pgnbgreunbgrlem4 - Mathbox for Alexander van der Vekens

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Theorem pgnbgreunbgrlem4 48099

Description: Lemma 4 for pgnbgreunbgr 48105. (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 11176	. . . . . 6 ⊢ 1 ∈ V
2		vex 3454	. . . . . 6 ⊢ 𝑦 ∈ V
3	1, 2	op2ndd 7981	. . . . 5 ⊢ (𝑋 = ⟨1, 𝑦⟩ → (2^nd ‘𝑋) = 𝑦)
4		oveq1 7396	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((2^nd ‘𝑋) + 2) = (𝑦 + 2))
5	4	oveq1d 7404	. . . . . . . . . 10 ⊢ ((2^nd ‘𝑋) = 𝑦 → (((2^nd ‘𝑋) + 2) mod 5) = ((𝑦 + 2) mod 5))
6	5	opeq2d 4846	. . . . . . . . 9 ⊢ ((2^nd ‘𝑋) = 𝑦 → ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ = ⟨1, ((𝑦 + 2) mod 5)⟩)
7	6	eqeq2d 2741	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐿 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ↔ 𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩))
8		opeq2 4840	. . . . . . . . 9 ⊢ ((2^nd ‘𝑋) = 𝑦 → ⟨0, (2^nd ‘𝑋)⟩ = ⟨0, 𝑦⟩)
9	8	eqeq2d 2741	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐿 = ⟨0, (2^nd ‘𝑋)⟩ ↔ 𝐿 = ⟨0, 𝑦⟩))
10		oveq1 7396	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝑋) = 𝑦 → ((2^nd ‘𝑋) − 2) = (𝑦 − 2))
11	10	oveq1d 7404	. . . . . . . . . 10 ⊢ ((2^nd ‘𝑋) = 𝑦 → (((2^nd ‘𝑋) − 2) mod 5) = ((𝑦 − 2) mod 5))
12	11	opeq2d 4846	. . . . . . . . 9 ⊢ ((2^nd ‘𝑋) = 𝑦 → ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ = ⟨1, ((𝑦 − 2) mod 5)⟩)
13	12	eqeq2d 2741	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐿 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ ↔ 𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩))
14	7, 9, 13	3orbi123d 1437	. . . . . . 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 2741	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐾 = ⟨1, (((2^nd ‘𝑋) + 2) mod 5)⟩ ↔ 𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩))
16	8	eqeq2d 2741	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐾 = ⟨0, (2^nd ‘𝑋)⟩ ↔ 𝐾 = ⟨0, 𝑦⟩))
17	12	eqeq2d 2741	. . . . . . . 8 ⊢ ((2^nd ‘𝑋) = 𝑦 → (𝐾 = ⟨1, (((2^nd ‘𝑋) − 2) mod 5)⟩ ↔ 𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩))
18	15, 16, 17	3orbi123d 1437	. . . . . . 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 632	. . . . . 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 2987	. . . . . . . . . . . 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 2730	. . . . . . . . . . . . 13 ⊢ ⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨1, ((𝑦 + 2) mod 5)⟩
24		eqneqall 2937	. . . . . . . . . . . . 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 4698	. . . . . . . . . . . . . . . 16 ⊢ {⟨1, 𝑏⟩, ⟨0, 𝑦⟩} = {⟨0, 𝑦⟩, ⟨1, 𝑏⟩}
30	29	eleq1i 2820	. . . . . . . . . . . . . . 15 ⊢ ({⟨1, 𝑏⟩, ⟨0, 𝑦⟩} ∈ 𝐸 ↔ {⟨0, 𝑦⟩, ⟨1, 𝑏⟩} ∈ 𝐸)
31		5eluz3 12848	. . . . . . . . . . . . . . . . 17 ⊢ 5 ∈ (ℤ_≥‘3)
32		pglem 48072	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ (1..^(⌈‘(5 / 2)))
33		eqid 2730	. . . . . . . . . . . . . . . . . 18 ⊢ (1..^(⌈‘(5 / 2))) = (1..^(⌈‘(5 / 2)))
34		eqid 2730	. . . . . . . . . . . . . . . . . 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 48046	. . . . . . . . . . . . . . . . 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 702	. . . . . . . . . . . . . . . 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 4840	. . . . . . . . . . . . . 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 4699	. . . . . . . . . . . . . . 15 ⊢ (𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩ → {𝐾, ⟨1, 𝑏⟩} = {⟨1, ((𝑦 + 2) mod 5)⟩, ⟨1, 𝑏⟩})
45	44	eleq1d 2814	. . . . . . . . . . . . . 14 ⊢ (𝐾 = ⟨1, ((𝑦 + 2) mod 5)⟩ → ({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ↔ {⟨1, ((𝑦 + 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸))
