Theorem gpg5nbgrvtx03star 48845

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 12905	. . 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 48841	. . 3 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (♯‘𝑈) = 3)
7	1, 6	sylanl1 692	. 2 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (♯‘𝑈) = 3)
8		eqid 2763	. . . . . . 7 ⊢ ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩
9	2	eleq2i 2855	. . . . . . . . . . 11 ⊢ (𝐾 ∈ 𝐽 ↔ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2))))
10	9	biimpi 219	. . . . . . . . . 10 ⊢ (𝐾 ∈ 𝐽 → 𝐾 ∈ (1..^(⌈‘(𝑁 / 2))))
11		gpgusgra 48822	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (𝑁 gPetersenGr 𝐾) ∈ USGraph)
12	3, 11	eqeltrid 2867	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → 𝐺 ∈ USGraph)
13	1, 10, 12	syl2an 607	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) → 𝐺 ∈ USGraph)
14	13	adantr 485	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 𝐺 ∈ USGraph)
15		gpgnbgr.e	. . . . . . . . . . 11 ⊢ 𝐸 = (Edg‘𝐺)
16	15	usgredgne 29556	. . . . . . . . . 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 2963	. . . . . . . . 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 417	. . . . . . . 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 18	. . . . . . 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 138	. . . . . 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 3065	. . . . . 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 237	. . . . 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 485	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) → 𝑁 ∈ (ℤ≥‘3))
24	23	adantr 485	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 𝑁 ∈ (ℤ≥‘3))
25		simplr 780	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 𝐾 ∈ 𝐽)
26	1	anim1i 626	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) → (𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽))
27		simpl 487	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0) → 𝑋 ∈ 𝑉)
28	26, 27	anim12i 624	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉))
29		eqid 2763	. . . . . . . 8 ⊢ (0..^𝑁) = (0..^𝑁)
30	29, 2, 3, 4	gpgvtxel2 48813	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → (2nd ‘𝑋) ∈ (0..^𝑁))
31	28, 30	syl 18	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (2nd ‘𝑋) ∈ (0..^𝑁))
32	2, 3, 4, 15	gpg5nbgrvtx03starlem1 48833	. . . . . 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 1398	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩} ∉ 𝐸)
34		simpll 778	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 𝑁 ∈ (ℤ≥‘4))
35		elfzoelz 13683	. . . . . . 7 ⊢ ((2nd ‘𝑋) ∈ (0..^𝑁) → (2nd ‘𝑋) ∈ ℤ)
36	28, 30, 35	3syl 19	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (2nd ‘𝑋) ∈ ℤ)
37	2, 3, 4, 15	gpg5nbgrvtx03starlem2 48834	. . . . . 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 1398	. . . . 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 5445	. . . . . 6 ⊢ ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ V
40		opex 5445	. . . . . 6 ⊢ ⟨1, (2nd ‘𝑋)⟩ ∈ V
41		opex 5445	. . . . . 6 ⊢ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ V
42		preq2 4700	. . . . . . 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 3070	. . . . . . 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 18	. . . . . 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 4700	. . . . . . 7 ⊢ (𝑦 = ⟨1, (2nd ‘𝑋)⟩ → {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, 𝑦} = {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩})
46		neleq1 3070	. . . . . . 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 18	. . . . . 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 4700	. . . . . . 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 3070	. . . . . . 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 18	. . . . . 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 4671	. . . . 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 1362	. . . 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 4698	. . . . . . 7 ⊢ {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} = {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩}
54		neleq1 3070	. . . . . . 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 237	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} ∉ 𝐸)
57		eqid 2763	. . . . . . 7 ⊢ ⟨1, (2nd ‘𝑋)⟩ = ⟨1, (2nd ‘𝑋)⟩
58	15	usgredgne 29556	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USGraph ∧ {⟨1, (2nd ‘𝑋)⟩, ⟨1, (2nd ‘𝑋)⟩} ∈ 𝐸) → ⟨1, (2nd ‘𝑋)⟩ ≠ ⟨1, (2nd ‘𝑋)⟩)
59	58	neneqd 2963	. . . . . . . . 9 ⊢ ((𝐺 ∈ USGraph ∧ {⟨1, (2nd ‘𝑋)⟩, ⟨1, (2nd ‘𝑋)⟩} ∈ 𝐸) → ¬ ⟨1, (2nd ‘𝑋)⟩ = ⟨1, (2nd ‘𝑋)⟩)
60	59	ex 417	. . . . . . . 8 ⊢ (𝐺 ∈ USGraph → ({⟨1, (2nd ‘𝑋)⟩, ⟨1, (2nd ‘𝑋)⟩} ∈ 𝐸 → ¬ ⟨1, (2nd ‘𝑋)⟩ = ⟨1, (2nd ‘𝑋)⟩))
61	14, 60	syl 18	. . . . . . 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 138	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ¬ {⟨1, (2nd ‘𝑋)⟩, ⟨1, (2nd ‘𝑋)⟩} ∈ 𝐸)
63		df-nel 3065	. . . . . 6 ⊢ ({⟨1, (2nd ‘𝑋)⟩, ⟨1, (2nd ‘𝑋)⟩} ∉ 𝐸 ↔ ¬ {⟨1, (2nd ‘𝑋)⟩, ⟨1, (2nd ‘𝑋)⟩} ∈ 𝐸)
64	62, 63	sylibr 237	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → {⟨1, (2nd ‘𝑋)⟩, ⟨1, (2nd ‘𝑋)⟩} ∉ 𝐸)
65	2, 3, 4, 15	gpg5nbgrvtx03starlem3 48835	. . . . . 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 1398	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∉ 𝐸)
67		preq2 4700	. . . . . . 7 ⊢ (𝑦 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ → {⟨1, (2nd ‘𝑋)⟩, 𝑦} = {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩})
68		neleq1 3070	. . . . . . 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 18	. . . . . 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 4700	. . . . . . 7 ⊢ (𝑦 = ⟨1, (2nd ‘𝑋)⟩ → {⟨1, (2nd ‘𝑋)⟩, 𝑦} = {⟨1, (2nd ‘𝑋)⟩, ⟨1, (2nd ‘𝑋)⟩})
