pgnbgreunbgrlem2lem1 - Mathbox for Alexander van der Vekens

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Theorem pgnbgreunbgrlem2lem1 48605

Description: Lemma 1 for pgnbgreunbgrlem2 48608. (Contributed by AV, 16-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
pgnbgreunbgrlem2lem1	⊢ ((((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐾, ⟨0, 𝑏⟩} ∈ 𝐸) → ¬ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸)

Distinct variable group: 𝑦,𝑏 Allowed substitution hints: 𝐸(𝑦,𝑏) 𝐺(𝑦,𝑏) 𝐾(𝑦,𝑏) 𝐿(𝑦,𝑏) 𝑁(𝑦,𝑏) 𝑉(𝑦,𝑏) 𝑋(𝑦,𝑏)

**Proof of Theorem pgnbgreunbgrlem2lem1**
Step	Hyp	Ref	Expression
1		5eluz3 12824	. . . . . . . 8 ⊢ 5 ∈ (ℤ_≥‘3)
2		pglem 48582	. . . . . . . 8 ⊢ 2 ∈ (1..^(⌈‘(5 / 2)))
3	1, 2	pm3.2i 471	. . . . . . 7 ⊢ (5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2))))
4		c0ex 11129	. . . . . . . 8 ⊢ 0 ∈ V
5		vex 3435	. . . . . . . 8 ⊢ 𝑦 ∈ V
6	4, 5	op1st 7939	. . . . . . 7 ⊢ (1^st ‘⟨0, 𝑦⟩) = 0
7		simpr 485	. . . . . . 7 ⊢ (((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) ∧ {⟨0, 𝑦⟩, ⟨0, 𝑏⟩} ∈ 𝐸) → {⟨0, 𝑦⟩, ⟨0, 𝑏⟩} ∈ 𝐸)
8		eqid 2739	. . . . . . . 8 ⊢ (1..^(⌈‘(5 / 2))) = (1..^(⌈‘(5 / 2)))
9		pgnbgreunbgr.g	. . . . . . . 8 ⊢ 𝐺 = (5 gPetersenGr 2)
10		pgnbgreunbgr.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
11		pgnbgreunbgr.e	. . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺)
12	8, 9, 10, 11	gpgvtxedg0 48554	. . . . . . 7 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ (1^st ‘⟨0, 𝑦⟩) = 0 ∧ {⟨0, 𝑦⟩, ⟨0, 𝑏⟩} ∈ 𝐸) → (⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) + 1) mod 5)⟩ ∨ ⟨0, 𝑏⟩ = ⟨1, (2^nd ‘⟨0, 𝑦⟩)⟩ ∨ ⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) − 1) mod 5)⟩))
13	3, 6, 7, 12	mp3an12i 1473	. . . . . 6 ⊢ (((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) ∧ {⟨0, 𝑦⟩, ⟨0, 𝑏⟩} ∈ 𝐸) → (⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) + 1) mod 5)⟩ ∨ ⟨0, 𝑏⟩ = ⟨1, (2^nd ‘⟨0, 𝑦⟩)⟩ ∨ ⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) − 1) mod 5)⟩))
14	13	ex 413	. . . . 5 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → ({⟨0, 𝑦⟩, ⟨0, 𝑏⟩} ∈ 𝐸 → (⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) + 1) mod 5)⟩ ∨ ⟨0, 𝑏⟩ = ⟨1, (2^nd ‘⟨0, 𝑦⟩)⟩ ∨ ⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) − 1) mod 5)⟩)))
15		vex 3435	. . . . . . . . . 10 ⊢ 𝑏 ∈ V
16	4, 15	op1st 7939	. . . . . . . . 9 ⊢ (1^st ‘⟨0, 𝑏⟩) = 0
17	8, 9, 10, 11	gpgvtxedg0 48554	. . . . . . . . 9 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ (1^st ‘⟨0, 𝑏⟩) = 0 ∧ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸) → (⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ ∨ ⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ ∨ ⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩))
18	3, 16, 17	mp3an12 1459	. . . . . . . 8 ⊢ ({⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸 → (⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ ∨ ⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ ∨ ⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩))
19		1ex 11131	. . . . . . . . . . 11 ⊢ 1 ∈ V
20		ovex 7389	. . . . . . . . . . 11 ⊢ ((𝑦 + 2) mod 5) ∈ V
21	19, 20	opth 5416	. . . . . . . . . 10 ⊢ (⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ ↔ (1 = 0 ∧ ((𝑦 + 2) mod 5) = (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)))
22		ax-1ne0 11098	. . . . . . . . . . . 12 ⊢ 1 ≠ 0
23		eqneqall 2945	. . . . . . . . . . . 12 ⊢ (1 = 0 → (1 ≠ 0 → (((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) ∧ (⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) + 1) mod 5)⟩ ∨ ⟨0, 𝑏⟩ = ⟨1, (2^nd ‘⟨0, 𝑦⟩)⟩ ∨ ⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) − 1) mod 5)⟩)) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