46		preq2 4700	. . . . . . . . . . . . . . 15 ⊢ (𝐿 = ⟨0, 𝑦⟩ → {⟨1, 𝑏⟩, 𝐿} = {⟨1, 𝑏⟩, ⟨0, 𝑦⟩})
47	46	eleq1d 2814	. . . . . . . . . . . . . 14 ⊢ (𝐿 = ⟨0, 𝑦⟩ → ({⟨1, 𝑏⟩, 𝐿} ∈ 𝐸 ↔ {⟨1, 𝑏⟩, ⟨0, 𝑦⟩} ∈ 𝐸))
48	45, 47	bi2anan9r 639	. . . . . . . . . . . . 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 4698	. . . . . . . . . . . . . . . 16 ⊢ {⟨1, ((𝑦 + 2) mod 5)⟩, ⟨1, 𝑏⟩} = {⟨1, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩}
54	53	eleq1i 2820	. . . . . . . . . . . . . . 15 ⊢ ({⟨1, ((𝑦 + 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ↔ {⟨1, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)
55		prcom 4698	. . . . . . . . . . . . . . . 16 ⊢ {⟨1, 𝑏⟩, ⟨1, ((𝑦 − 2) mod 5)⟩} = {⟨1, ((𝑦 − 2) mod 5)⟩, ⟨1, 𝑏⟩}
56	55	eleq1i 2820	. . . . . . . . . . . . . . 15 ⊢ ({⟨1, 𝑏⟩, ⟨1, ((𝑦 − 2) mod 5)⟩} ∈ 𝐸 ↔ {⟨1, ((𝑦 − 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸)
57	54, 56	anbi12ci 629	. . . . . . . . . . . . . 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 12273	. . . . . . . . . . . . . . . . 17 ⊢ 5 ∈ ℕ
59	58	nnzi 12563	. . . . . . . . . . . . . . . 16 ⊢ 5 ∈ ℤ
60		uzid 12814	. . . . . . . . . . . . . . . 16 ⊢ (5 ∈ ℤ → 5 ∈ (ℤ_≥‘5))
61	59, 60	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 5 ∈ (ℤ_≥‘5)
62		4t2e8 12355	. . . . . . . . . . . . . . . . . . 19 ⊢ (4 · 2) = 8
63	62	oveq1i 7399	. . . . . . . . . . . . . . . . . 18 ⊢ ((4 · 2) mod 5) = (8 mod 5)
64		8mod5e3 47351	. . . . . . . . . . . . . . . . . 18 ⊢ (8 mod 5) = 3
65	63, 64	eqtri 2753	. . . . . . . . . . . . . . . . 17 ⊢ ((4 · 2) mod 5) = 3
66		3ne0 12293	. . . . . . . . . . . . . . . . 17 ⊢ 3 ≠ 0
67	65, 66	eqnetri 2996	. . . . . . . . . . . . . . . 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 48048	. . . . . . . . . . . . . . 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 1454	. . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . 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 4700	. . . . . . . . . . . . . . 15 ⊢ (𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ → {⟨1, 𝑏⟩, 𝐿} = {⟨1, 𝑏⟩, ⟨1, ((𝑦 − 2) mod 5)⟩})
75	74	eleq1d 2814	. . . . . . . . . . . . . 14 ⊢ (𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ → ({⟨1, 𝑏⟩, 𝐿} ∈ 𝐸 ↔ {⟨1, 𝑏⟩, ⟨1, ((𝑦 − 2) mod 5)⟩} ∈ 𝐸))
76	45, 75	bi2anan9r 639	. . . . . . . . . . . . 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 1430	. . . . . . . 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 4699	. . . . . . . . . . . . . . 15 ⊢ (𝐾 = ⟨0, 𝑦⟩ → {𝐾, ⟨1, 𝑏⟩} = {⟨0, 𝑦⟩, ⟨1, 𝑏⟩})
85	84	eleq1d 2814	. . . . . . . . . . . . . 14 ⊢ (𝐾 = ⟨0, 𝑦⟩ → ({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ↔ {⟨0, 𝑦⟩, ⟨1, 𝑏⟩} ∈ 𝐸))
86		preq2 4700	. . . . . . . . . . . . . . 15 ⊢ (𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ → {⟨1, 𝑏⟩, 𝐿} = {⟨1, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩})
87	86	eleq1d 2814	. . . . . . . . . . . . . 14 ⊢ (𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ → ({⟨1, 𝑏⟩, 𝐿} ∈ 𝐸 ↔ {⟨1, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
88	85, 87	bi2anan9r 639	. . . . . . . . . . . . 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 2987	. . . . . . . . . . . 12 ⊢ ((𝐿 = ⟨0, 𝑦⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) → (𝐾 ≠ 𝐿 ↔ ⟨0, 𝑦⟩ ≠ ⟨0, 𝑦⟩))
96		eqid 2730	. . . . . . . . . . . . 13 ⊢ ⟨0, 𝑦⟩ = ⟨0, 𝑦⟩
97		eqneqall 2937	. . . . . . . . . . . . 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 2814	. . . . . . . . . . . . . 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 2814	. . . . . . . . . . . . . 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 632	. . . . . . . . . . . . 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 1430	. . . . . . . 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 2996	. . . . . . . . . . . . . . . 16 ⊢ (8 mod 5) ≠ 0
