Theorem gpg5nbgrvtx03star 47923

Description: In a generalized Petersen graph G(N,K) of order greater than 8 (3 < 𝑁), every vertex of the first kind has exactly three (different) neighbors, and none of these neighbors are connected by an edge (i.e., the (closed) neighborhood of every vertex of the first kind induces a subgraph which is isomorphic to a 3-star). (Contributed by AV, 31-Aug-2025.)

Hypotheses

Ref	Expression
gpgnbgr.j	⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))
gpgnbgr.g	⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)
gpgnbgr.v	⊢ 𝑉 = (Vtx‘𝐺)
gpgnbgr.u	⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)
gpgnbgr.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
gpg5nbgrvtx03star	⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((♯‘𝑈) = 3 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸))

Distinct variable groups:   𝑦,𝐺   𝑦,𝑉   𝑦,𝑋   𝑥,𝐽,𝑦   𝑥,𝐾,𝑦   𝑥,𝑁,𝑦   𝑥,𝑈,𝑦   𝑥,𝑉   𝑥,𝑋   𝑥,𝐸,𝑦

Allowed substitution hint:   𝐺(𝑥)

Proof of Theorem gpg5nbgrvtx03star

Step	Hyp	Ref	Expression
1		eluz4eluz3 12958	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘4) → 𝑁 ∈ (ℤ≥‘3))
2		gpgnbgr.j	. . . 4 ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))
3		gpgnbgr.g	. . . 4 ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)
4		gpgnbgr.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
5		gpgnbgr.u	. . . 4 ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)
6	2, 3, 4, 5	gpg3nbgrvtx0 47919	. . 3 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (♯‘𝑈) = 3)
7	1, 6	sylanl1 679	. 2 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (♯‘𝑈) = 3)
8		eqid 2740	. . . . . . 7 ⊢ ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩
9	2	eleq2i 2836	. . . . . . . . . . 11 ⊢ (𝐾 ∈ 𝐽 ↔ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2))))
10	9	biimpi 216	. . . . . . . . . 10 ⊢ (𝐾 ∈ 𝐽 → 𝐾 ∈ (1..^(⌈‘(𝑁 / 2))))
11		gpgusgra 47901	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (𝑁 gPetersenGr 𝐾) ∈ USGraph)
12	3, 11	eqeltrid 2848	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → 𝐺 ∈ USGraph)
13	1, 10, 12	syl2an 595	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) → 𝐺 ∈ USGraph)
14	13	adantr 480	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 𝐺 ∈ USGraph)
15		gpgnbgr.e	. . . . . . . . . . 11 ⊢ 𝐸 = (Edg‘𝐺)
16	15	usgredgne 29261	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USGraph ∧ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∈ 𝐸) → ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ≠ ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩)
17	16	neneqd 2951	. . . . . . . . 9 ⊢ ((𝐺 ∈ USGraph ∧ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∈ 𝐸) → ¬ ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩)
18	17	ex 412	. . . . . . . 8 ⊢ (𝐺 ∈ USGraph → ({⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∈ 𝐸 → ¬ ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩))
19	14, 18	syl 17	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ({⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∈ 𝐸 → ¬ ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩))
20	8, 19	mt2i 137	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ¬ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∈ 𝐸)
21		df-nel 3053	. . . . . 6 ⊢ ({⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∉ 𝐸 ↔ ¬ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∈ 𝐸)
22	20, 21	sylibr 234	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∉ 𝐸)
23	1	adantr 480	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) → 𝑁 ∈ (ℤ≥‘3))
24	23	adantr 480	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 𝑁 ∈ (ℤ≥‘3))
25		simplr 768	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 𝐾 ∈ 𝐽)