24	22, 23	mpi 20	. . . . . . . . . . 11 ⊢ (1 = 0 → (((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) ∧ (⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) + 1) mod 5)⟩ ∨ ⟨0, 𝑏⟩ = ⟨1, (2^nd ‘⟨0, 𝑦⟩)⟩ ∨ ⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) − 1) mod 5)⟩)) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
25	24	adantr 481	. . . . . . . . . 10 ⊢ ((1 = 0 ∧ ((𝑦 + 2) mod 5) = (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)) → (((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) ∧ (⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) + 1) mod 5)⟩ ∨ ⟨0, 𝑏⟩ = ⟨1, (2^nd ‘⟨0, 𝑦⟩)⟩ ∨ ⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) − 1) mod 5)⟩)) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
26	21, 25	sylbi 218	. . . . . . . . 9 ⊢ (⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ → (((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) ∧ (⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) + 1) mod 5)⟩ ∨ ⟨0, 𝑏⟩ = ⟨1, (2^nd ‘⟨0, 𝑦⟩)⟩ ∨ ⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) − 1) mod 5)⟩)) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
27	19, 20	opth 5416	. . . . . . . . . 10 ⊢ (⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ ↔ (1 = 1 ∧ ((𝑦 + 2) mod 5) = (2^nd ‘⟨0, 𝑏⟩)))
28	4, 15	op2nd 7940	. . . . . . . . . . . . 13 ⊢ (2^nd ‘⟨0, 𝑏⟩) = 𝑏
29	28	eqeq2i 2752	. . . . . . . . . . . 12 ⊢ (((𝑦 + 2) mod 5) = (2^nd ‘⟨0, 𝑏⟩) ↔ ((𝑦 + 2) mod 5) = 𝑏)
30		eqcom 2746	. . . . . . . . . . . 12 ⊢ (((𝑦 + 2) mod 5) = 𝑏 ↔ 𝑏 = ((𝑦 + 2) mod 5))
31	29, 30	bitri 276	. . . . . . . . . . 11 ⊢ (((𝑦 + 2) mod 5) = (2^nd ‘⟨0, 𝑏⟩) ↔ 𝑏 = ((𝑦 + 2) mod 5))
32	4, 5	op2nd 7940	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2^nd ‘⟨0, 𝑦⟩) = 𝑦
33	32	oveq1i 7366	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2^nd ‘⟨0, 𝑦⟩) + 1) = (𝑦 + 1)
34	33	oveq1i 7366	. . . . . . . . . . . . . . . . . 18 ⊢ (((2^nd ‘⟨0, 𝑦⟩) + 1) mod 5) = ((𝑦 + 1) mod 5)
35	34	opeq2i 4808	. . . . . . . . . . . . . . . . 17 ⊢ ⟨0, (((2^nd ‘⟨0, 𝑦⟩) + 1) mod 5)⟩ = ⟨0, ((𝑦 + 1) mod 5)⟩
36	35	eqeq2i 2752	. . . . . . . . . . . . . . . 16 ⊢ (⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) + 1) mod 5)⟩ ↔ ⟨0, 𝑏⟩ = ⟨0, ((𝑦 + 1) mod 5)⟩)
37	4, 15	opth 5416	. . . . . . . . . . . . . . . 16 ⊢ (⟨0, 𝑏⟩ = ⟨0, ((𝑦 + 1) mod 5)⟩ ↔ (0 = 0 ∧ 𝑏 = ((𝑦 + 1) mod 5)))
38	36, 37	bitri 276	. . . . . . . . . . . . . . 15 ⊢ (⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) + 1) mod 5)⟩ ↔ (0 = 0 ∧ 𝑏 = ((𝑦 + 1) mod 5)))
39		eqeq1 2743	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = ((𝑦 + 1) mod 5) → (𝑏 = ((𝑦 + 2) mod 5) ↔ ((𝑦 + 1) mod 5) = ((𝑦 + 2) mod 5)))
40	39	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏 = ((𝑦 + 1) mod 5) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (𝑏 = ((𝑦 + 2) mod 5) ↔ ((𝑦 + 1) mod 5) = ((𝑦 + 2) mod 5)))
41		eqcom 2746	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 + 1) mod 5) = ((𝑦 + 2) mod 5) ↔ ((𝑦 + 2) mod 5) = ((𝑦 + 1) mod 5))
42	41	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (0..^5) → (((𝑦 + 1) mod 5) = ((𝑦 + 2) mod 5) ↔ ((𝑦 + 2) mod 5) = ((𝑦 + 1) mod 5)))
43		elfzoelz 13604	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (0..^5) → 𝑦 ∈ ℤ)
44		2z 12550	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 2 ∈ ℤ
45	44	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (0..^5) → 2 ∈ ℤ)
46	43, 45	zaddcld 12628	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (0..^5) → (𝑦 + 2) ∈ ℤ)
47		1zzd 12549	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (0..^5) → 1 ∈ ℤ)
48	43, 47	zaddcld 12628	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (0..^5) → (𝑦 + 1) ∈ ℤ)
