Theorem gpg5nbgrvtx03star 48326

Description: In a generalized Petersen graph G(N,K) of order greater than 8 (3 < 𝑁), every outside vertex has exactly three (different) neighbors, and none of these neighbors are connected by an edge (i.e., the (closed) neighborhood of every outside vertex 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		uzuzle34 12799	. . 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 48322	. . 3 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (♯‘𝑈) = 3)
7	1, 6	sylanl1 680	. 2 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (♯‘𝑈) = 3)
8		eqid 2736	. . . . . . 7 ⊢ ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩
9	2	eleq2i 2828	. . . . . . . . . . 11 ⊢ (𝐾 ∈ 𝐽 ↔ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2))))
10	9	biimpi 216	. . . . . . . . . 10 ⊢ (𝐾 ∈ 𝐽 → 𝐾 ∈ (1..^(⌈‘(𝑁 / 2))))
11		gpgusgra 48303	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (𝑁 gPetersenGr 𝐾) ∈ USGraph)
12	3, 11	eqeltrid 2840	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → 𝐺 ∈ USGraph)
13	1, 10, 12	syl2an 596	. . . . . . . . 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 29279	. . . . . . . . . 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 2937	. . . . . . . . 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 3037	. . . . . 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 615	. . . . . . . 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 613	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉))
29		eqid 2736	. . . . . . . 8 ⊢ (0..^𝑁) = (0..^𝑁)
30	29, 2, 3, 4	gpgvtxel2 48294	. . . . . . 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 48314	. . . . . 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 1373	. . . . 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 13575	. . . . . . 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 48315	. . . . . 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 1373	. . . . 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 5412	. . . . . 6 ⊢ ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ V
40		opex 5412	. . . . . 6 ⊢ ⟨1, (2nd ‘𝑋)⟩ ∈ V
41		opex 5412	. . . . . 6 ⊢ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ V
42		preq2 4691	. . . . . . 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 3042	. . . . . . 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 4691	. . . . . . 7 ⊢ (𝑦 = ⟨1, (2nd ‘𝑋)⟩ → {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, 𝑦} = {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩})
46		neleq1 3042	. . . . . . 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 4691	. . . . . . 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 3042	. . . . . . 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 4662	. . . . 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 1344	. . . 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 4689	. . . . . . 7 ⊢ {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} = {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩}
54		neleq1 3042	. . . . . . 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 2736	. . . . . . 7 ⊢ ⟨1, (2nd ‘𝑋)⟩ = ⟨1, (2nd ‘𝑋)⟩
58	15	usgredgne 29279	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USGraph ∧ {⟨1, (2nd ‘𝑋)⟩, ⟨1, (2nd ‘𝑋)⟩} ∈ 𝐸) → ⟨1, (2nd ‘𝑋)⟩ ≠ ⟨1, (2nd ‘𝑋)⟩)
59	58	neneqd 2937	. . . . . . . . 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 3037	. . . . . 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 48316	. . . . . 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 1373	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∉ 𝐸)
67		preq2 4691	. . . . . . 7 ⊢ (𝑦 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ → {⟨1, (2nd ‘𝑋)⟩, 𝑦} = {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩})
68		neleq1 3042	. . . . . . 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 4691	. . . . . . 7 ⊢ (𝑦 = ⟨1, (2nd ‘𝑋)⟩ → {⟨1, (2nd ‘𝑋)⟩, 𝑦} = {⟨1, (2nd ‘𝑋)⟩, ⟨1, (2nd ‘𝑋)⟩})
71		neleq1 3042	. . . . . . 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 4691	. . . . . . 7 ⊢ (𝑦 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ → {⟨1, (2nd ‘𝑋)⟩, 𝑦} = {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩})
74		neleq1 3042	. . . . . . 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 4662	. . . . 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 1344	. . . 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 4689	. . . . . . 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 3042	. . . . . . 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 4689	. . . . . . 7 ⊢ {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩} = {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩}
83		neleq1 3042	. . . . . . 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 2736	. . . . . . 7 ⊢ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩
87	15	usgredgne 29279	. . . . . . . . . 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 2937	. . . . . . . . 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 3037	. . . . . 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 4691	. . . . . . 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 3042	. . . . . . 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 4691	. . . . . . 7 ⊢ (𝑦 = ⟨1, (2nd ‘𝑋)⟩ → {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, 𝑦} = {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩})
98		neleq1 3042	. . . . . . 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 4691	. . . . . . 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 3042	. . . . . . 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 4662	. . . . 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 1344	. . . 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 4690	. . . . . . 7 ⊢ (𝑥 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ → {𝑥, 𝑦} = {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, 𝑦})
106		neleq1 3042	. . . . . . 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 3159	. . . . 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 4690	. . . . . . 7 ⊢ (𝑥 = ⟨1, (2nd ‘𝑋)⟩ → {𝑥, 𝑦} = {⟨1, (2nd ‘𝑋)⟩, 𝑦})
110		neleq1 3042	. . . . . . 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 3159	. . . . 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 4690	. . . . . . 7 ⊢ (𝑥 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ → {𝑥, 𝑦} = {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, 𝑦})
114		neleq1 3042	. . . . . . 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 3159	. . . . 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 4662	. . . 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 1344	. . 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 48320	. . . . 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 680	. . . 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 3296	. . . 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 3304	. . 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 1541   ∈ wcel 2113   ∉ wnel 3036  ∀wral 3051  {cpr 4582  {ctp 4584  ⟨cop 4586  ‘cfv 6492  (class class class)co 7358  1^st c1st 7931  2^nd c2nd 7932  0cc0 11026  1c1 11027   + caddc 11029   − cmin 11364   / cdiv 11794  2c2 12200  3c3 12201  4c4 12202  ℤcz 12488  ℤ_≥cuz 12751  ..^cfzo 13570  ⌈cceil 13711   mod cmo 13789  ♯chash 14253  Vtxcvtx 29069  Edgcedg 29120  USGraphcusgr 29222   NeighbVtx cnbgr 29405   gPetersenGr cgpg 48286

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103  ax-pre-sup 11104

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-oadd 8401  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9345  df-inf 9346  df-dju 9813  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-div 11795  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-xnn0 12475  df-z 12489  df-dec 12608  df-uz 12752  df-rp 12906  df-fz 13424  df-fzo 13571  df-fl 13712  df-ceil 13713  df-mod 13790  df-hash 14254  df-dvds 16180  df-struct 17074  df-slot 17109  df-ndx 17121  df-base 17137  df-edgf 29062  df-vtx 29071  df-iedg 29072  df-edg 29121  df-upgr 29155  df-umgr 29156  df-usgr 29224  df-nbgr 29406  df-gpg 48287

This theorem is referenced by:  gpg5nbgr3star  48327