gpg5edgnedg - Mathbox for Alexander van der Vekens

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Theorem gpg5edgnedg 48160

Description: Two consecutive (according to the numbering) inside vertices of the Petersen graph G(5,2) are not connected by an edge, but are connected by an edge in a 5-prism G(5,1). (Contributed by AV, 29-Dec-2025.)

**Assertion**
Ref	Expression
gpg5edgnedg	⊢ ({⟨1, 0⟩, ⟨1, 1⟩} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {⟨1, 0⟩, ⟨1, 1⟩} ∉ (Edg‘(5 gPetersenGr 2)))

**Proof of Theorem gpg5edgnedg**
Step	Hyp	Ref	Expression
1		c0ex 11103	. . . . 5 ⊢ 0 ∈ V
2		eleq1 2819	. . . . . 6 ⊢ (𝑥 = 0 → (𝑥 ∈ (0..^5) ↔ 0 ∈ (0..^5)))
3		opeq2 4826	. . . . . . . . 9 ⊢ (𝑥 = 0 → ⟨0, 𝑥⟩ = ⟨0, 0⟩)
4		oveq1 7353	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (𝑥 + 1) = (0 + 1))
5	4	oveq1d 7361	. . . . . . . . . 10 ⊢ (𝑥 = 0 → ((𝑥 + 1) mod 5) = ((0 + 1) mod 5))
6	5	opeq2d 4832	. . . . . . . . 9 ⊢ (𝑥 = 0 → ⟨0, ((𝑥 + 1) mod 5)⟩ = ⟨0, ((0 + 1) mod 5)⟩)
7	3, 6	preq12d 4694	. . . . . . . 8 ⊢ (𝑥 = 0 → {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩} = {⟨0, 0⟩, ⟨0, ((0 + 1) mod 5)⟩})
8	7	eqeq2d 2742	. . . . . . 7 ⊢ (𝑥 = 0 → ({⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩} ↔ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 0⟩, ⟨0, ((0 + 1) mod 5)⟩}))
9		opeq2 4826	. . . . . . . . 9 ⊢ (𝑥 = 0 → ⟨1, 𝑥⟩ = ⟨1, 0⟩)
10	3, 9	preq12d 4694	. . . . . . . 8 ⊢ (𝑥 = 0 → {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} = {⟨0, 0⟩, ⟨1, 0⟩})
11	10	eqeq2d 2742	. . . . . . 7 ⊢ (𝑥 = 0 → ({⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ↔ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 0⟩, ⟨1, 0⟩}))
12	5	opeq2d 4832	. . . . . . . . 9 ⊢ (𝑥 = 0 → ⟨1, ((𝑥 + 1) mod 5)⟩ = ⟨1, ((0 + 1) mod 5)⟩)
13	9, 12	preq12d 4694	. . . . . . . 8 ⊢ (𝑥 = 0 → {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 5)⟩} = {⟨1, 0⟩, ⟨1, ((0 + 1) mod 5)⟩})
14	13	eqeq2d 2742	. . . . . . 7 ⊢ (𝑥 = 0 → ({⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 5)⟩} ↔ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 0⟩, ⟨1, ((0 + 1) mod 5)⟩}))
15	8, 11, 14	3orbi123d 1437	. . . . . 6 ⊢ (𝑥 = 0 → (({⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 5)⟩}) ↔ ({⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 0⟩, ⟨0, ((0 + 1) mod 5)⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 0⟩, ⟨1, 0⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 0⟩, ⟨1, ((0 + 1) mod 5)⟩})))
16	2, 15	anbi12d 632	. . . . 5 ⊢ (𝑥 = 0 → ((𝑥 ∈ (0..^5) ∧ ({⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 5)⟩})) ↔ (0 ∈ (0..^5) ∧ ({⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 0⟩, ⟨0, ((0 + 1) mod 5)⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 0⟩, ⟨1, 0⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 0⟩, ⟨1, ((0 + 1) mod 5)⟩}))))
17		5nn 12208	. . . . . . 7 ⊢ 5 ∈ ℕ
18		lbfzo0 13596	. . . . . . 7 ⊢ (0 ∈ (0..^5) ↔ 5 ∈ ℕ)