26	1	anim1i 614	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) → (𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽))
27		simpl 482	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0) → 𝑋 ∈ 𝑉)
28	26, 27	anim12i 612	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉))
29		eqid 2740	. . . . . . . 8 ⊢ (0..^𝑁) = (0..^𝑁)
30	29, 2, 3, 4	gpgvtxel2 47896	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → (2nd ‘𝑋) ∈ (0..^𝑁))
31	28, 30	syl 17	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (2nd ‘𝑋) ∈ (0..^𝑁))
32	2, 3, 4, 15	gpg5nbgrvtx03starlem1 47911	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ (2nd ‘𝑋) ∈ (0..^𝑁)) → {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩} ∉ 𝐸)
33	24, 25, 31, 32	syl3anc 1371	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩} ∉ 𝐸)
34		simpll 766	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 𝑁 ∈ (ℤ≥‘4))
35		elfzoelz 13727	. . . . . . 7 ⊢ ((2nd ‘𝑋) ∈ (0..^𝑁) → (2nd ‘𝑋) ∈ ℤ)
36	28, 30, 35	3syl 18	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (2nd ‘𝑋) ∈ ℤ)
37	2, 3, 4, 15	gpg5nbgrvtx03starlem2 47912	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ (2nd ‘𝑋) ∈ ℤ) → {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∉ 𝐸)
38	34, 25, 36, 37	syl3anc 1371	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∉ 𝐸)
39		opex 5485	. . . . . 6 ⊢ ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ V
40		opex 5485	. . . . . 6 ⊢ ⟨1, (2nd ‘𝑋)⟩ ∈ V
41		opex 5485	. . . . . 6 ⊢ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ V
42		preq2 4759	. . . . . . 7 ⊢ (𝑦 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ → {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, 𝑦} = {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩})
43		neleq1 3058	. . . . . . 7 ⊢ ({⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, 𝑦} = {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} → ({⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ↔ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∉ 𝐸))
44	42, 43	syl 17	. . . . . 6 ⊢ (𝑦 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ → ({⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ↔ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∉ 𝐸))
45		preq2 4759	. . . . . . 7 ⊢ (𝑦 = ⟨1, (2nd ‘𝑋)⟩ → {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, 𝑦} = {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩})
46		neleq1 3058	. . . . . . 7 ⊢ ({⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, 𝑦} = {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩} → ({⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ↔ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩} ∉ 𝐸))
47	45, 46	syl 17	. . . . . 6 ⊢ (𝑦 = ⟨1, (2nd ‘𝑋)⟩ → ({⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ↔ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩} ∉ 𝐸))
48		preq2 4759	. . . . . . 7 ⊢ (𝑦 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ → {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, 𝑦} = {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩})
49		neleq1 3058	. . . . . . 7 ⊢ ({⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, 𝑦} = {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} → ({⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ↔ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∉ 𝐸))
50	48, 49	syl 17	. . . . . 6 ⊢ (𝑦 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ → ({⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ↔ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∉ 𝐸))
51	39, 40, 41, 44, 47, 50	raltp 4730	. . . . 5 ⊢ (∀𝑦 ∈ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ↔ ({⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∉ 𝐸 ∧ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩} ∉ 𝐸 ∧ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∉ 𝐸))
