Theorem gpg5nbgrvtx03star 48440

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 12811	. . 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 48436	. . 3 ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (♯‘𝑈) = 3)
7	1, 6	sylanl1 681	. 2 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (♯‘𝑈) = 3)
8		eqid 2737	. . . . . . 7 ⊢ ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩
9	2	eleq2i 2829	. . . . . . . . . . 11 ⊢ (𝐾 ∈ 𝐽 ↔ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2))))
10	9	biimpi 216	. . . . . . . . . 10 ⊢ (𝐾 ∈ 𝐽 → 𝐾 ∈ (1..^(⌈‘(𝑁 / 2))))
11		gpgusgra 48417	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (𝑁 gPetersenGr 𝐾) ∈ USGraph)
12	3, 11	eqeltrid 2841	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → 𝐺 ∈ USGraph)
13	1, 10, 12	syl2an 597	. . . . . . . . 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 29291	. . . . . . . . . 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 2938	. . . . . . . . 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 3038	. . . . . 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 769	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 𝐾 ∈ 𝐽)
26	1	anim1i 616	. . . . . . . 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 614	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉))
29		eqid 2737	. . . . . . . 8 ⊢ (0..^𝑁) = (0..^𝑁)
30	29, 2, 3, 4	gpgvtxel2 48408	. . . . . . 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 48428	. . . . . 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 1374	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩} ∉ 𝐸)
34		simpll 767	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 𝑁 ∈ (ℤ≥‘4))
35		elfzoelz 13587	. . . . . . 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 48429	. . . . . 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 1374	. . . . 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 5419	. . . . . 6 ⊢ ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ ∈ V
40		opex 5419	. . . . . 6 ⊢ ⟨1, (2nd ‘𝑋)⟩ ∈ V
41		opex 5419	. . . . . 6 ⊢ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ ∈ V
42		preq2 4693	. . . . . . 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 3043	. . . . . . 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 4693	. . . . . . 7 ⊢ (𝑦 = ⟨1, (2nd ‘𝑋)⟩ → {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, 𝑦} = {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩})
46		neleq1 3043	. . . . . . 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 4693	. . . . . . 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 3043	. . . . . . 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 4664	. . . . 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 1345	. . . 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 4691	. . . . . . 7 ⊢ {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩} = {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩}
54		neleq1 3043	. . . . . . 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 2737	. . . . . . 7 ⊢ ⟨1, (2nd ‘𝑋)⟩ = ⟨1, (2nd ‘𝑋)⟩
58	15	usgredgne 29291	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USGraph ∧ {⟨1, (2nd ‘𝑋)⟩, ⟨1, (2nd ‘𝑋)⟩} ∈ 𝐸) → ⟨1, (2nd ‘𝑋)⟩ ≠ ⟨1, (2nd ‘𝑋)⟩)
59	58	neneqd 2938	. . . . . . . . 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 3038	. . . . . 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 48430	. . . . . 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 1374	. . . . 5 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩} ∉ 𝐸)
67		preq2 4693	. . . . . . 7 ⊢ (𝑦 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ → {⟨1, (2nd ‘𝑋)⟩, 𝑦} = {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩})
68		neleq1 3043	. . . . . . 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 4693	. . . . . . 7 ⊢ (𝑦 = ⟨1, (2nd ‘𝑋)⟩ → {⟨1, (2nd ‘𝑋)⟩, 𝑦} = {⟨1, (2nd ‘𝑋)⟩, ⟨1, (2nd ‘𝑋)⟩})
71		neleq1 3043	. . . . . . 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 4693	. . . . . . 7 ⊢ (𝑦 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ → {⟨1, (2nd ‘𝑋)⟩, 𝑦} = {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩})
74		neleq1 3043	. . . . . . 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 4664	. . . . 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 1345	. . . 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 4691	. . . . . . 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 3043	. . . . . . 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 4691	. . . . . . 7 ⊢ {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩} = {⟨1, (2nd ‘𝑋)⟩, ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩}
83		neleq1 3043	. . . . . . 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 2737	. . . . . . 7 ⊢ ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩
87	15	usgredgne 29291	. . . . . . . . . 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 2938	. . . . . . . . 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 3038	. . . . . 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 4693	. . . . . . 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 3043	. . . . . . 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 4693	. . . . . . 7 ⊢ (𝑦 = ⟨1, (2nd ‘𝑋)⟩ → {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, 𝑦} = {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, ⟨1, (2nd ‘𝑋)⟩})
98		neleq1 3043	. . . . . . 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 4693	. . . . . . 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 3043	. . . . . . 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 4664	. . . . 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 1345	. . . 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 4692	. . . . . . 7 ⊢ (𝑥 = ⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩ → {𝑥, 𝑦} = {⟨0, (((2nd ‘𝑋) + 1) mod 𝑁)⟩, 𝑦})
106		neleq1 3043	. . . . . . 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 3161	. . . . 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 4692	. . . . . . 7 ⊢ (𝑥 = ⟨1, (2nd ‘𝑋)⟩ → {𝑥, 𝑦} = {⟨1, (2nd ‘𝑋)⟩, 𝑦})
110		neleq1 3043	. . . . . . 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 3161	. . . . 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 4692	. . . . . . 7 ⊢ (𝑥 = ⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩ → {𝑥, 𝑦} = {⟨0, (((2nd ‘𝑋) − 1) mod 𝑁)⟩, 𝑦})
114		neleq1 3043	. . . . . . 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 3161	. . . . 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 4664	. . . 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 1345	. . 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 48434	. . . . 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 681	. . . 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 3298	. . . 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 3306	. . 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 1542   ∈ wcel 2114   ∉ wnel 3037  ∀wral 3052  {cpr 4584  {ctp 4586  ⟨cop 4588  ‘cfv 6500  (class class class)co 7368  1^st c1st 7941  2^nd c2nd 7942  0cc0 11038  1c1 11039   + caddc 11041   − cmin 11376   / cdiv 11806  2c2 12212  3c3 12213  4c4 12214  ℤcz 12500  ℤ_≥cuz 12763  ..^cfzo 13582  ⌈cceil 13723   mod cmo 13801  ♯chash 14265  Vtxcvtx 29081  Edgcedg 29132  USGraphcusgr 29234   NeighbVtx cnbgr 29417   gPetersenGr cgpg 48400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-oadd 8411  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9357  df-inf 9358  df-dju 9825  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-div 11807  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-n0 12414  df-xnn0 12487  df-z 12501  df-dec 12620  df-uz 12764  df-rp 12918  df-fz 13436  df-fzo 13583  df-fl 13724  df-ceil 13725  df-mod 13802  df-hash 14266  df-dvds 16192  df-struct 17086  df-slot 17121  df-ndx 17133  df-base 17149  df-edgf 29074  df-vtx 29083  df-iedg 29084  df-edg 29133  df-upgr 29167  df-umgr 29168  df-usgr 29236  df-nbgr 29418  df-gpg 48401

This theorem is referenced by:  gpg5nbgr3star  48441