19	17, 18	mpbir 231	. . . . . 6 ⊢ 0 ∈ (0..^5)
20		0p1e1 12239	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
21	20	oveq1i 7356	. . . . . . . . . 10 ⊢ ((0 + 1) mod 5) = (1 mod 5)
22		5re 12209	. . . . . . . . . . 11 ⊢ 5 ∈ ℝ
23		1lt5 12297	. . . . . . . . . . 11 ⊢ 1 < 5
24		1mod 13804	. . . . . . . . . . 11 ⊢ ((5 ∈ ℝ ∧ 1 < 5) → (1 mod 5) = 1)
25	22, 23, 24	mp2an 692	. . . . . . . . . 10 ⊢ (1 mod 5) = 1
26	21, 25	eqtr2i 2755	. . . . . . . . 9 ⊢ 1 = ((0 + 1) mod 5)
27	26	opeq2i 4829	. . . . . . . 8 ⊢ ⟨1, 1⟩ = ⟨1, ((0 + 1) mod 5)⟩
28	27	preq2i 4690	. . . . . . 7 ⊢ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 0⟩, ⟨1, ((0 + 1) mod 5)⟩}
29	28	3mix3i 1336	. . . . . 6 ⊢ ({⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 0⟩, ⟨0, ((0 + 1) mod 5)⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 0⟩, ⟨1, 0⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 0⟩, ⟨1, ((0 + 1) mod 5)⟩})
30	19, 29	pm3.2i 470	. . . . 5 ⊢ (0 ∈ (0..^5) ∧ ({⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 0⟩, ⟨0, ((0 + 1) mod 5)⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 0⟩, ⟨1, 0⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 0⟩, ⟨1, ((0 + 1) mod 5)⟩}))
31	1, 16, 30	ceqsexv2d 3488	. . . 4 ⊢ ∃𝑥(𝑥 ∈ (0..^5) ∧ ({⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 5)⟩}))
32		df-rex 3057	. . . 4 ⊢ (∃𝑥 ∈ (0..^5)({⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 5)⟩}) ↔ ∃𝑥(𝑥 ∈ (0..^5) ∧ ({⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 5)⟩})))
33	31, 32	mpbir 231	. . 3 ⊢ ∃𝑥 ∈ (0..^5)({⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 5)⟩})
34		5eluz3 12778	. . . 4 ⊢ 5 ∈ (ℤ_≥‘3)
35		2z 12501	. . . . . 6 ⊢ 2 ∈ ℤ
36	22	rehalfcli 12367	. . . . . . 7 ⊢ (5 / 2) ∈ ℝ
37		ceilcl 13743	. . . . . . 7 ⊢ ((5 / 2) ∈ ℝ → (⌈‘(5 / 2)) ∈ ℤ)
38	36, 37	ax-mp 5	. . . . . 6 ⊢ (⌈‘(5 / 2)) ∈ ℤ
39		2ltceilhalf 47358	. . . . . . 7 ⊢ (5 ∈ (ℤ_≥‘3) → 2 ≤ (⌈‘(5 / 2)))
40	34, 39	ax-mp 5	. . . . . 6 ⊢ 2 ≤ (⌈‘(5 / 2))
41		eluz2 12735	. . . . . 6 ⊢ ((⌈‘(5 / 2)) ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ (⌈‘(5 / 2)) ∈ ℤ ∧ 2 ≤ (⌈‘(5 / 2))))
42	35, 38, 40, 41	mpbir3an 1342	. . . . 5 ⊢ (⌈‘(5 / 2)) ∈ (ℤ_≥‘2)
43		fzo1lb 13610	. . . . 5 ⊢ (1 ∈ (1..^(⌈‘(5 / 2))) ↔ (⌈‘(5 / 2)) ∈ (ℤ_≥‘2))
44	42, 43	mpbir 231	. . . 4 ⊢ 1 ∈ (1..^(⌈‘(5 / 2)))
45		eqid 2731	. . . . 5 ⊢ (0..^5) = (0..^5)
46		eqid 2731	. . . . 5 ⊢ (1..^(⌈‘(5 / 2))) = (1..^(⌈‘(5 / 2)))
47		eqid 2731	. . . . 5 ⊢ (5 gPetersenGr 1) = (5 gPetersenGr 1)
48		eqid 2731	. . . . 5 ⊢ (Edg‘(5 gPetersenGr 1)) = (Edg‘(5 gPetersenGr 1))
49	45, 46, 47, 48	gpgedgel 48080	. . . 4 ⊢ ((5 ∈ (ℤ_≥‘3) ∧ 1 ∈ (1..^(⌈‘(5 / 2)))) → ({⟨1, 0⟩, ⟨1, 1⟩} ∈ (Edg‘(5 gPetersenGr 1)) ↔ ∃𝑥 ∈ (0..^5)({⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 5)⟩})))
50	34, 44, 49	mp2an 692	. . 3 ⊢ ({⟨1, 0⟩, ⟨1, 1⟩} ∈ (Edg‘(5 gPetersenGr 1)) ↔ ∃𝑥 ∈ (0..^5)({⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 5)⟩}))