52	22, 33, 38, 51	syl3anbrc 1343	. . . 4 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ∀𝑦 ∈ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, 𝑦} ∉ 𝐸)
53		prcom 4757	. . . . . . 7 ⊢ {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} = {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩}
54		neleq1 3058	. . . . . . 7 ⊢ ({⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} = {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩} → ({⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∉ 𝐸 ↔ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩} ∉ 𝐸))
55	53, 54	ax-mp 5	. . . . . 6 ⊢ ({⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∉ 𝐸 ↔ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩} ∉ 𝐸)
56	33, 55	sylibr 234	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∉ 𝐸)
57		eqid 2740	. . . . . . 7 ⊢ ⟨1, (2nd ‘𝑋)⟩ = ⟨1, (2nd ‘𝑋)⟩
58	15	usgredgne 29261	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USGraph ∧ {⟨1, (2nd ‘𝑋)⟩, ⟨1, (2nd ‘𝑋)⟩} ∈ 𝐸) → ⟨1, (2nd ‘𝑋)⟩ ≠ ⟨1, (2nd ‘𝑋)⟩)
59	58	neneqd 2951	. . . . . . . . 9 ⊢ ((𝐺 ∈ USGraph ∧ {⟨1, (2nd ‘𝑋)⟩, ⟨1, (2nd ‘𝑋)⟩} ∈ 𝐸) → ¬ ⟨1, (2nd ‘𝑋)⟩ = ⟨1, (2nd ‘𝑋)⟩)
60	59	ex 412	. . . . . . . 8 ⊢ (𝐺 ∈ USGraph → ({⟨1, (2nd ‘𝑋)⟩, ⟨1, (2nd ‘𝑋)⟩} ∈ 𝐸 → ¬ ⟨1, (2nd ‘𝑋)⟩ = ⟨1, (2nd ‘𝑋)⟩))
61	14, 60	syl 17	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ({⟨1, (2nd ‘𝑋)⟩, ⟨1, (2nd ‘𝑋)⟩} ∈ 𝐸 → ¬ ⟨1, (2nd ‘𝑋)⟩ = ⟨1, (2nd ‘𝑋)⟩))
62	57, 61	mt2i 137	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ¬ {⟨1, (2nd ‘𝑋)⟩, ⟨1, (2nd ‘𝑋)⟩} ∈ 𝐸)
63		df-nel 3053	. . . . . 6 ⊢ ({⟨1, (2nd ‘𝑋)⟩, ⟨1, (2nd ‘𝑋)⟩} ∉ 𝐸 ↔ ¬ {⟨1, (2nd ‘𝑋)⟩, ⟨1, (2nd ‘𝑋)⟩} ∈ 𝐸)
64	62, 63	sylibr 234	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → {⟨1, (2nd ‘𝑋)⟩, ⟨1, (2nd ‘𝑋)⟩} ∉ 𝐸)
65	2, 3, 4, 15	gpg5nbgrvtx03starlem3 47913	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ (2nd ‘𝑋) ∈ (0..^𝑁)) → {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∉ 𝐸)
66	24, 25, 31, 65	syl3anc 1371	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∉ 𝐸)
67		preq2 4759	. . . . . . 7 ⊢ (𝑦 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ → {⟨1, (2nd ‘𝑋)⟩, 𝑦} = {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩})
68		neleq1 3058	. . . . . . 7 ⊢ ({⟨1, (2nd ‘𝑋)⟩, 𝑦} = {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} → ({⟨1, (2nd ‘𝑋)⟩, 𝑦} ∉ 𝐸 ↔ {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∉ 𝐸))
69	67, 68	syl 17	. . . . . 6 ⊢ (𝑦 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ → ({⟨1, (2nd ‘𝑋)⟩, 𝑦} ∉ 𝐸 ↔ {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∉ 𝐸))
70		preq2 4759	. . . . . . 7 ⊢ (𝑦 = ⟨1, (2nd ‘𝑋)⟩ → {⟨1, (2nd ‘𝑋)⟩, 𝑦} = {⟨1, (2nd ‘𝑋)⟩, ⟨1, (2nd ‘𝑋)⟩})
71		neleq1 3058	. . . . . . 7 ⊢ ({⟨1, (2nd ‘𝑋)⟩, 𝑦} = {⟨1, (2nd ‘𝑋)⟩, ⟨1, (2nd ‘𝑋)⟩} → ({⟨1, (2nd ‘𝑋)⟩, 𝑦} ∉ 𝐸 ↔ {⟨1, (2nd ‘𝑋)⟩, ⟨1, (2nd ‘𝑋)⟩} ∉ 𝐸))
72	70, 71	syl 17	. . . . . 6 ⊢ (𝑦 = ⟨1, (2nd ‘𝑋)⟩ → ({⟨1, (2nd ‘𝑋)⟩, 𝑦} ∉ 𝐸 ↔ {⟨1, (2nd ‘𝑋)⟩, ⟨1, (2nd ‘𝑋)⟩} ∉ 𝐸))
73		preq2 4759	. . . . . . 7 ⊢ (𝑦 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ → {⟨1, (2nd ‘𝑋)⟩, 𝑦} = {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩})
74		neleq1 3058	. . . . . . 7 ⊢ ({⟨1, (2nd ‘𝑋)⟩, 𝑦} = {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} → ({⟨1, (2nd ‘𝑋)⟩, 𝑦} ∉ 𝐸 ↔ {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∉ 𝐸))
75	73, 74	syl 17	. . . . . 6 ⊢ (𝑦 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ → ({⟨1, (2nd ‘𝑋)⟩, 𝑦} ∉ 𝐸 ↔ {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∉ 𝐸))
76	39, 40, 41, 69, 72, 75	raltp 4730	. . . . 5 ⊢ (∀𝑦 ∈ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} {⟨1, (2nd ‘𝑋)⟩, 𝑦} ∉ 𝐸 ↔ ({⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∉ 𝐸 ∧ {⟨1, (2nd ‘𝑋)⟩, ⟨1, (2nd ‘𝑋)⟩} ∉ 𝐸 ∧ {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∉ 𝐸))