49		5nn 12258	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 5 ∈ ℕ
50	49	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (0..^5) → 5 ∈ ℕ)
51		difmod0 16247	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 + 2) ∈ ℤ ∧ (𝑦 + 1) ∈ ℤ ∧ 5 ∈ ℕ) → ((((𝑦 + 2) − (𝑦 + 1)) mod 5) = 0 ↔ ((𝑦 + 2) mod 5) = ((𝑦 + 1) mod 5)))
52	46, 48, 50, 51	syl3anc 1379	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (0..^5) → ((((𝑦 + 2) − (𝑦 + 1)) mod 5) = 0 ↔ ((𝑦 + 2) mod 5) = ((𝑦 + 1) mod 5)))
53	43	zcnd 12625	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (0..^5) → 𝑦 ∈ ℂ)
54		2cnd 12250	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (0..^5) → 2 ∈ ℂ)
55		1cnd 11130	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (0..^5) → 1 ∈ ℂ)
56	53, 54, 55	pnpcand 11533	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0..^5) → ((𝑦 + 2) − (𝑦 + 1)) = (2 − 1))
57		2m1e1 12293	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (2 − 1) = 1
58	56, 57	eqtrdi 2790	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (0..^5) → ((𝑦 + 2) − (𝑦 + 1)) = 1)
59	58	oveq1d 7371	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (0..^5) → (((𝑦 + 2) − (𝑦 + 1)) mod 5) = (1 mod 5))
60	59	eqeq1d 2741	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (0..^5) → ((((𝑦 + 2) − (𝑦 + 1)) mod 5) = 0 ↔ (1 mod 5) = 0))
61	42, 52, 60	3bitr2d 308	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (0..^5) → (((𝑦 + 1) mod 5) = ((𝑦 + 2) mod 5) ↔ (1 mod 5) = 0))
62		1re 11135	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ∈ ℝ
63		5rp 12940	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 5 ∈ ℝ⁺
64		0le1 11664	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ≤ 1
65		1lt5 12347	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 < 5
66		modid 13846	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1 ∈ ℝ ∧ 5 ∈ ℝ⁺) ∧ (0 ≤ 1 ∧ 1 < 5)) → (1 mod 5) = 1)
67	62, 63, 64, 65, 66	mp4an 699	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1 mod 5) = 1
68	67	eqeq1i 2744	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1 mod 5) = 0 ↔ 1 = 0)
69		eqneqall 2945	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1 = 0 → (1 ≠ 0 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
70	22, 69	mpi 20	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 = 0 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)
71	68, 70	sylbi 218	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1 mod 5) = 0 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)
72	61, 71	biimtrdi 254	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0..^5) → (((𝑦 + 1) mod 5) = ((𝑦 + 2) mod 5) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
73	72	ad2antll 735	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏 = ((𝑦 + 1) mod 5) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (((𝑦 + 1) mod 5) = ((𝑦 + 2) mod 5) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
74	40, 73	sylbid 241	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = ((𝑦 + 1) mod 5) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (𝑏 = ((𝑦 + 2) mod 5) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
75	74	ex 413	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = ((𝑦 + 1) mod 5) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (𝑏 = ((𝑦 + 2) mod 5) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
76	38, 75	simplbiim 509	. . . . . . . . . . . . . 14 ⊢ (⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) + 1) mod 5)⟩ → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (𝑏 = ((𝑦 + 2) mod 5) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
77	4, 15	opth 5416	. . . . . . . . . . . . . . 15 ⊢ (⟨0, 𝑏⟩ = ⟨1, (2^nd ‘⟨0, 𝑦⟩)⟩ ↔ (0 = 1 ∧ 𝑏 = (2^nd ‘⟨0, 𝑦⟩)))