51	33, 50	mpbir 231	. 2 ⊢ {⟨1, 0⟩, ⟨1, 1⟩} ∈ (Edg‘(5 gPetersenGr 1))
52		pglem 48121	. . . 4 ⊢ 2 ∈ (1..^(⌈‘(5 / 2)))
53		opex 5404	. . . . . . . . . . 11 ⊢ ⟨1, 0⟩ ∈ V
54		opex 5404	. . . . . . . . . . 11 ⊢ ⟨1, 1⟩ ∈ V
55	53, 54	pm3.2i 470	. . . . . . . . . 10 ⊢ (⟨1, 0⟩ ∈ V ∧ ⟨1, 1⟩ ∈ V)
56		opex 5404	. . . . . . . . . . 11 ⊢ ⟨0, 𝑥⟩ ∈ V
57		opex 5404	. . . . . . . . . . 11 ⊢ ⟨0, ((𝑥 + 1) mod 5)⟩ ∈ V
58	56, 57	pm3.2i 470	. . . . . . . . . 10 ⊢ (⟨0, 𝑥⟩ ∈ V ∧ ⟨0, ((𝑥 + 1) mod 5)⟩ ∈ V)
59	55, 58	pm3.2i 470	. . . . . . . . 9 ⊢ ((⟨1, 0⟩ ∈ V ∧ ⟨1, 1⟩ ∈ V) ∧ (⟨0, 𝑥⟩ ∈ V ∧ ⟨0, ((𝑥 + 1) mod 5)⟩ ∈ V))
60		ax-1ne0 11072	. . . . . . . . . . . . . 14 ⊢ 1 ≠ 0
61	60	orci 865	. . . . . . . . . . . . 13 ⊢ (1 ≠ 0 ∨ 0 ≠ 𝑥)
62		1ex 11105	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
63	62, 1	opthne 5422	. . . . . . . . . . . . 13 ⊢ (⟨1, 0⟩ ≠ ⟨0, 𝑥⟩ ↔ (1 ≠ 0 ∨ 0 ≠ 𝑥))
64	61, 63	mpbir 231	. . . . . . . . . . . 12 ⊢ ⟨1, 0⟩ ≠ ⟨0, 𝑥⟩
65	60	orci 865	. . . . . . . . . . . . 13 ⊢ (1 ≠ 0 ∨ 0 ≠ ((𝑥 + 1) mod 5))
66	62, 1	opthne 5422	. . . . . . . . . . . . 13 ⊢ (⟨1, 0⟩ ≠ ⟨0, ((𝑥 + 1) mod 5)⟩ ↔ (1 ≠ 0 ∨ 0 ≠ ((𝑥 + 1) mod 5)))
67	65, 66	mpbir 231	. . . . . . . . . . . 12 ⊢ ⟨1, 0⟩ ≠ ⟨0, ((𝑥 + 1) mod 5)⟩
68	64, 67	pm3.2i 470	. . . . . . . . . . 11 ⊢ (⟨1, 0⟩ ≠ ⟨0, 𝑥⟩ ∧ ⟨1, 0⟩ ≠ ⟨0, ((𝑥 + 1) mod 5)⟩)
69	68	a1i 11	. . . . . . . . . 10 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5)) → (⟨1, 0⟩ ≠ ⟨0, 𝑥⟩ ∧ ⟨1, 0⟩ ≠ ⟨0, ((𝑥 + 1) mod 5)⟩))
70	69	orcd 873	. . . . . . . . 9 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5)) → ((⟨1, 0⟩ ≠ ⟨0, 𝑥⟩ ∧ ⟨1, 0⟩ ≠ ⟨0, ((𝑥 + 1) mod 5)⟩) ∨ (⟨1, 1⟩ ≠ ⟨0, 𝑥⟩ ∧ ⟨1, 1⟩ ≠ ⟨0, ((𝑥 + 1) mod 5)⟩)))
71		prneimg 4806	. . . . . . . . 9 ⊢ (((⟨1, 0⟩ ∈ V ∧ ⟨1, 1⟩ ∈ V) ∧ (⟨0, 𝑥⟩ ∈ V ∧ ⟨0, ((𝑥 + 1) mod 5)⟩ ∈ V)) → (((⟨1, 0⟩ ≠ ⟨0, 𝑥⟩ ∧ ⟨1, 0⟩ ≠ ⟨0, ((𝑥 + 1) mod 5)⟩) ∨ (⟨1, 1⟩ ≠ ⟨0, 𝑥⟩ ∧ ⟨1, 1⟩ ≠ ⟨0, ((𝑥 + 1) mod 5)⟩)) → {⟨1, 0⟩, ⟨1, 1⟩} ≠ {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩}))
72	59, 70, 71	mpsyl 68	. . . . . . . 8 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5)) → {⟨1, 0⟩, ⟨1, 1⟩} ≠ {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩})
73	64	orci 865	. . . . . . . . . 10 ⊢ (⟨1, 0⟩ ≠ ⟨0, 𝑥⟩ ∨ ⟨1, 1⟩ ≠ ⟨1, 𝑥⟩)
74	73	a1i 11	. . . . . . . . 9 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5)) → (⟨1, 0⟩ ≠ ⟨0, 𝑥⟩ ∨ ⟨1, 1⟩ ≠ ⟨1, 𝑥⟩))
75	60	orci 865	. . . . . . . . . . . 12 ⊢ (1 ≠ 0 ∨ 1 ≠ 𝑥)
76	62, 62	opthne 5422	. . . . . . . . . . . 12 ⊢ (⟨1, 1⟩ ≠ ⟨0, 𝑥⟩ ↔ (1 ≠ 0 ∨ 1 ≠ 𝑥))
77	75, 76	mpbir 231	. . . . . . . . . . 11 ⊢ ⟨1, 1⟩ ≠ ⟨0, 𝑥⟩
78	77	olci 866	. . . . . . . . . 10 ⊢ (⟨1, 0⟩ ≠ ⟨1, 𝑥⟩ ∨ ⟨1, 1⟩ ≠ ⟨0, 𝑥⟩)
79	78	a1i 11	. . . . . . . . 9 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5)) → (⟨1, 0⟩ ≠ ⟨1, 𝑥⟩ ∨ ⟨1, 1⟩ ≠ ⟨0, 𝑥⟩))
80		opex 5404	. . . . . . . . . . . 12 ⊢ ⟨1, 𝑥⟩ ∈ V
81	56, 80	pm3.2i 470	. . . . . . . . . . 11 ⊢ (⟨0, 𝑥⟩ ∈ V ∧ ⟨1, 𝑥⟩ ∈ V)