77	56, 64, 66, 76	syl3anbrc 1343	. . . 4 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ∀𝑦 ∈ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} {⟨1, (2nd ‘𝑋)⟩, 𝑦} ∉ 𝐸)
78		prcom 4757	. . . . . . 7 ⊢ {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} = {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩}
79		neleq1 3058	. . . . . . 7 ⊢ ({⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} = {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} → ({⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∉ 𝐸 ↔ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∉ 𝐸))
80	78, 79	ax-mp 5	. . . . . 6 ⊢ ({⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∉ 𝐸 ↔ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∉ 𝐸)
81	38, 80	sylibr 234	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∉ 𝐸)
82		prcom 4757	. . . . . . 7 ⊢ {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩} = {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩}
83		neleq1 3058	. . . . . . 7 ⊢ ({⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩} = {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} → ({⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩} ∉ 𝐸 ↔ {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∉ 𝐸))
84	82, 83	ax-mp 5	. . . . . 6 ⊢ ({⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩} ∉ 𝐸 ↔ {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∉ 𝐸)
85	66, 84	sylibr 234	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩} ∉ 𝐸)
86		eqid 2740	. . . . . . 7 ⊢ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩
87	15	usgredgne 29261	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USGraph ∧ {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∈ 𝐸) → ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ≠ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩)
88	87	neneqd 2951	. . . . . . . . 9 ⊢ ((𝐺 ∈ USGraph ∧ {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∈ 𝐸) → ¬ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩)
89	88	ex 412	. . . . . . . 8 ⊢ (𝐺 ∈ USGraph → ({⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∈ 𝐸 → ¬ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩))
90	14, 89	syl 17	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ({⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∈ 𝐸 → ¬ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩))
91	86, 90	mt2i 137	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ¬ {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∈ 𝐸)
92		df-nel 3053	. . . . . 6 ⊢ ({⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∉ 𝐸 ↔ ¬ {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∈ 𝐸)
93	91, 92	sylibr 234	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∉ 𝐸)
94		preq2 4759	. . . . . . 7 ⊢ (𝑦 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ → {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, 𝑦} = {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩})
95		neleq1 3058	. . . . . . 7 ⊢ ({⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, 𝑦} = {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} → ({⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ↔ {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∉ 𝐸))
96	94, 95	syl 17	. . . . . 6 ⊢ (𝑦 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ → ({⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ↔ {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∉ 𝐸))
97		preq2 4759	. . . . . . 7 ⊢ (𝑦 = ⟨1, (2nd ‘𝑋)⟩ → {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, 𝑦} = {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩})
