Theorem gpg5nbgrvtx03star 48009

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		eluz4eluz3 12908	. . 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 48005	. . 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 2734	. . . . . . 7 ⊢ ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩
9	2	eleq2i 2825	. . . . . . . . . . 11 ⊢ (𝐾 ∈ 𝐽 ↔ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2))))
10	9	biimpi 216	. . . . . . . . . 10 ⊢ (𝐾 ∈ 𝐽 → 𝐾 ∈ (1..^(⌈‘(𝑁 / 2))))
11		gpgusgra 47985	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (𝑁 gPetersenGr 𝐾) ∈ USGraph)
12	3, 11	eqeltrid 2837	. . . . . . . . . 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 29152	. . . . . . . . . 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 2936	. . . . . . . . 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 3036	. . . . . 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 2734	. . . . . . . 8 ⊢ (0..^𝑁) = (0..^𝑁)
30	29, 2, 3, 4	gpgvtxel2 47979	. . . . . . 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 47997	. . . . . 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 1372	. . . . 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 13681	. . . . . . 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 47998	. . . . . 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 1372	. . . . 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 5449	. . . . . 6 ⊢ ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ V
40		opex 5449	. . . . . 6 ⊢ ⟨1, (2nd ‘𝑋)⟩ ∈ V
41		opex 5449	. . . . . 6 ⊢ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ V
42		preq2 4714	. . . . . . 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 3041	. . . . . . 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 4714	. . . . . . 7 ⊢ (𝑦 = ⟨1, (2nd ‘𝑋)⟩ → {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, 𝑦} = {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩})
46		neleq1 3041	. . . . . . 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 4714	. . . . . . 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 3041	. . . . . . 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 4685	. . . . 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 4712	. . . . . . 7 ⊢ {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} = {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩}
54		neleq1 3041	. . . . . . 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 2734	. . . . . . 7 ⊢ ⟨1, (2nd ‘𝑋)⟩ = ⟨1, (2nd ‘𝑋)⟩
58	15	usgredgne 29152	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USGraph ∧ {⟨1, (2nd ‘𝑋)⟩, ⟨1, (2nd ‘𝑋)⟩} ∈ 𝐸) → ⟨1, (2nd ‘𝑋)⟩ ≠ ⟨1, (2nd ‘𝑋)⟩)
59	58	neneqd 2936	. . . . . . . . 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 3036	. . . . . 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 47999	. . . . . 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 1372	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∉ 𝐸)
67		preq2 4714	. . . . . . 7 ⊢ (𝑦 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ → {⟨1, (2nd ‘𝑋)⟩, 𝑦} = {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩})
68		neleq1 3041	. . . . . . 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 4714	. . . . . . 7 ⊢ (𝑦 = ⟨1, (2nd ‘𝑋)⟩ → {⟨1, (2nd ‘𝑋)⟩, 𝑦} = {⟨1, (2nd ‘𝑋)⟩, ⟨1, (2nd ‘𝑋)⟩})
71		neleq1 3041	. . . . . . 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 4714	. . . . . . 7 ⊢ (𝑦 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ → {⟨1, (2nd ‘𝑋)⟩, 𝑦} = {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩})
74		neleq1 3041	. . . . . . 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 4685	. . . . 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 4712	. . . . . . 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 3041	. . . . . . 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 4712	. . . . . . 7 ⊢ {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩} = {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩}
83		neleq1 3041	. . . . . . 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 2734	. . . . . . 7 ⊢ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩
87	15	usgredgne 29152	. . . . . . . . . 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 2936	. . . . . . . . 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 3036	. . . . . 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 4714	. . . . . . 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 3041	. . . . . . 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 4714	. . . . . . 7 ⊢ (𝑦 = ⟨1, (2nd ‘𝑋)⟩ → {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, 𝑦} = {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩})
98		neleq1 3041	. . . . . . 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 4714	. . . . . . 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 3041	. . . . . . 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 4685	. . . . 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 4713	. . . . . . 7 ⊢ (𝑥 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ → {𝑥, 𝑦} = {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, 𝑦})
106		neleq1 3041	. . . . . . 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 3165	. . . . 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 4713	. . . . . . 7 ⊢ (𝑥 = ⟨1, (2nd ‘𝑋)⟩ → {𝑥, 𝑦} = {⟨1, (2nd ‘𝑋)⟩, 𝑦})
110		neleq1 3041	. . . . . . 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 3165	. . . . 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 4713	. . . . . . 7 ⊢ (𝑥 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ → {𝑥, 𝑦} = {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, 𝑦})
114		neleq1 3041	. . . . . . 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 3165	. . . . 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 4685	. . . 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 48003	. . . . 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 3309	. . . 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 3317	. . 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 1539   ∈ wcel 2107   ∉ wnel 3035  ∀wral 3050  {cpr 4608  {ctp 4610  ⟨cop 4612  ‘cfv 6541  (class class class)co 7413  1^st c1st 7994  2^nd c2nd 7995  0cc0 11137  1c1 11138   + caddc 11140   − cmin 11474   / cdiv 11902  2c2 12303  3c3 12304  4c4 12305  ℤcz 12596  ℤ_≥cuz 12860  ..^cfzo 13676  ⌈cceil 13813   mod cmo 13891  ♯chash 14352  Vtxcvtx 28942  Edgcedg 28993  USGraphcusgr 29095   NeighbVtx cnbgr 29278   gPetersenGr cgpg 47972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737  ax-cnex 11193  ax-resscn 11194  ax-1cn 11195  ax-icn 11196  ax-addcl 11197  ax-addrcl 11198  ax-mulcl 11199  ax-mulrcl 11200  ax-mulcom 11201  ax-addass 11202  ax-mulass 11203  ax-distr 11204  ax-i2m1 11205  ax-1ne0 11206  ax-1rid 11207  ax-rnegex 11208  ax-rrecex 11209  ax-cnre 11210  ax-pre-lttri 11211  ax-pre-lttrn 11212  ax-pre-ltadd 11213  ax-pre-mulgt0 11214  ax-pre-sup 11215

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4888  df-int 4927  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7870  df-1st 7996  df-2nd 7997  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-2o 8489  df-oadd 8492  df-er 8727  df-en 8968  df-dom 8969  df-sdom 8970  df-fin 8971  df-sup 9464  df-inf 9465  df-dju 9923  df-card 9961  df-pnf 11279  df-mnf 11280  df-xr 11281  df-ltxr 11282  df-le 11283  df-sub 11476  df-neg 11477  df-div 11903  df-nn 12249  df-2 12311  df-3 12312  df-4 12313  df-5 12314  df-6 12315  df-7 12316  df-8 12317  df-9 12318  df-n0 12510  df-xnn0 12583  df-z 12597  df-dec 12717  df-uz 12861  df-rp 13017  df-fz 13530  df-fzo 13677  df-fl 13814  df-ceil 13815  df-mod 13892  df-hash 14353  df-dvds 16274  df-struct 17167  df-slot 17202  df-ndx 17214  df-base 17231  df-edgf 28935  df-vtx 28944  df-iedg 28945  df-edg 28994  df-upgr 29028  df-umgr 29029  df-usgr 29097  df-nbgr 29279  df-gpg 47973

This theorem is referenced by:  gpg5nbgr3star  48010