82	55, 81	pm3.2i 470	. . . . . . . . . 10 ⊢ ((⟨1, 0⟩ ∈ V ∧ ⟨1, 1⟩ ∈ V) ∧ (⟨0, 𝑥⟩ ∈ V ∧ ⟨1, 𝑥⟩ ∈ V))
83		prneimg2 4807	. . . . . . . . . 10 ⊢ (((⟨1, 0⟩ ∈ V ∧ ⟨1, 1⟩ ∈ V) ∧ (⟨0, 𝑥⟩ ∈ V ∧ ⟨1, 𝑥⟩ ∈ V)) → ({⟨1, 0⟩, ⟨1, 1⟩} ≠ {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ↔ ((⟨1, 0⟩ ≠ ⟨0, 𝑥⟩ ∨ ⟨1, 1⟩ ≠ ⟨1, 𝑥⟩) ∧ (⟨1, 0⟩ ≠ ⟨1, 𝑥⟩ ∨ ⟨1, 1⟩ ≠ ⟨0, 𝑥⟩))))
84	82, 83	mp1i 13	. . . . . . . . 9 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5)) → ({⟨1, 0⟩, ⟨1, 1⟩} ≠ {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ↔ ((⟨1, 0⟩ ≠ ⟨0, 𝑥⟩ ∨ ⟨1, 1⟩ ≠ ⟨1, 𝑥⟩) ∧ (⟨1, 0⟩ ≠ ⟨1, 𝑥⟩ ∨ ⟨1, 1⟩ ≠ ⟨0, 𝑥⟩))))
85	74, 79, 84	mpbir2and 713	. . . . . . . 8 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5)) → {⟨1, 0⟩, ⟨1, 1⟩} ≠ {⟨0, 𝑥⟩, ⟨1, 𝑥⟩})
86		1ne2 12325	. . . . . . . . . . . . . . 15 ⊢ 1 ≠ 2
87	86	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((0 = 𝑥 ∧ ((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5))) → 1 ≠ 2)
88		2cn 12197	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℂ
89	88	addlidi 11298	. . . . . . . . . . . . . . . . . 18 ⊢ (0 + 2) = 2
90	89	oveq1i 7356	. . . . . . . . . . . . . . . . 17 ⊢ ((0 + 2) mod 5) = (2 mod 5)
91		2nn0 12395	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℕ₀
92		2lt5 12296	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 < 5
93		elfzo0 13597	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 ∈ (0..^5) ↔ (2 ∈ ℕ₀ ∧ 5 ∈ ℕ ∧ 2 < 5))
94	91, 17, 92, 93	mpbir3an 1342	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ (0..^5)
95		zmodidfzoimp 13802	. . . . . . . . . . . . . . . . . 18 ⊢ (2 ∈ (0..^5) → (2 mod 5) = 2)
96	94, 95	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (2 mod 5) = 2
97	90, 96	eqtr2i 2755	. . . . . . . . . . . . . . . 16 ⊢ 2 = ((0 + 2) mod 5)
98		oveq1 7353	. . . . . . . . . . . . . . . . 17 ⊢ (0 = 𝑥 → (0 + 2) = (𝑥 + 2))
99	98	oveq1d 7361	. . . . . . . . . . . . . . . 16 ⊢ (0 = 𝑥 → ((0 + 2) mod 5) = ((𝑥 + 2) mod 5))
100	97, 99	eqtrid 2778	. . . . . . . . . . . . . . 15 ⊢ (0 = 𝑥 → 2 = ((𝑥 + 2) mod 5))
101	100	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((0 = 𝑥 ∧ ((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5))) → 2 = ((𝑥 + 2) mod 5))
102	87, 101	neeqtrd 2997	. . . . . . . . . . . . 13 ⊢ ((0 = 𝑥 ∧ ((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5))) → 1 ≠ ((𝑥 + 2) mod 5))
103	102	olcd 874	. . . . . . . . . . . 12 ⊢ ((0 = 𝑥 ∧ ((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5))) → (0 ≠ 𝑥 ∨ 1 ≠ ((𝑥 + 2) mod 5)))
104	103	ex 412	. . . . . . . . . . 11 ⊢ (0 = 𝑥 → (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5)) → (0 ≠ 𝑥 ∨ 1 ≠ ((𝑥 + 2) mod 5))))
105		orc 867	. . . . . . . . . . . 12 ⊢ (0 ≠ 𝑥 → (0 ≠ 𝑥 ∨ 1 ≠ ((𝑥 + 2) mod 5)))
106	105	a1d 25	. . . . . . . . . . 11 ⊢ (0 ≠ 𝑥 → (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5)) → (0 ≠ 𝑥 ∨ 1 ≠ ((𝑥 + 2) mod 5))))