98		neleq1 3058	. . . . . . 7 ⊢ ({⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, 𝑦} = {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩} → ({⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ↔ {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩} ∉ 𝐸))
99	97, 98	syl 17	. . . . . 6 ⊢ (𝑦 = ⟨1, (2nd ‘𝑋)⟩ → ({⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ↔ {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩} ∉ 𝐸))
100		preq2 4759	. . . . . . 7 ⊢ (𝑦 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ → {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, 𝑦} = {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩})
101		neleq1 3058	. . . . . . 7 ⊢ ({⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, 𝑦} = {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} → ({⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ↔ {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∉ 𝐸))
102	100, 101	syl 17	. . . . . 6 ⊢ (𝑦 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ → ({⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ↔ {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∉ 𝐸))
103	39, 40, 41, 96, 99, 102	raltp 4730	. . . . 5 ⊢ (∀𝑦 ∈ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ↔ ({⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∉ 𝐸 ∧ {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩} ∉ 𝐸 ∧ {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∉ 𝐸))
104	81, 85, 93, 103	syl3anbrc 1343	. . . 4 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ∀𝑦 ∈ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, 𝑦} ∉ 𝐸)
105		preq1 4758	. . . . . . 7 ⊢ (𝑥 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ → {𝑥, 𝑦} = {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, 𝑦})
106		neleq1 3058	. . . . . . 7 ⊢ ({𝑥, 𝑦} = {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, 𝑦} → ({𝑥, 𝑦} ∉ 𝐸 ↔ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, 𝑦} ∉ 𝐸))
107	105, 106	syl 17	. . . . . 6 ⊢ (𝑥 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ → ({𝑥, 𝑦} ∉ 𝐸 ↔ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, 𝑦} ∉ 𝐸))
108	107	ralbidv 3184	. . . . 5 ⊢ (𝑥 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ → (∀𝑦 ∈ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} {𝑥, 𝑦} ∉ 𝐸 ↔ ∀𝑦 ∈ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, 𝑦} ∉ 𝐸))
109		preq1 4758	. . . . . . 7 ⊢ (𝑥 = ⟨1, (2nd ‘𝑋)⟩ → {𝑥, 𝑦} = {⟨1, (2nd ‘𝑋)⟩, 𝑦})
110		neleq1 3058	. . . . . . 7 ⊢ ({𝑥, 𝑦} = {⟨1, (2nd ‘𝑋)⟩, 𝑦} → ({𝑥, 𝑦} ∉ 𝐸 ↔ {⟨1, (2nd ‘𝑋)⟩, 𝑦} ∉ 𝐸))
111	109, 110	syl 17	. . . . . 6 ⊢ (𝑥 = ⟨1, (2nd ‘𝑋)⟩ → ({𝑥, 𝑦} ∉ 𝐸 ↔ {⟨1, (2nd ‘𝑋)⟩, 𝑦} ∉ 𝐸))
112	111	ralbidv 3184	. . . . 5 ⊢ (𝑥 = ⟨1, (2nd ‘𝑋)⟩ → (∀𝑦 ∈ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} {𝑥, 𝑦} ∉ 𝐸 ↔ ∀𝑦 ∈ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} {⟨1, (2nd ‘𝑋)⟩, 𝑦} ∉ 𝐸))
113		preq1 4758	. . . . . . 7 ⊢ (𝑥 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ → {𝑥, 𝑦} = {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, 𝑦})
114		neleq1 3058	. . . . . . 7 ⊢ ({𝑥, 𝑦} = {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, 𝑦} → ({𝑥, 𝑦} ∉ 𝐸 ↔ {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, 𝑦} ∉ 𝐸))
115	113, 114	syl 17	. . . . . 6 ⊢ (𝑥 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ → ({𝑥, 𝑦} ∉ 𝐸 ↔ {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, 𝑦} ∉ 𝐸))
116	115	ralbidv 3184	. . . . 5 ⊢ (𝑥 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ → (∀𝑦 ∈ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} {𝑥, 𝑦} ∉ 𝐸 ↔ ∀𝑦 ∈ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, 𝑦} ∉ 𝐸))
