿 → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))))
154	153	impd 410	. . . . . . . . . 10 ⊢ (𝐾 = 𝐿 → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩)))
155	152, 154	biimtrdi 253	. . . . . . . . 9 ⊢ (𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ → (𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))))
156	135, 150, 155	3jaoi 1431	. . . . . . . 8 ⊢ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∨ 𝐿 = ⟨0, 𝑦⟩ ∨ 𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩) → (𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩ → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))))
157	81, 112, 156	3jaod 1432	. . . . . . 7 ⊢ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∨ 𝐿 = ⟨0, 𝑦⟩ ∨ 𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩) → ((𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∨ 𝐾 = ⟨0, 𝑦⟩ ∨ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))))
158	157	imp 406	. . . . . 6 ⊢ (((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∨ 𝐿 = ⟨0, 𝑦⟩ ∨ 𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩) ∧ (𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∨ 𝐾 = ⟨0, 𝑦⟩ ∨ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩)) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩)))
159	19, 158	biimtrdi 253	. . . . 5 ⊢ ((2^nd ‘𝑋) = 𝑦 → (((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)⟩)) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))))
160	3, 159	syl 17	. . . 4 ⊢ (𝑋 = ⟨1, 𝑦⟩ → (((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)⟩)) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))))
161		eqeq1 2741	. . . . . 6 ⊢ (𝑋 = ⟨1, 𝑦⟩ → (𝑋 = ⟨1, 𝑏⟩ ↔ ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))
162	161	imbi2d 340	. . . . 5 ⊢ (𝑋 = ⟨1, 𝑦⟩ → ((({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩) ↔ (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩)))
163	162	imbi2d 340	. . . 4 ⊢ (𝑋 = ⟨1, 𝑦⟩ → (((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩)) ↔ ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → ⟨1, 𝑦⟩ = ⟨1, 𝑏⟩))))
164	160, 163	sylibrd 259	. . 3 ⊢ (𝑋 = ⟨1, 𝑦⟩ → (((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)⟩)) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
165	164	adantl 481	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑋 = ⟨1, 𝑦⟩) → (((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)⟩)) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ∧ {⟨1, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨1, 𝑏⟩))))
166	165	expdcom 414	1 ⊢ ((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⟩)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ w3o 1086 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 {cpr 4584 ⟨cop 4588 ‘cfv 6500 (class class class)co 7368 2^nd c2nd 7942 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 − cmin 11376 / cdiv 11806 2c2 12212 3c3 12213 4c4 12214 5c5 12215 8c8 12218 ℤcz 12500 ℤ_≥cuz 12763 ..^cfzo 13582 ⌈cceil 13723 mod cmo 13801 Vtxcvtx 29081 Edgcedg 29132 NeighbVtx cnbgr 29417 gPetersenGr cgpg 48400

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 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-oadd 8411 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9357 df-inf 9358 df-dju 9825 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-xnn0 12487 df-z 12501 df-dec 12620 df-uz 12764 df-rp 12918 df-ico 13279 df-fz 13436 df-fzo 13583 df-fl 13724 df-ceil 13725 df-mod 13802 df-hash 14266 df-dvds 16192 df-struct 17086 df-slot 17121 df-ndx 17133 df-base 17149 df-edgf 29074 df-vtx 29083 df-iedg 29084 df-edg 29133 df-umgr 29168 df-usgr 29236 df-gpg 48401

This theorem is referenced by: pgnbgreunbgrlem6 48484

W3C validator