107	104, 106	pm2.61ine 3011	. . . . . . . . . 10 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5)) → (0 ≠ 𝑥 ∨ 1 ≠ ((𝑥 + 2) mod 5)))
108	62, 1	opthne 5422	. . . . . . . . . . . . 13 ⊢ (⟨1, 0⟩ ≠ ⟨1, 𝑥⟩ ↔ (1 ≠ 1 ∨ 0 ≠ 𝑥))
109		neirr 2937	. . . . . . . . . . . . . 14 ⊢ ¬ 1 ≠ 1
110	109	biorfi 938	. . . . . . . . . . . . 13 ⊢ (0 ≠ 𝑥 ↔ (1 ≠ 1 ∨ 0 ≠ 𝑥))
111	108, 110	bitr4i 278	. . . . . . . . . . . 12 ⊢ (⟨1, 0⟩ ≠ ⟨1, 𝑥⟩ ↔ 0 ≠ 𝑥)
112	111	a1i 11	. . . . . . . . . . 11 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5)) → (⟨1, 0⟩ ≠ ⟨1, 𝑥⟩ ↔ 0 ≠ 𝑥))
113	62, 62	opthne 5422	. . . . . . . . . . . . 13 ⊢ (⟨1, 1⟩ ≠ ⟨1, ((𝑥 + 2) mod 5)⟩ ↔ (1 ≠ 1 ∨ 1 ≠ ((𝑥 + 2) mod 5)))
114	109	biorfi 938	. . . . . . . . . . . . 13 ⊢ (1 ≠ ((𝑥 + 2) mod 5) ↔ (1 ≠ 1 ∨ 1 ≠ ((𝑥 + 2) mod 5)))
115	113, 114	bitr4i 278	. . . . . . . . . . . 12 ⊢ (⟨1, 1⟩ ≠ ⟨1, ((𝑥 + 2) mod 5)⟩ ↔ 1 ≠ ((𝑥 + 2) mod 5))
116	115	a1i 11	. . . . . . . . . . 11 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5)) → (⟨1, 1⟩ ≠ ⟨1, ((𝑥 + 2) mod 5)⟩ ↔ 1 ≠ ((𝑥 + 2) mod 5)))
117	112, 116	orbi12d 918	. . . . . . . . . 10 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5)) → ((⟨1, 0⟩ ≠ ⟨1, 𝑥⟩ ∨ ⟨1, 1⟩ ≠ ⟨1, ((𝑥 + 2) mod 5)⟩) ↔ (0 ≠ 𝑥 ∨ 1 ≠ ((𝑥 + 2) mod 5))))
118	107, 117	mpbird 257	. . . . . . . . 9 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5)) → (⟨1, 0⟩ ≠ ⟨1, 𝑥⟩ ∨ ⟨1, 1⟩ ≠ ⟨1, ((𝑥 + 2) mod 5)⟩))
119		0re 11111	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
120		3pos 12227	. . . . . . . . . . . . . . . 16 ⊢ 0 < 3
121	119, 120	ltneii 11223	. . . . . . . . . . . . . . 15 ⊢ 0 ≠ 3
122	121	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((1 = 𝑥 ∧ ((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5))) → 0 ≠ 3)
123		1p2e3 12260	. . . . . . . . . . . . . . . . . 18 ⊢ (1 + 2) = 3
124	123	oveq1i 7356	. . . . . . . . . . . . . . . . 17 ⊢ ((1 + 2) mod 5) = (3 mod 5)
125		3nn0 12396	. . . . . . . . . . . . . . . . . . 19 ⊢ 3 ∈ ℕ₀
126		3lt5 12295	. . . . . . . . . . . . . . . . . . 19 ⊢ 3 < 5
127		elfzo0 13597	. . . . . . . . . . . . . . . . . . 19 ⊢ (3 ∈ (0..^5) ↔ (3 ∈ ℕ₀ ∧ 5 ∈ ℕ ∧ 3 < 5))
128	125, 17, 126, 127	mpbir3an 1342	. . . . . . . . . . . . . . . . . 18 ⊢ 3 ∈ (0..^5)
129		zmodidfzoimp 13802	. . . . . . . . . . . . . . . . . 18 ⊢ (3 ∈ (0..^5) → (3 mod 5) = 3)
130	128, 129	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (3 mod 5) = 3
131	124, 130	eqtr2i 2755	. . . . . . . . . . . . . . . 16 ⊢ 3 = ((1 + 2) mod 5)
132		oveq1 7353	. . . . . . . . . . . . . . . . 17 ⊢ (1 = 𝑥 → (1 + 2) = (𝑥 + 2))
133	132	oveq1d 7361	. . . . . . . . . . . . . . . 16 ⊢ (1 = 𝑥 → ((1 + 2) mod 5) = ((𝑥 + 2) mod 5))
134	131, 133	eqtrid 2778	. . . . . . . . . . . . . . 15 ⊢ (1 = 𝑥 → 3 = ((𝑥 + 2) mod 5))
135	134	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((1 = 𝑥 ∧ ((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5))) → 3 = ((𝑥 + 2) mod 5))
136	122, 135	neeqtrd 2997	. . . . . . . . . . . . 13 ⊢ ((1 = 𝑥 ∧ ((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5))) → 0 ≠ ((𝑥 + 2) mod 5))