117	39, 40, 41, 108, 112, 116	raltp 4730	. . . 4 ⊢ (∀𝑥 ∈ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩}∀𝑦 ∈ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} {𝑥, 𝑦} ∉ 𝐸 ↔ (∀𝑦 ∈ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, 𝑦} ∉ 𝐸 ∧ ∀𝑦 ∈ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} {⟨1, (2nd ‘𝑋)⟩, 𝑦} ∉ 𝐸 ∧ ∀𝑦 ∈ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, 𝑦} ∉ 𝐸))
118	52, 77, 104, 117	syl3anbrc 1343	. . 3 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ∀𝑥 ∈ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩}∀𝑦 ∈ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} {𝑥, 𝑦} ∉ 𝐸)
119	2, 3, 4, 5	gpgnbgrvtx0 47917	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 𝑈 = {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩})
120	1, 119	sylanl1 679	. . . 4 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 𝑈 = {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩})
121	120	raleqdv 3334	. . . 4 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸 ↔ ∀𝑦 ∈ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} {𝑥, 𝑦} ∉ 𝐸))
122	120, 121	raleqbidvv 3342	. . 3 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸 ↔ ∀𝑥 ∈ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩}∀𝑦 ∈ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} {𝑥, 𝑦} ∉ 𝐸))
123	118, 122	mpbird 257	. 2 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)
124	7, 123	jca 511	1 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((♯‘𝑈) = 3 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ∉ wnel 3052  ∀wral 3067  {cpr 4650  {ctp 4652  ⟨cop 4654  ‘cfv 6576  (class class class)co 7451  1^st c1st 8031  2^nd c2nd 8032  0cc0 11187  1c1 11188   + caddc 11190   − cmin 11524   / cdiv 11952  2c2 12353  3c3 12354  4c4 12355  ℤcz 12645  ℤ_≥cuz 12910  ..^cfzo 13722  ⌈cceil 13858   mod cmo 13936  ♯chash 14396  Vtxcvtx 29051  Edgcedg 29102  USGraphcusgr 29204   NeighbVtx cnbgr 29387   gPetersenGr cgpg 47889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5304  ax-sep 5318  ax-nul 5325  ax-pow 5384  ax-pr 5448  ax-un 7773  ax-cnex 11243  ax-resscn 11244  ax-1cn 11245  ax-icn 11246  ax-addcl 11247  ax-addrcl 11248  ax-mulcl 11249  ax-mulrcl 11250  ax-mulcom 11251  ax-addass 11252  ax-mulass 11253  ax-distr 11254  ax-i2m1 11255  ax-1ne0 11256  ax-1rid 11257  ax-rnegex 11258  ax-rrecex 11259  ax-cnre 11260  ax-pre-lttri 11261  ax-pre-lttrn 11262  ax-pre-ltadd 11263  ax-pre-mulgt0 11264  ax-pre-sup 11265

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4933  df-int 4972  df-iun 5018  df-br 5168  df-opab 5230  df-mpt 5251  df-tr 5285  df-id 5594  df-eprel 5600  df-po 5608  df-so 5609  df-fr 5653  df-we 5655  df-xp 5707  df-rel 5708  df-cnv 5709  df-co 5710  df-dm 5711  df-rn 5712  df-res 5713  df-ima 5714  df-pred 6335  df-ord 6401  df-on 6402  df-lim 6403  df-suc 6404  df-iota 6528  df-fun 6578  df-fn 6579  df-f 6580  df-f1 6581  df-fo 6582  df-f1o 6583  df-fv 6584  df-riota 7407  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7907  df-1st 8033  df-2nd 8034  df-frecs 8325  df-wrecs 8356  df-recs 8430  df-rdg 8469  df-1o 8525  df-2o 8526  df-oadd 8529  df-er 8766  df-en 9007  df-dom 9008  df-sdom 9009  df-fin 9010  df-sup 9514  df-inf 9515  df-dju 9973  df-card 10011  df-pnf 11329  df-mnf 11330  df-xr 11331  df-ltxr 11332  df-le 11333  df-sub 11526  df-neg 11527  df-div 11953  df-nn 12299  df-2 12361  df-3 12362  df-4 12363  df-5 12364  df-6 12365  df-7 12366  df-8 12367  df-9 12368  df-n0 12559  df-xnn0 12632  df-z 12646  df-dec 12766  df-uz 12911  df-rp 13067  df-fz 13579  df-fzo 13723  df-fl 13859  df-ceil 13860  df-mod 13937  df-hash 14397  df-dvds 16320  df-struct 17214  df-slot 17249  df-ndx 17261  df-base 17279  df-edgf 29042  df-vtx 29053  df-iedg 29054  df-edg 29103  df-upgr 29137  df-umgr 29138  df-usgr 29206  df-nbgr 29388  df-gpg 47890

This theorem is referenced by:  gpg5nbgr3star  47924