78		0ne1 12243	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≠ 1
79		eqneqall 2945	. . . . . . . . . . . . . . . . 17 ⊢ (0 = 1 → (0 ≠ 1 → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (𝑏 = ((𝑦 + 2) mod 5) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))))
80	78, 79	mpi 20	. . . . . . . . . . . . . . . 16 ⊢ (0 = 1 → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (𝑏 = ((𝑦 + 2) mod 5) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
81	80	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((0 = 1 ∧ 𝑏 = (2^nd ‘⟨0, 𝑦⟩)) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (𝑏 = ((𝑦 + 2) mod 5) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
82	77, 81	sylbi 218	. . . . . . . . . . . . . 14 ⊢ (⟨0, 𝑏⟩ = ⟨1, (2^nd ‘⟨0, 𝑦⟩)⟩ → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (𝑏 = ((𝑦 + 2) mod 5) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
83	32	oveq1i 7366	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘⟨0, 𝑦⟩) − 1) = (𝑦 − 1)
84	83	oveq1i 7366	. . . . . . . . . . . . . . . . 17 ⊢ (((2^nd ‘⟨0, 𝑦⟩) − 1) mod 5) = ((𝑦 − 1) mod 5)
85	84	opeq2i 4808	. . . . . . . . . . . . . . . 16 ⊢ ⟨0, (((2^nd ‘⟨0, 𝑦⟩) − 1) mod 5)⟩ = ⟨0, ((𝑦 − 1) mod 5)⟩
86	85	eqeq2i 2752	. . . . . . . . . . . . . . 15 ⊢ (⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) − 1) mod 5)⟩ ↔ ⟨0, 𝑏⟩ = ⟨0, ((𝑦 − 1) mod 5)⟩)
87	4, 15	opth 5416	. . . . . . . . . . . . . . . 16 ⊢ (⟨0, 𝑏⟩ = ⟨0, ((𝑦 − 1) mod 5)⟩ ↔ (0 = 0 ∧ 𝑏 = ((𝑦 − 1) mod 5)))
88		eqeq1 2743	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = ((𝑦 − 1) mod 5) → (𝑏 = ((𝑦 + 2) mod 5) ↔ ((𝑦 − 1) mod 5) = ((𝑦 + 2) mod 5)))
89	88	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑏 = ((𝑦 − 1) mod 5) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (𝑏 = ((𝑦 + 2) mod 5) ↔ ((𝑦 − 1) mod 5) = ((𝑦 + 2) mod 5)))
90	43, 47	zsubcld 12629	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (0..^5) → (𝑦 − 1) ∈ ℤ)
91		difmod0 16247	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑦 − 1) ∈ ℤ ∧ (𝑦 + 2) ∈ ℤ ∧ 5 ∈ ℕ) → ((((𝑦 − 1) − (𝑦 + 2)) mod 5) = 0 ↔ ((𝑦 − 1) mod 5) = ((𝑦 + 2) mod 5)))
92	91	bicomd 224	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 − 1) ∈ ℤ ∧ (𝑦 + 2) ∈ ℤ ∧ 5 ∈ ℕ) → (((𝑦 − 1) mod 5) = ((𝑦 + 2) mod 5) ↔ (((𝑦 − 1) − (𝑦 + 2)) mod 5) = 0))
93	90, 46, 50, 92	syl3anc 1379	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (0..^5) → (((𝑦 − 1) mod 5) = ((𝑦 + 2) mod 5) ↔ (((𝑦 − 1) − (𝑦 + 2)) mod 5) = 0))
94	53, 55, 53, 54	subsubadd23 11548	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (0..^5) → ((𝑦 − 1) − (𝑦 + 2)) = ((𝑦 − 𝑦) − (1 + 2)))
95	53	subidd 11484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (0..^5) → (𝑦 − 𝑦) = 0)
96		1p2e3 12310	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (1 + 2) = 3
97	96	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (0..^5) → (1 + 2) = 3)
98	95, 97	oveq12d 7374	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ (0..^5) → ((𝑦 − 𝑦) − (1 + 2)) = (0 − 3))
99		df-neg 11371	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ -3 = (0 − 3)
100	98, 99	eqtr4di 2792	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (0..^5) → ((𝑦 − 𝑦) − (1 + 2)) = -3)
101	94, 100	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0..^5) → ((𝑦 − 1) − (𝑦 + 2)) = -3)
102	101	oveq1d 7371	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (0..^5) → (((𝑦 − 1) − (𝑦 + 2)) mod 5) = (-3 mod 5))
103	102	eqeq1d 2741	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (0..^5) → ((((𝑦 − 1) − (𝑦 + 2)) mod 5) = 0 ↔ (-3 mod 5) = 0))
104		3re 12252	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 3 ∈ ℝ
105		negmod0 13828	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((3 ∈ ℝ ∧ 5 ∈ ℝ⁺) → ((3 mod 5) = 0 ↔ (-3 mod 5) = 0))