137	136	orcd 873	. . . . . . . . . . . 12 ⊢ ((1 = 𝑥 ∧ ((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5))) → (0 ≠ ((𝑥 + 2) mod 5) ∨ 1 ≠ 𝑥))
138	137	ex 412	. . . . . . . . . . 11 ⊢ (1 = 𝑥 → (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5)) → (0 ≠ ((𝑥 + 2) mod 5) ∨ 1 ≠ 𝑥)))
139		olc 868	. . . . . . . . . . . 12 ⊢ (1 ≠ 𝑥 → (0 ≠ ((𝑥 + 2) mod 5) ∨ 1 ≠ 𝑥))
140	139	a1d 25	. . . . . . . . . . 11 ⊢ (1 ≠ 𝑥 → (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5)) → (0 ≠ ((𝑥 + 2) mod 5) ∨ 1 ≠ 𝑥)))
141	138, 140	pm2.61ine 3011	. . . . . . . . . 10 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5)) → (0 ≠ ((𝑥 + 2) mod 5) ∨ 1 ≠ 𝑥))
142	62, 1	opthne 5422	. . . . . . . . . . . . 13 ⊢ (⟨1, 0⟩ ≠ ⟨1, ((𝑥 + 2) mod 5)⟩ ↔ (1 ≠ 1 ∨ 0 ≠ ((𝑥 + 2) mod 5)))
143	109	biorfi 938	. . . . . . . . . . . . 13 ⊢ (0 ≠ ((𝑥 + 2) mod 5) ↔ (1 ≠ 1 ∨ 0 ≠ ((𝑥 + 2) mod 5)))
144	142, 143	bitr4i 278	. . . . . . . . . . . 12 ⊢ (⟨1, 0⟩ ≠ ⟨1, ((𝑥 + 2) mod 5)⟩ ↔ 0 ≠ ((𝑥 + 2) mod 5))
145	144	a1i 11	. . . . . . . . . . 11 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5)) → (⟨1, 0⟩ ≠ ⟨1, ((𝑥 + 2) mod 5)⟩ ↔ 0 ≠ ((𝑥 + 2) mod 5)))
146	62, 62	opthne 5422	. . . . . . . . . . . . 13 ⊢ (⟨1, 1⟩ ≠ ⟨1, 𝑥⟩ ↔ (1 ≠ 1 ∨ 1 ≠ 𝑥))
147	109	biorfi 938	. . . . . . . . . . . . 13 ⊢ (1 ≠ 𝑥 ↔ (1 ≠ 1 ∨ 1 ≠ 𝑥))
148	146, 147	bitr4i 278	. . . . . . . . . . . 12 ⊢ (⟨1, 1⟩ ≠ ⟨1, 𝑥⟩ ↔ 1 ≠ 𝑥)
149	148	a1i 11	. . . . . . . . . . 11 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5)) → (⟨1, 1⟩ ≠ ⟨1, 𝑥⟩ ↔ 1 ≠ 𝑥))
150	145, 149	orbi12d 918	. . . . . . . . . 10 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5)) → ((⟨1, 0⟩ ≠ ⟨1, ((𝑥 + 2) mod 5)⟩ ∨ ⟨1, 1⟩ ≠ ⟨1, 𝑥⟩) ↔ (0 ≠ ((𝑥 + 2) mod 5) ∨ 1 ≠ 𝑥)))
151	141, 150	mpbird 257	. . . . . . . . 9 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5)) → (⟨1, 0⟩ ≠ ⟨1, ((𝑥 + 2) mod 5)⟩ ∨ ⟨1, 1⟩ ≠ ⟨1, 𝑥⟩))
152		opex 5404	. . . . . . . . . . . 12 ⊢ ⟨1, ((𝑥 + 2) mod 5)⟩ ∈ V
153	80, 152	pm3.2i 470	. . . . . . . . . . 11 ⊢ (⟨1, 𝑥⟩ ∈ V ∧ ⟨1, ((𝑥 + 2) mod 5)⟩ ∈ V)
154	55, 153	pm3.2i 470	. . . . . . . . . 10 ⊢ ((⟨1, 0⟩ ∈ V ∧ ⟨1, 1⟩ ∈ V) ∧ (⟨1, 𝑥⟩ ∈ V ∧ ⟨1, ((𝑥 + 2) mod 5)⟩ ∈ V))
155		prneimg2 4807	. . . . . . . . . 10 ⊢ (((⟨1, 0⟩ ∈ V ∧ ⟨1, 1⟩ ∈ V) ∧ (⟨1, 𝑥⟩ ∈ V ∧ ⟨1, ((𝑥 + 2) mod 5)⟩ ∈ V)) → ({⟨1, 0⟩, ⟨1, 1⟩} ≠ {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 2) mod 5)⟩} ↔ ((⟨1, 0⟩ ≠ ⟨1, 𝑥⟩ ∨ ⟨1, 1⟩ ≠ ⟨1, ((𝑥 + 2) mod 5)⟩) ∧ (⟨1, 0⟩ ≠ ⟨1, ((𝑥 + 2) mod 5)⟩ ∨ ⟨1, 1⟩ ≠ ⟨1, 𝑥⟩))))
156	154, 155	mp1i 13	. . . . . . . . 9 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5)) → ({⟨1, 0⟩, ⟨1, 1⟩} ≠ {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 2) mod 5)⟩} ↔ ((⟨1, 0⟩ ≠ ⟨1, 𝑥⟩ ∨ ⟨1, 1⟩ ≠ ⟨1, ((𝑥 + 2) mod 5)⟩) ∧ (⟨1, 0⟩ ≠ ⟨1, ((𝑥 + 2) mod 5)⟩ ∨ ⟨1, 1⟩ ≠ ⟨1, 𝑥⟩))))
157	118, 151, 156	mpbir2and 713	. . . . . . . 8 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5)) → {⟨1, 0⟩, ⟨1, 1⟩} ≠ {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 2) mod 5)⟩})