114	63, 113	eqnetri 2996	. . . . . . . . . . . . . . 15 ⊢ ((4 · 2) mod 5) ≠ 0
115		simpll 766	. . . . . . . . . . . . . . . 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 615	. . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . 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 702	. . . . . . . . . . . . . 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 2987	. . . . . . . . . . . . 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 631	. . . . . . . . . . 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 4699	. . . . . . . . . . . . . 14 ⊢ (𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩ → {𝐾, ⟨1, 𝑏⟩} = {⟨1, ((𝑦 − 2) mod 5)⟩, ⟨1, 𝑏⟩})
131	130	eleq1d 2814	. . . . . . . . . . . . 13 ⊢ (𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩ → ({𝐾, ⟨1, 𝑏⟩} ∈ 𝐸 ↔ {⟨1, ((𝑦 − 2) mod 5)⟩, ⟨1, 𝑏⟩} ∈ 𝐸))
132	131, 87	bi2anan9r 639	. . . . . . . . . . . 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 650	. . . . . . . . . . . . . . . . 17 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → ({⟨0, 𝑦⟩, ⟨1, 𝑏⟩} ∈ 𝐸 ↔ 𝑦 = 𝑏))
137		equcom 2018	. . . . . . . . . . . . . . . . 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 702	. . . . . . . . . . . . . 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 2814	. . . . . . . . . . . . 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 632	. . . . . . . . . . . 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 2742	. . . . . . . . . . 11 ⊢ (⟨1, ((𝑦 − 2) mod 5)⟩ = 𝐿 → (𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩ ↔ 𝐾 = 𝐿))
152	151	eqcoms 2738	. . . . . . . . . 10 ⊢ (𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ → (𝐾 = ⟨1, ((𝑦 − 2) mod 5)⟩ ↔ 𝐾 = 𝐿))
153		eqneqall 2937	. . . . . . . . . . 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 1430	. . . . . . . 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 1431	. . . . . . 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 2734	. . . . . 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 1085 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 {cpr 4593 ⟨cop 4597 ‘cfv 6513 (class class class)co 7389 2^nd c2nd 7969 0cc0 11074 1c1 11075 + caddc 11077 · cmul 11079 − cmin 11411 / cdiv 11841 2c2 12242 3c3 12243 4c4 12244 5c5 12245 8c8 12248 ℤcz 12535 ℤ_≥cuz 12799 ..^cfzo 13621 ⌈cceil 13759 mod cmo 13837 Vtxcvtx 28929 Edgcedg 28980 NeighbVtx cnbgr 29265 gPetersenGr cgpg 48021

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 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-pre-sup 11152

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-int 4913 df-iun 4959 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-om 7845 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-1o 8436 df-oadd 8440 df-er 8673 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-sup 9399 df-inf 9400 df-dju 9860 df-card 9898 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12188 df-2 12250 df-3 12251 df-4 12252 df-5 12253 df-6 12254 df-7 12255 df-8 12256 df-9 12257 df-n0 12449 df-xnn0 12522 df-z 12536 df-dec 12656 df-uz 12800 df-rp 12958 df-ico 13318 df-fz 13475 df-fzo 13622 df-fl 13760 df-ceil 13761 df-mod 13838 df-hash 14302 df-dvds 16229 df-struct 17123 df-slot 17158 df-ndx 17170 df-base 17186 df-edgf 28922 df-vtx 28931 df-iedg 28932 df-edg 28981 df-umgr 29016 df-usgr 29084 df-gpg 48022

This theorem is referenced by: pgnbgreunbgrlem6 48104

W3C validator