106	104, 63, 105	mp2an 698	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((3 mod 5) = 0 ↔ (-3 mod 5) = 0)
107		0re 11137	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 0 ∈ ℝ
108		3pos 12277	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 0 < 3
109	107, 104, 108	ltleii 11260	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 0 ≤ 3
110		3lt5 12345	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 3 < 5
111		modid 13846	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((3 ∈ ℝ ∧ 5 ∈ ℝ⁺) ∧ (0 ≤ 3 ∧ 3 < 5)) → (3 mod 5) = 3)
112	104, 63, 109, 110, 111	mp4an 699	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (3 mod 5) = 3
113	112	eqeq1i 2744	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((3 mod 5) = 0 ↔ 3 = 0)
114		3ne0 12278	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 3 ≠ 0
115		eqneqall 2945	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (3 = 0 → (3 ≠ 0 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
116	114, 115	mpi 20	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (3 = 0 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)
117	113, 116	sylbi 218	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((3 mod 5) = 0 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)
118	106, 117	sylbir 236	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((-3 mod 5) = 0 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)
119	103, 118	biimtrdi 254	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (0..^5) → ((((𝑦 − 1) − (𝑦 + 2)) mod 5) = 0 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
120	93, 119	sylbid 241	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (0..^5) → (((𝑦 − 1) mod 5) = ((𝑦 + 2) mod 5) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
121	120	ad2antll 735	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑏 = ((𝑦 − 1) mod 5) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (((𝑦 − 1) mod 5) = ((𝑦 + 2) mod 5) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
122	89, 121	sylbid 241	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏 = ((𝑦 − 1) mod 5) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (𝑏 = ((𝑦 + 2) mod 5) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
123	122	ex 413	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = ((𝑦 − 1) mod 5) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (𝑏 = ((𝑦 + 2) mod 5) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
124	87, 123	simplbiim 509	. . . . . . . . . . . . . . 15 ⊢ (⟨0, 𝑏⟩ = ⟨0, ((𝑦 − 1) mod 5)⟩ → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (𝑏 = ((𝑦 + 2) mod 5) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
125	86, 124	sylbi 218	. . . . . . . . . . . . . 14 ⊢ (⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) − 1) mod 5)⟩ → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (𝑏 = ((𝑦 + 2) mod 5) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
126	76, 82, 125	3jaoi 1436	. . . . . . . . . . . . 13 ⊢ ((⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) + 1) mod 5)⟩ ∨ ⟨0, 𝑏⟩ = ⟨1, (2^nd ‘⟨0, 𝑦⟩)⟩ ∨ ⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) − 1) mod 5)⟩) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → (𝑏 = ((𝑦 + 2) mod 5) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
127	126	com13 88	. . . . . . . . . . . 12 ⊢ (𝑏 = ((𝑦 + 2) mod 5) → ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → ((⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) + 1) mod 5)⟩ ∨ ⟨0, 𝑏⟩ = ⟨1, (2^nd ‘⟨0, 𝑦⟩)⟩ ∨ ⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) − 1) mod 5)⟩) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