158	72, 85, 157	3jca 1128	. . . . . . 7 ⊢ (((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) ∧ 𝑥 ∈ (0..^5)) → ({⟨1, 0⟩, ⟨1, 1⟩} ≠ {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩} ∧ {⟨1, 0⟩, ⟨1, 1⟩} ≠ {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∧ {⟨1, 0⟩, ⟨1, 1⟩} ≠ {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 2) mod 5)⟩}))
159	158	ralrimiva 3124	. . . . . 6 ⊢ ((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) → ∀𝑥 ∈ (0..^5)({⟨1, 0⟩, ⟨1, 1⟩} ≠ {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩} ∧ {⟨1, 0⟩, ⟨1, 1⟩} ≠ {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∧ {⟨1, 0⟩, ⟨1, 1⟩} ≠ {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 2) mod 5)⟩}))
160		ralnex 3058	. . . . . . 7 ⊢ (∀𝑥 ∈ (0..^5) ¬ ({⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 2) mod 5)⟩}) ↔ ¬ ∃𝑥 ∈ (0..^5)({⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 2) mod 5)⟩}))
161		3ioran 1105	. . . . . . . . 9 ⊢ (¬ ({⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 2) mod 5)⟩}) ↔ (¬ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩} ∧ ¬ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∧ ¬ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 2) mod 5)⟩}))
162		df-ne 2929	. . . . . . . . . 10 ⊢ ({⟨1, 0⟩, ⟨1, 1⟩} ≠ {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩} ↔ ¬ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩})
163		df-ne 2929	. . . . . . . . . 10 ⊢ ({⟨1, 0⟩, ⟨1, 1⟩} ≠ {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ↔ ¬ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩})
164		df-ne 2929	. . . . . . . . . 10 ⊢ ({⟨1, 0⟩, ⟨1, 1⟩} ≠ {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 2) mod 5)⟩} ↔ ¬ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 2) mod 5)⟩})
165	162, 163, 164	3anbi123i 1155	. . . . . . . . 9 ⊢ (({⟨1, 0⟩, ⟨1, 1⟩} ≠ {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩} ∧ {⟨1, 0⟩, ⟨1, 1⟩} ≠ {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∧ {⟨1, 0⟩, ⟨1, 1⟩} ≠ {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 2) mod 5)⟩}) ↔ (¬ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩} ∧ ¬ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∧ ¬ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 2) mod 5)⟩}))
166	161, 165	bitr4i 278	. . . . . . . 8 ⊢ (¬ ({⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 2) mod 5)⟩}) ↔ ({⟨1, 0⟩, ⟨1, 1⟩} ≠ {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩} ∧ {⟨1, 0⟩, ⟨1, 1⟩} ≠ {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∧ {⟨1, 0⟩, ⟨1, 1⟩} ≠ {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 2) mod 5)⟩}))
167	166	ralbii 3078	. . . . . . 7 ⊢ (∀𝑥 ∈ (0..^5) ¬ ({⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 2) mod 5)⟩}) ↔ ∀𝑥 ∈ (0..^5)({⟨1, 0⟩, ⟨1, 1⟩} ≠ {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩} ∧ {⟨1, 0⟩, ⟨1, 1⟩} ≠ {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∧ {⟨1, 0⟩, ⟨1, 1⟩} ≠ {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 2) mod 5)⟩}))