71		neleq1 3070	. . . . . . 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 18	. . . . . 6 ⊢ (𝑦 = ⟨1, (2nd ‘𝑋)⟩ → ({⟨1, (2nd ‘𝑋)⟩, 𝑦} ∉ 𝐸 ↔ {⟨1, (2nd ‘𝑋)⟩, ⟨1, (2nd ‘𝑋)⟩} ∉ 𝐸))
73		preq2 4700	. . . . . . 7 ⊢ (𝑦 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ → {⟨1, (2nd ‘𝑋)⟩, 𝑦} = {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩})
74		neleq1 3070	. . . . . . 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 18	. . . . . 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 4671	. . . . 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 1362	. . . 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 4698	. . . . . . 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 3070	. . . . . . 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 237	. . . . 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 4698	. . . . . . 7 ⊢ {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩} = {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩}
83		neleq1 3070	. . . . . . 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 237	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩} ∉ 𝐸)
86		eqid 2763	. . . . . . 7 ⊢ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩
87	15	usgredgne 29556	. . . . . . . . . 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 2963	. . . . . . . . 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 417	. . . . . . . 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 18	. . . . . . 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 138	. . . . . 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 3065	. . . . . 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 237	. . . . 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 4700	. . . . . . 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 3070	. . . . . . 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 18	. . . . . 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 4700	. . . . . . 7 ⊢ (𝑦 = ⟨1, (2nd ‘𝑋)⟩ → {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, 𝑦} = {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩})
98		neleq1 3070	. . . . . . 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 18	. . . . . 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 4700	. . . . . . 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 3070	. . . . . . 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 18	. . . . . 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 4671	. . . . 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 1362	. . . 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 4699	. . . . . . 7 ⊢ (𝑥 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ → {𝑥, 𝑦} = {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, 𝑦})
106		neleq1 3070	. . . . . . 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 18	. . . . . 6 ⊢ (𝑥 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ → ({𝑥, 𝑦} ∉ 𝐸 ↔ {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, 𝑦} ∉ 𝐸))
108	107	ralbidv 3188	. . . . 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 4699	. . . . . . 7 ⊢ (𝑥 = ⟨1, (2nd ‘𝑋)⟩ → {𝑥, 𝑦} = {⟨1, (2nd ‘𝑋)⟩, 𝑦})
110		neleq1 3070	. . . . . . 7 ⊢ ({𝑥, 𝑦} = {⟨1, (2nd ‘𝑋)⟩, 𝑦} → ({𝑥, 𝑦} ∉ 𝐸 ↔ {⟨1, (2nd ‘𝑋)⟩, 𝑦} ∉ 𝐸))
111	109, 110	syl 18	. . . . . 6 ⊢ (𝑥 = ⟨1, (2nd ‘𝑋)⟩ → ({𝑥, 𝑦} ∉ 𝐸 ↔ {⟨1, (2nd ‘𝑋)⟩, 𝑦} ∉ 𝐸))
112	111	ralbidv 3188	. . . . 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 4699	. . . . . . 7 ⊢ (𝑥 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ → {𝑥, 𝑦} = {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, 𝑦})
114		neleq1 3070	. . . . . . 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 18	. . . . . 6 ⊢ (𝑥 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ → ({𝑥, 𝑦} ∉ 𝐸 ↔ {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, 𝑦} ∉ 𝐸))
116	115	ralbidv 3188	. . . . 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 4671	. . . 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 1362	. . 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 48839	. . . . 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 692	. . . 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 3323	. . . 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 3331	. . 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 260	. 2 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)
124	7, 123	jca 520	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 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ∉ wnel 3064  ∀wral 3079  {cpr 4591  {ctp 4593  ⟨cop 4595  ‘cfv 6536  (class class class)co 7410  1^st c1st 7980  2^nd c2nd 7981  0cc0 11095  1c1 11096   + caddc 11098   − cmin 11436   / cdiv 11866  2c2 12290  3c3 12291  4c4 12292  ℤcz 12586  ℤ_≥cuz 12857  ..^cfzo 13678  ⌈cceil 13820   mod cmo 13898  ♯chash 14362  Vtxcvtx 29346  Edgcedg 29397  USGraphcusgr 29499   NeighbVtx cnbgr 29682   gPetersenGr cgpg 48805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172  ax-pre-sup 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-oadd 8453  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9398  df-inf 9399  df-dju 9883  df-card 9921  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-div 11867  df-nn 12229  df-2 12298  df-3 12299  df-4 12300  df-5 12301  df-6 12302  df-7 12303  df-8 12304  df-9 12305  df-n0 12500  df-xnn0 12573  df-z 12587  df-dec 12707  df-uz 12858  df-rp 13012  df-fz 13531  df-fzo 13679  df-fl 13821  df-ceil 13822  df-mod 13899  df-hash 14363  df-dvds 16306  df-struct 17202  df-slot 17237  df-ndx 17249  df-base 17265  df-edgf 29339  df-vtx 29348  df-iedg 29349  df-edg 29398  df-upgr 29432  df-umgr 29433  df-usgr 29501  df-nbgr 29683  df-gpg 48806

This theorem is referenced by:  gpg5nbgr3star  48846