128	127	impd 411	. . . . . . . . . . 11 ⊢ (𝑏 = ((𝑦 + 2) mod 5) → (((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) ∧ (⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) + 1) mod 5)⟩ ∨ ⟨0, 𝑏⟩ = ⟨1, (2^nd ‘⟨0, 𝑦⟩)⟩ ∨ ⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) − 1) mod 5)⟩)) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
129	31, 128	sylbi 218	. . . . . . . . . 10 ⊢ (((𝑦 + 2) mod 5) = (2^nd ‘⟨0, 𝑏⟩) → (((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) ∧ (⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) + 1) mod 5)⟩ ∨ ⟨0, 𝑏⟩ = ⟨1, (2^nd ‘⟨0, 𝑦⟩)⟩ ∨ ⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) − 1) mod 5)⟩)) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
130	27, 129	simplbiim 509	. . . . . . . . 9 ⊢ (⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ → (((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) ∧ (⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) + 1) mod 5)⟩ ∨ ⟨0, 𝑏⟩ = ⟨1, (2^nd ‘⟨0, 𝑦⟩)⟩ ∨ ⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) − 1) mod 5)⟩)) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
131	19, 20	opth 5416	. . . . . . . . . 10 ⊢ (⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩ ↔ (1 = 0 ∧ ((𝑦 + 2) mod 5) = (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)))
132	24	adantr 481	. . . . . . . . . 10 ⊢ ((1 = 0 ∧ ((𝑦 + 2) mod 5) = (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)) → (((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) ∧ (⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) + 1) mod 5)⟩ ∨ ⟨0, 𝑏⟩ = ⟨1, (2^nd ‘⟨0, 𝑦⟩)⟩ ∨ ⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) − 1) mod 5)⟩)) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
133	131, 132	sylbi 218	. . . . . . . . 9 ⊢ (⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩ → (((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) ∧ (⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) + 1) mod 5)⟩ ∨ ⟨0, 𝑏⟩ = ⟨1, (2^nd ‘⟨0, 𝑦⟩)⟩ ∨ ⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) − 1) mod 5)⟩)) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
134	26, 130, 133	3jaoi 1436	. . . . . . . 8 ⊢ ((⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) + 1) mod 5)⟩ ∨ ⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨1, (2^nd ‘⟨0, 𝑏⟩)⟩ ∨ ⟨1, ((𝑦 + 2) mod 5)⟩ = ⟨0, (((2^nd ‘⟨0, 𝑏⟩) − 1) mod 5)⟩) → (((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) ∧ (⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) + 1) mod 5)⟩ ∨ ⟨0, 𝑏⟩ = ⟨1, (2^nd ‘⟨0, 𝑦⟩)⟩ ∨ ⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) − 1) mod 5)⟩)) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
135	18, 134	syl 17	. . . . . . 7 ⊢ ({⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸 → (((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) ∧ (⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) + 1) mod 5)⟩ ∨ ⟨0, 𝑏⟩ = ⟨1, (2^nd ‘⟨0, 𝑦⟩)⟩ ∨ ⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) − 1) mod 5)⟩)) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
136		ax-1 6	. . . . . . 7 ⊢ (¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸 → (((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) ∧ (⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) + 1) mod 5)⟩ ∨ ⟨0, 𝑏⟩ = ⟨1, (2^nd ‘⟨0, 𝑦⟩)⟩ ∨ ⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) − 1) mod 5)⟩)) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
137	135, 136	pm2.61i 183	. . . . . 6 ⊢ (((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) ∧ (⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) + 1) mod 5)⟩ ∨ ⟨0, 𝑏⟩ = ⟨1, (2^nd ‘⟨0, 𝑦⟩)⟩ ∨ ⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) − 1) mod 5)⟩)) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)