168	160, 167	bitr3i 277	. . . . . 6 ⊢ (¬ ∃𝑥 ∈ (0..^5)({⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 2) mod 5)⟩}) ↔ ∀𝑥 ∈ (0..^5)({⟨1, 0⟩, ⟨1, 1⟩} ≠ {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩} ∧ {⟨1, 0⟩, ⟨1, 1⟩} ≠ {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∧ {⟨1, 0⟩, ⟨1, 1⟩} ≠ {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 2) mod 5)⟩}))
169	159, 168	sylibr 234	. . . . 5 ⊢ ((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) → ¬ ∃𝑥 ∈ (0..^5)({⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 2) mod 5)⟩}))
170		eqid 2731	. . . . . 6 ⊢ (5 gPetersenGr 2) = (5 gPetersenGr 2)
171		eqid 2731	. . . . . 6 ⊢ (Edg‘(5 gPetersenGr 2)) = (Edg‘(5 gPetersenGr 2))
172	45, 46, 170, 171	gpgedgel 48080	. . . . 5 ⊢ ((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) → ({⟨1, 0⟩, ⟨1, 1⟩} ∈ (Edg‘(5 gPetersenGr 2)) ↔ ∃𝑥 ∈ (0..^5)({⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 5)⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ {⟨1, 0⟩, ⟨1, 1⟩} = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 2) mod 5)⟩})))
173	169, 172	mtbird 325	. . . 4 ⊢ ((5 ∈ (ℤ_≥‘3) ∧ 2 ∈ (1..^(⌈‘(5 / 2)))) → ¬ {⟨1, 0⟩, ⟨1, 1⟩} ∈ (Edg‘(5 gPetersenGr 2)))
174	34, 52, 173	mp2an 692	. . 3 ⊢ ¬ {⟨1, 0⟩, ⟨1, 1⟩} ∈ (Edg‘(5 gPetersenGr 2))
175	174	nelir 3035	. 2 ⊢ {⟨1, 0⟩, ⟨1, 1⟩} ∉ (Edg‘(5 gPetersenGr 2))
176	51, 175	pm3.2i 470	1 ⊢ ({⟨1, 0⟩, ⟨1, 1⟩} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {⟨1, 0⟩, ⟨1, 1⟩} ∉ (Edg‘(5 gPetersenGr 2)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∨ w3o 1085 ∧ w3a 1086 = wceq 1541 ∃wex 1780 ∈ wcel 2111 ≠ wne 2928 ∉ wnel 3032 ∀wral 3047 ∃wrex 3056 Vcvv 3436 {cpr 4578 ⟨cop 4582 class class class wbr 5091 ‘cfv 6481 (class class class)co 7346 ℝcr 11002 0cc0 11003 1c1 11004 + caddc 11006 < clt 11143 ≤ cle 11144 / cdiv 11771 ℕcn 12122 2c2 12177 3c3 12178 5c5 12180 ℕ₀cn0 12378 ℤcz 12465 ℤ_≥cuz 12729 ..^cfzo 13551 ⌈cceil 13692 mod cmo 13770 Edgcedg 29023 gPetersenGr cgpg 48070

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080 ax-pre-sup 11081

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-oadd 8389 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-sup 9326 df-inf 9327 df-dju 9791 df-card 9829 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-div 11772 df-nn 12123 df-2 12185 df-3 12186 df-4 12187 df-5 12188 df-6 12189 df-7 12190 df-8 12191 df-9 12192 df-n0 12379 df-xnn0 12452 df-z 12466 df-dec 12586 df-uz 12730 df-rp 12888 df-fz 13405 df-fzo 13552 df-fl 13693 df-ceil 13694 df-mod 13771 df-hash 14235 df-struct 17055 df-slot 17090 df-ndx 17102 df-base 17118 df-edgf 28965 df-iedg 28975 df-edg 29024 df-gpg 48071

This theorem is referenced by: grlimedgnedg 48161

W3C validator