138	137	ex 413	. . . . 5 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → ((⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) + 1) mod 5)⟩ ∨ ⟨0, 𝑏⟩ = ⟨1, (2^nd ‘⟨0, 𝑦⟩)⟩ ∨ ⟨0, 𝑏⟩ = ⟨0, (((2^nd ‘⟨0, 𝑦⟩) − 1) mod 5)⟩) → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
139	14, 138	syld 47	. . . 4 ⊢ ((𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5)) → ({⟨0, 𝑦⟩, ⟨0, 𝑏⟩} ∈ 𝐸 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
140	139	adantl 482	. . 3 ⊢ (((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → ({⟨0, 𝑦⟩, ⟨0, 𝑏⟩} ∈ 𝐸 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
141		preq1 4665	. . . . . . 7 ⊢ (𝐾 = ⟨0, 𝑦⟩ → {𝐾, ⟨0, 𝑏⟩} = {⟨0, 𝑦⟩, ⟨0, 𝑏⟩})
142	141	eleq1d 2824	. . . . . 6 ⊢ (𝐾 = ⟨0, 𝑦⟩ → ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ↔ {⟨0, 𝑦⟩, ⟨0, 𝑏⟩} ∈ 𝐸))
143	142	adantl 482	. . . . 5 ⊢ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) → ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ↔ {⟨0, 𝑦⟩, ⟨0, 𝑏⟩} ∈ 𝐸))
144		preq2 4666	. . . . . . . 8 ⊢ (𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ → {⟨0, 𝑏⟩, 𝐿} = {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩})
145	144	eleq1d 2824	. . . . . . 7 ⊢ (𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ → ({⟨0, 𝑏⟩, 𝐿} ∈ 𝐸 ↔ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
146	145	notbid 319	. . . . . 6 ⊢ (𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ → (¬ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸 ↔ ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
147	146	adantr 481	. . . . 5 ⊢ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) → (¬ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸 ↔ ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸))
148	143, 147	imbi12d 345	. . . 4 ⊢ ((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 → ¬ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) ↔ ({⟨0, 𝑦⟩, ⟨0, 𝑏⟩} ∈ 𝐸 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
149	148	adantr 481	. . 3 ⊢ (((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 → ¬ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) ↔ ({⟨0, 𝑦⟩, ⟨0, 𝑏⟩} ∈ 𝐸 → ¬ {⟨0, 𝑏⟩, ⟨1, ((𝑦 + 2) mod 5)⟩} ∈ 𝐸)))
150	140, 149	mpbird 258	. 2 ⊢ (((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → ({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 → ¬ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸))
151	150	imp 407	1 ⊢ ((((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐾, ⟨0, 𝑏⟩} ∈ 𝐸) → ¬ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ w3o 1091 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 {cpr 4557 ⟨cop 4561 class class class wbr 5072 ‘cfv 6485 (class class class)co 7356 1^st c1st 7929 2^nd c2nd 7930 ℝcr 11028 0cc0 11029 1c1 11030 + caddc 11032 < clt 11170 ≤ cle 11171 − cmin 11368 -cneg 11369 / cdiv 11798 ℕcn 12165 2c2 12227 3c3 12228 5c5 12230 ℤcz 12515 ℤ_≥cuz 12779 ℝ⁺crp 12933 ..^cfzo 13599 ⌈cceil 13741 mod cmo 13819 Vtxcvtx 29083 Edgcedg 29134 NeighbVtx cnbgr 29419 gPetersenGr cgpg 48531

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-oadd 8399 df-er 8633 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-inf 9346 df-dju 9816 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-xnn0 12502 df-z 12516 df-dec 12636 df-uz 12780 df-rp 12934 df-fz 13453 df-fzo 13600 df-fl 13742 df-ceil 13743 df-mod 13820 df-hash 14284 df-dvds 16213 df-struct 17108 df-slot 17143 df-ndx 17155 df-base 17171 df-edgf 29076 df-vtx 29085 df-iedg 29086 df-edg 29135 df-umgr 29170 df-usgr 29238 df-gpg 48532

This theorem is referenced by: pgnbgreunbgrlem2 48608

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